Optimal Control Problems for Partial Differential Equations on Reticulated Domains: Approximation and Asymptotic Analysis

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Author: Peter I. Kogut | Günter Leugering

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Systems & Control: Foundations & Applications Series Editor Tamer Ba¸sar, University of Illinois at Urbana-Champaign Editorial Board Karl Johan Åström, Lund University of Technology, Lund, Sweden Han-Fu Chen, Academia Sinica, Beijing, China Bill Helton, University of California, San Diego, CA, USA Alberto Isidori, University of Rome, Rome, Italy; Washington University, St. Louis, MO, USA Miroslav Krstic, University of California, San Diego, CA, USA Alexander Kurzhanski, University of California, Berkeley, CA, USA; Russian Academy of Sciences, Moscow, Russia H. Vincent Poor, Princeton University, Princeton, NJ, USA Mete Soner, Koç University, Istanbul, Turkey

For further volumes: www.springer.com/series/4895

Peter I. Kogut r Günter R. Leugering

Optimal Control Problems for Partial Differential Equations on Reticulated Domains Approximation and Asymptotic Analysis

Peter I. Kogut Oles Honchar Dnipropetrovsk National University Faculty of Mathematics and Mechanics Department of Differential Equations Gagarin av., 72 49010 Dnipropetrovsk Ukraine [email protected]

Günter R. Leugering Friedrich-Alexander Universität Erlangen-Nürnberg Department of Mathematics Chair of Applied Mathematics II Martensstr. 3 91058 Erlangen Germany [email protected]

ISBN 978-0-8176-8148-7 e-ISBN 978-0-8176-8149-4 DOI 10.1007/978-0-8176-8149-4 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011912819 Mathematics Subject Classification (2010): 35B27, 35J25, 49J20, 93C20 © Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper www.birkhauser-science.com

To our wives, Ljuba and Barbara, with love and thanks

Preface

Optimal control of partial differential equations (PDEs) is by now, after more than 50 years of ever-increasing scientific interest, a well-established discipline in mathematics with many interfaces to science and engineering. During its development, the complexity of the systems to be controlled has also increased significantly, so that today, for example, fluid-structure interactions, magnetohydromechanical, and electromagnetical as well as chemical and civil engineering problems can be dealt with. However, the numerical realization of optimal controls based on, say, optimality conditions together with the simulation of states has also become an issue in scientific computing, as the number of variables involved may easily exceed a couple of millions and, hence, structureexploiting discretization and corresponding adaptive algorithms have to be developed. There are several trends to be observed in this discipline. One is to increase the complexity of the system description in terms of genuinely nonlinear partial differential equations and hybrid couplings to ordinary or even event-driven dynamics together with constraints not only on the controls but also on the states. This kind of investigation typically focuses on a more accurate modeling with respect to the physical description and typically subsumes simple domains for the components. On the other hand, we observe a development, in particular in multiscale modeling, where even classical equations are considered on very complicated domains such that the focus is on the increasing complexity of the geometry of the underlying domain. It obviously is desirable, but also mostly prohibitive, to ask for everything: a very detailed modeling of the process, the handling of very complex geometries, and a timely or even real-time capable numerical realization. However, in the context of modern industrial or science applications, this often turns out to be impossible if one insists on very high accuracy. Model reduction or effective modeling for optimal control problems involving systems of PDEs on complicated domains, therefore, has been the focus of many research initiatives in the last decade. The dynamical system describing the behavior of states may be replaced by a low-dimensional one using, for inVII

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stance, a technique that has come to be known as the reduced basis approach or, possibly greedy, proper orthogonal decomposition. Here, the geometry is built into the simulation tools that provide the empirical bases using “snapshots” which, in turn, makes this approach difficult for hierarchically ordered domains. On the other hand, one may use “surrogate models” on simple domains, models that are simplifying approximations of both the state equation and the domain. A third avenue, the one along which we will proceed in this monograph, is based on asymptotic analysis. The method we describe and further develop aims at combining techniques of homogenization and approximation in order to cover optimal control problems defined on reticulated domains, such as lattice structures, honeycomb structures, hierarchical structures, or networked domains in general. Here, error estimates are introduced in order to control the quality of the controls obtained on the approximation level. Our interest is mainly in problems where the control is exerted at, for example, highly oscillating boundaries or interfaces associated with such structures, but we also ask for “controls in the coefficients”, that is, for controlling material parameters on the microscopic ε-level. Our research is strongly motivated by recent developments in multiscale modeling and simulation in a variety of applications. However, from a mathematical point of view, only the aspect ratio – that is, the relation of, say, thickness versus length – is relevant (as long as one does not enter molecular dynamics). In that respect, we can also relate our research to networked structure mechanics in civil engineering, such as flexible structures, masts of all kinds, and gas, water, and traffic networks. As for material sciences, metallic, ceramic, or polymeric foams are particularly interesting because of their mechanical properties, such as being extremely lightweight and at the same time adequately stiff. Similarly, complex conductors on the micro-level exhibit graphlike structures, and carbon nanotubes are used in many applications. They themselves serve as gridlike domains supporting processes like electromagnetic wave propagation. However, properly assembled in thousands or millions on a waferlike substrate, they can be used as reactors for catalytic processes. Mathematically speaking, elliptic, parabolic, and hyperbolic systems on reticulated domains are used to model these applications which, in turn, exhibit a genuine multiscale character. Engineers define cost or merit functions that are to be optimized with respect to various control actions along the boundary of the structure and, even more challenging, with respect to material properties. Certainly, both states and controls have to be taken into account as being constrained. In particular, state constraints genuinely give rise to Lagrange-multipliers that are measures. As a result, in a general setting, such PDEs on reticulated domains should be dealt with in the context of descriptions allowing for measures on the “right-hand side” as well as in the system equations and the geometries. This means that providing an abstract approach to optimal

Preface

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control problems for PDEs in such domains is not the result of an intrinsically motivated mathematical desire to achieve the highest degree of generality—it follows the needs dictated by the applications! Moreover, the type of reticulated structures to be considered varies drastically with the application context or with the degree of approximation needed in a given context. Namely, in a gas network or a sewer system, civil engineers trust in one-dimensional models for the gas-flow, and consequently in network-flow problems on graphs. In a river system a fully three-dimensional treatment may be necessary, but, still, the problem would be one on a networked domain—a “fat graph”. The same rationale applies to elastic multistructures, foams, and all the way down the scales to nanotubes. Percolation networks carrying microflows can be modeled as perforated domains, but, in a further approximation, also as fat graphs. Obviously, the aspect ratio and the cell size for the periodic structure serve as scaling parameters. An optimal control problem defined on such structures, therefore, genuinely inherits a number of scales that may be separated or even not separable at all. The fundamental problem now is: How do such optimal control problems “behave” upon changing these parameters? More precisely, what happens if we go up the scales to a “continuum” description? However, why should we do that to begin with? This question brings us back to the question of model reduction and its interplay with approximation properties. For a small scale, for example, a fine gridstructure, the numerical effort related to a discretization in order to reveal a resolution according to the scale is typically prohibitive. On the other hand, a coarse graining may miss the effects looked for. Asymptotic analysis resolves this antagonism elegantly, in that the limiting problem and approximations thereof can be taken as a surrogate optimal control problem on a simple domain such that error estimates show how far the true fine-scale solution is away from the “homogenized” solution which, in turn, can be termed suboptimal. In order for all of this to become a mathematically sound theory, it is not sufficient to apply asymptotic expansions to all components of such an optimal control problem—namely the cost function, the state equation, the domains, and the control and observation instruments individually—and replace it by its low-order parts. An asymptotic analysis of optimal control problems as such is in order. Meanwhile, an ill-posed PDE problem—in the sense of Hadamard—due to the freedom of choosing proper control inputs may turn out to be well-posed as an optimization problem, while a well-posed PDE problem easily can exhibit ill-posedness, once integrated into an optimal control problem. Hence, well-posedness of optimal control problems is a new and interesting issue. The book focuses on all of these aspects from two perspectives. First, a rigorous and mostly self-contained mathematical theory of PDEs on reticulated domains together with well-posedness for the governing optimal control problems is described. The concept of optimal control problems for PDEs in varying such domains, and hence in varying Banach spaces, is developed, fol-

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lowed by appropriate concepts for convergence of optimal control problems in variable spaces. This comprises Part I of the book. Even though there are by now a number of textbooks for PDE-constrained optimal control available, this monograph contains a unique collection of results that are necessary to treat the optimal control problem on varying structures which are not available in a textbook otherwise. In Part II, particular examples and applications are investigated with the tools established in Part I of the book. These examples are strongly motivated by applications in mechanics and material sciences as explained above, but they can be understood without any knowledge from those fields. In order to accomplish this, the models are somewhat simplified such that, for example, only quasi-static flow in cylindrically perforated domains is considered instead of the fully time-dependent Navier–Stokes flow. Additionally, the elliptic second order problems on thin or fat graphs are scalar instead of being vectorial, which would be necessary in order to be directly applicable to problems of elasticity. Overall, the book’s first part can be seen as an advanced textbook for abstract optimal control problems, in particular on reticulated domains, which can be used in graduate courses, while its second part serves as a research monograph, using somewhat stratified applications in an exemplary manner. Part II can be of use also in seminars, building on the knowledge from a graduate course taught from Part I. For the reader’s convenience, in Part II, we sometimes reintroduce the basic concepts that are dealt with in Part I on an abstract level; however, they are explicitly geared towards the particular application. Admitting some potential redundancy, we thereby keep the chapters in the second part of the book self-contained for researchers in the field.

Dnipropetrowsk and Erlangen, May 2011

Peter I. Kogut G¨ unter R. Leugering

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Part I Asymptotic Analysis of Optimal Control Problems for Partial Differential Equations: Basic Tools 2

3

Background Material on Asymptotic Analysis of Extremal Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Measure theory and basic notation . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Hausdorff measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Sobolev spaces and boundary value problems . . . . . . . . . . . . . . . 2.2.1 Weak derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Vector-valued spaces of the type Lp (a, b; X) . . . . . . . . . 2.2.4 Lax–Milgram’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 General setting of the variational formulation of boundary value problems . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Spaces of periodic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Weak and weak-∗ convergence in Banach spaces . . . . . . . . . . . . 2.4.1 Weak convergence of measures . . . . . . . . . . . . . . . . . . . . . 2.4.2 Weak convergence in L1 (Ω) . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Elements of capacity theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 On the space W01,p (Ω) ∩ Lp (Ω, dμ) and its properties . . . . . . . 2.7 Sobolev spaces with respect to a measure . . . . . . . . . . . . . . . . . . 2.8 Boundary value problems in Sobolev spaces with measures . . . 2.9 On weak compactness of a class of bounded sets in Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 15 17 19 19 20 25 27 28 33 34 39 43 44 47 49 53 58

Variational Methods of Optimal Control Theory . . . . . . . . . . . 63 3.1 The general setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.2 Abstract extremal problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 XI

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3.3

3.4

3.5 3.6

3.7

Extremal problems for steady-state processes . . . . . . . . . . . . . . . 78 3.3.1 Dirichlet and Neumann boundary control problems . . . 78 3.3.2 Ill-posed control objects . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.3.3 Optimal control of the Cauchy problem for an elliptic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.3.4 Controls with hard constraints . . . . . . . . . . . . . . . . . . . . . 86 Optimal control problems for parabolic equations . . . . . . . . . . . 88 3.4.1 Distributed control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.4.2 Control in the initial conditions . . . . . . . . . . . . . . . . . . . . 91 3.4.3 Neumann boundary control . . . . . . . . . . . . . . . . . . . . . . . . 94 Optimal control problems for hyperbolic equations . . . . . . . . . . 98 Optimality system to optimal control problems . . . . . . . . . . . . . 100 3.6.1 The general setting of the Lagrange multiplier principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.6.2 Necessary optimality conditions in the form of variational inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Optimal control of distributed singular systems . . . . . . . . . . . . . 109

4

Suboptimal and Approximate Solutions to Extremal Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.1 The notion of suboptimal and approximate solutions . . . . . . . . 114 4.2 Regularization of optimal control problems . . . . . . . . . . . . . . . . 117 4.3 ε-Suboptimal solutions to optimal control problems . . . . . . . . . 123 4.4 Approximate solutions to distributed singular systems . . . . . . . 129

5

Introduction to the Asymptotic Analysis of Optimal Control Problems: A Parade of Examples . . . . . . . . . . . . . . . . . . 133 5.1 Component-by-component limit analysis . . . . . . . . . . . . . . . . . . . 134 5.2 Limit analysis of optimality conditions . . . . . . . . . . . . . . . . . . . . 143 5.3 Limit analysis of optimal control problems by Γ -convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.4 Direct variational convergence of optimal control problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

6

Convergence Concepts in Variable Banach Spaces . . . . . . . . . 161 6.1 General setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 6.2 Weak convergence in variable Lp -spaces . . . . . . . . . . . . . . . . . . . 167 6.3 Two-scale convergence in variable Lp -spaces . . . . . . . . . . . . . . . 171 6.4 Variable Sobolev spaces and two-scale convergence . . . . . . . . . . 181 6.4.1 p-Connected measures and their properties . . . . . . . . . . 181 6.4.2 Degenerate measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 6.4.3 Two-scale convergence in variable Sobolev spaces . . . . . 185 6.5 Approximation of singular measures by smoothing and its application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

Contents

6.6 6.7 6.8 6.9

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Two-scale convergence with respect to a variable measure . . . . 194 Some properties of the strong convergence in spaces L2 (, dμh ) associated with thin periodic structures . . . . . . . . . 198 On approximative properties of Hilbert spaces with respect to a periodic Borel measure μh . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 The homothetic mean value property on periodically perforated domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 6.9.1 A measure approach to the description of the sets Ωε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 6.9.2 The homothetic mean value property . . . . . . . . . . . . . . . 212

7

Convergence of Sets in Variable Spaces . . . . . . . . . . . . . . . . . . . . 217 7.1 Set convergence in Rn via the limit properties of characteristic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 7.2 On some limit properties of characteristic functions for nonperiodically perforated domains . . . . . . . . . . . . . . . . . . . . 224 7.3 Convergence of sets associated with thin periodic structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 7.4 Set convergence in the sense of Kuratowski and in the Hausdorff metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 7.5 Parametrical convergence of open sets and of associated mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 7.6 Kuratowski set convergence in variable spaces . . . . . . . . . . . . . 247 7.7 γp –Convergence of open sets and Mosco convergence of the associated Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 257

8

Passing to the Limit in Constrained Minimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 8.1 A short survey of Γ -convergence theory in metric spaces . . . . . 264 8.2 Γ -Convergence of functionals defined on a Banach space . . . . . 270 8.3 Variational convergence of constrained minimization problems in Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 8.4 Variational convergence of minimization problems in variable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 8.5 Asymptotic analysis of a Dirichlet optimal control problem . . 286 8.5.1 The statement of the optimal control problem and preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 8.5.2 On modifications of the optimal control problem with controllability constraints . . . . . . . . . . . . . . . . . . . . . 288 8.5.3 Passing to the limit in the modified optimal control problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 8.6 On homogenization of Dirichlet optimal control problems in perforated domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

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Part II Optimal Control Problems on Periodic Reticulated Domains: Asymptotic Analysis and Approximate Solutions 9

Suboptimal Control of Linear Steady-State Processes on Thin Periodic Structures with Mixed Boundary Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 9.1 A measure-theoretic approach to the description of the network Ωε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 9.2 Statement of the optimal control problem . . . . . . . . . . . . . . . . . . 320 9.3 Convergence in the variable space Z ε . . . . . . . . . . . . . . . . . . . . . 325 9.4 Definition of the limit problem and its properties . . . . . . . . . . . 340 9.5 Main convergence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 9.6 Identification of the limit optimal control problem . . . . . . . . . . 347 9.7 On suboptimal controls for Pε -problems . . . . . . . . . . . . . . . . . . . 350

10 Approximate Solutions of Optimal Control Problems for Ill-Posed Parabolic Problems on Thin Periodic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 10.1 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 10.2 On the solvability of Pε and its C-extension . . . . . . . . . . . . . . . . 362 10.3 On the description of CPε in terms of singular measures . . . . . 369 10.4 Convergence in the variable space Z ε . . . . . . . . . . . . . . . . . . . . . 374 10.4.1 Convergence formalism for Dirichlet boundary controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 10.4.2 w-Convergence of admissible solutions of the C-extended problems . . . . . . . . . . . . . . . . . . . . . . . . 380 10.5 The limiting optimal control problem and its properties . . . . . 381 10.6 Main convergence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 10.7 The limit analysis of the C-extended optimal control problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 10.8 Recovery of the limiting singular optimal control problem Phom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 10.9 On suboptimal controls for Pε -problems . . . . . . . . . . . . . . . . . . . 398 10.10 Optimal control problem for systems on thin lattice structures with blowup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 11 Asymptotic Analysis of Optimal Control Problems on Periodic Singular Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 11.1 ε-Periodic graphlike structures in R2 and their description . . . 410 11.2 Statement of an optimal control problem on ε-periodic graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 11.3 Convergence formalism in the variable spaces associated with ε-periodic graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416

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11.4 Variational convergence of constrained minimization problems on varying graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 11.5 Asymptotic analysis of optimal control problems on ε-periodic graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 11.6 Modeling of suboptimal controls . . . . . . . . . . . . . . . . . . . . . . . . . . 432 11.7 An example of an optimal control problem on ε-periodic square grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 12 Suboptimal Boundary Control of Elliptic Equations in Domains with Small Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 12.1 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 12.2 Reformulation of the original problem in terms of singular measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 12.3 Convergence in the variable space X ε . . . . . . . . . . . . . . . . . . . . . 449 12.4 Definition of a limit problem and its property . . . . . . . . . . . . . . 456 12.5 Convergence theorem and correctors . . . . . . . . . . . . . . . . . . . . . . 456 12.6 Identification of the limiting optimal control problem . . . . . . . . 467 12.7 Optimality conditions for the limit problem and suboptimal controls for Pε -problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 13 Asymptotic Analysis of Elliptic Optimal Control Problems in Thick Multistructures with Dirichlet and Neumann Boundary Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 13.1 Statement of the problem and basic notation . . . . . . . . . . . . . . . 478 13.2 Description of the optimal control problem in terms of singular measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 13.3 The choice of topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 13.4 Definition of the limit problem and its properties . . . . . . . . . . . 491 13.5 Analytical representation of the limit set of admissible solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 13.6 Identification of the limiting cost functional . . . . . . . . . . . . . . . . 502 13.7 Modeling of suboptimal controls . . . . . . . . . . . . . . . . . . . . . . . . . . 506 14 Gap Phenomenon in Modeling of Suboptimal Controls to Parabolic Optimal Control Problems in Thick Multistructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 14.1 On solvability of the original optimal control problems . . . . . . 518 14.2 Formalism of convergence in variable Banach spaces . . . . . . . . 522 14.2.1 The convergence concept for the Paε -problems . . . . . . . . 523 14.2.2 The convergence concept for the Pbε -problems . . . . . . . . 527 14.3 Definition of the limit problems and their properties . . . . . . . . 528 14.4 Analytical representation of the limit sets of admissible solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 14.4.1 Recovery of the set Ξa . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 14.4.2 Recovery of the set Ξb . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538 14.5 Identification of the cost functionals Ia and Ib . . . . . . . . . . . . . . 540

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15 Boundary Velocity Suboptimal Control of Incompressible Flow in Cylindrically Perforated Domains . . . . . . . . . . . . . . . . . 547 15.1 Preliminaries and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551 15.2 Admissible controls and regularity of solutions to the boundary value problem for Navier–Stokes equations . . 552 15.3 On solvability of the optimal boundary control problem . . . . . 554 15.4 Reformulation of the problem (Pε ) . . . . . . . . . . . . . . . . . . . . . . . . 555 15.5 Convergence in the variable space X ε . . . . . . . . . . . . . . . . . . . . . 559 15.6 Definition of suboptimal controls . . . . . . . . . . . . . . . . . . . . . . . . . 563 15.7 Convergence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565 15.8 Identification of the limit optimal control problem . . . . . . . . . . 576 15.9 Suboptimal controls and their approximation properties . . . . . 582 16 Optimal Control Problems in Coefficients: Sensitivity Analysis and Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 16.1 Notation and preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586 16.2 H-Convergence and a counterexample of Murat . . . . . . . . . . . . 590 16.3 Setting of the optimal control problem and existence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593 16.4 H c - and t-admissible domain perturbations for optimal control problems in coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 601 16.5 p-Perturbation of elliptic optimal control problems in coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609 16.6 Mosco-stability of optimal control problems . . . . . . . . . . . . . . . . 612 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635

1 Introduction

The aim of this work is to study the asymptotic behavior of various classes of optimal control problems governed by partial differential equations (PDEs) on domains with a lattice-type structure. The investigation of optimal control problems for such structures is important to researchers working with cellular materials (lightweight materials) such as honeycomb structures, ceramic, metal, or polymeric foams which are used in automotive industry, aircraft design, robotics, and micro-mechanics. Lattice-type materials are also important in life sciences. In particular, bones and, correspondingly, bio-mimetic materials are subject of major research initiatives. Other modern engineering applications are flexible multistructures and civil engineering technologies for transportation. Because of the complicated geometry of lattice-type structures, a direct numerical simulations of solutions, let alone of optimal controls, in such a domain is extremely demanding due to the excessive number of variables. Consequently, model reduction is in order. One way of thinking about model reduction is to consider geometrically motivated or material-related scale parameters that tend to zero in order to achieve a new so-called effective model. In the asymptotic limit, the problem may be lower dimensional, as for the 2D-plate models generated via asymptotic analysis out of 3D elasticity, or they can be higher dimensional, as in the case of thin graphlike structures when the spacing scale tends to zero. Regardless of the dimension of the limiting system versus the dimension of the original model, the limiting problem should be easier to access numerically. In this monograph, asymptotic analysis is the main approach to study optimal control of boundary value problems in such complex domains because it gives not only the possibility to replace the original problems by the corresponding limit problems defined in “simpler” domains but also to use an optimal solution to the limit problem as the basis for the construction of suboptimal controls for the original control problem. As we will see later, asymptotic analysis of optimal control problems is typically not a trivial task. One of main reasons for this is the fact that there is a principle difference between the theory of optimal control problems for systems with distributed parameters and the theory of partial differential P.I. Kogut, G.R. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains, Systems & Control: Foundations & Applications, DOI 10.1007/978-0-8176-8149-4 1, © Springer Science+Business Media, LLC 2011

1

2

1 Introduction

equations. Indeed, let us consider the following example. Let Ω be a bounded open subset of Rn , and let ∂Ω be its boundary. The optimal control problem we consider can be described as follows: minimize the cost functional I= |y(x)|p dx + uL∞ (Ω) (1.1) Ω

subject to the constraints n ∂ ∂y u(x) = f (x), − ∂xi ∂xi i=1

x ∈ Ω,

(1.2)

y|∂Ω = 0,

(1.3)

u ∈ U = { v ∈ L (Ω) | 0 < α ≤ v(x) ≤ β a.e. on Ω} .

(1.4)

∞

Here, p ≥ 2,

f ∈ Lq (Ω),

and p−1 + q −1 = 1.

It is well known that for every fixed control function u ∈ U , the boundary value problem (1.2)–(1.3) admits a unique weak solution y(u) ∈ W01,p (Ω). Hence, the problem (1.2)–(1.3) is well-posed in the Hadamard sense from the theory of boundary value problems point of view. However, the corresponding optimal control problem (1.1)–(1.4) has no solution, in general (see Lions [172] and Murat [193]). In [172], this fact was formulated as an example of an illposed optimization problem. Thus, a well-posed problem in the sense of PDEs turns out to be ill-posed in the sense of optimization theory. On the other hand, let us consider in the same domain Ω the following optimal control problem: minimize the cost functional |y(x)|p dx + δ |u|q dHn−1 (δ > 0) (1.5) I= Ω

∂Ω

subject to the constraints −

n ∂ ∂y = f (x), x ∈ Ω, ∂xi ∂xi i=1 ∂y = u(x), x ∈ ∂Ω, ∂ν ∂Ω u ∈ U = Lq (∂Ω),

(1.6)

(1.7) (1.8)

where ∂y ∂y νi (x), = ∂ν ∂xi i=1 n

p ≥ 2,

f ∈ Lq (Ω), and p−1 + q −1 = 1.

We recall that a pair (u, y) ∈ U × W 1,p (Ω) satisfying the conditions (1.6)– (1.7) is called admissible, and the problem (1.5)–(1.8) is regular if its set of admissible pairs is nonempty.

1 Introduction

3

It is easy to see that the boundary value problem (1.6)–(1.7) is noncoercive unless certain compatibility conditions hold for the data. Hence, we cannot assert that this problem has a solution y(u) for every u ∈ U . So, we deal with a potentially ill-posed problem from the the point of view of PDEs. Nevertheless, the corresponding optimization problem (1.5)–(1.8) is regular, coercive, and, hence, solvable. Indeed, to verify the regularity property, it is enough to consider the problem (1.6) with the Dirichlet boundary condition (1.3). In this case, the existence of a unique solution y ∗ ∈ W01,p (Ω) is well known. Further, ∂ and, taking the corresponding trace on the acting on y ∗ by the operator ∂νA boundary ∂Ω, we obtain ∂y ∗ ∗ u = ∈ W −1,q (∂Ω) ⇒ u∗ ∈ Lq (∂Ω), ∂νA ∂Ω and, hence, the pair (u∗ , y ∗ ) is admissible for the problem (1.5)–(1.8). Thus, the examples given above show that well-posedness for optimal control problems governed by PDEs and well-posedness for PDEs themselves are different notions, and consequently, the corresponding theories are not mutually included. It is expected that this difference also surfaces when it comes to asymptotic analysis of optimal control problems governed by PDEs versus asymptotic analysis of PDEs without controls. Asymptotic analysis of optimal control problems for PDEs is a comparatively modern theory that, as a mathematical discipline, took shape only in the last three decades, whereas the physically motivated ideas of homogenization for PDEs date back at least to [69, 183, 192, 215, 218]. A very good historical record of works related to homogenization until 1975 can be found in [13] and the references therein. Recently, there is renewed interest in such asymptotic results due to the growing importance of multiscale models. Let us now focus on the geometry of the lattice-type structures. Typically, these depend on two parameters mutually related to each other; namely, ε defines the periodicity cell and εh is the thickness of constituting elements of the structure. We have several types of such structures that consist of identical cells, periodically distributed in all directions, reticulated structures (see Fig. 1.1(a)); or in two dimensions, gridworks (see Fig. 1.2(a)); or in only one direction, tall structures (see Fig. 1.1(b)). If the material is concentrated along layers, then we speak about honeycomb structures (Fig. 1.2(b)), and if it is along bars, then we speak of reinforced structures (see Fig. 1.3). Concerning methods for handling such structures in a classical engineering sense, we refer to Gibson and Ashby [114]. The above mentioned 3D lattice-type structures can be described as follows: Let F be a periodic box or rod graph structure in R3 with zero thickness (h = 0) for its components. We assume that F consists of the coordinate planes or axes and their shifts by integer-valued vectors. We call F a singular 3 structure. Let the cube = [0, 1) be the cell of periodicity for F .

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1 Introduction

(a) A lattice plate

(b) A tall structure

Fig. 1.1. Examples of 2D and 1D lattice-type structures

(a) A gridwork

(b) A honeycomb structure

Fig. 1.2. Examples of 3D lattice-type structures

Let F h be the -periodic structure composed of infinite plates (or rods) whose thickness is h > 0, having the corresponding plane (or straight line) from the structure F as the middle plane or median. Fragments of the structures F and F h within the limits of the cell of periodicity are depicted in Fig. 1.2(b) (see also Fig. 1.3). The structure F h is said to be thin. Let ε be a small parameter. We will always suppose that ε varies in a strictly decreasing sequence of positive numbers which converges to 0. Then the lattice-type structure Fεh in R3 can be represented as the homothetic contraction of the structure F h with a factor of ε−1 (i.e., Fεh = εF h ). In the 2D case, we call Fεh the periodic networks (see Fig. 1.1(a)). In this work, we suppose that the thickness εh (0 < εh < ε/2) of constituting elements of Fεh is always supposed to be dependent on ε. As a result,

1 Introduction

5

Fig. 1.3. A reinforced structure

we come to the following definition: We say that a domain Ωε has a periodic lattice-type structure if Ωεh(ε) = Ω ∩ F h (ε),

0 < h(ε)
0. ε→0

At the same time, for the thin domains the parameters ε and h = h(ε) are related by the supposition h(ε) → 0 as ε → 0. As we will see later, the asymptotic behavior of the optimal control problems essentially depends on how h(ε) tends to h∗ as ε → 0. In view of this, we introduce the classifications of lattice-type structures as they were proposed by Zhikov [257]: If h(ε) is said to be a domain with thin periodic struclimε→0 h(ε) = 0, then Ωε h(ε) ture; otherwise, Ωε is a periodically perforated domain. A periodic structure is characterized as follows: (i)

Sufficiently thick, when lim h(ε)ε−1 = ∞.

ε→0

(ii) Sufficiently thin, when lim h(ε)ε−1 = 0.

ε→0

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1 Introduction

(iii) A structure of critical thickness, when lim h(ε)ε−1 = θ > 0.

ε→0

Given this characterization, the main problem is to relate the geometry of lattice-type structures to the properties of the solutions to different classes of optimal control problems that can be considered in these domains. Note that, as the small parameter ε varies, it is often possible to foresee some “limit” behavior and “guess” that we may substitute the complex, degenerate problems we started with by a new one, simpler and with a more comprehensible behavior, possibly of a completely different type. Since the numerical computation of the optimal states and controls on each ε-level is very costly due to the complexity of the geometry of lattice-type structures, the main question is to study the asymptotic behavior of optimal solutions as the small parameter ε tends to 0 and to view the approximation as a suboptimal control for the original problem. Hence, the crucial point of this book is the development of methods for the regularization and approximation of optimal solutions in lattice-type structures using the properties of the limit optimal control problems. We decompose the content of this monograph in two parts. The first part gives an up-to-date and in-depth, mathematically self-contained account on asymptotic analysis of optimal control problems on a general level, whereas the second part concentrates on more specific problems on periodic latticetype structures. In the first part, we dispense with providing proofs for most of the results that are considered well known, providing detailed proofs only for those results that are typically not contained in a textbook, whereas in the second part, we do provide proofs for most of the results, some of which have not been published elsewhere. We now describe the content of this book in more detail. Part I. Basic Tools of Asymptotic Analysis of Optimal Control Problems. In Part I, we follow, as a guideline, the variational treatment of the main common boundary value problems and related optimal control problems. We introduce basic elements of variational analysis which allow one to study these problems for thin and reticulated structures and the closely related questions of their asymptotic analysis. Chapter 2 contains an in-depth exposition of the main notions, concepts, and results which play a fundamental role in the variational approach to the asymptotic analysis of optimal control problems. To this end, we recall some main concepts and basic results from functional analysis, measure theory, Sobolev spaces, boundary value problems, and capacity theory, which are important constituents of modern mathematical PDE control theory. Chapter 3 provides an exposition of the mathematical theory of parameterized optimal control problems. We discuss the general setting of optimal control theory for PDEs, or distributed parameter systems, in variable spaces.

1 Introduction

7

We show the importance of the Direct Method of Calculus of Variation and emphasize its basic topological ingredients: lower semicontinuity, coercivity, and inf-compactness. We discuss the topological properties of solutions, optimality conditions for a wide class of extremal problems in the form of variational inequalities, and also questions related to the construction of approximative solutions to the different classes of optimal control problems. Our main interest in Chap. 4 is the regularization and approximation of solutions to general optimal control problems. We provide precise definitions of ε-approximate solutions, weak ε-approximate solutions, weakened ε-approximate solutions, ε-suboptimal solutions, and approximate solutions of such problems. We also provide illustrative examples. Chapter 5 deals with the main object of this book, namely parameterized optimal control problems depending on some small parameter ε. We discuss different approaches to the study of the asymptotic behavior of a wide class of optimal control problems when the parameter ε tends to 0. In Chap. 6, we describe the main convergence concepts in variable Banach spaces. The aim of this chapter is to give a systematic exposition of the main properties of weak and strong convergence in variable Lp - and W 1,p -spaces for p > 1 with respect to Radon measures which can be associated with different classes of thin and reticulated structures. The objective of Chap. 7 is to study problems where the knowledge of “the limit set” is important. We discuss different analytical frameworks and concepts that form the basis of set convergence in variable spaces. In this context, we introduce different notions of limits for sequences of nonempty sets and give some applications to the theory of thin periodic and reticulated structures. Chapter 8 provides an exposition of variational methods for constrained minimization problems in varying spaces. We show that in many practical situations it is more appropriate to study the asymptotic behavior of the family of minimum problems by defining an appropriate limit minimum problem, which also serves as a “good approximation” to the original problem, rather than via the study of the “limit” properties of their solutions. As an example, we study the asymptotic behavior of an optimal control problem for linear elliptic equation with partial exact controllability constraints. We show that this problem can be viewed as a particular example covered by the approach described in this chapter. Part II. Approximate Solutions of Optimal Control Problems on Periodic Lattice Structures. The second part of this monograph includes Chaps. 9 through 16 and deals with the second objective which is to present new results in optimal control theory for PDEs on reticulated structures and networks. In Chap. 9, we study an optimal control problem for a linear steady-state process on a thin periodic structure with mixed boundary controls. We focus our attention on optimal control problems with different types of control and state constraints. Having provided the asymptotic analysis of such problems

8

1 Introduction

as the parameter of periodicity ε tends to 0, we construct so-called asymptotically suboptimal controls with respect to the original problems. For the sake of simplicity, we restrict our analysis to the case of a 2D lattice structure. It is well known that homogenization of most of the boundary value problems on thin structures with Dirichlet and Neumann boundary conditions reveals a classical form: If strong convergence of the corresponding solutions holds, the limit function does not contain an oscillating variable and the limit problem does not depend on the type of the thin structure. In contrast to this, we show that for the considered optimal control problems, the structure of the corresponding limit problem depends essentially on how the thickness parameter h(ε) tends to 0 as ε → 0; we have a so-called “scaling effect.” The next observation deals with the “structural stability” of the limiting optimal control problem. We show that the passage to the limit in the original problem, with boundary controls of two different types, leads to an optimal control problem with only Dirichlet controls. The Neumann boundary control can only appear in the limit problem if the thin domain has a critical thickness ξ ∗ = θ > 0. In Chap. 10, we focus on control objects which are described by singular parabolic equations with Robin boundary conditions at the boundary of holes, and with two different types of boundary controls – Dirichlet and Neumann controls – on the external boundary of the thin periodic structure Ωε . Having admitted that a blowing-up phenomenon can appear in the original problem, we provide its asymptotic analysis, as the parameter ε tends to 0. It is shown that the structure of the limiting problem depends essentially on how the thickness h(ε) tends to zero as ε → 0 (“scaling effect”). The characteristic feature of this control problem is the fact that the original initial-boundary value problem has no global solution for some admissible controls. This is in contrast to the assumption which usually plays a fundamental role in classical approaches of the homogenization theory for PDEs. In this chapter, we propose an “indirect approach” to the asymptotic analysis of boundary optimal control problems for singular systems. The main idea is to introduce a couple of new optimal control problems for parabolic equations on the reticulated structure Ωε . The first of them is a virtual extension of the original problem. Having introduced a fictitious “distributed” control into the original problem and having modified its cost functional, we prove that the new problem always has a nonempty set of admissible solutions (in contrast to the original one), and moreover that for every ε > 0, there is a oneto-one correspondence between the sets of optimal solutions for the original and virtual problems. The second problem is the so-called C-extension of the virtual problem. The consideration of this problem is motivated by the fact that the state equation of the C-extended problem has a linear structure. It is clear that in the linear case, the C-extended problem always coincides with the original one. As a result, we propose a new scheme for the asymptotic analysis of optimal control problems for parabolic equations with potential blowup.

1 Introduction

9

By analogy with the previous chapter, we show that for optimal control problems with Robin conditions at the boundary of holes, the structure of the corresponding limit problem depends essentially on how the parameter h(ε) tends to 0 as ε → 0; in particular for critical thickness, some additional terms appear in the limiting state equation. At the same time, we do not have any such artifacts if the thin structure Ωε is sufficiently thick. However, as for sufficiently thin domains Ωε , a limiting optimal control problem does not exist, in general. However, in this case, the asymptotic analysis is possible only if homogeneous Neumann conditions are given instead of the Robin conditions ε along the boundary Sint . We also derive conditions under which in the limit we do not obtain an optimal control problem, but rather some initial-boundary value problem with or without controls. In conclusion, we construct asymptotically suboptimal controls for the original problem and show an approximation property of such controls for small enough ε. Another important aspect of asymptotic methods concerns the study of optimal control problems on singular structures, like periodic graphs. In Chap. 11, we examine the statements and properties of different classes of optimization problems for 1D PDEs on periodic graphs. We concentrate on the asymptotic analysis of optimal control problems for 1D elliptic equations on periodic graphs with “distributed” and “boundary” controls, as the period ε of the graph tends to 0. Note that if the small parameter ε is changed, then all components of the original control problem, including the ε-periodic graph Ωε , the control constraint sets, the cost functional, and the set, where we seek its infimum, are changed as well. This means that the original problem (on the graph) for different values of the parameter ε “lives” in different function spaces. Moreover, the Lebesgue measure of the “material” included in the periodic graph Ωε is equal to 0 for every ε > 0, whereas there exists a set Ω which is filled up by this planar graph in the limit, as ε → 0. Using approaches of variational convergence of minimization problems in variable spaces and relying on the description of optimal control problems for planar networks in terms of singular measures, we show that the problem on the periodic graph tends to some standard optimal control problem for a 2D system, and its solution can be used as suboptimal control to the original one. Our prime interest in Chap. 12 deals with the construction of suboptimal solutions to a class of boundary optimal control problems in ε-periodically perforated domains with small holes. We suppose that the support of the controls is contained in the boundaries of the holes. This set is divided into two parts: On one part the controls are of Dirichlet type, whereas on the other part the controls are of Neumann type. Using the ideas of the theory of Γ -convergence and the concept of variational convergence of constrained minimization problems, we show that the limit problem can be recovered in an explicit analytical form. However, in contrast to the case of thin periodic structures (see Chap. 9), the control through the boundary of small holes leads us in the limit to an optimal control problem with drastically different properties.

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1 Introduction

In particular, we show that in this case, a “strange term” appears both in the limit equation and in the cost functional, and these terms depend on the geometry of the holes. Moreover, the characteristic feature of the obtained limiting control problem is also the fact that it contains two independent distributed control functions which can be used as suboptimal controls for the original one. In Chap. 13, we study the asymptotic behavior of an optimal control problem for an elliptic equation in a thick multistructure with different types and classes of admissible boundary controls. This thick multistructure consists of a domain (the domain of a basis structure) and a large number of ε-periodically situated thin cylinders. We mentioned in the Preface that such structures are very important in certain applications where processes are studied (e.g., in nanoreactors). We consider two types of boundary controls – namely Dirichlet H 1/2 -controls on the bases Γε of thin cylinders, and Neumann L2 -controls on their vertical sides. We present some ideas and results concerning the asymptotic analysis for such problems as ε → 0 and derive conditions under which the homogenized problem can be recovered in an explicit form. We show that the mathematical description of the limit optimal boundary control problem is different from the original one. These differences appear not only in the control constraints, limit cost functional, state equations, and boundary conditions, but also in the type of admissible controls for the limit problem. In Chap. 14, we study the asymptotic behavior of a parabolic optimal control problem in a domain Ωε ⊂ Rn whose boundary ∂Ωε contains a highly oscillating part. We consider this problem with two different classes of Dirichlet boundary controls and provide its asymptotic analysis with respect to different topologies of variational convergence. It is shown that the mathematical descriptions of the limiting optimal control problems have different forms and these differences appear not only in the state equation and boundary conditions, but also in the control constrains and the limit cost functional. In Chap. 15, we focus on suboptimal controls for an optimal boundary control problem for the 3D steady-state Navier–Stokes equation in a cylindrically perforated domain Ωε . The control is the boundary velocity field supported on the “vertical” sides of the thin cylinders. We minimize the vorticity of a viscous flow through the perforated domain. We show that an optimal solution to the limit problem in a nonperforated domain can be used as a basis for the construction of suboptimal controls for the original control problem. It is worth noticing that the limit problem may take the form of either a variational calculation problem or an optimal control problem for Brinkman’s law with another cost functional, depending on the cross-section of the thin cylinders. The last topic, considered in Chap. 16, is a classical Dirichlet optimal control problem for an elliptic equation with controls in the coefficients taken in L∞ (Ω). Since problems of this type have no solutions in general, we make a special assumption on the coefficients of the state equation and introduce the class of the so-called solenoidal controls. Using the direct method of cal-

1 Introduction

11

culus of variations, we prove an existence result for this problem. We also provide the sensitivity analysis of this problem with respect to domain perturbations. With this aim, we introduce the concept of the Mosco-stability for such problems and study the variational properties of Mosco-stable problems with respect to different types of domain perturbations. We show that this is one of the possible ways for the approximation of optimal solutions in domains with a complicated geometry. It is clear that such problems are of great importance in material optimization, which is an emerging field for PDE-constrained optimization.

Part I

Asymptotic Analysis of Optimal Control Problems for Partial Differential Equations: Basic Tools

2 Background Material on Asymptotic Analysis of Extremal Problems

This chapter is intended to provide various facts, notions, and concepts which play a fundamental role in modern asymptotic analysis of optimization problems. We recall some main concepts and basic results of measure theory, Sobolev spaces, and boundary value problems which are used later. We include proofs only if the line of arguments is of importance for the understanding of subsequent remarks. For a deeper insight in the subject, we refer to the books of Adams [2], Bucur and Buttazzo [38], Evans and Gariepy [106], Kantorovich and Akilov [128], Lions and Magenes [173], Maz’ya [185], Yosida [251], Ziemer [267], and so on.

2.1 Measure theory and basic notation Let Ω ⊂ Rn be a nonempty set. We say that a collection E of subsets of Ω is a σ-algebra on Ω if

∅ ∈ E,

Ω \ A ∈ E whenever A ∈ E,

Ak ∈ E whenever Ak ∈ E for every k ∈ N.

k∈N

Given a σ-algebra on Ω, we say that the pair (Ω, E) is a measure space. We denote by B(Ω) the intersection of all σ-algebras on Ω containing the open subsets of Ω. It turns out that B(Ω) is actually the smallest σ-algebra on Ω containing the open subsets of Ω, and it is called the σ-algebra of Borel subsets of Ω and its elements are called Borel sets. Let (Ω, B(Ω)) be a Borel measure space. We define measures as set functions. Definition 2.1. A function μ : B(Ω) → R is a Borel measure on Ω (or simply a measure) if μ(∅) = 0 and μ is countably additive in the sense that Ak , Ak ∩ Aj = ∅ if k = j ⇒ μ(A) = μ(Ak ). (2.1) A= k∈N

k∈N

P.I. Kogut, G.R. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains, Systems & Control: Foundations & Applications, DOI 10.1007/978-0-8176-8149-4 2, © Springer Science+Business Media, LLC 2011

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16

2 Background Material on Extremal Problems

The set of such measures will be denoted by M(Ω). We also say that a Borel measure is positive if it takes its values in [0, ∞). The set of positive Borel measures will be denoted by M+ (Ω). We observe that, in the case of measures, the series in (2.1) must necessary converge absolutely, since the union on the left-hand side of (2.1) does not depend on the order in which the sets A1 , A2 , . . . are listed. Let μ : 2Ω → R be a set function. We define the restriction μLA of μ to A ⊂ Ω by μLA(B) = μ(A ∩ B) for all B ∈ 2Ω . Definition 2.2. A positive Borel measure on Ω that is finite on each compact subset of Ω is said to be a Radon measure on Ω. The restriction of the Lebesgue measure to B(Rn ) is a classical example of a Radon measure on Rn . Note also that the Lebesgue measure on Rn can be defined as the unique positive Radon measure Ln on Rn satisfying Ln ([0, 1]n ) = 1 and Ln (a + tA) = tn Ln (A)

∀ a ∈ Rn , A ∈ B(Rn ), t > 0. (2.2)

We denote |A| = Ln (A). Definition 2.3. For μ ∈ M(Ω) and A ∈ B(Ω), we define the total variation of μ on A by |μ(Ak )| : A = Ak , Ak ∩ Aj = ∅ if k = j . |μ|(A) = sup k∈N

k∈N

It is well known that the total variation of a measure is a positive measure taking only finite values |μ(A)| ≤ |μ|(A) for all A ∈ B(Ω), and the total variation can be viewed as a norm on the set of measures on Ω. We will indicate by Mb (Ω) the space of Radon measures on Ω with finite total variation. Note that Mb (Ω) is a Banach space (i.e., a complete linear normed space) when equipped with the norm μ Mb (Ω) := |μ|(Ω). Definition 2.4. The support of μ ∈ M(Ω) is defined as spt μ = {x ∈ Ω : |μ|(Br (x)) > 0 for all open balls Br (x) ⊂ Ω} . Theorem 2.5. Every measure μ ∈ M+ (Ω) is regular in the following sense: μ(A) = inf {μ(B) : A ⊂ B, B open} , μ(A) = sup {μ(C) : C ⊂ A, C closed} for all A ∈ B(Ω). Note that approximating closed sets with compact sets, we also have μ(A) = sup {μ(C) : K ⊂ A, K compact} .

(2.3) (2.4)

2.1 Measure theory and basic notation

17

Definition 2.6. Let μ ∈ M+ (Ω) and λ ∈ M(Ω). We say that λ is absolutely continuous with respect to μ (and we write λ μ) if λ(A) = 0 for every A ∈ B(Ω) with μ(A) = 0. We say that λ is singular with respect to μ, if there exists a set E ∈ B(Ω) such that μ(E) = 0 and λ(A) = 0 for all A ∈ B(Ω) with A ∩ E = ∅ (in this case, we say that λ is concentrated on E). For μ ∈ M(Ω) we adopt the usual notation Lp (Ω, dμ) to indicate the space of p-summable functions with respect to μ on Ω. We omit μ if it is the Lebesgue measure. Let us observe that if f ∈ L1 (Ω, dμ) and μ ∈ M(Ω) then we can define the measure f μ ∈ M(Ω) by f μ(A) = f dμ. A

Hence, f μ |μ| and |f μ| = |f ||μ|. In particular, if λ μ, then λ = f μ for some f ∈ L1 (Ω, dμ). Theorem 2.7. (Radon–Nikodym) For λ ∈ M(Ω) and μ ∈ M+ (Ω) there exists a function f ∈ L1 (Ω, dμ) and a measure λs , singular with respect to λ, such that λ = f μ + λs . This relation is called the Radon–Nikodym decomposition of λ with respect to μ. Definition 2.8. Let μ = Ln and let f ∈ L1 (Ω). Then each point x ∈ Ω with 1 lim |f (x) − f (y)| dμ(y) = 0 r→0+ μ(Br (x)) B (x) r is called a Lebesgue point for f . Note that the set of Lebesgue points of f depends on the particular choice of the representative in the equivalence class of L1 (Ω). Hence, we will always take a particular choice of the representative of f whenever we consider Lebesgue points. The expression “μ a.e.” means “almost everywhere with respect to the measure μ” – that is, except possibly on a set A with μ(A) = 0. 2.1.1 Hausdorff measures We next introduce certain “lower-dimensional” measures on Rn , which allow one to measure certain “very small” subsets of Rn . The idea is that A is an “s-dimensional subset” of Rn if there is a so-called Hausdorff measure Hs such that 0 < Hs (A) < +∞ even if A is very complicated geometrically. To do so, we will construct a positive measure Hs on Rn from set functions, following a procedure due to Carathéodory, which we briefly recall.

18


Definition 2.9. A set function λ : 2Ω → [0, +∞] is an outer measure if λ(∅) = 0 and λ is countably subadditive, that is, λ(Bk ) for B ⊆ Bk . λ(B) = k∈N

k∈N

We say that a set M is measurable for λ if λ(B) = λ(B ∩ M ) + λ(B \ M )

∀ B ⊆ Ω.

Let Mλ be a family of all measurable sets for λ. If λ is an outer measure, then Mλ is a σ-algebra and λ|Mλ is countably additive (see Evans [106]). So, if B(Ω) ⊆ Mλ and λ(Ω) < +∞, then μ = λ|Mλ ∈ M+ (Ω). In order to see that an outer measure generates a measure by the above construction, we have to prove that Borel sets are measurable. The following proposition provides a simple criterion for measurability. Proposition 2.10. Let λ be an outer measure. Then Borel sets are measurable for λ if and only if dist (A, B) > 0

⇒

λ(A) + λ(B) = λ(A ∪ B)

(2.5)

for all A, B ⊆ Ω, where dist (A, B) = inf {|x − y| : x ∈ A, y ∈ B}. We now apply Carathéodory’s construction to define the measure which will be of use in the sequel. Definition 2.11. Let α ≥ 0 and δ > 0. For all E ⊂ Rn we define the preHausdorff measure Hδα of E as ωα α α (diam Ek ) : diam Ek < δ, E ⊆ Ek , Hδ (E) = α inf 2 k∈N

k∈N

+∞ where ωα = π α/2 /Γ (α/2 + 1) and Γ (α) = 0 sα−1 exp(−s) ds is the Euler function, which coincides with the Lebesgue measure of the unit ball in Rn if α is integer. Note that Hδα (E) is decreasing in δ. Consequently, the limit Hα (E) = lim Hδα (E) = sup Hδα (E) δ→0+

δ>0

exists and is in [0, +∞]. The value Hα (E) is called the α-dimensional Hausdorff measure of E. Remark 2.12. Let us note the following: (a) Hα is a regular Borel measure (0 ≤ α < +∞). (b) Hα is the null measure if α > n.

2.2 Sobolev spaces and boundary value problems

19

(c) Hα (λA) = λα Hα (A) for all λ > 0, A ⊂ Rn . (d) If Hα (A) < ∞, then Hβ (A) = 0 for all 0 ≤ α < β < ∞. (e) If Hβ (A) > 0, then Hα (A) = +∞ for all 0 ≤ α < β < ∞. Thus, if α < n, then Hα (A) = +∞ for all nonempty open sets A ⊂ Rn , and Hα (A) agrees with the ordinary “k-dimensional surface area” on nice sets A ∈ B(Rn ). Thus, Hα is not a positive Radon measure on Rn if α < n, since Rn is not σ-finite with respect to Hα . However, if Hα (B) < +∞ for some B ∈ B(Ω), then Hα LB ∈ M+ (Ω). Moreover, it can be proved that Hn = Ln in Rn .

2.2 Sobolev spaces and boundary value problems Here, we briefly outline some basic facts from the theory of Sobolev spaces and the theory of boundary value problems, which are widely used throughout the book. Most of these result are well known; therefore, we just formulate them without proof. For details and complete proofs, the reader may turn to the numerous textbooks on the subject – e.g., Adams [2], Kantorovich and Akilov [128], Lions and Magenes [173], Maz’ya [185], Sobolev [232], Smirnov [231], and so forth. 2.2.1 Weak derivatives Let Ω be a bounded open subset of Rn , k ∈ N, let C k (Ω) be the space of k-times continuously differentiable functions u : Ω → R, and let C0k (Ω) be the functions in C k (Ω) with compact support in Ω. Hence, C0∞ (Ω) is the set of all real-valued infinitely differentiable functions with compact support in Ω. In this case, for any x0 ∈ Ω and ε > 0 small enough, the function ⎧

1 ⎨ x − x0 if x Rn < 1, exp − ε (x) = , where (x) = 1 − x 2Rn ⎩ ε 0, otherwise, is in C0∞ (Ω). The support of ε is the ball with center x0 and radius ε. We define the set C0 (Ω) as the closure of C0∞ (Ω) in the uniform topology. It is a separable Banach space if equipped with the · ∞ -norm. Denote by Lp (Ω), 1 ≤ p < +∞, the space of functions defined on Ω and pth-power summable in the sense of Lebesgue. An element of Lp (Ω) is an equivalence class of functions different only on a set of zero Ln -measure. The space Lp (Ω) equipped with norm u Lp (Ω) = Ω

is a Banach space.

1/p |u| dx p

20


Introduce the space L∞ (Ω) of essentially bounded (i.e., bounded by a constant almost everywhere) real-valued functions in Ω that are Lebesgue measurable. The space L∞ (Ω) is equipped with the norm u L∞ (Ω) = esssup |u(x)| = x∈Ω

inf

sup |u(x)|;

A⊂Ω, |A|=0 x∈Ω\A

that is, “ess sup” means the supremum up to a set of zero measure. In the case of a bounded domain Ω, we have u L∞ (Ω) = lim u Lp (Ω) , p→∞

∞

which justifies the notation L (Ω). Suppose α = (α1 , . . . , αn ) is a multi-index of order |α| = α1 + · · ·+ αn = k. We call a function ϕ belonging to C0∞ (Ω) a test function. We say that v ∈ L1 (Ω) is the αth-weak partial derivative of u ∈ L1 (Ω), denoted D α u = v, provided uDα ϕ dx = (−1)|α| vϕ dx (2.6) Ω

Ω

for all test functions ϕ ∈ C0∞ (Ω). If a vector v = {v1 , . . . , vn }, vi ∈ L1 (Ω), is the gradient of a function u ∈ L1 (Ω) in the weak sense, then we denote it either by ∇u or by ∂u/∂x. 2.2.2 Sobolev spaces Having fixed 1 ≤ p < +∞ and k, we denote by W k,p (Ω) the Sobolev space formed by all functions u ∈ Lp (Ω) such that for each multi-index α with |α| ≤ k, Dα u exists in the weak sense and the norm u W k,p (Ω)

⎛ =⎝

⎞1/p p |Dα u(x)| dx⎠

(2.7)

Ω |α|≤k

is finite. If p = 2, we usually write H k (Ω) = W k,2 (Ω) (k = 0, 1, . . . ). By convention we set H 0 (Ω) = L2 (Ω) and D0 v = v. The letter H is used, since H k (Ω) is a Hilbert space equipped with the scalar product Dα u1 (x)Dα u2 (x) dx. (u1 , u2 )H k (Ω) = |α|≤k

Ω

The space W0k,p (Ω) is the closure of the set C0∞ (Ω) in W k,p (Ω). The space = W01,2 (Ω) is naturally associated with the Dirichlet problem, since the inclusion u ∈ H01 (Ω) represents an equivalent formulation of the boundary condition u|∂Ω = 0. To clarify the presentation further, we will consider only the Sobolev space W 1,p (Ω), which is a separable Banach space for the norm (2.7) and reflexive if 1 < p < +∞.

H01 (Ω)


21

We recall that a Banach space X is said to be compactly embedded in a Banach space Y provided (i) x Y ≤ C x X (x ∈ X) for some constant C and (ii) each bounded sequence in X is precompact in Y . The following definition expresses in precise terms the intuitive notion of a regular boundary (C ∞ , C k , or Lipschitz). Definition 2.13. (i) Let Ω ⊂ Rn be open and bounded. We say that Ω is a bounded open set with C k , k ≥ 1, boundary, if for every x ∈ ∂Ω, there exist a neighborhood U ⊂ Rn of x and a one-to-one and onto map T : Q → U , where Q = {x ∈ Rn : |xj | < 1, j = 1, 2, . . . , n} , T ∈ C k (Q), T −1 ∈ C k (U ), T (Q+ ) = U ∩ Ω, T (Q0 ) = U ∩ ∂Ω with Q+ = {x ∈ Q : xn > 0} and Q0 = {x ∈ Q : xn = 0}. (ii) If T is in C k,α , 0 < α ≤ 1, we will say that Ω is a bounded open set with C k,α boundary. (iii) If T is in C 0,1 , we will say that Ω is a bounded open set with Lipschitz boundary. Thus, an open set Ω ⊂ Rn has Lipschitz boundary if for every x ∈ ∂Ω, there exists a neighborhood U ⊂ Rn of x such that U ∩ ∂Ω is a graph, in a suitable coordinate system, of a Lipschitz continuous function whose epigraph contains U ∩ Ω; see Fig. 2.1. Note that every polyhedron has a Lipschitz boundary, whereas the unit ball in Rn has a C ∞ boundary. If Ω is an open convex set, then Ω has a Lipschitz boundary as well. If Ω has a Lipschitz boundary, then for Hn−1 -a.e. x ∈ ∂Ω, there exists the outward unit vector normal to ∂Ω, which we denote by nΩ (see Neˇcas [205]).

Fig. 2.1. A regular boundary

22


Suppose that Ω ⊂ Rn is a bounded open set with Lipschitz boundary. Then, by the Sobolev embedding theorem and the Rellich–Kondrachov theorem (see Adams [2], Maz’ya [185]), we have the following: • If 1 ≤ p < n, W 1,p (Ω) → Lq (Ω) with – compact embedding for q ∈ [1, p∗ ), where

1 1 1 = − , and ∗ p p n

– continuous embedding for q = p∗ ; • If p = n, W 1,p (Ω) → Lq (Ω) with compact embedding for q ∈ [1, ∞); • If p > n, W 1,p (Ω) → C 0 (Ω) with compact embedding. Hereafter we call the constant p∗ = np/(n − p) the Sobolev conjugate of p. Note that in all cases (i.e., 1 ≤ p ≤ ∞), the embedding of W 1,p (Ω) in p L (Ω) is compact. These results are still valid for W01,p (Ω) without any regularity assumption on ∂Ω. However, examples of domains Ω between two spirals can be constructed where the embedding W 1,p (Ω) in Lp (Ω) is not compact [91]. If Ω is unbounded, then the compactness of that embeddings is lost. Indeed, as in P. L. Lions [174, 175], we concentrate on the case when Ω = Rn . Theorem 2.14. Assume that for n ≥ 3, fk → f strongly in L2loc (Rn ),

Dfk Df in

n

L2 (Rn )

.

Suppose further that |Dfk |2 μ in M(Rn ),

∗

|fk |2 ν in M(Rn ).

Then there exist an at most countable index set J, distinct points {xj }j∈J ⊂ Rn , and non-negative weights {μj , νj } such that ∗ νj δxj , μ ≥ |Df |2 + μj δxj , ν = |f |2 + j∈J

j∈J

where δxj ∈ Mb (Rn ) denotes the Dirac measure located at the point xj . Note that a typical function y ∈ W 1,p (Ω) is not, in general, continuous and is only defined almost everywhere on Ω. Since ∂Ω has n-dimensional Lebesgue measure 0, there is no direct meaning we can give to the expression “u restricted to ∂Ω.” The notion of a trace operator resolves this problem – namely if ∂Ω is Lipschitz continuous, then there exists a unique linear continuous map γ : W 1,p (Ω) → Lp (∂Ω) such that for any y ∈ W 1,p (Ω) ∩ C(Ω), one has γ(y) = y|∂Ω . The function γ(y) is called the trace of y on ∂Ω. Remark 2.15. It is worth noting that when Ω is an open set with cusps on the boundary, then the existence of the trace operator with the above properties may fail. Indeed, consider, for instance, in R2 the domain


23

Ω = (x1 , x2 ) : 0 < x1 < 1, 0 < x2 < x51 . Then for the function u(x1 , x2 ) = 1/x1 , we have ∂u 1 = − 2, ∂x1 x1

∂u = 0 in Ω ∂x2

in the distributional sense. So 2 1 1 2 u + |∇u| dx = + 4 dx1 dx2 x22 x1 Ω Ω x51 1 1 1 = dx1 + 4 dx2 x22 x1 0 1 1 3 3 = x1 + x1 dx1 = . 4 0 Thus, u ∈ H 1 (Ω), but the trace of u on ∂Ω does not belong to L2 (∂Ω) since 1 2 x1 ∈ L (0, 1). It is clear now that the space H01 (Ω) can be defined as H01 (Ω) = u ∈ H 1 (Ω), γ(u) = 0 . However, we note that γ is not onto Lp (∂Ω) (i.e., there exist functions in Lp (∂Ω) which are not traces of any element of W 1,p (Ω)). In particular, if p = 2, then this leads us to the following set H 1/2 (∂Ω) := γ(H 1 (Ω)), where H 1/2 (∂Ω) is a Banach space with respect to the norm u H 1/2 (∂Ω) =

u 2L2 (∂Ω) +

∂Ω

1/2

∂Ω

|u(x) − u(y)|2 dx dy |x − y|n+1

1/2

2

with compact embedding H (∂Ω) → L (∂Ω). As mentioned earlier, if ∂Ω is Lipschitz continuous, then the unit outward normal vector n = (n1 , . . . , nn ) to Ω is well defined almost everywhere. As a result, the well-known Green formula for smooth functions can be extended to Sobolev spaces. Indeed, in this case we have ∂v ∂u u dx = − v dx + γ(u)γ(v) ni ds, i = 1, . . . , n, Ω ∂xi Ω ∂xi ∂Ω for any u, v ∈ H 1 (Ω). For any bounded domain Ω, the Friedrichs inequality 2 y dx ≤ C |∇y|2 dx, ∀ y ∈ H01 (Ω), Ω

(2.8)

Ω

holds with a constant C independent of y. Inequality (2.8) implies that the 1/2 can be taken as an equivalent norm in functional y 1 = Ω |∇y|2 dx H01 (Ω), and, indeed, we will always consider y 1 as a norm in this space.

24


Remark 2.16. It should be stressed that we need to impose a condition of the type y = 0 on ∂Ω (which comes from the hypothesis y ∈ H01 (Ω)) to avoid constant functions y (which imply ∇y = 0), otherwise inequality (2.8) would be trivially false. If u ∈ H 1 (Ω) and Ω is a bounded connected open domain with Lipschitz boundary, then the Poincaré inequality 2

y 2 dx ≤ C Ω

y dx

|∇y|2 dx ,

+

Ω

∀ y ∈ H 1 (Ω),

(2.9)

Ω

is valid, and, by the Rellich–Kondrachov compactness theorem, the imbedding H 1 (Ω) → L2 (Ω) is compact. Sometimes the Poincaré inequality appears in the following form. If 1 ≤ p ≤ ∞ and if we set 1 y(x) dx, MΩ (y) = |Ω| Ω then there exists a constant C(Ω, p) > 0 so that y − MΩ (y) Lp (Ω) ≤ C(Ω, p) ∇y Lp (Ω) ,

∀ y ∈ W1,p (Ω).

The dual space of H01 (Ω) (i.e., the set of all continuous linear functionals on H01 (Ω)) is denoted by H −1 (Ω). If f is an element of H −1 (Ω), then f, yH −1 (Ω),H 1 (Ω) stands for the value of the functional f applied to the ele0 ment y ∈ H01 (Ω). Note that for any element F∈ H −1 (Ω), there exist n + 1 n functions f0 , f1 , . . . , fn such that F = f0 + k=1 ∂fk /∂xk in the sense of distributions, that is, F, yH −1 (Ω),H 1 (Ω) = 0

f0 y dx − Ω

n k=1

Moreover, F 2H −1 (Ω) = inf

n

fk Ω

∂y dx. ∂xk

fk 2L2 (Ω) ,

k=0

where the infimum is taken over all vectors n+1 (f0 , f1 , . . . , fn ) ∈ L2 (Ω) such that the representation for F given above holds true. One can give an example of a linear functional on H01 (Ω), setting f0 ϕ dx, f0 ∈ L2 (Ω). f0 , yH −1 (Ω),H 1 (Ω) = 0

Ω

Similarly, if ϕ ∈ H 1/2 (∂Ω) and f ∈ L2 (∂Ω), one also has


25

f, y(H 1/2 (∂Ω))∗ ,H 1/2 (∂Ω) =

f ϕ ds. ∂Ω

Suppose that ∂Ω is Lipschitz continuous. Then one has L2 (Ω) ⊂ H −1 (Ω) with compact injection. Hence, the following embeddings are compact (socalled Gelfand–Lions triplet of Sobolev spaces): H01 (Ω) → L2 (Ω) → H −1 (Ω). Remark 2.17. It is clear that the restriction of any element of (H 1 (Ω))∗ to H01 (Ω) is in H −1 (Ω). However, the dual space (H 1 (Ω))∗ is not contained in H −1 (Ω) since (H 1 (Ω))∗ can be identified with the direct sum H −1 (Ω) ⊕ ∗ H −1/2 (∂Ω), where H −1/2 (∂Ω)= (H 1/2 (∂Ω)) if ∂Ω is Lipschitz 2 .Moreover, n continuous, y ∈ H(Ω, div) = y : y ∈ L (Ω) , div y ∈ L2 (Ω) , and w ∈ H 1 (Ω), then y · n ∈ H −1/2 (∂Ω), the map H(Ω, div) y → y · n ∈ H −1/2 (∂Ω) is linear and continuous, and − (divy)w dx = y · ∇w dx + y · n, wH −1/2 (∂Ω),H 1/2 (∂Ω) . Ω

(2.10)

Ω

n For any vector p ∈ L2 (Ω) = L2 (Ω) , the divergence is an element of the space H −1 (Ω) defined by p · ∇ϕ dx, ∀ ϕ ∈ H01 (Ω), (2.11) div p, ϕH −1 (Ω),H 1 (Ω) = − 0

Ω

where “·” denotes the scalar product of two vectors. The following estimate is evident: −1 div p H (Ω) = sup p · ∇ϕ dx ≤ p L2 (Ω) . (2.12) ϕ1 =1

Ω

A vector field p is said to be solenoidal if div p = 0. We say that a vector field v ∈ L2 (Ω) is potential if v can be represented in the form v = ∇u, where u ∈ H01 (Ω). 2.2.3 Vector-valued spaces of the type Lp (a, b; X) Let X be a Banach space, Ω ⊂ Rn , and p such that 1 ≤ p ≤ +∞. We denote by Lp (Ω; X) the set of measurable functions y : Ω → X such that u(·) X ∈ Lp (Ω). Similarly, one can also define the set of distributions D (Ω; X) on Ω with values in X. Lp (Ω; X) is a Banach space with respect to the norm y Lp (Ω;X) = Ω

u(x) pX

1/p dx .

26


If X is reflexive and 1 < p < ∞, the space Lp (Ω; X) is reflexive, too. Moreover, if X is separable and 1 ≤ p < ∞, then Lp (Ω; X) is separable. This type of spaces is well adapted to the study of problems where one of the variables plays a special role. For instance, this occurs for the variable “time” in time-dependent problems. For various results on vector-valued functions, we refer to Schwartz [224] and Lions and Magenes [173]. Let B0 and B be two Banach spaces such that the embedding B0 → B is compact. A natural question is whether the embedding Lp (a, b; B0 ) → Lp (a, b; B) is also compact. Actually, one can prove that this is not true, in general. However, if we have three Banach spaces B0 → B → B1 such that B0 and B1 are reflexive and the embedding B0 → B is compact, then the embedding W → Lp0 (a, b; B) is compact too, where

∂y p0 p1 ∈ L (a, b; B1 ) , 1 < p0 , p1 < +∞, W = y : y ∈ L (a, b; B0 ), ∂t is a Banach space with respect to the graph norm ∂y y W = y Lp0 (a,b;B0 ) + . ∂t p L 1 (a,b;B1 ) Here, the derivative ∂y/∂t is the distribution in D (a, b; B1 ) defined by ∂y (ϕ) = − ∂t

b

y a

∂ϕ dt, ∂t

∀ ϕ ∈ D(a, b).

The following theorem plays an important role in the study of PDEs. Theorem 2.18. Let us define the Banach spaces

∂y ∈ L2 (a, b; H −1 (Ω)) , W = y : y ∈ L2 (a, b; H01 (Ω)), ∂t

∂y ∈ L2 (a, b; H −1 (Ω)) , W1 = y : y ∈ L2 (a, b; L2 (Ω)), ∂t equipped with the norm of the graph. Then the following properties holds true: (a) The embeddings W → L2 (a, b; L2 (Ω)), W1 → L2 (a, b; H −1 (Ω)) are compact. (b) One has the embedding W → C([a, b]; L2 (Ω)),

W1 → C([a, b]; H −1 (Ω)),

where, for X = L2 (Ω) or X = H −1 (Ω), one denotes by C([a, b]; X) the space of measurable functions on [a, b] × Ω such that y(t, ·) ∈ X for any t ∈ [a, b] and such that the map t ∈ [a, b] → y(t, ·) ∈ X is continuous.


27

(c) For any u, v ∈ W, one has d u(t, x)v(t, x) dx = u (t, ·), v(t, ·)H −1 (Ω),H 1 (Ω) 0 dt Ω + v (t, ·), u(t, ·)H −1 (Ω),H 1 (Ω) . 0

Let y ∈ L2 a, b; H01 (Ω) ∩ C [a, b]; L2 (Ω) . Then the following density result holds: For any δ > 0, there exists Φ ∈ C ∞ ([a, b]; D(Ω)) such that y − Φ C([a,b];L2 (Ω)) ≤ δ,

∇y − ∇Φ L2 ((a,b)×Ω) ≤ δ.

To end this subsection, we recall one property, useful in the sequel, concerning the space L2 (Ω; Cper (Y )), where Cper (Y ) denotes the subset of C(Y ) of Y -periodic functions – namely the space L2 (Ω; Cper (Y )) is separable and dense in L2 (Ω; L2 (Y )) = L2 (Ω × Y ). 2.2.4 Lax–Milgram’s lemma Consider a real Hilbert space V . Let a(u, v) be a bilinear form on V (i.e., a : V × V → R is linear with respect to each argument). Suppose that the form a(·, ·) is continuous and coercive – that is, the following inequalities are satisfied: a(u, v) ≤ ν1 u V v V , ∀ u, v ∈ V, ν1 > 0, (2.13) a(u, u) ≥ ν2 u 2V , ∀ u ∈ V, ν2 > 0. Let V ∗ be the dual space of V . For any fixed u ∈ V , the linear form a(u, v) is continuous with respect to v ∈ V and represents an element of V ∗ denoted by Au. Thus, we obtain a linear operator defined by the formula Au, vV ∗ ,V = a(u, v),

A : V → V ∗.

Inequalities (2.13) show that the operator A is bounded and coercive: A ≤ ν2 , Au, uV ,V ≥ ν1 u 2V . For any given f ∈ V ∗ , consider the following problem: Find an element u ∈ V such that a(u, v) = f, vV ∗ ,V ,

∀ v ∈ V.

(2.14)

In fact, this problem is equivalent to establishing solvability of the equation Au = f . The following assertion generalizes the Riesz representation theorem. Lemma 2.19. (Lax–Milgram) The problem (2.14) has a solution u which is unique and satisfies the estimate u V ≤ ν1−1 f V ∗ . In other words, the bounded coercive operator A is an isomorphism between the spaces V and V ∗ and the norm of the inverse operator is bounded by ν1−1 .

28


If the bilinear form a(u, v) is symmetric, then Au = f is the Euler equation associated with the following variational problem: E = inf F (v), v∈ V

F (v) =

1 a(v, v) − f, vV ∗ ,V . 2

Indeed, let u = A−1 f . Then 2(F (v) − F (u)) = Av, vV ∗ ,V − Au, uV ∗ ,V − 2 f, vV ∗ ,V +2 f, uV ∗ ,V = Av, vV ∗ ,V − Au, uV ∗ ,V + 2 f, uV ∗ ,V −2 Au, vV ∗ ,V = Av, vV ∗ ,V − 2 Au, vV ∗ ,V + Au, uV ∗ ,V = A(v − u), v − uV ∗ ,V > 0,

∀ v = u.

So, u = A−1 f is the unique minimizer for F (v). 2.2.5 General setting of the variational formulation of boundary value problems To begin with, we recall the notation of a well-posed problem introduced by Hadamard. Let P be a boundary value problem and let Y and F be two Banach spaces. We say that P is well-posed (with respect to Y and F) if the following hold: (i) For any element f ∈ F there exists a solution y ∈ Y of P. (ii) The solution is unique. (iii) The map F f → y ∈ Y is continuous. Obviously, the well-posedness of a problem depends on the choice of the spaces Y and F. As a matter of fact, the examples of boundary value problems with this property, which we treat in the sequel, are related to an equation of the form Ay = f , where the operator A is given as follows: A = −div(A(x)∇) = −

n ∂ ∂ aij (x) . ∂xi ∂xj

(2.15)

i,j=1

Here, A(x) = {aij (x)} is a matrix (not necessarily symmetric) with bounded measurable elements (i.e., aij L∞ (Ω) ≤ β) satisfying the ellipticity condition ∃ α > 0 such that

n

aij (x)λi λj ≥ α

i,j=1

n

λ2i a.e. on Ω ∀λ ∈ Rn .

(2.16)

i=1

The matrix will always be associated with the bilinear form ∇ϕ · A(x)∇y dx. a(y, ϕ) = Ω

(2.17)


29

The Dirichlet problem Let f ∈ H −1 (Ω) and consider the problem − div(A∇y) = f in Ω,

y = 0 on ∂Ω.

The corresponding variational formulation is Find y ∈ H01 (Ω) such that a(y, ϕ) = f, ϕH −1 (Ω),H 1 (Ω) , 0

∀ ϕ ∈ H01 (Ω).

(2.18)

(2.19)

Because of the Friedrichs inequality (2.8), the form a(y, ϕ) is coercive on H01 (Ω). Therefore, for any f ∈ H −1 (Ω), there exists a unique element y ∈ H01 (Ω) satisfying (2.19). Moreover, y H01 (Ω) ≤ α−1 f H −1 (Ω) . This element is called a weak solution of the Dirichlet problem (2.18). It is clear that (2.18) is a well-posed problem (in Hadamard’s sense) for the choice Y = H01 (Ω) and F = H −1 (Ω) (or F = L2 (Ω)). Assume that ∂Ω is Lipschitz continuous. Suppose we are given f in H −1 (Ω) and g in H 1/2 (∂Ω). Consider the nonhomogeneous Dirichlet problem −div(A∇y) = f in Ω,

y = g on ∂Ω.

Using the trace notion, we say that y is a weak solution of this problem iff − div(A∇y) = f in D (Ω),

γ(y) = g in H 1/2 (∂Ω).

(2.20)

Then the problem (2.20) has a unique solution y in H 1 (Ω) such that y H 1 (Ω) ≤ C f H −1 (Ω) + g H 1/2 (∂Ω) , where C is a positive constant depending on Ω, α, and β. Moreover, we have the following result (see Lions and Magenes [173]). Theorem 2.20. Suppose that N s ≥ 2. Then for any f ∈ H s−2 (Ω) and g ∈ H s−1/2 (∂Ω), there exists a unique solution y ∈ H s (Ω) of the problem − y = f in Ω,

γ(y) = g.

Moreover, in this case, y 2H s (Ω) ≤ C f 2H s−2 (Ω) + g 2H s−1/2 (∂Ω) , where the constant C is independent of f and g.

(2.21)

30


The Neumann problem Let f ∈ (H 1 (Ω))∗ and g ∈ H −1/2 (∂Ω). Let us introduce the following notation: n ∂ ∂ = aij (x)ni , (2.22) ∂νA ∂x j i,j=1 where ni denotes ith component of the unit outward normal vector to Ω. We consider the Neumann problem − div(A∇y) + y = f in Ω,

∂y = 0 on ∂Ω. ∂νA

The corresponding variational formulation is ⎧ Find y ∈ H 1 (Ω) such that ⎨ ⎩ f ϕ dx, ∀ ϕ ∈ H 1 (Ω), a(y, ϕ) =

(2.23)

(2.24)

Ω

where now a is defined by a(y, ϕ) = ∇ϕ · A(x)∇y dx + yϕ dx, Ω

∀y, ϕ ∈ H 1 (Ω).

Ω

Let us observe that if y is a solution of the problem (2.24), then (2.23) holds in D (Ω). Then, due to Remark 2.17 (see (2.10)), A∇y belongs to H 1 (Ω, div), and therefore, ∂y/∂νA is well-defined as an element of H 1/2 (∂Ω). This is the sense to be given to the boundary condition in (2.23). If g = 0 and the domain Ω is sufficiently smooth (for instance, such that the Poincaré inequality holds in Ω), then for any f ∈ (H 1 (Ω))∗ , there exists a unique solution y ∈ H 1 (Ω) of the problem (2.24). Moreover, y H 1 (Ω) ≤

1 1 f (H 1 (Ω))∗ and y H 1 (Ω) ≤ f L2 (Ω) , min{1, α} min{1, α}

provided f ∈ L2 (Ω). As for the nonhomogeneous Neumann problem in domains with a Lipschitz continuous boundary, namely − div(A∇y) + y = f in Ω,

∂y = g on ∂Ω, ∂νA

(2.25)

we have the following result: For any f ∈ L2 (Ω) and for any g ∈ H −1/2 (∂Ω) there exists a unique solution y ∈ H 1 (Ω) of the problem (2.25). Moreover, in this case, y H 1 (Ω) ≤

1 f L2 (Ω) + C(Ω) g H −1/2 (∂Ω) , min {1, α}

(2.26)


31

where C(Ω) is a positive constant. If we consider, instead of (2.25), the nonhomogeneous Neumann problem − div(A∇y) = f in Ω,

∂y = g on ∂Ω ∂νA

(2.27)

under the same hypotheses on f and g as earlier, then the corresponding bilinear form a(y, ϕ) is no longer coercive on H 1 (Ω), but it is coercive in the quotient space W (Ω) = H 1 (Ω)/R. This space is the space of classes of equivalence with respect to the relation y1 y2 ⇔ y1 − y2 is a constant, ∀ y1 , y2 ∈ H 1 (Ω). Let us denote by y˙ the class of equivalence represented by y. Then the natural variational formulation of (2.27) is ⎧ Find y˙ ∈ W (Ω) such that ∀ ϕ˙ ∈ W (Ω) ⎨ (2.28) ⎩ a( ˙ y, ˙ ϕ) ˙ = f ϕ dx + g, ϕH −1/2 (∂Ω),H 1/2 (∂Ω) , ∀ ϕ ∈ ϕ, ˙ Ω

where a˙ is defined by ∇ϕ · A(x)∇y dx, a( ˙ y, ˙ ϕ) ˙ =

∀y ∈ y, ˙ ϕ ∈ ϕ, ˙ ∀ϕ, ˙ y˙ ∈ W (Ω).

Ω

It is clear that this problem makes sense if the right-hand side of (2.28) is independent of ϕ ∈ ϕ˙ – namely suppose that f ∈ L2 (Ω) and for any g ∈ H −1/2 (∂Ω) satisfy the compatibility condition f dx + g, 1H −1/2 (∂Ω),H 1/2 (∂Ω) = 0, Ω

then there exists a unique solution y˙ ∈ W (Ω) of the problem (2.28) with the estimate 1 (2.29) y ˙ W (Ω) ≤ f L2 (Ω) + C(Ω) g H −1/2 (∂Ω) . min {1, α} We refer to Lions and Magenes [173] for the following result. Theorem 2.21. Suppose that N s ≥ 2 and that f ∈ H s−2 (Ω) and g ∈ H s−1−1/2 (∂Ω) satisfy the compatibility condition: f (x) dx + g(s) dHn−1 = 0. (2.30) Ω

∂Ω

Then, there exists a unique solution y˙ ∈ H s (Ω) \ R to the problem (2.27). Moreover, y ˙ 2H s (Ω)\R ≤ C f 2H s−2 (Ω) + g 2H s−1/2 (∂Ω) + y 2L2 (Ω) , where the constant C is independent of f and g.

32


The Robin problem As in the previous case, suppose that ∂Ω is Lipschitz continuous and let f ∈ L2 (Ω) and g ∈ H −1/2 (∂Ω). The Robin boundary value problem we consider can be stated as follows: − div(A∇y) + y = f in Ω,

∂y + dy = 0 on ∂Ω, ∂νA

(2.31)

where d ∈ R is such that d ≥ 0. The variational formulation of this problem is then ⎧ ⎨ Find y ∈ H 1 (Ω) such that (2.32) f ϕ dx + g, ϕH −1/2 (∂Ω),H 1/2 (∂Ω) , ∀ ϕ ∈ H 1 (Ω), a(y, ϕ) = ⎩ Ω

where for all y, ϕ ∈ H 1 (Ω), a(y, ϕ) = ∇ϕ · A(x)∇y dx + yϕ dx + d Ω

Ω

yϕ ds.

(2.33)

∂Ω

In view of our suppositions, the linear form F (y) = f y dx+ < g, ϕ >H −1/2 (∂Ω),H 1/2 (∂Ω) Ω

∗ is bounded on H 1 (Ω). Hence, F ∈ H 1 (Ω) . Then, having observed that the bilinear form a(y, ϕ) given by (2.33) is continuous on H 1 (Ω) × H 1 (Ω) and coercive, since d is positive, we can apply the Lax–Milgram Lemma 2.19 with V = H 1 (Ω) to get a unique element y ∈ H 1 (Ω) satisfying (2.32). Moreover, in this case we have the same estimate for y H 1 (Ω) as in (2.26). To end this section, we consider the case where one has a Dirichlet condition on a part of the boundary ∂Ω and a homogeneous Robin one on the rest of the boundary. Let Ω be a bounded connected domain with a Lipschitz boundary ∂Ω = Γ1 ∪ Γ2 , where Γ1 and Γ2 are two disjoint closed sets and Γ1 is of positive measure. Let V be the closure of C0∞ (Rn \ Γ1 ) with respect to norm of H 1 (Ω), that is, V = y | y ∈ H 1 (Ω), y|Γ1 = 0 . We say that y ∈ V is a weak solution to the mixed Dirichlet–Robin problem ⎧ ⎨ −div(A∇y) = f in Ω, (2.34) ∂y ⎩ y = 0 on Γ1 , + dy = 0 on Γ2 , ∂νA where d ≥ 0 if ∇ϕ · A(x)∇y dx + d Ω

Γ2

f ϕ dx ∀ ϕ ∈ V.

yϕ ds = Ω

2.3 Spaces of periodic functions

33

The existence of a unique solution to this problem immediately follows from the Lax–Milgram lemma and the inequality ϕ2 (x) dx ≤ C0 |∇ϕ|2 dx, ∀ ϕ ∈ C0∞ (Rn \ Γ1 ), Ω

Ω

which, in analogy to (2.8), will also be called the Friedrichs inequality.

2.3 Spaces of periodic functions In this section, we provide a notion of periodicity for functions in the Sobolev space H 1 . Let us consider measurable functions defined on Rn and periodic in each argument x1 , x2 , . . . , xn with periods l1 , l2 , . . . , ln , respectively. Let be the parallelepiped in Rn defined by = (0, l1 ) × · · · × (0, ln ). We will refer to as the reference period. Then the function f is called -periodic iff f (x + kli ei ) = f (x)

a.e. on Rn , ∀ k ∈ Z, ∀ i ∈ {1, 2, . . . , n},

where {e1 , . . . , en } is the canonical basis of Rn . The mean value of a periodic functions is essential when studying periodic oscillating functions. Let us recall that for any -periodic function f the mean value of f is the real number M (f ) given by 1 M (f ) = f (y) dy, || where || = l1 l2 · · · ln is the volume of the parallelepiped . The Lebesgue 1/α space of periodic measurable functions with a finite norm M (|f |α ) for α α ≥ 1 we denote by L (). The following property of periodic functions is frequently used in the asymptotic analysis. A property of the mean value. Let f be a -periodic function, f ∈ Lα (), α ≥ 1. Then −1 f (ε x)ϕ(x) dx −→ M (f ) ϕ(x) dx as ε → 0 (2.35) Ω

Ω

for every ϕ ∈ Lα (Ω), where 1/α+1/α = 1, α ∈ (0, ∞), and Ω is an arbitrary bounded domain in Rn . ∞ Let Cper () be the subset of C ∞ (Rn ) of -periodic functions. We denote 1 ∞ () with respect to the H 1 -norm. by Hper () the completion of the space Cper 1 It should be stressed that Hper () does not coincide with the entire Sobolev 1 space H 1 (Ω) for Ω = . Functions in Hper () as well as all other periodic n 1 (), then u is functions, are assumed to be defined on R – namely if u ∈ Hper 1 n in H (D) for any bounded open subset D of R and u satisfies the following condition:

34


u(x + kli ei ) = u(x), ∀ k ∈ Z, ∀ i ∈ {1, 2, . . . , n}. 1 1 () = u ∈ Hloc (Rn ), u is -periodic . The space Thus, we may write Hper 1 1 Hper () possesses the following obvious property: If u ∈ Hper (), then u has the same trace on the opposite site faces of . 1 ()/R In the sequel, we will make use of the quotient space Wper () = Hper defined as the space of equivalent classes with respect to the relation 1 u∼ (). = v ⇔ u − v is a constant, ∀ u, v ∈ Hper

As usual, we denote by u˙ the equivalence class represented by u. Then the quantity ˙ u˙ ∈ Wper () u ˙ Wper () = ∇u L2 () , ∀ u ∈ u, defines a norm on Wper () for which Wper () is a Banach space. Moreover, ∗ the dual space Wper () can be identified with the set ∗ () F (c) = 0, ∀ c ∈ R , F ∈ Wper with F, u ˙ W ∗ (),Wper () = F, u(H 1 ())∗ ,H 1 () for all u ∈ u˙ and ∀ u˙ ∈ per per per Wper (). We define also the space of solenoidal periodic vector fields, by setting (2.36) L2sol () := p ∈ L2 (), div p = 0 in Rn . Clearly, L2sol () is a closed subspace of L2 (), and, by definition, p ∈ L2sol () if p ∈ L2 () and the identity ∞ p · ∇ϕ dx = 0, ∀ ϕ ∈ Cper ()

is valid (see [261]). Moreover, in this case, the following orthogonal representation holds: 1 () . L2 () = L2sol () ⊕ V 2pot (), V 2pot () = ∇u, u ∈ Hper It should be pointed out that because of the Poincaré inequality, V 2pot () is a closed subspace of L2 (). We also introduce the space of potential periodic vector fields L2pot () = Rn ⊕ V 2pot (). Then any vector field v ∈ L2pot () is potential by definition and can be represented in the form v = M (v) + ∇u.

2.4 Weak and weak-∗ convergence in Banach spaces In this section, we recall without proofs the basic facts from functional analysis concerning the weak and weak-∗ convergence in Banach spaces.

2.4 Weak and weak-∗ convergence in Banach spaces

35

Let F be a Banach space equipped with the norm · F and let F ∗ be its dual. We set E = F ∗ . The norm of u in E is defined by the formula u E = sup u, f E,F , f F =1

where u, f E,F denotes the value of u ∈ E at f ∈ F . ∞

Definition 2.22. A sequence {uk }k=1 ⊂ E is said to be strongly convergent to u ∈ E as k → ∞ if uk − u E → 0 as k → ∞. In this case, we write uk → u strongly in E as k → ∞. Definition 2.23. A sequence {uk }∞ k=1 ⊂ E is said to converge weakly to u, written uk u in E, provided u∗ , uk E ∗ ,E → u∗ , uE ∗ ,E ,

∀ u∗ ∈ E ∗ .

For instance, let Ω be an open bounded domain in Rn . We write wk w in Lp (Ω) if for any ϕ ∈ Lp (Ω) (1 ≤ p < ∞, p = p/(p − 1)) wk ϕ dx = wϕ dx. lim k→∞

Ω

Ω

A weakly convergent sequence is necessarily bounded in the norm. Theorem 2.24. (Boundedness of weakly convergent sequences) Assume uk u in E. Then the following hold: ∞

(i) {uk }k=1 is bounded in E, (ii) u E ≤ lim inf k→∞ uk E , that is, the norm in E is lower semicontinuous with respect to the weak convergence in E. Moreover, a refinement of (ii) holds: If E is reflexive, uk u in E, and limk→∞ uk E = u E , then uk → u strongly in E. For the case of Lebesgue spaces, Brezis and Lieb [31] obtained the following: Let uk u in Lp (Ω) (1 ≤ p < ∞) and uk → u almost everywhere in Ω. Then lim uk pLp (Ω) − uk − u pLp (Ω) = u pLp (Ω) . k→∞

The following theorem states one of the main properties of the weak convergence in reflexive Banach spaces. For the proof, which is rather technical, we refer to Yosida [251] or Kantorovich and Akilov [128]. ˆ Theorem 2.25. (Eberlein-Smuljan) Assume that E is a reflexive Banach ∞ space and let {uk }k=1 be a bounded sequence in E. Then the following hold:

36


∞ ∞ (i) There exist a subsequence ukj j=1 of {uk }k=1 and an element u ∈ E such that ukj u in E as k → ∞. ∞

(ii) If each weakly convergent subsequence of {uk }k=1 has the same limit u, then the whole sequence {uk }∞ k=1 weakly converges to u. This theorem is definitely false if we take F = L1 (Ω) and E = L∞ (Ω) as a simple example illustrates. Indeed, let us consider the following sequence in L1 (Ω) (Ω = (−1, 1)): ⎧ 0 if t ∈ (−1, −1/n) ∪ (1/n, 1); ⎪ ⎨ 2 n t + n if t ∈ [−1/n, 0); k = 1, 2, . . . . xk = ⎪ ⎩ 2 −n t + n if t ∈ [0, 1/n], It is clear that supk∈ N xk L1 (Ω) = 1. However, this sequence is not compact with respect to the weak convergence in L1 (Ω). In order to prove it, we note that xk (t)ϕ(t) dt = ϕ(0), ∀ ϕ ∈ C0∞ (Ω), (2.37) lim k→∞

Ω

by the mean value theorem. Indeed, due to the continuity of ϕ, for any η > 0 there exists an ε0 > 0 such that |ϕ(t) − ϕ(0)| < η, provided |t| < ε0 . Then 1/n 1/n n xk (t)ϕ(t) dt − ϕ(0) = x (t)ϕ(t) dt − ϕ(0) dt k 2 −1/n −1/n Ω 1/n 1/n n n ≤ |ϕ(t) − ϕ(0)| dt < η dt = η 2 −1/n 2 −1/n for all ε < ε0 . Let us suppose that there is a function x ∈ L1 (Ω) such that x(t)ϕ(t) dt = ϕ(0), for all ϕ ∈ C0∞ (Ω).

(2.38)

Ω

Since tϕ(t) ∈ C0∞ (Ω), from (2.37) it follows that x(t)tϕ(t) dt = tϕ(t)|t=0 = 0 ∀ ϕ ∈ C0∞ (Ω). Ω

Hence, by Raymond’s lemma (see, for instance, [49]), we conclude that tx(t) = 0 a.e. in Ω. So, x(t) = 0 for almost all t ∈ Ω, and we come into conflict with (2.38). This means that the above sequence does not converge weakly in L1 (Ω). We will also need a more general form of the weak convergence, which is usually called the weak-∗ convergence.


37

∞

Definition 2.26. A sequence {uk }k=1 ⊂ E is said to converge weakly-∗ to u ∗ (written uk u in E = F ∗ ), provided uk , u∗ F ∗ ,F → u, u∗ F ∗ ,F ,

∀ u∗ ∈ F. ∞

In particular, if uk , u ∈ L1 (Ω) and the sequence {uk }k=1 is bounded in L1 (Ω), ∗ then uk u, provided the relation uk ϕ dx = uϕ dx lim k→∞

Ω

Ω

holds for any ϕ ∈ C0∞ (Ω). The main properties of weak convergence are still valid for weak-∗ convergence. In particular, any weakly-∗ convergent sequence in E is bounded and the norm in E is lower semicontinuous with respect to the weak-∗ convergence in E. It is very important for what follows that a weak-∗ limit is uniquely defined. It is also clear that the weak convergence in E (in L1 (Ω), in particular) implies the weak-∗ convergence. However, in general, the converse statement does not hold. Indeed, let Ω = (0, 1),

1 uk = √ exp(−t2 k −1 ). πk

∗

Then uk 0 in L1 (Ω). However, 1 uk dt = 1, uk L∞ (Ω),L1 (Ω) −→ 2 Ω

as k → ∞.

The equivalent of Theorem 2.25 for weak-∗ convergence read as follows. Theorem 2.27. Let F be a separable Banach space and let E = F ∗ . If {uk }∞ k=1 is a bounded sequence in E, then following hold: ∞ ∞ (i) There exist a subsequence ukj j=1 of {uk }k=1 and an element u ∈ E such that ∗ ukj u in E as k → ∞. ∞

(ii) If each weakly-∗ convergent subsequence of {uk }k=1 has the same limit u, ∞ then the whole sequence {uk }k=1 weakly-∗ converges to u. One often has to find the limit of the product fk , uk E ∗ ,E as k → ∞. In the trivial case, when either uk u in E and fk → f in E ∗ or uk → u in E ∗ and fk f in E ∗ , we have fk , uk E ∗ ,E → f, uE ∗ ,E . However, in general, one cannot pass to the limit in the product fk , uk E ∗ ,E as k → ∞ when both of these sequences are only known to be weakly convergent. The following classical example is very significant. Let v(t) be the periodic function of period 1, defined on R by v(t) = sin(2πt), and set = [0, 1), Ω = (a, b), and

38


vk (t) := v (kt) = sin (2πkt) ,

t ∈ (a, b),

where a, b ∈ R. Applying the mean value property (2.35), we have vk M (v) =

1

sin(2πt) dt = 0

in L2 (a, b) as k → ∞.

0

On the other hand, vk , vk L2 (Ω),L2 (Ω) does not converge to 0. Indeed, vk , vk L2 (Ω),L2 (Ω) =

b

sin2 (2πkt) dt = a

1 2πk

2πkb

sin2 (τ ) dτ 2πka

2πkb 1 − cos(2τ ) 1 dτ 2πk 2πka 2 1 b−a + [− sin(4πkb) + sin(4πka)] , = 2 8πk =

so that, as k → ∞, vk , vk L2 (Ω),L2 (Ω) −→

b−a = 0. 2

To understand the ways in which a weakly convergent sequence of functions ∞ can fail to be strongly convergent, we consider a sequence {uk }k=1 in Lp (Ω) (1 < p < ∞), where Ω is a bounded smooth open subset of Rn , such that uk u in Lp (Ω) as k → ∞. Let us observe that even if we know the functions ∞ {uk }k=1 to be bounded in the supremum norm, so that uk converges weakly to u in Lq (Ω) for all 1 ≤ q < ∞, we still cannot deduce strong convergence in Lq (Ω) for any 1 ≤ q < ∞. The difficulty is with the possibility of very rapid fluctuations in the functions uk (see the previous example). This is the problem of wild oscillations. Second, observe that even if we know additionally that uk → u a.e. in Ω, so that wild oscillations are excluded, we still cannot deduce strong convergence in Lp (Ω). The obstruction is that the mass of |uk − u|p may somehow coalesce onto a set of zero Lebesgue measure. This is the problem of concentration. To characterize the concentration effects, let us suppose that uk u in Lp (Ω) and introduce the following measure: θk (E) = |uk − u|p dt, k = 1, 2, . . . , E

where E is a Borel subset of Ω. Thus, θk (E) controls how close the function uk is to u in the Lp -norm restricted to the set E. Following DiPerna and Majda [95], we call the value θ(E) = lim supk→∞ θk (E) the reduced defect measure associated with the weak convergence uk u in Lp (Ω). The idea is that θ(E)


39

encodes an information about the extent to which strong convergence fails. In particular, uk → u

in Lp (E) if and only if θ(E) = 0 (see [105]).

For instance, let Ω = (−1, 1), u = 0, and

k if − k −1 ≤ t ≤ k −1 , uk (t) = 0 otherwise. Then uk 0 in L2 (Ω) and θ is concentrated on E = {0}. In this case θ(Ω \ V ) = 0 for each open set V ⊃ E. The following theorem, taken from Valadier’s work [245], gives necessary and sufficient conditions under which weak convergence implies strong convergence. Theorem 2.28. Suppose that uk u in L1 (Ω). Then uk → u strongly if and only if the following criterion is satisfied: ∀ε > 0, ∀A ⊂ Ω with meas(A) > 0, ∃N ∈ N, ∃B ⊂ A with meas(B) > 0, such that, ∀k ≥ N , the inequality 1 1 uk (t) − dt < ε u (t) dt k meas(B) B meas(B) B holds true. 2.4.1 Weak convergence of measures Let M(Ω) be the set of Borel measures on Ω. For all μ ∈ M(Ω), we define the functional Lμ (ϕ) = ϕ dμ, ∀ ϕ ∈ L1 (Ω, dμ). (2.39) Ω

It is clear that Lμ is linear and continuous on C0 (Ω). Following Riesz’s theorem [28, 221], for any linear continuous functional Lμ : C0 (Ω) → R there exists a unique measure μ ∈ M(Ω) such that the representation Lμ (ϕ) =

ϕ dμ Ω

holds true for every ϕ ∈ C0 (Ω). Thus, the map μ → Lμ is a bijection between ∗ M (Ω) and (C0 (Ω)) . Moreover, in this case, we have Lμ = |μ|(Ω). Indeed,

Lμ = sup ϕ dμ : ϕ ∈ C0 (Ω), |ϕ| ≤ 1

Ω |ϕ| d|μ| : ϕ ∈ C0 (Ω), |ϕ| ≤ 1 = d|μ| = |μ|(Ω). ≤ sup Ω

Ω

Thus, measures can be identified as elements of the dual space of continuous functions vanishing on ∂Ω. Hence, they inherit a notion of weak-∗ convergence which was defined earlier. In view of this, we can define a weak topology on M(Ω) as follows.

40

2 Background Material on Extremal Problems ∞

Definition 2.29. We say that a sequence {μk }k=1 ⊂ M(Ω) converges weakly ∗ to μ (and we write μk μ) if Lμk Lμ in the weak-∗ topology of C0 (Ω), that is, ϕ dμk = ϕ dμ, ∀ ϕ ∈ C0 (Ω). lim k→∞

Ω

Ω

By the Banach–Steinhaus theorem, we have that if μk μ, then sup |μk |(Ω) < +∞. k

Note, moreover, that by the lower semicontinuity of the dual norm with respect to weak-∗ convergence, we have that μ → |μ|(Ω) is weakly lower semicontinuous (i.e., |μ|(Ω) ≤ lim inf k→∞ |μk |(Ω), provided μk μ). Example 2.30. Let be the parallelepiped in Rn defined by = (0, l1 ) × · · · × (0, ln ). We will refer to as the reference period. Let F be a -periodic connected domain in Rn . Let Ω be as usual a bounded smooth open subset of Rn . So, we may always suppose that Ω is a measurable set in the sense of Jordan. Let us introduce the following measure on Ω, |F ∩ Ω|−1 if x ∈ Ω, dμ = ρ(x)dx, ρ(x) = (2.40) 0 if x ∈ Rn \ Ω. It is clear that μ ∈ M(Ω) as a periodic Borel positive measure with periodicity cell and

dμ = 1. This measure is absolutely continuous with respect to

∞

the Lebesgue measure Ln . Let us consider the sequence {μk }k=1 in M(Ω), where μk (B) = k −n μ(kB), ∀ k ∈ N, for every Borel set B ⊂ Rn . Then dμk = ρ(kx) dx and dμk = k−n dμ = k−n . k−1

We wish to prove that μk Ln as k → ∞. Indeed, let i = + i, where i is a vector in Rn with integer components. Then k −1 i is a partition of Rn for every fixed k ∈ N and ϕ(x) dμk = ϕ(x) dμk + ϕ(x) dμk (2.41) Ω

k−1 i

k−1 i ∩Ω

for every ϕ ∈ C0 (Ω), where the first sum is taken over all i such that k −1 i is inside Ω and the second sum is over all i such that k−1 i and ∂Ω have


41

common points. Let us first consider the first sum in (2.41). Since ϕ ∈ C0 (Ω), there exist points xi ∈ k−1 i such that ϕ(x) dμk = ϕ(xi ) dμk = ϕ(xi )k −n dμ = k −n ϕ(xi ). k−1 i

k−1 i

This, together with the fact that sum, implies that lim

k→∞

k −n ϕ(xi ) is the construction of a Riemann

k −n ϕ(xi ) →

ϕ(x) dx.

(2.42)

Ω

Let us consider the second sum in (2.41). We have ϕ(x) dμk ≤ max |ϕ(x)|k−n M (k), x∈Ω −1 k

i ∩Ω

where M (k) is the number of cubes k −1 i containing the boundary of Ω. Since k−n M (k) → 0 by the Jordan measurability property of Ω, we conclude ϕ(x) dμk = 0. (2.43) lim k→∞

k−1 i ∩Ω

Thus, the desired property follows by taking (2.42)–(2.43) into account. Theorem 2.31. Let {μk }∞ k=1 be a sequence in M(Ω) with supk |μk |(Ω) < +∞. Then there exists a subsequence of {μk }∞ k=1 weakly converging to some μ ∈ M(Ω). To characterize this result, which can be found in Federer [109] and Evans ∞ and Gariepy [106], let us consider a bounded sequence {uk }k=1 in L1 (Ω, dμ), where μ is a positive Borel measure on Ω. Then Theorem 2.31 applied with ∞ μk = uk μ yields the relative compactness of {μk }k=1 only in the weak-M(Ω) topology. Consequently, in general, its cluster points need not be in L1 (Ω). Let us list some general properties of the weak convergence in the space of Radon measures Mb (Ω) which we apply below. We recall that a sequence ∞ {μk }k=1 of Radon measures on Rn is said to be bounded if sup μk (K) < +∞ for each compact set K ⊂ Rn . k∈N

Lemma 2.32. Let {μk }∞ k=1 and μ be Radon measures on Ω such that μk μ in Mb (Ω) as k → ∞. Then (see Zhikov [258]) the following hold: 1. ημk ημ in Mb (Ω) for every positive function η ∈ C(Ω). 2. lim inf μk (A) ≥ μ(A) for every open set A ⊂ Ω. k→∞

3. lim sup μk (K) ≤ μ(K) for every compact set K ⊂ Ω. k→∞

42


4. If lim μk (Ω) = μ(Ω), then the weak convergence μk μ implies the k→∞

following convergence: lim ϕ dμk = ϕ dμ, k→∞

Ω

∀ ϕ ∈ C(Ω).

Ω

Since Mb (Ω) is the dual space of C0 (Ω), it follows that every Radon measure on Ω can be identified with an element of the space of distributions D (Ω) in the usual way. Therefore, we say that a Radon measure μ belongs to Mb (Ω) ∩ W −1,q (Ω) if and only if there exist functions f0 , f1 , . . . , fn ∈ Lq (Ω) such that n ∂ϕ ϕ dμ = f0 ϕ dx − fk dx, ϕ ∈ C0∞ (Ω). ∂xk Ω Ω Ω k=1

In other words, we say that a Radon measure μ on Ω belongs to W −1,q (Ω) if there exists f ∈ W −1,q (Ω) such that ϕ dμ, ϕ ∈ C0∞ (Ω). f, ϕW −1,q (Ω),W 1,p (Ω) = 0

Ω

−1,q

In this case, we can identify f ∈ W (Ω) and μ ∈ Mb (Ω). Note also that, by the Riesz theorem, every non-negative element of W −1,q (Ω) is a Radon measure. ∞

Theorem 2.33 ([105]). Assume that a sequence of Radon measures {μk }k=1 ∞ is bounded in Mb (Ω). Then {μk }k=1 is precompact in W −1,q (Ω) for each ∗ 1 ≤ q < 1 = n/(n − 1). ∞ Proof. In view of Theorem 2.31, we may extract a subsequence μkj j=1 ⊂ ∞ {μk }k=1 so that μkj μ in M(Ω) for some measure μ ∈ Mb (Ω). Let us

set q = q/(q − 1) and denote by B the closed unit ball in W01,q (Ω). Since 1 ≤ q < 1∗ , we have q > n. So, by the Sobolev embedding theorem, B is a compact set in C0 (Ω). Hence, for a given δ > 0, there exist functions N (δ) {φi }i=1 ⊂ C0 (Ω) such that sup 1≤i≤N (δ)

ϕ − φi C(Ω) < δ,

∀ ϕ ∈ B.

Thus, if ϕ ∈ B, then ϕ dμk − ϕ dμ ≤ 2δ sup μkj (Ω) + φi dμkj − φi dμ j j Ω

Ω

Ω

for some index 1 ≤ i ≤ N (δ). Consequently, lim sup ϕ dμkj − ϕ dμ = 0, j→∞ ϕ∈B

and so μkj → μ in W −1,q (Ω).

Ω

Ω

Ω


43

The notion of a Radon measure can be easily extended to the case when Ω is a locally compact Hausdorff space. For more results about weak convergence of measures and its consequences, we refer to Evans and Gariepy [106], Federer [109], and Rudin [221]. 2.4.2 Weak convergence in L1 (Ω) Let Ω be a bounded open domain in Rn . Let uk ∈ L1 (Ω) (k ∈ N) and ∞ u ∈ L1 (Ω) be given functions. The sequence {uk }k=1 is said to be weakly 1 convergent to u in L (Ω) if lim uk ϕ dx = uϕ dx, ∀ ϕ ∈ L∞ (Ω). k→∞

Ω

Ω

Since L1 (Ω) cannot be characterized as the dual of some Banach space, the notion of weak-∗ convergence is not interesting in this space. However, as ∞ was mentioned earlier, for any bounded sequence {uk }k=1 in L1 (Ω) its clus1 ter points need not be in L (Ω). Indeed, having applied Theorem 2.31 with ∞ dμk = uk dx, we obtain the relative compactness of {μk }k=1 only in the weakM(Ω) topology. At this point, one can ask under which conditions a bounded sequence in L1 (Ω) is weakly compact. To answer this question, we need the following definitions. ∞

Definition 2.34. A sequence {uk }k=1 in L1 (Ω) is said to be equi-integrable if, for any η > 0, there exists δ > 0 such that |uk (x)| dx < η, for any E ⊂ Ω with |E| < δ, ∀ k ∈ N, E

where |E| stands for the Lebesgue measure of E. Definition 2.35. A function h(t) (t ≥ 0) is said to be coercive, if it is nonnegative, non-decreasing, and satisfies the condition lim t−1 h(t) dt = +∞.

t→∞

Then the answer to the above question is as in the following proposition. Proposition 2.36. (Dunford–Pettis) The following statements are equivalent: ∞

(a) The sequence {uk }k=1 is weakly compact in L1 (Ω). (b) The sequence {uk }∞ k=1 is equi-integrable. (c) There is a coercive function h(t) such that supk∈N (d) Given δ > 0, there is λ = λ(δ) such that supk∈N

h(|uk |) dx < +∞. Ω

{|uk |>λ}

|uk | dx < δ.

44

2 Background Material on Extremal Problems ∞

Theorem 2.37. (Generalized Lebesgue’s Theorem) Let {uk }k=1 be an equiintegrable sequence in L1 (Ω) such that uk → u almost everywhere on Ω as k → ∞. Then u ∈ L1 (Ω) and uk → u in L1 (Ω). The proof of the above statements can be found in Dunford and Schwartz [99], Ekeland and Temam [104], and Natanson [198]. One of the useful implication of the condition (c) is that for any u ∈ L1 (Ω), there exists a coercive function ϕ such that ϕ(|u|) ∈ L1 (Ω). Of course, this result is very simple and can be proved quite easily ∞ on the basis of the following observation: For any given numerical series k=1 ak < +∞, where ak ≥ 0, ∞ there exists a sequence λk → ∞ such that k=1 λk ak < +∞.

2.5 Elements of capacity theory In this section, we give the notion of capacity as a way to study certain “small” subsets of Rn . Let Ω ⊂ Rn be a bounded open set and let 1 < p < +∞. The p-capacity of a subset E in Ω is

|∇u|p dx : u ∈ UE , capp (E, Ω) = inf Ω

where UE is the set of all functions of the Sobolev space W01,p (Ω) such that u ≥ 1 almost everywhere in a neighborhood of E. We say that a property P(x) holds quasi everywhere (abbreviated as q.e.) in a set E if it holds for all x ∈ E except for a subset N of E with capp (N, E) = 0. The expression almost everywhere refers, as usual, to the Lebesgue measure. A subset A of Ω is said to be p-quasi-open if for every ε > 0, there exists an open subset Aε of Ω, such that A ⊆ A and capp (Aε \ A, Ω) < . The class of all p-quasi-open subsets of Ω we denote by A(Ω). A function f : Ω → R is said to be p-quasi-continuous (resp., quasi-lower semicontinuous) if for every ε > 0, there exists a continuous (resp., lower semicontinuous) function fε : Ω → R such that capp ({f = f } , Ω) < ε, where {f = f } = {x ∈ Ω : f (x) = f (x)}. It is well known that (see, e.g., Ziemer [267]) every function u ∈ W01,p (Ω) has a p-quasi-continuous representative, which is uniquely defined up to a set of p-capacity 0. We will always identify the function u with its quasi-continuous representative, so that a pointwise condition can be imposed on u(x) for p-quasi-every x ∈ Ω. Note that with this convention we can write

1,p p capp (E, Ω) = inf |∇u| dx : u ∈ W0 (Ω), u ≥ 1 q.e. on E (2.44) Ω

for every subset E of Ω. Since p is fixed, the index p may be dropped when speaking about p-quasiopen sets, p-quasi-continuity, and so forth. It is clear that a set of zero capacity

2.5 Elements of capacity theory

45

has zero measure, but the converse is not true. Moreover, when p > n, the p-capacity of a point is strictly positive and every W 1,p -function has a continuous representative. Therefore, a property which holds p-quasi-everywhere, with p > n, holds in fact everywhere. We refer to [106, 120] for a review of the main properties of the p-capacity. We recall the following key results. Theorem 2.38. Assume A, B ⊂ Ω. Then the following hold: 1. capp (A, Ω) = inf capp (U, Ω) : U is open, A ⊂ U ⊆ Ω . 2. capp (λA, Rn ) = λn−p capp (A, Rn ), λ > 0. 3. capp (B(x, r), Rn ) = rn−p capp (B(0, 1), Rn ). 4. capp (A, Ω) ≤ CHn−p (A) for some constant C depending only on p and n. 5. Ln (A) ≤ Ccapp (A, Ω)n/(n−p) for some constant C depending only on p and n. 6. capp (A ∪ B, Ω) + capp (A ∩ B, Ω) ≤ capp (A, Ω) + capp (B, Ω). Theorem 2.39. Let u ∈ H 1 (Rn ). Then for q.e. x ∈ Rn , u(y) dy = u (x), lim |B(x, ε)|−1 ε→0

B(x,ε)

where u is a quasi-continuous representative of u. Theorem 2.40. Every strongly converging sequence in H 1 (Rn ) has a subsequence converging q.e. in Rn . Theorem 2.41. Let A and Ω be two bounded open subsets of Rn such that A ⊂ Ω and consider an element u of H01 (Ω). Then u|A ∈ H01 (A) if and only if u = 0 quasi-everywhere on Ω \ A, where u = 0 is a quasi-continuous representative of u. In view of this, we note that the following two spaces: Ho1 (A; Ω) = ϕ ∈ H01 (Ω) : ϕ = 0 a.e. in Ω \ A and H01 (A; Ω) = ϕ ∈ H01 (Ω) : ϕ = 0 q.e. in Ω \ A are not equal, in general. Indeed, ϕ ∈ H01 (A; Ω) cannot be characterized by merely saying that this function and its derivatives are 0 almost everywhere in Ω \ A. By definition, a function ϕ ∈ H 1 (Ω) is said to be 0 quasi-everywhere in a subset E of Ω if there exists a quasi-continuous representative of ϕ which is 0 quasi-everywhere in E. This makes sense, since any two quasi-continuous representatives of an element ϕ of H 1 (Ω) are equal quasi-everywhere. So, we have H01 (A; Ω) ⊂ Ho1 (A; Ω). However, as can be seen from the following example, in general the reverse inclusion is not true. Let B(0, r) be the open ball of radius r > 0 in Rn . Let us set Ω = B(0, 3) and A = B(0, 2) \ ∂B(0, 1). It is well known that the circular crack ∂B(0, 1) in A has nonzero capacity but zero Lebesgue measure Ln . Since ∂B(0, 1) has

46


zero measure, it follows that Ho1 (A; Ω) contains functions ψ ∈ H01 (B(0, 2)) whose restriction to B(0, 2) are not 0 on the sphere ∂B(0, 1). Hence, those functions ψ do not belong to H01 (A, Ω). So, Ho1 (A; Ω) H01 (A; Ω). However, if A has a Lipschitz boundary, then the spaces Ho1 (A; Ω) and H01 (A; Ω) can be identified (see Delfour and Zolésio [91]). In view of this, it is natural to introduce the following terminology (see Rauch and Taylor [217]). Definition 2.42. A set A is said to be stable with respect to Ω if Ho1 (A; Ω) = H01 (A; Ω). Following Buttazzo and Dal Maso [45], let us denote by Mp0 (Ω) the set of all non-negative Borel measures μ on Ω such that the following hold: 1. 2.

μ(B) = 0 for every Borel set B ⊂ Ω with capp (B, Ω) = 0. μ(B) = inf {μ(U ) : U quasi-open, B ⊆ U } for every Borel set B ⊂ Ω.

When p = 2, we use the notation M0 (Ω) instead of M20 (Ω). As examples of measures in the class Mp0 (Ω) we can quote the following: ϕ Ln ∈ Mp0 (Ω) for every ϕ ∈ L∞ (Ω), where, as usual, Ln is the ndimensional Lebesgue measure. (ii) If n − 2 < α ≤ n, then the α-dimensional Hausdorff measure Hα belongs to Mp0 (Ω). This is a consequence of the two following implications:

(i)

Hn−2 (B) < +∞ ⇒ capp (B, Ω) = 0, capp (B, Ω) = 0 ⇒ Hn−2+δ (B) = 0,

∀δ > 0.

(iii) The measure

∞S (B) =

0 +∞

if capp (B ∩ S, Ω) = 0, otherwise

(2.45)

belongs to Mp0 (Ω) for every quasi-closed set S, and so does the measure μA = ∞Ω\A for every open set A ⊂ Ω, that is,

0 if capp (B \ A, Ω) = 0 μA (B) = +∞ otherwise. Let B ∗ (Ω) be the σ-field generated by the Borel subsets of Ω. It is well known that a subset E of Ω belongs to B ∗ (Ω) if and only if there exists B ∈ B(Ω) with capp (EB) = 0, where denotes the symmetric difference of sets. Therefore, each measure μ ∈ Mp0 (Ω) can be extended in a unique way to a countably additive set function, still denoted by μ, defined on the larger σ-field B ∗ (Ω). We say that A(μ) is a regular set for the measure μ ∈ Mp0 (Ω) if A(μ) is defined as the union of all open subsets A of Ω such that μ(A) < +∞. The singular set S(μ) is defined as the complement of A(μ) in Ω. It is easy to see that A(μ) is also open, and if A is a open subset of Ω which intersects S(μ), then μ(A) = +∞.

2.6 On the space W01,p (Ω) ∩ Lp (Ω, dμ) and its properties

47

2.6 On the space W01,p(Ω) ∩ Lp(Ω, dμ) and its properties Let Ω be a bounded open subset of Rn with n ≥ 2. Let us fix μ ∈ Mp0 (Ω) and denote by Xpμ (Ω) the vector space of all functions u ∈ W01,p (Ω) such that |u|p dμ < +∞. Note that this definition makes sense because μ vanishes Ω

1,p on all sets of capacity 0 and every function u ∈ W0 (Ω) is defined up to a set

|u|p dμ is unambiguously defined. On Xpμ (Ω)

of capacity 0. So, the integral Ω

we consider the norm

"

#1/p |u| dμ .

|Du| dx +

u Xpμ (Ω) =

p

Ω

p

Ω

Theorem 2.43. Xpμ (Ω) is a Banach space. ∞

∞

Proof. Let {ui }i=1 be a Cauchy sequence in Xpμ (Ω). Then {ui }i=1 is a Cauchy ∞ sequence both in W01,p (Ω) and in Lp (Ω, dμ). Therefore, {ui }i=1 converges to 1,p p a function u in W0 (Ω) and to a function v in L (Ω, dμ). Taking into account the fact that every convergent sequence in W01,p (Ω) is relatively compact with respect to the pointwise convergence q.e. on Ω, we can extract a subsequence ∞ {uik }k=1 converging to u q.e. in Ω. Since μ vanishes on all sets with capacity 0, ∞ {uik }k=1 converges to u μ-a.e. in Ω. On the other hand, a further subsequence ∞ of {uik }k=1 converges to v μ-a.e. in Ω. Hence, u = v μ-a.e. in Ω and, therefore, ∞ u ∈ Xpμ (Ω) and {ui }i=1 converges to u both in W01,p (Ω) and in Lp (Ω, dμ). ∞ This implies that {ui }i=1 converges to u in Xpμ (Ω). Thus, the normed space Xpμ (Ω) is complete. It is clear now that in the case when p = 2, X2μ (Ω) = H01 (Ω) ∩ L2 (Ω, dμ) is a Hilbert space with respect to the scalar product, DuDv dx + uv dμ. (2.46) (u, v)X2 (Ω) = μ

Ω

Ω

Let us consider now some examples which illustrate the structure of the space Xpμ (Ω) under some special assumptions on the measure μ. Example 2.44. Assume that μ = gLn with g ∈ Lq (Ω), where

q ∈ [1, +∞) if p ≥ n, q ∈ [1, p∗ = pn/(n − p)] if p < n. By the Sobolev embedding theorem, we have that W01,p (Ω) → Lq (Ω). Hence, Xpμ (Ω) = W01,p (Ω) with equivalent norm.

48


Example 2.45. Let A be an open subset of Ω and let S = Ω \A. If μ is equal to the measure μ = ∞S + gLn , where g ∈ Lq (Ω), q satisfies the conditions of the previous example, and ∞S is defined by (2.45), then Xpμ (Ω) = W01,p (A), and the norms in Xpμ (Ω) and W01,p (Ω) are equivalent by the Poincaré inequality. ∗ Consider now a measure μ ∈ Mp0 (Ω). By Xpμ (Ω) we denote the dual space of Xpμ (Ω), with duality pairing ·, ·(Xpμ (Ω))∗ ,Xpμ (Ω) . Note that even in the case when p = 2, the space X2μ (Ω) is not dense in L2 (Ω). Therefore, we do not ∗ identify the isomorphic spaces X2μ (Ω) and X2μ (Ω) . Let us show that the spaces Lp (Ω), W −1,p (Ω), and Lp (Ω, dμ) can be viewed as linear subspaces ∗ of (Xμ (Ω)) (here p = p/(p − 1)). Let i : Xpμ (Ω) → W01,p (Ω) be the natural embedding defined by i(u) = u ∗ for every u ∈ Xpμ (Ω). The transpose map t i : W −1,p (Ω) → Xpμ (Ω) allows ∗ us to consider W −1,p (Ω) as a subspace of (Xμ (Ω)) . With a little abuse of notation, which is discussed in a moment, we write f instead t i(f ) for every f ∈ W −1,p (Ω). With this convention we have f, v(Xpμ (Ω))∗ ,Xpμ (Ω) = f, vW −1,p (Ω),W 1,p (Ω) , 0

∀ v ∈ Xpμ (Ω).

(2.47)

In particular, for f ∈ Lp (Ω), we have f v dx, f, v(Xpμ (Ω))∗ ,Xpμ (Ω) =

∀ v ∈ Xpμ (Ω).

Ω

The the map t i : W −1,p (Ω) → ∗ in our notation consists in the fact that p abuse p Xμ (Ω) is, in general, not injective, because Xμ (Ω) is, in general, not dense in W01,p (Ω). Therefore, there may exist two elements f and g of W −1,p (Ω) ∗ such that f = g in W −1,p (Ω) but f = g in Xpμ (Ω) , where the last equality t t means i(f ) = i(g), according to our convention (2.47). Example 2.46. Assume that μ is the measure ∞Ω defined in (2.45) taking S = Ω. Then Xpμ (Ω) = {0}; hence, t i(f ) = 0 for every f ∈ W −1,p (Ω). Therefore, p ∗ in view of (2.47), we have f = 0 in Xμ (Ω) for every f ∈ W −1,p (Ω). Let j : Xpμ (Ω) → Lp (Ω, dμ) be the natural embedding defined by j(u) = u ∗ for every u ∈ Xpμ (Ω). Then the transpose map t j : Lp (Ω, dμ) → Xpμ (Ω) ∗ allows us to consider Lp (Ω, dμ) as a subspace of Xpμ (Ω) . For every g ∈ Lp (Ω, dμ), the image t j(g) is denoted by gμ. With this convention we have gμ, v(Xpμ (Ω))∗ ,Xpμ (Ω) = vg dμ, ∀ v ∈ Xpμ (Ω). (2.48) Ω

Since Xpμ (Ω) is, in general, not dense in Lp (Ω, dμ), the map t j : Lp (Ω, dμ) → p ∗ Xμ (Ω) is, in general, not injective. Therefore, there may exist two elements f and g of Lp (Ω, dμ) such that f = g in Lp (Ω, dμ), that is,

2.7 Sobolev spaces with respect to a measure

49

μ({x ∈ Ω : f (x) = g(x)}) > 0, but f μ = gμ in Xpμ (Ω). Example 2.47. Let E be the set of all points x = (x1 , x2 , . . . , xn ) in Ω whose first coordinate x1 is rational. Let μ = ∞E +Ln . Then it is clear that Xpμ (Ω) = {0}. Therefore, taking g = χΩ\E , where χΩ\E is the characteristic function of the set Ω \ E, we haveg ∈ Lp (Ω, dμ) and g = 0 in Lp (Ω, dμ), whereas ∗ t p gμ = j(g) = 0 in Xμ (Ω) . 2 Let us fix a measure μ ∈2 M0 (Ω). ∗ Then by the Riesz–Fréchet representation theorem, for every F ∈ Xμ (Ω) , there exists a unique u ∈ X2μ (Ω) such that

∀ v ∈ X2μ (Ω).

F, v(X2 (Ω))∗ ,X2 (Ω) = (u, v)X2μ (Ω) , μ

μ

(2.49)

By definition (2.46) of the scalar product in X2μ (Ω), (2.49) is equivalent to DuDv dx + uv dμ = F, v(X2 (Ω))∗ ,X2 (Ω) , ∀ v ∈ X2μ (Ω). Ω

μ

Ω

μ

However, according to our conventions (2.47) and (2.48), the last equality can be written in the form −u, v(X2 (Ω))∗ ,X2 (Ω) + uμ, v(X2 (Ω))∗ ,X2 (Ω) μ

μ

μ

μ

= F, v(X2 (Ω))∗ ,X2 (Ω) , μ

μ

∀ v ∈ X2μ (Ω). (2.50)

∗ This shows that each element F of X2μ (Ω) can be represented as F = f +gμ with f ∈ H −1 (Ω) and g ∈ L2 (Ω, dμ). On the other hand, because of (2.50), we refer to the solution of (2.49) as the solution of the problem ∗ u ∈ X2μ (Ω), −u + uμ = F in X2μ (Ω) .

2.7 Sobolev spaces with respect to a measure Let Ω be an open domain in Rn and let μ be a finite positive (e.g., probability) measure of Mp0 (Ω). We introduce the Sobolev space W 1,p (Ω, dμ) as follows. Definition 2.48.We say that a function u belongs to W 1,p (Ω, dμ) if there ∞ exist a sequence uk ∈ C ∞ (Ω) k=1 and a vector-function z ∈ Lp (Ω, dμ) := n [Lp (Ω, dμ)] such that uk → u in Lp (Ω, dμ) and ∇uk → z in Lp (Ω, dμ).

(2.51)

In this case we say that z is a gradient or μ-gradient of u and denote it by ∇μ u.

50


In other words, to define the space W 1,p (Ω, dμ), we construct the space W = W (Ω, dμ) as the closure in Lp (Ω, dμ) × Lp (Ω, dμ) of the set of pairs ∞ (u, ∇u) : ∀ u ∈ C (Ω) . Thus, the elements of W are pairs (u, z), where the vector z is denoted by ∇μ u, and said to be a gradient of u. As a result, the collection of the first components u is called the Sobolev space W 1,p (Ω, dμ). Note that if we put in (2.51) functions u ∈ C0∞ (Ω), we just obtain the definition of the Sobolev space W01,p (Ω, dμ). Note also that we do not introduce a norm in these spaces. If p = 2, we usually write H 1 (Ω, dμ) = W 1,2 (Ω, dμ). Remark 2.49. In the above definition, the strong convergence in Lp (Ω, dμ) and Lp (Ω, dμ) can be replaced by the weak convergence in the same spaces. In general, the gradient of a W 1,p (Ω, dμ) function is not unique since a function u in W 1,p (Ω, dμ) can have many gradients. Let us denote by Γ μ (u) the set of all gradients of a fixed function u ∈ W 1,p (Ω, dμ). It is clear that Γ μ (u) has the structure Γ μ (u) = ∇μ u + Γ μ (0), where ∇μ u is some gradient and Γ μ (0) is the set of gradients of 0. By definition, z ∈ Γ μ (0) if there exist uk ∈ C ∞ (Ω) (k = 1, 2, . . . ) such that p p lim |uk | dμ = 0 and lim |∇uk − z|Rn dμ = 0. k→∞

k→∞

Ω

Ω

Obviously, Γ μ (0) is a closed subspace of the vector space Lp (Ω, dμ). So, the gradient of an arbitrary W 1,p (Ω, dμ) function can be viewed as the corresponding equivalence class. As an illustration of nonuniqueness of the gradient, we consider the following example. Example 2.50. The case of a singular measure μ concentrated on the segment. Let I = {x | a ≤ x1 ≤ b; x2 = 0} be a segment in R2 and suppose that a bounded domain Ω ⊂ R2 contains I. Let μ be a probability measure concentrated on this segment, uniformly distributed on it, and coinciding with 1D Lebesgue measure on I – namely we set dμ =

1 χ(x1 ) dx1 × δ(x2 ), b−a

(2.52)

where χ(t) is the characteristic function of the segment [a, b] and δ(t) is the Dirac mass concentrated at 0. It is clear that μ is a singular measure with respect to L2 , and, by definition of the class M0 (Ω), we have μ ∈ M0 (Ω). Note also that μ(Ω \ I) = 0. Therefore, any functions taking the same values on the segment I coincide as elements of L2 (Ω, dμ). By Definition 2.48, a function u is an element of H 1 (Ω, dμ) if there are a sequence of smooth functions uk ∈ C ∞ (Ω) and z = (z1 , z2 ) ∈ L2 (Ω, dμ) such that 2 2 ∂uk ∂uk 2 |u − uk | dx1 → 0, ∂x1 − z1 dx1 → 0, ∂x2 − z2 dx1 → 0. I I I

2.7 Sobolev spaces with respect to a measure

51

Thus, due to (2.51) each element of H 1 (Ω, dμ) is uniquely defined by the respective element of the 1D Sobolev space H 1 ([a, b]) and z1 = ∂u/∂x1 . Thus, ∇μ u = (∂u/∂x1 , z2 ). We now show that the component z2 can be an arbitrary element of L2 (I). In other words, Γ μ (0) = {(0, α)} ,

α ∈ L2 (I) = L2 (Ω, dμ).

Indeed, since Γ μ (0) is closed in L2 (Ω, dμ) and C ∞ (I) is dense in L2 (I), it is sufficient to verify that (0, α) ∈ Γ μ (0) for α ∈ C ∞ (I). To do so, we set uk (x1 , x2 ) = x2 α(x1 ). Then uk → 0 strongly in L2 (Ω, dμ) as k → ∞ and, moreover, ∂uk ∂uk = 0, = α(x1 ). ∂x1 x2 =0 ∂x2 x2 =0 Hence, the required conclusion is obtained. To conclude this example, we note that a Borel function u = u(x1 , x2 ) belongs to the space H 1 (Ω, dμ) if and only if u ∈ H 1 (Ω) and the restriction (trace) of u to the segment I is an H 1 function of a single variable. Note also that the trace of a function in H 1 (Ω) is defined on I and is an element of the space H 1/2 (I), in general! Let us consider several examples of the Sobolev spaces H 1 (Ω, dμ). Example 2.51. Network node. Consider the segments I1 , I2 , . . . , IN starting at the origin and directed along vectors v1 , v2 , . . . , vN . Suppose that vi /|vi | = vj /|vj | for i = j, and a bounded open domain Ω ⊂ R2 contains this star structure. Let μ1 , μ2 , . . . , μN be the 1D measures on the segments I1 , I2 , . . . , IN , respectively (see, for instance, (2.52)). Let λ1 , λ2 , . . . , λN be arbitrary positive numbers. We set μ=

N

λi μi .

i=1

Then it is easy to verify that a Borel function u = u(x1 , x2 ) belongs to the space H 1 (Ω, dμ) if and only if its restriction to each segment Ij is an H 1 function of a single variable and the values of the restricted functions at the origin coincide for all segments (recall that, by the Sobolev embedding theorem, an H 1 function of a single variable is continuous). 3 Example 2.52. Junction. Let Ω = (−1, 1)3 , G = − 18 , 18 , #3 # " # " " 1 1 1 1 3 × − , , Π= x∈ − , : x3 = 0, (x1 , x2 ) ∈ 0, 2 2 8 8 8 " #3 # " 1 1 1 1 I= x∈ − , , , : x2 = x3 = 0, x1 ∈ 2 2 4 2

52


μ1 is the standard Lebesgue measure on the segment I, μ2 is the planar Lebesgue measure on Π, and dx is the spatial Lebesgue measure in R3 restricted to the cube G. Introduce the measure on Ω as follows: dμ = dμ1 + dμ2 + dx. Note that the trace of an H 1 (Ω) function is defined on Π and is an element of the space H 1/2 (Π). The trace of an H 1 (Π) function on the segment I is also well defined. However, unlike the 2D case, in the case of dimension 3, the traces of an H 1 (Ω) function on 1D segments are not defined. Therefore, we have to use the following obvious fact: The space H 1 (Ω, dμ) is isomorphic to the direct sum of the spaces H 1 (Ω, χG dx) and H 1 (Ω, dμ1 + dμ2 ). Thus, the function u belongs to H 1 (Ω, dμ) if u = u $+u , where u $ ∈ H 1 (G), u ∈ 1 1 1 |I is an element of H (I). For the other properties H (Π) ∩ H (I, dμ2 ), and u of the space H 1 (Ω, dμ), we refer to [58, 59]. We now describe some properties of the subspace Γ μ (0). (i)

If g ∈ Γ μ (0) and a ∈ L∞ (Ω, dμ), then ag ∈ Γ μ (0). To see this, it suffices to consider the case a ∈ C ∞ (Ω). Then if follows from the definition of Γ μ (0) that auk

L2 (Ω,dμ)

−→

0, ∇μ (auk ) = ∇μ a uk + a∇uk

L2 (Ω,dμ)

−→

ag.

Hence, ag ∈ Γ μ (0). (ii) Let Π be the orthogonal projection of L2 (Ω, dμ) onto Γ μ (0). Then Π(ag) = aΠ(g),

∀ g ∈ L2 (Ω, dμ), a ∈ L∞ (Ω, dμ).

$⊥Γ μ (0). Then, $ = Π(g), we obtain g $ ∈ Γ μ (0) and g − g Indeed, setting g by property (i), we have a$ g ∈ Γ μ (0),

$), h) = 0, ∀ h ∈ Γ μ (0), (a(g − g

as required. (iii) The set Γ μ (0) ∩ L∞ (Ω, dμ) is dense in Γ μ (0). In fact, if ak is the characteristic function of the set {x ∈ Ω : |g| ≤ k}, then ak g ∈ Γ μ (0) and ak g → g as k → ∞. (iv) There exists a μ-measurable subspace D(x) ⊂ Rn such that (2.53) Γ μ (0) = g ∈ L2 (Ω, dμ) : g(x) ∈ D(x) . Indeed, let e1 , e2 , . . . , en be the natural basis of Rn . We set ξi = Πei and define D(x) as the linear span of the vectors {ξ1 (x), . . . , ξ n (x)}. We denote by B the subspace of L2 (Ω, dμ) defined by the right-hand side of (2.53) and show that Γ μ (0) = B. Let g ∈ Γ μ (0) ∩ L∞ (Ω, dμ). Then, by property (ii), we obtain

2.8 Boundary value problems in spaces with measures

53

g = g1 e1 + · · · + gn en = Πg = g1 ξ1 + · · · + gn ξn ∈ B. In view of property (iii) and the fact that B is closed, we have the inclusion Γ μ (0) ⊆ B. To verify the reverse inclusion B ⊆ Γ μ (0), we note that a(x)

n

λi ξ i ∈ Γ μ (0),

∀ a ∈ L∞ (Ω, dμ), ∀ λi ∈ R1

i=1

because of properties (i) and (ii) and the fact that ξ 1 , . . . , ξ n ∈ Γ μ (0). The set of such elements is dense in B. Let us assume that there exists a b ∈ B such that b = 0 and % & n ab, λi ξ i = 0. i=1

L2 (Ω,dμ)

Since a is an arbitrary function, it follows that % & n b(x), λi ξ i (x) =0 i=1

Rn

for μ-almost all x ∈ Ω, which means that b(x) = 0. The proof is complete. Definition 2.53. We say that a gradient ∇μ u is tangential for u ∈ H 1 (Ω, dμ) if ∇μ u⊥Γ μ (0) (or in the equivalent form, if ∇μ u(x) ∈ T (x) for μ-almost all x ∈ Ω, where T (x) = (D(x))⊥ ). It is clear that each function in the Sobolev space H 1 (Ω, dμ) has a unique tangential gradient. It is also obvious that if ∇μ u(x) is some gradient of u and P (x) is the orthogonal projection Rn → T (x), then P (x)∇μ u(x) is the tangential gradient. Combining these results, we come to the following conclusion (see [256]): There exists a μ-measurable subspace T (x) such that the set of gradients of each function in H 1 (Ω, dμ) has the representation ∇μ u(x) + g(x), where ∇μ u(x) ∈ T (x) and g is an arbitrary vector in L2 (Ω, dμ) such that g(x) ∈ T ⊥ (x). The subspace T (x) is called the tangential space at the point x and ∇μ u is called the tangential gradient. For more results concerning the Sobolev spaces with respect to a measure and their applications, we refer to Chechkin, Zhikov, Lukkassen and Piatnitski [58], Bouchitté, Buttazzo and Seppecher [25], and Fragal` a and Mantegazza [110].

2.8 Boundary value problems in Sobolev spaces with measures The asymptotic behavior of thin and reticulated structures such as, for example, shells, plates, thin films, rod structures, skeletons, and so on is widely

54


discussed in the mathematical and engineering literatures. As a rule, the goal of an asymptotic analysis is to reduce dimension (i.e., to reduce the original problem to a problem on some structure of smaller dimension). For instance, the equation on a thin 3D plate is replaced with an equation on a 2D domain [257], the system describing a rod construction is reduced to a family of ordinary differential equations [265], and so on. The reduced system is usually simpler, especially from the point of view of numerical analysis. Asymptotic methods for such problems are well known in the literature, but, as a rule, they are derived under strict restrictions on the geometry and smoothness of the structure (see, for instance, [14, 22, 66, 200, 209, 216]). There are numerous works in mechanics where asymptotics for different special models were obtained at the physical level of rigor. From mathematical point of view, it is of great interest to construct a general approach to the asymptotic analysis of different classes of boundary value problems on thin and reticulated structures, that would be associated with the corresponding periodic measures, singular measures, partially singular measures, as well as measures converging to singular ones. A successful attempt to create such a theory was made in [25, 26, 254, 256, 257, 265]. In this section, following these works, we define the meaning of boundary value problems in spaces with measures and clarify the idea of Sobolev spaces with an arbitrary measure. The main motivation of our intention can be clarified by a simple example. Let μ be a positive finite Borel measure defined on a smooth bounded domain Ω. We have the following variational problem: A(x)∇ϕ(x) · ∇ϕ(x) + ϕ2 (x) − 2f (x)ϕ(x) dμ(x), inf ∞ ϕ∈C0 (Ω)

Ω

where A ∈ C(Ω, Rn×n ) is a coercive matrix of positive elements and f ∈ C(Ω) is a given function. Our aim is to describe a minimizer to this problem as an element of an appropriate Sobolev space with respect to the measure μ and associate it with a solution to the corresponding Euler equation. Throughout this section, we assume that Ω is an open domain in Rn and μ is a periodic Borel (e.g., probability) measure of Mp0 (Ω) such that dμ = 1, where = [0, 1)n is the cell (or the torus) of periodicity for μ. Let 1 1,2 Hper (, dμ) = Wper (, dμ) be the periodic Sobolev space with respect to the measure μ. Since μ can be identified with the corresponding periodic measure in Rn , we will also make use the Sobolev space H 1 (Rn , dμ). Let A = [aij (x)]i,j=1,...,n be a continuous function with values in the space of symmetric n × n matrices satisfying the uniform ellipticity condition α|ξ|2Rn ≤ (A(x)ξ, ξ)Rn ≤ α−1 |ξ|2Rn ,

α > 0, ξ ∈ Rn

(2.54)

for all x ∈ Rn . Let f be a given element of L2 (Rn , dμ) and let λ > 0. To begin with, we define the notion of divergence with respect to the measure.


55

n Definition 2.54. Suppose that g ∈ L2 (Rn , dμ) and v ∈ L2 (Rn , dμ) . We say that g(x) = divμ v(x) if g(x)ϕ(x) dμ(x) = − v(x) · ∇ϕ(x) dμ(x), ∀ ϕ ∈ C0∞ (Ω). Rn

Rn

It is easy to see that, in this definition, instead of smooth functions ϕ one can take functions ϕ ∈ H 1 (Rn , dμ). Definition 2.55. We say that a pair (u, ∇μ u), where u ∈ H 1 (Rn , dμ) and ∇μ u is a μ-gradient of u, satisfies the equation (2.55) − divμ A(x)∇μ u + λu = f in L2 (Rn , dμ) if for any v ∈ H 1 (Rn , dμ) and any gradient ∇μ v of v, we have μ μ A(x)∇ u · ∇ v dμ + λ uv dμ = f v dμ. (2.56) Rn

Rn

Rn

Definition 2.56. A function u ∈ H 1 (Rn , dμ) is called a solution to (2.55) if the integral identity (2.56) holds for some of the gradients of u and for any v ∈ H 1 (Rn , dμ) and any gradient ∇μ v of v. Note that in Definitions 2.55 and 2.56, instead of functions v ∈ H 1 (Rn , dμ), one can take the test functions v in C0∞ (Rn ). Remark 2.57. In the special case when the matrix A(x) in (2.55) is identity, relation (2.56) takes the form μ μ ∇ u · ∇ v dμ + λ uv dμ = f v dμ. Rn

Rn

Rn

Then the corresponding expression div ∇μ u is usually denoted by Δμ u and called the μ-Laplacian of u. μ

The main result of this section can be formulated as follows: Lemma 2.58. Let A ∈ C(Ω, Rn×n ) be a symmetric matrix satisfying conditions (2.54). Then for every f ∈ L2 (Rn , dμ), (2.55) has a unique solution (u, ∇μ u), u ∈ H 1 (Rn , dμ). Moreover, the choice of a μ-gradient of u is uniquely determined by the condition of orthogonality of the vector A(x)∇μ u and the subspace Γμ (0) of the gradients of 0. Proof. Since the matrix A = A(x) is positive definite, the left-hand side of (2.56) is the inner product in W 2 (Rn , dμ) (see the definition of the space W 2 (Rn , dμ) in Sects. 2.7 and 6.4), whereas the right-hand side of (2.56) is a continuous linear functional on W 2 (Rn , dμ). By the Riesz representation theorem, there exists a pair (u, ∇μ u) ∈ W 2 (Rn , dμ) satisfying relation (2.56). Taking for a test function in (2.56) the pair (0, z), where z ∈ Γμ (0), we see

56


that the vector A(x)∇μ u is orthogonal to Γ μ (0). Using the Riesz theorem again, we conclude that there is a unique element z of the set of μ-gradients of u satisfying the orthogonality condition. Thus, the solution u ∈ H 1 (Rn , dμ) to (2.55) is unique. Remark 2.59. By Lemma 2.58, the gradient ∇μ u chosen by a solution u to the problem (2.55) satisfies the condition ∇μ u⊥Γμ (0). Hence, in view of Definition 2.53, ∇μ u is the tangential gradient for a function u ∈ H 1 (Rn , dμ). Moreover, in the case when the measure μ is defined as in Example 2.51, the orthogonality A(x)∇μ u|x=0 ⊥Γμ (0) is equivalent to the classical Kirchoff condition. Let us apply Lemma 2.58 to establish some additional properties of the solutions to the elliptic equation (2.55). Lemma 2.60. Under the assumptions of Lemma 2.58, the set of solutions to (2.55), for all f ∈ L2 (Rn , dμ), is dense in the space L2 (Rn , dμ). Proof. Denote by D the set of solutions to the (2.55) when f runs over the entire space L2 (Rn , dμ). Assume that there is a nontrivial element g ∈ L2 (Rn , dμ) which is orthogonal to D. For a fixed f ∈ L2 (Rn , dμ), we denote by uf ∈ H 1 (Rn , dμ) the corresponding solution to (2.55). Then ug ∈ H 1 (Rn , dμ) is a solution to the equation −divμ A(x)∇μ u + λu = g. Taking ug as a test function in (2.56) and uf for a test function in the last equation and taking the difference of the obtained integral identities, we come to the relation f ug dμ = 0, ∀ f ∈ L2 (Rn , dμ). Rn

g

Hence, u = 0 and g = 0, which contradicts the assumption that g is nontrivial. The proof is complete. Further, we note that (2.55) can be written in the operator form A u+λu = f , where A is a self-adjoint symmetric operator. Indeed, let uf be any element of D (i.e., uf is a solution to (2.55) for some f ∈ L2 (Rn , dμ)). We set A uf = f − λuf . Then, in view of Lemma 2.58, the operator (A + λI)−1 sends a function f ∈ L2 (Rn , dμ) to the corresponding unique solution uf of (2.55). Since this operator is nonnegative, bounded, and symmetric, we come to the required conclusion. The following assertion can be proved in the same way as in the case of classical variational problems in the space H 1 (Rn ) (see, for instance, [169]).


57

Proposition 2.61. Let f ∈ L2 (Rn , dμ) and let A ∈ C(Ω, Rn×n ) be a symmetric matrix satisfying conditions (2.54). Then for every λ > 0, the variational problem

μ μ 2 u · ∇ u + λu 2f u dμ (2.57) A(x)∇ dμ − inf 1 n ϕ∈H (R ,dμ)

Rn

Rn

has a unique minimum point in H 1 (Rn , dμ) and it is a solution to (2.55). Thus, (2.55) or the equivalent equation A u + λu = f is an Euler equation for the variation problem (2.57). The developed technique allows us to study not only problems in the entire space Rn but also various boundary value problems. We illustrate this by an example of a Dirichlet problem (we consider other types of boundary problems in the second part of this book). Let Ω be a bounded Lipschitz domain in Rn and let μ be a positive finite Borel measure on Ω. Definition 2.62. We say that a function u ∈ L2 (Ω, dμ) belongs to the space 2 n 1 2 H0 (Ω, dμ) and z ∈ L (Ω, dμ) = L (Ω, dμ) is a μ-gradient of u if there ∞ ∞ exists a sequence {uk ∈ C0 (Ω)}k=1 such that uk → u in L2 (Ω, dμ) as k → ∞, ∇uk → z in L2 (Ω, dμ) as k → ∞. Consider the Dirichlet problem −divμ A(x)∇μ u + λu = f

in L2 (Ω, dμ),

u = 0 on ∂Ω.

(2.58) (2.59)

As before, we assume that the matrix A ∈ C(Ω, Rn×n ) is symmetric and satisfies the uniform ellipticity condition in Ω like (2.54). Definition 2.63. We say that u ∈ H01 (Ω, dμ) is a solution to the Dirichlet problem (2.58)–(2.59) if the integral identity A(x)∇μ u · ∇μ v dμ + λ uv dμ = f v dμ Ω

Ω

Ω

holds true for any v ∈ H01 (Ω, dμ). The existence and uniqueness of a solution to the problem (2.58)–(2.59) can be established in the same way as in the proof of Lemma 2.58. We recall that a gradient of the solution that satisfies the above integral identity is chosen in a unique way.

58


2.9 On weak compactness of a class of bounded sets in Banach spaces Let X be a Banach space and X∗ its dual. Let us recall that the weak-∗ topology of X∗ is the locally convex topology σ(X∗ , X) in which the linear functionals X ∗ f → f, uX∗ ,X , u ∈ X are continuous. Due to the Banach–Alaoglu theorem, it is well known that any bounded weakly-∗ closed subset B ⊂ X∗ is compact with respect to the weak-∗ topology of X∗ . However, it is hard to verify the property of weak-∗ closure for sets which differ from a ball, even for relatively simple sets B ⊂ X∗ . Therefore, the main goal of this section is to study the compactness property for other types of subsets in X∗ (see, for instance, [151, 152]). k k Let X and {Zj }j=1 be Banach spaces and let X∗ and Z∗j j=1 be their duals, respectively. Let k

{Λj : (D(Λj ) ⊂ Zj ) → X}j=1 be linear mappings defined on the domains D(Λj ) which are dense subsets of the corresponding spaces Zj . Let ∗ k Λj : D(Λ∗j ) ⊂ X∗ → Z∗j j=1 be their dual mappings, respectively. We set D∗ =

k '

D(Λ∗j )

j=1

and endow this set with the graph norm y ∗ ∗ = y ∗ X∗ +

k ∗ ∗ Λj y j=1

Z∗ j

.

(2.60)

By Y∗ we denote the normed space D ∗ equipped with the norm · ∗ . Let X∗σ and Z∗j,σ be the spaces X∗ and Z∗j , respectively, endowed with the topologies σ(X∗ , X) and σ(Z∗j , Zj ). The graphs gr Λ∗j of operators Λ∗j are closed in X∗σ × Z∗j,σ for every j = 1, 2, . . . , k (see [241]). Hence, Y∗ is a Banach space. By Xw we denote the Banach space X endowed with the σ(X, X∗ ) topology. Let us consider the following class of subsets in Y∗ : ( ) K∗ = ξ ∈ Y∗ : y ∗ X∗ ≤ l0 , Λ∗j y ∗ Z∗j ≤ lj , j = 1, 2, . . . , k , (2.61) where l0 , l1 , . . . , lk are positive numbers.

2.9 On weak compactness of bounded sets in Banach spaces

59

*k Having used the notation D = X× j=1 D(Λj ), we note that the set D can be associated with the family of linear continuous functionals Gφ (·) on Y∗ . Indeed, let φ = (φ0 , φ1 , . . . , φk ) be a fixed element of D. Then the functional Gφ (·) can be defined as follows: Gφ (y ∗ ) = y∗ , φ := y ∗ , φ0 X∗ ,X +

k +

Λ∗j y ∗ , φj

j=1

, Z∗ j ,Zj

.

Remark 2.64. Let σ(Y∗ , D) be the weakest topology on Y∗ with respect to which all functionals Gφ (·) are continuous. Then the pair (Y∗ , σ(Y∗ , D)) is a locally convex topological space. ∞

Definition 2.65. We say that a sequence {yn∗ }n=1 ⊂ Y∗ converges D-weakly D

to an element y ∗ ∈ Y∗ , written yn∗ y ∗ , provided lim yn∗ , φ0 X∗ ,X = y ∗ , φ0 X∗ ,X ,

n→∞

+ , , + lim Λ∗j yn∗ , φj Z∗ ,Z = Λ∗j y ∗ , φj Z∗ ,Z

n→∞

j

j

j

j

for all φ = (φ0 , φ1 , . . . , φk ) ∈ D. Note that this concept can be easy extended to the case of D-weakly convergent nets (or generalized sequences) {yα∗ }α∈A ⊂ Y∗ , where A is a directed set of indices. The following result deals with the compactness property of the set K∗ . k Theorem 2.66. Let X and {Zj }kj=1 be Banach spaces and let X∗ , Z∗j j=1 be their duals, respectively. Let k

{Λj : (D(Λj ) ⊂ Zj ) → X}j=1 be a given family of linear mappings with dense domains D(Λj ) in Zj for each j = 1, . . . , k. Then the set K∗ , defined by (2.61), is D-weakly compact in Y∗ for every collection of positive numbers l0 , l1 , . . . , lk . Proof. Let l0 , l1 , . . . , lk be given positive numbers. Let {yα∗ }α∈A be an arbitrary net in K∗ . To prove this theorem, we have to show that there is a subnet {x∗β }β∈B of the original net {yα∗ }α∈A , which D-weakly converges in Y∗ to some element of K∗ . Taking into account the structure of the set K∗ , we see that {yα∗ }α∈A belongs to the close ball BX∗ (0, l0 ) centered at the origin with radius l0 . Hence, by the Banach–Alaoglu theorem (see Yosida [251] or Kantorovich and Akilov [128]), there exists a subnet of {yα∗ }α∈A , denoted by {x∗β }β∈B0 , such that x∗β → y ∗ in X∗σ and y ∗ X∗ ≤ l0 . Setting now j = 1, we consider the net {Λ∗1 x∗β }β∈B0 . Since the elements Λ∗1 x∗β belong to the closed ball BZ∗1 (0, l1 ),

60


∗ it follows that there exists a subnet zγ∗ γ∈B such that Λ∗1 zγ∗ → μ1 in Z1,σ 1 and μ1 Z∗1 ≤ l1 . Repeating this iteration process under j = 2, . . . , k, finally, we can extract a subnet {d∗ν }ν∈Bk of the origin net {yα∗ }α∈A such that d∗ν , φ0 X∗ ,X −→k y ∗ , φ0 X∗ ,X , + ∗ ∗ , ν∈B Λj dν , φj Z∗ ,Z −→k μj , φj Z∗ ,Zj , ν∈B

j

∗

y X∗ ≤ l0 ,

j

j

μj Z∗j ≤ lj , j = 1, 2, . . . , k,

for each φ = (φ0 , φ1 , . . . , φk ) ∈ D. In view of the properties of the domains D(Λj ), the operators Λ∗j : D(Λ∗j ) ⊂ X∗ → Z∗j are closed in X∗σ × Z∗j,σ . Hence, μj = Λ∗j y ∗ for evD

ery j = 1, 2, . . . , k. As a result, we have d∗ν y ∗ , where y∗ X∗ ≤ l0 and Λ∗j y ∗ Z∗j ≤ lj for j = 1, . . . , k. Thus, y ∗ ∈ K∗ , which concludes the proof. To illustrate the possible applications of this theorem, we give the following example. Example 2.67. Let Ω be open bounded subset of Rn with a Lipschitz boundary. Let X = Zj = L1 (Ω) for all j = 1, . . . , k. We define the collection of linear k mappings {Λj : (D(Λj ) ⊂ Zj ) → X}j=1 as follows: Λj y := −∂y/∂xj ,

j = 1, 2, . . . , k,

where D(Λj ) is the closure of C0∞ (Ω) with respect to the norm y j = y L1 (Ω) + ∂y/∂xj L1 (Ω) . Thus, if y ∈ C(Ω) ∩ D(Λj ), then y ∈ L1 (Ω), supp y(x) is a compact in Ω and ∂y/∂xj ∈ L1 (Ω), where the partial derivatives ∂y/∂xj we mean in the weak sense (see (2.6)). It is easy to see that the following chain of embeddings holds: C01 (Ω) ⊂ W01,1 (Ω) ⊂ D(Λj ) ⊂ L1 (Ω),

∀ j = 1, . . . , k.

Since C01 (Ω) is dense in L1 (Ω) with respect to the topology induced by the norm · L1 (Ω) , it follows that the weak closure of the set C01 (Ω) coincides with the entire space L1 (Ω). Therefore, the domains D(Λj ) are weakly dense in L1 (Ω) for all j = 1, . . . , k. In view of this, the dual operators

k Λ∗j : D(Λ∗j ) ⊂ L∞ (Ω) → L∞ (Ω) j=1

are well defined and closed. Moreover, in this case, we have Λ∗j = ∂/∂xj . As a result, we obtain

2.9 On weak compactness of bounded sets in Banach spaces

D ∗ :=

k '

61

D(Λ∗j ) = W 1,∞ (Ω),

j=1

where W 1,∞ (Ω) =

∂y y(x) : y ∈ L∞ (Ω), ∈ L∞ (Ω), j = 1, . . . , k ∂xj

is the Banach space equipped with the norm ∂y(x) . = ess sup |y(x)| + ess sup ∂x x∈Ω x∈Ω j j=1 k

y W 1,∞ (Ω)

Hence Y∗ = W 1,∞ (Ω) and, therefore, due to the Theorem 2.66, the bounded set ∂y ∗ 1,∞ (Ω) : y L∞ (Ω) ≤ l0 , ≤ lj , j = 1, . . . , k K = y∈W ∂xj ∞ L (Ω) is D-weakly compact in W 1,∞ (Ω) for every positive numbers l0 , l1 , . . . , lk . Following Definition 2.65 and using the fact that L1 (Ω) is a separable Banach space, it means that for any net {yα∗ }α∈A ⊂ K∗ , there exists a sequence {x∗i }i∈N (which is a subnet of {yα∗ }α∈A ) such that lim x∗i , φY∗ ,Y = y, φY∗ ,Y ,

i→∞

∀ φ = (φ0 , . . . , φk ) ∈ L1 (Ω) ×

k j=1

where y ∈ K∗ and y, φY∗ ,Y =

y(x)φ0 (x) dx + Ω

k j=1

φj (x)

Ω

Thus, the set K∗ is sequentially D-weakly compact.

∂y(x) dx. ∂xj

D(Λj ),

3 Variational Methods of Optimal Control Theory

Since our main interest is related to the mathematical theory of the parameterized optimal control problems (OCPε ) for partial differential equations (PDEs), we discuss in this chapter general questions of optimal control theory, different settings of optimal control problems (OCPs) for distributed systems in variable spaces, the direct method of Calculus of Variation, the topological properties of solutions to ill-posed problems, optimality conditions for a wide class of extremal problems in the form of variational inequalities and also questions related to the construction of approximative solutions to different classes of OCPs. We refer to Alekseev, Tikchomirov and Fomin [4], Egorov [103], Fursikov [111], Ivanenko and Mel’nik [126], Lions [171], Zgurovsky and Mel’nik [253], Bonnans and Shapiro [24], etc., or the textbooks by Tr¨ oltzsch [242] and Mordukhovich [190] for more results concerning this topic. A recent account on PDE-constrained OCP’s can also be found in Hinze, Pinnau, Ulbrich and Ulbrich [122]. It is well known that OCPs are extremal problems which describe the behavior of a system that can be modified by the action of an operator. Many problems in applied sciences can be modeled by means of OCPs. Two kinds of variables (or sets of variables) are then involved: One of them, called the state variable, describes the state of the system and cannot be modified directly by the operator; the second one, called the control variable, is under direct control of the operator who may choose his strategy among a given set of admissible ones. The operator is allowed to modify the state of the system indirectly, acting directly on admissible control variables. A link “control–state” is normally put into the format of what is called the state equation. Typically, the size of the admissible control is constrained. Finally, the operator, acting directly on controls and indirectly on states through the state equation, must achieve a goal, usually written as a minimization of a functional, the so-called cost functional, which depends on the control that has been chosen as well as on the corresponding state. The typical ingredients of an OCP are as follows: P.I. Kogut, G.R. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains, Systems & Control: Foundations & Applications, DOI 10.1007/978-0-8176-8149-4 3, © Springer Science+Business Media, LLC 2011

63

64

(A1 ) (A2 ) (A3 ) (A4 )


A space of states Y; A space of controls U; A cost (objective) functional I : U × Y → R (CF); A set Ξ of admissible pairs, that is, a subset of pairs (u, y) ∈ U × Y such that I(u, y) < +∞, y is linked to u through the state equation (SE) and u satisfies the control constraints u ∈ U (CCs).

As a result, the OCP can be stated in the form of a constrained minimization problem: (OCP) : min {I(u, y) : (u, y) ∈ Ξ} . (3.1)

3.1 The general setting We begin this section with the following notion. Definition 3.1. Let (X, τ ) be a topological space, let R := R ∪ {+∞} and let F : X → R be a functional. (1) We say that F is τ -lower semicontinuous (τ -lsc) if for every t ∈ R the set {x ∈ X : F (x) ≤ t} is τ -closed in X. We say that F is sequentially τ -lsc if F (x) ≤ lim inf F (xn ) n→∞

for every x ∈ X and every sequence {xn } converging to x in (X, τ ). (2) We say that F is τ -coercive if for every t ∈ R, there exists a τ -compact subset Kt of X such that {x ∈ X : F (x) ≤ t} ⊂ Kt . We say that F is sequentially τ -coercive if for every t ∈ R, every sequence in {x ∈ X : F (x) ≤ t} has a subsequence that τ -converges to a point of {x ∈ X : F (x) ≤ t}. Let Y and Z be linear normed spaces and let Y1 be a reflexive separable Banach space such that Y1 is continuously and densely embedded in Y. Let V be a separable Banach space and U = V∗ be its dual. Let U∂ be a closed convex subset of U. In particular, the convexity of U means that for any u1 ∈ U∂ and u2 ∈ U∂ , the condition αu1 + (1 − α)u2 ∈ U∂ holds true for all α ∈ (0, 1). We consider the following OCP: Find a pair (u0 , y0 ) ∈ U∂ × Y1 such that a functional I(u, y) → inf (3.2) subject to the restrictions L(u, y) + F (y) = 0, and u ∈ U∂ ,

(3.3)

where L : U × Y1 → Z is a linear continuous mapping, F : Y1 → Z is a nonlinear mapping and I : U × Y → R is a proper cost functional which is always supposed to be bounded below and sequentially lower semicontinuous

3.1 The general setting

65

with respect to the product of the weak-∗ topology for U and of the weak topology for Y. Hereinafter, we associate u ∈ U∂ with an admissible control and y ∈ Y1 with a state of the control process. Let us note also that in the applications, the state space Y1 endowed with its weak topology and the control space U endowed with its weak-∗ topology are not metrizable, in general. Nevertheless, due to some growth assumptions on the cost functional I, we may often restrict ourselves to work on a bounded subset of Y1 ×U which is, thanks to the initial assumptions, metrizable. However, the current point of view leaves room for ill-posed boundary value problems. Indeed, we suppose that ∃u ∈ U, ∃ y ∈ Y1 : L(u, y) + F (y) = 0. Thus, the corresponding state y may be not unique, in general. So, we admit the situation when there exist controls u ∈ U such that y ∈ Y1 satisfying the equation L(u, y) + F (y) = 0. It is clear that in this case, it is not possible to write y = y(u). To avoid this situation, it is convenient to introduce the set of admissible pairs for the problem (3.2)–(3.3): Ξ = {(u, y) ∈ U × Y1 : L(u, y) + F (y) = 0, u ∈ U∂ , I(u, y) < +∞} . (3.4) We say that a pair (u0 , y0 ) ∈ U × Y1 is optimal for the problem (3.2)–(3.3) if (u0 , y 0 ) ∈ Ξ and I(u0 , y 0 ) =

inf

I(u, y).

(u,y)∈ Ξ

In what follows, we assume that the following hypotheses hold true: (H1) (Regularity condition) Ξ = ∅. (H2) (Coerciveness condition) For any λ > 0, the set Ξλ = {(u, y) ∈ Ξ : I(u, y) ≤ λ} is bounded in U × Y1 . (H3) (Compactness condition) There exists a normed space Y−1 containing Y such that the embedding Y1 → Y−1 is compact and there exists a dense subset S of Z∗ such that for any z ∗ ∈ S, the mapping Y1 y → z ∗ , F (y)Z∗ , Z can be extended to a continuous functional defined on the whole space Y−1 . In order to explain the role of these assumptions, we give the following results (see Fursikov [111]). Theorem 3.2. Assume that the following conditions hold true: (i) The embedding Y1 → Y is continuous. (ii) There is an element F0 ∈ Z such that F (y) = F0 for all y ∈ Y1 .

66


(iii) L : U × Y1 → Z is a linear continuous mapping with respect to the weak topology of Z and the product of the weak-∗ topology for U and of the weak topology for Y. Then, under hypotheses (H1)–(H2), the OCP (3.2)–(3.3) admits at least one solution. In addition, if I is a strictly convex functional, that is, the inequality I(αu1 + (1 − α)u2 , αy1 + (1 − α)y2 ) < αI(u1 , y1 ) + (1 − α)I(u2 , y2 ) is valid whenever (u1 , y1 ) = (u2 , y2 ) and α ∈ (0, 1), then the OCP (3.2)–(3.3) has a unique solution. Proof. Since the original problem is regular, it follows that there exists a minimizing sequence {(uk , yk )}k∈ N ⊂ Ξ for the original problem, that is, lim I(uk , yk ) =

k→∞

inf (u,y)∈ Ξ

I(u, y) > −∞.

Hence, supk∈N I(uk , yk ) ≤ C, where the constant C is independent of k. Then in view of (H2), we obtain that supn∈N [uk U + yk Y1 ] ≤ C. Hence, using the Banach–Alaoglu theorem and passing to a subsequence, we may assume that ∗ uk u0 in U, yk y 0 in Y1 . Since the set U∂ is convex and closed, it follows that U∂ is sequentially closed with respect to the weak-∗ topology. Therefore, u0 ∈ U∂ . In addition, for any z ∗ ∈ Z∗ , we have z ∗ , L(uk , yk )Z∗ , Z = L∗ z ∗ , (uk , yk )U∗ ×Y∗ , U×Y1 1 −−−−→ L∗ z ∗ , (u0 , y 0 ) U∗ ×Y∗ , U×Y1 = z ∗ , L(u0 , y0 ) Z∗ , Z , k→∞

1

where L∗ z ∗ ∈ U∗ × Y∗ . Thus, summing up the results obtained above – namely, L(u0 , y0 )+F0 = 0, 0 u ∈ U∂ and I(u0 , y 0 ) < +∞ – we conclude that (u0 , y 0 ) is an admissible pair to the problem (3.2)–(3.3). Since the cost functional I is lower semicontinuous with respect to the product of the weak-∗ topology for U and of the weak topology for Y convergence, we get −∞ < I(u0 , y 0 ) ≤ lim I(un , yn ) = n→∞

inf

I(u, y).

(u,y)∈ Ξ

Thus, the pair (u0 , y 0 ) is optimal for the problem (3.2)–(3.3). This pair is unique in the case when the cost functional is strictly convex. Indeed, let (u01 , y10 ) and (u02 , y20 ) be two different solutions to the original problem. Since the set equation L(u, y)+F0 = 0 is linear, it follows U∂ is convex and the state that (u01 + u02 )/2, (y10 + y20 )/2 ∈ Ξ. Hence,


67

1 0 0 I(u1 , y1 ) + I(u02 , y20 ) I (u01 + u02 )/2, (y10 + y20 )/2 < 2 = inf I(u, y), (u,y)∈ Ξ

and we come to a contradiction. Hence, our assumption was wrong, and this concludes the proof. Theorem 3.3. Under hypotheses (H1)–(H3), OCP (3.2)–(3.3) admits at least one solution. Proof. Since Ξ = ∅ and the cost functional is bounded below on Ξ, it follows that there exists a minimizing sequence {(uk , yk )}k∈ N ⊂ Ξ for the (3.2)–(3.3), that is, I(uk , yk ) −−−−→ Imin ≡ inf I(u, y) > −∞. n→∞

(u,y)∈ Ξ

Using the arguments of the previous theorem, we may extract a subsequence of {(uk , yk )}k∈ N ⊂ Ξ (still indexed by k) such that ∗

uk u0 in U, yk y 0 in Y1 , z ∗ , L(uk , yk )Z∗ , Z −−−−→ z ∗ , L(u0 , y0 ) Z∗ , Z ∀ z ∗ ∈ Z∗ . k→∞

(3.5)

Hence, u0 ∈ U∂ . By (H3), the embedding Y1 → Y−1 is compact and, hence, yn → y 0 strongly in Y−1 . Therefore, (3.6) z ∗ , F (yk )Z∗ , Z −−−−→ z ∗ , F (y 0 ) Z∗ , Z , ∀ z ∗ ∈ S ⊂ Z∗ . k→∞

Then, taking relations (3.5) and (3.6) and the density of the embedding S ⊂ Z∗ into account and passing to the limit in L(uk , yk ) + F (yk ) = 0 as k → ∞, we obtain L(u0 , y0 ) + F (y 0 ) = 0. As a result, (u0 , y 0 ) ∈ Ξ. Since the embedding Y1 ⊂ Y is continuous, it follows that yn y 0 in Y. Using the property of lower semicontinuity for I with respect to the product of the weak-∗ topology for U and of the weak topology for Y convergence, we get −∞ < I(u0 , y0 ) ≤ lim I(un , yn ) = Imin . n→∞

Thus, the pair (u0 , y 0 ) is optimal for the problem (3.2)–(3.3). Remark 3.4. It is well known (see, for instance, Ekeland and Temam [104]) that a convex lower semicontinuous functional is also lower semicontinuous with respect to the weak-∗ topology. So, instead of the suppositions concerning the cost functional I given by Theorem 3.2, we can take the following ones:

68


(a) I is lower semicontinuous on Ξ ⊂ U×Y with respect to the strong topology of U × Y. (b) I : U × Y → R is a convex functional, that is, for any (u1 , y1 ) ∈ Ξ and (u2 , y2 ) ∈ Ξ, the inequality I(αu1 + (1 − α)u2 , αy1 + (1 − α)y2 ) ≤ αI(u1 , y1 ) + (1 − α)I(u2 , y2 ) (3.7) holds true for every α ∈ [0, 1]. In addition, if I is a strictly convex functional (i.e., the inequality (3.7) is strict whenever (u1 , y1 ) = (u2 , y2 ) and α ∈ (0, 1)) then the OCP (3.2)–(3.3) with a nonlinear state equation has a unique solution (see Lions [169]). ∞

Definition 3.5. A sequence of pairs {(uk , yk )}k=1 ⊂ U × Y is said to be a ∗ w-convergent to a pair (u, y) ∈ U × Y if uk u in U and yk y in Y. As a consequence of the assumptions given above, we have the following result. Lemma 3.6. Under the hypotheses of Theorem 3.2 and the supposition that the cost functional I is bounded on any bounded subset of U∂ × Y, the set of admissible solutions Ξ to the problem (3.2)–(3.3) is sequentially closed with respect to the product of the weak-∗ topology for U and of the weak topology for Y. ∞

Proof. Let {(uk , yk )}k=1 be any w-convergent sequence of admissible pairs, ∗ that is, (uk , yk ) ∈ Ξ for all k ∈ N and uk u in U and yk y in Y. Hence, sup uk U + yY < +∞. k∈N

Then, by the initial supposition, we have supk∈ N I(uk , yk ) < +∞. By virtue of the coerciveness property (H2), we obtain ∗

uk u in U and yk y in Y1 . Thus, having repeated all arguments of the proof of Theorem 3.3, we conclude u, y) ∈ Ξ. that L( u, y) + F ( y ) = 0 and u ∈ U∂ , that is, ( As an application of Theorem 3.3, take an OCP as follows: 3 |y(x) − z∂ (x)| dx + N |u(x)|2 dx → inf, I(u, y) = Ω Ω

−Δy(x) − y |y(x)| = f (x) + u(x), x ∈ Ω, y|∂Ω = 0, u ∈ U∂ , U∂ = u ∈ L2 (Ω) : α1 (x) ≤ u(x) ≤ α2 (x) a.e. in Ω ,

(3.8) (3.9) (3.10)

where Ω is a bounded open domain in Rn with a smooth boundary ∂Ω, z∂ ∈ L3 (Ω),

f ∈ H −1 (Ω),

N > 0,

αi ∈ C(Ω), i = 1, 2,


69

and α1 (x) ≤ α2 (x) − δ in Ω with some δ > 0. To give the variational formulation of the boundary value problem (3.9), we say that y is a weak solution to that problem with a fixed right-hand side f (x) + u(x) if y ∈ H01 (Ω) and the integral identity

∇y(x) · ∇ϕ(x) − y(x) |y(x)|ϕ(x) dx = f + u, ϕH −1 (Ω), H 1 (Ω) 0

Ω

holds true for every ϕ ∈ C0∞ (Ω). However, there is a good reason to believe that we do not, in general, have the existence of a weak solution y(x) to the above boundary value problem for any functions f ∈ H −1 (Ω) and u ∈ U∂ (see Lions [172]). So, we have to begin with the following assumption: There exists a pair (u, y) ∈ L2 (Ω) × H01 (Ω) ∩ L3 (Ω) satisfying relations (3.9) and (3.10). Evidently, this supposition is fulfilled if f ∈ L2 (Ω) and −f ∈ U∂ . In this case, the pair (−f, 0) is admissible and we have the following result. Theorem 3.7. Assume that the set of admissible pairs for the problem (3.8)– (3.10) is nonempty. Thenthis problem admits at least one solution (u0 , y 0 ) ∈ L2 (Ω) × H01 (Ω) ∩ L3 (Ω) . Proof. In order to prove the desired result, we make use of Theorem 3.3. To do so, we set Y = L3 (Ω),

Y1 = L3 (Ω) ∩ H01 (Ω),

U = L2 (Ω),

Z = H −1 (Ω), L(u, y) = −Δy − u, F (y) = −|y|3/2 − f. Now we observe that the cost functional I : L2 (Ω) × L3 (Ω) → R is convex and lower semicontinuous of L2 (Ω)×L3 (Ω). 1 with ∗ respect to the weak topology −1 1 Since H (Ω) = H0 (Ω) and the operator −Δ : H0 (Ω) → H −1 (Ω) is continuous [168], it follows that the mappings L(u, y) = −Δy − u : L2 (Ω) × H01 (Ω) ∩ L3 (Ω) → H −1 (Ω), F (y) = −|y|3/2 − f : H01 (Ω) ∩ L3 (Ω) → H −1 (Ω) are continuous as well. Let us show that the coerciveness property (H2) is valid. Indeed, for any λ > 0, we have Ξλ = (u, y) ∈ L2 (Ω) × H01 (Ω) ∩ L3 (Ω) : u ∈ U∂ ,

−Δy = y |y| + f + u, I(u, y) ≤ λ

⊂ (u, y) ∈ L2 (Ω) × H01 (Ω) ∩ L3 (Ω) : −Δy = y |y| + f + u,

√ 3 uL2 (Ω) ≤ λ/N , yL3 (Ω) ≤ λ + z∂ L3 (Ω) , u ∈ U∂ .

70


Since yH01 (Ω) ≤ f H −1 (Ω) + CuL2 (Ω) + CyL3 (Ω) , we have

Ξλ ⊂ (u, y) ∈ L2 (Ω) × H01 (Ω) ∩ L3 (Ω) : uL2 (Ω) ≤ λ/N , √ 3 yL3 (Ω) ≤ λ + z∂ L3 (Ω) , √

3 yH01 (Ω) ≤ f H −1 (Ω) + C λ + z∂ L3 (Ω) + C λ/N .

Hence, Ξλ is a bounded subset of L2 (Ω) × H01 (Ω) ∩ L3 (Ω) . It remains only to verify the compactness property (H3). With this aim, we set Y−1 = L3/2 (Ω). Since Ω is a bounded domain, it follows that L3 (Ω) ⊂ L3/2 (Ω), that is, the embedding Y1 → Y−1 is continuous. On the other hand, using the compactness of the embedding H01 (Ω) ∩ L3 (Ω) → L2 (Ω) and the 3/2 estimate yL3/2 (Ω) ≤ CyL2 (Ω) that is fulfilled for every y ∈ L2 (Ω), we obtain that H01 (Ω) ∩ L3 (Ω) → L3/2 (Ω) is compact. It is easy to see that the functional |y(x)|3/2 ϕ(x) dx− < f, ϕ >H −1 (Ω),H01 (Ω) y → F (y), ϕH −1 (Ω),H 1 (Ω) = − 0

Ω

is continuous for every ϕ ∈ C0∞ (Ω). Since C0∞ (Ω) is dense in −1on 1 H0 (Ω) = H (Ω) , we obtain the desired property (H3). Thus, all suppositions of Theorem 3.3 hold true. Hence, the OCP (3.8)–(3.10) is solvable, and we obtain the required conclusion. L3/2 (Ω) ∗

To illustrate the significance of the main hypotheses to Theorem 3.3, let us consider the following OCP for which the compactness condition (H3) fails. As we will see later, it is an explicit example of an OCP for which an optimal solution does not exist: 1 2 (3.11) I(u, y) = y (t) + u2 (t) dt → inf, 0

y˙ 4 (t) + u(t) − 1 = 0,

y(0) = y(1) = 0,

(3.12)

where y(t) ˙ = dy/dt. Let us set Y = U = Z = L2 (0, 1), U∂ = L2 (0, 1), Y1 = W01,8 (0, 1) = y ∈ W 1,8 (0, 1) : y(0) = y(1) = 0 , L(u, y) = u(t),

F (y) = y˙ 4 (t) − 1.

Then under such a choice of the operators and spaces, all assumptions of Theorem 3.3, with the exception of the compactness property (H3), hold true.


71

Indeed, since (u, y) = 1 − π 4 cos4 πt, sin πt ∈ L2 (0, 1) × W01,8 (0, 1), it follows that the set of admissible pairs Ξ = (u, y) ∈ L2 (0, 1) × W01,8 (0, 1) : u = 1 − y˙ 4 (t) is nonempty. Let us show the coerciveness property (H2). To do so, we make use of the H¨ older inequality

1

y˙ 4 (t) dt ≤ 0

Then

1/2

1

y˙ 8 (t) dt

.

0

1 2 1,8 2 4 y (t) + 1 − y˙ (t) dt ≤ λ y ∈ W0 (0, 1) : 0 1 1 y˙ 8 (t) dt ≤ λ − 1 + 2 y˙ 4 (t) dt ⊂ y ∈ W01,8 (0, 1) : 0 0 1,8 8 4 ˙ L8 (0,1) − 2y ˙ L8 (0,1) + 1 − λ ≤ 0 ⊂ y ∈ W0 (0, 1) : y √ 2 = y ∈ W01,8 (0, 1) : y ˙ 8L8 (0,1) ≤ 1 + λ .

As a result, for any λ > 0, we have {(u, y) ∈ Ξ : I(u, y) ≤ λ} ⎧ 1 ⎫ 2 ⎨ ⎬ 2 4 y dt ≤ λ, (t) + 1 − y ˙ (t) = (u, y) 0 ⎩ u = 1 − y˙ 4 , y(0) = y(1) = 0 ⎭ ⎫ ⎧ √ 2 ⎪ ⎪ y ˙ 8L8 (0,1) ≤ 1 + λ , ⎬ ⎨ . ⊂ (u, y) 1 2 ⎪ u2 ⎪ 4 ⎭ ⎩ 1 − y ˙ = (t) dt ≤ λ 2 L (0,1) 0

Since the set on the right-hand side of this inequality is bounded in L2 (0, 1) × W01,8 (0, 1), it follows that the OCP (3.11)–(3.12) is coercive. It remains to prove that the compactness property (H3) fails to hold. To do so, we show that no pair (u, y) ∈ Ξ can solve the OCP (3.11)–(3.12). Then, by Theorem 3.3, it means that (H3) cannot be valid. Indeed, let us rewrite the original problem as follows: 1 2 I(y) = y 2 (t) + 1 − y˙ 4 (t) dt −→ inf, y(0) = y(1) = 0. (3.13) 0 ∞

Let {yk }k=1 be the following sequence in W01,8 (0, 1): t sign sin(2πkt) dt, ∀ k ∈ N. yk (t) = 0

72


It is clear that yk (0) = yk (1) = 0, |y˙ k (t)|4 = 1 for almost all t ∈ (0, 1) and yk (t) → y ∗ (t) ≡ 0 as k → ∞ uniformly with respect to t ∈ [0, 1]. Then I(yk ) → 0 as k → ∞. Hence, 1 2 2 4 I(y) = inf (t) + 1 − y ˙ (t) inf y dt = 0. 1,8 1,8 y∈W0

(0,1)

y∈W0

(0,1)

0

On the other hand, as immediately follows from (3.13), if y(t) = y ∗ (t) ≡ 0 then I(y ∗ ) = 1. Moreover, in this case, we have 1 I(y) ≥ y 2 (t) dt > 0 ∀ y(·) ≡ y ∗ (·). 0

Thus, the OCP (3.11)–(3.12) has no optimal solutions.

3.2 Abstract extremal problems In this section, we study the OCPs in Banach spaces for essentially nonlinear operator equations with additional control and state constraints. This topic has been widely studied by many authors. We only mention Barbu [19], Fattorini [108], Fursikov [111], Ivanenko and Mel’nik [126], Lions [172], Zgurovsky and Mel’nik [253]. Our prime interest is to study these problems in the case when the control and state constraints take the form of some inequalities and inclusions in Banach spaces. Let Y be a reflexive Banach space, Y∗ be its dual, and Z be a Banach space partially ordered by closed reproducing pointed cone Λ ⊂ Z. Let U be a control space which is assumed to be dual to some separable Banach space V (U = V∗ ). Let U∂ be a subset of admissible controls in U and let K be a subset of Y. The OCPs we consider can be described in a general manner as follows: I(u, y) −→ inf (3.14) subject to

where

A(u, y) = f,

(3.15)

F (u, y) ≥Λ 0, y ∈ K, u ∈ U∂ ⊂ U,

(3.16) (3.17)

A : U × Y → Y∗ ,

F :U×Y→Z

are nonlinear mappings, I : U × Y → R is a cost functional and f is a given element of Y∗ . Definition 3.8. We say that the problem (3.14)–(3.17) is regular if for every f ∈ Y∗ , there exists a pair (u, y) ∈ U × Y, where y = y(u) is the corresponding solution of (3.15), such that (u, y) satisfies the restrictions (3.16), (3.17) and I(u, y) < +∞. In this case, the pair (u, y(u)) is said to be admissible.

3.2 Abstract extremal problems

73

As usual, we denote by Ξ the set of all admissible pairs to the problem (3.14)– (3.17). Let us note that, in general, the mapping u → y(u) can be multivalued. Remark 3.9. As an example of the situation, when the mapping u → y(u) is multivalued, we have the following OCP Minimize I(u, y) = |y(x) − yd (x)|2 dx + |∇y(x)|2RN u dx (3.18) Ω

Ω

subject to the constraints −div (u(x)∇y) + y = f in Ω, y = 0 on ∂Ω, u dx = m, ξ1 (x) ≤ u(x) ≤ ξ2 (x) a.e. in Ω, u ∈ L1 (Ω),

(3.19) (3.20)

Ω

where f ∈ L2 (Ω) is a given function and ξ1 and ξ2 are given elements of L1 (Ω) satisfying the condition ξ1 (x) ≤ ξ2 (x) for a.e. x ∈ Ω,

ξ1−1 ∈ L1 (Ω).

(3.21)

We can associate with each admissible control u (i.e., u satisfies conditions (3.20)–(3.21)), the weighted Sobolev space Wu = W (Ω, u dx), where 1/2 2 1,1 2 y + u |∇y| dx < +∞ . Wu = y ∈ W0 (Ω) : yu = Ω

It is reasonable to say that a function y = y(ρ, f ) ∈ Wu is a weak solution to the boundary value problem (3.19) for a fixed control u ∈ Rad if the integral identity f ϕ dx (3.22) (∇y, ∇ϕ)RN u + yϕ dx = Ω

Ω

holds for any test function ϕ ∈ C0∞ (Ω). It is clear that the question of uniqueness of a weak solution leads us to the question of whether C0∞ (Ω) is dense in Wu . However, as was indicated in [266], for a “typical” admissible control u, the space C0∞ (Ω) is not dense in Wu ; hence, there is no uniqueness of weak solutions (for more details and other types of solutions, we refer to [23, 159, 255, 266]). Thus, for OCP (3.18)–(3.21) the mapping u → y(u) is multivalued, in general. We begin with the following suppositions: (A1) U∂ is a bounded sequentially weakly-∗ closed subset of U. (A2) The operator A : U × Y → Y∗ is coercive, that is, inf

u∈G

A(u, y), yY∗ ;Y yY

for any bounded subset G ⊂ U.

→ +∞

as

yY → +∞

74


(A3) The operator A : U × Y → Y∗ possesses the property (M), that is, for ∞ any w-convergent sequence {(uk , yk )}k=1 , the conditions A(uk , yk ) d in Y∗ ,

lim sup A(uk , yk ), yk Y∗ ;Y ≤ d, yY∗ ;Y k→∞

imply the relation d = A(u, y). (A4) The operator F : U × Y → Z is sequentially continuous in the following w sense: F (uk , yk ) F (u, y) in Z whenever (uk , yk ) → (u, y). (A5) K is a weakly closed subset of Y. (A6) The cost functional I : U × Y → R is sequentially lower semi-continuous with respect to the w-convergence in U × Y, that is, lim inf I(uk , yk ) ≥ I(u, y) i→∞

w

whenever (uk , yk ) → (u, y). Before proceeding further, we note that the cone Λ ⊂ Z is uniquely defined by its conjugate semigroup Λ∗ , that is, Λ = ξ ∈ Z : ϕ, ξZ∗ ;Z ≥ 0 ∀ ϕ ∈ Λ∗ . (3.23) Let us denote by P+ (Λ∗ ) the set of all equivalence classes with respect to the following binary relation: ϕ1 ϕ2 ⇐⇒ ∃ t ∈ R \ {0} : ϕ1 = tϕ2 , ϕ1 , ϕ2 ∈ Λ∗ . Let Π ∗ : Λ∗ \ 0 → P+ (Λ∗ ) be the corresponding canonical quotient mapping. We assume that P+ (Λ∗ ) is endowed with the quotient topology. It is clear that, in this case, the mapping Π ∗ |S ∗ ∩Λ∗ : S1∗ ∩ Λ∗ → P+ (Λ∗ ) 1

is a continuous surjection, that is, every equivalence class can be interpreted as the image of some element of Λ∗ belonging to the unit sphere S1∗ in Z∗ . Hence, if F (v, ξ) ∈ Λ for some pair (v, ξ) ∈ U∂ ×Y, then there is an element ψ ∈ S1∗ ∩Λ∗ such that ψ, F (v, ξ)Z∗ ;Z < 0. Taking into account these observations, we introduce the following penalized version of the original problem (3.14)–(3.17):

Iε (u, y) = I(u, y) + ε−1 sup μ ψ, F (u, y)Z∗ ;Z → inf, (3.24) ψ∈S1∗ ∩Λ∗

A(u, y) = f, y ∈ K, u ∈ U∂ ⊂ U,

(3.25) (3.26)

where μ : R → R+ is a lower semicontinuous monotone decreasing function such that μ(0) = 0 and μ is strictly monotone on R− . We denote by F(R, R+ ) the set of all functions μ with the properties mentioned above. To begin with, we assert the following results.


75

Lemma 3.10. For every ε > 0 and every fixed μ ∈ F(R, R+ ), the problem (3.24)–(3.26) has a nonempty set of solutions. Proof. Let Ξ be a set of admissible solutions to the penalized problem (3.24)– (3.26). It is clear that Ξ ⊂ Ξ for every ε > 0. First, we show that the cost functional Iε : U∂ × Y → R is bounded below on the set Ξ. Let us assume the converse. Then there exists a sequence {(uk , yk )}∞ k=1 ⊂ Ξ such that Iε (uk , yk ) < −k for all k ∈ N. Due to the initial assumptions, we have ∞ {uk }∞ k=1 ⊂ U∂ ; hence, the sequence {uk }k=1 is bounded in U. Thus, we may ∗ assume that uk u in U and u ∈ U∂ . Since A(uk , yk ), yk Y∗ ;Y = f, yk Y∗ ;Y ≤ f Y∗ yk Y ,

∀k ∈ N

and the operator A is coercive (see (A2)), it follows that sup

A(uk , yk ), yk Y∗ ;Y

k∈N

yk Y

≤ f Y∗ .

∞

Hence, {yk }k=1 ⊂ K is a bounded sequence in Y and there exists an element y ∈ K such that, passing to a subsequence if necessary, we obtain yk y in Y. Then, having used the sequential lower semicontinuity of I with respect to the w-convergence and the non-negativeness of the term

μ ψ, F (u, y)Z∗ ;Z , sup ψ∈S1∗ ∩Λ∗

we come to the contradiction I(u, y) ≤ lim inf I(uk , yk ) ≤ lim inf Iε (uk , yk ) < −∞. k→∞

k→∞

Thus, the cost functional Iε : U∂ × Y → R is bounded below on the set Ξ. ∞ Let {(uk , yk )}k=1 ⊂ Ξ be a minimizing sequence of admissible pairs to the problem (3.24)–(3.26). By the previous arguments, this sequence is bounded in U × Y. Since Ξ ⊂ U∂ × K and the set U∂ × K is sequentially closed with respect to the w-convergence, we may assume that there exists a pair w (u0ε , yε0 ) ∈ U∂ × K such that (uk , yk ) → (u0ε , yε0 ). Then, taking into account that A(uk , yk ) = f for all k ∈ N and passing to the limit in the equality A(uk , yk ), yk Y∗ ;Y = f, yk Y∗ ;Y as k → ∞, we obtain lim A(uk , yk ), yk Y∗ ;Y = f, yε0 Y∗ ;Y .

k→∞

Hence, A(u0ε , yε0 ) = f by the (M) property of the operator A : U × Y → Y∗ . Thus, the limit pair (u0ε , yε0 ) is admissible for the problem (3.24)–(3.26) (i.e., (u0ε , yε0 ) ∈ Ξ).

76


Let us show that (u0ε , yε0 ) ∈ Ξ is an optimal pair to this problem. Indeed, in view of the continuity property of the operator F : U × Y → Z, we have F (uk , yk ) F (u0ε , yε0 ) in Z. Since the cone Λ is reproducing, it follows that the elements of conjugate semigroup Λ∗ are linear continuous functionals on Z. Hence, ψ, F (uk , yk )Z∗ ;Z → ψ, F (u0ε , yε0 ) Z∗ ;Z as k → ∞ ∀ ψ ∈ λ∗ . By our assumptions, Z ξ → μ ψ, ξZ∗ ;Z ∈ R is a convex lower semicontinuous function. Hence, this function is semi-continuous with respect to the weak topology of Z. Therefore, inf

Iε (u, y) = lim inf Iε (uk , yk ) k→∞

(u,y)∈Ξ

≥ I(u0ε , yε0 ) + ε−1 lim inf

sup

k→∞ ψ∈S ∗ ∩Λ∗ 1

≥

μ ψ, F (uk , yk )Z∗ ;Z

Iε (u0ε , yε0 ),

and we obtain the required conclusion: (u0ε , yε0 ) is an optimal pair to the penalized problem (3.24)–(3.26). 0 0 Lemma 3.11. Let (uε , yε ) ∈ Ξ ε>0 be a sequence of optimal pairs to the problem (3.24)–(3.26) when the parameter ε > 0 varies in a strictly decreasing sequence of positive numbers which converge to 0. Then there is a subsequence of (u0ε , yε0 ) ε>0 , still denoted by ε, such that w

(u0ε , yε0 ) → (u0 , y 0 ) in U × Y as ε → 0, 0

0

A(u , y ) = f,

F (u , y ) ≥Λ 0, 0

0

0

0

I(u , y ) =

y ∈ K,

u ∈ U∂ ⊂ U,

0

inf

0

I(u, y).

(3.27) (3.28) (3.29)

(u,y)∈Ξ

Proof. In thesame way as in the proof of Lemma 3.10, we can conclude that the sequence (u0ε , yε0 ) ∈ Ξ ε>0 is relatively w-compact in U × Y and, passing to a subsequence when the occasion requires, we get ∗

u0ε u0 in U,

yε0 y 0 in Y,

(u0 , y0 ) ∈ Ξ.

Let us prove that F (u0 , y0 ) ≥ 0. Let (u, y) be any admissible pair to the original problem, that is, (u, y) ∈ Ξ. Then, by our assumptions, μ ψ, F (u, y)Z∗ ;Z = 0. Therefore, Iε (u0ε , yε0 ) ≤ Iε (u, y) ≡ I(u, y), from which it follows that


sup

ψ∈S1∗ ∩Λ∗

77

μ ψ, F (u0ε , yε0 ) Z∗ ;Z ≤ εC,

where a constant C is independent of both ε and ψ ∈ S1∗ ∩ Λ∗ . Then, using the w-lower semicontinuity property, we have lim inf Iε (u0ε , yε0 ) ≥ lim inf I(u0ε , yε0 ) ≥ I(u0 , y 0 ), ε→0

ε→0

and 0≤

sup

ψ∈S1∗ ∩Λ∗

≤ lim inf ε→0

μ ψ, F (u0 , y 0 ) Z∗ ;Z sup

ψ∈S1∗ ∩Λ∗

μ ψ, F (u0ε , yε0 ) Z∗ ;Z = 0.

Since this is equivalent to the inequality F (u0 , y0 ) ≥ 0, it follows that the limit pair (u0 , y0 ) is admissible for the original problem (3.14)–(3.17). It remains to prove that this pair is optimal. Let us assume the converse – namely, suppose there is a pair ( u, y) ∈ Ξ such that I( u, y) < I(u0 , y0 ). Then this pair is admissible to the penalized problem (i.e., ( u, y) ∈ Ξ). Hence, u, y) ≥ I( u, y) ≡ Iε (

inf (u,y)∈Ξ

Iε (u, y) = Iε (u0ε , yε0 )

∀ ε > 0,

I( u, y) ≥ lim inf Iε (u0ε , yε0 ) ≥ I(u0 , y 0 ), ε→0

and it leads us to the contradiction. The proof is complete. As a direct consequence of the previous lemmas, we have the following result. Theorem 3.12. Assume that the properties (A1)–(A6) hold true. Then the set of optimal solutions to the problem (3.14)–(3.17) is nonempty if and only if this problem is regular. Remark 3.13. Note that Theorem 3.12 is still true if instead of assumption (A3), we require that the operator A : U × Y → Y∗ is quasi-monotone. We recall that an operator A : U × Y → Y∗ is said to be quasi-monotone if for ∞ any sequence {(uk , yk )}k=1 which is w-convergent to some pair (u, y), the condition (3.30) lim sup A(uk , yk ), yk − yY∗ ,Y ≤ 0 k→∞

implies the relation lim inf A(uk , yk ), yk − ξY∗ ,Y ≥ A(u, y), y − ξY∗ ,Y k→∞

∀ ξ ∈ Y.

(3.31)

Indeed, let us prove the implication: “A is quasi-monotone” =⇒ “A possesses ∞ the property (M).” Let {(uk , yk )}k=1 be a sequence such that

78


(uk , yk ) (u, y), A(uk , yk ) ζ in Y∗ , lim sup A(uk , yk ), yk Y∗ ,Y ≤ ζ, yY∗ ,Y . w

k→∞

This immediately leads us to the inequality (3.30). Hence, by quasi-monotonicity, we have A(u, y), y − ξY∗ ,Y ≤ lim inf A(uk , yk ), yk − ξY∗ ,Y k→∞

≤ lim sup A(uk , yk ), yk − ξY∗ ,Y k→∞

≤ ζ, y − ξY∗ ,Y

∀ ξ ∈ Y.

Thus, A(u, y) = ζ, and we come to the conclusion that quasi-monotone operators A possess the property (M).

3.3 Extremal problems for steady-state processes In this section, we consider the application of the previous results to OCPs governed by elliptic PDEs. We show that there is a principle difference between the theory of OCPs for systems with distributed parameters and the theory of PDEs. Before we give the formal settings of various OCPs, we note that, for both physical and mathematical reasons, the size of the admissible controls should be constrained and that the cost of effecting controls should be accounted for in the optimization process. Limits on the size of the control are also needed in order to obtain a mathematically meaningful problem (e.g., to guarantee the existence of an optimal solution in a certain function class). There are two common ways of constraining the control. The first one is to impose an explicit bound on the control — u ∈ U∂ in our case. The second way of constraining the control is to add some norm of the control to the cost functional. Both ways of constraining the control allow one to use different norms to measure the control. The physical problem does not always tell us which norm to use. The choice of a norm is also influenced by the need to establish the wellposedness of the problem (i.e., the existence of an optimal solution in some functional class, the regularity of the optimal solution, etc.). 3.3.1 Dirichlet and Neumann boundary control problems We begin this subsection with the following optimal boundary control problem. Let Ω be an open bounded domain in Rn with a Lipschitz continuous boundary ∂Ω. Let z∂ ∈ L2 (Ω), f ∈ L2 (Ω) and N > 0 be given data. Let U∂ ⊂ L2 (∂Ω) be a convex closed set of admissible Neumann controls. The OCP is to minimize the cost functional 2 |y(x) − z∂ (x)| dx + N |u(s)|2 dHn−1 (3.32) I(u, y) = Ω

∂Ω

3.3 Extremal problems for steady-state processes

79

subject to y(x) = f (x), ∂y/∂ν = u(s),

x ∈ Ω,

s ∈ ∂Ω,

u ∈ U∂ ,

(3.33) (3.34)

!n

where = i=1 ∂ 2 /∂x2i is the Laplace operator and ν is the unit outward normal vector to Ω. From a physical point of view, we want to find a boundary control u ∈ U∂ such that the corresponding state y is as close as possible to a desired state z∂ and with minimal cost. The typical examples of the sets of admissible controls U∂ with the above properties are U∂1 = u ∈ L2 (∂Ω) : u2L2 (∂Ω) ≤ R2 , U∂2 = u ∈ L2 (∂Ω) : u ≥ 0 almost everywhere on ∂Ω , U∂3 = u ∈ L2 (∂Ω) : α(x) ≤ u ≤ β(x) almost everywhere on ∂Ω . Note that if we choose the control variable u in the form u(x) =

α(x) + β(x) β(x) − α(x) + sin v(x), 2 2

(3.35)

where v is a “free” control mapping and replace (3.35) in the boundary value problem (3.33)–(3.34), we obtain a new minimization problem where the control constraints expressed by U∂3 are automatically satisfied. Such techniques (which are basically due to Miele; see [239]) may be easily implemented in a large class of control constraints. A thorough exposition along these lines may be found in the book by Banichuk [18]. In order to apply the direct method of the calculus of variations (see Theorem 3.2) to the optimal boundary control problem (3.32)–(3.34), we make use of the space s (Ω) = y ∈ H s (Ω) : y ∈ L2 (Ω) , s ∈ [0, 2), (3.36) HΔ s and endow HΔ (Ω) with the norm

1/2 s (Ω) = yHΔ y2H s (Ω) + y2L2 (Ω) . Remark 3.14. It is worth noticing that, in this case (see Lions and Magenes [173]), the trace operator y → ( y|∂Ω , ∂y/∂ν|∂Ω ) is well defined and continuous as the mapping s (Ω) → H s−1/2 (∂Ω) × H s−3/2 (∂Ω). Π : HΔ

(3.37)

Moreover, assume that f ∈ L2 (Ω) and g ∈ H s−1−1/2 (∂Ω) satisfy the compatibility condition (2.30), where the second integral in (2.30) should be interpreted as g, 1H s−1−1/2 (∂Ω),H −s+1+1/2 (∂Ω) in the case when s − 1 − 1/2 < 0.

80


Then the Neumann problem (2.28) has a unique solution y in the quotient s (Ω)/R with the estimate space HΔ y 2L2 (Ω) y 2H s (Ω) ≤ C f 2L2 (Ω) + g2H s−1−1/2 (∂Ω) + (3.38) Δ

for all y such that y − y is a constant. Taking the compatibility condition to the boundary value problem (3.33)– (3.34) into account, we set u(s) dHn−1 = f (x) dx (3.39) Lf = u ∈ L2 (∂Ω) : ∂Ω

Ω

and prove the following result. Theorem 3.15. Assume that U∂ ∩ Lf = ∅. Then the optimal Neumann boundary control problem (3.32)–(3.34) admits a unique solution (u0 , y 0 ) ∈ 3/2 L2 (∂Ω) × HΔ (Ω). Proof. To prove this assertion, we make use of Theorem 3.2. With this aim, we set Y = L2 (Ω), Z = (L2 (Ω), L2 (∂Ω)),

3/2

Y1 = HΔ (Ω),

U = L2 (∂Ω),

L(u, y) = (y, ∂y/∂ν − u), and F (y) = (−f, 0).

Then Y1 is continuously and densely embedded in Y and, as follows from Remark 3.14, the operator L : U × Y1 → Z is continuous. Let us show that, in this case, the hypotheses (H1)–(H2) are valid. Indeed, as was indicated in Remark 3.14, for every fixed f ∈ L2 (Ω) and u ∈ L2 (∂Ω) the boundary value problem (3.33)–(3.34) has a solution y ∈ 3/2 HΔ (Ω), provided u(s) dHn−1 − f (x) dx = 0. ∂Ω

Ω

Moreover, this solution satisfies the estimate (see (3.38) with s = 3/2) y2H 3/2 (Ω) + y2L2 (Ω) ≤ C f 2L2 (Ω) + u2L2 (∂Ω) + y2L2 (Ω) , (3.40) Δ

where a constant C is independent of f and u. Since U∂ ∩ Lf = ∅, it follows that there is an admissible control u∗ ∈ U∂ such that u∗ ∈ Lf . As a result, the boundary value problem (3.33)–(3.34) is 3/2 solvable under u = u∗ . Let y ∗ ∈ HΔ (Ω) be a solution. Then the pair (u∗ , y ∗ ) is admissible to the problem (3.32)–(3.34). Thus, hypothesis (H1) is satisfied. To verify the coerciveness condition (H2), we make use of the estimate (3.40). 3/2 Then, for any admissible pair (u, y) ∈ Ξ ⊂ L2 (∂Ω) × HΔ (Ω), we have


81

(u, y)2U×Y1 : = y2H s (Ω) + y2L2 (Ω) + u2L2 (∂Ω)

≤ (1 + C) f 2L2 (Ω) + u2L2 (∂Ω) + y2L2 (Ω) ≤ C1 I(u, y) + C2 , where the constants C1 and C2 are independent of (u, y). Therefore, the property (H2) holds true. So, to conclude the proof, it remains to apply Theorem 3.2. In contrast to the above problem, for given functions z∂ ∈ L2 (∂Ω) and f ∈ L2 (Ω), let us consider the following Dirichlet boundary control problem 2 ∂y(s) n−1 +N |u(s)|2 dHn−1 −→ inf, I(u, y) = ∂ν − z∂ (s) dH ∂Ω ∂Ω (3.41)

y(x) = f (x), x ∈ Ω, y|∂Ω = u, u ∈ U∂ ,

(3.42) (3.43)

where N > 0, ∂ν = ∂/∂ν is the outward normal derivative to ∂Ω and U∂ is a convex closed subset of L2 (∂Ω). The distinguishing feature of this problem is the fact that the cost functional (3.41) includes the boundary observations of the outward normal derivative of the state system. Theorem 3.16. Assume that U∂ ∩ H 1 (∂Ω) = ∅. Then the optimal boundary control problem (3.41)–(3.43) admits a unique solution (u0 , y 0 ) ∈ L2 (∂Ω) × 3/2 HΔ (Ω). Proof. Let us set 3/2

Y = Y1 = HΔ (Ω),

U = L2 (∂Ω),

Z = (L2 (Ω), L2 (∂Ω)), L(u, y) = (y, y|∂Ω − u), F (y) = (−f, 0). It is clear that the operator L : U × Y1 → Z is continuous. In addition, due 3/2 to Remark 3.14, the operator ∂ν : HΔ (Ω) → L2 (∂Ω) is continuous. Hence, 3/2 the cost functional I : L2 (∂Ω) × HΔ (Ω) → R is continuous as well. To verify hypothesis (H1), we note that the Dirichlet boundary value prob3/2 lem (3.42)–(3.43) has a unique solution y ∈ HΔ (Ω) for every u ∈ H 1 (∂Ω) 1 (see [173]). Since U∂ ∩ H (∂Ω) = ∅ by our assumptions, it follows that the set of admissible pairs Ξ is nonempty in this case. Thus, regularity property (H1) is valid. 3/2 Let (u, y) ∈ L2 (∂Ω) × HΔ (Ω) be any admissible pair. Then, using the estimate (3.40), we can write y2H 3/2 (Ω) ≤ C1 f 2L2 (Ω) + ∂ν y2L2 (∂Ω) + y2L2 (Ω) . (3.44) Δ

82


On the other hand, for a given pair (u, y), related to the Dirichlet boundary value problem (3.42)–(3.43), we have the estimate (see [173])

y2H 1/2 (Ω) + y2L2 (Ω) ≤ C2 f 2L2 (Ω) + u2L2 (∂Ω) . Since the embedding H 1/2 (Ω) → L2 (Ω) is continuous, we apply the last estimate to the term y2L2 (Ω) on the right-hand side of inequality (3.44). As a result, we get

y2H 3/2 (Ω) + u2L2 (∂Ω) ≤ C f 2L2 (Ω) + ∂ν y2L2 (∂Ω) + u2L2 (∂Ω) Δ

≤ C3 I(u, y) + C4 . Thus, the coerciveness property (H2) is also valid. So, by Theorem 3.2, the optimal boundary control problem (3.41)–(3.43) has a nonempty set of solutions. 3/2 However, the cost functional (3.41) is not strictly convex on L2 (∂Ω)×HΔ (Ω). Therefore, to prove the uniqueness of an optimal solution, we assume the converse – namely, suppose the original problem admits two different solutions: (u01 , y10 ) ∈ Ξ and (u02 , y20 ) ∈ Ξ, respectively. Then ∂ν y10 ∂Ω = ∂ν y20 ∂Ω , and u01 = u02 . Since u0i = yi0 ∂Ω , it implies that y10 ∂Ω = y20 ∂Ω . As a result, for the function 3/2

z = y10 − y20 ∈ HΔ (Ω), we have z(x) = 0 in Ω, z|∂Ω = 0, ∂ν z|∂Ω = 0.

(3.45)

Hence, z ∈ C ∞ (Ω) (see [173]) and z(x) ≡ 0 as a unique solution of the Cauchy problem (3.45) (see [189]). Thus, y10 ≡ y20 and u01 ≡ u02 . This concludes the proof. The last OCP that we are going to consider in this subsection can be stated as follows. Let Ω ⊂ Rn be a bounded connected domain with a Lipschitz boundary ∂Ω = Γ1 ∪ Γ2 , where Γ1 and Γ2 are two disjoint closed subsets and Γ1 has a positive Hausdorff measure Hn−1 . For given functions z∂ ∈ L2 (Ω) and f ∈ L2 (Ω), we consider the following problem I(u, y) = |y(x) − z∂ (x)|2 dx + N |u(s)|2 dHn−1 −→ inf, (3.46) Ω

Γ0

y(x) = f (x), y|Γ0 = u,

x ∈ Ω,

y|Γ1 = 0,

u ∈ U∂ ,

where N > 0 and U∂ is a convex closed subset of L2 (Γ0 ). Having taken

(3.47) (3.48)


Y = L2 (Ω),

83

1/2

Y1 = y ∈ HΔ (Ω) : y|Γ1 = 0 ,

U = L2 (Γ0 ),

Z = L2 (Ω) × L2 (Γ0 ), L(u, y) = (y, y|Γ0 − u), F = (−f, 0) and applying Theorem 3.2 (see also Remark 3.14), we come to the following conclusion. 1/2

Theorem 3.17. There exists a unique solution (u0 , y 0 ) ∈ L2 (Γ0 ) × HΔ (Ω) to the optimal boundary control problem (3.46)–(3.48). 3.3.2 Ill-posed control objects This subsection deals with the OCPs for incorrect (ill-posed) boundary value problems. However, as we will see later, there are extremal problems for such objects which have a particular sense and they are well-posed. To begin with, we consider the following OCP with distributed controls: 2 I(u, y) = |y(x) − z∂ (x)| dx + N |u(x)|2 dx −→ inf, (3.49) Ω

Ω

y(x) = u(x) in Ω,

u ∈ U∂ ,

(3.50)

where z∂ ∈ L2 (Ω) is a given function, N > 0, U∂ is a nonempty convex closed subset of L2 (Ω). Since there are no boundary conditions for the control system (3.50), it means that this is an underdetermined problem and hence, this problem is ill-posed. Nevertheless, we have the following result concerning the correctness of the corresponding OCP. Theorem 3.18. The OCP (3.49)–(3.50) admits a unique solution (u0 , y 0 ) ∈ L2 (Ω) × L2 (Ω). Proof. To make use of Theorem 3.2, we set Y = Y1 = U = L2 (Ω), Z = H −2 (Ω), F = 0, and L(u, y) = y − u. It is easy to see that, in this case, all suppositions of Theorem 3.2 hold true with the exception of hypothesis (H1). However, if we take u as an arbitrary admissible control (i.e., u ∈ U∂ ) and y = y(u) as the corresponding solution of the Dirichlet problem y(x) = u(x) in Ω,

y|∂Ω = 0,

then y ∈ H 2 (Ω) ∩ H01 (Ω) (see Theorem 2.20). Hence, (u, y) ∈ Ξ, and we obtain the required conclusion. The another example of an ill-posed extremal problem can be stated as follows: I(u, y) = |y(x) − z∂ (x)|2 dx + N |u(x)|2 dx −→ inf, (3.51) Ω

Ω

y(x) = u(x) in Ω, y|∂Ω = 0, ∂ν y|∂Ω = 0 u ∈ U∂ ,

(3.52)

84


where ∂ν = ∂/∂ν is the outward normal derivative to ∂Ω. As is obvious, the boundary value problem (3.52) is an overdetermined problem; hence, this problem is ill-posed as well. Following Fursikov (see [111]), we introduce the following set: ΔH02 (Ω) = z : z ∈ H02 (Ω) . Theorem 3.19. Under the assumption that U∂ ∩ ΔH02 (Ω) = ∅, the OCP (3.51)–(3.52) has a unique solution (u0 , y 0 ) ∈ L2 (Ω) × H02 (Ω). Proof. Let us set Y = Z = U = L2 (Ω), Y1 = H02 (Ω), F = 0 and L(u, y) = y − u. Then the condition U∂ ∩ ΔH02 (Ω) = ∅ immediately implies the fulfillment of the regularity assumption (H1). To apply Theorem 3.2, it remains only to verify the coerciveness property (H2). In order to do it, we note that the set of admissible pairs Ξ to the problem (3.51)–(3.52) can be represented in the form Ξ := (u, y) ∈ L2 (Ω) × H02 (Ω) : y = u in Ω, u ∈ U∂ ⊂ (u, y) ∈ L2 (Ω) × H 2 (Ω) ∩ H01 (Ω) : y = u, u ∈ U∂ =: Ξ1 . (3.53) However, in view of the a priori estimate (2.21), we have y2H 2 (Ω) + u2L2 (Ω) ≤ (C + 1)u2L2 (Ω) ≤ C1 I(u, y) + C2 . Hence, the set {(u, y) ∈ Ξ1 : I(u, y) ≤ R} is bounded in L2 (Ω) × H 2 (Ω) for every R > 0. So, taking into account inclusion (3.53), we come to the same conclusion with respect to the set Ξ. This concludes the proof. Thus, the examples of OCPs cited above illustrate the fundamental difference between the notion of well-posedness of the boundary value problems and the corresponding OCPs. 3.3.3 Optimal control of the Cauchy problem for an elliptic equation Let Ω be a bounded connected open domain with a smooth boundary ∂Ω = Γ1 ∪ Γ2 , where Γ1 and Γ2 are two disjoint closed sets, both of positive measure and Γ1 ∩ Γ2 = ∅. We consider the following control problem: 2

I(u1 , u2 , y) = y|∂Ω − z∂ L2 (Γ2 ) +

N1 N2 u1 − f1 2L2 (Γ1 ) + u2 − f2 2L2 (Γ1 ) −→ inf, 2 2

(3.54)

y(x) = 0 in Ω, y|Γ1 = u1 , ∂ν y|Γ1 = u2 ,

(3.55)

u = (u1 , u2 ) ∈ U∂ ,

(3.56)


85

where z∂ ∈ L2 (Γ2 ), fi ∈ L2 (Γ1 ), i = 1, 2, are given functions, N1 > 0, N2 > 0, U∂ is a convex closed subset of L2 (Γ1 ) × L2 (Γ1 ). By analogy with [111], we define the space 1/2 Y1 = y ∈ HΔ (Ω) : ∂ν y|Γ1 ∈ L2 (Γ1 ) (3.57) endowed with the norm y2Y1 = y2H 1/2 (Ω) + ∂ν y|Γ1 L2 (Γ1 ) . Δ

Let us show that y ∈ Y1

implies

y|Γ1 ∈ H 1 (Γ1 ).

(3.58)

Indeed, let ϕ be a smooth function of C ∞ (Ω) such that ϕ(x) = 1 in a neighborhood of the manifold Γ1 and ϕ(x) = 0 in a neighborhood of Γ2 . Assume that a function y ∈ Y1 is such that y(x) = 0 in Ω. Then (ϕy(x)) ∈ L2 (Ω) and y ∈ H 2 (Ω1 ) for any subset Ω1 of Ω satisfying the condition Ω1 ⊂ Ω. In addition, in view of the properties of ϕ, we have ∂ν (ϕy)|Γ2 = 0 and

∂ν (ϕy)|Γ1 = ∂ν y|Γ1 ∈ L2 (Γ1 ). 3/2

Then, in view of Remark 3.14 (see also [173]), we can conclude ϕy ∈ HΔ (Ω). Thereby, by the trace theorem, we just obtain the desired property: y|Γ1 ∈ H 1 (Γ1 ). We are now in a position to prove the main result of this subsection. Theorem 3.20. Assume that there is a triplet (u1 , u2 , y) such that (u1 , u2 ) ∈ U∂ and y ∈ Y1 satisfies the conditions (3.55). Then the OCP (3.54)–(3.56) has a unique solution (u01 , u02 , y 0 ) ∈ H 1 (Γ1 ) × L2 (Γ2 ) × Y1 . Proof. To make use of Theorem 3.2, we set Y = Y1 ,

U = L2 (Γ1 ) × L2 (Γ1 ), Z = L2 (Γ1 ) × L2 (Γ1 ) × L2 (Ω), L(u, y) = (y, y|Γ1 − u1 , ∂ν y|Γ1 − u2 ), and F (y) = (0, 0, 0), ∀ y ∈ Y1 . In view of the initial suppositions, the problem (3.54)–(3.56) is regular. Let us prove the coerciveness property (H2). To do so, we note that for any u1 ∈ L2 (Γ1 ) and v ∈ L2 (Γ2 ), the Dirichlet boundary value problem y(x) = 0 in Ω, y|Γ1 = u1 , y|Γ2 = v 1/2

has a unique solution y ∈ HΔ (Ω) with the estimate (see [173]) y2H 1/2 (Ω) ≤ C u1 2L2 (Γ1 ) + v2L2 (Γ2 ) . Δ

(3.59)

86


It now follows that y2Y1 + u1 2L2 (Γ1 ) + u2 2L2 (Γ1 ) ≤ y2H 1/2 (Ω) + u1 2L2 (Γ1 ) + 2u2 2L2 (Γ1 ) Δ

≤ C y|Γ2 2L2 (Γ2 ) + (C + 1)u1 2L2 (Γ1 ) + 2u2 2L2 (Γ1 ) ≤ C1 I(u1 , u2 , y) + C2 . Hence, the set {(u1 , u2 , y) ∈ Ξ : I(u, y) ≤ R} is bounded in L2 (Γ1 )×L2 (Γ1 )× Y1 for every R > 0. Now, by Theorem 3.2, the problem (3.54)–(3.56) has a solution (u01 , u02 , y 0 ) ∈ L2 (Γ1 )×L2 (Γ2 )×Y1 . Combining this with the property (3.58), we get (u01 , u02 , y0 ) ∈ H 1 (Γ1 ) × L2 (Γ2 ) × Y1 . To conclude the proof, we should repeat the arguments of Theorem 3.16 concerning the proof of the uniqueness of the optimal solution. 3.3.4 Controls with hard constraints Before we consider the OCPs with the so-called rigid controls, we note that, typically, the size of the admissible controls should be constrained. There are two common ways of constraining the control. The first one is to impose an explicit bound on the control — u ∈ U∂ in our case. Such constraints are called “hard”. The second way of constraining the control is to add some norm of the control to the cost functional. Those constraints are termed “soft”. To study the correctness of the extremal problems for elliptic equations with hard constrained controls, we begin with the following OCP I(u, y) = y|∂Ω − z∂ 2H 1 (∂Ω) −→ inf,

(3.60)

y(x) = 0 in Ω, ∂ν y|∂Ω = u on ∂Ω, u ∈ U∂ ,

(3.61) (3.62)

where z∂ ∈ H 1 (∂Ω) is a given function and U∂ is a convex closed subset of L2 (∂Ω) such that n−1 U∂ ∩ u ∈ L2 (∂Ω) : u dH = 0 = ∅. (3.63) ∂Ω

It should be stressed here that the set of admissible controls with hard constraints to the problem (3.60)–(3.63) can be unbounded with respect to the L2 (∂Ω)-norm. Theorem 3.21. For any z∂ ∈ H 1 (∂Ω), the OCP (3.60)–(3.63) admits a 3/2 unique solution (u0 , y 0 ) ∈ L2 (∂Ω) × HΔ (Ω).


87

3/2

Proof. In order to apply Theorem 3.2, we set Y = Y1 = HΔ (Ω), U = L2 (∂Ω), Z = L2 (Ω) × L2 (∂Ω), L(u, y) = (y, ∂ν y|∂Ω − u) and F (y) = (0, 0) for all y ∈ Y1 . Since the trace operator y → ∂y/∂ν|∂Ω is well defined and continuous as the mapping 3/2

Π : HΔ (Ω) → L2 (∂Ω), it follows that the mappings L : U×Y1 → Z and I : U×Y1 → R are continuous as well. Moreover, as indicated in Remark 3.14, the Neumann problem (3.61) 3/2 has a unique solution y˙ in the quotient space HΔ (Ω)/R with the estimate (3.64) y2H 3/2 (Ω) ≤ C u2L2 (∂Ω) + y2L2 (Ω) . Δ

Thus, the regularity assumption (H1) holds true. To prove the coerciveness property (H2), we note that due to the properties of the Dirichlet problem for the Laplace operator y = 0 in Ω,

y|∂Ω = g ∈ H 1 (∂Ω),

we have [173] y2H 3/2 (Ω) ≤ C y|∂Ω 2H 1 (∂Ω) ≤ C1 I(u, y) + C2 .

(3.65)

Δ

On the other hand, u2L2 (∂Ω) := ∂ν y|∂Ω 2L2 (∂Ω) ≤ C3 y2H 3/2 (Ω) .

(3.66)

Δ

So, combining (3.65) and (3.66), we obtain u2L2 (∂Ω) + y2H 3/2 (Ω) ≤ C1 I(u, y) + C2 + C3 [C1 I(u, y) + C2 ] .

(3.67)

Δ

This implies the fulfilment of the hypothesis (H2). As for the uniqueness of the optimal solution, we assume the converse. Let (u01 , y10 ) ∈ Ξ and (u02 , y20 ) ∈ Ξ be two different solutions to the problem (3.60)–(3.63). Since the cost functional (3.60) is strictly convex with respect to y|∂Ω y and the set of admissible pairs Ξ is convex as well, it follows that 0 u1 + u02 y10 + y20 1 0 0 I , < I(u1 , y1 ) + I(u02 , y20 ) = inf I(u, y). (3.68) 2 2 2 (u,y)∈ Ξ This contradiction implies y10 ∂Ω = y20 ∂Ω . To conclude the proof, it remains to note that the corresponding Dirichlet problem has a unique solution. So, y10 (x) ≡ y20 (x) in Ω, and thereby u01 = u02 . In the next example, we carry out the analysis of an OCP with hard control constraints for the case when the determining norm in the cost functional is replaced by a weaker one, namely we consider the following problem

88


I(u, y) = y|∂Ω − z∂ 2L2 (∂Ω) −→ inf,

(3.69)

y(x) = 0 in Ω, ∂ν y|∂Ω = u on ∂Ω, u ∈ U∂ ,

(3.70) (3.71)

where z∂ ∈ L2 (∂Ω) is a given function and U∂ is a convex closed subset of L2 (∂Ω) satisfying the condition (3.63). Theorem 3.22. Assume that the set of admissible controls U∂ is bounded in L2 (∂Ω). Then for any z∂ ∈ L2 (∂Ω), the OCP (3.69)–(3.71) admits a unique 3/2 solution (u0 , y 0 ) ∈ L2 (∂Ω) × HΔ (Ω). Proof. Indeed, having defined the spaces Y, Y1 , Z and U, and the operators L and F as in Theorem 3.21, in other respects it remains only to repeat all arguments of the previous proof with the exception of the verification of hypothesis (H2). In this case, instead of (3.67), it is enough to use another estimate: y2H 3/2 (Ω) ≤ C ∂ν y|∂Ω 2L2 (∂Ω) + y2L2 (Ω) Δ ≤ C1 u2L2 (∂Ω) + y2H 1/2 (Ω) 2 2 ≤ C1 sup uL2 (∂Ω) + C2 y|∂Ω L2 (∂Ω) u∈U∂

≤ C3 (I(u, y) + C4 ) , which immediately follows from the estimates (3.38) and (3.65).

3.4 Optimal control problems for parabolic equations In this section, we study the existence of solutions and their uniqueness to some classes of OCPs associated with linear evolution equations of the parabolic type. We carry out an analysis of such problems for both well-possed and ill-possed initial boundary value problems. 3.4.1 Distributed control Let Ω be an open bounded subset of Rn with a smooth boundary ∂Ω. Let T > 0, N > 0, Q = (0, T ) × Ω and Σ = (0, T ) × ∂Ω. Let z∂ ∈ L2 (Q) and y0 ∈ L2 (Ω) be given functions. We begin with the following OCP N 1 y − z∂ 2L2 (Q) + u2L2 (Q) −→ inf, 2 2 y(t, ˙ x) − ηy(t, x) = u(t, x), in Q, y|Σ = 0, y|t=0 = y0 , u ∈ U∂ , I(u, y) =

(3.72) (3.73) (3.74)

3.4 Optimal control problems for parabolic equations

89

where U∂ is a nonempty convex closed subset of L2 (Q) and y(t, ˙ x) = ∂y(t, x)/∂t, η = 0. It is well known that for a fixed u ∈ U∂ and a positive η > 0, the initial boundary value problem (3.73) is well-posed, whereas the condition η < 0 corresponds to the so-called inverse problem for the heat equation, which is always an ill-posed problem from the point of view of PDEs. Typical examples of the sets of admissible controls U∂ to the problem (3.72)–(3.74) are (3.75) U∂ = u ∈ L2 (Q) : u − u0 2L2 (Q) ≤ R , 2 U∂ = u ∈ L (Q) : u(t, x) ≥ 0 a.e. on Q , (3.76) U∂ = u ∈ L2 (Q) : α(t, x) ≤ u(t, x) ≤ β(t, x) a.e. on Q , (3.77) where α ∈ L∞ (Q), α(t, x) < β(t, x) for almost all (t, x) ∈ Q. To begin, we make use of the space V = v ∈ L2 (Q) : v˙ − ηv ∈ L2 (Q)

(3.78)

endowed with the norm v2V = v2L2 (Q) + v˙ − ηv2L2 (Q) .

(3.79)

Let us define the mappings γτ : v → v|t=τ ,

0 γΣ : v → v|Σ ,

1 γΣ : v → ∂ν v|Σ ,

where ∂ν = ∂/∂ν is the outward normal derivative to ∂Ω. In this case, the following result takes a place. Lemma 3.23 ([111]). The trace operators γτ : V → H −1 (Ω), 0 γΣ 1 γΣ

:V→H

−1/2

:V→H

−3/2

(∂Ω; H (∂Ω; H

−1 −1

(3.80) (0, T )),

(3.81)

(0, T )).

(3.82)

are well defined and continuous. We are now in a position to prove the first result of this section. Theorem 3.24. Assume that the parameter η in (3.73) is positive. Then there exists a unique solution (u0 , y0 ) ∈ L2 (Q) × V1 of OCP (3.72)–(3.74), where V1 = {v ∈ V : v|Σ = 0} ,

vV = vV1 .

(3.83)

Proof. We set Y = U = L2 (Q), Y1 = V1 , Z = L2 (Q) × H −1 (Ω), L(u, y) = (y˙ − ηy − u, y|t=0 ), and F = (0, −y0 ). Then, in view of the definition of the space V1 and Lemma 3.23 (see (3.80)), the operator L : U × Y1 → Z is continuous. To verify hypothesis (H1), we note that the initial-boundary

90


value problem (3.72)–(3.74) has a unique solution y ∈ L2 (0, T ; H01 (Ω)) for every u ∈ L2 (Q) and y0 ∈ L2 (Ω) (see [169]). So, y ∈ L2 (Q). On the other hand, y˙ − ηy = u ∈ U∂ ⊂ L2 (Q). Then y ∈ V1 , and hence Ξ = ∅. Thus, hypothesis (H1) is valid. Let us verify the coercivity condition (H2). To do so, we fix an arbitrary admissible pair (u, y) ∈ Ξ ⊂ U × Y1 and estimate its norm in U × Y1 . One gets u2U + y2Y1 = y2L2 (Q) + y˙ − ηy2L2 (Q) + u2L2 (Q) = y2L2 (Q) + 2u2L2 (Q) ≤ CI(u, y) = C1 . Thereby, the set {(u, y) ∈ Ξ : I(u, y) ≤ R} is bounded in U × Y1 for every R > 0. So, applying Theorem 3.2 and taking into account the strict convexity of the cost functional (3.72), we come to the required conclusion. Regarding the case when η < 0 (i.e., when the corresponding initial boundary value problem is ill-posed), we have the following result. Theorem 3.25. Assume that y0 ∈ H01 (Ω) and the L2 -interior of the set of admissible controls U∂ is nonempty (i.e., int U∂ = ∅). Then, for any η < 0, there exists a unique solution (u0 , y 0 ) ∈ L2 (Q) × V1 of OCP (3.72)–(3.74). Remark 3.26. In particular, the main assumption of this theorem will be satisfied if we take the set U∂ in the form (3.75). Proof. Let us define the spaces Y, Y1 , Z, and U, and the operators L and F as in Theorem 3.24. We have to prove only the regularity property (H1) because in other respects we should repeat all the arguments of the previous proof. As indicated by Lions and Magenes in [173], if y0 ∈ H01 (Ω) then there exists a function y ∈ H 1,2 (Q) such that y|Σ = 0 and y|t=0 = y0 . Let us set g = y˙ − ηy. Then g ∈ L2 (Q). As a result, we have that for a fixed u ∈ U∂ , the initial-boundary value problem (3.73) is solvable if and only if there exists a solution to the modified problem v(t, ˙ x) − ηv(t, x) = p(t, x) in Q,

v|Σ = 0,

v|t=0 = 0,

(3.84)

where p = u − g. Assume that u∗ ∈ int U∂ . Then p∗ = u∗ − g ∈ int P∂ , where P∂ = {p = u − g : u ∈ U∂ } ⊂ L2 (Q). ∞

∞

By {ek }k=1 and {λk }k=1 let us denote the eigenfunctions and the corresponding eigenvalues to the spectral problem −e = λe,

e|∂Ω = 0. ∞

Then we may suppose that the sequence {λk }k=1 is nondecreasing. Following the main idea of the Fourier expansion method, we set


pk (t, x) =

k "

i

p (t)ei (x),

vk (t, x) =

p∗ (t, x)ei (x) dx,

i

p (t) = Ω

i=1 k "

91

i

v (t)ei (x),

t

exp [ηλi (τ − t)]pi (t) dt.

i

v (t) = 0

i=1

It is clear that v˙ k (t, x) − ηvk (t, x) = pk (t, x) in Q,

vk |Σ = 0,

vk |t=0 = 0

for any k ∈ N. Since p∗ − pk L2 (Q) → 0 as k → ∞, it follows that the pair (pk , vk ) satisfies relations (3.84) with p = p∗ := u∗ − g for sufficiently large k ∈ N. As a result, we can assert that there is a pair (u∗ , y) ∈ U∂ ×V1 satisfying the initial-boundary value problem (3.73). Thus, the regularity property (H1) holds true, and this concludes the proof. 3.4.2 Control in the initial conditions In this subsection, we consider an OCP for which a control action appears in the initial condition. Following Lions’ terminology [169], such controls are called “starting”. Let z∂ ∈ L2 (Ω), f ∈ L2 (Q), y0 ∈ L2 (Ω), N > 0, η = 0 and Q = (0, T ) × Ω. The corresponding control problem can be stated as follows: N 1 y(T, ·) − z∂ 2L2 (Ω) + u − y0 2L2 (Ω) −→ inf, 2 2 y(t, ˙ x) − ηy(t, x) = f (t, x) in Q, y|Σ = 0, y|t=0 = u, I(u, y) =

u ∈ U∂ ,

(3.85) (3.86) (3.87)

where U∂ is nonempty convex closed subset of L2 (Ω). In what follows, we will need the following result concerning the solvability of the initial-boundary value problem (3.86) (see Lions and Magenes [173]). Theorem 3.27. Let Y1 be the vector space defined by Y1 = y ∈ L2 (0, T ; H01 (Ω)) : y˙ ∈ L2 (0, T ; H −1 (Ω))

(3.88)

equipped with the norm of the graph. Then the following hold: (a) For any τ ∈ [0, T ) the mapping γτ : Y1 → L2 (Ω) is a continuous epimorphism, where γτ : v → v|t=τ is the trace operator. (b) The mapping (∂/∂t − η, γ0 ) : Y1 → L2 (Q) × L2 (Ω) is an isomorphism. We are now in a position to carry out the analysis of the original starting control problem. Theorem 3.28. If η > 0 then there exists a unique solution (u0 , y0 ) ∈ L2 (Ω) × Y1 of the OCP (3.85)–(3.87).

92


Proof. Having set Y = Y1 , U = L2 (Ω), Z = L2 (0, T ; H −1 (Ω)) × L2 (Ω), L(u, y) = (y˙ − ηy, y|t=0 − u), F = (−f, 0), we note that the operator L : U × Y1 → Z is continuous. Moreover, for any fixed control u ∈ U∂ , there exists a unique solution y ∈ Y1 to the problem (3.86). Hence, hypothesis (H1) is obviously true in this case. Let us verify condition (H2). To do so, we multiply the state equation (3.86) by y(t, x) and integrate with respect to (t, x) over Q. One gets 1 1 y(T, ·)2L2 (Ω) − y(0, ·)2L2 (Ω) + η 2 2 T = (f, y)L2 (Ω) dt.

T

∇y(t, ·)2L2 (Ω) dt 0

(3.89)

0

Then an application of the Cauchy–Buniakowsky and Friedrichs inequalities to the right-hand side of this equality yields T 1 η ∇y(t, ·)2L2 (Ω) dt ≤ y(0, ·)2L2 (Ω) + f L2 (Q) yL2 (Q) 2 0 1 y(0, ·)2L2 (Ω) + Cη f 2L2 (Ω) 2 η T + ∇y(t, ·)2L2 (Ω) dt. (3.90) 2 0 Using the fact that −|η| ∈ L L2 (0, T ; H01 (Ω)), L2 (0, T ; H −1 (Ω)) , we obtain ≤

y ˙ L2 (0,T ;H −1 (Ω)) ≤ f L2 (Q) + |η|yL2 (0,T ;H −1 (Ω)) ≤ f L2 (Q) + |η|yL2 (0,T ;H01 (Ω)) . As a result, there is a constant C > 0 depending on η such that y2Y1 ≤ C f 2L2 (Q) + y(0, ·)2L2 (Ω) .

(3.91)

(3.92)

Therefore, for any admissible pair (u, y) ∈ L2 (Ω) × Y1 , we have y2Y1 + u2L2 (Ω) ≤ C y(T, ·)2L2 (Ω) + f 2L2 (Q) + u2L2 (Ω) ≤ C1 I(u, y) + C2 ,

(3.93)

which implies the fulfilment of hypothesis (H2). As for the uniqueness of the optimal solution, we may use the standard approach. Let (u01 , y10 ) ∈ Ξ and (u02 , y20 ) ∈ Ξ be two different solutions to the


93

problem (3.85)–(3.87). Having set z = y10 − y20 and v = u01 − u02 and applied inequality (3.68), it can be shown that y10 (T, x) = y20 (T, x) almost everywhere in Ω. Then z˙ − ηz = 0 in Q,

z|Σ = 0,

z|t=0 = v = 0,

z|t=T = 0,

(3.94)

and hence z(t, x) ≡ 0. This completes the proof. We pass now to the case when η < 0. Theorem 3.29. Assume that η < 0 and int U∂ = ∅. Then there exists a unique solution (u0 , y 0 ) ∈ L2 (Ω) × Y1 of the OCP (3.85)–(3.87). Proof. Let us define the spaces Y, Y1 , Z, and U, and the operators L and F as in Theorem 3.28. First, we show that the regularity assumption (H1) holds true. To do so, we use the arguments of the proof of Theorem 3.25. With this aim, we introduce the problem v(t, ˙ x) − ηv(t, x) = f (t, x) in Q,

v|Σ = 0,

v|t=T = 0.

(3.95)

Let v ∈ Y1 be a solution to (3.95). Let us set P∂ = {p = u − v(0, ·) : u ∈ U∂ }. Then, in view of the initial assumptions, we have int P∂ = ∅. Let p be any ∞ ∞ element of int P∂ and let {ek }k=1 and {λk }k=1 be the collections defined as in the proof of Theorem 3.25. For every k ∈ N, we set vk (t, x) =

k "

exp (−ηλi t) pi ei (x),

i=1

where

pi =

(u(x) − v(0, x))ei (x) dx.

p(x)ei (x) dx = Ω

Ω

It is clear that, in this case, we have limk→∞ vk (0, ·) − pL2 (Ω) = 0. Hence, vk (0, ·) ∈ P∂ for all sufficiently large k ∈ N. Therefore, the pair (u(x), y(t, x)) = (p(x) + vk (0, x), v(t, x) + vk (t, x)) satisfies relations (3.86), that is, this is an admissible pair to the original problem (3.85)–(3.87). Thus, Ξ = ∅, and the regularity property is established. In order to prove hypothesis (H2), we take into account the relation (3.89). Since η < 0, it follows that instead of the estimates (3.90) and (3.92), we will have T ∇y(t, ·)2L2 (Ω) dt |η| 0

y2Y1

|η| 1 ≤ y(T, ·)2L2 (Ω) + Cη f 2L2 (Ω) + 2 2 ≤ C f L2 (Q) + y(T, ·)2L2 (Ω) .

T

∇y(t, ·)2L2 (Ω) dt, 0

94


As a result, supposition (H2) is the direct consequence of the following inequality y2Y1 + u2L2 (Ω) ≤ C y(T, ·)2L2 (Ω) + f L2 (Q) + u2L2 (Ω) ≤ C1 I(u, y) + C2 ,

∀ (u, y) ∈ L2 (Ω) × Y1 .

(3.96)

To conclude the proof, it remains to note that the restriction z|t=T = 0 in (3.94) can be seen as the initial condition to the problem (3.94) (the so-called reverse problem). In this case, the function z(t, x) = y10 − y20 ≡ 0 is the unique solution to (3.94). Therefore, z|t=0 = u01 − u02 ≡ 0, and this implies the uniqueness of the solution to the original OCP (3.85)–(3.87). The proof is complete. 3.4.3 Neumann boundary control Consider the optimal boundary control problem N 1 y(T, ·) − z∂ 2L2 (Ω) + u2L2 (Σ) −→ inf, 2 2 y(t, ˙ x) − ηy(t, x) = f (t, x) in Q, ∂ν y|Σ = u, y|t=0 = y0 , u ∈ U∂ , I(u, y) =

(3.97) (3.98) (3.99)

where z∂ ∈ L2 (Ω), f ∈ L2 (Q), y0 ∈ H 1/2 (Ω), N > 0, η = 0, Σ = (0, T ) × ∂Ω, ∂ν = ∂/∂ν is the outward normal derivative to ∂Ω, and U∂ is a nonempty convex closed subset of L2 (Σ). Due to a result of Lions and Magenes [173], it is well known that for every u ∈ L2 (Σ) and η > 0, there exists a unique solution y to the initial-boundary value problem (3.98) in the space L2 (0, T ; H 3/2 (Ω)) ∩ H 3/4 (0, T ; L2 (Ω)). So, following the line of the proof of the theorems cited above, the reader can easily show that the OCP (3.97)–(3.99) has a unique solution (u0 , y0 ) in the class L2 (Σ) × Y1 = L2 (Σ) × y ∈ L2 (0, T ; H 3/2 (Ω)) ˙ x) − ηy(t, x) ∈ L2 (Q) , provided this problem is ∩ H 3/4 (0, T ; L2 (Ω)) : y(t, regular. However, as we will see later, the same result remains valid in a less regular space of admissible pairs – namely, in the case when we take L2 (Σ) × L2 (0, T ; H 1 (Ω)) instead of L2 (Σ) × Y1 . To show this, we begin with the following notion. Definition 3.30. We say that a pair (u, y) ∈ L2 (Σ) × L2 (0, T ; H 1 (Ω)) satisfies relations (3.98) in the weakened sense if the integral identity τ [y(t, x) (ϕ(t, ˙ x) + ηϕ(t, x)) + f (t, x)ϕ(t, x)] dx dt 0 Ω = (y(τ, x)ϕ(τ, x) − y0 (x)ϕ(0, x)) dx Ω τ u(t, s)ϕ(t, s) dHn−1 dt (3.100) −η 0

∂Ω


95

holds true for all τ ∈ [0, T ] and ϕ ∈ H 1,2 (Q) such that ∂ν ϕ|Σ = 0. The main question is: In which sense does the trace y(t, x)|t=τ exist? Having put ϕ(t, x) ≡ ϕ(x), where ϕ ∈ H02 (Ω), in (3.100), we can rewrite this relation in the form τ y(τ, x)ϕ(x) dx = y0 (x)ϕ(x) dx + η u(t, s)ϕ(s) dHn−1 dt 0 Ω Ω ∂Ω τ + [f (t, x)ϕ(x) − η∇y(t, x) · ∇ϕ(x)] dx dt. (3.101) 0

Ω

It is clear that the right-hand side of (3.101) can be continuously extended by ϕ from H02 (Ω) onto H01 (Ω). Therefore, the left-hand side of (3.101) can be associated with a linear continuous functional on H01 (Ω), that is, y(τ, x)ϕ(x) dx = y(τ, ·), ϕH −1 (Ω);H 1 (Ω) . 0

Ω

Using the fact that the right-hand side of (3.101) is a continuous function with respect to the τ , we can conclude that any function y ∈ L2 (0, T ; H 1 (Ω)) satisfying (3.101) has a trace y(τ, ·)|t=τ ∈ H −1 (Ω) for every τ ∈ [0, T ]. In fact, as immediately follows from Lemma 3.31, we have more precise result y(τ, ·)|t=τ ∈ L2 (Ω). Lemma 3.31 ([111]). If a pair (u, y) satisfies relations (3.98) in the weakened sense, then 1 2

τ 2 2 y (τ, x) − y0 (x) dx + η |∇y(t, x)|2 dx dt 0 Ω Ω τ τ f (t, x)y(t, x) dx dt + η u(t, s)y(t, s) dHn−1 dt. =

0

Ω

0

(3.102)

∂Ω

The set of all pairs (u, y) ∈ U∂ × L2 (0, T ; H 1 (Ω)) satisfying (3.98) in the weakened sense is called the set of admissible solutions to the problem (3.97)– (3.99), and we denote it by Ξ. Due to the well-known solvability result [173], it is clear that for given f ∈ L2 (Q), y0 ∈ H 1/2 (Ω) and η > 0, the set Ξ is nonempty. As for the case η > 0, the verification of the condition Ξ = ∅ is not trivial, in general. For instance, if f = 0, y0 = 0 and 0 ∈ U∂ , then Ξ = ∅ because the pair (0, 0) is admissible to (3.98). Theorem 3.32. Assume that there exists a pair (u, y) ∈ U∂ ×L2 (0, T ; H 1 (Ω)) satisfying (3.98) in the weakened sense. Then the OCP (3.97)–(3.99) has a unique solution (u0 , y 0 ) ∈ U∂ × L2 (0, T ; H 1 (Ω)). Proof. Let us fix an arbitrary pair (u, y) ∈ Ξ and suppose that η < 0. Then, taking into account Lemma 3.31, we have

96


1 2

τ2 2 2 |∇y(t, x)|2 dx dt y (τ2 , x) − y (τ1 , x) dx + η Ω τ1 Ω τ2 τ2 = f (t, x)y(t, x) dx dt + η u(t, s)y(t, s) dHn−1 dt

τ1

Ω

τ1

(3.103)

∂Ω

for any 0 ≤ τ1 < τ2 ≤ T . Then, replacing τ1 by τ and τ2 by T in (3.103) and using the the Cauchy–Buniakowsky and Friedrichs inequalities, we get T 2 y (τ, x) dx + 2|η| |∇y(t, x)|2 dx dt Ω

τ

≤

Ω

Ω

τ

T

+2|η| τ

T

f (t, ·)L2 (Ω) y(t, ·)L2 (Ω) dt

y 2 (T, x) dx + 2

u(t, ·)L2 (∂Ω) y(t, ·)L2 (∂Ω) dt # T

≤

f (t, ·)2L2 (Ω)

2

y (T, x) dx + Cδ Ω

u(t, ·)2L2 (∂Ω)

dt +

τ

$

T

dt

τ

T

y(t, ·)2H 1 (Ω) dt =: J0 .

+δ

(3.104)

τ

Hence,

T

T

y 2 (t, x) dx dt ≤ T J0 , 0

|∇y(t, x)|2 dx dt ≤ 0

Ω

Ω

1 J0 . 2|η|

As a result, we obtain T T 1 y(t, ·)2H 1 (Ω) dt ≤ δ T + y(t, ·)2H 1 (Ω) dt 2|η| τ τ # T

f (t, ·)2L2 (Ω) dt +

+ C1δ

$

T

τ

1 + T+ y(T, x)2 dx. 2|η| Ω

u(t, ·)2L2 (∂Ω) dt τ

(3.105)

−1

If we take δ ≤ 1/2 (T + 1/(2|η|)) , then it follows that y2L2 (0,T ;H 1 (Ω)) ≤ C f 2L2 (Q) + u2L2 (Σ) + y(T, ·)2L2 (Ω) ,

(3.106)

where the constant C is independent of u and y. By analogy, it can be shown that in the case when η > 0, we have a similar estimate y2L2 (0,T ;H 1 (Ω)) ≤ C f 2L2 (Q) + u2L2 (Σ) + y0 2L2 (Ω) . (3.107) Combining (3.106), (3.107) and (3.97), we obtain


97

y2L2 (0,T ;H 1 (Ω)) + u2L2 (Σ)

≤ C1 I(u, y) + C2 f 2L2 (Q) + y0 2L2 (Ω) + C3 . (3.108)

∞

Let {(uk , yk )}k=1 ⊂ Ξ be a minimizing sequence of admissible pairs to the problem (3.97)–(3.99), that is, lim I(uk , yk ) =

k→∞

inf

I(u, y).

(3.109)

(u,y)∈Ξ

Then, in view of (3.109) and (3.108), we conclude that there is a constant C > 0 independent of k ∈ N such that yk 2L2 (0,T ;H 1 (Ω)) + uk 2L2 (Σ) ≤ C,

∀ k ∈ N.

(3.110)

uk u0 in L2 (Σ).

(3.111)

So, passing to subsequences, we may suppose that yk y0 in L2 (0, T ; H 1 (Ω)),

Moreover, substituting y for yk in (3.101), it is easy to see that the function t → (yk (t, ·, ϕ)L2 (Ω) belongs to H 1 (0, T ) and satisfies the estimate (yk (t, ·, ϕ)L2 (Ω) H 1 (0,T ) ≤ C, where the constant C is independent of k. Therefore, due to the Sobolev embedding theorem, we may suppose that (3.112) (yk (t, ·), ϕ)L2 (Ω) −→ y 0 (t, ·), ϕ L2 (Ω) in C[0, T ]. Thus, plugging yk into the integral identity (3.100), passing there to the limit as k → ∞ and taking into account (3.111) and (3.112), we see that the limit pair (u0 , y 0 ) satisfies relations (3.98) in the weakened sense. Moreover, since U∂ is a convex closed subset of L2 (Σ), it follows that U∂ is closed with respect to the weak convergence in L2 (Σ). Hence, (3.111) implies that u0 ∈ U∂ . Thus, (u0 , y 0 ) ∈ Ξ. It remains to show that this pair is optimal for (3.97)–(3.99). Indeed, as follows from (3.110) and (3.102) with τ = T , we have the estimate yk (T, ·)L2 (Ω) ≤ C, where C does not depend on k. So, in view of (3.112), we may suppose that yk (T, ·) y 0 (T, ·) in L2 (Ω).

(3.113)

Finally, using the lower semicontinuity for I with respect to the product of the weak topologies for L2 (Σ) and L2 (Ω), we get inf (u,y)∈Ξ

I(u, y) = lim inf I(uk , yk ) ≥ I(u0 , y 0 ). k→∞

Thus, (u0 , y 0 ) is an optimal pair to the original problem (3.97)–(3.99). To conclude the proof, it remains only to note that the uniqueness of the optimal solution (u0 , y 0 ) can be easily proved by the standard approach (see Theorems 3.28 and 3.29).

98


3.5 Optimal control problems for hyperbolic equations Let Ω be a bounded connected open subset of Rn with a Lipschitz boundary ∂Ω. Let, as usual, T > 0, N > 0, Q = (0, T ) × Ω, and Σ = (0, T ) × ∂Ω. For given functions z∂ ∈ L2 (Q), y0 ∈ H01 (Ω) and y1 ∈ L2 (Ω), we consider the following OCP: 1 N y − z∂ 2L2 (Q) + u2L2 (Q) −→ inf, 2 2 y¨(t, x) − y(t, x) + ηy(t, x) = u(t, x) in Q, ˙ t=0 = y1 , y|Σ = 0, y|t=0 = y0 , y| u ∈ U∂ ,

I(u, y) =

(3.114) (3.115) (3.116) (3.117)

where U∂ is nonempty convex closed subset of L2 (Q), η ∈ R and y¨(t, x) = ∂ 2 y(t, x)/∂t2 . We make use of the spaces V = v ∈ L2 (Q) : v¨ − v ∈ L2 (Q), v|Σ = 0 , (3.118) 2 2 (3.119) V1 = v ∈ L (Q) : v¨ − v ∈ L (Q) endowed with the graph norm v2V = v2V1 = v2L2 (Q) + ¨ v − v2L2 (Q) .

(3.120)

In order to show that the space V is well defined, we recall that a function v belongs to the space H −1/2 (∂Ω; H −2 (0, T )) if for any ϕ ∈ H02 (0, T ) the distribution T v(t, x)ϕ(t) dt := v(·, x), ϕH −2 (0,T ),H 2 (0,T ) , x ∈ ∂Ω, vϕ (x) = 0

0

is an element of H −1/2 (∂Ω), and we cite the following result. 0 : v → v|Σ , the mapping Proposition 3.33 ([111]). For the trace operator γΣ 0 γΣ : V1 → H −1/2 (∂Ω; H −2 (0, T ))

(3.121)

is well defined and continuous. By this proposition, the equality y|Σ = 0 has a meaning. So, the definition of the space V is correct. As a result, the boundary condition y|Σ = 0 in (3.116) should be interpreted as an equality in H −1/2 (∂Ω; H −2 (0, T )). As for the correctness of the initial conditions y|t=0 = y0 and y| ˙ t=0 = y1 , we note that y¨ = h + y ∈ L2 (0, T ; H −2 (Ω)), provided y ∈ L2 (Q) and h := y¨ − y ∈ L2 (Q). Hence, the mappings γ0

∂ : V1 → H −2 (Ω), ∂t

γ0 : V1 → H −2 (Ω)

3.5 Optimal control problems for hyperbolic equations

99

are well defined and continuous. Thus, in the case when y0 ∈ H01 (Ω) and ˙ t=0 = y1 are meaningful. y1 ∈ L2 (Ω), the conditions y|t=0 = y0 , y| Let us define the space Y1 = y ∈ L2 (0, T ; H01 (Ω)) ∩ V : y˙ ∈ L2 (0, T ; L2 (Ω)) (3.122) and make use of the following result (see Lions [172]). Lemma 3.34. Under the conditions that y0 ∈ H01 (Ω), y1 ∈ L2 (Ω) and f ∈ L2 (Q), there exists a unique solution y ∈ Y1 to the initial-boundary value problem y¨ − y + ηy = f, y|Σ = 0,

y|t=0 = y0 ,

y| ˙ t=0 = y1 ,

with the estimate yL∞ (0,T ;H01 (Ω)) + yL∞ (0,T ;L2 (Ω)) ≤ C y0 H01 (Ω) + y1 L2 (Ω) + f L2 (Q) , (3.123) where the constant C is independent of y0 , y1 and f . We are now in a position to state the main result of this section. Theorem 3.35. Assume that the set of admissible controls U∂ possesses the property that there exists a pair (u, y) ∈ U∂ × Y1 satisfying relations (3.115) and (3.116). Then the OCP (3.114)–(3.117) has a unique solution (u0 , y 0 ) ∈ U∂ × Y1 . Remark 3.36. It is clear that the regularity assumption of this theorem holds true in the case when U∂ = L2 (Q) and y0 , y1 ∈ C ∞ (Ω). Indeed, if we take a function y ∈ C ∞ (Q) satisfying the boundary condition y|Σ = 0, then the pair (¨ y − y + ηy, y) is admissible for the problem (3.114)–(3.117). So, Ξ = ∅. As for the case U∂ = L2 (Q), the fulfilment of the condition Ξ = ∅, in general, is an open question. However, if y0 = y1 ≡ 0 and 0 ∈ U∂ , then Ξ = ∅ since (u, y) = (0, 0) satisfies relations (3.115) and (3.116). Proof. In order to apply Theorem 3.2, we set Y = L2 (Q), U = L2 (Q), Z = y − y + ηy − L2 (Q) × H −2 (Ω) × H −2 (Ω), Y1 as in (3.122), L(u, y) = (¨ u, y|t=0 , y| ˙ t=0 ) and F (y) = (0, −y0 , −y1 ) for all y ∈ Y1 . Since y − y2L2 (Q) + y2L2 (0,T ;H 1 (Ω)) + y ˙ 2L2 (Q) , (3.124) y2Y1 = y2L2 (Q) + ¨ 0

it follows that the embedding Y1 → Y is continuous and L : U × Y1 → Z is a linear continuous operator. By our assumptions, the original problem is regular. So, it remains to verify hypothesis (H2). To do so, we make use of Lemma 3.34. We obtain

100


y2Y1 + u2U = y2L2 (Q) + ¨ y − y2L2 (Q) +y2L2 (0,T ;H 1 (Ω)) + y ˙ 2L2 (Q) + u2U 0

≤ y2L2 (0,T ;H 1 (Ω)) + y ˙ 2L2 (Q) + u − ηy2L2 (Q) + y2L2 (Q) + u2U 0 ≤ C y0 H01 (Ω) + y1 L2 (Ω) + uL2 (Q) +C1 y2L2 (Q) + u2U ≤ C2 I(u, y) + C3 , (3.125) which implies the fulfillment of hypothesis (H2). As for the uniqueness of the optimal solution, this fact immediately follows from the strict convexity of the cost functional I and Theorem 3.2. The proof is complete.

3.6 Optimality system to optimal control problems Since the optimality system serves as the basis for computing numerical approximations to optimal solutions, our prime interest in this section is to discuss different ways and procedures for the derivation of such systems to a wide class of OCPs. The main attention will be paid to the Lagrange multiplier principle adapted even to the control systems that are described by ill-posed boundary value problems. 3.6.1 The general setting of the Lagrange multiplier principle Let Y and Z be linear normed spaces and let Y1 be a reflexive, separable Banach space such that Y1 is continuously and densely embedded in Y. Let V be a separable Banach space and U = V∗ be its dual. Let U∂ be a closed convex subset of U. Now we discuss the Lagrange multiplier principle with regard to the following OCP: Minimize I(u, y)

(3.126)

over all (u, y) ∈ U × Y1 subject to the state equation and control constraint L(u, y) + F0 = 0 and u ∈ U∂ ,

(3.127)

where L : U ×Y1 → Z is a linear continuous mapping, F0 ∈ Z and I : U ×Y → R is a proper cost functional. As usual, by Ξ we denote the set of admissible pairs for the problem (3.126)–(3.127), that is, Ξ = {(u, y) ∈ U × Y1 : L(u, y) + F0 = 0, u ∈ U∂ , I(u, y) < +∞} . (3.128) In this subsection, we assume that all suppositions of Theorem 3.2 hold true. Let (u0 , y0 ) ∈ Ξ be an optimal pair to the problem (3.126)–(3.127),

3.6 Optimality system to optimal control problems

101

which obviously exists due to Theorem 3.2. We associate with this problem the Lagrange function L(u, y, λ, ψ) = λI(u, y) + ψ, L(u, y) + F0 Z∗ ,Z ,

(3.129)

where λ ∈ R1+ and ψ ∈ Z∗ . In accordance with the Lagrange multiplier principle, there is a pair (λ, ψ) = 0 such that (u0 , y 0 ) ∈ Ξ is a minimizer to the extremal problem (3.130) L(u, y, λ, ψ) −→ inf, u ∈ U∂ . In order to give a rigorous substantiation of this principle, we begin with the case when the set of admissible pairs Ξ has a nonempty interior. Obviously, this assumption immediately implies a very restrictive condition from the practical point of view: int U∂ = ∅. Let us introduce the set Ker L = {(u, y) ∈ U × Y1 : L(u, y) = 0} . As a result, the standard Lagrange multiplier principle can be quoted in the following particular form (see [24, 111, 125]) Theorem 3.37. Let (u0 , y 0 ) ∈ Ξ be a solution of the OCP (3.126)–(3.127). Assume that the following conditions hold true: The cost functional I is Gˆ ateaux differentiable at (u0 , y0 ), int Ξ = ∅ and the set Im L ≡ L (U × Y1 ) is closed in Z. Then there exists a pair (λ, ψ) ∈ R+ × Z∗ \ {0} such that

Dy L(u0 , y 0 , λ, ψ), y − y0

Y∗ 1 ,Y1

+ Du L(u0 , y 0 , λ, ψ), u − u0 U∗ ,U ≥ 0, ∀ (u, y) ∈ U∂ × Y1 , (3.131)

where Du L(·, ·, ·, ·) and Dy L(·, Gˆ ateaux derivatives ·, ·, ·) are the corresponding of L. Moreover, if int Ξ ∩ (u0 , y 0 ) + Ker L = ∅, then λ = 1. Conversely, if I is a convex functional, (u0 , y 0 ) ∈ Ξ and there exists a ψ ∈ Z∗ such that the collection (u0 , y0 , 1, ψ) satisfies relations (3.131), then (u0 , y0 ) is an optimal pair to the problem (3.126)–(3.127). Let us consider the case when the condition int U∂ = ∅ and, hence, int Ξ = ∅ are too restrictive. Let U0 and Y0 be two Banach spaces such that U0 ⊂ U and Y0 ⊂ Y. Instead of the condition int Ξ = ∅, we consider the following one: (3.132) intU0 ×Y0 (U0 × Y0 ) ∩ (Ξ − (u0 , y 0 )) = ∅, where intU0 ×Y0 M isthe interior of M with respect to the topology of U0 × Y0 and Ξ − (u0 , y0 ) = (u − u0 , y − y 0 ) : ∀ (u, y) ∈ Ξ . In what follows, by lin Ξ − (u0 , y0 ) we denote the linear span of the set Ξ − (u0 , y 0 ) and by lin (Ξ − (u0 , y 0 )) its closure with respect to the space U × Y1 . Since the condition (3.132) is weaker than int Ξ = ∅, the following lemma guarantees its fulfillment.

102


Lemma 3.38 ([111]). Let (u0 , y0 ) be a fixed pair of Ξ. Assume that Ξ is a convex closed separable subset of U × Y1 . Then there exist Banach spaces U0 and Y0 such that the embeddings U0 → U and Y0 → Y are continuous, U0 × Y0 is a dense subset of lin (Ξ − (u0 , y 0 )) and condition (3.132) holds true. Definition 3.39. We say that the cost functional I : U × Y → R is Gˆ ateaux differentiable at (u0 , y 0 ) along the space U0 ×Y0 if there exist linear functionals Iu : U0 → R and Iy : Y0 → R such that I(u0 + λv, y 0 + λh) − I(u0 , y 0 ) − λ Iu (u0 , y 0 ), v U∗ ,U − λ Iy (u0 , y0 ), h Y∗ ,Y = o(λ), 0

0

0

0

where |o(λ)|/|λ| → 0 as λ → 0 for every v ∈ U0 and h ∈ Y0 . Let Z0 be a Banach space such that L : Ξ ⊂ Z0 , L : Ξ → Z0 is a continuous mapping and LΞ is a closed set with respect to the topology of Z0 . Then the following result can be viewed as a natural generalization of Theorem 3.37 to the case of (3.132) (see Fursikov [111]). Theorem 3.40. Let (u0 , y 0 ) ∈ Ξ be a solution of the OCP (3.126)–(3.127). Assume that there exist Banach spaces U0 and Y0 satisfying condition (3.132) and the cost functional I is Gˆ ateaux differentiable at (u0 , y 0 ) along U0 × Y0 . Then there exists a pair (λ, ψ) ∈ R+ × Z∗0 \ {0} such that

Dy L(u0 , y 0 , λ, ψ), y

Y∗ 0 ,Y0

+ Du L(u0 , y 0 , λ, ψ), u U∗ ,U ≥ 0, 0

0

∀ (u, y) ∈ (U0 × Y0 ) ∩ (Ξ − (u0 , y 0 )),

(3.133)

where Du L(·, ·, ·, ·) and Dy L(·,·, ·, ·) are the correspondingGˆ ateaux derivatives of L. Moreover, if intU0 ×Y0 (U0 × Y0 ) ∩ (Ξ − (u0 , y 0 )) ∩ Ker L = ∅ then λ = 1. Conversely, if (u0 , y0 ) ∈ Ξ, I is a convex functional, I is Gˆ ateaux differentiable at (u0 , y 0 ) along U0 × Y0 , there exists a ψ ∈ Z∗ such that the collection (u0 , y 0 , 1, ψ) satisfies relations (3.133) and (U0 × Y0 ) ∩ Ker L ∩ (Ξ − (u0 , y0 )) is dense in Ker L ∩ (Ξ − (u0 , y 0 )) with respect to the topology where I is continuous, then (u0 , y 0 ) is an optimal pair to the problem (3.126)–(3.127). In fact, the principal feature of Theorems 3.37 and 3.40 is the fact that each of these results can be viewed as a key point to the study of OCPs for illposed processes. As for the control systems described by well-posed boundary value problems, we have the following result (see Ioffe and Tikhomirov [125]) Theorem 3.41. Let (u0 , y 0 ) ∈ U × Y1 be a solution to the problem


I(u, y) → inf, L(u, y) + F (y) = 0, u ∈ U∂ .

103

(3.134) (3.135)

Assume that for every u ∈ U∂ the mappings y → I(u, y), y → L(u, y) and y → F (y) are continuously differentiable at y ∈ O(y0 ), where O(y 0 ) is some neighborhood of y0 ∈ Y1 . Assume that the image of the mapping Dy L(u0 , y 0 ) + Dy F (y 0 ) : U × Y1 → Z

(3.136)

is closed in Z and has a finite codimension. Assume also that for every y ∈ O(y0 ) the mapping u → I(u, y) is convex and Gˆ ateaux differentiable at (u0 , y 0 ) and the mapping u → L(u, y) is affine and continuous. Then there exists a pair (λ, ψ) ∈ R1+ × Z \ {0} such that Dy L(u0 , y 0 , λ, ψ), h Y∗ ,Y1 = 0, ∀ h ∈ Z, 1 (3.137) Du L(u0 , y0 , λ, ψ), u − u0 U∗ ,U ≥ 0, ∀ u ∈ U∂ , where the Lagrange function L is defined in (3.129). Furthermore, if the mapping (3.136) is an epimorphism, that is, Im Dy L(u0 , y 0 ) + Dy F (y 0 ) = Z, (3.138) then the parameter λ can be taken as 1. To illustrate the application of the results given above, we consider a few examples. Let us begin with the following OCP N 1 |∇y(x) − ∇z∂ (x)|2 dx + |u(x)|2 dx −→ inf, (3.139) I(u, y) = 2 Ω 2 Ω y(x) = u(x), x ∈ Ω, y|∂Ω = 0, (3.140) u ∈ U∂ ,

(3.141)

where Ω is an open bounded domain in Rn with a Lipschitz continuous boundary ∂Ω, z∂ ∈ H 1 (Ω), N > 0, and U∂ is a convex closed subset of L2 (Ω). Theorem 3.42. A pair (u0 , y 0 ) ∈ L2 (Ω) × (H 2 (Ω) ∩ H01 (Ω)) is optimal for the problem (3.139)–(3.141) if and only if for some ψ ∈ L2 (Ω) the triplet (u0 , y 0 , ψ) satisfies the relations (3.142) y 0 (x) = u0 (x), x ∈ Ω, y 0 ∂Ω = 0, ψ(x) − y 0 (x) = z∂ (x), x ∈ Ω, ψ|∂Ω = 0, (N u0 (x) − ψ(x))(u(x) − u0 (x)) dx ≥ 0 ∀ u ∈ U∂ . u0 ∈ U∂ , Ω

Proof. Let us set

(3.143) (3.144)

104


U = L2 (ω),

Y = H01 (Ω),

Y1 = H 2 (Ω) ∩ H01 (Ω),

Z = L2 (Ω),

L(u, y) = y − u,

F (y) = F0 = 0.

First, we note that by Theorem 3.2 the OCP (3.139)–(3.141) has a unique solution (u0 , y 0 ) ∈ L2 (Ω)×(H 2 (Ω)∩H01 (Ω)). Indeed, it remains only to verify hypotheses (H1) and (H2) because the remaining conditions of Theorem 3.2 are obvious. Since the Dirichlet boundary value problem (3.140) has a unique solution y = y(u) ∈ H 2 (Ω) ∩ H01 (Ω) for every u ∈ L2 (Ω) with the estimate (see Theorem 2.20) y2H 2 (Ω) ≤ Cu2L2 (Ω) , it follows that hypothesis (H1) is obviously true. As for the coerciveness condition (H2), this immediately follows from (3.139) and the fact that |∇y(x)|2 dx, y2H 1 (Ω) = 0

Ω

by Friedrichs inequality. To derive the optimality system for the OCP, we make use of Theorem 3.41. To do so, we have to verify condition (3.138). However, since the boundary value problem (3.140) has a unique solution in H 2 (Ω) ∩ H01 (Ω) for every u ∈ L2 (Ω), it means that Im Dy L(u0 , y 0 ) + Dy F (y 0 ) = Im Dy L(u0 , y 0 ) = Z = L2 (Ω). Thus, by Theorem 3.41, we take a Lagrange function as follows: L(u, y, ψ) =

N 1 ∇y − ∇z∂ |2L2 (Ω) + u2L2 (Ω) + (y − u, ψ)L2 (Ω) , (3.145) 2 2

where ψ ∈ L2 (Ω). It is clear that, in this case, condition (3.137)2 implies inequality (3.144), whereas from (3.137)1 , it follows that 0 z(x)ψ(x) dx = 0 (3.146) ∇y − ∇z∂ · ∇z dx + Ω

Ω

for all z ∈ H 2 (Ω) ∩ H01 (Ω). Let us show that the last relation leads us to (3.143). Indeed, by Friedrichs inequality, we have yH01 (Ω) = ∇yL2 (Ω) . So, from (3.146) we deduce that (z, ψ)L2 (Ω) ≤ C∇zL2 (Ω) . Therefore, the mapping z → (z, ψ)L2 (Ω) can be extended by continuity from H 2 (Ω)∩H01 (Ω) onto H01 (Ω). Due to the Riesz theorem, an element q ∈ H01 (Ω) can be found such that (z, ψ)L2 (Ω) = (∇z, ∇q)L2 (Ω) = z, qH −1 (Ω),H 1 (Ω) . 0

(3.147)


105

So, q = ψ ∈ H01 (Ω), and hence the boundary condition in (3.143) is valid. In addition, combining (3.146) and (3.147), we obtain the conjugate equation (3.143) which should be interpreted in the distributional sense. This concludes the proof. We close this subsection with the following optimal boundary control problem: I(u, y) =

N 1 ∂ν y − z∂ 2L2 (∂Ω) + u2L2 (∂Ω) −→ inf, 2 2 y(x) = f (x), x ∈ Ω, y|∂Ω = u, u ∈ U∂ ,

(3.148) (3.149) (3.150)

where z∂ ∈ L2 (∂Ω), f ∈ L2 (Ω), N > 0, ∂ν = ∂/∂ν is the outward normal derivative to ∂Ω and U∂ is a convex closed subset of L2 (∂Ω) such that U∂ ∩ H 1 (∂Ω) = ∅. As follows from Theorem 3.16, there exists a unique solu3/2 tion (u0 , y 0 ) ∈ L2 (∂Ω) × HΔ (Ω) to the optimal boundary control problem 3/2 (3.148)–(3.150). We recall that HΔ (Ω) = y ∈ H 3/2 (Ω) : y ∈ L2 (Ω) and the trace operator y → ( y|∂Ω , ∂y/∂ν|∂Ω ) is well defined and continuous as the mapping 3/2 (3.151) Π : HΔ (Ω) → H 1 (∂Ω) × L2 (∂Ω). 3/2

Theorem 3.43. A pair (u0 , y0 ) ∈ H 1 (∂Ω) × HΔ (Ω) is optimal for the 1/2 problem (3.148)–(3.150) if and only if for some ψ ∈ HΔ (Ω) the triplet 3/2 1/2 (u0 , y 0 , ψ) ∈ H 1 (∂Ω) × HΔ (Ω) × HΔ (Ω) satisfies the relations (3.152) y 0 (x) = f (x), x ∈ Ω, y 0 ∂Ω = u0 , 0 ψ(x) = 0 x ∈ Ω, ψ|∂Ω + ∂ν y ∂Ω = z∂ , (3.153) (N u0 − ∂ν ψ)(u − u0 ) dHn−1 ≥ 0, ∀ u ∈ U∂ . (3.154) u0 ∈ U∂ , Ω

Proof. In order to prove this assertion, we make use of Theorem 3.2. To this end, we set 3/2

Y = Y1 = HΔ (Ω),

U = H 1 (∂Ω),

Z = L2 (Ω) × H01 (∂Ω),

L(u, y) = (y − f, y|∂Ω − u), F (y) = F0 = (0, 0). 3/2

As in the previous case, the existence of a unique solution z ∈ HΔ (Ω) to the problem z = h1 , z|∂Ω = h2 for every h1 ∈ L2 (Ω) and h2 ∈ H 1 (∂Ω) implies the fulfillment of the condition (3.138) of Theorem 3.41. Since the remaining assumptions of this theorem are obviously true, we can apply relations (3.137) to the case when the Lagrange function is

106


L(u, y, ψ) =

1 N ∂ν y − z∂ 2L2 (∂Ω) + u2L2 (∂Ω) 2 2 + (y − f, ψ)L2 (Ω) + q, y|∂Ω − uH −1 (∂Ω),H 1 (∂Ω) , (3.155) 0

−1

where ψ ∈ L (Ω) and q ∈ H (∂Ω). Then, using (3.155) in (3.137)1 , we get ∂ν y 0 ∂Ω − z∂ , ∂ν z|∂Ω L2 (∂Ω) + (z, ψ)L2 (Ω) 2

+ q, z|∂Ω (H 1 (∂Ω))∗ ,H 1 (∂Ω) = 0,

3/2

∀ z ∈ HΔ (Ω). (3.156)

0 This implies that ψ = 0 in D (Ω). Hence, ψ ∈ HΔ (Ω). Then if we combine ψ = 0 with (3.156), we get ∂ν y 0 ∂Ω − z∂ + ψ|∂Ω , ∂ν z|∂Ω L2 (∂Ω)

+ q − ∂ν ψ|∂Ω , z|∂Ω (H 1 (∂Ω))∗ ,H 1 (∂Ω) = 0

3/2

∀ z ∈ HΔ (Ω).

As a result, we deduce ψ|∂Ω + ∂ν y 0 ∂Ω = z∂ ,

(3.157)

∂ν ψ|∂Ω = q.

(3.158)

Hence, the conditions ψ = 0 in D (Ω) and (3.157) imply the required relations (3.153). As for the inequality (3.154), it remains only to apply (3.137)2 to (3.155) and take into account the condition (3.158). This concludes the proof. 3.6.2 Necessary optimality conditions in the form of variational inequalities The question we will discuss in this subsection is the optimality condition to essentially nonlinear OCPs in Banach spaces with control constraints. Typically, these optimality conditions consist of some operator equations and variational inequalities. We refer to Barbu [19], Barbu and Precupanu [20], Lions [171, 172], Tiba [239] as some basic textbooks on various questions and results related to this subdomain of control theory. Let Y be a reflexive Banach space, Y∗ be its dual, and U be a control space which is assumed to be dual to some separable Banach space V (U = V∗ ). Let U∂ be a subset of admissible controls in U. The OCPs we consider can be stated as follows: I(u, y) −→ inf, A(u, y) = f, u ∈ U∂ ⊂ U,

(3.159) (3.160)

where A : U×Y → Y∗ is nonlinear mapping, I : U×Y → R is a cost functional, f is a given element of Y∗ and U∂ is a subset of admissible controls.


107

We assume that there exists at least one admissible pair (u, y) for the problem (3.159)–(3.160) (i.e., satisfying (3.160) and such that I(u, y) is finite). As indicated in Theorem 3.12, under assumptions (A1)–(A3) and (A6), the OCP (3.159)–(3.160) has a nonempty set of solutions. Let (u0 , y0 ) ∈ Ξ be an optimal pair and let V = O(u0 , y0 ) ⊂ U × Y be an open neighborhood of this pair. To derive the optimality system, we begin with the following assumptions: ateaux derivatives in V , (B1) The mapping A : U × Y → Y∗ has partial Gˆ Dy A : V → L(Y, Y∗ ) and Du A : V → L(U, Y∗ ), which are continuous with respect to the uniform operator topology. (B2) The cost functional I : U × Y → R is Gâteaux differentiable in V and its partial derivatives Dy I : V → Y∗ and Du I : V → U∗ are continuous. (B3) If a pair (u, y) ∈ V is admissible then Ker Dy A(u, y) = {0} and Dy A(u, y)ξ, ξY∗ ,Y ξY

→ +∞ as ξY → +∞.

Before proceeding further, we recall some basic notation of nonlinear analysis. Definition 3.44. We say that the cost functional I : U×Y → R is sequentially ∞ lower semicompact if for any sequences {uk }∞ k=1 ⊂ U and {yk }k=1 ⊂ Y such ∗ that uk u in U and yk y in Y as k → ∞, there exist subsequences ∞ {uki }i=1 ⊂ U and {yki }∞ i=1 ⊂ Y satisfying the inequality lim inf I(uki , yki ) ≥ I(u, y). i→∞

An operator A(u, ·) : Y → Y∗ is said to be bounded if for any u ∈ U, the mapping A(u, ·) : Y → Y∗ transforms a bounded subset of Y to a bounded subset of Y∗ . We also say that an operator A(u, ·) : Y → Y∗ is monotone if for any u ∈ U A(u, y1 ) − A(u, y2 ), y1 − y2 Y∗ ,Y ≥ 0,

∀ y1 , y2 ∈ Y, y1 = y2 .

Now we are in a position to state the main result of this subsection. Theorem 3.45. Assume that (i) U∂ is a sequentially weakly-∗ closed bounded convex subset of U; (ii) I : U × Y → R is a sequentially lower semicompact functional; (iii) the operator A : U × Y → Y∗ is coercive and possesses the property (M) (see (A2)–(A3)); (iv) for any u ∈ U the operator A(u, ·) : Y → Y∗ is monotone and bounded. Then the OCP (3.159)–(3.160) is solvable under conditions (B1)–(B3) and any optimal pair (u0 , y 0 ) ∈ U∂ × Y satisfies the relations A(u0 , y 0 ) = f, ∗ Dy A(u0 , y0 ) p = Dy I(u0 , y0 ), & % ∗ − Du A(u0 , y 0 ) p + Du I(u0 , y0 ), u − u0 ∗ ≥ 0, U ,U

(3.161) (3.162) ∀ u ∈ U∂ .

(3.163)

108


Proof. Since A(u, ·) is a bounded coercive operator satisfying the property (M), it follows that for any u ∈ U∂ ⊂ U the operator equation A(u, y) = f has at least one solution (see [126, 132]). Moreover, due to Theorem 3.12 and the fact that the lower semicompactness property of I is essentially weaker than property (A6) (see Sect. 3.2), the OCP (3.159)–(3.160) is solvable. Let (u0 , y 0 ) ∈ U∂ ×Y be some optimal pair and let (u, y) be any admissible pair belonging to the neighborhood V = O(u0 , y 0 ) ⊂ U × Y. First, we show that the inequality (Dy A(u, y)) ξ, ξY∗ ,Y ≥ 0,

∀ ξ ∈ Y,

(3.164)

holds true for every u ∈ U∂ . Indeed, let s ∈ (0, 1]. Then, for any u ∈ U∂ , y ∈ Y and ξ ∈ Y, there exists τ ∈ [0, s] such that, by the monotonicity property of A(u, ·) and the definition of the Gˆ ateaux derivative Dy A(u, y), we have 0 ≤ A(u, y + sξ) − A(u, y), sξY∗ ,Y s = (Dy A(u, y + tξ)) ξ, sξY∗ ,Y dt 0

= s2 (Dy A(u, y + τ ξ)) ξ, ξY∗ ,Y .

(3.165)

Since the mapping τ → (Dy A(u, y + τ ξ)) ξ, ξY∗ ,Y is continuous on [0, s], after dividing both sides of (3.165) by s2 and letting s → 0, we get (3.164). As a result, inequality (3.164) together with the coerciveness property of the operator Dy A(u, y) (i.e., ξ−1 Y (Dy A(u, y)) ξ, ξY∗ ,Y → +∞ as ξY → ∞) implies the fact that the operator Dy A(u, y) is surjective at the pair (u, y) ∈ V . Since Ker Dy A(u, y) = {0}, it means that the operator Dy A(u, y) is invertible. In fact, [Dy A(u, y)]−1 is a bounded operator. Indeed, let us set Dy A(u, y)ξ = η, where ηY∗ ≤ M . Then ξY γ(ξY ) = Dy A(u, y)ξ, ξY∗ ,Y ≤ ηY∗ ξY , where γ(ξY ) = Dy A(u, y)ξ, ξY∗ ,Y /ξY . Hence, γ(ξY ) ≤ ηY∗ . However, by the initial assumptions, we have γ(s) → +∞ as s → ∞. So, there exists a constant C such that ξY = −1 [Dy A(u, y)] ηY = C and C is independent of M . We may conclude that (see [126]) for every (u, y) ∈ O(u0 , y 0 ) the corresponding solutions y = y(u) of the operator equation A(u, y) = f are differentiable with respect to u ∈ U∂ and yu (u)h = − [Dy A(u, y)]−1 [Du A(u, y)] h, ∀ h ∈ U∂ . (3.166) For a given optimal pair (u0 , y 0 ), we set (u, y) ∈ V and (w, z) = (1 − τ )(u0 , y 0 ) + τ (u, y), where τ ∈ (0, 1). Then I(u0 , y 0 ) ≤ I(w, z) = I (u0 , y 0 ) + τ (u − u0 , y − y 0 )

3.7 Optimal control of distributed singular systems

109

for all τ ∈ (0, 1) and (u, y) ∈ V . Passing to the limit in the last inequality as τ → +0, we obtain Du I(u0 , y 0 ), (u − u0 ) U∗ ,U + Dy I(u0 , y 0 ), (y − y 0 ) Y∗ ,Y ≥ 0, ∀(u, y) ∈ V. Using (3.166), we get Du I(u0 , y0 ), u − u0 U∗ ,U + Dy I(u0 , y 0 ), y − y0 Y∗ ,Y = Du I(u0 , y 0 ), u − u0 U∗ ,U % & −1 Du A(u0 , y 0 ) (u − u0 ) − Dy I(u0 , y 0 ), Dy A(u0 , y 0 ) Y∗ ,Y ' −1 ∗ ∗ = Du I(u0 , y0 ) − Du A(u0 , y 0 ) Dy A(u0 , y 0 ) × Dy I(u0 , y 0 ), u − u0 U∗ ,U ≥ 0 for all u ∈ U∂ . Let us introduce the adjoint state p ∈ Y as a solution of the problem ∗ Dy A(u0 , y0 ) p = Dy I(u0 , y0 ). ∗ Since the operator Dy A(u0 , y0 ) is invertible at the point (u0 , y 0 ), we have ∗ −1 p = Dy A(u0 , y 0 ) Dy I(u0 , y 0 ). Taking this into account, the last inequality just implies the required relation (3.163). The proof is complete.

3.7 Optimal control of distributed singular systems Let Ω be an open bounded domain in Rn with Lipschitz boundary ∂Ω. Let T > 0, ν > 0, r > 1, ρ > 0 and q > 1 be given values. The OCPs we consider in this section can be described as follows: Find a control function u and a corresponding state y such that a functional I(u, y) =

ν 1 y − z∂ rLr (Ω) + uqLq (Ω) r q

(3.167)

is minimized subject to the constraints y˙ − Δy − |y|ρ y − u = 0 y|t=0 = y

0

on Ω,

in Q = (0, T ) × Ω,

y = 0 on Σ = (0, T ) × ∂Ω, u ∈ U∂ .

(3.168) (3.169) (3.170)

Here, y(t, ˙ x) = ∂y(t, x)/∂t, U∂ is a convex closed subset of Lq (Ω), y 0 ∈ H01 (Ω) and z∂ ∈ Lr (Q) are given functions. Let us note that the state equation (3.168) with the sign “–” before the nonlinear term has no a priori estimates. Hence, this equation is singular in the

110


sense that for a given control u ∈ U∂ , the boundary value problem (3.168)– (3.169) has no global solution, in general (see [172]). So, we deal with the OCP for an ill-posed (singular) initial-boundary value problem. However, for some controls, such a solution exists, and we can suppose that hypothesis (H1) holds true. Let us set U = Lq (Ω), Y = Lr (Q), Y1 = Lr (Q) ∩ y ∈ W (1,2), s (Q) : y|t=0 = y 0 , L(u, y) = (y˙ − y − u, y|t=0 ), F (y) = (−|y|ρ y, −y 0 ), where by W (1,2),s (Q) we denote the Sobolev space formed by all functions y ∈ Ls (Q) such that ∂y/∂t and Dxα y exist in the weak sense for each multi– index α with |α| ≤ 2 and the norm ⎛

T

yW (1,2),s (Q) = ⎝

0

⎡

⎤ ⎞1/s " ∂y(t, x) s s ⎣ |Dxα y(t, x)| ⎦ dx dt⎠ ∂t + Ω |α|≤2

is finite. Here, s = min{q, r/(ρ + 1)}, for r > ρ + 1 and s = q if r < ρ + 1. According to the general theory of singular control systems [172], we define the set of admissible pairs Ξ ⊂ U × Y1 to the above problem in the form (3.4). Then we have the following result. Theorem 3.46 ([228]). If r > ρ + 1, then the OCP (3.167)–(3.170) admits at least one solution. Proof. Since the cost functional I is bounded below, the extremal problem inf (u,y)∈Ξ I(u, y) possesses a minimizing sequence, i.e., there exists a sequence {(uk , yk ) ∈ Ξ}∞ k=1 of admissible pairs such that I(uk , yk ) −→

inf

(u,y)∈Ξ

I(u, y).

(3.171)

By coerciveness of the functional in the space Lq (Q)×Lr (Q), condition (3.171) implies that the sequence under consideration is bounded on the set Ξ. Since the pair (uk , yk ) is admissible, its components are connected by the equality y˙k = Δyk + |yk |ρ yk + uk . Thereby the function yk is a solution to the following initial-boundary value problem y˙ k − Δyk = fk yk |t=0 = y

0

on Ω,

in Q = (0, T ) × Ω,

(3.172)

yk = 0 on Σ = (0, T ) × ∂Ω,

(3.173) ∞

where fk = |yk |ρ yk + uk . Using the fact that the sequence {fk }k=1 is bounded in Ls (Q) for r > ρ + 1, we can establish the boundedness of the solutions

3.7 Optimal control of distributed singular systems

111

∞

{yk }k=1 to the problem (3.172)–(3.173) in W (1,2), s (Q), and hence in Y1 . Dropping down to a subsequence (keeping the former notation), we obtain the convergence uk u in Lq (Q),

yk y in Y1 .

(3.174)

By compactness of the embedding of W (1,2),s (Q) into Lβ−α (Q) (see [168], Theorem 5.1]), where α > 0 and 1/β = 1/s + 1/(n + 2), we establish that yk → y in Lβ−α (Q) and a.e. in Q. By the convergence |yk |ρ yk → |y|ρ y a.e. ∞ in Q, the boundedness of the sequence {|yk |ρ yk }k=1 in Lα (Q), we see that (see [168]) |yk |ρ yk |y|ρ y in Lα (Q). As a result, passing to the limit in the problem (3.172)–(3.173) as k tends to ∞, we come to relations (3.168)– (3.169). Recalling the convexity and closure properties of U∂ , we conclude that the limit pair (u, y) is admissible to the original problem (3.167)–(3.170). It remains to prove that this pair is optimal. Using (3.174), we obtain the inequality lim inf I(uk , yk ) ≥ I(u, y). k→∞

Then it follows by (3.171) that the pair (u, y) is optimal. This concludes the proof. The particular case of this OCP (with ρ = 2, q = 2 and r = 6) was studied by Lions in [172]. So, Theorem 3.46 can be viewed as a natural generalization of Theorem 1.3 in [172]. Note that in [172] this problem was later regularized by an adaptive penalty method and the necessary optimality conditions were derived for the regularized problem. However, these results can be easily extended to the general case, provided that the problem (3.167)–(3.17) is solvable (i.e., when r > ρ + 1). The main reason is the fact that in the proof of convergence of the penalty method (see [172]), these constraints on the parameters of the problem play a principle role and the smoothness of the functions under consideration is insufficient for passing to the limit in the optimality conditions for the regularized problem as the regularization parameter tends to 0. As we will see in the next chapter, it can be done through the application of a somewhat different form of the regularization method and by weakening the notion of an approximate solution to the problem.

4 Suboptimal and Approximate Solutions to Extremal Problems

In this chapter, we continue the discussion of the optimal control problems (OCPs) described in Chap. 3. Our main focus here is a proper regularization and approximation of these problems. In order to clarify the significance of these notions, we begin with the following abstract extremal problem. Let Y be a reflexive Banach space, Y∗ be its dual, and Z be a Banach space partially ordered by a closed reproducing pointed cone Λ ⊂ Z. Let U be a control space which is assumed to be dual to some separable Banach space V (U = V∗ ). Let U∂ be a subset of admissible controls in U and let K be a subset of admissible states in Y. The OCPs we consider can be described in a general manner as follows (see (3.14)–(3.17)): I(u, y) −→ inf

(4.1)

A(u, y) = f, F (u, y) ≥Λ 0, y ∈ K, u ∈ U∂ ⊂ U,

(4.2)

subject to

(4.3) (4.4)

where A : U × Y → Y∗ , F : U × Y → Z are nonlinear mappings, I : U × Y → R is a cost functional, and f is a given element of Y∗ . Let us recall that, in general, the mapping u → y(u) can be multivalued. As usual, by Ξ we denote the set of all admissible pairs for the problem (4.1)–(4.4), that is, Ξ = {(u, y) ∈ U∂ × K : A(u, y) = f, F (u, y) ≥Λ 0, I(u, y) < +∞} . (4.5) We associate with the problem (4.1)–(4.4) the constrained minimization problem inf

I(u, y) ,

(4.6)

(u,y)∈ Ξ

which, for the sake of simplicity, we will identify with the pair (I, Ξ). P.I. Kogut, G.R. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains, Systems & Control: Foundations & Applications, DOI 10.1007/978-0-8176-8149-4 4, © Springer Science+Business Media, LLC 2011

113

114


As we said earlier, the problem (4.1)–(4.4) is regular if its set of admissible pairs is nonempty. Thus, we ask: Does there exist a pair (u, y) in U∂ × K ⊂ U × Y satisfying the state equation (4.2) and the inequality constraint (4.3) such that I(u, y) < +∞? In fact, one needs the set of admissible pairs to be sufficiently rich in some sense; otherwise the OCP (4.1)–(4.4) becomes trivial. However, from a mathematical point of view, to deal directly with all constraints presented above is typically very difficult and, except for some special cases, this question is largely open. Nevertheless, in many applications and, for example, in “inexact” numerical methods, it is important to find at least an approximately admissible – in a sense to be made precise – pair when both state and control constraints are given. On the other hand, since U∂ and K are closed and convex subsets, they may be very “thin” and it is possible that the original problem has no solutions. In view of this, it is reasonable to weaken the requirements on approximate solutions to the OCPs. In particular, we may accept that the cost functional is minimized with some error. However, it would also be reasonable to assume that the relations A(u, y) = 0 and F (u, y) ≥Λ 0 and the inclusions u ∈ U∂ and y ∈ K hold not strictly but rather with some (possibly high) accuracy. Moreover, the set of solutions of the extremal problem (4.6) can possibly be empty, that is, the greatest lower bound of the cost functional is often unattainable on the given set Ξ. Nevertheless, the absence of a minimum of the functional does not mean that the problem does not make any sense (see, e.g., [252]), since its greatest lower bound exists and hence can be approached with some accuracy. Thus, an extremal problem may have an approximate or suboptimal solution even if it is not solvable.

4.1 The notion of suboptimal and approximate solutions Let τ be the product of the weak-∗ topology of U and the weak topology of Y. Let (I, Ξ) be a constrained minimization problem on U × Y which is associated with some OCP. We assume that this control problem can be described in a general manner as the problem (4.1)–(4.4). If the cost functional I is bounded below on the set Ξ and Ξ = ∅, then I has the greatest lower bound μ = inf (u,y)∈ Ξ I(u, y). Hence, in view of the “direct method” in the calculus of variations, we can find a minimizing sequence {(uk , yk )}∞ k=1 in Ξ (see, e.g., [169]) such that I(uk , yk ) → μ. As a result, we can introduce the following notion. Definition 4.1. For a given ε > 0, the OCP (4.1)–(4.4) is said to be εsolvable if there exists a pair (uε , yε ) ∈ Ξ such that I(uε , yε ) ≤

inf

I(u, y) + ε.

(u,y)∈ Ξ

In this case, the pair (uε , yε ) is called the ε-approximate solution to the problem.

4.1 The notion of suboptimal and approximate solutions

115

∞

Indeed, having a minimizing sequence {(uk , yk )}k=1 ⊂ Ξ, we can take an ε-approximate solution to the problem (4.1)–(4.4) as an element of this sequence with a sufficiently large index. Thus, any ε-approximate solution is an admissible pair such that the corresponding value of the cost functional is arbitrarily close to its greatest lower bound on the set Ξ. However, even if an optimal pair (uopt , y opt ) is available, we cannot guarantee the proximity of an ε-approximate solution (uε , yε ) to (uopt , y opt ) with respect to the τ topology of U × Y. Definition 4.2. The OCP (4.1)–(4.4) is said to be well-posed in the Tikhonov ∞ sense if it is solvable and every minimizing sequence {(uk , yk )}k=1 ⊂ Ξ conopt opt verges to a solution (u , y ) ∈ Ξ. In this case, a minimizing sequence is called a Tikhonov minimizing sequence. Note that having a Tikhonov minimizing sequence, we can guarantee both the proximity of the corresponding cost functional to its greatest lower bound and the proximity of the approximation itself to the solution (uopt , y opt ) of the problem. As a result, for every ε > 0 and every neighborhood O(uopt , y opt ) of (uopt , y opt ) in U×Y, we can define a Tikhonov approximate solution (u∗ε , yε∗ ) ∈ Ξ to the problem (4.1)–(4.4), for which we have (u∗ε , yε∗ ) ∈ O(uopt , y opt ) and I(u∗ε , yε∗ ) ≤

inf

I(u, y) + ε.

(u,y)∈ Ξ

Nevertheless, it should be stressed that even in simple applications, the construction of Tikhonov minimizing sequences and corresponding Tikhonov approximate solutions is very difficult. We introduce the notion of a weakly minimizing sequence. ∞

Definition 4.3. We say that a sequence {(uk , yk )}k=1 in U × Y is a weakly ∗ minimizing sequence, if (uk , yk ) ∈ U∂ × K for every k ∈ N, uk uopt in U, yk y opt in U, A(uk , yk ) → f in Y∗ , limk→∞ F (uk , yk ) ≥Λ 0 and I(uopt , yopt ) ≤ lim inf I(uk , yk ) ≤ k→ ∞

inf

I(u, y).

(u,y)∈ Ξ

Moreover, if for every ε > 0, every neighborhood O(uopt , yopt ) of (uopt , yopt ) in the τ -topology of U × Y and every neighborhood OZ (0) of the origin in Z, w there is a pair (uw ε , yε ) ∈ U∂ × K such that the conditions w opt opt w (uw , y ), A(uw ε , yε ) ∈ O(u ε , yε ) − f Y∗ < ε, w F (uw ε , yε ) ∈ OZ (0) + Λ w hold true, then (uw ε , yε ) is said to be a weak ε-approximate solution to the problem (4.1)–(4.4).

It is clear that any Tikhonov approximate solution to the problem (4.1)– (4.4) is also a weak approximate solution. However, the original OCP may fail to have an exact solution (uopt , y opt ) ∈ Ξ, that is, the greatest lower

116


bound of the cost functional is often unattainable on the set of admissible pairs Ξ. An extremal problem may have an approximate solution even in the absence of solvability. However, in this case, the notions of Tikhonov and weak approximate solutions make no sense, for they explicitly appeal to the notion of an exact solution. Removing the optimality of the pair (uopt , y opt ) from the definitions of a weak minimizing sequence and a weak approximate solution problem, we arrive at the weakened forms of these notions. ∞

Definition 4.4. We say that a sequence {(uk , yk )}k=1 in U × Y is a weakened ∗ in U, minimizing sequence if (uk , yk ) ∈ U∂ × K for every k ∈ N, uk u yk y in U, A(uk , yk ) → f in Y∗ , limk→∞ F (uk , yk ) ≥Λ 0 and I( u, y) ≤ lim inf I(uk , yk ) ≤ k→ ∞

inf

I(u, y).

(u,y)∈ Ξ

Moreover, if for some ( u, y) ∈ U × Y, every ε > 0, every neighborhood O( u, y) of ( u, y) in the τ -topology of U × Y and every neighborhood OZ (0) of the origin in Z, there is a pair ( uε , yε ) such that the conditions u, y) ∩ (U∂ × K) , A( uε , yε ) − f Y∗ < ε, ( uε , yε ) ∈ O( F ( uε , yε ) ∈ OZ (0) + Λ hold true, then ( uε , yε ) is said to be a weakened ε-approximate solution to the problem (4.1)–(4.4). Thus, a weakened approximate solution is a pair arbitrarily close to the set of admissible solutions, guaranteeing in a sense the proximity of the cost functional to its greatest lower bound. Since the only difference from the above notion (see Definition 4.3) is the absence of the optimality requirement on ( u, y), a weak approximate solution to the OCP is certainly a weakened approximate solution. Observe also that the convex closed subsets U∂ and K may be very “thin‘” with rather complicated structures. Because of this, a direct numerical computation of solutions to the corresponding OCPs with such constraints can be extremely difficult. Usually a very fine discretization mesh is needed, which means a significant computation time. Therefore, it is reasonable to introduce the following notion. sub Definition 4.5. If for every ε > 0 there exists a pair (usub ε , yε ) ∈ U × Y such that sub sub I(uε , yε ) − inf I(u, y) ≤ ε, (4.7) (u,y)∈ Ξ sub (usub (4.8) ε , yε ) ∈ Ξε = (u, y) ∈ U × Y : dist ((u, y), Ξ)U×Y ≤ ε , sub then the pair (usub ε , yε ) is said to be an ε-suboptimal solution to the problem (4.1)–(4.4).

4.2 Regularization of optimal control problems

117

In the next sections, we consider a large class of OCPs for nonlinear operator equations. Their solvability is established only under certain additional assumptions. Nevertheless, even for the general statement, we describe an algorithm, following in many aspects Serova˘iski˘i [228] and Ivanenko and Mel’nik [126], that enables us to construct weak and weakened minimizing sequences and hence to find the corresponding approximate solutions to these problems.

4.2 Regularization of optimal control problems The main object of our consideration in this section is the OCP (4.1)–(4.4). We begin with the following observation: The set of admissible pairs Ξ to the original problem can be presented in the form Ξ = Ξ1 ∩ Ξ2 , where Ξ1 = {(u, y) ∈ U∂ × Y : A(u, y) = f, y ∈ K} , Ξ2 = {(u, y) ∈ U∂ × Y : F (u, y) ≥Λ 0} . Let us assume that Ξ = ∅, provided that Ξ1 = ∅. In this case, we associate with the set Ξ2 a functional JΞ2 : Ξ1 → R characterizing a “deviation measure” from the set Ξ2 . We suppose that JΞ2 (u, y) = 0 if and only if (u, y) ∈ Ξ2 . In what follows, we will call each of the sets Ξi the corresponding constraint system. Definition 4.6. Assume that Ξ1 = ∅. We say that the OCP (4.1)–(4.4) is u, y) in U∂ × Y such that JΞ2 ( u, y) = JΞ2 -regularized if there exists a pair ( inf (u,y)∈Ξ1 JΞ2 (u, y). We also say that Ξ1 is a JΞ2 -attainable restriction sysu, y) = 0. tem if there exists a pair ( u, y) ∈ U∂ ×Y satisfying the condition JΞ2 ( Before proceeding further, for every (u, y) ∈ Ξ1 we define the set L∗ (u, y) = ξ ∈ Λ∗ : ξ, F (u, y)Z∗ ,Z < 0 ∪ {0}, where Λ∗ ⊂ Z∗ is the conjugate semigroup with respect to the cone Λ (see (3.23)). It is clear that L∗ (u, y) is a convex pointed cone in Z∗ . Let S1∗ be the unit sphere in Z∗ . We set JΞ2 (u, y) = (4.9) sup ξ, F (u, y)Z∗ ,Z . ξ∈L∗ (u,y)∩S1∗

A pair ( u, y) ∈ Ξ1 is said to be a regularizer of the constraint system Ξ2 to the OCP (4.1)–(4.4) if u, y) = inf sup JΞ2 ( ξ, F (u, y)Z∗ ,Z . (u,y)∈Ξ1 ξ∈L∗ (u,y)∩S ∗ 1

By R we denote a set of all regularizers of Ξ2 .

118


Theorem 4.7. Assume that Ξ1 = ∅ and conditions (A1)–(A6) of Sect. 3.2 hold true. Then the OCP (4.1)–(4.4) is JΞ2 -regularized. Moreover, there exists a pair (u, y) ∈ R (optimal regularizer to this problem) such that I(u, y) ≤ I(u, y) for all (u, y) ∈ R. Proof. We divide our proof into three steps. Step 1. By analogy with notation of Sect. 3.2, we denote by F (R, R+ ) the set of lower semicontinuous monotone decreasing functions μ : R → R+ such that μ(0) = 0 and μ is strictly monotone on R− . To be specific, we set

|ξ| if ξ < 0, μ(ξ) = 0 if ξ ≥ 0. Let us consider the following approximating control problem

μ ψ, F (u, y)Z∗ ;Z → inf, Iμ (u, y) = sup ψ∈S1∗ ∩Λ∗

A(u, y) = f, y ∈ K, u ∈ U∂ ⊂ U,

(4.10) (4.11) (4.12)

where μ ∈ F(R, R+ ). It is easy to see that (4.10)–(4.12) is a special case of the penalized problem studied in Sect. 3.2. Since Ξ1 = ∅ and suppositions (A1)–(A6) are valid, the existence at least one optimal pair ( uμ , yμ ) ∈ Ξ1 to the problem (4.10)– (4.12) immediately follows by Lemma 3.10. Note that this optimal pair is not necessarily unique. To begin, we show that ( uμ , yμ ) is a regularizer for the OCP (4.1)–(4.4) (i.e., ( uμ , yμ ) ∈ R). Indeed, if Iμ ( uμ , yμ ) = 0 then F ( uμ , yμ ) ∈ Λ and hence JΞ2 ( uμ , yμ ) = 0. So, in this case, ( uμ , yμ ) ∈ R, and moreover, ( uμ , yμ ) ∈ Ξ is an optimal regularizer to the original problem. uμ , yμ ) > 0. This implies F ( uμ , yμ ) ∈ Λ. For now we assume that Iμ ( Therefore, sup

ψ∈S1∗ ∩Λ∗

μ ψ, F ( uμ , yμ )Z∗ ;Z = =

sup

ψ∈S1∗ ∩L∗ (b uμ ,b yμ )

sup

ξ∈L∗ (b uμ ,b yμ )∩S1∗

μ ψ, F ( uμ , yμ )Z∗ ;Z

uμ , yμ )Z∗ ,Z = JΞ2 ( uμ , yμ ). ξ, F (

(4.13)

Assume that there is a pair (u∗ , y∗ ) ∈ Ξ1 such that 0 < JΞ2 (u∗ , y∗ ) < JΞ2 ( uμ , yμ ). Taking into account (4.13), we come to the contradiction that Iμ (u∗ , y∗ ) < uμ , yμ ). Thus, ( uμ , yμ ) ∈ Ξ1 is an regularizer of the OCP (4.1)–(4.4). Iμ (


119

Step 2. Existence of an optimal regularizer. Having set d=

inf (u,y)∈Ξ1

JΞ2 (u, y),

we consider the following penalized OCP: 2

−1 Iε (u, y) = I(u, y) + ε sup μ ψ, F (u, y)Z∗ ;Z − d −→ inf ψ∈S1∗ ∩Λ∗

(4.14) subject to the restrictions (4.11)–(4.12). Following Lemma 3.10, for every ε > 0, the problem (4.11)–(4.12), (4.14) has a nonempty set of solutions. Let {(uε , yε ) ∈ Ξ1 }ε>0 be a sequence of optimal pairs to this problem when the small parameter ε > 0 varies in a strictly decreasing sequence of positive numbers which converge to 0. In the same way as in the proof of Lemma 3.10, we can conclude that the sequence {(uε , yε ) ∈ Ξ1 }ε>0 is relatively w-compact in U × Y and, passing to a subsequence, we get ∗

uε u in U,

yε y in Y,

Let us prove sup

ξ∈L∗ (u,y)∩S1∗

(u, y) ∈ U∂ × K,

A(u, y) = f.

ξ, F (u, y)Z∗ ,Z = d.

(4.15)

To do so, we observe that, as follows from the previous step, the extremal problem (4.11)–(4.12), (4.14) with the additional restriction

sup μ ψ, F (u, y)Z∗ ;Z = d ψ∈S1∗ ∩Λ∗

is regular. Let (v, q) be any admissible pair to this problem. Then Iε (uε , yε ) ≤ I(v, q). Hence, 2

sup μ ψ, F (uε , yε )Z∗ ;Z − d ≤ Cε. ψ∈S1∗ ∩Λ∗

So using the lower semicontinuity property of Iε with respect to the wconvergence, we have

sup

ψ∈S1∗ ∩Λ∗

2

μ ψ, F (u, y)Z∗ ;Z − d

≤ lim inf ε→0

sup

ψ∈S1∗ ∩Λ∗

2

μ ψ, F (uε , yε )Z∗ ;Z − d

≤ C lim ε = 0. ε→0

Therefore, lim inf ε→0 Iε (uε , yε ) ≥ I(u, y), and in view of the (4.13), we obtain the required relation (4.15).

120


Step 3. It remains to prove that (u, y) is an optimal regularizer. Let us assume the converse: There exists an admissible pair (v, q) to the problem (4.11)–(4.12), (4.14), (4.15) such that I(v, q) < I(u, y). Then due to the construction of the penalized cost functional (4.14), we come to the opposite inequality I(u, y) ≤ lim inf I(uε , yε ) ≤ lim inf Iε (uε , yε ) ≤ I(v, q). ε→0

ε→0

This concludes the proof. We now proceed with the discussion of the JΞ1 -regularization of the OCP (4.1)–(4.4). We assume that the constraint system Ξ1 = {(u, y) ∈ U∂ × Y : A(u, y) = f, y ∈ K} is not considered as level constraints and we may satisfy the state equation A(u, y) = f and the corresponding state constraint y ∈ K with some accuracy. For this purpose, we make use the following observation: If a pair (u, y) is admissible to the original problem, then this pair satisfies the relation

A(u, y), ζ − yY∗ ,Y ≥ f, ζ − yY∗ ,Y ,

∀ ζ ∈ K.

(4.16)

Note that the reverse statement is not true, in general. In fact, we discuss a variant of the penalization approach, called the “variational inequality method”. This idea was first studied in Mel’nik [186]. Further references and a complete theoretical analysis may be found in Tiba [238, 239], Neittaanm¨ aki and Tiba [204], Ivanenko and Mel’nik [126]. We reconsider the problem (4.1)–(4.4) with the given assumptions and we apply the variational inequality procedure. For every fixed u ∈ U∂ , the operator A(u, ·) : Y → Y∗ is pseudo-monotone bounded and coercive. So, there exists at least one solution y = y(u) of the variational inequality (4.16) such that y ∈ K (see [19, 104]). Note that it is not necessary that the corresponding pair (u, y) satisfy the operator equation A(u, y) = f . In view of this, we can use the penalized term ε−1 A(u, y) − f Y∗ as a deviation measure in an associated cost functional. As a result, we consider the following penalized OCP: Iε (u, y) = I(u, y) + ε−1 A(u, y) − f Y∗

μ ψ, F (u, y)Z∗ ;Z → inf + ε−1 sup ψ∈S1∗ ∩Λ∗

u∈U∂

(4.17)

subject to the restriction (4.16). Our aim is to show that each cluster pair of any sequence of optimal pairs to the problem (4.16)–(4.17), when the small parameter ε > 0 varies in a strictly decreasing sequence of positive numbers converging to 0, is a weak ε-approximate solution to the original problem (4.1)–(4.4). We begin with the following result. Lemma 4.8. Assume that A : U × Y → Y∗ is a quasi-monotone operator (see Remark 3.13). Then, under suppositions (A1), (A2), and (A4)–(A6) of


121

Sect. 3.2, the OCP (4.16)–(4.17) admits at least one solution for every fixed ε > 0. ∞

Proof. Let {(uk , yk )}k=1 ⊂ U∂ × K be a minimizing sequence of admissible pairs to the problem (4.16)–(4.17). The coerciveness property (A2) and the estimate

A(uk , yk ), yk − ζY∗ ,Y ≤ f, yk − ζY∗ ,Y ≤ f Y∗ yk − ζY

(4.18)

∞

immediately imply that the sequence {yk }k=1 is bounded. Since the set U∂ ×K is sequentially closed with respect to the w-convergence, we may assume that w there exists a pair (u0ε , yε0 ) ∈ U∂ × K such that (uk , yk ) → (u0ε , yε0 ). Then passing to the limit in (4.18) as k → ∞, we obtain lim sup A(uk , yk ), yk − ζY∗ ,Y ≤ 0

for all

ζ ∈ K.

k→∞

Hence,

lim inf A(uk , yk ), yk − ζY∗ ,Y ≥ A(u0ε , yε0 ), yε0 − ζ Y∗ ,Y , k→∞

∀ ζ ∈ K,

by the quasi-monotonicity property of the operator A. Combining this inequality with lim sup A(uk , yk ), yk − ζY∗ ,Y ≤ f, yε0 − ζ Y∗ ,Y , ∀ ζ ∈ K, k→∞

we come to the relation A(u0ε , yε0 ), ζ − yε0 Y∗ ,Y ≥ f, ζ − yε0 Y∗ ,Y

∀ ζ ∈ K.

Thus, (u0ε , yε0 ) ∈ U∂ × K is an admissible pair to the problem (4.16)–(4.17). Let us show that (u0ε , yε0 ) is an optimal pair to this problem. Let d be a weak limit in Y∗ of the sequence {A(uk , yk )}∞ k=1 as k → ∞. Then lim sup A(uk , yk ), yk Y∗ ,Y k→∞ ≤ f, yε0 − ζ Y∗ ,Y + d, ζY∗ ,Y = d, yε0 Y∗ ,Y + d − f, ζ − yε0 Y∗ ,Y

∀ ζ ∈ K.

Substituting yε0 for ζ in the last inequality, we get lim sup A(uk , yk ), yk Y∗ ,Y ≤ d, yε0 Y∗ ,Y . k→∞

Since the quasi-monotone operator possesses the (M)-property (see Remark 3.13), it follows that d = A(u0ε , yε0 ). Moreover, it is easy to see that for every ψ ∈ Λ∗ , we have

ψ, F (uk , yk )Z∗ ;Z → ψ, F (u0ε , yε0 ) Z∗ ;Z as k → ∞. As a result, using the w-lower semicontinuity property of the cost functional (4.17), we finally obtain

122


lim inf Iε (uk , yk ) ≥ I(u0ε , yε0 ) + ε−1 A(u0ε , yε0 ) − f Y∗ k→∞

+ ε−1 sup μ ψ, F (u0ε , yε0 ) Z∗ ;Z = Iε (u0ε , yε0 ). ψ∈S1∗ ∩Λ∗

Thus, (u0ε , yε0 ) is an optimal pair to the penalized problem (4.16)–(4.17). next step of our analysis is to consider a sequence of optimal pairs The (u0ε , yε0 ) ε>0 ⊂ U∂ × K in the limit as ε tends to 0. Theorem 4.9. In addition to the assumptions of Lemma 4.8, assume that K is a bounded subset of the state space Y and that the OCP (4.1)–(4.4) is regular. Let (u0ε , yε0 ) ε>0 be a sequence of optimal pairs to the penalized problem (4.16)–(4.17). Then (u0ε , yε0 ) ε>0 is a weak minimizing sequence for the original OCP (4.1)–(4.4). Moreover, each element of this sequence can be taken as the corresponding weak ε-approximate solution to the problem (4.1)– (4.4). Proof. In view of our assumptions, the sequence (u0ε , yε0 ) ε>0 is bounded in U∂ × K. Hence, we may assume that there exists a pair (u0 , y 0 ) ∈ U∂ × K w such that (u0ε , yε0 ) → (u0 , y 0 ) as ε → 0. Since A(u0ε , yε0 ), yε0 − ζ Y∗ ,Y ≤ f, yε0 − ζ Y∗ ,Y , ∀ ζ ∈ K, ∀ ε > 0. and the operator A is quasi-monotone and bounded, it follows that we can use the arguments of the proof of Lemma 4.8. As a result, we have A(u0ε , yε0 ) d in Y∗ , d = A(u0 , y 0 ) and f, ζ − y 0 Y∗ ,Y ≤ lim sup A(u0ε , yε0 ), ζ − yε0 Y∗ ,Y ε→0 ≤ A(u0 , y 0 ), ζ − y 0 Y∗ ,Y , ∀ζ ∈ Y. Thus, the limit pair (u0 , y 0 ) ∈ U∂ ×K satisfies the variational inequality (4.16). Let us show that this pair is admissible to the original problem (4.1)–(4.4). Let (u, y) ∈ Ξ be any admissible pair. Then Iε (u, y) = I(u, y). On the other hand, since (u0ε , yε0 ) is an optimal pair to the problem (4.16)–(4.17), this yields Iε (u0ε , yε0 ) ≤ Iε (u, y) = I(u, y) for every ε > 0. So, the sequence {Iε (u0ε , yε0 )}ε>0 is uniformly bounded with respect to ε. Hence, in view of the structure of the cost functional (4.17), we deduce

A(u0ε , yε0 ) − f Y∗ ≤ C1 ε, μ ψ, F (u0ε , yε0 ) Z∗ ;Z ≤ C2 ε. sup ψ∈S1∗ ∩Λ∗

From this, we immediately conclude that A(u0 , y0 ) = f,

F (u0 , y0 ) ≥ 0.

4.3 ε-Suboptimal solutions to optimal control problems

123

Thus, (u0 , y 0 ) ∈ Ξ. It remains to prove that (u0 , y 0 ) is an optimal pair. If, on the contrary, we assume that the exists a pair (u, y) ∈ Ξ such that I(u, y) < I(u0 , y 0 ), then I(u0ε , yε0 ) ≤ I(u0ε , yε0 ) + ε−1 A(u0ε , yε0 ) − f Y∗

+ ε−1 sup μ ψ, F (u0ε , yε0 ) Z∗ ;Z ≤ I(u, y), ψ∈S1∗ ∩Λ∗

∀ ε > 0.

Therefore, passing to the limit in this inequality as ε → 0 and using the w-lower semicontinuity property of the cost functional (4.1), we get I(u0 , y0 ) ≤ lim inf ≤ I(u, y). ε→0

This contradiction immediately leads us to the conclusion: The sequence of optimal pairs (u0ε , yε0 ) ε>0 satisfies all the properties of Definition 4.3. So, the proof is complete.

4.3 ε-Suboptimal solutions to optimal control problems As was noted in Sect. 4.1, many problems coming from applications are lacking solvability. Nevertheless, as discussed above, an inexact or relaxed formulation may seem reasonable. Thus, an extremal problem may have an approximate solution even in the absence of solvability. Taking as an example the OCP (4.1)–(4.4), we note that one of the most restrictive assumptions is the coerciveness property of the operator A : U × Y → Y∗ . Below we give a particular result concerning a noncoercive OCPs. We show that in spite of the absence of optimal solutions to such problems, for every fixed ε > 0, there exists an ε-suboptimal pair in the sense of Definition 4.5. We begin with the following result. Theorem 4.10. Assume that the hypotheses (A1) and (A3)–(A6) of Sect. 3.2 and the regularity property Ξ = ∅ for the problem (4.1)–(4.4) hold true. Assume also that the sets U∂ ⊂ U and K ⊂ Y are not necessarily bounded. Then for every ε > 0 an ε-suboptimal solution to the problem (4.1)–(4.4) can be taken as a solution to the following extremal problem: Jε (u, y) = I(u, y) + ε (uU + yY ) −→ inf, A(u, y) = f,

(4.19) (4.20)

F (u, y) ≥Λ 0, y ∈ K, u ∈ U∂ ⊂ U.

(4.21) (4.22)

Proof. It is clear that for every fixed ε > 0, the cost functional Jε : U×Y → R possesses the following property: ∀ C > 0, the set {(u, y) ∈ U∂ × K : Jε (u, y) ≤ C}

124


is bounded in U × Y. Hence, the solution set to the problem (4.19)–(4.22) is nonempty for every ε > 0. ∞ Let {εk }k=1 be a strictly decreasing sequence of positive numbers converg∞ ing to 0 as k → ∞. Let {(uε , yε ) ∈ U∂ × K}k=1 be the corresponding solutions of the extremal problem (4.19)–(4.22) with ε = εk . Let us show that this sequence is ε-minimizing to the problem (4.1)–(4.4), that is, for every ε > 0 a value j ∈ N can be found such that I(uεk , yεk ) ≤

inf

I(u, y) + ε,

(u,y)∈ Ξ

∀ k ≥ j.

Indeed, let (u, y) be any admissible pair to the original noncoercive problem (4.1)–(4.4). Then for any εi and εk such that εi < εk , we have Jεi (u, y) ≤ Jεk (u, y). Hence, d≤

inf (u,y)∈Ξ

Jεi+1 (u, y) ≤

inf (u,y)∈Ξ

Jεi (u, y) ≤ · · · ≤

inf (u,y)∈Ξ

Jεi−k (u, y) ≤ · · · .

Using the fact that the problem (4.19)–(4.22) is solvable for all ε > 0, we deduce d ≤ Jεi+1 (uεi+1 , yεi+1 ) ≤ Jεi (uεi , yεi ) ≤ · · · ≤ Jεi−k (uεi−k , yεi−k ) ≤ · · · . ∞

Thus, the non-negative numerical sequence {Jεi (uεi , yεi )}i=1 is monotonically decreasing and bounded below. As a result, for any ε > 0, a value j(ε) can be found such that I(uεi , yεi ) ≤ Jεi (uεi , yεi ) ≤ d + ε

for all

i ≥ j(ε).

This concludes the proof. As a special case of the OCP (4.1)–(4.4), let us consider the OCP of the distributed singular system (3.167)–(3.170). To rewrite it in the form of the OCP (4.1)–(4.4), we make use of the following notation: Let Ω be an open bounded domain in Rn with Lipschitz boundary ∂Ω and let T > 0, ν > 0, r > 1, ρ > 0 and q > 1 be given values. Let U∂ be a convex closed subset of Lq (Ω) and let y 0 ∈ H01 (Ω) and z∂ ∈ Lr (Q) be given functions, where Q = (0, T ) × Ω. Let us set U = Lq (Ω), Y = Lr (Q) ∩ y ∈ W (1,2), s (Q) : y|t=0 = y 0 , A(u, y) = y˙ − y − |y|ρ y − u, I(u, y) =

1 ν y − z∂ rLr (Ω) + uqLq (Ω) , r q

where s = min{q, r/(ρ + 1)} for r > ρ + 1 and s = q for r < ρ + 1. We define the set of admissible pairs as follows: Ξ = {(u, y) ∈ U × Y : A(u, y) = 0, u ∈ U∂ } . As a result, the OCP (3.167)–(3.170) can be reduced to the abstract extremal problem


inf

I(u, y).

125

(4.23)

(u,y)∈Ξ

Let us note that this problem admits at least one solution in the case when r > ρ + 1 (see Theorem 3.46). So, our prime interest in this section is to consider the general statement of the problem (4.23), that is, the case in which there are no constraints (as in [172]) on the parameters, the inequality r > ρ+1 may be violated, and the problem (4.23) may be unsolvable. Following Serova˘iski˘i [227], to construct ε-suboptimal solutions for this problem, we will use the penalty method. In view of this, we introduce the following auxiliary cost functional: Iε (u, y) = I(u, y) +

εγ 1 ybLb (Q) + A(u, y)qLq (Q) , b εq

(4.24)

where ε > 0 is the regularization parameter, γ = 0 for r > ρ + 1, γ = 1 for r ≤ ρ + 1, b = ρ + 1 + α and α > 0. Let us note that if the problem (4.23) is solvable, then we have the standard penalty method (see [172]), whereas for r ≤ ρ + 1 the cost functional involves one more summand which guarantees the availability of an additional a priori estimate and tends to 0 as ε → 0. We begin with the following result. Lemma 4.11. The approximated problem Minimize the functional Iε on the set U∂ × Y

(4.25)

is solvable for every ε > 0. Proof. Since the cost functional Iε is bounded below on the set U∂ × Y, it ∞ follows that for every ε > 0, there is a sequence {(uk , yk )}k=1 in U∂ × Y such that the condition lim Iε (uk , yk ) =

k→∞

inf

(u,y)∈U∂ ×Y

Iε (u, y) ∞

holds true. Then, as follows from (4.24), the sequence {yk }k=1 is bounded in ∞ q Lc (Q) and the sequences {uk }∞ k=1 and {gk }k=1 are bounded in L (Q), where

r for r > ρ + 1, ρ gk = y˙ k − yk − |yk | yk − uk , c = b for r ≤ ρ + 1. ∞

From this we can deduce that the sequence {fk = gk + |yk |ρ yk + uk }k=1 is bounded in Ld (Q), where d = min{q, c/(ρ + 1)}. As a result, it turns out that each of the functions yk (t, x) is a solution to the problem y˙ k − Δyk = fk yk |t=0 = y

0

on Ω,

in Q = (0, T ) × Ω, yk = 0

on Σ = (0, T ) × ∂Ω.

(4.26) (4.27)

126

4 Suboptimal and Approximate Solutions to Extremal Problems ∞

Hence, due to Theorem 3.46, the sequence {yk }k=1 is uniformly bounded in W (1,2),d (Q). So, passing to subsequences, we may suppose that uk u in Lq (Q) and yk y in W (1,2), d (Q) ∩ Lc (Q). Repeating the arguments of the proof of Theorem 3.46, it is easy to derive the relation lim Iε (uk , yk ) ≥ Iε (u, y).

k→∞

Hence, the limit pair (u, y) is a solution to the problem (4.25). The proof is complete. Since the approximated problem (4.25) has a solution (uε , yε ) ∈ U∂ × Y for every ε > 0, the next step of our consideration is to consider the asymptotic behavior of the sequence of such solutions {(uε , yε )}ε>0 as ε tends to 0. Having supposed that the small parameter ε discretely diminishes in the sequence {ε > 0}, we can state the following result: Theorem 4.12. Assume that Ξ = ∅. Then for every δ > 0, there exists εδ > 0 such that an optimal pair (uεδ , yεδ ) to the corresponding problem (4.25) (with ε = εδ ) is δ-suboptimal to the original OCP (3.167)–(3.170). Proof. By our assumptions, the set of admissible pairs Ξ to the problem (3.167)–(3.170) is nonempty. Since the cost functional I is bounded below on Ξ, it has the greatest lower bound inf (u,y)∈Ξ I(u, y) on the set Ξ. Then ∞ we can find a minimizing sequence {(uk , yk )}k=1 in Ξ such that I(uk , yk ) → inf (u,y)∈Ξ I(u, y). Having a minimizing sequence, we can take an approximate ∞ solution to the problem as an element of the sequence {(uk , yk )}k=1 with a sufficiently large index. Thereby, for every δ > 0, we can define a pair (uδ , y δ ) ∈ Ξ, called an approximate solution to the problem, such that the inequality I(uδ , y δ ) < inf (u,y)∈Ξ I(u, y) + δ is valid. Since Ξ ⊂ U∂ × Y and the pair (uε , yε ) is optimal for the problem (4.23), we come to the relation Iε (uε , yε ) =

min

(u,y)∈U∂ ×Y

Iε (u, y) ≤ Iε (uδ , yδ )

εγ δ b y Lb (Q) b εγ δ b ≤ inf I(u, y) + δ + y Lb (Q) . (u,y)∈Ξ b = I(uδ , yδ ) +

(4.28)

Taking into account the structure of the functional Iε (see (4.24)) and passing to the limit in (4.28) as ε → 0, we get lim inf I(uε , yε ) ≤ lim Iε (uε , yε ) ≤ ε→0

ε→0

inf

I(u, y) + δ.

(4.29)

(u,y)∈Ξ

From this, we immediately deduce: For a given δ, there exist εδ > 0 and a constant C > 0 independent of ε and δ > 0 such that (uεδ , yεδ ) ∈ U∂ × Y, I(uεδ , yεδ ) − inf I(u, y) < δ, (u,y)∈Ξ

A(uεδ , yεδ )Lq (Q) ≤

1/q εδ C

< δ.


127

Thus, (uεδ , yεδ ) is the δ-suboptimal pair in the sense of Definition 4.5. As was noted earlier, the approximated problem (4.25) has a solution (uε , yε ) ∈ U∂ × Y for every ε > 0. To define these solutions, we derive the optimality system for the OCP (4.25). However, before we do so, we should emphasize that the necessary conditions for optimality in the approximated problem (4.25) will differ slightly in form from those in [172] for the case p = 2, q = 2 and r = 6. Using equivalent transformations, we can nevertheless rewrite them in the same form. Theorem 4.13. Let (uεδ , yεδ ) ∈ U∂ × Y be an optimal pair to the problem (4.25). Then there is a function pε ∈ Lq (Q), where 1/q + 1/q = 1, satisfying the relations T ν|uε |q−2 uε + pε (w − uε ) dx dt ≥ 0, ∀ w ∈ U∂ , (4.30) 0

Ω

p˙ ε + Δpε + (ρ + 1)|yε |ρ pε = −|yε − z∂ |r−2 (yε − z∂ ) − γε|yε |

yε

on Ω,

pε = 0

b−2

pε |t=T = 0

(4.31)

in Q = (0, T ) × Ω, on Σ = (0, T ) × ∂Ω.

(4.32)

where yε is a solution to the problem

y˙ ε − Δyε − |yε |ρ yε − uε = ε1/(q−1) |pε |q −1 pε yε |t=0 = y

0

on Ω,

yε = 0

on Σ.

in Q,

(4.33) (4.34)

Proof. Following [169] (see also Theorems 3.37 and 3.41) and taking into account the structure of the set of admissible pairs U∂ ×Y to the problem (4.25), we see that the necessary conditions for this problem take the form of the variational inequality in the first argument and the stationary state condition in the second argument, that is, Du Iε (uε , yε )[w − uε ] ≥ 0, ∀ w ∈ U∂ , Dy Iε (uε , yε ) = 0,

(4.35) (4.36)

where by Du Iε (uε , yε ) and Dy Iε (uε , yε ) we denote the corresponding Gâteaux derivatives of Iε calculated at the pair (uε , yε ). Naturally, the existence of these derivatives needs justification. As Lemma 4.11 indicates, the optimal state yε for the problem (4.25) turns out to be an element of the set Z = W (1,2), d (Q) ∩ Lc (Q). Therefore, by Dy Iε we mean the corresponding derivative over the subspace Z of Y. It means (see [11]) that, varying the functional in the second argument, we can choose the increment in Z. In view of the conditions (3.169), this increment must satisfy zero initial conditions, which agrees again with the definition of the derivative over the subspace Z0 of Z constituted by the functions vanishing

128


at t = 0. Thus, by Dy Iε in (4.36) we mean the partial Gˆ ateaux derivative over the subspace Z0 . To find the derivative of the cost functional Iε in the first argument, we note that the following equality is valid for arbitrary functions u and g in Lq (Q), y in Z and a number σ: Iε (u + σg, y) − Iε (u, y) ν T [|u + σg|q − |u|q ] dx dt = q 0 Ω T 1 + [|A(u + σg, y)|q − |A(u, y)|q ] dx dt qε 0 Ω T σ T q−2 |u| ug dx dt + |A(u, y)|q−2 A(u, y)g dx dt = σν ε 0 Ω 0 Ω + o(σ). As a result, we have

T

|u|q−2 ug dx dt

Du Iε (u, y)[g] = ν 0

1 + ε

Ω

T

|A(u, y)|q−2 A(u, y)g dx dt, 0

∀ g ∈ Lq (Q).

Ω

Inserting this value into (4.35), we just establish (4.30), where pε = ε−1 |A(uε , yε )|q−2 A(uε , yε ). Since the value of the operator A at a solution to the approximated problem (4.25) belongs to Lq (Q), we obtain pε ∈ Lq (Q), where 1/q + 1/q = 1. The preceding equality also implies the validity of (4.33). The boundary conditions (4.34) hold because the solution to the problem (4.25) satisfies (3.169). Similarly, we can deduce the validity of the following equality for arbitrary functions u in Lq (Q), y in Z, h in Z0 and a number σ: Iε (u, y + σh) − Iε (u, y) 1 T [|y + σh − z∂ |r − |y − z∂ |r ] dx dt = r 0 Ω εγ T + |y + σh|b − |y|b dx dt b 0 Ω T 1 + [|A(u, y + σh)|q − |A(u, y)|q ] dx dt qε 0 Ω T T |y − z∂ |r−2 (y − z∂ )h dx dt + σεγ |y|b−2 yh dx dt =σ 0

σ + ε

Ω T

0

Ω

|A(u, y)|q−2 A(u, y) h˙ − Δh − (ρ + 1)|y|ρ h

0

Ω

dx dt + o(σ).

4.4 Approximate solutions to distributed singular systems

129

From this, we obtain the relation Dy Iε (u, y)[h] T = |y − z∂ |r−2 (y − z∂ )h dx dt + εγ Ω

0

+

1 ε

T

0

0

T

|y|b−2 yh dx dt Ω

|A(u, y)|q−2 A(u, y) h˙ − Δh − (ρ + 1)|y|ρ h dx dt,

Ω

∀ h ∈ Z. Consequently, condition (4.36) takes the form

T 0

Ω

|yε − z∂ |r−2 (yε − z∂ ) + εγ|yε |b−2 yε h + pε h˙ − Δh − (ρ + 1)|yε |ρ h dx dt = 0,

∀ h ∈ Z.

As a result, we see that the function pε is a solution to the boundary value problem (4.31)–(4.32) in Lq (Q). Thus, for every fixed ε > 0, a solution (uε , yε ) to the problem (4.25) can be found by solving the relations (4.30)–(4.34). This concludes the proof.

4.4 Approximate solutions to distributed singular systems This section deals with the design of weak and weakened ε-approximate solutions to one class of OCPs for distributed singular systems. In many aspects, we follow Serova˘iski˘i (see [228, 229]). Let Ω be an open bounded domain in Rn with Lipschitz boundary ∂Ω. Let T > 0, ν > 0, r > 1, ρ > 0 and q > 1 be given values. Let U∂ be a convex closed subset of Lq (Ω) and let y0 ∈ H01 (Ω) and z∂ ∈ Lr (Q) be given functions. We consider the following OCP: ν 1 y − z∂ rLr (Ω) + uqLq (Ω) −→ inf, r q y˙ − Δy − |y|ρ y − u = 0 in Q = (0, T ) × Ω,

I(u, y) =

y|t=0 = y

0

on Ω,

y = 0 on Σ = (0, T ) × ∂Ω, u ∈ U∂ .

(4.37) (4.38) (4.39) (4.40)

By analogy with the previous section, we set U = Lq (Ω), Y = Lr (Q) ∩ y ∈ W (1,2), s (Q) : y|t=0 = y 0 , A(u, y) = y˙ − y − |y|ρ y − u, I(u, y) =

1 ν y − z∂ rLr (Ω) + uqLq (Ω) , r q

130


where s = min{q, r/(ρ + 1)} for r > ρ + 1 and s = q for r < ρ + 1. We define the set of admissible pairs as follows: Ξ = {(u, y) ∈ U × Y : A(u, y) = 0, u ∈ U∂ } . Let us recall that (see Theorem 3.46) the problem (4.37)–(4.40) is solvable in U × Y if Ξ = ∅ and r > ρ + 1. In the general case, this question remains open. However, as we will see later, this problem has weakened ε-approximate solutions, which in the case r > ρ + 1 become weak approximate solutions. We turn our attention to the approximate problem Iε (u, y) = I(u, y) +

εγ 1 ybLb (Q) + A(u, y)qLq (Q) → inf , b εq (u,y)∈U∂ ×Y

(4.41)

where the regularization parameter ε > 0 takes its values in a sequence which tends to 0, γ = 0 for r > ρ + 1, γ = 1 for r ≤ ρ + 1, b = ρ + 1 + α and α > 0. Let us point out that this problem has a solution (uε , yε ) ∈ U∂ × Y for every ε > 0 (see Lemma 4.11). Let {(uε , yε )}ε>0 be a sequence of such solutions. We begin with the following result: Theorem 4.14. Assume that Ξ = ∅. Then the sequence {(uε , yε )}ε>0 is a weakened minimizing sequence (in the sense of Definition 4.4) for the problem (4.37)–(4.40). ∞

Proof. Let {(uk , yk )}k=1 ⊂ Ξ be a minimizing sequence for the cost functional I on the set Ξ. Then for every δ > 0, we can define an approximate solution (uδ , y δ ) ∈ Ξ such that the inequality I(uδ , y δ )
0 (keeping the former notation for simplicity) such that yε y0 in Lr (Q),

uε u0 in Lq (Q), q

A(uε , yε ) 0 in L (Q).

(4.45) (4.46)

However, conditions (4.45) are equivalent to the relation (uε , yε ) (u0 , y0 ) in Lq (Q) × Lr (Q). Then, using convexity and continuity of the cost functional I in Lq (Q) × Lr (Q), we obtain lim I(uε , yε ) ≥ I(u0 , y0 ).

(4.47)

ε→0

From the definition of the regularized functional (see (4.41)), we deduce the inequality Iε (u, y) ≥ I(u, y) for all (u, y) ∈ U∂ × Y. Hence, Iε (uε , yε ) ≥ I(uε , yε ) and, therefore, lim Iε (uε , yε ) ≥ lim I(uε , yε ). ε→0

ε→0

Using (4.42) and (4.45), we then find that I(uε , yε ) ≤ lim I(uε , yε ) ≤ lim Iε (uε , yε ) ≤ ε→0

ε→0

inf

I(u, y).

(4.48)

(u,y)∈Ξ

Then, as follows from (4.45), (4.46) and (4.48), {(uε , yε )}ε>0 is a weakened minimizing sequence for the problem (4.37)–(4.40), and the proof is complete. Let us show that under the condition r > ρ + 1 (see Theorem 3.46), the sequence {(uε , yε )}ε>0 defined by the relations (4.30)-(4.34) is also a weak minimizing sequence. Theorem 4.15. If Ξ = ∅, then for r > ρ + 1 the sequence {(uε , yε )}ε>0 is a weak minimizing sequence for the problem (4.37)–(4.40). Proof. First, we note that, as follows from (4.44), for every ε > 0 the function yε is a solution to the boundary value problem y˙ε − Δyε = |yε |ρ yε + uε + fε yε |t=0 = y

0

on Ω,

in Q = (0, T ) × Ω,

yε = 0 on Σ = (0, T ) × ∂Ω,

(4.49) (4.50)

where the function fε is such that fε 0 in Lq (Q).

(4.51)

132


When the inequality r > ρ + 1 holds true, by (4.43) and (4.51), the right-hand side of (4.49) is bounded in Ls (Q) (see the proof of Theorem 3.46). Then the sequence {yε }ε>0 is bounded in W (1,2), s (Q) and hence in Y. Thus, the following convergence holds: yε y0 in Y.

(4.52)

Repeating the arguments of the proof of Theorem 3.46, we establish that |yε |ρ yε |y0 |ρ y0 in La (Q), where a = r/(ρ + 1).

(4.53)

Passing to the limit in (4.49) and using (4.43) and (4.51)–(4.53), we arrive at the relation y˙0 = y0 − |y0 |ρ y0 − u0 in Q. Since the embedding of Y into C(0, T ; L2 (Ω)) is continuous (see [113, Theorem 1.17]), using (4.53) we obtain the convergence yε |t=0 y0 |t=0 in L2 (Ω). Passing to the limit in (4.50), we conclude that the function y0 satisfies the boundary conditions y0 |t=0 = y 0

on Ω,

y0 = 0 on Σ.

By convexity and closure properties of U∂ , we have u0 ∈ U∂ . From the above results, we conclude that the pair (u0 , y0 ) is admissible for the problem (4.37)– (4.40). Thereby the equality A(u0 , y0 ) = 0 is valid. Together with (4.43) and (4.44), this means continuity of the operator A : Lq (Q) × Lr (Q) → W at (u0 , y0 ), where W denotes the space Lq (Q) furnished with the weak topology. As mentioned earlier, the continuity of A at the pair (u0 , y0 ) guarantees that {(uε , yε )}ε>0 is a weak minimizing sequence for the problem (4.37)– (4.40). Indeed, in the proof of Theorem 4.14, we established relation (4.48). Hence, since (u0 , y0 ) ∈ Ξ, it follows that (u0 , y0 ), as the limit of the sequence {(uε , yε )}ε>0 , is a solution to the problem (4.37)–(4.40). Thus, we have shown that for sufficiently small values of the regularization parameter ε, the approximate solution (uε , yε ) satisfies the state equation with any accuracy and, in a sense, approximates the greatest lower bound of the cost functional I. Moreover, if the inequality r > ρ + 1 is valid, (uε , yε ) is also sufficiently close to a solution to the original problem (4.37)–(4.40) (in the topology of Lq (Q) × Lr (Q)).

5 Introduction to the Asymptotic Analysis of Optimal Control Problems: A Parade of Examples

The main interest in this book is in mathematical models of optimal control problems (OCPs) that depend on some small parameter ε. In many mathematical problems, which come from natural or engineering sciences, industrial applications, or abstract mathematical questions all by themselves, some parameters appear (small or large, of geometric or constitutive origin, coming from approximation processes or discretization arguments) and make those control problems increasingly complex or degenerate. For instance, if we deal with OCPs on reticulated structures or perforated domains, this small parameter comes from the geometry of such domains. This dependence can be defined both by a regular and a singular occurrence of the parameter ε (the presence of ε-periodic coefficients, ε-periodically perforated zone of controls, quickly oscillating boundaries, etc.). In order to proceed with a mathematical description, we formulate the following parameterized OCP (OCPε ), where ε is a small parameter: (OCPε ) :

min {Iε (u, y) : (u, y) ∈ Ξε } ,

(5.1)

where (B1 ) (B2 ) (B3 ) (B4 )

Iε : Uε × Yε → R is a cost functional (CFε ), Yε is a space of states, Uε is a space of controls, Ξε ⊂ {(uε , yε ) ∈ Uε × Yε : u ∈ Uε , Iε (u, y) < +∞} is a set of all admissible pairs linked by some state equation (SEε ).

Note that, as the small parameter ε varies, it is sometimes possible to foresee some “limiting” behavior and “guess” which effective model we may substitute for the complex model. In view of this, the main questions we are going to answer are: (i) What is the behavior of the OCP under the variation of the parameter ε? (ii) How does one pass to the limit in such problems, as ε tends to 0? (iii) What should be expected as a limit problem? P.I. Kogut, G.R. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains, Systems & Control: Foundations & Applications, DOI 10.1007/978-0-8176-8149-4 5, © Springer Science+Business Media, LLC 2011

133

134

5 Introduction to the Asymptotic Analysis of OCPs

(iv) Can the limit problems be classified? (v) Is there a characterization of the variational properties of the limiting problem? It is reasonable to consider optimal solutions to the “limit” problem as approximate solutions for the original problem. We begin our consideration with the OCP that can be described as follows: ⎧ ⎫ (CFε ) : Iε (u, y) → inf ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ subject to (OCPε ) : . (5.2) : (u, y) ∈ Uε × Yε , u ∈ Uε ⊂ Uε ⎪ (CSCε ) ⎪ ⎪ ⎪ ⎩ ⎭ (SEε ) : Lε (u, y) + Fε (y) = 0 In what follows, we always suppose that all ingredients of the mathematical model (5.2) satisfy the assumptions of the previous section, including hypotheses (H1)–(H3). Our prime interest in this chapter is to discuss different ways and procedures for asymptotic analysis of a wide class of OCPs as the parameter ε tends to 0. We note that without going into details of the definition of “asymptotic behavior” of OCPs, independently of what kind of “limit passage” in (OCPε ) we take, this procedure has to guarantee the following variational properties: (V1) If (u0ε , yε0 ) is an optimal pair of (OCPε ) and if (u0ε , yε0 ) tends (in some sense) to (u0 , y 0 ), then (u0 , y0 ) is an optimal solution for the limit problem. (V2) limε→0 inf (u,y)∈ Ξε Iε (u, y) = inf (u,y)∈ Ξ0 I0 (u, y). To give a precise meaning to this statement, we begin with the discussion of the different examples and procedures for the definition of an appropriate limit problem (OCP0 ) to the family (5.2).

5.1 Component-by-component limit analysis As follows from the description (5.2), the typical OCP consists of the independent mathematical objects such as a state equation (SEε ), control and state constraints (CSCε ) and a cost functional (CFε ). Therefore, the scheme of component-by-component limit analysis is based on the passage to the limit in each of the ingredients of the mathematical model (5.2) separately as ε → 0. For this, the well-known concepts of the asymptotic analysis such as epi-convergence or Γ -convergence of functionals, G-convergence of differential operators and Painlevé–Kuratowski convergence of sets and spaces can be applied. In this case, the structure of a limit OCP is well given by ⎧ ⎫ : I0 (u, y) → inf (CF0 ) ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ subject to . (5.3) (OCP0 ) : (CSC0 ) : (u, y) ∈ U × Y, u ∈ U0 ⊂ U ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ : L0 (u, y) + F0 (y) = 0 (SE0 )

5.1 Component-by-component limit analysis

135

In spite of the apparent simplicity of this approach, it has rather limited application (see Akulenko [3], Lions [170] and Kokotovic [154]). The main reason is the fact that the limiting problem (5.3) does not preserve, in general, the variational properties (V1)–(V2). As an illustrative example, let us consider the following OCP governed by a nonlinear PDE: Example 5.1. Let Ω be a bounded open subset of Rn (n < 12) with a boundary ∂Ω ∈ C ∞ and let us consider the following OCP in Ω (see Haraux and Murat [117]): |y(x) − zd (x)|6 dx + N u2 (x) dx −→ inf, (5.4) I(u, y) = Ω

Ω

−εΔy(x) − y 3 (x) = u(x), x ∈ Ω, y |∂Ω = 0,

2 u ∈ U∂ = u ∈ L (Ω) : λ ≤ u ≤ ν a.e. on Ω ,

(5.5) (5.6) (5.7)

where zd ∈ L6 (Ω) is a fixed function, ε > 0 is a small parameter, U∂ is a convex closed subset of L2 (Ω) and 0 < λ < ν. In order to give a sense of the problem (5.4)–(5.7), we make the folassumptions: For every ε > 0, there exists a pair (uε , yε ) ∈ U∂ × lowing L6 (Ω) ∩ H 2 (Ω) ∩ H01 (Ω) satisfying the conditions (5.5)–(5.7). To fulfill the conditions of Theorem 3.3, we put Y = L6 (Ω), U = L2 (Ω),

Y1 = L6 (Ω) ∩ H01 (Ω) ∩ H 2 (Ω), Z = L2 (Ω),

F (y) = −y , 3

L(u, y) = −Δy − u,

Y−1 = L3 (Ω),

and make use the compactness property of the embedding H 2 (Ω) → L3 (Ω) (n < 12). Since the functional y → F (y), ϕ H 1 (Ω) = − y 3 (x)ϕ(x) dx 0

Ω

and C0∞ (Ω) is dense in H01 (Ω), is continuous on L (Ω) for every ϕ ∈ it follows that, by Theorem 3.3, the OCP (5.4)–(5.7) admits at least one solution. Now, we deal with the question of the asymptotic behavior of this problem as ε tends to 0. Taking into account the procedure of component-bycomponent limit analysis, we pass to the limit in the state equation (5.5) as ε → 0. Let τ be the product of the weak topology for L6 (Ω) and the strong topology for L2 (Ω). Let {(uε , yε )}ε→0 be any τ -convergent sequence such that 3

C0∞ (Ω),

−εΔyε (x) = uε (x) + yε3 (x) in Ω,

yε |∂Ω = 0

for every ε > 0. Multiplying this relation by yε and integrating by parts, we obtain

136


|∇yε |2 dx = (uε , yε )L2 (Ω) + yε 4L4 (Ω) .

ε

(5.8)

Ω

Since (uε , yε )L2 (Ω) → (u, y)L2 (Ω) (as the product of weakly and strongly convergent sequences), it follows that there exists a constant C > 0 such that √ ε Ω |∇yε |2 dx ≤ C. Hence, the sequence { ε ∇yε } is bounded in (L2 (Ω))n . Then, passing to the limit in the integral identity √ √ ε ( ε ∇yε , ϕ) dx − (yε3 , ϕ)L2 (Ω) = (uε , ϕ)L2 (Ω) , ∀ ϕ ∈ H01 (Ω), Ω

we get (y 3 + u, ϕ)L2 (Ω) = 0. Since H01 (Ω) is a dense subset of L2 (Ω), we conclude that u = −y 3 a.e. on Ω. As a result, we obtain the following limit OCP 6 I0 (u, y) = |y(x) − zd (x)| dx + N u2 (x) dx −→ inf, (5.9) Ω

Ω

−y 3 (x) = u(x), x ∈ Ω, y |∂Ω = 0

2 u ∈ U∂ = u ∈ L (Ω) λ ≤ u ≤ ν a.e. on Ω .

(5.10) (5.11) (5.12)

However, this problem does not preserve the main variational properties (V1)– (V2). Indeed, let us substitute in (5.4)–(5.7) the following data: ε = 1/n2 , Ω = (0, T ), 0 ≤ N < +∞, 0 < λ ≤ ν ≤ +∞, and zd = −C, where −1/3 5/6

1

2

T =λ

0

dx √ , 1 − t4

1/3 λ C≥3 , 2

and define a T -periodic function ϕ ∈ H01 (0, T ) as follows: ϕ + ϕ3 = λ,

ϕ(0) = ϕ(T ) = 0.

√ Then, as was shown in [117], the pair (u0ε , yε0 ) = (λ, ϕ (x/ ε)) is an optimal one for the problem (5.4)–(5.7). Hence, for every ε = n−2 , we have

T

inf I(u, y) = I(u0ε , yε0 ) =

0 T

=

6 ϕ √x dx + N λ2 T ε T 6 2 ϕ (nx) dx + N λ T = ϕ6 (x) dx + N λ2 T.

0

As a result,

0

T

lim inf I(u0ε , yε0 ) = ε→0

ϕ6 (x) dx + N λ2 T. 0

On the other hand, the limit problem (5.9)–(5.12) is solvable as well. Moreover, (u0 , y 0 ) = (λ, λ1/3 ) is an optimal pair for (5.9)–(5.12). Finally, it is easy to see that


137

inf I0 (u, y) = I0 (λ, λ1/3 ) = T |λ1/3 + C|6 + N λ2 T. However,

T

6 ϕ6 (x) dx < T λ1/3 + C ,

0

and we obtain the required result. Let us represent the OCP (5.2) in the form of a constrained minimization problem, namely (OCPε ) : inf Iε (u, y) , (5.13) (u,y)∈ Ξε

where the set of admissible solutions Ξε is defined as (SASε ) : Ξε = {(u, y) ∈ Uε × Yε , Uε ⊂ Uε , Lε (u, y) + Fε (y) = 0} . (5.14) It is clear that the set of OCPs (OCPε ) correlates well with the set of pairs Iε , Ξε . However, we cannot assert that there is a one-to-one correspondence between these sets. Actually, this would be true only when each of the pairs Iε , Ξε could be interpreted as some optimal control problem (i.e., when the sets Ξε have the representation Ξε = {(u, y) : (SEε ) ∩ (CSCε )}). However, this is by no means always the case. Thus, we come to the following scheme of a limit analysis that can be viewed as one more variant of the procedure of component-by-component passage to the limit (see Muthukumar [197]): 1. Taking into account the concepts of Γ -convergence of functionals and Painlevé–Kuratowski convergence of sets, find a limit of the sequence { Iε , Ξε ; ε → 0} – namely Lim Iε , Ξε = I0 , Ξ0 , where ε→0

Ξ0 = K− Lim Ξε ε→0

and

I0 (u, y) = Γ − Lim inf Iε (u, y). ε→0

2. If the set Ξ0 can be represented in the form Ξ0 = {(u, y) : (u, y) ∈ (SE0 ) ∩ (CSC0 )} , then the OCP (5.2) should be taken as a limit to (5.2). As in the previous case, let us show that this scheme does not guarantee the fulfillment of the main variational properties for the limit problems. For this, we consider a particular model OCP in the 1D periodic media. Example 5.2. Let β > α > 0, γ > 0, and c and d (c < d) be given values and f ∈ L2 (c, d). Let a ∈ L∞ (c, d) be a positive, 1-periodic function such that 0 < α ≤ a(x) ≤ β < +∞ almost everywhere in Ω = (c, d). In Ω, we consider the following OCP

138


d d 2 I(u, y) = [y (x)] dx + u2 (x) dx −→ inf, c c x y = u(x) + f (x), x ∈ (c, d), y(c) = y(d) = 0, − a ε

u ∈ U∂ = u ∈ L2 (c, d) : u L2 (c,d) ≤ γ ,

(5.15) (5.16) (5.17)

where ε = 1/n is a small parameter and u ∈ L2 (c, d) is a control. For each ε, this problem has a unique solution. We denote it by (u0ε , yε0 ) ∈ L2 (c, d) × H01 (c, d). So, I(u0ε , yε0 ) =

inf

I(u, y),

(u,y)∈Ξε

where Ξε is the set of admissible solutions for the original problem, that is, ⎧ ⎫ − a x y = u(x) + f (x), x ∈ (c, d), ⎬ ⎨ ε . (5.18) Ξε = (u, y) y ∈ H01 (c, d), u ∈ L2 (c, d), ⎩ ⎭ u L2 (c,d) ≤ γ Let τ be the product of the weak topologies for L2 (c, d) and H01 (c, d), respectively. In accordance with the procedure of limit analysis given above, we begin with the identification of the Painlevé–Kuratowski limit of the sequence {Ξε ; ε → 0} with respect to the τ -topology of L2 (c, d) × H01 (c, d). As immediately follows from (5.16), the estimate (5.19) yε H01 (c,d) ≤ α−1 uε L2 (c,d) + f L2 (c,d) is valid for every ε > 0. Hence, the sets {Ξε ; ε → 0} are uniformly bounded. Indeed, in this case, we have y H01 (c,d) + u L2 (c,d) ≤ γ(α−1 + 1) + α−1 f L2 (c,d) . (5.20) sup sup ε>0 (u,y)∈Ξε

Since the space H01 (c, d) is separable and reflexive, there exists a metric ρ on H01 (c, d) such that the τ -topology for ∪ε>0 Ξε ⊂ L2 (c, d) × H01 (c, d) is associated to the metric ρ on ∪ε>0 Ξε . Hence, by the property of Painlevé– Kuratowski convergence (see Kuratowski [161] and Dal Maso [78]), we can use its sequential version for the sets {Ξε ; ε → 0} (see also Definition 7.21). Let {(uε , yε ) ∈ Ξε } be any sequence of admissible pairs. Let us define x y . νε = a ε ε ε>0 From the initial supposition and the estimates (5.19)–(5.20), it follows that there exists a subsequence, still denoted by ε, such that yε y∗ in H01 (c, d), uε u∗ in L2 (c, d), νε ν in H 1 (c, d), νε → ν in L2 (c, d), 1 ∗ a(·/ε) a∗ = a(ξ) dξ in L∞ (c, d). 0

(5.21)


139

Using the continuity of the trace operator, we have y∗ (c) = y∗ (d) = 0. Now, passing to the limit as ε → 0 in the integral identity

d

a

x ε

c

yε ϕ

d

dx =

(uε + f )ϕ dx,

ϕ ∈ H01 (c, d),

c

we get

d

νϕ dx =

d

(u∗ + f )ϕ dx.

c

(5.22)

c

To establish the relation between y∗ and ν, we note that νε is the product of two weakly converging quantities. In general, this does not imply that the limit ν is the product of the limits a∗ and dy∗ /dx. To identify it, we recall that x (5.23) yε = a−1 νε , ε where a−1 (·/ε) ∈ L∞ (0, 1), a−1 (·/ε) is the 1-periodic function, 1 < 1/α ≤ 1/a(·/ε) ≤ 1/β < +∞ and, hence, −1

a

(·/ε) a−1 ≡ ∗

1

a−1 (ξ) dξ

in L∞ (c, d).

0

Since the term a−1 (·/ε)νε is the product of the weakly convergent quantity a−1 (·/ε) and the strongly convergent one νε , its limit is the product of the limits of a−1 (·/ε) and νε . Thus, a−1 (·/ε)νε a−1 ν in L2 (c, d). Making use of (5.22), we come to the following “homogenized” state equation −1 −1 a

d c

y∗ ϕ

d

dx =

(u∗ + f )ϕ dx,

∀ ϕ ∈ H01 (c, d).

c

Note that for each fixed u∗ ∈ L2 (c, d), this problem has a unique solution y∗ ∈ H01 (c, d). Thus, taking into account the fact that uε L2 (c,d) ≤ γ for every ε > 0 and the norm · L2 (c,d) is lower semicontinuous with respect to the weak convergence in L2 (c, d), we have γ ≥ lim uε L2 (c,d) ≥ lim uε = u∗ L2 (c,d) . ε→0

τ

ε→0

L2 (c,d)

Hence, (uε , yε ) → (u∗ , y∗ ), where the τ -limit pair (u∗ , y∗ ) belongs to the set ⎧ ⎫ −1 −1 ⎪ ⎪ − a y = u(x) + f (x), x ∈ (c, d), ⎪ ⎪ ⎨ ⎬ 1 2 Ξ0 = (u, y) . (5.24) y ∈ H0 (c, d), u ∈ L (c, d), ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ u L2 (c,d) ≤ γ

140


Let us consider the opposite situation. Let (v, y) be any pair from the set uε , yε )}ε→0 be a sequence such that u ε = v for every ε > 0 Ξ0 and let {( and the functions yε are the corresponding solutions of the problem (5.16) with u = v. Then, in a similar manner as earlier, it is easy to prove that τ uε , yε ) ∈ Ξε for every ε > 0. Hence, the set Ξ0 is ( uε , yε ) → (v, y) and ( the sequential τ -limit of {Ξε }ε>0 in the Painlevé–Kuratowski sense (shortly, K(τ )-limit) [161] (see also Sect. 7.4). Thus, in accordance with the above scheme, the limit OCP can be written as d d 2 [y (x)] dx + u2 (x) dx −→ inf, (5.25) I(u, y) =

− a

−1 −1

c

c

y = u(x) + f (x), x ∈ (c, d), y(c) = y(d) = 0,

u ∈ U∂ = u ∈ L2 (c, d) u L2 (c,d) ≤ γ .

(5.26) (5.27)

However, for this kind of “homogenized” problem, the variational properties (V1)–(V2) do not hold. Indeed, let {(uε , yε ) ∈ Ξε } be any τ -convergent sequence of admissible pairs and let (u∗ , y∗ ) be its τ -limit. Then, by the property of K-limits (see Proposition 7.23), the pair (u∗ , y∗ ) belongs to the set Ξ0 . Taking into account (5.23) and the properties (5.21), one has −1 y∗ in H 1 (c, d), a (x/ε) yε = νε ν = a−1 νε2 → ν 2 ∗ a−2 (·/ε) a−2 =

in L1 (c, d), 1

a−2 (ξ) dξ

in L∞ (c, d).

0

Then, passing to the limit in the sequence {I(uε , yε )}ε>0 as ε tends to 0, we obtain d d 1 x 2 a y dx + lim inf u2ε dx lim inf I(uε , yε ) = lim inf 2 x ε→0 ε→0 ε→0 ε ε c a (ε) c ! d d 1 −2 2 a (ξ) dξ ν dx + lim inf u2ε dx = 0 1

≥

a−2 (ξ) dξ

0

= + c

!

d

1

−2

! (ξ) dξ

0 d

u2∗ dx.

1

a 0

c

−1 2 y∗ dx + a−1

c

a

ε→0

c

−1

d

c

!−2

d

(ξ) dξ c

u2∗ dx

(y∗ ) dx 2

(5.28)

On the other hand, for any pair ( u, y) ∈ Ξ0 , we can consider a sequence ε = u for every ε > 0 and yε are the solutions {( uε , yε )}ε→0 as follows: u


141

of (5.16) provided u = u . Then, by analogy with the previous case, it can τ uε , yε ) → ( u, y). be easily proved that ( uε , yε ) ∈ Ξε for all ε > 0 and ( Consequently, d 1 x 2 a y dx + u 2 dx 2 x ε ε ε→0 c a (ε) c ! d 1 d −2 2 = a (ξ) dξ ν dx + u 2 dx

d

lim sup I( uε , yε ) = lim sup ε→0

0

c

1

=

a−2 (ξ) dξ

0

a

−2

!

−1 2 y dx + a−1

1

(ξ) dξ

a

0 d

c

d

c

1

=

!

−1

!−2 (ξ) dξ

0

d

u 2 dx c

d

( y ) dx

c

u 2 dx.

+

2

(5.29)

c

Introducing the functional

d

u 2 dx

Ihom (u, y) = c

1

a

+

−2

! (ξ) dξ

0

1

a

−1

!−2 (ξ) dξ

0

d

(y ) dx, 2

(5.30)

c

we can rewrite the relations (5.28) and (5.29) as lim inf I(uε , yε ) ≥ Ihom (u∗ , y∗ ),

(5.31)

lim sup I( uε , yε ) ≤ Ihom ( u, y).

(5.32)

ε→0

ε→0

The functional Ihom : Ξ0 → R with the properties (5.31) and (5.32) is called in [143] (see also [140], [141, 142, 144, 147]) the S-limit of the sequence {I : Ξε → R}ε>0 . It does not coincide in general with the Γ -closure of the original cost functional I : L2 (c, d) × H01 (c, d) → R. However, it can be easily on Ξ0 shown that the functional Ihom : Ξ0 → R coincides with the restriction of the Γ -limit of the sequence Iε : L2 (c, d) × H01 (c, d) → R ε>0 (see [147]), where ⎧ d d ⎨ 2 [y (x)] dx + u2 (x) dx if (u, y) ∈ Ξε , Iε (u, y) = c c ⎩ +∞ if (u, y) ∈ Ξε . Let us prove that the limit functional (5.30) possesses the fine variational properties (V1) and (V2). To begin, we note that the OCP (5.26)–(5.27), (5.30) admits a unique solution (u0 , y 0 ).

142


Theorem 5.3. Let (u0ε , yε0 ) be an optimal pair to the original problem (5.15)– (5.17). Then τ

inf (u, y)∈ Ξ0

(u0ε , yε0 ) −→ (u0 , y 0 ), Ihom (u, y) = Ihom u0 , y 0 = lim I(u0ε , yε0 ). ε→0

(5.33) (5.34)

Proof. In view of the estimate (5.20), the sequence of optimal pairs (u0ε ,yε0 ) is uniformly bounded. Hence, we may extract a subsequence (u0εk , yε0k ) k∈ N τ -converging in L2 (c, d) × H01 (c, d) to some pair (u∗ , y ∗ ). By the arguments given above, the pair belongs to the set Ξ0 . Moreover, taking (5.28) into account, we have lim inf

min

k→∞ (u,y)∈ Ξεk

I(u, y) = lim inf I(u0εk , yε0k ) ≥ Ihom (u∗ , y∗ ) k→∞

≥

min (u, y)∈ Ξ0

I0 (u, y) = Ihom (u0 , y 0 ).

(5.35)

On the other hand, since (u∗ , y∗ ) ∈ Ξ0 , it follows that there exists a sequence {(uε , yε )} τ -converging to (u0 , y 0 ) such that (uε , yε ) ∈ Ξε for every ε > 0 and Ihom (u0 , y 0 ) ≥ lim supε→0 I(uε , yε ) (see (5.29)). Consequently, min (u, y )∈Ξ0

Ihom (u, y) = Ihom (u0 , y 0 ) ≥ lim sup I(uε , yε ) ε→0

≥ lim sup ε→0

min

(u,y) ∈Ξε

I(u, y) ≥ lim sup

min

k→∞ (u,y) ∈ Ξεk

I(u, y)

= lim sup Iεk (u0εk , y0εk ).

(5.36)

k→∞

However, by (5.35), we get lim inf I(u0εk , yε0k ) ≥ lim sup I(u0εk , yε0k ). k→∞

k→∞

As a result, combining (5.35) and (5.36), we come to the following conclusion: Ihom (u∗ , y ∗ ) = Ihom (u0 , y 0 ) = Ihom (u0 , y 0 ) = lim

min (u, y)∈ Ξ0

min

k →∞ (u,y) ∈ Ξεk

Ihom (u, y),

Iεk (u, y).

Using these relations and the fact that the OCP (5.26)–(5.27) and (5.30) admits a unique solution (u0 , y 0 ), we obtain (u∗ , y ∗ ) = (u0 , y0 ). Since this equality holds for τ -limits of all subsequences (u0εk , yε0k ) k∈ N of the sequence

0 0 (uε , yε ) ε>0 , it follows that these limits coincide and therefore (u0 , y0 ) is

the τ -limit of the original sequence (u0ε , yε0 ) ε>0 . Therefore, using the same argument for the sequence of minimizers as for the subsequence, we have

5.2 Limit analysis of optimality conditions

lim inf ε →0

min (u,y)∈ Ξε

143

I(u, y) = lim inf I(u0ε , yε0 ) ε →0

≥ Ihom (u0 , y 0 ) =

min

Ihom (u, y)

(u, y)∈ Ξ0

≥ lim sup I(uε , yε ) ≥ lim sup ε→0

ε→0

min

(u,y) ∈Ξε

I(u, y)

= lim sup I(u0ε , yε0 ). ε →0

Thus, we obtain the required result. The proof is complete. The question is now how to compare the original cost functional I with the limit functional Ihom . First, we note that each of them has a sense only on the set of admissible pairs Ξ0 . Let (" u0 , y"0 ) ∈ Ξ0 be an optimal pair to the problem (5.25)–(5.27). The existence of this pair is an obvious fact. Moreover, in general, we have (" u0 , y"0 ) = (u0 , y 0 ). Then u0 , y"0 ), I(" u0 , y"0 ) ≤ Ihom ("

I(u0 , y 0 ) ≤ Ihom (u0 , y 0 ).

The correctness of this conclusion immediately follows from the following inequality

1

1=

−1

a 0

≤

0

1

!2

1

(ξ) dξ

1 dx

a

−1

!−2 (ξ) dξ

0 1

−2

a

!

1

(ξ) dξ

0

−1

a

!−2 (ξ) dξ

.

0

Consequently, taking into account the structure of I, it leads us to the inequality I(u, y) ≤ Ihom (u, y) for every (u, y) ∈ Ξ0 . Moreover, if the function a ∈ L∞ (0, 1) is such that a(ξ) = a2 (ξ) and a(ξ) = const almost everywhere on the interval (0, 1), then I(" u0 , y"0 ) ≤ I(u0 , y 0 ) < Ihom (u0 , y0 ) =

min (u, y)∈ Ξ0

Ihom (u, y).

Thus, the variational properties (V1)–(V2) are lost if we take the limit OCP for the family (5.15)–(5.17) in the form (5.25)–(5.27).

5.2 Limit analysis of optimality conditions An asymptotic analysis of OCPs for the systems with distributed parameters has been considered in [10, 48, 61, 64, 117, 118, 220]. As follows from pioneering works in this field and more recent results, the most typical approach to asymptotic analysis of OCPs consists in the application of the homogenization theory of boundary value problems for investigation of the limiting behavior of necessary optimality conditions and the corresponding optimal solutions. Thus, the algorithm of asymptotic analysis consists of the

144


following steps: First, write down the necessary optimality conditions for the original problem; further, look for the corresponding limiting relations and interpret them as necessary optimality conditions for some optimal control problem; finally, using the limiting necessary optimality conditions, recover an OCP which is called the limiting control problem to the initial one (see, e.g., [21, 22, 71, 88, 129, 130, 134, 135, 136, 137, 197, 222, 268]). Thus, if we denote by OCP ε , NOC ε , HOCP, and HNOC the original OCP on the ε-level, the corresponding necessary optimality conditions on the ε-level, the limit OCP and the limit necessary optimality system, respectively, then the above-mentioned algorithm can be represented in the form of the following diagram: ?

OCP ε =⇒ HOCP ↓ ↑

(5.37)

ε→0

NOC ε −→ HNOC. However, it is possible that this diagram is not commutative. Moreover, it should be stressed that such approach is suitable only for simple enough (from the control theory point of view) OCPs in which there are no control and state constraints. Attempts to transfer this approach to a wider class of OCPs was realized in [71, 72], where it was shown that the construction of the limit OCP is possible only under restrictive assumptions. Moreover, the diagram (5.37) does not imply, in general, the main variational properties (V1) and (V2) for the limit problems. So, the verification of these properties is an independent problem in each concrete case (see [71, 135, 197]). Along with this, necessary optimality conditions for OCPs with constraints, as a rule, contain variational inequalities or equations in measure terms that complicate the procedure of limit analysis [98, 126, 169, 171, 250]. On the other hand, the optimal pair (i.e., the optimal control and the corresponding state of system) cannot completely characterize the entire problem. As a rule, the full identification of an OCP (including a cost functional, a state equation and existing restrictions on the control functions and state functions) is impossible by means of the optimal solution only. Moreover, the diagram (5.37) is unacceptable when the existence of solutions to the problem (5.2) may fail [38, 48] or when these solutions have nonclassical character (solutions satisfying only the requirements of Tonelli’s partial regularity theorem and having no “variational character”) [17, 235]. Indeed, as is well known, the necessary conditions of optimality can be derived by the abstract Lagrange multiplier principle [111, 169] or the Pontryagin maximum principle [4]. In particular, the so-called direct analogue of Pontryagin maximum principle [219] leads us (under some natural assumptions for the problem (5.2)) to the following relation (see [4]): Iε (u0ε , yε0 ) − Ψε , Lε (u0ε , yε0 ) + Fε (yε0 ) Z∗ , Z ≤ Iε (uε , yε0 ) − Ψε , Lε (uε , yε0 ) + Fε (yε0 ) Z∗ , Z , ∀ uε ∈ Uε , (5.38)


145

where (u0ε , yε0 ) is an optimal pair to the problem (5.2) and the Lagrange multiplier Ψε satisfies some adjoint equation. Let us show that even in the 1D case, this principle may fail. Example 5.4. Let Ω = {x ∈ R : −1 < x < 1}, let a ∈ R be a given positive value and let U∂ be a set of 1-periodic function on R such that U∂ = {u ∈ L∞ (0, 1) : u(x) = a or u(x) = 1/a a.e. on (0, 1)} . In Ω, we consider the following OCP 2 I(u, y) = [yx − z∂ ] dx −→ inf,

(5.39)

(5.40)

Ω

d x u yx − z∂ = 0, dx ε

x ∈ Ω,

y(−1) = y(1) = 0,

u ∈ U∂ .

(5.41) (5.42)

Here, ε = 1/n is a small parameter, u(·/ε) ∈ L2 (Ω) is an ε-periodic control influence and the function z∂ is given by # x − γ if x < 0, z∂ = z∂ (x) = x + γ if x ≥ 0. We always suppose that the constants a and γ satisfy the conditions: 0 < a < 1, γ > 0. To begin, we prove that the OCP (5.40)–(5.42) has a solution. For this, we observe that the state equation (5.41) has a unique solution yε ∈ H01 (Ω) for every admissible control u ∈ U∂ , and yε can be defined as follows x , (5.43) (yε )x (x) = (z∂ (x) − c0 ) u−1 ε where

c0 =

1

−1

u−1

x ε

−1 dx

1 −1

z∂ (x)u−1

x ε

dx.

Hence, substituting these values into the cost functional I, it takes the form I2 I3 z∂2 dx + I22 2 + I6 − 2 (I4 − I2 ) − 2I5 , I(u(·/ε), yε ) = I I1 Ω 1 where we used the notations 1 1 1 I1 = dx, I2 = −1 u(x/ε) −1 1 1 z∂ (x) I4 = dx, I5 = 2 −1 u (x/ε) −1

1 z∂ (x) 1 dx, I3 = dx, 2 u(x/ε) −1 u (x/ε) 1 z∂2 (x) z∂2 (x) dx, I6 = dx. 2 u(x/ε) −1 u (x/ε)

(5.44)

Let us introduce the auxiliary 1-periodic function v ∈ L∞ (0, 1) satisfying

146


v(x) = a − 1 or v(x) = Then

x

x =1+v , ε ε x 1 v +1 . = −1 + a + a ε

u−1 x −2 u ε

1 − 1 a.e. on (0, 1). a

(5.45) (5.46)

Interpreting the function v as a new control, we see that each of the functionals (5.44) is affine with respect to v. Now, let us consider the mapping I : V∂ → R6 , where V∂ = {v ∈ L∞ (0, 1) : v = u − 1 ∀ u ∈ U∂ } , x x I(v) = I1 v , . . . , I6 v . ε ε By the Lyapunov theorem (see [57]), Aε = {I(v) : ∀ v ∈ V∂ } is a convex closed subset of R6 . Since the functional I continuously depends on I1 , . . . , I6 , it follows that I reaches its infimum on the set of admissible pairs. Thus, the OCP(5.40)–(5.42) is solvable. Let (u0ε , yε0 ) = (u0 ε· , yε0 ) ∈ U∂ × H01 (Ω) be its solution. Then, following the direct analogue of Pontryagin Maximum Principle [219], the necessary optimality conditions take the form x x 0 ≥ yε0 x (x) (ψε )x (x)u0 , x∈Ω (5.47) yε x (x) (ψε )x (x)u ε ε for all u ∈ U∂ , where ψε ∈ H01 (Ω) is the solution of the problem d 0 x u ψx − 2(yε0 − z∂ ) = 0, dx ε

x ∈ Ω, ψ(−1) = ψ(1) = 0.

(5.48)

By analogy with the original system, we get x −1 (ψε )x (x) = 2(yε0 (x) − z∂ (x)) − c1 u0 , ε

(5.49)

where c1 =

1

0

u −1

x −1 ε

−1 dx

x −1 2(yε0 (x) − z∂ (x)) u0 dx. ε −1 1

Then, substituting (5.43) and (5.49) in (5.47), we come to the inequality $ # x −1 x 0 x δε (x) − u0 ≤ 0, (5.50) u u − 1 δε (x) − c2 ε ε ε


147

which has to be true for all u ∈ U∂ and almost every x ∈ Ω. Here, 1 −1 1 x −1 −1 0 x dx z∂ (x) u0 dx, δε (x) = z∂ (x) − u ε ε −1 −1 c2 =

−1 1 x −2 x −1 0 dx δε (x) u0 dx. u ε ε −1 −1 1

However, the direct calculations show that (5.50) does not hold for any u0 ∈ U∂ if γa2 > 1. Thus, we have an example of a solvable OCP for which its solution does not satisfy the necessary optimality condition in the form of direct analogue of the Pontryagin maximum principle. So, we cannot derive relations (5.41), (5.47) and (5.48) on the basis of the asymptotic analysis for the OCP (5.40)–(5.42). Another example concerns the case when the existence of solutions to the problem (5.2) may fail. Following Luc Tartar, we consider an elementary model OCP which has no solution. However, this fact by itself does not tell much about the existence of solutions for the system of necessary conditions of optimality. Indeed, let us consider the problem & % T 2 2 |y| − |u| dt , (5.51) Minimize I(u, y) = 0

where the control u belongs to the set

U∂ = u ∈ L2 (0, T ) | − 1 ≤ u(t) ≤ 1, a.e. t ∈ (0, T ) ,

(5.52)

and the state y is defined by the equation of state dy =u dt

a.e. on (0, T ),

y(0) = 0.

(5.53)

Before we consider the necessary conditions of optimality, we show directly that the OCP (5.51)–(5.53) has no solution. To do so, we first notice that I(u, y) > −T for all u ∈ U∂ ,

(5.54)

because |y|2 − |u|2 ≥ −1 a.e. on (0, T ) for every u ∈ U∂ implies I(u, y) ≥ −T and because one cannot have I(u, y) = −T , which would require both y = 0 and |u| = 1 a.e. on (0, T ), in contradiction to the fact that y = 0 a.e. on (0, T ) implies u = 0 a.e. on (0, T ). Let us consider the following sequence {uk }k∈N , where uk = sign cos kt on (0, T ). Then uk ∈ U∂ for all k ∈ N and, moreover,

∗

|uk | = 1

∀ k ∈ N,

2π

uk

sign(cos t) dt = 0 0

in L∞ (0, T ).

(5.55) (5.56)

148


Since the weak-∗ convergence uk 0 in L∞ (0, T ) implies that yk = y(uk ) converges uniformly to 0, it follows that I(uk , yk ) → −T . However, as was shown earlier, the cost functional I does not achieve the value −T on the set of admissible pairs # $ t Ξ = (u, y) : u ∈ U∂ , y(t) = u(τ ) dτ, t ∈ (0, T ) . 0

Having formally used the classical scheme, we can derive the necessary optimality conditions to the problem (5.51)–(5.53) – namely, we note that U∂ is a convex set, the map u → y is affine continuous, and therefore, the map (u, y) → I is quadratic continuous, and so Fréchet differentiable. Let u0 ∈ U∂ , let y 0 be the corresponding state and let δu ∈ L∞ (0, T ) be an admissible direction at u0 (i.e., u = u0 + εδu ∈ U∂ for ε > 0 small). Then y = y 0 + εδy and I(u, y) = I(u0 , y0 ) + εδI + o(ε), where d(δy) = δu, δy(0) = 0, dt T δI = 2 (y 0 δy − u0 δu) dt,

(5.57) (5.58)

0

and classical necessary conditions of optimality consist in writing δI ≥ 0 for all admissible δu and the corresponding δy. In order to eliminate δy so that δI is expressed only in terms of δu, we introduce the adjoint state p 0 by the relation dp 0 = y 0 , p 0 (T ) = 0. (5.59) − dt Then a simple integration by parts gives T T T T dp 0 dδy y 0 δy dt = − p0 p 0 δu dt, (5.60) δy dt = dt = dt dt 0 0 0 0 and therefore,

T

(p 0 − u0 )δu dt.

δI = 2

(5.61)

0

The admissibility of δu means that δu ≥ 0 provided u0 = −1, δu ≤ 0 provided u0 = +1, and δu is arbitrary if −1 < u0 < +1. From this, we immediately deduce the classical necessary conditions of optimality: ⎧ 0 implies u0 = +1, ⎨ p < −1 −1 ≤ p 0 ≤ +1 implies u0 ∈ −1, p 0 , +1 , (5.62) ⎩ 0 p > +1 implies u0 = −1. Note that the system of these classical necessary conditions of optimality (i.e., (5.53), (5.59) and (5.62)) has at least one solution u0 = 0, y 0 = 0, p0 = 0 on (0, T ).

(5.63)


149

On the other hand, using the Pontryagin maximum principle, we can obtain the necessary conditions of optimality if we note that any control which jumps from u0 to another control w ∈ U∂ is admissible as well. It means that u = (1 − χ)u0 + χw ∈ U∂ for every characteristic function χ of a measurable subset of (0, T ). Therefore, it is natural to consider a sequence of characteristic functions {χk }k∈N such that ∗ χk θ in L∞ (0, T ), where 0 ≤ θ ≤ 1 a.e. in (0, T ). Then the corresponding functions yk , which satisfy a uniform Lipschitz condition, converge uniformly to y ∞ , the solution of −

dy ∞ = (1 − θ)u∗ + θw dt

in (0, T ),

y ∞ (0) = 0,

and, using the fact that F ((1 − χ)u∗ + χw) = (1 − χ)F (u∗ ) + χF (w) for every function F and every characteristic function χ, we deduce that given by I(uk , yk ) converges to I(θ) T ∞2 = I(θ) |y | − (1 − θ)|u0 |2 − θ|w|2 dt. 0

Assume that I attains its minimum on Ξ at (u0 , y 0 ). Then = I(u0 , y 0 ) ≤ lim I(uk , yk ) = I(θ), I(0) k→∞

and therefore, I attains its minimum at 0. Then, noticing that the admissibility for δθ means δθ ≥ 0, we can write the classical necessary conditions of optimality for I as dδy = (w − u0 )δθ, dt

δy(0) = 0

(5.64)

and, as θ = 0 corresponds to y ∞ = y 0 , that T 0 2y δy + |u0 |2 − |w|2 δθ dt. δ I = 0

Using the same p 0 as defined in (5.59), integration by parts gives a different result because the equation for δy is different (see (5.60)): T T T dp 0 dδy 2y 0 δy dt = −2 2p 0 δy dt = dt dt dt 0 0 0 T 2p 0 (w − u0 )δθ dt, = 0

150


and therefore, δ I =

T

2p 0 (w − u0 ) + |u0 |2 − |w|2 δθ dt.

0

Hence, the necessary conditions of optimality for I become 2p 0 (w − u0 ) + |u0 |2 − |w|2 ≥ 0 a.e. on (0, T ).

(5.65)

Using the fact that p 0 is independent of the choice of w, (5.65) means 0 2p w − |w|2 a.e. on (0, T ) 2p 0 u0 − |u0 |2 = inf −1≤w≤+1

= −2p 0 − 1 a.e. on (0, T ). As a result, we come to the following necessary conditions of optimality: ⎧ 0 ⎨ p < 0 implies u0 = +1, p 0 = 0 implies u0 = ±1, (5.66) ⎩ 0 p > 0 implies u0 = −1, which are obviously more restrictive than (5.62). Moreover, in this case, the triplet (u0 , y 0 , p 0 ) = (0, 0, 0) does not satisfy the necessary conditions of optimality (5.53), (5.59), and (5.66).

5.3 Limit analysis of optimal control problems by Γ -convergence Among the different ways and approaches for studying the asymptotic behavior of different classes of OCPs depending on a small parameter, the primary one relies on the notion of Γ -convergence [29, 33, 41, 42, 43, 44, 92, 93, 94]. Γ -convergence has become, over more than 30 years after its introduction by Ennio De Giorgi, the commonly recognized notion of convergence for variational problems in a general setting. A complete introduction to the general theory of Γ -convergence is the by now classical book by Dal Maso [78] and the book by Braides [29]. To illustrate this approach, we rewrite the OCP (5.2) as follows min {Jε (u, y) = Iε (u, y) + 1Ξε (u, y) : (u, y) ∈ Uε × Yε } .

(5.67)

Here, we denote by 1Ξε the indicator function of the set of admissible pairs Ξε defined by # 0 if (u, y) ∈ Ξε , 1Ξε (u, y) = (5.68) +∞ if (u, y) ∈ Ξε . Following this representation, we say that a good definition of a “pass to the limit” in (5.67) is a convergence of functionals, which guarantees the

5.3 Limit analysis of optimal control problems by Γ -convergence

151

convergence of optimal pairs and minimum values of problems (5.67) to those of a limit problem. Note that the space Uε × Yε and the cost functional Jε at level ε may vary with ε, so that a preliminary problem is to define the convergence of a sequence of functionals which belong to different spaces. The way this convergence is defined must be quite flexible. This is usually done by choosing U × Y large enough so that it contains the domain of the limiting cost functional and all Uε × Yε [10, 29]. As a rule, one considers all functionals Jε as defined on the space U × Y by identifying them with the functionals # Jε (u, y) if (u, y) ∈ Uε × Yε , J ε (u, y) = +∞ if (u, y) ∈ U × Y \ (Uε × Yε ) . This type of identification is common in dealing with minimum problems and sometimes it is very useful to include constraints directly into the functional. Hence, one may therefore suppose that all Uε = U and Yε = Y. Under this supposition, the next theorem gives a definition of a limit problem such that the variational properties (V1)–(V2) hold (for the proof, see [43, Proposition 2.1]). Theorem 5.5. Let Y and U be two topological spaces and let Jε : U × Y → R be a sequence of functionals. Let (u0ε , yε0 ) be a minimum point for Jε , or simply a pair such that ! 0 0 lim Jε (uε , yε ) = lim Jε (u, y) . inf ε→0 (u,y)∈ U×Y

ε→0

Assume that (u0ε , yε0 ) converges to (u0 , y 0 ) in U × Y and there exists J(u, y) = Γ (E, U− , Y− ) lim Jε (u0ε , yε0 ). ε→0

(5.69)

Then we have the following: (i) (u0 , y 0 ) is a minimum point for J on U × Y. (ii) limε→0 inf (u,y)∈ U×Y Jε (u, y) = min(u,y)∈ U×Y J(u, y). Here, E = {ε = 1/n : n ∈ N}, and in this case, min {J(u, y) : (u, y) ∈ U × Y}

(5.70)

is called a limit problem, where J is defined by the multiple Γ -limit (5.69). A multiple Γ -limit is in general defined as follows: Definition 5.6. Let Y and U be two topological spaces and let Jε : U × Y → R be a sequence of functionals. By Z(+) we will denote the “sup” operator and by Z(−) the “inf ” one. For every u ∈ U and y ∈ Y, we define: Γ Eα , Uβ , Yγ lim Jε (u, y) = Z(β) Z(γ) Z(−α) Z(α) Jε (uε , yε ), ε→0

uε ∈ S(u) yε ∈ S(y)

ε∈ E

ε ≤ε

where α, β, and γ are signs + or − and S(u) and S(y) respectively denote the set of all sequences uε → u in U and that of yε → y in Y. Note that when the Γ -limit does not depend on the sings + or −, this sing will be omitted.

152


For the complete theory of Γ -limits, we refer to Dal Maso [78] and Braides [29]. The limit problem (5.70) can be calculated explicitly in some particular cases (see [33, 38, 41, 42, 43, 44, 47, 48]). Indeed, let us suppose that the set of admissible pairs for the problem (5.2) has the form Ξε = {(u, y) ∈ U × Y : Lε (u) + Fε (y) = 0} , where for every fixed ε ∈ E, the operators Lε and Fε map respectively U and Y to a third topological space Z. The collection of OCPs is then min {Iε (u, y) + 1Ξε (u, y) : (u, y) ∈ U × Y} ,

ε ∈ E.

(5.71)

The main trick of this calculation is the introduction of an auxiliary variable v = Lε (u). Then the functional Jε = Iε + 1Ξε can be split as Jε (u, y) = Iε (u, y) + 1{v=Lε (u)} (u, v) + 1{v=−Fε (y)} (v, y) := Gε (u, v, y) + 1{v=−Fε (y)} (v, y),

(5.72)

and the identification of (5.70) is given by the following theorem (for the proof, see [47]). Theorem 5.7. Assume the following: HFε : −Fε G-converges to −F . HLε : The condition uε → u in U implies that the sequence {Lε (uε )} is relatively compact in Z.

HGε : For the sequence of functionals Gε (u, y, v) : U × Y × Z → R there exists a Γ -limit G(u, v, y) := Γ (E, U × Z− , Y− ) lim Gε (u, v, y). ε→ 0

(5.73)

Then J(u, y) = G(u, −F (y), y). In view of this result, the calculation of the limit (5.69) can be reduced to the calculation of (HGε ) and to the identification of the G-limit of the operators {Fε }. We recall the definition of G-convergence of operators (see [210, 261, 262]). Definition 5.8. We say that the sequence {Fε } G-converges to F if Γ (E, Z, Y− ) lim 1{v=Fε (y)} (v, y) = 1{v=F (y)} (v, y) ε→ 0

for every v ∈ Z and y ∈ Y, that is, if the following conditions hold true (i) If yε → y in Y, vε → v in Z and Fε (yε ) = vε for infinitely many ε ∈ E, then F (y) = v. (ii) If y ∈ Y and v ∈ Z are such that F (y) = v and vε → v in Z, then there exists yε → y in Y satisfying Fε (yε ) = vε for every ε ∈ E large enough.


153

A lot of results have been established about the characterization of those conditions (see, for instance, [29, 33, 38, 41, 42, 47, 48]). The identification of the limit usually can be resumed in three steps: Step 1. Introduce the auxiliary variable v ∈ Z. Step 2. Calculate the operator −F given by (HFε ) and the Γ -limit in (HGε ). Step 3. Eliminate the variable v in order to write the final expression of the limit problem setting J(u, y) = G(u, −F (y), y). Having assumed that −Fε G-converge to an operator −F , one focuses on the behavior of the input operators Lε . It is clear that hypothesis (HLε ) is valid when the operators Lε converge continuously to an operator L, that is, uε → u in U ⇒ Lε (uε ) → L(u) in Z.

(5.74)

In order to characterize condition (HGε ), one requires some kind of uniform continuity on the cost functional. Usually, it is assumed that there exist a function Ψ : U → R bounded on the U-bounded sets and a function ω : Y × Y → R with limz→ y ω(y, z) = 0 for every y ∈ Y such that for every u ∈ U and z, y ∈ Y (5.75) Iε (u, y) ≤ Iε (z, u) + Ψ (u)ω(y, z). In this case, the limit problem has the same structure as the one of the original OCP and can be represented as min {I(u, y) : L(u) + F (y) = 0, (u, y) ∈ U × Y} . In order to describe the limit problems under more general assumptions on the operator Lε , we consider a case in which the verification of the hypotheses and the corresponding calculations can be made explicitly. Example 5.9. We deal with the following OCP governed by an ordinary differential equations (see Briani [33] for details): ! 1 2 2 Iε (u, y) = |u| + |y − φ(t)| dt −→ inf, 2 0 $ # t Ξε = (u, y) : y (t) = aε (t, y) + sin u(t), y(0) = y0 , ε

T

(5.76) (5.77)

Ξε ⊂ L2 (0, T ) × W 1,1 (0, T ), where φ ∈ L2 (0, T ) and the functions aε are Lipschitz continuous with respect to y and measurable with respect to t, such that aε (·, y) −→ a(·, y)

weakly in L1 (0, T ), ∀ y ∈ R.

(5.78)

Then assumption (HFε ) is fulfilled with Fε (y) = −y + aε (t, y) and F (y) = −y + a(t, y). Moreover, having set

154


Gε (u, v, y) = 0

T

! 1 2 |u| + |y − φ(t)|2 + 1{sin(t/ε)u=v} (t, u, v) dt 2

and closely following along the line of the calculation scheme of the Γ -limit G(u, v, y) := Γ (E, U × Z− , Y− ) limε→ 0 Gε (u, v, y) in [33, 42], one shows that the limit problem takes the form min I0 (u, y) ! T 1 2 |u| + |y − φ(t)|2 + |y − a(t, y)|2 dt, y(0) = y0 . (5.79) := 2 0 Note that in this case, the state equation has disappeared in the limit problem and it is replaced by an integral contribution in the cost functional as an additional extra term. Thus, we do not have an OCP in the limit as ε tend to 0, but rather a minimization problem. To characterize the main distinctive features of this approach, we give a few comments relating to the asymptotic analysis of the OCP (5.2). First, since the basic spaces Uε × Yε may differ substantially from level to level, we have no guarantee in saying that in the general case, a space U × Y can be found, containing the domains Uε × Yε for every ε ∈ E and guaranteeing the following compactness property: The existence of a limit of optimal pairs to (5.2) – assuming that they exist – is ensured beforehand. As follows from Example 5.9, this property may not hold. Second, as Theorem 5.7 indicates (see also (5.75) and Example 5.9), one always assumes that the cost functional Iε : Ξε → R is well defined on the whole space Uε ×Yε . However, taking into account specific OCPs coming from applications, cost functionals have a sense (economical, engineering, physical, etc.) only on the set of admissible solutions and they may be undefined outside of this set. As was mentioned earlier, one of the primary goals of the asymptotic analysis of the OCPs (5.2) is to obtain the full identification of the structure of a limiting problem, including the precise form of the state equations, cost functional and possible restrictions on managing parameters. The next observation deals with the fact that for both physical and mathematical reasons, we always have some control (and state) constraints in the original problem. In particular, if u ∈ Uε ⊂ Uε , then instead of the functional (5.72), we have to consider Jε (u, y) = Iε (u, y) + 1{v=Lε (u)} (u, v) + 1{u∈Uε } (u) + 1{v=−Fε (y)} (v, y) := Gε (u, v, y) + 1{u∈Uε } (u) + 1{v=−Fε (y)} (v, y). However, it is a nontrivial matter to find its Γ (E, U × Z− , Y− )-limit as ε tends to 0 because, in general, the equality


155

Γ (E, U × Z− , Y− ) lim Gε (u, v, y) + 1{u∈Uε } (u) ε→ 0

= Γ (E, U × Z− , Y− ) lim Gε (u, v, y) + Γ (E, U) lim 1{u∈Uε } (u) ε→ 0

ε→ 0

is not valid [10, 78]. Finally, we note that – returning to Example 5.9 – in spite of the good variational properties of the limit problem (5.79), it cannot be recovered in the form of some optimal control problem. However, if we consider the case, where the control space L2 (0, T ) has a topology associated with the weak twoscale convergence, then this limit problem may exhibit another mathematical 2 description. Indeed, the weak two-scale convergence uε (t) u(t, ξ) means, by definition, that the following relation holds: T T 2π −1 uε (t)b(tε )ϕ(t) dt = u(t, ξ)b(ξ)ϕ(t) dξ dt, lim ε→ 0

0

0

∀ ϕ ∈ C0∞ (0, T ),

0

∞ ∀ b ∈ Cper (0, 2π),

∞ where Cper (0, 2π) is the space of infinitely differentiable 2π-periodic functions. Many properties of weak two-scale convergence have been established which make it possible to pass to the limit in the integral identity ! T t uε (t) ϕ(t) dt = 0, (5.80) yε (t) − aε (t, yε ) + sin ε 0

where yε ∈ W 1,1 (0, T ) is a solution of (5.77) with u = uε . One of these properties is as follows [6, 256, 259]: If uε is bounded in L2 (0, T ), then (after a possible transition to a subsequence) 2

uε (t) u(t, ξ) ∈ L2 (0, T ; L2per (0, 2π)). As we will see later, using the Γ -convergence technique and the scheme of the direct variational convergence, the limit OCP to the family (5.76) and (5.77) with respect to the product of the weak two-scale convergence in L2 (0, T ) and the weak convergence in W 1,1 (0, T ) as ε → 0 can be recovered in the following form: T 1 T 2π I0hom (u, y) = |u(t, ξ)|2 dξ dt + |y − φ(t)|2 dt −→ inf, (5.81) 2 0 0 0 Ξ0hom = (u, y) ∈ L2 (0, T ; L2per (0, 2π)) × W 1,1 (0, T ) : T 2π T [y (t) − a(t, y)] ϕ(t) dt + sin(ξ)u(t, ξ)ϕ(t) dξ dt = 0, 0 0 0 y(0) = y0 , ∀ ϕ ∈ C0∞ (0, T ) . (5.82) It should be emphasized that in this case, the limit problem (5.81)–(5.82) is some OCP for an object the state equation of which takes the form of an

156


integral identity (5.82). As it is easy to see, the problem (5.81)–(5.82) admits a unique solution (u0 , y 0 ) ∈ L2 (0, T ; L2per (0, 2π)) × W 1,1 (0, T ). Moreover, following the variational properties of Γ -convergence, it can be shown that mε :=

inf (u,y)∈ Ξε

Iε (u, y) = Iε (u0ε , yε0 ) → I0 (u0 , y 0 ) =

yε0 y 0 in W 1,1 (0, T ),

inf (u,y)∈ Ξ0

I0 (u, y) =: m0 ,

2

u0ε (t) u0 (t, ξ) in L2 (0, T ).

5.4 Direct variational convergence of optimal control problems The main idea of our approach in this work is to study the asymptotic behavior of a family of OCPs ⎧ ⎫ (CFε ) : Iε (u, y) → inf, ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ subject to (OCPε ) : (5.83) : (u, y) ∈ Uε × Yε , u ∈ Uε ⊂ Uε , ⎪ (CSCε ) ⎪ ⎪ ⎪ ⎩ ⎭ (SEε ) : Lε (u, y) + Fε (y) = 0 not through the study of the limit properties of the solutions (u0ε , yε0 ) ∈ Ξε , but rather by defining a limit OCP that would be a “good approximation” of the previous one as ε → 0 and has a clearly defined structure, including the limit form of a state equation, control constraints and a limit cost functional. For this, we apply the so-called scheme of the direct variational convergence, which was developed in [140, 141, 144, 147]. The idea is still to be in the framework of Γ -convergence. The principal feature is the representation of the original OCP as a constrained minimization problem and subsequent study of its asymptotic behavior. As we will see later, such an approach allows reducing the procedure of asymptotic analysis to consecutive identification of the set of admissible pairs for the limit OCP and then its cost functional. We will always associate to every OCP (5.83) the corresponding constrained minimization problem: inf Iε (u, y) , (5.84) (CMPε ) : (u,y)∈ Ξε

where the set of admissible solutions Ξε = {(u, y) ∈ Uε × Yε : Lε (u, y) + Fε (y) = 0, u ∈ Uε , Iε (u, y) < +∞} features a specific structure, namely Ξε = {(u, y) : (SEε ) ∩ (CSCε ) ∩ {Iε (u, y) < +∞}} .

(5.85)

5.4 Direct variational convergence of OCPs

157

Having observed that the cost functional is well defined and has a sense only on the set of admissible solutions leads us to the object epi(Iε ; Ξε ) = {(u, y, λ) ∈ Uε × Yε × R | (u, y) ∈ Ξε ,

Iε (u, y) ≤ λ} , (5.86)

and we call it the epigraph of the cost functional. Note that the set epi(Iε ; Ξε ) coincides with the epigraph of the extended cost functional I"ε : Uε × Yε → R, namely (5.87) epi(Iε ; Ξε ) ≡ epi(I"ε ) := (u, y, λ) ∈ Uε × Yε × R : I"ε (x) ≤ λ , where I"ε (u, y) =

#

Iε (u, y) if (u, y) ∈ Ξε , +∞ if (u, y) ∈ Uε × Yε \ Ξε ,

∀ ε > 0.

(5.88)

Obviously, epi(I"ε ) = Uε × Yε × R if I"ε ≡ −∞ and epi(I"ε ) = ∅ if I"ε ≡ +∞. However, the set epi(Iε ; Ξε ) differs considerably from the classical definition of an epigraph epi(Iε ) = {(u, y, λ) ∈ Uε × Yε × R | Iε (u, y) ≤ λ} – namely, epi(Iε ; Ξε ) is the restriction of epi(Iε ) on the set Ξε × R. Thus, in view of this construction, the original OCP can be associated with the subset epi(Iε , Ξε ) of U × Y × R and this correspondence is one-to-one. It means that the examination of the asymptotic behavior of (OCPε ) as ε → 0 can be done through the study of the limit properties for the corresponding sequence of sets {epi(Iε ; Ξε )}ε→0 . In fact, this suggestion is motivated by the following result that will be proved in Chap. 8. Theorem 5.10. Assume that there is a set C = ∅ such that K

epi(Iε ; Ξε ) −−−→ C ε→ 0

in the Painlevé–Kuratowski sense with respect to some appropriate convergence in a variable space Uε × Yε × R. Then C ⊂ U × Y has the structure C = epi(Ihom ; Ξhom ). As a result, the set C = epi(Ihom ; Ξhom ) can be associated with the corresponding constrained minimization problem inf Ihom (u, y) , (5.89) (CMPhom ) : (u,y)∈ Ξhom

and we call it the variational limit of the sequence # $ inf Iε (u, y) ; ε ∈ E . (u,y) ∈Ξε

It leads us to the following definition.

(5.90)

158


Definition 5.11. We say that the family of OCPs (5.83) has a limit problem as ε tends to 0 if for the corresponding sequence of constrained minimization problems (5.90), there exists a variational limit which can be recovered in the form of some OCP. Resuming this approach, we come to the following scheme of the limit analysis (SLA) of OCPs: (OCPε )→ inf Iε (u, y) inf Ihom (u, y) → (OCPhom ) (u,y)∈ Ξε (u,y)∈ Ξhom ⏐ ) ⏐ ⏐ . ( ⏐ ε→0

−−−−→

epi(Iε ; Ξε )

epi(Ihom ; Ξhom )

It is worth noticing that in spite of apparent differences in the formulation of the limit problem compared with the analogous one by the Γ -convergence approach, it can be proved that these notions are closely connected. Theorem 5.12 ([140, 143]). Assume that there exist spaces U and Y large enough such that Ξε ⊂ Uε × Yε ⊂ U × Y for all ε ∈ E. Let I = Γ − lim inf PΞε Iε ε→0

and

I = Γ − lim sup PΞε Iε ε→0

be the lower and upper Γ -limits of the sequence

PΞε Iε : U × Y → R = R ∪ {+∞} ε→0 , respectively, where # PΞε Iε (x) =

Iε (u, y) if (u, y) ∈ Ξε , +∞ if (u, y) ∈ Ξε .

Then epi(I ; K− lim sup Ξε ) = K− lim sup [epi(Iε ; Ξε )] , ε→0

ε→0

epi(I ; K− lim inf Ξε ) = K− lim inf [epi(Iε ; Ξε )] . ε→0

ε→0

In particular, the sequence {epi(Iε ; Ξε )}ε→0 has a limit in the sense of Painlevé–Kuratowski if and only if the following conditions hold: K− lim sup Ξε = K− lim inf Ξε and I = I on the set K− lim inf Ξε . ε→0

ε→0

ε→0

In view of this, there is a good reason to believe that the limit OCP in the sense of Definition 5.11 enjoys the following important properties: — (SLA) implies the convergence of minimum values and the convergence of (sub)sequence of (almost-)minimizers of (CMPε ) to minimizers of the variational limit (CMPhom ).

5.4 Direct variational convergence of OCPs

159

— It is stable under continuous perturbations. This means that this analysis valid if we add to the cost functional Iε any fixed continuous term. — The cost functional Ihom is lower semicontinuous and its domain Ξhom is closed, which usually implies the existence of optimal pairs to the limit problem. As we will see later, a crucial role for the realization of the scheme of direct variational convergence will be played by the type of convergence in variable spaces Uε × Yε . To identify the clearly defined structure of a limit optimal problem including the limit form of a state equation, control constraints and a limit cost functional, it will be a matter of balance between the convenience of a stronger notion of such convergence and a weaker one. Because our interest is in the asymptotic analysis of OCPs, we note that sometimes it is convenient to rewrite (3.2)–(3.3) in the form of the minimization problem (5.91) inf I(u, y), (u,y)∈ Ξ

where the set of admissible pairs is chosen to be Ξ = {(u, y) ∈ U × Y : u ∈ U∂ , y ∈ argmin G(u, ·)} .

(5.92)

Here, G : U × Y → R is a functional of the calculus of variations whose integrand depends on the control and its Euler–Lagrange equation takes the form L(u, y) + F (y) = 0. Following Bucur and Buttazzo [38], we call G the state functional. It should be stressed that the set Ξ can be always written in the form (5.92) by choosing # 0 if (u, y) ∈ Ξ, (5.93) G(u, y) = 1Ξ = +∞ otherwise. For instance, when a state equation is −Δy + y = f + u in Ω,

y ∈ H01 (Ω),

u ∈ U∂ ⊂ L2 (Ω),

the corresponding state functional G : L2 (Ω) × H01 (Ω) → R can be taken as 1 |∇y|2 + y 2 − 2(f + u)y dx + 1U∂ (u). G(u, y) = 2 Ω In this case, the natural topology on U that takes into account the convergence minimizers of G is the one related to Γ -convergence of the mappings G(u, ·) [48]. Indeed, as soon as the convergence of controls implies the convergence of associated states, it would be enough to have the compactness of minimizing sequences in U and the lower semicontinuity of the cost functional I in U × Y to obtain the existence of an optimal pair for (5.91)–(5.92).

6 Convergence Concepts in Variable Banach Spaces

The aim of this chapter is to provide a systematic exposition of the main properties of weak and strong convergence in variable Lp - and W 1,p -spaces for p > 1. It will always be assumed that p and q are conjugate indices (i.e., 1 = 1/p + 1/q) and that p > 1. The main objectsof our consideration are sequences of the types yεh ∈ Lp (Ω, dμhε ) ε>0 and yεh ∈ W 1,p (Ω, dμhε ) ε>0 , where μhε is a two-parametric Borel measure related to the geometry of thin periodic structures. Typically, the parameter ε defines the periodicity cell and εh is the thickness of constituting elements of such structures. However, there is a principal difference between perforated domains and thin structures. For perforated domains, the typical case is when the parameter h is either independent of ε or such that lim inf ε→0 h(ε) = h∗ > 0, whereas the principle feature of thin structures is the fact that the parameters ε and h = h(ε) are related by the supposition h(ε) → 0 as ε → 0. Therefore, our main intension in this chapter is to shed some light on convergence properties in described spaces.

6.1 General setting We begin with the following assumptions and notations. For any small positive value ε > 0, let Xε be a real reflexive separable Banach space equipped with the norm · Xε . Let θε be 0 in Xε and let · 0,Xε be an auxiliary norm in Xε . As usual, by X∗ε we denote the dual space of Xε . We also assume that the small parameter ε > 0 varies in a strictly decreasing sequence of positive numbers which converge to 0. Further, let X be a real reflexive separable Banach space equipped with the norm ·X , let θ be 0 in X, and let ·0,X be an auxiliary norm in X relative to which the ball BX (0, 1) = {u ∈ X : uX ≤ 1} is precompact. For any ε > 0, we suppose that there exists a linear mapping Qε : X → Xε such that P.I. Kogut, G.R. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains, Systems & Control: Foundations & Applications, DOI 10.1007/978-0-8176-8149-4 6, © Springer Science+Business Media, LLC 2011

161

162

6 Convergence Concepts in Variable Spaces

if u ∈ X, then Qε uXε ≤ uX , Qε u0,Xε ≤ u0,X ; if u ∈ X and lim inf Qε u0,Xε = 0, then u = θ. ε→0

(6.1) (6.2)

By Lε we denote the set of all continuous linear mappings of Xε into X. Further, let P be the set of all sequences {Pε }ε>0 satisfying the following conditions: Pε ∈ Lε for every ε > 0;

sup Pε Lε < +∞;

(6.3)

ε>0

if u ∈ Xε , then Qε (Pε u) = u. Hereinafter it is assumed that P = ∅ and κ = inf sup Pε Lε : {Pε }ε>0 ∈ P .

(6.4)

(6.5)

ε>0

We note that κ ≥ 1. Indeed, taking {Pε }ε>0 ∈ P, u ∈ Xε , and u = θε , by (6.1) we have uXε = Qε (Pε u)Xε ≤ Pε uX ≤ Pε Lε uXε . Hence, supε>0 Pε Lε ≥ 1. Since here {Pε }ε>0 ∈ P is arbitrary, it follows that κ ≥ 1. Definition 6.1. Assume that uε ∈ Xε for any ε > 0. We say that the sequence {uε }ε>0 is bounded if the sequence of norms {uε Xε }ε>0 is bounded. Simiof the sequence {fε }ε>0 larly, if fε ∈ X∗ε for every ε > 0, then boundedness means boundedness of the sequence of norms fε X∗ε ε>0 . Definition 6.2. Assume that uε ∈ Xε for any ε > 0 and that u ∈ X. We say that the sequence {uε }ε>0 converges weakly to u (in the scale of variable spaces {Xε }) if the sequence {uε }ε>0 is bounded and limε→0 uε − Qε u0,Xε = 0. Let us note that if u ∈ X, then the sequence {uε = Qε u}ε>0 converges weakly to u in the above sense. Definition 6.3. Assume that uε ∈ Xε for any ε > 0 and that u ∈ X. We say that the sequence {uε ∈ Xε }ε>0 converges strongly to u (in the scale of variable spaces {Xε }) if limε→0 uε − Qε uXε = 0. Definition 6.4. Assume that fε ∈ X∗ε for any ε > 0 and that f ∈ X∗ . We say that the sequence {fε ∈ X∗ε }ε>0 converges strongly to f if the fact that uε ∈ Xε for any ε > 0, u ∈ X and the sequence {uε ∈ Xε }ε>0 converges weakly to u implies lim fε , uε X∗ε ,Xε = f, uX∗ ,X . ε→0

6.1 General setting

163

As follows from condition (6.1), the strong convergence of the sequence {uε ∈ Xε }ε>0 to u ∈ X implies the weak convergence of this sequence to the same element u. Next, following Kovalevskiˇi [155], we proceed in this section with the main properties of the convergence in variable spaces (see also [156, 157, 158]). Proposition 6.5. Let {uε ∈ Xε }ε>0 be a bounded sequence. Then from this sequence it is possible to extract a weakly convergent subsequence. Proof. It is easy to see that for any {Pε }ε>0 ∈ P, the sequence {Pε uε }ε>0 is bounded in X. Hence, by the reflexivity of X, there exist an decreasing sequence of positive numbers {εk }∞ k=1 converging to 0 and u ∈ X such that Pεk uεk u in X. Using the assumption regarding the norm · 0,X , we find that Pεk uεk − u0,X → 0. Hence, from (6.1) we conclude that the sequence {uεk ∈ Xε }∞

k=1 converges weakly to u. The proposition is proved. Proposition 6.6. Assume that uε ∈ Xε for any ε > 0, u ∈ X and the sequence {uε ∈ Xε }ε>0 converges weakly to u. Then Pε uε u in X for any sequence {Pε }ε>0 ∈ P. Proof. Let {Pε }ε>0 ∈ P be a fixed sequence. We assume that the sequence {Pε uε }ε>0 does not converge weakly to u in X. Then there exist a sequence ∞ {εk }k=1 converging to 0 and v ∈ X, v = u, such that Pεk uεk v in X. From this and the weak convergence of {uε ∈ Xε }ε>0 to u it follows that lim inf k→∞ Qε (u − v)0,Xε = 0. Then, by condition (6.2), we obtain u = v. However, this contradicts the inequality v = u. Thus, Pε uε u in X. The proposition is proved.

Proposition 6.7. Assume that uε ∈ Xε for any ε > 0, u ∈ X and the sequence {uε ∈ Xε }ε>0 converges weakly to u. Then lim inf uε Xε ≥ κ−1 uX . ε→0

(6.6)

Proof. Having chosen an arbitrary δ > 0 and a sequence {Pε }ε>0 ∈ P such that (6.7) sup Pε Lε ≤ κ + δ, ε→0

by Proposition 6.6, we have lim inf ε→0 Pε uε X ≥ uX . From this and (6.7), it follows that lim inf uε Xε ≥ (κ + δ)−1 uX . ε→0

Passing to the limit in this inequality as δ → 0, we obtain (6.6). This concludes the proof.

It is useful to note that, as follows from Proposition 6.6, if f ∈ X∗ and {Pε }ε>0 ∈ P, then the sequence {f ◦ Pε }ε>0 converges strongly to f .

164


Proposition 6.8. Assume that fε ∈ X∗ε for any ε > 0, f ∈ X∗ and the sequence {fε ∈ X∗ε }ε>0 converges strongly to f . Then, for any sequence {Pε }ε>0 ∈ P, the following relation holds true: lim fε − f ◦ Pε X∗ε = 0.

(6.8)

ε→0

Proof. Let {Pε }ε>0 ∈ Pbe a fixed sequence. We assume that the sequence of norms fε − f ◦ Pε X∗ε ε>0 does not converge to 0. Then there exist δ > 0 ∞ and a sequence of indices {εk }k=1 converging to 0 such that fεk − f ◦ Pεk X∗ε ≥ δ,

k ∈ N.

k

(6.9)

For any k ∈ N, we set gk = fεk − f ◦ Pεk . From (6.9), it follows that there exists a bounded sequence {wk ∈ Xεk }∞ k=1 such that

gk , wk X∗ε

k

,Xεk

≥ δ/2,

k ∈ N.

(6.10)

∞

Since the sequence {vk = Pεk wk }k=1 is bounded in X, there exist an increasing sequence {ki }∞ i=1 and v ∈ X such that vki v in X. From this and the strong convergence of fε ∈ X∗ε to f follows that gki , wki X∗ε ,Xε → 0. However, this contradicts (6.10). Thus, (6.8) holds.

ki

ki

Proposition 6.9. Under the conditions of Proposition 6.8, the following inequalities hold true: lim inf fε X∗ε ≥ f X∗ ,

(6.11)

lim sup fε X∗ε ≤ κf X∗ .

(6.12)

ε→0

ε→0

Proof. We fix an arbitrary δ > 0. Suppose that u ∈ X, uX ≤ 1, and f X∗ ≤ f, uX∗ ,X + δ.

(6.13)

Using the strong convergence of fε to f , the weak convergence of Qε u to u and (6.1), we find

f, uX∗ ,X = lim fε , Qε uX∗ ,Xε ≤ lim inf fε X∗ε . ε ε→0

ε→0

From this and (6.13), we come to the inequality lim inf fε X∗ε ≥ f X∗ − δ. ε→0

Passing to the limit in this inequality as δ → 0, we obtain (6.11). We now prove (6.12). We choose an arbitrary δ > 0 and a sequence {Pε }ε>0 ∈ P such that (6.14) sup Pε Lε ≤ κ + δ. ε>0

6.1 General setting

165

For every ε > 0, we have fε X∗ε ≤ f ◦ Pε X∗ε + fε − f ◦ Pε X∗ε . From this, using (6.14), the strong convergence of fε to f and Proposition 6.8, we deduce that lim sup fε X∗ε ≤ (κ + δ)f X∗ . ε→0

Passing to the limit in this quality as δ → 0, we obtain the desired relation (6.12). This concludes the proof.

The following example illustrates the application of the results given above to the convergence formalism in the variable Sobolev spaces connected with nonperiodically perforated domains. Example 6.10. Let Ω be a bounded open domain in Rn (n ≥ 2) with Lipschitz boundary, and let {Ωε }ε>0 be a sequence of open domains in Rn contained in Ω. It is assumed that there exist a constant ν > 1, a sequence of finite sets {Jε }ε>0 , points xjε ∈ Ω and numbers rεj > 0 (j ∈ Jε , ε > 0) such that Bεj , (6.15) Ω \ Ωε = j∈Jε

lim max rεj = 0,

ε→0 j∈Jε

2(ν − 1)rεj ≤ ρjε ,

∀ j ∈ Jε , ∀ ε > 0,

(6.16) (6.17)

j j j where Bεj is the closed ball centered at thel point xε with radius rε and ρε is j the distance from Bε to the set Jε l=j Bε ∪ ∂Ω. Let us show that the spaces Xε = W 1,p (Ωε ) and X = W 1,p (Ω) satisfy all conditions which are essential to the application of the convergence concept in the variable spaces in the sense of Definitions 6.2 and 6.3. Note that conditions (6.1)–(6.4) are the same as the so-called condition of “strong connectedness” of the domains Ωε (see [182]). Let · 0,Xε be an auxiliary norm in W 1,p (Ωε ) coinciding with the restriction to W 1,p (Ωε ) of the norm in Lp (Ωε ) and let · X∗ε be the norm in (W 1,p (Ωε ))∗ . By analogy we set

· 0,W 1,p (Ω) = · Lp (Ω) ,

· X∗ = · (W 1,p (Ω))∗ .

For every ε > 0, we define an operator Qε : W 1,p (Ω) → W 1,p (Ωε ) as a linear mapping of W 1,p (Ω) to W 1,p (Ωε ) such that Qε u = u|Ωε for any u ∈ W 1,p (Ω). It is clear that in this case condition (6.1) is valid. We are now in a position to establish the fulfillment of condition (6.2). Lemma 6.11. Suppose that u ∈ W 1,p (Ω) and lim inf Qε uLp (Ωε ) = 0. ε→0

Then u = 0 almost everywhere on Ω.

(6.18)

166


Proof. In view of conditions (6.15)–(6.17), it follows that for any cube ⊂ Ω, we have (6.19) lim inf | ∩ Ωε | ≥ (1 − ν −n )||. ε→0

From this we conclude that for any non-negative function ϕ ∈ L1 (Ω) the inequality lim inf ϕ dx ≥ (1 − ν −n ) ϕ dx ε→0

Ωε

Ω

is valid. Applying this inequality to ϕ = |u|p and using (6.18), we come to the conclusion that u = 0 almost everywhere in Ω. The lemma is proved.

In what follows, we make use the following notation: If ε > 0 is a fixed j we denote the open ball with center at the value and j ∈ Jε , then by Bε,ν j j j j point xε and radius νrε . Let Eε = Bε,ν \Bεj . We note that, by condition (6.17), j j l ∩ Bε,ν = ∅ for j = l. Therefore, the balls Bε,ν are contained in Ω and Bε,ν j E ⊂ Ω for all ε > 0. ε j∈Jε ε Lemma 6.12. There exist a sequence of linear mappings Pε ∈ L(W 1,p (Ωε ), W 1,p (Ω)) ε>0 ∈ P and a constant C > 0 independent of ε > 0 such that ∇(Pε u)Lp (Bεj ) ≤ C∇uLp (Eεj )

(6.20)

for any u ∈ W 1.m (Ωε ), j ∈ Jε and ε > 0. Proof. Let B1 be the closed ball with center at 0 and radius 1, let Bν be the open ball with center at 0 and radius ν, let E = Bν \ B1 and for any function u ∈ W 1,p (E), let u be the mean value of u on E. Because of the smoothness of the boundary of the domain E, there exists a continuous linear mapping P0 : W 1,p (E) → W 1,p (Bν ) such that (P0 u)|E = u for all u ∈ W 1,p (E). We denote the norm of the mapping P0 by ν0 . Suppose now that P is a mapping of W 1,p (E) into W 1,p (Bν ) such that P u = P0 (u − u) + u ,

∀ u ∈ W 1,p (E).

It is easy to see that the mapping P is linear and continuous and that (P u)|E = u for all u ∈ W 1,p (E). Since, according to [162], there exists ν1 > 0 such that for any function u ∈ W 1,p (E) with 0 mean value on E, uW 1,p (E) ≤ ν1 ∇uLp (E) , it follows that for any u ∈ W 1,p (E) and i ∈ {1, . . . , n}, we have ∂(P u) ≤ ν0 u − u W 1,p (E) ≤ ν0 ν1 ∇uLp (E) . ∂xi p L (Bν )

(6.21)

6.2 Weak convergence in variable Lp -spaces

167

We denote by ν2 the norm of the mapping P . We define the mappings fεj and gεj as follows: If j ∈ Jε , then fεj is a mapping of W 1,p (Eεj ) into W 1,p (E) such that for any u ∈ W 1,p (Eεj ) and x ∈ E, (fεj u)(x) = u(xjε + rεj x); j gεj is a mapping of W 1,p (Bν ) into W 1,p (Bε,ν ) such that for any u ∈ W 1,p (Bν ) j , and x ∈ Bε,ν

(gεj u)(x) = u (rεj )−1 (x − xjε ) .

For any j ∈ Jε , we set Pεj = gεj ◦ P ◦ fεj . It is not hard to see that Pεj is a j continuous linear mapping of W 1,p (Eεj ) into W 1,p (Bε,ν ) and (Pej u) E j = u ε for all u ∈ W 1,p (Eεj ). Moreover, for j ∈ Jε and u ∈ W 1,p (Eεj ), we have j j Pεj uLp (Bε,ν ) ≤ ν2 (1 + rε )uW 1,p (Eεj ) ,

and by (6.21), n

∂(Pεj u) ∂u ≤ ν ν 0 1 ∂xi p j ∂xk p j , L (Bε,ν ) L (Eε )

i = 1, . . . , n.

(6.22)

(6.23)

k=1

Suppose now that Pε is a mapping of W 1,p (Ωε ) to W 1,p (Ω) such that, for 1,p u ∈ W (Ωε ), (Pε u)(x) = u(x) if x ∈ Ωε and (Pε u)(x) = Pε ( u|Eεj ) (x) if x ∈ Bεj . Using (6.22) and (6.23) and condition (6.16), we establish that {Pε }ε>0 ∈ P. We set C = n2 ν0 νn . Then, using (6.23), we find that for any u ∈ W 1,p (Ωε ), and j ∈ Jε , (6.20) holds true. The lemma is proved.

As a result, by Lemma 6.12, we obtain P = ∅ and sup Pε Lε (W 1,p (Ωε ),W 1,p (Ω)) < +∞. ε>0

Moreover, from this, Lemma 6.11, and the obvious inequalities Qε uW 1,p (Ωε ) ≤ uW 1,p (Ω) ,

Qε uLp (Ωε ) ≤ uLp (Ω)

valid for any u ∈ W 1,p (Ω) and ε > 0, it follows that the spaces W 1,p (Ωε ) and W 1,p (Ω) satisfy all conditions that we started with in this section.

6.2 Weak convergence in variable Lp-spaces Let με and μ be Radon measures. Let us recall that a non-negative Borel measure μ on Ω (μ ∈ M+ (Ω)) is said to be a Radon measure if μ(K) < +∞ for any compact subset K of Ω. As usual, we will indicate by Mb (Ω) the space of Radon measures on Ω with finite total variation. Note that Mb (Ω)

168


is a Banach space when equipped with the norm μMb (Ω) := |μ|(Ω). For any μ ∈ Mb (Ω), we adopt the usual notation Lp (Ω, dμ) to indicate the space of p-summable functions with respect to μ on Ω. We omit μ if it is the Lebesgue measure. Hereinafter in this section, we suppose that με μ as ε → 0 in the sense of Definition 2.29, that is, lim ϕ dμε = ϕ dμ, ∀ ϕ ∈ C0 (Ω). (6.24) ε→0

Ω

Ω

We say that a sequence {vε ∈ L (Ω, dμε )}ε>0 is bounded if |vε |p dμε < +∞. lim sup p

ε→0

Ω

Remark 6.13. Note that, in contrast to the situation indicated in the previous section, the spaces Xε = Lp (Ω, dμε ) do not possess properties (6.1)– (6.5). However, as will be shown in Chaps. 11, 13, 14 and 15, the spaces like Lp (Ω, dμε ) are natural for a wide class of OPCs in reticulated and perforated domains. Therefore, in order to provide an asymptotic analysis of such OCPs as ε → 0, we give in this section a convergence formalism in the scale of variable spaces Xε = Lp (Ω, dμε ). Definition 6.14. A bounded sequence {vε ∈ Lp (Ω, dμε )}ε>0 is weakly convergent in the variable space Lp (Ω, dμε ) to a function v ∈ Lp (Ω, dμ) if vε ϕ dμε = vϕ dμ, ∀ ϕ ∈ C0∞ (Ω). (6.25) lim ε→0

Ω

Ω

The main property concerning the weak convergence in Lp (Ω, dμε ) can be expressed as follows. Proposition 6.15. If a sequence {vε ∈ Lp (Ω, dμε )}ε>0 is bounded, then it is compact in the sense of weak convergence in Lp (Ω, dμε ). vε ϕ dμε , ∀ ϕ ∈ C0∞ (Ω), and making use of Proof. Having set Lε (ϕ) = Ω

Hölder inequality, we get

1/p

|Lε (ϕ)| ≤

|vε |p dμε Ω

1/q

≤C

|ϕ|q dμε

1/q |ϕ|q dμε

Ω

≤ C max |ϕ|με (K),

Ω

where K = supp ϕ. From this and Lemma 2.32, we conclude that |Lε (ϕ)| ≤ 2Cμ(K) max |ϕ|

(6.26)

6.2 Weak convergence in variable Lp -spaces

169

for ε > 0 small enough, (i.e., ε ≤ ε0 = ε0 (K)). Since the set T (K) = {ϕ ∈ C0∞ (Ω) : ϕ ⊆ K} is separable with respect to the norm max |ϕ| and {Lε (ϕ)}ε>0 is a uniformly bounded sequence of linear functionals, it follows that (see [258]) there exists ∞ a subsequence of positive numbers {εj }j=1 for which the limit (in the sense of point-by-point convergence) lim Lεj (ϕ) = L(ϕ)

(6.27)

j→∞

is well defined for every ϕ ∈ T (K). Using the fact that the domain Ω can be represented as the combination of a sequence of compact sets Ω = Ki , one gets C0∞ (Ω) = T (Ki ). Therefore, following the diagonal method, we may always suppose that the limit in (6.27) is well defined for every ϕ ∈ C0∞ (Ω). As a result, using (6.26), we have 1/q 1/q |L(ϕ)| ≤ C lim |ϕ|q dμε =C |ϕ|q dμ . ε→0

Ω

Ω

Hence, L(ϕ) is a continuous functional on Lq (Ω, dμ) and it admits the following representation: L(ϕ) =

vϕ dμ, Ω

where v is some element of Lp (Ω, dμ). Thus, taking into account Definition 6.14, the element ∞ v can be taken as the weak limit of the sequence

vεj ∈ Lp (Ω, dμεj ) j=1 . The next property of weak convergence in Lp (Ω, dμε ) shows that the variable Lp -norm is lower semicontinuous with respect to the above convergence. Proposition 6.16. If the sequence {vε ∈ Lp (Ω, dμε )}ε>0 converges weakly to v ∈ Lp (Ω, dμ), then lim inf |vε |p dμε ≥ |v|p dμ. (6.28) ε→0

Ω

Ω

Proof. Indeed, using the Young inequality ab ≤ |a|p /p + |b|q /q and the initial assumptions, we have 1 1 p |vε | dμε ≥ vε ϕ dμε − |ϕ|p dμε , ∀ϕ ∈ C0∞ (Ω), p Ω q Ω Ω 1 1 |vε |p dμε ≥ vϕ dμ − |ϕ|p dμ. lim inf p ε→0 Ω q Ω Ω Since the last inequality is valid for all ϕ ∈ C0∞ (Ω) and C0∞ (Ω) is a dense subset of Lq (Ω, dμ), it follows that this inequality also holds true for ϕ ∈

Lq (Ω, dμ). So, taking ϕ = |v|p−2 v, we come to the required relation (6.28).

170


Definition 6.17. A sequence {vε ∈ Lp (Ω, dμε )}ε>0 is said to be strongly convergent to a function v ∈ Lp (Ω, dμ) if lim vε bε dμε = vb dμ (6.29) ε→0

Ω

Ω

whenever bε b in L (Ω, dμε ) as ε → 0. p

Remark 6.18. Note that in (6.29) the boundedness of the sequence {vε }ε>0 ⊂ Lp (Ω, dμε ) is not required because it is ensured by relation (6.29). Indeed, let us suppose the converse: ξε = vε Lp (Ω,dμε ) → ∞ as ε → 0. Let bε = −(p−1) p−1 ξε vε . Then vε bε dμε = ξε → ∞ as ε → 0. (6.30) lim ε→0

Ω

On the other hand, the sequence {bε }ε>0 is bounded in Lq (Ω, dμ) with q = p/(p − 1) since Ω

(p−1)q

vε

(p−1)q

vε Lp (Ω,dμε )

dμε =

vεp p Ω vε Lp (Ω,dμε )

dμε = 1, ∀ ε > 0.

Therefore, taking into account the compactness criterium, we may assume that {bε }ε>0 is weakly convergent in Lq (Ω, dμε ). Then, due to (6.29), the left-hand side of (6.30) must be finite, and we come into conflict with the initial supposition. We have the following property of the strong convergence in the variable Lp -spaces. Lemma 6.19. The weak convergence of the sequence {vε ∈ Lp (Ω, dμε )}ε>0 to u ∈ Lp (Ω, dμ) and the equality |vε |p dμε = |v|p dμ (6.31) lim ε→0

Ω

Ω

are equivalent to the strong convergence of {vε }ε>0 in Lp (Ω, dμε ) to u ∈ Lp (Ω, dμ). Proof. We begin with the case p = 2. It is easy to verify that the strong convergence implies the weak one and (6.31). Indeed, to do so, we should suppose that bε = ϕ ∈ C0∞ (Ω) in (6.29) and then substitute bε = vε in this relation. Thus, the problem is to prove the converse statement. In view of Proposition 6.15, we may suppose that there exist two values α and β such that (up to the subsequences) vε bε dμε = α, lim |bε |2 dμε = β. lim ε→0

Ω

ε→0

Ω

6.3 Two-scale convergence in variable Lp -spaces

171

Using the lower semicontinuity property (6.28) and (6.31), we obtain 2 [vε + tbε ] dμε = lim |vε |2 dμε + 2tα + t2 β lim ε→0 Ω ε→0 Ω (v + tb)2 dμ ≥ Ω 2 2 = |v| dμ + 2t vb dμ + t |b|2 dμ. Ω

Ω

Ω

From this, we conclude that

2tα + t2 β ≥ 2t

|b|2 dμ ∀ t ∈ R1 .

vb dμ + t2 Ω

Ω

Hence, α = Ω vb dμ. Thereby the strong convergence of vε ∈ L2 (Ω, dμε ) ε>0 is established. As for the case p = 2, it is more technical. For the details, we refer to Zhikov [258].

In general, it is false to assert that the strong convergence of vε in the variable spaces Lp (Ω, dμε ) to v ∈ Lp (Ω, dμ) implies |v − vε |p dμε = 0. lim ε→0

Ω

Nevertheless, the following result is valid. Proposition 6.20. Assume that lim με (Ω) = μ(Ω) > 0;

ε→0

v is a bounded continuous function on Ω(v ∈ C(Ω)). Then

(6.32) (6.33)

vε → v in Lp (Ω, dμε ) =⇒ lim

ε→0

|v − vε |p dμε = 0.

(6.34)

Ω

Proof. Note that in view of Definition 6.17 and our initial assumptions (6.32) and (6.33), the sequence {gε = v}ε>0 is strongly convergent in Lp (Ω, dμε ) to v. Therefore, vε − v → 0 in Lp (Ω, dμε ). Hence, to obtain the required conclusion (6.34), it remains only to apply Lemma 6.19.

6.3 Two-scale convergence in variable Lp-spaces Let Ω be a bounded open subset of Rn . Let = [0; 1)n be the semiopen cube in Rn and {ε} be a sequence of positive numbers converging to 0. In 1989, Nguetseng (see [206]) proved that for each bounded sequence {vε }ε>0

172


in L2 (Ω), there exists a subsequence, still indexed by ε, and a v ∈ L2 (Ω × ) such that x dx → vε (x)φ x, v(x, y)φ(x, y) dydx (6.35) ε Ω Ω for every sufficiently smooth function φ(x, y) which is -periodic in y. In fact, this idea opened a new way for the substantiation of the limit properties of quickly oscillating functions. Following Allaire (see [6]), the convergence defined by (6.35) is called two-scale convergence. Allaire also developed the theory further by studying some general properties of two-scale convergence. Two-scale convergence is now a well-known concept among people who work with homogenization. The concept of two-scale convergence associated with a fixed periodic Borel measure μ ∈ M(Ω) was introduced by Zhikov in [256]. In the case when dμ = dx is the Lebesgue measure on the cell of periodicity, this convergence coincides with the two-scale convergence in the sense of Allaire. In this section, we describe the main properties of weak and strong two-scale convergence with respect to measures in variable Lp -spaces following the results of Zhikov [258]. These convergence concepts will be used in Chap. 11, where we study the asymptotic behavior of an OCP on singular graphs. Let μ be a periodic Borel measure on Rn such that dμ = 1, where

= [0, 1)n is the cell of periodicity for μ. For any Borel B ⊂ Rn , we define the scaling measure με as με (B) = εn μ(ε−1 B),

ε−1 B = {x/ε : ∀ x ∈ B} .

(6.36)

It is clear that με is a ε-periodic measure on Rn and that n dμε = ε dμ = εn μ() = εn .

ε

Hereinafter we suppose that the open bounded domain Ω ⊂ Rn is measurable in the Jordan’s sense. It means that its boundary ∂Ω has a zero n-dimensional Lebesgue measure. For the beginning, we state the following useful mean value property. Theorem 6.21. (Mean value property) Let φ(x, y) = ψ(x)b(y), where ψ is a bounded continuous function on Ω and b is a periodic μ-measurable function on Rn such that b ∈ L1 () = L1per (, dμ). Then x dμε = φ x, φ(x, y) dμ(y)dx = b ψ dx. (6.37) lim ε→0 Ω ε Ω Ω In particular,

b

lim

ε→0

Ω

x ε

dμε = |Ω|

b(y) dμ.

(6.38)


173

Proof. Assume that b ≥ 0. Let k = + k, where k is a vector in Rn with integer components. Then {εk } is a partition of Rn and x x

dμε = dμε ψ(x)b ψ(x)b ε ε Ω εk x

dμε , + ψ(x)b (6.39) ε εk ∩Ω where the first sum is taken over all k such that εk is inside and the second sum is over all k such that εk and ∂Ω have common points. Let us first consider the first sum in (6.39). Since ψ is continuous and bounded, there exist points xk ∈ εk such that x x n dμε = ψ(xk ) dμε = ψ(xk )ε ψ(x)b b b(y) dμ. ε ε εk εk

n This together with the fact ψ(xk )ε → ψ(x) dx as ε → 0 implies that lim

ε→0

ψ(x)b εk

x ε

Ω

dμε =

ψ(x)b(y) dμ(y)dx Ω

= b

ψ(x) dx.

(6.40)

Ω

Let us now consider the second sum in (6.39): x n ψ(x)b |ψ(x)|ε |b(y)| dμM (ε), ≤ max dμ ε x∈Ω ε εk ∩Ω

where M (ε) is the number of cubes εk containing the boundary of Ω. Moreover, εn M (ε) → 0, since the boundary of Ω has Lebesgue measure 0. Thus, x

ψ(x)b (6.41) dμε = 0. lim ε→0 ε εk ∩Ω As a result, the desired relation (6.37) follows by taking (6.39)–(6.41) into account. The proof for b of arbitrary sign follows by dividing b into positive and negative parts and repeating the arguments above.

Let D be the set of functions defined as

∞ ∞ D = φ : φ(x, y) = ψi (x)bi (y), ψi ∈ C0 (Ω), bi ∈ Cper () . (6.42) finite

Then Theorem 6.21 implies that x dμε −→ φ x, φ(x, y) dμ(y) dx as ε → 0 ε Ω Ω

(6.43)

for any φ ∈ D. It is important to obtain a wide class of functions which satisfy the mean value property (6.37). One such important class is defined below.

174


Definition 6.22. Let B be the class of functions φ : Ω × Rn → R satisfying the following conditions: (i) The function x → φ(x, y) is continuous for μ-almost every y. (ii) The function y → φ(x, y) is μ-measurable and -periodic for every x ∈ Ω. (iii) The function y → supx∈Ω |φ(x, y)| is in L1per (, dμ). Theorem 6.23. Let φ ∈ B. Then the limit passage (6.43) holds true. For the proof of this theorem, we refer to Zhikov [258] and Lukkassen and Wall [179]. Note also that as follows from the counterexample in [6], property (i) of Definition 6.22 is essential for the validity of Theorem 6.23. We are now in the position to introduce the two-scale convergence concept in variable Lp -spaces. Definition 6.24. We say that a bounded sequence {uε ∈ Lp (Ω, dμε )} weakly 2 two-scale converges to u ∈ Lp (Ω×, dx×dμ) = Lp (Ω×) (we write uε u) if x lim dμε = uε (x)φ x, u(x, y)φ(x, y) dμ(y) dx (6.44) ε→0 Ω ε Ω for every test function φ of the form φ(x, y) = ψ(x) b(y), where ψ ∈ C0∞ (Ω) ∞ (). and b ∈ Cper Definition 6.25. A bounded sequence {uε ∈ Lp (Ω, dμε )} is said to be strongly 2 two-scale convergent to a function u ∈ Lp (Ω × , dx × dμ) (we write uε → u) if uε vε dμε = u(x, y)v(x, y) dμ(y) dx (6.45) lim ε→0

Ω

Ω

2

whenever vε v in Lq (Ω, dμε ) as ε → 0. We note that we have an equivalent definition of two-scale convergence if we replace the set of test functions by D (D defined as in (6.42)). Moreover, since the set D is dense in Lq (Ω × ), it follows that a weak two-scale limit is uniquely defined. As a direct consequence of the definition of weak two-scale convergence, we have that if {uε ∈ Lp (Ω, dμε )}ε>0 is a bounded sequence weakly two-scale converging to u ∈ Lp (Ω × , dx × dμ), then {uε }ε>0 p p weakly converges in L (Ω, dμε ) to v ∈ L (Ω) in the sense of Definition 6.14, where v(x) = u(x, y) dμ(y). This fact follows by choosing test functions independent of y in (6.44). Moreover, in view of the mean value property, we have 2 η(x) b(x/ε) = uε (x) η(x)b(y), provided η ∈ C(Ω) and b ∈ Lpper (, dμ). Let us study the main properties of such a convergence.


175

Proposition 6.26. If {uε ∈ Lp (Ω, dμε )}ε>0 is a bounded sequence, then it is compact with respect to the weak two-scale convergence. Proof. Let φ ∈ D be a fixed test function. By H¨ older inequality and the fact that {uε ∈ Lp (Ω, dμε )}ε>0 is a bounded sequence, we get that x uε (x)φ x, x dμε ≤ C φ x, ε ε Lq (Ω,dμε ) Ω ≤ Cμε (Ω)1/q φC(Ω×) .

(6.46)

This means that uε can be identified with an element u∗ε in the dual space D∗ of D through the formula x uε (x)φ x,

u∗ε , φD∗ ,D = dμε . ε Ω From (6.46), it follows that u∗ε D∗ =

sup φ C(Ω×) =1

u∗ε , uε D∗ ,D ≤ Cμε (Ω)1/q .

Applying lim supε→0 on both sides in this inequality and taking Theorem 6.21 into account on the right-hand side, we get lim supε→0 u∗ε D∗ ≤ C|Ω|1/q . Further, we note that there exists a u∗ ∈ D∗ such that x dμε = u∗ε , φD∗ ,D → u∗ , φD∗ ,D , ∀ φ ∈ D. uε (x)φ x, (6.47) ε Ω To end of the proof, we have to show that there exists an element u ∈ Lp (Ω × , dx × dμ) such that ∗ u(x, y)φ(x, y) dμ(y) dx.

u , φD∗ ,D = Ω

By (6.46), we have x

u∗ε , φD∗ ,D ≤ C φ x, . ε Lq (Ω,dμε ) Then taking (6.47) and Theorem 6.23 into account and passing to the limit in the last inequality as ε → 0, we get

u∗ , φD∗ ,D ≤ CφLq (Ω×) ,

∀ φ ∈ D.

Since D is dense in Lq (Ω ×), it follows that for a given φ ∈ Lq (Ω ×, dx×μ), ∞ there exists a sequence {φk }k=1 ⊂ D such that φk → φ in Lq (Ω × , dx × μ). This means that we can define an extension u ∗ of u∗ to Lq (Ω × , dx × μ) as

u∗ , φ(Lq (Ω×))∗ ,Lq (Ω×) = lim u∗ , φk D∗ ,D . k→∞

176


Then Riesz representation theorem guarantees the existence of an u ∈ Lp (Ω × , dx × μ) such that ∗ u(x, y)φ(x, y) dμ(y) dx

u , φLp (Ω×),Lq (Ω×) = Ω

for each φ ∈ L (Ω × , dx × μ). In particular, for φ ∈ D we have u∗ , φLp (Ω×),Lq (Ω×) = u(x, y)φ(x, y) dμ(y) dx

u∗ , φD∗ ,D = q

Ω

and this concludes the proof.

Proposition 6.27. If a bounded sequence {uε ∈ Lp (Ω, dμε )}ε>0 weakly twoscale converges to u ∈ Lp (Ω × , dx × dμ), then lim inf uε Lp (Ω,dμε ) ≥ uLp (Ω×) ≥ vLp (Ω) , where v(x) =

ε→0

u(x, y) dμ(y).

∞ {φk }k=1

Proof. Let be a sequence in D such that φk → |u|p−2 u in Lq (Ω × ). By Young’s inequality, we have that ab ≤ |a|p /p + |b|q /q for any real numbers a and b, where q = p/(p − 1). Hence, x x q dμε − (p − 1) |uε |p dμε ≥ p uε φk x, φk x, dμε . ε ε Ω Ω Ω Passing to the limit in this inequality as ε → 0 and taking into account our assumptions, we get u(x, y)φk (x, y) dμ(y) dx lim inf uε pLp (Ω,dμε ) ≥ p ε→0 Ω q − (p − 1) |φk (x, y)| dμ(y) dx. Ω

Since this relation is valid for each k ∈ N, we can pass to the limit as k tends to +∞. As a result, we have p p |u(x, y)| dμ(y) dx lim inf uε Lp (Ω,dμε ) ≥ p ε→0 Ω p |u(x, y)| dμ(y) dx = upLp (Ω×) . − (p − 1) Ω

Finally, by Jensen’s inequality, we obtain the required result p p vLp (Ω) = u(x, y) dμ(y) dx Ω ≤ |u(x, y)|p dμ(y) dx = upLp (Ω×) . Ω


177

Proposition 6.28. Weak two-scale convergence of {uε ∈ Lp (Ω, dμε )}ε>0 to u ∈ Lp (Ω × , dx × dμ) together with the relation lim |uε |p dμε = |u(x, y)|p dμ(y) dx (6.48) ε→0

Ω

Ω

is equivalent to strong two-scale convergence of {uε ∈ Lp (Ω, dμε )} to u. Proof. (i) We start by proving that weak two-scale convergence together with ∞ (6.48) imply strong two-scale convergence. Let {φk }k=1 be a sequence in D such that φk → u in Lp (Ω × ). Let {vε ∈ Lq (Ω, dμε )} be a weakly two-scale convergent sequence to a function v ∈ Lq (Ω × , dx × dμ). Then x dμε = lim lim vε (x)φk x, v(x, y)u(x, y) dμ(y) dx. (6.49) k→∞ ε→0 Ω ε Ω We also have that v(x, y)u(x, y) dμ(y) dx vε (x)uε (x) dμε − Ω Ω x ≤ uε (x) − φk x, vε (x) dμε ε Ω x v(x, y)u(x, y) dμ(y) dx . + φk x, vε (x) dμε − ε Ω Ω

(6.50)

From (6.49) and (6.50), we can obtain lim sup vε (x)uε (x) dμε − v(x, y)u(x, y) dμ(y) dx ε→0 Ω Ω x ≤ lim sup lim sup uε (x) − φk x, vε (x) dμε . (6.51) ε ε→0 k→∞ Ω

Hence, in order to prove (6.45), we have to show that x uε (x) − φk x, vε (x) dμε = 0. lim sup lim sup ε ε→0 k→∞ Ω

(6.52)

To do so, we note that Hölder inequality and the fact that each weakly twoscale convergent sequence is bounded imply x uε (x) − φk x, vε (x) dμε ε Ω · ≤ uε (·) − φk ·, vε (x)Lq (Ω,dμε ) ε Lp (Ω,dμε ) · ≤ C uε (·) − φk ·, . (6.53) ε Lp (Ω,dμε )

178


Then using the Clarkson inequalities for p ≥ 2 and for p ≤ 2, respectively, we have · p u (·) + φ ·, · p · p 1 1 ε k ε uε p + φk ·, − , uε (·) − φk ·, ≤ 2p ε 2 2 ε 2 · q · p 1/(p−1) 1 1 uε p + φk ·, uε (·) − φk ·, ≤ 2p−1 ε 2 2 ε u (·) + φ ·, · q k ε ε − , 2 where all norms are the usual norm in Lp (Ω, dμε ). By (6.48) and

an application of the mean value property to the sequence uε (x) + φk x, xε (which weakly two-scale converges to u + φk ∈ Lp (Ω × ), it follows for p ≥ 2 that · p lim sup uε (·) − φk ·, ε ε→0 u + φk p 1 p p p 1 ≤2 uLp (Ω×) + φk Lp (Ω×) − 2 p 2 2 L (Ω×)

(6.54)

and, for 1 ≤ p ≤ 2, that · q lim sup uε (·) − φk ·, ε ε→0 1/(p−1) u + φk q 1 1 p p−1 p u + φk ≤2 − 2 . (6.55) 2 2 In view of the choice of φk , we have that {(u + φk )/2} converges to u in Lp (Ω × ). By passing to lim sup as k → ∞ on both sides of (6.54)) and (6.55), we come to the conclusion that x lim sup lim sup (6.56) uε (x) − φk x, vε (x) dμε = 0. ε ε→0 k→∞ Ω Thus, the required relation (6.45) follows from (6.53) and (6.56). (ii) It remains to prove that strong two-scale convergence implies a weak two-scale convergence and relation (6.48). Assume that {uε ∈ Lp (Ω, dμε )}ε>0 strongly two-scale converges to u ∈ Lp (Ω × , dx × dμ). By choosing vε (x) = ∞ () in (6.45), we see that strong ψ(x)b(x/ε), where ψ ∈ C0∞ (Ω) and b ∈ Cper two-scale convergence implies a weak two-scale convergence. Now, we will show that strong two-scale convergence implies (6.48). In fact, let vε = |uε |p−1 uε . By the initial assumptions, the sequence {uε ∈ Lp (Ω, dμε )}ε>0 is uniformly bounded, and hence {vε ∈ Lq (Ω, dμε )}ε>0 is a bounded sequence as well. By Proposition 6.26, there exists a subsequence of {vε ∈ Lq (Ω, dμε )}ε>0 (still


179

denoted by ε) which weakly two-scale converges to some v ∈ Lq (Ω × ). According to Definition 6.25, we have |uε |p dμε = uε vε dμε → u(x, y)v(x, y) dμ(y) dx. Ω

Ω

Ω

From this we see that we are done if we show that v = |u|p−2 u. In fact, the function f (t) = |t|p−2 t is monotone, which means that [f (t1 )−f (t2 )](t1 −t2 ) ≥ 0 for every t1 , t2 ∈ Rn . Let φ(x, y) = finite ψk (x)bk (y), where ψk ∈ C0∞ (Ω) ∞ and bk ∈ Cper (). Then [f (φ(x, x/ε)) − f (uε (x))] (φ(x, x/ε) − uε (x)) dμε ≥ 0. Ω

Hence, in the limit we get [f (φ(x, y)) − v(x, y)] (φ(x, y) − u(x, y)) dμ(y) dx ≥ 0. Ω

By density and continuity, this inequality holds for any φ ∈ Lp (Ω ×). Having put φ = u + tw, where w ∈ Lp (Ω × ), for t > 0 we get |u + tw|p−2 (u + tw) − v w dμ(y) dx ≥ 0. (6.57) Ω

Whereas for t < 0, we obtain |u + tw|p−2 (u + tw) − v w dμ(y) dx ≤ 0. Ω

(6.58)

By taking (6.57) and (6.58) into account and letting t tend to 0, we find that |u(x, y)|p−2 (u(x, y) − v(x, y)) w(x, y) dμ(y) dx = 0, ∀ w ∈ Lp (Ω × ). Ω

Thus, v = |u|p−2 u and this concludes the proof.

To extend the class of test functions D in Definition 6.24, we give the following motivation. Let Aq be the set of functions φ ∈ B (B defined as in Definition 6.22) such that the function y → supx∈Ω |φ(x, y)| is in Lqper (, dμ). It is easy to see that Aq ⊂ Lq (Ω × ) and that if φ ∈ Aq , ∞ then φ(x, y)ψ(x)b(y) ∈ Aq ⊂ B for any ψ ∈ C0∞ (Ω) and any b ∈ Cper (). q Moreover, if φ ∈ Aq , then |φ| ∈ Aq and, thus, |φ| ∈ B. As a result, we have the following property. Lemma 6.29. If φ ∈ Aq , then x 2 → φ(x, y) as ε → 0. φ x, ε

(6.59)

180


Proof. Indeed, in view of the of the class A and property (6.38), definition we see that the sequence φ ·, ε· ε>0 as a sequence in the variable space Lq (Ω, dμε ) is bounded. Then, by Theorem 6.23, we have x x ψ(x)b dμε = lim φ x, φ(x, y)ψ(x)b(y) dμ(y) dx, ε→0 Ω ε ε Ω ∞ (), ∀ ψ ∈ C0∞ (Ω), ∀ b ∈ Cper q x lim |φ(x, y)|q dμ(y) dx. φ x, dμε = ε→0 Ω ε Ω

2 The first of this equalities means the weak two-scale convergence φ x, xε φ(x, y). As for the second one, it gives (6.48). Hence, by Proposition 6.28, we just come to the desired conclusion.

As an obvious consequence of this lemma and Proposition 6.28, we have the following result. Lemma 6.30. If a bounded sequence {uε ∈ Lp (Ω, dμε )}ε>0 weakly two-scale converges to u ∈ Lp (Ω × , dx × dμ), then x dμε = uε (x)φ x, u(x, y)φ(x, y) dμ(y) dx, ∀ φ ∈ Aq . lim ε→0 Ω ε Ω Under some additional conditions, strong two-scale convergence can be expressed in a more convenient form. Lemma 6.31. Let {uε ∈ Lp (Ω, dμε )}ε>0 be a bounded sequence which strongly two-scale converges to u ∈ Lp (Ω×, dx×dμ). If the limit function u belongs to the class Ap , then x p (6.60) uε (x) − u x, dμε = 0. ε Ω Proof. We have 2

uε → u(x, y) by the initial assumptions, x 2 → u(x, y) by Lemma 6.29. u x, ε

2 Hence, uε (x) − u x, xε → 0. Therefore, (6.60) follows by taking Proposition 6.28 into account.

To end this section, we compare the usual strong convergence in the sense of Definition 6.17 and strong two-scale convergence. Proposition 6.32. If the sequence {uε ∈ Lp (Ω, dμε )}ε>0 converges strongly 2

to u ∈ Lp (Ω, dμ), then uε → u, that is, in this case, the strong two-scale limit is independent of y.

6.4 Variable Sobolev spaces and two-scale convergence

181

2

Proof. Let uε (x) u(x, y). Then, in view of our assumptions, we have that Lp (Ω, dμ) u = u(x, y) dμ is the strong limit of {uε ∈ Lp (Ω, dμε )}ε>0 in the sense of Definition 6.17. Therefore, due to Lemma 6.19 and Proposition 6.27, we have p p |u| dx = lim |uε | dμε ≥ |u(x, y)|p dμ(y) dx ε→0 Ω Ω Ω |u|p dx. (6.61) ≥ Ω

We clearly have the strict equality here; hence, u(x, y) = 2

u(x, y) dμ. So,

u = u and to obtain the strong two-scale convergence uε → u, it remains only to take (6.61) and Proposition 6.28 into account. This concludes the proof.

6.4 Variable Sobolev spaces and two-scale convergence In this section, we present some definitions and additional facts concerning variable Sobolev spaces and two-scale convergence in such spaces. For details, we refer to Zhikov [256, 257], Bouchitté and Fragala [26] and Lukkassen and Wall [179]. As will be shown in the second part of this book, these notions are essential for the asymptotic analysis of OCPs for PDEs on reticulated domains. Let Ω be an open domain in Rn and let μ be a periodic Borel (e.g., probability) measure of Mp0 (Ω) such that dμ = 1, where = [0, 1)n is 1,p the cell of periodicity for μ. Let Wper (, dμ) be the periodic Sobolev space with respect to the measure μ. Let us recall that (see Definition 2.48) the 1,p (, dμ) is defined as the collection of the first component u of space Wper the closure in Lp (, dμ) × Lp (, dμ) of the set (u, ∇u) : ∀ u ∈ C ∞ () . We denote this closure by W p = W p (, dμ). Thus, the elements of W p are pairs (u, z), where we denote the vector z by ∇μ u and call it a gradient or 1,p μ-gradient of u. As Example 2.50 indicates, the gradient of the Wper (, dμ) μ function is not unique, in general. We recall that Γp (u) denotes the set of all 1,p gradients of a fixed function u ∈ Wper (, dμ), and Γpμ (u) has the structure μ μ μ μ Γp (u) = ∇ u + Γp (0), where ∇ u is some gradient and Γpμ (0) is the set of gradients of 0. In what follows, for the case p = 2 we will use the notation 1,2 1 (, dμ) = Hper (, dμ). Wper 6.4.1 p-Connected measures and their properties As we will see later, the following ergodicity property is important (see [254, 256]).

182


Definition 6.33. A measure μ is said to be p-connected (or ergodic on the 1,p (, dμ) with 0 gradient (0 ∈ Γpμ (u)) torus ) if each function u ∈ Wper is constant μ-almost everywhere. In other words, if there exists a sequence ∞ {uk ∈ Cper ()}k=1 such that |uk − u|p dμ → 0 and |∇uk |p dμ → 0 as k → ∞, (6.62)

then u ≡ const μ-everywhere in . To illustrate this property, we give the following two examples. Example 6.34. Assume that a measure μ is absolutely continuous with respect 1 to the Lebesgue measure in Rn (i.e., dμ = w(x) dx, where w ∈ L ()). We show that this measure is p-connected provided

w−1/(p−1) dx < ∞. Indeed,

let {uk ∈ Cper ()}∞ k=1 be a sequence with properties (6.62). Then

|∇uk | dx ≤ p

w

−1/(p−1)

(p−1)/p 1/p p dx |∇uk | w dx → 0.

By analogy, it can be shown that

(6.63)

|uk − u|p dx → 0. Thus, u belongs to the

usual Sobolev space W 1,1 () and from (6.63), it follows that u is constant a.e. with respect to the Lebesgue measure. Hence, u ≡ const μ-everywhere in . Example 6.35. Let Q be an open, connected in the usual sense set on the torus . Let χQ be its characteristic function. We define the periodic Borel measure μ as 1 dμ = χQ (x) dx. |Q ∪ | It is easy to see that in this case, Γpμ (0) = {0}. Moreover, having taken an arbitrary sequence {uk ∈ Cper ()}∞ k=1 with properties (6.62), we have u is constant a.e. on Q with respect to the Lebesgue measure. Since Q is connected set in the usual sense, it follows that u ≡ const μ-everywhere in . (with respect Definition 6.36. We say that a ∈ L1 (, dμ) is the divergence n to the measure μ) of a vector function b ∈ L1 (, dμ) = L1 (, dμ) in the weak sense and use the notation a = divμ b if the following identity holds: ∞ aϕ dμ = − b · ∇ϕ dμ, ∀ ϕ ∈ Cper (). (6.64)

Note that in view of the periodicity of the measure μ, (6.64) can be equivalently rewritten as aϕ dμ = − b · ∇ϕ dμ, ∀ ϕ ∈ C0∞ (Rn ). (6.65) Rn

Rn


183

Then it is easy to note that each function a ∈ L1(, dμ) admitting the representation a = divμ b has the mean value 0 (i.e.,

a dμ = 0).

As an important property following from ergodicity of the measure μ we have the following result. Theorem 6.37. If the measure μ is p-connected, then the set of functions a ∈ Lq (, dμ) which can be represented as a = divμ b, where b ∈ Lq (, dμ), is dense in the subspace of functions in Lq (, dμ) with mean value 0. Here, 1/p + 1/q = 1. Proof. Let us denote by A the set of functions a ∈ Lq (, dμ) which can be represented as a = divμ b with b ∈ Lq (, dμ). Define the annihilator A⊥ as the set of functions f ∈ Lp (, dμ) such that f dμ = 0 and f a dμ = 0, ∀ a ∈ A.

As follows from Theorem 4.7 in [221], in order to prove that A is dense in Lq (, dμ), it is enough to derive the equality A⊥ = {0}. For this, let us fix f ∈ A⊥ and consider the periodic problem: Find a pair (u, ∇μ u) such that 1,p u ∈ Wper (, dμ),

∇μ u ∈ Γpμ (u),

μ p−2 μ |∇ u| ∇ u · ∇ϕ + |u|p−2 uϕ dμ ∞ = |f |p−2 f ϕ dμ, ∀ ϕ ∈ Cper ().

(6.66)

(6.67)

First, we note that this problem is known to be solvable. Then it is obviously that |f |p−2 f − |u|p−2 u ∈ A, and since f ∈ A⊥ , we have

p |f | − |u|p−2 f u dμ = 0.

Hölder inequality implies that 1/q 1/p |f |p dμ = |u|p−2 f u dμ ≤ |u|(p−1)q dμ |f |p dμ ,

that is,

|f |p dμ ≤

|u|p dμ.

(6.68)

∞ 1,p () in Wper (, dμ), we may choose ϕ = u in (6.67). As By the density of Cper a result, we obtain μ p p (|∇ u| + |u| ) dμ = |f |p−2 f u dμ

≤

1/q 1/p |f |p dμ |u|p dμ ,

184


and hence p p μ p p |u| dμ ≤ |f | dμ and (|∇ u| + |u| ) dμ ≤ |f |p dμ.

This together with (6.68) gives |u|p dμ = |f |p dμ and |∇μ u|p dμ = 0.

By the p-connectedness of μ, it follows that u is constant μ-almost everywhere. This fact together with (6.67) implies that f is constant μ-almost everywhere. f dμ = 0, we deduce that f = 0. The proof is complete.

Since

For other properties of p-connected measures, we refer to [179, 256]. 6.4.2 Degenerate measures p By analogy to Sect. 2.3, we define two spaces: the space Vpot of potential ∞ vectors as the closure of the set ∇ϕ : ϕ ∈ Cper () in Lp (, dμ), and the q of solenoidal vectors as the collection of vectors b ∈ Lq (, dμ) such space Vsol that ∞ b · ∇ϕ dμ = 0, ∀ ϕ ∈ Cper ().

p ⊥ It is clear that b ∈ Vpot whenever b ∈ Lq (, dμ) is a solenoidal vector. Since every norm-closed subspace of Lp (, dμ) is an annihilator of its annihilator (see [221, p. 96]), we have

p ⊥ q Vsol = Vpot ,

⊥

p q Vpot = (Vsol ) .

(6.69)

In particular, for the Hilbert space case, p = 2, we have the orthogonal decomposition (see Sect. 2.3) 2 2 L2 (, dμ) = Vsol ⊕ Vsol . p ) be defined as the closure in Lp (Ω × ) of the linear span of Let Lp (Ω, Vpot ∞ (). Correspondingly, vectors f (x)∇ϕ(y), where f ∈ C0∞ (Ω) and ϕ ∈ Cper q q q L (Ω, Vsol ) is defined as the closure in L (Ω × ) of the linear span of vectors q . By (6.69), it follows that f (x)b(y), where f ∈ C0∞ (Ω) and b ∈ Vsol q ⊥ p Lq (Ω, Vsol ) = Lp (Ω, Vpot ),

p ⊥ q Lp (Ω, Vpot ) = Lq (Ω, Vsol ).

(6.70)

Note that for some measures, a nonzero constant vector may be a potential vector (see the example given below). Let E be the subspace of Rn of such potential vectors and E ⊥ its orthogonal complement. We say that the measure μ is nondegenerate if E = {0}.


185

Example 6.38. Let n = 2 and let μ be the probability -periodic measures on R2 , concentrated on the interval I = [0, 1) × {1/2} and proportional there to 1D Lebesgue measure. Let g = g(x2 ) be a smooth 1-periodic function such that g(1/2) = 0 and g (1/2) = 1. For every k ∈ N, we define the test functions ∞ () as ϕk (x1 , x2 ) = νg(x2 ). Here, ν = const. Then ∇ϕk ∈ Lp () ϕk ∈ Cper ∀ k ∈ N and 1 ∂ϕk p ∂ϕk (x1 , 1/2) p dμ = dx1 = 0, ∀ k ∈ N, ∂x1 0 ∂x1 p p 1 ∂ϕk ∂ϕk (x1 , 1/2) − ν dx1 = 0, ∀ k ∈ N. ∂x2 − ν dμ = ∂x 2 0 p . Thus, μ is an example of a degenerate Hence, the constant vector (0, ν) ∈ Vpot Borel measure.

Now, we formulate some results following from nondegeneration of the measure μ (their proof can be found in [254]). Theorem 6.39. Let m : Rn → R be the functional given by m(ξ) = minp |ξ + v|p dμ. v∈Vpot

(6.71)

Then the following assertions hold true: (i) m is a convex functional. (ii) m(ξ + η) = m(ξ) for any ξ ∈ Rn and η ∈ E. (iii) There exists a constant C > 0 such that m(ξ) ≥ C|ξ|p for ξ ∈ E ⊥ (i.e., m is coercive on E ⊥ ). q if and only if its mean value beTheorem 6.40. A vector b belongs to Vsol longs to E ⊥ .

6.4.3 Two-scale convergence in variable Sobolev spaces Let {uε ∈ C0∞ (Ω)}ε>0 be a sequence of smooth functions such that {uε }ε>0 is bounded in Lp (Ω, dμε ),

(6.72)

{∇uε }ε>0 is bounded in L (Ω, dμε ),

(6.73)

p

2

2

uε (x) u(x, y), ∇uε p (x, y).

(6.74)

We are going to discuss the following questions: When is the two-scale limit u(x, y) independent of y? Is u(x, y) = u(x) in the Sobolev space W 1,p (Ω)? What is the relation between ∇μ u and p (x, y)? As we will see later, the pconnectedness of the measure μ is essential for the answers to these questions.

186


Theorem 6.41. Let μ be a p-connected nondegenerate measure and let {uε ∈ C0∞ (Ω)}ε>0 be a sequence with properties (6.72)–(6.74). Then the weak two-scale limit u is independent of y and belongs to the Sobolev space W01,p (Ω). Moreover, in this case, p (x, y) = ∇μ u(x) + v(x, y),

p where v ∈ Lp (Ω, Vpot ).

(6.75)

Proof. Let functions b ∈ Lqper (, dμ) and a ∈ Lqper (, dμ) be such that a = divμ b. Then (6.65) implies that x x ∇ϕ(x) · b ϕ(x)a (6.76) ε dμε = − dμε , ϕ ∈ C0∞ (Ω). ε ε Ω Ω On the other hand, for ψ ∈ C0∞ (Ω), the partial differentiation gives ε

x dμε ∇ (ψ(x)uε (x)) · b ε Ω x x dμε + ε dμε . =ε uε ∇ψ(x) · b ψ(x)∇uε (x) · b ε ε Ω Ω

This together with (6.76) means that −

x dμε ψ(x)uε (x)a ε Ω x x dμε + ε dμε . =ε uε ∇ψ(x) · b ψ(x)∇uε (x) · b ε ε Ω Ω

By our assumptions, the right-hand side tends to 0 as ε → 0 since {uε }ε>0 and {∇uε }ε>0 are weakly two-scale convergent sequences. Thus, by passing to the limit, we obtain ψ(x)u(x, y)a(y) dμ(y) dx = 0. (6.77) Ω

We note that

a(y) dμ = 0 and by Theorem 6.37 the set of functions a ∈

Lq (, dμ) which can be represented as a = divμ b, where b ∈ Lq (, dμ), is dense in the subspace of functions in Lq (, dμ) with mean value 0. Hence, in view of (6.77), we just conclude that u is independent of y. μ Next, we prove that u ∈ W01,p (Ω) and that p(x, y) = ∇ u(x) + v(x, y), p q ). For b ∈ Vsol with where v ∈ Lp (Ω, Vpot

φ ∈ C ∞ (Ω), we have the identity

b dμ = η = (η1 , . . . , ηn ) and


x

∇(φuε ) · b

0=

ε

Ω

uε ∇φ · b

dμε =

x ε

Ω

φ∇uε · b

dμε + Ω

x ε

187

dμε .

By passing to the limit in the weak two-scale sense, we get u(x)∇φ · b(y) dμ(y) dx + φp (x, y) · b(y) dμ(y) dx. (6.78) 0= Ω

Thus,

Ω

u(x)∇φ · η dx = − Ω

where vb (x) =

vb (x) φ dx

∀ φ ∈ C ∞ (Ω),

Ω

p (x, y) · b(y) dμ(y).

We see that η·∇μ u ∈ Lp (Ω) in the sense of distributions and that η·∇μ u = vb (x). We now use the fact that μ is a nondegenerate measure. In this case, q such that for each η ∈ Rn there exists a vector bk ∈ Vsol u(x)∂φ/∂xk dx = − vbk (x) φ dx, ∀ φ ∈ C ∞ (Ω), k = 1, . . . , n. Ω

Ω

(6.79) Thus, the distributional partial derivatives ∂u/∂xk = vbk of u are in Lp (Ω), that is, u ∈ W 1,p (Ω). Moreover, (6.79) together with the integration by parts formula gives that u ∈ W01,p (Ω) for Ω with a Lipschitz boundary. As a result, (6.78) can now be rewritten as φ(x)p (x, y) · b(y) dμ(y) dx = − u(x)∇φ(x) · b(y) dμ(y) dx Ω

Ω

= Ω

Hence,

φ∇μ u(x) · b(y) dμ(y) dx.

Ω

[p (x, y) − ∇μ u(x)] · φ(x)b(y) dμ(y) dx = 0.

q ) is defined as closure in Lq (Ω × ) of the linear span of Since Lq (Ω, Vsol q vectors f (x)b(y), where f ∈ C0∞ (Ω) and b ∈ Vsol , we obtain that q [p (x, y) − ∇μ u(x)] · w(x, y) dμ(y) dx = 0, ∀ w ∈ Lq (Ω, Vsol ). Ω

To conclude the proof, it remains only to note that from (6.70) we have that p ) or, in other words, [p (x, y) − ∇μ u(x)] belongs to Lp (Ω, Vpot p (x, y) = ∇μ u(x) + v(x, y), p ).

where v ∈ Lp (Ω, Vpot

188


As an obvious consequence of this theorem, we have the following result. Corollary 6.42. Let μ be a p-connected non-degenerate measure and assume that for a sequence {uε ∈ C0∞ (Ω)}ε>0 , the following conditions hold: 2

(i) uε (x) u(x, y) as ε → 0. (ii) ε∇uε L2 (Ω,dμε ) → 0 as ε → 0. Then the weak two-scale limit u(x, y) is independent of y. Note that for the results given above (see Theorem 6.41 and its corollary) the following supposition is essential: μ is a p-connected nondegenerate periodic Borel measure on Rn . In order to analyze the situation without these conditions, we introduce another (dual) definition of Sobolev spaces connected with the Borel measure μ (see [260] for the case p = 2). Definition 6.43. We say that a function u ∈ Lp (, dμ) belongs to the space p 1,p per (, dμ) and a vector v ∈ Vpot is its gradient if W ua dμ = − v · b dμ (6.80)

provided a vector b ∈ Lq (, dμ) and a function a ∈ Lq (, dμ) are connected by the relation divμ b = a with respect to the measure μ (see (6.64)). 1,p (, dμ) is a closed It is clear that the set of pairs (u, v) such that u ∈ W per p p subset of L (, dμ) × L (, dμ). Let us show that this space can be viewed 1,p (, dμ). as the dual definition of the Sobolev space Wper 1,p 1,p (, dμ) coincide. Theorem 6.44. The spaces Wper (, dμ) and W per

Proof. Let (u, ∇μ u) be any pair such that 1,p (, dμ), u ∈ Wper

∇μ u ∈ Γpμ (u). ∞

Then, there exists a sequence {ϕk ∈ Cper ()}k=1 satisfying |ϕk − u|p dμ → 0 and |∇ϕk − ∇μ u|p dμ → 0 as k → ∞.

(6.81)

Hence, (6.81) and (6.64) imply b · ∇ϕk dμ = − aϕk dμ,

b · ∇ u dμ = − μ

au dμ

1,p per (i.e., u ∈ W (, dμ) and ∇μ u is its gradient in the sense of Definition 6.43). 1,p 1,p per (, dμ) = W (, dμ). Find a non-zero pair (u, v) such Assume that Wper

6.5 Approximation of singular measures by smoothing and its application

1,p (, dμ), (6.80) holds true, and that u ∈ W per p−2 |u| uϕ + |v|p−2 v · ∇ϕ dμ = 0,

189

∞ ∀ ϕ ∈ Cper ().

Then, by Definition 6.36, this means that div (|v|p−2 v) = |u|p−2 u. Hence, setting in (6.80) b = |v|p−2 v and a = |u|p−2 u, we obtain p |u| dμ = − |v|p dμ. (6.82)

Since (6.82) implies that u = 0 and v = 0, this concludes the proof.

As a result, Corollary 6.42 can be generalized as follows. Theorem 6.45. Let {uε ∈ C0∞ (Ω)}ε>0 be a sequence such that {uε }ε>0 is bounded in Lp (Ω, dμε ), {ε∇uε }ε>0 is bounded in Lp (Ω, dμε ). Then (up to considering subsequences) we have the following: 2

1,p (, dμ)) as ε → 0. (i) uε (x) u(x, y) ∈ Lp (Ω, Wper 2

(ii) ε∇uε ∇μy u(x, y) as ε → 0.

6.5 Approximation of singular measures by smoothing and its application Due to the dual definition of Sobolev spaces (see Definition 6.43), we can approximate singular measures by smoothing. Let K(x) be a non-negative C0∞ -function such that Rn K(x) dx = 1 and K(x) = K(−x) for all x ∈ Rn . For an arbitrary Radon measure μ ∈ Mb (Rn ), we set dμδ ≡ dKδ (μ) = ρδ (x) dx, where ρδ (x) = δ −n

K Rn

x−y δ

(6.83)

dμ(y).

(6.84)

Note that the measures μδ locally weakly converge in Rn to the measure μ. Indeed, for an arbitrary continuous function ϕ with compact support, we have

190


Rn

ϕ(x) dμ n R x−y dμ(y) dx δ −n ϕ(x)K = δ Rn Rn x−y − dμ(y) dx δ −n ϕ(y)K δ Rn Rn x−y ≤ dμ(y) dx δ −n |ϕ(x) − ϕ(y)|K δ Rn Rn ≤ max |ϕ(x) − ϕ(y)| χspt ϕ (y) dμ(y)

ϕ(x) dμ − δ

|x−y|≤CK δ

≤C

max

|x−y|≤CK δ

Rn

|ϕ(x) − ϕ(y)|

as δ → 0, where the constant CK is chosen in such a way that K(y) = 0 for |y| > CK . The right-hand side of the last formula converges to 0 as δ → 0 since ϕ(x) is uniformly continuous. We also introduce an ordinary smoothing operator by the equality y δ ϕ (x) = δ −n ϕδ (x) = K ϕ(x − y) dy. (6.85) K δ Rn It is easy to verify that in this case, we have the equality δ ϕ (x) dμ(x) = ϕ(x) dμδ (x). Rn

(6.86)

Rn

The following assertion allows us to define a special smoothing operator associated with the measure μ. Lemma 6.46. For every v ∈ L2 (Rn , dμ), there is vδ ∈ L2 (Rn , dμδ ) such that δ vδ (x)ϕ(x) dμ (x) = v(x)ϕδ (x) dμ(x) (6.87) Rn

Rn

for all ϕ ∈ C0 (Rn ). Moreover, vδ → v in L2 (Rn , dμδ ) as δ → 0. Proof. Let us denote Fδ (ϕ) =

v(x)ϕδ (x) dμ(x),

Rn

v2 (x) dμ.

C(v) = Rn

Then for all ϕ ∈ C0 (Rn ), we have Fδ2 (ϕ)

≤ C(v)

Rn

δ 2 ϕ (x) dμ.


191

Jensen’s inequality and property

K(x) dx = 1 and K(x) = K(−x) imply Rn

Rn

2 ϕδ (x) dμ ≤

δ ϕ2 (x) dμ.

Rn

Hence, by (6.86), we get Fδ2 (ϕ)

≤ C(v)

δ ϕ (x) dμ = C(v)

2

Rn

ϕ2 (x) dμδ (x). Rn

Therefore, the linear functional Fδ (ϕ) is bounded on the space L2 (Rn , dμδ ) and, by the Riesz representation theorem, Fδ (ϕ) can be represented in the form vδ (x)ϕ(x) dμδ (x), where vδ ∈ L2 (Rn , dμδ ). Fδ (ϕ) = Rn

Moreover, in this case, we have 2 δ vδ dμ ≤ Rn

v 2 ϕ(x) dμ.

(6.88)

Rn

Since for every ϕ ∈ C0 (Rn ), the functions ϕδ ∈ C0∞ (Rn ) uniformly converge to ϕ by properties of the classical smoothing, it follows that vδ (x)ϕ(x) dμδ (x) = vϕδ dμ → vϕ dμ as δ → 0 Rn

Rn

Rn

(i.e., vδ v in L2 (Rn , dμδ )). The strong convergence in L2 (Rn , dμδ ) immediately follows from (6.88) and Lemma 6.19. This concludes the proof.

In the following assertions, the measure μδ is also defined by n (6.83) and (6.84), but the functions gδ ∈ L2 (Rn , dμδ ), vδ ∈ L2 (Rn , dμδ ) , and uδ ∈ H 1 (Rn , dμδ ) are arbitrary and are not related to the smoothing of any fixed function. Lemma 6.47. Let g = divμ v. Then there are sequences gδ ∈ L2 (Rn , dμδ ) δ→0 and vδ ∈ L2 (Rn , dμδ ) δ→0 such that

δ

divμ vδ = gδ

(6.89)

and gδ → g

in L2 (Rn , dμδ ) as δ → 0,

(6.90)

vδ → v

in L (R , dμ ) as δ → 0.

(6.91)

2

n

δ

192


Proof. Let us show that it suffices to take the same gδ and vδ as in Lemma 6.46. As a result, the strong convergence in (6.90) and (6.91) is guaranteed by this lemma. To do so, we check (6.89). Indeed, in this case, for every ϕ ∈ C0∞ (Rn ), by (6.87), we have vδ · ∇ϕ dμδ = v · (ϕδ ) dμ = gϕδ dμ = gδ ϕ dμδ . Rn

Rn

Rn

Rn

Theorem 6.48. Let uδ ∈ H 1 (Rn , dμδ ) for every δ > 0 and let in L2 (Rn , dμδ ) as δ → 0,

uδ u δ

∇μ uδ z

in L2 (Rn , dμδ ) as δ → 0.

(6.92) (6.93)

Then u ∈ H 1 (Rn , dμ) and z = ∇μ u. Proof. For g and v such that g = divμ v, we use the approxi functions 2 n δ mations gδ ∈ L (R , dμ ) δ→0 and vδ ∈ L2 (Rn , dμδ ) δ→0 constructed in δ

Lemma 6.46, so that divμ vδ = gδ . Then δ δ gδ uδ dμ = vδ · ∇μ uδ dμδ . Rn

Rn

Passing to the limit in this equality as δ → 0 and taking into account the properties of strong convergence of gδ and vδ , we come to the relation gu dμ = v · z dμ. Rn

Rn

To end of this proof, it remains to use the dual definition of Sobolev spaces

(see Definition 6.43). As a result, we have u ∈ H 1 (Rn , dμ) and z = ∇μ u. As a natural application of the above assertions, we provide the following important result on the convergence and approximation of solutions of elliptic equations. Let μ be a Radon measure in Rn and let μδ be a smoothed measure given by (6.83). Let A(·) = [aij (·)]i,j=1,...,n be a function with values in the space of symmetric n × n matrices satisfying the uniform ellipticity condition α|ξ|2Rn ≤ (A(x)ξ, ξ)Rn ≤ α−1 |ξ|2Rn ,

α > 0, ξ ∈ Rn ,

(6.94)

for all x ∈ Rn . Let f be a given element of L2 (Rn , dμ) and let λ > 0. We consider the elliptic equation in Rn − divμ A(x)∇μ u + λu = f in L2 (Rn , dμ) (6.95) and the family of approximating equations of the form


δ δ − divμ Aδ (x)∇μ u + λu = fδ

in L2 (Rn , dμδ ),

193

(6.96)

where Aδ and fδ are defined by the rule (6.87). The main result of this section we formulate as follows. Theorem 6.49. Assume that the approximations {Aδ }δ→0 and {fδ }δ→0 are such that n×n , fδ ∈ L2 (Rn , dμδ ), Aδ ∈ L2 (Rn , dμδ ) α|ξ|2Rn ≤ (Aδ (x)ξ, ξ)Rn ≤ α−1 |ξ|2Rn , ∀ ξ ∈ Rn , ∀ x ∈ Rn , n×n , Aδ → A in L2 (Rn , dμδ ) fδ → f

in L2 (Rn , dμδ ).

Then the solutions uδ of (6.96) strongly converge in the variable L2 (Rn , dμδ ) to the solution u of (6.95) as δ → 0.

δ Proof. Since each of the pairs uδ , ∇μ uδ satisfies the estimate δ

uδ L2 (Rn ,dμδ ) + ∇μ uδ (L2 (Rn ,dμδ ))n ≤ C, it follows that (see Proposition 6.15) for some subsequence δk → 0, we have weak convergence n+1 δ uδ , ∇μ uδ (u0 , z0 ) in L2 (Rn , dμδ ) . By Theorem 6.48, we deduce u0 ∈ H 1 (Rn , dμ) and z0 = ∇μ u0 . It is easy to verify that n δ Aδ (x)∇μ uδ A(x)∇μ u0 in L2 (Rn , dμδ ) . Then, passing to the limit in the integral identity δ Aδ (x)∇μ uδ · ∇ϕ dμδ + λ uδ ϕ dμδ = fδ ϕ dμδ , Rn

Rn

Rn

∀ ϕ ∈ C0∞ (Rn ),

we conclude that (u0 , ∇μ u0 ) is a solution to (6.95). Strong convergence uδ → u0 in L2 (Rn , dμδ ) can be proved by using the trick based on the en

δ ergy convergence. Taking uδ , ∇μ uδ for a test function in the integral identity for (6.96) and taking into account the strong convergence of fδ to f in L2 (Rn , dμδ ), we obtain the chain of equalities δ δ Aδ (x)∇μ uδ · ∇μ uδ + λu2δ dμδ lim δ→0 Rn δ = lim fδ uδ dμ = f u0 dμ δ→0 Rn Rn

= A(x)∇μ u0 · ∇μ u0 + λu20 dμ. (6.97) Rn

194


However, both quadratic terms on the left-hand side of this equality are lower semi-continuous with respect to weak convergence in L2 (Rn , dμδ ) (see Proposition 6.16). Hence, δ δ lim inf Aδ (x)∇μ uδ · ∇μ uδ dμδ ≥ A(x)∇μ u0 · ∇μ u0 dμ, δ→0 n n R R 2 δ λuδ dμ ≥ λu20 dμ. lim inf δ→0

Rn

Rn

Therefore, (6.97) can be valid only if the two last inequalities are, in fact, equalities. Strong convergence uδ → u0 in L2 (Rn , dμδ ) immediately follows from this and Lemma 6.19.

6.6 Two-scale convergence with respect to a variable measure In the previous sections, we supposed that μ was a non-negative periodic fixed Borel measure on Rn such that dμ = 1. Having defined the scaling measure με by (6.36), we introduced the concept of convergence in the variable spaces Lp (Ω, dμε ) and two-scale convergence. However, a typical feature of thin structures is the fact that their geometry depends on two parameters mutually related to one another – namely, ε defines the periodicity cell and εh is the thickness of constituting elements of the structure (see Chaps. 9 and 10). Moreover, for the thin structures the parameters ε and h = h(ε) are usually related by the supposition h(ε) → 0 as ε → 0. In view of this, we begin this section with the following assumptions: Let μh be a -periodic normalized Borel measure depending on the parameter h, that is, dμh = 1 for all h n and μh ∈ M(Ω), where Ω is an open bounded domain in R . Hereinafter, h we suppose that the sequence μ ⊂ M(Ω) converges weakly to a Borel measure μ, that is, h h ∞ μ μ ⇐⇒ lim ψ dμ = ψ dμ, ∀ ψ ∈ Cper (). h→0

For every fixed h, we define ε-periodic scaling measure μhε as follows: μhε (B) = εn μh (ε−1 B) for any Borel set B ⊂ Rn .

(6.98)

Further, we assume that there is a relation h = h(ε) between the parameters ε and h such that limε→0 h(ε) = 0. Then n h(ε) n dμh(ε) = ε dμ (x/ε) = ε dμh(ε) = εn . ε ε

ε

This leads us to the following conclusion.

6.6 Two-scale convergence with respect to a variable measure

195

Lemma 6.50. Assume that an open bounded domain Ω ⊂ Rn is measurable h(ε) converges weakly in M(Ω) to in the Jordan sense. Then the measure με h(ε) the Lebesgue measure on Rn as ε tends to 0: dμε dx, that is, lim ϕ dμh(ε) = ϕ dx, ∀ ϕ ∈ C0 (Ω). (6.99) ε ε→0

Ω

Ω

Proof. Let i = + i, where i is a vector in Rn with integer components. Then {εi } is a partition of Rn for every fixed ε > 0 and

h(ε) h(ε) ϕ(x) dμε = ϕ(x) dμε + ϕ(x) dμh(ε) (6.100) ε Ω

εi ∩Ω

εi

for every ϕ ∈ C0 (Ω), where the first sum is taken over all i such that εi is inside Ω and the second sum is over all i such that εi and ∂Ω have common points. Let us first consider the first sum in (6.100). Since ϕ ∈ C0 (Ω), there exist points xi ∈ εi such that h(ε) h(ε) n ϕ(x) dμε = ϕ(xi ) dμε = ϕ(xi )ε dμh = εn ϕ(xi ). εi

εi

n

This together with the fact that ε ϕ(xi ) is the construction of a Riemann sum implies that

n lim ε ϕ(xi ) → ϕ(x) dx. (6.101) ε→0

Ω

Let us consider the second sum in (6.100). We have h(ε) ϕ(x) dμ |ϕ(x)|εn M (ε), ε ≤ max x∈Ω εi ∩Ω

where M (ε) is the number of cubes εi containing the boundary of Ω. Since εn M (ε) → 0 by the Jordan measurability property of Ω, we conclude that

lim ϕ(x) dμh(ε) = 0. (6.102) ε ε→0

εi ∩Ω

Thus, (6.99) immediately follows by taking (6.100)–(6.102) into account.

Two-scale convergence with respect to a measure – in particular, a variable measure μhε – was introduced and studied by Zhikov [256, 257, 258] and was also considered by Bouchitté and Fragala [26]. In the works of Zhikov and Pastukhova [211, 213, 265], this convergence is crucial for studying homogenization problems of elasticity for thin structures. It is easy to see that for a 2 sequence {uε ∈ Lp (Ω, dμε )}ε→0 , we have uε u(x, y) ∈ Lp (Ω × , dx × dμ) h(ε) provided that με = με . The strong two-scale convergence in this case can be defined in a similar manner. However, as indicated in [258], the main properties of two-scale convergence (see Lemmas 6.29–6.31 and Propositions 6.26–6.28) with respect to the variable two-parametric measure μhε can be established in a sharp form. Indeed, to begin, we consider the mean value property in two-parametric spaces Lp (, dμh ).

196


Theorem 6.51. If a sequence ah ∈ Lp (, dμh ) is such that ah a in Lp (, dμh ) as h → 0, then x lim dμεh(ε) = ϕ(x)ah(ε) ϕ(x)a(y) dμ(y) dx (6.103) ε→0 Ω ε Ω for any continuous bounded function ϕ. Proof. We start with the case when ah ≥ 0 for each h → 0. Then closely following the proof of Theorem 6.21 and using the fact that ah dμh = a dμ (by the weak convergence ah a), lim h→0

we see that (6.40) can be rewritten in the form x

lim dμhε = lim ϕ(x)ah ϕ(x)ah (y) dμh (y) dx ε→0 h→0 Ω ε εk = ϕ(x)a(y) dμ(y) dx. (6.104) Ω

So, proceeding as in the proof of Theorem 6.21, we come to the desired relation (6.103). h ah = As for the general h case, we can represent each of a in the form h h p a1 − a2 , where ai ≥ 0 are uniformly bounded sequences in L (, dμh ). Having assumed that ah1 a1 , ah2 a2 in Lp (, dμh ), where a = a1 − a2 and applying the previous arguments, we obtain (6.103).

Definition 6.52. We say that a bounded sequence uε,h ∈ Lp (Ω, dμhε ) , that is, sup uε,h Lp (Ω,dμhε ) < +∞, ε→0

weakly two-scale converges to a function u(x, y) ∈ Lp (Ω × , dx × dμ) if x h uε,h (x)ϕ(x)b u(x, y)ϕ(x)b(y) dμ(y) dx (6.105) lim dμε = ε→0 Ω ε Ω ∞ for every test function ϕ ∈ C0∞ (Ω) and b ∈ Cper (). Definition 6.53. We say that a sequence uε,h ∈ Lp (Ω, dμhε ) strongly twoscale converges to a function u(x, y) ∈ Lp (Ω × , dx × dμ) if uε,h (x)vε,h (x) dμhε = u(x, y)v(x, y) dμ(y) dx (6.106) lim ε→0

Ω

Ω

2

whenever vε,h (x) v(x, y). Let us list here certain general properties of the two-scale convergence, which are the obvious consequence of the results of Sect. 6.3.

6.6 Two-scale convergence with respect to a variable measure

197

Any sequence bounded in Lp (Ω, dμhε ) contains a weakly two-scale convergent subsequence. 2 (ii) If uε,h (x) u(x, y), then lim inf |uε,h |p dμhε ≥ |u(x, y)|p dμ(y) dx. (i)

ε→0

Ω

Ω 2

(iii) Strong two-scale convergence uε,h (x) → u(x, y) is equivalent to 2

uε,h (x) u(x, y) and |uε,h |p dμhε = |u(x, y)|p dμ(y) dx. lim

ε→0

Ω

Ω

2

(iv) If uε,h (x) u(x, y), then uε,h

(6.107)

u(x, y) dμ in Lp (Ω, dμhε ).

However, this collection can be extended by some additional properties of two-scale convergence in Lp (, dμhε ). 2

Proposition 6.54. If ch c in Lp (, dμh ), then ch (x/ε) c(y). ∞ Proof. Let ϕ ∈ C0∞ (Ω) and b ∈ Cper () be any fixed functions. Then, in view h of Definition 6.14, we have c b cb in Lp (, dμh ). Hence, by the mean value property (see (6.103)), we obtain x h x h b ϕ(x) dμε = ϕ(x)c c(y)b(y)ϕ(x) dμ(y) dx. lim ε→0 Ω ε ε Ω

It remains to take Definition 6.52 into account.

2

Proposition 6.55. If ch → c in Lp (, dμh ), then ch (x/ε) → c(y). 2

Proof. By the previous proposition, we have ch (x/ε) c(y). So, it remans to verify the equality p h x lim |c(y)|p dμ(y) dx. c dμhε = ε→0 Ω ε Ω However, this fact immediately follows from (6.103) since lim |ch |p dμh = |c|p dμ, h→0

by the property of strong convergence ch → c in Lp (, dμh ). To conclude, we note that the class of test functions in Definition 6.52 is minimal. However, as follows from the proposition given below, this class can be essentially extended with preservation of the convergence in (6.105).

198

6 Convergence Concepts in Variable Spaces 2

Proposition 6.56. Let uε,h (x) u(x, y) in the sense of Definition 6.52 and let bh → b in Lq (, dμh ), 1/p + 1/q = 1. Then x lim dμhε uε,h (x)ϕ(x)bh ε→0 Ω ε = u(x, y)ϕ(x)b(y) dμ(y) dx, ∀ ϕ ∈ C0∞ (Ω). (6.108) Ω

The proof of this result can be found in [258].

6.7 Some properties of the strong convergence in spaces L2 (, dμh) associated with thin periodic structures To illustrate some specific properties of the strong convergence in variable spaces Lp (, dμh ), we consider a particular case of the scaling measure μhε associated with periodic thin structures in the plane. This highly contrasting medium depends on two small related parameters ε and h which control the size of a periodicity cell and thickness of the reinforcement. We assume that the periodic thin structure is contained in an open bounded domain Ω ⊂ R2 with a Lipschitz boundary ∂Ω. Let F be a periodic graph in the plane that consists of segments. Let = [0, 1)2 be its periodicity cell. We call this graph a singular (or an infinitely thin) grid. We define a thin grid F h applying the procedure of h-thickening to the graph F . It means that F h can be obtained if any component I of F is replaced by a strip I h of width h whose median coincides with I. Sometimes it is convenient to include into a thin grid F h disks of radius h/2 centered at nodes of the grid F . The procedure of hthickening becomes simpler if the grid F is composed by several families of parallel straight lines. In this case, we obtain the grid F h by replacing all of the lines l of the grid F by the infinite strip of width h with median l (see Fig. 6.1). Denote by λ a measure concentrated on F and proportional to the linear Lebesgue measure. Denote by λh a measure concentrated on F h and proportional to the planar Lebesgue measure. It is assumed that the measures λ h and λ are normed by the following condition: dλ = dλh = 1. Clearly, λh λ as h → 0, that is, lim ϕ dλh = ϕ dλ h→0

∞ for any function ϕ ∈ Cper ().

h The homothetic contraction of the plane by a factor of ε−1 takes the grids hF h h h and F to Fε and Fε , respectively. In particular, Fε = εF = εx : x ∈ F . Note that in this case, Fεh consists of strips whose thickness is εh. Each of the grids Fεh and Fε can be associated with ε-periodic measure λhε and λε , respectively, such that

6.7 Some properties of the strong convergence in L2 (, dμh )

199

Fig. 6.1. The procedure of h-thickening for the graph F

λhε (B) = ε2 λh (B) and λε (B) = ε2 λ(B)

for any planar Borel set B.

As usual, we relate parameters h and ε assuming that h = h(ε) → 0 as ε → 0. Since λhε (ε) = λε () = ε2 , it follows that (see Lemma 6.50) the measures λhε and λε converge weakly to the planar Lebesgue measure as ε → 0, that is, ϕ dλhε → ϕ dx, ϕ dλε → ϕ dx, ∀ ϕ ∈ C0∞ (R2 ). R2

R2

R2

R2

We define a thin structure Ωε,h as Ωε,h = Ω ∩ Fεh . Following in many aspects Pastuchova (see [213]), we give some additional properties of twoscale convergence variable spaces L2 (Ω, dλhε ). ∞ (), the Poincaré inequalLemma 6.57. For any periodic functions ϕ ∈ Cper ity ϕ2 dλh ≤ C |∇ϕ|2 dx, ϕ dλh = 0, ∀ h > 0, (6.109)

holds true, where the constant C depends only on the geometry of the grid F . Proof. Let I h = (0, h)×(0, 1) be any strip in the square . Then the following inequality (see [213]) is obvious:

2 1 ∞ ϕ + |∇ϕ|2 dx, ϕ ∈ Cper ϕ2 dx ≤ C1 (), C1 = C1 (). (6.110) h Ih This inequality combined with the standard Poincaré inequality ∞ ϕ2 dx ≤ C2 |∇ϕ|2 dx, ϕ ∈ Cper (), ϕ dx = 0

200


implies that 1 ϕ2 dx ≤ C3 |∇ϕ|2 dx, h Ih

∞ ϕ ∈ Cper (),

ϕ dx = 0

(6.111)

uniformly in h. Since the set F h ∩ can be presented as a collection of strips in , from (6.111) we deduce that ∞ ϕ2 dλh ≤ C3 |∇ϕ|2 dx, ϕ ∈ Cper (), ϕ dx = 0,

only on the geometry of the set F h ∩. If we set ϕ = v −m, where C3 depends where m = v dx, then

2

(v − m) dλh ≤ C

|∇v|2 dx,

∞ v ∈ Cper ().

This proves the Poincaré inequality (6.109).

Lemma 6.58. There exists a constant C0 depending only on the geometry of the grid F such that for any compactly supported function v ∈ C0∞ (R2 ) the following inequality holds true:

2 2 h v dλε ≤ C0 (6.112) v + ε2 |∇v|2 dx, v ∈ C0∞ (R2 ). R2

R2

Proof. In view of (6.110) and the fact that the measure λh can be represented as the weighed sum of planar Lebesgue measures supported on different strips I h in , we have

2 ϕ + |∇ϕ|2 dx, ϕ ∈ C ∞ (), C0 = C0 (F). (6.113) ϕ2 dλh ≤ C0

After a homothetic contraction, the latter is transformed into the following inequality for the square ε :

2 ϕ2 dλhε ≤ C0 ϕ + ε2 |∇ϕ|2 dx, ϕ ∈ C ∞ (ε). ε

ε

Decomposing the plane into disjoint semiopen squares, R2 = i εi , we obtain

ϕ2 + ε2 |∇ϕ|2 dx v2 dλhε = ϕ2 dλhε ≤ C0 R2

i

≤ C0 Thus, (6.112) is valid.

ε

R2

i

ε

2 ϕ + ε2 |∇ϕ|2 dx,

∀ v ∈ C0∞ (R2 ).

6.7 Some properties of the strong convergence in L2 (, dμh )

201

Let us apply Lemmas 6.57 and 6.58 to establish some specific properties of convergence in the space L2 (Ω, dλhε ). Proposition 6.59. Let v be any element of H01 (Ω). Then v 2 dλhε → v 2 dx as ε → 0. Ω

(6.114)

Ω

Proof. Indeed, in the case when v ∈ C0∞ (Ω), (6.114) is the direct consequence of the weak convergence dλhε dx. For an arbitrary v ∈ H01 (Ω), we have the following estimate: 2 2 v 2 dλhε − v − vδ2 dλhε v dx ≤ Ω Ω Ω 2 h 2 2 2 + vδ dλε − vδ dx + vδ dx − v dx , Ω

Ω

Ω

Ω

where an element vδ ∈ C0∞ (Ω) is such that v − vδ H01 (Ω) < δ. Using (6.112) and H¨ older inequality, we get 2 v − v 2 dλh ≤ v + vδ L2 (Ω,dλh ) v − vδ L2 (Ω,dλh ) δ ε ε ε Ω

≤ Cv − vδ H 1 (Ω) ≤ C1 δ. Moreover, by weak convergence dλhε dx, we conclude 2 ≤ δ for ε small enough. vδ2 dλhε − v dx δ Ω

Ω

2 2 Since vδ dx − v dx < Cδ by the estimate v −vδ H01 (Ω) < δ, it follows Ω Ω that v 2 dx ≤ Cδ lim sup v 2 dλhε − ε→0 Ω

Ω

for an arbitrary δ > 0, where the constant C does not depend on δ. This completes the proof of (6.114).

Proposition 6.60. Let vε ∈ H01 (Ω) ε>0 be a sequence such that in H01 (Ω) as ε → 0.

(6.115)

in L2 (Ω, dλhε ) as ε → 0.

(6.116)

vε v Then vε → v

Proof. By strong convergence in L2 (Ω, dλhε ) (see Proposition 6.16) and the property of lower semicontinuity (see Lemma 6.19), to establish convergence (6.116) it is enough to show that

202


lim

ε→0

vϕ dx, ∀ ϕ ∈ C0∞ (Ω), |vε |2 dλhε ≤ v2 dx. lim

vε ϕ dλhε Ω

=

(6.117)

Ω

ε→0

Ω

(6.118)

Ω

Let vδ be an element of C0∞ (Ω) such that v − vδ H01 (Ω) < δ for a given δ > 0. Then vε ϕ dλhε − vϕ dx = (vε − v)ϕ dλhε + (v − vδ )ϕ dλhε Ω Ω Ω Ω h + vδ ϕ dλε − vδ ϕ dx Ω Ω + (vδ − v)ϕ dx. (6.119) Ω

Applying H¨ older inequality and (6.112), we can estimate the second and fourth terms in (6.119) as follows: (v − vδ )ϕ dλhε ≤ Cδ δ, (vδ − v)ϕ dx ≤ Cδ δ. Ω

Ω

It follows from (6.115) and the facts that

|vε − v|2 dλhε ≤ C0 (vε − v)2 + ε2 |∇(vε − v)|2 dx 2 Ω R (6.120) 2 2 (vε − v) dx + O(ε ) = o(1), = C0 R2 vδ ϕ dλhε −→ vδ ϕ dx due to the convergence dλhε dx Ω

Ω

that the limits of the first and third terms in (6.119) as ε → 0 are equal to 0. Hence, h vε ϕ dλε − vϕ dx ≤ Cδ δ. lim sup ε→0

Ω

Ω

Since δ > 0 is an arbitrary value, (6.117) is proved. As for the validity of (6.117), we apply (6.120) and (6.114). One gets |vε |2 dλhε ≤ lim |vε2 − v 2 | dλhε + lim v 2 dλhε lim ε→0 Ω ε→0 Ω ε→0 Ω 2 ≤ C lim vε − vL2 (Ω,dλhε ) + v dx = v 2 dx. ε→0

Ω

Ω

This concludes the proof.

We are now in the position to establish the compactness principle for the strong convergence in L2 (Ω, dλhε ).

6.8 On approximative properties of Hilbert spaces

203

Theorem 6.61. Let {vε,h ∈ C0∞ (Ω)}ε→0 be a sequence such that sup vε,h L2 (Ω) < +∞,

sup ∇vε,h L2 (Ω) < +∞.

ε→0

ε→0

Then this sequence is relatively compact with respect to the strong convergence in L2 (Ω, dλhε ). Proof. Due to the Banach–Alaoglu theorem, there exist a subsequence of {vε,h ∈ C0∞ (Ω)}ε→0 (still indexed by ε) and an element v ∈ H01 (Ω) such that vε v in H01 (Ω) as ε → 0. Then by Proposition 6.60, we come to the required conclusion.

To end this section, we cite a couple of results which turn out to be useful for applications. Lemma 6.62. If we replace H01 (Ω) by H 1 (Ω) in (6.114) and Proposition 6.60, then the conclusions of Propositions 6.59 and 6.60 remain valid. Proof. Taking into account Lemma 1.1 in [258], we have that the weak convergence of measures dλhε dx possesses the property lim ϕ dλhε = ϕ dx, ∀ ϕ ∈ C(Ω). (6.121) ε→0

Ω

Ω

It follows that it is enough to consider vδ ∈ C ∞ (Ω) to deduce (6.114) for v ∈ H 1 (Ω). The remaining reasoning is preserved. The lemma is proved.

Lemma 6.63. If vε → v in L2 (Ω, dλhε ) and v ∈ H 1 (Ω), then |vε − v|2 dλhε = 0. lim ε→0

Ω

Proof. We note that due to Lemma 6.62, the still sequence gε = v ∈ H 1 (Ω) is strongly convergent to v in L2 (Ω, dλhε ). Therefore, (vε − v) → 0 in L2 (Ω, dλhε ). Hence, to obtain the required conclusion, it remains only to use the criterium of strong convergence in L2 (Ω, dλhε ) (see Lemma 6.19).

6.8 On approximative properties of Hilbert spaces with respect to a periodic Borel measure μh In this section, we discuss the approximation properties connecting a periodic Borel measure μh with its weak-limit measure μ as h → 0. In the case p = 2, Zhikov showed that (see [257]) these properties are necessary for the averaging of a wide class of variational problems in W01,2 (Ω, dμhε ) with two small parameters arising in thin periodic structures with thickness tending to 0. To make this consideration more precise, let us assume that μ is a periodic Borel measure on Rn , = [0, 1)n is its periodicity cell and dμ = 1.

204


1,2 We associate with the measure μ the Sobolev space Wper (, dμ) defined in Definition 2.48 as the collection of the first components u of the closure in Lp (, dμ) × Lp (, dμ) of the set (u, ∇u) : ∀ u ∈ C ∞ () . Hereinafter, we 1 will use the following standard notation for this space: Hper (, dμ). Let μh be a -periodic normalized Borel measure depending on the parameter h such that dμh = 1 for all h and suppose μh converges weakly to the Borel measure μ as h → 0. Now, we formulate the approximation conditions necessary for the asymptotic analysis of the variational problems with two small parameters.

Definition 6.64. We say that the family of -periodic normalized Borel measures μh h>0 with limit property μh μ as h → 0 possesses the approximative conditions if the following hold: ∞ () with 0 mean value (a) For any function a ∈ Cper

a = a dμ = 0 ,

there are a function ah ∈ L2 (, dμh ) and a vector bh ∈ L2 (, dμh ) such that h ∞ divμ bh = ah ⇐⇒ ah ϕ dμh = − bh · ∇ϕ dμh ∀ ϕ ∈ Cper (), 2

a −→ a in L (, dμ ), h

h

b −→ b in L2 (, dμh ). h

2 2 (b) For any vector b ∈ Vsol (, dμ), there is a vector bh ∈ Vsol (, dμh ) such that bh → b in L2 (, dμh ).

It should be emphasized that the main point of these approximative conditions is to apply the strong convergence in variable spaces L2 (, dμh ) and L2 (, dμh ). Note also that the approximative properties are closely connected with the possibility of passing to the limit in the variable Sobolev space 1 1 (, dμh ). This means that if uh ∈ Hper (, dμh ) for all h > 0 and Hper uh u in L2 (, dμh ),

h

∇μ uh v in L2 (, dμh ),

(6.122)

then 1 (, dμ) and v = ∇μ u. u ∈ Hper

(6.123)

As was shown in [257] if the limit measure μ satisfies the Poincaré inequality 2 2 ∞ ϕ dμ ≤ C |∇ϕ| dμ, ϕ ∈ Cper (), ϕ dμ = 0,

then the approximate conditions (a) and (b) guarantee the passage to the 1 1,2 limit in the Sobolev space Hper (, dμh ) = Wper (, dμh ). The main result of this section can be stated as follows (see [230]):

6.8 On approximative properties of Hilbert spaces

Theorem 6.65. Assume that the Poincaré inequality ∞ |ϕ|2 dμh ≤ C |∇ϕ|2 dμh , ϕ ∈ Cper (), ϕ dμh = 0

205

(6.124)

holds true uniformly with respect to the parameter h and that the passage 1 (, dμh ) is possible. Then the to the limit in the variable Sobolev space Hper approximative conditions (a) and (b) are valid. ∞ Proof. Let us prove (a). For any function a ∈ Cper () such that a dμ = 0, a dμh . Then, taking into account the we take the sequence ah = a −

h>0

weak convergence μh μ and Lemma 6.62, we have lim ah ϕ dμh = lim aϕ dμh h→0 h→0 h − lim a dμ lim ϕ μh h→0 h→0 ∞ = aϕ dμ, ∀ ϕ ∈ Cper (),

ah 2L2 (,dμh )

=

a2L2 (,dμh )

−

2 h

a dμ

→ a2L2 (,dμ) .

Hence, due to Lemma 6.19, we conclude that ah → a strongly in L2 (Ω, dμh ). Let us consider the following parameterized periodic problem: Find a pair h (uh , ∇μ uh ) such that h

h

1 (, dμh ), ∇μ uh ∈ Γ2μ (u), uh ∈ Hper h

divμ ∇uh = ah .

The variational formulation of this problem is h ∞ ∇μ uh · ∇ϕ dμh = ah ϕ dμh , ∀ ϕ ∈ Cper ().

(6.125) (6.126)

(6.127)

It is well known that this problem is solvable [261]. On the other hand, the h uniform Poincaré inequality (6.124) implies that uh h>0 and ∇μ uh h>0 are bounded in L2 (, dμh ) and L2 (, dμh ), respectively. Without loss of gen1 erality, we can assume that there exists u ∈ Hper (, dμ) such that conditions h (6.122) and (6.123) hold true. Let us show that ∇μ uh → ∇μ u strongly in 2 h L (, dμ ). For this, we pass to the limit in (6.127). As a result, we obtain ∞ ∇u · ∇ϕ dμ = aϕ dμ, ∀ ϕ ∈ Cper (). (6.128)

206


Now, setting the test function ϕ = uh in (6.127) and using the strong convergence ah → a, we get lim ∇μ uh · ∇uh dμh = au dμ. h→0

If we compare this identity with (6.128) for ϕ = u, we obtain h |∇μ uh |2 dμh = |∇μ u|2 dμ. lim h→0

h

This together with (6.122) implies the strong convergence ∇μ uh → ∇μ u in h L2 (, dμh ). Setting bh = ∇μ uh and b = ∇μ u, we deduce the validity of the approximative property (a). h 2 ˜ ∈ (, dμ) be any fixed vector and let b Now we prove (b). Let b ∈ Vsol h ˜ → b in L2 (, dμh ). For every L2 (, dμh ) h>0 be a sequence such that b h > 0, we consider the variational problem 1 (, dμh ), uh ∈ Hper

h h h ˜ . divμ ∇uh = divμ b

Then the integral identity μh h h ˜ h · ∇ϕ dμh , b ∇ u · ∇ϕ dμ =

∞ ∀ ϕ ∈ Cper ()

(6.129)

h and the uniform Poincaré inequality imply that uh h>0 and ∇μ uh h>0 are bounded sequences in L2 (, dμh ) and L2 (, dμh ), respectively. By the initial suppositions, h

∇μ uh ∇μ u in L2 (, dμh ) and

1 u ∈ Hper (, dμ).

2 2 (, dμ) and ∇μ u ∈ Vpot (, dμ), Therefore, in view of the fact that b ∈ Vsol we obtain h h ˜ h · ∇μh uh dμh b lim ∇μ uh · ∇μ uh dμh = lim h→0 h→0 = b · ∇μ u dμ = 0. (6.130)

˜ h −∇μh uh . Indeed, To obtain the desired property (b) it remains to set bh = b in this case, we have the following: 2 bh · ∇ϕ dμh = 0 by (6.129). Hence, bh ∈ Vsol (, dμ). (i)

6.9 On the homothetic mean value property

207

h

(ii) Since ∇μ uh ∇μ u in L2 (, dμh ) and h 0 = lim |∇μ uh |2 dμh ≥ |∇μ u|2 dμ h→0

(by (6.130)), it follows that ∇ u = 0. Hence, bh b in L2 (, dμh ). Moreover, due to the equality h 2 h 2 ˜ ˜ h · ∇μh uh dμh b lim b L2 (,dμh ) = lim b L2 (,dμh ) + 2 lim μ

h→0

h→0

+ lim

h→0

=

h→0

uh 2L2 (,dμh )

˜ h 2 2 = lim b L (,dμh ) h→0

b2L2 (,dμ) ,

we conclude that bh → b in L2 (, dμh ). This concludes the proof.

6.9 The homothetic mean value property on periodically perforated domains In this section, we study the limit properties of one class of periodic functions defined on ε-periodically perforated domain as ε tends to 0. The principal feature is the fact that the holes have a critical size with respect to ε-sized mesh of periodicity. We will use these results in Chap. 12 to the construction of suboptimal controls for a class of boundary OCPs in ε-periodically perforated domains with small holes. Let Ω ⊂ Rn , n ≥ 2, be a bounded open domain and let ε be a small positive parameter. We define a perforated domain Ωε as follows: Let Y = [−1/2, +1/2)n ; Q and K be compact subsets of Y such that 0 ∈ intK ∩ ∂Q, Θε = {k = (k1 , k2 , . . . , kn ) ∈ Zn : (εY + ε k) ∩ Ω = ∅} , Yε = εn/(n−1) Q + ε k , {ε(Y + k)} , Tε = k∈Θε

Sε = ΓεD =

k∈Θε n/(n−2)

ε

K ∩ ∂(ε

Q),

n ≥ 3,

n/(n−1)

Q), n = 2,

n/(n−1)

exp (−1/ε )K ∩ ∂(ε 2

{Sε + ε k} ∩ Ω,

ΓεN = ∂Tε \ ΓεD ∩ Ω.

k∈Θε

Then we set Ωε = Ω \ Tε . As was mentioned earlier, the principal feature of this perforated domain is the fact that the size of the holes Qε + ε k tends to 0 as ε → 0 and their boundaries ΓεD , ΓεN are not proportional to the size of the periodicity cell εY .

208


Fig. 6.2. Example of perforation scheme

Our aim is to study the limit properties of the classes of periodic functions x x λ,h λ,h dμε dνε ϕ(x)g ϕ(x)g , ε λ(ε) ε h(ε) Ω Ω as ε tends to 0 and to obtain an explicit form of these limits. Here, 0 < λ 1, where Qς = {ςx, ∀ x = (x1 , . . . , xn ) ∈ Q} is the homothetic stretching of Q by a factor of ς. Hence, |K ∩ ∂Qς |H = 0 for all ς > 1.


209

6.9.1 A measure approach to the description of the sets Ωε We will describe of the geometry of the perforated domain Ωε in terms of singular periodic Borel measures on Rn . Let us denote by K λ and Qh the homothetic contractions of the sets K and Q by factors λ−1 and h−1 , respectively. In what follows, it is always assumed that 0 < λ 0, and lim ς(ε) = +∞. ε→0

We are interested in the limit behavior of the sequence {|K ∩ ∂Qς (ε)|H } as ε → 0. We recall that each of the sets Qς (ε) = {ς(ε)x, ∀ x = (x1 , . . . , xn ) ∈ Q} is the homothetic stretching of Q by a factor of ς(ε). Proposition 6.66. There exist an open cone Λ ⊂ {x = (x1 , . . . , xn ) ∈ Rn : x1 > 0} such that lim |K ∩ ∂Qς (ε)|H = |K ∩ ∂Λ|H .

ε→0


211

Proof. Indeed, by initial assumptions, the origin is a Lipschitz point of the boundary ∂Q and int Q is a strongly connected set in classical sense. Hence, there is a neighborhood U(0) such that U(0) ∩ intQ is a convex set [53, 106]. Hence, the set Λ = {x ∈ Rn : x ∈ t [ U(0) ∩ intQ] ∀ t ∈ (0, +∞)} is a nonempty open cone. Assume that the origin is a Lipschitz point of ∂Q, but does not belong to a smooth part of the boundary ∂Q. Then the inclusion K ∩ Λ ⊂ K ∩ Qς(ε) holds true for ε small enough, and it immediately implies the existence of a value ε0 > 0 such that |K ∩ ∂Λ|H = |K ∩ ∂Qς (ε)|H ,

∀ ε < ε0 .

If a part of the boundary ∂Q containing the origin is smooth, it follows that there is a neighborhood U(0) of the origin such that U(0) ∩ ∂Q is a graph of a smooth function whose epigraph contains U(0) ∩ Q. Thus, we may always suppose that there is a function Ψ : Rn−1 → R≥ such that Ψ ∈ C0∞ (Rn−1 ) and x1 = Ψ (x2 , . . . , xn ) for every x = (x1 , x2 , . . . , xn ) ∈ U(0) ∩ ∂Q. Let Λ = {x ∈ Rn : x1 > 0}. Then ∂Λ = {x ∈ Rn : x1 = 0} and, taking into account the homothetic transform, we have that x = (x1 , x2 , . . . , xn ) ∈ K ∩ ς(ε)∂Q for small enough ε if and only if x ∈ ς(ε) (U(0) ∩ ∂Q). Hence, x1 =

1 Ψ (ς(ε)x2 , . . . , ς(ε)xn ) ς(ε)

for every

x ∈ K ∩ ς(ε)∂Q.

Since limε→0 ς −1 (ε)Ψ (ς(ε)x2 , . . . , ς(ε)xn ) = 0 by definition of the Hausdorff measure Hn−1 , we immediately obtain the required result.

Remark 6.67. As follows from the proof of the last proposition, the cone Λ can be recovered in an explicit form for the case when the origin belongs to a smooth part of the boundary ∂Q. Moreover, as an evident consequence of Proposition 6.66 and (6.137), we have lim σ(ε)/εn = |∂Q|H .

ε→0

(6.138)

In a similar way, the following statement can be proved. Proposition 6.68. Let {ρε ∈ R}ε>0 be a numerical sequence such that ρε = |A \ Qς(ε) |/|A|,

∀ ε > 0.

(6.139)

Then {ρε }ε>0 is the monotone sequence and there exists a value ρ∗ ∈ [1/2, 1) such that limε→0 ρε = ρ∗ . Remark 6.69. The example in Fig. 6.3 indicates the case when the origin in not a Lipschitz’s point of ∂Q, and hence ρ∗ can be equal to 1.

212


Fig. 6.3. Example of the set Q with non-Lipschitz boundary at the origin

6.9.2 The homothetic mean value property To begin, let us recall the main types of convergence in variable spaces with that was introduced in respect to the family of periodic Borel measures μλ,h ε (6.134), where the parameters λ = λ(ε) and h = h(ε) are defined by (6.135). Let vελ,h ∈ L2 (Ω, dμλ,h be a bounded sequence, that is, ε < +∞. lim sup (vελ,h )2 dμλ,h ε ε→0

Ω

1. The weak convergence vελ,h v in L2 (Ω, dμλ,h ε ) means that vελ,h ϕ dμλ,h = vϕ dx for any ϕ ∈ C0∞ (Ω). v ∈ L2 (Ω) and lim ε ε→0

Ω

Ω

2. The strong convergence vελ,h → v in L2 (Ω, dμλ,h ε ) means that 2 λ,h λ,h λ,h vε zε dμε = vz dx v ∈ L (Ω) and lim ε→0

Ω

Ω

if zελ,h z inL2 (Ω, dμλ,h ε ). 3. If a sequence bλ,h ∈ L2 (Y, dμλ,h ) is bounded, then the weak convergence bλ,h b in L2 (Y, dμλ,h ) means that b ∈ L2 (Y, dδ) and ∞ bλ,h ψ dμλ,h = bψ dδ for any ψ ∈ Cper (Y ), lim max(λ,h)→0

Y

Y

where dδ is the Y -periodic Dirac measure concentrated at the origin. We are now in a position to state the main result of this section. Theorem 6.70. Let g : Rn → R be a Y -periodic function such that g ∈ L2 (∂Q, dHn−1 ). Then


lim

ε→0

ϕ(x)g Ω

x ε h(ε)

dνελ,h =

1 |∂Q|H

213

g dHn−1 ∂Q

ϕdx Ω

for any ϕ ∈ C(Ω). In particular, x 1 λ,h n−1 . lim dνε = |Ω| g g dH ε→0 Ω ε h(ε) |∂Q|H ∂Q Proof. It is evident that we can restrict our attention to the case when g ≥ 0. Let us partition the set Ω into cubes εY with edges ε and denote these cubes by the symbols εY j . Then

x x λ,h dνε = dνελ,h ϕ(x)g ϕ(x) g ε h(ε) ε h(ε) Ω εY j

x dνελ,h + ϕ(x) g ε h(ε) Ω∩ εY j

x λ,h dνελ,h = ϕ(xj ) g ε h(ε) εY j

x dνελ,h , (6.140) + ϕ(x) g ε h(ε) Ω∩ εY j is a point of the cube εY j and the second sum is calculated over where xλ,h j the set of “boundary” cubes. Taking into account the definition of the scaling measure νελ,h and Y -periodicity of g, we have x x x g g dνελ,h = εn dν λ(ε),h(ε) ε h(ε) ε h(ε) ε εY j εY n x ε g (6.141) dHn−1 , = σ(ε) Λλ(ε),h(ε) h(ε) where the set Λλ(ε),h(ε) is defined by (6.131). Since Λλ(ε),h(ε) = ∂Qh(ε) \ Γ λ(ε),h(ε) , it follows that n−1 n−1 g dH = g dH − Λλ(ε),h(ε)

∂Qh(ε)

g dHn−1 .

Γ λ(ε),h(ε)

Then due to the definition of the homothetic contraction and (6.135), we have x dHn−1 = hn−1 (ε) g g dHn−1 = εn g dHn−1 , (6.142) h(ε) ∂Qh(ε) ∂Q ∂Q x dHn−1 = hn−1 (ε) g g dHn−1 . (6.143) h(ε) Γ λ(ε),h(ε) h−1 (ε)Γ λ(ε),h(ε) Using the definition of the set Γ λ(ε),h(ε) (see (6.131)), we obtain h−1 (ε)Γ λ(ε),h(ε) = K λ(ε)/h(ε) ∩ ∂Q = λ(ε) K ∩ ∂Qh(ε)/λ(ε) .

214


In view of Proposition 6.66, we have lim λ(ε) K ∩ ∂Qh(ε)/λ(ε) ε→0

H

= |K ∩ ∂Λ|H lim λn−1 (ε) = 0. ε→0

Thus, combining (6.141)–(6.143), we conclude εn n x dνελ,h = ε g g dHn−1 + J(ε) , ε h(ε) σ(ε) εY j ∂Q 1/2 ε→0

J(ε) ≤ gL2 (∂Q,dHn−1 ) |K λ(ε)/h(ε) ∩ ∂Q|H

−→ 0.

(6.144) (6.145)

Substituting (6.144) and (6.145) into (6.140), we have x εn λ,h n−1 n ϕ(x)g g dH ϕ(xλ,h dνε − j )ε ε h(ε) σ(ε) Ω ∂Q n

ε λ,h J(ε) ϕ(xj ) εn ≤ σ(ε) εn n−1 + g dH + J(ε) sup |ϕ| εn D(ε), (6.146) σ(ε) x∈ Ω ∂Q where D(ε) is a quantity of “boundary” cubes and εn D(ε) → 0 by Jordan’s measurability property of the set ∂Ω. Since

n ϕ(xλ,h ) ε = ϕ dx by construction of Riemann sum, lim j ε→0

Ω

εn lim = |∂Q|−1 H ε→0 σ(ε) lim J(ε) = 0 ε→0 lim εn D(ε) ε→0

=0

by (6.138), by (6.145), by Jordan’s measurability property of ∂Ω,

it follows that estimate (6.146) immediately leads us to the required result.

In a similar way, the following result can be proved. Theorem 6.71. Let Λ be the cone defined in Proposition 6.66 and let g : Rn → R be a Y -periodic such that g ∈ H 1 (K). Then x 1 λ,h n−1 lim dμε = ϕ(x)g g dH ϕ dx ε→0 Ω ε λ(ε) |K ∩ ∂Λ|H K∩∂Λ Ω for any ϕ ∈ C(Ω). In particular, x 1 n−1 lim dμλ,h . g = |Ω| g dH ε ε→0 Ω ε λ(ε) |K ∩ ∂Λ|H K∩∂Λ


215

Remark 6.72. Theorems 6.70 and 6.71 provide examples of bounded sequences in variable spaces whose weak limits can be recovered in an explicit form. To illustrate the application of these results, let us consider the following example. Let Ω be a bounded open subset of R2 and let Y = [−1/2, +1/2)2 . We set the following: K is the cube with the center at the origin and sides 1/2, Q is the circle with the center (1/6, 0) in radius 1/6, h(ε) = ε2 , and λ(ε) = exp(−ε−2 ). Let v : Y → R be a function such that v(x, y) = x2 + y 2 if (x, y) ∈ K,

v(x, y) = 0 in Y \ K.

For every positive α ∈ (0, 1], let us introduce the Y -periodic function Vα ∈ L1per (Y ) as follows: x y if (x, y) ∈ Y α , , v Vα (x, y) = α α 0 otherwise. Then the support of this function satisfies the inclusion Y ∩ supp Vα ⊆ K α . Now, we consider the following sequence of Y -periodic functions: x y ∀ ε > 0. , gε ∈ L1per (Y ) ∩ H 1 (K) , where gε (x, y) = Vλ(ε) ε ε The problem is to find lim ε→0

ϕ(x, y)gε (x, y)dH1 ,

∀ ϕ ∈ C(Ω),

ΓεD

where Sε = exp (−1/ε2 )K ∩ ∂(ε2 Q), D Γε = {Sε + ε k} ∩ Ω. k∈Θε

Since Λ = (x1 , x2 ) ∈ R2 : x1 > 0 , K ∩ ∂Λ = {(0, x2 ) : x2 ∈ (−1/4, +1/4)} , it follows that (see Theorem 6.71) x y dμλ,h lim , ϕ(x, y)gε (x, y) dH1 = lim ϕ(x, y)Vλ(ε) ε ε→0 Γ D ε→0 Ω ε ε ε 1 = (x2 + y 2 )dH1 ϕ dx dy |K ∩ ∂Λ|H K∩∂Λ Ω 1/4 1 2 y dy ϕ dx dy = ϕ dx dy. =2 48 Ω Ω −1/4

7 Convergence of Sets in Variable Spaces

The objective of this chapter is to give a presentation of the mathematical construction and tools that can be used to study problems for which the knowledge of “the limit set” in some appropriate sense is important. We discuss different analytical frameworks and concepts that can be formed on the basis of set convergence in fixed and varying spaces. In this context, we introduce different notions of limits for sequences of nonempty sets and give some applications to the theory of thin periodic and reticulated structures.

7.1 Set convergence in Rn via the limit properties of characteristic functions The main object of our concern in this section is a family of subsets {Ωε }ε>0 of a finite-dimensional Euclidean space with nonempty boundary and interior. We assume that each of the sets Ωε is Lebesgue measurable and can be associated with its characteristic function χΩε (x) = 1 if x ∈ Ωε and χΩε (x) = 0 if x ∈ Ωε . In most applications, it is interesting to consider only domains Ωε that are contained in a fixed bounded measurable subset D of Rn . In what follows, we associate with a subset D the set X(D) = {χΩ : ∀ Ω measurable in D} .

(7.1)

Clearly, X(D) ⊂ L∞ (D), and for all p ≥ 1, X(D) ⊂ Lp (D). This induces a complete metric space structure on X(D) via the bijection [Ω] → χΩ ∈ X(D) ⊂ Lp (D), where by [Ω] we denote the equivalence classes of Lebesgue measurable subsets Ω. Recall that two sets Ω1 and Ω2 are said to be equivalent if the set {x : χΩ1 (x) = χΩ2 (x)} has zero Lebesgue measure Ln . P.I. Kogut, G.R. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains, Systems & Control: Foundations & Applications, DOI 10.1007/978-0-8176-8149-4 7, © Springer Science+Business Media, LLC 2011

217

218


Theorem 7.1. Let 1 ≤ p < ∞ be an integer. Then for a nonempty measurable subset D of Rn , X(D) is closed and bounded in Lp (D) and ρD ([Ω1 ], [Ω2 ]) = χΩ1 − χΩ2 Lp (D) defines a complete metric structure on the set of equivalence classes of measurable subsets of D. Proof. Let {Ωk }∞ k=1 be a sequence of Lebesgue measurable subsets of D such ∞ that {χΩk }k=1 is a Cauchy sequence in Lp (D). It converges to some function f in Lp (D) and there exists a subsequence such that χΩk (x) → f (x) in D except for a subset Z of zero Lebesgue measure. Hence, 0 = χΩk (x)(1 − χΩk (x)) → f (x)(1 − f (x)). Define the set Ω = {x ∈ D \ Z : f (x) = 1} . Clearly, Ω is measurable and χΩ = f on D \ Z since f (x)(1 − f (x)) = 0 on D \ Z. Hence, f = χΩ a.e. on D, χΩk → χΩ , and χΩ ∈ X(D).

Remark 7.2. As an obvious consequence of this theorem, we have the following result: For p = 1 and Ω1 and Ω2 in D, the metric ρD ([Ω1 ], [Ω2 ]) is the Lebesgue measure of the symmetric difference Ω1 Ω2 = (Ω1 \ Ω2 ) ∪ (Ω2 \ Ω1 ). In other words, it means that the distance μ(A, B) between two open sets A ⊂ D and B ⊂ D defined as μ(A, B) = Ln (AB),

(7.2)

coincides with the well known Ekeland metric in L∞ (D) applied to the characteristic functions: dE (χA , χB ) = Ln {x ∈ D | χA (x) = χB (x)} = μ(A, B).

(7.3)

However, relations (7.2) and (7.3) are defined up to sets of measure 0 (see Fig. 7.1). The next question concerns the comparison between the topologies on X(D) induced by Lp (D) for different values p ∈ [1, ∞). From Delfour and Zolésio (see [91]), we cite the following result. Theorem 7.3. Let D be a nonempty bounded measurable subset of Rn . Then the topologies induced by Lp (D) on X(D) are equivalent for 1 ≤ p < ∞.

7.1 Set convergence in Rn

219

Proof. For a bounded domain D, any p ≥ 1, 1/p + 1/q = 1, and any functions χ and χ in X(D), we have χ − χL1 (D) = |χ(x) − χ(x)| dx ≤ χ − χLp (D) |D|1/q D

since χ(x) and χ(x) are either 0 or 1 in D. On the other hand, |χ(x) − χ(x)|p dx = |χ(x) − χ(x)| dx = χ − χL1 (D) . χ − χpLp (D) = D

D

Therefore, for any ε > 0, selecting δ = εp , we have ∀ χ such that χ − χL1 (D) < δ,

χ − χLp (D) < ε.

Hence, the metrics on X(D) for L1 (D) and Lp (D) are equivalent.

In view of this property, we introduce the following notion. Definition 7.4. Let D be a nonempty bounded measurable subset of Rn . We say that a sequence of subsets {Ωε }ε>0 in D strongly χ-converges to some Ω ⊆ D as ε → 0 if the corresponding sequence of characteristic functions {χΩε }ε>0 ⊂ X(D) converges strongly to χΩ in Lp (D) for some p, 1 ≤ p < ∞. We have the following approximating result. Theorem 7.5. Let Ω be an arbitrary Lebesgue measurable subset of D ⊂ Rn . ∞ n Then there exists a sequence {Ωk }∞ k=1 of open C -domains in R such that χΩk → χΩ

in L1 (D). ∞

Proof. The construction of the family of C ∞ domains {Ωk }k=1 can be found in many places (see, for instance, Giusti [115, Sect. 1.14]). Associate with the characteristic function χ = χΩ the sequence of convolutions fk = χ ∗ ρk for a sequence {ρk }∞ k=1 of mollifiers (see [115]). By construction, we have 0 ≤ fk ≤ 1 for all k ∈ N. For t (0 < t < 1), define the sets Fkt = {x ∈ Rn : fk (x) > t} . By definition, fk − χ > t in Fkt \ Ω, and |χ − χFkt | ≤

χ − fk > 1 − t in Ω \ Fkt ,

1 |χ − fk | a.e. in Rn . min {t, 1 − t}

By Sard’s theorem (see Theorem 4.3 in [91]), the set Fkt is a C ∞ -domain for ∞ almost all t in (0, 1). Fix α (0 < α < 1/2) and choose a sequence {tk }k=1 such that α ≤ tk ≤ 1 − α for all k, and define Ωk = Fktk . As a result,

220


|χΩ − χΩk | ≤

1 1 |χΩ − fk | ≤ |χΩ − fk | a.e. in Rn . min {tk , 1 − tk } α

Therefore, for a bounded measurable subset D of Rn , we have χΩ − χΩk L1 (D) ≤ α−1 χΩ − fk L1 (D) , which goes to 0 as k tends to infinity. The proof is complete.

It should be stressed here that even if the set X(D) is closed and bounded in Lp (D), it is not strongly compact. Hence, the strong χ-convergence of the sequences {Ωε }ε>0 in D needs to be relaxed to a weaker convergence. To this end, we note that for 1 < p < ∞, the closed convex hull co X(D) is weakly compact in the reflexive Banach space Lp (D). In fact, for the set co X(D), we have the representation co X(D) = {χ ∈ Lp (D) : χ(x) ∈ [0, 1] a.e. in D} .

(7.4)

Indeed, by definition, co X(D) ⊆ {χ ∈ Lp (D) : χ(x) ∈ [0, 1] a.e. in D} . On the other hand, any χ which belongs to the right-hand side of (7.4) can be approximated by a sequence of convex combinations of elements of X(D). Choose k m 1 χBkm , Bkm = x : χ(x) ≥ χk = k k m=1 for which |χk (x) − χ(x)| < 1/k. However, the elements of the set co X(D) are not necessary characteristic functions of a subdomain of D, that is, the identity χ(x)(1 − χ(x)) = 0 a.e. in D is not necessary satisfied. This fact leads to apparent paradoxes. That is why Definition 7.4 cannot be relaxed to the notion of weak χ-convergence of subsets. Before we introduce the appropriate terminology, we first give a few basic results (see [91] for the proof) Lemma 7.6. Let D be a bounded open subset of Rn , G a bounded subset of R, and G = {g : D → R : g is measurable and g(x) ∈ G a.e. in D} . Then the following hold: ∞

(i) For any p, 1 ≤ p < ∞, and any sequence {gk }k=1 ⊂ G, the following statements are equivalent: ∞ (a) {gk }k=1 weakly-∗ converges in L∞ (D),


221

∞

(b) {gk }k=1 weakly converges in Lp (D), ∞ (c) {gk }k=1 converges in D (D), where D (D) is the space of scalar distributions on D. (ii) If G is bounded, closed, and convex, then G is convex and compact with respect to the weak-∗ topology in L∞ (D), the weak topology in Lp (D), and the convergence in D (D). Due to these results, we adopt the following terminology. Definition 7.7. We say that a sequence of characteristic functions {χΩε }ε>0 is weakly convergent in X(D) if it converges for some topology between the weak-∗ topology of L∞ (D) and the topology of D (D). It is interesting to observe that working with the weak convergence in X(D) only makes sense when the limit element of {χΩε }ε>0 is not a characteristic function. Theorem 7.8. Let D be a bounded open subset of Rn and let {Ωε }ε>0 be a sequence of subsets in D. Let χ be element of X(D) such that χΩε χ in L2 (D). Then there exists a subset Ω ⊂ D such that the sequence {Ωε }ε>0 strongly χ-converges to Ω in the sense of Definition 7.4. Proof. As follows from Theorem 7.3, it is sufficient to prove that χΩε → χΩ strongly in Lp (D) for some p, 1 ≤ p < ∞, where we set χΩ = χ. Let p = 2. Then the strong L2 (D)-convergence follows from the the fact that 2 |χΩε | dx = χΩε dx → χΩ dx = |χΩ |2 dx D

D

D

D

implies

|χΩε − χΩ | dx = 2

D

D

→

|χΩε |2 − 2χΩε χΩ + |χΩ |2 dx

|χΩ |2 − 2χΩ χΩ + |χΩ |2 dx = 0.

D

This proof is complete.

Let us note that working with the weak convergence in X(D) creates new phenomena and difficulties. For instance, when a characteristic function is present in the coefficients of the highest-order term of a differential equation, ∞ the weak convergence of a sequence of characteristic functions {χk }k=1 to some element χ of co X(D), ∗

χk χ in L2 (D) =⇒ χk χ in L∞ (D), ∞

does not imply the weak convergence in H 1 (D) of the sequence {y(χk )}k=1 of solutions to the solution of the differential equation corresponding to y(χ), that is,

222


y(χk ) y(χ) weakly in H 1 (D). Since D is supposed to be a smooth bounded domain, if follows that compactness of the injection of H 1 (D) into L2 (D) would have implied strong convergence in L2 (D): y(χk ) → y(χ) in L2 (D). This fact was pointed out in 1971 by Murat [193] in the following example, which will be rewritten to emphasize the role of the characteristic function in the asymptotic analysis of boundary value problems. Example 7.9. Let us consider the following problem: ⎧ dy ⎨ d − g + gy = 0 in D = (0, 1), dx dx ⎩ y(0) = 1, y(1) = 2,

(7.5)

where G = {g = g1 (x)χ + g2 (x)(1 − χ) : χ ∈ X(D)}

with g1 (x) = 1 −

1 x2 − , 2 6

g2 (x) = 1 +

1 x2 − . 2 6

This is equivalent to G = {g ∈ L∞ (0, 1) : g(x) ∈ {g1 (x), g2 (x)}

a.e. in (0, 1)} .

Associate with each integer p ≥ 1 the function g p , ⎧

⎪ 1 x2 m 2m + 1 ⎪ ⎪ − if <x≤ , ⎨1 − 2 6 p 2p

g p (x) = ⎪ 1 x2 2m + 1 m+1 ⎪ ⎪ ⎩1 + − if <x≤ , 2 6 2p p

0 ≤ m ≤ p − 1,

and the corresponding function χp (gp = g1 χp + g2 (1 − χp )), ⎧ 2m + 1 m ⎪ ⎪ <x≤ , ⎨ 1 if p 2p χp (x) = 0 ≤ m ≤ p − 1. 2m + 1 m+1 ⎪ ⎪ <x≤ , ⎩ 0 if 2p p Murat showed that for each p, the following statements hold true: gp ∈ G, χp ∈ X(D), ∗ ∗ 1 g p g∞ = 1 in L∞ (0, 1), resp., χp 2 1 1 1 ∗ 1 1 in L∞ (0, 1). = + gp 2 g1 g2 1/2 + x2 /6


223

Moreover, yp y

in H 1 (0, 1),

where yp denotes the solution of (7.5) corresponding to g = g p and y denotes the solution of the boundary value problem ⎧ 1 x2 dy ⎨ d + y = 0 in D = (0, 1), + − (7.6) dx 2 6 dx ⎩ y(0) = 1, y(1) = 2, Let us define the function gH (x) =

1 x2 + , 2 6

which corresponds to

1 1 x2 1+ ∈ co X(D). − χH (x) = 2 2 6 Notice that gH appears in the second-order term and g∞ appears in the zerothorder term in (7.6): dy d − gH + g∞ y = 0 in (0, 1). (7.7) dx dx It is easy to check that y(x) = 1 + x2 in (0, 1), which is not equal to the solution y∞ of (7.5) for the weak-∗ limit g∞ = 1: y∞ (x) =

2(ex − e−x ) + e1−x − e−(1−x) . e − e−1

The above example uses space-varying coefficients g1 (x) and g2 (x). However, it is still valid for two constants g1 > 0 and g2 > 0. In that case, it is easy to show that g1 + g2 1 1 1 1 , g∞ = = + , 2 gH 2 g1 g2 1 g2 χH = χ∞ = , , 2 g1 + g2 and the solution y of the boundary value problem (7.7) is given by y(x) = where

2 sinh cx + sinh c(1 − x) , sinh c

224


g1 + g2 c= √ ≥ 1, 2 g1 g2 and the solution y∞ is given by y∞ (x) =

2 sinh x + sinh(1 − x) . sinh 1

Thus, for g1 = g2 , or equivalently, c > 1, we have y = y∞ . In fact, using the same sequence of χp s, the sequence {yp = y(χp )}∞ p=1 of the solutions of (7.5) weakly converges to yH = y(χH ), which is different from the solution y∞ = y(χ∞ ) for g1 = g2 . This is readily seen by noticing that since χ∞ and χH are constant, −Δ(gH yH ) = χ∞ f = −Δ(g∞ y∞ ) =⇒ gH yH = g∞ y∞ =⇒ yH =

(g1 + g2 )2 y∞ = y∞ for g1 = g2 . 4g1 g2

Remark 7.10. Since the strong and weak χ-convergence are defined to within the equivalence classes [Ω] of (Lebesgue) measurable subsets Ω of Rn , it should be noted that such a type of convergence is rather hard. As an illustration of what we meant by “hard convergence”, consider two sets: the smiling hedgehog and the expressionless circular segment in Fig. 7.1. The circular segment Ω2 is obtained by adding missing points and curves “inside” of Ω1 and removing the needles “outside” Ω1 . Hence, these sets belong to the same equivalence class. However, from a geometrical point of view, we have absolutely different junction structures.

Fig. 7.1. Examples of sets belonging to the same equivalence class

7.2 On some limit properties of characteristic functions for nonperiodically perforated domains Much of this section will be devoted to the study of some specific properties of the weak convergence in X(D) with respect to the weak-∗ topology of L∞ (D). Attention will be focused on the case when the limit element of

7.2 On limit properties of characteristic functions

225

{χΩε }ε>0 ⊂ X(D) is not a characteristic function. We will follow results obtained by Kesavan and Saint Jean Paulin [135, 136]. Let Ω ⊂ Rn be a bounded open domain and let Sε ⊂ Ω be a closed subset for each ε > 0. We set Ωε = Ω \ Sε and call it the perforated domain. Let χΩε be the characteristic function of the perforated domain Ωε . Following Braine, Damlamian, and Donato [32], we say that {Sε }ε>0 is an admissible family of holes in Ω if the following conditions are fulfilled: (H1) There exists, for each ε > 0, an extension operator Pε : Vε → H01 (Ω), where

Vε = u ∈ H 1 (Ωε ) : u = 0 on ∂Ω ,

(7.8)

such that for every u ∈ Vε , Pε u|Ωε = u,

∇Pε uL2 (Ω) ≤ C∇uL2 (Ωε ) .

(7.9)

Here, as usual, the symbol C denotes a positive constant independent of ε. (H2) Every weak-∗ limit point in L∞ (Ω) of the sequence {χΩε }ε>0 is positive a.e. on Ω. ∗ ∞ (H3) If χΩε χ0 in L∞ (Ω), then χ−1 0 ∈ L (Ω). In addition to the extension operator Pε described above, we always have the trivial extension by 0 of a function f ∈ Lp (Ωε ). We denote this extension by f. Cioranescu and Saint Jean Paulin [67] have shown that when Ω is perforated periodically, hypotheses (H1)–(H3) are satisfied. In particular, χ0 will be a positive constant. However, we present below an example where hypotheses (H2) and (H3) hold true, whereas hypothesis (H1) is not fulfilled. Example 7.11. Let us consider a planar, thick multi-structure Ωε , which consists of the junction body Ω0 = {x = (x1 , x2 ) ∈ R2 : 0 < x1 < a,

0 < x2 < γ(x1 ) },

−1 j and a large number N of the thin rods Gε = ∪N j=0 Gε ,

Gjε = {x ∈ R2 : |x1 − ε (j + 1/2)| < εh/2, x2 ∈ (−l, 0]}, j = 0, 1, . . . , N − 1, situated along the boundary Γ0 = {x2 = 0, 0 < x1 < a}, that is, Ωε = Ω0 ∪Gε (see Fig. 7.2). Here, γ ∈ C 1 ([0, a]), 0 < γ0 = min[0,a] γ, the fixed number h ∈ (0, 1) and N is a large positive integer; therefore, ε = a/N is a small discrete parameter, which characterizes the distance between the rods and their thickness. Let Ω be the interior of Ω0 ∪ D, where D = (0, a) × (−l, 0) is a rectangle that is filled up by the thin rods in the limit passage.

226


Fig. 7.2. Thick multi-structure Ωε

Let χΩε be the characteristic function of the domain Ωε . Let us define a 1-periodic function χh (·) as follows: 1 if |t − 1/2| < h/2, χh (t) = 0 if h/2 ≤ |t − 1/2| ≤ 1/2. Since

1

χh (x1 /ε) →

weakly-* in L∞ (0, a)

χh (t) dt = h

as ε → 0

0

(see the mean value property of periodic functions (2.35)), it follows that for any element g of L1 (Ω), we have

gχΩε dx = Ω

0 χh (x1 /ε) g(x1 , x2 ) dx2 dx1 0 −l g dx + h g dx = χ0 g dx, ε → 0. a

g dx + Ω0

→ Ω0

D

Ω

∗

Thus, χΩε χ0 in L∞ (Ω) and, clearly, χ0 (x) = 1 if x ∈ Ω0 ∪ Γ and χ0 (x) = h if x ∈ D. So, hypotheses (H2) and (H3) are satisfied. However, one of the principal futures of domain Ωε (see also Fig. 7.3 for a 3D example) is the fact that hypothesis (H1) fails (see [187]). Proposition 7.12. Let 1 ≤ p < ∞. Assume that hypotheses (H2) and (H3) are satisfied. Let fε : Ωε → R be functions in Lp (Ωε ) such that fε f0 in Lp (Ω). Then, we have


227

Fig. 7.3. 3D thick multi-structure

lim inf ε→0

|fε |p dx ≥

Ω

|f0 |p Ω

χ0p−1

dx.

(7.10)

Proof. Inequality (7.10) is obvious if p = 1 by virtue of the lower semicontinuity of the norm in Lp (Ω) with respect to the weak convergence. To prove (7.10) in the general case, we consider the convex functional Φ : Lp (Ω) → R defined by |ϕ|p dx.

Φ(ϕ) = Ω

Note that hypothesis (H3) implies the inclusion (χΩε /χ0 )f0 ∈ Lp (Ω). Then, taking into account the well-known inequality for convex differentiable functionals (see, e.g., [16]) Φ(ϕ) − Φ(ψ) ≥ DΦ(ψ), ϕ − ψLq (Ω),Lp (Ω) ,

∀ϕ, ψ ∈ Lp (Ω),

we have χΩε p−2 χΩε χΩε χΩε f0 ≥ p f0 f0 fε − f0 dx. Φ fε − Φ χ0 χ0 χ0 Ω χ0 However, χΩε fε = fε and χrΩε = χΩε for all r > 0. Hence, the right-hand side of the above inequality becomes f0 |f0 |p−2 χΩ ε fε − p f0 dx, χ0 χ0p−1 Ω which tends to 0 as ε → 0 by the weak convergence. Further, by the definition of Φ, we have

228


Φ

χΩε f0 χ0

|f0 |p = χΩε p dx → χ0 Ω

|f0 |p

Ω

χ0p−1

dx.

Hence, (7.10) is an immediate consequence of these observations.

Remark 7.13. Let {fε : Ω → R}ε>0 be a bounded sequence in Lp (Ω) such that fε = χΩε fε f0

in Lp (Ω).

Then, by the lower semicontinuity of the Lp -norm with respect to the weak convergence, we have |fε |p dx = lim inf χΩε |fε |p dx ≥ |f0 |p dx. (7.11) lim inf ε→0

ε→0

Ω

Ω

Ω

However, as follows from Proposition 7.12, the fact that the functions fε = χΩε fε vanish on the holes (i.e., where χΩε vanishes) gives the improved estimate (7.11). On the other hand, inequality (7.10) has a simple explanation. Let us consider the sequence {fε : Ω → R}ε>0 in the variable space Lp (Ω, dμε ) (1 ≤ p < ∞), where the Radon measure με is defined as dμε = χΩε dx for every ε > 0. In view of supposition (H3), we conclude that

fε f∗

dμε χ0 dx in the sense of Definition 2.29, in variable space Lp (Ω, dμε ) in the sense of Definition 6.14.

Then Proposition 6.16 implies lim inf |fε |p dx = lim inf |fε |p dμε ≥ |f∗ |p χ0 dx. ε→0

ε→0

Ω

Ω

(7.12)

Ω

However, due to hypothesis (H3) and the definition of the weak convergence in the variable spaces (see (6.25)), we have f∗ = f0 /χ0 . Combining this fact with (7.12), we obtain the estimate (7.10). Proposition 7.14. Assume that hypotheses (H2) and (H3) are satisfied. Let ∗ fε : Ωε → R be functions in L∞ (Ωε ) such that fε f0 in L∞ (Ω). Then, we have f0 lim inf fε ≥ . (7.13) χ0 ∞ ε→0 L∞ (Ω) L (Ω) Proof. Since for any 1 < p < ∞,

fε Lp (Ω) ≤ fε

L∞ (Ω)

it follows that

|Ω|1/p ,

lim inf fε Lp (Ω) ≤ |Ω|1/p lim inf fε ε→0

ε→0

L∞ (Ω)

,


and hence,

lim lim inf fε Lp (Ω) ≤ lim inf fε

p→∞

ε→0

ε→0

L∞ (Ω)

.

229

(7.14)

On the other hand, by (7.10) we have lim inf fε Lp (Ω) ε→0

1/p |f0 |p ≥ p−1 dx Ω χ0 p 1/p f0 1/p f0 = ≥α , χ0 χ0 dx χ0 Lp (Ω) Ω

where χ0 ≥ α > 0, as per hypothesis (H3). Hence, f0 f0 p lim lim inf fε L (Ω) ≥ lim = . χ0 ∞ p→∞ ε→0 p→∞ χ0 Lp (Ω) L (Ω) As a result, (7.13) follows from (7.14) and (7.15).

(7.15)

In the case when fε = χΩε f , we also have the following result, which is more precise than (7.13). Proposition 7.15. Let f be a fixed element of L∞ (Ω). Then for the sequence fε = χΩε f ε>0 , we have lim fε L∞ (Ω) = f L∞ (Ω) .

ε→0

(7.16)

∗ Proof. Since fε χ0 f in L∞ (Ω), by Proposition 7.14, we have ≥ f0 L∞ (Ω) . lim inf fε ∞ ε→0

L

(Ω)

On the other hand, χΩε f L∞ (Ω) ≤ f L∞ (Ω) , and hence, lim sup fε L∞ (Ω) = lim sup χΩε f L∞ (Ω) ≤ f L∞ (Ω) . ε→0

ε→0

Combining this with the previous inequality, we obtain the required result. This completes the proof.

Remark 7.16. Note that in Proposition 7.15, the limit of fε ε>0 is χ0 f , not f . Thus, (7.16) does not imply the strong convergence in L∞ (Ω). In fact, if χ0 = 1, then for any 1 ≤ p ≤ ∞, given a sequence fε = χΩε fε ε>0 that converges weakly to f0 in Lp (Ω) (weakly-∗ in L∞ (Ω) if p = ∞), we cannot, in general, have the strong convergence if f0 = 0. To see this, assume that fε → f0 strongly in Lp (Ω). If we take any element g ∈ Lq (Ω), where 1/p + 1/q = 1, then

230


fε g dx =

Ω

χΩε fε dx,

∀ ε > 0.

Ω

Passing to the limit (using the strong convergence of fε to f0 and the weak convergence of χΩε g to χ0 g for the term on the right-hand side), we get f0 g dx = χ0 f0 g dx. Ω

Ω

Since g was arbitrarily chosen in Lq (Ω), it follows that f0 = χ0 f0 in Lp (Ω), which is not generally true (except in the case when f0 = 0) if χ0 = 1.

7.3 Convergence of sets associated with thin periodic structures To illustrate some specific properties of set convergence in Rn , we consider a particular case when each of the sets Ωε can be associated with a periodic thin structure. In general, this highly contrasting medium depends on two small related parameters ε and h which control the size of a periodicity cell and thickness of the reinforcement. To describe this structure in a general manner, we assume that there is a singular structure F in Rn with Ln (F) = 0 such that F is a generating skeleton of the thin structure. For instance, for thin grids, F can be associated with a periodic singular graph in the corresponding Euclidean space that consists of straight segments (see Figs. 1.1 and 1.3). For 3D box structures, the singular structure F can be represented as some family of planes (see Fig. 1.2). We refer to Chechkin and Zhikov [58, 257] for the details. Let = [0, 1)n be a periodicity cell of F . We define a thin structure F h applying the procedure of h-thickening to the skeleton F . In the case n = 2, it means that F h can be obtained if any component I of F is replaced by a strip I h of width h whose median coincides with I. For the 3D case, each plane of the structure F turns out to be the middle of the corresponding plate from the structure F h whose thickness is h > 0. Having applied the procedure of homothetic contraction of Rn with the h factor of ε−1 , we transform the structures F h and F into Fε and Fε , respec h h h tively. In particular, Fε = εF = εx : x ∈ F . Note that in this case, Fεh consists of components (rods, planes, slabs, etc.) whose thicknesses are εh. Let Ω be an open bounded domain Ω ⊂ Rn with a Lipschitz boundary ∂Ω. We define the thin periodic structure Ωε,h on Ω as Ωε,h = Ω ∩ Fεh . In spite of the fact that the sets Ωε,h fill up the whole of the domain Ω in the limit as ε → 0, the sequence {Ωε,h }ε>0 does not strongly χ-converge to Ω in the sense of Definition 7.4. Indeed, let χhε and χh be the characteristic functions of the sets Ωε,h and F h , respectively. Then we have the following result.

7.3 Set convergence of thin periodic structures

231

Theorem 7.17. Let Ω be an open bounded subdomain of Rn and let Ω be measurable in the Jordan sense. Let Ωε,h be a thin periodic structure in Ω, the geometry of which depends on two small parameters ε and h(ε) related to each other by the relation h = h(ε) → 0 as ε → 0. Then χΩε,h 0 = χ∅

in L2 (Ω).

(7.17)

Proof. Since C0∞ (Ω) is a dense subset of L2 (Ω), it is sufficient to show that ψ(x)χΩε,h (x) dx = 0, ∀ ψ ∈ C0∞ (Ω). lim ε→0

Ω

C0∞ (Ω)

Let ψ ∈ be a fixed function and let k = + k, where k is a vector in Rn with integer components. Then {εk } is a partition of Rn . Hence, ψ(x)χΩε,h (x) dx = ψ(x)χΩε,h (x) dx Ω εk + ψ(x)χΩε,h (x) dx, (7.18) εk ∩Ω

where the first sum is taken over all k such that εk is inside and the second sum is over all k such that εk and ∂Ω have common points. Let us first consider the second sum in (7.18). We have the following estimate: ψ(x)χΩε,h (x) dx ≤ max |ψ(x)| χFεh (x) dxM (ε), (7.19) x∈Ω

εk ∩Ω

ε

where M (ε) is the number of cubes εk containing the boundary of Ω. However, n χFεh (x) dx = ε χF h dx ≤ εn .

ε

Combining this relation with (7.19) and using the fact that εn M (ε) → 0, since the boundary of Ω has Lebesgue measure zero, we obtain lim ψ(x)χΩε,h (x) dx = 0. (7.20) ε→0

εk ∩Ω

Let us now consider the first sum in (7.18). Since ψ is continuous and bounded, there exist points xk ∈ εk such that ψ(x)χΩε,h dx = ψ(xk ) χFεh (x) dx = ψ(xk )εn χF h (x) dx. εk

ε

This together with the fact ψ(x) dx as ε → 0 and χF h (x) dx → 0 as h → 0 ψ(xk )εn → Ω

232


implies that lim ε→0

ψ(x)χΩε,h dx =

εk

ψ(x) dx lim

ε→0

Ω

χF h(ε) (x) dx = 0. (7.21)

As a result, the desired conclusion (7.17) follows by taking (7.18)–(7.21) into account.

As an evident consequence of this result and Theorem 7.8, we admit the following conclusion. Corollary 7.18. The empty set is the strong χ-limit of the thin periodic structures {Ωε,h }ε>0 as ε → 0 in the sense of Definition 7.4. At the same time, it is interesting to observe that for any point x ∈ Ω and its arbitrary neighborhood U (x), there exists ε0 > 0 such that U (x)∩Ωε,h = ∅ for every ε < ε0 . This provides good reasons to take a new view of Ω as a limit set for the sequence {Ωε,h }ε>0 in some appropriate sense. To do so, we make use of the geometrical approach. Denote by λ a measure concentrated on F and proportional to the corresponding Lebesgue measure. Denote by λh a measure concentrated on F h and proportional to the Lebesgue measure h that Ln . It is assumed the measures λ and λ are normed by the following dλ =

condition:

h

lim

h→0

dλh = 1. Clearly, λh λ as h → 0, that is,

ϕ dλ =

ϕ dλ

∞ for any function ϕ ∈ Cper ().

(7.22)

Moreover, each of the structures Fεh and Fε can be associated with the socalled ε-periodic scaling measures λhε and λε , respectively. Here, λhε (B) = εn λh (B) and λε (B) = εn λ(B)

for any Borel set B ⊂ Rn .

As usual, we relate parameters h and ε assuming that h = h(ε) → 0 as ε → 0. Since λhε (ε) = λε () = εn , it follows that (see Lemma 6.50) the measures λhε and λε converge weakly to the Lebesgue measure Ln as ε → 0, that is, h ϕ dλε → ϕ dx, ϕ dλε → ϕ dx, ∀ ϕ ∈ C0∞ (Rn ). Rn

Rn

Rn

Rn

We are now in a position to state the main result of this section. Theorem 7.19. Let χΩε,h and χF h be the characteristic functions of the sets Ωε,h and F h , respectively. Then χF h −→ 1 in L2 (, dλh ), 2

(7.23)

χΩε,h −→ χΩ in L2 (Ω, dλhε ),

(7.24)

χΩε,h −→ χΩ in L

(7.25)

2

(Ω, dλhε ).

7.3 Set convergence of thin periodic structures

Proof. Since λh λ as h → 0 and ϕ dλh = ϕχF h dλh ,

233

∞ ϕ ∈ Cper (),

it follows that (7.22) implies χF h 1 L2 (, dλ) in the variable space L2 (, dλh ) (see Definition 6.14). Therefore, to prove assertion (7.23), it remains only to take into account Lemma 6.19 and the fact that 2 dλh = |χF h | dλh , ∀ h > 0.

Due to this result and Theorem 6.51, we have x dλh(ε) ϕ(x)χΩε,h dλh(ε) = lim ϕ(x)χF h(ε) lim ε ε ε→0 Ω ε→0 Ω ε = ϕ(x) 1 dλ(y) dx = ϕ(x)χΩ dx Ω

Ω

for any continuous bounded function ϕ. Hence, 2

χΩε,h χΩ in L2 (Ω, dλhε ),

χΩε,h χΩ in L2 (Ω, dλhε ).

Further, we note that 2 h(ε) h(ε) ϕ(x) dλε = ϕ(x)χΩε,h dλε = ϕ(x) χΩε,h dλh(ε) . ε Ω

Ω

Ω

Since the weak convergence of measures dλhε dx possesses the property h ϕ dλε = ϕ dx, ∀ ϕ ∈ C(Ω) lim ε→0

Ω

Ω

(see Lemma 1.1 in [258]), it follows that 2 2 χΩ 2 dλh(ε) lim = |χ | dx = |χΩ | dλ(y) dx. Ω ε ε,h ε→0

Ω

Ω

Ω

It remains only to apply the criterium of the strong convergence in variable spaces (see (6.107) and Lemma 6.19).

We can, therefore, draw the following conclusion: The fact that the thin periodic structure Ωε,h fills up the domain Ω in the limit as ε → 0 (in symbols, Ωε,h Ω) means that its characteristic function χΩε,h converges to χΩ in the variable space L2 (Ω, dλhε ). To end of this section, we observe that the weak convergence dλhε dx as ε → 0 implies

234


Ω

where

χΩε,h

ϕ

dx ≡

χF h(ε) dx

ϕ dλhε Ω

−→

ϕ dx as ε → 0,

∀ ϕ ∈ C0 (Ω),

Ω

χF h(ε) dx → 0 by supposition that h(ε) → 0 as ε → 0.

However, this fact can be always interpreted as weak-∗ convergence in −1 χF h(ε) dx χΩε,h to χΩ . Thus, for comparison with the L∞ (Ω) of

strong χ-convergence, we have the following implication: −1 ∗ Ωε,h Ω =⇒ χF h(ε) dx χΩε,h χΩ in L∞ (Ω).

7.4 Set convergence in the sense of Kuratowski and in the Hausdorff metric To begin, we recall a general concept of the set convergence, named Kuratowski convergence [161]. Let (X, τ ) be an arbitrary topological space and let ∞ {Ek }k=1 be a sequence of subsets of X. Definition 7.20. The K-lower limit of the sequence {Ek }∞ k=1 , denoted by K− lim inf k→∞ Ek , is the set of all points x ∈ X with the following property: For every neighborhood U of x, there exists k0 ∈ N such that U ∩ Ek = ∅ if k ≥ k0 . The K-upper limit, denoted by K− lim supk→∞ Ek , is the set of all points x ∈ X with the following property: For every neighborhood U of x and for every k0 ∈ N there exists k ≥ k0 such that U ∩ Ek = ∅. If there exists a subset E of X such that E = K− lim inf k→∞ Ek = K− lim supk→∞ Ek , then we write E = K− limk→∞ Ek and say that {Ek }∞ k=1 converges to the set E in the sense of Kuratowski, or K-converges to E. We also need the sequential version of such convergence. Definition 7.21. We define the sequential K-lower and K-upper limits of ∞ {Ek }k=1 by Ks − lim inf Ek = {x ∈ X : ∃ xk → x, ∃ k0 ∈ N, ∀ k ≥ k0 : xk ∈ Ek } , k→∞ Ks − lim sup Ek = x ∈ X : ∃ σ(k) → +∞, ∃ xk → x, ∀ k ∈ N : xk ∈ Eσ(k) , k→∞ ∞

respectively. Now, we say that the sequence {Ek }k=1 Ks -converges to E if Ks − lim inf Ek = Ks − lim sup Ek = E. k→∞

k→∞

Sometimes we will write Ks (τ )− lim, K(τ )− lim, and so forth, to indicate the topology τ explicitly.

7.4 Set convergence in the sense of Kuratowski and Hausdorff

235

Remark 7.22. We note that K-convergence coincides with Ks -convergence if the space X satisfies the first axiom of countability [161]. It is also clear that ∞ Ks − lim inf k→∞ Ek ⊆ Ks − lim supk→∞ Ek ; hence, {Ek }k=1 Ks -converges to E if and only if Ks − lim sup Ek ⊆ E ⊆ Ks − lim inf Ek . k→∞

k→∞ ∞

Let {xk }k=1 be a sequence in X. If Ek = {xk } for every k ∈ N, then ∞ ∞ the K-upper limit of {Ek }k=1 is the set of all cluster points of {xk }k=1 ∞ and the K-lower limit of {Ek }k=1 is the (possibly empty) set of all lim∞ its of {xk }k=1 (recall that we do not assume that X satisfies the Haus∞ dorff separation axiom, so {xk }k=1 may have more that one limit). If Ek = ∞ {xi : i ≥ k}, then {Ek }k=1 K-converges to the set of all cluster points of ∞ {xk }k=1 in X. In the following proposition, we list the main properties of K-convergence (see, for instance, [161]). Proposition 7.23. (i) If E ⊂ X and Ek = E for every k ∈ N, then K− lim Ek = E is the closure of E in X. ∞ (ii) For any subsequence {σ(k)}k=1 , we have Ks − lim inf Ek ⊆ Ks − lim inf Eσ(k) , k→∞

k→∞

Ks − lim sup Ek ⊇ Ks − lim sup Eσ(k) . k→∞

k→∞

(iii) If Ek ⊆ Fk for every k ∈ N, then Ks − lim inf Ek ⊆ Ks − lim inf Fk , Ks − lim sup Ek ⊆ Ks − lim sup Fk . k→∞

k→∞

k→∞

k→∞

(iv) The sets K− lim inf k→∞ Ek and K− lim supk→∞ Ek are closed. We also have K− lim inf Ek = K− lim inf E k , K− lim sup Ek = K− lim sup E k . k→∞

k→∞

k→∞

k→∞

∞

(v) A sequence {Ek }k=1 Ks -converges to E if and only if any subsequence of ∞ {Ek }k=1 has a further subsequence Ks -converging to E. The main result on K-convergence is the following Kuratowski compactness theorem. Theorem 7.24. Assume that X satisfies the second axiom of countability ∞ (i.e., X has a countable base). ∞ Then for every sequence {Ek }k=1 , there ex ists a subsequence Eσ(k) k=1 such that it K-converges to a subset E ⊂ X (possibly, empty). For the proof and more details, we refer to [161].

236


Example 7.25. Let X = R and let Ek = [0, 1/k] ∪ [k, +∞). Then the sequence ∞ of sets {Ek }k=1 K-converges to {0}. Example 7.26. Let X = R2 and let Ek = {(1/k, y) : 0 < y < 1}. Then K− lim Ek = {(0, y) : 0 ≤ y ≤ 1} . k→∞

∞ {xk }k=1

be a sequence in X. If Ek = {xk } for every Example 7.27. Let ∞ k ∈ N, then K− lim supk→∞ Ek is the set of all cluster points of {xk }k=1 ∞ and K− lim inf k→∞ Ek is the (possibly empty) set of all limits of {xk }k=1 (we do not assume X to be a Hausdorff space). If Ek = {xj : j ≥ k}, then K− limk→∞ is the set of all cluster points of {xk }∞ k=1 . In many cases, the natural topology on the class of closed sets is given by the so-called Hausdorff distance. Definition 7.28. The Hausdorff distance between two nonempty closed subsets E1 and E2 of Rn (taking possibly the value +∞) is defined by (7.26) dH (E1 , E2 ) = max sup dist(x, E2 ), sup dist(x, E1 ) , x∈E1

x∈E2

where dist(x, E1 ) = inf {|x − y| : y ∈ E1 }. Definition 7.29. We say that a sequence {Ek }∞ k=1 of nonempty closed subsets of Rn converges to a nonempty closed subset E in the Hausdorff metric (in H symbols, Ek → E) if dH (Ek , E) converges to 0 as k → ∞. Remark 7.30. The convergence in the Hausdorff metric implies the convergence in the sense of Kuratowski and, in general, the converse is false. However, if {Ek }∞ k=1 is a uniformly bounded sequence of nonempty closed sets (i.e., there exists a bounded subset of Rn which contains all sets of the sequence), ∞ then {Ek }k=1 converges to a closed set E in the Hausdorff metric if and only ∞ if {Ek }k=1 converges to E in the sense of Kuratowski. Remark 7.31. It is well known (see [107]) that the class of all closed subsets of a given compact set is compact with respect to the Hausdorff distance. H Moreover, the convergence Ek → E (induced by the Hausdorff distance) is equivalent to the so-called uniform convergence, which occurs if for every ε > 0, there exists kε ∈ N such that Ek ⊂ E + B0,ε

and E ⊂ Ek + B0,ε

∀ k ≥ kε .

Here B0,ε is the ball in Rn centered at the origin and of radius ε. Further, we consider the concept of set convergence adapted to the case of open subsets of Rn . Let D be a bounded open set in Rn . Let A and B be two open sets containing in the bounded domain D of Rn .


237

Definition 7.32. We say that δ∞ (A, B) is the distance between open sets A and B if it is the Pompeiu–Hausdorff distance between the closed sets D \ A and D \ A, that is, δ∞ (A, B) = max{ρ(A, B), ρ(B, A)},

(7.27)

where ρ(A, B) = sup

sup x − yRn .

x∈D\A y∈D\B

Definition 7.33. A sequence of open sets {Ak ⊂ D}∞ k=1 is said to be δ∞ convergent to an open set A ⊂ D if δ∞ (Ak , A) → 0 as k → ∞. δ

∞ As a result, we have: Ak −→ A if and only if D \ Ak converges to D \ A in the sense of Pompeiu–Hausdorff (in symbols, D \ A = P H− limk→∞ D \ Ak ).

Remark 7.34. Note that without supplementary regularity assumptions on the boundaries of the sets, there is no connection between the distance δ∞ and distance μ defined in (7.2). For instance, let B(0, 1) be the open unit ball in Rn . Add k (closed) rays of length 2, starting from the origin, into the ball such that the union of the rays is dense in B(0, 2) as k → ∞ and denote the resulting (closed) sets by Ak . Then P H− lim Ak = B(0, 2),

(7.28)

μ(Ak , B(0, 1)) → 0 as k → ∞.

(7.29)

k→∞

In Chenais [60], it was proved that for uniformly Lipschitz domains, convergence in the metric (7.27) yields convergence in the metric (7.2) with the same limit up to a set of zero measure. Note that the convergence with respect to the Pompeiu–Hausdorff distance is closely related to the convergence in the Kuratowski sense, butin some ∞ important respects, it is distinct from it. It particular, if a sequence Ak k=1 is uniformly bounded in Rn , then its convergence in the sense of Kuratowski is equivalent to the convergence with respect to Pompeiu–Hausdorff distance. However, the HP -convergence is not equivalent to K-convergence without this boundedness restriction. Note also that, in general, the ∞ P H-limits of ∞ Ω k k=1 , where Ωk are open subsets of D, and of D \ Ωk k=1 may be not complementary to each other (see the above example with the sets Ωk := Ak ). As for the topology on the class of open sets, we introduce several useful notions. We set A = {Ω : Ω ⊆ D, Ω is open} . Definition 7.35. We say that τ is the Hausdorff complementary topology on A if it given by the metric dH c (Ω1 , Ω2 ) = dH (Ω1c , Ω2c ), where Ωkc are the complements of Ωk in Rn .

(7.30)

238

7 Convergence of Sets in Variable Spaces ∞

Definition 7.36. We say that a sequence {Ωk }k=1 of open subsets of Rn converges to an open Ω in the Hausdorff complementary topology (in symbols, Hc

Ωk → Ω) if dH c (Ωk , Ω) converges to 0 as k → ∞. Remark 7.37. Since every uniformly bounded sequence of nonempty closed sets is compact with respect to the Hausdorff convergence, it follows that every uniformly bounded sequence of open sets is compact with respect to the Hausdorff complementary topology. Moreover, a uniformly bounded sequence of open sets {Ωk }∞ k=1 converges to an open set Ω in the Hausdorff comple∞ mentary topology if and only if the sequence {Ωkc }k=1 converges to Ω c in the sense of Kuratowski. Proposition 7.38. The following properties of the Hausdorff convergence hold: 1. (A, dH c ) is a compact metric space. Hc

2. If Ωk → Ω, then for every compact set K ⊆ Ω, there exists NK ∈ N such that for every k ≥ NK we have K ⊂ Ωk . 3. The Lebesgue measure is lower semicontinuous in the H c topology. 4. The number of connected components of the complement of an open set is ∞ lower semicontinuous in the H c topology, that is, if {Ωk }k=1 and Ω are open n subsets of R and Hc

Ωk → Ω, then lim inf Ωkc ≥ Ω c , k→∞

where by we denote the number of connected components. 5. A sequence of open sets {Ωε }ε>0 ⊂ D H c -converges to an open set Ω if and only if the sequence of complements {Ωεc }ε>0 converges to Ω c in the sense of Kuratowski. Proof. The proof of this proposition is quite simple; we refer to [121] for further details. For the convenience of the reader, we only recall that property 1 is a consequence of the Ascoli–Arzelˆ a theorem.

For these and other properties on H c -topology, we refer to [107]. We recall here the well known fact (see [38]) that in the case when p > n, the H c convergence of open sets {Ωε }ε>0 ⊂ D is equivalent to the convergence in the sense of Mosco of the associated Sobolev spaces W01, p (Ωε ). Definition 7.39. We say that a sequence of Sobolev spaces W01, p (Ωε ) ε>0 converges in the sense of Mosco to W01, p (Ω) if the following conditions are satisfied (see [191]): (M1 ) For every y ∈ W01, p (Ω), there exists a sequence yε ∈ W01, p (Ωε ) ε>0 such that yε → y strongly in W 1, p (Rn ).


239

(M2 ) fI {εk }k∈N is a sequence converging to 0, {yk }k∈N is a sequence such that yk ∈ W01, p (Ωεk ) for every k ∈ N and yk → ψ weakly in W 1, p (Rn ), then there exists a function y ∈ W01, p (Ω) such that y = ψ|Ω . Here, we denote by yε (respectively, y) the zero-extension to Rn of a function defined on Ωε (respectively, on Ω), that is, yε = yε χΩε and y = yχΩ . Remark 1,p 7.40. As follows from Definition 7.39, the sequence of the spaces W0 (Ωε ) ε>0 converges in the sense of Mosco to W01,p (Ω) if the following two inclusions hold: W01,p (Ω) ⊆ Ks (s)− lim inf W01,p (Ωε ),

(7.31)

W01,p (Ω) ⊇ Ks (w)− lim sup W01,p (Ωε ).

(7.32)

ε→0

ε→0

Here, by Ks (s)− lim inf ε→0 W01,p (Ωε ) and Ks (w)− lim supε→0 W01,p (Ωε ) we respectively denote the strong lower limit and the weak in the sense upper limit of Kuratowski of the sequence of Sobolev spaces W01,p (Ωε ) ε>0 , which are defined by (see Definition 7.20) Ks (s)− lim inf W01,p (Ωε ) ε→0 = u ∈ W01,p (D) ∃ uε ∈ W01,p (Ωε ), and uε

Ks (w)− lim sup W01,p (Ωε ) ε→0 = u ∈ W01,p (D) ∃ uεi ∈ W01,p (Ωεi ), and uεi

W01,p (D)

→

u , (7.33)

W01,p (D)

u . (7.34)

Note that, in general, Ks (s)− lim inf W01,p (Ωε ) ⊆ Ks (w)− lim sup W01,p (Ωε ). ε→0

ε→0

Therefore, if W01,p (Ωε ) ε>0 converges in the sense of Mosco to W01,p (Ω), then Ks (s)− lim inf W01,p (Ωε ) = W01,p (Ω) = Ks (w)− lim sup W01,p (Ωε ). ε→0

(7.35)

ε→0

As for the general case (p ≤ n), we have the following (see [39]): Theorem 7.41. Let {Ωε }ε>0 be a sequence of open subsets of D such that Hc

Ωε −→ Ω and Ωε ∈ Ww (D) for every ε > 0, where the class Ww (D) is defined as

240


Ww (D) = {Ω ⊆ D : ∀ x ∈ ∂Ω, ∀ 0 < r < R < 1; ⎫ 1 R ⎬ capp (Ω c ∩ B(x, t); B(x, 2t)) p−1 dt ≥ w(r, R, x) , (7.36) ⎭ t capp (B(x, t); B(x, 2t)) r where B(x, t) is the ball with center at x and radius t, capp (K; D) is the capacity of K in D, and w : (0, 1) × (0, 1) × D → R+ is such that 1. limr→0 w(r, R, x) = +∞, locally uniformly on x ∈ D; 2. w is lower semicontinuous function in the third argument. Then Ω ∈ Ww (D) and the sequence of Sobolev spaces converges in the sense of Mosco to W01, p (Ω).

W01, p (Ωε )

ε>0

Theorem 7.42. Let n ≥ p > n − 1 and let {Ωε }ε>0 be a sequence of open Hc

subsets of D such that Ωε −→ Ω and Ωε ∈ Ol (D) for every ε > 0, where the class Ol (D) is defined as Ol (D) = {Ω ⊆ D : Ω c ≤ l}

(7.37)

(here, as usual, by one denotes the number of connected components). Then Ω ∈ Ol (D) and the sequence of Sobolev spaces W01, p (Ωε ) ε>0 converges in the sense of Mosco to W01, p (Ω). However, the perturbation in H c topology (without additional assumptions) may be very irregular (see, for instance, [38, 80]). In view of this, we make use once more of the concept of the set convergence. Following Dancer [83] (see also [84]), we have the following: Definition 7.43. A sequence {Ωε }ε>0 of open subsets of D topologically contop

verges to an open set Ω ⊆ D (in symbols, Ωε −→ Ω) if there exists a compact set K0 ⊂ Ω of p-capacity zero capp (K0 , D) = 0 and a compact set K1 ⊂ Rn of Lebesgue measure zero such that the following hold: (D1 ) Ω ⊂⊂ Ω \ K0 implies that Ω ⊂⊂ Ωε for ε small enough. (D2 ) For any open set U with Ω ∪ K1 ⊂ U , we have Ωε ⊂ U for ε small enough. Note that without supplementary regularity assumptions on the sets, there is no connection between this type of set convergence and the set convergence in the Hausdorff complementary topology. Moreover, the topological set convergence allows that certain parts of the subsets Ωε degenerate and are deleted in the limit. For instance, assume that Ω consists of two disjoint

7.5 Parametrical convergence of open sets

241

Fig. 7.4. Example of the set convergence in the sense of Definition 7.43

balls and that Ωε is a dumbbell with a small hole on each side. By shrinking the holes and the handle, we can approximate the set Ω by sets Ωε in the sense of Definition 7.43 as shown in Fig. 7.4. It is obvious that in this case, dH c (Ωε , Ω) does not converge to 0 as ε → 0. However, as an estimate of an top “approximation” of Ω by elements of the above sequence Ωε −→ Ω, we can take the Lebesgue measure of the symmetric set difference Ωε Ω, that is, μ(Ω, Ωε ) = Ln (Ω \ Ωε ∪ Ωε \ Ω). It should be noted that in this case, the distance μ coincides with the well-known Ekeland metric in L∞ (D) applied to characteristic functions dE (χ Ω , χ Ωε ) = Ln {x ∈ D : χ Ω (x) = χ Ωε (x)} = μ(Ω, Ωε ). Another example of subsets which are H c -convergent but have no limit in the sense of Definition 7.43 is the sets {Ωε }ε>0 containing an oscillating crack with vanishing amplitude ε (see Fig. 7.5).

Fig. 7.5. The p-unstable sets which are compact with respect to the H c topology

We refer to [37] for a detailed description of this topic.

7.5 Parametrical convergence of open sets and of associated mappings Our main interest in this section is to introduce the so-called distance-type mappings and discuss their application to the convergence of associated sets. Let Ω be a given open subset of D. For an open subset Ω ⊂ D, we introduce

242


the so-called distance-type mapping dΩ : D → R based on the Euclidean distance functions associated with the domain Ω and its complement ⎧ ⎨ dist (x, D \ Ω) if x ∈ Ω, if x ∈ ∂Ω, dΩ (x) = 0 (7.38) ⎩ −dist(x, Ω) if x ∈ D \ Ω. It is well known that the mapping dΩ is uniformly Lipschitzian for any open subset Ω ⊂ D (see Clarke [68]). Let {Ωk ⊂ D}∞ k=1 be a sequence of open sets, not necessarily connected. Let dk = dΩk be the associated mapping via (7.38). By the Ascoli theorem, on a subsequence again indexed by k, we have dk → d" uniformly in D. However, d" is not a function of the necessary ∞ ∞ same type since, in general, the P H-limits of Ω k k=1 and of D \ Ωk k=1 may not be complementary to each other (see Definition 7.32). The following result indicates that the well known compactness property of the Pompeiu– Hausdorff distance is a direct consequence of the Ascoli compactness criterion (see [178]). " > 0 (possibly void). Then " = x ∈ D | d(x) Proposition 7.44. Let Ω " = D \ P H− lim D \ Ωk . Ω k→∞

$ # Proof. Let us fix an element x of D \ P H− limk→∞ D \ Ωk such that x ∈ P H− limk→∞ D \ Ωk . Then, by definition of the Pompeiu–Hausdorff distance, we have limk→∞ dist(x, D \ Ωk ) > 0. Hence, limk→∞ dk (x) > 0 (i.e., " Thus, we come to the inclusion x ∈ Ω). " D \ P H− lim D \ Ωk ⊆ Ω. k→∞

To obtain the reverse one, we assume that " x ∈ Ω and x ∈ D \ P H− lim D \ Ωk . k→∞

" " Then d(x) > 0 and x ∈ P H− limk→∞ D \ Ωk , that is, d(x) > 0 and there " are elements xk ∈ D \ Ωk such that xk → x. This means that d(x) > 0, " dk (xk ) ≤ 0, and xk → x. By the uniform convergence, we have d(x) > 0 and " ≤ 0, which leads to a contradiction. It follows that d(x) " ⊆ D \ P H− lim D \ Ωk , Ω k→∞


As a consequence of this proposition, we have the following result.


243

" then for any compact Proposition 7.45. If P H− limk→∞ D \ Ωk = D \ Ω, set K ⊂ Ω there is an integer kK = k(K) ∈ N such that K ⊂ Ωk

for k ≥ kK .

Proof. Using the same notation as in Proposition 7.44, we note that since the mapping d" is continuous on D and strictly positive on K, it follows that there exists a constant cK > 0 satisfying the inequality " ≥ cK > 0, d(x)

∀ x ∈ K.

By the uniform convergence, for any k ≥ kK , we obtain dk (x) ≥ cK /2 > 0 for

all x ∈ K, that is, K ⊂ Ωk for all k ≥ kK , as required. Remark 7.46. This property is called the Γ -property by Liu [177], and as we will see later, it plays an essential role in the local convergence theory for the solutions of PDEs defined in a sequence of bounded domains. The same property is also proved in Pironneau [214], by different methods, together with other domain convergence results. We are now in a position to introduce the following concept. Definition 7.47. A sequence of open sets {Ωk ⊂ D}∞ k=1 is called parametri ⊂ D if there is a sequence of continuous cally convergent to an open set Ω ∞ mappings p k : D → R k=1 such that p k → p uniformly in D and Ωk D \ Ωk Ω D\Ω

= {x ∈ D | p k (x) > 0} , = {x ∈ D | p k (x) < 0} , = {x ∈ D | p (x) > 0} , = {x ∈ D | p (x) < 0} .

= p − lim Ωk . We denote this limit by Ω k→∞

Remark 7.48. It is possible to weaken the conditions of this definition by replacing the uniform convergence ∞with other types of functional convergence for the mappings p k : D → R k=1 . Note that the “parameterization” p k associated with the domain Ωk is not unique and the distance mapping dk is just one example of such a parameterization. It means that, in general, the p-limit and the convergence properties depend on the parameterization. If it is different from the function dΩ , then the convergence may differ from the Pompeiu–Hausdorff convergence. Example 7.49. Let us choose the mapping p : R → R as follows: ⎧ ⎨ −(x − 1)2 + 1/2 if x ≥ 1/2, if |x| ≤ 1/2, p(x) = x2 ⎩ −(x + 1)2 + 1/2 if x ≤ −1/2.

(7.39)

244


We rotate its graph to define a continuous mapping p : R2 → R. Let p k : R2 → R be defined as p k (x) = p(x) + 1/k. Then the corresponding domains are % Ωk = x ∈ R2 | xR2 < 1 + (n + 2)/2n , √ Ω = x ∈ R2 | 0 < xR2 < 1 + 1/ 2 . Notice that in this case, Ω is a nonsmooth domain, Ω = p − lim Ωk , and Ω = P H− limk→∞ Ω k . Moreover, if p is 0 around x = 0 on some interval, then p − lim Ωk will be a circular crown, and so forth. By taking sup(p k , p m ) or inf(p k , p m ), one can easily parameterize the domains Ωk ∪ Ωm or Ωk ∩ Ωm , respectively. Following [178], we list the fine properties of the set convergence in the sense of Definition 7.47 and its applications. Proposition 7.50. The parametric convergence has the Γ -property for any parameterization. Proof. This is similar to the proof of Proposition 7.45.

Proposition 7.51. If Ω = p − limk→∞ Ωk and the closed set C = x ∈ D | p(x) = 0 has zero Lebesgue measure, then χΩk → χΩ a.e. in D. Proof. If x ∈ Ω, then, by definition, p (x) > 0. Thus, there is an integer kx = k(x) ∈ N such that p k (x) > 0 for all k ≥ kx . Hence, χΩ k (x) = χΩ (x) = 1, ∀ k ≥ kx . If x ∈ D\Ω, then p (x) < 0 and there exists an integer kx = k(x) ∈ N such that p k (x) < 0 for all k ≥ kx . Consequently, χΩ k (x) = χΩ (x) = 0 for k ≥ kx . Since the set C has zero measure, we get χΩ k (x) → χΩ (x) almost everywhere in D.

Proposition 7.50 motivates us to introduce the following concept. ∞ Definition 7.52. Let Ω = p − limk→∞ Ωk and let yk ∈ H 1 (Ωk ) k=1 be a uniformly bounded sequence (i.e., supk∈N yk H 1 (Ωk ) < +∞). We say that ∞ the sequence {yk }k=1 is locally convergent to y ∈ H 1 (Ω), and we write y = L−lim yk if for any open set G ⊂⊂ Ω (compactly embedded in Ω), we have yk |G → y|G

weakly in H 1 (G).

(7.40)

Notice that the limit mapping y = L−lim yk , given by Definition 7.52, is uniquely determined (see [178]). Moreover, in this case, we have the following compactness result.


245

∞ Theorem 7.53. Let Ω = p − limk→∞ Ωk . Suppose that yk ∈ H 1 (Ωk ) k=1 1 is a uniformly bounded sequence in the scale of the spaces H (Ωk ) . Then ∞ there are a function y ∈ H 1 (Ω) and a subsequence of yk ∈ H 1 (Ωk ) k=1 still indexed by k such that y = L−lim yk . ∞

Proof. Let {Gj ⊂⊂ Ω}j=1 be a sequence of open subsets such that Gj ⊂ ˆ Gj+1 and ∪j∈N Gj = Ω. Following the Eberlein–Smuljan ∞ theorem, for each j ∈ N we can choose a subsequence yk ∈ H 1 (Ωk ) k=1 (one after another and still indexed by k) such that yk |Gj → y j Gj weakly in H 1 (Gj ). Using ∞ the properties of the family of subsets {Gj ⊂⊂ Ω}j=1 , we define the function y on Ω by y(x) y j (x) a.e. x ∈ Gj . Then it is clear that y ∈ L2 (Ω) since = ∞ j the sequence y L2 (Gj ) j=1 is uniformly bounded. Let ϕ ∈ C0∞ (Ω) be any fixed test function. Then, by the initial supposition, there is an index j0 such that ϕ ∈ C0∞ (Gj ) for all j ≥ j0 . Therefore, ∇y ϕ dx = ∇y ϕ dx = − y ∇ϕ dx Gj Ω Ω y j ∇ϕ dx = ∇y j ϕ dx. =− Gj

Gj

∞ Hence, ∇y = ∇y j in Gj for all j ≥ j0 . Since yk ∈ H 1 (Ωk ) k=1 is a uniformly 1 bounded sequence in the scale of the spaces H (Ωk ) , it follows that ∇y ∈ [L2 (Ω)]n (i.e., y ∈ H 1 (Ω)). As easy to see in this case, (7.40) holds true. This concludes the proof.

Theorem 7.54. Let J : Rn × R × Rn → R be a non-negative measurable mapping such that J(x, ·, ·) is continuous on R × Rn and J(x, s, ·) is convex on Rn . Let Ω = p − limk→∞ Ωk . Then lim inf J(x, yk , ∇yk ) dx ≥ J(x, y, ∇y) dx (7.41) k→∞

Ωk

Ω

provided that y = L−lim yk . ∞

Proof. Let {Gj ⊂⊂ Ω}j=1 be a sequence of open sets selected as in the previous proof. Then we have χGj → χΩ a.e. in D. By the initial assumptions, we have that yk → y weakly in H 1 (Gj ) for any fixed Gj . Then, due to the property of the weak lower semicontinuity for the convex integrals in fixed domains, we obtain lim inf J(x, yk , ∇yk ) dx ≥ J(x, y, ∇y) dx. k→∞

Gj

Gj

As a result, using the positivity of J, Fatou’s lemma, and the properties of ∞ the family of subsets {Gj ⊂⊂ Ω}j=1 , we get

246


J(x, y, ∇y) dx =

Ωj

lim χGj J(x, y, ∇y) dx

Ω j→∞

≤ lim inf j→∞

J(x, y, ∇y) dx Gj

≤ lim inf lim inf J(x, yk , ∇yk ) dx j→∞ k→∞ Ωk J(x, yk , ∇yk ) dx. = lim inf k→∞

Ωk

The proof is complete.

Let G be a family of equibounded and equiuniformly continuous functions g : B(0, l) → R with l > 0 fixed and B(0, l) ⊂ Rn−1 an open ball centered at the origin with radius l. To indicate a class of open sets which is compact with respect to the parameterical convergence, we adopt the following uniform segment property (see [178]). Definition 7.55. We say that an open set Ω ⊂ D belongs to the class O (or equivalently, has the segment property) if the following conditions are fulfilled: (G1) There is a subset GΩ ⊂ G and, for any g ∈ GΩ , it can be associated to an orthogonal system of axes of center og ∈ ∂Ω, “vertical” vector bg ∈ Rn of unit length, and a rotation Rg in Rn such that bg = Rg (0, . . . , 0, 1) and & {Rg (s, 0) + og + g(s)bg | ∀ s ∈ B(0, l)} = ∂Ω. g∈GΩ

(G2) There is an a > 0 such that for any g ∈ GΩ , the uniform segment property is valid: Rg (s, 0) + og + (g(s) + t) bg ∈ Rn \ Ω,

∀ s ∈ B(0, l), ∀ t ∈ (0, a),

Rg (s, 0) + og + (g(s) − t) bg ∈ Ω,

∀ s ∈ B(0, l), ∀ t ∈ (0, a).

(G3) There is a constant r ∈ (0, l) such that & Rg (s, 0) + og + g(s)bg | ∀ s ∈ B(0, r) = ∂Ω. g∈GΩ

Remark 7.56. Let us note that conditions (G1) and (G2) represent the usual definition of boundaries of class C with added uniformity assumptions. Moreover, due to the compactness of the boundary ∂Ω, it can be covered by a finite number of local charts; therefore, both conditions are automatically satisfied, and the only real requirement is the uniformity with respect to the whole family G, which does not allow the local charts to shrink. As for condition (G3), this is a uniform restriction property for the whole family G of the local charts. It avoids, for instance, clustering of singularities (like cusps) near the

7.6 Kuratowski convergence in variable spaces

247

boundary of any local chart. Notice also that due to the finite number of the local charts, a positive number 0 < rΩ < l can be found in each Ω with the above property. However, condition (G3) ensures that this number can be chosen independent of Ω. Theorem 7.57. Let {Ωk }∞ k=1 ⊂ O be a sequence of open subsets of D. Let ∞ dk : D → R k=1 be the corresponding sequence of the distance-type functions " ⊂ D of the class O, a introduced in (7.38). Then there are an open set Ω ∞ " function d ∈ C(D), and a subsequence of {Ωk }k=1 ⊂ O still indexed by k such that the following hold: # $ " = D \ P H− limk→∞ D \ Ωk . " = p − limk→∞ Ωk and Ω (i) Ω (ii) dk → d" in C(D). (iii) χΩk → χΩb a.e. in D, and χΩk → χΩb strongly in any Lq (D) for all q ≥ 1. " " = x ∈ D | d(x) (iv) ∂ Ω = 0 has zero Lebesgue measure and it can be represented as a finite union of graphs of continuous functions. Proof. We refer to Propositions 7.44 and 7.51, and to [178] for the details of this proof.

7.6 Kuratowski set convergence in variable spaces Under suitable additional assumptions, the convergence of sets in the sense of Kuratowski can be extended to the case of variable Banach spaces Xε . To do so, we make use the initial suppositions of Sect. 6.1. Let the small parameter ε > 0 vary in a strictly decreasing sequence of positive numbers which converges to 0. For any ε > 0, let Xε be a real reflexive separable Banach space, equipped with the norm · Xε . Let · 0,Xε be an auxiliary norm in Xε . Further, let X be a real reflexive separable Banach space equipped with the norm · X and · 0,X be an auxiliary norm in X relative to which the ball BX (0, 1) = {u ∈ X : uX ≤ 1} is precompact. For any ε > 0, we suppose that there exist linear mappings Qε : X → Xε and Pε : Xε → X with properties (6.1)–(6.5). Moreover, by the weak convergence in the scale of variable spaces {Xε }ε>0 , we mean the convergence in the sense of Definition 6.2. Definition 7.58. We say that a sequence of sets {Eε ⊂ Xε }ε>0 weakly Ks converges in the scale of variable spaces {Xε }ε>0 to a set E ⊂ X if the following conditions are satisfied: (a) For every x ∈ E, there exist a constant ε0 > 0 and a sequence {xε }ε>0 weakly converging to x in the scale of variable spaces {Xε } such that xε ∈ Eε for every ε < ε0 . (b) If {Eεk }k∈N is a subsequence of {Eε ⊂ Xε }ε>0 and {xk }k∈N is a sequence converging to x in the scale of variable spaces {Xε } such that xk ∈ Eεk for every k ∈ N, then x ∈ E.

248


We are now in a position to prove the compactness property of weak Ks convergence in variable spaces provided the suppositions of Sect. 6.1 hold true. Theorem 7.59. Let {Eε ⊂ Xε }ε>0 be an equibounded sequence of sets, that is, sup sup xXε < +∞. ε>0 x∈Eε

Assume that for any ε > 0 there exist linear mappings Qε : X → Xε and Pε : Xε → X with properties (6.1)–(6.5). Then {Eε ⊂ Xε }ε>0 has a subsequence which weakly Ks -converges a subset E ⊂ X in the sense of Definition 7.58. Proof. To begin, for every ε > 0 we define the sets "ε = Pε (Eε ) = { x = Pε x, ∀ x ∈ Eε } ⊂ X. E

(7.42)

Since the sequence {Eε ⊂ Xε }ε>0 is equibounded and operators {Pε }ε>0 satisfy property (6.3), it follows that there exists a norm bounded closed set ' "ε ⊆ M . M ⊂ X such that ε>0 E By definition, the weak topology on M is the topology induced by the weak topology of X. Since the space X is separable and reflexive, there exists a metric d on X such that the weak topology on a norm bounded subset M ⊂ X coincides with the topology induced on M by the metric d. In other words, it means that for any sequence {xk }k∈N in X, the following statements are equivalent: (j) xk x in X. (jj) {xk }k∈N is bounded in X and d(xk , x) → 0 as k → ∞ (see, e.g. [99]). We denote by τ the topology associated to the metric d on X. Since the topology τ has a countable base, by the Kuratowski compactness ε theorem (Theorem 7.24), there exists a subsequence of E ⊂ X, still ε>0 denoted by Eε ε>0 , which Ks (τ )-converges to a set E ⊂ X. In view of (7.42) and the initial assumptions, it is clear that the set E is nonempty. Now, we prove that E is the Ks -limit of {Eε }ε>0 in the scale of variable spaces {Xε }ε>0 . First, let us verify condition (a) of Definition 7.58. Let x be any element of E. Since E ⊂ M and (jj) implies (j), it follows that there exist a constant xε = Pε (xε )}ε>0 weakly converging to x in X such ε0 > 0 and a sequence { that x ε ∈ Eε for every ε < ε0 . Using the assumption regarding the norm · 0,X , we find that Pεk uεk − u0,X → 0. Hence, from (6.4) and (7.42), we conclude that xε ∈ Eε for every ε < ε0 , Pε (xε − Qε x) 0,X = xε − Qε x0,Xε → 0,


249

∞

that is, by Definition 6.2, the sequence {xε ∈ Xε }k=1 converges weakly to x (in the scale of variable spaces {Xε }). Thus, condition (a) is valid. The verification of condition (b) can be done in a similar manner. This concludes the proof.

Remark 7.60. As an evident consequence of Theorem 7.59, we have the following result: Under suppositions (6.15)–(6.17) of Example 6.10, any equibounded sequence of sets Eε ⊂ W 1,p (Ωε ) ε>0 is relatively compact with respect to Ks -convergence. Moreover, it is easy to see that in this case, the sequence of periodically perforated domains {Ωε }ε>0 ⊂ Ω converges to Ω in the sense of Kuratowski. Hence, suppositions (6.15)–(6.17) imply the following limit property for Sobolev spaces: K

K

(7.43) Ωε → Ω =⇒ W 1,p (Ωε ) ⊃ Eε →s E ⊂ W 1,p (Ω). 1,p However, in general, a sequence of the entire Sobolev spaces W (Ωε ) ε>0 can fail to satisfy the stability property (7.43) (see, for instance, [36, 249]). Our next step is to study the Ks -convergence property of sets in the variable spaces Xε = L2 (Ω, dμε ), where {με }ε>0 are some Radon measures on Ω ⊂ Rn and X = L2 (Ω). In contrast to the suppositions of Theorem 7.59, we make no assumptions on the existence of linear mappings Qε : X → Xε and Pε : Xε → X and their properties. Moreover, we assume that each of the Radon measures με can be singular with respect to the Lebesgue measure Ln (see Definition 2.6). It means that Qε : X → Xε are linear mappings like “trace operators”. So, in this case, there are no well-defined extension operators Pε : Xε → X with properties (6.3)–(6.5). For simplicity, we consider the case when each of the Radon measures με is associated with some ε-periodic graphlike structure on Ω ⊂ R2 . To be more specific, we introduce the graphlike structure on the domain Ω as follows: We say that the set = [0, 1)2 = [0, 1) × [0, 1) is the cell of periodicity for some graph F on R2 if contains a “star”-structure such that (see Fig. 7.6) the following hold: (i) All edges of this structure have a common point M ∈ int , each edge is a line segment and all end points of these edges belong to the boundary of . (ii) In the set of end points (vertices), there exist pairs (Mi ; Mk ) such that Mk Mk i i or xM xM 1 = x1 2 = x2 . As follows from condition (ii), we admit the existence of isolated vertices in the -periodic graph F on R2 . Let ε ∈ E = (0, ε0 ] be a small parameter. Definition 7.61. We say that Fε is an ε-periodic graph on R2 if Fε = εF = {εx : x ∈ F}.

250


Fig. 7.6. The cell of periodicity

It is clear that the cell of periodicity for Ωε is ε. Let I ed = {Ij , j = 1, 2, . . . , K}

(7.44)

be the set of all edges on . Let Ω be an open bounded domain in R2 with a Lipschitz boundary such that Ω = {(x1 , x2 ) : x1 ∈ Γ1 , 0 < x2 < γ(x1 )} ,

(7.45)

where Γ1 = (0, a), γ ∈ C 1 ([0, a]), and 0 < γ0 = inf x1 ∈[0,a] γ(x1 ). Then ∂Ω = Γ1 ∪ Γ2 , where Γ2 = ∂Ω \ Γ1 . Definition 7.62. We say that Ωε has an ε-periodic graphlike structure if Ωε = Ω ∩ Fε (see Fig. 7.7). For every segment Ii ∈ I ed , i = 1, 2, . . . , K, we denote by μi its corresponding Lebesgue measure. Now, we define the -periodic Borel measure μ in R2 as follows: K gi μi on , (7.46) μ= i=1

where g1 , g2 , . . . , gK are non-negative weights such that

dμ = 1.

Thus, the support of the measure μ is the union of all edges Ii ∈ I ed , each of which is a 1D manifold in R2 . Since the homothetic contraction of the plane with a factor of ε−1 takes the grid F to Fε = εF, we introduce a “scaling” ε-periodic measure με as follows: με (B) = ε2 μ(ε−1 B) for every Borel set B ⊂ R2 .

(7.47)


251

Fig. 7.7. Periodic grid on Ω

Then

dμε = ε2 ε

dμ = ε2 .

Hence, the measure με is weakly convergent to the Lebesgue measure L2 , that is, dμε dx ⇔ lim ϕ dμε = ϕ dx, ∀ ϕ ∈ C0∞ (R2 ). (7.48) ε→0

R2

R2

In the scale of spaces L (Ω, dμε ) , let us consider the following sequence of sets: (7.49) Eε = h ∈ L2 (Ω, dμε ) : |h| ≤ C με -a.e. in Ω .

2

It is clear that the spaces Xε = L2 (Ω, dμε ) and the “limit” space X = L2 (Ω) do not satisfy the key initial suppositions of Theorem 7.59. Nevertheless, the following result shows that the sequence (7.49) is Ks -convergent as ε → 0 and its Ks -limit set can be recovered in an explicit form. Theorem 7.63. Let C be a positive constant independent of ε. Let Ω be an open bounded measurable set in the sense of Jordan. Then the sequence (7.49) Ks -converges in the sense of Definition 7.58 to the set E0 , where E0 = h ∈ L2 (Ω) : |h| ≤ C a.e. in Ω . (7.50) Proof. We divide our proof into two steps. Step 1. We show that condition (a) of Definition 7.58 is valid. Let u0 be any element of E0 . Since the space of smooth functions C ∞ (Ω) is dense in L2 (Ω), it follows that there is a sequence {uε ∈ C ∞ (Ω)}ε>0 satisfying the conditions |uε | ≤ C on Ω for every ε > 0, Therefore,

uε −u0 L2 (Ω) → 0 as ε → 0. (7.51)

252


lim

ε→0

ϕu0 dx for every ϕ ∈ C 0 (Ω).

ϕuε dx = Ω

(7.52)

Ω

Further, we note that uε ∈ L2 (Ω, dμε ) (as a smooth function) and hence |uε | ≤ C με -almost everywhere in Ω. We have to show that uε u0 weakly in L2 (Ω, dμε ), that is, ϕuε dμε → ϕu0 dx for every ϕ ∈ C0∞ (R2 ). (7.53) Ω

Ω

We partition the domain Ω into the sets εj , where {j } is a periodic covering of R2 by the cell . Then ϕuε dμε = ϕuε dμε + ϕuε dμε , (7.54) Ω

εj

j

Ω∩εj

where the second sum is calculated over the set of the “boundary” squares such that εj ∩ ∂Ω = ∅. By the mean value theorem, for each index j there exist points xj in the cells εj such that ϕuε dμε = ϕ(xj ) uε (xj ) dμε εj εj = ϕ(xj ) uε (xj ) ε2 dμ = ϕ(xj ) uε (xj ) ε2 , ∀ j. (7.55)

Then, in view of (7.54), we get ⎛ ⎞ ϕuε dμε = ⎝ ϕ(xj ) uε (xj ) ε2 − ϕuε dx⎠ j

Ω

+

ϕuε dμε +

Ω∩εj

Ω

ϕuε dx Ω

= I1 + I2 +

ϕuε dx.

(7.56)

Ω

Note that

|I2 | = ϕuε dμε Ω∩εj ≤ sup j∈ D(ε)

sup

|ϕ| |uε | ε2 D(ε) ≤ CϕC(Ω) ε2 D(ε),

x∈ Ω∩εj

where D(ε) is the quantity of the “boundary” squares and ε2 D(ε) → 0 by Jordan’s measurability property of the set ∂Ω. Hence, I2 → 0 as ε tends to 0. Now, we show that I1 → 0. For this, we note that


253

|I1 | = ϕ(xj ) uε (xj ) ε2 − ϕuε dx Ω j 1 2 ϕuε dx ε + ϕuε dx ϕ(xj ) uε (xj ) − 2 ≤ ε εj Ω∩εj j 1 ϕ(x ≤ )u (x ) − ϕuε dx ε2 + CϕC(Ω) ε2 D(ε). j ε j 2 ε εj j

Let us suppose the converse, that is, −2 ϕuε dx ε2 > 0. lim ϕ(xj )uε (xj ) − ε ε→0 εj j

Since Ω is bounded, it is contained in a number of squares εj smaller " does not depend on ε. So, there exist a constant " 2 , where C than C/ε ∗ C > 0 and a value ε > 0 such that −2 ϕuε dx ≥ C ∗ (7.57) ϕ(xj )uε (xj ) − ε εj (for an infinite number of indices j for every fixed ε). Hence, wild oscillations are present in the sequence {ϕuε }. However (see [105, 245]), if we have rapid fluctuations in the functions {ϕuε }, then the convergence ϕuε → ϕu0 almost everywhere in Ω is excluded. This fact immediately reflects the lack of strong convergence ϕuε → ϕu0 in L2 (Ω) as ε → 0. Indeed, by the initial assumptions, we have |uε | ≤ C for every ε ∈ E, ϕuε ϕu0 in L1 (Ω), and ϕuε − ϕu0 L1 (Ω) → 0 as ε → 0 for any ϕ ∈ C0∞ (R2 ). Let A be any subset of Ω with |A| = 0. Then, by Valadier’s theorem [245] (see also Theorem 2.28), ϕuε → ϕu0 strongly if and only if the following criterion is satisfied: ∀ δ > 0 ∃ ε0 > 0, ∃ B ⊂ A with |B| = 0 such that ∀ ε < ε0 , −1 −1 ϕuε dx dx < δ. |B| ϕuε − |B| B

B

Hence, for any ε < ε0 , there is a square εj ⊂ B such that ϕuε dx dx < δ. ε−2 ϕuε − ε−2 εj εj Since the functions ϕuε are continuous and uniformly bounded, it follows that, for any point xj of εj satisfying the condition ϕuε dx = 0, ϕ(xj )uε (xj ) − ε−2 εj

254


there is a constant A∗ > 0 satisfying ε−2 ϕuε − ε−2 εj

ϕuε dx dx

εj

= A∗ ϕ(xj )uε (xj ) − ε−2

ϕuε dx.

εj

Hence,

−2 ϕuε dx < A−1 ϕ(xj )uε (xj ) − ε ∗ δ, εj

and we come into conflict with (7.57). Hence, our assumption was wrong, and we get −2 ϕ(x ϕuε dxε2 = 0. lim j )uε (xj ) − ε ε→0

εj

j

As a result, we have I1 → 0. Thus, summing up the results obtained above and relations (7.56) and (7.52), we come to the desired identity (7.53). Thereby the following inclusion Ks − limε→0 (Eε ) ⊇ E0 is established (see Definition 7.58). Step 2. We show that the reverse inclusion to the above one holds true. To do so, we have to verify condition (b) of Definition 7.58. With this aim in view, we consider a bounded sequence {uk }k∈N with the following properties: (a) uk ∈ Eεk for every k ∈ N, where {εk } is some subsequence of indices converging to 0 as k tends to ∞. (aa) uk u∗ in the variable space L2 (Ω, dμεk ). It remains to show that the limit function u∗ ∈ L2 (Ω) satisfies the following constraint: (7.58) |u∗ | ≤ C a.e. in Ω. Indeed, for every k ∈ N and every positive function ϕ ∈ C ∞ (Ω), we have ϕ(C − uk ) dμεk ≥ 0 and ϕ(uk + C) dμεk ≥ 0. (7.59) Ω

Ω

Using the fact that dμε dx, property (6.121), and passing to the limit in (7.59) as k → ∞, one gets ∗ ϕ(C − u ) dx ≥ 0 and ϕ(u∗ + C) dx ≥ 0. (7.60) Ω

Ω

Since ϕ in (7.60) is an arbitrary positive function, it follows that inequalities (7.58) hold true. As a result, we deduce that u∗ ∈ E0 . Thus, we have obtained the required inclusion Ks − limε→0 Eε ⊆ E0 , which concludes the proof.


255

Remark 7.64. We provide a few comments on the scheme of the proof of Theorem 7.63. First, we note that the choice of points xj in (7.55) is not arbitrary. These points {xj } must be chosen in such way that ϕuε dμε = ϕ(xj )uε (xj ) dμε . εj

εj

In particular, the position of xj in εj depends on ϕ and ε, in general. At the same time, the choice of the sequence {uε ∈ C ∞ (Ω)}ε>0 is only restricted by approximated condition (7.51). Neglecting these assumptions can lead us to inconsistent results. Indeed, if {uε }ε>0 is a sequence in C ∞ (Ω) which converges to a function u0 in L2 (Ω), ϕ ∈ C ∞ (Ω), and {xj } are arbitrary points in εj , then, by properties of Riemann integrals, we have lim ϕ(xj )uε (xj )ε2 = ϕu0 dx. (7.61) ε→0

Ω

j

However, this assertion is not true, in general. For example, let us consider the following counterexample. Assume that xj is the center of the cube εj for every j and consider a C ∞ (Ω) function uε such that for every j, uε = 1 in the cube of center xj and side ε2 , uε = 0 in εj outside the cube of center xj and side 2ε2 and 0 ≤ uε ≤ 1 in Ω. Then the sequence {uε }ε>0 clearly converges to 0 in L2 (Ω) strongly; however, for every ϕ ∈ C ∞ (Ω), we have ϕ(xj )uε (xj )ε2 = lim ϕ(xj )ε2 → ϕ dx, (7.62) lim ε→0

ε→0

j

j

Ω

The strong convergence to 0 of uε in L2 (Ω) can be motivated as follows. Note that in each square εj , u2ε is smaller than the characteristic function of the square of center xj and side 2ε2 . Thus, the integral of u2ε in εj is smaller than the measure of the square of center xj and side 2ε2 (i.e., (2ε2 )2 = 4ε4 ). On the other hand, as Ω is bounded, it is contained in a number of squares " does not depend on ε. So, we have " 2 and C smaller that C/ε u2ε dx ≤ Ω

" C " 2 → 0. (4ε4 ) = 4Cε ε2

Thus, uε → 0 in L2 (Ω) and we come to relation (7.62) which contradicts (7.61). To end of this section, we present one observation which is a direct consequence of Theorem 7.63 and which, we feel, will be interesting per se. It is well known that the space C0∞ (Ω) is not dense in L∞ (Ω). However, as we will see later, any element of y ∈ L∞ (Ω) can be approximated by a sequence of smooth functions {yε ∈ C0∞ (Ω)}ε>0 for which y is the weak limit in appropriate variable spaces.

256


For every ε > 0, let us define the space L∞ (Ω, dμε ) in the following way: yε ∈ L∞ (Ω, dμε ) if and only if yε is a με -measurable function on Ω and there exists a constant M > 0 such that |yε (x)| ≤ M με -everywhere in Ω. For a comparison, we refer to Sect. 6.2. Definition 7.65. We say that a sequence {yε ∈ L∞ (Ω, dμε )}ε→0 is uniformly bounded if supε>0 yε L∞ (Ω, dμε ) < +∞. Definition 7.66. A uniformly bounded sequence {yε ∈ L∞ (Ω, dμε )}ε→0 is said to be weakly-∗ convergent in the variable space L∞ (Ω, dμε ) to y ∈ L∞ (Ω) if lim ϕyε dμε = ϕy dx for every ϕ ∈ C0∞ (Ω) ε→0

Ω

Ω

∗

(in symbols, yε y). We begin with the following result. Proposition 7.67. Let {yε }ε→0 be any bounded sequence in the variable space L∞ (Ω, dμε ). Then this sequence is relatively compact with respect to the weak∗ convergence in L∞ (Ω, dμε ). Proof. Let us set

lε (ϕ) =

yε ϕ dμε

ϕ ∈ C0∞ (Ω).

Ω

Then, by H¨ older inequality, we have |yε ||ϕ| dμε ≤ yε L∞ (Ω, dμε ) |ϕ| dμε . |lε (ϕ)| ≤ Ω

(7.63)

Ω

Hence, |lε (ϕ)| ≤ yε L∞ (Ω, dμε ) ϕC(Ω) με (K), where by K we denote the support of ϕ in Ω. Since dμε dx = dL2 in the space of Radon measures and lim sup με (K) ≤ L2 (K)

for every compact subset of Ω

ε→0

(see Zhikov [259]), it follows that |lε (ϕ)| ≤ 2ϕC(Ω) μ(K) sup yε L∞ (Ω, dμε ) ε>0

for ε > 0 small enough. On the other hand, the set T (K) = {ϕ ∈ C0∞ (Ω), supp ϕ ⊆ K} is separable with respect to the norm ϕC(Ω) . Then, due to the Cantor diagonal method, it can be easy proved that the sequence {lε (·)}ε→0 contains

7.7 γp -Convergence of sets and Mosco convergence of Sobolev spaces

257

a subsequence which is pointwise convergent on T (K). As a result, there exists a subsequence of values εj → 0 such that lim lεj (ϕ) = l(ϕ),

j→∞

∀ ϕ ∈ C0∞ (Ω).

(7.64)

Taking into account inequality (7.63), we conclude |l(ϕ)| ≤ sup yε L∞ (Ω, dμε ) lim |ϕ| dμε = sup yε L∞ (Ω, dμε ) |ϕ| dx. ε→0

ε>0

ε>0

Ω

Ω

So, l(·) is the linear continuous functional on L1 (Ω). Hence, the following representation holds true: l(ϕ) = vϕ dx, Ω ∞

where v is some element of∞L (Ω). Thus, in view of (7.64), v is a weak-∗ limit

of the subsequence yεj j=1 in the variable space L∞ (Ω, dμε ). Combining this result with Theorem 7.63, we have the following conclusion which can be interpreted as the density concept of the locally convex space C0∞ (Ω) in L∞ (Ω): Theorem 7.68. For any element y ∈ L∞ (Ω), there is a sequence of smooth functions {yε ∈ C0∞ (Ω)}ε>0 satisfying the conditions |yε | ≤ yL∞ (Ω) for every ε ∈ E, ∗

yε y in L∞ (Ω, dμε ) as ε → 0. Note that, in general, the set C0∞ (Ω) is not dense in L∞ (Ω), that is, the assertion ∞

. . . for any f ∈ L∞ (Ω) a sequence {uk ∈ C0∞ (Ω)}k=1 can be found such that uk → f strongly in L∞ (Ω) as k → ∞. . . is not true. So, Theorem 7.68 shows that it can be done through the concept of the weak-∗ convergence in the variable spaces.

7.7 γp–Convergence of open sets and Mosco convergence of the associated Sobolev spaces We begin this section with the following concept (see [38]). Definition 7.69. Let {Ωk }k∈N be a sequence of open subsets of a bounded smooth domain D ⊂ Rn and let 1 < p < ∞. We say that Ωk γp -converges to Ω if for every f ∈ W −1,q (D) and g ∈ W01,p (D), the solutions uΩk ,f,g of the Dirichlet problem

258


−Δp uΩk ,f,g = f in Ωk , uΩk ,f,g = g on ∂Ωk

(7.65)

extended by g on D \ Ωk converge weakly in W01,p (D) to the function uΩ,f,g of the same equation with Ωk replaced by Ω. Here, 1/p + 1/q = 1 and by Δp we denote the p-Laplacian in W01,p (D) (i.e., Δp u = div (|∇u|p−2 ∇u)). As follows from this definition, the γp -convergence is precisely the topology on the space of open sets making the mapping Ω → uΩ,f,g continuous. This type of convergence is especially important in shape optimization problems [38]. The difficulty is to relate this convergence of domains to a geometrical one. The continuity with respect to the geometric domain, for the solution of the boundary value problem (7.65) or other related problems, has been studied by many authors (see, e.g., [36, 38, 40, 76, 121, 133, 217, 234, 249].) A ∞ natural question is: Given a sequence of open sets {Ωk }k=1 , does there exist an open set Ω such that the solution of the Dirichlet problem (7.65) on Ω is the weak limit of a subsequence of solutions on Ωk ? In general, the answer is negative – the “limit” domain being of relaxed type. ∞ Let us consider a sequence {Ωk }k=1 of quasi-open sets contained in a bounded design region D. We want to find a domain Ω ⊆ D such that (for a subsequence labeled using the same indices) we have uΩk ,f,g uΩ,f,g in W01,p (D). To do so, we associate with each domain Ωk the corresponding Sobolev space W01,p (Ωk ). a result, ∞ we have (see Definition 7.39) that the sequence of spaces As W01,p (Ωk ) k=1 converges in the sense of Mosco to W01,p (Ω) if the following conditions are satisfied: ∞ (M1) For all u ∈ W01,p (Ω), there exists a sequence uk ∈ W01,p (Ωk ) k=1 such that uk converges strongly in W01,p (D) to u. ∞ (M2) For every sequence uki ∈ W01,p (Ωki ) i=1 which is weakly convergent in W01,p (D) to a function u, we have u ∈ W01,p (Ω). It is easy to see (see [36]) that the γp -convergence of the sets Ωk to Ω is equivalent to the Mosco convergence of the Sobolev spaces W01,p (Ωk ) to W01,p (Ω) (i.e., to both relations (7.31) and (7.32)). Even more, it is still equivW 1,p (D)

alent with the continuity of the solutions uΩk ,1,0 0 uΩ,1,0 (see [82]). In view of this, we cite the main properties of the γp -convergence following the brilliant book [38]: 1. The γp -convergence does not depend on f ∈ W −1,q (D) and g ∈ W01,p (D) in (7.65). γp 2. The γp -convergence is local, that is, Ωk −→ Ω if and only if there exists δ > 0 such that for every x ∈ D and, for every r ∈ (0, δ), we have that γp Ωk ∩ Bx,r −→ Ω ∩ Bx,r , where by Bx,r we denote the ball in Rn centered at x and of radius r.

7.7 γp -Convergence of sets and Mosco convergence of Sobolev spaces ∞

259

∞

3. For any two sequences of quasi-open sets {Ak }k=1 and {Bk }k=1 such that γp γp γp Ak −→ A and Bk −→ B, we have Ak ∩ Bk −→ A ∩ B. Hc 4. The first Mosco condition (7.31) is fulfilled for every sequence Ωk −→ Ω with respect to the Hausdorff complementary topology on the set A = {Ω : Ω ⊆ D, Ω open} (see Definition 7.36). However, the second Mosco condition (7.32) does not hold, in general, for sequences converging in H c . So, the geometrical constraints play a crucial role for this case. Indeed, the following counterexample shows that, in general, H c -convergent sequences are not γp -convergent. Let {x1 , x2 , . . . } be an enumeration of points of rational coordinates of the square D = (0, 1) × (0, 1) in R2 . Defining Ωk = D \ {x1 , x2 , . . . , xn }, Hc

γp

we get that Ωk −→ ∅ and Ωk −→ D since capp (D \ Ωk = 0 (see (2.44)). 4. Let Oc,r (D) be the class of open subsets of D having the so-called (r, c) capacity density condition, that is, Ω ∈ Oc,r (D) if ∀ x ∈ ∂Ω, ∀ t ∈ (0, r),

capp (Ω c ∩ Bx,t , Bx,2t ) ≥ c. capp (Bx,t , Bx,2t )

∞

Let {Ωk }k=1 be a sequence in Oc,r (D) which converges in the H c -topology to an open set Ω. Then Ωk γp -converges to Ω as well. Note that each of the Mosco conditions (7.31) and (7.32) gives separately interesting information equation (7.65). For p = 2, relation (7.31) (respectively, (7.32)), gives the upper (respectively, lower) semicontinuity of the jth eigenvalue (counted with its multiplicity) of the Laplace operator with homogeneous Dirichlet boundary conditions on the moving domain, for any j. If Ωk “converges” in some “uniform” sense to Ω, then relations (7.31) and (7.32) can be easily established. A natural question is to give the minimal conditions on Ωk and on Ω such that the Mosco convergence holds. Using the capacities and the framework of the Γ -convergence theory, Dal Maso [76, 77] and Dal Maso and Defranceschi [79] gave a characterization of the Mosco convergence. Essentially, W01,p (Ωk ) converges in the sense of Mosco to W01,p (Ω) if and only if there exists a family of sets A ⊆ P(D) which is rich or dense in P(D) such that capp (Ω c ∩ X, D) = lim capp (Ωkc ∩ X, D), k→∞

∀ X ∈ A.

As a result, it was shown in [79] that the convergence in the sense of Mosco is still equivalent with the following two relations, which have to be satisfied for all p-quasi open sets A ⊆ D and p-quasi compact sets F ⊆ A ⊆ D: capp (Ω c ∩ A, D) ≥ lim sup capp (Ωkc ∩ F, D),

(7.66)

capp (Ω c ∩ A, D) ≤ lim inf capp (Ωkc ∩ A, D).

(7.67)

k→∞

k→∞

260


However, the direct verification of these inequalities in the most practical situation is rather difficult. So, the question is to reduce the family of “test” sets to more simple ones. In [36, 38], the following results were proved: 1. Instead of all p-quasi open (respectively, p-quasi compact) sets, one can consider only open (respectively, closed) balls – namely, it is sufficient to consider only A = F = B x,r in (7.66) and A = Bx,r in (7.67), ∀ x ∈ Rn , ∀ r > 0. 2. Relation (7.66) (respectively, (7.67)), on open (respectively, closed balls), represents a characterization of the strong lower (respectively, weak upper) limit in the sense of Kuratowski of the associated Sobolev spaces, being equivalent to (7.31) (respectively, (7.32)). We also recall in this section some results of [45, 79, 82], where relaxed versions of the γp -convergence are studied. Let {Ωk }∞ k=1 be a sequence of open sets containing in D. For every f ∈ W −1,q (D) and g ∈ W01,p (D), we consider the solutions uΩk ,f,g of the Dirichlet problem (7.65). We extend uΩk ,f,g to the ∞ whole of Ω by setting uΩk ,f,g = 0 on D \ Ωk , and we regard {uΩk ,f,g }k=1 as ∞ 1,p a sequence in W0 (D). Since the sequence {Ωk }k=1 does not γp -converge in ∞ general, the problem is to describe the asymptotic behavior of {uΩk ,f,g }k=1 as k tends to infinity. The main idea is to rewrite the Dirichlet problem in a relaxed form. For this, each of the domains Ωk can be associated with the non-negative Borel measure ∞D\Ωk defined by 0 if capp (D ∩ S, Ω) = 0, ∀ S ⊆ D. (7.68) ∞S (D) = +∞ otherwise, If u and g are the functions in W01,p (D) such that u − g = 0 quasi-everywhere in D \ Ωk , then the restriction of u − g to Ωk belongs to W01,p (Ωk ) (see [15]). Conversely, if g ∈ W01,p (D) and we extend a function u ∈ W 1,p (Ωk ) by setting u = g in D \ Ωk , then u is quasi-continuous on D and belongs to W01,p (D). Therefore, if μk = ∞D\Ωk , then a function u in W01,p (D) is the solution of the relaxed problem −Δp u + |u − g|p−2 (u − g)μk = f in W01,p (D) ∩ L2 (D, dμk ), (7.69) u − g ∈ W01,p (D) ∩ L2 (D, dμk ) if and only if g ∈ W01,p (D) ∩ Lp (D, dμk ) and the restriction of u to Ωk is the solution of the classical Dirichlet problem (7.65) and, in addition, u − g = 0 quasi-everywhere on D \ Ωk . In view of this, we have a good reason to introduce the following concept (see [38]). ∞

Definition 7.70. We say that a sequence of Borel measures {μk }k=1 in Mp0 (D) γ-converges to a measure μ ∈ Mp0 (D) if and only if the corresponding solutions uμk ,f,g of relaxed Dirichlet problem (7.69) satisfy the condition uμk ,f,g uμ,f,g in W01,p (D),

∀ f ∈ W −1,q (D), ∀ g ∈ W01,p (D).

7.7 γp -Convergence of sets and Mosco convergence of Sobolev spaces

261

The main result concerning an explicit identification of the γ-limit measure μ ∈ Mp0 (D) and its formal computation can be stated as follows (see [45, 79, 82]). ∞

Theorem 7.71. Given a sequence of open sets {Ωk ⊆ D}k=1 for which the solutions uΩk ,1,0 of (7.65) with f = 1 and g = 0 converge weakly in W01,p (D) to a function w, there exists a Borel measure μ ∈ Mp0 (D) which does not charge sets of p-capacity zero defined by ⎧ ⎨ ω 1−p dν if capp (A ∩ {w = 0}) = 0, μ(A) = ⎩ A +∞ if cap (A ∩ {w = 0}) > 0, p

where ν = 1 + Δp w, such that for any f ∈ W −1,q (D) and g ∈ W01,p (D), we have uΩk ,f,g uμ,f,g , weakly in W01,p (D), where uμ,f,g ∈ W01,p (D) ∩ Lp (D, dμ) and

−Δp u + |u − g|p−2 (u − g)μ = f in W01,p (D) ∩ L2 (D, dμ), u − g ∈ W01,p (D) ∩ L2 (D, dμ).

8 Passing to the Limit in Constrained Minimization Problems

The main object of our consideration in this chapter is the following parameterized minimization problem (8.1) inf Iε (x) , x∈Ξε

where Iε : (Ξε ⊆ Xε ) → R is an objective functional, Ξε ⊆ Xε is a set of admissible solutions, and Xε is some Banach space. In what follows, we make a difference between inf Iε (x) and inf Iε (x) . x∈Ξε

x∈Ξε

The first of one means the infimum mε = inf {Iε (x), x ∈ Ξε } of Iε over the set Ξε , whereas by the second one, we mean the constrained minimization problem as an object being defined by the triplet (Iε , Ξε , Xε ). Note that the basic space Xε , the set Ξε , and the objective functional Iε can be essentially dependent on ε. Hence, as the small parameter ε varies, we have a sequence of constrained minimization problems (8.2) inf Iε (x) , ε → 0 x∈Ξε

living in the corresponding scale of functional spaces. In the sequel, each of the spaces Xε will be associated with some thin or reticulated ε-periodic structure. In view of this, it is convenient in many practical situations to study the asymptotic behavior of the family of problems (8.2), not through the study of the “limit” properties of their solutions x0ε or minimum values mε = inf {Iε (x), x ∈ Ξε }, but by defining a limit functional I0 and its domain Ξ0 such that as ε → 0 the problem inf I0 (x) with m0 = inf {I0 (x), x ∈ Ξ0 } (8.3) x∈Ξ0


263

264

8 Passing to the Limit in Minimum Problems

is a “good approximation” to the family (8.2), that is, mε → m0 and x0ε → x0 in variable space Xε , where x0 ∈ Ξ0 is itself a solution of the “limit” problem (8.3). Our prime interest in this chapter is to discuss the formalism of variational convergence of the minimization problems (8.1) as ε tends to 0, which would possess the approximated property indicated above.

8.1 A short survey of Γ -convergence theory in metric spaces This section is devoted to the definition of De Giorgi’s Γ -convergence and its main properties. We do not want here to enter into the details of that theory, but only to use it in order to characterize the notion of limit minimization problem to the family (8.2). We refer the reader for all the details to [10, 29, 78], many of which are, nevertheless, simple applications of the definitions. Let (X, ρ) be a separable metric space, endowed with a distance ρ. In applications, we will usually deal with metric spaces (as Lp -spaces) or metrizable spaces (as bounded subsets of Sobolev spaces or of spaces of measures, equipped with the weak topology) that in addition are also separable. In what follows in this section, we will consider a family of functionals Iε : Ξε → R indexed by the positive parameter ε under the supposition that Ξε ⊂ X for all ε > 0 and the sequential K-lower limit Ks − lim inf ε→0 Ξε is a nonempty set (see Definition 7.21). We also suppose that each of the functionals Iε : Ξε → R is defined on its domain Ξε only and it may be undefined outside of this set. In this case, we identify such functionals by Iε (x) if x ∈ Ξε , Iε (x) = (8.4) +∞ if x ∈ X \ Ξε . Definition 8.1. Given a sequence of functionals Iε : Ξε → R ε>0 , we say that Iε : X → R Γ -converges to a functional I : X → R as ε → 0 if for all x ∈ X, we have the following: (i) (lim inf inequality) For every sequence {xε }ε>0 converging to x, I(x) ≤ lim inf Iε (xε ). ε→0

(8.5)

(ii) (lim sup inequality) For all η > 0, there exists a sequence {xε }ε>0 converging to x such that I(x) ≥ lim sup Iε (xε ) − η. ε→0

(8.6)

8.1 A short survey of Γ -convergence theory

265

If (i) and (ii) hold, we write I(u) = Γ − limε→0 Iε (x) and I is the Γ -limit of Iε . Sometimes we will write Γ (ρ)− limε→0 to indicate the topology associated with the metric ρ explicitly. We also introduce the notation

I (x) = Γ − lim inf Iε (x) = inf lim inf Iε (xε ) : xε → x , ε→0

ε→0

I (x) = Γ − lim sup Iε (x) = inf lim sup Iε (xε ) : xε → x , ε→0

ε→0

so that the equality I = I is equivalent to the existence of the limit Γ − lim Iε (x). ε→0

We list the main properties of Γ -convergence. • Lower semicontinuity. Every Γ -limit is lower semicontinuous on X. • Convergence of minima. If Iε : X → R ε>0 Γ -converges to I and is equicoercive on X, that is, for every t ∈ R there exists a compact set Kt ⊂ X such that

x ∈ X : Iε (x) ≤ t ≡ {x ∈ Ξε : Iε (x) ≤ t} ⊂ Kt , ∀ ε > 0, then I is coercive too, and so it attains its minimum on X. Moreover, in this case, we have min I(x) = lim inf Iε (x) . x∈X

ε→0 x∈ Ξε

• Convergence of minimizers. Let Iε : X → R ε>0 be an equicoercive sequence of functionals which Γ -converges to a functional I. If xε ∈ arg min Iε (x) x∈Ξε

ε>0

is a sequence with xε → x in X, then we have x ∈ arg minx∈X I(x). Moreover, if I is not identically +∞ and if xε ∈ arg minx∈Ξε Iε (x), then there exists a subsequence of {xε }ε>0 which converges to an element of arg minx∈X I(x). In particular, if I has a unique minimum point x on X, then every sequence xε ∈ arg minx∈Ξε Iε (x) ε>0 converges to x in X. • Compactness. From every sequence Iε : Ξε → R ε>0 of functionals, it is possible to extract a subsequence Γ -converging to a functional I on X. • Metrizability. The Γ -convergence, considered on the family S(X) of all lower semicontinuous functions on X, does not come from a topology unless the

266


space X is locally compact, which never occurs in the infinite-dimensional case. However, if instead of considering the whole family S(X), we take the smaller classes

SΨ (X) = I : Ξ → R : I ≥ Ψ on Ξ, Ξ ⊂ X, I l.s.c. on X , where Ψ : X → R is lower semicontinuous and coercive (and non-negative, for simplicity), then the Γ -convergence on SΨ (X) is metrizable. More precisely, it turns out to be equivalent to the convergence associated to the distance dΓ (I, J) =

∞

2−i−j |arctan(Ij (xi ) − arctan(Jj (xi )| ,

i,j=1 ∞

where {xi }i=1 is a dense sequence in X and Hj denotes the Moreau–Yosida : X → R, defined by transform of a functional H

+ jρ(x, y) : x ∈ X . Hj (x) = inf H(x) According to the compactness property seen above, the family SΨ (X) endowed with the distance dΓ turns out to be a compact metric space. • Semiadditivity. Let Iε : Ξε → R ε>0 and Jε : X → R ε>0 be any two sequences of functionals. Then each of the inequalities (8.7) Γ − lim inf Iε + Jε ≥ Γ − lim inf Iε + Γ − lim inf Jε , ε→0 ε→0 ε→0 Γ − lim sup Iε + Jε ≥ Γ − lim inf Iε + Γ − lim sup Jε (8.8) ε→0

ε→0

ε→0

is true, provided that the sums occurring in it are well defined on X. In particular, if Iε : X → R Γ -converges to I, Jε : X → R Γ -converges to J, and the sum Iε + Jε Γ -converges to H, then I + J ≤ H, provided that the functionals Iε + Jε and I + J are well defined on X. • Stability property. If G : X → R is a continuous functional and I = Γ − limε→0 Iε then I + G = Γ − limε→0 Iε + G . • Relaxation. If H(x) ≤ Iε (x), ∀ x ∈ Ξε , then sc− H(x) ≤ Γ − lim Iε (x) ≤ lim inf Iε (x), ε→0

ε→0

∀ x ∈ X,

(8.9)

where by sc− H : X → R we denote the lower semicontinuous envelope (or relaxed functional) of H : X → R. • Convexity. If Iε : Ξε → R ε>0 is a sequence of convex functionals and if I = Γ − limε→0 Iε , then I : X → R is a convex closed functional.


267

Example 8.2. Let X = R and let Iε : R → R for all ε > 0 be defined as Iε (x) = sin(x/ε). It is easy to calculate that in this case we have Γ − lim Iε (x) = −1. ε→0

Example 8.3. Let ε = 1/n, where n ∈ N. For every n ∈ N, we define the functional Iε : Ξε → R as follows: 1 x2 1 1 if x ≤ 0, , Iε (x) = + 1 if ε = ; Iε (x) = if ε = 2 x + 1 if x > 0, 2n 2 2n + 1 (−1, 1) if ε = 1/(2n), Ξε = (−2, 2) if ε = 1/(2n + 1). Then, following Definition 8.1, straightforward calculations show that ⎧ 2 x /2 + 1 if x ∈ [−2, −1), ⎪ ⎪ ⎨ 1 if x ∈ [−1, 0), Γ − lim inf Iε (x) = 2 /2 + 1 if x ∈ [0, 2], x ε→0 ⎪ ⎪ ⎩ +∞ if x ∈ [−2, 2], ⎧ 2 ⎨ x /2 + 1 if x ∈ [−1, 0), if x ∈ [0, 1], Γ − lim sup Iε (x) = x2 + 1 ⎩ ε→0 +∞ if x ∈ [−1, 1]. Disregarding the domains Ξε in Definition 8.1 leads us to the following analytic representation of Γ -limits of Iε : R → R: 1 if x ≤ 0, Γ − lim inf Iε (x) = x2 /2 + 1 if x > 0, ε→0 2 x /2 + 1 if x ≤ 0, Γ − lim sup Iε (x) = if x > 0. x2 + 1 ε→0 Thus, in general, we have Γ − lim inf Iε = Γ − lim inf Iε and Γ − lim sup Iε = Γ − lim sup Iε . ε→0

ε→0

ε→0

ε→0

Further, we focus on a topological aspect of the Γ -convergence. For this, we make use of the following result which shows that the K-convergence of a sequence of sets (see Definition 7.21) is equivalent to the Γ -convergence of the corresponding indicator functions (we refer to (5.68) for the definition of the indicator function). Proposition 8.4 ([10]). Let {Ξε }ε>0 be a sequence of subsets of X and let {χΞε }ε>0 be the corresponding indicator functions. Also let Ξ = K− lim inf Ξε , ε→0

Ξ = K− lim sup Ξε . ε→0

268


Then 1Ξ = Γ − lim sup 1Ξε , ε→0

1Ξ = Γ − lim inf 1Ξε , ε→0

(8.10)

where 1Ξε denotes the indicator function of the set Ξε : 0, x ∈ Ξε , 1 Ξε = +∞, x ∈ Ξε . In particular, {Ξε }ε>0 K-converges to Ξ if and only if {1Ξε }ε>0 Γ -converges to 1Ξ in X. Having observed that each of the functionals Iε : Ξε → R is well defined and has a sense only on the set of admissible solutions Ξε yields the following object: (8.11) epi(Iε ; Ξε ) = {(x, λ) ∈ X × R | x ∈ Ξε , Iε (x) ≤ λ} ; we call it the epigraph of Iε : Ξε → R. Note that the set epi(Iε ; Ξε ) coincides with the epigraph of the extended functional Iε : X → R, namely

epi(Iε ; Ξε ) ≡ epi(Iε ) := (x, λ) ∈ X × R : Iε (x) ≤ λ , ∀ ε > 0. (8.12) Obviously, epi(Iε ) = X × R if Iε ≡ −∞ and epi(Iε ) = ∅ if Iε ≡ +∞. A functional Iε : Ξε → R is said to be closed, or lower semicontinuous, if epi(Iε ) is a closed set. A notable feature of the set epi(Iε ; Ξε ) is that for a fixed x ∈ Ξε , the set of all λ ∈ R such that (x, λ) ∈ epi(Iε ; Ξε ) can either be empty, or coincide with a closed half-line λ ≥ λε , or coincide with the whole line of real numbers. A converse statement can also be made: Assume that a given subset Fε ⊂ X×R satisfies the above property for every x ∈ ΠrX Fε = {x ∈ X : ∃λ ∈ R, (x, λ) ∈ Fε }. Then there exists a functional Iε with domain Ξε = ΠrX Fε for which epi(Iε ; Ξε ) = Fε . Taking this into account, it is important to note that the K-lower and K-upper limits of a sequence of epigraphs epi Iε ε>0 are always closed epigraphs (see [10]). Moreover, the following theorem shows that Γ -convergence of functionals Iε : X → R can be defined in terms of the K-convergence of their epigraphs. This is the reason why Γ -convergence is sometimes called epi-convergence. Theorem 8.5 ([10, 78]). Let Iε : Ξε → R ε>0 be a sequence of functionals and let J (x) = Γ − lim inf Iε (x), ε→0

Then

epi(J ) = K− lim sup epi Iε , ε→0

J (x) = Γ − lim sup Iε (x). ε→0

epi(J ) = K− lim inf epi Iε , ε→0

(8.13)


269

where K-limits are taken in the product metric of X × R. In particular, Iε : X → R ε>0 Γ -converges to J in X if and only if epi (Iε ) ε>0 Kconverges to epi (J) in X × R. According to (8.12), we have the obvious relations K− lim sup epi Iε = K− lim sup epi Iε ; Ξε , ε→0

ε→0

ε→0

ε→0

(8.14)

K− lim inf epi Iε = K− lim inf epi Iε ; Ξε .

(8.15)

Further, we make the observation: Iε (x) = Iε (x) + 1Ξε (x) for every ε > 0. From this, using (8.7) and (8.8), Proposition 8.4, and Theorem 8.5, we deduce that there exist functionals (8.16) I : K− lim sup Ξε → R and I : K− lim inf Ξε → R ε→0

ε→0

such that epi(J ) = epi I ; K− lim sup Ξε ,

epi(J ) = epi I ; K− lim inf Ξε . ε→0

ε→0

Definition 8.6. Given a sequence of functionals Iε : Ξε → R ε>0 , we state the following: (i) A functional I : (K− lim supε→0 Ξε ) → R is the constrained Γ -lower limit of the sequence Iε : Ξε → R ε>0 as ε → 0 (we write I = Γ − liε→0 Iε ) if epi I ; K− lim sup Ξε = K− lim sup epi Iε ; Ξε . (8.17) ε→0

ε→0

(ii) A functional I :(K− lim inf ε→0 Ξε ) → R is the constrained Γ -upper limit of the sequence Iε : Ξε → R ε>0 as ε → 0 (we write I = Γ − lsε→0 Iε ) if epi I ; K− lim inf Ξε = K− lim inf epi Iε ; Ξε . (8.18) ε→0

ε→0

If there exist a set Ξ ⊆ X and a functional I : Ξ → R such that K− lim inf Ξε = K− lim sup Ξε = Ξ, ε→0

(8.19)

ε→0

I (x) = I (x) = I(x)

on Ξ,

(8.20)

then we write I(x) = Γ − lmε→0 Iε(x) and we say that I : Ξ → R is the constrained Γ -limit of the sequence Iε : Ξε → R ε>0 .

270


Remark 8.7. Coming back to ⎧ 2 ⎨ x /2 + 1 Γ − li Iε (x) = 1 ε→0 ⎩ 2 x /2 + 1 Γ − ls Iε (x) = ε→0

Example 8.3, it is easy to see that if x ∈ [−2, −1), if x ∈ [−1, 0), if x ∈ [0, 2]

x2 /2 + 1 if x ∈ [−1, 0), if x ∈ [0, 1] x2 + 1

on K− lim sup Ξε = [−2, 2], ε→0

on K− lim inf Ξε = [−1, 1]. ε→0

Remark 8.8. Taking into account (8.5) and (8.6) and the sequential version of Kuratowski convergence (see Definition 7.21 ), the definition of the constrained Γ -limit can be rewritten in a more elaborate form. We say that a functional I : Ξ → R is the constrained Γ -limit of a sequence Iε : Ξε → R ε>0 if for all x ∈ Ξ, the following conditions are satisfied: (a) For all η > 0, there exist a constant ε0 > 0 and a sequence {xε }ε>0 converging to x in X such that xε ∈ Ξε for every ε ≤ ε0 ,

I(x) ≥ lim sup Iε (xε ) − η.

(8.21)

ε→0

∞

(aa) For every sequence {xk }k=1 converging to x as k → ∞ in X, there exists a subsequence {Ξεk }∞ k=1 of the sequence {Ξε }ε>0 such that xk ∈ Ξεk for all k ∈ N,

I(x) ≤ lim inf Iεk (xk ). k→∞

(8.22)

In what follows, the following notation for the constrained we will use Γ -convergence of Iε : Ξε → R ε>0 to I : Ξ → R: Γ

Iε ; Ξε −→ I; Ξ .

8.2 Γ -Convergence of functionals defined on a Banach space Now, let X be a Banach space, and let Iε : Ξε → R ε>0 be a given sequence of functionals, where Ξε ⊂ X for every ε > 0. It is clear that the properties of the Γ -limit can be considerably affected by the choice of the metric ρ. The natural metric x − yX in X happens to be not very effective in many situations. Moreover, the existence of the Γ -limit in one metric does not imply the existence of Γ -limit in another. Therefore, in each particular case, the application of general results is always a separate problem. So, it is natural to use the notion of weak convergence in X. Thus we come to the following concept.

8.2 Γ -Convergence in a Banach space

271

Definition 8.9. Let Iε : Ξε → R ε>0 be a sequence of functionals defined on the subsets of a Banach space X. Afunctional I : Ξ → R is said to be the constrained Γ -limit of Iε : Ξε → R ε>0 with respect to the weak topology of Γ (w)

X (we write I = Γ (w)− lmε→0 Iε or Iε ; Ξε −→ I; Ξ) if for any x ∈ Ξ, the following two conditions are satisfied: (b) There exist a constant ε0 > 0 and a sequence xε x as ε → 0 in X (called a Γ -realizing sequence) such that xε ∈ Ξε for every ε ≤ ε0 and I(x) = lim sup Iε (xε ).

(8.23)

ε→0

(bb) For every sequence xk x as k → ∞ in X, there exists a subsequence ∞ {Ξεk }k=1 of {Ξε }ε>0 such that xk ∈ Ξεk for all k ∈ N and I(x) ≤ lim inf Iεk (xk ). k→∞

(8.24)

To illustrate Definition 8.9, consider the following example. Example 8.10. Let Ω ⊂ Rn be a bounded open domain and let Sε ⊂ Ω be a closed subset for each ε > 0. We set Ωε = Ω \Sε . Let χΩε be the characteristic function of the perforated domain Ωε . Let X = L2 (Ω). For every ε > 0, we define the functional Iε : Ξε → R as follows: aε (x)y 2 (x) dx, Ξε = χΩε z : ∀ z ∈ L2 (Ω) , Iε (y) = Ω

where aε ∈ L∞ (Ω). Our aim is to calculate the constrained Γ -limit of Iε : Ξε → R ε>0 with respect to the weak topology of X as ε → 0. Assume that the following conditions are satisfied: ∗

(i) χΩε χ0 in L∞ (Ω) as ε → 0; ∗ ∞ ∞ (ii) a−1 ε ∈ L (Ω) ∀ ε > 0 and χΩε /aε g0 in L (Ω) as ε → 0; −1 ∞ (iii) g0 ∈ L (Ω). Let us show that, in this case, K− lim epi Iε ; Ξε = epi I0 ; Ξ0 , ε→0

where

g0−1 y 2 dx,

I(y) =

Ξ0 = L2 (Ω).

(8.25)

Ω

First, we verify condition (b) of Definition 8.9. For η = 0 and y ∈ L2 (Ω), we construct a sequence {yε }ε>0 weakly converging to y in L2 (Ω) by setting

272


yε = g0−1 χΩε a−1 ε y, ∀ ε > 0. Then using the fact that χΩε χΩε = χΩε and g0−2 y 2 ∈ L1 (Ω), we have yε ∈ Ξε for each ε > 0, χΩ g0−2 ε y 2 dx −→ g0−1 y 2 dx = I(y). Iε (yε ) = a ε Ω Ω Ω

Next, we show that condition (bb) is satisfied, too. Let y be an element of 2 L2 (Ω) and let {zk }∞ k=1 be an arbitrary sequence in L (Ω) such that zk y 2 in L (Ω) as k → ∞ and zk ∈ Ξεk for all k ∈ N, where εk → 0 as k tends to ∞. Then, we obviously have yεk = g0−1 aεk Ω

zk2

−

yε2k

χΩεk y y in L2 (Ω) as k → ∞, aεk

dx ≥ 2 Ω =2

aεk (zk − yεk ) yεk dx g0−1 (zk − yεk ) y dx −→ 0 as k → ∞.

Ω

Hence, lim inf Iεk (zk ) ≥ lim inf Iεk (yεk ) = I(y), k→∞

k→∞

and we obtain the required result. In particular, as follows from (8.25), if aε = 1 for all ε > 0, then y2 dx, ∀ y ∈ L2 (Ω) I(y) = Γ (w)− lm Iε (y) = ε→0 χ 0 Ω (see Proposition 7.12 for comparison). It should be noted that Definition 8.9 cannot be reduced to the general conditions (8.21) and (8.22) because the weak convergence in X cannot be specified in terms of a metric. However, there exists a wide class of functionals for which Definition 8.9 can be formulated in terms of a metric; moreover, within that class, the constrained Γ -convergence can be expressed in “variational terms”. Let us consider a sequence of functionals Iε : Ξε → R ε>0 for which the following conditions hold true: Ks (w)− lim inf Ξε = ∅, ε→0

{Ξε }ε>0 are uniformly bounded in X, i.e., sup sup xX < ∞.

(8.26) (8.27)

ε>0 x∈Ξε

The following result establishes the compactness property of the constrained Γ -convergence in a Banach space.

8.2 Γ -Convergence in a Banach space

273

Theorem 8.11. Assume that X is a Banach space endowed with its weak topology and that the dual X∗ is separable. Then a sequence of functionals Iε : Ξε → R ε>0 satisfying properties (8.26)–(8.27) is compact with respect to the constrained Γ -convergence in the sense of Definition 8.9. Proof. To prove the compactness of the class (8.26)–(8.27), let us choose a countable dense set in X∗ and denote its points by x∗j . Set ρ(x, y) =

j

| x∗j , x − y X∗ ,X | . j 2 + | x∗j , x − y X∗ ,X |

∞

Then for any sequence {xk }k=1 in X, the following conditions are equivalent (see [99]): (i) xk x in X; ∞ (ii) {xk }k=1 is bounded in X and ρ(xk , x) → 0. Since the space X is separable with respect to the above metric, it follows that thegeneral compactness principle holds. Therefore, there exist a subsequence of Iε : Ξε → R ε>0 , still denoted by Iε : Ξε → R ε>0 , a set Ξ ⊂ X, and a functional I : Ξ → R such that (see [78]) Ξ = K(ρ)− lim Ξε , epi I; Ξ = K(ρ × τ )− lim epi Iε ; Ξε , ε→0

ε→0

where by τ we denote the topology of pointwise convergence in R. Moreover, as follows from (8.26), the following condition is satisfied: Ξ = ∅. that the functional I : Ξ → R Thus, I = Γ (ρ)− lmε→0 I ε . Now, we prove coincide with the Γ -limit of Iε : Ξε → R ε>0 in the sense of Definition 8.9. ∞ First, we verify condition (bb). Let {xk }k=1 be a sequence in X such that xk x in X, xk ∈ Ξεk for all k ∈ N, where

lim εk = 0.

k→∞

ρ

Then xk → x, and therefore inequality (8.24) is satisfied. Next, we show that condition (b) holds, too. For any x ∈ Ξ, denote by {xε }ε>0 a Γ (ρ)-realizing sequence: ρ Iε (xε ) → I(x), xε → x, xε ∈ Ξε , ∀ ε ≤ ε0 . Since the norms of xε are uniformly bounded with respect to ε, it follows that xε x, and therefore condition (b) is satisfied. Thus, the compactness of class (8.26)–(8.27), with respect to the constrained Γ (w)-convergence in X, is established. This completes the proof. For every ε > 0, we introduce the conjugate functionals

− Iε∗ (x∗ ) = inf Iε (x) − x∗ , xX∗ ,X . x∈Ξε

(8.28)

274


According to the results of Sect. 3.1, each of the constrained minimization problems (8.28) is solvable, provided Ξε is a bounded closed set and Iε is sequentially lower semicontinuous with respect to the weak topology of X. Then there exists a sequence of minimizers x0ε ∈ Ξε ε>0 such that inf

x∈Ξε

Iε (x) − x∗ , xX∗ ,X = Iε (x0ε ) − x∗ , x0ε X∗ ,X ,

∀ ε > 0.

The following result establishes a relation between constrained Γ -convergence and variational problems. Theorem 8.12. Assume that X is a separable reflexive Banach space. Let Iε : Ξε → R ε>0 be a sequence of functionals satisfying properties (8.26) and (8.27). Assume also that for every ε > 0, Ξε is a closed set and Iε is sequentially lower semicontinuous with respect to the weak topology of X. Then the Γ constrained Γ -convergence Iε ; Ξε −→ I; Ξ implies the pointwise convergence of the conjugate functionals: Iε∗ (x∗ ) → I ∗ (x∗ ) as ε → 0,

∀ x∗ ∈ X∗ .

(8.29)

Γ

Moreover, Γ -convergence Iε ; Ξε −→ I; Ξ is accompanied by the weak convergence of minimizers: If x0ε is a minimizer for problem (8.28), then the 0 sequence xε ε>0 is bounded and any weak cluster point of x0ε ε>0 is a minimizer of the limit variational problem

− I ∗ (x∗ ) = inf I(x) − x∗ , xX∗ ,X . (8.30) x∈Ξ

Proof. Assume that I : Ξ → R is the constrained Γ -limit of Iε : Ξε → R ε>0 0 in the sense of Definition 8.9. Let xε be a minimizer for problem (8.28). Then 0 xε ε>0 is the bounded sequence in X. So, we can assume, without loss of generality, that x0ε x0 in X, where x0 ∈ Ξ by the properties of K(w)convergence. It then follows from inequality (8.24) that lim inf Iε (x0ε ) ≥ I(x0 ). ε→0

Therefore,

lim sup Iε∗ (x∗ ) = lim sup sup x∗ , xX∗ ,X − Iε (x) ε→0 ε→0 x∈Ξε

= lim sup x∗ , x0ε X∗ ,X − Iε (x0ε ) ε→0 ≤ x∗ , x0 X∗ ,X − I(x0 ) ≤ I ∗ (x∗ ).

(8.31)

Let x be a minimizers of the limit problem (8.30) and consider a Γ (w)-realizing sequence xε x in X. Then

8.3 Variational convergence of constrained minimization problems

275

lim inf Iε∗ (x∗ ) ≥ lim inf x∗ , xε X∗ ,X − Iε (xε ) ε→0

ε→0

= x∗ , xX∗ ,X − I(x) = I ∗ (x∗ ). Thus, we establish the pointwise convergence (8.29) and we also see that (8.31) holds an equality, that is, x0 is a minimizer for the limit problem (8.30). This concludes the proof.

8.3 Variational convergence of constrained minimization problems in Banach spaces As mentioned in Chap. 5, it is often necessary to have a reasonable definition of the “limit problem” as ε tends to zero for a given sequence of constrained minimization problems inf Iε (x) , ε → 0 . (8.32) x∈Ξε

For instance, the homogenization theory of OCPs deals with a special sequence of minimization problems (OCPε ) : inf Iε (u, y) , (8.33) (u,y)∈ Ξε

where the set of admissible solutions Ξε is defined as (SASε ) : Ξε = (u, y) ∈ Uε∂ × Y, Uε∂ ⊂ U, Lε (u, y) + Fε (y) = 0 , (8.34) and dependence on a small parameter ε in (8.33) and (8.34) usually has the form (·/ε). As a result, its object consists in passing to the limit, as ε → 0, in problem (8.33). The expression “passing to the limit” means that we have to find a kind of “limit cost functional” I and a “limit set of constraints” Ξ with clearly defined structure Ξ = {(u, y) : (SE) ∩(CSC)} (SE=state equation) such that the limit object inf (u,y)∈Ξ I(u, y) could be interpreted as some OCP ⎫ ⎧ (CF) : I(u, y) → inf, ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ subject to . (8.35) (OCP) : ∂ (CSC) : (u, y) ∈ U × Y, u ∈ U ⊂ U, ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ (SE) : L(u, y) + F (y) = 0 In addition, the definition, used for the convergence of constrained minimization problems, has to satisfy two natural requirements: (V1) If xε0 is a minimizer for inf x∈Ξε Iε (x), and if xε0 tends (in some sense) to x 0 then x 0 is a minimizer for the limit problem inf I(x) . x∈Ξ

276


(V2) limε→0 inf x∈ Ξε Iε (x) = inf x∈ Ξ I(x). In the present section, we discuss the definition of a “limit minimization problem” for sequence (8.32) in the case when all elements of this sequence are defined on a fixed Banach space X. In view of this, we begin with the following assumptions: Let X be a Banach space endowed with its weak topology. For simplicity, we assume that X is reflexive and separable. Let {Ξε }ε>0 be a given sequence of subsets of X satisfying the following property: There exist a sequence {xε ∈ Ξε }ε>0 and element x ∈ X such that xε x in X. Hence, by the properties of set convergence in the sense of Kuratowski, the sequential K-lower limit Ks (w)− lim inf ε→0 Ξε with respect to the weak topology of X is nonempty. Taking into account the concept of constrained Γ -convergence, we adopt the following definition for the convergence of minimization problems. Definition 8.13. Let Iε : Ξε → R ε>0 be a sequence of functionals defined on the subsets of a Banach space X. A problem inf x∈Ξ I(x) is said to be the variational limit of sequence (8.32) with respect to the weak topology of X, we write inf I(x) = Γ (w)− Lim inf Iε (x) , (8.36) x∈Ξ

ε→0

x∈Ξε

Γ (w)

if Iε ; Ξε −→ I; Ξ, or in other words, if the following conditions are satisfied: (c) For any x ∈ Ξ, there exist a constant ε0 > 0 and a sequence xε x as ε → 0 in X (called a Γ -realizing sequence) such that xε ∈ Ξε for every ε ≤ ε0 and I(x) = lim sup Iε (xε ).

(8.37)

ε→0

(cc) For any x ∈ Ξ and for every sequence xk x as k → ∞ in X, there exists a subsequence {Ξεk }∞ k=1 of {Ξε }ε>0 such that xk ∈ Ξεk for all k ∈ N and I(x) ≤ lim inf Iεk (xk ). k→∞

(8.38)

Remark 8.14. As immediately follows from the definition of Ks -convergence, conditions (c) and (cc) imply the equality Ξ = Ks (w) − limε→0 Ξε . In the following theorem, we prove that the variational convergence of a sequence (8.32) to problem (8.36) implies the convergence of the minimum values of Iε on Ξε to the minimum value of I on Ξ and, moreover, in this case, every weakly cluster point of the sequence of the minimizers of Iε is a minimizer of I. Theorem 8.15. Assume that the minimization problem inf x∈Ξ I(x) is the variational limit of the sequence (8.32) in the sense of Definition 8.13 and this problem has a nonempty set of solutions. For every ε > 0, let x0ε ∈ Ξε


277

be a solution of the corresponding problem (8.32). If the sequence x0ε ε>0 is relatively weakly compact in X, then every weakly cluster point x0 of this sequence is a minimizer of I on Ξ and I(x0 ) = min I(x) = lim min Iε (x). ε→0 x∈Ξε

x∈Ξ

(8.39)

Proof. Let {xεk }k∈N be a subsequence of the sequence of minimizers weakly converging to some element x∗ . Then by properties of K-convergence, we have x0 ∈ Ξ. Moreover, due to property (ccc) of Definition 8.13, one gets lim inf min Iεk (x) = lim inf Iεk (x0εk ) ≥ I(x∗ ) ≥ min I(x). k→∞ x∈Ξεk

k→∞

x∈Ξ

(8.40)

Let x0 be a minimizer for the limit problem (8.36). Since x0 ∈ Ξ, it follows that there exist a constant ε0 > 0 and a sequence {xε }ε>0 weakly converging to x0 such that xε ∈ Ξε for every ε < ε0 and I(x0 ) ≥ lim sup Iε (xε ). ε→0

Using this fact, we have min I(x) = I(x0 ) ≥ lim sup Iε (xε ) ≥ lim sup min Iε (x) x∈Ξ

ε→0

ε→0

x∈Ξε

≥ lim sup min Iεk (x) = lim sup Iεk (x0εk ). k→∞ x∈Ξεk

(8.41)

k→∞

Therefore, in view of (8.40), we come to the following inequality: lim inf Iεk (x0εk ) ≥ lim sup Iεk (x0εk ). k→∞

k→∞

As a result, combining (8.40) and (8.41), we conclude I(x∗ ) = I(x0 ) = min I(x), x∈Ξ

∗

I(x ) = lim min Iεk (x). k→∞ x∈Ξεk

(8.42)

Hence, x∗ is a minimizer for the limit problem inf x∈Ξ I(x). On the other hand, 0 since (8.42) holds for the weak limits of all subsequences {xεk }k∈N of xε ε>0 , it follows that the limits limk→∞ minx∈Ξεk Iεk (x) are coincident and, therefore, I(x∗ ) = lim min Iε (x). ε→0 x∈Ξε

This concludes the proof. As an obvious consequence of Definition 8.13 and Theorem 8.11, we have the following compactness result.

278


Theorem 8.16. Let X be a Banach space with the separable dual X∗ . Let {inf x∈Ξε Iε (x) , ε → 0} be a sequence of constrained minimization problems satisfying properties (8.26) and (8.27). Then this sequence is relatively compact with respect to the variational convergence in the sense of Definition 8.13. Our next intention is to study some aspects of the concept of variational convergence applied to different classes of constrained minimization problems. Let X be a Banach space endowed with its weak topology μ. We define the class P(X) of all constrained minimization problems as inf I(x) x∈Ξ

such that I : Ξ → R ∪ {+∞} = R are lower μ-semicontinuous functionals and Ξ are μ-closed subsets of X. Let us introduce the following binary relation L; P(X) on P(X): inf I1 (x) L inf I2 (x) x∈Ξ1

x∈Ξ2

if and only if there exists a set Q ⊂ X such that (R1) (R2)

Dom(I1 ) ∩ Ξ1 = Q = Dom(I2 ) ∩ Ξ2 , I1 (x) = I2 (x) for every x ∈ Q.

(8.43) (8.44)

Here, Dom(I) = {x ∈ X : I(x) < +∞}. It is easily seen that L; P(X) is an equivalence relation. Definition 8.17. We say that two sequences of constrained minimization problems inf Iε (x), ε → 0 and inf Jε (x), ε → 0 (8.45) x∈ Ξε

x∈ Λε

are jointly L-equivalent if inf Iε (x) L inf Jε (x) , x∈Ξε

x∈Λε

∀ ε > 0.

For the sake of convenience, we make use of the following notations: I(x), x ∈ Ξ, 0, x ∈ Ξ, PΞ I(x) = 1Ξ (x) = +∞, x ∈ X\Ξ, +∞, x ∈ X\Ξ. We are now in a position to state the following result.


279

Theorem 8.18. Assume that the sequences (8.45) are jointly L-equivalent. Moreover, assume that for the sequence inf Iε (x) , ε → 0 , (8.46) x∈ Ξε

there exists a variational limit {inf x∈ Ξ0 I0(x)} (with respect to the μtopology) and the sequence of functionals Iε : Ξε → R ε>0 is uniformly bounded below, that is, there exists a constant γ > −∞ such that Iε (x) ≥ γ

∀ x ∈ Ξε ,

∀ ε > 0.

Then the constrained minimization problems inf x∈Ks (μ)− lim inf Λε

Γ (μ)− lim sup[PΛε Jε ](x) ,

ε→0

inf x∈Ks (μ)− lim sup Λε ε→0

Γ (μ)− lim inf[PΛε Jε ](x) ,

inf I0 (x)

x∈ Ξ0

belong to the same class of L-equivalence. Here, Γ (μ)− lim sup PΛε Jε and Γ (μ)− lim inf PΛε Jε ε→0

ε→0

denote the Γ -lower and Γ -upper limits of {PΞε Iε }ε>0 , respectively. Proof. By our assumptions, for every x ∈ X we have Γ (μ)− lim sup[PΞε Iε ](x) = Γ (μ)− lim inf [PΞε Iε ](x) = PΞ0 I0 (x). ε→0

ε→0

Moreover, since the sequences (8.45) are jointly L-equivalent, it follows that PΞε Iε (x) = PΛε Jε (x), ∀ x ∈ X, ∀ ε > 0. (8.47) Therefore, the sequence PΛε Jε : X → R ε>0 is Γ -convergent as well. Moreover, in this case we have Γ (μ)− lim PΞε Iε = Γ (μ)− lim PΛε Jε ε→0

ε→0

on X.

(8.48)

It remains to verify that there exists a set Q0 satisfying the condition Q0 = Dom(I0 ) ∩ Ξ0 = Dom Γ (μ)− lim [PΛε Jε ] ∩ Ks (μ)− lim inf Λε . (8.49) ε→0

ε→0

Using the uniform boundedness of the family Iε : Ξε → R the properties of Γ -limits (see [78]), we obtain

ε>0

, (8.48), and

280


Γ (μ)− lim [PΛε Jε ] (x) ≥ γ + Γ (μ)− lim sup 1Qε (x), ∀x ∈ X, ε→0

ε→0

where the sets Qε are defined as Qε = Dom(Iε ) ∩ Ξε = Dom(Jε ) ∩ Λε , ∀ε > 0.

(8.50)

As follows from the Γ -limit properties, we have Γ (μ)− lim sup 1Qε (x) = 1Ks (μ)− lim inf ε→0

ε→0

Qε (x).

Hence, ∅ = Dom Γ (μ)− lim [PΛε Jε ] ε→0

= Dom(PΞ0 I0 ) ⊂ Ks (μ)− lim inf Qε . ε→0

(8.51)

Using (8.50), we see that Ks (μ)− lim inf Qε ⊆ Ks (μ)− lim Ξε ∩ Ks (μ)− lim sup (Dom(Fε )) ε→0

ε→0

ε→0

= Λ0 ∩ μ− lim sup (Dom(Iε )) , ε→0

Ks (μ)− lim inf Qε ⊂ Ks (μ)− lim inf Λε ∩ Ks (μ)− lim sup (Dom(Jε )) . ε→0

ε→0

ε→0

Then, from (8.51), we deduce that Dom Γ (μ)− lim [PΛε Jε ] ⊆ Ks (μ)− lim inf Λε , ε→0

ε→0

Dom (PΞ0 I0 ) ⊆ Q0 . Thus, in view of (8.48), equality (8.49) follows immediately. Let us consider the following application of this theorem. Let Iε : X → R ε>0 be a sequence of functionals uniformly bounded below and let {Ξε }ε>0 and {Λε }ε>0 be two sequences of subsets of X such that Ξε = Λε ∩ W

for every ε > 0,

where W is some fixed subset. Then it is easy to see that the sequences of constrained minimization problems and inf (Iε (x) + 1W (x)) , ε → 0 inf Iε (x) , ε → 0 x∈ Ξε

x∈ Λε

are jointly L-equivalent. Assume that for the sequence {inf x∈ Λε (Iε (x) + 1W (x)) , ε → 0}, there exists a variational limit

8.4 Variational convergence in variable spaces

inf J 0 (x) .

x∈ Λ0

281

(8.52)

Then, by Theorem 8.18, there exists a variational limit for the sequence inf Iε (x) , ε → 0 x∈ Ξε

which is L-equivalent to (8.52).

8.4 Variational convergence of minimization problems in variable spaces In this section, we study the limit properties of a sequence of constrained minimization problems inf Iε (x) , ε → 0 (8.53) x∈Ξε

in the case when for every ε > 0, the sets of admissible solutions Ξε belong to the corresponding Banach spaces Xε . Our aim is to give for (8.53) a reasonable definition of the “limit problem” as ε tends to 0. In view of this, we begin with the following suppositions. For any small positive value ε > 0, let Xε be a real Banach space equipped with the norm · Xε . Following Khruslov’s classification (see [138]), we say that the spaces Xε are strongly connected with a real Banach space X if there exists a sequence of continuous linear mappings {Pε : Xε → X}ε>0 satisfying the property supε>0 Pε < +∞. In contrast to this concept, the weak connectedness of the above spaces means the absence of the corresponding extension operators Pε : Xε → X with properties prescribed above. In what follows in this section, we make no assumptions of strong connectedness of the variable spaces Xε with X. Moreover, in our next applications, each of the spaces Xε is usually associated with some reticulated structure. So, in general, we admit that Xε can be taken as Sobolev spaces with respect to some variable Borel measure. Let τ be a “convergence concept” in the scale of variable spaces {Xε }ε>0 . As follows from our previous considerations, this convergence can be introduced in a variety of ways (see Chap. 6 for the details). It should be stressed here that when we speak about “choice of the convergence concept in the scale of variable spaces”, we should look at this problem from two opposite points of view. On the one hand, this convergence has to be weak enough in order to have compactness result for bounded sequences {uε ∈ Xε }ε>0 . Still, this type of convergence has to be strong enough in order to have the so-called strong approximation property for the “limit” space X. This means that for every τ element u ∈ X, one can find a sequence {uε ∈ Xε }ε>0 such that uε −→ u

282


in Xε . Note that, in general, this is not easily attained. However, the choice of the convergence in the scale of variable spaces {Xε }ε>0 is completely free, provided the compactness and approximation occur. For further analysis, we suppose that the τ -convergence in Xε is given and X is the τ -“limit” space for the scale {Xε }ε>0 . For any fixed u ∈ X, let us denote by Wτ (u) the set of all sequences {uε }ε>0 such that uε ∈ Xε , ∀ ε > 0,

sup uε Xε < +∞, ε>0

τ

uε −→ u in Xε . So, hereinafter the inclusion {uε }ε>0 ∈ Wτ (u) means that {uε }ε>0 is a bounded τ -convergent sequence in Xε and an element u is its τ -limit. Let {Ξε ⊂ Xε }ε>0 be a given sequence of subsets satisfying the following properties: (Ai ) {Ξε ⊂ Xε }ε>0 is uniformly bounded, that is, supε>0 supu∈Ξε uXε < +∞. (Aii ) There exist a sequence {uε ∈ Ξε }ε>0 and element u ∈ X such that {uε }ε>0 ∈ Wτ (u). Hence, by properties of set convergence in the sense of Kuratowski, the sequential K-lower limit Ks (τ )− lim inf ε→0 Ξε with respect to the τ -convergence in variable space Xε is nonempty. Definition 8.19. We say that a sequence of sets {Ξε ⊂ Xε }ε>0 Ks -converges with respect to the τ -convergence in the variable spaces {Xε }ε>0 to a set Ξ ⊂ X if the following conditions are satisfied: (a) For every u ∈ Ξ, there exist a constant ε0 > 0 and a sequence {uε }ε>0 ∈ Wτ (u) such that uε ∈ Ξε for every ε < ε0 . (b) If {Ξεk }k∈N is a subsequence of {Ξε ⊂ Xε }ε>0 and {uk }k∈N is a sequence τ -converging to u in the variable spaces {Xε } such that uk ∈ Ξεk for every k ∈ N, then u ∈ Ξ. For our next analysis, we make use the following observation. Lemma 8.20. Let {Ξε ⊂ Xε }ε>0 be a sequence of subsets satisfying properties (Ai ) and (Aii ). Let {Iε : Ξε → R}ε>0 be a sequence of functionals such that sup sup |Iε (x)| ≤ C

(8.54)

ε>0 x∈ Ξε

for some constant C > 0. Assume that there exists a nonempty set D ⊂ X × R such that D = Ks (τ × σ)− lim epi(Iε ; Ξε ) e→0

= Ks (τ × σ)− lim {(x, λ) ∈ X × R | x ∈ Ξε , e→0

Iε (x) ≤ λ}

(8.55)


283

in the sense of Definition 8.19, where by σ we denote the topology of pointwise convergence in R. Then there exist a nonempty set Ξ ⊂ X and a functional I : Ξ → R such that D = epi(I; Ξ),

Ξ = Ks (τ )− lim Ξε . ε→0

(8.56)

Remark 8.21. In context of Definition 8.6, this result can be rephrased as follows: Under assumptions given above, the functional I : Ξ → R is the constrained Γ -limit of the sequence {Iε : Ξε → R}ε>0 with respect to the τ convergence in the variable space Xε . Proof. We first prove that the set D can be represented in the form D = epi(I; Ξ). To do so, we note that in view of supposition (8.54), a notable feature of the set epi(Iε ; Ξε ) is the fact that for a fixed uε ∈ Ξε , the set of all λ ∈ R such that (uε , λ) ∈ epi(Iε ; Ξε ) coincides with a closed half-line λ ≥ Iε (uε ). Let (u0 , λ0 ) be an arbitrary pair belonging to the set D. To begin, we show that (u0 , λ∗ ) ∈ D for any λ∗ ≥ λ0 . Due to the sequential properties of Ks convergence (see Definition 8.19), for the given pair (u0 , λ0 ) ∈ D there exist a constant ε0 > 0 and a sequence {(uε , λε )}ε>0 such that {uε }ε>0 ∈ Wτ (u0 ), λε → λ0 in R, and (uε , λε ) ∈ epi(Iε ; Ξε ) for every ε < ε0 . Hence, Iε (uε ) ≤ λε for all ε < ε0 . Let us define the sequence {λ∗ε }ε>0 ⊂ R as follows: ∗ λ if λ∗ ≥ λε , ∗ λε = λε otherwise. Since λε → λ0 as ε → 0 and λ∗ ≥ λ0 , it follows that λ∗ε converge to λ∗ as ε tends to 0. Moreover, for every ε < ε0 , we have (uε , λ∗ε ) ∈ epi(Iε ; Ξε ) and (uε , λ∗ε ) → (u0 , λ∗ ) with respect to the τ × σ-convergence in variable space Xε ×R. Then, by property (b) of Definition 8.19, we conclude that (u0 , λ∗ ) ∈ D for all λ∗ ≥ λ0 . Since this property is valid for any pair (u0 , λ0 ) ∈ D, it follows that there exists a functional I with the domain Ξ = ΠrX D = {x ∈ X : ∃λ ∈ R, (x, λ) ∈ D} , for which epi(I; Ξ) = D. To prove the second equality in (8.56), it remains to apply Definition 8.19 to the case when each of the τ × σ-convergent sequences {(uε , λε )}ε>0 has the ≥ C. Here, C is defined in (8.54). This concludes , where λ form (uε , λ) ε>0 the proof. It is reasonable now to introduce the following concept. Definition 8.22. Let (8.53) be a given sequence of constrained minimization problems in variable spaces Xε . Let X be a Banach space with respect to which the τ -convergence in the scale {Xε }ε>0 is defined. A problem inf x∈Ξ I(x),

284


where Ξ ⊆ X and I : Ξ → R, is said to be the variational limit of the sequence (8.53) with respect to the τ -convergence in Xε , we write inf I(x) = Γ (τ )− Lim inf Iε (x) , (8.57) ε→0

x∈Ξ

x∈Ξε

if Ks (τ )

epi(Iε ; Ξε ) −→ epi(I; Ξ). Before we give a more elaborate form of this definition, we introduce the following concept. Definition 8.23. We say that the space X possesses the weak τ -approximation property with respect to the scale of spaces {Xε }ε>0 if for every δ > 0 and every u ∈ X, there exist an element u∗ ∈ X and a sequence {uε ∈ Xε }ε>0 such that τ (8.58) u − u∗ X ≤ δ and uε −→ u∗ in Xε . In this case, the sequence {uε ∈ Xε }ε∈E is called δ-realizing. Hereinafter in this framework we suppose that the τ -approximation property for the limit space X holds true. We are now in a position to introduce a “natural” notion of the “limiting” minimization problem to the family (8.53) as ε tends to 0. Definition 8.24. A problem inf x∈Ξ I(x) is the variational limit of the sequence (8.53) as ε → 0 if and only if the following conditions are satisfied: (d) If sequences {εk }k∈N and {uk }k∈N are such that εk → 0 as k → ∞, τ uk ∈ Ξεk ∀ k ∈ N, and uk −→ u in Xε , then u ∈ Ξ,

I(u) ≤ lim inf Iεk (uk ). k→∞

(8.59)

(dd) For every element u ∈ Ξ ⊂ X and any δ > 0, there are a constant ε0 > 0 and a δ-realizing sequence {uε }ε>0 such that uε ∈ Ξε , ∀ ε ≤ ε0 ,

τ

uε −→ u in Xε ,

u − uX ≤ δ,

I(u) ≥ lim sup Iε (uε ) − Cδ,

(8.60) (8.61)

ε→0

with some constant C > 0 independent of δ. The following result can be regarded as a natural generalization of Theorem 8.15 to the case when the original sequence of constrained minimization problems is defined in the scale of variable spaces.


Theorem 8.25. Assume that the constrained minimization problem " ! inf I0 (u) u∈Ξ0

285

(8.62)

is the variational limit of the sequence (8.53) in the sense of Definition 8.24 and this problem has a unique solution u0 ∈ Ξ0 . For every ε > 0, let u0ε ∈ Ξε be a minimizer of Iε on the corresponding sets Ξε . If the sequence {u0ε }ε>0 is relatively compact with respect to the τ -convergence in Xε , then τ

inf I0 (u) = I0 u

0

u∈ Ξ0

u0ε −→ u0 , = lim

ε→0

Iε (u0ε )

(8.63) = lim inf Iε (uε ). ε→0 uε ∈ Ξε

(8.64)

Proof. Let {u0εk }k∈N be a subsequence of the sequence of minimizers τ converging to some element u∗ ∈ X. Then due to property (ii) of Definition 8.24, we have u∗ ∈ Ξ0 and lim inf min Iεk (u) = lim inf Iεk (u0εk ) ≥ I0 (u∗ ) ≥ min I0 (u). k→∞ u∈Ξεk

k→∞

u∈Ξ0

(8.65)

Let u0 ∈ Ξ0 be a minimizer of the limit problem (8.62). Let us fix a value δ > 0. Then, by property (ii) of Definition 8.24, there exist a δ-realizing sequence {uε ∈ Ξε }ε>0 such that τ

uε −→ u,

u0 − uX ≤ δ,

and I0 (u0 ) ≥ lim sup Iε (uε ) − Cδ. ε→0

Using this fact, we have min I0 (u) + Cδ = I0 (u0 ) + Cδ

u∈Ξ0

≥ lim sup Iε (uε ) ≥ lim sup min Iε (u) ε→0

ε→0

u ∈Ξε

≥ lim sup min Iεk (u) = lim sup Iεk (u0εk ). k→∞ u ∈ Ξεk

(8.66)

k→∞

From this and (8.65), we deduce that lim inf Iεk (u0εk ) ≥ lim sup Iεk (u0εk ) − Cδ. k→∞

k→∞

Since this inequality holds true for any δ > 0 small enough, combining the (8.65) and (8.66), we get I0 (u∗ ) = I0 (u0 ) = min I0 (u), (u∈ Ξ0

I0 (u0 ) = lim

min Iεk (u).

k →∞ u ∈ Ξεk

Using these relations and the fact that a minimizer of problem (8.62) is ∗ 0 unique, we obtain 0u = u . Since this equality holds for the τ -limits of all subsequences of uε ε∈E , it follows that these limits coincide and, therefore,

286


u0 is the τ -limit of the whole sequence u0ε ε>0 . Hence, accomplishing for the sequence of minimizers what we did earlier with a subsequence u0εk k∈ N , we have lim inf min Iε (u) = lim inf Iε (u0ε ) ε →0 u∈ Ξε

ε →0

≥ I0 (u0 ) = min I0 (u) ≥ lim sup Iε (uε ) − δ u∈ Ξ0

ε→0

≥ lim sup min Iε (u) − Cδ ε→0

u ∈Ξε

= lim sup Iε (u0ε ) − Cδ, ε →0

∀ δ > 0.

Thus, we have obtained the required conclusion. This proof is complete.

8.5 Asymptotic analysis of a Dirichlet optimal control problem The aim of this section is to study the asymptotic behavior of an OCP with partial exact controllability constraints. More precisely, for the sake of simplicity, the main control object is a Dirichlet boundary value problem for the linear Laplacian with bounded controls in the space of Radon measures M(Ω). We suppose that for every ε, where ε takes its values in a sequence of positive numbers tending to 0, there are some closed “holes” Tiε , 1 ≤ i ≤ n(ε), such that on the subdomain Sε = ∪Tiε the state is supposed to exactly match a certain profile Ψεi on each Tiε . We do not make any assumption on the holes other than meas Tiε > 0, ∀ε > 0. The problem is to describe the asymptotic behavior of the parameterized OCP as ε tends to 0, using the variational convergence approach given above. We show that this problem can be viewed as a particular example covered by the theory developed in this chapter. In particular, we consider holes having a critical size depending on their number and distribution. We show that the state equation and the cost functional in the homogenized OCP include additional terms, the so-called “strange terms,” that are associated with the holes. The analysis is very much in the spirit of Dal Maso and Murat [82], where, however, OCPs have not been considered. It is to be noted that the scheme of asymptotic analysis, used in this section, does not provide estimates of the convergence of solutions with respect to some distance measure between the domains. Convergence estimates have been investigated by Savaré and Schimperna [223], again for problems without control. 8.5.1 The statement of the optimal control problem and preliminary results Let Ω be a bounded connected and open subset of Rn with Lipschitz boundary ∂Ω. Let E = (0; ε0 ] be an index set and let {Sε }ε∈E be a family of closed

8.5 Asymptotic analysis of a Dirichlet optimal control problem

287

subsets such that Sε ⊂ Ω with an nonempty interior and Lipschitz boundary for every ε ∈ E. Let {Ψε ∈ H01 (Ω) ∩ H 2 (Ω)}ε∈E and {fε ∈ L2 (Ω)}ε∈E be given functions such that Ψε → Ψ0

weakly in H01 (Ω),

fε → f0

weakly in L2 (Ω).

As usual, by cap (B, Ω) we will denote the capacity of a set B with respect |∇y|2 dx over the set of all functions

to Ω which is defined as the infimum of Ω

y ∈ H01 (Ω) such that y ≥ 1 almost everywhere (a.e.) in a neighborhood of B. Using the terminology of Sect. 2.5 (see also [267]), we say that some property P(α) holds quasi-everywhere (q.e.) in the set Ω, if it holds for all x ∈ Ω except for a subset B of Ω with cap (B, Ω) = 0. We recall that every function y ∈ H01 (Ω) has a quasi-continuous representative which is uniquely defined up to a set of capacity 0. Here, a function y : Ω → R is said to be quasicontinuous if for every δ > 0, there exists a subset B ⊆ Ω with cap (B, Ω) < δ such that the restriction of y to Ω \ B is continuous. We will always identify elements of H01 (Ω) with their quasi-continuous representatives. We will frequently use the notion of a filter on the set E. Definition 8.26. The Fréchet filter on E is the family H of all subsets of E which contain an interval of the form (0, δ) for some δ > 0. The family of all subsets of E that meet all sets H in H we denote by H# . Now, for every fixed ε ∈ E, we define the control problem in the class of bounded Radon measures as follows: −y = fε + ν y∈

H01 (Ω)

y = Ψε I(ν, y) =

in D (Ω),

(8.67)

ν ∈ M(Ω),

(8.68)

quasi-everywhere in Sε , |y − zd |2 dx+ ν M(Ω) → inf,

(8.69) (8.70)

Ω\Sε

where zd is some fixed function of L2 (Ω) and by ν M(Ω) we denote the norm of ν in the space of Radon measures M(Ω), that is, ν M(Ω) =

sup ϕC0 (Ω)≤1

|ν(ϕ)|.

It should be stressed here that, in general, this problem has no optimal solutions. The reason for this is as follows. First, for a Radon measure ν ∈ M(Ω), in general, the Dirichlet problem for the equation −Δy = fε + ν has no solution in H01 (Ω). Second, even if the set of admissible pairs Ξε = (ν, y) ∈ M(Ω) × H01 (Ω) : −Δy = fε + ν in D (Ω), y = Ψε q.e. on Sε }

288


is nonempty for a fixed value ε, it is not clear that this set is closed with respect to the product of the weak-∗ topology for M(Ω) and the topology of the weak convergence in H01 (Ω) (see [106, 169]). Therefore, in order to solve this problem, we have to use some regularization approaches. The simplest way for this is penalizing the controllability constraints on Sε and choosing a narrower class of admissible controls compared with the class of all bounded Radon measures M(Ω). 8.5.2 On modifications of the optimal control problem with controllability constraints We recall that a subset U of Ω is said to be quasi-open if for every δ > 0, there exists an open subset V of Ω with cap (V, Ω) < δ such that U ∪ V is open. If μ is a non-negative Borel measure on Ω, we will use L2μ (Ω) to denote the Lebesgue space with respect to the measure μ. We introduce the space Vμ = H01 (Ω)∩L2μ (Ω) which is well defined for every μ ∈ M20 (Ω) since all functions in H01 (Ω) are defined μ-almost everywhere in Ω (for the details, we refer to Sect. 2.6). In general, this space is not dense in L2 (Ω). Moreover, this space is a Hilbert space with respect to the scalar product ∇y · ∇g dx + y g dμ (see [45]). (y, g)μ = Ω

Ω

Vμ

Let denote the dual space of Vμ and let ·, ·μ be the duality pairing. We note that even if the transposed mappings to the embeddings of Vμ into H01 (Ω) and into L2 (Ω) are not injective, the spaces H −1 (Ω) and L2 (Ω) can be considered as the linear subspaces of Vμ∗ and we have f, yμ = f, yH 1 (Ω) 0

for f ∈ H −1 (Ω), y ∈ Vμ

and, in particular, f, yμ =

f y dx

for f ∈ L2 (Ω), y ∈ Vμ ,

Ω

where we denote by ·, ·H 1 (Ω) the duality pairing between H0−1 (Ω) and 0 H01 (Ω). Let f ∈ H −1 (Ω), Ψ ∈ H01 (Ω), and μ ∈ M20 be fixed elements. We introduce the following Dirichlet problem: Find y ∈ H01 (Ω) such that y − Ψ ∈ Vμ and (∇y, ∇φ) dx + (y − Ψ )ϕ dμ = f, ϕ ∀ϕ ∈ Vμ . (8.71) Ω

Ω

It is easy to see that this problem has a unique solution since the operator B : Vμ → Vμ∗ , which is defined by


Bz, ϕ =

289

(∇z, ∇ϕ) dx +

Ω

∀ ϕ ∈ Vμ ,

zϕ dμ, Ω

is monotone, continuous, and coercive. Moreover, if we take ϕ = y − Ψ as the test function in (8.71), we obtain |∇(y − Ψ )|2 dx + (y − Ψ )2 dμ = f + Ψ, y − Ψ . Ω

Ω

Therefore, by Young’s inequality, we have y − Ψ Vμ ≤ f + Ψ H −1 (Ω) ≤ f H −1 (Ω) + Ψ H −1 (Ω) .

(8.72)

Now, we note that whenever f ∈ L2 (Ω) and Ψ ∈ H01 (Ω) ∩ H 2 (Ω), the solution of (8.71) is actually the solution of a new equation involving a Radon measure λ. Indeed, let y be the corresponding solution of problem (8.71). Since (f + Ψ ) ∈ L2 (Ω), it follows that the positive part of f + Ψ is well defined. So, we define the element λ1 of H −1 (Ω) such that − (y − Ψ )+ + λ1 = (f + Ψ )+ ,

(8.73)

where f + = max(f, 0), f − = max(−f, 0) and f = f + − f − . Let v ∈ H01 (Ω) be any function so that v ≥ 0 q.e. in Ω. Following the line of the proof of Proposition 2.8 from [82], we put vn =

1 v ∧ (y − Ψ )+ , n

where by y ∧ v we denote the minimum of {y, v}. Then it is easy to see that vn ≥ 0

q.e. in Ω and vn ∈ Vμ ,

∀ n ∈ N.

Since (y − Ψ ) · vn ≥ 0

q.e. in Ω,

(f + Ψ )vn ≤ (f + Ψ )+ vn

a.e. in Ω,

it follows that by taking vn as the test function in (8.71), we obtain 1 + (∇(y − Ψ ), ∇vn ) dx ≤ (f +Ψ ) vn dx ≤ (f +Ψ )+ v dx. (8.74) n Ω Ω Ω We note that

1 + , v < (y − Ψ ) ∇vn = ∇v/n a.e. in the domain n 1 + + ∇vn = ∇(y − Ψ ) v ≥ (y − Ψ ) . a.e. in n

290


Therefore, from (8.74) we have 1 (∇(y − Ψ ), ∇v) dx + |∇(y − Ψ )+ |2 dx n {v 0} and ∇(y − Ψ ) = 0 a.e. in {(y − Ψ )+ = 0}, it follows that (∇(y−Ψ )+ , ∇v) dx ≤ (f +Ψ )+ v dx, ∀v ∈ H01 (Ω) with v ≥ 0 q.e. in Ω. Ω

Ω

Hence, this implies that the element λ1 , which was defined in (8.73), is nonnegative, that is, λ1 is a Radon measure. By analogy, we deduce that if λ2 ∈ H −1 (Ω) is defined as # $ (8.76) − −(y − Ψ )− − λ2 = −(f + Ψ )− , then λ2 is a non-negative Radon measure as well. Since −y + Ψ = −(y − Ψ )+ − [−(y − Ψ )− ], it follows that the element λ = λ1 − λ2 of H −1 (Ω) satisfies the equality − (y − Ψ ) + λ = f + Ψ,

(8.77)

and it is a Radon measure with |λ| ≤ λ1 + λ2 . Thus, we have obtained the following result. Lemma 8.27. Let μ ∈ M20 (Ω), Ψ ∈ H01 (Ω) ∩ H 2 (Ω), and f ∈ L2 (Ω) and let y ∈ H01 (Ω) be the corresponding solution of problem (8.71). Let λ1 , λ2 , and λ be the elements of H −1 (Ω) defined by (8.73), (8.76), and (8.77), respectively. Then λ1 , λ2 , and λ are Radon measures such that λ = λ1 − λ2 , |λ| ≤ λ1 + λ2 , λ1 ≥ 0, and λ2 ≥ 0 and for every compact set K ⊆ Ω, the following inequality holds: % |λ|(K) ≤ 2 cap (K, Ω) ∇y − ∇Ψ L2 (Ω) + f L2 (Ω) + ΔΨ L2 (Ω) , (8.78) where

|∇z|2 dx, z ≥ 1 a.e. in the neighborhood of K .

cap(K, Ω) = inf Ω


291

In order to prove (8.78), we note that from (8.73) we have the following upper bound: λ1 H −1 (Ω) ≤ ∇y − ∇Ψ L2 (Ω) + f L2 (Ω) + ΔΨ L2 (Ω) . By analogy with (8.76), we have λ2 H −1 (Ω) ≤ ∇y − ∇Ψ L2 (Ω) + f L2 (Ω) + ΔΨ L2 (Ω) . For every δ > 0, we fix a function z ∈ H01 (Ω) such that z 2H 1 (Ω) ≤ cap(K, Ω) + δ 0

and z ≥ 0 q.e. in Ω and z ≥ 1 q.e. in a neighborhood of K, where K ⊆ Ω is any compact set. Then |λ|(K) ≤ z dλ1 + z dλ2 Ω

Ω

≤ z H01 (Ω) λ1 H −1 (Ω) + λ2 H −1 (Ω)

≤ 2(cap(K, Ω) + δ)1/2 ( ∇y − ∇Ψ L2 (Ω) + f L2 (Ω) + ΔΨ L2 (Ω) ). Passing to the limit as δ → 0, we obtain (8.78). Let us consider the following modification of the state constrained OCP (8.67)–(8.70): −y + με (y − Ψε ) = fε + u in Vμ∗ε , y − Ψε ∈ Vμε

u ∈ L2 (Ω),

Ω\Sε

|∇(y − Ψε )|2 dx

u2 dx + Ω

(y − Ψε )2 dμε −→ inf,

+

(8.80)

|y − zd |2 dx +

Iε (u, y) =

(8.79)

Ω

(8.81)

Ω

where fε ∈ L2 (Ω), Ψε ∈ H01 (Ω) ∩ H 2 (Ω), zd ∈ L2 (Ω), a Borel measure με is defined in such a form that με ∈ M20 (Ω), and for any u ∈ L2 (Ω), the corresponding solution of the problem (8.79)–(8.80) satisfies condition (8.69) (see [82]). For instance, this can be achieved by the rule 0 if cap(B ∩ Sε , Ω) = 0, (8.82) με (B) = +∞ otherwise, As we will see later, this problem always has a solution and good stability properties. In particular, in view of the results mentioned above and the wellknown Lax–Milgram lemma, we may conclude the following:

292


Proposition 8.28. Assume that for a fixed ε ∈ E, fε ∈ L2 (Ω), Ψε ∈ H01 (Ω)∩ H 2 (Ω), the Borel measure με ∈ M20 (Ω) is defined in (8.82). Then for every u ∈ L2 (Ω), there exists a unique solution of the problem y − Ψε ∈ Vμε , (y − Ψε )ϕ dμε = (fε + u) ϕ dx,

(∇y, ∇ϕ) dx + Ω

Ω

∀ϕ ∈ Vμε ,

Ω

such that yε = Ψε quasi-everywhere in Sε and λ = yε + fε is a Radon measure satisfying (8.78) for every compact set K ⊆ Ω. 8.5.3 Passing to the limit in the modified optimal control problem The aim of this subsection is to study the limiting behavior of the modified problem (8.79)–(8.81) as ε → 0. To do so, we represent the OCP (8.79)–(8.81) in the following form: ! " ε∈E , (8.83) inf Iε (u, y) , (u,y)∈Ξε

where the cost functionals Iε : Ξε → R and their domains Ξε are ' ( & ' −y + με (y − Ψε ) = fε + u in Vμ∗ε ' Ξε = (u, y) ' , ' y − Ψε ∈ Vμε , u ∈ L2 (Ω)

(8.84)

(y − zd ) dx + u2 dx Ω\Sε Ω |∇(y − Ψε )|2 dx. + (y − Ψε ) dμε + 2

Iε (u, y) =

(8.85)

Ω

Ω

First, we emphasize the fact that for every ε ∈ E, each of the sets of admissible pairs Ξε is nonempty, convex and closed in the following sense: We say that a sequence{(un , yn ) ∈ Ξε }n∈N τε -converges to some pair (u, y) if un → u

(∇yn , ∇ϕ) dx +

Ω

weakly in L2 (Ω),

(yn − Ψε )ϕ dμε → Ω

(∇y, ∇ϕ) dx Ω

(y − Ψε )ϕ dμε ,

+

∀ ϕ ∈ Vμε .

Ω

It should be remarked that for different values ε ∈ E, the sets Ξε belong to different functional spaces. We proceed to define the concept of convergence in “varying functional spaces.” For this, we use the notion of γ Δ -convergence in M20 (Ω) that was introduced in [81] (see also Sect. 7.7).


293

Definition 8.29. Let {με }ε∈E be a sequence of Borel measures of M20 (Ω) and let μ ∈ M20 (Ω). We say that {με }ε∈E γ -converges to μ in Ω if for every f ∈ H −1 (Ω), the solution yε of the problem − yε , ϕ + yε ϕ dμε = f, ϕ , ∀ ϕ ∈ Vμε , (8.86) yε ∈ Vμε , Ω

as ε → 0 to the solution y ∈ Vμ of the problem − y, ϕ + yϕ dμ = f, ϕ , ∀ ϕ ∈ Vμ . (8.87)

converges weakly in

H01 (Ω)

Ω

We quote the following compactness result by Dal Maso and Murat [82]. Theorem 8.30. Every sequence of measures of M20 (Ω) contains a γ -convergent subsequence, the γ -limit of which is unique. Remark 8.31. Note also that since the solutions of (8.86) depend continuously on f uniformly with respect to the measure με (see [82]), it follows that if μ ∈ M20 (Ω) is the γ -limit of {με }ε∈E and {fε ∈ H −1 (Ω)}ε∈E is strongly convergent to an f ∈ H −1 (Ω), then the functions yε ∈ Vμε defined as the solutions of the problem − yε , ϕ + yε ϕ dμε = fε , ϕ for every ϕ ∈ Vμε Ω

converge weakly in lem (8.87).

H01 (Ω)

to the solution y ∈ H01 (Ω) of the limit prob-

Due to this remark, we immediately have the following result. Lemma 8.32. Let {με ∈ M20 (Ω)}ε∈E , {fε ∈ L2 (Ω)}ε∈E , {uε ∈ L2 (Ω)}ε∈E , and {Ψε ∈ H01 (Ω) ∩ H 2 (Ω)}ε∈E be sequences such that ⎫ in the sense of γ -convergence, ⎪ με → μ0 ⎪ ⎪ ⎪ fε → f0 weakly in L2 (Ω), ⎬ 2 weakly in L (Ω), uε → u0 (8.88) ⎪ ⎪ Ψε → Ψ0 weakly in H01 (Ω), ⎪ ⎪ ⎭ Ψε → Ψ0 strongly in H −1 (Ω). Then the solution yε of the problem yε − Ψε ∈ Vμε , (fε + uε )ϕ, dx, − yε , ϕ + (yε − Ψε )ϕ dμε = Ω

converges weakly in

∀ ϕ ∈ Vμε , ⎭

(8.89)

Ω

H01 (Ω)

to the solution y0 of the problem

y0 − Ψ0 ∈ Vμ0 , − y0 , ϕ + (y0 − Ψ0 )ϕ dμ0 = (f0 + u0 )ϕ dx, Ω

⎫ ⎬

Ω

⎫ ⎬

∀ ϕ ∈ Vμ 0 ⎭

. (8.90)

294


In view of this lemma, we give the definition of the limit object for the sequence of constrained minimization problems (8.83) (see Definition 8.24 for comparison). Definition 8.33. We say that a constrained minimization problem " ! inf I0 (u, y)

(8.91)

(u,y)∈Ξ0

is the variational limit of the sequence (8.83) with respect to the γ -convergence if the following conditions are satisfied: (V1) ⎧ ⎪ ⎪ ⎨

' ⎫ ' u ∈ L2 (Ω), y − Ψ0 ∈ Vμ0 , ⎪ ' ⎪ ⎬ ' ' Ξ0 = (u, y) ' −y, ϕ + (y − Ψ0 )ϕ dμ0 = , (f + u)ϕ dx, 0 ⎪ ⎪ ⎪ ' ⎪ Ω Ω ⎩ ⎭ ' ∀ ϕ ∈ Vμ0 (8.92) where the elements f0 , Ψ0 , and μ0 are defined by (8.88). (V2) For every pair (u, y) ∈ Ξ0 , every index set H ∈ H# , and every sequence {(uε , yε )}ε∈H such that (uε , yε ) ∈ Ξε , ∀ ε ∈ H, uε → u weakly in L2 (Ω) and yε → y weakly in H01 (Ω), we have I0 (u, y) ≤ lim inf Iε (uε , yε ). H ε→0

(V3) For every pair (u, y) ∈ Ξ0 , one can find an index set H ∈ H and a sequence {(¯ uε , y¯ε )}ε∈H such that (¯ uε , y¯ε ) ∈ Ξε for all ε ∈ H, u ¯ε → u weakly in L2 (Ω), y¯ε → y weakly in H01 (Ω), and uε , y¯ε ). I0 (u, y) ≥ lim sup Iε (¯ H ε→0

Remark 8.34. As follows from condition (V1), the set Ξ0 is the sequential version of the limit for {Ξε }ε∈E in Kuratowski’s sense. Indeed, since μ0 ∈ M20 (Ω) is a γ -limit of {με }ε∈E , it follows that for any weakly in L2 (Ω) convergent subsequence of controls {uε }ε∈H to some u0 , H ∈ H# , the corresponding sequence of solutions {yε = y(uε , με )}ε∈H , in accordance with Lemma 8.32, converges weakly in H01 (Ω) to the solution y0 = y(u0 , μ0 ) of the problem (8.90). Hence, the limit pair (u0 , y0 ) of {(uε , yε )} belongs to Ξ0 . Conversely, for any pair (u0 , y0 ) ∈ Ξ0 , the sequence {u0 , yε = y(u0 , με )}ε∈H is such that (u0 , yε ) ∈ Ξε for every ε ∈ H,

yε → y0 weakly in H01 (Ω) (Lemma 8.32).

Thus, Ξ0 satisfies all sequential properties of the Kuratowski’s limit (see [10]). Now, we are in a position to prove the main result of this subsection.


295

Theorem 8.35. Let {fε ∈ L2 (Ω)}ε∈E , {με ∈ M20 (Ω)}, and {Ψε ∈ H01 (Ω) ∩ H 2 (Ω)}ε∈E be arbitrary sequences such that fε → f0 weakly in L2 (Ω), Ψε → Ψ0 weakly in H01 (Ω), Ψε → Ψ0 strongly in H −1 (Ω), and με (B) = +∞ for every B ⊆ Ω such that cap (B ∩ Sε , Ω) > 0. Then there exist a non-negative measure μ0 ∈ M20 (Ω) and a subsequence of (8.83) for which the variational limit exists, and it can be recovered in the form of the following OCP in Vμ∗0 ,

−y + (y − Ψ0 )μ0 = f0 + u y − Ψ0 ∈ Vμ0 ,

u ∈ L2 (Ω),

χ∗ (y − zd )2 dx +

I0 (u, y) = Ω

(8.94)

|∇(y − Ψ0 )|2 dx

u2 dx + Ω

Ω

|y − Ψ0 |2 dμ0 → inf .

+

(8.93)

(8.95)

Ω

Proof. Let {χε }ε∈E be the sequence of the characteristic functions of the sets Ω\Sε . We denote by χ∗ the weak-∗ limit point in L∞ (Ω) of this sequence. For the sake of simplicity, we always assume that με → μ0 in the sense of γ convergence and χε → χ∗ weakly-∗ in L∞ (Ω). For every {yε }ε∈E such that yε → y0 weakly in H01 (Ω), we have yε2 → y02 strongly in L1 (Ω). Therefore, (yε − zd )2 dx = χε (yε − zd )2 dx Ω\Sε

Ω

χ∗ (y0 − zd )2 dx

→

as ε → 0.

(8.96)

Ω

To prove representation (8.95), we have to verify conditions (V2) and (V3). Let {(uε , yε ) ∈ Ξε }ε∈H be any sequence satisfying condition (V2), that is, (uε , yε ) → (u, y) ∈ Ξ0

weakly in L2 (Ω) × H01 (Ω).

Then by (8.96) and the lower semi-continuity property of the norm in L2 (Ω) with respect to the weak convergence, we have ) 2 χε (yε − zd ) dx + u2ε dx lim inf H ε→0

Ω

Ω

* |∇(yε − Ψε )|2 dx

(yε − Ψε )2 dμε +

+ Ω

Ω

χ∗ (y − zd )2 dx +

≥ Ω

(yε − Ψε )2 dμε +

+ lim inf

H ε→0

u2 dx Ω

Ω

Ω

|∇(yε − Ψε )|2 dx .

(8.97)

296


Since for every ε ∈ H, each of the pairs (uε , yε ) satisfies the equality ! " − yε , ϕ + (yε − Ψε )ϕ dμε = (fε + uε )ϕ dx, ∀ ϕ ∈ Vμε , (8.98) Ω

Ω

one obtains, upon putting ϕ = yε − Ψε into (8.98) and integrating by parts,

|∇(yε − Ψε )| dx +

(yε − Ψε )2 dμε

2

Ω

Ω

=

! " (fε + uε )(yε − Ψε ) dx + Ψε , yε − Ψε . (8.99)

Ω

As a result, taking into account that fε → f0

weakly in L2 (Ω),

uε → u

Ψε → Ψ0

weakly in H01 (Ω), Ψε → Ψ0

weakly in L2 (Ω), strongly in H −1 (Ω),

we obtain ⎧ ⎪ ⎨ (fε + uε )(yε − Ψε ) dx → (f0 + u)(y − Ψ0 ) dx, Ω Ω ! " ! " ⎪ ⎩ Ψε , yε − Ψε → Ψ0 , y − Ψ0 .

(8.100)

Now, we use the fact that (u, y) ∈ Ξ0 . We take as a test function ϕ = y−Ψ0 in (8.92) and integrate by parts. Then

|∇(y − Ψ0 )|2 dx + Ω

(y − Ψ0 ) dμ0 " ! = (f0 + u)(y − Ψ0 ) dx + Ψ0 , y − Ψ0 . (8.101) Ω

Ω

Using (8.100) and (8.101), from (8.99) we conclude ) * 2 lim |∇(yε − Ψε )| dx + (yε − Ψε )2 dμε ε→0 Ω Ω ) * (fε + uε )(yε − Ψε ) dx + Ψε , yε − Ψε = lim ε→0 Ω = (f0 + u)(y − Ψ0 ) dx + Ψ0 , y − Ψ0 Ω 2 |∇(y − Ψ0 )| dx + (y − Ψ0 )2 dμ0 . (8.102) = Ω

Ω

Inserting this equality into (8.97), we immediately obtain lim inf Iε (uε , yε ) ≥ I0 (u, y),

H ε→0


297

that is, condition (V2) holds true. In order to prove condition (V3), we take any pair (u, y) of Ξ0 and construct the corresponding sequence {(¯ uε , y¯ε )}ε∈E by the rule u ¯ε = u

for every ε ∈ E,

¯ε = u. and y¯ε = y(u, με ) are the solutions of problems (8.89) with uε = u It is easy to see that (¯ uε , y¯ε ) ∈ Ξε for every ε ∈ E. Moreover, due to Theorem 6.8 in [82] and the fact that με → μ0 in the sense of γ -convergence, we have weakly in H01 (Ω). y¯ε → y Using (8.96) and (8.102), we obtain ) uε , y¯ε ) = lim sup χε (¯ yε − zd )2 dx lim sup Iε (¯ ε→0 ε→0 Ω * u2 dx + |∇(¯ yε − Ψε )|2 dx + (¯ yε − Ψε )2 dμε + Ω

Ω

Ω

= I0 (u, y), that is, condition (V3) holds true as well. This concludes the proof. In view of this theorem, we will assume that the sequence of OCPs (8.83) is convergent in the sense of Definition 8.33. Remark 8.36. Conditions (V2) and (V3) coincide with the definition of the Γ -limit for the following sequence of functionals which are uniformly coercive in L2 (Ω) × H01 (Ω) (see [10, 45])

¯ , PΞε Iε : L2 (Ω) × H01 (Ω) → R ε∈E

where the extension operators PΞε are defined as I(u, y) if (u, y) ∈ Ξε , PΞε I(u, y) = +∞ otherwise. Therefore, the variational limit (8.91) inherits all the variational properties of Γ -limits, namely if (u∗ , y ∗ ) ∈ Ξ0 is a weak limit in L2 (Ω) × H01 (Ω) of the sequence of optimal pairs for the original problem (8.83), then we have the following: (A1) (u∗ , y ∗ ) is an optimal solution for the limit problem (8.91). (A2) I0 (u∗ , y ∗ ) = limε→0 inf (u,y)∈Ξε Iε (u, y). Indeed, the following result holds.

298


Corollary 8.37. The unique optimal pair (u0 , y 0 ) for the limit OCP (8.93)– (8.95) is the weak limit in L2 (Ω)×H01 (Ω) of the sequence of optimal solutions {(u0ε , yε0 )}ε∈E for the original problem (8.83) and, moreover, lim Iε (u0ε , yε0 ) = I0 (u0 , y 0 ).

(8.103)

ε→0

Proof. Using the direct method of Calculus of Variations, it is easy to show that each of the OCPs (8.83) and (8.93)–(8.95) admits a unique solution (u0ε , yε0 ) ∈ Ξε and (u0 , y0 ) ∈ Ξ0 , respectively. Moreover, taking into account the coerciveness property of the cost functional (8.81), the sequence of optimal pairs {(u0ε , yε0 )}ε∈E is uniformly bounded in L2 (Ω)×H01 (Ω) with respect to the small parameter from {(u0ε , yε0 )}ε∈E a weakly convergent 0ε. So, we may extract 0 # subsequence (vσ , pσ ) σ∈H (H ∈ H ). Let (u∗ , y∗ ) ∈ L2 (Ω) × H01 (Ω) be its limit. Then, by Lemma 8.32, we have (u∗ , y ∗ ) ∈ Ξ0 . Moreover, due to part (V2) of Definition 8.33, lim inf

min

H σ→0 (u,y)∈ Ξσ

Iσ (u, y) = lim inf Iσ (vσ0 , p0σ ) H σ→0

≥ I0 (u∗ , y∗ ) ≥

min (u, y)∈ Ξ0

I0 (u, y) = I0 (u0 , y 0 ). (8.104)

At the same time, as follows from part (V3) of Definition 8.33, there exist an index set H ∈ H and a sequence {(uε , yε )} such that (uε , yε ) ∈ Ξε for all values ε ∈ H, (uε , yε )−→(u0 , y0 ) as H ε → 0, and I0 (u0 , y0 ) ≥ lim sup Iε (uε , yε ). ε→0

Using this fact, we get min (u, y)∈Ξ0

I0 (u, y) = I0 (u0 , y 0 ) ≥ lim sup Iε (uε , yε ) H ε→0

≥ lim sup ε→0

min

(u,y) ∈Ξε

Iε (u, y) ≥ lim sup

min

H σ→0 (u,y) ∈ Ξσ

= lim sup Iσ (vσ0 , p0σ ).

(8.105)

H σ→0

From (8.104), it follows that lim inf Iσ (vσ0 , p0σ ) ≥ lim sup Iσ (vσ0 , p0σ ).

H σ→0

H σ→0

Combining (8.104) and (8.105), we conclude I0 (u∗ , y∗ ) = I0 (u0 , y0 ) = I0 (u0 , y0 ) =

lim

Iσ (u, y)

min (u, y)∈ Ξ0

min

H σ →0 (u,y) ∈ Ξσ

I0 (u, y),

Iσ (u, y).

8.6 On homogenization of Dirichlet OCP in perforated domains

299

Taking into account these relations and the uniqueness of the solution of problem (8.93), we obtain (u∗ , y ∗ ) = (u0 , y0 ). Since this equality holds for the limits of any converging subsequences of (u0ε , yε0 ) ε∈E , (u0 , y 0 ) is the weak limit of the sequence (u0ε , yε0 ) ε∈E . Accomplishing for the sequence of minimizers what we did earlier with the subsequence (vσ0 , p0σ ) H σ→0 , we obtain lim inf

min

E ε→0 (u,y)∈ Ξε

Iε (u, y) = lim inf Iε (u0ε , yε0 ) ≥ I0 (u0 , y 0 ) E ε→0

=

min (u, y)∈ Ξ0

≥ lim sup = Thus, relations (8.103) hold.

I0 (u, y) ≥ lim sup Iε (uε , yε ) ε→0

min

Iε (u, y)

E ε→0 (u,y) ∈Ξε lim sup Iε (u0ε , yε0 ). E ε →0

8.6 On homogenization of Dirichlet optimal control problems in perforated domains In this section, we will study the homogenization of an optimal control problem involving a linear elliptic equation with Dirichlet boundary conditions in a perforated domain. The problem is to describe the asymptotic behavior of the sequence of such OCPs as a small parameter tends to 0. We will show that the study of this problem can be traced by the above approach. Let Ω be a bounded open subset of Rn . Let ε ∈ E = (0; ε0 ] be a small parameter and let {Ωε }ε∈E be a sequence of open sets contained in Ω. For given zd ∈ L2 (Ω) and fε ∈ L2 (Ω), we define the OCP in Ωε as follows: Find a pair “control state” (u0ε , yε0 ) ∈ L2 (Ωε ) × H01 (Ωε ) such that Iε (u0ε , yε0 ) = where

inf (u,y)∈Ξε

|∇y|2 dx +

Iε (u, y) = Ωε

Iε (u, y),

(8.106)

(y − zd )2 dx +

Ωε

u2 dx,

(8.107)

Ωε

' ( ' −yε = fε + uε in D (Ωε ) ' . (8.108) (uε , yε ) ∈ L2 (Ωε ) × H01 (Ωε ) ' 'u ˜ε ∈ U ε

& Ξε =

Here, u ˜ε is the trivial extension of uε ∈ L2 (Ωε ) by 0 to the hole of Ω; U ε is a weakly closed convex subset of L2 (Ω) such that U ε −→ U 0 , (U 0 = ∅) in the Kuratowski’s sense

300


with respect to the weak topology of L2 (Ω); χε is the characteristic function of the set Ωε , that is, χε = 1 in Ωε and χε = 0 otherwise. It is well known for every ε ∈ E, there exists a unique solution of the problem (8.106)–(8.108) (see [169]). In view of the uniform coerciveness of the cost functionals Iε : Ξε → R, the sequence of optimal pairs {(u0ε , yε0 ) ∈ L2 (Ωε ) × H01 (Ωε )}ε∈E is uniformly bounded, that is, ) * sup u0ε L2 (Ωε ) + yε0 H01 (Ωε ) < ∞. ε∈E

Therefore, it is natural to choose as a basic topology for the homogenization procedure the product of the weak topologies for L2 (Ω) and H01 (Ω), respectively. Since yε0 is equal to 0 on the boundary ∂Ωε , it is reasonable to extend yε0 by 0 to the hole of Ω. We denote this extension by y˜ε0 . It belongs to H01 (Ω) and, moreover, by the Poincaré inequality, it is easy to show that y˜ε0 H01 (Ω) ≤ C

(8.109)

for all fε ∈ L2 (Ω), zd ∈ L2 (Ω), and u0ε ∈ L2 (Ωε ). Note the contrast with the case of Neumann boundary conditions on ∂Ωε (see [135]). The extension by 0 of the solution of the Dirichlet problem (8.108) now belongs to the space H01 (Ω), whereas the same extension for the solution of the Neumann problem is only an element of L2 (Ω). That is why in that case, one needs to construct extension operators. Further, we note that due to estimate (8.109), one may assume that y˜ε0 y weakly in H01 (Ω). Let us denote by χ∗ the weak-∗ limit in L∞ (Ω) of the sequence {χε ∈ L (Ω)}ε∈E . ∞

Remark 8.38. We will always assume that supp(χ∗ ) = closure(Ω), that is, χ∗ > 0 almost everywhere on Ω. It is clear that in this case, Ω \ Ωε has to be ∞ small enough. Suppose also that χ−1 ∗ ∈ L (Ω). Remark 8.39. We also note that if y is a function of H01 (Ω) such that y = 0 quasi-everywhere in Ω\Ωε , then the restriction of y to Ωε belongs to H01 (Ωε ) (see [15]). Conversely, if we extend a function y ∈ H01 (Ωε ) by setting y = 0 in Ω\Ωε , then y is quasi-continuous and belongs to H01 (Ω). Therefore, if we define the non-negative Borel measure με ∈ M20 (Ω) as & 0 if cap(B ∩ (Ω\Ωε ), Ω) = 0, (8.110) με (B) = +∞ if cap(B ∩ (Ω\Ωε ), Ω) > 0, then a function y ∈ H01 (Ω) is the solution of the problem


y ∈ Vμε = H01 (Ω) ∩ L2με (Ω), yϕ dμε = (fε + χε u)ϕ dx, − y, ϕ + Ω

301

⎫ ⎬ ∀ ϕ ∈ Vμε ⎭

(8.111)

Ω

if and only if the restriction of y to Ωε is the solution of the Dirichlet boundary value problem −y = fε + u in D (Ωε )

y ∈ H01 (Ωε ),

and, in addition, y = 0 q.e. in Ω\Ωε (see [82]). Thanks to this result, we may rewrite the original OCP (8.106)–(8.108) as (8.83)–(8.85). We thus consider the sequence

(8.112) inf I˜ε (u, y) , ε ∈ E ˜ε (u,y)∈Ξ

with

|∇y|2 dx +

I˜ε (u, y) =

Ω

(y − χε zd )2 dx Ω

χε u2 dx , ∀ (u, y) ∈ Ξ˜ε ,

+

(8.113)

Ω

' ⎫ ' − y, ϕ + yϕ dμε = (fε + χε u)ϕ dx ⎪ ' ⎪ ⎪ ' ⎬ Ω Ω ' ˜ Ξε = (u, y) ' , (8.114) ∀ ϕ ∈ V με ⎪ ' ⎪ ⎪ ⎪ ⎪ ' ⎪ ⎩ ⎭ ' y ∈ Vμε , u ∈ L2 (Ω), χε u ∈ U ε ⊆ L2 (Ω) ⎧ ⎪ ⎪ ⎪ ⎨

where the Borel measures με ∈ M20 (Ω) are defined in (8.110). By virtue of the compactness of M20 (Ω) with respect to the γ -convergence and the fact that for the sequence of constrained sets {U ε }ε∈E there exists a nonempty limit U 0 (in the sense of Kuratowski with respect to the weak topology of L2 (Ω)), we have the following result. Lemma 8.40. Assume that fε −→ f0 weakly in L2 (Ω). Then there exists a non-negative Borel measure μ0 ∈ M20 (Ω) such that the set ⎫ ⎧ ' ' ⎨ ' − y, ϕ + yϕ dμ0 = f0 + u, ∀ ϕ ∈ Vμ0 , ⎬ Ξ0 = (u, y) '' (8.115) Ω ⎭ ⎩ ' y ∈ Vμ , u ∈ U 0 ⊆ L2 (Ω) 0 ˜ε ⊂ L2 (Ω)×Vμ }ε∈E in the Kuratowski sense, is the τ -limit of the sequence {Ξ ε where τ is the product of the weak topologies for L2 (Ω) and H01 (Ω). ˜ε and K(τ )−Ls Ξ˜ε the lower and the Proof. Let us denote by K(τ )−Li Ξ ˜ upper sequential limits of {Ξε } in the Kuratowski’s sense, respectively. Let μ0

302


be a γ -limit of the sequence of measures {με }ε∈E . Then due to Remark 8.31 and Lemma 8.32, K(τ )−Ls Ξ˜ε ⊆ Ξ0 . Hence, in order to prove the lemma, it is sufficient to show that Ξ0 ⊆ K(τ )−Li Ξ˜ε . Let (u∗ , y ∗ ) be any pair of Ξ0 . Then u∗ ∈ U 0 ; hence, there is a sequence {uε }ε∈E such that uε → u∗ weakly in L2 (Ω), uε ∈ U ε for every ε ∈ E and uε = χε uε , by the definition of U ε . We denote by yε = y(uε , με ) the corresponding solutions of the problem (8.111) with u = uε . Let yε∗ ∈ Vμε be the solution of (8.111) corresponding to fε = f0 and u = u∗ . By the definition of γ -convergence, {yε∗ } converges to y ∗ weakly in H01 (Ω). However, as follows immediately from (8.114), we have

yε∗

− yε H01 (Ω) +

(yε∗ Ω

1/2

− yε ) dμε √ ≤ 2 fε − f0 H −1 (Ω) + uε − u∗ H −1 (Ω) . 2

In view of the standing assumptions, we have yε∗ − yε −→ 0 strongly in H01 (Ω). Hence, yε → y ∗ weakly in H01 (Ω). As a result, the sequence of τ admissible pairs {(uε , yε ) ∈ Ξ˜ε }ε∈E has been constructed such that (uε , yε ) → (u∗ , y ∗ ), that is, by the definition of the Kuratowski limit, we have (u∗ , y ∗ ) ∈ ˜ε . This implies that K(τ )−Li Ξ Ξ0 ⊆ K(τ )−Li Ξ˜ε ⊆ K(τ )−Ls Ξ˜ε ⊆ Ξ0 . This completes the proof.

In order to recover the limit functional I0 : Ξ0 → R with (V2)–(V3), we note that due to the construction (8.110), we have y 2 dμε = 0 for any (u, y) ∈ Ξ˜ε . Ωε

Therefore, for every ε ∈ E, the cost functionals I˜ε : Ξ˜ε → R can be represented in the form |∇y|2 dx + y 2 dμε I˜ε (u, y) = Ω Ω 2 + (y − χε zd ) dx + χε u2 dx, ∀(u, y) ∈ Ξ˜ε . (8.116) Ω

Ω

Taking (8.102) into account, we have ) * |∇yε |2 dx + yε2 dμε = |∇y|2 dx + y 2 dμ0 , lim H ε→0

Ω

Ω

Ω

(yε − χε zd ) dx =

(y − χ∗ zd )2 dx

2

lim

h ε→0

(8.117)

Ω

Ω

Ω

(8.118)


303

for any index set H ∈ H# and any τ -convergent sequence {(uε , yε ) ∈ Ξ˜ε }ε∈H . In order to recover the limit value of the last term in (8.116), we assume that the constrained sets U ε have the form ' & ( ' ξ1 χε ≤ u ≤ ξ2 χε , ' 2 U ε = u ∈ L (Ω) ' , (8.119) ' u ∈ Λ ⊂ L2 (Ω) where ξ1 , ξ2 ∈ L2 (Ω), ξ1 ≤ ξ2 almost everywhere, and Λ is a closed convex subset of L2 (Ω) for which the following property holds: If u ∈ Λ, then χε u ∈ Λ for every ε > 0. It is easy to see that for the sequence {U ε } there exists a Kuratowski limit U0 such that '

' U 0 = u = χ∗ v ∈ L2 (Ω)' ξ1 ≤ v ≤ ξ2 , v ∈ Λ . (8.120) As a result, we have (see [135]) the following: 2 (i) lim inf χε (uε )2 dx ≥ χ−1 ∗ u dx for every sequence {uε ∈ U ε }ε∈H H ε→0

Ω

Ω

weakly converging in L2 (Ω) to some u ∈ U 0 . (ii) For every u ∈ U 0 , the sequence {uε = χε χ−1 ∗ u}ε∈E is such that uε → u weakly in L2 (Ω), uε ∈ U ε , ∀ ε ∈ E, and 2 2 (uε )2 dx = lim χε , χ−2 u = χ−1 lim ∗ ∗ u dx. (L∞ ,L1 ) ε→0

ε→0

Ω

Ω

Hence, for the sequence of functionals 2 χε u dx Ω

ε∈E

properties (V2)–(V3) hold. Combining this fact with relations (8.117) and (8.118), we obtain the following result. Lemma 8.41. Assume that the sets U ε ⊂ L2 (Ω) have the form (8.119). Then under assumptions of Lemma 8.40, there exists a variational limit ! " ! " Γ (τ )− Lim inf I˜ε (u, y) = inf I0 (u, y) (8.121) ε→0

˜ε (u,y)∈Ξ

(u,y)∈Ξ0

such that the set Ξ0 has the representation (8.115) with U 0 given by (8.120) and 2 2 |∇y| dx + y dμ0 + (y − χ∗ zd )2 dx I0 (u, y) = Ω Ω Ω −1 2 + χ∗ u dx. (8.122) Ω

304


Now, we are in the position to state the main result of this section. We recall that in the framework presented here the variational limit (8.121) is called a limit OCP whenever it corresponds to some control problem that can be recovered from the limiting process. Theorem 8.42. Let ξ1 , ξ2 , zd ∈ L2 (Ω) be such that ξ1 ≤ ξ2 a.e. in Ω and let {fε ∈ L2 (Ω)}ε∈E be a sequence weakly converging to some function f0 ∈ L2 (Ω). Assume that the sequence of open subsets {Ωε ⊂ Ω}ε∈E is such that χε → χ∗ weakly- ∗ in L∞ (Ω),

∞ χ−1 ∗ ∈ L (Ω).

Then there exists a non-negative Borel measure μ0 ∈ M20 (Ω) such that for the family of OCPs ⎫ −y = fε + u in D (Ωε ), ⎪ ⎪ ⎪ ⎬ 1 y ∈ H0 (Ωε ), χε ξ1 ≤ u ≤ χε ξ2 , u ∈ Λ, (8.123) ⎪ ⎪ ⎭ |∇y|2 dx + (y − zd )2 dx + u2 dx → inf ⎪ Iε (u, y) = Ωε

Ωε

Ωε

there exists a limit problem as ε → 0 with the following representation: ⎫ − y + μ0 y = f0 + u in Vμ∗0 ⎪ ⎪ ⎪ ⎪ ⎪ −1 ⎪ y ∈ Vμ0 , χ∗ ξ1 ≤ u ≤ χ∗ ξ2 a.e. on Ω, χ∗ u ∈ Λ ⎪ ⎪ ⎪ ⎬ . (8.124) 2 2 2 |∇y| dx + y dμ0 + (y − χ∗ zd ) dx⎪ I0 (u, y) = ⎪ ⎪ Ω Ω Ω ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −1 2 ⎭ χ∗ u dx → inf + Ω

In addition, if (u0ε , yε0 ) and (u0 , y0 ) are the optimal solutions of problems (8.123) and (8.124), respectively, then y˜ε0 → y 0 weakly in H01 (Ω),

u ˜0ε → u0 weakly in L2 (Ω),

(8.125)

0 2 u0ε − χ−1 χε (˜ ∗ u ) → 0 strongly in L (Ω),

(8.126)

Iε (u0ε , yε0 ) → I0 (u0 , y 0 ).

(8.127)

Proof. The validity of these statements including the variational properties (8.125) and (8.127) immediately follows from Lemma 8.40, Lemma 8.41, Definition 8.33, and Theorem 8.30. As for the property (8.126), we note that as χε → χ∗ weakly-∗ in L∞ (Ω) and 0 2 (u0ε )2 dx = χ−1 lim ∗ (u ) dx, ε→0

one gets

Ωε

Ω


Ω

305

0 2 (χε (˜ u0ε − χ−1 ∗ u )) dx 0 0 2 (˜ u0ε )2 dx − 2 u ˜0ε χ−1 u dx + χ−2 = ∗ ∗ (u ) χε dx → 0. Ω

Ω

Ω

This concludes the proof. We conclude this section with the example of an OCP on a perforated domain with Dirichlet conditions on the boundary and show that the γ -limit measure μ0 ∈ M02 (Ω) in the limit problem (8.124) can be recovered. We consider the case where the domain Ω is perforated by an increasing number of holes of critical size. This is the type of perforated domains which was proposed by Cioranescu and Murat in their celebrated paper [65]. We define the perforated domain as Ω ε = Ω ∩ Qε ,

n(ε)

Qε = Rn \ ∪i=1 Tiε .

Here, we assume that Rn (n ≥ 2) is covered by cubes Piε of size 2ε and Tiε is a ball of radius aε = exp(−C0 /ε2 ) if n = 2 and aε = C0 εn/n−2 if n ≥ 3, centered at the very center of the cube Piε . For given f ∈ L2 (Ω) and each value of ε ∈ (0; ε0 ], we consider the following OCP on Ωε : ⎫ −y = f + u in D (Ωε ), ⎪ ⎪ ⎪ ⎪ 1 ⎪ y ∈ H0 (Ωε ), u L2 (Ω) ≤ 2, ⎪ ⎬ 1 −2 . (8.128) −(meas Ω) χε ≤ χε u a.e. on Ω, ⎪ ⎪ ⎪ ⎪ ⎪ |∇y|2 dx + u2 dx → inf ⎪ Iε (u, y) = ⎭ Ωε

Ωε

It is easy to see that the weak-∗ limit in L∞ (Ω) of the characteristic functions χΩε is equal to χ∗ = 1 a.e. in Ω. Thanks to Theorem 8.42, there is a limit problem for the family (8.128) which can be represented in the form −y + μ0 y = f + u

in Vμ∗0 ,

y ∈ Vμ0 , u L2 (Ω) ≤ 2, −(meas Ω)−1/2 ≤ u a.e. in Ω, |∇y|2 dx + y 2 dμ0 + u2 dx → inf. I0 (u, y) = Ω

Ω

(8.129)

Ω

In order to recover the measure μ0 ∈ M20 (Ω), we use the following result (see [55, 65]). Theorem 8.43 (Casado-Diaz [55], Cioranescu and Murat [65]). If there exists a sequence {ωε ∈ H 1 (Ω)}ε∈E such that ωε = 0 on the holes Tiε , 1 ≤ i ≤ n(ε),

ωε → 1 weakly in H 1 (Ω),

306


then there is a measure μ∗ ∈ M20 (Ω) satisfying the following: For any vε ∈ H01 (Ωε ) and for any v ∈ H01 (Ω) such that v˜ε = χε v we have

and !

v˜ε → v

weakly in H01 (Ω),

" v dμ∗ . − ωε , vε →

(8.130)

Ω

The sequence {ωε }ε∈E was constructed in [65] for the above-mentioned case of perforated domains {Ωε } and it was proven that condition (8.130) holds true if μ∗ is defined as dμ∗ =

π 1 dx if n = 2, 2 C0

dμ∗ = Sn (n − 2)C0n−2 /2n dx if n ≥ 3,

where Sn is the surface of the unit sphere in Rn and C0 is a positive constant. A different approach for the construction of the functions {ωε }ε∈E can be found in [5, 63, 182, 225]. Now, we prove that μ∗ = μ0 in (8.128). For any fixed f ∈ L2 (Ω) and u ∈ L2 (Ω), we consider the variational formulation of the original problem taking as the test function ωε ϕ, where ϕ ∈ D(Ω). One gets (f + u)ωε ϕ dx = ϕ(∇˜ yε , ∇ωε ) dx + ωε (∇˜ yε , ∇ϕ) dx Ω Ω Ω ωε (∇˜ yε , ∇ϕ) dx + − ωε , ϕ˜ yε = Ω − y˜ε (∇ωε , ∇ϕ) dx. Ω

Then, using (8.130) and the facts that ωε → 1 ∇ωε → ∇1 = 0 y˜ε → y

strongly in L2 (Ω), weakly in (L2 (Ω))n , weakly in H01 (Ω)

and passing to the limit as ε → 0, we get (f + u)ϕ dx = (∇y, ∇ϕ) dx + yϕ dμ∗ Ω

Ω

∀ ϕ ∈ D(Ω).

Ω

Hence, the limit function y ∈ H01 (Ω) satisfies the equation − y + μ∗ y = f + u

in D (Ω).

(8.131)

Since the function y ∈ H01 (Ω) also satisfies the state equation (8.129), it follows from the uniqueness result (see Lemma 5.4 in [82]) that we have


307

μ0 = μ∗ . Thus, in order to summarize, the homogenized OCP for the family (8.128) is unique and has the following form (for n = 2): ⎫ −y + π/(2C0 )y = f + u in D (Ω), ⎪ ⎪ ⎪ ⎬ 1 −1/2 a.e. on Ω, y ∈ H0 (Ω), u L2 (Ω) ≤ 2, u ≥ −(meas Ω) . (8.132) ⎪ ⎪ 2 2 2 ⎪ ⎭ |∇y| dx + π/(2C ) y dx + u dx → inf I (u, y) = 0

0

Ω

Ω

Ω

In view of the variational properties of limit problems (see Theorem 8.42), we may add that the unique solution (u0 , y 0 ) ∈ L2 (Ω) × H01 (Ω) of the limit problem (8.132) satisfies the following conditions: y˜ε0 → y0 weakly in H01 (Ω), u ˜0ε → u0 weakly in L2 (Ω), (u0ε − u0 )2 dx → 0, Iε (u0ε , yε0 ) → I0 (u0 , y 0 ), Ωε

where {(u0ε , yε0 ) ∈ L2 (Ωε ) × H01 (Ωε )} is the sequence of the optimal pairs for the family (8.128).

Part II

Optimal Control Problems on Periodic Reticulated Domains: Asymptotic Analysis and Approximate Solutions

9 Suboptimal Control of Linear Steady-State Processes on Thin Periodic Structures with Mixed Boundary Controls

Our main interest in this chapter is in approximate solutions to a class of optimal control problems (OCPs) on thin periodic structures. Typically, thin structures are characterized by two properties: periodicity and small thickness of the material. More precisely, we suppose that the geometry of such structures depends on two small parameters, ε and h(ε), related to each other by the assumption h(ε) → 0 as ε → 0 and determining the cell of periodicity and thickness of constituting components, respectively. This is in contrast to the periodically perforated domain for which h = h(ε) → const ∈ (0, 1] as ε → 0. In view of this, it should be noted that the asymptotic analysis of boundary value problems in a perforated domain with small holes (without controls) has been intensively studied by many authors. We mainly mention Cioranescu and Saint Jean Paulin [66], Cioranescu and Murat [65], Dal Maso and Murat [82], Marchenko and Khruslov [182], Zhikov, Kozlov, and Oleinik [261]. The interesting effect of homogenization of the Poisson equation with (zero) Dirichlet conditions on the boundary of the holes is well known when a “strange term” appears in the limit equation. This effect first was discovered and studied by Marchenko and Khruslov in [182], and Cioranescu and Murat [65]. Another effect of homogenization of the same equation with a critical size of the holes, when nonhomogeneous Neumann conditions on the boundary of the holes are assumed, was studied by Conca and Donato [70]. In this case, some constant that is proportional to the limit of the total flux of the solution through the boundary of the holes appears in the limit equation. In Cardone, D’Apice, and De Maio [54], and Corbo Esposito, D’Apice, and Gaudiello [73], the authors examined a homogenization problem with both Dirichlet and Neumann conditions on the boundary of the holes. It was shown in [73] for a very simple geometry of holes that an interference phenomenon in the homogenization of the Poisson equation is present when a nonhomogeneous Neumann condition is imposed on one part of the boundary of the holes and Dirichlet condition is imposed on the remaining part. The asymptotic analysis of boundary value problems on thin periodic structures has been the subject of a large number of publications, among which P.I. Kogut, G.R. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains, Systems & Control: Foundations & Applications, DOI 10.1007/978-0-8176-8149-4 9, © Springer Science+Business Media, LLC 2011

311

312

9 Suboptimal Control of Steady-State Processes on Thin Structures

we mention the monographs of Ciarlet [62], Nazarov [200], and Panasenko [209]. There are several recent works devoted to the homogenization of thin structures and other singular media. We quote here only the works Argatov and Nazarov [8, 9], Bouchitté and Fragala [26], Braides and Defranceschi [30], Chechkin, Jikov, Lukkassen, and Piatnitski [58], Cioranescu and Saint Jean Paulin [66], Nazarov [199], and Panasenko [207, 208]. Elasticity problems on rod junctures were studied in the interesting works of Bouchitté and Fragala [27], Nazarov and Slutskii [201, 202, 203], Zhikov [257], Pastuchova [211, 213], and Zhikov and Pastuchova [265, 264]. We focus our attention on OCPs with different types of controls as well as control and state constraints. Having provided the asymptotic analysis of such problems as the small parameter ε tends to 0, we construct the so-called asymptotically suboptimal controls for the original objects. For the sake of simplicity, we restrict the analysis tothe case when a 2D lattice structure Ωε has the representation Ωε = Ω ∩ εF h , where F h denotes a 1-periodic thin grid with a small but same thickness h. It means that hereinafter by thin grid we always mean a 1-periodic planar “fattened”graph after applying the procedure of h-thickening to some singular periodic graph F. It is well known (see Cioranescu and Saint Jean Paulin [66], Panasenko [209], and Zhikov [257]) that the homogenization limit of most of the boundary value problems on thin structures with Dirichlet and Neumann boundary conditions is always of the following form: Strong convergence of the corresponding solutions holds, the limit function does not contain an oscillating variable, and the limit problem is the same for any h(ε) → 0, that is, it does not depend on the particular type of the thin structures. However, as we see later, for OCPs with Robin boundary conditions, the structure of the corresponding limit problem depends essentially on how the parameter h tends to 0 as ε → 0 (the so-called “scaling effect”). Let Ω be a bounded open domain in R2 with a Lipschitz boundary ∂Ω. By L2 (Ω, dμ) we denote the Banach space of square integrable functions on Ω with respect to the measure μ. If the measure μ is the Lebesgue, we abbreviate the notation to L2 (Ω). The space H 1 (Ω, S) is the set of all functions from H 1 (Ω) such that γ(y) = 0 on S. Let χE be the characteristic function of a set E (i.e., χE (x) = 1 if x ∈ E and χE (x) = 0 if x ∈ E). Assume that a periodic thin structure is contained in an open bounded domain Ω ⊂ R2 . This highly contrasting medium in the plane depends on two small related parameters, ε and h, which control the size of a periodicity cell and the thickness of the reinforcement. To describe this structure, we suppose that there is a specified periodic graph F in the plane such that F consists of segments. For simplicity, we assume that all the end points of each segment are the intersection points of two or more segments. Let = [0, 1)2 be the periodicity cell of F. We call this graph a singular (or an infinitely thin) grid. Following Zhikov [257] (see also Bouchitté and Fragala [27]), we define a thin grid F h applying the procedure of h-thickening to the graph F . This means that F h can be obtained if any component I of F is replaced by a strip I h of


313

Fig. 9.1. The network of thin wires Ωε

width h whose median coincides with I. Sometimes, it is convenient to include into a thin grid F h , disks of radius h/2 centered at the nodes of the grid F. The procedure of h-thickening becomes simpler if the grid F is composed of several families of parallel straight lines. In this case, we obtain the grid F h by replacing any of the lines l of the grid F by the infinite strip of width h with median l. For the sake of simplicity, we consider a particular case of 2D reticulated structures, namely the network (or gridworks) that consists of regular arrays of thin wires and is defined by Fig. 9.1 (for the more general case, see Lagnese and Leugering [163], and Leugering and Schmidt [166]). We assume that it is contained in the domain Ω ⊂ R2 , where Ω = (0, L 1 ) × (0, L 2 ),

L1 ∈ Q. L2

We describe this network as follows. Let F be the periodic graph in the plane with the cell of periodicity Y = [−0.5, 0.5) × [0, 1) that consists of the following segments: 1 1 1 1 I1 = x1 ∈ − , 0 , x2 = , I2 = x1 ∈ 0, , x2 = , 2 2 2 2 1 1 , I4 = x1 = 0, x2 ∈ ,1 . I3 = x1 = 0, x2 ∈ 0, 2 2 We define a thin grid F h applying the procedure of h-thickening to the graph F (see Fig. 9.2). In this case,

F h = (x1 , x2 ) ∈ F h + k, ∀ k ∈ R2 , h 1 1 1−h 1+h h F h = − ≤ x1 < , < x2 < ∨ − < x1 < , 0 ≤ x2 < 1 . 2 2 2 2 2 2

314


Fig. 9.2. The procedure of h-thickening for the graph F

Let ε be a small parameter. We will always suppose that ε varies in a strictly decreasing sequence of positive numbers which converges to 0 and such that the following numbers are integers: Nε1 = L 1 /ε, Nε2 = 2L 2 + 1 /ε. So, when we write ε > 0, we consider only the elements of this sequence. contraction of the grid F h with a factor of Let Fεh be the homothetic

−1 h h h ε (i.e., Fε = εF = εx : x ∈ F ). Note that the structure Fεh consists of strips whose thickness is εh. For every fixed ε, let us define the sets

Qε = (x1 , x2 ) ∈ R2 : x1 ∈ (0, L 1 ), x2 ∈ (0, ε/2) ⊂ Ω, (9.1) ΓεD = ∂Ω ∩ Fεh ∩ {(x1 , x2 ) : x2 = 0} , Sext = ∂Ω \ (ΓεD ∪ ΓεN ),

ΓεN = ∂Fεh ∩ Qε ,

Sint = [Ω \ Qε ] ∩ ∂Fεh .

(9.2) (9.3)

As a result, the network Ωε is defined as Ωε = Ω ∩ Fεh ,

(9.4)

ε ∪ Sext . ∂Ωε = ΓεD ∪ ΓεN ∪ Sint

(9.5)

Essentially, for fixed ε and h, the network Ωε can be viewed as a perforated domain with rapidly oscillating part of boundary. However, we suppose that the parameters ε and h(ε) are related to each other: h(ε) → 0 as ε → 0. So, according to Zhikov [257] and Cioranescu and Saint Jean Paulin [66], all thin structures can be divided into three classes:

9.1 A measure-theoretic approach to the description of the network Ωε

315

(A1 ) Sufficiently thick, when limε→0 ε/h(ε) = 0; (A2 ) Sufficiently thin, when limε→0 ε/h(ε) = +∞; (A3 ) Structures of critical thickness, when limε→0 ε/h(ε) = θ ∈ (0, ∞). Moreover, it should be noted that the domain Ωε has such a type of the oscillating boundary that the ratio of the amplitude and the period of the oscillation is bounded as the parameter ε tends to 0. As it will be seen later, this fact is very essential for the research method of various processes in such domains.

9.1 A measure-theoretic approach to the description of the network Ωε To begin, we introduce the periodic framework in which we will work in this section. Note that there are many possibilities for the choice of a representative period Y on the structure Ωε . However, as was mentioned in Cioranescu and Saint Jean Paulin [66] and in Bachvalov and Panasenko [14], the limit analysis of many boundary value problems on reticulated structures is independent of the particular choice of this period. For convenience, we consider three different periodicity cells for the network of wires Ωε (see Fig. 9.3). Each part of Fig. 9.3 indicates zones where the corresponding boundary conditions will be located, namely, we suppose that the segment (A, B) is the influence zone ΛhD of Dirichlet control (see Fig. 9.3(b)). The polyline ABCD (as it is indicated in Fig. 9.3(a)) coincides with the Neumann control zone ΛhN . Additionally, the boundary ΛhR of the hole ABCD is a zone where Robin condition is concentrated (see Fig. 9.3(c)). Here, the index h indicates the dependence of these sets on the thickness of the thin grid F h . Let us consider three periodic, finite positive Borel measures ν h , μh , and h λ in R2 with the toruses of periodicity Y1 = [0, 1)2 ,

Y2 = [−1/2, 1/2) × [0, 1),

and

Y3 = [0, 1) × [−1/2, 1/2),

respectively. Let ν h , μh , and λh be the probability measures, concentrated, and uniformly distributed on the sets ΛhN , ΛhD , and ΛhR , respectively. So, these measures are proportional to 1D Lebesgue measures and are normalized by the conditions dν h = dμh = dλh = 1. Y1

Y2

Y3

= μ (Y2 \ = λh (Y3 \ ΛhR ) = 0, that is, these It is clear that ν (Y1 \ measure are singular with respect to the Lebesgue measure L2 . Moreover, for any smooth function ϕ ∈ C ∞ (R2 ), we have h

ΛhN )

h

ΛhD )

316


Fig. 9.3. The periodicity cells for Ωε

ϕ dH = 2(1 − h) 1

Λh N

ϕ dμh ,

ϕ dx1 = h Λh R

ϕ dν h ,

Λh D

(9.6)

Y1

Y2

(9.7)

ϕ dH1 = 4(1 − h)

ϕ dλh .

(9.8)

Y3

Since the homothetic contraction of the plane with a factor of ε−1 takes the grid F h to Fεh = εF h , we introduce the so-called “scaling”measures νεh , μhε , and λhε by the following rules: ⎫ λhε (B) = ε2 λh (ε−1 B), ⎪ ⎪ ⎪ h −1 ⎪ ⎪ ⎪ εν (ε (B ∩ O )) if B ∩ O = ∅, ε ε ⎬ h νε (B) = 0 otherwise (9.9) ⎪ ⎪ h −1 ⎪ ⎪ εμ (ε (B ∩ Oε )) if B ∩ Oε = ∅, ⎪ ⎪ μhε (B) = ⎭ 0 otherwise,

where B is any planar Borel set and Oε = (x1 , x2 ) ∈ R2 : 0 ≤ x2 < ε .


μhε

317

Clearly, λhε is an ε-periodic measure on R2 , whereas the measures νεh and are ε-periodic along the x1 -axis. Then 2 h(ε) 2 dλh(ε) = ε dλ (x/ε) = ε dλh(ε) = ε2 , ε εY3 εY3 Y3 h(ε) h(ε) h(ε) dμε = ε dμ = ε, dνε = ε dν h(ε) = ε. εY2

Y2

εY1

Y1

h(ε)

weakly converges to the planar Lebesgue measure Hence, the measure λε h(ε) as ε → 0 (in symbols, dλε dx), that is, ϕ dλh(ε) = ϕ dx (9.10) lim ε ε→0

R2

R2

for any ϕ ∈ C0∞ (R2 ) (see Lemma 6.50). h(ε) h(ε) As for the limit properties of the measures νε and με , we have the following result. Proposition 9.1. ε→0

dνεh(ε) δ{x2 =0} dx,

ε→0

dμh(ε)

δ{x2 =0} dx, ε

where by δ{x2 =0} dx we denote the product of the linear Lebesgue measure dx1 and the Dirac measure δ{x2 =0} dx2 . Proof. Let us partition the plane R2 into squares εY1 with edges ε and denote these squares by the symbols εY1i,j . Then for any test function ϕ ∈ C0∞ (R2 ), we have ϕ dνεh(ε) = ϕ dνεh(ε) R2

i

=

i

=

j

i

εY1i,j

εY1i,0

ϕ dνεh(ε) =

ϕ(xi, 0 )

i

dν h(ε) (x/ε) =

ϕ(xi, 0 )ε εY1

εY1i,0

dνεh(ε)

ϕ(xi, 0 )ε,

i

where xi, 0 are some points of the corresponding squares εY1i,0 = ε(Y1 + (i, 0)) ⊂ Oε . Hence, for a given function ϕ ∈ C0∞ (R2 ), there are a constant Mϕ > 0 independent of i and some points (xi1 , 0) on the boundary of the torus εY1i,0 such that ϕ(xi, 0 ) = ϕ(xi1 , 0) + (ϕ(xi, 0 ) − ϕ(xi1 , 0)), |ϕ(xi, 0 ) − ϕ(xi1 , 0)| ≤ Mϕ xi, 0 − (xi1 , 0) R2 ≤ Mϕ ε.

318


Taking the construction of the Riemann sum into account, we conclude that lim ϕ(xi, 0 )ε = lim ϕ(xi1 , 0)ε + lim D(ε) = ϕ(x1 , 0) dx1 + lim D(ε), ε→0

ε→0

i

ε→0

i

ε→0

R

where |D(ε)| ≤ i ε|ϕ(xi, 0 ) − ϕ(xi1 , 0)| ≤ Mϕ i ε2 → 0 as ε → 0. Thereby we obtain the required result. It is clear that the same conclusion is true for h(ε) the measure με . Let us define the following sets:

ΘεR = k = (k1 , k2 ) ∈ Z2 : (εY3 + ε k) ∩ Ω = ∅ ,

ΘεN = k = (k1 , 0) ∈ Z2 : (εY1 + ε k) ∩ Ω = ∅ .

(9.11) (9.12)

ε ) be a fixed function. Then for any ϕ ∈ C0∞ (R2 ), the integral Let zε ∈ L2 (Sint zε ϕ dH1 can be rewritten it in the following equivalent form: term ε Sint

zε ϕ dH1 = ε Sint

zε ϕ dH1 ε(Λh R +k)

k∈ΘεR

= [4ε (1 − h(ε))]

k∈ΘεR

=

4 (1 − h(ε)) ε

k∈ΘεR

zε ϕ dλh (x/ε)

ε(Y3 +k)

zε ϕ ε2 dλh (x/ε) ε(Y3 +k)

4 zε ϕ dλhε (1 − h(ε)) ε ε(Y +k) 3 k∈ΘεR 4 = (1 − h(ε)) zε ϕ dλhε . ε Ω =

h(ε)

Here, zε is a function of L2 (Ω, dλε zε ∈ L

2

ε (Sint )

on

ε Sint .

(9.13)

) taking the same values with the function zε ϕ dλεh(ε) is well defined.

We note that the integral Ω

h(ε)

Indeed, since the set Ω is bounded and zε dλε is a Radon measure, it follows zε ϕ dλεh(ε) is a linear continuous functional on C0∞ (R2 ). Thus, that Ω

4 zε ϕ dH = (1 − h(ε)) ε ε Sint

zε ϕ dλhε .

1

(9.14)

Ω h(ε)

Remark 9.2. It is easy to see that the scaling measure λε belongs to the (Ω). In view of our initial supposition, cap(S ) = 0, so λh (B) = 0 class M+ int 0 for every Borel set B ⊆ Ω with cap(B, Ω) = 0. Hence, the first property


319

h(ε)

of the cone M+ . As for the second property, it is true 0 (Ω) is valid for λε because the measure λh is concentrated on Sint and proportional there to the linear Lebesgue measure. Thus, λh ∈ M+ 0 (Ω). Since the scaling measure h(ε) h(ε) inherits these properties from λh , it follows that λε ∈ M+ λε 0 (Ω) for h(ε) 1 every ε > 0. Thus, every function y in H (Ω) is defined λε -everywhere and h(ε) is λε -measurable in Ω (see Dal Maso and Murat [82]). Hence, the space h(ε) H01 (Ω) ∩ L2 (Ω, dλε ) is well defined. Taking these facts into account and proceeding by analogy with (9.13), we can obtain the relation wε ϕ dH1 = wε ϕ dH1 ΓεN

k∈ΘεN

ε(Λh N +k)

= 2 ε (1 − h(ε))

k∈ΘεN

= 2 (1 − h(ε))

w ε ϕ dν h (x/ε) ε(Y1 +k)

w ε ϕ dνεh ,

∀ ϕ ∈ C0∞ (R2 ).

(9.15)

Ω

Let χhε = χΩε be the characteristic function of the network Ωε (it is clear that this function is two-parametric, in general). We are now in a position to introduce the non-negative Y -periodic Borel measure η h such that dη h = |h(2 − h)|−1 χh dx,

(9.16)

where by χh we denote the characteristic function of the grid F h (here, Y is the sell of periodicity for the grid F h ; see Fig. 9.2(b)). If we define the scaling measure ηεh in the usual way, then we immediately obtain |h(ε)(2 − h(ε))|dηεh = χhε dx.

(9.17)

It is clear that this measure is uniformly distributed and concentrated on the thin grid Fεh = εF h , normalized, and proportional there to the planar Lebesgue measure. Moreover, in this case we have ε2 L2 (F h ∩ ε−1 B), h(2 − h) ε2 L2 (F h ∩ Y ) = ε2 ηεh (εY ) = h(2 − h) ηεh (B) =

(9.18) (9.19)

for any planar Borel set B. Hence, ηεh (ε) converges weakly to the planar Lebesgue measure as ε → 0, and for every fixed ε > 0 this measure is absolutely continuous with respect to L2 (in symbols, ηεh (ε) L2 ), that is, L2 (B) = 0 implies ηεh (ε)(B) = 0 for all B ⊂ R2 .

320


9.2 Statement of the optimal control problem We consider the following boundary value problem in Ωε : −div (A(x/ε)∇yε ) + yε ∂νA yε ∂νA yε yε ∂ νA y ε

⎫ in Ωε , ⎪ ⎪ ⎪ on Sext , ⎪ ⎬ ε on Sint , ⎪ on ΓεD , ⎪ ⎪ ⎪ N ⎭ on Γε ,

= fε =0 = ε2 (−dyε + gε ) = uε = εpε

(9.20)

ε where d is a positive constant, gε ∈ L2 (Sint ) and fε ∈ L2 (Ω) are given functions, ∂νA yε = (A(x/ε)∇yε ) · ν, ν is the unit outward normal vector with respect to Ωε , and A(x) is a Y -periodic measurable matrix such that

A(·) ∈ L∞ (Y, R2×2 ), α0 ξ ≤ (A(x)ξ, ξ) ≤ α0−1 ξ 2 for a.e. x ∈ Y, 2

(9.21)

for some fixed constant α0 > 0 and every ξ ∈ R2 . Hereinafter we interpret the functions uε and pε as Dirichlet and Neumann boundary controls, respectively. There are two common ways of constraining the control. The first one is to impose an explicit bound on the control. The second way is to add some norm of the control to the cost functional. We will use both of these approaches. We say that the control functions uε and pε are admissible if the following conditions hold: pε ∈ L2 (ΓεN ), where the set Uε is defined as

Uε = u|ΓεD : u ∈ H 1 (Γ0 ),

uε ∈ Uε ,

u H 1 (Γ0 ) ≤ C0 ,

(9.22)

(9.23)

Γ0 = {0 < x 1 < L 1 , x2 = 0}, and the constant C0 is independent of ε. We suppose that the cost functional takes the form 1 (yε (x) − yεT )2 dx Iε (uε , pε , yε ) = h Ωε u2 dx1 + p2ε dH1 , (9.24) + κ(ε) ΓεD

ΓεN

where yεT ∈ L2 (Ω) is a given function and κ(ε) is a given value. Thus, the OCP we consider in this section can be stated as follows: Find a triplet (u0ε , pε0 , yε0 ) ∈ H 1 (ΓεD ) × L2 (ΓεN ) × H 1 (Ωε ) such that Iε (u0ε , pε0 , yε0 ) =

inf (uε ,pε ,yε )∈ Ξε

Iε (uε , pε , yε ),

where the set of admissible triplets Ξε is defined as

(9.25)

9.2 Statement of the optimal control problem

Ξε = (uε , pε , yε ) ∈ H 1 (ΓεD ) × L2 (ΓεN ) × H 1 (Ωε ) : (uε , pε , yε ) satisfies (in some weak sense) (9.20)–(9.23)} .

321

(9.26)

In the sequel, the OCP (9.20)–(9.26) will be called the Pε -problem. It is well known that for every fixed ε and for given ε ), u ∈ H 1 (ΓεD ), and pε ∈ L2 (ΓεN ), f ∈ L2 (Ω), gε ∈ L2 (Sint

the boundary value problem (9.20) admits a unique solution yε ∈ H 1 (Ωε ) such that (see Lions and Magenes [173]) yε = uε a.e. on ΓεD and the integral identity A(x/ε)∇yε · ∇ϕ dx + Ωε

yε ϕ dx + d ε2

Ωε

=

(9.27)

fε ϕ dx + ε

2

yε ϕ dH1 ε Sint

gε ϕ dH1 ε Sint

Ωε

pε ϕ dH1 ,

+ε ΓεN

∀ ϕ ∈ C0∞ (R2 ; ΓεD ),

(9.28)

is valid, where we denote by C0∞ (R2 ; ΓεD ) the set of all functions from C0∞ (R2 ) such that ϕ|ΓεD = 0. In the sequel, the function yε is called the weak solution to the problem (9.20). It should be emphasized that since the domain Ωε is nonconvex with nonsmooth boundary ∂Ωε , any solution of the initial-boundary value problem (9.20) can have only minimal smoothness for ε small enough. It is the main reason for the choice of H 1 (Ωε ) as the state space for the original problem. In addition, if u and pε are admissible controls, then the weak solution satisfies a priori norm estimate

yε H 1 (Ωε ) ≤ c1 fε L2 (Ωε ) + uε H 1 (ΓεD ) √ ε ) . + ε pε L2 (ΓεN ) + ε gε L2 (Sint It is quite obvious that for every ε > 0, the set of admissible solutions Ξε to the Pε -problem is a nonempty and convex subset of H 1 (ΓεD ) × L2 (ΓεN ) × H 1 (Ωε ). As a result, this problem has a unique solution (u0ε , p0ε , yε0 ) ∈ Ξε for every ε > 0 (see Sect. 3.3). Moreover, as follows from the a priori norm estimate given above, the Pε -problem is uniformly regular: (H1) Ξε = ∅ for every ε > 0. (H2) There is a sequence of admissible solutions {(u∗ε , p∗ε , yε∗ ) ∈ Ξε }ε>0 such that lim supε→0 u∗ε 2H 1 (Γ D ) + p∗ε 2L2 (Γ N ) + yε∗ 2H 1 (Ωε ) < +∞. ε

ε

322


Let us observe that we touch the OCP with two different types of boundary controls. Moreover, the influence zones each of controls are imposed on the different parts of the boundary (ΓεD ∩ ΓεN = ∅). Therefore, one of the distinctive feature of this problem is the fact that the control zones ΓεD and ΓεN have not only a highly oscillating geometrical form and are nonconnected, but also they coincide in the limit, that is, lim ΓεD = Γ0 = lim ΓεN

ε→0

ε→0

in the Hausdorff metric. It produces the situation when one of the boundary controls can disappear in the limit as ε → 0. On the other hand, the dependence of the Pε -problem on the small parameter ε has a rather complicated nature. Indeed, if the parameter ε is changed, then all the components of this control problem (the domain Ωε , the constraint set Uε , the cost functional Iε , and the set of admissible solutions) and the main functional space H 1 (ΓεD ) × L2 (ΓεN ) × H 1 (Ωε ) are changed as well. So, in spite of the fact that this problem is rather simple from a control theory point of view, the numerical simulation of its optimal characteristics is very complicated for ε small enough. Therefore, one of the main questions is how to obtain an appropriate approximation for the optimal solutions to the Pε -problem for small values of ε. Taking this into account, we reformulate the original OCP in terms of measures for the thin structure Ωε . By analogy with (9.13) and (9.15), it is easy to show that the following representations hold true: 2 1 yε ϕ dH = 4εd (1 − h(ε)) yε ϕ dλhε , (9.29) dε

ε Sint

Ω

pε ϕ dH1 = 2ε (1 − h(ε))

ε ΓεN

u2 dx1 = κ(ε)h(ε)

κ(ε) ΓεD

ΓεN

2

pε ϕ dνεh , u 2 dμhε ,

(9.31)

p2ε dνεh ,

(9.32)

Ω

p2ε dH1 = 2 (1 − h(ε)) Ω

gε ϕ dH = 4 ε (1 − h(ε))

gε ϕ dλhε .

1

ε

ε Sint

(9.30)

Ω

(9.33)

Ω

Let y˘ε ∈ L2 (0, T ; H 1 (Ω)) be some extension of the weak solution yε to the problem (9.27)–(9.28) on the whole of domain Ω. Then summing up the representations (9.29)–(9.33), the boundary value problem (9.20) can be rewritten in the following variational form: y˘ε = u ε

μhε -a.e. in Ω,

(9.34)

9.2 Statement of the optimal control problem

Ω

323

χhε (A(x/ε)∇˘ yε · ∇ϕ) dx + χhε y˘ε ϕ dx + 4d ε (1 − h(ε)) y˘ε ϕ dλhε Ω Ω = χhε fε ϕ dx + 2 ε (1 − h(ε)) 2 gε ϕ dλhε + pε ϕ dνεh (9.35) Ω

Ω

Ω

for any ϕ ∈ C0∞ (R2 ; Γ0 ). It is easy to see that using the scaling measure η h with properties (9.16)– (9.19), the corresponding terms in (9.35) can be rewritten as χhε y˘ε ϕ dx = h(ε)(2 − h(ε)) yε ϕ dηεh , Ω Ω h χε (A(x/ε)∇˘ yε · ∇ϕ) dx = h(ε)(2 − h(ε)) (A(x/ε)∇ yε · ∇ϕ) dηεh , Ω Ω χhε fε ϕ dx = h(ε)(2 − h(ε)) fε ϕ dηεh , Ω

Ω

where, as usual, we denote by yε a function of L2 (Ω, dηεh ) taking the same values as yε on Ωε . Let us define the Sobolev space H 1 (Ω, dηεh ) as we did it in Sect. 2.7. We say that a function y belongs to H 1 (Ω, dηεh ) if there exist a vector z ∈ (L2 (Ω, dηεh ))2 and a sequence {ym ∈ C ∞ (Ω)}m∈N such that ∇ym − z2 dηεh = 0. lim (ym − y)2 dηεh = 0, lim m→∞

m→∞

Ω

Ω

In this case, one says that z is a gradient of y and denotes it as ∇ y (i.e., z = ∇ y ). In view of the ergodicity property of the measure ηεh , this gradient is unique for any y ∈ H 1 (Ω, dηεh ) (see [254]). Moreover, we may always suppose that for any function y ∈ H 1 (Ω, dηεh ), one can point out its prototype y in H 1 (Ω) such that y = y, ηεh -a.e. in Ω. Thus, we may reformulate the Pε -optimal control problem as follows: Find a triplet ( uε0 , pε0 , yε0 ) such that ( uε0 , pε0 , yε0 ) ∈ Zε ≡ H 1 (Γ0 ) ∩ L2 (Ω, dμhε ) × L2 (Ω, dνεh ) × H 1 (Ω, dηεh ) ∩ L2 (Ω, dλhε ) , Iε ( uε0 , pε0 , yε0 ) =

inf

bε (b uε ,b pε ,b yε )∈ Ξ

uε , pε , yε ), Iε (

(9.36)

ε have the where the cost functional Iε and the set of admissible solutions Ξ following analytical representation: uε , pε , yε ) = (2 − h(ε)) ( yε − yεT )2 dηεh Iε ( Ω + κ(ε)h(ε) u 2ε dμhε + 2 (1 − h(ε)) p2ε dνεh , (9.37) Ω

Ω

324


⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

⎫ ε μhε -a.e. in Ω, yε = u ⎪ ⎪ ⎪ ⎪ ⎪ 1

≤ C ,

u ⎪ ε H (Γ0 ) 0 ⎪ ⎪ ⎪ ⎪ h h ⎪ ⎪ ⎪ (A(x/ε)∇ y · ∇ϕ) dη + y ϕ dη ε ε ε ε ⎪ ⎬ Ω Ω ε = ( Ξ uε , pε , yε ) . ⎪ yε ϕ dλhε = fε ϕ dηεh ⎪ ⎪ +4d β(ε) ⎪ ⎪ ⎪ ⎪ ⎪ Ω ⎪ ⎪ Ω ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ h h ⎪ ⎪ +4 β(ε) ⎪ gε ϕ dλε +2 β(ε) pε ϕ dνε , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Ω Ω ⎪ ⎪ ⎩ ⎭ ∞ 2 D ∀ ϕ ∈ C0 (R ; Γε ) Here,

(9.38)

ε 1 − h(ε) · . β(ε) = 2 − h(ε) h(ε)

ε -problem. It is In what follows, the OCP (9.36)–(9.38) will be called the P quite obvious that this problem has a unique solution ( uε0 , pε0 , yε0 ) for every ε > 0 (see Theorem 3.2; see also Fursikov [111] and Lions [169]). This solution can be interpreted as a prototype of an optimal triplet to the Pε -problem. Moreover, in this case, the a priori norm estimate takes the form fε L2 (Ω,dηh ) + yε L2 (Ω,dλhε ) ≤ C pε L2 (Ω,dνεh )

yε H 1 (Ω,dηεh ) + ε + uε H 1 (Γ0 )∩L2 (Ω,dμhε ) + gε L2 (Ω,dλhε ) ,

(9.39)

is independent of ε > 0. where the constant C ε as ε → 0. We proceed to the asymptotic behavior of the problem P Note that, in contrast to the usual algorithm of homogenization of OCPs (see Conca, Osses, and Saint Jean Paulin [72], Kesavan and Saint Jean Paulin [135], and Saint Jean Paulin and Zoubairi [222]), we do not look for the limit of optimal solutions and do not study the asymptotic behavior of the corresponding necessary optimality conditions as ε tends to 0. Our approach to the homogenization of OCPs is based on the ideas of the epi-convergence theory ε -problem in variable spaces (see Sect. 8.4). With this aim, we represent the P for various values of ε in the form of the sequence of corresponding constrained minimization problems uε , pε , yε ) ; ε > 0 , (9.40) Iε ( inf bε (b uε ,b pε ,b yε ) ∈ Ξ

where the cost functional Iε : Σε → R and the sets of admissible solutions ε are defined by (9.37) and (9.38), respectively. Then the definition of an Ξ appropriate limiting OCP to the family (9.36) can be reduced to the analysis of the limit properties of the sequences (9.40) as ε → 0. To obtain this limit in the form of some constrained minimization problem, we follow Definition 8.22

9.3 Convergence in the variable space Z ε

325

with respect to the scale of spaces {Zε }ε>0 . In view of this, one has to define a concept of convergence in variable spaces Zε such that the sequence of optimal solutions is relatively compact. This question is discussed in the next section.

9.3 Convergence in the variable space Z ε We recall that the measure ηεh introduced earlier is a non-degenerate 2connected Borel measure in R2 , absolutely continuous with respect to the planar Lebesgue measure (see Therefore, we may suppose that

Definition 6.33). for a bounded sequence yεh ∈ H 1 (Ω, dηεh ) , there is an element y ∈ H 1 (Ω) such that yεh z in L2 (Ω, dηεh ) in the 6.14. Moreover,

sense of Definition a sequence of the so-called prototypes yεh ∈ H 1 (Ω) ε>0 can be found such that (9.41) yεh y in H 1 (Ω) and yεh = yεh , ηεh -a.e. in Ω. Occasionally, we will identify the elements of H 1 (Ω, dηεh ) with their prototypes in H 1 (Ω). Remark 9.3. A representation like (9.41) can be explained also by results of Bachvalov and Panasenko [14], Cioranescu and Saint Jean Paulin [66], and Donato and Nabil [96]. It is proved in [14] and [96] that for every fixed exists a uniformly bounded family of extension operators

h h, there Pε ∈ L(H 1 (Ω, dηεh ), H 1 (Ω)) such that Pεh yεh y h

weakly in H 1 (Ω) as ε → 0.

As a result, we have that if yεh y in L2 (Ω, dηεh ) and y ∈ H 1 (Ω), then y h y weakly in H 1 (Ω) as h → 0. In view of this, we introduce the following concept.

Definition 9.4. Let yεh ε>0 be a bounded sequence in the variable space Mε = H 1 (Ω, dηεh ) ∩ L2 (Ω, dλhε ), that is, lim sup yεh(ε) 2H 1 (Ω,dηh(ε) ) + yεh(ε) 2L2 (Ω,dλh(ε) ) < +∞. ε

ε→0

ε

We say that this sequence is weakly convergent in Mε if there exists a function y ∈ H 1 (Ω) such that ⎧ h yε → y in L2 (Ω, dηεh ), ⎪ ⎪ ⎪ ⎨ h in L2 (Ω, dλhε ), yε y (9.42) ⎪ ⎪ h 2 h ⎪ ⎩ |yε − y| dηε → 0 as ε → 0. Ω

326


We have the following lemma. Lemma 9.5. The inequality 2 h y dηε ≤ C1 (y 2 + ε2 |∇y|2 ) dx Ω

(9.43)

Ω

holds for every y ∈ C0∞ (R2 ) uniformly in h > 0, where the constant C1 depends on the geometry of the singular grid F and on the diameter of the domain Ω. Inequality (9.43) can be established in full analogy with Lemma 6.58. Remark 9.6. Since ηεh is absolutely continuous with respect to the planar Lebesgue measure (ηεh (ε) L2 ) and C0∞ (Ω) is dense in H01 (Ω), it follows that inequality (9.43) can be extended to functions y ∈ H01 (Ω). To begin, let us apply this inequality in order to establish some specific properties of the convergence in the spaces L2 (Ω, dλhε ) and L2 (Ω, dηεh ). Lemma 9.7. If y ∈ H 1 (Ω), then lim y 2 dηεh = y 2 dx, ε→0 Ω Ω yϕ dηεh = yϕ dx, ∀ ϕ ∈ C0∞ (Ω). lim ε→0

Ω

(9.44)

Ω

Moreover, y 2 dλhε < +∞, then y y in L2 (Ω, dλhε ).

if sup ε>0

(9.45)

Ω

Proof. If y ∈ C(Ω), then the weak convergence of the measures dλhε dx and dηεh dx immediately implies relations (9.44) and (9.45) (see Definition 2.29). Let y be an arbitrary element of H01 (Ω). Then due to Lusin’s theorem, a sequence y δ can be found such that y δ ∈ C0∞ (R2 ),

y − y δ H01 (Ω) < δ.

(9.46)

We have 2 δ 2 2 ≤ y 2 dηεh − y dx − y y dηεh Ω Ω Ω δ 2 h δ 2 + (y ) dηε − (y ) dx Ω Ω 2 δ 2 + y dx − (y ) dx = J1 + J2 + J3 . Ω

Ω

It is clear that due to the weak convergence of the measures dηεh dx and condition (9.46), we have J2 + J3 ≤ d1 δ for some constant d1 > 0.


327

As for the value J1 , we apply inequality (9.43) to the function y − y δ . As a result, ! 2 2 y − y δ dηεh ≤ C1 y − y δ + ε2 |∇y − ∇y δ |2 dx, Ω

Ω

1/2 1/2 2 δ 2 h δ 2 h δ 2 h |y + y | dηε |y − y | dηε < C2 δ, y − y dηε ≤ Ω

Ω

and hence,

Ω

2 h 2 y dx ≤ Cδ lim sup y dηε − ε→0 Ω

Ω

for an arbitrary δ, where the constant C does not depend on δ. This implies (9.44)1 . In order to prove (9.44)2 , we apply similar arguments: δ h yϕ dηεh − yϕ dx ≤ (y − y )ϕ dηε Ω Ω Ω δ h δ y ϕ dx + y ϕ dηε − Ω Ω + (y − y δ )ϕ dx = I1 + I2 + I3 , Ω

where

1/2

I1 ≤ C

(y − y )

δ 2

Ω

I3 ≤ C

dηεh

1/2 (y − y δ )2 dx

by (9.44)1

≤

Cδ,

by (9.46)

≤

Cδ,

Ω

I2 → 0 is the weak convergence dηεh dx. In order to replace H01 (Ω) by H 1 (Ω) in relation (9.44), we observe that the weak convergence dηεh dx implies ϕ dηεh = ϕ dx, ∀ ϕ ∈ C(Ω) lim ε→0

Ω

Ω

(see Lemma 6.62). It follows that it is enough to consider y δ ∈ C ∞ (Ω) in (9.46) to deduce relation (9.44) for y ∈ H 1 (Ω). The remaining reasoning is preserved. As for property (9.45), we prove it by contradiction. Indeed, by the criterium of the weak convergence in L2 (Ω, dλhε , there exists en element z ∈ L2 (Ω) such that, up to a subsequence, yϕ dλhε = zϕ dx, ∀ ϕ ∈ C0∞ (Ω). (9.47) lim ε→0

Ω

Ω

328


Let us assume that for a fixed ϕ ∈ C0∞ (Ω), we have zϕ dx − yϕ dx = δ > 0. Ω

(9.48)

Ω

Following Ziemer [267], we can identify the function y ∈ H 1 (Ω) with its quasi-continuous representative (for the details, we refer to Sect. 2.5). Since ε , as a support of the measure λhε , has nonzero capacity but zero the set Sint Lebesgue measure L2 , it follows that y ∈ H 1 (Ω) is a continuous function for all x ∈ Ω except for a subset with dλhε -measure zero. Then for a given ϕ ∈ C0∞ (Ω), there exist an ε1 > 0 and a function uδ ∈ C0∞ (Ω) such that δ h yϕ dλhε − u ϕ dλε < δ/3, ∀ ε ∈ (0, ε1 ). (9.49) Ω

Ω

Moreover, by the smooth property of uδ and the weak convergence of the measures dλhε dx, there exists an ε2 > 0 such that the estimate δ h uδ ϕ dx − u ϕ dλε < δ/3 (9.50) Ω

Ω

holds true for all ε ∈ (0, ε2 ). Further, we note that (9.47) implies the following inequality: h zϕ dx − yϕ dλ (9.51) ε < δ/3, Ω

Ω

which holds true for every ε ∈ (0, ε3 ) with some ε3 > 0. As a result, combining the estimates (9.49)–(9.51), we obtain h zϕ dx − ≤ zϕ dx − yϕ dx yϕ dλ ε Ω Ω Ω Ω δ δ h u ϕ dλε + u ϕ dx − Ω Ω h δ h + yϕ dλε − u ϕ dλε Ω

< δ,

Ω

∀ ε < min{ε1 , ε2 , ε3 },

which comes into conflict with (9.48). Thus, z = y a.e. in Ω, and this completes the proof. We now consider a more delicate situation.

Lemma 9.8. Let yε ∈ H 1 (Ω) ε>0 and y ∈ H 1 (Ω) be such that yε y in " H 1 (Ω) and supε>0 Ω y 2 dλhε < +∞. Then yε y in L2 (Ω, dλhε ),

yε → y in L2 (Ω, dηεh ).

(9.52)


329

Proof. By the criterium of strong convergence in L2 (Ω, dηεh ) and the property of lower semicontinuity, it is enough to show that yε ϕ dηεh = yϕ dx, ∀ ϕ ∈ C0∞ (Ω), (9.53) lim ε→0 Ω Ω |yε |2 dηεh ≤ |y|2 dx. (9.54) lim ε→0

Ω

Ω

Let y δ be the sequence with properties (9.46). Then yε ϕ dηεh − yϕ dx = (yε − y)ϕ dηεh Ω Ω Ω δ h δ h δ + (y − y )ϕ dηε + y ϕ dηε − y ϕ dx Ω Ω Ω + (y δ − y)ϕ dx = J1 + J2 + J3 + J4 . Ω

Applying estimate (9.46) and inequality (9.43) to J2 and J4 , we obtain (y − y δ )ϕ dηεh < Cϕ δ, (y δ − y)ϕ dx < Cϕ δ. Ω

Ω

At the same time, the limits limε→0 J1 (ε) and limε→0 J3 (ε) are equal to 0 since 2 h (yε − y) dηε ≤ C1 (yε − y)2 + ε|∇(yε − y)|2 dx Ω Ω (9.55) = C1 (yε − y)2 dx + O(ε2 ), Ω

and

y δ ϕ dηεh → Ω

y δ ϕ dx due to the weak convergence dηεh → dx. Ω

Thus,

lim sup yε ϕ dηεh − yϕ dx ≤ Cϕ δ. ε→0

Ω

Ω

Since the parameter δ is arbitrary, relation (9.53) is valid. As for (9.54), inequality (9.55) and Lemma 9.7 imply that |yε |2 dηεh ≤ lim |yε − y|2 dηεh + lim |y|2 dηεh = |y|2 dx, lim ε→0

Ω

ε→0

Ω

ε→0

Ω

Ω

and we obtain the required result. The first relation in (9.52) immediately follows from (9.45) and the strong convergence yε → y in L2 (Ω).

330


As an obvious consequence of these lemmas, we have the following assertion. Corollary 9.9. If yε → y in L2 (Ω, dηεh ) and y ∈ H 1 (Ω), then lim |yε − y|2 dηεh = 0. ε→0

(9.56)

Ω

Indeed, to prove this statement, we note that (yε − y) → 0 in L2 (Ω, dηεh ). Then it is enough to refer to the criterium of the strong convergence in L2 (Ω, dηεh ).

Lemma 9.10. Let yε ∈ L2 (Ω, dλhε ) and ϕε ∈ C0∞ (R2 ) be sequences satisfying the condition yε y in L2 (Ω, dλhε ) and ϕε → ϕ in C(Ω). Then yε ϕε dλhε = yϕ dx. (9.57) lim ε→0

Ω

Ω

Proof. Let us rewrite the left-hand side of (9.57) in the form h h yε ϕε dλε = yε (ϕε − ϕ) dλε + yε ϕ dx. Ω

Ω

Ω

Since dλhε dx and # # 1/2 # # 2 h # yε (ϕε − ϕ) dλhε # ≤ yε dλε

ϕε − ϕ C(Ω) # # Ω

Ω

1/2 dλhε

,

Ω

we immediately obtain the required conclusion.

Note that by analogy one can prove similar results with respect to the measure ηεh . Further, we make use the concept of the two-scale convergence in the variable space L2 (Ω, dηεh ), where h = h(ε) and h(ε) → 0 as ε → 0 (for general properties of the two-scale convergence, we refer to Sect. 6.3). Following the main properties of the two-scale convergence and the representative formula (9.16), we have the following obvious result (see Theorem 7.19 for comparison). Proposition 9.11. Let χhε and χh be the characteristic functions of the network Ωε and the grid F h , respectively. Then χh 1 in L2 (Y, dηh ),

2

χhε −→ 1,

χhε −→ 1 in L2 (Ω, dηεh ).

By analogy, similar results can be obtained with respect to the limiting behavior of the characteristic functions χΓεD and χΓεN . Indeed, using the facts that the measures ν h and μh are Y1 - and Y2 -periodic, respectively, and ν h ν and μh μ, where the limit measures ν and μ are defined as

1 ϕ dν = 2 Y1


$

1/2

1

1 ϕ(x1 , ) dx1 + 2

ϕ(0, x2 ) dx2 + 0

0

ϕ dμ =

Y2

Y2

ϕ δ{x1 =0,

x2 =0}

1/2

331

%

ϕ(1, x2 ) dx2 , 0

dx = ϕ(0, 0),

we have the following conclusion. Proposition 9.12. We have χΓεD −→ 1 in L2 (Ω, dμhε ),

χΓεN −→ 1 in L2 (Ω, dνεh ).

We now observe that since dηεh dx as ε → 0, it follows that χΩε h dx ≡ ϕ ϕ dηε −→ ϕ dx as ε → 0, ∀ ϕ ∈ C0 (Ω). h(ε)(2 − h(ε)) Ω Ω Ω However, this fact can be always interpreted as the property of the weak-∗ −1 convergence in L∞ (Ω) of [h(ε)(2 − h(ε))] χΩε ∈ L∞ (Ω) to 1. Thus, taking this proposition into account, we can supplement it in the following way. Proposition 9.13. −1

[h(ε)(2 − h(ε))] h

−1

∗

χΩε 1

(ε)χΓεD 1

−1

ϕ [2(1 − h(ε))] Ω

in Ω,

∗

in Γ0 ,

χΩε dx −→

ϕ dx1 as ε → 0, ∀ ∈ ϕ ∈ C(Ω), Γ0

ΓεD ∪ΓεN

ϕ dH1 −→

ϕ dx1 as ε → 0, ∀ ∈ ϕ ∈ C(Ω). Γ0

Let us introduce the spaces of periodic vector functions with singular Y periodic measure η: L2 (Ω, Vpot ) and L2 (Ω, Vsol ) (see Sect. 6.4.2). In this case, the following orthogonal expansion holds true (see properties (6.70)): (9.58) L2 Ω, [L2 (Y, dη)]2 = L2 (Ω, Vpot ) ⊕ L2 (Ω, Vsol ). As a result one has the following.

Theorem 9.14. Let zεh ε>0 be a bounded sequence in H 1 (Ω, dηεh ), that is, sup zεh 2L2 (Ω,dηεh ) + ∇zεh 2[L2 (Ω,dηh )]2 < ∞. ε→0

ε

h

Then there are a subsequence zε ε>0 (still indexed by ε) and functions 2 r(x, y) ∈ L2 (Ω × Y ) and z(x) ∈ H 1 (Ω) such that 2

zεh z(x), 2 ∇zεh

r(x, y),

r(x, y) − ∇z(x) ∈ L (Ω, Vpot ). 2

(9.59) (9.60) (9.61)

332


Proof. In view of the compactness property of two-scale convergence (see Proposition 6.26 and its generalization in Sect. 6.6), we may assume, without loss of generality, the existence of elements z ∈ L2 (Ω × Y ; dx × dη) and 2 r ∈ L2 (Ω × Y ; dx × dη) such that 2

2

and ∇zεh r(x, y). (9.62) 2 Let b be a vector-valued function such that b ∈ L2 (Y, dη) . We may assume that for this function there is an element a ∈ L2 (Y, dη) satisfying the relation ∞ aϕ dη = (b, ∇ϕ) dη, ∀ ϕ ∈ Cper (Y ). (9.63) − zεh z(x, y)

Y

Y

Then taking ϕ ∈ C0∞ (Ω) as a test function and using the equality ∇(ϕzεh ) = ϕ∇zεh + zεh ∇ϕ, we have the following relations: ε Ω

∇zεh ϕ, b(ε−1 x) dηεh ∇(ϕzεh ), b(ε−1 x) dηεh − ε =ε zεh ∇ϕ, b(ε−1 x) dηεh , (9.64) Ω

a(ε−1 x)ϕ dηεh = ε

− Ω

Ω

(b(ε−1 x), ∇ϕ) dηεh .

(9.65)

Ω

Therefore, applying (9.65) to (9.64), we obtain ε

ϕ ∇zεh , b(ε−1 x) dηεh Ω −1 h h =− a(ε x)zε ϕ dηε − ε zεh ∇ϕ, b(ε−1 x) dηεh . (9.66) Ω

Ω

However, in view of the initial assumptions and the definition of two-scale convergence, we obtain lim ε ∇zεh (L2 (Ω,dηεh ))2 = 0, lim ε ϕ ∇zεh , b(ε−1 x) dηεh = 0, ε→0 ε→0 Ω zεh ∇ϕ, b(ε−1 x) dηεh = 0. lim ε ε→0

Ω

Therefore, passing to the limit in (9.66), we obtain −1 h h a(ε x)zε ϕ dηε = a(y)z(x, y) dη(y)ϕ dx = 0. lim ε→0

Ω

Ω

(9.67)

Y

Due to Theorem 6.41 and the ergodicity property of the measure η, the set of all functions a ∈ L2 (Y, dη) satisfying condition (9.63) with b as an element 2 of L2 (Y, dη) is dense in the subspace of functions in L2 (Y, dη) with mean value 0. Thus, from (9.67) we immediately conclude that


333

z(x, y) = z(x), that is, the weak two-scale limit z(x, y) in (9.62) is independent of y. It remains to show that condition (9.61) is satisfied. To this end, we consider (9.64) with a vector b ⊥ Vpot . This yields h −1 ϕ ∇zε , b(ε x) dηε = − zεh ∇ϕ, b(ε−1 x) dηεh (9.68) Ω

Ω

for every ϕ ∈ C0∞ (Ω). Passing to the limit as ε → 0 in this relation (in the sense of two-scale convergence), we obtain (r(x, y), b(y)) dη(y) ϕ dx

Ω

Y

=− b(y) dη(y), ∇ϕ dx. z(x) Ω

2 Since r ∈ L2 (Ω, Y ) , it follows that s =

(9.69)

Y

(r(x, y), b(y)) dη(y) belongs to Y

L2 (Ω). Let us define the function Υ (t, x) = z(x)

b(y) dη(y). Y

Then rewriting (9.69) in the form sϕ dx = − (Υ, ∇ϕ) dx Ω

Ω

and integrating the right-hand side by parts, we deduce b(y) dη(y) . s(x) = ∇z(x), Y

Using the non-degeneracy property of the measure η, we see that there 2 2 exists a vector c ∈ Vpot Vpot ⊂ [L (Y )] such that c(y) dη(y) = ϑ, where Y b(y) dη(y). Then s(x) = ∇z(x), c(y) dη(y) and, therefore, ϑ= Y

(i) (∇z, (ii) c ∈

Y

c dη) ∈ L2 (Ω) in the sense of distributions,

Y Vpot .

2 Hence, ∇z ∈ L2 (Ω) . Summing up the results obtained above, we deduce 2 =⇒ z ∈ H 1 (Ω). z ∈ L2 (Ω), ∇z ∈ L2 (Ω)

334


Rewriting (9.69) in the form ∇z(x), (r(x, y), b(y)) dη(y) ϕ dx = b(y) dη(y) ϕ dx, Ω

Y

we observe that

Ω

Y

(r(x, y) − ∇z(x), b(y)) dη(y)ϕ dx = 0 Ω

Y

for every ϕ ∈ C0∞ (Ω) and b ⊥ Vpot . Hence, taking into account that the linear span of the vector-valued functions ϕ(x)b(y) is dense in L2 (Ω, Vsol ) and (9.58) holds, we obtain the required property r(x, y) − ∇z(x) ∈ L2 (Ω, Vpot ). The proof is complete.

h Corollary 9.15. Let zε ε>0 be a bounded sequence in H 1 (Ω, dηεh ) and let z ∈ H 1 (Ω) be its weak limit in L2 (Ω, dηεh ). Then zεh → z in L2 (Ω, dηεh ), lim |zεh − z|2 dηεh = 0. (9.70) ε→0

Ω

Proof. As follows from Theorem 9.14, 2

zεh z and zεh z

in L2 (Ω, dηεh ).

In order to prove the first assertion in (9.70), it is enough to show that (see the criterium of strong convergence in L2 (Ω, dηεh ) and the property of lower semicontinuity) |zεh |2 dηεh ≤ |z|2 dx. (9.71) lim ε→0

Ω

Ω

In accordance with inequality (9.43), we have h (zεh − z)2 dηεh ≤ C1 zεh − z)|2 dx ( zε − z)2 + ε|∇( Ω Ω = C1 ( zεh − z)2 dx + O(ε2 ),

(9.72)

Ω

where by zεh we denote functions of H 1 (Ω) such that zεh = zεh , ηεh -almost everywhere on Ω, and zεh z in H 1 (Ω). Note that such a choice is always possible. Indeed, let us assume the converse, that is, zεh = zεh ηεh -almost everywhere on Ω for every ε > 0, but zεh z ∗ in H 1 (Ω) where z ∗ = z. Then taking into account the definitions of the measure ηεh and of the weak limit in L2 (Ω, dηεh ), we get


1 h(2 − h)

zεh ϕ dηεh = Ω

ε→0

335

χΩε zεh ϕ dx −→ Ω

zϕ dx. Ω −1

However, using the facts that zεh → z ∗ in L2 (Ω) and [h(2 − h)] weakly-∗ in L∞ (Ω), we deduce 1 ε→0 χΩε zεh ϕ dx −→ z ∗ ϕ dx. h(2 − h) Ω Ω

χΩε → 1

Hence, z = z ∗ and inequality (9.72) leads us to the following conclusion: lim (zεh − z)2 dηεh = 0. (9.73) ε→0

Ω

Then in view of Lemma 9.7, we have h 2 h h 2 h 2 h |zε | dηε ≤ lim |zε − z| dηε + lim |z| dηε = |z|2 dx. lim ε→0

Ω

ε→0

ε→0

Ω

Ω

Ω

Thus, zεh → z strongly in L2 (Ω, dηεh ). In conclusion, we note that to prove the second statement in (9.70), it remains only to use Corollary 9.9. Theorem 9.16. Every bounded sequence

h zε ∈ H 1 (Ω, dηεh ) ∩ L2 (Ω, dλhε ) ε>0 is relatively compact with respect to the weak convergence in the sense of Definition 9.4.

Proof. Since the sequence zεh is bounded in H 1 (Ω, dηεh ) ∩ L2 (Ω, dλhε ), we may suppose that there exist elements z and m such that z, m ∈ L2 (Ω) and zεh → z weakly in L2 (Ω, dηεh ), zεh → m weakly in L2 (Ω, dλhε ). However, from Theorem 9.14, Lemma 9.8, and Corollary 9.15, we conclude that z ∈ H 1 (Ω), zεh → z

zεh z

strongly in L2 (Ω, dηεh ),

in

L2 (Ω, dλhε ), lim |zεh − z|2 dηεh = 0.

ε→0

Ω

Hence, z = m a.e. in Ω and we obtain the required conclusion. The proof is complete.

336


Further, we proceed to the convergence for Dirichlet boundary controls ε -optimal control problem. As follows from (9.36)–(9.38), the class for the P of admissible Dirichlet controls for every fixed value of ε can be described as follows:

u ε ∈ Uε = u ∈ H 1 (Γ0 ) ∩ L2 (Ω, dμhε ) : u H 1 (Γ0 ) ≤ C0 . (9.74) Hence, for every ε > 0 and for every admissible control function u ε ∈ Uε , there exists an extension operator Pε : H 1 (Γε ) → H 1 (Γ0 ) such that Pε ( uε ) H 1 (Γ0 ) ≤ C0 . In view of this, we can give two different convergence concepts for such controls. Definition 9.17. We say that afunction u∗ is an sa -limit for the sequence of uε )}ε>0 Dirichlet controls u ε ∈ H 1 (Γε ) ε>0 if the sequence of its images {Pε ( ∗ 1 converges to u weakly in H (Γ ). So, we say that the sequence of controls 0

u ε ∈ L2 (Ω, dμhε ) ε>0 is sb -convergent to an element u∗∗ ∈ L2 (Γ0 ) if this sequence is uniformly bounded, that is, | uε |2 dμhε < +∞ sup ε→0

Ω

and u ε u∗∗ in the variable space L2 (Ω, dμhε ). Below we show that these concepts are the same for admissible controls. Let us fix some sequence of controls { u } and assume that there are two

(1) ε

(2) different sequences of its images Pε ( uε ) ε>0 and Pε ( uε ) ε>0 such that uε ) u∗1 in H 1 (Γ0 ) and Pε(2) ( uε ) u∗2 in H 1 (Γ0 ). Pε(1) ( Let χhε ∈ L∞ (Γ0 ) be the characteristic function of the set ΓεD , on which the corresponding controls u ε are located. By Proposition 9.12, we have that h−1 (ε)χhε → 1 weakly-∗ in L∞ (Γ0 ) as ε → 0. Then passing to the limit in the following integral identity as ε → 0, h−1 (ε) χhε Pε(1) ( uε )ϕ dx1 = h−1 (ε) χhε Pε(2) ( uε )ϕ dx1 , Γ0

Γ0

which holds true for every ϕ ∈ H 1 (Γ0 ), we get u∗1 ϕ dx1 = u∗2 ϕ dx1 , ∀ ϕ ∈ H 1 (Γ0 ). Γ0

Γ0

Hence, u∗1 = u∗2 almost everywhere in Γ0 . As a result, we arrive at the following conclusion.


337

Lemma 9.18. Every sequence of admissible controls u ε ∈ Uε ε>0 is relatively compact with respect to the sa -convergence introduced earlier. Moreover, its sa -weak limit u∗ belongs to the set

U = u ∈ H 1 (Γ0 ) | u H 1 (Γ0 ) ≤ C0 . Indeed, the fact u∗ ∈ U is readily seen from the lower semicontinuity property of the norm in H 1 (Γ0 ) with respect to the weak convergence in this space.

Theorem 9.19. Let u ε ∈ Uε be a sequence of admissible Dirichlet conε>0

uε }ε>0 for which trols. Then one can extract an sb -convergent subsequence of { its sa - and sb -limits coincide.

Proof. As follows from the definition of the class Uε (see (9.74)), the representation formula (9.31), and Proposition 9.12, we have 1 lim sup

uε 2L2 (ΓεD ) | uε |2 dμhε = lim sup h(ε) ε→0 ε→0 Ω ≤ lim sup ε→0

1 uε ) 2H 1 (ΓεD ) χ D Pε ( h(ε) Γε

= lim sup Pε ( uε ) 2H 1 (ΓεD ) ≤ C20 . ε→0

Consequently, every sequence of admissible controls u ε ∈ Uε ε>0 is uniformly bounded in L2 (Ω, dμhε ). Therefore, taking the compactness criterium of weak convergence in variable spaces into account, we can always suppose without loss of generality that for a given sequence { uε }ε>0 , there are functions u∗ ∈ 1 ∗∗ 2 H (Γ0 ) and u ∈ L (Γ0 ) such that lim Pε ( uε )ϕ dx1 = u∗ ϕ dx1 , ∀ ϕ ∈ C0∞ (R), ε→0

Γ0

u∗∗ ϕ dx1 ,

u ε ϕ dμhε =

lim

ε→0

Since

Γ0

Ω

∀ ϕ ∈ C0∞ (Γ0 ).

Γ0

u ε ϕ dμhε ,

Pε ( uε )ϕ dx1 = Γ0

∀ ε > 0,

Ω

it follows that u∗ = u∗∗ almost everywhere on Γ0 . The proof is complete. Remark 9.20. Taking into account this result, we can assume that for any

sequence of Dirichlet controls u ∈ Uε , there is a sequence of its prototypes ε

1 ε |ΓεD = uε |ΓεD and uε u in H 1 (Γ0 ). Then, by uε ∈ H (Γ0 ) such that u analogy with Lemma 9.8, it can be proved that, in this case, u ε converges to u|Γ0 weakly in L2 (Ω, dμhε ) and u|Γ0 = u∗∗ by virtue of the uniqueness of weak

338


limits in L2 (Ω, dμhε ). Therefore, in what follows, we will assume that every

sequence of admissible Dirichlet controls u ε ∈ Uε is compact with respect to the weak convergence in L2 (Ω, dμhε ) and its limit belongs to H 1 (Γ0 ).

ε be a sequence of admissible solutions for the Let (uε , pε , yε ) ∈ Ξ ε>0 ε -problem. Each of the sets Ξ ε is defined in (9.38) and Ξ ε ⊂ Zε for every P ε > 0, where Zε ≡ H 1 (Γ0 ) ∩ L2 (Ω, dμhε ) × L2 (Ω, dνεh ) (9.75) × H 1 (Ω, dηεh ) ∩ L2 (Ω, dλhε ) . We assume that this sequence is bounded, that is, lim sup (uε , pε , yε ) Z ε < +∞. ε>0

Then summing up the above given arguments, we may introduce the following concept of weak convergence in the variable space Z ε . Definition 9.21. We say that a bounded sequence {(uε , pε , yε ) ∈ Z ε }ε>0 is w-convergent to a triplet (u, p, y) ∈ H 1 (Γ0 ) × L2 (Γ0 ) × H 1 (Ω) in the variable w space Z ε as ε tends to 0 (in symbols, (uε , pε , yε ) −→ (u, p, y) ), if uε u

in L2 (Ω, dμhε ),

pε p

in

u ∈ H 1 (Γ0 ),

L2 (Ω, dνεh ),

yεh → y in L2 (Ω, dηεh ), yεh y in L2 (Ω, dλhε ), |yεh − y|2 dηεh → 0 as ε → 0.

y ∈ H 1 (Ω),

Ω

As an evident consequence of Theorems 9.16 and 9.19, we have the following conclusion. Theorem 9.22. Let {(uε , pε , yε ) ∈ Z ε }ε>0 be a bounded sequence of admissiε . Then this sequence is relatively compact with ble solutions to the OCPs P respect to the w-convergence in the variable space Z ε . To conclude this section, we provide the following result.

Proposition 9.23. Let H 1 (Ω; Γ0 ) = y ∈ H 1 (Ω) : y = 0 on Γ0 . Then for every y ∈ H 1 (Ω; Γ0 ), there is a sequence ϕε ∈ C0∞ (R2 ; ΓεD ) such that ϕε → χΩ y strongly in L2 (R2 ), ∇ϕε → χΩ ∇y strongly in L2 (R2 , R2 ).

Vice versa, if ϕε ∈ C0∞ (R2 , ΓεD ) is a sequence such that ϕε |Ω converges to y weakly in H 1 (Ω), then y ∈ H 1 (Ω; Γ0 ).


339

Proof. Let us introduce the sequence of sets Bε ⊂ R2 with

Bε = Ω ∪ (x1 , x2 ) ∈ R2 : (x1 , −x2 ) ∈ Qε \ Ωε , ∀ ε > 0. Here, Qε = {(x1 , x2 ) : x1 ∈ [0, L1 ], x2 ∈ [0, ε/2]}. Let XBε and XΩ be closed linear subspaces of L2 (R2 ) × L2 (R2 , R2 ) such that

XBε = (χBε y, χBε ∇y) : y ∈ H 1 (Bε ) ,

XΩ = (χΩ y, χΩ ∇y) : y ∈ H 1 (Ω) . It is clear that {Bε } is uniformly bounded sequence of open subsets of R2 which converges to Ω in the Hausdorff complementary topology, that is, limε→0 dH (Bεc , Ω c ) = 0, where the Hausdorff distance between the complements of Bε and Ω in R2 is defined by (see Definition 7.28; see also Falconer [107] for the details) dH (Bεc , Ω c ) = max

sup dist(x, Ω c ), sup dist(x, Bεc ) .

x∈Bεc

x∈Ω c

Then we conclude (see Dal Maso, Ebobisse, and Ponsiglione [80], Theorem 4.2): The following sequence of subspaces XBε converges to XΩ in the sense of Mosco (see Definition 7.39):

(M1 ) For every y ∈ H 1 (Ω), there exists a sequence yε ∈ H 1 (Bε ) such that χBε yε converges to χΩ y strongly in L2 (R2 ) and χBε ∇yε converges to χΩ ∇y strongly in L2 (R2 , R2 ). (M2 ) If {εk } is a sequence of indices converging to 0, {yk } is a sequence such that yk ∈ H 1 (Bεk ) for every k and χBεk yεk converges weakly in L2 (R2 ) to a function φ, and χBεk ∇yεk converges weakly in L2 (R2 , R2 ) to a function ψ, then there exists y ∈ H 1 (Ω) such that φ = χΩ y and ψ = χΩ ∇y a.e. in R2 . (χΩ y, χΩ∇y) ∈ XΩ , and, hence, Let y ∈ H 1 (Ω; Γ0 ) be any function. Then

by property (M1 ), there is a sequence yε ∈ H 1 (Bε ) such that χBε yε converges to χΩ y strongly in L2 (R2 ). For every fixed ε, let us choose a function ϕε ∈ C0∞ (R2 , ΓεD ) such that ϕε − χBε yε L2 (R2 ) ≤ ε2 . Then

χΩ y − ϕε L2 (R2 ) ≤ ϕε − χBε yε L2 (R2 ) + χBε yε − χΩ y L2 (R2 ) −→ 0 as ε → 0, and hence, ϕε → χΩ y strongly in L2 (R2 ). To prove the converse case, we observe that by the trace theorem we have the following: If ϕε |Ω converges to y weakly in H 1 (Ω), then ϕε |Γ0 ∈ H 1/2 (Γ0 ) for every ε > 0 and ϕε |Γ0 → y|Γ0 strongly in L2 (Γ0 ). Consequently,

340


ΓεD

ϕε |Γ0 ψ dx1 = 0,

However, ϕ ψ dx 0= ε |Γ 0 1 = ΓεD

Γ0

∀ ψ ∈ C0∞ (R2 ), ∀ ε > 0.

h−1 (ε)χΓεD

ϕε |Γ0 ψ dx1 ,

∀ ε > 0, (9.76)

where χΓεD is the characteristic functions of the sets ΓεD . Using Proposition 9.13 and passing to the limit on the right-hand side of (9.76), which is the product of strongly and weakly converging sequences, we immediately obtain −1 lim h (ε)χΓεD ϕε |Γ0 ψ dx1 = yψ dx1 = 0, ∀ ψ ∈ C0∞ (R2 ). ε→0

Γ0

Γ0

Thus, y = 0 on Γ0 and we establish the required result.

9.4 Definition of the limit problem and its properties In this section, we aim to characterize a “limit” minimization problem of the ε -problem is the sequence (9.40) as ε tends to 0. The main feature of the P specific construction of the solution space Z ε and the absence of the strong approximation property for the “w-limit space” (see Pastuchova [212]) Y = H 1 (Γ0 ) × L2 (Γ0 ) × H 1 (Ω). This means that not for every triplet (u, p, y) ∈ Y there is a sequence w {(uε , pε , yε ) ∈ Z ε }ε>0 such that (uε , pε , yε ) −→ (u, p, y). Because of this, we accept the following concept of a “limit” minimization problem. Definition 9.24. We say that a minimization problem ' & I 0 (u, p, y) inf

(9.77)

(u,p,y)∈ Ξ0

is the weak variational limit of the sequence (9.40) with respect to the wconvergence in the variable space Z ε (or the variational w-limit) if conditions (d)–(dd) of Definition 8.24 hold true, with Xε = H 1 (Γ0 ) ∩ L2 (Ω, dμhε ) × L2 (Ω, dνεh ) × H 1 (Ω, dηεh ) ∩ L2 (Ω, dλhε ) , X = H 1 (Γ0 ) × L2 (Γ0 ) × H 1 (Ω), Remark 9.25. Note that in spite of the lack of the strong approximation property for the “w-limit space” Y, the weak variational limit in the sense of Definition 9.24 possesses the fine variational properties that are similar to that of the Γ -limit (in contrast to Theorem 8.25, we do not assume that the limit problem has a unique solution).

9.4 Definition of the limit problem and its properties

341

Theorem 9.26. Assume that the constrained minimization problem (9.77) is the weak variational limit of the sequence (9.40) and assume that this problem is solvable. Assume also that the initial data for the original problem (9.20) satisfy the condition

ε ) sup max fε L2 (Ω) , gε L2 (Sint < +∞. ε>0

ε be a sequence of optimal solutions to the corre(u0ε , p0ε , yε0 ) ∈ Ξ ε>0 ε -problems. Then this sequence is relatively w-compact and all wsponding P cluster triplets are optimal solutions to problem (9.77); more precisely, if

Let

w

(u0εk , p0εk , yε0k ) −→ (u0 , p0 , y0 ),

(9.78)

then inf (u,p,y)∈ Ξ0

I 0 (u, p, y) = I 0 u0 , p0 , y0

= lim Iεk (u0εk , p0εk , yε0k ) = lim

inf

ε→0 (uε ,pε ,yε )∈ Ξ bε

k→∞

Iε ((uε , pε , yε ).

(9.79)

Proof. for a moment that the sequence of optimal solutions

0 0Let0 us assume ε to the Pε -problems is such that (uε , pε , yε ) ∈ Ξ ε>0 lim sup Iε (u0ε , p0ε , yε0 ) = ∞. ε>0

Then for any given value D > 0 there is a sequence {εk } such that limk→∞ εk = 0 and min

bε (uεk ,pεk ,yεk )∈ Ξ k

Iεk (uεk , pεk , yεk ) = Iεk (u0εk , p0εk , yε0k ) ≥ D,

∀ k ∈ N.

On the other hand, it is easy to see that the triplet (0, 0, yε∗ ) is admissible for every ε (here, ye∗ is the weak solution of the boundary value problem (9.20) under the condition uε = 0 and pε = 0). Hence, taking the a priori estimate (9.39) into account, we immediately conclude lim sup ε→0

min

bε (uε ,pε ,yε )∈ Ξ

( < +∞. Iε (uε , pε , yε ) ≤ lim sup Iε (0, 0, yε∗ ) ≤ C ε→0

The obtained contradiction proves that there exists a constant D0 > 0 such that lim sup Iε (u0ε , p0ε , yε0 ) ≤ D0 . ε→0

This fact implies the boundedness of the sequence pε0 in L2 (Ω, dνεh ). Then using the boundedness of the Dirichlet type controls u0ε and estimate (9.39), we deduce

342


lim sup (u0ε , p0ε , yε0 ) Z ε = lim sup u0ε H 1 (Γ0 )∩L2 (Ω,dμhε ) ε>0

ε>0

+ p0ε L2 (Ω,dνεh ) + yε0 H 1 (Ω,dηεh ) < +∞.

Then, taking

Theorem 9.22 into account, we may suppose that there exist a subsequence (u0εk , p0εk , yε0k ) k∈ N of the sequence of optimal solutions and a w

triplet (u∗ , p∗ , y ∗ ) such that (u0εk , p0εk , yε0k ) −→ (u∗ , p∗ , y ∗ ) as εk → 0. Hence, property (d) of Definition 8.24 leads us to (u∗ , p∗ , y ∗ ) ∈ Ξ0 , and lim inf

min

k→∞ (u,p,y)∈ Ξ bε k

Iεk (u, p, y) = lim inf Iεk (u0εk , p0εk , yε0k ) ≥ I0 (u∗ , p∗ , y ∗ ) k→∞

≥

min

(u,p,y)∈ Ξ0

I 0 (u, p, y) = I 0 (u0 , p0 , y 0 ).

(9.80)

Let us fix a value δ > 0. Then, by property (d) of Definition 8.24, there

ε such that exists a δ-realizing sequence ( uε , pε , yε ) ∈ Ξ ε>0 w

u, p, y), ( uε , pε , zε ) −→ (

(u0 , p0 , y0 ) − ( u, p, y) Y ≤ δ,

I 0 (u0 , p0 , y 0 ) ≥ lim sup Iε ( uε , pε , yε ) − δ. ε→0

Using this fact, we have min (u,p,y)∈Ξ0

I 0 (u, p, y) + δ = I 0 (u0 , p0 , y0 ) + δ uε , pε , yε ) ≥ lim sup ≥ lim sup Iε ( ε→0

≥ lim sup

ε→0

min

bε k→∞ (u,p,y) ∈ Ξ k

min

bε (u,p,y) ∈Ξ

Iε (u, p, y)

Iεk (u, p, y) = lim sup Iεk (u0εk , p0εk , yε0k ). (9.81) k→∞

From this and (9.80), we deduce lim inf Iεk (u0εk , p0εk , yε0k ) ≥ lim sup Iεk (u0εk , p0εk , p0εk ) − δ. k→∞

k→∞

Since this inequality holds true for all sufficiently small δ > 0, it follows that combining the above obtained relations (9.80) and (9.81), we obtain the required conclusion I0 (u∗ , p∗ , y∗ ) = I 0 (u0 , p0 , y 0 ) = I 0 (u0 , p0 , y 0 ) = lim

min (u,p,y)∈ Ξ0

Iεk (u, p, y).

min

k →∞ (u,p,y) ∈ Ξ bε

I 0 (u, p, y),

(9.82) (9.83)

k

However, we observe that (9.82) and (9.83) hold true for all w-convergent

subsequences of (u0ε , p0ε , yε0 ) ε>0 . It follows that these limits in (9.82) and

9.5 Main convergence theorem

343

(9.83) coincide and, therefore, I 0 (u0 , p0 , y 0 ) is the limit of the whole sequence of minimal values 0 0 0 Iε ((uε , pε , yε ) Iε (uε , pε , yε ) = inf . bε (uε ,pε ,yε )∈ Ξ

ε>0

This concludes the proof.

9.5 Main convergence theorem The main question that we are going to discuss in this section concerns the limit analysis of the following boundary value problem (see (9.38)): yε = uε

μhε -a.e. in Ω, uε H 1 (Γ0 ) ≤ C0 ,

(A(x/ε)∇yε · ∇ϕ) dηεh +

Ω

(9.84)

yε ϕ dηεh + 4d β(ε)

Ω

yε ϕ dλhε

Ω

dηεh

fε ϕ + 2 β(ε) pε ϕ dνεh Ω + 4 β(ε) gε ϕ dλhε , ∀ ϕ ∈ C0∞ (R2 ; ΓεD ), =

Ω

(9.85)

Ω

where β(ε) = (1 − h(ε))/(2 − h(ε))ξ(ε), and ξ(ε) = ε/h(ε).

ε , which is supposed In other words, for a given sequence (uε , pε , yε ) ∈ Ξ to be w-convergent (in this section only), we look for the relations in terms of which a w-limit of this sequence can be determined. We proceed to the passage to the limit in the integral identity (9.85). We note that in view of Proposition 9.23, the class of test functions C0∞ (R2 ; ΓεD ) for integral identity (9.85) “tends in the limit” to C0∞ (R2 ; Γ0 ). So, in what follows, we suppose that ϕ ∈ C0∞ (R2 ; Γ0 ) in (9.85).

ε be an equibounded sequence of admissible solutions Let (uε , pε , yε ) ∈ Ξ ε -problem. By Theorem 9.22, this sequence is relatively compact with for the P respect to the w-convergence in the variable space Zε . So we may suppose that w there exists a triplet (u, p, y) ∈ Y such that (uε , pε , yε ) (u, p, y). The main question is: By what kind of relation can this triplet be determined? In order to give an answer, we provide the following result. Theorem 9.27. Assume that Aε (x) = A(x/ε) in (9.85) is a Y -periodic measurable matrix satisfying the initial conditions (9.21). Assume also that lim ξ(ε) = ξ ∗ < +∞

ε→0

and that there exist functions f 0 ∈ L2 (Ω) and g ∈ L2 (Ω) such that

344


fε f 0 in L2 (Ω, dηεh ) and gε g in L2 (Ω, dλhε ).

(9.86)

pε ∈ L2 (Ω, dνεh ) and uε ∈ H 1 (Γ0 ) ∩ L2 (Ω, dμhε )

(9.87)

Let

ε -problems such that be any bounded sequences of admissible controls for P pε p in L2 (Ω, dνεh ), uε u in L

2

(9.88)

(Ω, dμhε ).

(9.89) Let yε = yε (uε , pε ) ∈ H 1 (Ω, dηεh ) ∩ L2 (Ω, dλhε ) ε>0 be the corresponding so

w

lutions to the problem (9.84)–(9.85). Then (uε , pε , yε ) (u, p, y) as ε → 0, where y is the unique solution in H 1 (Ω) of the following boundary value problem: (9.90) z = u a.e. in Γ0 , u H 1 (Γ0 ) ≤ C0 ,

(Ahom ∇y · ∇ϕ) dx + Ω

yϕ dx + 2d ξ ∗ yϕ dx Ω Ω ∗ = f0 ϕ dx + 2 ξ gϕ dx. Ω

Here, the matrix Ahom is defined by the rule A(y)(ζ + r0 ) dη(y), Ahom ζ =

(9.91)

Ω

(9.92)

Y

where υ 0 ∈ L2 (Ω, Vpot ) is the unique solution of the minimum problem (ζ + r, A(ζ + r)) dη = (ζ + r0 , A(ζ + r0 )) dη. (9.93) min r∈Vpot

Y

Y

Proof. To begin, we note that by the a priori estimate (9.39), the sequence of

ε is uniformly bounded in Zε . Hence, admissible solutions (uε , pε , yε ) ∈ Ξ ε>0 due to Theorem 9.22, we may suppose that there is a function y ∈ H 1 (Ω) satisfying the conditions ⎧ y → y in L2 (Ω, dηεh ), ⎪ ⎪ ⎨ ε yε y in L2 (Ω, dλhε ), (9.94) ⎪ ⎪ 2 ⎩ ∇yε ∇y + r, where r ∈ L2 (Ω, Vpot ). On the other hand, in view of Theorem 9.19 and Remark 9.20, the limit sa element u in (9.89) is such that u ∈ H 1 (Γ0 ) and uε −→ u. Then passing to the limit in (9.84) as ε → 0, we come to relation (9.90). Our aim now is to prove the correctness of the integral equality (9.91).


345

For this, we consider the integral identity (9.85) with the test function ∞ (Y ), and Ψ ∈ C0∞ (R2 , ΓεD ). ϕ(x) = εΨ (x)ω(ε−1 x), where ω ∈ Cper This yields ε (A(x/ε)∇yε · ∇Ψ )ω(ε−1 x) dηεh + ε yε Ψ ω(ε−1 x) dηεh Ω Ω h + (A(x/ε)∇yε · ∇ω)Ψ dηε + 4εd β(ε) yε Ψ ω(ε−1 x) dλhε Ω Ω = ε fε Ψ ω(ε−1 x) dηεh + 2ε β(ε) pε Ψ ω(ε−1 x) dνεh Ω Ω + 4ε β(ε) gε Ψ (x)ω(ε−1 x) dλhε . (9.95) Ω

Taking the mean value property (see Theorem 6.51), which holds for all periodic Borel measures ηεh , λhε , and νεh , into account, we pass to the limit in (9.95) as ε → 0. We infer (9.96) lim A(ε−1 x)∇yε , ∇ω(ε−1 x) Ψ dηεh = 0. ε→0

Ω

Then, in view of (9.94)3 and the definition of the weak two-scale limit, we have lim Ψ ω(ε−1 x)A(ε−1 x) ∇yε dηε ε→0 Ω = Ψ (x)ω(z)A(z) [∇y(x) + r(x, z)] dη(z) dx, Ω

or

Y

(9.97) A(ε−1 x)∇yε A(z) [∇y + r(x, z)] , 2 A(ε−1 x)∇yε (x) A(z) [∇y(x) + r(x, z)] dη(z) in L2 (Ω, dηεh , (9.98) 2

Y

that is, (9.96) can be rewritten in the explicit form as

lim

ε→0

Ω

A(ε−1 x)∇yε , ∇ω(ε−1 x) Ψ dηεh = (A(z) [∇y(x) + r(x, z)] , ∇ω) dη(z) Ψ dx = 0. Ω

Y

Since this equality holds true for every Ψ ∈ C0∞ (R2 , Γ0 ), it follows that ∞ 2 (A(∇y + r), ζ) dη = 0, ∀ ζ ∈ Cper (Y ) . (9.99) Y

However, following Zhikov [256], (9.99) can be viewed as Euler’s equation for the minimum problem (9.93). Since this problem has a unique solution

346


r0 ∈ L2 (Ω, Vpot ), it follows that r0 is the unique solution of (9.99) as well. Let us define the matrix Ahom as in (9.92). Then, having put r = r0 in (9.98) and using formula (9.92), we immediately deduce that A ε−1 x ∇yε Ahom ∇y

in

2 2 L (Ω, dηεh ) .

(9.100)

Our next intention is to pass to the limit in (9.85) as ε → 0. We choose the test functions in (9.85) as follows: ϕ = ϕε ∈ C0∞ (R2 , ΓεD ) for every ε > 0 and ϕε → ϕ in C(R2 ), where ϕ ∈ C 1 (Ω, Γ0 ). Note that, due to Proposition 9.12, such choice is always possible. Then the passage to the limit in (A(x/ε)∇yε · ∇ϕε ) dηεh + yε ϕε dηεh + 4d β(ε) yε ϕε dλhε Ω Ω Ω h h = fε ϕε dηε + 2 β(ε) pε ϕε dνε + 4 β(ε) gε ϕε dλhε

Ω

Ω

Ω

is justified by relations (9.86)–(9.89), (9.92), (9.94), lim β(ε) = (1/2) lim ξ(ε) = (1/2)ξ ∗ ,

ε→0

(9.101)

ε→0

and Lemma 9.10. Hence, we obtain

hom

(A Ω

∇y · ∇ϕ) dx +

∗

yϕ dx + 2d ξ yϕ dx Ω = f0 ϕ dx + ξ ∗ pϕ dx1 + 2 ξ ∗ gϕ dx

Ω

Ω

Γ0

Ω

1 for every ϕ ∈ C 1 (Ω, Γ0 ). By the density of C 1 (Ω, Γ0 ) in H (Ω, Γ0 ), this p ϕ dx1 = 0 relation can be extended to any function ϕ ∈ H 1 (Ω, Γ0 ). Since Γ0

provided ϕ ∈ H 1 (Ω, Γ0 ), it follows that (9.91) holds true. To conclude, we note that relations (9.90) and (9.91) can always be interpreted as the variational formulation of the following boundary value problem: ⎫ in Ω, −div Ahom ∇y + (1 + 2dξ ∗ )y = f0 + 2ξ ∗ g ⎬ ∂νAhom y = 0 on ∂Ω \ Γ0 , (9.102) ⎭ y=u on Γ0 , for which we have the following: For every f0 , g ∈ L2 (Ω) and u ∈ H 1 (Γ0 ), there exists a unique solution of (9.102) (see Lions [168] and Michajlov [188]). The next statement is a direct consequence of well-known results of the theory of boundary value problems (see Lions [168]). Corollary 9.28. Let (f01 , u1 , y1 ), (f02 , u2 , y1 ) ∈ L2 (Ω) × H 1 (Γ0 ) × H 1 (Ω) be triplets satisfying relation (9.102). Then there exists a constant

9.6 Identification of the limit optimal control problem

347

ˆ (Cˆ = C(Ω, Γ0 , g, ξ ∗ , α0 ))

Cˆ > 0 such that

y1 − y2 H 1 (Ω) ≤ Cˆ f01 − f02 L2 (Ω) + u1 − u2 H 1 (Γ0 ) .

(9.103)

9.6 Identification of the limit optimal control problem In this section, we show that for the sequence of constrained minimization problems (9.40), there exists a weak variational limit with respect to the wconvergence in the variable space Zε , and this limit problem can be recovered in an explicit form. We begin with the following result. Theorem 9.29. Let yεT ∈ L2 (Ω, dηεh ), gε ∈ L2 (Ω, dλhε ), and fε ∈ L2 (Ω, dηεh ) be given functions such that T 2 h y dηε < ∞, yεT → yT in L2 (Ω, dηεh ), (9.104) sup ε ε→0 Ω 2 sup (fε ) dηεh < ∞, fε → f in L2 (Ω, dηεh ), (9.105) ε→0 Ω 2 sup (gε ) dλhε < ∞, gε g in L2 (Ω, dλhε ). (9.106) ε→0

Ω

Assume that the correction factor κ(ε) in the cost functional (9.37) is such that (9.107) lim κ(ε)h(ε) = κ∗ < +∞. ε→0

Then for the sequence (9.40), there exists a unique weak variational limit with respect to the w-convergence which has the following representation & ' inf I 0 (u, p, y) , (9.108) (u,p,y)∈ Ξ0

where the cost functional I 0 and the set of admissible solutions Ξ0 are defined as T 2 ∗ 2 I 0 (u, p, y) = 2 (y(x) − y ) dx + κ u dx1 + 2 p2 dx1 , (9.109) Ω

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ Ξ0 = (u, p, y) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Γ0

Γ0

⎫ y ∈ H 1 (Ω), u ∈ H 1 (Γ0 ), p ∈ H 1 (Γ0 ), ⎪ ⎪ ⎪ ⎪ u H 1 (Γ0 ) ≤ C0 , y = u a.e. on Γ0 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ hom (A ∇y · ∇ϕ) dx + yϕ dx . Ω Ω ⎪ ⎪ ∗ ⎪ ⎪ yϕ dx = f ϕ dx +2d ξ ⎪ ⎪ ⎪ Ω Ω ⎪ ⎪ ⎪ ∗ ∞ ⎪ +2 ξ gϕ dx ∀ ϕ ∈ C (Ω, Γ ) ⎭ 0 Ω

(9.110)

348


Proof. The proof of this theorem is divided into two steps concerning the verification of the correspondent items of Definition 8.24. Step 1: Statement (d) of Definition 8.24 holds. Let {(uk , pk , yk ) ∈ Z εk }ε>0 be a bounded w-convergent sequence, and let (u, p, y) ∈ Y be its w-limit. Here, {εk } is a subsequence of {ε > 0} such that ε for all k ∈ N. Then, due εk → 0 as k → ∞. We suppose that (uk , pk , yk ) ∈ Ξ k to Theorems 9.22 and 9.27, we have that the w-limit triplet (u, p, y) belongs to the space Y = H 1 (Γ0 ) × L2 (Γ0 ) × H 1 (Ω) and satisfies conditions (9.90) and (9.91). Hence, the first inclusion in (8.59) holds true. It remains only to verify inequality (8.59) in which the limit cost functional I0 has the representation (9.109). Taking into account the property of lower semicontinuity of the weak convergence in variable spaces, we have lim (2 − h(εk )) (yεk (x) − yεTk )2 dηεhk k→∞ Ω + κ(εk )h(εk ) u2εk dμhεk +2 (1 − h(εk )) p2εk dνεhk Ω Ω T 2 ∗ 2 u dx1 + 2 p2 dx1 . (9.111) ≥ 2 (y(x) − y ) dx + κ Ω

Γ0

Γ0

Step 2: Statement (dd) of Definition 8.24 holds true. Let (u, p, y) ∈ Ξ0 be any admissible solution for the limiting minimization problem (9.108). Let p( ∈ H 3/2 (Ω) be a given function. Then p( |Γ0 ∈ H 1 (Γ0 ) and, therefore, (u, p( |Γ0 , y) ∈ Ξ0 . Moreover, it is readily seen from (9.109) that there is a constant D1 > 0 such that p |Γ0 − p L2 (Γ0 ) . |I 0 (u, p, y) − I 0 (u, p( |Γ0 , y)| ≤ D1 ( Let 0 < δ < 1 be a given value. Using the density of the embedding ( ∈ H 3/2 (Ω) such that H 1 (B) → L2 (B), we can choose an element w

w − w ( L2 (Γ0 )
0 as follows: h 2 ε→0 u2 dx1 , (9.114) uε dμhε −→ uhε u in L2 (Ω, dμhε ) and Ω Γ0 h 2 h ε→0 phε p( in L2 (Ω, dνεh ) and pε dνε −→ p(2 dx1 , (9.115) Ω

Γ0


349

u H 1 (Γ0 ) ≤ C0 , and yεh is the correspondent solutions of the initial-boundary ε for every ε > 0. It is value problem (9.84)–(9.85). Hence, (uhε , phε , yεh ) ∈ Ξ clear that this sequence is equibounded in Zε .

Then, applying Theorem 9.22, we conclude that the sequence (uhε , phε , yεh ) is compact with respect to w-convergence. Let (u, p(, y(∗ ) be its w-limit. Due to Theorem 9.27, we have (u, p(, y(∗ ) ∈ Ξ0 . Since the boundary value problem (9.102) has a unique solution for every fixed u, it follows that y = y(∗ and, w hence, (uhε , phε , yεh ) (u, p(, y) as ε → 0. Thus, properties (8.60) of Definition 8.24 are fulfilled. It remains to verify inequality (8.61). To this end, we take into account properties (9.114) and (9.115) and Theorem 9.22. Then limε→0 Iε (uhε , phε , yεh ) = I 0 (u, p(, y). To conclude, we apply inequality (9.112). This yields the required relation I 0 (u, p, y) ≥ lim Iε (uhε , phε , yεh ) − δ. ε→0

Remark 9.30. We emphasize that as follows from (9.109) and (9.110), the Neumann boundary control p ∈ L2 (Γ0 ) has a so-called “passive” influence on the limit object (9.108). The limit cost functional (9.109) (in contrast to the limit problem (9.102)) does not depend on this control. Moreover, it can be represented in the form I 0 (u, p, y) = 2 (y(x) − y T )2 dx + κ∗ u2 dx1 + C(p). (9.116) Ω

Γ0

In fact, taking into account the variational properties of the problem (9.108) 0 2 (p ) dx1 , stated in Theorem 9.26, the constant C(p) should be defined as 2 Γ0

where p 0 ∈ L2 (Γ0 ) is a weak limit in L2 (Ω, dνεh ) of the optimal control pε0 ε -problem (see (9.36)–(9.38)). On the other hand, it is clear that for the P an optimal solution (u0 , p 0 , y 0 ) to the problem (9.108)–(9.110) is always such that p 0 = 0. So, in the sequel we will omit the constant C(p) in the (9.116). Moreover, in view of the variational properties of the problem (9.108), it ε -problem converges to means that the Neumann optimal control pε0 for the P 2 h 0 weakly in L (Ω, dνε ). Note that since the constrained minimization problem (9.108) admits interpretation in the form of an OCP, this means that for the OCP (9.36)–(9.38), there exists a unique limit with respect to w-convergence as ε → 0 and it can be represented in the form

350


⎫ −div Ahom ∇y + y + 2dξ ∗ y = f + 2ξ ∗ g in Ω,⎪ ⎬ ∂νAhom y = 0 on ∂Ω \ Γ0 , ⎪ ⎭ y = u on Γ0 ,

u ∈ H 1 (Γ0 ),

u H 1 (Γ0 ) ≤ C0 , T 2 ∗ (y(x) − y ) dx + κ u2 dx1 −→ inf .

I hom (u, y) = 2 Ω

(9.117)

(9.118) (9.119)

Γ0

In what follows, this OCP will be called the Phom -problem. To examine the variational properties for the Phom -problem, we apply Theorem 9.26 and make use of the following obvious result (see [169]). Proposition 9.31. The limit OCP (9.117)–(9.119) has a unique solution. Combining the results of Theorem 9.26, Proposition 9.31, and Remark 9.30, we can provide the following conclusion concerning the variational properties of the Phom -problem.

ε ε Theorem 9.32. Let (uε0 , pε0 , yε0 ) ∈ Ξ be optimal solutions to the P ε>0 0 0 1 1 problem. Then there exists a pair (u , y ) ∈ H (Γ0 ) × H (Ω) such that lim Iε (uε0 , pε0 , yε0 ) = lim

ε→ 0

inf

ε→ 0 (uε ,pε ,yε )∈ Ξ bε

=

inf (u,p,y)∈Ξ0

Iε (uε , pε , yε )

I 0 (u, 0, y) = I hom (u 0 , y 0 ),

w

(uε0 , pε0 , yε0 ) −→ (u0 , 0, y 0 ) in variable space Zε .

(9.120)

(9.121)

In contrast to Theorem 9.26, (9.121) is valid for the whole sequence of optimal triplets. Indeed, since the limit OCP (9.117)–(9.119) has a unique solution, it follows that the limits of all w-convergent subsequences (uε0k , pε0k , yε0k ) k∈N

ε of the sequence (uε0 , pε0 , yε0 ) ∈ Ξ are coincident. Therefore, (u0 , 0, y 0 ) ε>0 is the w-limit triplet of the whole sequence. So, it remains to apply the arguments of Theorem 9.26 to the original sequence of optimal solutions

ε (uε0 , pε0 , yε0 ) ∈ Ξ . ε>0

9.7 On suboptimal controls for Pε-problems This section deals with the approximation of optimal solutions to the original problem Pε for small enough values of ε. It is clear that the numerical computation of optimal solutions to Pε -problems is very complicated for high accuracy; therefore, we will focus our attention on the definition of suboptimal solutions

9.7 On suboptimal controls for Pε -problems

351

sub sub (usub ε , pε , yε ) which should be admissible triplets guaranteeing the closesub sub ness of the corresponding value of the cost functional Iε (usub ε , pε , yε ) to its minimal one. In view of this, we introduce the following concept.

Definition 9.33. We say that a sequence of pairs

0 0 (( uε , p(ε ) ∈ H 1 (ΓεD ) × L2 (ΓεN ) ε>0 is asymptotically suboptimal for the problem Pε if for every δ > 0, there is ε0 > 0 such that 0 0 inf ∀ ε < ε0 . I (u , p , y ) − I (( u , p ( , y ( ) (9.122) ε ε ε ε ε ε ε < δ, ε (uε ,pε ,yε )∈ Ξε

Here, by y(ε = y(ε (( uε0 , p(ε0 ) we denote the corresponding solutions of the boundary value problem (9.20). The main result can be stated as follows. Theorem 9.34. Let u0 ∈ H 1 (Γ0 ) be an optimal Dirichlet control for the Phom -problem (9.117)–(9.119). Then the sequence of pairs

0 (9.123) (u |ΓεD , 0) ∈ H 1 (ΓεD ) × L2 (ΓεN ) ε>0 is asymptotically suboptimal for the original optimal control problem Pε .

Proof. Let us consider the sequence of triplets (u0 |ΓεD , 0, y(ε ) ε>0 , where y(ε = yε (u0 |ΓεD , 0) are the corresponding solutions of the boundary value problem (9.20). It is easy to see that each of these triplets is admissible for the original control problem Pε . Hence, due to estimate (9.39), we conclude that the sequence of prototypes * ) ε ( u 0 , 0, yε ) ∈ Ξ ε>0

is uniformly bounded in Zε and relatively compact with respect to the wconvergence (see Theorem 9.22). Here, u 0 is a prototype in L2 (Ω, dμhε ) of the 0 control function u |ΓεD . Let (u, 0, y) ∈ Y be some w-cluster triplet of this sequence. Applying Definition 9.21, Theorem 9.22, and the fact that u 0 |ΓεD = u 0 |ΓεD for every ε > 0, we have yεk → y in L2 (Ω, dηεh ), u 0 u in L2 (Ω, dμhε ),

s

a u 0 −→ u 0.

Thus, (u, 0, y) = (u0 , 0, y), and due to Theorems 9.27 and 9.32, we obtain (u0 , 0, y) ∈ Ξ0

=⇒

(u0 , y) ∈ Ξhom , u0 = u,

352


where by Ξhom we denote the set of admissible solutions for the limit problem (9.117)–(9.119). Hence, y = y(u0 ) is the corresponding solution of the limit boundary value problem (9.102). Thus, in view of Theorem 9.29, we have a more precise result: The pair (u0 , y 0 ) ≡ (u0 , y) ∈ Ξhom is optimal for the limit problem Phom . Let (u0ε , pε0 , yε0 ) ∈ Ξε ε>0 be optimal solutions to the problems Pε . We observe that inf Iε (uε , pε , yε ) − Iε (u0 |ΓεD , 0, y(ε ) (uε ,pε ,yε )∈ Ξε = Iε (u0ε , pε0 , yε0 ) − Iε (u0 |ΓεD , 0, y(ε ) ≤ Iε (u0 , p 0 , y0 ) − I0 (u0 , y0 ) + I0 (u0 , y0 ) − Iε (u0 |Γ D , 0, y(ε ) ε

ε

ε

ε

≤ J1 + J2 + J3 , where J1 = Iε (u0ε , pε0 , yε0 ) − I0 (u0 , y0 ) , J2 = 2 (y 0 (x) − y T )2 dx − (2 − h(ε)) (yε0 (x) − y T )2 dηεh , Ω

Ω

∗ 0 2 0 2 h J3 = κ u ε dμε . (u ) dx1 − κ(ε)h(ε) Γ0

Ω

To conclude the proof, it remains to note that for a given δ > 0, one can find the following: (1) ε1 > 0 such that J1 < δ/3 for all ε < ε1 by Theorem 9.32. (2) ε2 > 0 such that J2 < δ/3 for all ε < ε2 by Theorem 9.22. (3) ε3 > 0 such that J3 < δ/3 for all ε < ε3 by Theorem 9.19 (see also Remark 9.20). Thus, estimate (9.122) holds true for all ε < min{ε1 , ε2 , ε3 }. Our next intention is to derive the optimality conditions for the limit problem (9.117)–(9.119) from which an optimal control for the Phom -problem can be determined. To this end, we use the Lagrange multiplier principle. We obtain the weak form of the optimal system that an optimal pair (u0 , y0 ) and Lagrange multipliers must satisfy. As Theorem 9.34 indicates, this optimality system can serve as a basis for the construction of suboptimal solutions to the original problem on the thin periodic structures Ωε . We recall a few notions of the abstract Lagrange multiplier principle. Let Y, U, and V be Banach spaces. Let I : Y × U → R be a cost functional and let F (y, u) : Y × U → V be a map. Let U∂ be a closed subset of U with a nonempty interior. We have the following minimization problem: I(y, u) −→ inf,

F (y, u) = 0,

u ∈ U∂ .

(9.124)


353

The Lagrange functional for the problem (9.124) is defined by L(y, u, λ, Ψ ) = λI(y, u) + F (y, u), Ψ ,

(9.125)

where λ ∈ R+ and Ψ ∈ V ∗ . Then each of the solutions to the constrained minimization problem (9.124) can be characterized as follows (see Theorem 3.41). Theorem 9.35. Let (y0 , u0 ) ∈ Y × U be a solution of (9.124). Assume that the mappings y → I(y, u) and y → F (y, u) are continuously differentiable at y ∈ O(y 0 ) and Im Fy (y 0 , u0 ) = V. Assume that the mapping u → I(y, u) is convex, I is differentiable at (y 0 , u0 ), and the mapping u → F (y, u) is continuous and affine. Then λ can be taken as 1 and there exists a Ψ ∈ V ∗ such that + 0 0 , Ly (y , u , 1, Ψ ), h = 0, ∀ h ∈ V, (9.126) and

+

, Lu (y 0 , u0 , 1, Ψ ), u − u0 ≥ 0,

∀ u ∈ U∂ .

(9.127)

We are now in a position to apply the Lagrange multiplier principle to optimal control problem (9.117)–(9.119). Theorem 9.36. If a pair (u0 , y 0 ) ∈ H 1 (Γ0 ) × H 1 (Ω) is an optimal solution to the problem (9.117)–(9.119), then there exists a function Ψ ∈ H01 (Ω) (9.128) such that the triplet (u0 , y 0 , Ψ ) satisfies the following optimality system: ⎫ −div Ahom ∇y 0 + y 0 + 2dξ ∗ y 0 = f + 2ξ ∗ g in Ω,⎪ ⎪ ⎬ 0 ∂νAhom y = 0 on ∂Ω \ Γ0 , (9.129) ⎪ ⎪ ⎭ 0 0 on Γ0 , y =u ⎫ ! t −div Ahom ∇Ψ + Ψ + 2dξ ∗ Ψ = 4(y 0 − y T ) in Ω,⎬ (9.130) ⎭ Ψ = 0 on ∂Ω, Γ0

⎫ ! ⎪ 2κ∗ u0 − ∂ν Ahom t Ψ (u − u0 ) dx1 dt ≥ 0, ⎬ [

]

∀ u ∈ H (Γ0 ) : u H 1 (Γ0 ) ≤ C0 1

⎪ ⎭

.

(9.131)

354


Proof. Let (u0 , y0 ) be an optimal solution to the limit problem (9.117)– (9.119). To apply the Lagrange multiplier principle, we set

Y = z ∈ H 1 (Ω) , U = H 1 (Γ0 ), V = (H 1 (Ω))∗ , and

F (u, y) = −div Ahom ∇y + y + 2dξ ∗ y − f − 2ξ ∗ g .

Then Fy (u0 , y 0 ) : Y → V is an isomorphism. Hence, all suppositions of Theorem 9.35 are fulfilled. We define the Lagrange function as T 2 ∗ u2 dx1 L(u, y, Ψ ) = 2 (y(x) − y ) dx + κ Ω

Γ0

(Ahom ∇y · ∇Ψ ) dx +

+ Ω

+ 2d ξ ∗

zΨ dx −

Ω

yΨ dx Ω

f Ψ dx − 2 ξ ∗

Ω

gΨ dx Ω

for all Ψ ∈ H01 (Ω). Then in accordance with Theorem 9.35, there exists a function Ψ ∈ H01 (Ω) such that relations (9.126) and (9.127) are valid. It is easy to see that in this case, (9.126) takes the form (9.129)–(9.130), whereas (9.127) can be written as (9.131). We note that this approach to the construction of suboptimal controls for the problem Pε is acceptable if the following hold: (A) The thin structure Ωε is sufficiently thick, that is, ξ ∗ = 0. Then the limit problem from which a suboptimal control can be determined takes the form ⎫ −div Ahom ∇y = f in Ω, ⎪ ⎬ ∂νAhom y = 0 on ∂Ω \ Γ0 , (9.132) ⎪ ⎭ y = u on Γ0 ,

u H 1 (Γ0 ) ≤ C0 , (y(x) − yT )2 dx + κ∗ u2 dx1 −→ inf.

u ∈ H 1 (Γ0 ),

I hom (u, y) = 2 Ω

(9.133) (9.134)

Γ0

(B) The thin structure Ωε has a critical thickness ξ ∗ = θ ∈ (0, +∞). Then the corresponding limit problem for suboptimal control takes another form which depends on the thickness θ, namely


355

⎫ −div Ahom ∇y + y + 2dθy = f + 2θg in Ω,⎪ ⎬ ∂νAhom y = 0 on ∂Ω \ Γ0 , ⎪ ⎭ y = u on Γ0 , u ∈ H 1 (Γ0 ), u H 1 (Γ0 ) ≤ C0 , T 2 ∗ u2 dx1 −→ inf. I hom (u, y) = 2 (y(x) − y ) dx + κ Ω

Γ0

(C) The thin structure Ωε is sufficiently thin (i.e., ξ ∗ = +∞) and instead of ε the Robin and Neumann conditions along the boundaries Sint and ΓεN , respectively, we have the homogeneous Neumann condition ∂νA yε = 0

on ΓεN ∪ ΓεN .

In this case, as it immediately follows from the integral identity (9.85) and Theorem 9.27, the limit problem from which a suboptimal Dirichlet control can be determined has the form (9.132)–(9.134).

10 Approximate Solutions of Optimal Control Problems for Ill-Posed Parabolic Problems on Thin Periodic Structures

In this chapter, we focus our attention on control objects which are described by singular parabolic equations with Robin boundary conditions at the boundary of the holes, and with two different types of boundary controls – Dirichlet and Neumann controls – on the external boundary of the thin periodic structure Ωε . We allow for a blowing-up phenomenon in the original problem and we provide its asymptotic analysis as the small parameter ε tends to 0. It is shown that the structure of the limit problem depends essentially on how h tends to 0 as ε → 0 (the so-called “scaling effect”). We derive conditions under which in the limit we do not obtain an optimal control problem (OCP), but rather some initial-boundary value problems with or without controls. Furthermore, we construct asymptotically suboptimal controls for the original problem and show an approximation property of such controls for small enough ε near the optimal characteristics. Let Ωε ⊂ R2 be a periodic open set with a reticulated structure contained in a fixed bounded open set Ω = (0, L 1 ) × (0, L 2 ). For the sake of simplicity, we restrict our analysis to the case when set Ωε has the representation Ωε = Ω ∩ εF h , where F h denotes a 1-periodic thin grid with a small but same thickness h. So, we preserve all suppositions of Chap. 9 concerning the definition of the thin periodic structure Ωε and its measure description (see Fig. 9.1). The OCPs we consider can be described in a general manner as follows: Seek a triplet (u0ε , pε0 , yε0 ) such that the functional Iε (uε , pε , yε ) =

1 h

(yε (T, ·) − yεT )2 dx + Ωε

T

1 h T

T

0

2

+ κ(ε)

u dx1 dt + 0

ΓεD

0

ΓεN

(yε − yε∗ )6 dx dt

Ωε

p2ε dH1 dt −→ inf

(10.1)

is minimized, subject to the constraints


357

358

10 Approximate Solutions of OCPs for Ill-Posed Objects

∂t yε − div (A(x/ε)∇yε ) − yε3 = fε ∂νA yε = 0 ∂νA yε = ε2 (−d yε + gε ) y ε = uε ∂νA yε = εpε yε (0, x) = yε,0

pε ∈ L2 (0, T ; L2 (ΓεN )),

Uε = u|ΓεD : u ∈ L2 (0, T ; H 1 (Γ0 )),

⎫ in (0, T ) × Ωε , ⎪ ⎪ ⎪ ε on (0, T ) × Sext ,⎪ ⎪ ⎪ ⎬ ε on (0, T ) × Sint , (10.2) D on (0, T ) × Γε , ⎪ ⎪ N ⎪ on (0, T ) × Γε , ⎪ ⎪ ⎪ ⎭ for a.a. x ∈ Ωε ,

uε ∈ Uε ,

uL2 (0,T ; H 1 (Γ0 )) ≤ C0 ,

(10.3) (10.4)

Here, pε ∈ L2 (0, T ; L2 (ΓεN )) and uε ∈ L2 (0, T ; H 1 (ΓεD )) are the controls which are realized via the Neumann and Dirichlet boundary conditions posed on different parts ΓεD and ΓεN of the quickly oscillating part of the external boundary ∂Ωε . The main object of our consideration is the control problem for the singular distributed system (10.2) for which such properties as instabilities, multiple solutions, and blowing-up phenomenon may occur. We provide an asymptotic analysis of the OCP for which not for every admissible controls uε and pε there exists the corresponding global solution yε = yε (uε , pε ) of the original initialboundary value problem (10.2) in (0, T ) × Ωε , in contrast to fundamental assumptions in classical approaches to the homogenization theory (see, for instance, Attouch [10], Buttazzo and Dal Maso [43], Cioranescu and Saint Jean Paulean [66], Dal Maso [78], Kapustyan and Panik [129], Pastuchova [213], and Zhikov, Kozlov, and Olienik [261]). Second, the thin domains Ωε we consider here are always such that their thickness h = h(ε) is related with the parameter of periodicity ε by the supposition h(ε) → 0 as ε → 0. It is the principal difference between this type of domain and the periodically perforated ones for which the another rule takes place, namely h = h(ε) → const ∈ (0, 1] as ε → 0. Third, in this chapter we deal with the OCP rather than with the initialboundary value problem. In spite of the fact that the distributed system (10.2) is singular, the OCP (10.1)–(10.4), that we are considering here is well defined and solvable for every ε > 0. However, because of the complexity of the numerical computation of the optimal solutions to the original problem for small enough values of ε (through the thick perforation of the domain Ωε and mainly because of the singularity of the corresponding optimality system), we proceed to pass to the limit as ε → 0 and use the limiting problem as a reduced model with preservation of the main variational property: Both optimal solution and minimal value of the cost functional for the original problem converge to the corresponding characteristics of a limit OCP as ε tends to 0. We propose the so-called “indirect approach” to the limit analysis of the boundary OCP for the singular system. The main idea is to introduce a couple of new OCPs for parabolic equations on the reticulated structure Ωε . The first of them is a so-called virtual extension of the original problem Pε . Having


359

entered a fictitious “distributed” control into the Pε -problem and modified its ε ) cost functional, we prove that the new problem (hereafter referred to as P always has a non-empty set of admissible solutions (in contrast to the original one); thus, this one is solvable, and moreover for every ε > 0, there is a oneto-one correspondence between the sets of optimal solutions for the Pε - and ε -problems. The second problem is the so-called C-extension of the virtual P one. We denote it by CPε . The consideration of this problem is motivated by ε - and the following observations: First, the sets of admissible states for the P CPε -problems coincide, whereas their sets of admissible solutions are different. Second, the state equation of the CPε -problem has a linear structure. Finally, ε there is a one-to-one correspondence between optimal solutions for the P and CPε -problems. It is clear that in the linear case the C-extended problem always coincides with the original one. As a result, we propose the following scheme for asymptotic analysis of optimal control problems for the blowing-up parabolic equation: ε =⇒ CPε – passage to the limit – CPhom =⇒ P hom =⇒ Phom . Pε =⇒ P hom which, in We show that the CPhom -problem is a C-extension for P turn, can be interpreted as a virtual extension of the OCP Phom . However, the main advantage of such an approach is the fact that it preserves the well-known variational property: “If (u0ε , p0ε , yε0 ) ∈ Ξε is an optimal triplet of the Pε -problem and if (u0ε , p0ε , yε0 ) tends (in some sense) to (u0 , p0 , y 0 ), then (u0 , p0 , y 0 ) is an optimal solution for the limit problem.” Thus, in contrast to the well-known approach (see, for instance, Kesavan and Saint Jean Paulin [134, 135], Saint Jean Paulin and Zoubairi [222], and Muthukumar [197]), we do not just look for a limit of optimal solutions and for a limit of minimal values of the cost functionals for the problem (10.1)–(10.4). We propose primarily to look for a CPhom -problem as some variational limit of the C-extended ones and after that to pass from CPhom for the recovery of the Phom -problem. Of course, this limiting problem should be unique (as a result of some passage to the limit) and should have a clearly defined structure including the limit form of a state equation, control constraints, and a limit cost functional. For this we follow the main ideas of the theory of Γ -convergence and the concept of variational convergence of constrained minimization problems (see Chap. 8; see also Kogut [140], Kogut and Leugering [147, 145, 146, 148, 149, 150], and D’Apice, De Maio, and Kogut [86, 87, 85]). As we will see later, for the OCP (10.1)–(10.4) with Robin conditions on the boundary of holes, the structure of the corresponding limit problem depends essentially on how h tends to 0 as ε → 0 (the so-called “scaling effect”), namely we show that such effects hold only for the thin structures of critical thickness. In this case, some additional terms appear in the limiting state equation. At the same time, we do not have any artifacts, if the thin structure Ωε is sufficiently thick. However, as for the sufficiently thin domains Ωε , the limit problem for (10.1)–(10.4) does not exist. However, the asymptotic anal-

360


ε ysis of this object is possible only if, along the boundary Sint , homogeneous Neumann conditions are given instead of the Robin conditions. The circumstances we would like to concentrate on concern the so-called “stability of homogenization” with respect to the Dirichlet boundary control. The point is that the original problem Pε consists of two types of boundary controls uε and pε , respectively. However, the passage to the limit in Pε always leads us to an OCP with only one boundary control, namely the Dirichlet control. Moreover, this conclusion holds true in spite of a “volume” of the boundary ΓεD ⊂ ∂Ωε occupied by the Dirichlet control zone. We show that the optimal solution for this limit problem can be used for the construction of the so-called asymptotically suboptimal controls. Moreover, we prove an approximation property of such controls for small enough ε. We also show that an OCP with the Neumann boundary control can appear as a result of the homogenization of Pε only if ΓεD = ∅ for every ε > 0 and the domain Ωε is of critical thickness. This result can be explained by the fact that the Dirichlet boundary control is more stable with respect to the homogenization procedure than the Neumann one. In contrast to this case, we consider also the problem given above on fattened graphs which do not satisfy the definition of thin grids. It is clear that in this case, the F h may have rather complicated geometry with non-rectilinear form of their edges and with components of different thickness. We show that in this case, the “scaling effect” in the limit may appear not only for the reticulated structures of critical thickness but also for the sufficiently thin domains. Moreover, the homogenization on fattened graphs can be characterized by “non-classical” artifacts. To explain this, we give several concrete examples when the original object (10.1)–(10.4) in the limit leads us to atypical results from the control theory point of view. For instance, in the limit as ε → 0, we do not have an OCP but only some initial-boundary value problems with or without controls.

10.1 Statement of the problem Let Ωε be a thin structure for which description (9.4) and suppose all suppositions of Chap. 9 hold true. For a given T > 0, we consider the following controlled initial-boundary value problem in Ωε × (0, T ): ⎫ in (0, T ) × Ωε , ⎪ ∂t yε − div (A(x/ε)∇yε ) − yε3 = fε ⎪ ⎪ ∂νA yε = 0 on (0, T ) × Sext , ⎪ ⎪ ⎪ ⎬ ε , ∂νA yε = ε2 (−dyε + gε ) on (0, T ) × Sint (10.5) yε = uε on (0, T ) × ΓεD , ⎪ ⎪ ⎪ ∂νA yε = εpε on (0, T ) × ΓεN , ⎪ ⎪ ⎪ ⎭ for a.a. x ∈ Ωε , yε (0, x) = yε,0 ε )), fε ∈ L2 (0, T ; L2 (Ω)), where d is a positive constant, gε ∈ L2 (0, T ; L2 (Sint 1 and yε, 0 ∈ H (Ω) are given functions, ∂νA yε = (A(x/ε)∇yε ) · ν, ν is the

10.1 Statement of the problem

361

unit outward normal vector with respect to Ωε , and A(x) is a Y -periodic measurable matrix such that A(·) ∈ L∞ (Y, R2×2 ), α0 ξ2 ≤ (A(x)ξ, ξ) ≤ α0−1 ξ2 for a.a. x ∈ Y,

(10.6)

for some fixed constant α0 > 0 and every ξ ∈ R2 . Hereinafter we interpret the functions uε and pε as Dirichlet and Neumann boundary controls, respectively. We say that the control functions uε and pε are admissible if the following conditions hold: pε ∈ L2 (0, T ; L2 (ΓεN )), where the set Uε is defined as Uε = u|ΓεD : u ∈ L2 (0, T ; H 1 (Γ0 )),

uε ∈ Uε ,

uL2 (0,T ; H 1 (Γ0 )) ≤ C0 ,

(10.7)

(10.8)

Γ0 = {0 < x 1 < L 1 , x2 = 0}, and the constant C0 is independent of ε. The initial-boundary value problem (10.5) is a singular one in the following sense: Because of the term −yε3 , a global solution yε = yε (uε , pε ) of (10.5) may not exist for some admissible controls uε and pε (see Lions [172]). Taking into account the compactness of the embedding H 1 (ΓεD ) → 2 L (ΓεD ), we associate with (10.5) the following cost functional: Iε (uε , pε , yε ) =

1 h

1 T (yε − yε∗ )6 dx dt h 0 Ωε T u2 dx1 dt + p2ε dH1 dt (10.9)

(yε (T, x) − yεT )2 dx + Ωε

T

+ κ(ε) 0

ΓεD

0

ΓεN

defined over all uε ∈ L2 (0, T ; H 1 (ΓεD )), pε ∈ L2 (0, T ; L2 (ΓεN )), and yε ∈ L2 (0, T ; H 1 (Ωε )) given by (10.5), subject to the control constraints (10.7). Here, yε∗ ∈ L6 ((0, T ) × Ωε ) and yεT ∈ L2 (Ω) are given functions, κ(ε) is a given value, and H1 is the 1-dimensional Hausdorff measure on ΓεN . Thus, the OCP we consider in this chapter can be stated as follows: Find a triplet (u0ε , pε0 , yε0 ) ∈ L2 (0, T ; H 1 (ΓεD )) × L2 (0, T ; L2 (ΓεN )) × L2 (0, T ; H 1 (Ωε )) such that inf Iε (uε , pε , yε ), (10.10) Iε (u0ε , pε0 , yε0 ) = (uε ,pε ,yε )∈ Ξε

where the set of admissible triplets Ξε is defined as Ξε = (uε , pε , yε ) ∈ L2 (0, T ; H 1 (ΓεD )) ×L2 (0, T ; L2 (ΓεN )) × L2 (0, T ; H 1 (Ωε )) : (uε , pε , yε ) satisfies (in some weak sense) (10.5)–(10.8)} . In the sequel, the OCP (10.5)–(10.11) will be called the Pε -problem.

(10.11)

362


10.2 On the solvability of Pε and its C-extension We begin this section with the variational formulation of the initial-boundary value problem (10.5): Find a function yε = yε (uε , pε ) ∈ L2 (0, T ; H 1 (Ωε )) such that (see Ladyzhenskaya and Ural’tseva [162], and Michajlov [188]) yε = uε a.e. on ΓεD ,

yε (0, ·) = yε, 0 a.e. on Ωε

(10.12)

and the following identity is valid:

T 0

−yε ϕ ψ + A(x/ε)∇yε · ∇ϕ ψ − yε3 ϕ ψ dx dt

Ωε

T

+ d ε2

T

yε ϕ ψ dH1 dt = 0 T

ε Sint

+ ε2

fε ϕ ψ dx dt

0 T

Ωε

gε ϕ ψ dH1 dt + ε 0

ε Sint

pε ϕ ψ dH1 dt, 0

ΓεN

∀ ϕ ∈ C0∞ (R2 ; ΓεD ), ∀ ψ ∈ C0∞ (0, T ),

(10.13)

where by C0∞ (R2 ; ΓεD ) we denote the set of all functions from C0∞ (R2 ) such that ϕ|ΓεD = 0. In the sequel, the function yε is called a weak solution to the problem (10.5). Since the domain Ωε is nonconvex with nonsmooth boundary ∂Ωε , a solution of initial-boundary value problem (10.5) can have only minimal smoothness for ε small enough. It is the main reason for the choice of L2 (0, T ; H 1 (Ωε )) as the main state space for the original problem. Moreover, for given functions ε f ∈ L2 ((0, T ) × Ω), gε ∈ L2 (0, T ; L2 (Sint )), u ∈ L2 (0, T ; H 1 (ΓεD )),

pε ∈ L2 (0, T ; L2 (ΓεN )), the problem (10.5), in general, may not have a global solution at the time, that is, there does not exist a function yε ∈ L2 (0, T ; H 1 (Ωε )) satisfying (10.12)– (10.13). Nevertheless, we suppose that for every ε > 0, there exists at least one triplet

(u∗ε , p∗ε , yε∗ ) ∈ Uε × L2 (0, T ; L2 (ΓεN )) × L6 (0, T ; L6 (Ωε )) ∩ L2 (0, T ; H 1 (Ωε )) such that u∗ε , p∗ε , and yε∗ satisfy (10.5)–(10.8), that is, Ξε = ∅

for every ε > 0.

(10.14)

We make use now of the following result (cf. Dubinskiy [97] and Lions [168, Theorem 5.1, Chap. 1]. Lemma 10.1. Let B0 ⊂ B1 ⊂ B2 be Banach spaces such that B0 and B1 are reflexive, the injection B0 → B1 is compact, and B1 → B2 is continuous. Let

10.2 On the solvability of Pε and its C-extension

363

W = {ϕ : ϕ ∈ Lp 0 (0, T ; B0 ), ∂ϕ/∂t ∈ Lp 2 (0, T ; B2 )} , where 1 < p 0 , p 1 , p 2 < +∞. Then W is a Banach space with respect to the graph norm and the injection W → Lp 1 (0, T ; B 1 ) is compact. Having put B 0 = H 1 (Ωε ), B 1 = L6 (Ωε ), B 2 = L2 (Ωε ), p 0 = 2, p 1 = 6, and p 2 = 2, the following statement is the direct consequence of the Sobolev embedding theorem and Lemma 10.1. Lemma 10.2. The injection

Yε = H 1 (0, T ; L2 (Ωε )) ∩ L2 (0, T ; H 1 (Ωε )) → L6 (0, T ; L6 (Ωε )) is compact. We are now in a position to prove the following result. Theorem 10.3. For every value ε > 0, there exists a solution (u0ε , p0ε , yε0 ) ∈ Ξε for the OCP Pε . Proof. We have assumed (in order to make the problem Pε meaningful) that supposition holds true. Then inf Iε (uε , pε , yε ) ≤ Iε (u∗ε , p∗ε , yε∗ ). Hence,

k k (10.14) k ∞ if (uε , pε , yε ) k=1 is a minimizing sequence, we have Iε (ukε , pεk , yεk ) ≤ const,

ukε L2 (0,T ;H 1 (ΓεD )) ≤ C0

for all k ∈ N.

Then, as follows from (10.9), we conclude that yεk L6 ((0,T )×Ωε ) ≤ C,

pkε L2 ((0,T )×ΓεN ) ≤ C.

(10.15)

Consequently, 3 ∂t yεk − div A(x/ε)∇yεk = yεk + fε ∈ L2 (0, T ; L2 (Ω)),

∀ k ∈ N.

Therefore, using standard techniques, the following a priori estimate can be easily obtained from the integral identity (10.13) (see Lions [172]): yεk L2 (0,T ;H 1 (Ωε )) + yεk H 1 (0,T ;L2 (Ωε )) ≤ C(ε) fε L2 ((0,T )×Ω) + ukε L2 (0,T ;H 1 (ΓεD )) + pεk L2 ((0,T )×ΓεN ) + yεk L6 ((0,T )×Ωε )

(10.16) + yε0 L2 (Ωε ) . k

In view of (10.15), we conclude that yε is bounded in Yε , and for every k ∈ N, the triplets (ukε , pεk , yεk ) are admissible for the Pε -problem.

364


Hence, we can extract a subsequences (still indexed by k) such that pεk p0ε in L2 ((0, T ) × ΓεN ), yεk yε0 in L2 (0, T ; H 1 (Ωε )),

ukε u0ε in L2 (0, T ; H 1 (ΓεD )), yεk → yε0 in L6 ((0, T ) × Ωε )

(here we have used Lemma 10.2). Taking these results into account and passing to the limit in (10.13) as k → ∞, we deduce that (u0ε , p0ε , yε0 ) is admissible for the Pε -problem. Finally, using the lower semicontinuity of the cost functional Iε with respect to the weak topology of L2 (0, T ; H 1 (ΓεD )) × L2 ((0, T ) × ΓεN ) × L2 (0, T ; H 1 (Ωε )), we obtain Iε (u0ε , p0ε , yε0 ) ≤ lim inf Iε (ukε , pεk , yεk ) = k→∞


which proves that (u0ε , p0ε , yε0 ) is an optimal triplet.


Now, we formulate a couple of new OCPs for parabolic equations on the ε of the original network Ωε . The first of them, the so-called virtual extension P Pε -problem, is formulated as follows: Find a quadruple ε0 , pε0 , yε0 ) ∈ Xε ≡ L2 ((0, T ) × Ωε ) × L2 (0, T ; H 1 (ΓεD )) ( qε0 , u × L2 ((0, T ) × ΓεN ) × L2 (0, T ; H 1 (Ωε )) such that ε : P

Iε ( qε0 , u ε0 , pε0 , yε0 ) =

inf

bε (qε ,uε ,pε ,yε )∈ Ξ

Iε (qε , uε , pε , yε ),

(10.17)

ε have the where the cost functional Iε and the set of admissible solutions Ξ following analytical representations: Iε (qε , uε , pε , yε ) =

+

+

1 h(ε) 1 h(ε) γ(ε) h(ε)

(yε (T, x) −

yεT )2

u2 dx1 dt 0

T

0

(yε − yε∗ )6 dx dt +

T

ΓεD

ΓεN

p2ε dH1 dt

(qε − fε )2 dx dt,

0

0

Ωε T

dx + κ(ε)

Ωε

T

Ωε

(10.18)


365

ε = {(qε , uε , pε , yε ) : Ξ ⎫ yε = uε on Γ D , uε ∈ Uε , yε (0, ·) = yε,0 on Ωε , ⎪ ε ⎪ ⎪ ⎪ ⎪ ⎪ 2 N 2 ⎪ pε ∈ L ((0, T ) × Γε ), qε ∈ L ((0, T ) × Ωε ), ⎪ ⎪ ⎪ ⎪ ⎪ T ⎪ ⎪ ⎪ ⎪ (−yε ϕ ψ + A(x/ε)∇yε · ∇ϕ ψ) dx dt ⎪ ⎪ ⎪ 0 Ω ε ⎪ ⎪ ⎪ T T ⎬ 3 2 1 yε ϕ ψ dx dt + d ε yε ϕ ψ dH dt . (10.19) − 0 0 Ωε Sε ⎪ ⎪ T int T ⎪ ⎪ ⎪ = 2 1 ⎪ qε ϕ ψ dxdt + ε gε ϕ ψ dH dt ⎪ ⎪ ⎪ ε ⎪ Ωε Sint 0 0 ⎪ ⎪ ⎪ T ⎪ ⎪ 1 ⎪ ⎪ pε ϕ ψ dH dt, +ε ⎪ ⎪ ⎪ 0 ΓεN ⎪ ⎪ ∞ 2 D ∞ ∀ ϕ ∈ C0 (R ; Γε ), ∀ ψ ∈ C0 (0, T ) ⎭ Here, γ(ε) > 0 is a penalizing coefficient. It is clear that the corresponding initial-boundary value problem for (10.19) can be formally written in the form ⎫ in (0, T ) × Ωε , ⎪ ∂t yε − div (A(x/ε)∇yε ) − yε3 = qε ⎪ ⎪ ⎪ ∂νA yε = 0 on (0, T ) × Sext , ⎪ ⎪ ⎪ ∂ y = ε2 (−dy + g ) on (0, T ) × S ε , ⎬ νA ε

ε

ε

int

on (0, T ) × ΓεD , on (0, T ) × ΓεN , for a.a. x ∈ Ωε ,

yε = uε ∂νA yε = εpε yε (0, x) = yε,0

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(10.20) where qε ∈ L2 (0, T ; L2 (Ωε )) denotes a virtual control. ε problem, Additionally, the second OCP, the so-called C-extension of the P takes the following form: Find a quadruple (a0ε , vε0 , wε0 , zε0 ) ∈ Xε such that (CPε ) :

Jε (a0ε , vε0 , wε0 , zε0 ) =

inf (aε ,vε ,wε ,zε )∈ Σε

Jε (aε , vε , wε , zε ),

(10.21)

where the cost functional Jε and the set of admissible solutions Σε have the following analytical representations: 1 Jε (aε , vε , wε , zε ) = h(ε) + +

1 h(ε) γ(ε) h(ε)

(zε (T, x) −

Ωε T 0 T

yεT )2

T 0

(aε − zε3 − fε )2 dx dt, 0

Ωε

dx + κ(ε)

(zε − yε∗ )6 dx dt + Ωε

T

0

ΓεN

ΓεD

vε2 dx1 dt

wε2 dH1 dt (10.22)

366


Σ ε = {(aε , vε , wε , zε ) : ⎫ zε = vε on ΓεD , vε ∈ Uε , zε (0, ·) = yε, 0 on Ωε , ⎪ ⎪ ⎪ ⎪ ⎪ wε ∈ L2 ((0, T ) × ΓεN ), aε ∈ L2 ((0, T ) × Ωε ), ⎪ ⎪ ⎪ ⎪ ⎪ T ⎪ ⎪ (−zε ϕ ψ + A(x/ε)∇zε · ∇ϕ ψ) dx dt ⎪ ⎪ ⎪ ⎪ 0 Ω ⎪ ε ⎪ T ⎪ ⎬ 2 1 + d ε z ϕ ψ dH dt ε . (10.23) ε 0 Sint ⎪ ⎪ T T ⎪ ⎪ ⎪ = aε ϕ ψ dx dt + ε2 gε ϕ ψ dH1 dt ⎪ ⎪ ⎪ ε ⎪ 0 0 Ωε Sint ⎪ ⎪ T ⎪ ⎪ ⎪ 1 ⎪ w ϕ ψ dH dt, +ε ⎪ ε ⎪ ⎪ 0 ΓεN ⎪ ⎭ ∞ 2 D ∞ ∀ ϕ ∈ C0 (R ; Γε ), ∀ ψ ∈ C0 (0, T ) In this case, the corresponding initial-boundary value problem has the form ⎫ in (0, T ) × Ωε , ⎪ ∂t zε − div (A(x/ε)∇zε ) = aε ⎪ ⎪ ∂νA zε = 0 on (0, T ) × Sext , ⎪ ⎪ ⎪ ⎬ 2 ε on (0, T ) × Sint , ∂νA zε = ε (−dzε + gε ) (10.24) D zε = vε on (0, T ) × Γε , ⎪ ⎪ N ⎪ ∂νA zε = εwε on (0, T ) × Γε , ⎪ ⎪ ⎪ ⎭ for a.a. x ∈ Ωε . zε (0, x) = yε, 0 ε with a virtual control qε ∈ Remark 10.4. The consideration of the problem P L2 ((0, T ) × Ωε ) can be motivated as follows. First, in contrast to the original ε are always nonempty for every problem Pε , the sets of admissible solutions Ξ ε > 0. Hence, without any additional suppositions, it can be proved (by ε is always solvable. Moreover, analogy with Theorem 10.3) that the problem P due to the structure of the cost functional Iε , one can take the penalizing ε -problem coefficient γ(ε) > 0 in (10.18) such that any optimal solution of the P takes the form (fε , u0ε , p0ε , yε0 ), where (u0ε , p0ε , yε0 ) is an optimal triplet for the original problem Pε . As we will see later, this is the main reason to consider ε -problem instead of the original one. the P εY (ΣεY , resp.) the projection of the set of admissible Let us denote by Ξ ε (ΣεY , resp.) into the state space Y = L2 (0, T ; H 1 (Ω)) and call it solutions Ξ the set of admissible states for the problem Pε (CPε , resp.). Definition 10.5. We say that an optimal control problem is the C-extension ε , if their sets of admissible states coincide. of P Note that the same property for the control objects was introduced by Serova˘iski˘i in [227] and this notion was named an equivalence of such objects. However, we accept the notion of “C-extension” or “extension by control” ε . These since not every admissible control for the CPε -problem is that for P


367

problems may have different sets of admissible solutions, in general. For different realizations of this idea, we refer to Zgurovsky and Mel’nik in [253]. Our next assertion justifies the role of the problem CPε introduced above. ε . Lemma 10.6. CPε is a C-extension of P Proof. Let (qε , uε , pε , yε ) be any admissible solution to the problem (10.17)– (10.19). Let us define vε = uε , wε = pε , and aε = qε + yε3 . By Lemma 10.2, we have aε ∈ L2 ((0, T ) × Ωε ). Hence, ae , vε , and wε are admissible controls for the CPε -problem. Let zε be the corresponding solution of the problem (10.24). It is well known that for given aε , vε , wε , and ε, this problem admits a unique solution in L2 (0, T ; H 1 (Ωε )) (see Michajlov [188]). From this, we immediately conclude that zε = yε and hence (aε , vε , wε , zε ) is an admissible quadruple to Y ⊆ Σ Y holds. the CPε -problem. Thereby we can deduce that the inclusion Ξ ε ε To prove the reverse inclusion, we take an arbitrary admissible solution (aε , vε , wε , zε ) to CPε and define the corresponding quadruple (qε , uε , pε , yε ) as follows: qε = aε − zε3 , uε = vε , pε = wε , and yε = zε . Then it is a matter of direct verification to show that (qε , uε , pε , yε ) is an admissible solution for ε . Hence, Σ Y ⊆ Ξ Y , and we are done. P ε ε Note now that the C-extended problem (10.21)–(10.23) is always solvable; however, its optimal solution is, generally speaking, nonunique (see Fursikov [111] and Lions [169]). Nevertheless, as an obvious consequence of the previous lemma, we have the following conclusion. Theorem 10.7. (a0ε , vε0 , wε0 , zε0 ) ∈ Σε is an optimal solution to the CPε ε . problem if and only if the quadruple (a0ε − (zε0 )3 , vε0 , wε0 , zε0 ) is optimal for P In view of Remark 10.4, we give the following characteristic of the Cextended problems CPε . Lemma 10.8. Assume that the original problem (10.5)–(10.11) is uniformly regular, that is, (H1) Ξε = ∅ for every ε > 0; (H2) There is a sequence of admissible solutions {(u∗ε , p∗ε , yε∗ ) ∈ Ξε }ε>0 such that T lim sup u∗ε 2H 1 (ΓεD ) + p∗ε 2L2 (ΓεN ) + yε∗ 6L6 (Ωε ) dt < +∞. (10.25) ε→0

0

Then there is a constant γ ∗ > 0 such that, having assumed γ = γ ∗ in (10.18) and (10.22), one has for every ε > 0 that (u0ε , p0ε , yε0 ) ∈ Ξε is an optimal solution to the original problem (10.5)–(10.11) if and only if the quadruple (fε + (yε0 )3 , u0ε , p0ε , yε0 ) is optimal for CPε . Proof. First, we note that due to hypothesis (H1), the

original OCP (10.5)– (10.11) is solvable for every ε > 0. Let (u0ε , pε0 , yε0 ) ∈ Ξε ε>0 be a sequence of optimal solutions to {Pε }ε>0 . Then for every ε, the quadruple (fε , u0ε , pε0 , yε0 ) is

368


ε . Hence, in view of (10.25), admissible for the corresponding virtual problem P there exists a constant D > 0 such that inf


Iε (qε , uε , pε , yε ) ≡ Iε ( qε0 , u ε0 , pε0 , yε0 ) ≤ Iε (fε , u0ε , pε0 , yε0 ) = Iε (u0ε , pε0 , yε0 ) Iε (uε , pε , yε ) ≤ Iε (u∗ε , p∗ε , yε∗ ) = inf (uε ,pε ,yε )∈ Ξε

≤ D < +∞.

(10.26)

ε → R is strictly monotone increasing with Since each of the functionals Iε : Ξ respect to the parameter γ(ε), it follows that, passing to the limit in (10.26) as γ(ε) → ∞, we have Iε (u0ε , pε0 , yε0 ) ≤ lim inf Iε ( qε0 , u ε0 , pε0 , yε0 ) ≤ Iε (u0ε , pε0 , yε0 ) ≤ D, γ(ε)→∞

∀ ε > 0.

Hence, there exists a constant γ ∗ such that Iε (u0ε , pε0 , yε0 ) =

inf


Iε (qε , uε , pε , yε ) ∀ γ(ε) ≥ γ ∗ , ∀ ε > 0. (10.27)

Then from (10.18) we immediately conclude that equalities (10.27) hold true if only qε0 = fε for every ε > 0. ε has Thus, from now, we can always suppose that any optimal solution to P ε , where (u0ε , pε0 , yε0 ) is the solution to the original the form (fε , u0ε , pε0 , yε0 ) ∈ Ξ problem Pε . To get the required conclusion, we have to apply Theorem 10.7. The proof is complete. As was mentioned earlier, our main intention is to study the asymptotic behavior of the OCP Pε as ε tends to 0 and to identify the limit (or homogenized) problem that should exhibit the following variational property: “If (u0ε , pε0 , yε0 ) ∈ Ξε is an optimal triplet of the Pε -problem, and if (u0ε , pε0 , yε0 ) tends (in some sense) to (u0 , p0 , y 0 ), then (u0 , p 0 , y 0 ) is an optimal solution for the limit problem.” Therefore, taking the above results into account, we may say that the following diagram is quite reasonable for the limit analysis of the singular OCP in our case

ε ) (Pε ) =⇒ (P ⏐ ⏐

hom ) (Phom ) ⇐= (P ⏐ ⏐

(CPε ) ⏐ ⏐

(CPhom ) ⏐ ⏐

inf (aε ,vε ,wε ,zε )∈ Σε

ε→0 Jε (aε , vε , wε , zε ) −−−−→

inf

(a,v,w,z)∈ Σ 0

J 0 (a, v, w, z) .

(10.28) Moreover, in the next sections we prove that this scheme preserves the abovementioned variational property.

10.3 On the description of CPε in terms of singular measures

369

10.3 On the description of CPε in terms of singular measures In this section, we deal with the construction of measure-theoretical tools for the description of the class of admissible solutions to the C-extended problem CPε in terms of singular periodic Borel measures on R2 . It helps us to give a sharp formulation of the limit problem for the considered heat processes on the reticulated structure Ωε . By analogy with Sect. 9.1, we introduce three periodic finite positive Borel measures ν h , μh , and λh in R2 , concentrated and uniformly distributed on the sets ΛhN , ΛhD , and ΛhR , respectively, with the correspondent toruses of periodicity Y1 = [0, 1)2 ,

Y2 = [−1/2, 1/2) × [0, 1),

and

Y3 = [0, 1) × [−1/2, 1/2)

(see Fig. 9.3 and relations (9.6)–(9.8)). Recall that each of the sets ΛhN , ΛhD , and ΛhR indicates a zone where the corresponding boundary conditions for the CPε -problem are located. We also introduce the so-called “scaling” measures νεh , μhε , and λhε by rules (9.9). Clearly, λhε is an ε-periodic measure on R2 , whereas the measures νεh and μhε are ε-periodic along the x1 -axis. As usual, we relate the parameters h and ε assuming that h = h(ε) → 0 as ε → 0. As a result, we have (for the details, we refer to Sect. 9.1) the following. Proposition 10.9. ε→0

dλh(ε) dx, ε

ε→0

dνεh(ε) δ{x2 =0} dx,

ε→0

dμh(ε) δ{x2 =0} dx, ε

where by δ{x2 =0} dx we denote the product of the linear Lebesgue measure dx1 and the Dirac measure δ{x2 =0} dx2 . We are now in a position to give the equivalent formulation of the Cextended OCP (10.21)–(10.23). To begin, we make use of the sets ΘεR and ΘεN defined by (9.11) and (9.12) and consider the second term in the left part of the integral identity (10.23). One has

T

dε2

zε ϕψ dH1 dt 0

ε Sint

⎛

T

= dε2 0

ψ⎝

⎞

k∈ΘεR

zε ϕ dH1 ⎠ dt

ε(Λh R +k)

⎛

T

= dε2 [4ε (1 − h(ε))] 0

ψ⎝

⎞

k∈ΘεR

zε ϕ dλh (x/ε)⎠ dt ε(Y3 +k)

370


⎛

T

= 4εd (1 − h(ε))

ψ⎝

0

k∈ΘεR

⎛ T

= 4εd (1 − h(ε))

ψ⎝

0

T

zε ϕ ε2 dλh (x/ε)⎠ dt

ε(Y3 +k)

⎞

k∈ΘεR

⎞

zε ϕ dλhε ⎠ dt

ε(Y3 +k)

zε ϕψ dλhε dt.

= 4εd (1 − h(ε))

(10.29)

Ω

0

h(ε)

Here, zε is a function of L2 (0, T ; L2 (Ω, dλε )) taking the same values as 2 2 ε ε the function zε ∈ L (0, T ; L (Sint )) on Sint . We note that the integral h(ε)

zε ϕ dλh(ε) is well defined. Indeed, since the set Ω is bounded and zε dλε ε is a Radon measure, it follows that zε ϕ dλh(ε) is a linear continuous funcε Ω

Ω

tional on C0∞ (R2 ; ΓεD ). Thus, T 2 1 zε ϕψ dH dt = 4εd (1 − h(ε)) dε 0

ε Sint

T

0

zε ϕψ dλhε dt.

(10.30)

Ω h(ε)

Remark 10.10. It is easy to see that the scaling measure λε belongs to the h class M+ 0 (Ω). In view of our initial supposition, cap(Sint ) = 0, so λ (B) = 0 for every Borel set B ⊆ Ω with cap(B, Ω) = 0. Hence, the first property h(ε) is valid. As for the second propof the cone M+ 0 (Ω) for the measure λε erty, the measure λh is concentrated on Sint and proportional there to the linear Lebesgue measure. Thus, λh ∈ M+ 0 (Ω). Since the scaling measure h(ε) h(ε) h λε inherits these properties from λ , it follows that λε ∈ M+ 0 (Ω) for h(ε) every ε > 0. Thus, every function y in H 1 (Ω) is defined λε -everywhere and h(ε) is λε -measurable on Ω (see Dal Maso and Murat [82]). Hence, the space h(ε) 1 H0 (Ω) ∩ L2 (Ω, dλε ) is well defined. Taking these facts into account and proceeding by analogy with (10.29), we may obtain the following relation: T wε ϕψ dH1 dt ε 0

ΓεN

⎛

T

=ε 0

ψ⎝

⎞

wε ϕ dH1 ⎠ dt

ε(Λh N +k)

k∈ΘεN

⎛

T

= ε [2 ε (1 − h(ε))] 0

T

ψ⎝

k∈ΘεN

0

Ω

w ε ϕ dν h (x/ε)⎠ dt

ε(Y1 +k)

w ε ϕψ dνεh dt.

= 2 ε (1 − h(ε))

⎞

(10.31)


371

Hence, applying this approach for the rest of the terms in (10.21)–(10.23), we have T T κ(ε) v 2 dx1 dt = κ(ε)h(ε) v2 dμhε dt, (10.32)

T

ΓεN

0

T

ε2 ε Sint

0

ΓεD

0

0

Ω

T

wε2 dH1 dt = 2 (1 − h(ε))

w ε2 dνεh dt, 0

(10.33)

Ω

T

gε ϕ ψ dH1 dt = 4 ε (1 − h(ε))

gε ϕψ dλhε dt. 0

(10.34)

Ω

Let z˘ε ∈ L2 (0, T ; H 1 (Ω)) be some extension of the weak solution zε to the problem (10.24) onto the whole domain Ω and let χhε = χΩε be the characteristic function of the network Ωε (it is clear that this function is twoparametric, in general). As a result, summing up representations (10.30)– (10.34), the initial-boundary value problem (10.24) can be rewritten in the following variational form: z˘ε = vε

T

χhε z˘ε ϕ ψ dx dt +

− 0

χhε [˘ zε (0, ·) − yε, 0 ] = 0 a.e. on Ω;

μhε -a.e. on Ω,

Ω

T

χhε (A(x/ε)∇˘ zε · ∇ϕ)ψ dx dt

0

T

Ω

+ 4d ε (1 − h(ε))

T

(10.35)

z˘ε ϕψ dλhε dt 0

Ω

χhε aε ϕ ψ dx dt

= 0

Ω

T

+ 2 ε (1 − h(ε)) 0

2 gε ϕψ dλhε

w ε ϕψ dνεh

dt +

Ω

dt

(10.36)

Ω

for any ϕ ∈ C0∞ (R2 ; Γ0 ) and ψ ∈ C0∞ (0, T ). Let ηεh be the uniformly distributed normalized measure concentrated on the thin grid Fεh = εF h , which is defined in (9.16) with properties (9.17)– (9.19). It is clear that this measure is absolutely continuous with respect to L2 (in symbols, ηεh (ε) L2 ), that is, L2 (B) = 0 implies ηεh (ε)(B) = 0 for all B ⊂ R2 . By analogy with (10.29) and (10.34), it is easy to show that the first terms in the both parts of the integral equality (10.36) can be rewritten as

T

T

0

χhε z˘ε ϕ ψ dx dt = h(ε)(2 − h(ε))

Ω

T

T

0

Ω

χhε aε ϕ ψ dx dt = h(ε)(2 − h(ε)) 0

Ω

zε ϕ ψ dηεh dt, aε ϕ ψ dηεh dt,

0

Ω

372


where, as usual, zε ∈ L2 (0, T ; L2 (Ω, dηεh )) denotes a function taking the same values as zε on Ωε . To define the Sobolev space H 1 (Ω, dηεh ), we say that a function zˆ belongs to H 1 (Ω, dηεh ) if there exist a vector d ∈ (L2 (Ω, dηεh ))2 and a sequence {zm ∈ C ∞ (Ω)}m∈N such that 2 h ∇zm − d2 dηεh = 0. (zm − zˆ) dηε = 0, lim lim m→∞

m→∞

Ω

Ω

In this case, one says that d is a gradient of zˆ and denotes it as ∇ˆ z (i.e., d = ∇ˆ z ). In view of the ergodicity property of the measure dηεh , this gradient is unique for any zˆ ∈ H 1 (Ω, dηεh ) (see Sect. 2.7). Moreover, we suppose that for any function zˆ ∈ H 1 (Ω, dηεh ), one can point out its prototype z in H 1 (Ω) such that z = zˆ ηεh -a.e. in Ω. Thus, taking the above-mentioned facts into account, we may reformulate aε0 , vε0 , w ε0 , zε0 ) such the C-extended OCP CPε as follows: Find a quadruple ( that ( aε0 , vε0 ,w ε0 , zε0 ) ∈ Zε

≡ L2 (0, T ; L2 (Ω, dηεh )) × L2 (0, T ; H 1 (Γ0 ) ∩ L2 (Ω, dμhε ) )

× L2 (0, T ; L2 (Ω, dνεh ))

× L2 (0, T ; H 1 (Ω, dηεh )) ∩ L2 (0, T ; L2 (Ω, dλhε )) , Jε ( aε0 , vε0 , w ε0 , zε0 ) =

inf

bε (b aε ,b vε ,w bε ,b zε )∈ Σ

aε , vε , w ε , zε ), Jε (

(10.37) (10.38)

ε have the where the cost functional Jε and the set of admissible solutions Σ following analytical representations: aε , vε , w ε , zε ) = (2 − h(ε)) ( zε (T, x) − yεT )2 dηεh Jε ( Ω

T

( zε − yε∗ )6 dηεh dt

+ (2 − h(ε))

0

Ω

T

vε2 dμhε dt

+ κ(ε)h(ε) 0

Ω T

w ε2 dνεh dt

+ 2 (1 − h(ε)) 0

+ γ ∗ (2 − h(ε))

Ω

T

( aε − zε3 − fε )2 dηεh dt, 0

Ω

(10.39)


373

ε = {( aε , vε , w ε , zε ) : Σ ⎫ zε = vε μhε -a.e. on (0, T ) × Ω, ⎪ ⎪ ⎪ ⎪ ⎪ h ⎪ z (0, ·) = y η -a.e. on Ω; ε ε, 0 ⎪ ε ⎪ ⎪ ⎪ ⎪ vε L2 (0,T ;H 1 (Γ0 )) ≤ C0 , ⎪ ⎪ ⎪ ⎪ ⎪ T T ⎪ ⎪ h h zε ϕ ψ dηε dt + (A(x/ε)∇ zε · ∇ϕ)ψ dηε dt ⎪ ⎬ − 0 0 Ω Ω . (10.40) T T ⎪ ⎪ ⎪ +4d β(ε) h h ⎪ zε ϕψ dλε dt = aε ϕ ψ dηε dt ⎪ ⎪ ⎪ Ω Ω 0 0 ⎪ ⎪ ⎪ T T ⎪ ⎪ +4 β(ε) h h ⎪ gε ϕψ dλε dt+2 β(ε) w ε ϕψ dνε dt, ⎪ ⎪ ⎪ ⎪ 0 0 Ω Ω ⎪ ⎪ ⎭ ∞ 2 D ∞ ∀ ϕ ∈ C0 (R ; Γε ), ∀ ψ ∈ C0 (0, T ) Here

1 − h(ε) ε β(ε) = · . 2 − h(ε) h(ε)

! ε -problem. In what follows, the OCP (10.38)–(10.40) will be called the CP 0 ε0 , zε0 ) It is quite obvious that this problem has a unique solution ( aε , vε0 , w for every ε > 0 (see Fursikov [111] and Lions [169]). In view of Lemma 10.6 and Remark 10.4, this solution can be interpreted as a prototype of an optimal triplet to Pε . Moreover, in this case, an a priori norm estimate (10.16) takes a form (see Kalantarov and Ladizhenskaja [127]) zε /∂tL2 (0,T ;L2 (Ω,dηεh )) zε L2 (0,T ;H 1 (Ω,dηεh )) + ∂ + zε L2 (0,T ;L2 (Ω,dλhε )) + zε L6 (0,T ;L6 (Ω,dηεh )) ≤C aε L2 (0,T ;L2 (Ω,dηεh )) + w ε L2 (0,T ;L2 (Ω,dνεh )) + yε, 0 H 1 (Ω)

+ gε L2 (0,T ;L2 (Ω,dλhε )) + vε L2 (0,T ;H 1 (Γ0 )∩L2 (Ω,dμhε )) , (10.41) is independent of ε > 0. where the constant C In fact, the Pε -problem with a virtual distributed control qε (see (10.17)– (10.19)) can be rewritten in the same manner as (10.38)–(10.40). To investigate the asymptotic behavior of the C-extended OCP, we apply the scheme of the direct variational convergence (see Sect. 8.4). With this ! ε -problem for various values of ε in the form of the aim, we represent the CP sequence of corresponding constrained minimization problems $ % "# aε , vε , w ε , zε ) ; ε > 0 , (10.42) Jε ( inf bε (b aε ,b vε ,w bε ,b zε ) ∈ Σ

ε where the cost functional Jε : Σε → R and the sets of admissible solutions Σ are defined in (10.39) and (10.40), respectively. Then the definition of an appropriate homogenized OCP to the family (10.38) can be reduced to the anal-

374


ysis of the limit properties of the sequences (10.42) as ε → 0. In order to obtain this limit in the form of some constrained minimization problem, we reduce the procedure of the “passage to the limit” in (10.42) to the consecutive identification of the set of admissible solutions for the limit minimization problem and its cost functional. We prove that the definition of the “pass to the limit” given below will guarantee the following property: “If ( aε0 , vε0 , w ε0 , zε0 ) is an 0 0 0 0 0 ! aε , vε , w ε , zε ) tends to ( a , v 0 , w 0 , z 0 ) optimal quadruple of CPε and if ( 0 0 0 0 (in some sense), then ( a , v , w , z ) is an optimal solution for the limit problem.” ε , zε ) belongs Since for every fixed ε, the admissible quadruple ( aε , vε , w to the variable functional space Zε , one has to define a natural topology for the variational convergence that preserves the above-mentioned variational property and such that the sequence of optimal solutions is relatively compact. These questions are discussed in the next section.

10.4 Convergence in the variable space Z ε We begin this section with the following concept (for the details, we refer to Sect. 6.6).

Definition 10.11. Let zεh ε>0 be a bounded sequence in the variable space Mε = L2 (0, T ; H 1 (Ω, dηεh )) ∩ L2 (0, T ; L2 (Ω, dλhε )), that is,

T

lim sup ε→0

0

zεh(ε) 2H 1 (Ω,dηh(ε) ) + zεh(ε) 2L2 (Ω,dλh(ε) ) dt < +∞. ε

ε

We say that this sequence is weakly convergent in the variable space Mε if there is a function z ∈ L2 (0, T ; H 1 (Ω)) such that ⎧ h zε → z in L2 (0, T ; L2 (Ω, dηεh )), ⎪ ⎪ ⎪ ⎨ z h z in L2 (0, T ; L2 (Ω, dλh )), ε ε T ⎪ ⎪ ⎪ ⎩ |zεh − z|2 dηεh dt → 0 as ε → 0. 0

(10.43)

Ω

Further, we note that similar to Sect. 9.3, the following results concerning some specific properties of the convergence in the time-dependent variable spaces L2 (0, T ; L2 (Ω, dλhε )), L2 (0, T ; L2 (Ω, dηεh )), and Mε can be established (for the details, we refer to the proofs of Lemmas 9.7, 9.8, 9.10, Theorem 9.14, and Corollaries 9.9 and 9.15). Lemma 10.12. If z ∈ L2 (0, T ; H 1 (Ω)), then


T

ε→0

0 T

Ω

0

0

T 0

Ω

z 2 dx dt,

(10.44)

zϕψ dx dt,

(10.45)

Ω

zϕψ dηεh dt =

lim

ε→0

T

z 2 dηεh dt =

lim

375

Ω

∀ ϕ ∈ C0∞ (Ω) and ∀ψ ∈ C0∞ (0, T ). Moreover, if T sup z 2 dλhε dt < +∞, ε>0

Ω

0

then (10.46) z z in L2 (0, T ; L2 (Ω, dλhε )).

Lemma 10.13. Let zε ∈ L2 (0, T ; H 1 (Ω)) and z ∈ L2 (0, T ; H 1 (Ω)) be such that zε z in L2 (0, T ; H 1 (Ω)), zε → z in L2 ((0, T ) × Ω), and T sup z 2 dλhε dt < +∞. ε>0

0

Ω

zε z

in

L2 (0, T ; L2 (Ω, dλhε )),

zε → z

in

L2 (0, T ; L2 (Ω, dηεh )).

Then

% (10.47)

Corollary 10.14. If zε → z in L2 (0, T ; L2 (Ω, dηεh )) and z ∈ L2 (0, T ; H 1 (Ω)), then T |zε − z|2 dηεh dt = 0. (10.48) lim ε→0

0

Ω

Lemma 10.15. Let zε ∈ L2 (0, T ; L2 (Ω, dλhε )) and ϕε ∈ C0∞ (R2 ) be sequences such that zε z in L2 (0, T ; L2 (Ω, dλhε )) and ϕε → ϕ in C(Ω). Then T T h lim zε ϕε ψ dλε dt = zϕψ dx dt ∀ ψ ∈ C0∞ (0, T ). (10.49) ε→0

0

0

Ω

Ω

Theorem 10.16. Let zεh ε>0 be a bounded sequence in L2 (0, T ; H 1 (Ω, dηεh )), that is, T sup zεh (t, ·)2L2 (Ω,dηεh ) + ∇zεh (t, ·)2[L2 (Ω,dηh )]2 dt < ∞. ε→0

ε

0

h

Then there are a subsequence zε ε>0 (still indexed by ε) and elements

2 r(t, x, y) ∈ L2 (0, T ; L2 (Ω, Y ) ) and z(t, x) ∈ L2 (0, T ; H 1 (Ω)) such that 2

zεh z(t, x) 2 ∇zεh

r(t, x, y)

L2 (0, T ; L2 (Ω, dηεh )),

2 in L2 (0, T ; L2 (Ω, dηεh ) ),

in

r(t, x, y) − ∇z(t, x) ∈ L (0, T ; L (Ω, Vpot )). 2

2

(10.50) (10.51) (10.52)

376


Corollary 10.17. Let zεh ∈ L2 (0, T ; H 1 (Ω, dηεh )) ε>0 be a bounded sequence and let z ∈ L2 (0, T ; H 1 (Ω)) be its weak limit in L2 (0, T ; L2 (Ω, dηεh )). Then T |zεh −z|2 dηεh dt = 0. (10.53) zεh → z in L2 (0, T ; L2 (Ω, dηεh )), lim ε→0

0

Ω

Theorem 10.18. Every bounded sequence

h zε ∈ L2 (0, T ; H 1 (Ω, dηεh )) ∩ L2 (0, T ; L2 (Ω, dλhε )) ε>0 is relatively compact with respect to the weak convergence in the sense of Definition 10.11. The following result is crucial for the identification of the limit OCP.

Theorem 10.19. Let zεh ε>0 be a bounded sequence in Mε = L2 (0, T ; H 1 (Ω, dηεh )) ∩ L2 (0, T ; L2 (Ω, dλhε )) and let z ∈ L2 (0, T ; H 1 (Ω)) be its weak limit in the sense of Definition 10.11. Then zεh → z in L6 (0, T ; L6 (Ω, dηεh )) and (zεh )3 → z 3 in L2 (0, T ; L2 (Ω, dηεh )), namely T T h 3 h lim zε uε dηε dt = z 3 u dx dt (10.54) ε→0

0

0

Ω

Ω

∀ uε u in L2 (0, T ; L2 (Ω, dηεh )),

T

lim

ε→0

|zεh |6

0

dηεh

T

z 6 dx dt.

dt = 0

Ω

(10.55)

Ω

Proof. First, we note that in accordance with Lemma 10.2, the injection Mε → L8 (0, T ; L8 (Ω, dηεh )) is compact for every fixed ε > 0. To see this, it is enough to take B 0 = H 1 (Ωε ), B 1 = L8 (Ωε ), B 2 = L2 (Ωε ), p 0 = 2, p 1 = 8, and p 2 = 2 in Lemma 10.1. Hence, in view of the property of uniform boundedness of zεh ε>0 in Mε , we have

T

|zεh |8 dηεh dt < +∞.

lim sup ε→0

0

(10.56)

Ω

4

) is compact with respect to Thus, the sequence zεh ε>0 (resp., zεh ε>0 8 8 h 2 2 h the weak convergence in L (0, T ; L (Ω, dηε )) (resp.,

L (0, T8 ; L (Ω, 8dηε ))). hLet ∗ h z be a weak limit of the original sequence zε ε>0 in L (0, T ; L (Ω, dηε )). Then zεh z ∗ in L2 (0, T ; L2 (Ω, dηεh )) as well. However, since the weak limit in L2 (0, T ; L2 (Ω, dηεh )) must be unique, it follows that zεh z = z ∗ in L8 (0, T ; L8 (Ω, dηεh )).


377

From (10.56) we may deduce that this sequence is uniformly bounded in Lp (0, T ; Lp (Ω, dηεh )) for any p ∈ {3, . . . , 7} and its weak limits in the variable space Lp (0, T ; Lp (Ω, dηεh )) coincide with z. On the other hand, the element z ∈ L2 (0, T ; H 1 (Ω)) is the strong limit of the sequence zεh ε>0 in L2 (0, T ; L2 (Ω, dηεh )) (see Definition 10.11). Then by the definition of the strong convergence in variable spaces, we have

T

ε→0

T

(zεh · zεh ) ϕψ dηεh dt

(zεh )2 ϕψ dηεh dt = lim

lim

0

ε→0

Ω

T

0

Ω

z 2 ϕψ dx dt.

= 0

(10.57)

Ω

Indeed, in this case, we have the following estimate: T T h 2 h 2 (zε ) ϕψ dηε dt − z ϕψ dx dt 0 Ω 0 Ω T T h h h h h zε (zε − z) ϕψ dηε dt + (zε − z)z ϕψ dηε dt ≤ 0 Ω 0 Ω T T + z 2 ϕψ dηεh dt − z 2 ϕψ dx dt 0 Ω 0 Ω

h ≤ 2 sup max zε L2 (0,T ;L2 (Ω,dηεh )) , zL2 (0,T ;L2 (Ω,dηεh )) ε→0 ) × ϕC(Ω) ψC(0,T ) Jε T T 2 h 2 + z ϕψ dηε dt − z ϕψ dx dt , 0 Ω 0 Ω

T

where Jε =

(zεh − z)2 dηεh dt. 0

Ω

Passing to the limit as ε → 0, we see that limε→0 Jε = 0 and T T z 2 ϕψ dηεh dt − z 2 ϕψ dx dt −→ 0 0 Ω 0 Ω (by Lemma 10.12). Thereby, we have shown that relation (10.57) holds true. However, this can be

interpreted as a definition of the weak convergence of the sequence (zεh )2 ε>0 to z 2 in L2 (0, T ; L2 (Ω, dηεh )). As a matter of fact, we can establish a stronger result, namely (zεh )3 z 3 in L2 (0, T ; L2 (Ω, dηεh )). Indeed, in order to prove this, it is enough to use the definition of strong convergence in L2 (0, T ; L2 (Ω, dηεh )). Making an a priori estimate as above, we can obtain the assertion

378


T

T

(zεh )2 · zεh ϕψ dηεh dt

(zεh )3 ϕψ dηεh dt = lim

lim

ε→0

0

ε→0

Ω

T

0

Ω

z 3 ϕψ dx dt.

= 0

Ω

By analogy with the previous motivation, we conclude from this that the sequence {(zεh )3 } is weakly convergent to z 3 in L2 (0, T ; L2 (Ω, dηεh )). Therefore,

T

lim

ε→0

0

T

4

h 4

ε→0

zε

0

ε→0

zεh

3

z 4 dxdt,

0

h 3 h h zε zε dηε dt =

Ω

0

T

zεh dηεh dt =

Ω

T

ϕψ dηεh dt = lim

Ω

0

T

dηεh dt = lim

Ω

lim

ε→0

zεh

Ω

T

z 4 ϕψ dx dt Ω

0

(as the limit of the product of weakly and strongly convergent sequences in L2 (0, T ; L2 (Ω, dηεh ))). Hence, by the criterium of the strong convergence in variable spaces, we deduce that zεh → z in L4 (0, T ; L4 (Ω, dηεh )), h 2 zε → z 2 in L2 (0, T ; L2 (Ω, dηεh )), h 4 zε z 4 in L8 (0, T ; L8 (Ω, dηεh )). Using a similar procedure, we obtain

T

h 6 h zε dηε dt = lim

lim

ε→0

0

ε→0

Ω

T 0

h 4 h 2 h zε zε dηε dt =

T

z 6 dx dt 0

Ω

Ω

(as the limit of the product of weakly and strongly convergent sequences in L2 (0, T ; L2 (Ω, dηεh ))). Thus, zεh z in L6 (0, T ; L6 (Ω, dηεh )), T T h 6 h zε dηε dt −→ z 6 dx dt. 0

0

Ω

Ω

and we come to the required conclusion. 10.4.1 Convergence formalism for Dirichlet boundary controls As follows from the statement of the C-extended OCP (10.38)–(10.40), the class of admissible Dirichlet controls for every fixed value of ε can be described as

vε ∈ Uε = v ∈ L2 (0, T ; H 1 (Γ0 ) ∩ L2 (Ω, dμhε ) ) :

vL2 (0,T ;H 1 (Γ0 )) ≤ C0 . (10.58)


379

Hence, for every ε > 0 and for every admissible control function vε ∈ Uε , there exists an extension operator Pε : L2 (0, T ; H 1 (Γε )) → L2 (0, T ; H 1 (Γ0 )) such that Pε ( vε )L2 (0,T ;H 1 (Γ0 )) ≤ C0 . In view of this, we can give two different convergence concepts for such controls. Definition 10.20. We say that a function u∗ is an sa -limit for the sequence of Dirichlet controls vε ∈ L2 (0, T ; H 1 (Γε )) ε>0 if some sequence of its images vε )}ε>0 converges to u∗ weakly in L2 (0, T ; H 1 (Γ0 )). We also say that {Pε ( 2 the sequence vε ∈ L (0, T ; L2 (Ω, dμhε )) ε>0 is sb -convergent to an element u∗∗ ∈ L2 (0, T ; L2 (Γ0 )) if this sequence is uniformly bounded, that is, T sup | vε |2 dμhε dt < +∞, ε→0

and vε u

∗∗

0

Ω

in the variable space L2 (0, T ; L2 (Ω, dμhε )).

Below we show that these concepts are the same for the admissible controls. However, to begin, we prove the correctness of the first one. Indeed, let us fix some sequence of the controls { vε } and assume that there are two different (1)

(2)

sequences of its images Pε ( vε ) ε>0 and Pε ( vε ) ε>0 such that vε ) u∗1 Pε(1) (

and Pε(2) ( vε ) u∗2 in L2 (0, T ; H 1 (Γ0 )).

Let χhε ∈ L∞ (Γ0 ) be the characteristic function of the set ΓεD where the corresponding controls vε are located. By Proposition 9.12, we have that h−1 (ε)χhε → 1 weakly-∗ in L∞ (Γ0 ) as ε → 0. Then, passing to the limit in the integral identity T T h−1 (ε) χhε Pε(1) ( vε )ϕψ dx1 dt = h−1 (ε) χhε Pε(2) ( vε )ϕψ dx1 dt, 0

0

Γ0

Γ0

as ε → 0, which holds true for every ϕ ∈ H (Γ0 ) and ψ ∈ C0∞ (0, T ), we get T T ∗ u1 ϕψ dx1 dt = u∗2 ϕψ dx1 dt, ∀ ϕ ∈ H 1 (Γ0 ), ∀ ψ ∈ C0∞ (0, T ). 1

0

0

Γ0

u∗1

u∗2

Γ0

Hence, = almost everywhere in (0, T ) × Γ0 . Thus, if for a given sequence of controls its sa -limit exists, then it is unique and does not depend on the choice of the extension operators. As an obvious consequence of this remark, we have the following result.

Lemma 10.21. Every sequence of admissible controls vε ∈ Uε ε>0 is relatively compact with respect to the sa -convergence introduced above. Moreover, if u∗ is its sa -weak limit point, then u∗ belongs to the set

U = u ∈ L2 (0, T ; H 1 (Γ0 )) | uL2 (0,T ;H 1 (Γ0 )) ≤ C0 .

380


Indeed, the fact u∗ ∈ U is apparent from the lower semicontinuity property of the norm in L2 (0, T ; H 1 (Γ0 )) with respect to the weak convergence in this space. The following result can be proved in much the same way as Theorem 9.19.

Theorem 10.22. Let vε ∈ Uε ε>0 be a sequence of admissible Dirichlet controls. Then one can extract an sb -convergent subsequence of { vε }ε>0 for which its sa - and sb -limits coincide. Remark of Dirichlet con 10.23. Hereinafter we suppose that for any sequence

trols vε ∈ Uε , there is a sequence of its prototypes vε ∈ L2 (0, T ; H 1 (Γ0 )) such that vε |ΓεD = vε |ΓεD and vε u in L2 (0, T ; H 1 (Γ0 )). Then, by analogy with Lemma 10.13, it can be proved that vε converging to u|Γ0 weakly in L2 (0, T ; L2 (Ω, dμhε )) and u|Γ0 = u∗∗ by virtue of the uniqueness of weak limits in L2 (0, T ; L2 (Ω, dμhε )). Therefore, in what follows, we will suppose that

every sequence of admissible Dirichlet controls vε ∈ Uε is compact with respect to the weak convergence in L2 (0, T ; L2 (Ω, dμhε )) and its limit belongs to L2 (0, T ; H 1 (Γ0 )). 10.4.2 w-Convergence of admissible solutions of the C-extended problems

ε Let (aε , vε , wε , zε ) ∈ Σ be a sequence of admissible solutions for the ε>0 ε ⊂ Zε for every ! CPε -problem. Each of the sets Σ ε is defined in (10.40) and Σ ε > 0, where Zε ≡ L2 (0, T ; L2 (Ω, dηεh ))

× L2 (0, T ; H 1 (Γ0 ) ∩ L2 (Ω, dμhε ) ) × L2 (0, T ; L2 (Ω, dνεh ))

× L2 (0, T ; H 1 (Ω, dηεh )) ∩ L2 (0, T ; L2 (Ω, dλhε )) . (10.59) We assume that this sequence is bounded, that is, lim sup (aε , vε , wε , zε )Z ε ] < +∞. ε>0

Now, summing up the above given arguments, we may introduce the following concept of the weak convergence in the variable space Z ε . Definition 10.24. We say that a bounded sequence {(aε , vε , wε , zε ) ∈ Z ε }ε>0 is w-convergent to a quadruple (a, v, w, z) ∈ L2 ((0, T ) × Ω) × L2 (0, T ; H 1 (Γ0 )) × L2 ((0, T ) × Γ0 ) × L2 (0, T ; H 1 (Ω))

10.5 The limiting optimal control problem and its properties

381 w

in the variable space Z ε as ε tends to 0 (in symbols, (aε , vε , wε , zε ) −→ (a, v, w, z) ) if aε a

in L2 (0, T ; L2 (Ω, dηεh )),

vε v

in L2 (0, T ; L2 (Ω, dμhε )),

wε w

v ∈ L2 (0, T ; H 1 (Γ0 )),

L2 (0, T ; L2 (Ω, dνεh )),

in

zεh → z in L2 (0, T ; L2 (Ω, dηεh )), zεh z in L2 (0, T ; L2 (Ω, dλhε )), T |zεh − z|2 dηεh dt → 0 as ε → 0. 0

Ω

Then, taking into account Theorems 10.18 and 10.22, we come to the following conclusion. Theorem 10.25. Let {(aε , vε , wε , zε ) ∈ Z ε }ε>0 be a bounded sequence of ad! ε . Then this sequence is relamissible solutions to the C-extended problems CP tively compact with respect to the w-convergence in the variable space X ε and, w moreover, if (aε , vε , wε , zε ) −→ (a, v, w, z), then zε → z in L6 (0, T ; L6 (Ω, dηεh )), (zε ) → z 3

3

2

in L (0, T ; L

2

(Ω, dηεh )).

(10.60) (10.61)

10.5 The limiting optimal control problem and its properties By analogy with the case of steady-state processes on thin periodic structures, we begin this section with the definition of a “limit” minimization problem for the sequence (10.42) as ε tends to 0. To this end, we make use of the concept of variational convergence of constrained minimization problems in variable spaces. Note that, in view of the specific construction of the solution space Z ε , the strong approximation property for the “w-limit space” Y = L2 ((0, T ) × Ω) × L2 (0, T ; H 1 (Γ0 )) × L2 ((0, T ) × Γ0 ) × L2 (0, T ; H 1 (Ω)) does not hold, in general. The last sentence means that not for every quadruple (a, v, w, z) ∈ Y can a sequence {(aε , vε , wε , zε ) ∈ Z ε }ε>0 be found such that w (aε , vε , wε , zε ) −→ (a, v, w, z). Because of this, we give the following concept of a “limit” minimization problem. Definition 10.26. We say that a minimization problem J 0 (a, v, w, z) inf (a,v,w,z)∈ Σ0

(10.62)

382


is the weak variational limit of the sequence (10.42) with respect to the wconvergence in the variable space Z ε (or variational w-limit) if the conditions (d)–(dd) of Definition 8.24 hold true with Xε = L2 (0, T ; L2 (Ω, dηεh ))

× L2 (0, T ; H 1 (Γ0 ) ∩ L2 (Ω, dμhε ) ) × L2 (0, T ; L2 (Ω, dνεh ))

× L2 (0, T ; H 1 (Ω, dηεh )) ∩ L2 (0, T ; L2 (Ω, dλhε )) , X = L2 ((0, T ) × Ω) × L2 (0, T ; H 1 (Γ0 )) × L2 ((0, T ) × Γ0 ) × L2 (0, T ; H 1 (Ω)). Theorem 10.27. Assume that the constrained minimization problem (10.62) is the weak variational limit of the sequence (10.42), and is solvable. Assume also that the initial data for the original problem (10.5) satisfy the condition

ε )) < +∞. sup max yε, 0 H 1 (Ω) , gε L2 (0,T ;L2 (Sint ε>0

! εε be a sequence of optimal solutions for the CP Let (a0ε , vε0 , wε0 , zε0 ) ∈ Σ ε>0 problems. Then this sequence is relatively w-compact and its every w-cluster quadruple is an optimal solution to the problem (10.62); more precisely, if w

(a0εk , vε0k , wε0k , zε0k ) −→ (a0 , v 0 , w0 , z 0 ),

(10.63)

then inf (a,v,w,z)∈ Σ0

J 0 (a, v, w, z) = J 0 a0 , v 0 , w0 , z 0 = lim Jεk (a0εk , vε0k , wε0k , zε0k ) k→∞

= lim

inf

ε→0 (aε ,vε ,wε ,zε )∈ Σ bε

Jε (aε , vε , wε , zε ).

(10.64)

Proof. Let us assume that the sequence of optimal solutions + * ε (a0ε , vε0 , wε0 , zε0 ) ∈ Σ ε>0

to the C-extended problems CPε is such that lim sup Jε (a0ε , vε0 , wε0 , zε0 ) = ∞. ε>0

Then for any given value D > 0, there is a sequence {εk } such that εk → 0 and min

bε (aεk ,vεk ,wεk ,zεk )∈ Σ k

Jεk (aεk , vεk , wεk , zεk ) = Jεk (a0εk , vε0k , wε0k , zε0k ) ≥ D,

∀ k ∈ N.


383

On the other hand, it is easy to see that the quadruple (0, 0, 0, zε∗ ) is admissible for every ε (here, ze∗ is the weak solution of the initial-boundary value problem (10.24) under the condition aε = 0, vε = 0, and wε = 0). Hence, taking the a priori estimate (10.41) into account, we immediately conclude lim sup ε→0

, < +∞. Jε (aε , vε , wε , zε ) ≤ lim sup Jε (0, 0, 0, zε∗ ) ≤ C

min

bε (aε ,vε ,wε ,zε )∈ Σ

ε→0

The obtained contradiction proves that there is a constant D0 > 0 such that lim sup Jε (a0 , v 0 , w0 , z 0 ) ≤ D0 . ε

ε→0

ε

ε

ε

This fact implies the boundedness of the sequences aε0 and wε0 in Then using the L2 (0, T ; L2 (Ω, dηεh )) and L2 (0, T ; L2 (Ω, dνεh )), respectively.

boundedness of the Dirichlet type controls vε0 and estimate (10.41), we deduce lim sup(a0ε , vε0 , wε0 , zε0 )Z ε ε>0 = lim sup a0ε L2 (0,T ;L2 (Ω,dηεh )) + vε0 L2 (0,T ;H 1 (Γ0 )∩L2 (Ω,dμhε ) ε>0

+ wε0 L2 (0,T ;L2 (Ω,dνεh )) + zε0 L2 (0,T ;H 1 (Ω,dηεh )) < +∞. Then, taking Theorem 10.25 into account, we assume that there exist a subsequence (a0εk , vε0k , wε0k , zε0k ) k∈ N of the sequence of optimal solutions and w

a quadruple (a∗ , v ∗ , w ∗ , z ∗ ) such that (a0εk , vε0k , wε0k , zε0k ) −→ (a∗ , v ∗ , w∗ , z ∗ ) as εk → 0. In order to conclude this proof, it remains to apply the rest of the arguments of the proof of Theorem 9.26.

10.6 Main convergence theorem The main question that we are going to discuss in this section concerns the asymptotic behavior as ε → 0 of the initial-boundary value problem μhε -a.e. on (0, T ) × Ω, vε L2 (0,T ;H 1 (Γ0 )) ≤ C0 ,

zε = vε

zε (0, ·) = yε, 0

T

−

zε ϕ ψ 0

dηεh

Ω

T

(10.66)

(A(x/ε)∇zε · ∇ϕ)ψ dηεh dt 0

Ω

zε ϕψ dλhε dt

+ 4d β(ε) T

in Ω,

0

Ω

T

aε ϕ ψ dηεh dt + 2 β(ε)

= 0

(10.65)

dt +

T

ηεh -a.e.

Ω

T

wε ϕψ dνεh dt 0

Ω

gε ϕψ dλhε dt,

+ 4 β(ε) 0

Ω

∀ ϕ ∈ C0∞ (R2 ; ΓεD ), ∀ ψ ∈ C0∞ (0, T ),

(10.67)

384


where β(ε) =

ε (1 − h(ε)) · ξ(ε), and ξ(ε) = . (2 − h(ε)) h(ε)

As follows from Proposition 9.23, the sequence of sets of test functions ∞ C0 (R2 ; ΓεD ) ε>0 for the integral identity (10.67) converges in the sense of Kuratowski to C0∞ (R2 ; Γ0 ) as ε tends to 0. So, in the sequel, we suppose that ϕ ∈ C0∞ (R2 ; Γ0 ) in (10.67).

ε be any equibounded sequence of admissible soLet (aε , vε , wε , zε ) ∈ Σ ! ε -problem. By Theorem 10.25, this sequence is relatively lutions for the CP compact with respect to the w-convergence in the variable space Zε . So, we may suppose that there exists a quadruple (a, v, w, z) ∈ Y such that w (aε , vε , wε , zε ) (a, v, w, z). The main question is: By what kind of relation can this quadruple be determined? We give the following result. Theorem 10.28. Suppose that Aε (x) = A(x/ε) in (10.67) is a Y -periodic measurable matrix satisfying conditions (9.21). Assume also that lim ξ(ε) = ξ ∗ < +∞

ε→0

and there exist functions y 0 ∈ L2 (Ω) and g ∈ L2 (0, T ; L2 (Ω)) such that

Let

yε, 0 y 0 in H 1 (Ω) and gε g in L2 (0, T ; L2 (Ω, dλhε )).

(10.68)

wε ∈ L2 (0, T ; L2 (Ω, dνεh )) , aε ∈ L2 (0, T ; L2 (Ω, dηεh )) ,

vε ∈ L2 (0, T ; H 1 (Γ0 ) ∩ L2 (Ω, dμhε ) )

(10.69)

! ε -problems such be any bounded sequences of admissible controls for the CP that aε a in L2 (0, T ; L2 (Ω, dηεh )), wε w in vε → v in

2

2

L (0, T ; L (Ω, dνεh )), L2 (0, T ; L2 (Ω, dμhε )).

(10.70) (10.71)

(10.72)

Let zε = zε (aε , vε , wε ) ∈ L2 (0, T ; H 1 (Ω, dηεh )) ∩ L2 (0, T ; L2 (Ω, dλhε )) ε>0 be the corresponding solutions to the problem (10.65)–(10.67). Then w

(aε , vε , wε , zε ) (a, v, w, z) as ε → 0, where z is a unique solution in L2 (0, T ; H 1 (Ω)) of the initial-boundary value problem z = v a.e. on (0, T ) × Γ0 , vL2 (0,T ;H 1 (Γ0 )) ≤ C0 , z(0, ·) = y0 a.e. in Ω,

(10.73) (10.74)


T

− 0

zϕ ψ dx dt +

T

(Ahom ∇z · ∇ϕ)ψ dx dt

0

Ω

+ 2d ξ

∗

T

T

Ω

zϕψ dx dt

0

Ω

a ϕ ψ dx dt + 2 ξ

= 0

385

∗

gϕψ dx dt.

0

Ω

T

(10.75)

Ω

Here, the matrix Ahom is defined by the rule Ahom ζ = A(y)(ζ + υ 0 ) dη(y),

(10.76)

Y

where υ 0 ∈ L2 (Ω, Vpot ) is the unique solution of the minimum problem (9.93). Proof. First, we note that by the a priori estimate (10.41), the sequence of

ε is uniformly bounded in Zε . admissible solutions (aε , vε , wε , zε ) ∈ Σ ε>0 Hence, due to Theorem 10.25, we may suppose that there is a function z ∈ L2 (0, T ; H 1 (Ω)) satisfying the conditions ⎧ zε → z in L2 (0, T ; L2 (Ω, dηεh )), ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ zε z in L2 (0, T ; L2 (Ω, dλhε )), ⎪ zε → z in L6 (0, T ; L6 (Ω, dηεh )), ⎪ ⎪ ⎪ ⎪ ⎩ 2 ∇zε ∇z + r, where r ∈ L2 (0, T ; L2 (Ω, Vpot )).

(10.77)

On the other hand, in view of Theorem 10.22 and Remark 10.23, the limit sa v. Moreover, element v in (10.72) is such that v ∈ L2 (0, T ; H 1 (Γ0 )) and vε −→ taking the initial suppositions (10.68) and Lemma 10.13 into account, we can conclude yε, 0 → y0 in L2 (Ω, dηεh ). Passing to the limit in (10.65) and (10.66) as ε → 0, we come to relations (10.73). Our aim is to prove that the integral equality (10.75) holds true. For this, we consider the integral identity (10.67) with the test func∞ tions ϕ(x) = εΨ (x)ω(ε−1 x) and ψ ∈ C0∞ (0, T ), where ω ∈ Cper (Y ) and ∞ 2 D Ψ ∈ C0 (R , Γε ). This yields

386


T

zε Ψ ω(ε−1 x) ψ dηεh dt

−ε 0

Ω

T

(A(x/ε)∇zε · ∇Ψ )ω(ε−1 x)ψ dηεh dt

+ε 0 Ω T

(A(x/ε)∇zε ·

+ 0

Ω T

−1

aε Ψ ω(ε

= ε 0

Ω

T

0

dt + 2ε β(ε)

zε Ψ ω(ε−1 x)ψ dλhε dt

Ω

wε Ψ ω(ε−1 x)ψ dνεh dt

Ω

gε Ψ (x)ω(ε−1 x)ψ dλhε dt.

+ 4ε β(ε) 0

x) ψ dηεh

T

dt + 4εd β(ε)

0

T

∇ω)Ψ ψ dηεh

(10.78)

Ω

Taking the mean value property which holds for every periodic Borel measure ηεh , λhε , and νεh into account, we pass to the limit in (10.78) as ε → 0 and get T (10.79) A(ε−1 x)∇zε , ∇ω(ε−1 x) Ψ ψ dηεh dt = 0. lim ε→0

0

Ω

Then, in view of (10.77)4 and the definition of the weak two-scale limit, we have

T

Ψ ω(ε−1 x)A(ε−1 x) ∇zε dηε ψ dt

lim

ε→0

0

Ω

T

=

Ψ (x)ω(y)A(y) [∇z(t, x) + r(t, x, y)] dη(y) dx ψ dt. 0

Ω

Y

Thus, A(ε−1 x)∇zε A(y) [∇z + r(t, x, y)] , 2

A(ε−1 x)∇zε (t, x)

(10.80)

A(y) [∇z(t, x) + r(t, x, y)] dη(y) Y

2 in L2 (0, T ; L2 (Ω, dηεh ), (10.81)

that is, equality (10.79) can be rewritten in explicit form as

A(ε−1 x)∇zε , ∇ω(ε−1 x) Ψ ψ dηεh dt ε→0 0 Ω T = Ψ (A(y) [∇z(t, x) + r(t, x, y)] , ∇ω) dη(y) dx ψ dt = 0. T

lim

0

Ω

Y

Since this equality holds true for every Ψ ∈ C0∞ (R2 , Γ0 ) and every ψ ∈ C0∞ (0, T ), it follows that


∞

2 ∀ ζ ∈ Cper (Y ) .

(A(∇z + r), ζ) dη = 0,

387

(10.82)

Y

However, following Zhikov [256], (10.82) can be viewed as Euler’s equation for the minimum problem (ζ + υ, A(ζ + υ)) dη = (ζ + υ 0 , A(ζ + υ 0 )) dη. (10.83) min υ∈Vpot

Y

Y

Since this problem has a unique solution υ 0 ∈ L2 (Ω, Vpot ), it follows that υ is the unique solution of (10.82) as well. Let us define the matrix Ahom as in (10.76). Then, having put υ = υ 0 in (10.81) and used formula (10.76), we immediately deduce that 0

A ε−1 x ∇zε Ahom ∇z

2 in L2 (0, T ; L2 (Ω, dηεh ) ).

(10.84)

Our intention next is to pass to the limit in (10.67) as ε → 0. For this, following Proposition 9.23, we choose the test functions in (10.67) as follows: ϕ = ϕε ∈ C0∞ (R2 , ΓεD ) for every ε > 0, where ϕε → ϕ0 in C(R2 ), and ϕ0 is an element of C 1 (Ω, Γ0 ). Then the passage to the limit in

T

− 0

zε ϕε ψ dηεh dt +

Ω

T

T

(A(x/ε)∇zε · ∇ϕε )ψ dηεh dt

0

Ω

zε ϕε ψ dλhε dt

+ 4d β(ε)

T

Ω

0

aε ϕε ψ dηεh

= 0

Ω

wε ϕε ψ dνεh dt

dt + 2 β(ε) 0

T

T

Ω

gε ϕε ψ dλhε dt

+ 4 β(ε) 0

Ω

is justified by relations (10.70)–(10.72), (10.76), and (10.77), lim β(ε) = (1/2) lim ξ(ε) = (1/2)ξ ∗ ,

ε→0

(10.85)

ε→0

and Lemma 10.15. Hence, we obtain T T − zϕ ψ dx dt + (Ahom ∇z · ∇ϕ)ψ dx dt 0

Ω

Ω

0

+ 2d ξ ∗

T

zϕψ dx dt 0

Ω

T

Ω T

a ϕ ψ dx dt

= 0

+ ξ∗

0

Γ0

wϕψ dx1 dt + 2 ξ ∗

T

gϕψ dx dt.

0

Ω

388


T

However, given ϕ ∈ H 1 (Ω, Γ0 ) have

wϕψ dx1 dt = 0. As a result, we 0

Γ0

come to the required equality (10.75). To conclude, we note that relations (10.73)–(10.75) can be interpreted as the variational formulation of the initial-boundary problem ⎫ ∂t z − div Ahom ∇z + 2dξ ∗ z = a + 2ξ ∗ g in (0, T ) × Ω, ⎪ ⎪ ⎪ on (0, T ) × ∂Ω \ Γ0 , ⎬ ∂νAhom z = 0 (10.86) ⎪ z=v on (0, T ) × Γ0 , ⎪ ⎪ ⎭ for a.a. x ∈ Ω, z(0, x) = y0 for which the following result is well known: For every a, g ∈ L2 (0, T ; L2 (Ω)), y0 ∈ H 1 (Ω), and v ∈ L2 (0, T ; H 1 (Γ0 )), there exists a unique solution of (10.86) (see Lions [168] and Michajlov [188]). This completes the proof. The following statement is a direct consequence of the well-known results of the theory of boundary value problems (see Lions [168]). Corollary 10.29. Let (a1 , v1 , z1 ), (a2 , v2 , z1 ) ∈ L2 ((0, T ) × Ω) × L2 (0, T ; H 1 (Γ0 )) × L2 (0, T ; H 1 (Ω)) be any triplets satisfying relation (10.86). Then there exists a constant Cˆ > 0,

ˆ Cˆ = C(Ω, Γ0 , v 0 , g, ξ ∗ , α0 )

such that z1 − z2 L2 (0,T ;H 1 (Ω)) + z1 − z2 L6 ((0,T )×Ω)

≤ Cˆ a1 − a2 L2 ((0,T )×Ω) + v1 − v2 L2 (0,T ;H 1 (Γ0 )) . (10.87)

10.7 The limit analysis of the C-extended optimal control problem In this section, we show that for the sequence of constrained minimization problems (10.42), there exists a weak variational limit with respect to the w-convergence in the variable space Zε and that this limit problem can be recovered in an explicit form. We begin with the following result. Theorem 10.30. Let yεT ∈ L2 (Ω, dηεh ), yε∗ ∈ L6 (0, T ; L6 (Ω, dηεh )), yε, 0 ∈ H 1 (Ω), gε ∈ L2 (0, T ; L2 (Ω, dλhε )), and fε ∈ L2 (0, T ; L2 (Ω, dηεh )) be given

10.7 The limit analysis of the C-extended optimal control problem

functions such that T 2 h dηε < ∞, sup yε ε→0

Ω

-

T

6 (yε∗ )

0

< ∞,

dηεh dt

Ω

yε∗ -

T

(fε )

0

6

in L (0, T ; L

2

dηεh dt

< ∞,

Ω

fε → f in L (0, T ; L

T

2

dλhε dt

(gε ) 0

2

⎪

⎭ (Ω, dηεh )),

yε, 0 y0 in H 1 (Ω), .

sup ε→0

⎪

⎭ (Ω, dηεh )),

(10.89)

⎫ ⎪ ⎬ 2

-

6

.

sup ε→0

→y

∗

(10.88) ⎫ ⎪ ⎬

.

sup ε→0

yεT → yT in L2 (Ω, dηεh ),

389

(10.90)

(10.91) ⎫ ⎪ ⎬

< ∞,

⎪

Ω 2

gε g in L (0, T ; L

⎭ (Ω, dλhε )).

2

(10.92)

Assume that the correction factor κ(ε) in the cost functional (10.39) is such that lim κ(ε)h(ε) = κ∗ < +∞. (10.93) ε→0

Then, for the sequence (10.42), there exists a unique weak variational limit in the sense of Definition 10.26 which has the representation inf J 0 (a, v, w, z) , (10.94) (a,v,w,z)∈ Σ0

where the cost functional J 0 and the set of admissible solutions Σ0 are defined as follows:

T

(z − y ∗ )6 dx dt

(z(T, x) − y T )2 dx + 2

J 0 (a, v, w, z) = 2 Ω

+ κ∗

0 T

T

v 2 dx1 dt + 2γ ∗

Γ0

w2 dx1 dt,

+2 0

Γ0

(a, v, w, z) ∈ Σ0 if and only if

0 T

Ω

(a − z 3 − f )2 dx dt 0

Ω

(10.95)

390


⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ Σ0 =

z ∈ L2 (0, T ; H 1 (Ω)), v ∈ L2 (0, T ; H 1 (Γ0 )),

a ∈ L2 ((0, T ) × Ω), w ∈ L2 (0, T ; H 1 (Γ0 )),

vL2 (0,T ;H 1 (Γ0 )) ≤ C0 ,

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

z = v a.e. on (0, T ) × Γ0 , z(0, ·) = y0 a.e. on Ω; T T . ⎪ ⎪ zϕ ψ dx dt + (Ahom ∇z · ∇ϕ)ψ dx dt − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Ω Ω 0 0 ⎪ ⎪ ⎪ ⎪ T T ⎪ ⎪ ⎪ ⎪ ∗ ⎪ ⎪ ⎪ ⎪ +2d ξ zϕψ dx dt = a ϕ ψ dx dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 0 Ω Ω T ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∗ ∞ ∞ ⎭ ⎩ +2 ξ gϕψ dx dt ∀ ϕ ∈ C (Ω, Γ0 ), ∀ ψ ∈ C0 (0, T ) ⎪ 0

Ω

(10.96) Proof. By analogy with Theorem 9.29, we divide the proof of this theorem into two steps for the verification of the correspondent items of Definition 10.26. Step 1: Statement (d) of Definition 8.24 is valid. Let {(ak , vk , wk , zk ) ∈ Z εk }ε>0 be a bounded sequence, w-convergent to a quadruple (u, v, w, z) ∈ Y. Here, {εk } is a subsequence of {ε > 0} such ε for all k ∈ N. Then due that εk → 0 as k → ∞ and (ak , vk , wk , zk ) ∈ Σ k to Theorems 10.25 and 10.28, we have that the w-limit quadruple (u, v, w, z) belongs to the space Y = L2 ((0, T ) × Ω) × L2 (0, T ; H 1 (Γ0 )) × L2 ((0, T ) × Γ0 ) × L2 (0, T ; H 1 (Ω)) and satisfies conditions (10.73)–(10.75). Thus, the first inclusion in (8.59) holds true. We now verify inequality (8.59), where the limit cost functional J0 has representation (10.95). Then, taking into account the property of lower semicontinuity of the weak convergence in variable spaces, we have lim (2 − h(εk )) (zεk (T, x) − yεTk )2 dηεhk k→∞ Ω / T

T

+ κ(εk )h(εk ) 0

≥2

Ω

vε2k dμhεk dt +2 (1 − h(εk )) ∗

T

(z(T, x) − y ) dx + κ T 2

Ω T

w2 dx1 dt. 0

Ω

wε2k dνεhk dt

v 2 dx1 dt 0

+2

0

Γ0

(10.97)

Γ0

In order to pass to the limit in the rest of the terms of the cost functional (10.39), we make use of the strong convergence properties (10.60), (10.89), and (10.90), and the criterion of strong convergence in variable spaces. As a result, we get


0

T

lim (2 − h(εk ))

k→∞

Ω

0

+γ

T

≥2 0

∗

391

(zεk − yε∗k )6 dηεhk dt

T

/

(aεk −

0

Ω

zε3k

(z − y ∗ )6 dx dt + 2γ ∗

− fεk )

dηεhk

dt

(a − z 3 − f )2 dx dt.

0

Ω

T

2

(10.98)

Ω

To conclude this step, it remains only to unite the relations (10.97)–(10.98) obtained above. Step 2: Statement (dd) of Definition 8.24 holds true. Let (a, v, w, z) ∈ Σ0 be any admissible solution for the minimization problem (10.94). Let , a ∈ L2 (0, T ; H 1 (Ω)) and w , ∈ L2 (0, T ; H 3/2 (Ω)) be given functions and let z, = z(, a, v) be the corresponding solution of the problem a, v, w| , Γ0 , z,) ∈ Σ0 . Moreover, as (10.86). Then w| , Γ0 ∈ L2 (0, T ; H 1 (Γ0 )) and (, follows from (10.95) and the a priori estimate (10.87), there are constants D1 > 0 and D2 > 0 such that J 0 (a, v, w, z) − J 0 (, a, v, w| , Γ0 , z,) ≤ D1 a − , a L2 ((0,T )×Ω) + w − w , L2 ((0,T )×Γ0 ) + , z − zL2 (0,T ;H 1 (Ω)) + , z − zL6 ((0,T )×Ω) , , z − zL2 (0,T ;H 1 (Ω))∩L6 ((0,T )×Ω) ≤ D2 a − , , L2 ((0,T )×Γ0 ) . a L2 ((0,T )×Ω) + w − w Let 1 > δ > 0 be a given value. Using the density of the embedding a ∈ L2 (0, T ; H 1 (Ω)) H 1 (B) → L2 (B), we can always choose the elements , 2 3/2 and w , ∈ L (0, T ; H (Γ0 )) such that , L2 ((0,T )×Γ0 ) < = max{D1−1 , D2−1 } δ. a − , aL2 ((0,T )×Ω) + w − w Due to the above estimates, we have a, v, w, , z, )| ≤ δ. |J 0 (a, v, w, z) − J 0 (, , z − zL2 (0,T ;H 1 (Ω))∩L6 ((0,T )×Ω) ≤ δ, (a, v, w, z) − (, a, v, w, , z, )Y
0 as follows:

ahε → , a in L2 (0, T ; L2 (Ω, dηεh ) T h 2 h ε→0 aε dηε dt −→ and 0

Ω

T

a2 dx dt,

0

Ω

(10.102)

392


vεh v in L2 (0, T ; L2 (Ω, dμhε ), vεh L2 (0,T ;H 1 (Γ0 )) ≤ C0 T T h 2 ε→0 h and v 2 dx1 dt, (10.103) vε dμε dt −→ 0

0

Ω

wεh w , in L2 (0, T ; L2 (Ω, dνεh ) T h 2 h ε→0 and wε dνε dt −→ 0

T

Γ0

w ,2 dx1 dt. (10.104)

0

Ω

Γ0

Let zεh be the correspondent solutions of the initial-boundary value probε for every ε > 0. It is clear lem (10.65)–(10.67). Hence, (ahε , vεh , wεh , zεh ) ∈ Σ that this sequence is equibounded in Zε . Then

applying Theorem 10.25, we conclude that the sequence (ahε , vεh , wεh , zεh ) is compact with respect to the w-convergence. Let (, a, v, w, , z,∗ ) be its w-limit. Due to Theorem 10.28, we ∗ have (, a, v, w, , z, ) ∈ Σ0 . Since the initial-boundary value problem (10.86) has a unique solution for every fixed , a and v, it follows that z, = z,∗ and, hence, w h h h h a, v, w, , z,) as ε → 0. (aε , vε , wε , zε ) (, Thus, we have shown that the first part of property (dd) in Definition 8.24 holds true. It remains only to verify the validity of inequality (8.59). For this, we take into account properties (10.102)–(10.104) and Theorem 10.25. Then a, v, w, , z,). lim Jε (ahε , vεh , wεh , zεh ) = J 0 (,

ε→0

Hence, applying inequality (10.99), we obtain the required inequality J 0 (a, v, w, z) ≥ lim Jε (ahε , vεh , wεh , zεh ) − δ. ε→0

Remark 10.31. As follows from (10.95) and (10.96), the Neumann boundary control w ∈ L2 ((0, T ) × Γ0 ) has a “passive” influence on the limit object (10.94). At the same time, the limit cost functional (10.95) (in contrast to the limit problem (10.86)) is not indifferent to this control. Moreover, it can be represented in the form J 0 (a, v, w, z) = 2 (z(T, x) − y T )2 dx

Ω T

+2 0

+ 2γ ∗

(z − y ∗ )6 dxdt + κ∗

Ω T

T

v 2 dx1 dt

0

Γ0

(a − z 3 − f )2 dxdt + C(w). 0

(10.105)

Ω

Taking into account the variational properties of the problem (10.94) stated in Theorem 10.27, the constant C(w) should be defined as


T

393

(w0 )2 dx1 dt,

C(w) = 2 0

Γ0

where w 0 ∈ L2 ((0, T ) × Γ0 ) is a weak limit in L2 (0, T ; L2 (Ω, dνεh )) of the opti! ε -problem (see (10.38)–(10.40)). On the other hand, mal control w ε0 for the CP it is clear that an optimal solution (a0 , v 0 , w0 , z 0 ) to the problem (10.94)– (10.96) is always such that w0 = 0. So, in the sequel, we will omit the constant C(w) in (10.105). Moreover, in view of the variational properties of the prob! ε converges to 0 weakly lem (10.94) the Neumann optimal control w ε0 for CP 2 2 h in L (0, T ; L (Ω, dνε )). Thus, since the constrained minimization problem (10.94) admits interpretation in the form of an OCP, it follows that for the C-extended OCP ! ε -problem) there exists a unique limit with (10.38)–(10.40) (the so-called CP respect to w-convergence as ε → 0 and it can be represented in the form ⎫ ∂t z − div Ahom ∇z + 2dξ ∗ z = a + 2ξ ∗ g in (0, T ) × Ω,⎪ ⎪ ⎪ ⎪ ∂νAhom z = 0 on (0, T ) × ∂Ω \ Γ0 , ⎬ (10.106) ⎪ z = v on (0, T ) × Γ0 , ⎪ ⎪ ⎪ ⎭ z(0, x) = y for a.a. x ∈ Ω, 0

a ∈ L2 ((0, T ) × Ω), v ∈ L2 (0, T ; H 1 (Γ0 )), vL2 (0,T ;H 1 (Γ0 )) ≤ C0 ,

(10.107)

(z(T, x) − y T )2 dx

J 0 (a, v, z) = 2 Ω

T

∗ 6

(z − y ) dx dt + κ

+2 0

Ω

+ 2γ ∗

T

∗

T

v 2 dx1 dt 0

Γ0

(a − z 3 − f )2 dx dt −→ inf . (10.108) 0

Ω

This OCP will be called the CPhom -problem. In order to examine the variational properties for the CPhom -problem, we apply Theorem 10.27 and make use of the following result (for the proof see [172]). Proposition 10.32. The limit OCP (10.106)–(10.108) admits at least one solution. Then, combining the results of Theorem 10.27, Proposition 10.32, and Remark 10.31, we can give the conclusion concerning the variational properties of the CPhom -problem.

ε be the optimal solutions of Theorem 10.33. Let (a0ε , vε0 , wε0 , zε0 ) ∈ Σ ε>0 ! the C-extended problem CPε . Then a subsequence

394


+ *

ε (a0εk , vε0k , wε0k , zε0k ) k∈N of (a0ε , vε0 , wε0 , zε0 ) ∈ Σ

ε>0

and a triplet (a0 , v 0 , z 0 ) ∈ Y = L2 ((0, T ) × Ω) × L2 (0, T ; H 1 (Γ0 )) × L2 (0, T ; H 1 (Ω)) can be found such that lim Jε (a0ε , vε0 , wε0 , zε0 ) = lim

ε→ 0

inf

ε→ 0 (aε ,vε ,wε ,zε )∈ Σ bε

Jε (aε , vε , wε , zε )

J 0 (a, v, z) = J 0 (a0 , v 0 , z 0 ),

(10.109)

(a0εk , vε0k , wε0k , zε0k ) −→ (a0 , v 0 , 0, z 0 ) in the variable space Zε .

(10.110)

=

inf (a, v,z)∈Σ0

w

10.8 Recovery of the limiting singular optimal control problem Phom Our intention in this section is to show that the homogenized OCP with respect to the original one exists, preserves the fine variational properties, and can be recovered in an explicit form. For this, we will follow the diagram (10.28) and essentially use Lemmas 10.6–10.8 and Theorem 10.7. Let us begin by considering a couple of new OCPs. For the first of them, hom -problem as follows: we formulate the P I 0 (q, u, y) = 2 (y(T, x) − y T )2 dx

Ω T

+2 0

+ 2γ

∗

(y − y ∗ )6 dx dt + κ∗

Ω T

T

u2 dx1 dt

0

Γ0

(q − f )2 dx dt −→ inf, 0

(10.111)

Ω

⎫ ∂t y − div Ahom ∇y + 2dξ ∗ y − y3 = q + 2ξ ∗ g in (0, T ) × Ω,⎪ ⎪ ⎪ ⎪ ∂νAhom y = 0 on (0, T ) × ∂Ω \ Γ0 , ⎬ (10.112) ⎪ y = u on (0, T ) × Γ0 , ⎪ ⎪ ⎪ ⎭ for a.a. x ∈ Ω, y(0, x) = y 0

q ∈ L2 ((0, T ) × Ω), u ∈ L2 (0, T ; H 1 (Γ0 )), uL2 (0,T ;H 1 (Γ0 )) ≤ C0 .

(10.113)

The second problem is called the Phom -problem and has the following description:

10.8 Recovery of the limiting singular optimal control problem Phom

Ω

+ κ∗

T

(y − y ∗ )6 dx dt

(y(T, x) − y T )2 dx + 2

I 0 (u, y) = 2

T

0

Ω

u2 dx1 dt −→ inf, 0

395

(10.114)

Γ0

⎫ ∂t y − div Ahom ∇y + 2dξ ∗ y − y 3 = f + 2ξ ∗ g in (0, T ) × Ω,⎪ ⎪ ⎪ ⎪ ∂νAhom y = 0 on (0, T ) × ∂Ω \ Γ0 , ⎬ ⎪ y = u on (0, T ) × Γ0 , ⎪ ⎪ ⎪ ⎭ y(0, x) = y for a.a. x ∈ Ω,

(10.115)

0

u ∈ L2 (0, T ; H 1 (Γ0 )),

uL2 (0,T ;H 1 (Γ0 )) ≤ C0 .

(10.116)

Here, the functions y T ∈ L2 (Ω), y∗ ∈ L6 (0, T ; L6 (Ω)), y0 ∈ H 1 (Ω), and f ∈ L2 (0, T ; L2 (Ω)) and the values κ∗ , ξ ∗ , and γ ∗ are as in Theorems 10.28 and 10.30. hom can be interpreted as a virtual extension of the OCP It is clear that P hom P with the virtual control function q ∈ L2 (0, T ; L2 (Ω)). On the other hom (see Lemma 10.6). hand, CPhom can be viewed as the C-extension for P However, the validity of these conclusions is essentially based on the regularity hom and Phom , namely the sets of admissible solutions for these property of P problems have to be nonempty. In spite of the fact that it is still a challenging open problem in general, we have the following result. Lemma 10.34. Assume that conditions (10.88)–(10.93) are fulfilled. Assume also that the original OCP (10.5)–(10.11) is uniformly regular in the sense of Lemma 10.8. Then there are functions y ∈ L2 (0, T ; H 1 (Ω)) and u ∈ L2 (0, T ; H 1 (Γ0 )) such that hom , (f, u, y) ∈ Ξ

(u, y) ∈ Ξhom ,

(10.117)

hom and Ξhom denote the sets of admissible solutions for the P hom where Ξ hom and P -problems, respectively. Proof. Since the original problem Pε is uniformly regular, it follows that this

problem is solvable for every ε > 0 (see Theorem 10.3). Let (u0ε , p0ε , yε0 ) ∈ Ξε be a sequence of optimal solutions to the problem (10.5)–(10.11). Then, by Lemma 10.8, we have (fε + (yε0 )3 , u0ε , p0ε , yε0 ) ∈ Σε

for every ε > 0,

and, moreover, these quaternaries (fε + (yε0 )3 , u0ε , p0ε , yε0 ) are optimal solutions for the corresponding C-extended problems CPε . Hence, due to Theorem 10.27, we conclude that the sequence of prototypes + * ε yε0 )3 , u ε0 , pε0 , yε0 ) ∈ Σ (fε + ( ε>0

396


is uniformly bounded in Zε , and therefore, is relatively compact with respect to the w-convergence (see Theorem 10.25). Let (q, u, 0, y) ∈ Y be some w-cluster “quadruple” of it (see Theorem 10.33). Making use of Definition 10.24, the initial assumptions (10.88), and Theorem 10.25, we get yε0k )3 q fεk + (

in L2 (0, T ; L2 (Ω, dηεh )),

fε → f

in L2 (0, T ; L2 (Ω, dηεh )),

yε0k → y 0 3 yεk → y 3

in L2 (0, T ; L2 (Ω, dηεh )), in L2 (0, T ; L2 (Ω, dηεh )).

Thus, (q, u, 0, y) = (f + y 3 , u, 0, y), and due to Theorem 10.33, we obtain (f + y 3 , u, 0, y) ∈ Σ0

=⇒ =⇒

hom (f, u, y) ∈ Ξ (u, y) ∈ Ξhom .

(10.118)

This concludes the proof. Taking this assertion into account and using the well-known results of optimal control theory for distributed singular systems (see Lions [172]), we can give the following conclusion. Corollary 10.35. Under the assumptions of Lemma 10.34, there exist soluhom and Phom . tions to the OCPs P We are now in a position to establish the main result of this section concerning the variational properties of the Phom -problem. Theorem 10.36. Assume that the assumptions of Lemma 10.34 hold true. Let (u0ε , p0ε , yε0 ) ∈ Ξε be an optimal solution to the problem (10.5)–(10.11) for every ε > 0. Then there is a sequence of indices {εk } converging to 0 as k → ∞, and functions y 0 ∈ L2 (0, T ; H 1 (Ω)) and u0 ∈ L2 (0, T ; H 1 (Γ0 )) such that u ε0k u0 pε0k yε0k 0 3 yεk

weakly in L2 (0, T ; L2 (Ω, dμhεk )),

0

weakly in

2

2

L (0, T ; L (Ω, dνεhk )); L2 (0, T ; L2 (Ω, dηεhk )),

−→ y strongly in 3 −→ y 0 strongly in L2 (0, T ; L2 (Ω, dηεhk )), 2 0 ∇ yεk − ∇y 0 r ∈ L2 (0, T ; L2 (Ω, Vpot )), 0

lim

ε→0

Iε (u0ε , p0ε , yε0 )

0

0

= I0 (u , y ),

(10.119) (10.120) (10.121) (10.122) (10.123) (10.124)

where the pair (u0 , y0 ) is an optimal solution to the nonlinear problem (10.114)–(10.116).

10.8 Recovery of the limiting singular optimal control problem Phom

397

Remark 10.37. We say that a sequence of triplets {(uε , pε , yε )} converges weakly to (u, p, y) in the variable space L2 (0, T ; H 1 (ΓεD )) × L2 (0, T ; L2 (ΓεN )) × L2 (0, T ; H 1 (Ωε )) if (u, p, y) ∈ L2 (0, T ; H 1 (Γ0 )) × L2 (0, T ; L2 (Γ0 )) × L2 (0, T ; H 1 (Ω)) and conditions similar to (10.119)–(10.123) are satisfied. Proof. Using the same arguments as in the proof of Lemma 10.34, we can is conclude that the sequence of quaternaries (fε + ( yε0 )3 , u ε0 , pε0 , yε0 ) ∈ Σ relatively compact with respect to the w-convergence. Moreover, its every wcluster quadruple can be represented in the form (f + (y 0 )3 , u0 , 0, y0 ), where the limits elements u0 and y 0 are determined by relations (10.119)–(10.123) (see Definition 10.24). Then by Theorem 10.25, we have yε0 )3 , u ε0 , pε0 , yε0 ) = lim lim Jε (fε + (

ε→ 0

inf

ε→ 0 (aε ,vε ,wε ,zε )∈ Σ bε

=

inf (a, v,z)∈Σ0

Jε (aε , vε , wε , zε )

J 0 (a, v, z)

= J 0 (f + y 3 , u, y).

(10.125)

However, in view of the relations (10.9), (10.22), and (10.39), we immediately obtain Jε (fε + ( yε0 )3 , u ε0 , pε0 , yε0 ) = Jε (fε + (yε0 )3 , u0ε , p0ε , yε0 ) = Iε (u0ε , p0ε , yε0 ), J 0 (f − y 3 , u, y) = I0 (u, y). Thus, summing up the above results, we conclude that the variational property (10.124) holds true for (u, 0, y), and, hence, (u, y) = (u0 , y 0 ) is an optimal solution to the limit problem (10.114)–(10.116). It is clear now that the problem (10.114)–(10.116) can be interpreted as a “limit” OCP for the original OCP (10.5)–(10.11) as ε tends to 0. In order to formalize this notion, we give the obvious conclusion, which immediately follows from Theorems 10.28, 10.30, 10.7, and 10.36. Theorem 10.38. Assume that the OCP (10.5)–(10.11) is uniformly regular (see Lemma 10.8) and assumptions (10.88)–(10.93) hold true. Then the OCP (10.114)–(10.116) is the variational limit of the original one as ε → 0 in the following sense: (i) If a sequence of triplets (uε , pε , yε ) ∈ L2 (0, T ; H 1 (ΓεD )) × L2 (0, T ; L2 (ΓεN ))

×L2 (0, T ; H 1 (Ωε )) ε>0

is weakly convergent to (u, 0, y) in the space

398


L2 (0, T ; H 1 (ΓεD )) × L2 (0, T ; L2 (ΓεN )) × L2 (0, T ; H 1 (Ωε )) (see Remark 10.37) and there exists a subsequence of indices {εk } such ε for all k ∈ N, then that εk → 0 as k → ∞ and (uεk , pεk , yεk ) ∈ Ξ k the pair (u, y) is admissible for the Phom -problem, I0 (u, y) ≤ lim inf Iεk (uεk , pεk , yεk ). k→∞

(10.126) (10.127)

(ii) For every admissible pair (u, y) and any value δ > 0, there exist a δrealizing sequence ( uε , pε , yε ) ∈ L2 (0, T ; H 1 (ΓεD )) × L2 (0, T ; L2 (ΓεN ))

×L2 (0, T ; H 1 (Ωε )) ε>0

such that ( uε , pε , yε ) ∈ Ξε ∀ ε > 0,

weakly

( uε , pε , yε ) −→ ( u, p, y),

(u, 0, y) − ( u, p, y) ≤ δ, uε , pε , yε ) − δ. I 0 (u, 0, y) ≥ lim sup Iε (

(10.128) (10.129) (10.130)

ε→0

10.9 On suboptimal controls for Pε-problems In this section, we will derive the optimality conditions for the limit Cextended problem (10.106)–(10.108) from which an optimal pair for the Phom problem may be determined. For this, we use the Lagrange multiplier principle. We obtain the weak form of optimal system equations that an optimal solution (u0 , y 0 ) and Lagrange multipliers must satisfy. This optimality system can serve as a basis for the construction of suboptimal solutions to the original problem on the periodic structure Ωε . We are now in a position to apply the Lagrange principle to the OCP (10.106)–(10.108). Theorem 10.39. If a triplet (a0 , v 0 , z 0 ) ∈ L2 (0, T ; L2 (Ω)) × L2 (0, T ; H 1 (Γ0 )) × L2 (0, T ; H 1 (Ω)) is an optimal solution to the problem (10.106)–(10.108), then there exists a function (10.131) Ψ ∈ L2 (0, T ; H01 (Ω)) ∩ H 1 (0, T ; L2 (Ω)) such that the quadruple (a0 , v 0 , z 0 , Ψ ) satisfies the following optimality system:


399

0

⎫ ∂t z 0 − div Ahom ∇z + 2dξ ∗ z 0 = a0 + 2ξ ∗ g in (0, T ) × Ω,⎪ ⎪ ⎪ ⎪ ∂νAhom z 0 = 0 on (0, T ) × ∂Ω \ Γ0 , ⎬ (10.132) ⎪ z 0 = v 0 on (0, T ) × Γ0 , ⎪ ⎪ ⎪ ⎭ z 0 (0, x) = y0 for a.a. x ∈ Ω, ⎫

t ⎪ −∂t Ψ − div Ahom ∇Ψ + 2dξ ∗ Ψ = 12(z 0 − y ∗ )5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ∗ 0 0 3 0 2 −12γ a − z −f z in (0, T ) × Ω, (10.133) ⎪ ⎪ Ψ = 0 on (0, T ) × ∂Ω,⎪ ⎪ ⎪ ⎪ ⎭ for a.a. x ∈ Ω, Ψ (T, x) = −4 z 0 (T, x) − y T 3 −1 a0 = z 0 + f + [4γ ∗ ] Ψ a.e. in (0, T ) × Ω, ⎫ T ⎪ ∗ 0 0 ⎬ 2κ v − ∂ν Ahom t Ψ (v − v ) dx1 dt ≥ 0 [ ] Γ0 0 ⎪ ⎭ ∀v ∈ L2 (0, T ; H 1 (Γ0 )) : vL2 (0,T ;H 1 (Γ0 )) ≤ C0 .

(10.134)

(10.135)

Proof. Let (a0 , v 0 , z 0 ) be an optimal solution to the problem (10.106)–(10.108). To apply the Lagrange multiplier principle, we set

Y = z ∈ L2 (0, T ; H 1 (Ω)), z ∈ L2 (0, T ; (H 1 (Ω))∗ ) , U = L2 (0, T ; L2 (Ω)) × L2 (0, T ; H 1 (Γ0 )), V = L2 (0, T ; (H 1 (Ω))∗ ) × H 1 (Ω), and F (a, v, z) = z − div Ahom ∇z + 2dξ ∗ z − a − 2ξ ∗ g, z|t=0 − y0 . Then Fz (a0 , v 0 , z 0 ) : Y → V is an isomorphism. Hence, all assumptions of Theorem 9.35 are fulfilled. We define the Lagrange function as follows: T L(a, v, z, Ψ ) = 2 (z(T, x) − y T )2 dx + 2 (z − y ∗ )6 dx dt Ω

+ κ∗

T

0

T

Ω

+ 2d ξ ∗ − 2 ξ∗

T

T

T

Ω

(Ahom ∇z · ∇Ψ ) dx dt

0

Ω

T

zΨ dx dt − 0

T

Ω

gΨ dx dt 0

Ω

Ω

(a − z 3 − f )2 dx dt 0

z Ψ dx dt +

+ 0

v2 dx1 dt + 2γ ∗ Γ0

0

a Ψ dx dt 0

Ω

400


for all Ψ ∈ L2 (0, T ; H01 (Ω)) ∩ H 1 (0, T ; L2 (Ω)). Then, in accordance with Theorem 9.35, there exists a function Ψ ∈ L2 (0, T ; H01 (Ω)) ∩ H 1 (0, T ; L2 (Ω)) such that relations (9.126)–(9.127) are valid. It is easy to see that in this case, relation (9.126) takes the form (10.132)– (10.133), whereas (9.127) can be written as (10.134)–(10.135). The next question we are going to consider in this section concerns the approximation of optimal solutions of the original problem Pε for the values of the parameter ε small enough. It is clear that the numerical computation of optimal solutions to Pε -problems is very complicated due to the geometry of the original domain Ωε and the nonlinearity and large dimension of the corresponding optimality system. Hence, it is very difficult to keep an acceptable level of accuracy for the optimal solutions. Therefore, we will focus our attention on the possibility of defining approximate solutions which have to guarantee the closeness of the corresponding value of the cost functional (i.e., sub sub Iε (usub ε , pε , yε )) to its minimal one for ε small enough. In view of this, we make use the following concept (for a comparison, see Definition 9.33). Definition 10.40. We say that a sequence of pairs 0 0

(, uε , p,ε ) ∈ L2 (0, T ; H 1 (ΓεD )) × L2 (0, T ; L2 (ΓεN )) ε>0 is asymptotically suboptimal for the problem Pε if for every δ > 0, there is ε0 > 0 such that 0 0 inf I (u , p , y ) − I (, u , p , , y , ) ∀ ε < ε0 . (10.136) ε ε ε ε ε ε ε < δ, ε (uε ,pε ,yε )∈ Ξε

uε0 , p,ε0 ) we denote the corresponding solutions of the initialHere by y,ε = y,ε (, boundary value problem (10.5). Taking this concept into account, we get the following final result. Theorem 10.41. We suppose that the assumptions of Lemma 10.34 hold true. Let u0 ∈ L2 (0, T ; H 1 (Γ0 )) be an optimal Dirichlet control for the CPhom problem (10.106)–(10.108). Then the sequence of pairs

0 (10.137) (u |ΓεD , 0) ∈ L2 (0, T ; H 1 (ΓεD )) × L2 (0, T ; L2 (ΓεN )) ε>0 is asymptotically suboptimal for the original OCP Pε if and only if the se

quence (u0 |ΓεD , 0) is a reproducing one for problem (10.5) (i.e., provided this problem is solvable on this sequence of controls). Proof. We prove only the sufficient conditions of this assertion. Suppose that the sequence (10.137) is a reproducing one for the problem (10.5). Note that in view of recent results on the global steady-state boundary controllability of the blowing-up heat equations (see Coron and Trélat [75]), this supposition


401

has

to be true. Let us now consider the sequence of triplets 0every reason (u |ΓεD , 0, y,ε ) ε>0 , where y,ε = yε (u0 |ΓεD , 0) are the corresponding solutions of the initial-boundary value problem (10.5). It is easy to see that each of this triplet is admissible for the original control problem Pε . Then, by Lemma 10.8, we see that the quaternaries yε )3 , u0 |ΓεD , 0, y,ε ) (fε + (, are admissible for the C-extended problems CPε for every ε > 0. Hence, due to Theorem 10.27, we conclude that the sequence of prototypes * + ε (fε + ( yε )3 , u 0 , 0, yε ) ∈ Σ ε>0

is uniformly bounded in Zε , and it means that this one is relatively compact with respect to the w-convergence (see Theorem 10.25). Here, u 0 is a 2 2 h 0 prototype in L (0, T ; L (Ω, dμε )) of the control function u |ΓεD . Let (q, u, 0, y) ∈ Y be some w-cluster “quadruple” of this sequence (see Theorem 10.33). Applying Definition 10.24, the initial suppositions (10.88), Theorem 10.25, and the fact that u 0 |ΓεD = u 0 |ΓεD for every ε > 0, we have fεk + ( yε0k )3 q in L2 (0, T ; L2 (Ω, dηεh )), fε → f in L2 (0, T ; L2 (Ω, dηεh )), yεk → y in L2 (0, T ; L2 (Ω, dηεh )), ( yεk )

3

→ y 3 in L2 (0, T ; L2 (Ω, dηεh )),

u 0 u in L2 (0, T ; L2 (Ω, dμhε )), s

a u 0 −→ u 0.

Thus, (q, u, 0, y) = (f + y 3 , u0 , 0, y), and due to Theorem 10.33, we obtain (f + y 3 , u0 , 0, y) ∈ Σ0

=⇒

hom (f, u0 , y) ∈ Ξ

=⇒

(u0 , y) ∈ Ξhom .

Hence, taking Theorem 10.30 into account, we have a more precise result: The pair (u0 , y 0 ) ≡ (u0 , y) ∈ Ξ hom is optimal for the limit problem Phom . Let (u0ε , pε0 , yε0 ) ∈ Ξε ε>0 be optimal solutions to the problems Pε . Then

it is obvious that (fε + (yε0 )3 , u0ε , pε0 , yε0 ) ∈ Ξε ε>0 is the sequence of optimal solutions to CPε . We now observe that

402


Iε (uε , pε , yε ) − Iε (u0 , 0, y,ε ) (uε ,pε ,yε )∈ Ξε = Iε (u0ε , pε0 , yε0 ) − Iε (u0 , 0, y,ε ) ≤ Iε (u0ε , pε0 , yε0 ) − I0 (u0 , y 0 ) + I0 (u0 , y 0 ) − Iε (u0 , 0, y,ε ) ≤ Iε (u0ε , pε0 , yε0 ) − I0 (u0 , y 0 ) + 2 (y 0 (T, x) − y T )2 dx − (2 − h(ε)) (y 0 (T, x) − yεT )2 dηεh Ω Ω T T ∗ 6 0 ∗ 6 h + 2 (y − y ) dxdt − (2 − h(ε)) (yε − yε ) dηε dt 0 Ω 0 Ω T T 0 2 u2 dx1 dt − κ(ε)h(ε) dμhε dt u + κ∗ Γ0 Ω 0 0 inf

= J1 + J2 + J3 + J4 . To conclude the proof, we note that for a given δ > 0 the following can always be found: (i) (ii) (iii) (iv)

ε1 > 0 such that J1 < δ/4 for all ε < ε1 by Theorem 10.36; ε2 > 0 such that J2 < δ/4 for all ε < ε2 by Theorem 10.25; ε3 > 0 such that J3 < δ/4 for all ε < ε3 by Theorem 10.25; ε4 > 0 such that J4 < δ/4 for all ε < ε4 by Theorem 10.22 (see also Remark 10.23).

Thus, estimate (10.136) holds true for all ε < min{ε1 , ε2 , ε3 , ε4 }.

10.10 Optimal control problem for systems on thin lattice structures with blowup This section deals with some generalizations of the blowing-up OCP considered above. First, we show that the 2D lattice structure Ωε can be taken in much more general form than was indicated in Fig. 9.1 – namely let us define the set Ωε as follows: Ωε = Ω ∩ Fεh , where Fεh = εF h and F h denotes a thin grid that is the h-thickening of some singular Y -periodic graph F in the plane. Thus, Ωε ⊂ R2 can be viewed as a periodic open set with a reticulated-like structure contained in a fixed bounded open set Ω = (0, L 1 ) × (0, L 2 ). ε ε , and Sext In contrast to (9.1)–(9.3), we define the sets Qε , ΓεD , ΓεN , Sint as follows:

(10.138) Qε = (x1 , x2 ) ∈ R2 : x1 ∈ (0, L 1 ), x2 ∈ (0, ε) ⊂ Ω, ΓεD ⊂ ∂Ω ∩ Fεh ∩ {(x1 , x2 ) : x2 = 0} , ε Sint ⊆ [Ω \ Qε ] ∩ ∂Fεh ,

ε Sext

ΓεN ⊂ ∂Fεh ∩ Qε , = ∂Ωε \ ΓεD ∪ ΓεN ∪ Sint .

(10.139) (10.140)

10.10 Optimal control problem for blowing up systems

403

As usual, we relate the parameters h and ε assuming that h = h(ε) → 0 as ε → 0. The case when h = h(ε) → 1 as ε → 0 is not considered in this chapter because it indicates periodically perforated domains rather than thin structures. Let us denote by D(h) the area of the part of Y occupied by the graph F h . Let us define also the following values: LR (h) is the length of a boundary ∂F h in Y where the Robin boundary conditions are given and LD (h) and LN (h) are the lengths of the parts of boundaries of the set F h in Y , where the Dirichlet and Neumann control zones are located, respectively. In the domain Ωε , we consider the following nonlinear OCP Pε : 1 Iε (uε , pε , yε ) = (yε (T, ·) − yεT )2 dx h Ωε T 1 T ∗ 6 (yε − yε ) dx dt + κ(ε) u2 dx1 dt + h 0 Ωε 0 ΓεD T p2ε dH1 dt −→ inf (10.141) + ΓεN

0

subject to the constraints ⎫ in (0, T ) × Ωε , ⎪ ⎪ ⎪ ε on (0, T ) × Sext ,⎪ ⎪ ⎪ ⎬ ε on (0, T ) × Sint , D on (0, T ) × Γε , ⎪ ⎪ ⎪ on (0, T ) × ΓεN , ⎪ ⎪ ⎪ ⎭ for a.a. x ∈ Ωε , (10.142) pε ∈ L2 (0, T ; L2 (ΓεN )), uε ∈ Uε , (10.143)

2 1 : u ∈ L (0, T ; H (Γ0 )), uL2 (0,T ; H 1 (Γ0 )) ≤ C0 . (10.144)

∂t yε − div (A(x/ε)∇yε ) − yε3 = fε ∂ νA y ε = 0 ∂νA yε = ε2 (−d yε + gε ) yε = uε ∂νA yε = εpε yε (0, x) = yε,0 Uε = u|ΓεD

Here, pε ∈ L2 (0, T ; L2 (ΓεN )) and uε ∈ L2 (0, T ; H 1 (ΓεD )) are the boundary ε controls, gε ∈ L2 (0, T ; L2 (Sint )), fε ∈ L2 (0, T ; L2 (Ω)), yε, 0 ∈ H 1 (Ω), yε∗ ∈ 6 T 2 L ((0, T )×Ωε ), and yε ∈ L (Ω) are given functions, and κ(ε) is a given value. As for the matrix A, we suppose that A(x) is a Y -periodic measurable matrix such that A(·) ∈ L∞ (Y, R2×2 ), α0 ξ2 ≤ (A(x)ξ, ξ) ≤ α0−1 ξ2 for a.a. x ∈ Y

(10.145)

for some fixed constant α0 > 0 and every ξ ∈ R2 . Following the approach given above, our main intention is to construct the ! ε to (10.141)–(10.144) and then study its asymptotic C-extended problem CP behavior as ε tends to 0. For this, we make relations (9.6)–(9.8), (10.30)– (10.34), and (9.17) more precise. Having introduced the singular scaling measures μhε , νεh , λhε , and ηεh as we did it earlier, we have

404


1

N

ϕ dH = L (h) Λh N

D

ϕ dν ,

1

h

R

ϕ dx1 = L (h) Λh D

Y1

h

ϕ dH = L (h)

ϕ dλ ,

Λh R

(10.146)

ϕ dη h .

(10.147)

ϕ dx = D(h)

Y ∩F h

Y3

ϕ dμh , Y2

Y

Then, after the transformations were applied in Sect. 10.3, formulas (10.30)– (10.34) take the form

T

ε Sint

0

T

zε ϕψ dH1 dt = ε−1 LR (h(ε))

T

zε ϕψ dλhε dt, 0

Ω

T

w ε ϕψ dνεh dt,

wε ϕψ dH1 dt = LN (h(ε)) 0

ΓεN T

0

Ω

T

v2 dμhε dt,

v 2 dx1 dt = LD (h(ε)) ΓεD

0

T 0

T

ΓεN

0

0

T

w ε2 dνεh dt,

wε2 dH1 dt = LN (h(ε)) 0

zε ϕ ψ dx dt = D(h(ε))

Ωε

T

Ω

T 0

Ω

T

aε ϕ ψ dηεh dt,

aε ϕ ψ dx dt = D(h(ε)) 0

Ωε

zε ϕ ψ dηεh dt,

Ω

0

Ω

! ε -problem now as follows: It means that we can formulate the CP * + ε , inf Jε ( aε , vε , w ε , zε ) : ( aε , vε , w ε , zε ) ∈ Σ where D(h(ε)) Jε ( aε , vε , w ε , zε ) = ( zε (T, x) − yεT )2 dηεh h(ε) Ω T D(h(ε)) + ( zε − yε∗ )6 dηεh dt h(ε) Ω 0 T vε2 dμhε dt + LN (h(ε)) + κ(ε)LD (h(ε)) + γ∗

D(h(ε)) h(ε)

0

T 0

Ω

T

w ε2 dνεh dt

0

Ω

ε , (10.148) ( aε − zε3 − fε )2 dηεh dt −→ inf on Σ Ω


405

ε = {( aε , vε , w ε , zε ) : Σ ⎫ zε = vε μhε − a.e. on (0, T ) × Ω, ⎪ ⎪ ⎪ ⎪ ⎪ h ⎪ z (0, ·) = y η − a.e. in Ω; ε ε, 0 ⎪ ε ⎪ ⎪ ⎪ ⎪ vε L2 (0,T ;H 1 (Γ0 )) ≤ C0 , ⎪ ⎪ ⎪ ⎪ T ⎪ ⎪ ⎪ h ⎪ zε · ∇ϕ)ψ dηε dt − zε ϕ ψ + (A(x/ε)∇ ⎪ ⎪ ⎪ 0 Ω ⎪ ⎬ T T R ε L (h(ε)) h h +d zε ϕψ dλε dt = aε ϕ ψ dηε dt ⎪ . (10.149) ⎪ D(h(ε)) ⎪ Ω Ω 0 0 ⎪ T ⎪ R ⎪ ⎪ ε L (h(ε)) ⎪ h ⎪ g ϕψ dλ dt + ε ⎪ ε ⎪ D(h(ε)) ⎪ 0 Ω ⎪ ⎪ T ⎪ N ⎪ ε L (h(ε)) ⎪ h ⎪ w ε ϕψ dνε dt, + ⎪ ⎪ ⎪ D(h(ε)) 0 Ω ⎪ ⎪ ⎭ ∞ 2 D ∞ ∀ ϕ ∈ C0 (R ; Γε ), ∀ ψ ∈ C0 (0, T ) It is clear that for any thin grid which can be generated by the h-thickening of the corresponding singular graphs F, we have lim D(h(ε))/h(ε) = β1 > 0 and 0 ≤ βN , βR < +∞, βD = 0,

ε→0

where

(10.150)

βN = lim LN (h(ε)), βD = lim LD (h(ε)), ε→0

ε→0

(10.151)

and βR = lim LR (h(ε)). ε→0

Under the assumptions of Theorem 10.38, the limiting OCP can be recovered in the following explicit form: T I 0 (u, y) = β1 (y(T, x) − y T )2 dx + β1 (y − y ∗ )6 dx dt Ω

D + lim κ(ε)L (h(ε))

T

0

Ω

u2 dx1 dt −→ inf,

(10.152)

⎫ βR ⎪ ⎪ ∂t y − div Ahom ∇y +ξ ∗ d y − y3 ⎪ ⎪ β1 ⎪ ⎪ ⎪ ⎪ ⎪ β R ∗ ⎬ =f +ξ g in (0, T ) × Ω,⎪ β1 ⎪ ⎪ ∂νAhom y = 0 on (0, T ) × ∂Ω \ Γ0 , ⎪ ⎪ ⎪ ⎪ ⎪ y = u on (0, T ) × Γ0 , ⎪ ⎪ ⎪ ⎭ y(0, x) = y0 a.e. x ∈ Ω,

(10.153)

u ∈ L2 (0, T ; H 1 (Γ0 )), uL2 (0,T ;H 1 (Γ0 )) ≤ C0 .

(10.154)

ε→0

0

Γ0

subject to the constraints

406


Thus, the additional terms in the state equation (10.153) can appear only if the following conditions are valid: a “fattened” graph F h is a thin grid in the above given sense, the value βR is a strict positive constant, and the domain Ωε has a critical thickness, that is, ξ ∗ = limε→0 ε/h(ε) = θ > 0. It is interesting to note that for the domains Ωε which are sufficiently thick (i.e., when ξ ∗ = limε→0 ε/h(ε) = 0) the limit state equation in (10.153) is similar to the original one in (10.142). However, it should be emphasized that in the case of sufficiently thin domains Ωε , the variational limit for the problem (10.141)–(10.144) does not exist! Nevertheless, the passage to the limit in the OCP in such domains is possible as ε → 0 if only this problem instead ε of the Robin and Neumann conditions along the boundaries Sint and ΓεN , respectively, takes the homogeneous Neumann condition ∂νA yε = 0

on ΓεN ∪ ΓεN .

In this case, d = 0, gε = 0, and pε = 0 in (10.142). Then, as immediately follows from (10.153), the limiting form of such problem will be the same as in the previous case. Our next observation deals with the structural “stability” of the limit OCP (10.152)–(10.154). Indeed, as follows from Proposition 9.23 and Theorem 10.28, the passage to the limit in Pε containing boundary controls of two different types leads us to the OCP (10.152)–(10.154) with only one boundary control function: the Dirichlet control. Moreover, this conclusion holds true in spite of a “volume” of the boundary ∂Ωε occupied by the Dirichlet control zone. Thus, the Neumann boundary control in the limit problem can appear only if the domain Ωε has a critical thickness ξ ∗ = θ > 0 and LD (h) = 0 for every h > 0, that is, in the case when the Dirichlet boundary control is absent in the original statement. In particular, applying Theorems 10.28 and 10.30 provided the conditions LD (h) = 0, (10.138)–(10.140), and (10.150) hold true, it can be easily shown that for the Neumann optimal boundary control problem 1 1 T Iε (pε , yε ) = (yε (T, ·) − yεT )2 dx + (yε − yε∗ )6 dx dt h Ωε h 0 Ωε T p2ε dH1 dt −→ inf, (10.155) + 0

ΓεN

x ⎫ ∇yε − yε3 = fε in (0, T ) × Ωε ,⎪ ∂t yε − div A ⎪ ⎪ ε ⎪ ⎪ ⎪ ε N ⎪ ∂νA yε = 0 on (0, T ) × ∂Ω \ (Sint ∪ Γε ), ⎬ ∂νA yε = ε2 (−dyε + gε ) ∂νA yε = εpε yε (0, x) = yε,0

ε on (0, T ) × Sint ,

on (0, T ) × ΓεN , for a.a. x ∈ Ωε , pε ∈ L2 (0, T ; L2 (ΓεN )),

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(10.156)

(10.157)


407

there exists a limit which can be represented in the form

T

(y − y ∗ )6 dx dt

(y(T, x) − y ) dx + β1 T 2

I 0 (u, y) = β1 Ω

T

0

Ω

p2 dx1 dt −→ inf,

+ βN 0

(10.158)

Γ0

⎫ βR ⎪ ⎪ y − y3 ∂t y − div Ahom ∇y +ξ ∗ d ⎪ ⎪ β1 ⎪ ⎪ ⎪ ⎪ ⎪ β ∗ R ⎪ =f +ξ g in (0, T ) × Ω,⎪ ⎪ ⎬ β1 ∂νAhom y = 0 on (0, T ) × ∂Ω \ Γ0 , ⎪ ⎪ ⎪ ⎪ ⎪ β N ⎪ ∗ ∂νAhom y = ξ p on (0, T ) × Γ0 , ⎪ ⎪ ⎪ ⎪ β1 ⎪ ⎪ ⎭ y(0, x) = y0 for a.a. x ∈ Ω, p ∈ L2 (0, T ; L2 (Γ0 )).

(10.159)

(10.160)

It is quite obvious now (see (10.158)–(10.160)) that for the problem (10.155)–(10.157) in a sufficiently thick domain Ωε (i.e., in the case when ξ ∗ = 0), we have no OCP in the limit as ε → 0, but rather the initial-boundary value problem of the following structure: ⎫ ∂t y − div Ahom ∇y − y 3 = f in (0, T ) × Ω, ⎪ ⎬ ∂νAhom y = 0 on (0, T ) × ∂Ω, (10.161) ⎪ ⎭ y(0, x) = y0 for a.a. x ∈ Ω. It is interesting to note that in this case, in view of Theorem 10.36, the variational properties of the limit problem (10.161) can be formulated as follows: Let (p0ε , yε0 ) ∈ Ξε be an optimal pair for the Neumann boundary control problem (10.155)–(10.157). Then there is a sequence of indices {εk } converging to 0 as k → ∞, and functions y 0 ∈ L2 (0, T ; H 1 (Ω)) and p 0 ∈ L2 (0, T ; L2 (Γ0 )) such that pε0k −→ p 0

weakly in

L2 (0, T ; L2 (Ω, dνεhk )),

yε0k −→ y 0 strongly in L2 (0, T ; L2 (Ω, dηεhk )), 3 3 yε0k −→ y 0 strongly in L2 (0, T ; L2 (Ω, dηεhk )), 2 0 ∇ yεk − ∇y 0 r ∈ L2 (0, T ; L2 (Ω, Vpot )),

(10.162) (10.163) (10.164) (10.165)

408


lim

ε→0

Iε (p0ε , yε0 )

= const ≡ β1

(y 0 (T, x) − y T )2 dx Ω T

(y 0 − y ∗ )6 dx dt

+ β1 0

Ω

T

(p 0 )2 dx1 dt,

+ βN 0

(10.166)

Γ0

where the function y 0 is a solution of the Neumann boundary value problem (10.161).

11 Asymptotic Analysis of Optimal Control Problems on Periodic Singular Graphs

Partial differential equations (PDEs) on graphs have frequently been considered in the literature, beginning with the works of Lumer [180] and A. Vol’pert [247]. We also mention Lagnese, Leugering, and Schmidt [164], Lagnese and Leugering [163], Pokornyi et al. [216], and von Below [248]. Applications of the theory of PDEs on multi-link structures are ubiquitous. For instance, we encounter problems on graphs, or, more generally, on networked domains, in structural control, civil engineering, infrastructures, mechatronics, biology, medicine, traffic flow, electrical and communication networks, and fluid flow through a network of pipes or canals. For more details, we refer the reader to a recent review [160] on these topics. In this book, we will concentrate on the asymptotic analysis of optimal control problems for 1D elliptic equations on periodic graphs, as the period of the graph tends to 0. We focus on optimal control problems (OCPs) for linear elliptic equations with distributed and boundary controls. Using the variational approach, we show that the original problem on the periodic graph tends to a standard linear quadratic optimal control problem for a 2D homogenized system, and its solution can be used as a suboptimal control for the original problem. Even though PDEs on networked domains and their homogenized substitutes are very important, only a few papers [56, 163, 165, 184, 246] deal with the homogenization problem on periodic networks. Typically, not only the process on the graph itself but also the optimization and control of processes on such networks are of great importance. On the one hand, as one goes down the scales of the periodic structures, the numerical computation of the solution of these problems is very costly due to the singular behavior for small scales and the complexity of large networks, whereas on the other hand, local computations on a regular locally 1D grid are much easier than 2D or 3D calculations. It may thus happen, as in sparse-grid computations, that an originally 2D or 3D problem is approximated by a semidiscretization on a suitable grid, whereas on the other hand, as in carbon-nanotube technology, photonic lattice devices and in infrastructural problems involving water P.I. Kogut, G.R. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains, Systems & Control: Foundations & Applications, DOI 10.1007/978-0-8176-8149-4 11, © Springer Science+Business Media, LLC 2011

409

410

11 Asymptotic Analysis of OCPs on Thick Periodic Graphs

and gas networks, the vascular system, to mention just a few applications, a problem on a planar (or 3D) graph may be substituted by a homogenized one on a simple but higher-dimensional domain. Thus, the asymptotic analysis for OCPs on graphs appears to be a major tool in order investigate these questions. Moreover, in such an asymptotic analysis we may require both the optimal solution and the minimal value of the cost functional for the original problem to converge to the corresponding characteristics of a limit OCP, as ε tends to 0. It should be stressed that the original problem (on the graph) and the homogenized one (in a 2D domain) live in different function spaces. Moreover, if the small parameter ε is changed, then all components of the original control problem, including the ε-periodic graph Ωε , the control constraint sets, the cost functional, and the set where we seek its infimum are changed as well. Let us observe also that the Lebesgue measure of the “material” included in the periodic graph Ωε is equal to 0 for every ε > 0, whereas there exists a set Ω which is filled up by this planar graph in the limit, as ε → 0. In view of this, our approach is based on the description of OCPs for planar networks in terms of singular measures as proposed by Zhikov, Bouchitte, and Fragala in their recent works [26, 58, 255, 256]. Our emphasis is not on the homogenization of the underlying system of PDEs on graphs as such, but rather on the homogenization of the OCP with distributed controls and Neumann boundary controls on planar ε-periodic graphs as ε → 0.

11.1 ε-Periodic graphlike structures in R2 and their description In this section, we recall the main notions concerning ε-periodic graphlike structures in R2 , the description of the geometry of periodic graph structures and their boundaries in terms of singular measures. It is important to note that we introduce two types of singular measures. One of them we use for the representation of a ε-periodic bounded graph, and the second one is a “boundary” measure for the description of boundary conditions on a graphlike domain. Definition 11.1. We say that the set = [0, 1)2 is the cell of periodicity for some planar graph F on R2 if contains a “star” structure such that (see Fig. 7.6) the following hold: (i) All edges of this structure have a common point M ∈ int . (ii) Each edge is a line segment and all end points of these edges belong to the boundary of . (iii) In the set of end points (vertices) there exist pairs (Mi ; Mk ) such that Mk Mk i i or xM xM 1 = x1 2 = x2 . So, by analogy with Sect. 7.6, we admit the existence of isolated vertices.

11.1 ε-Periodic graphlike structures in R2 and their description

411

As usual, we always suppose that the small parameter ε ∈ E = (0, ε0 ] varies in a strictly decreasing sequence of positive numbers which converges to 0. Definition 11.2. We say that Fε is an ε-periodic graph on R2 if Fε = εF = {εx : x ∈ F}. Let S = {(x1 , x2 ) : x1 ∈ [0, 1), x2 = 0}, I ed = {Ij , j = 1, 2, . . . , K} be the set of all edges on and let M = {Mi , i = 1, 2. . . . , L} be the set of all vertices on which belong to S . Let Ω be an open bounded domain in R2 such that Ω = {(x1 , x2 ) : x1 ∈ Γ1 , 0 < x2 < γ(x1 )} , (11.1) where Γ1 = (0, a), γ ∈ C 1 ([0, a]), and 0 < γ0 = inf x1 ∈[0,a] γ(x1 ). It is clear that in this case, ∂Ω = Γ1 ∪ Γ2 , where Γ2 = ∂Ω \ Γ1 . Definition 11.3. We say that Ωε has an ε-periodic graphlike structure (see Fig. 7.7) if Ωε = Ω ∩ Fε . Following the approach of Sect. 7.6, we will describe the geometry of the set Ωε in terms of singular measures in R2 . With this aim, for every segment Ii ∈ I ed , i = 1, 2, . . . , K, we denote by μi its corresponding Lebesgue measure. Now, we define the -periodic Borel measure μ in R2 as K

μ=

gi · μi ,

(11.2)

i=1

where g1 , g2 , . . . , gK are non-negative weights such that dμ = 1. Thus, the support of the measure μ is the union of all edges Ii ∈ I ed . Since the homothetic contraction of the plane with a factor of ε−1 takes the grid F to Fε = εF, we introduce a “scaling” ε-periodic measure με as (11.3) με (B) = ε2 μ(ε−1 B) for every Borel set B ⊂ R2 . As a result, ε dμε = ε2 dμ = ε2 . Then the measure με weakly converges to the Lebesgue measure on R2 , that is, lim ϕ dμε = ϕ dx ∀ ϕ ∈ C0∞ (R2 ). (11.4) ε→0

R2

R2

Let μS be the S -periodic measure in R1 defined as μS =

L

ρj δMj ,

(11.5)

j=1

L where the non-negative weights ρj satisfy j=1 ρj = 1 and δMj are the Dirac measures located at the vertices Mj . It is the easy to see that μS is a Radon

412


measure, S dμS = 1. So, we may then define the corresponding scaling measure by μSε (B) = εμS (ε−1 B)

for any Borel set B ⊂ R1 .

(11.6)

Since εS dμSε = ε, it follows that dμSε converges weakly to d as ε → 0, where by d we denote the linear Lebesgue measure on R1 .

11.2 Statement of an optimal control problem on ε-periodic graphs To formulate the OCP on singular structures such as the ε-periodic graph Ωε = Ω ∩ εF, we introduce a certain associated Sobolev space V (Ω, Γ2 , dμ), where μ is the -periodic Borel non-negative measure on R2 that was defined in (11.2). Let us denote by C ∞ (Ω, Γ2 ) the class of smooth functions ϕ ∈ C ∞ (Ω) such that ϕΓ2 = 0. Here, Γ2 is the second “part” of the boundary ∂Ω = Γ1 ∪ Γ2 . Let us define the spaces L2 (Ω, dμε ) and L2 (Γ1 , dμSε ) in the usual way. Definition 11.4. We say that a function y(x) belongs to the functional class V (Ω, Γ2 , dμ) if there exist a vector z ∈ (L2 (Ω, dμ))2 and a sequence {ym ∈ C ∞ (Ω, Γ2 )}m∈N such that ∇ym − z 2 dμ = 0. (ym − y)2 dμ = 0, lim (11.7) lim m→∞

m→∞

Ω

Ω

In this case, we say that z is a gradient of y and denote it as ∇y (i.e., z = ∇y). Note that every function y ∈ V (Ω, Γ2 , dμ) may have many gradients z = ∇y. Moreover, if we denote by Γ (y) the set of gradients for a fixed function y ∈ V (Ω, Γ2 , dμ), then this set has the structure Γ (y) = ∇y + Γ (0) (see Sect. 6.4), where ∇y ∈ L2 (Ω, dμ)2 is some fixed gradient and Γ (0) is the set of gradients of 0, that is, by definition, g ∈ Γ (0) if there exists a sequence {ϕm ∈ C ∞ (Ω)} such that 2 ∇ϕm − g 2 dμ −→ 0 as m → ∞. ϕm dμ −→ 0 as m → ∞, Ω

Ω

From this, it follows immediately that the set of gradients Γ (y) is a closed subspace of L2 (Ω, dμ)2 . Let us recall also that any gradient ∇y can be represented as a sum of two orthogonal terms: ∇y = ∇t y + g, where g ∈ Γ (0) and ∇t y ⊥ Γ (0). Hence, the first term is also a gradient of y, and this gradient is minimal in the following sense: |∇t y|2 dμ = min |∇y|2 dμ. Ω

∇y∈Γ (y)

Ω

11.2 Statement of an optimal control problem

413

The crucial fact is that the space Γ (0) admits a pointwise description; namely there is a μ-measurable subspace T (x) ⊂ R2 such that Γ (0) = g ∈ L2 (Ω, dμ)2 : g(x) ∈ T ⊥ (x) . So, the “minimal” gradient is determined by the tangential condition ∇y(x) ∈ T (x) μ-a.e. The following result can be viewed as the necessary and sufficient conditions for the inclusion y ∈ V (Ω, Γ2 , dμ), where μ is defined in (11.3). Proposition 11.5. Let F be a -periodic unbounded graph on R2 , let μ be the -periodic Borel measure in R2 defined by (11.3), and let y be any function of V (Ω, Γ2 , dμ). Then the following hold: (i) y Ii ∈ H 1 (Ii ) for any edge Ii ∈ Ω ∩ F, where H 1 (Ii ) is the 1D Sobolev space on the segment Ii . (ii) The values of y I coincide at the vertices of the graph. i

The validity of Proposition 11.5 immediately follows from Lemma 3 in [58]. Remark 11.6. Due to the classical Sobolev embedding theorem, we have the obvious consequence of the previous statement: If y ∈ V (Ω, Γ2 , dμ), then its restriction on the set Ω ∩ F is a continuous function. Let {Aε (x) : Aε (·) ∈ L∞ (Ω, R2×2 ; dμε )}ε∈N be a family of matrices such that α0 ξ 2 ≤ (Aε (x)ξ, ξ) ≤ α0−1 ξ 2

for με -a.e. x ∈ Ω,

(11.8)

where α0 > 0 is some constant which is independent of ε. We define the OCP on the ε-periodic graphlike domain Ωε = Ω ∩ εF as follows: Find a “boundary” control h0ε ∈ L2 (Γ1 , dμSε ), a “distributed” control u0ε ∈ L2 (Ω, dμε ), and a corresponding state yε0 ∈ V (Ω, Γ2 , dμε ) such that the cost functional 2 2 uε dμε + k3 h2ε dμSε (11.9) Iε (yε , uε , hε ) = k1 (yε − zd ) dμε + k2 Ω

Ω

Γ1

is minimized subject to the following constraints: yε ∈ V (Ω, Γ2 , dμε ), uε ∈ L2 (Ω, dμε ), hε ∈ L2 (Γ1 , dμSε ),

(11.10)

(Aε (x)∇yε , ∇ϕ) dμε + Ω

=

α yε ϕ dμε Ω

hε ϕ dμSε ,

uε ϕ dμε + Ω

Γ1

∀ ϕ ∈ C ∞ (Ω, Γ2 ) (α > 0), (11.11)

414


|uε | ≤ cu

με -a.e. in Ω,

|hε | ≤ ch

μSε -a.e. on Γ1 .

(11.12)

Here, cu and ch are some positive constants, zd ∈ C 0 (Ω) is a given function, με and μSε are the “scaling” measures defined by (11.3) and (11.6), respectively, and Aε (x) ∈ L(R2 , R2 ) is a με -measurable symmetric matrix satisfying inequality (11.8), k1 , k2 , k3 > 0 are given constants. Note that by the vectorfunction ∇yε in (11.11), we mean only that ∇yε ∈ Γ (yε ) is some gradient of yε in the sense of Definition 11.4 and ∇yε satisfies, with yε , the integral identity (11.11). We emphasize also that the notion “boundary” and “distributed” controls should be understood with respect to the corresponding measures and the integral identity (11.11). For example, the inclusion h ∈ L2 (Γ1 , dμSε ) implies that this function is uniquely defined by the respective set of values at the points Kε = Γ1 ∩ (∪n∈Z (εM + nε)). Here, M = {Mi , i = 1, 2, . . . , L} is the set of all vertices on which belong to S . Note that by definition of the μS we have μSε (Γ1 \Kε ) = 0. First, we show that for every uε ∈ L2 (Ω, dμε ), hε ∈ L2 (Γ1 , dμSε ), and ε ∈ E, there exists a unique pair (yε , ∇yε ) that satisfies identity (11.11). Lemma 11.7. Under the standing assumption with respect to the measures μ and μS and with respect to the matrix Aε , there exists a unique function yε ∈ V (Ω, Γ2 , dμε ) and a unique gradient of this function ∇yε ∈ Γ (yε ) that satisfy (11.11). Proof. Since the set C ∞ (Ω, Γ2 ) is dense in the class V (Ω, Γ2 , dμε ), the lefthand side of (11.11) induces a new scalar product on L2 (Ω, dμε )×L2 (Ω, dμε )2 , and the corresponding norm is equivalent to the usual norm in this space. Thus, the existence and uniqueness of the solution regarded as the pair (yε , ∇yε ) is an easy consequence of the Lax–Milgram lemma. However, the uniqueness is twofold here: There exists a unique function yε in the Sobolev space V (Ω, Γ2 , dμε ) such that only one of its gradients satisfies the identity (11.11). The uniqueness and existence of such a gradient was proved in [256]. It is interesting to note that the gradient ∇yε in this identity is defined only by matrix Aε alone and it is not related to the equation itself (for details, we refer to Zhikov [256]). Remark 11.8. It is easy to see that the solution of (11.11) satisfies the following estimate: yε L2 (Ω,dμε ) + ∇yε (L2 (Ω,dμε ))2

(11.13) ≤ α−1 uε L2 (Ω,dμε ) + hε L2 (Γ1 ,dμSε ) for every ε ∈ E, uε ∈ L2 (Ω, dμε ), and hε ∈ L2 (Γ1 , dμSε ), where the constant α is independent of ε. Indeed, if we take ϕ = yε as a test function in (11.11) and use Young’s inequality, we immediately obtain relation (11.13).

11.2 Statement of an optimal control problem

415

Definition 11.9. The triplet (y, u, h) ∈ Zε (Ω, Γ1 ) is called admissible if (y, u, h) satisfies the restrictions (11.10)–(11.12). We denote by Ξε the set of all admissible triplets for the OCP (11.9)–(11.12). Remark 11.10. In view of (11.12) and estimate (11.13), we see that the sequence of sets {Ξε }ε∈E is uniformly bounded in the following sense: There is a constant C > 0 such that sup

sup

ε∈E (yε ,uε ,hε )∈Ξε

yε L2 (Ω,dμε ) + uε L2 (Ω,dμε ) + hε L2 (Γ1 ,dμSε ) + ∇yε (L2 (Ω,dμε ))2 ≤ C. (11.14)

Moreover, each of the sets Ξε is convex and closed with respect to the weak convergence in the space L2 (Ω, dμε ) × L2 (Ω, dμε ) × L2 (Γ1 , dμSε ). Indeed, let {uk }k∈N ⊂ L2 (Ω, dμε ) and {hk }k∈N ⊂ L2 (Γ1 , dμSε ) be any bounded sequences such that uk → u weakly in L2 (Ω, dμε ),

hk → h weakly in L2 (Γ1 , dμSε ).

(11.15)

We define the sequence {yk ∈ H 1 (Ω, dμε )} as the corresponding solutions of the problem (11.11) with u = uk and h = hk . Then by inequality (11.13), there exists a constant C > 0 such that yk L2 (Ω,dμε ) ≤ C,

∇yk (L2 (Ω,dμε ))2 ≤ C,

where, for every k ∈ N, ∇yk is the unique gradient of yk satisfying the integral identity (11.11) for suitable uk and hk . Hence, we may assume that there is a function y ∈ L2 (Ω, dμε ) and a vector p ∈ L2 (Ω, dμε )2 such that yk → y weakly in L2 (Ω, dμε ), in

∇yk → p weakly in L2 (Ω, dμε )2 . (11.16)

Therefore, using (11.15) and (11.16), we can pass to the limit as k → ∞ [(Aε ∇yk , ∇ϕ) + αyk ϕ] dμε = uk ϕ dμε + hk ϕ dμSε Ω

Ω

Ω

and obtain [(Aε p, ∇ϕ) + αyϕ] dμε = uϕ dμε + Ω

Ω

hϕ dμSε

∀ ϕ ∈ C ∞ (Ω, Γ2 ).

Γ1

This proves that p ∈ Γε (y), that is, p = ∇y is some gradient of the function y ∈ H 1 (Ω, dμε ). As a result, we have obtained that for every ε ∈ E, the set Ξε is sequentially closed with respect to the weak convergence. Taking this remark into account and also the fact that the norm in L2 (Ω, dμε ) × L2 (Ω, dμε ) × L2 (Γ1 , dμSε ) is sequentially weakly lower semicontinuous, we come to the following conclusion (due to Tonelli’s direct method).

416


Lemma 11.11. For every ε ∈ E, the OCP (11.9)–(11.12) has a unique solution, that is, there exists a unique triplet (yε0 , u0ε , h0ε ) ∈ Ξε such that Iε (yε0 , u0ε , h0ε ) =

inf (yε ,uε ,hε )∈Ξε

Iε (yε , uε , hε ).

Note that the uniqueness of this solution is a consequence of the convexity property for Ξε and strict convexity of the cost functional Iε . As was mentioned earlier, the main question of this chapter is to study the asymptotic behavior of the OCP (11.9)–(11.12) as ε → 0. With this aim, we represent this problem for varying values of ε ∈ E in the form of the sequence of corresponding constrained minimization problems

Iε (yε , uε , hε ) ; ε → 0 , (11.17) inf (yε ,uε ,hε )∈Ξε

where the cost functional Iε : Ξε → R and the sets of admissible triplets are defined in (11.9) and (11.10)–(11.12), respectively. Then the definition of an appropriate limiting OCP to the family (11.9)–(11.12) as ε → 0 can be reduced to the analysis of the limit properties of the sequences (11.17). This will be done through the concept of variational convergence of constrained minimization problems (see Sect. 8.4). Definition 11.12. We say that the OCP (11.9)–(11.12) has a limiting problem as ε tends to 0, if the following hold: (i) For the sequence of the corresponding constrained minimization problems (11.17), there exists a variational limit as ε → 0 in the sense of Definition 8.22, that is,

inf (yε ,uε ,hε )∈Ξε

Iε (yε , uε , hε )

as →0

−→

inf (y,u,h)∈Ξ0

I0 (y, u, h) .

(11.18)

(ii) The minimization problem (11.18) can be recovered in the form of some OCP.

11.3 Convergence formalism in the variable spaces associated with ε-periodic graphs In this section, we focus our attention on the convergence formalism in variable spaces associated with boundary OCPs on ε-periodic graphs. For the definitions and main properties of the weak and two-scale weak convergence in the variable spaces L2 (Ω, dμε ) and L2 (Γ1 , dμSε ), we refer to Sects. 6.2 and 6.3. Note that the “scaling” measure μSε is singular with respect to με and associated with a “boundary condition” on the set ∂(Ω ∩ Fε ). Moreover, at the heart of the two-scale convergence in L2 (Γ1 , dμSε ), there is the following mean value property of periodic functions (see Theorem 6.21 for the proof).

11.3 Convergence formalism in spaces associated with ε-periodic graphs

417

Proposition 11.13. Assume that η is any μS -measurable S -periodic function on R1 . Then for any a ∈ C0∞ (R1 ), the following equality holds true: S lim a(l)η(ε−1 l) dμSε = η a(l) dl, ε→0

where ηS =

S

Γ1

Γ1

η(ξ) dμS (ξ) is the mean value of η on the cell S .

We define the sets of potential and solenoidal vectors on the cell of period∞ ∞ icity following Sect. 6.4. Let Cper = Cper () be the space of infinitely differ2 entiable periodic functions and let Lper (, dμ)2 be the space of μ-measurable periodic functions f = [f1 , f2 ] such that (f12 (x) + f22 (x)) dμ < ∞. Ω

We say that a vector function g belongs to the space Vpot of potential vectors ∞ } such that if there exists a sequence {ϕm ∈ Cper ∇ϕm − g2 dμ → 0 as m → ∞. that is, ∇ϕm → g in L2per (, dμ)2 ,

We also say that a vector function b belongs to the space Vsol of solenoidal vectors if b is orthogonal to all potential vectors (i.e., (b, g)R2 dμ = 0 for every g ∈ Vpot ). In view of this, we always have the decomposition L2 (, dμ)2 = Vpot ⊕ Vsol .

(11.19)

Let us denote by L2 (Ω, ) the space L2 (Ω × , dx × dμ), that is, y(x, z) ∈ L2 (Ω; ) if y is dx × dμ-measurable on Ω × and y 2 (x, z) dx dμ(z) < ∞. Ω

Then from (11.19), we immediately have that

L2 (Ω, )2 ≡ L2 Ω, L2 ()2 = L2 (Ω, Vpot ) ⊕ L2 (Ω, Vsol ).

(11.20)

Here, by L2 (Ω, Vpot ) and L2 (Ω, Vsol ) we understand the following spaces: (i) L2 (Ω, Vpot ) is the closure in L2 (Ω, ) of the linear span of the vectors ∞ f (x)∇z ϕ(z), where f ∈ C0∞ (Ω) and ϕ ∈ Cper . (ii) L2 (Ω, Vsol ) is the closure in L2 (Ω, ) of the linear span of the vectors f (x)b(z), where f ∈ C0∞ (Ω) and b ∈ Vsol . In what follows, we need the following properties of the -periodic Borel measure μ (see Sect. 6.4). Theorem 11.14. Let μ be the non-negative periodic Borel measure in R2 which is defined in (11.3). Then μ is nondegenerate and ergodic, that is,

418


(i) (nondegeneratedness) For every vector η ∈ R2 , there exists a vector b ∈ Vpot such that b(z) dμ = η; (ii) (ergodicity) Every function y ∈ H 1 (, dμ) with gradient 0 is constant μ-everywhere. Remark 11.15. The validity of this theorem can also be illustrated by the following observation. In view of (11.3), we have μ(Ω\F) = 0. Therefore, any functions taking the same values on the graph Ω ∩ F coincide as elements of L2 (Ω, dμ). Thus, every element of the space H 1 (Ω, dμ) is uniquely defined by the respective element of the space ∪H 1 (Ij ), where {Ij } is the set of all edges on the graph Ω ∩ F. Let {(yε , uε , hε ) ∈ Ξε }ε∈E be any sequence of admissible triplets. For every fixed ε, each of these triplets (yε , uε , hε ) belongs to the corresponding functional space Zε (Ω, Γ1 ) ≡ V (Ω, Γ2 , dμε ) × L2 (Ω, dμε ) × L2 (Γ1 , dμSε ) depending on the small parameter ε. Definition 11.16. We say that the sequence of triplets {(yε , uε , hε )}ε∈E is weakly convergent in the variable space Zε (Ω, Γ1 ), or, in short, is w-convergent, if there are functions y ∈ H 1 (Ω), u ∈ L2 (Ω), h ∈ L2 (Γ1 ), p ∈ L2 (Ω, )2 , and there exists a sequence of gradients {∇yε ∈ L2 (Ω, dμε )2 }ε∈E such that uε u in L2 (Ω, dμε ), 2

2

yε y(x) in L (Ω, dμε ),

hε h in L2 (Γ1 , dμSε ), 2

(11.21)

∇yε p(x, z) in L (Ω, dμε ) , 2

2

p(x, y) − ∇y(x) ∈ L (Ω, Vpot ). 2

(11.22) (11.23)

Remark 11.17. Note that in Definition 11.16, a two-scale limit y has to be independent of the second variable z. Moreover, as follows from (11.23), we do not conjecture that for the family of H 1 -functions {yε ∈ V (Ω, Γ2 , dμε )}, there exists a sequence of gradients {∇yε }ε∈E such that ∇yε → ∇y μ-weakly or two-scale weakly. As we will see later, conditions (11.21)–(11.23) will be sufficient in order to identify the limit OCP on the graph. In the following theorem, we establish sufficient conditions of the relative w-compactness of uniformly bounded sequences. Theorem 11.18. Let {(yε , uε , hε ) ∈ Zε (Ω, Γ1 )}ε∈E be a sequence for which the following conditions hold: (i) There exists a constant C > 0 such that sup yε L2 (Ω,dμε ) , uε L2 (Ω,dμε ) , hε L2 (Ω,dμSε ) ≤ C. ε

(11.24)

11.3 Convergence formalism in spaces associated with ε-periodic graphs

419

(ii) There exists a bounded sequence of gradients {∇yε ∈ L2 (Ω, dμε )2 }, that is, |∇yε |2 dμε < +∞. lim sup ε→0

Ω

Then the sequence {(yε , uε , hε )}ε∈E is relatively compact with respect to the w-convergence. Proof. First, we note that in view of the properties of the weak (two-scale) convergence in the variable spaces, it may be supposed without loss of generality that ⎫ 2 ⎪ yε y(x, z) ∈ L2 (Ω, ), ⎪ ⎪ ⎪ ⎪ ⎪ S 2 S ⎪ ⎪ h(l, ξ) dμ (ξ) in L (Γ1 , dμε ), ⎬ hε h(l) = S (11.25) 2 ⎪ ∇yε p (x, z) ∈ L2 (Ω, ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 uε u(x) = u(x, z) dμ(z) in L (Ω, dμε ). ⎪ ⎭

Let b ∈ L2 (, dμ)2 be a vector-valued function for which there is an element a ∈ L2 (, dμ) such that ∞ a(z) ϕ(z) dμ = (b(z), ∇ϕ(z)) dμ, ∀ ϕ ∈ Cper (). (11.26) −

Taking ϕ ∈ C0∞ (Ω) as a test function and using the equality ∇(ϕyε ) = ϕ∇yε + yε ∇ϕ, we have

ε ∇yε (x)ϕ(x), b(ε−1 x) dμε = ε ∇(ϕyε ), b(ε−1 x) dμε Ω Ω

−ε yε (x) ∇ϕ(x), b(ε−1 x) dμε (11.27) Ω

and

a(ε−1 x)ϕ(x) dμε = ε

− Ω

(b(ε−1 x), ∇ϕ) dμε .

(11.28)

Ω

Therefore, applying (11.28) to (11.27), we obtain

−1 ε ϕ(x) ∇yε (x), b(ε x) dμε = − a(ε−1 x)yε (x)ϕ(x) dμε Ω Ω

−ε yε (x) ∇ϕ(x), b(ε−1 x) dμε . Ω

(11.29) Taking (11.25) and supposition (ii) into account, one gets

420


lim ε ∇yε (L2 (Ω,dμε ))2 = 0,

lim ε

ε→0

ε→0

ϕ(x) ∇yε (x), b(ε−1 x) dμε = 0,

Ω

and by definition of the weak two-scale convergence, we have

lim ε yε (x) ∇ϕ(x), b(ε−1 x) dμε = 0. ε→0

Ω

Therefore, passing to the limit in (11.29), we conclude lim a(ε−1 x)yε (x)ϕ dμε = a(z)y(x, z) dμ(z)ϕ(x) dx = 0. ε→0

Ω

Ω

(11.30)

Due to the approximation lemma [257] and the ergodicity property of the measure μ (see Theorem 11.14), the set of all functions a ∈ L2 (, dμ) satisfying condition (11.26), where b is a vector from (L2 (, dμ))2 , is dense in the subspace of functions in L2 (, dμ) with mean value 0. Thus, from (11.30), we immediately conclude that y(x, z) = y(x), that is, the weak two-scale limit y(x, z) in (11.25) is independent of z. Let us show now that condition (11.23) is satisfied. For this, we consider equality (11.27) with any vector b ∈ Vsol (i.e., b ⊥ Vpot ). This yields

ϕ(x) ∇yε (x), b(ε−1 x) dμε Ω

=− yε (x) ∇ϕ, b(ε−1 x) dμε ,

∀ ϕ ∈ C0∞ (Ω). (11.31)

Ω

Passing in (11.31) to the limit (in the sense of two-scale convergence) and using (11.25), we obtain p(x, y), b(z) dμ(z) ϕ(x) dx Ω y(x) =− b(z) dμ(z), ∇ϕ(x) dx.

(11.32)

Ω

Note that since p ∈ L2 (Ω, ), it follows that the function (p(x, z), b(z)) dμ(x) v(x) =

belongs to L2 (Ω). Let us set Θ(x) = y(x) (11.32) in the form

b(z) dμ(z). Then rewriting

vϕ dx = − Ω

(Θ, ∇ϕ) dx Ω

11.4 Variational convergence on varying graphs

421

and integrating by parts the expression on the right-hand side, we conclude that v(x) = ∇y(x), b(z) dμ(z) .

Now, we may use the nondegeneracy property of the measure μ (Theo b(z) dμ(z), there exists a vector q ∈ Vpot rem 11.14). As a result, for η = (Vpot ⊂ L2 ()) such that η = q(z) dμ(z). Therefore,

v(x) = ∇y(x), q(z) dμ(z) ,

and we can draw the following conclusion: Since (∇y,

q dμ) ∈ L2 (Ω) in

the sense of distributions and q ∈ Vpot , it follows that ∇y ∈ L2 (Ω)2 in the sense of distributions as well. Hence, y ∈ L2 (Ω), ∇y ∈ L2 (Ω)2 ⇒ y ∈ H 1 (Ω). In view of this, (11.32) can be rewritten in the form ϕ(x) (p(x, z), b(z)) dμ(z) dx = ϕ(x) ∇y, b(z) dμ(z) dx, Ω

that is,

Ω

Ω

(p(x, z) − ∇y(x), b(z)) dμ(z)ϕ(x) dx = 0

for every ϕ ∈ C0∞ (Ω) and b ∈ Vsol . Since the linear span of the vector-valued functions ϕ(x)b(z) is dense in L2 (Ω, Vsol ) and the orthogonal decomposition (11.20) holds, it leads to the inclusion p(x, z) − ∇y(x) ∈ L2 (Ω, Vpot ). This concludes the proof.

11.4 Variational convergence of constrained minimization problems on varying graphs The main object in this section is the sequence of constrained minimization problems

Iε (y, u, h) , ε → 0 , (11.33) inf (y,u,h)∈Ξε

where Iε : Ξε → R and Ξε ⊂ Zε (Ω, Γ1 ), ∀ ε ∈ E. To begin, we show that the space L2 (Ω) possesses a weak approximation property in the sense of Definition 8.23.

422


Lemma 11.19. The weak approximation property for L2 (Ω) with respect to the family of Borel measures {με }ε∈E holds true, that is, u − u∗ L2 (Ω) ≤ δ and uε u∗ in L2 (Ω, dμε ) (11.34) (in this case, the sequence uε ∈ L2 (Ω, dμε ) ε∈E is called δ-realizing). Proof. Let u be any element of L2 (Ω). Since the inclusion C0∞ (R2 ) ⊂ L2 (Ω) is dense with respect to the strong topology for L2 (Ω), it follows that for a given value δ > 0, there is an element u∗ ∈ C0∞ (R2 ) such that u − u∗ L2 (Ω) ≤ δ. Let us construct the δ-realizing sequence as follows: uε = u∗ for every ε > 0. Then due to the weak convergence of the measures dμε dx, we have uε ϕψ dμε = u∗ ϕψ dx, ∀ ϕ ∈ C0 (Ω), (11.35) lim ε→0 Ω Ω lim (uε )2 dμε = (u∗ )2 dx. (11.36) ε→0

Ω

Ω

Hence, by the criterium of strong convergence in L2 (Ω, dμε ), we obtain uε → u∗ in L2 (Γ1 , dμSε ). It is clear that the same assertion holds for the space L2 (Γ1 ), that is, this space possesses the weak approximation property with respect to the family of measures μSε . Taking into account these results, we give the following notion of set convergence in the space with variable measure (for comparison, we refer to Definition 7.58). Definition 11.20. We say that a set Ξ0 ⊂ Y0 = H 0 (Ω) × L2 (Ω) × L2 (Γ1 ) is the sequential two-scale limit, or K-limit, of the sequence {Ξε ⊂ Zε (Ω, Γ1 ) ≡ V (Ω, Γ2 , dμε ) × L2 (Ω, dμε ) × L2 (Γ1 , dμSε )}ε∈E (11.37) if the following conditions are satisfied: (i) For every triplet (y, u, h) ∈ Ξ0 and any value δ > 0, there exist a constant ε0 ∈ E and a δ-realizing sequence {(yε , uε , hε )}ε∈E such that w (yε , uε , hε ) −→ ( y, u , h), (y, u, h) − ( y, u , h)Y0 ≤ δ.

(yε , uε , hε ) ∈ Ξε , ∀ ε ≤ ε0 ,

(ii) If {Ξεk } is a subsequence of {Ξε }ε∈E and {(yk , uk , hk )} is a sequence wconverging to (y, u, h) such that (yk , uk , hk ) ∈ Ξεk for every k ∈ N, then (y, u, h) ∈ Ξ0 . Remark 11.21. Note also that if dμε = dx, δ = 0, and dμSε = dl in (11.37), then the notion of the K-limit set coincides with the well-known notion of the sequential topological limit in the sense of Kuratowski with respect to the product of weak topologies in H 1 (Ω), L2 (Ω), and L2 (Γ1 ), respectively (see Definition 7.21).

11.5 Asymptotic analysis of OCPs on ε-periodic graphs

423

Now, we are in the position to clarify the convergence property (11.18). Following the concept of variational convergence of constrained minimization problem in variable spaces (see Definition 8.24), we make use of the following notion. Definition 11.22. We say that a minimization problem I0 (y, u, h) inf

(11.38)

(y,u,h)∈Ξ0

is the variational limit of the sequence (11.33) with respect to w-convergence if Ξ0 ⊂ Y0 is a nonempty K-limit of the sets {Ξε }ε∈E and the conditions (d)–(dd) of Definition 8.24 hold true with Xε = V (Ω, Γ2 , dμε ) × L2 (Ω, dμε ) × L2 (Γ1 , dμSε ), X = H 0 (Ω) × L2 (Ω) × L2 (Γ1 ). As a result, Theorem 8.25 (see also Corollary 8.37 of Theorem 8.35) implies the following properties of variational limit in the sense of Definition 11.22. Theorem 11.23. Assume that the constrained minimization problem (11.38) is the variational limit of the sequence (11.33) in the sense of Definition 11.20 and this problem has a unique solution (y 0 , u0 , h0 ). For every ε ∈ E, let (yε0 , u0ε , h0ε ) ∈ Ξε be an optimal solution of the corresponding problem (11.9)– (11.12). If the sequence {(yε0 , u0ε , h0ε )}ε∈E is relatively w-compact, then w

(yε0 , u0ε , h0ε ) −→ (y 0 , u0 , h0 ), inf (y,u,h)∈ Ξ0

(11.39)

I0 (y, u, h) = I0 y 0 , u0 , h0 = lim Iε (yε0 , u0ε , h0ε ) ε→0

= lim

inf

ε→0 (yε ,uε ,hε )∈ Ξε

Iε (yε , uε , hε ).

(11.40)

11.5 Asymptotic analysis of optimal control problems on ε-periodic graphs We begin with the so-called “convergence property of gradients of arbitrary solutions.” This property can be viewed as a natural requirement on the homogenized matrix to the family of matrices Aε in (11.11). Let {Aε (x) ∈ L(R2 , R2 )}ε∈N be a family of square με -measurable matrices satisfying inequality (11.8) for every ε ∈ E. Definition 11.24. (Convergence of gradients of arbitrary solutions). We say that a matrix Ahom (x) ∈ L(R2 , R2 ) is the homogenized matrix with respect to the family {Aε (x)} as ε tends to 0, if the following hold:

424


(a) Ahom ∈ [L∞ (Ω)] , Ahom (x) is coercive. (aa) Aε ∇yε Ahom ∇y0 in L2 (Ω, dμε )2 for every sequence {(yε , uε , hε ) ∈ Zε (Ω, Γ1 )}ε∈E such that (yε , ∇yε ) satisfy relation (11.11) and 2×2

w

(yε , uε , hε ) → (y0 , u0 , h0 ). In order to find a liming problem to the family (11.9)–(11.12), we introduce two auxiliary sets

[(Aε (x)∇y, ∇ϕ) + αyϕ] dμε Graph(Pε ) = (y, u, h) ∈ Zε (Ω, Γ1 ) : Ω S ∞ = uϕ dμε + hϕ dμε , ∀ ϕ ∈ C (Ω, Γ2 ) , (11.41) Ω

Γ1

Graph(Phom ) = (y, u, h) ∈ Z0 (Ω, Γ1 ) ≡ H 1 (Ω, Γ2 ) × L2 (Ω) × L2 (Γ1 ) : hom (A ∇y, ∇ϕ) + αyϕ dx Ω uϕ dx + hϕ dl, ∀ C ∞ (Ω, Γ2 ) , = Ω

Γ1

and make use of the following lemma. Lemma 11.25. Assume that for the family {Aε ∈ L(R2 , R2 )} there exists a homogenized matrix in the sense of Definition 11.24. Then the sequence of sets {Ξε }ε∈E is K-convergent to the set Ξ0 , where Ξ0 = (y, u, h) ∈ Z0 (Ω, Γ1 ) : |u| ≤ cu a.e. in Ω, |h| ≤ ch a.e. on Γ1 , hom (A ∇y, ∇ϕ) + αyϕ dx Ω uϕ dx + hϕ dl, ∀ C ∞ (Ω, Γ2 ) . (11.42) = Ω

Γ1

Proof. Let (y 0 , u0 , h0 ) be any triplet of Ξ0 and let us fix a value δ > 0. Since the space of smooth functions C ∞ (Ω) is dense in L2 (Ω), it follows that for a given value δ > 0, there is an element u∗ ∈ C0∞ (R2 ) such that u0 − u∗ L2 (Ω) ≤ δ and |u∗ | ≤ cu in Ω. Let us construct a δ-realizing sequence as follows: uε = u∗ for every ε > 0. Then due to the weak convergence of the measures dμε dx, we have ϕuε dμε → ϕu∗ dx for every ϕ ∈ C0∞ (R2 ), Ω

Ω

that is, {uε } is a δ-realizing sequence of admissible controls for u0 . By analogy, we may choose the sequence {hε ∈ C ∞ (Γ1 )} such that


hε = h∗ , ∀ ε, hε ∈ L2 (Γ1 , dμε ),

425

|hε | ≤ ch μS -a.e. on Γ1 , hε h∗ in L2 (Γ1 , dμSε ),

where h0 − h∗ L2 (Γ1 ) ≤ δ and |h∗ | ≤ ch on Γ1 . Let {yε ∈ V (Ω, Γ2 , dμε )} be the sequence of solution of the boundary value problem (11.10)–(11.11) with the corresponding control functions u = uε and h = hε . Using estimate (11.13), we find that yε L2 (Ω,dμε ) ≤ 2α−1 C , ∇yε (L2 (Ω,dμε ))2 ≤ 2α−1 C. Then, thanks to Remark 11.8 and Theorem 11.18, we may suppose that the sequence of the triplets {(yε , uε , hε ) ∈ Ξε }ε∈E is w-convergent. Let (y ∗ , u∗ , h∗ ) be its w-limit. Our aim is to prove that (y ∗ , u∗ , h∗ ) ∈ Graph(Phom ). In view of our assumptions and Definition 11.24, we have uε u∗ in L2 (Ω, dμε ), yε y∗ (x), 2

hε h∗ in L2 (Γ1 , dμSε ),

yε y∗ in L2 (Ω, dμε ),

Aε (·)∇yε Ahom ∇y ∗ in L2 (Ω, dμε )2 . Therefore, passing to the limit as ε → 0 in the integral identity Ω

[(Aε (x)∇yε , ∇ϕ) + αyε ϕ] dμε = uε ϕ dμε + Ω

we obtain

hom ∗ ∇y , ∇ϕ + αy ∗ ϕ dx A Ω ∗ u ϕ dx + = Ω

hε ϕ dμSε , ∀ ϕ ∈ C ∞ (Ω, Γ2 ), Γ1

h∗ ϕ dl,

∀ ϕ ∈ C ∞ (Ω, Γ2 ). (11.43)

Γ1

Hence, we have (y ∗ , u∗ , h∗ ) ∈ Graph(Phom ). In view of the coerciveness property of the homogenized matrix Ahom , we have the standard a priori estimate y 0 − y ∗ H 1 (Ω) ≤ C u0 − u∗ L2 (Ω) + h0 − h∗ L2 (Γ1 ) ≤ 2δC. Thus,

(y 0 , u0 , h0 ) − (y∗ , u∗ , h∗ )Y0 ≤ 2 max{C, 1}δ,

that is, {(yε , uε , hε ) ∈ Ξε }ε∈E is a δ-realizing sequence with the required properties. Therefore, the following inclusion K − lim(Ξε ) ⊇ Ξ0 is established. In order to obtain the inverse inclusion, we consider a w-convergent sequence {(yk , uk , hk )}k∈N with the following properties:

426


(yk , uk , hk ) ∈ Ξεk for every k ∈ N, where {εk } is some subsequence of indices converging to 0 as k tends to ∞. (aa) (yk , uk , hk ) → (y∗ , u∗ , h∗ ) in the w-sense.

(a)

Then, proceeding as in the previous part of this lemma, we can show that ∇yk ∇y∗ + v(x, z), 2

v ∈ L2 (Ω, Vpot ),

Aεk (·)∇yk → Ahom ∇y ∗ in L2 (Ω, dμε )2 . Therefore, passing to the limit in the integral identity Ω

[(Aεk (x)∇yk , ∇ϕ) + αyk ϕ] dμεk = uk ϕ dμεk + Ω

Γ1

hk ϕ dμSεk ,

∀ ϕ ∈ C ∞ (Ω, Γ2 ),

leads us to relation (11.43). Hence, (y∗ , u∗ , h∗ ) ∈ Graph(Phom ).

(11.44)

Hence, it remains to show that the limit control functions u∗ ∈ L2 (Ω) and h∗ ∈ L2 (Γ1 ) satisfy the corresponding constraints |u∗ | ≤ cu a.e. in Ω,

|h∗ | ≤ ch a.e. on Γ1 .

(11.45)

Indeed, for every k ∈ N and every positive function ϕ ∈ C ∞ (Ω, Γ2 ), we have ⎧ ⎪ ⎪ ϕ(cu − uk ) dμεk ≥ 0 and ϕ(uk + cu ) dμεk ≥ 0, ⎨ Ω Ω (11.46) ⎪ ⎪ ϕ(ch − hk ) dμSεk ≥ 0 and ϕ(hk + ch ) dμSεk ≥ 0. ⎩ Γ1

Γ1

Using the facts that dμε dx and dμSε dl and passing the limit in (11.46) as k → ∞, one gets ⎫ ⎪ ϕ(cu − u∗ ) dx ≥ 0 and ϕ(u∗ + cu ) dx ≥ 0, ⎪ ⎬ Ω Ω (11.47) ⎪ ϕ(ch − h∗ ) dl ≥ 0 and ϕ(h∗ + ch ) dl ≥ 0. ⎪ ⎭ Γ1

Γ1

Since ϕ in (11.47) is an arbitrary positive function, it follows that inequalities (11.45) hold true. As a result, combining (11.44) and (11.45), we deduce (y ∗ , u∗ , h∗ ) ∈ Ξ0 . Thus, we have obtained the required inclusion K − lim Ξε ⊆ Ξ0 , which concludes the proof.


427

Corollary 11.26. Suppose that the matrix Aε (x) is defined as Aε (x) = A(ε−1 x), where A(z) is a -periodic μ-measurable matrix satisfying condition (11.8). Then the limiting matrix Ahom can be defined as

Ahom ξ = (11.48) A(z) ξ + v 0 dμ(z),

where v 0 ∈ Vpot is the solution of the problem (ξ + v, A(ξ + v)) dμ = (ξ + v 0 , A(ξ + v 0 )) dμ. min v∈Vpot

(11.49)

Proof. Let {(yε , uε , hε ) ∈ Ξε }ε∈E be a w-convergent sequence and let (y ∗ , u∗ , h∗ ) be its w-limit. Then ∇yε ∇y ∗ + v(x, z), 2

(11.50)

where v ∈ L2 (Ω, Vpot ). We emphasize that the gradients in (11.50) are uniquely defined (Lemma 11.7). For this, we consider the integral identity (11.11) with the test function ϕ(x) = εΨ (x)ω(ε−1 x), where Ψ ∈ C ∞ (Ω, Γ2 ) ∞ and ω ∈ Cper (). This yields (A(ε−1 x)∇yε ,∇z ω)Ψ dμε Ω + ε (A(ε−1 x)∇yε , ∇Ψ )ω dμε + ε αyε Ψ ω dμε Ω Ω =ε uε Ψ ω dμε + ε hε Ψ ω dμSε . Ω

Γ1

It is easy to see that after passing to the limit as ε → 0, we obtain

A(ε−1 x)∇yε , ∇z ω(ε−1 ) Ψ (x) dμε = 0. lim ε→0

(11.51)

Ω

In view of (11.50) and the definition of the weak two-scale limit, we have Ψ (x) ω(ε−1 x)A(ε−1 x) ∇yε (x) dμε lim ε→0 Ω = Ψ (x)ω(z)A(z) [∇y∗ (x) + v(x, z)] dμ(z) dx. Ω

Therefore, 2

A(ε−1 x)∇yε (x) A(z) [∇y ∗ (x) + v(x, z)] , −1 A(ε x)∇yε (x) A(z) [∇y ∗ (x) + v(x, z)] dμ(z)

in L2 (Ω, dμε )2 .

(11.52)

428


As a result, we can rewrite (11.51) in the explicit form

lim

ε→0

Ω

A(ε−1 x)∇yε , ∇z ω Ψ dμε = (A(z) [∇y ∗ + v(x, z)] , ∇z ω) dμ(z)Ψ (x) dx = 0. Ω

Since this equality holds true for every Ψ ∈ C ∞ (Ω, Γ2 ), this means that ∞ (A(∇y∗ + v), p) dμ = 0, ∀ p ∈ Cper ()2 . (11.53)

However, as follows from (11.49), equality (11.53) can be viewed as Euler’s equation for the minimum problem (11.49). Since this problem has a unique solution v0 , it follows that v0 is the unique solution of (11.53) as well. Thus, putting v = v0 in (11.52) and using formula (11.48), we immediately deduce that

in L2 (Ω, dμε )2 . A ε−1 x ∇yε Ahom ∇y ∗ Lemma 11.27. Under the assumptions of Lemma 11.25, the limit functional I0 : Ξ0 → R in (11.18) has the representation I0 (y, u, h) = k1 (y − zd )2 dx + k2 u2 dx + k3 h2 dl. (11.54) Ω

Ω

Γ1

Proof. To prove representation (11.54), we have to verify conditions (ii)–(iii) of Definition 11.20. Let (y, u, h) be any triplet of Ξ0 and let {(yk , uk , hk )}k∈N be a w-convergent sequence such that w

(yk , uk , hk ) → (y, u, h),

(yk , uk , hk ) ∈ Ξεk for every k ∈ N,

(11.55)

where {εk } is a subsequence of E converging to 0. Then, the following inequalities hold (see Sect. 6.6): 2 2 2 S uk dμεk ≥ u dx, lim inf hk dμεk ≥ h2 dl, lim inf k→∞ k→∞ Ω Ω Ω Γ1 (11.56) yk2 dμεk ≥ y 2 dx. lim inf k→∞

Ω

Ω

First, we note that the property of the weak compactness (11.4) for the sequence of measures {με } holds automatically in a wider class of test functions, when ϕ ∈ C0 (R2 ). Thus, 2 (zd ) dμε = (zd )2 dx, lim k→∞

Ω

Ω

where zd ∈ C(Ω) by the standing assumptions.


429

Therefore, using this, (11.56), and the definition of the weak two-scale convergence, it follows that 2 2 lim inf (yk − zd ) dμεk = lim inf yk dμεk − 2 yzd dx k→∞ k→∞ Ω Ω Ω (y − zd )2 dx. (11.57) + (zd )2 dx ≥ Ω

Ω

Thus, summing up (11.56) and (11.57), we get lim inf Iεk (yk , uk , hk ) ≥ I0 (y, u, h), k→∞

that is, property (d) of Definition 8.24 is valid. We now verify the correctness of the reverse inequality (see (8.61)). Let (y, u, h) be any triplet of the limit set Ξ0 and let δ > 0 be a fixed small value. We construct the δ-realizing sequence {(yε , uε , hε ) ∈ Ξε }ε∈E such that uε = u #, # # u, h) is the corresponding solution of the boundary value hε = h, and yε = yε (# problem (11.10)–(11.11) with u = u # and h = # h. Here, u # and # h are the functions with the following properties: (i) u # ∈ C ∞ (Ω), |# u| ≤ cu on Ω, and # u − uL2 (Ω) ≤ δ. ∞ # # h − hL2 (Γ1 ) ≤ δ. (ii) h ∈ C (Γ1 ), |h| ≤ ch on Γ1 , and # It is clear that, in this case, we have (see (11.35)) uε → u # in L2 (Ω, dμε ),

hε → # h in L2 (Γ1 , dμSε ). 2

Moreover, as follows from Lemma 11.25, yε y#, where y# is the unique solution of the problem (11.10)–(11.12) with u = u # and h = # h. Hence, w (yε , uε , hε ) −→ (# y, u #, # h). In view of the coerciveness property of the homogenized matrix Ahom , we have the standard a priori estimate % $ y − y#H 1 (Ω) ≤ C u − u (11.58) #L2 (Ω) + h − # hL2 (Γ1 ) ≤ 2δC. Thus, (y, u, h) − (# y, u #, # h)Y ≤ 2 max{C, 1}δ, that is, {(yε , uε , hε )}ε∈E is the δ-realizing sequence with the required properties. By the initial construction, we have # u #2ε dμε = u2 dx, lim sup h2 dl. (11.59) lim sup h2ε dμSε = ε→0

Ω

ε→0

Ω

Γ1

Γ1

In order to obtain the convergence (# y − zd )2 dx, lim sup (yε − zd )2 dμε = ε→0

Ω

(11.60)

Ω

we make use the idea of Cioranescu, Murat, and Zhikov (see [65, 256]). For this, we introduce the following auxiliary problem: Find pε ∈ V (Ω, Γ2 , dμε ) such that

430


yε Ψ dμε , ∀ Ψ ∈ C ∞ (Ω, Γ2 ).

[(Aε ∇pε , ∇Ψ ) + αpε Ψ ] dμε = Ω

(11.61)

Ω

Note that the linear span of the test functions Ψ ∈ C ∞ (Ω, Γ2 ) is dense in L2 (Ω); hence, we may take pε as a test function in (11.11) with u = uε and h = hε and yε as a test function in (11.61). Then, using the symmetry property of the matrix Aε , we get yε2 dμε = uε pε dμε + hε pε dμSε , ∀ ε ∈ E. (11.62) Ω

Ω

Γ1

Since uε = u # ∈ C ∞ (Ω), hε = # h ∈ C ∞ (Γ1 ), and pε p in L2 (Ω, dμε ), it follows that lim uε pε dμε = lim u #pε dμε = u # p dx, ε→0 Ω ε→0 Ω Ω # h p dl, hε pε dμSε = lim ε→0

Γ1

Γ1

where p ∈ H 1 (Ω) is the solution of a limit problem to (11.61). Note that since 2 2 yε y#(x), it follows that pε p(x), where hom (A (x)∇p, ∇Ψ ) + αpΨ dx = y#Ψ dx, ∀ Ψ ∈ C ∞ (Ω, Γ2 ). Ω

Ω

Hence, returning to (11.62), we have yε2 dμε = lim uε pε dμε + hε pε dμSε lim ε→0 Ω ε→0 Ω Γ1 # h p dl = = u # p dx + y#2 dx, Ω

Γ1

Ω

y strongly in L2 (Ω, dμε ). Using this fact, we immediately obtain that is, yε →# (yε − zd )2 dμε = (# y − zd )2 dx. (11.63) lim ε→0

Ω

Ω

As a result, combining (11.59) and (11.63), we deduce lim sup Iε (yε , uε , hε ) = I0 (# y, u #, # h). ε→0

Since

y, u #, # h) − I0 (y, u, h) ≤ Cδ I0 (#

independent of δ, this concludes the proof. with some constant C


431

Remark 11.28. Note that the result of the Lemma 11.25 remains correct without assuming Aε to be symmetric, if we assume that the homogenized matrix in Definition 11.24 satisfies the condition ATε (x)∇yε (Ahom )T ∇y0 in L2 (Ω, dμε )2 for every {yε } such that

T (Aε (x)∇y, ∇ϕ) + αyϕ dμε (y, u, h) ∈ Zε (Ω, Γ1 ) : Ω = uϕ dμε + hϕ dμSε , ∀ ϕ ∈ C ∞ (Ω, Γ2 ) Ω

Γ1

(yε , uε , hε ) → (y0 , u0 , h0 ) in the sense of w-convergence. Furthermore, we emphasize that this property holds automatically for a μmeasurable symmetric periodic matrix Aε (x) = A(ε−1 x) satisfying the conditions of ellipticity and boundedness (11.8). Now, we are in a position to prove the main result of this section. Theorem 11.29. Under the supposition of Lemma 11.11 for the family of problems (11.9)–(11.12), there exists a unique limiting OCP which has the representation −div(Ahom (x)∇y) + αy = u in Ω, y = 0 on Γ2 ∂y/∂νAhom = h on Γ1 , |h| ≤ ch a.e. on Γ1 , |u| ≤ cu a.e. on Ω, I0 (y, u, h) = k1 (y − zd )2 dx + k2 u2 dx + k3 h2 dl → inf, Ω

where

Ω

(11.64) (11.65) (11.66) (11.67)

Γ1

2 ∂y ∂y = ahom cos(n, xi ), ij (x) ∂νAhom ∂x ij i,j=1

cos(n, xi ) is ith direction cosine of n, and n is the normal of Γ1 exterior to Ω. Moreover, the sequence of optimal solutions {(yε0 , u0ε , h0ε ) ∈ Ξε } for the original problems (11.9)–(11.12) and corresponding minimal values of the cost functional (11.9) satisfy the following variational properties: lim

inf

ε→0 (y,u,h)∈Ξε 2

yε0 → y 0 ,

Iε (y, u, h) = I0 (y 0 , u0 , h0 ) =

u0ε → u0 in L2 (Ω, dμε ),

inf

I0 (y, u, h),

(11.68)

h0ε → h0 in L2 (Γ1 , dμSε ),

(11.69)

(y,u,h)∈Ξ0

where (y 0 , u0 , h0 ) is the unique solution of the limiting problem (11.64)– (11.67). Proof. As immediately follows from Lemmas 11.11 and 11.25 for the sequence of constrained minimization problems (11.17), there exists a variational limit

432


(11.18) the main components of which can be recovered in the form (11.42) and (11.54), respectively. Moreover, from (11.42), we have the implication (y, u, h) ∈ Ξ0 ⇒ (y, u, h) ∈ Graph(Phom ). It is clear now that the limit problem (11.18) can be written in the form of the OCP (11.64)–(11.67). Therefore, in view of Definition 11.12, the problem (11.64)–(11.67) is the homogenized optimal control problem for the original family (11.9)–(11.12). Applying now Theorem 11.23, we come to the following variational properties of the limit problem (11.64)–(11.67): Let {(yε0 , u0ε , h0ε )} be the sequence of optimal triplets for the problems (11.9)–(11.12), then lim Iε (yε0 , u0ε , h0ε ) = lim

ε→ 0

inf

ε→ 0 (yε ,uε ,hε )∈ Ξε

=

inf (y,u,h)∈Ξ0

Iε (yε , uε , hε )

I0 (y, u, h) = I0 (y 0 , u0 , h0 ),

w

(yε0 , u0ε , h0ε ) −→ (y 0 , u0 , h0 ).

(11.70)

Hence, Iε (yε0 , u0ε , h0ε ) → I0 (y 0 , u0 , h0 ). Rewriting this in the explicit form, we get

(yε0

lim k1

ε→0

Ω

− zd ) dμε + k2 dμε + k3 Ω Γ1 0 2 0 2 = k1 (y − zd ) dx + k2 (u ) dx + k3 (h0 )2 dl. 2

(u0ε )2

(h0ε )2

Ω

Ω

dμSε

Γ1

Therefore, the validity of this equality for every ki ≥ 0 implies 0 2 0 2 0 2 (yε ) dμε = (y ) dx, lim (uε ) dμε = (u0 )2 dx, lim ε→0 Ω ε→0 Ω Ω Ω 0 2 S lim (hε ) dμε = (h0 )2 dl. ε→0

Γ1

Ω

Combining these properties with (11.70) and using Lemma 6.19, we obtain the required assertions (11.68) and (11.69).

11.6 Modeling of suboptimal controls The main question we are going to consider here concerns the approximation of the optimal solution of the original problem (11.9)–(11.12) for ε small sub enough. We define suboptimal controls (usub ε , hε ) which approximate the optimal value of the original problem (11.9)–(11.12). To do so, we introduce the following concept (for comparison, see Definition 4.5).

11.6 Modeling of suboptimal controls

433

Definition 11.30. We say that a sequence of admissible controls sub sub (uε , hε ) ε>0 is asymptotically suboptimal to the problem (11.9)–(11.12) if there exists a > 0 independent of ε such that for every δ (0 < δ < δ0 ) there is constant C ε0 > 0 satisfying sub sub sub inf I (y, u, h) − I (y , u , h ) (11.71) ε ε ε ε ε < Cδ, ∀ ε < ε0 , (y,u,h)∈Ξε

sub where yεsub = yεsub (usub ε , hε ) denotes the corresponding solutions of the boundary value problem (11.10)–(11.11).

Theorem 11.29 leads to the following final result. Theorem 11.31. Let u0 ∈ L2 (Ω) and h0 ∈ L2 (Γ1 ) be the optimal controls for the homogenized problem (11.64)–(11.67). Then any δ-realizing sequence to the pair (u0 , h0 ) is asymptotically suboptimal for the original OCP (11.9)– (11.12). Proof. Let (y 0 , u0 , h0 ) be the unique solution of the homogenized problem (11.64)–(11.67). For a given δ > 0, we construct the δ-realizing sequence {(yε , uε , hε ) ∈ Ξε }ε∈E in the usual way, that is, uε = u #, hε = # h, and yε = # u, h) is the corresponding solution of the boundary value problem (11.10)– yε (# (11.11) with u = u # and h = # h. Here, u # ∈ C ∞ (Ω), |# u| ≤ cu on Ω, # u − u0 L2 (Ω) ≤ δ, (11.72) # h| ≤ ch on Γ1 , # h − h0 L2 (Γ1 ) ≤ δ. h ∈ C ∞ (Γ1 ), |# It is clear that (see Lemma 11.27) 2 lim sup (yε − zd ) dμε = (# y − zd )2 dx, ε→0

Ω

Ω

y 0 − y#H 1 (Ω) ≤ 2δC (see (11.58)), lim sup u #2ε dμε = (u0 )2 dx, ε→0 Ω Ω 2 S # (h0 )2 dl, lim sup hε dμε = ε→0

Γ1

(11.73)

(11.74)

Γ1

where y# is the unique solution of the problem (11.64)–(11.65) with u = u # and # h = h. In addition, in view of these relations, we have the following obvious estimates:

k2 (u0 )2 dx − (# u)2 dx ≤ k2 δ0 + 2u0 L2 (Ω) δ = C2 δ, (11.75) Ω Ω

h)2 dx ≤ k3 δ0 + 2h0 L2 (Γ1 ) δ = C3 δ, (11.76) k3 (h0 )2 dx − (# Ω

Ω

434


0 2 2 k1 (y − zd ) dx − (# y − zd ) dx Ω Ω

≤ 4k1 δ0 C + y 0 − zd L2 (Ω) Cδ = C1 δ. (11.77) We now observe that inf Iε (y, u, h) − Iε (yε , uε , hε ) (y,u,h)∈Ξε #, # h) = Iε (yε0 , u0ε , h0ε ) − Iε (yε , u y, u #, # h) − Iε (yε , u #, # h) ≤ Iε (yε0 , u0ε , h0ε ) − I0 (y 0 , u0 , h0 ) + I0 (# + I0 (y 0 , u0 , h0 ) − I0 (# y, u #, # h) ≤ Iε (yε0 , u0ε , h0ε ) − I0 (y 0 , u0 , h0 ) + I0 (# y, u #, # h) − Iε (yε , u #, # h) 0 2 2 + k1 (y − zd ) dx − (# y − zd ) dx Ω Ω 0 2 2 0 2 2 # u) dx + k3 (h ) dl − (h) dl + k2 (u ) dx − (# Ω

Ω

Γ1

Γ1

= J1 + J2 + J3 + J4 + J5 . To conclude the proof, we note that for a given δ > 0, one can always find (i) ε1 > 0 such that J1 < δ/2 for all ε < ε1 by Theorem 11.29 and (ii) ε2 > 0 such that J2 < δ/2 for all ε < ε2 by Lemma 11.27. Additionally, J3 < C1 δ, J4 < C2 δ, and J5 < C3 δ by estimates (11.75)–(11.77). As a result, we have ' & 3 inf Ci δ, ∀ ε < min{ε1 , ε2 }. (y,u,h)∈Ξε Iε (y, u, h) − Iε (yε , uε , hε ) ≤ 1 + i=1

Thus, we have obtained the required estimate (11.71).

11.7 An example of an optimal control problem on ε-periodic square grid On the domain Ω that was defined in (11.1), we consider the ε-periodic square grid εF with the cell of periodicity ε. Here, the set = [0, 1)2 contains the “cross”-structure such as indicated in Fig. 11.1(a). Following the notation of Sect. 11.6, we say that Ωε has ε-periodic grid-like ¯ x2 = structure if Ωε = Ω ∩ εF. We also set ∂Ω = Γ1 ∪ Γ2 , where Γ1 = x ∈ Ω 0, 0 < x1 < a . Now, we recall the standard notations on graphs (see Lagnese and Leuger ing [163]). Let V ε = vJ : J ∈ Jε be the set of vertices of our ε-periodic

11.7 An example of an OCP on ε-periodic square grid

435

Fig. 11.1. (a) The cell of grid periodicity; (b) periodic grid on Ω

graph (grid) Ωε and let E ε = ei : i ∈ I˙ ε be the index set of corresponding edges. Here, by Jε and I˙ ε , we denote the index sets for vertices and edges, respectively. For a given vertex vJ , we consider the set of edges that are incident at vJ . The corresponding set of edge indices is denoted by I˙ J = i ∈ I˙ ε : ei is incident at vJ . The cardinality of I˙ J is the edge degree at vJ , that is, dJ = |I˙ J |. It is easy to see that dJ ≤ 4 in our case. Note that as follows from Fig. 11.1(b), every edge ei on the graph Ωε can be parameterized by x ∈ [0, li ], where li ≤ ε/2 denotes the length of the edge ei . With every edge ei we will associate a so-called “state function” yi : [0, li ] → R1 , i ∈ I˙ ε .

436


Since we are going to consider an OCP on a bounded periodic graph, it is necessary to specify the boundary and the so-called transmission conditions at the vertices V ε of Ωε . To this end, we subdivide the set of vertices (nodes) as follows: ε , V ε = VSε ∪ VM ε signifies where VSε denotes the set of simple modes such that dJ = 1 and VM the set of multiple nodes where 4 ≥ dJ > 1. The set of simple nodes VSε will be divided as V ε = VΓε1 ∪ VΓε2 ,

where VΓε1 represents the set of simple nodes belonging to the Γ1 -boundary and VΓε2 signifies those simple nodes which belong to the Γ2 -boundary. It is easy to see that, in the case of ε-periodic grid on Ω, there is no simple node lying in the interior of the domain Ω. Further, we will look at the set VΓε1 as the set of control-active Neumann nodes and at VΓε2 as the set of nodes with zero Dirichlet conditions. On all the edges ei , we consider the differential operator Li of the form Li yi = −Ri yi + αyi , where Ri ≥ α > 0. Moreover, using the ε-periodic structure of Ωε , we will always suppose that for every ε-cell εj , we have (see Fig. 11.3(a)) Ri = β, Ri+1 = γ,

Ri+2 = β,

Ri+3 = γ,

(11.78)

where α−1 ≥ β, γ ≥ α > 0. We now define the classes of admissible controls U ε and H ε where U ε = u : Ωε → R1 : uli ∈ L2 (0, li ); |u(x)| ≤ cu (11.79) for almost every x ∈ Ωε } , H ε = h = (h1 , h2 , . . . , hLε ) ∈ RLε : Lε = |VΓε1 |, Lε (11.80) h2K < ∞, |hK | ≤ ch . K=1

Here, cu and ch are some positive constant; by |VΓε1 | we denote the number of all simple nodes belonging to Γ1 . Let k1 , k2 , and k3 (ki > 0) be penalty terms and let zd , ud , and hd be given functions of C(Ω). We consider the following OCP on the grid Ωε : − Ri yi + αyi = ui , x ∈ (0, li ), i ∈ I˙ ε ,

(11.81)

yi (vJ ) = 0, i ∈ I˙ Jε , vJ ∈ VΓε2 , ε Ri yi (vJ ) = 0, vJ ∈ VM ,

(11.82) (11.83)

i∈IJε

yi (vJ ) = hk(J) ,

∀ vJ ∈ VΓε1 , i ∈ IJε ; k(J) ∈ {1, 2, . . . , Lε },

(11.84)


u ∈ U ε, Iε (y, u, h) =

0

i∈I˙ε

+ k3

li

k1

Lε

h ∈ H ε,

437

(11.85)

(yi − zd e )2 dx + k2 i

i∈I˙ ε

0

li

(ui − ud e )2 dx

(hK − hd (vJ ))2 → inf .

i

(11.86)

K=1

Using the properties of the sets U ε and H ε and invoking the standard arguments, it is easy to prove that for every ε ∈ E, the problem (11.81)– (11.86) admits a unique optimal triplet (yε0 , u0ε , h0ε ) which can be characterized by some adjoint system. Our aim is to study the asymptotic behavior of this problem as ε tends to 0. For this, we reformulate the problem (11.81)–(11.86) in terms of some variational control problem defined on spaces with singular measures. We introduce the -periodic Borel measure μ in R2 as follows: 1 (μ1 + μ2 + μ3 + μ4 ), 2 where μi are the 1D Lebesgue measures on the corresponding line segments (edges) Ii (see Fig. 11.1). Also, we define the S -periodic Radon measure μS on R1 as μS = δ(1/2,0) , where δ(1/2,0) is the Dirac measure at the point (1/2, 0) ∈ S . It is easy to see that dμ = 1 and dμS = 1. μ=

S

Therefore, we may define the “scaling” measures με (B) = ε2 μ(ε−1 B),

μSε (B1 ) = εμS (ε−1 B1 ),

where B and B1 are corresponding Borel sets in R2 and R1 , respectively. Obviously, each of these measures με and μSε converges weakly to the corresponding Lebesgue measure: dμε dx and dμSε dl. Here, dx and dl are the Lebesgue measures on R2 and R1 , respectively. Now, we define the matrix Aε (x) = A(ε−1 x) as a11 (z) 0 A(z) = , 0 a22 (z) where a11 (z) = β and a22 (z) = γ. It is easy to see that the matrix so defined is symmetric, μ-measurable, and satisfying property (11.8). As a result, the original OCP can be presented in the form (A(ε−1 x)∇y, ∇ϕ) + αyϕ dμε Ω = uϕ dμε + hϕ dμSε , ∀ ϕ ∈ C ∞ (Ω, Γ2 ), (11.87) Ω

Γ1

438


y ∈ V (Ω, Γ2 , dμε ), |u| ≤ cu

με -a.e. in Ω,

|h| ≤ ch

(11.88)

μSε -a.e.

(11.89)

(y − zd )2 dμε + k2

Iε (y, u, h) = k1

on Γ1 ,

Ω

(u − ud )2 dμε Ω

(h − hd )2 dμSε → inf .

+ k3

(11.90)

Γ1

The validity of this representation immediately follows from Proposition 11.5 and Remark 11.15. Then, due to Theorem 11.29, for the control problem (11.87)–(11.90) there exists a limit problem which can be recovered in the form −div(Ahom ∇y) + αy = u in Ω, y = 0 on Γ2 , ∂y/∂νAhom = h on Γ1 , |u| ≤ cu a.e. on Ω, |h| ≤ ch a.e. on Γ1 , (y − zd ) dx + k2

(11.93)

(u − ud )2 dx

2

I0 (y, u, h) = k1

(11.91) (11.92)

Ω

Ω

(h − hd )2 dl −→ inf .

+ k3

(11.94)

Γ1

The identification of the matrix Ahom can be done either by definition, that is, as the solution of the minimum problem (Ahom ξ, ξ) = min (A(ξ + p), ξ + p) p∈Vpot

(see [256]) or using a more classical method: First, we homogenize the problem on the grid with nonzero thickness in the usual way (see [66, 261]) and then pass to the limit as thickness goes to 0. As for the second approach, it was shown in [58] that in this case the corresponding diagram of homogenization (see Fig. 11.2) is commutative.

Fig. 11.2. Homogenization diagram

Here, δ denotes a small parameter which characterizes the fixed “thickness” of graphs. The corresponding structures will be called δ-grids F δ .


439

Fig. 11.3. (a) εJ -cell; (b) periodicity cell for F δ

We will follow the second approach. Therefore, we define the ε-periodic δ-grids Fεδ by setting Fεδ = ε−1 F δ ; the cell of periodicity δ for F δ has the form that Fig. 11.3(b) shows. Let μεδ (B) = ε2 μδ (ε−1 B) and let μδ be the measure on F δ that can be defined as the probability measure on Ω supported by the cross-bar in δ and is uniformly distributed on it. The weak limit of this measure μδ as δ → 0 is the singular measure μ which was defined earlier. Assume that the matrix Aε,δ = A(ε−1 x) (i.e., Aε,δ = diag(β, γ).) Then using the standard technique of homogenization (see [66, 209]), we ε→0 get Aε,δ −→ Ahom , where δ δ $ % 1 b11 0 hom β + o(δ 1/2 ) , = Aδ , bδ11 = δ 0 b22 (2 − δ) $ % 1 γ + o(δ 1/2 ) . bδ22 = (2 − δ) Furthermore, due to the results of Cioranescu and Saint Jean Paulin ([66]), δ→0 −→ Ahom . As a result, we obtain (for details, we we may take the limit Ahom δ hom refer to [22, 66, 207, 209]) A = diag( 12 β, 12 γ). Thus, the homogenized state equation for the limit problem (11.91)– (11.94) has the form −β

∂2y ∂2y − γ 2 + 2αy = 2u 2 ∂x1 ∂x2

in Ω, y = 0 on Γ2 ,

γ

∂y = 2h on Γ1 . ∂x2

Furthermore, in view of the result of Theorem 11.29, the homogenized OCP (11.91)–(11.94) satisfies the variational properties (11.68)–(11.69). Note also that other approaches for the identification of homogenized operators

440


−div(Ahom ∇y) have been discussed in the literature (see, for instance, Mazja and Slutskij [184]). In the case when

( a = 2, α = β = γ = 1, Ω = (x1 , x2 ) : x1 ∈ (0, 2), 0 < x2 < 2x1 − x21 ,

hd = 0, zd = exp (A), ud = exp(A) 1 − 2A2 − 6A3 − 2A4 , A = (x21 + x22 − 2x1 )−1 , we have that

u0 = exp(A) 1 − 2A2 − 6A3 − 2A4

and

h0 = 0

are the optimal controls to the homogenized problem (11.91)–(11.94). Hence, in view of Theorem 11.29, the restriction of these functions on the sets Ωε and VΓε1 , respectively, can be taken as suboptimal controls to the original problem on an ε-periodic square grid (11.81)–(11.85).

12 Suboptimal Boundary Control of Elliptic Equations in Domains with Small Holes

Our prime interest in this chapter concerns the construction of suboptimal solutions to a class of boundary optimal control problems (OCPs) in ε-periodically perforated domains with small holes. We suppose that the support of controls is contained in the set of boundaries of the holes. This set is divided into two parts: On one part, the controls are of Dirichlet type; on the other one, the controls are of Neumann type. Using the ideas of the Γ -convergence theory and the concept of the variational convergence of constrained minimization problems, we show that the limit problem, as ε tends to 0, can be recovered in an explicit analytical form. However, in contrast to the case of thin periodic structures (see Chap. 9), the control through the boundary of small holes leads us in the limit to an OCP with drastically different properties and structures. In particular, we show that in this case, a “strange term” appears both in the limit equation and the cost functional, and that this term depends on the geometry of the holes. Moreover, the characteristic feature of the obtained limiting control problem is the fact that it contains two independent distributed control functions which can be used as suboptimal controls to the original one.

12.1 Statement of the problem Let Ω ⊂ Rn , n ≥ 2, be a bounded open domain and let ε be a small positive parameter. In order to define a perforated domain Ωε , we introduce the following sets: Y = [−1/2, +1/2)n ; Q and K are compact subsets of Y such that 0 ∈ int K ∩ ∂Q (see Fig. 12.1),


441

442

12 Suboptimal Boundary Control in Domains with Small Holes

Fig. 12.1. The perforation scheme

Θε = {k = (k1 , k2 , . . . , kn ) ∈ Zn : (εY + ε k) ∩ Ω = ∅} ; εn/(n−1) Q + ε k ; {ε(Y + k)} ; Tε = Yε = k∈Θε

Sε =

ΓεD

(12.1) (12.2)

k∈Θε

εn/(n−2) K ∩ ∂(εn/(n−1) Q),

n ≥ 3,

exp (−1/ε2 )K ∩ ∂(εn/(n−1) Q), n = 2; = {Sε + ε k} ∩ Ω, ΓεN = ∂Tε \ ΓεD ∩ Ω.

(12.3)

(12.4)

k∈Θε

Then, we set Ωε = Ω \ Tε . Notice that (in contrast to the case of thin periodic structures) the size of the holes Qε + ε k tends to 0 as ε → 0 and their boundaries ΓεD and ΓεN are not proportional to the size of the periodicity cell εY . In Ωε , we consider the following boundary value problem: ⎫ −Δ yε + yε = fε in Ωε , ⎪ ⎪ ⎪ ⎪ ⎬ ∂ν yε = −k0 yε + pε on ΓεN , (12.5) ⎪ on ΓεD , yε = uε ⎪ ⎪ ⎪ ⎭ yε = 0 on Σε = ∂Ω ∩ ∂Ωε , where fε ∈ L2 (Ω) is a given function, k0 is a positive constant, and ∂ν = ∂/∂ν is the outward normal derivative. In (12.5), uε and pε are the control functions which act on the system through the set of boundaries of the holes. We interpret the functions uε and pε as Dirichlet and Neumann boundary controls, respectively. We say that the control functions uε and pε are admissible if the following conditions hold:

12.1 Statement of the problem

uε ∈ Uε =

pε ∈ L2 (ΓεN ),

a|ΓεD : a ∈ H01 (Ω) ∩ H 2 (Ω), a H 2 (Ω) ≤ C0 .

443

(12.6)

Now, the OCP can be formulated as follows: Given zε ∈ L2 (Ω) and C0 > 0, find a triple (u0ε , pε0 , yε0 ) ∈ Ξε such that Iε (u0ε , pε0 , yε0 ) =



(12.7)

where the cost functional Iε and the set of admissible triplets Ξε are defined as Iε = |∇yε |2 dx + |yε − zε |2 dx Ω Ωε ε (12.8) 2 n−1 pε dH + β(ε) u2ε dHn−1 , + ΓεN

ΓεD

× L (ΓεN ) × H 1 (Ωε ; Σε ) : Ξε = {(uε , pε , yε ) ∈ H (uε , pε , yε ) satisfies (12.5)–(12.6)} . 1

(ΓεD )

2

Here, H 1 (Ωε ; Σε ) = yε ∈ H 1 (Ωε ) : yε = 0 on Σε , β(ε) = εn/(2−n) if n ≥ 3,

β(ε) = ε2 exp(ε−2 ) if n = 2.

In the sequel, the OCP (12.5)–(12.7) will be called the Pε -problem. Throughout this chapter, we suppose that Ω is a measurable set in the sense of Jordan; the small parameter ε varies in a strictly decreasing sequence of positive numbers which converges to 0; Q and K are compact subsets of Y such that 0 ∈ int K ∩ ∂Q; the set Q has Lipschitz boundary ∂Q, int Q is a strongly connected set, Q ⊂ {x = (x1 , . . . , xn ) ∈ Rn : x1 ≥ 0}, and its boundary ∂Q contains the origin; A = B(0, r0 ) is an open ball centered at the origin with a radius r0 < 1/2, so that A ⊂⊂ Y and K ⊂⊂ A (see Fig. 6.2 for a 2D example); C0 > 0 is a constant independent of ε; the functions fε ∈ L2 (Ω), and zε ∈ L2 (Ω) are such that fε f and zε → z ∂ in L2 (Ω) as ε → 0. For any subset E ⊂ Ω, we denote by |E| its n-dimensional Lebesgue measure Ln (E), whereas |∂E|H denotes the (n − 1)-dimensional Hausdorff measure of the manifold ∂E on Rn . We suppose that the sets K ∩∂Qς and ∂Q\(K ∩∂Qς ) have nonzero capacity for any ς > 1, where Qς = {ςx, ∀ x = (x1 , . . . , xn ) ∈ Q} is the homothetic stretching of Q by a factor of ς. Hence, |K ∩ ∂Qς |H = 0 for all ς > 1. Let Mb (Ω) be the space of bounded Borel measures on Ω with values in [0, +∞]. Let M+ 0 (Ω) be the cone of all non-negative Borel measures μ on Ω such that μ(B) = 0 for every set B ⊆ Ω with cap(B, Ω) = 0 and μ(B) =

444


inf{μ(U ) : U quasi open, B ⊆ U } for every Borel set B ⊆ Ω. Note that if 1 μ ∈ M+ 0 (Ω), then the functions of H (Ω) are defined μ-almost everywhere and are μ-measurable in Ω; hence, the space H 1 (Ω)∩L2 (Ω, dμ) is well defined. We now give the variational formulation of the boundary value problem (12.5). For this, we set H 1 (Ωε ; ΓεD ∪ Σε ) = yε ∈ H 1 (Ωε ) : yε = 0 on ΓεD ∪ Σε and fix a function aε ∈ H01 (Ω)∩H 2 (Ω) such that aε H 2 (Ω) ≤ C0 and aε |ΓεD = uε (the so-called prototype of a control function uε ∈ Uε ). It is clear that for every admissible control uε ∈ Uε , there exists at least one prototype. Then the variational version of the problem (12.5) consists in looking for a function yε = yε (uε , pε ) such that yε − aε ∈ H 1 (Ωε ; ΓεD ∪ Σε ), (∇yε · ∇ϕ) dx + yε ϕ dx + k0 yε ϕ dHn−1

Ωε

=

Ωε

pε ϕ dHn−1 ,

fε ϕ dx + Ωε

ΓεN

ΓεN

∀ ϕ ∈ H 1 (Ωε ; ΓεD ∪ Σε ). (12.9)

Note that the trace of yε equals to uε almost everywhere on ΓεD . In the sequel, we call the function yε the weak solution to the problem (12.5) and identify yε with its quasi-continuous representative [82]. For the weak solution of the problem (12.5), we have the following existence and uniqueness result. Theorem 12.1. For every fixed ε and for any control functions uε ∈ H 1 (ΓεD ) and pε ∈ L2 (ΓεN ) the boundary value problem (12.5) has a unique weak solution yε ∈ H 1 (Ωε ; Σε ) such that the a priori norm-estimate yε H 1 (Ωε ) ≤ C fε L2 (Ω) + D1 (ε) uε H 1 (ΓεD ) + D2 (ε) pε L2 (ΓεN ) (12.10) holds true, where C, D1 (ε), and D2 (ε) are some positive constants and C is independent of ε. Proof. Since ∂Ωε is Lipschitz continuous and there exists a constant γ > 0 depending only on Ωε such that yε H 1/2 (∂Ωε ) ≤ γ yε H 1 (Ωε ) , it follows that the bilinear form < ∇yε , ∇vε >L2 (Ωε ) + < yε , vε >L2 (Ωε ) +k0 < yε , vε >H 1/2 (∂Ωε ) is coercive on the space H 1 (Ωε ). Hence, the same assertion holds for the space H 1 (Ωε ; ΓεD ∪ Σε ). Thus, the required result immediately follows from [167, Theorems 1.1 and 2.2, Chap. IV].

12.2 Reformulation of the original problem in terms of singular measures

445

We are now in a position to prove the existence of a solution to the Pε problem. Theorem 12.2. For every value of ε, the OCP Pε has a unique solution (u0ε , p0ε , yε0 ) ∈ Ξε . Proof. Let τε = wH 1 (ΓεD ) × wL2 (ΓεN ) × wH 1 (Ωε ) be the basic topology on the set of admissible triplets Ξε ⊂ H 1 (ΓεD ) × L2 (ΓεN ) × H 1 (Ωε ), where by w(·) we denote the weak topologies of the corresponding Banach spaces. It is easy to see that this set is nonempty, convex, and τε -closed for every ε. Hence, we ∞ may choose a minimizing sequence (ukε , pkε , yεk ) k=1 ⊂ Ξε such that lim Iε (ukε , pkε , yεk ) =

k→∞


Iε (uε , pε , yε ).

∞ The boundedness of the sequence Iε (ukε , pkε , yεk ) k=1 implies the boundedness k ∞ of pε ∈ L2 (ΓεN ) k=1 . Let akε ⊂ H01 (Ω) ∩ H 2 (Ω) be a sequence of the corresponding prototypes for ukε ⊂ Uε . Since this sequence is bounded, we may always suppose that akε a0ε in H 2 (Ω). Then the compact imbedding H 2 (Ω) → H 1 (ΓεD ) implies that ukε → u0ε = a0ε |ΓεD strongly in H 1 (ΓεD ). As ∞ a result, using (12.10), we see that the sequence (ukε , pkε , yεk ) k=1 ⊂ Ξε is relatively sequentially τε -compact. Hence, we may extract subsequences (still indexed by k) such that pkε p0ε in L2 (ΓεN ), ukε → u0ε in H 1 (ΓεD ), yεk → yε0 in L2 (Ωε ) for some (u0ε , p0ε , yε0 ) ∈ H 1 (ΓεD ) × L2 (ΓεN ) × H 1 (Ωε ). Taking this fact into account and passing to the limit in (12.9) as k → ∞, we deduce (u0ε , p0ε , yε0 ) ∈ Ξε . Finally, using the lower τε -semicontinuity of the cost functional Iε , we obtain Iε (u0ε , p0ε , yε0 ) ≤ lim inf Iε (ukε , pkε , yεk ) = k→∞


Iε (uε , pε , yε ).

Thus, we have shown that the triplet (u0ε , p0ε , yε0 ) ∈ Ξε is a solution for the problem Pε . The uniqueness of such a solution is an obvious consequence of the property of strict convexity of Iε . This concludes the proof.

12.2 Reformulation of the original problem in terms of singular measures We begin this section with the description of the geometry of the perforated domain Ωε as we did it in Sect. 6.9. We describe the class of admissible solutions to the problem Pε in terms of singular periodic Borel measures on Rn . Let us denote by K λ and Qh the homothetic contractions of the sets K and Q, respectively, by factors of λ−1 and h−1 . In what follows, it is assumed

446


that 0 < λ 0. The same conclusion is true for the measure ν λ,h . Hence, the spaces 1 2 λ,h H01 (Ω)∩H 2 (Ω)∩L2 (Ω, dμλ,h ε ) and H (Ω; Σε )∩L (Ω, dνε ) are well defined. Now, we turn back to the definition of the set of admissible solutions of the problem Pε (see (12.1)). We see that ΓεD = K λ(ε) ∩ ∂Qh(ε) + εk , k∈Θε

ΓεN

=

k∈Θε

∂Qh(ε) \ (K λ(ε) ∩ ∂Qh(ε) ) + εk .

12.2 Reformulation of the original problem in terms of singular measures

Then, using properties (6.132) and (6.133) and setting σ(ε) = h(ε)n−1 |∂Q|H − λ(ε)n−1 |K ∩ ∂Qh(ε)/λ(ε) |H ,

447

(12.14)

pε ϕ dHn−1 of the integral identity (12.9) can be rewritten in

the term ΓεN

the form

pε ϕ dHn−1 = ΓεN

k∈Θε

= σ(ε)

∂Qh(ε) \Γ λ(ε),h(ε) +εk

k∈Θε

= ε−n σ(ε)

pε ϕ dHn−1

pε ϕ dν λ(ε),h(ε) (x/ε)

ε(Y +k)

pε ϕ dνελ,h

(12.15)

Ω

for every function ϕ ∈ C0∞ (Rn ; Σε ∪ ΓεD ) = ψ ∈ C0∞ (Rn ) : ψ = 0 on Σε ∪ ΓεD . Here, pε is a function of L2 (Ω, dνελ,h ) taking the same values as pε ∈ L2 (ΓεN ) on ΓεN . It is clear that for every boundary control pε ∈ L2 (ΓεN ), one can find a function pε ∈ L2 (Ω, dνελ,h ) such that pε = pε on ΓεN . Hence, pε2 dHn−1 = ε−n σ(ε) pε2 dνελ,h , (12.16) ΓεN

Ω

where

⎧ if n ≥ 3, ⎨ |∂Q|H − εn/(n−2) |K ∩ ∂Qh(ε)/λ(ε) |H −n ε σ(ε) = ⎩ |∂Q|H − 1 exp(− 1 )|K ∩ ∂Qh(ε)/λ(ε) |H if n = 2. ε2 ε2

By analogy, we obtain k0 yε ϕ dHn−1 = ε−n σ(ε)k0 y˘ε ϕ dνελ,h , ΓεN

ΓεD

Ω

u2ε dHn−1 = ε−n λ(ε)n−1 |K ∩ ∂Qh(ε)/λ(ε) |H

(12.17)

(12.18)

a2ε dμλ,h ε .

(12.19)

Ω

Here, y˘ε ∈ H 1 (Ω; Σε ) is an extension of the weak solution yε to the problem (12.5) to the whole of domain Ω, and the function aε ∈ H01 (Ω) ∩ H 2 (Ω) ∩ L2 (Ω, dμλ,h ε ) is a prototype of the Dirichlet control uε ∈ Uε (see (12.6)). Remark 12.4. In view of our initial assumptions, the measure νελ,h is supported on a set with nonzero capacity for every ε > 0. Since every element v of the space H 1 (Ω; Σε ) ∩ L2 (Ω, dνελ,h ) can be interpreted as a quasi-continuous

448


function, it is reasonable to suppose that for every element v ∈ H 1 (Ω), one can find a sequence vk ∈ C(Ω) k∈N such that sup lim sup (v − vk )2 dνελ,h < +∞ and lim cap(Vk , Ω) = 0, k∈N

ε→0

k→∞

Vk

where Vk = {x ∈ Ω : v = vk }. We assume that the same property is valid for the elements of the space H01 (Ω) ∩ L2 (Ω, dμλ,h ε ). So the original boundary value problem (12.5) can be rewritten in the form χε (∇˘ yε · ∇ϕ) dx + χε y˘ε ϕ dx + k0 ε−n σ(ε) y˘ε ϕ dνελ(ε),h(ε) Ω Ω Ω −n = ε σ(ε) pε ϕ dνελ(ε),h(ε) Ω + χε fε ϕ dx, ∀ ϕ ∈ H 1 (Ω; ΓεD ∪ Σε ), (12.20) Ω

y˘ε − aε ∈ H 1 (Ω; ΓεD ∪ Σε ),

(12.21)

whereχε denotes the characteristic function of the domain Ωε . We can reformulate the original OCP Pε (12.7)–(12.1) as follows: Find some (aε0 , pε0 , yε0 ) ∈ Xε such that Iε (aε0 , pε0 , yε0 ) = where

inf

bε (aε ,pε ,yε )∈ Ξ

Iε (aε , pε , yε ),

(12.22)

Xε = H01 (Ω) ∩ H 2 (Ω) ∩ L2 (Ω, dμλ,h ε ) × L2 (Ω, dνελ,h ) × H 1 (Ω; Σε ) ∩ L2 (Ω, Dνελ,h ) , Iε (aε , pε , yε ) =

χε |∇˘ yε |2 dx

Ω

|χε y˘ε −

−n

zε∂ |2

dx + ε σ(ε) p2ε dνελ,h Ω + |K ∩ ∂Qh(ε)/λ(ε) |H a2ε dμλ,h ε ,

+

Ω

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ε = (aε , pε , yε ) Ξ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(12.23)

Ω

⎫ y˘ε − aε ∈ H 1 (Ω; ΓεD ∪ Σε ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 (Ω) ≤ C0 , a ⎪ H ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ χε (∇˘ yε · ∇ϕ) dx + χε y˘ε ϕ dx ⎬ Ω Ω . +k ε−n σ(ε) ⎪ y˘ε ϕ dνελ,h − χε fε ϕ dx ⎪ 0 ⎪ ⎪ ⎪ Ω Ω ⎪ ⎪ ⎪ −n λ,h ⎪ = ε σ(ε) pε ϕ dνε , ⎪ ⎪ ⎪ Ω ⎪ ⎭ 1 D ∀ ϕ ∈ H (Ω; Γε ∪ Σε )

(12.24)

12.3 Convergence in the variable space X ε

449

ε the OCP (12.22)–(12.24). It is clear that P ε has a unique We denote by P 0 0 0 solution (aε , pε , yε ) for every ε ([169, 111]). This solution can be viewed as a prototype of the optimal triplet to the Pε -problem. Moreover, in this case, the a priori norm estimate (12.10) takes the form ˘ yε H 1 (Ω, χε dx) + ε−n σ(ε) yε L2 (Ω,dνελ,h ) ≤C ε−n σ(ε) pε L2 (Ω,dνελ,h ) + fε L2 (Ω) h(ε)/λ(ε) + |K ∩ ∂Q |H aε H 1 (Ω)∩L2 (Ω,dμλ,h ) . (12.25) ε 0

We set

⎧ ⎪ ⎪ ⎨ exp

!

−n ln ε n2 − ! 3n + 2 ς(ε) = h(ε)/λ(ε) = 1 ⎪ ⎪ ⎩ ε2 exp 2 ε

⎫ ⎪ if n ≥ 3, ⎪ ⎬ ⎪ if n = 2. ⎪ ⎭

(12.26)

Then ς(ε) ∈ (1, +∞), ∀ ε, and limε→0 ς(ε) = +∞. Let Qς (ε) = {ς(ε)x, ∀ x = (x1 , . . . , xn ) ∈ Q} be the homothetic stretching of the set Q by a factor of ς(ε). We recall some results of Sect. 6.9 (see Propositions 6.66 and 6.68) that will be useful later. Proposition 12.5. There exists an open cone Λ ⊂ {x ∈ Rn : x1 > 0} such that (12.27) lim |K ∩ ∂Qς (ε)|H = |K ∩ ∂Λ|H . ε→0

This cone can be recovered in an explicit form for the case when the origin belongs to a smooth part of the boundary ∂Q. Moreover, in view of (12.14), we have limε→0 σ(ε)/εn = |∂Q|H . Proposition 12.6. Let {ρε ∈ R}ε>0 be numerical sequence such that (12.28) ρε = Ln A \ Qς(ε) /Ln (A), ∀ ε > 0. Then the sequence {ρε }ε>0 is monotone and there exists a value ρ∗ ∈ [1/2, 1) such that limε→0 ρε = ρ∗ .

12.3 Convergence in the variable space X ε Assume that the parameters λ = λ(ε) and h = h(ε) are defined by (12.12). We begin with the following concept. Definition 12.7. Let vελ,h ∈ H01 (Ω) ∩ L2 (Ω, dμλ,h ε ) be a bounded sequence. We say that this sequence converges weakly in H01 (Ω) ∩ L2 (Ω, dμλ,h ε ) to v ∈ H 1 (Ω) if vελ,h v in H01 (Ω) and vελ,h v in L2 (Ω, dμλ,h ε ).

(12.29)

450


In order to check the correctness of this definition, we make use of the following auxiliary statements. Lemma 12.8. If v ∈ H01 (Ω), then λ,h lim v ϕ dμε = v ϕ dx, ε→0

Ω

∀ ϕ ∈ C0∞ (Ω).

Ω

dx of the measures Proof. If v ∈ C0 (Ω), then the weak convergence dμλ,h ε immediately implies this relation. Let v be an arbitrary element of H01 (Ω). Then for every δ > 0, there exist a set Aδ ⊂ Ω and a function v δ ∈ C0 (Ω) such that cap(Aδ , Ω) < δ and v δ = v on Ω \ Aδ . In what follows, we suppose that the set Aδ is closed. We now consider the following estimate: λ,h v ϕ dμε − vϕ dx Ω Ω δ λ,h δ v ϕ dμ − v ϕ dx ≤ (v − v δ ) ϕ dμλ,h + ε ε Ω Ω Ω + (v − v δ ) ϕ dx = J1 + J2 + J3 . Ω

Owing to the weak convergence μλ,h dx, we have J2 → 0 as ε → 0. By ε Lusin’s theorem, we may suppose that there is a constant d1 > 0 such that v − v δ L2 (Ω) ≤ d1 δ. Hence, J3 ≤ d1 ϕ L2 (Ω) δ. As for the value J1 , we note that each measure μλ,h is supported on a set with nonzero capacity. So, there ε is a constant d2 > 0 such that v − v δ L2 (Ω,dμλ,h ) ≤ d2 for any δ > 0 small ε enough (see Remark 12.4). It implies the following estimate: δ λ,h δ 1/2 J3 = (v − v ) ϕ dμε ≤ d2 ϕ C(Ω) μλ,h . ε (A ) Ω

As a result, we have the following: (i)

δ δ λ,h lim supε→0 μλ,h dx implies the ε (A ) ≤ |A | since the convergence με following (see [258]): λ,h ϕ dμε = ϕ dx, ∀ ϕ ∈ C(Ω), (12.30) lim ε→0 Ω λ,h lim sup με (F ) ≤ ε→0

Ω

|F |

for any compact set F ⊂ Ω.

(12.31)

(ii) |Aδ | ≤ d3 capn/(n−2) (Aδ ) by the properties of capacity (see [106]). (iii) cap(Aδ ) < δ by the initial assumption. Hence, summing up all estimates that were obtained earlier, we conclude v ϕ dμλ,h − vϕ dx ≤ dδ ε Ω

Ω

for any δ > 0 small enough (here, a constant d does not depend on δ). This completes the proof.


451

We now consider a more delicate situation. and v ∈ L2 (Ω) be such Lemma 12.9. Let vελ,h ∈ H01 (Ω) ∩ L2 (Ω, dμλ,h ε ) λ,h 1 λ,h 2 that vε v in H0 (Ω), and hence vε → v in L (Ω). Then " λ,h lim vελ,h dμλ,h − v dx = 0. (12.32) ε ε ε→0

Ω

Ω

Proof. As in Lemma 12.8, we introduce two functions v#ελ,h ∈ C(Ω) and v# ∈ C(Ω) such that v#ελ,h = vελ,h and v# = v quasi-everywhere. Let us partition the set Ω into cubes εY with edges ε and denote these cubes by εY j . Then there are points xλ,h ∈ εY j such that j λ,h λ,h λ,h λ,h v#ε dμε = v#ε (x) dμε + v#ελ,h (x) dμλ,h ε j j Ω εY Ω∩ εY = v#ελ,h (xλ,h dμλ,h + v#ελ,h (x) dμλ,h ε ε , j ) εY j

Ω∩ εY j

where the second sum is calculated over the set of the “boundary” cubes. By the definition of the measure μλ,h ε , we have λ,h n dμε = ε dμλ,h = εn . ε εY j

Hence, v#ελ,h

dμλ,h ε

=

Y

n v#ελ,h (xλ,h j )ε

+

Ω

Ω∩ εY j

v#ελ,h (x) dμλ,h ε .

(12.33)

It is clear that an analogous representation takes place for the second term in (12.32), namely v#ελ,h dx = v#ελ,h (xj ) dx + v#ελ,h (x) dx Ω εY j Ω∩ εY j λ,h n v#ελ,h (x) dx (12.34) = v#ε (xj ) ε + Ω∩ εY j

for some xj ∈ εY j . Note that λ,h λ,h v#ε (x) dμε ≤ sup j∈D(ε) Ω∩ εY j v#(x) dx ≤ sup j Ω∩ εY

j∈D(ε)

! sup x∈ Ω∩ εY j

sup

|# vελ,h (x)|

|# v (x)|

εn · D(ε),

! εn · D(ε),

x∈ Ω∩ εY j

where D(ε) is the quantity of the “boundary” cubes, and εn D(ε) → 0 by Jordan’s measurability property of the set ∂Ω. Moreover, since vελ,h , v ∈ H01 (Ω), it follows that

452


! sup

sup

j∈D(ε)

x∈ Ω∩ εY j

!

|# v (x)|

< +∞ and

sup

sup

j∈D(ε)

x∈ Ω∩ εY j

|# vελ,h (x)|

< +∞

for ε small enough. Then, substituting (12.33) and (12.34) into (12.32), we come to the following relation: ! λ,h λ,h λ,h vε dμε − vε dx lim ε→0 Ω Ω λ,h v#ελ,h (xλ,h ) − v # (x ) εn ≤ lim j ε j ε→0 ! ! λ,h + max sup vε (x)| , sup v (x)| sup |# sup |# j∈D(ε)

x∈ Ω∩ εY j

j∈D(ε)

x∈ Ω∩ εY j

× lim sup (εn D(ε)) + lim (J1 + J2 ) ε→0 ε→0 λ,h v#ελ,h (xj ) − v#ελ,h (xj ) εn , = lim ε→0

where

(vελ,h

J1 =

(12.35)

−

v#ελ,h ) dμλ,h ε ,

Ω

(vελ,h − v#ελ,h ) dx

J2 = Ω

and by the arguments of Lemma 12.8 and Remark 12.4, we may suppose that lim (J1 + J2 ) = 0. ε→0

We now use the fact that vελ,h → v in L2 (Ω). One has (vελ,h − v)2 dx ≤ 2 lim (# vελ,h − v#)2 dx + 2 lim J0 (ε) lim ε→0 Ω ε→0 Ω ε→0 λ,h ∗ ∗ 2 n = 2 lim v#ε (xj ) − v#(xj ) ε = 0, ε→0

lim

ε→0

Ω

2 vελ,h

dx −

v 2 dx λ,h 2 v#ε dx − v#2 dx + lim J(ε) = lim ε→0 Ω ε→0 Ω λ,h ∗ 2 n = lim v#ε (xj ) ε − lim (# v (xj ))2 εn ε→0 ε→0 2 2 v#ελ,h (x∗j ) − (# = lim v (xj )) εn , (12.37) Ω

ε→0

where, as usual, we suppose that the values (# vελ,h − vελ,h + v − v#)2 dx J0 (ε) = Ω

and

(12.36)

J(ε) =


vελ,h

2

2 dx + − v#ελ,h

Ω

453

v 2 − v#2 dx

Ω

are arbitrarily small. Hence, by (12.35), the construction of the Riemann sum, and the fact that v ∈ H01 (Ω), we conclude that λ,h λ,h λ,h λ,h v#ε dx ≤ lim ) − v # (x ) v#ελ,h (xλ,h lim v#ε dμε − j j ε→0 ε→0 Ω Ω + v#(xλ,h #(xj ) + v#(xj ) − v#ελ,h (xj ) εn j )−v $ $ v#(xλ,h ≤ 2 |Ω| lim $vελ,h − v $L2 (Ω) + lim sup ) − v#(xj ) εn j ε→0 ε→0 λ,h n n v#(xj )ε − ≤ lim sup v#(xj )ε ε→0 = v#dx − v#dx = 0. Ω

Ω

Taking into account the proofs of the previous lemmas and relations (12.36) and (12.37), the following statement is readily ascertained. 1 Lemma 12.10. Let vελ,h ∈ H01 (Ω) ∩ L2 (Ω, dμλ,h ε ) and v ∈ H0 (Ω) be such λ,h 1 that vε v in H0 (Ω). Then " λ,h 2 λ,h 2 lim vε v dμλ,h − dx = 0, (12.38) ε ε ε→0 Ω Ω lim v2 dμλ,h = v 2 dx, ∀ v ∈ H01 (Ω). (12.39) ε ε→0

Ω

Ω

Remark 12.11. Since the set Ω is bounded and |∂Ω \Σε |H ∼ ε1−n hn−1 (ε) = ε, it follows that |Σε |H → |∂Ω|H as ε → 0. Hence, by properties (12.30) and (12.31), the statements of Lemmas 12.8–12.10 remain valid if the space H01 (Ω) is changed to H 1 (Ω, Σε ). Theorem 12.12. Every bounded sequence vελ,h ∈ H01 (Ω) ∩ L2 (Ω, dμλ,h ε ) is relatively compact with respect to the weak convergence in the variable space H01 (Ω) ∩ L2 (Ω, dμλ,h ε ). Proof. Since the sequence vελ,h is bounded in H01 (Ω), we may suppose that there is an element v ∈ H01 (Ω) such that vελ,h → v weakly in H01 (Ω). Then the compact imbedding H01 (Ω) → L2 (Ω) implies the strong convergence vελ,h → v in L2 (Ω). Hence, for every ϕ ∈ C0∞ (Ω), we have λ,h λ,h λ,h vελ,h ϕ dμλ,h ≤ − vϕ dx v ϕ dx − v ϕ dμ ε ε ε ε Ω Ω Ω Ω λ,h + vε ϕ dx − vϕ dx . Ω

Ω

454


Passing to the limit as ε → 0 on the right-hand part of this inequality, we obtain λ,h vελ,h ϕ dμλ,h − vε ϕ dx → 0 (by Lemma 12.9) ε Ω

and

Ω

(vελ,h − v) ϕ dx → 0 as a weak limit in L2 (Ω). Ω

The proof is complete. Using the above results, we introduce the concept of the weak convergence for the following sequences (for comparison, see Definition 12.7): yε ∈ H 1 (Ωε ; Σε ) : y˘ε ∈ H 1 (Ω; Σε ) ∩ L2 (Ω, dνελ,h ) ε>0 . Here, y˘ε is some extension of the function yε on the whole of Ω. Let us recall that the perforated domain Ωε considered here satisfies the so-called “condition of strong connectedness” (see [182]). It means that there exist a family {Pε }ε>0 of extension operators Pε : H 1 (Ωε ; Σε ) → H 1 (Ω; Σε ) and a constant C independent of ε such that ∇(Pε yε ) L2 (Ω) ≤ C yε H 1 (Ωε ) for every yε ∈ H 1 (Ωε ; Σε ). So, we can assume that y˘ε := Pε yε for some extension operator with the above properties. Definition 12.13. We say that a sequence yε ∈ H 1 (Ωε ; Σε ) ε>0 is weakly convergent in H 1 (Ω; Σε ) ∩ L2 (Ω, dνελ,h ) if there exists an element y ∈ H01 (Ω) such that y˘ε y in H 1 (Ω) and yε y in L2 (Ω, dνελ,h ). We are now in a position to verify the correctness of this definition. Theorem 12.14. Every bounded sequence yε ∈ H 1 (Ωε ; Σε ) ε>0 is relatively compact with respect to the weak convergence in the variable space H 1 (Ω; Σε )∩ L2 (Ω, dνελ,h ). Proof. Taking into account Remark 12.11 and the fact that ε→0 y˘ε ϕχΩε dx −→ y ϕ dx, Ω

yε ϕ dνελ,h = Ω

Ω

y˘ε ϕ dνελ,h ,

∀ ϕ ∈ C0∞ (Ω),

Ω

this theorem can be established in complete analogy with the proof of Theorem 12.12. The main difference is the addition of the property v ∈ H01 (Ω). However, |Σε |H → |∂Ω|H as ε → 0, and we obtain the required result. In fact, we can prove a more precise result. Theorem 12.15. Let yελ,h ∈ H 1 (Ωε ; Σε ) ∩ L2 (Ω, dνελ,h ) ε>0 be a bounded sequence such that y˘ελ,h y in H 1 (Ω; Σε ) ∩ L2 (Ω, dνελ,h ). Then y ∈ H01 (Ω) and yελ,h → y strongly in L2 (Ω, dνελ,h ).


455

Proof. By the criterium of strong convergence in L2 (Ω, dνελ,h ), to establish the convergence yελ,h → y in L2 (Ω, dνελ,h ) it is enough to show that yελ,h y in L2 (Ω, dνελ,h ), lim (yελ,h )2 dνελ,h = y 2 dx. (12.40) ε→0

Ω

Ω

The first statement in (12.40) is valid by Definition 12.13. In order to prove the second one, we apply the following estimate: 2 ≤ (yελ,h )2 dνελ,h − (yελ,h )2 dx (yελ,h )2 dνελ,h − y dx Ω Ω Ω Ω yελ,h )2 dx − y 2 dx . (12.41) + (˘ Ω

Ω

The second term on the right-hand side of (12.41) tends to 0 as ε → 0 by the strong convergence of y˘ελ,h to y in L2 (Ω). The first one is equal to 0 at the limit as ε → 0 by applying Lemma 12.10, and this concludes the proof. Let {(aε , pε , yε ) ∈ X ε }ε>0 be a sequence of admissible solutions for the original problem. We assume that this sequence is bounded. Then summing up the above given reasonings, we may introduce the following concept of the weak convergence in the variable space X ε . Definition 12.16. We say that a bounded sequence {(a ε , pε , yε ) ∈ X ε }ε>0 is w-convergent to a triplet (a, p, y) ∈ H 2 (Ω) ∩ H01 (Ω) × L2 (Ω) × H01 (Ω) in w the variable space X ε as ε tends to 0 (in symbols, (aε , pε , yε ) (a, p, y)) if the following hold: (i) aε a in H 2 (Ω) and aε a in L2 (Ω, dμλ,h ε ); (ii) pε p in L2 (Ω, dνελ,h ); (iii) y˘ε y in H 1 (Ω) and yε y in L2 (Ω, dνελ,h ). In view of Theorems 12.12 and 12.14, we come to the following conclusion: Theorem 12.17. Every bounded sequence of admissible solutions {(aε , pε , yε )}ε>0 ε is relatively compact with respect to the w-convergence to the problems P in X ε . To conclude this section, we observe that for the characteristic function χε of the perforated domain Ωε , the following result is obvious [73]. Lemma 12.18. χε converges strongly to 1 both in L2 (Ω) and in the variable space L2 (Ω, χε dx) as ε → 0.

456


12.4 Definition of a limit problem and its property ε as Since our main goal is to study the asymptotic behavior of the problem P ε -problem for various values of ε in the form of the ε → 0, we represent the P following sequence: % & ' inf (12.42) Iε (aε , pε , yε ) ; ε > 0 . bε (aε ,pε ,yε ) ∈Ξ

Then the definition of an appropriate limiting OCP to the family (12.22) can be reduced to the analysis of the limit properties of the sequence (12.42) as ε → 0. Definition 12.19. We say that a minimization problem ) ( I0 (a, p, y) inf

(12.43)

(a,p,y)∈ Ξ0

is the weak variational limit of the sequence (12.42) with respect to the wconvergence in the variable space X ε (or variational w-limit) if the conditions (d)–(dd) of Definition 8.24 hold true with Xε = H01 (Ω) ∩ H 2 (Ω) ∩ L2 (Ω, dμλ,h ε ) × L2 (Ω, dνελ,h ) × H 1 (Ω; Σε ) ∩ L2 (Ω, Dνελ,h ) , X = H 2 (Ω) ∩ H01 (Ω) × L2 (Ω) × H01 (Ω). As a result, Theorem 8.25 (see also Corollary 8.37 of Theorem 8.35) implies the following properties of a weak variational limit in the sense of Definition 12.19. Theorem 12.20. Assume that the constrained minimization problem (12.43) is the weak variational limit of the sequence (12.42) and has a unique solu0 0 0 ε be the sequence of optimal tion (a , p , y ) ∈ Ξ0 . Let (a0ε , pε0 , yε0 ) ∈ Ξ ε>0

ε -problems. Then triplets for the P w

inf (a, p, y)∈ Ξ0

(a0ε , pε0 , yε0 ) −→ (a0 , p 0 , y 0 ), I0 (a, p, y) = I0 a0 , p 0 , y 0 = lim

inf

ε→0 (aε ,pε ,yε )∈ Ξ bε

Iε (aε , pε , yε ).

(12.44) (12.45)

12.5 Convergence theorem and correctors The main question of this section is the homogenization of the boundary value 1 problem (12.5). Let Hper (Y ) be the Sobolev space of Y -periodic functions. We begin with the following result.

12.5 Convergence theorem and correctors

457

Lemma 12.21. There exists a sequence of functions wλ,h h>λ>0 satisfying the following: 1 (H1) wλ,h ∈ Hper (Y ), wλ,h = 0 on K λ ∩ ∂Qh , 0 ≤ wλ,h ≤ 1; (H2) w λ,h = 1 in Y \ Ah ; (H3) wλ,h (x1 , x2 , . . . , xn ) = wλ,h (−x1 , x2 , . . . , xn ) ∀ x ∈ Ah , ∀ h > λ > 0; 1 (Y ) and strongly in L2per (Y ). (H4) wλ,h 1 weakly in Hper

Proof. Let us define A=

λ,h : v λ,h = 0 on K λ ∩ ∂Qh , v v λ,h (x1 , x2 , . . . , xn ) = v λ,h (−x1 , x2 , . . . , xn ) 1 (Y ), v λ,h = 1 in Y \ Ah , ∀ x ∈ Ah , ∀ h > λ > 0, v λ,h 1 in Hper * α = inf

lim inf (h>λ)→0

+ |∇v λ,h |2 dx : {v λ,h } ∈ A .

Y

Note that the set A is not empty. Indeed, if we define the functions v λ,h as 1 vλ,h ∈ Hper (Y ), Δvλ,h = 0 in Ah \ Aλ ,

v λ,h = 0 in Aλ , v λ,h = 1 in Y \ Ah , one has {v λ,h } ∈ A immediately. For any k ∈ N, we consider a sequence {vkλ,h } ∈ A such that |∇vkλ,h |2 dx < α + 1/k. lim inf (h>λ)→0

Y

Let vkλ,h = T (vkλ,h ), where T (s) = |s| if −1 ≤ s ≤ 1 and T (s) = 1 otherwise. Then vkλ,h ∈ A, 0 ≤ vkλ,h ≤ 1, and 1 |∇ vkλ,h |2 dx ≤ lim inf |∇vkλ,h |2 dx < α + . lim inf (h>λ)→0 Y (h>λ)→0 Y k By Rellich–Kondrashov’s compactness and Lebesgue’s dominated conver1 (Y ) ∩ L∞ (Y ) → Lq (Y ) gence theorems, we conclude that the embedding Hper λ,h (1 ≤ q < +∞) is compact. As a result, the sequence vk converges strongly to 1 in L2 (Y ) as (h > λ) → 0 for every fixed k. Then it is possible to define a subsequence (λk , hk ) of (λ, h) which is decreasing and tends to 0 such that |∇ vkλk ,hk |2 dx < α + 2/k, vkλk ,hk − 1 L2 (Y ) < 1/k. Y

Then the desired sequence wλ,h h>λ>0 is defined by w λ,h = vkλk ,hk .

458


From now on, we suppose that each of the functions wλ,h satisfying conditions (H1)–(H4) is extended by Y -periodicity onto Rn . We set wε (x) = wλ(ε),h(ε) (x/ε) ,

∀ x ∈ Ω, ∀ ε > 0.

From Lemma 12.21, we have the following: wε ≤ 1; wε ∈ H 1 (Ω), 0 ≤, wε = 0 on ΓεD = k∈Θε K λ(ε) ∩ ∂Qh(ε) + εk ; , wε = 1 in Ω \ k∈Θε Ah(ε) + εk ; wε (x1 , x2 , . . . , xn ) = wε (−x1 , x2 , . . . , xn ), ∀ x ∈ Aλ(ε) , ∀ k ∈ Θε , and ε > 0; (P5) wε 1 weakly in H 1 (Ω) and strongly in L2 (Ω) as ε → 0. Note that the sequence |∇wε |2 is bounded in L1 (Ω). So that extracting, if necessary, a subsequence, we can suppose the existence of a bounded nonnegative Radon measure μ∗ such that |∇wε |2 converges to μ∗ in the weak sense of the space Mb (Ω). Following in many aspects Casado-D´ıaz [55, Theorem 2.1], the following quite similar result can be proved. Theorem 12.22. Let wε ∈ H 1 (Ω) be a sequence satisfying the properties (P1)–(P5). Then the following hold:

(P1) (P2) (P3) (P4)

(L1) |∇wε |2 → μ∗ weakly in Mb (Ω), where μ∗ ∈ M+ 0 , that is, ϕ |∇wε |2 dx → ϕ dμ∗ for any ϕ ∈ C0∞ (Ω); Ω

Ω

(L2) For any functions vε ∈ H 1 (Ω; ΓεD ∪ Σε ) and for any v ∈ H01 (Ω) such that vε v in H 1 (Ω), we have v ∈ L2 (Ω, dμ∗ ), ϕ (∇vε · ∇wε ) dx → ϕ v dμ∗ ,

Ω

∀ ϕ ∈ C0∞ (Ω).

(12.46)

Ω

2 In fact, the measure μ∗ ∈ M+ 0 that appeared as the weak limit of |∇wε | in the space Mb (Ω) can be recovered in an explicit form. For this, we recall some properties of capacity (see Theorem 2.38).

Lemma 12.23. Let D be an open subset of Rn and let B be a compact subset of D. Then the following hold: (i) If {Di }i∈N is an increasing sequence of open sets such that ∪i∈N Di = D, then limi→∞ cap(B, Di ) = cap(B, D). (ii) If {Di ⊂ E}i∈N is a decreasing sequence of compact sets such that ∩i∈N Di = clD, then limi→∞ cap(Di , E) = cap(D, E). (iii) If D1 ⊂ D2 , then cap(D1 , E) ≤ cap(D2 , E). (iv) If t > 0, then cap(tB, tD) = tn−2 cap(B, D).


459

We now give the recovery result for the measure μ∗ . Lemma 12.24. Assume that the origin belongsto a smooth part of the boundary ∂Q (∂Q(0) ∈ C ∞ ). Then for a sequence wε ∈ H 1 (Ω) which satisfies properties (P1)–(P5), we have |∇wε |2 μ∗ weakly in Mb (Ω), where μ∗ = cap(K ∩ {x ∈ Rn : x1 = 0}) if n ≥ 3, μ∗ = 2π if n = 2. (12.47) Proof. The proof follows standard techniques in such situations (see [106]), and in some aspects, it is similar to the one given in [73]. First, we note that for any function ϕ ∈ C0∞ (Ω), ε > 0, and every k ∈ Θε we have the following inequality: |∇wε |2 dx ≤ |∇wε |2 ϕ dx ϕ(xεk ) εY +εk εY +εk ≤ ϕ(ykε ) |∇wε |2 dx, (12.48) εY +εk

∈ εY + εk. where Let us begin with the case n ≥ 3. From the definition of the capacity and Theorem 12.22, it readily follows that |∇wε |2 dx = cap K λ(ε) ∩ ∂Qh(ε) , Ah(ε) . xεk , ykε

εY +εk

Then, taking into account property (iv) of Lemma 12.23 and relation (12.12), we have |∇wε |2 dx = cap λ(ε) K ∩ ∂Qς(ε) , Ah(ε) εY +εk = λn−2 (ε)cap K ∩ ∂Qς(ε) , Aς(ε) (12.49) = εn cap K ∩ ∂Qς(ε) , Aς(ε) , where ς(ε) = h(ε)/λ(ε) = exp −n ln ε/(n2 − 3n + 2) for n ≥ 3. Now we interpret the sequence cap K ∩ ∂Qς(ε) , Aς(ε) ε>0 as a two parametric one: Λδ,ε = cap K ∩ ∂Qς(δ) , Aς(ε) δ,ε>0 . Since this sequence is monotone with respect to the parameter ε, it follows that limδ,ε→0 Λδ,ε = limε→0 Λδ(ε),ε for every sequence {δ(ε)} converging to 0. Then due to the inequality cap K ∩ ∂Qς(δ) , Aς(ε) − cap K ∩ D − cap K ∩ D ≤ cap K ∩ ∂Qς(δ) + cap K ∩ ∂Qς(δ) , Aς(ε) − cap K ∩ ∂Qς(δ) = J (δ) + J (δ, ε)

(12.50)

460


and using property (i) of Lemma 12.23, we have limε→0 J (δ, ε) = 0 for every δ > 0. In order to examine the limit properties of the sequence {J (δ)}δ>0 , we have to perform its analysis in a more precise form, namely since a part of boundary ∂Q containing the origin is smooth, it follows that there is a neighborhood U(0) of the origin such that U(0) ∩ ∂Q is a graph of a smooth function whose epigraph contains U(0) ∩ Q. So, we may suppose that there is a function Ψ : Rn−1 → R≥ such that Ψ ∈ C0∞ (Rn−1 ) and x1 = Ψ (x2 , . . . , xn ) for every x = (x1 , x2 , . . . , xn ) ∈ U(0) ∩ ∂Q. Then the following conclusion is valid: x = (x1 , x2 , . . . , xn ) ∈ K ∩ ε−1 ∂Q for ε small enough if and only if x ∈ ε−1 (U(0) ∩ ∂Q) and, hence, x1 = εΨ (x2 /ε, . . . , xn /ε). As a result, for any sufficiently small ε0 , there exists a constant C > 0 such that K ∩ ε−1 ∂Q ⊂ K ∩ Πr

∀ ε ≤ ε0 , with r = C ε0 Ψ C(Rn−1 ∩ U (0))

where Πr = {x ∈ Rn : 0 ≤ x1 ≤ r}. Then by properties (ii)–(iii) of Lemma 12.23, we have the following implication: lim cap(K ∩ Πr ) = cap(K ∩ D),

r→0

K ∩ ε−1 ∂Q ⊂ K ∩ ΠεC Ψ C(Rn−1 ∩

U (0))

,

∀ ε > 0,

implies that limε→0 cap K ∩ ε−1 ∂Q = cap(K ∩ D). Hence, − cap K ∩ D J (δ) = cap K ∩ ∂Qς(δ) ≤ C Ψ C(Rn−1 )∩U(0) ς(δ)

(12.51)

for δ small enough. Summing up relations (12.48) for every k ∈ Θε and taking into account (12.49)–(12.51), we come to cap K ∩ D −C ς(δ) Ψ C(Rn−1 ) − J (δ, ε) εn ϕ(xεk ) ≤

k∈Θε

k∈Θε

|∇wε |2 ϕ dx εY +εk

≤ cap (K ∩ D) + C ς(δ) Ψ C(Rn−1 ) + J (δ, ε) εn ϕ(ykε ). (12.52) ×

k∈Θε

Therefore, if we consider the construction of the Riemann sum for

ϕ dx, Ω

setting δ = ε, and passing to the limit in (12.52) as ε → 0, we immediately obtain the required result


|∇wε |2 ϕ dx = cap K ∩ D

lim

ε→0

Ω

461

ϕ dx. Ω

If n = 2, then we have a situation similar to the previous one. The only difference concerns the following obvious equality: |∇wε |2 dx = cap exp − 1/ε2 K ∩ ∂Q, A εY +εk = cap K ∩ exp 1/ε2 ∂Q, exp 1/ε2 A . For the sequence Λδ,ε = cap K ∩ exp 1/δ 2 ∂Q, exp 1/ε2 A

,

δ>0,ε>0

we can apply the above arguments. Therefore, there is a constant C > 0 such that for ε small enough, (12.53) Λδ,ε − cap K ∩ D, exp 1/ε2 A < C exp − 1/δ 2 . However, as follows from [73] (see Lemma 12.3.3), we have cap K ∩ D, exp 1/ε2 A = 2πε2 (1 + cε ), where lim cε = 0. ε→0

(12.54)

Then, summing up relations (12.49) for all k ∈ Θε and taking into account (12.53) and (12.54), we obtain 2 ε ϕ(xεk ) 2π(1 + cε ) − C ε−2 exp −1/δ2 ≤

k∈Θε

k∈Θε

|∇wε |2 ϕ dx

εY +εk

2 ≤ 2π(1 + cε ) − C ε−2 exp −1/δ 2 ε ϕ(ykε ).

(12.55)

k∈Θε

Setting δ = ε and passing to the limit as ε → 0, we get |∇wε |2 ϕ dx = 2π. lim ε→0

Ω

Corollary 12.25. Under the assumptions of Lemma 12.24 concerning the local smoothness property of the boundary ∂Q, item (L2) of Theorem 12.22 can be made more precise in the following way: For any vε ∈ H 1 (Ω; ΓεD ∪ Σε ) and for any v ∈ H01 (Ω) such that vε v in H 1 (Ω), we have ∗ ϕ (∇vε · ∇wε ) dx → μ ϕ v dx, ∀ ϕ ∈ C0∞ (Ω), (12.56) Ω

Ω ∗

where the multiplier μ is defined by (12.47). Moreover, in this case we have (see Proposition 12.5) |K ∩ ∂Λ|H = |K ∩ D|H .

462


The following result is crucial in this section. Theorem 12.26. Let vε ∈ H 1 (Ωε ; ΓεD ∪ Σε ) be a bounded sequence such that vε v in the variable space H 1 (Ωε ; ΓεD ∪Σε )∩L2 (Ω, dνελ,h ). Let {ρε }ε>0 ∗ be the sequence of numbers that was defined in Proposition 12.6 and ρ be its 1 limit. Then for the sequence wε ∈ H (Ω) with properties (P1)–(P5), we have v ∈ L2 (Ω, dμ∗ ), ∗ ϕ (∇vε · ∇wε ) dx → ρ ϕ v dμ∗ ,

Ωε

(12.57)

∀ ϕ ∈ C0∞ (Ω).

Ω

Proof. Denote by v˘ε ∈ H and define the following sets: 1

(Ω; ΓεD

∪ Σε ) some extensions of the functions vε

Jε = {k = (k1 , . . . , kn ) ∈ Zn : (εY + εk) ⊂ Ω} , Ah(ε) + εk Aε = k∈Θε

-

=

. ⎛ Ah(ε) + εk ∪ ⎝

=

∪

⎞

Ah(ε) + εk ⎠

k∈Θε \Jε

k∈Jε

Aε

Aε .

It is clear that for any bounded sequence zε ∈ H 1 (Ω) , we have ϕ (∇zε · ∇wε ) dx Ω ϕ (∇zε · ∇wε ) dx = Ω∩Aε ϕ (∇zε · ∇wε ) dx + ϕ (∇zε · ∇wε ) dx = (Ω\Ωε )∩Aε Ωε ∩Aε = ϕ (∇zε · ∇wε ) dx + ϕ (∇zε · ∇wε ) dx

(Ω\Ωε )∩Aε

+ (Ω\Ωε )∩A ε

ϕ (∇zε · ∇wε ) dx +

Ωε ∩Aε

Ωε ∩A ε

ϕ (∇zε · ∇wε ) dx.

(12.58)

Since each of the sets (Ω \ Ωε ) ∩ Aε and Ωε ∩ Aε is located along the boundary ∂Ω, it follows that Ln ((Ω \ Ωε ) ∩ Aε ) → 0 and Ln (Ω ε ∩ Aε ) → 0 1as ε→ 0. 1 Hence, in view of the boundedness of zε ∈ H (Ω) and wε ∈ H (Ω) , we conclude ⎫ ε→0 ⎪ ϕ (∇zε · ∇wε ) dx −→ 0,⎪ ⎪ ⎬ (Ω\Ωε )∩Aε (12.59) ⎪ ε→0 ⎪ ϕ (∇zε · ∇wε ) dx −→ 0,⎪ ⎭ Ωε ∩A ε


" ϕ (∇zε · ∇wε ) dx −

lim

ε→0

463

Ω

(Ω\Ωε )∩Aε

ϕ (∇zε · ∇wε ) dx

−

Ωε ∩Aε

ϕ (∇zε · ∇wε ) dx = 0. (12.60)

For vε ∈ H 1 (Ωε ; ΓεD ∪ Σε ) and ϕ ∈ C0∞ (Ω), let us define a function ψ ∈ C0∞ (Ω) in the following way (which is always possible by property (P4) of wε and some freedom of choosing of the extension operators v˘ε = Pε (vε ) [54, 73]): ρε ϕ (∇vε · ∇wε ) dx = ρε ψ (∇˘ vε · ∇wε ) dx, ∀ ε > 0. Ωε ∩Aε

(Ω\Ωε )∩Aε

Here, ρε =

Ln (A \ Qh(ε)/λ(ε) ) Ln (Aλ(ε) \ Qh(ε) ) , = Ln (A) Ln (Aλ(ε) )

ρε =

Ln (A ∩ Qh(ε)/λ(ε) ) Ln (Aλ(ε) ∩ Qh(ε) ) = . λ(ε) Ln (A) Ln (A )

It is clear that ρε + ρε = 1 for every ε > 0. Then ϕ (∇vε · ∇wε ) dx Ωε ∩Aε

=

ρε

= ρε =

ρε

+ ρε

Ωε ∩Aε

Ωε ∩Aε

Ωε ∩Aε

ϕ (∇vε · ∇wε ) dx + ϕ (∇˘ vε · ∇wε ) dx + ϕ (∇˘ vε · ∇wε ) dx +

(Ω\Ωε )∩Aε

(Ω\Ωε )∩Aε

(Ω\Ωε )∩Aε

ψ (∇˘ vε · ∇wε ) dx ψ (∇˘ vε · ∇wε ) dx ϕ (∇˘ vε · ∇wε ) dx

.

-

(Ω\Ωε )∩Aε

(ψ − ϕ) (∇˘ vε · ∇wε ) dx .

(12.61)

Since the sequences {˘ vε } and {wε } are equibounded in H 1 (Ω), ρε tends to ρ∗ as ε → 0, and dϑ = max{ |ψ(x) − ϕ(y)| : |x − y| < ϑ} tends to 0 as ϑ → 0, we easily obtain (ψ − ϕ) (∇˘ vε · ∇wε ) dx ρε (Ω\Ωε )∩Aε ε→0

≤ ∇˘ vε L2 (Ω) ∇wε L2 (Ω) d2λ(ε)r0 −→ 0. (12.62) As a result, taking properties (12.59)–(12.62) into account, we come to the following relation:

464


lim

ε→0

Ωε

ϕ (∇vε · ∇wε ) dx

= lim

ε→0

Ωε ∩Aε

= lim

ε→0

= lim

ε→0

Ωε ∩Aε

ρε

∗

· lim

ε→0

ϕ (∇vε · ∇wε ) dx +

Ωε ∩A ε



Ωε ∩Aε

ϕ (∇˘ vε · ∇wε ) dx +

(Ω\Ωε )∩Aε

ϕ (∇˘ vε · ∇wε ) dx

ϕ (∇˘ vε · ∇wε ) dx.

= ρ lim

ε→0

(12.63)

Ω

In order to complete this proof, it only remains to apply property (12.46) of Theorem 12.22 and the fact that v ∈ H01 (Ω) due to Theorem 12.14. Remark 12.27. As follows from Proposition 12.6, the condition ρ∗ ∈ [1/2, 1) is always valid. In particular, using the suppositions of Lemma 12.24 concerning the smoothness of the boundary ∂Q in a neighborhood of the origin, we have ρε =

|A \ Qh(ε)/λ(ε) | ε→0 ∗ −→ ρ = 1/2. |A|

So, the main result of Theorem 12.26 can be viewed as ϕ (∇vε · ∇wε ) dx → (1/2)μ∗ ϕ v dx, ∀ ϕ ∈ C0∞ (Ω), Ωε

Ω

where μ∗ is defined by (12.47). We are now in a position to prove the main result of this section concerning the passage to the limit as ε → 0 in the integral identity χε (∇˘ yε · ∇ϕ) dx + χε y˘ε ϕ dx + k0 ε−n σ(ε) y˘ε ϕ dνελ,h Ω Ω Ω −n = χε fε ϕ dx + ε σ(ε) pε ϕ dνελ,h , Ω

∀ ϕ ∈ H 1 (Ω; ΓεD ∪ Σε ).

Ω

(12.64)

Here, {(aε , pε , y˘ε ) ∈ Xε }ε>0 is an equibounded sequence of admissible triplets and σ(ε) is defined by (12.17). By Theorem 12.14, this sequence is relatively compact with respect to the weak convergence in the variable space Xε . So, we may suppose that there exists a triplet (a, p, y) ∈ H 2 (Ω) ∩ H01 (Ω) × L2 (Ω) × H01 (Ω) such that w (aε , pε , yε ) (a, p, y).


465

Theorem 12.28. Let ρ∗ be a limit of the sequence (12.28) as ε → 0 and let and pε ∈ L2 (Ω, dνελ,h ) (12.65) aε ∈ H01 (Ω) ∩ H 2 (Ω) ∩ L2 (Ω, dμλ,h ε ) ε -problems such that be any bounded sequences of admissible controls for the P aε a in H 2 (Ω) ∩ H01 (Ω) ∩ L2 (Ω, dμλ,h ε ), pε p in L

2

(Ω, dνελ,h ).

(12.66) (12.67)

Let yε = yε (aε , pε ) ∈ H 1 (Ω, Σε ) ∩ L2 (Ω, dνελ,h ) ε>0 be the corresponding sow

lutions to the problem (12.5). Then (aε , pε , yε ) (a, p, y) as ε → 0, y − a ∈ L2 (Ω, dμ∗ ), and y is the unique function in H01 (Ω) satisfying the following integral identity: (∇y · ∇ϕ) dx + (1 + k0 |∂Q|H ) yϕ dx + ρ∗ (y − a) ϕ dμ∗ Ω Ω Ω 1 = f ϕ dx + |∂Q|H p ϕ dx, ∀ ϕ ∈ H0 (Ω) ∩ L2 (Ω, dμ∗ ). (12.68)

Ω

Ω

Proof. Let wε ∈ H 1 (Ω) ε>0 be a sequence defined by Theorem 12.22. Let ϕ ∈ C0∞ (Ω) be a fixed function. It is clear that wε ϕ ∈ H 1 (Ω; ΓεD ∪ Σε ) for every ε > 0. Take wε ϕ as test functions in (12.64). Then the following integral identity holds true for every ε > 0: χε (∇(˘ yε − aε ) · ∇(wε ϕ)) dx + χε (∇aε · ∇(wε ϕ)) dx Ω Ω −n + χε y˘ε wε ϕ dx + k0 ε σ(ε) y˘ε wε ϕ dνελ,h Ω Ω −n = χε fε wε ϕ dx + ε σ(ε) pε wε ϕ dνελ,h , Ω

Ω

∀ ϕ ∈ H 1 (Ω; ΓεD ∪ Σε ). (12.69) Observe that in view of the boundedness of {fε ∈ L2 (Ω)}, by using estimate (12.25) and Theorem 12.17, we may suppose that there is a function w y ∈ H01 (Ω) such that (aε , pε , yε ) (a, p, y) as ε → 0. We now pass to the limit in (12.69) as ε → 0. We do it for each term of (12.69) separately. Observe first that χε (∇(˘ yε − aε ) · ∇(wε ϕ)) dx = χε wε (∇(˘ yε − aε ) · ∇ϕ) dx Ω Ω χε ϕ(∇(˘ yε − aε ) · ∇wε ) dx. + Ω

Take into account that χε wε → 1 strongly in L2 (Ω) (see Lemma 12.18), y˘ε − aε v = y − a in H01 (Ω), and vε = y˘ε − aε ∈ H 1 (Ωε ; ΓεD ∪ Σε ) for every ε > 0. Then, by Theorem 12.26 we have

466


ε→0

χε wε (∇(˘ yε − aε ) · ∇ϕ) dx −→

(∇(y − a) · ∇ϕ) dx, ε→0 χε ϕ(∇(˘ yε − aε ) · ∇wε ) dx −→ ρ∗ ϕ (y − a) dμ∗ ,

Ω

(12.70)

Ω

Ω

(12.71)

Ω ∗

y − a ∈ L2 (Ω, dμ ). By (12.24), it follows that {aε } is bounded in H 2 (Ω) ∩ H01 (Ω), so aε → a weakly in H 2 (Ω) and hence ∇aε → ∇a strongly in [L2 (Ω)]n . Then, due to (12.5) and (12.66)–(12.67) and since ∇wε 0 in [L2 (Ω)]n , we obtain ε→0 ε→0 χε (∇aε · ∇(wε ϕ)) dx −→ 0, χε y˘ε wε ϕ dx −→ y ϕ dx, (12.72) Ω

k0 ε−n σ(ε) −→ k0 |∂Q|H , ε→0

Ω ε→0

χε fε wε ϕ dx −→ Ω

ε→0

Ω

f ϕ dx,

(12.73)

Ω

y˘ε wε ϕ dνελ(ε),h(ε) −→ Ω

y ϕ dx as a weak limit in L2 (Ω, dνελ,h ), Ω

ε−n σ(ε)

ε→0

pε wε ϕ dνελ(ε),h(ε) −→ |∂Q|H Ω

(12.74)

p ϕ dx.

(12.75)

Ω

Thus, the required relation (12.68) is established for any function ϕ ∈ C0∞ (Ω). Moreover, from the fact that y˘ε − aε ∈ H 1 (Ω, ΓεD ∪ Σε ) and aε → a in H01 (Ω), we conclude that (˘ yε − aε ) (y − a) in H 1 (Ω), and hence 1 y ∈ H0 (Ω). To conclude, we note that the integral identity (12.68) can always be interpreted as the variational formulation of the problem ' −Δ y + (1 + k0 |∂Q|H )y + ρ∗ (y − a)μ∗ = f + p |∂Q|H , (12.76) y ∈ H01 (Ω), y − a ∈ L2 (Ω, dμ∗ ), with respect to which the following result is well known: For every a ∈ H 2 (Ω)∩ H01 (Ω)∩L2 (Ω, dμ∗ ), p ∈ L2 (Ω), and f ∈ L2 (Ω), there exists a unique solution of (12.76) (see [82]). This completes the proof. The following statement is a direct consequence of well known results of the theory of boundary value problems [167]. Corollary 12.29. Let (a1 , p1 , y1 ), (a2 , p2 , y2 ) ∈ H 2 (Ω) ∩ H01 (Ω) × L2 (Ω) × H01 (Ω) be any triplets satisfying relation (12.76). Then there exists a constant ˆ Cˆ > 0 (Cˆ = C(Ω, |∂Q|, ρ∗ )) such that

12.6 Identification of the limiting optimal control problem

y1 − y2 H01 (Ω)∩L2 (Ω,dμ∗ ) ≤ Cˆ a1 − a2 H 2 (Ω)∩H01 (Ω) + p1 − p2 L2 (Ω) + a1 − a2 L2 (Ω,dμ∗ ) .

467

(12.77)

12.6 Identification of the limiting optimal control problem In this section, we show that for the sequence (12.42), there exists a weak variational limit with respect to the w-convergence and it can be recovered in an explicit form. We begin with the following result. Lemma 12.30. Let {(aε , pε , yε ) ∈ X ε }ε>0 be a bounded sequence of admissible solutions, assumed to be w-convergent to a triplet (a, p, y) ∈ H 2 (Ω) ∩ H01 (Ω) × L2 (Ω) × H01 (Ω). Then

|∇yε | dx =

lim

ε→0

|∇y| dx + ρ

2

Ωε

2

∗

Ω

(y − a)2 dμ∗ ,

(12.78)

Ω

where the measure μ∗ and value ρ∗ are defined in Theorems 12.22 and 12.26, respectively. Proof. We first observe that 2 |∇yε | dx = χε |∇˘ yε − ∇aε |2 dx Ωε Ω χε (∇˘ yε · ∇aε ) dx − χε |∇aε |2 dx. +2 Ω

(12.79)

Ω

Then taking into account the facts that ∇aε → ∇a in [L2 (Ω)]n , y˘ε y in H 1 (Ω), and χε → 1 in L2 (Ω), we have ε→0 χε (∇˘ yε · ∇aε ) dx −→ ∇y · ∇a dx, (12.80) Ω

ε→0

Ω

χε |∇aε |2 dx −→ Ω

|∇a|2 dx.

(12.81)

Ω

Since (˘ yε − aε ) ∈ H 1 (Ω, ΓεD ∪ Σε ) for every ε > 0, it follows that we can take y˘ε − aε as a test function ϕ in (12.64). Then the following equality is ensured:

468


Ω

χε |∇˘ yε − ∇aε |2 dx χε ∇aε · (∇˘ yε − ∇aε ) dx =− Ω −n χε y˘ε (˘ yε − aε ) dx − k0 ε σ(ε) y˘ε (˘ yε − aε ) dνελ,h − Ω Ω −n + χε fε (˘ yε − aε ) dx + ε σ(ε) pε (˘ yε − aε ) dνελ,h . Ω

(12.82)

Ω

By properties (12.70)–(12.75), we obtain χε |∇˘ yε − ∇aε |2 dx lim ε→0 Ω ∇a · (∇y − ∇a) dx − y(y − a) dx =− Ω Ω y˘ε (˘ yε − aε ) dνελ,h + f (y − a) dx − k0 |∂Q|H lim ε→0 Ω Ω pε (˘ yε − aε ) dνελ,h . + |∂Q|H lim ε→0

Ω

Since (˘ yε − aε ) ∈ H 1 (Ω, ΓεD ∪ Σε ) ∩ L2 (Ω, dνελ,h ), we have y˘ε − aε → y − a strongly in L2 (Ω, dνελ,h ) (by Theorem 12.15) and y˘ε y in L2 (Ω, dνελ,h ) (by Theorem 12.14). Hence, in view of the definition of the strong convergence in variable spaces, we conclude that λ,h lim y˘ε (˘ yε − aε ) dνε = y (y − a) dx, ε→0 Ω Ω pε (˘ yε − aε ) dνελ,h = p (y − a) dx. lim ε→0

Ω

Ω

As a result, we get χε |∇˘ yε −∇aε |2 dx lim ε→0 Ω ∇a · (∇y − ∇a) dx − y(y − a) dx =− Ω Ω − k0 |∂Q|H y (y − a) dx + f (y − a) dx Ω Ω p (y − a) dx. (12.83) + |∂Q|H Ω

Let us consider the integral identity (12.68) with the test function ϕ = y − a. By rearrangement, we have


Ω

469

|∇y−∇a|2 dx + ρ∗ y (y − a) dμ∗ − ρ∗ a (y − a) dμ∗ Ω Ω =− ∇a · (∇y − ∇a) dx − (1 + k0 |∂Q|H ) y(y − a) dx Ω Ω f (y − a) dx + |∂Q|H p (y − a) dx. (12.84) + Ω

Ω

The comparison of (12.83) with (12.84) leads to the following equality: 2 2 ∗ lim χε |∇˘ yε − ∇aε | dx = |∇y − ∇a| dx + ρ (y − a)2 dμ∗ . ε→0

Ω

Ω

Ω

which, together with (12.79)–(12.81), concludes the proof.

We are now in position to establish the identification result of the weak variational limit for the sequence (12.42). Theorem 12.31. For the sequence of constrained minimization problems (12.42), there exists a unique weak variational limit with respect to w-convergence which can be represented in the form (12.43), where the cost functional I0 and the set of admissible solutions Ξ0 are defined as follows: |∇y|2 dx + |y − z ∂ |2 dx + ρ∗ (y − a)2 dμ∗ I0 (a, p, y) = Ω Ω Ω + |∂Q|H p2 dx + |K ∩ ∂Λ|H a2 dx, (12.85) Ω

Ω

⎫ y ∈ H01 (Ω), p ∈ L2 (Ω), ⎪ ⎪ ⎪ ⎪ 2 1 2 ∗ ⎪ a ∈ H (Ω) ∩ H0 (Ω), y − a ∈ L (Ω, dμ ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ a H 2 (Ω) ≤ C0 , ⎪ ⎪ ⎬ ∗ ∗ . (12.86) Ξ0 = (a, p, y) (∇y · ∇ϕ) dx + ρ (y − a) ϕ dμ ⎪ ⎪ ⎪ ⎪ Ω Ω ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + (1 + k0 |∂Q|H ) y ϕ dx = f ϕ dx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Ω Ω ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 2 ∗ ⎭ ⎩ p ϕ dx, ∀ ϕ ∈ H (Ω) ∩ L (Ω, dμ ) +|∂Q| H 0 ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

Ω

Here, Λ is a cone in Rn (see Proposition 12.5). Proof. The proof of this theorem is divided into two steps; each of them concerns the verification of the corresponding item of Definition 8.24. Step 1: Statement (d) of Definition 8.24 is valid. sequence which is w-convergent Let {(ak , pk , y˘k ) ∈ Xε }ε>0 be a bounded to a triplet (u, p, y) ∈ H 2 (Ω) ∩ H01 (Ω) × L2 (Ω) × H01 (Ω). Let {εk } be a ε for all subsequence of {ε} such that εk → 0 as k → ∞ and (ak , pk , yk ) ∈ Ξ k k ∈ N. Then, due to Theorem 12.28, we have that the w-limit triplet (u, p, y) satisfies the integral identity (12.68), and, moreover,

470


C0 ≥ lim inf ak H 2 (Ω) ≥ a H 2 (Ω) , k→∞

by the lower semicontinuity of · H 2 (Ω) with respect to the weak convergence in H 2 (Ω). So, the first inclusion in (8.59) holds true. We now turn back to the test of inequality (8.59). By the lower semicontinuity of the weak convergence in variable spaces, Proposition 12.5, and relation (12.5), we have " k ),h(εk ) σ(ε ) p2εk dνελ(ε lim ε−n k k k k→∞ Ω k ),h(εk ) +|K ∩ ∂Qς(εk ) |H a2εk dμελ(ε k Ω p2 dx + |K ∩ ∂Λ|H a2 dx. ≥ |∂Q|H Ω

Ω

To conclude, it remains only to apply Lemma 12.30. Step 2: Statement (dd) of Definition 8.24 holds true. Let (a, p, y) ∈ Ξ0 be an admissible triplet for the minimization problem (12.43). As readily follows from (12.85), for any triplet (a, pˆ, yˆ) ∈ Ξ0 there exists a constant γ > 0 depending on Ω, z ∂ , ρ∗ , p , y, and |∂Q|H such that |I0 (a, p, y) − I0 (a, pˆ, yˆ)| ≤ γ y − yˆ 2H 1 (Ω)∩L2 (Ω,dμ∗ ) + p − pˆ 2L2 (Ω) . (12.87) 0

Let 1 > δ > 0 be a given value. Using the density of the embedding H 1 (Ω) → L2 (Ω), we can choose an element pˆ ∈ H 1 (Ω) such that * + ˆ ˆ p − pˆ L2 (Ω) < , where < max δ/ γ(1 + C), δ/C . (12.88) Let yˆ = yˆ(a, pˆ ) be the corresponding solution of the boundary value problem (12.76). Then due to estimates (12.77) and (12.87), we have ˆ |I0 (a, p, y) − I0 (a, pˆ, yˆ)| ≤ γ(1 + C) p − pˆ 2L2 (Ω) < δ,

(12.89)

ˆ − pˆ L2 (Ω) < δ. y − yˆ H01 (Ω)∩L2 (Ω,dμ∗ ) ≤ C p λ,h λ,h λ,h We now construct the δ-realizing sequence ( aε , pε , yε ) ∈ X ε ε>0 as aλ,h = a, pελ,h = pˆ , and we take yελ,h as the corresponding solution of the ε ε for ελ,h , yελ,h ) ∈ Ξ original boundary value problem (12.5). Hence, ( aλ,h ε ,p every ε > 0. It is clear that this sequence in equibounded in Xε . Moreover, by Lemma 12.8 and Theorems 12.14 and 12.15, we have λ,h 2 λ,h 2 λ,h λ,h ε→0 aε aε a in L (Ω, dμε ) and dμε −→ a2 dx, (12.90) Ω Ω λ,h 2 λ,h ε→0 2 λ,h pλ,h p p ˆ in L (Ω, dν ) and dν −→ pˆ 2 dx. (12.91) ε ε ε ε Ω

Ω


471

Then, applying Theorem 12.17, we conclude that the sequence λ,h λ,h λ,h ( aε , pε , yε ) is compact with respect to the w-convergence. Let (a, pˆ, yˆ∗ ) be its w-limit. Due to Theorem 12.28, we have (a, pˆ, yˆ∗ ) ∈ Ξ0 . Since the boundary value problem (12.76) has a unique solution for every fixed a and pˆ , it follows that yˆ∗ = yˆ, w and hence ( aλ,h ελ,h , yελ,h ) (a, pˆ, yˆ) as ε → 0. ε ,p Consequently, the first part of property (dd) of Definition 8.24 is fulfilled. It remains only to verify inequality (8.61). To do so, we use properties (12.90) and (12.91) and Lemma 12.30. Then aλ,h ελ,h , yελ,h ) = I0 (a, pˆ, yˆ). lim Iε ( ε ,p

ε→0

To conclude, we apply inequality (12.89). This yields the required result I0 (a, p, y) ≥ lim Iε ( aλ,h ελ,h , yελ,h ) − δ. ε ,p ε→0

It is now clear that the constrained minimization problem (12.43) can be interpreted as the following OCP:

(∇y · ∇ϕ) dx + ρ∗ (y − a) ϕ dμ∗ + (1 + k0 |∂Q|H ) y ϕ dx Ω Ω Ω f ϕ dx + |∂Q|H p ϕ dx, ∀ ϕ ∈ H01 (Ω) ∩ L2 (Ω, dμ∗ ), (12.92) = Ω

Ω

y ∈ H01 (Ω),

p ∈ L2 (Ω), ∗

y − a ∈ L (Ω, dμ ), 2

a ∈ H 2 (Ω) ∩ H01 (Ω),

(12.93)

a H 2 (Ω) ≤ C0 ,

(12.94)

|∇y|2 dx + |y − z ∂ |2 dx + ρ∗ (y − a)2 dμ∗ Ω Ω Ω +|∂Q|H p2 dx + |K ∩ ∂Λ|H a2 dx −→ inf . (12.95)

I0 (a, p, y) =

Ω

Ω

Note that the parameters ρ∗ , |∂Q|H , |K ∩ ∂Λ|H ∈ R, and the Borel measure μ∗ are coming from the geometry of control zones. Moreover, in contrast to Pε , the limit control problem (12.92)–(12.95) contains two independent distributed control functions. Proposition 12.32. The limit OCP (12.92)–(12.95) has a unique solution.

472


Proof. The proof is quite similar to that given in Theorem 12.2. The main difference is the choice of the topology for the space of admissible solutions 2 H (Ω) ∩ H01 (Ω) × L2 (Ω) × H01 (Ω) with respect to which the set Ξ0 and the cost functional I0 possess the required topological properties, one of which has to guarantee the inclusion (12.94). It is clear that this topology can be taken as τ = (wH 2 (Ω)∩H01 (Ω) ) × (wL2 (Ω) ) × (wH01 (Ω) ), where w(·) denotes the weak topology of the corresponding Banach space. Indeed, due to the fact 1 2 ∗ that μ∗ ∈ M+ 0 (Ω), the space H0 (Ω) ∩ L (Ω, dμ ) is well defined (see Remark 1 12.3). Hence, if yn y in H0 (Ω) and an a in H 2 (Ω) ∩ H01 (Ω), then (yn − an ) (y − a) in L2 (Ω, dμ∗ ). Moreover, it can be easily checked (by passing to the limit in (12.92) and (12.94)) that the set Ξ0 is τ -closed and the cost functional I0 is τ -lower semicontinuous. The proof of Theorem 12.2 is then valid without any more modifications. Thus, combining the results of Theorem 12.2, Theorem 12.20, and Proposition 12.32, we come to the following conclusion concerning the variational properties of the limit OCP (12.92)–(12.95). ε Theorem 12.33. Let (a0ε , pε0 , yε0 ) ∈ Ξ be the optimal solutions of the ε>0

ε . Then problems P

lim Iε (a0ε , pε0 , yε0 ) =

I0 (a, p, y) = I0 (a0 , p 0 , y0 )

(12.96)

(a0ε , pε0 , yε0 ) −→ (a0 , p 0 , y 0 ) in the variable space Xε .

(12.97)

ε→ 0

inf (a, p,y)∈Ξ0

and w

12.7 Optimality conditions for the limit problem and suboptimal controls for Pε-problem In this section, we derive the optimality conditions for the problem (12.92)– (12.95) from which an optimal triplet may be determined. For this, we use the Lagrange multiplier principle and make use of Theorem 9.35 (for more details, we refer to Ioffe and Tikhomirov [125]). We obtain the weak form of the optimality system equations that an optimal triplet (a0 , p 0 , y 0 ) and Lagrange multipliers must satisfy. As we see later, this optimality system can serve as a basis for the construction of suboptimal solutions to the original problem in perforated domains. Theorem 12.34. A triplet (a0 , p 0 , y0 ) ∈ H 2 (Ω) ∩ H01 (Ω) × L2 (Ω) × H 2 (Ω) ∩ H01 (Ω) , y 0 − a0 ∈ L2 (Ω, dμ∗ )

12.7 Suboptimal controls for Pε -problem

473

in an optimal solution to the problem (12.92)–(12.95) if and only if there exists a function ψ ∈ H 2 (Ω) ∩ H01 (Ω) ∩ L2 (Ω, dμ∗ ) such that the quadruple (a0 , p 0 , y 0 , ψ) satisfies the following optimality system: (∇y 0 · ∇ϕ) dx + ρ∗ (y 0 − a0 ) ϕ dμ∗ + (1 + k0 |∂Q|H ) y 0 ϕ dx Ω Ω Ω f ϕ dx + |∂Q|H p0 ϕ dx, ∀ ϕ ∈ H01 (Ω) ∩ L2 (Ω, dμ∗ ). (12.98) = Ω

Ω

0 2(y − z ∂ ) + (1 + k0 |∂Q|) ψ φ dx (∇ψ + 2∇y ) · ∇φ dx + Ω Ω ∗ 0 0 +ρ ψ + 2(y − a ) φ dμ∗ = 0, ∀ φ ∈ H01 (Ω) ∩ L2 (Ω, dμ∗ ), (12.99) 0

Ω

p 0 = ψ/2 a.e. in Ω, (12.100) a0 (a − a0 ) dx − ρ∗ ψ(a − a0 ) dμ∗ 2|K ∩ ∂Λ|H Ω Ω ∗ 0 0 − 2ρ (y − a )(a − a0 ) dμ∗ ≥ 0, Ω ∀ a ∈ a ∈ H 2 (Ω) ∩ H01 (Ω) : a H 2 (Ω) ≤ C0 . (12.101) Proof. Let (a0 , p 0 , y 0 ) be an optimal solution to the problem (12.92)–(12.95). To apply the Lagrange principle, we set (see Theorem 9.35) Y = H 2 (Ω) ∩ H01 (Ω), U = H 2 (Ω) ∩ H01 (Ω) × L2 (Ω), V = L2 (Ω), and F (a, p, y) = −Δ y + (1 + k0 |∂Q|H )y + ρ∗ (y − a)μ∗ − f − |∂Q|H p. Since f ∈ L2 (Ω), it follows that the boundary value problem (12.76) has the unique solution y ∈ H 2 (Ω) ∩ H01 (Ω) for any a ∈ H 2 (Ω) ∩ H01 (Ω) and p ∈ L2 (Ω), and, moreover, in this case y − a ∈ L2 (Ω, dμ∗ ) (see [82, 167]). Hence, Im Fy = V. Thus, all the assumptions of Theorem 9.35 are fulfilled. We now define the Lagrange function as follows: |∇y|2 dx + |y − z ∂ |2 dx L(a, p, y, ψ) = Ω ∗

Ω

(y − a) dμ + |∂Q|H

+ρ

∗

2

Ω

(∇y · ∇ψ) dx −

+ Ω

p dx + |K ∩ ∂Λ|H 2

Ω ∗

+ (1 + k0 |∂Q|H )

Ω

y ψ dx − |∂Q|H Ω

(y − a) ψ dμ∗

f ψ dx + ρ Ω

a2 dx Ω

p ψ dx, Ω

∀ ψ ∈ H01 (Ω) ∩ L2 (Ω, dμ∗ ).

474


In accordance with Theorem 9.35, there exists a function ψ ∈ H01 (Ω) ∩ L2 (Ω, dμ∗ ) such that relations (9.125) and (9.126) are valid. In this case, relation (9.126) takes the form (12.100)–(12.101), whereas (9.125) can be written as (12.99). Since y 0 ∈ H 2 (Ω) ∩ H01 (Ω) and z ∂ ∈ L2 (Ω), it follows that the bilinear form ∇ψ, ∇φL2 (Ω) + (1 + k0 |∂Q|H ) ψ, φL2 (Ω) + ρ∗ ψ, φL2 (Ω,dμ∗ ) is coercive on the space H01 (Ω) ∩ L2 (Ω, dμ∗ ). So, by the Riesz representation theorem, we immediately conclude that there exists a unique function ψ ∈ H01 (Ω) ∩ L2 (Ω) satisfying equality (12.99) and such that ψ ∈ H 2 (Ω). Thus, the first part of Theorem 9.35 is proved, that is, (12.98)–(12.101) are the necessary optimality conditions. Since the mapping y → I0 (a, p, y) is convex and the mapping (a, p ) → F (a, p, y) is continuous and affine, relations (12.98)– (12.101) are also sufficient optimality conditions for the problem (12.92)– (12.95). As this problem is uniquely solvable, the proof is complete. As an evident consequence of this theorem, we have the following result. Corollary 12.35. If (a0 , p 0 , y 0 ) is an optimal solution to (12.92)–(12.95), then p 0 ∈ H 2 (Ω) ∩ H01 (Ω) ∩ L2 (Ω, dμ∗ ). (12.102) Now, using (12.102) and applying Lemmas 12.8 and 12.10, we immediately establish the following approximation property for the optimal controls. Proposition 12.36. If p 0 ∈ H 2 (Ω) ∩ H01 (Ω) ∩ L2 (Ω, dμ∗ ) and a0 ∈ H 2 (Ω) ∩ H01 (Ω) are the optimal controls to the homogenized problem (12.92)–(12.95), then 2 λ,h ∀ ε > 0, (a0 , p 0 ) ∈ L2 (Ω, dμλ,h ε ) × L (Ω, dνε ), 0 0 2 λ,h 0 2 λ,h (a ) dμε = (a0 )2 dx, a a in L (Ω, dμε ), lim ε→0

0

p p

0

in L

2

(Ω, dνελ,h ),

Ω

(p )

Ω

(12.104)

Ω

0 2

lim

ε→0

(12.103)

dνελ,h

(p 0 )2 dx.

=

(12.105)

Ω

The next question we are going to consider in this section concerns the approxε for ε small enough. imation of the optimal solutions of the original problem P We focus our attention on suboptimal solutions which guarantee the closeness sub sub of the corresponding value of the cost functional Iε (asub ε , pε , yε ) to its minimum if ε is small enough (see Definition 11.30). Then Proposition 12.36 leads to the following final result. Theorem 12.37. Let p 0 ∈ H 2 (Ω) ∩ H01 (Ω) ∩ L2 (Ω, dμ∗ ) and a0 ∈ H 2 (Ω) ∩ H01 (Ω)

12.7 Suboptimal controls for Pε -problem

475

be the optimal controls for the homogenized problem (12.92)–(12.95). Then the sequence of the pairs (a0 , p 0 ) ε>0 is asymptotically suboptimal for the ε . original OCP P

Proof. Let us consider the sequence of triplets (a0 , p 0 , y˘#ε ) ∈ Xε ε>0 , where y#ε = y#ε (a0 , p 0 ) are the corresponding solutions of the boundary value problem ε . Moreover, due to (12.5). Each of these triplets is admissible for the problem P estimate (12.25), this sequence is equibounded in Xε . By Theorem 12.17, it is relatively compact with respect to the w-convergence in Xε . Hence, taking into account Proposition 12.36 and Theorem 12.28, we deduce that this sequence ε→0 is w-compact and (a0 , p 0 , y#ε ) −→ (a0 , p 0 , y 0 ), where (a0 , p 0 , y 0 ) is an optimal solution to the homogenized problem (12.92)–(12.95). ε . We observe that ε be the optimal solutions to P Let (a0ε , pε0 , yε0 ) ∈ Ξ ε>0

inf

bε (aε ,pε ,yε )∈ Ξ

Iε (aε , pε , yε ) − Iε (a0 , p 0 , y#ε )

= Iε (a0ε , pε0 , yε0 ) − Iε (a0 , p 0 , y#ε ) ≤ Iε (a0ε , pε0 , yε0 ) − I0 (a0 , p 0 , y 0 ) + I0 (a0 , p 0 , y0 ) − Iε (a0 , p 0 , y#ε ) ≤ Iε (a0ε , pε0 , yε0 ) − I0 (a0 , p 0 , y 0 ) 0 2 ∗ 0 0 2 ∗ 2 ˘ (y − a ) dμ − χε |∇y#ε | dx + |∇y | dx + ρ Ω Ω Ω + |y 0 − z ∂ |2 dx − χε |y˘#ε − z ∂ |2 dx Ω Ω 0 2 −n 0 2 λ,h (p ) dx − ε σ(ε) (p ) dνε + |∂Q|H Ω Ω 0 2 ς(ε) 0 2 λ,h (a ) dx − |K ∩ ∂Q |H (a ) dμε + |K ∩ ∂Λ|H Ω

Ω

= J1 + J2 + J3 + J4 + J5 . To conclude the proof, we note that for a given δ > 0 one can always find the following: (i) ε1 > 0 such that J1 < δ/5 for all ε < ε1 by Theorem 12.33; (ii) ε2 > 0 such that J2 < δ/5 for all ε < ε2 by Lemma 12.30; (iii) ε3 > 0 such that J3 < δ/5 for all ε < ε3 by the w-convergence (a0 , p 0 , y#ε ) to (a0 , p 0 , y 0 ); (iv) ε4 > 0 such that J4 < δ/5 for all ε < ε4 by (12.17) and (12.105); (v) ε5 > 0 such that J5 < δ/5 for all ε < ε5 by (12.104) and (12.27). Thus, as expected, estimate (11.71) is valid for all ε < min1≤i≤5 {εi }. It should be stressed that a sequence of asymptotically suboptimal controls ε has a particularly simple and attractive form if the optimal for the problem P

476


ε is such that control pair (a0ε , pε0 ) for the problem P ! ! x x 0 ∗ 0 ∗ , pε (x) = p , aε (x) = a ε λ(ε) ε h(ε) where a∗ ∈ H 1 (K) and p ∗ ∈ L2 (∂Q, dHn−1 ) are some Y -periodic functions. Indeed, in this case by Theorems 6.70 and 12.33, we have ! 1 0 ∗ n−1 a dH aε = a0 , |K ∩ ∂Λ|H K∩∂Λ ! 1 0 ∗ n−1 pε p dH = p0 |∂Q|H ∂Q 2 λ,h 0 0 in L2 (Ω, dμλ,h ε ) and L (Ω, dνε ), respectively, where (a , p ) are optimal controls for the homogenized problem (12.92)–(12.95). Hence, the conclusion of Theorem 12.37 can be reformulated as follows: The sequence of constant pairs (a0 , p 0 ) ∈ R2 ε>0 , where 0

p =

1 |∂Q|H

∗

p dH ∂Q

! n−1

0

and a =

1 |K ∩ ∂Λ|H

∗

a dH

! n−1

K∩∂Λ

ε . is a sequence of asymptotically suboptimal controls for the problem P

,

13 Asymptotic Analysis of Elliptic Optimal Control Problems in Thick Multistructures with Dirichlet and Neumann Boundary Controls

In this chapter, we study a class of optimal control problems (OCPs) for a linear elliptic equation in a domain Ωε ⊂ Rn (thick multistructure), whose boundary ∂Ωε contains a very highly oscillating part with respect to ε, as ε → 0. We consider this problem assuming that there are two types of the controls active via Neumann and Dirichlet boundary conditions posed on the different parts of the oscillating boundary (for a comparison, see [73, 88]). By a thick multistructure Ωε we mean a domain in Rn , which consists of some domain Ω + and a large number of cylinders with axes parallel to Oxn and ε-periodically distributed along some manifold Σ on the boundary of Ω + (see Fig. 13.1). This manifold is called the joint zone and the domain Ω + is called the junction’s body. Here, ε is a small positive parameter which characterizes the distance between the neighboring cylinders and their thickness. Thus, each attached cylinder has a small cross-section of size ε and its limiting dimension (as ε → 0) is equal to 1. In view of this, such cylinders will be called thin domains. For the study of boundary value problems in thick multistructures, we refer to [35, 58, 89, 90, 187], where, however, OCPs have not been considered. Thick multistructures (or thick junctions) are prototypes of widely used engineering constructions as well as many other physical and biological systems with very distinct characteristic scales (microscopic radiators, ferritefilled rod radiators, and others). The computational calculation of the solutions of boundary value problems is very complicated due to singularities associated with the thick junctions. Asymptotic analysis is one of the main approaches to study boundary value problems in such domains. However, only few works deal with the problem of optimal control of partial differential equations (PDEs) in thick multistructures (see [85, 88, 100, 153]). This can be explained as follows. Thick junctions have a special character of the connectedness: There are points in a thick junction which are at a short distance of order O(ε), but the length of all curves, which connect these points in the junction, is of order O(1). As a result, there are no extension operators 1,p (Rn ) that would be uniformly bounded in ε. AdditionP : W 1,p (Ωε ) → Wloc P.I. Kogut, G.R. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains, Systems & Control: Foundations & Applications, DOI 10.1007/978-0-8176-8149-4 13, © Springer Science+Business Media, LLC 2011

477

478

13 Elliptic Optimal Control Problems in Thick Multi-Structures

ally, boundary value problems lose coercivity in the limit passage as ε → 0 and this creates special difficulties in the asymptotic investigation.

13.1 Statement of the problem and basic notation Let B = (0, a)n−1 and C be bounded open smooth domain in Rn−1 (n ≥ 2) and C ⊂⊂ (0, 1)n−1 , let γ ∈ C 1 (B) and β ∈ C 1 (B) be functions such that γ(x ), 0 < γ0 = inf x ∈B

0 < β0 = inf β(x ), x ∈B

and let {ε} be a sequence of positive numbers converging to 0. In the sequel, we will always assume that ε = a/N , where N is a large positive integer. When we write ε > 0, we consider only the elements of this sequence. Let us denote a point of Rn by x = (x1 , x2 , . . . , xn ) = (x , xn ) and introduce the following sets: = {(x , xn ) ∈ Rn : −β(x ) < xn < γ(x ), x ∈ B},

Ω

= B × {0},

Σ Ω

+

= {(x , xn ) ∈ Rn : x ∈ B, 0 < xn < γ(x )},

Ω − = {(x , xn ) ∈ Rn : x ∈ B, −β(x ) < xn < 0}, Γ0 = {(x , xn ) ∈ Rn : x ∈ B, xn = −β(x )},

(13.1)

θε = {k = (k1 , k2 , . . . , kn−1 ) ∈ Nn−1 : εC + εk ⊂⊂ B}, Ωε = Ω + ∪ {(x , xn ) : x ∈ k∈θε (εC + εk) , −β(x ) < xn ≤ 0}, Ωε− = Ωε ∩ Ω − . Then, in view of our previous description, the set Ωε is a thick multistructure which consists of the junction’s body Ω + and a large number N n−1 of the thin cylinders Gkε with axis Oxn and ε-periodically distributed on the basis Σ of Ω + (see Fig. 13.1 for a 3D example). Here, each cylinder Gkε is obtained with an ε-homothety in the first (n − 1) variables, that is, Gkε = {(x , xn ) : x ∈ εC + εk, −β(x ) < xn ≤ 0}. k It is easy to see that Ωε = Ω + ∪ k∈θε Gε . Denote by Γε the union of the lower bases Γεk = {(x , xn ) : x ∈ ε · C + εk, xn = −β(x )} of the thin cylinders Gkε when k ∈ θε and by Sε the union of their boundaries along the axis Oxn : Sεk = {(x , xn ) : x ∈ ε∂C + εk, −β(x ) < xn < 0}.

13.1 Statement of the problem and basic notation

479

Fig. 13.1. 3D thick multistructure

In Ωε , we consider the following optimal boundary control problem: 2 2 n−1 (yε − q0 ) dx + uε dH +ε p2ε dHn−1 −→ inf, (13.2) Iε = Ω+

Γε

Sε

⎫ x ∈ Ωε , −Δx yε (x) + yε = fε (x), ⎪ ⎪ ⎬ ∂ν yε (x) = ε pε (x), x ∈ Sε , yε (x) = uε (x), x ∈ Γε , ⎪ ⎪ ⎭ x ∈ ∂Ωε \ (Γε ∪ Sε ) , ∂ν yε (x) = 0,

uε ∈ Uε = u ∈ H 1/2 (Γε ; Hn−1 ), u L2 (Γε ;Hn−1 ) ≤ Cu ,

pε ∈ Pε = p ∈ L2 (Sε ; Hn−1 ), ε p 2L2 (Sε ;Hn−1 ) ≤ Cp ,

(13.3)

(13.4) (13.5)

where Hn−1 is the (n − 1)-dimensional Hausdorff measure on Rn , which coincides with ordinary (n − 1)-dimensional surface area, q0 ∈ L2 (Ω + ) and fε ∈ L2 (Ω) are given functions such that fε −→ f0

strongly in L2 (Ω) as ε → 0,

(13.6)

∂ν = ∂/∂ν is the outward normal derivative, Cu > 0 and Cp > 0 are some fixed constants that are independent of ε, uε and pε , Uε and Pε are the sets of admissible controls, uε and pε are control functions, and yε = yε (uε , pε ) ∈ H 1 (Ωε ) is the corresponding weak solution of the boundary value problem (13.3). In the sequel, for simplicity, we will use the following notation: H 1/2 (Γε ) = H 1/2 (Γε ; Hn−1 ), L2 (Γε ) = L2 (Γε ; Hn−1 ), and L2 (Sε ) = L2 (Sε ; Hn−1 ).

480


It is well known that for every fixed ε and for any control functions uε ∈ H 1/2 (Γε ) and pε ∈ L2 (Sε ), the boundary value problem (13.3) admits a unique solution yε ∈ H 1 (Ωε ) such that the following integral identity ∇yε · ∇ϕ dx + yε ϕ dx = fε (x) ϕ(x) dx Ωε Ωε Ωε pε ϕ dHn−1 , ∀ ϕ ∈ H 1 (Ωε ; Γε ) +ε Sε

(13.7) holds and the trace of yε equals uε on Γε . The function yε is called the weak solution to the problem (13.3). Here, H 1 (Ωε ; Γε ) = {ϕ ∈ H 1 (Ωε ) : ϕ = 0 on Γε }. In addition, if u ∈ H 1/2 (Γε ) and pε ∈ L2 (Sε ), then the weak solution satisfies the a priori norm estimate

√ yε H 1 (Ωε ) ≤ c1 fε L2 (Ωε ) + uε H 1/2 (Γε ) + ε pε L2 (Sε ) . (13.8) Remark 13.1. Since the thick junction Ωε is a nonconvex domain with a nonsmooth boundary, it follows that solutions of boundary value problems in such a domain have only minimal H 1 -smoothness. So it makes no sense to take admissible boundary controls with more smoothness than u ∈ H 1/2 (Γε ; Hn−1 ) and p ∈ L2 (Sε ; Hn−1 ). For a comparison, we refer to [101], where the class of admissible Dirichlet controls was taken as

Θε = u = ϑ|Γε : ϑ ∈ H 3/2 (Γ0 ; Hn−1 ), u H 3/2 (Γ0 ;Hn−1 ) ≤ Cu . The aim of our research in this chapter is to study the asymptotic behavior of this OCP as ε → 0, that is, when the number of attached thin cylinders infinitely increases and their thickness vanishes. The approach we propose gives the possibility of replacing the original OCP by some limit problem defined in a more simple domain. We show that an optimal control for the limit problem can be taken as a prototype for the modeling of suboptimal controls to the original one. It should be stressed here that if the small parameter ε is changed, then all components of this control problems (the domain Ωε , the constraint sets Uε and Pε , the cost functional Iε , and the set, where we seek its infimum) are changed as well. Let us observe also that the volume of the material included in the set Ωε− does not converge to 0 as ε → 0. Moreover, limε→0 Ωε− = Ω − in the Hausdorff metric and |Ωε− | → |Ω − |, that is, the set Ω − is filled up by the thin cylinders in the limit passage as ε → 0. It produces the fact that the Neumann boundary controls pε will transform (as ε → 0) to some distributed control function p ∈ L2 (Ω − ) on the right-hand side of the homogenized equation.

13.2 On measure description of the optimal control problem

481

13.2 Description of the optimal control problem in terms of singular measures Following the standard approach, we will describe the geometry of the set Sε , and hence the class of admissible boundary controls, in the terms of singular measures on Rn which do not satisfy the regularity property with respect to the corresponding Lebesgue measures. Let μ0 be a periodic finite positive Borel measure on Rn−1 . Let = [0, 1)n−1 be the cell or torus of periodicity for μ0 . We assume that the Borel measure μ0 is the probability measure on Rn−1 , concentrated and uniformly distributed on the set ∂C, so dμ0 = 1. Remark 13.2. By definition, we have μ0 (\∂C) = 0. Therefore, any functions, taking the same values on the set ∂C, coincide as elements of L2 (, dμ0 ). Here, the Lebesgue space L2 (, dμ0 ) with respect to the measure μ0 is defined in a usual way with the corresponding norm |f (x)|2 dμ0 f 2L2 (, dμ0 ) =

(we adopt the standard notation L2 () when μ0 is the Lebesgue measure). We set n = × [0, 1) = [0, 1)n and consider the measure dμ = dμ0 × dxn on n . It is easy to see that this measure is concentrated on the set ∂C ×[0, 1), and for any smooth function g, we have 1 n−1 −1 g dμ = g dxn dμ0 = H (∂C × [0, 1)) g dHn−1 . n

0

∂C×[0,1)

However, as follows from the properties of the Hausdorff measure, we have Hn−1 (∂C × [0, 1)) = Hn−2 (∂C) (see [106]). In what follows, we use the notation |∂C|H = Hn−2 (∂C) for (n − 2)-dimensional Hausdorff measure of the set ∂C. Then the previous relation can be rewritten in the form 1 −1 g dμ = g dxn dμ0 = |∂C|H g dHn−1 . (13.9) n

0

∂C×(0,1)

In order to clarify relation (13.9), we consider the planar thick multistructure Ωε ⊂ R2 . Then n = 2 and the set C is some part of the segment (0, 1). For instance, let C = {x1 ∈ (0, 1) : |x1 − 1/2| < h/2}, where h ∈ (0, 1) is a fixed value. In this case, |∂C|H = 2 and the 1-periodic measure μ0 on R1 can be defined as 1 1 (δM1 + δM2 ) (B) = (δM1 + δM2 ) (B) μ0 (B) = |∂C|H 2 1 for any Borel set B ⊆ [0, 1), where Mi = 2 + i − 32 h, i = 1, 2. Here, δMi is the Dirac measures located at the points Mi . Thus, the multiplier |∂C|−1 H in (13.9) is equal to 1/2.

482


Let Λ be any Borel set of Rn . We introduce the scaling measure με by the rule με (Λ) = εn μ(ε−1 Λ). This measure has a period ε, and, moreover, since μ(εn ) = ε μ0 (ε), it follows that ε 1 dμ0 (x /ε) d(xn /ε) = εn dμ0 dxn = εn . με (εn ) = εn 0

0

ε

As a result, the measure με weakly converges to Lebesgue measure on Rn as ε → 0 (in symbols dμε dx), that is, ϕ dμε = ϕ dx for all functions ϕ ∈ C0∞ (Rn ). lim ε→0

Rn

Rn

Now, we turn back to the definition of a weak solution yε ∈ H 1 (Ωε ) of the boundary value problem (13.3) (see (13.7)). Since the Sobolev space H 1 (Ωε ) can be viewed as the closure of C0∞ (Rn ) with respect to the norm 1/2 2 2 (y + |∇y| ) dx , Ωε

it follows that a function yε ∈ H 1 (Ωε ) is a weak solution of the abovementioned problem whenever ∇yε · ∇ϕ dx + yε ϕ dx Ωε Ωε fε (x) ϕ(x) dx + ε pε ϕ dHn−1 (13.10) = Ωε

C0∞ (Rn ; Γε ),

Sε

for all ϕ ∈ where is the set of all functions of C0∞ (Rn ) such that ϕ|Γε = 0. Let us consider the last term on the left-hand side of identity (13.10). Using the notation of Sect. 13.1, we get pε ϕ dH

ε Sε

n−1

=ε

C0∞ (Rn ; Γε )

n−1 N

ε(∂C+kj )

j=1

= ε|∂C|H

n−1 N

N

= |∂C|H

= |∂C|H

j=1

= |∂C|H

Ω−

ε(+kj )

ε(+kj )

j=1 n−1 N

−β(x )

j=1 n−1

0

pε ϕ dHn−2 dxn 0

−β(x ) 0 −β(x )

pε ϕ εn−2 dμ0 (x /ε) dxn

pε ϕ εn dμ0 (x /ε) d(xn /ε)

j

˛ ff ˛ x ∈ε(+kj ) (x ,xn ) :˛˛ −β(x )<xn 0, sible control functions for the P independent of ε, such that (13.17) pε L2 (Ω − ,dμε ) yε H 1 (Ωε ) ≤ c fε L2 (Ωε ) + uε H 1/2 (Γε ) + for all ε > 0. Definition 13.5. The triplet (uε , pε , yε ) ∈ Xε is called admissible for the probε if uε ∈ Uε , pε ∈ Pε , and yε = yε (uε , pε ) is the corresponding weak lem P solution of the problem (13.3) whose trace on Γε is equal to uε . ε is It is easy to see that the set Ξε of all admissible solutions for P nonempty, convex, and τε -closed for every ε > 0, where by τε we denote the product of the weak topologies of H 1/2 (Γε ), L2 (Ω − , dμε ), and H 1 (Ωε ). Note that the cost functional (13.15) is convex and lower τε -semicontinuous, but not τε -coercive. So, in general, we cannot assert the solvability of this problem [111, 169]. Thus, the OCPs we consider are such that the existence of an opε must be assumed. This is untypical timal solution for the control problem P for the majority of investigations in this field. However, if an optimal solution to the Pε problem exists, then this solution is unique. Additionally, for every δ > 0, a triplet (uδε , pεδ , yεδ ) ∈ Ξε (the so-called approximate solution) can be always found such that Iε (uδ , p δ , y δ ) − ≤ δ. inf I (u, p, y) (13.18) ε ε ε ε (u,p,y) ∈Ξε

13.3 The choice of topology ε -problem Following the concept of variational convergence, we represent the P for various values of ε in the form of the sequence of corresponding constrained minimization problems

13.3 The choice of topology

inf

(u,p,y) ∈Ξε

Iε (u, p, y) ; ε = a/N → 0 ,

485

(13.19)

where the cost functional Iε : Ξε → R and the sets of admissible solutions are defined in (13.15) and (13.16), respectively. Then the study of asymptotic behavior of the OCP (13.15)–(13.16) as ε → 0 can be reduced to the analysis of the limit properties of the sequences (13.19). However, since for every fixed ε the admissible solution (uε , pε , yε ) belongs to the functional space Xε = H 1/2 (Γε ) × L2 (Ω − , dμε ) × H 1 (Ωε ) depending on the small parameter ε, we focus our attention on the convergence formalism in these variable spaces. Let us denote by u ε the trivial extension by 0 of a function uε ∈ L2 (Γε ) ε ∈ L2 (Γ0 ). to the set Γ0 (see (13.1)). Then u ! Definition 13.6. We say that a sequence uε ∈ H 1/2 (Γε ) ε>0 weakly converges to a function u∗ with respect to the space L2 (Γ0 ) if u ε u∗ in L2 (Γ0 ). Using the ideas of Sect. 7.2 (see Proposition 7.12) and the convergence concept in variable spaces (see Sect. 6.2), we state the following result. Remark 13.7. For brevity, hereinafter in this chapter we restrict ourself to the case when the function β ∈ C 1 (B) is kept constant β(x) ≡ β0 , ∀ x ∈ B. Then Γ0 = {(x , xn ) ∈ Rn : x ∈ B, xn = −β0 }, " {(x , xn ) : x ∈ εC + εk, xn = −β0 }, Γε = k∈θε

Gε =

"

{(x , xn ) : x ∈ εC + εk, −β0 < xn < 0}.

k∈θε

Proposition 13.8. For every bounded sequence {uε ∈ H 1/2 (Γε )}ε>0 such that u ε u∗ in L2 (Γ0 ), the following inequality holds: u2ε dHn−1 ≥ |C|−1 u2∗ dHn−1 . (13.20) lim inf ε →0

Γε

Γ0

Proof. Let χΓ0 and χGε be the characteristic functions of the sets Γ0 and Gε , respectively. Then Γε = Γ0 ∩ Gε . Let χΓε ∈ L∞ (Γ0 ) be the characteristic function of the Dirichlet control zone Γε , that is, 1 if x ∈ Γε , ∀ x ∈ Γ0 . χΓε (x ) = 0 otherwise, To begin, let us show that ∗

χΓε |C| in L∞ (Γ0 ) as ε → 0.

(13.21)

486


Indeed, in view of ε [0, 1)n−1 -periodicity of this characteristic function and the mean value property, we have ϕ(x )χΓε (x ) dHn−1 = lim ϕ(x )χ∪k∈θε (εC+εk) (x ) dx lim ε→0 Γ ε→0 B 0 # $ ϕ(x ) dx

= B

χC (x ) dx

(0,1)n−1

ϕ(x ) dx

= |C|

B

= |C|

ϕ(x ) dHn−1 ,

∀ ϕ ∈ C0∞ (Γ0 ).

Γ0

Thus, statement (13.21) holds true. Let νε be a Radon measure on Γ0 defined as the surface measure dνε = χΓε dHn−1 . Then, in view of (13.21), we have dνε dν = |C| dHn−1 in the sense of Definition 2.29.

(13.22)

Let {uε ∈ H 1/2 (Γε )}ε>0 be a bounded sequence of admissible controls such that u ε u∗ in L2 (Γ0 ). Hence, u ε ϕ dHn−1 −→ u∗ ϕ dHn−1 as ε → 0, ∀ ϕ ∈ C0∞ (Γ0 ). (13.23) Γ0

Γ0

On the other hand, since uε ∈ Uε for every ε > 0, it follows that u 2ε dνε = sup u 2ε χΓε dHn−1 = sup uε 2L2 (Γε ) ≤ Cu 2 , sup ε>0

Γ0

ε>0

Γ0

ε>0

! that is, the sequence u ε ∈ L2 (Γ0 , dνε ) ε>0 is bounded. So, by Proposition 6.15, we can suppose that this sequence is compact with respect to the weak convergence in the variable space L2 (Γ0 , dνε ). Hence, there exists an element u0 ∈ L2 (Γ0 , dν) such that passing if necessary to a subsequence, we get u ε ϕ dHn−1 = u ε ϕ dνε −→ u0 ϕ dν Γ0 Γ0 Γ0 u0 ϕ dHn−1 as ε → 0, ∀ ϕ ∈ C0∞ (Γ0 ). (13.24) = |C| Γ0

Having compared the results of limit passages (13.23) and (13.24), we deduce that (13.25) u0 = |C|−1 u∗ . As a result, inequality (13.20) is a direct consequence of Proposition 6.16. Indeed, by (6.28) and (13.24), we have


lim inf ε →0

Γε

487

u2ε dHn−1 = lim inf u 2ε dνε ≥ u20 dν ε →0 Γ0 Γ0 |C|−2 u2∗ |C| dHn−1 = |C|−1 = Γ0

Γ0

u2∗ dHn−1 .

This concludes the proof. In fact, in view Proposition 13.8 and representation (13.4), we can deduce the following lemma. Lemma 13.9. Every sequence of admissible controls {uε ∈ Uε }ε>0 is relatively compact with respect to the convergence in the sense of Definition 13.6. Moreover, its weak L2 -limit u∗ satisfies the following condition: % u∗ L2 (Γ0 ) ≤ |C|Cu < Cu . Now, we give the convergence formalism of the sequence of boundary con! trols pε ∈ L2 (Ω − , dμε ) ε>0 . First, we recall that the sequence of Borel measures {dμε } weakly converges to Lebesgue measure dx. Let ! pε ∈ L2 (Ω − , dμε ) ε>0 be any bounded sequence, that is, lim sup ε→0

Ω−

pε2 dμε < +∞. !

Definition 13.10. We say that a bounded sequence pε ∈ L2 (Ω − , dμε ) is weakly convergent if there exists an element p∗ ∈ L2 (Ω − ) such that ϕ pε dμε = ϕp∗ dx for any function ϕ ∈ C0∞ (Rn ). lim ε→0

Ω−

ε>0

Ω−

Note that the class of test functions in Definition 13.10 can be essentially extended (see Sect. 6.2) Proposition 13.11. If a bounded sequence { pε ∈ L2 (Ω − , dμε )}ε>0 converges ∗ 2 − 2 weakly to p ∈ L (Ω ) and a function ψ ∈ L (Ω − ) is such that |ψ(x)| ≤ const almost everywhere on Ω − , then ∗ ψ pε dμε = ψp dx, lim ψ dμε = ψ dx. lim ε→0

Ω−

Proposition 13.12. Let Then the following hold:

Ω−

ε→0

Ω−

!

pε ∈ L2 (Ω − , dμε )

ε>0

(i) This sequence is relatively weakly compact.

Ω−

be any bounded sequence.

488


(ii) If pε p∗ in L2 (Ω − , dμε ) as ε → 0, then pε2 dμε ≥ (p∗ )2 dx. lim inf ε→0

Ω−

Ω−

Then, in view of representations (13.5) and (13.14), the following result is obvious. ! Lemma 13.13. Every sequence of admissible controls pε ∈ Pε ε>0 is relatively weakly compact, and its weak limit p∗ ∈ L2 (Ω − ) satisfies the condition p∗ 2L2 (Ω − ) ≤ |∂C|−1 H Cp . ! Now, we consider the following type of sequences yε ∈ H 1 (Ωε ) ε>0 . Following the idea of Brizzi and Chalot [35] (see also [74, 88]), we extend each of the functions yε by 0 onto the whole of domain Ω, namely & yε (x) if x ∈ Ωε , yε (x) := (13.26) 0 if x ∈ Ω \ Ωε . We set yε+ (x) := yε (x) if x ∈ Ω + , yε− (x) := yε (x) if x ∈ Ωε− and yε− (x) := yε (x) if x ∈ Ω − . Then, using the fact that the boundaries of the thin rods Sε are rectilinear with respect to xn we have yε− ) = ∂x (yε− ) ∂xn ( n

in Ω − .

(13.27)

This means that yε− ∈ W2 (Ω − ), where W2 (Ω − ) is the anisotropic Sobolev space {v ∈ L2 (Ω − ) : ∂xn v ∈ L2 (Ω − )}. Let χC be the characteristic function of the set C. We suppose that this function is -periodically extended on the entire space Rn−1 . It is well known that ∗ χC (·/ε) |C| in L∞ (B) as ε → 0. (0,1)

(0,1)

Here, B = (0, a)n−1 and |C| is the (n − 1)-dimensional Lebesgue measure of C. Then the following facts are valid: ∗

χΩε− |C| ∗

χΩε ∩Σ |C|

in L∞ (Ω − ) ∞

in L (Σ; H

n−1

as ε → 0, )

as ε → 0,

(13.28) (13.29)

where χΩε− and χΩε ∩Σ are the characteristic functions of the sets Ωε− and Ω ε ∩ Σ, respectively. ! Thus, if for a sequence yε ∈ H 1 (Ωε ) ε>0 there exists a constant C > 0 independent of ε such that yε H 1 (Ωε ) ≤ C, then yε+ H 1 (Ω + ) + y− ε W (0,1) (Ω − ) ≤ C 2


489

and there exist a subsequence {ε } of {ε} (still denoted by ε) and elements y0+ ∈ H 1 (Ω + ), y0− ∈ L2 (Ω − ) such that ⎫ yε+ y0+ in H 1 (Ω + ), ⎪ ⎬ (13.30) yε− v = |C|(|C|−1 v) =: |C|y0− in L2 (Ω − ), ⎪ ⎭ − − 2 − in L (Ω ). ∂xn yε |C| ∂xn y0 The last limit is an evident consequence of (13.27). Moreover, similarly as in [35], it is easy to prove the relation y0+ = y0−

a.e. on Σ,

(13.31)

that is, the traces of the limit functions coincide on Σ. ! Definition 13.14. We say that a sequence yε ∈ H 1 (Ωε ) ε>0 is weakly convergent to a function y∗ = (y∗+ , y∗− ) with respect to the space H 1 (Ω + ) × (0,1) W2 (Ω − ) as ε tends to 0 (in symbols, yε y∗ = (y∗+ , y∗− )) if the following hold: (a) yε+ y∗+ in H 1 (Ω + ); (0,1) (b) yε− |C| y∗− in W2 (Ω − ). Let us introduce the space V(Ω) = y ∈ L2 (Ω) : and endow it with the norm # y 2V(Ω)

= |C|

∂y ∈ L2 (Ω − ), y ∈ H 1 (Ω + ) ∂xn

y 2L2 (Ω − )

' ' ' ∂y '2 ' +' ' ∂xn ' 2 L

(13.32)

$ (Ω − )

+ y 2H 1 (Ω + ) .

It is easy to see that each limit function y∗ in the sense of Definition 13.14 satisfies the condition y∗ ∈ V(Ω). Moreover, as follows from [74], any function of this space admits a trace on Σ, and the space H 1 (Ω) is dense in V(Ω). As a result, each element of the space V(Ω) has the following ! properties: Let y∗ = (y∗+ , y∗− ) ∈ V(Ω) be a weak limit of yε ∈ H 1 (Ωε ) ε>0 with respect to the space H 1 (Ω + ) × W2 (Ω − ) and let γn ∈ L2 (Ω − ) be a weak limit in L2 (Ω − ) of the corresponding sequence of partial derivatives {∂xn yε− }ε>0 . Then (0,1)

γn = |C| · ∂y∗− /∂xn a.e. in Ω − , y∗+

=

y∗− ,

H

n−1

-a.e. on Σ.

(13.33) (13.34)

490


Remark 13.15. Note that assertions (13.30), (13.31), (13.33), and (13.34) can be proved without the assumption that yε is a solution of the boundary value problem (13.3). Therefore, for any pair (y∗+ , y∗− ), which is a weak limit in the sense of Definition 13.14, we have y∗+ (·) = y∗− (·) Hn−1 -almost everywhere on Σ. Moreover, any function y∗ = (y∗+ , y∗− ) from V(Ω) has a trace on the manifold Γ0 = {(x , xn ) ∈ Rn : x ∈ B, xn = −β(x )} such that y∗− |Γ0 ∈ B2 space [111].

(0,1/2)

(0,1/2)

(Γ0 ) (see [243, 244]). Here, B2

is the Besov

Definition 13.16. We say that a sequence of triplets {(uε , pε , yε ) ∈ Xε } ε>0 w is w-convergent to (u, p, y + , y − ) (in symbols, (uε , pε , yε ) −→ (u, p, y + , y − )) with respect to the space Y0 ≡ L2 (Γ0 ) × L2 (Ω − ) × H 1 (Ω + ) × W2

(0,1)

(Ω − )

as ε tends to 0 if the following hold: (i) uε → u in the sense of Definition 13.6; (ii) pε → p in the sense of Definition 13.10; (iii) yε (y + , y − ) in the sense of Definition 13.14. As a result, we can give the following conclusion: Proposition 13.17. Let {(uε , pε , yε ) ∈ Ξε } ε>0 be any sequence of admissib ε -problem such that sup le triplets for the P ε>0 yε H 1 (Ωε ) < +∞. Then there exist a subsequence {(uε , pε , yε )} ε >0 and a quadruple (u, p, y + , y− ) ∈ Y0 w 0 , p ∈ P0 , and (uε , pε , yε ) −→ such that u ∈ U (u, p, y+ , y− ), where the sets U0 and P0 are defined as

% 0 = u ∈ L2 (Γ0 ) : u L2 (Γ ) ≤ |C| · Cu , U 0

P0 = p ∈ L2 (Ω − ) : p 2L2 (Ω − ) ≤ |∂C|−1 H · Cp . Remark 13.18. As follows from Proposition 13.17, ! any uniformly bounded sequence of suboptimal triplets (uδε , pεδ , yεδ ) ∈ Ξε ε>0 in Yε ≡ L2 (Γε ) × L2 (Ω − , dμε ) × H 1 (Ωε ) ε is relatively compact with respect to the topology associated with wfor P convergence on Y0 . So, this topology can be chosen as the most natural one for the limit analysis of the original optimal boundary control problem.

13.4 Definition of the limit problem and its properties

491

13.4 Definition of the limit problem and its properties We begin this section with some notions that will be useful for the definition of a limit boundary control problem. Let u be any element of L2 (Γ0 ). Since the embedding H 1/2 (Γ0 ) → L2 (Γ0 ) is dense with respect to the strong topology for L2 (Γ0 ), it follows that for a given value γ > 0, there is an element q ∈ H 1/2 (Γ0 ) such that u − q L2 (Γ0 ) ≤ γ. Let us consider the sequence ! qε = |C|−1 χε q ε>0 . It is clear that qε = |C|−1 q Γ ∈ H 1/2 (Γε ) for every ε ε > 0, and, moreover, qε |C|−1 |C|q = q in L2 (Γ0 ). Thus, the space L2 (Γ0 ) possesses the weak approximation property with respect to the family of sets {Γε }ε>0 , that is, for every γ > 0 and any u ∈ L2 (Γ0 ), there exist an element ! q ∈ H 1/2 (Γ0 ) and a sequence qε ∈ H 1/2 (Γε ) ε>0 such that sup qε H 1/2 (Γε ) < +∞,

(13.35)

ε>0

u − q L2 (Γ0 ) ≤ γ

and

qε q

with respect to L2 (Γ0 ).

(13.36)

! In this case, the sequence qε ∈ H 1/2 (Γε ) ε>0 is called γ-realizing. Taking this observation into account, we introduce the following concept of a limit minimization problem for the family (13.19) (for a comparison, see Definition 8.24). Definition 13.19. We say that a minimization problem on Y0 I0 (u, p, y+ , y− ) inf + −

(13.37)

(u,p,y ,y )∈ Ξ0

is the variational limit of the sequence (13.19) with respect to the w-convergence (or variational w-limit) if the following conditions are satisfied: (i) If a sequence {(uε , pε , yε ) ∈ Xε }ε>0 w-converges to a (u, p, y + , y − ), and there exists a subsequence {εk } of {ε > 0} such that εk → 0 as k → ∞ and (uεk , pεk , yεk ) ∈ Ξεk for all k, then (u, p, y + , y − ) ∈ Ξ0 ,

(13.38)

I0 (u, p, y+ , y− ) ≤ lim inf Iεk (uεk , pεk , yεk ).

(13.39)

k→∞

(ii) For every quadruple (u, p, y+ , y− ) ∈ Ξ0 and any value γ > 0, there exists a γ-realizing sequence {(uε , pε , yε )}ε>0 such that (uε , pε , yε ) ∈ Ξε , ∀ ε > 0,

(uε , pε , yε ) → (v, q, z + , z − ), w

(13.40)

(u, p, y + , y − ) − (v, q, z + , z − ) Y0 ≤ γ,

(13.41)

I0 (u, p, y + , y − ) ≥ lim sup Iε (uε , p ε , yε ) − γ.

(13.42)

ε→0

492


In what follows, conditions (i)–(ii) together with relations (13.38), (13.40) and (13.41) will be used as a basis for the recovery of the set of admissible solutions Ξ0 , whereas inequalities (13.39) and (13.42) will be exploited for the identification of the limit cost functional I0 : Ξ0 → R. Note that in this case, the set Ξ0 is the sequential w-limit of the sequence {Ξε ⊂ Xε }ε>0 in the Kuratowski sense (or K(w)-limit, for simplicity). Following the arguments of the proof of Theorem 8.25, it is easy to show that the variational w-convergence of the sequence (13.19) to the problem (13.37) implies the convergence of the minimum values of Iε on Ξε to the minimum of I0 on Ξ0 , and moreover in this case, every w-cluster point of any sequence of subminimizers for Iε is a minimizer of I0 . Theorem 13.20. Assume that the constrained minimization problem (13.37) is the variational w-limit of the sequence (13.19) and this problem ! has a unique solution (u0 , p 0 , (y 0 )+ , (y 0 )− ) in Ξ0 . Let (uδεε , pεδε , yεδε ) ∈ Ξε ε>0 be a sequence of approximate solutions to the Pε problems (see (13.18)) such that supε>0 yεδε H 1 (Ωε ) < +∞ and δε → 0 as ε → 0. Then (uδεε , pεδε , yεδε ) −→ (u0 , p 0 , (y 0 )+ , (y 0 )− ), w

inf

(u, p, y + , y − )∈ Ξ0

(13.43)

I0 (u, p, y + , y − ) = I0 u0 , p 0 , (y 0 )+ , (y 0 )− = lim Iε (uδεε , pεδε , yεδε ) ε→0

= lim

inf

ε→0 (u,p,y)∈ Ξε

Iε (u, p, y).

(13.44)

Definition 13.21. We say that the family of optimal control problems (13.2)– (13.5) has a limit, as ε tends to 0, with respect to the w-convergence if for the corresponding sequence of constrained minimization problems (13.19), there exists a variational limit which can be recovered in the form of some OCP.

13.5 Analytical representation of the limit set of admissible solutions The main object of our consideration in this section is the sequence of the sets of admissible solutions

Ξε ∈ Xε ≡ H 1/2 (Γε ) × L2 (Ω − , dμε ) × H 1 (Ωε ) (13.45) ε>0

ε . Our aim is to study its w-limiting properties in the to the original problem P sense of Definition 13.19. In spite of the fact that we have no result concerning the K(w)-compactness of the sequences of sets in the scale {Xε }, we will show that the K(w)-limit set exists for the sequence (13.45) and it can be recovered in an implicit analytical form. As it follows from Definition 13.19, the main difficulty is connected with the passage to the limit in the integral identity (13.12) for the problem (13.3) as ε → 0.

13.5 Limit set of admissible solutions

493

Proposition 13.22. Let w∗ ∈ L∞ (Ω − ) and p∗ ∈ L2 (Ω − ) be elements such that L∞ (Ω − ) wε −→ w∗ −

L (Ω , dμε ) pε −→ p∗ 2

lim

ε→0

Ω−

(13.46)

weakly in the sense of Definition 13.10. (13.47)

Then

strongly in L∞ (Ω − ),

wε pε ϕ dμε =

Ω−

w∗ p∗ ϕ dx,

∀ϕ ∈ C0∞ (Rn ).

Proof. First, we note that for every ε > 0, each of the integrals Ω− wε pε ϕ dμε is well defined. Indeed, using the Cauchy–Schwartz inequality, we have − 1/2 − wε pε ϕ dμε ≤ wε L∞ (Ω − ) |με (Ω )| pε L2 (Ω − ,dμε ) · sup− |ϕ(x)|. x∈Ω

Ω

Since the sequence of Borel measures {dμε } is weakly convergent to the Lebesgue measure dx, the sequences {wε ∈ L∞ (Ω − )}ε>0 ,

{pε ∈ L2 (Ω − , dμε )}ε>0

are bounded, and for every compact set K ⊂ Rn , the inequality lim sup με (K) ≤ |K| ε→0

holds (see Lemma 2.32); it follows that w p ϕ dμ ε ≤ C sup |ϕ(x)|. − ε ε − Ω

x∈Ω

Thus, (wε pε ) dμε is a Radon measure. Moreover, from this, we immediately have that pε w −→ p∗ w weakly in {L2 (Ω − , dμε )} for any function w ∈ L∞ (Ω − ). Then taking (13.46) and (13.47) into account and passing to the limit in the inequality w p dμ − w p dx ≤ p (w − w ) dμ ∗ ∗ ε ε ∗ ε − ε ε ε Ω Ω− Ω− w∗ pε dμε − w∗ p∗ dx + − Ω− Ω ≤ wε − w∗ L∞ (Ω − ) |pε ϕ| dμε Ω− + w∗ pε dμε − w∗ p∗ dx Ω−

Ω−

as ε tends to 0, we immediately obtain the required result.

494


Let us introduce the anisotropic Sobolev space V(Ω; Γ0 ) = v ∈ L2 (Ω) : ∂xn v ∈ L2 (Ω), v ∈ H 1 (Ω + ), v = 0 Hn−1 a.e. on Γ0

!

with the scalar product (y, v)V(Ω;Γ0 ) =

(y v + ∇y · ∇v) dx (y v + ∂xn y ∂xn v) dx. + |C| Ω+

(13.48)

Ω−

Note that each function of this space has a trace on the manifold Σ (see [243, 244]), and the space H 1 (Ω; Γ0 ) is dense in V(Ω; Γ0 ). Moreover, for any function v ∈ V(Ω), relation (13.31) holds true. pε ∈ Pε }ε>0 be any sequences Theorem 13.23. Let {uε ∈ Uε }ε>0 and { of admissible boundary controls for the Pε problems. Let u0 ∈ L2 (Γ0 ) and p0 ∈ L2 (Ω − ) be their weak limits ! in the sense of Definitions 13.6 and 13.10, respectively. Let yε ∈ H 1 (Ωε ) be the corresponding weak solutions to the w problem (13.3) such that supε>0 yε H 1 (Ωε ) < +∞. Then (uε , pε , yε ) −→ (u0 , p 0 , v0+ , v0− ) as ε → 0, where & v0+ (x) if x ∈ Ω + , (13.49) v0 (x) = v0− (x) if x ∈ Ω − is the unique weak solution in V(Ω) of the following limit problem: ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ − ∂x2n xn v0− (x) + v0− = f0 (x) + |∂C|H |C|−1 · p 0 (x), x ∈ Ω − , ⎪ ⎪ ⎪ + + x ∈ ∂Ω \ Σ, ⎬ ∂ v (x) = 0,

−Δx v0+ (x) + v0+ (x) = f0 (x),

x ∈ Ω+,

ν 0

v0− = |C|−1 u0 ,

on Γ0 ,

=

v0− ,

on Σ,

=

|C| ∂xn v0− ,

on Σ.

v0+ ∂xn v0+

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ (13.50)

Proof. As follows from Proposition 13.17, there exist a subsequence {ε } of {ε} (still denoted by {ε}) and a quadruple (u0 , p 0 , v0+ , v0− ) ∈ L2 (Γ0 ) × L2 (Ω − ) × H 1 (Ω + ) × W2

(0,1)

such that

(uε , pε , yε ) −→ (u0 , p 0 , v0+ , v0− ) w

(Ω − )

as ε → 0.

Similarly as in Sect. 13.3 (see relations (13.30) and (13.31)), we prove that


yε+ v0+

in H 1 (Ω + ),

yε− |C| v0−

in L2 (Ω − ),

v0+ = v0−

Hn−1 -almost everywhere on Σ,

495

(13.51)

∂xn v0+ = |C|∂xn v0− Hn−1 -almost everywhere on Σ. In addition, there exist functions γi ∈ L2 (Ω − ) (i = 1, . . . , n − 1) such that n ∂v0− − y γ , . . . , γ , |C| (13.52) in L2 (Ω − ) . ∇ x ε 1 n−1 ∂xn Let us prove that the function v0− satisfies the boundary condition v0− = |C|−1 u0

Hn−1 -almost everywhere on Γ0 ,

(13.53)

and give a sense to this equality. In view of our initial suppositions, we have ε yε− = u

Hn−1 -a.e. on Γ0 , ∀ ε > 0, yε− |C| v0−

in

u ε u0

in L2 (Γ0 ),

(0,1) W2 (Ω − ).

Then, as follows from the integral identity ∂ yε− yε− ψ dHn−1 = − ψ dx Γ0 Ω − ∂xn ∂ψ yε− dx, − ∂xn Ω−

(13.54) (13.55)

∀ψ ∈ C0∞ (Rn ; ∂Ω − \ Γ0 ),

where C0∞ (Rn ; ∂Ω − \ Γ0 ) = {ϕ ∈ C0∞ (Rn ) : ϕ = 0 on ∂Ω − \ Γ0 }, we get ( yε− )|Γ0 −→ |C| v0− Γ0 weakly in L2 (Γ0 ). Hence,

Γ0

yε− ψ dHn−1

u ε ψ dHn−1 = lim

u0 ψ dHn−1 = lim

ε→0

ε→0

Γ0

= −|C| = |C|

Ω−

∂v0− ψ dx − |C| ∂xn

v0− ψ dHn−1 ,

Γ0

Ω−

v0−

∂ψ dx ∂xn

∀ψ ∈ C0∞ (Rn ; ∂Ω − \ Γ0 ), (13.56)

Γ0

and we obtain the required relation (13.53). Now, let us show that the function v0 is the unique weak solution of the boundary value problem (13.50). With this aim, we rewrite the integral identity (13.12) in the following way:

496


Ω+

∇yε+

· ∇ϕ dx +

− ∇y ε · ∇ϕ dx +

Ω−

Ω+

fε ϕ dx + |∂C|H

= Ω

yε+ ϕ dx

+

+

χΩε− fε ϕ dx,

Ω−

Ω−

yε− ϕ dx

+ Ω−

pε ϕ dμε

∀ ϕ ∈ C0∞ (Rn ; Γ0 ).

(13.57)

Passing to the limit in (13.57) as ε → 0 and taking (13.51) into account (see also (13.28)), we obtain n−1 + ∇v0 · ∇ϕ dx + γi (∂ϕ/∂xi ) dx Ω+

Ω − i=1

+ |C| + |C|

Ω−

Ω−

(∂v0− /∂xn ) (∂ϕ/∂xn ) dx + v0− ϕ dx p 0 ϕ dx +

= |∂C|H Ω− f0 ϕ dx, + |C| Ω−

Ω+

v0+ ϕ dx

f0 ϕ dx

Ω+

∀ ϕ ∈ C0∞ (Rn ; Γ0 ).

(13.58)

In order to prove that γi = 0 a.e. on Ω − for all i ∈ {1, . . . , n − 1}, let us fix i ∈ {1, . . . , n − 1} and let wεi be a sequence in W 1,∞ (Ω − ) satisfying the following conditions: wεi −→ xi D wεi

strongly in L∞ (Ω − ), = 0 a.e. on

Ωε−

(13.59) (13.60)

for every ε. The existence of such a sequence is proved in [35, 74]. Further, we take the test functions ϕ = wεi φ and ϕ = x i φ with φ ∈ ∞ C0 (Ω − ) in (13.57). By virtue of (13.60), we get − i ∇y yε− φ wεi dx ε · ∇φ wε dx + Ω− Ω− pε φ wεi dμε = |∂C|H Ω− + χΩε− fε φ wεi dx, ∀ φ ∈ C0∞ (Ω − ), (13.61) Ω−

Ω−

− ∇y ε · ∇(φ xi ) dx +

yε− φ xi dx pε φ xi dμε = |∂C|H Ω− + χΩε− fε φ xi dx, ∀ φ ∈ C0∞ (Ω − ), Ω−

Ω−

(13.62)


497

for every ε. Then, passing to the limit in (13.61) and (13.62) as ε → 0 and using the properties (13.28), (13.51), and (13.59) and Proposition 13.22, we have n−1 ∂φ ∂v0− ∂φ γk xi dx + |C| xi dx + |C| v0− φ xi dx ∂xk Ω − k=1 Ω − ∂xn ∂xn Ω− = |∂C|H p 0 φ xi dx Ω− f0 φ xi dx, ∀ φ ∈ C0∞ (Ω − ), (13.63) + |C| Ω−

n−1

Ω − k=1

∂v0− ∂φ xi dx + |C| v0− φ xi dx ∂x ∂x − − n n Ω Ω p 0 φ xi dx = |∂C|H Ω− + |C| f0 φ xi dx, ∀ φ ∈ C0∞ (Ω − ). (13.64)

∂(φ xi ) γk dx + |C| ∂xk

Ω−

Making comparison (13.63) with (13.64), we conclude that γk φ dx = 0, ∀k ∈ {1, . . . , n − 1}, ∀φ ∈ C0∞ (Ω − ), Ω−

that is, γi = 0 a.e. on Ω − . Thus, the function v0 satisfies the identity ∂v0− ∂ϕ + ∇v0 · ∇ϕ dx + |C| dx Ω+ Ω − ∂xn ∂xn v0+ ϕ dx + |C| v0− ϕ dx + + − Ω Ω p 0 ϕ dx + f0 ϕ dx = |∂C|H Ω− Ω+ + |C| f0 ϕ dx, ∀ ϕ ∈ C0∞ (Rn ; Γ0 ). Ω−

(13.65)

If we take into account that v0 = v0+ , v0− , where v0+ ∈ H 1 (Ω + ) and (0,1) v0− ∈ W2 (Ω − ), and make use of the definition of the space V(Ω; Γ0 ) (see (13.48)), then identity (13.65) can be rewritten in the form f0 ϕ dx + |C| f0 ϕ dx v0 , ϕ V = Ω+ Ω− + |∂C|H p 0 ϕ dx ∀ ϕ ∈ V(Ω; Γ0 ). (13.66) Ω−

498


As a result, we say that a function v0 ∈ V(Ω) is a weak solution to the problem (13.50) if it satisfies identity (13.66) and the trace of v0− on Γ0 equals |C|−1 u0 in the sense of (13.56). Then, following the standard Hilbert space methods, we obtain the required result: The function v0 ∈ V(Ω) is the unique weak solution to the problem (13.50). Due to the uniqueness of the solution to the problem (13.50), the above reasoning holds for any subsequence of {ε} chosen at the beginning of the proof. This concludes the proof. We are now in a position to state the main result of this section, which deals with the K(w)-limit analysis of the set sequence (13.45). Theorem 13.24. For the sequence of the sets of admissible solutions for the ε problems {Ξε } , there exists a no-empty K(w)-limit set Ξ0 ⊂ L2 (Γ0 ) × P ε>0 (0,1) L2 (Ω − ) × H 1 (Ω + ) × W2 (Ω − ) with the following representation: Ξ0 = (u, p, v + , v − )

⎫ % = u ∈ L2 (Γ0 ) : u L2 (Γ0 ) ≤ |C| · Cu , ⎪ u ∈ U 0 ⎪ ⎪ ⎪

⎪ ⎪ ⎪ p ∈ P0 = p ∈ L2 (Ω − ) : p 2L2 (Ω − ) ≤ |∂C|−1 ⎪ , · C ⎪ p H ⎪ ⎪ ⎪ ⎪ + + + ⎪ v (x) + v = f (x), x ∈ Ω , −Δ ⎪ x 0 ⎪ ⎪ ⎪ ⎪ 2 − ⎪ ∂ v (x) ⎪ − ⎬ − + v (x) 2 ∂x n . (13.67) ⎪ = f0 (x) + |∂C|H |C|−1 · p(x), x ∈ Ω − , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ν v + (x) = 0, x ∈ ∂Ω + \ Σ, ⎪ ⎪ ⎪ ⎪ ⎪ − −1 ⎪ = |C| u on Γ , v ⎪ 0 ⎪ ⎪ ⎪ + − v = v on Σ, ⎪ ⎪ ⎪ ⎪ ⎭ + − ∂xn v = |C| ∂xn v on Σ Proof. To obtain representation (13.67), we have to verify conditions (13.38) and (13.40)–(13.41) of Definition 13.19. We also show that a K(w)-limit set exists for the whole sequence (13.45). For this, we divide the proof into several steps. Step 1. Since the set Ξ0 is nonempty (see the previous lemma), let (u, p, y + , y − ) be any of its!representatives. We begin with the construction of a sequence u ε ∈ H 1/2 (Γε ) ε>0 such that u ε → u in the sense of Definition 13.6 and u ε ∈ Uε for every ε < ε0 , where ε0 > 0 is some positive value. For % this, we consider two cases: The given quadruple is such that u L2 (Γ0 ) < |C| · Cu , % and, in the second case, u L2 (Γ0 ) = |C| · Cu . In the first case, we construct a weakly convergent sequence { uε }ε>0 to u as follows: Let uε ∈ L2 (Γε ) be the restriction of |C|−1 u on Γε . Then (see the proof of Proposition 13.8) u ε = |C|−1 χΓε u u

in L2 (Γ0 ).

(13.68)


499

Additionally, uε L2 (Γε ) = |C|

−1

1/2 2

χΓε u dH

n−1

Γ0

−→

%

1/2

|C|−1

2

u dH

n−1

< Cu

Γ0

as ε →0 (recall that χ2Γε = χΓε , and χΓε → |C| weakly-∗ in L∞ (Γ0 ) and weakly in L2 (Γ0 )). Further, using the fact that the Sobolev space H 1/2 (Γε ) is dense in L2 (Γε ), we get the following: For every ε > 0, there is an element u ε ∈ H 1/2 (Γε ) such 2 that uε − u ε L2 (Γε ) < ε and u ε ∈ Uε for ε sufficiently small. Then taking into account (13.68), we obtain

n−1 ( ( g ) − u dH ( u uε ) − u > ε ≡ < g, ( Γ0

≤ g L2 (Γ0 ) uε − uε L2 (Γε ) + |< g, u ε − u >| −→ 0

as ε → 0

for every g ∈ L2 (Γ0 ), that is, u ε → u weakly in the sense of Definition 13.6 and u ε ∈ Uε for ε sufficiently small. % Now, we consider the second case when u L2 (Γ0 ) = |C|·Cu . Then there exists a control sequence {uε ∈ L2 (Γ0 )} satisfying the condition: % uε u in L2 (Γ0 ) and uε L2 (Γ0 ) < |C| · Cu , ∀ ε > 0. Since the weak topology of L2 (Γ0 ) is metrizable on the set

% 0 = u ∈ L2 (Γ0 ) : u L2 (Γ ) ≤ |C| · Cu , U 0 one can construct a sequence {wε ∈ L2 (Γ0 )}ε>0 such that each of wε is in a convex envelope of a finite number of elements {uε }ε>0 , and wε −→ u strongly % in L2 (Γ0 ). Note that in this case we also have wε L2 (Γ0 ) < |C| · Cu for every ε > 0. So, we construct a weakly to u convergent sequence { uε }ε>0 as 1/2 −1 uε − |C| wε L2 (Γε ) < ε2 . follows: u ε ∈ H (Γε ) are elements such that Then, since |C|−1 χΓε wε u in L2 (Γ0 ) and

) −1

|C|

wε L2 (Γε ) =

|C|−2

) −→

χΓε wε2 dHn−1 Γ0

|C|−1

u2 dHn−1 < Cu Γ0

500


(as the limit of the product of weakly and strongly convergent sequences), it ( ε ∈ Uε for ε sufficiently small. follows that ( uε ) → u weakly in L2 (Γ0 ) and u Step 2. For the above given element p ∈ P0 , we have p ∈ L2 (Ω − ) and p 2L2 (Ω− ) ≤ |∂C|−1 H · Cp . Let {ζε } be any sequence of smooth functions such that ζε → p strongly in L2 (Ω − ) as ε → 0. The following result will be useful later. Lemma 13.25. lim ε→0

Ω−

ζε ϕ dμε =

p ϕ dx, Ω−

∀ ϕ ∈ C0∞ (Rn ).

(13.69)

Proof. Indeed, let us partition Ω − into cubes with edges ε and denote these cubes by the symbols εjn . Then ζε ϕ dμε = ζε (x) ϕ(x) dμε + ζε (x) ϕ(x) dμε Ω−

j

=

εjn

Ω − ∩ εjn

ζε (xj ) ϕ(xj )

j

εjn

dμε +

Ω − ∩ εjn

ζε (x) ϕ(x) dμε , (13.70)

where xj is a Lebesgue point of p in the cube εjn and the second sum is calculated over cubes. By the definition of the measure the set of boundary με , we have

εjn

dμε = εn

Ω − ∩ εjn

dμ = εn . Moreover, n

ζε (x) ϕ(x) dμε ≤ sup |ζε (x)ϕ(x)|εn · D(ε), x∈ Ω −

where D(ε) is the number of boundary cubes and εn D(ε) → 0 by Jordan’s measurability of the set ∂Ω − . Thus, summarizing the above-cited facts, we have ⎡ ⎤ lim ζε ϕ dμε = lim ⎣ ζε (xj ) ϕ(xj )εn − ζε ϕ dx⎦ ε→0

Ω−

+ lim

ε→0

Ω−

ε→0

Ω−

j

ζε ϕ dx = lim ζε (xj ) ϕ(xj ) − ε−n ε→0

+ lim

j

ε→0

Ω − ∩ εjn

ζε (x) ϕ(x) dx +

−n ζ = lim ε (xj ) ϕ(xj ) − ε ε→0

+

j

p ϕ dx. Ω−

εjn

εjn

ζε ϕ dx εn p ϕ dx ζε ϕ dx εn Ω−


501

Let us suppose that ζε (xj ) ϕ(xj ) − ε−n ε→0 lim

εjn

j

ζε ϕ dx εn > 0.

Then there exist a constant C ∗ > 0 and a value ε∗ > 0 such that ζε (xj ) ϕ(xj ) − ε−n ζε ϕ dx ≥ C ∗ , ∀ ε ∈ (0, ε∗ ) j εn

(possibly except for a finite number of indices j for every fixed ε). Hence, wild oscillations are present in the sequence {ζε ϕ}. However (see [106, 245]), if we have very rapid fluctuations in the functions {ζε ϕ}, then the convergence ζε ϕ → p ϕ almost everywhere on Ω − is excluded. This fact immediately reflects the failure of the strong convergence ζε ϕ → p ϕ in L2 (Ω − ) as ε → 0 (see Valadier’s theorem [245]). So, our supposition was wrong, and we get −n lim ζε ϕ dx εn = 0. ζε (xj ) ϕ(xj ) − ε ε→0

εjn

j

As a result, we have lim

ε→0

Ω−

ζε ϕ dμε =

p ϕ dx, Ω−

which completes the proof of this lemma. Taking property (13.69) into account, we note that for the selected sequence {ζε }, there exist a numerical sequence {dε ∈ R}ε>0 and a value ε0 > 0 such that √ lim dε = 1 and dε ζε 2L2 (Ω− ) ≤ |∂C|−1 for all ε < ε0 . H · Cp ε→0

Thus, the desired sequence of admissible boundary controls { pε ∈ L2 (Ω − , dμε )}ε>0 , √ which is weakly convergent to p , can be constructed as follows: pε = dε · ζε for every ε > 0. Step 3. Let yε be a weak solution of the boundary value problem (13.3) in the sense of Definition 13.3 corresponding to u ε and pε . If this sequence satisfies the condition supε>0 yε H 1 (Ωε ) < +∞, then, due to Theorem 13.23, w we have ( uε , pε , yε ) −→ (u, p, v0− , v0+ ), where v = (v0+ , v0− ) is a weak solution in V(Ω) of the limit problem (13.50). Since this problem has a unique solution we immediately deduce that (u, p, v0− , v0+ ) = (u, p, y − , y + ). Thus, putting γ = 0, we see that for a given quadruple (u, p, y + , y − ) ∈ Ξ0 , properties (13.40) and (13.41) of Definition 13.19 hold true.

502


! However, if the sequence yε ∈ H 1 (Ωε ) ε>0 is unbounded, then the sequence {( uε , pε , yε )}ε>0 does not need to be relatively w-compact. Let us fix 2 a value γ > 0. Then by the weak approximation property of the space ! L (Γ0 ), 1/2 1/2 there exists an element q ∈ H (Γ0 ) and a sequence qε ∈ H (Γε ) ε>0 such that relations (13.35) and (13.36) are valid. Let { pε ∈ Pε }ε>0 be the sequence that has been constructed in Step 2. Then pε p in L2 (Ω − , dμε ). Let yε be the corresponding weak solution of the boundary value problem (13.3) with prescribed qε and pε . Then, in view of estimate (13.17), we immediately conclude that supε>0 yε H 1 (Ωε ) < +∞. Hence, due to Theorem 13.23, we have w (qε , pε , yε ) −→ (q, p, v − , v + ), where v = (v + , v − ) is a weak solution in V(Ω) of the limit problem (13.50) with u0 = q and p0 = p. Let y = (y + , y − ) be a weak solution of the problem (13.50) with u0 = u and p0 = p. Then using standard techniques, we obtain y − v V(Ω) ≤ C∗ u − q L2 (Γ0 ) < C∗ γ.

(13.71)

As a result, we have (see (13.32)) (uε , pε , yε ) → (q, p, v − , v + ), √ (u, p, y + , y − ) − (q, p, v− , v + ) Y0 ≤ γ 1 + C∗ . w

Therefore, properties (13.40) and (13.41) of Definition 13.19 hold true. Step 4. We now verify property (13.38) of Definition 13.19. Let {(uk , pk , yk )}k ∈N be a w-convergent sequence for which there exists a sequence {εk → 0} such that (uk , pk , yk ) ∈ Ξεk for all k ∈ N. Let (u, p, y + , y − ) be its w-limit. Then, by Proposition 13.8, we immediately have % C0 ≥ lim inf uk L2 (Γεk ) ≥ |C|−1 · u L2 (Γ0 ) , k→∞

0 . that is, u ∈ U Due to Proposition 13.12 and the definition of constrained set Pε (see (13.13)), we also have 2 C ≥ lim inf p dμ ≥ p2 dx; |∂C|−1 p ε k k H k→0

Ω−

Ω−

that is, p ∈ P0 . In conclusion, it remains only to apply Theorem 13.23. Thus, (u, p, y + , y − ) ∈ Ξ0 , and we have obtained the required result.

13.6 Identification of the limiting cost functional In this section, we show that the cost functional in the limit constrained minimization problem (13.37) can be recovered in the explicit form, and moreover, its analytical representation is different from the original one (13.2).

13.6 Identification of the limiting cost functional

503

Theorem 13.26. For the sequence of constrained minimization problems (13.19), there exists a variational w-limit (13.37) (in the sense of Definition 13.19) as ε tends to 0, where the set Ξ0 is defined in (13.67), and I0 (u, p, y+ , y− ) = (y + − q0 )2 dx + |C|−1 u2 dHn−1 Ω+ Γ0 + |∂C|H p 2 dx. (13.72) Ω−

Proof. In order to obtain the result of identification for the limit cost func(0,1) tional I0 : L2 (Γ0 ) × L2 (Ω − ) × H 1 (Ω + ) × W2 (Ω − ) → R in the form (13.72), we have to verify conditions (i) and (ii) of Definition 13.19. Let (u, p, y+ , y− ) be any representative of Ξ0 , and let {(uk , pk , yk )}k∈N w be a w-convergent sequence such that (uk , pk , yk ) −→ (u, p, y + , y− ) and (uk , pk , yk ) ∈ Ξεk for every k ∈ N, where {εk } is a subsequence of {ε} converging to zero. Then, using Propositions 13.12 and 13.8 and the properties of w-convergence, we get u2k dHn−1 = lim inf u 2k dHn−1 ≥ |C|−1 u2 dHn−1 , lim inf k→∞

k→∞

Γεk

Γ0

Γ0

lim inf k→∞

Ω−

Therefore,

pk2 dμεk ≥

p 2 dx. Ω−

lim inf Iεk (uk , pk , yk ) = lim k→∞

k→∞

Ω+

(yk+ − q0 )2 dx + lim inf k→∞

u2k dHn−1 Γεk

+ |∂C|H lim inf pk2 dμεk k→∞ − Ω + 2 ≥ (y − q0 ) dx + |C|−1 u2 dHn−1 Ω+ Γ0 + |∂C|H p 2 dx, Ω−

that is, property (i) of Definition 13.19 is valid. To verify inequality (13.42) for an arbitrary quadruple (u, p, y+ , y − ) ∈ Ξ0 and a positive value γ, we have to construct a special γ-realizing sequence {(uε , pε , yε )} that satisfies condition (ii) of Definition 13.19. Let q ! ∈ H 1/2 (Γ0 ) be an element of the open ball w ∈ L2 (Γ0 ) : w − u L2 (Γ0 ) < γ for which there exists a sequence of admissible Dirichlet controls {qε ∈ Uε }ε>0 such that qε2 dHn−1 = |C|−1 q 2 dHn−1 . lim ε→0

Γε

Γ0

In view of Lusin’s theorem, such a choice is always possible.

504


As for the realizing sequence for the function p ∈ L2 (Ω − ), we take any collection of smooth functions {pε }ε>0 such that pε → p strongly in L2 (Ω − ). Then lim pε2 dx = p 2 dx, ε→0

Ω−

Ω−

and we may always suppose that there exists a value ε0 > 0 such that pε2 dx ≤ p 2 dx for every ε < ε0 . Ω−

Ω−

So, using the same arguments as in the proof of Lemma 13.25, we obtain 2 pε dμε = p 2 dx and pε ∈ Pε , ∀ ε < ε0 . lim ε→0

Ω−

Ω−

Let yε be the corresponding weak solutions of the boundary value problem (13.3) in the sense of Definition 13.3. Then, in view of the a priori estimate w (13.17) and Theorem 13.23, we have (qε , pε , yε ) −→ (q, p, v + , v − ). By analogy with Theorem 13.24, we can easily obtain that √ (13.73) (u, p, y + , y − ) − (q, p, v− , v + ) Y0 ≤ γ 1 + C∗ , where the constant C∗ is defined in (13.71). Therefore, lim sup Iε (qε ,pε , yε ) ε→0 = lim ε→0

Ω+

(yε+ − q0 )2 dx + lim

ε→0

qε2 dHn−1 Γε

+ |∂C|H lim inf pε2 dμε ε→0 Ω− + 2 −1 = (y − q0 ) dx + |C| q 2 dHn−1 Ω+ Γ0 + |∂C|H p 2 dx = I0 (q, p, v − , v + ).

(13.74)

Ω−

However, as immediately follows from (13.72) and (13.17), there is a constant C∗ such that I0 (u, p, y − , y + ) − I0 (q, p, v − , v + ) ≤ C∗ (u, p, y+ , y − ) − (q, p, v − , v + ) Y . 0 Hence, taking (13.73) into account, we conclude that √ I0 (u, p, y − , y+ ) − I0 (q, p, v − , v + ) ≤ γC∗ 1 + C∗ . Combining the last estimate and relation (13.74), we obtain the required inequality (13.42). This completes the proof.

13.6 Identification of the limiting cost functional

505

It is easy to see that the constrained minimization problems (13.37) can be recovered in the form of some OCP. So, in view of Theorems 13.24 and 13.26 and Definition 13.21, we can give the following conclusion. Theorem 13.27. For the OCP (13.2)–(13.5) under condition (13.6), there exists the unique limit with respect to w-convergence and it can be represented in the form + − + 2 −1 (y − q0 ) dx + |C| u2 dHn−1 I0 (u, p, y , y )∗ = Ω+ Γ0 ∗ + |∂C|H p 2 dx −→ inf, (13.75) Ω−

% 0 = u ∈ L2 (Γ0 ) : u L2 (Γ ) ≤ |C| Cu , u∈U 0

2 − 2 p ∈ P0 = p ∈ L (Ω ) : p L2 (Ω − ) ≤ |∂C|−1 H Cp ,

(13.76)

(13.77) ⎫ x ∈ Ω+, −Δx y + (x) + y + (x) = f0 (x), ⎪ ⎪ ⎪ ⎪ ⎪ − ∂ 2 y − (x)/∂x2n + y − (x) = f0 (x) + |∂C|H |C|−1 p (x), x ∈ Ω − , ⎪ ⎪ ⎪ + + ∂ν y (x) = 0, x ∈ ∂Ω \ Σ, ⎬ ⎪ y − = |C|−1 u, on Γ0 , ⎪ ⎪ ⎪ ⎪ + − ⎪ y =y , on Σ, ⎪ ⎪ ⎭ + − ∂xn y = |C| ∂xn y , on Σ. (13.78) To end of this section, we give the following observation concerning the variational properties of the homogenized problem (13.75)–(13.78). Since the cost functional I0 : Ξ0 → R is strictly convex and lower semicontinuous with respect to the product of the weak topologies for L2 (Γ0 ), L2 (Ω − ), H 1 (Ω + ), (0,1) and W2 (Ω − ) and its domain Ξ0 is a bounded, convex, and closed subset of (0,1) Y0 ≡ L2 (Γ0 ) × L2 (Ω − ) × H 1 (Ω + ) × W2 (Ω − ), we conclude that the limiting OCP (13.75)–(13.78) has a unique solution. Let us denote by (u0 , p 0 , (y 0 )+ , (y 0 )− ) ∈ Y0 the optimal quadruple for the limit problem (13.75)–(13.78). Then applying Theorem 13.20, we immediately obtain the following result. ! Theorem 13.28. Let (uδεε , pεδε , yεδε ) ∈ Ξε ε>0 be a sequence of admissible solutions for the Pε problems such that Iε (uδεε , pεδε , yεδε ) −

inf

(u,p,y) ∈Ξε

Iε (u, p, y) ≤ δε ,

sup yεδε H 1 (Ωε ) < +∞, ε>0

(13.79) (13.80)

506


and δε → 0 as ε → 0. Assume that fε → f0 strongly in L2 (Ω). Then (uδεε , pεδε , yεδε ) −→ (u0 , p 0 , (y 0 )+ , (y 0 )− ), w

I0 (u, p, y+ , y− ) = I0 u0 , p 0 , (y 0 )+ , (y 0 )−

inf

(u, p, y + , y − )∈ Ξ

0

= lim Iε (uδεε , pεδε , yεδε ) ε→0

= lim

inf

ε→0 (u,p,y)∈ Ξε

Iε (u, p, y).

Note that this result holds true for any bounded sequence of approximate solutions. Additionally, if the original OCP (13.2)–(13.5) has an optimal solution (u0ε , pε0 , yε0 ) for every ε > 0 such that supε>0 yε0 H 1 (Ωε ) < +∞, then, having put δε = 0 in Theorem 13.28, we obtain the classical variational property for the limit problems.

13.7 Modeling of suboptimal controls In this section, we deal with the construction of suboptimal controls to the original optimal boundary control problem (13.2)–(13.5). As was mentioned in Sect. 13.2, we cannot assert the solvability of this problem for each ε-level. However, for every δ > 0, there is always a triplet (uδε , pεδ , yεδ ) ∈ Ξε (the so-called approximate solution) such that ≤ δ. Iε (uδε , pεδ , yεδ ) − inf I (u, p, y) ε (u,p,y) ∈Ξε

As follows from Theorem 13.27, the limiting OCP (13.75)–(13.78) has a unique solution. Let (u0 , p 0 , (y 0 )+ , (y 0 )− ) ∈ Y0 be an optimal one to this limit problem. Since p 0 ∈ L2 (Ω − ), we cannot just use the restriction of this function as an admissible Neumann boundary control for the original problem (13.2)– (13.5) because a trace of an L2 -function on the (n−1)-dimensional surface Sε is not defined. As forthe Dirichlet control u0 , we have a similar situation. Indeed, the restriction u0 Γε is an element of L2 (Γε ), whereas admissible Dirichlet controls must be elements of H 1/2 (Γε ). In view of this, we make use of the idea of classical smoothing. Namely (see Sect. 6.5), let Ku ∈ C0∞ (Rn−1 ) and Kp ∈ C0∞ (Rn ) be nonnegative functions such that Ku (x ) dx = 1 and Kp (x) dx = 1, Rn−1

Rn

Ku (x ) = Ku (−x ) ∀ x ∈ Rn−1 and Kp (x) = Kp (−x) ∀ x ∈ Rn . Let u 0 and p0 be the trivial extensions by zero of functions u0 ∈ L2 (Γ0!) and 0 p ∈ L2 (Ω − ) to the sets O = (x , xn ) ∈ Rn : x ∈ Rn−1 , xn = −β0 and


507

Rn , respectively. For every fixed ε > 0, we define the following smoothing operators ε 0 0 x − y −n+1 Ku u ε = Ku u (x ) = ε u 0 (y , yn ) dHn−1 ε O x − y −n+1 u 0 (y ) dy , (13.81) =ε Ku ε Rn−1

p

0

ε

= Kpε p 0 (x) = ε−n

Rn

Kp

x−y ε

p0 (y) dy.

(13.82)

We begin with the following result: Proposition 13.29. For every δ > 0, there exists ε(δ) = a/N , where N ∈ N, such that usub = |C|−1 u0 δ Γ ∈ Uε , (13.83) ε ε

that is, usub is an admissible Dirichlet control to the problem (13.2)–(13.5). ε Moreover, 0 as δ → 0 in the sense of Definition 13.6. usub ε(δ) u

(13.84)

Proof. First, we note that usub ∈ H 1/2 (Γε ) by the properties of the smoothing ε ∈ Uε for some ε > 0, it remains to prove operator (13.81). To show that usub ε the estimate ' sub ' 'uε ' 2 ≤ Cu . (13.85) L (Γε )

Following the definition of the smoothing operator (13.81) and using the Cauchy–Bunyakovski˘i inequality, we have ' 0 '2 ' u ' 2 n−1 ) δ L (R 2 x − y δ −n+1 u 0 (y ) dy = Ku dx δ Rn−1 Rn−1 x − y δ −n+1 dy ≤ Ku δ Rn−1 Rn−1 x − y 0 2 × δ −n+1 u (y ) dy dx Ku δ Rn−1 x − y 0 2 u (y ) dy dx δ −n+1 = Ku δ Rn−1 Rn−1 0 2 x − y × Ku δ −n+1 (y ) dy dx u δ Rn−1 Rn−1 0 2 0 2 u (y ) dy = u (y ) dy = u 0 2L2 (Γ0 ) . (13.86) = Rn−1

Γ0

508


Due to this estimate, we come to the inequality ' ' 0 ' ' ' u ' 2 = 'χΓε u0 δ 'L2 (Γ ) δ L (Γε ) ε % ' ' ' 0 ' ' ' ≤ χΓ0 u δ L2 (Rn−1 ) ≤ 'u 0 'L2 (Γ ) ≤ |C| Cu , 0

(13.87)

which holds true for every ε > 0. Further, for a given δ > 0, we define a parameter ε = a/N so that the following inequality would be valid: 0 2 0 2 1 1 u δ dx ≤ u δ dx . |Γε | Γε |Γ0 | Γ0 Then, from (13.87) we deduce ' −1 0 '2 '|C| u δ 'L2 (Γ

ε)

≤

' |Γε | ' |Γε | −1 2 '|C|−1 u0 '2 |C| Cu . ≤ δ L2 (Γ0 ) |Γ0 | |Γ0 |

(13.88)

Since |Γ0 | = |B| = an−1 and a n−1 dx = εn−1 |C| = εn−1 |C| = an−1 |C|, |Γε | = ε Γεk k∈θε

k∈θε

it follows from (13.88) that ' sub ' 'u ' 2 ε L (Γ

ε)

' ' ≡ '|C|−1 u0 δ 'L2 (Γ

ε)

≤ Cu .

(13.89)

Thus, for the chosen value of ε, inequality (13.85) is valid, and hence usub is ε an admissible Dirichlet control for the Pε problem. To conclude the proof, we note that 0 (13.90) u δ → u0 strongly in L2 (Γ0 ) as δ → 0, by the properties of the classical smoothing. Hence in view of (13.21), (13.90), and the fact that the value of ε depends on a given parameter δ, we have −1 u sub χΓε(δ) u0 δ |C|−1 |C|u0 = u0 as δ → 0, ε(δ) = |C| as a limit of the product of the weakly and strongly convergent sequences in L2 (Γ0 ). This implies the required conclusion (13.84). For the Neumann controls, we have a similar result. Proposition 13.30. Let p 0 ∈ L2 (Ω − ) be an optimal distributed control for the limit problem (13.75)–(13.78). Then there exists ε0 > 0 such that (13.91) pεsub = p 0 ε S ∈ Pε , ∀ ε ∈ (0, ε0 ), ε

pεsub

is an admissible Neumann boundary control to the problem that is, (13.2)–(13.5) for ε > 0 small enough. Moreover, pεsub p 0 in L2 (Ω − , dμε ) as ε → 0.

(13.92)


509

Proof. Since p 0 ε ∈ C(Ω − ), it is clear that pεsub = p 0 ε S ∈ L2 (Sε , Hn−1 ). ε By analogy with (13.86), it is easy to show the validity of the inequality ' 0 '2 0 2 ' p ' 2 − ≤ dx = p 0 2L2 (Ω− ) ≤ |∂C|−1 (13.93) p H Cp . ε L (Ω ) Ω−

Thus, as follows from (13.93), in order to assert that pεsub is an admissible control for the problem (13.2)–(13.5), we have to be sure that the estimate ' '2 ε 'pεsub 'L2 (S

ε ;H

n−1 )

≤ Cp

(13.94)

is valid at least for ε small enough. In order to verify this fact, we will show that the value ' 0 '2 ' 0 '2 ' p ' ' p ' σ = ε|∂C|−1 − 2 n−1 2 − H ε L (S ;H ε L (Ω ) ) ε

can be made as small as we wish for ε small enough. To begin, we note that ' 0 '2 ' p ' 2

ε L (Ω − )

=

n−1 N

j=1

j

˛ ff ˛ x=(x ,xn ) :˛˛x ∈ε(+kj ) −β0 <xn 0, there exist an εη > 0, an integer J ∈ N, and a collection of points ! xkε,i ∈ Δi Sεk such that J 0 2 0 2 k η , p ε dx − p ε (xε,i )Ln (ε)Δ < 2N n−1 Lkε i=1 Taking into account the chain of transformations

∀ ε ∈ (0, εη ).

510


J 0 2 k p ε (xε,i )Ln (ε)Δ i=1

=

J

p0

2

(xkε,i )εn Δ = ε

i=1

=

J

p0

2 ε

(xkε,i )εn

i=1 J

ε H n−2 (∂C)

= ε|∂C|−1 H

J

p0

2 ε

Hn−1 (Δi Sεk ) H n−2 (ε∂C)

(xkε,i )Hn−1 (Δi Sεk )

i=1

p0

2 ε

(xkε,i )Hn−1 (Δi Sεk )

i=1

and the fact that J 0 2 k 0 2 η|∂C|H n−1 k n−1 (Δi Sε ) − p ε (xε,i )H p ε dH < 2εN n−1 Sεk i=1 for J large enough, we deduce the following: For a given η > 0, there exists an εη > 0 such that 0 2 2 n−1 p 0 ε dx − ε|∂C|−1 p dH H ε Lk ε

Sk

ε J 0 2 k −1 0 2 n−1 k ≤ p ε dx − ε|∂C|H p ε (xε,i )H (Δi Sε ) Lkε i=1 J 0 2 k p ε (xε,i )Hn−1 (Δi Sεk ) + ε|∂C|−1 H i=1 0 2 η n−1 − p ε dH (13.96) < n−1 N k Sε

for all ε ∈ (0, εη ). As a result, combining this estimate with representation (13.95), we come to the conclusion that ' ' ' 0 '2 ' 0 '2 ' p ' p ε L2 (Ω − ) − ε|∂C|−1 2 n−1 H ε L (Sε ;H ) N n−1 0 2 2 = p ε dx − ε|∂C|−1 p 0 ε dHn−1 < η. (13.97) H kj kj Lε Sε j=1 Since relation (13.93) turns into an equality only in the case when p 0 is a constant, it follows that choosing an appropriate value η > 0, we can find an εη > 0 such that for every ε ∈ (0, εη ), the inequality ' '2 ' '2 ≤ 'p 0 ' 2 − ≤ |∂C|−1 Cp ε|∂C|−1 ' p 0 ' H

ε

L2 (Sε ;Hn−1 )

L (Ω )

H


511

holds true. Hence, estimate (13.94) is valid for ε > 0 small enough. To conclude the proof, it remains to use the main property of the classical smoothing, 0 p ε → p 0 strongly in L2 (Ω − ) as ε → 0, and apply Lemma 13.25. As a direct consequence of the proof of Proposition 13.30, which we feel to be interesting per se, we present one observation concerning L2 -functions defined on thick junctions. Corollary 13.31. Let Ωε be a thick multistructure in Rn , which consists of some domain Ω + and a large number of thin cylinders Gkε = {(x , xn ) : x ∈ εC + εk, −β0 < xn ≤ 0} with a small cross-section of the size εC and ε-periodically distributed along some manifold Σ on the boundary of Ω + (see Fig. 14.1 for a 3D example). Let Ω − = Σ × (a, b) be a domain which is filled up by thin cylinders in the limit passage as ε → 0. Let f ∈ L2loc (Rn ) be a given function. Then 2

f L2 (Ω − ) ≥

ε 2 (f )ε L2 (Sε ;Hn−1 ) Hn−2 (∂C)

for ε small enough, where (f )ε denotes the direct smoothing of the function χΩ − f , that is, x−y (f )ε = ε−n χΩ − (y)f (y) dy. K ε Rn Now we are in a position to give the main result of this section. Theorem 13.32. Let (u0 , p 0 , (y 0 )+ , (y 0 )− ) ∈ Y0 be an optimal solution to the limit OCP (13.75)–(13.78). Then for every η > 0, there exist δ > 0 and ε > 0 such that sub sub sub inf I (u, p, y) − I (u , p , y ) (13.98) ε ε ε ε < η, (u,p,y) ∈Ξε

where usub = |C|−1 u0 δ Γ ∈ Uε , ε ε pεsub = p 0 ε S ∈ Pε ,

(13.99) (13.100)

ε

sub and yεsub = yε (usub ε , pε ) is the corresponding solution of the boundary value sub problem (13.3), that is, the pair (usub ε , pε ) is a suboptimal control to the original problem (13.2)–(13.5).

512


Proof. For a given sequence {δ(ε)}ε>0 with limε→0 δ(ε) ! = 0, we construct sub sub the following sequence of triplets (usub , p , y ) ∈ X . By Proposiε ε ε ε ε>0 tions 13.29 and 13.30, we can suppose that each of these triplets is admissible ε (to be more precise, it is so for ε small enough). Moreover, for the problem P due to estimate (13.17), this sequence is equibounded in Xε . Then by Proposition 13.17, this one is relatively compact with respect to the w-convergence in Xε . Hence, taking into account Propositions 13.29 and 13.30 and Theorem 13.24, we deduce the following: This sequence is w-compact and sub sub 0 0 0 + 0 − (usub ε , pε , yε ) −→ (u , p , (y ) , (y ) ), w

where (u0 , p 0 , (y 0 )+ , (y 0 )− ) ∈ Y0 is a unique optimal solution to the limit problem (13.75)–(13.78). Let {(uηε ε , pεηε , yεηε ) ∈ Ξε }ε>0 be a sequence of approximate solutions to the Pε problems such that η Iε (uηε ε , pεηε , yεηε ) − (13.101) inf Iε (u, p, y) ≤ ηε ≤ 3 (u,p,y) ∈Ξε and supε>0 yεηε H 1 (Ωε ) < +∞, where ηε → 0 as ε → 0 and η > 0 is a given value. Then, by Theorem 13.20, we have (uηε ε , pεηε , yεηε ) −→ (u0 , p 0 , (y 0 )+ , (y 0 )− ) as ε → 0. w

Further, we observe that sub sub Iε (u, p, y) − Iε (usub , p , y ) inf ε ε ε (u,p,y) ∈Ξε

≤

inf

(u,p,y) ∈Ξε

Iε (u, p, y) − Iε (uηε ε , pεηε , yεηε )

+ Iε (uηε ε , pεηε , yεηε ) − I0 u0 , p 0 , (y 0 )+ , (y 0 )− sub sub + I0 u0 , p 0 , (y 0 )+ , (y 0 )− − Iε (usub ) ε , pε , yε = J1 + J2 + J3 . (13.102) We note that for a given η > 0 one can always find the following: (i) ε1 > 0 such that J1 < η/3 for all ε < ε1 by property(13.101); (ii) ε2 > 0 such that J2 < η/3 for all ε < ε2 by Theorem 13.20. As for the estimate for the term J3 , we make use of the following observations: (i)

By the definition of the w-convergence (see Definition 13.16), we have sub + (y 0 )+ in H 1 (Ω + ). yε + Hence, yεsub → (y0 )+ in L2 (Ω + ). Therefore, there exists ε3 > 0 such that


L1 =

η sub 2 0 + 2 (yε − q0 ) dx − ((y ) − q0 ) dx < 12 Ω+ Ω+

513

(13.103)

for all ε < ε3 . (ii) As follows from Propositions 13.8 and 13.29, the following properties take place: ∗

χΓε |C| in L∞ (Γ0 ) as ε → 0, 0 u δ(ε) → u0 strongly in L2 (Γ0 ) as ε → 0, 2 2 strongly in L1 (Γ0 ) as ε → 0. u0 δ(ε) → u0 Hence,

sub 2 dHn−1 = |C|−2 uε Γε

χΓε Γ0

−2

u0

2

δ(ε)

dHn−1

2 |C| u0 dHn−1

−→ |C|

Γ0

as a product of the weakly and strongly convergent sequences. Therefore, there exists ε4 > 0 such that 2 χΓε |C|−1 u0 δ (ε) dHn−1 L2 = Γ0 η , ∀ ε < ε4 . −|C|−1 (u0 )2 dHn−1 < (13.104) 12 Γ0 (iii) Estimate (13.97) implies the existence of ε5 > 0 such that sub 2 0 2 η pε p ε dx < L3 = ε dHn−1 − |∂C|H 12 − Sε

(13.105)

Ω

for every ε < ε5 . (iv) Since p 0 ε → p 0 strongly in L2 (Ω − ) as ε → 0 by the properties of the smoothing operator, it follows that there exists ε6 > 0 such that 0 2 0 2 η L4 = , ∀ ε < ε6 . p ε dx − p dx < (13.106) 12 − − Ω Ω To conclude the proof, we note that the term J3 in (13.102) can be estimated as follows:

514


sub sub J3 = I0 u0 , p 0 , (y 0 )+ , (y 0 )− − Iε (usub ) ε , pε , yε ≤ (yεsub − q0 )2 dx − ((y 0 )+ − q0 )2 dx Ω+

Ω+

sub 2 uε dHn−1 − |C|−1 (u0 )2 dHn−1 + Γε Γ0 sub 2 0 2 n−1 + ε pε p dH − |∂C|H dx Ω−

Sε

sub 2 0 + 2 ≤ (yε − q0 ) dx − ((y ) − q0 ) dx + + Ω Ω

2 0 −1 n−1 −1 0 2 n−1 u δ(ε) dH + χΓε |C| − |C| (u ) dH Γ Γ0 0 sub 2 0 2 + ε pε p ε dx dHn−1 − |∂C|H Sε

+ |∂C|H

Ω−

2 p0 ε

dx −

Ω−

Ω−

p

0 2

dx

= L1 + L2 + L3 + L4 .

(13.107)

As a result, combining estimates (13.103)–(13.107), we obtain J3 < η/3 for ε < min{ε3 , ε4 , ε5 , ε6 }. Hence, in view of (13.102), we just come to the required conclusion: Inequality (13.98) holds true for all ε < min{ε1 , ε2 , ε3 , ε4 , ε5 , ε6 }. This concludes the proof.

14 Gap Phenomenon in Modeling of Suboptimal Controls to Parabolic Optimal Control Problems in Thick Multistructures

In this chapter, we study the asymptotic behavior of the following class of the parabolic optimal control problems (OCPs) T T (yε − q0 )2 dx dt + u2ε dx dt −→ inf, (14.1) Iε (uε , yε ) = 0

yε − Δx yε + yε ∂ν yε yε ∂ ν yε yε (0, x)

Ω+

0

= fε = −ε k0 yε = uε =0 = yε0

Γε

⎫ in (0, T ) × Ωε , ⎪ ⎪ ⎪ ⎪ on (0, T ) × Sε , ⎬ on (0, T ) × Γε , ⎪ on (0, T ) × ∂Ωε \ (Γε ∪ Sε ) , ⎪ ⎪ ⎪ ⎭ a.e. x ∈ Ωε

(14.2)

as a small parameter ε tends to 0. Here, Ωε ⊂ Rn denotes a thick multistructure for which the following representation holds true (see Fig. 14.1 for a 3D example) + k Ωε = (B × (0, c)) ∪ (εC + εk) × (−d, 0] = Ω ∪ Gε , k∈θε

k∈θε

where B = (0, a)n−1 and C are bounded open smooth domains in Rn−1 (n ≥ 2), C ⊂⊂ (0, 1)n−1 , θε = {k = (k1 , k2 , . . . , kn−1 ) ∈ Nn−1 : εC + εk ⊂⊂ B}, Ω = B × (−d, c), Σ = B × {0},

Gkε = {(x , xn ) : x ∈ εC + εk, −d < xn ≤ 0}, Ω + = B × (0, c),

Γ0 = B × {−d},

Ωε−

Ω − = B × (−d, 0),

(14.3)

−

= Ωε ∩ Ω ,

Γε is the union of the lower bases Γεk = {(x , xn ) : x ∈ ε · C + εk, xn = −d} of the thin cylinders Gkε when k ∈ θε (i.e., Γε = Γ0 ∩ ∂Ωε ), Sε is the union of their boundaries along the axis Oxn , P.I. Kogut, G.R. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains, Systems & Control: Foundations & Applications, DOI 10.1007/978-0-8176-8149-4 14, © Springer Science+Business Media, LLC 2011

515

516

14 Gap Phenomenon in Modeling of Suboptimal Controls

Sεk = {(x , xn ) : x ∈ ε · ∂C + εk, −d < xn < 0}, k0 is a positive constant, ∂ν = ∂/∂ν is the outward normal derivative, and q0 ∈ L2 (0, T ; L2 (Ω + )), yε0 ∈ L2 (Ωε ), and fε ∈ L2 (0, T ; L2 (Ω)) are given functions. Throughout this chapter, we suppose that ε = a/N , where N is a large positive integer. For this kind of domains and boundary value problems in Ωε , we refer to Brizzi and Chalot [35] and Mel’nyk and Nazarov [187].

Fig. 14.1. Thick multistructure Ωε

We consider the OCP (14.1)–(14.2) assuming that there are two different classes of admissible boundary controls Uεa (the so-called regular controls) and Uεb (the so-called contrast controls) which are realized via the Dirichlet boundary conditions posed on the lower bases Γε of the thin cylinders Gkε , where

uε ∈ Uεa = u|Γε : u ∈ L2 (0, T ; H 1 (Γ0 )), u L2 (0,T ; H 1 (Γ0 )) ≤ C0 , (14.4)

(14.5) uε ∈ Uεb = u ∈ L2 (0, T ; H 1 (Γε )), u L2 (0,T ;L2 (Γε )) ≤ C0 . We denote problems (14.1), (14.2), (14.4) and (14.1), (14.2), (14.5) by Paε and Pbε , respectively. It is well known that the computational calculation of optimal solutions to these problems is very complicated due to singularities of the thick junctions Ωε . Therefore, we study the asymptotic behavior of these problems as the parameter ε tends to 0. We want to obtain appropriate asymptotic limits for the OCPs Paε and Pbε as the parameter ε tends to 0 and to show that their optimal solutions can be used as suboptimal controls to the corresponding problems Paε and Pbε . For comparison, we note that the asymptotic analysis of an optimal boundary control problem for linear elliptic equations in the thick multistructures with Uεa -admissible boundary controls was given in Chap. 12. At the same


517

time the OCPs for linear parabolic and hyperbolic equations in the same domains with unconstrained distributed L2 -controls were studied in [88, 100]. However, the characteristic feature of these OCPs is the fact that each of these problems has a unique optimal solution for every ε. The OCPs Paε and Pbε we consider are such that the existence of an optimal solution for the control problem Pbε has to be assumed. This is not typical for the majority of investigations in this field. We show that the result of homogenization for these problems as ε → 0 (i.e., when the number of attached thin cylinders increases without bound and their thicknesses vanish) essentially depends on the classes of admissible controls, namely let |C| be the (n − 1)dimensional Lebesgue measure of the set C, let v ε be the zero-extension to Ω of a function v defined on Ωε , and let χΩ + and χΩ − be the characteristic functions of the sets Ω + and Ω − , respectively. Having assumed y ε0 (|C|χΩ − + χΩ + ) y 0 in L2 (Ω) as

ε → 0,

f ε (|C|χΩ − + χΩ+ ) f0 in L (0, T ; L (Ω)) as ε → 0, 2

2

(14.6) (14.7)

we prove that for the Paε problem there exists a unique variational limit (Pahom ) as ε → 0 that can be represented in the form ⎫ (y + ) − Δx y + + y + = f0 in (0, T ) × Ω + , ⎪ ⎪ ⎪ ⎪ ⎪ |∂C| |C| + k ⎪ 0 H − − 2 − − ⎪ (y ) − ∂xn y + y = f0 in (0, T ) × Ω , ⎪ ⎪ ⎪ |C| ⎬ ∂ν y + = 0 in (0, T ) × ∂Ω + \ Σ, (14.8) ⎪ ⎪ y − = u on (0, T ) × Γ0 , ⎪ ⎪ ⎪ ⎪ ⎪ y + = y − , ∂xn y + = |C| ∂xn y − on (0, T ) × Σ, ⎪ ⎪ ⎪ ⎭ 0 y(0, x) = y (x) a.e. x ∈ Ω,

u ∈ Ua = u ∈ L2 (0, T ; H 1 (Γ0 )) : u L2 (0,T ;H 1 (Γ0 )) ≤ C0 , Ia (u, y + , y − ) =

T

(14.9)

(y + − q0 )2 dx dt

0

Ω+ T

u2 dx dt −→ inf .

+ |C| 0

Γ0

(14.10)

Whereas for the OCP Pbε , the limit problem Pbhom has another analytical representation ⎫ (y + ) − Δx y + + y + = f0 in (0, T ) × Ω + , ⎪ ⎪ ⎪ ⎪ ⎪ |C| + k0 |∂C|H − ⎪ − 2 − − ⎪ (y ) − ∂xn y + y = f0 in (0, T ) × Ω , ⎪ ⎪ ⎪ |C| ⎬ + + ∂ν y = 0 in (0, T ) × ∂Ω \ Σ, (14.11) ⎪ ⎪ v0− = |C|−1 u on (0, T ) × Γ0 , ⎪ ⎪ ⎪ ⎪ ⎪ y + = y − , ∂xn y + = |C| ∂xn y − on (0, T ) × Σ, ⎪ ⎪ ⎪ ⎭ 0 y(0, x) = y (x) a.e. x ∈ Ω,

518


u ∈ Ub = u ∈ L2 ((0, T ) × Γ0 ) : u L2 ((0,T )×Γ0 ) ≤ |C|C0 ,

Ib (u, y + , y − ) =

T

(14.12)

(y + − q0 )2 dx dt

0

1 + |C|

Ω+

0

T

u2 dx dt −→ inf .

(14.13)

Γ0

Here, by v + and v − we denote the restrictions of a function v : (0, T )×Ω → R to the sets (0, T ) × Ω + and (0, T ) × Ω − , respectively. Hence, the suboptimal controls for the original problems Paε and Pbε will be different.

14.1 On solvability of the original optimal control problems We begin this section with the description of the geometry of the set Sε in terms of a singular measure on Rn as we did in Sect. 13.2. Let μ0 be a periodic finite positive Borel measure on Rn−1 with the torus of periodicity = [0, 1)n−1 . We assume that the Borel measure μ0 is the probability mea sure, concentrated and uniformly distributed on the set ∂C, so dμ0 = 1. Now, we set n = × [0, 1) = [0, 1)n and consider the measure dμ = dμ0 × dxn on n . It is easy to see that this measure concentrated on the set ∂C × [0, 1) and for any smooth function g, we have 1 −1 g dμ = g dxn dμ0 = Hn−1 (∂C × [0, 1)) g dHn−1 . n

0

∂C×[0,1)

However, Hn−1 (∂C × [0, 1)) = Hn−2 (∂C). Then, using in the sequel the notation |∂C|H = Hn−2 (∂C), the previous relation can be rewritten in the form 1 −1 g dμ = g dxn dμ0 = |∂C|H g dHn−1 . (14.14) n

0

∂C×(0,1)

Let Λ be any Borel set of Rn . We introduce the so-called “scaling” measure με by the rule με (Λ) = εn μ(ε−1 Λ). This measure has period ε. Since μ(εn ) = ε μ0 (ε) by the definition of μ, it follows that ε 1 n n με (εn ) = ε dμ0 (x /ε) d(xn /ε) = ε dμ0 dxn = εn . 0

0

ε

Hence, the measure με weakly converges to the Lebesgue measure on Rn as ε → 0, that is, ϕ dμε = ϕ dx for all functions ϕ ∈ C0∞ (Rn ). lim ε→0

Rn

Rn

14.1 On solvability of the original optimal control problems

519

It is clear that yε ∈ L2 (0, T ; H 1 (Ωε )) is the weak solution of the abovementioned problem whenever (see Lions [169]) T − yε ϕ ψ + ∇yε · ∇ϕ ψ + yε ϕ ψ dx dt Ωε

0

T

yε ϕ ψ dx dt

+ k0 ε 0

T

Sε

fε ϕ ψ dx dt, ∀ ϕ ∈ C0∞ (Rn ; Γε ), ∀ ψ ∈ C0∞ (0, T ),

= 0

Ωε

(14.15) where by C0∞ (Rn ; Γε ) we denote the set of all functions of C0∞ (Rn ) such that ϕ|Γε = 0. Let us consider the last term on the left-hand side of identity (14.15). Using the notations introduced above, we may write T k0 ε yε ϕψ dx dt 0

Sε

T

= k0 ε

n−1 N

0

ε(∂C+kj )

j=1

T

= k0 ε|∂C|H

T

T

T

= k0 |∂C|H 0

ε(+kj )

j=1

N

j=1

Ω−

0 −d

0 −d

ε(+kj )

N n−1

0

n−1

j=1

= k0 |∂C|H

−d

yε ϕ dHn−2 dxn ψ dt

N

0

0

n−1

0

= k0 |∂C|H

ε(+kj )

0 −d

yε ϕ εn−2 dμ0 (x /ε) dxn ψ dt

yε ϕ εn dμ0 (x /ε) d(xn /ε) ψ dt yε ϕ dμε ψ dt

yε ϕψ dμε dt.

− Here, by yε we denote a function of L2 (0, T ; L2 (Ω , dμε )) taking the same

values as yε on the set Sε . Note that the integral

Ω−

yε ϕ dμε is well defined

for every function ϕ ∈ C0∞ (Rn ; Γε ). Indeed, since the set Ω − is bounded and yε ϕ dμε is a linear continuous yε dμε is a Radon measure, it follows that Ω−

functional on C0∞ (Rn ; Γε ). Let Xμε be the vector space of functions yε ∈ L2 (0, T ; H 1 (Ωε )) such that y ∈ L2 (0, T ; L2 (Ω − , dμε )), that is, for any function yε ∈ Xμε , the integral εT y 2 dμε is well defined. It is easy to see that Xμε is a Hilbert space with 0 Ω− ε respect to the following scalar product

520


(yε , vε )Xμε =

T

T

(∇yε · ∇vε + yε vε ) dx dt + 0

0

Ωε

Ω−

yε vε dμε dt.

As a result of this motivation, we give the following variational formulation of the initial boundary value problem (14.2). Definition 14.1. We say that a function yε = yε (uε ) is a weak solution of the parabolic problem (14.2) for a given function uε ∈ L2 (0, T ; H 1 (Γε )) if

T 0

(−yε ϕψ + ∇yε · ∇ϕψ + yε ϕψ) dx dt Ωε

+ k0 |∂C|H 0

T

Ω−

T

yε ϕψ dμε dt =

fε ϕψ dx dt, 0

(14.16)

Ωε

yε ∈ Xμε , yε (0, x) = yε0 a.e. x ∈ Ωε , yε |Γε = uε a.e. t ∈ (0, T ) (14.17) holds for every ϕ ∈ C0∞ (Rn ; Γε ) and ψ ∈ C0∞ (0, T ). Then using the standard Hilbert space method, we have the following result. Proposition 14.2. For any given function uε ∈ L2 (0, T ; H 1 (Γε )), the problem (14.2) admits a unique weak solution in the sense of Definition 14.1 such that

yε Xμε

yε ∈ L2 (0, T ; (H 1 (Ωε )) ),

≤ c fε L2 ((0,T )×Ωε ) + uε L2 (0,T ;H 1 (Γε )) ,

∀ ε > 0,

(14.18)

where the constant c > 0 is independent of ε (see Lions [169]). Now, we can return to the question on solvability of the OCPs Paε and Pbε . For this we rewrite the original problems as follows:

b (Paε ) : inf inf a Iε (uε , yε ) , Pε : Iε (uε , yε ) , (14.19) (uε ,yε )∈ Ξε

(uε ,yε )∈ Ξεb

where by Ξεa and Ξεb we denote the sets of admissible pairs for the corresponding control problems (i.e., (uε , yε ) ∈ Ξ iε (i = 1, 2)) if ⎫ ⎧ uε ∈ Uεi , yε ∈ Xμε , yε |Γε = uε ,⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ (0, x) = y a.e. x ∈ Ω , y ⎪ ⎪ ε ε ε ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ T ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ (−yε ϕψ + ∇yε · ∇ϕψ + yε ϕψ) dx dt i . (14.20) Ξε = Ωε 0 ⎪ ⎪ ⎪ ⎪ T T ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ yε ϕψ dμε dt = fε ϕψ dx dt,⎪ ⎪ ⎪ + k0 |∂C|H ⎪ ⎪ ⎪ ⎪ − Ω Ωε 0 0 ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ ∞ n ∞ ∀ ϕ ∈ C0 (R ; Γε ), ∀ ψ ∈ C0 (0, T )

14.1 On solvability of the original optimal control problems

521

Using the direct method of the Calculus of Variations, it can be easily shown that Paε has a unique solution for every value ε > 0: Iε (uaε , yεa ) =

inf

(uε ,yε )∈Ξεa

Iε (uε , yε ).

As for the Pbε -problem, we observe that its set of admissible pairs Ξεb is convex and closed in L2 (0, T ; H 1 (Ωε )) × Xμε and the cost functional (14.1) is strictly convex and lower semicontinuous with respect to the weak topology of L2 (0, T ; H 1 (Ωε )) × L2 (0, T ; H 1 (Ωε )). Hence, we cannot expect solvability of this problem, in general, that is, the existence of an optimal solution (ubε , yεb ) ∈ Ξεb for the Pbε -problem must be taken as an assumption. However, if the Pbε -problem is solvable, then its solution is unique. Let τεa be the product of the weak topologies of L2 (0, T ; H 1 (Γε )) and L2 (0, T ; H 1 (Ωε )) and let τεb be the product of the weak topologies of L2 (0, T ; L2 (Γε )) and L2 (0, T ; H 1 (Ωε )). Let us denote by clτεb Ξεb the closure of the set Ξεb with respect to the τεb topology and consider the following constrained minimization problem (the so-called τεb -relaxed problem for the OCP Pbε ): inf

(uε ,yε ) ∈clτ b Ξεb

Iε (uε , yε ) .

ε

It is clear that this problem is solvable for every ε. Indeed, clτεb Ξεb is a convex, closed and bounded subset of L2 (0, T ; L2 (Γε )) × Xμε , and Iε : L2 (0, T ; L2 (Γε )) × Xμε → R is a strictly convex τεb -lower semicontinuous functional. This means that this problem has a unique solution (u∗ε , yε∗ ) ∈ clτεb Ξεb . Theorem 14.3. If (ubε , yεb ) is an optimal pair for the Pbε problem, then (ubε , yεb ) is the unique solution of the τεb -relaxed problem. Proof. Let ε be any fixed value (we recall that ε = a/N ). Since Ξεb ⊂ clτεb Ξεb , we have Iε (uε , yε ) ≤ inf Iε (uε , yε ). inf (uε ,yε )∈ clτ b Ξεb ε


Let (u∗ε , yε∗ ) be a solution of the τεb -relaxed problem. Assume that Iε (u∗ε , yε∗ )
0: Iε (uaε , yεa ) =

min

(uε ,yε )∈ Ξεa

Iε (uε , yε ) >

min


Iε (uε , yε ).

Hence, in the “limit”as ε tends to 0, we can obtain one homogenized problem for the (a) case, and another one for the (b) case. To study the asymptotic behavior of the problems Paε and Pbε , we adopt the concept of the variational convergence of constrained minimization problems (see Sect. 8.4). Then this procedure can be reduced to the limit analysis of the sequences ! ! inf Iε (u, y) : ε → 0 , (14.23) inf a Iε (u, y) : ε → 0 , (u,y) ∈Ξε

(u,y) ∈Ξεb

where the cost functionals Iε : Ξεi → R, i = a, b, and the corresponding sets of admissible pairs are defined in (14.1) and (14.20), respectively. Note that because of the specific construction of the domains Ωε , we have a rather delicate situation with the limit passage in (14.19) as ε → 0. Indeed, each of the admissible pairs (uε , yε ) belongs to the corresponding space Yε := L2 (0, T ; H 1 (Γε )) × Xμε

(14.24)

and this fact is as common for the Paε -problem as for the Pbε -problem. Therefore, we focus our attention in this section on working up the convergence formalism in such spaces.

14.2 Formalism of convergence in variable Banach spaces

523

14.2.1 The convergence concept for the Pa ε -problems Let {(uε , yε )}ε>0 be a sequence of pairs such that uε ∈ Uεa , yε ∈ Xμε ∀ ε > 0, and lim sup yε 2Xμε ε>0 "

T

= lim sup ε>0

0

|∇yε |2 dx + yε2 dx dt +

T

0

Ωε

#

Ω−

yε2 dμε < +∞.

It is clear that any sequence of admissible pairs {(uε , yε ) ∈ Ξεa }ε>0 satisfies these assumptions. As the definition of the sets Uεa indicates (see (14.4)), for every ε > 0 and for every control function uε ∈ Uεa , there exists an extension operator Pε : L2 (0, T ; H 1 (Γε )) → L2 (0, T ; H 1 (Γ0 )) such that Pε (uε ) L2 (0,T ;H 1 (Γ0 )) ≤ C0 . However, the weak limits of any two weakly convergent sequences {Pε(1) (uε )}ε>0 and {Pε(2) (uε )}ε>0 are the same. Indeed, let us assume that Pε(1) (uε ) u∗1 and Pε(2) (uε ) u2 in L2 (0, T ; H 1 (Γ0 )). Let χΓε be the characteristic function of the set Γε . Since χΓε is a -periodic function, it follows that χΓε → |C| weakly-∗ in L2 (B) as ε → 0. Then passing to the limit in the integral identity

T

T

χΓε Pε(1) (uε )ϕ(x)ψ(t) dx dt = 0

χΓε Pε(2) (uε )ϕ(x)ψ(t) dx dt, 0

Γ0

Γ0

∀ ψ ∈ C0∞ (0, T ), ∀ ϕ ∈ H 1 (Γ0 ) as ε tends to 0, we conclude

T

|C| 0

Γ0

u∗1 ϕ(x)ψ(t) dx dt = |C|

T

0

u∗2 ϕ(x)ψ(t) dx dt, Γ0

∀ ψ ∈ C0∞ (0, T ), ∀ ϕ ∈ H 1 (Γ0 ). Hence, u∗1 = u∗2 and we are done. In view of this, we give the following Definition 14.4. We say that a sequence of controls

uε ∈ L2 (0, T ; H 1 (Γε )) ε>0 is weakly convergent to a function u∗ with respect to the space L2 (0, T ; H 1 (Γ0 )) if some sequence of its images {Pε (uε )}ε>0 ⊂ L2 (0, T ; H 1 (Γ0 )) converges to u∗ weakly in L2 (0, T ; H 1 (Γ0 )).

524


As a consequence, we have the following obvious result. Lemma 14.5. Any sequence of admissible controls {uε ∈ Uεa }ε>0 is relatively compact with respect to the weak convergence introduced above. Moreover, its weak limit u∗ belongs to the set

Ua = u ∈ L2 (0, T ; H 1 (Γ0 )) | u L2 (0,T ;H 1 (Γ0 )) ≤ C0 . Now, we give the convergence formalism for the sequences of the type {yε ∈ Xμε }. By analogy with Brizzi and Chalot [35], we extend each of the functions yε by 0 onto the whole domain Ω, namely $ yε (x) if x ∈ Ωε , (14.25) y ε (x) := 0 if x ∈ Ω \ Ωε , and introduce the functions yε+ (x) = yε (x) if x ∈ Ω + and y ε− (x) = y ε (x) if x ∈ Ω − . Thanks to the rectilinear boundaries of Sε with respect to xn , we have ∂xn ( yε− ) = ∂x (yε− ) on Ω − . (14.26) n This means that y ε− ∈ L2 (0, T ; W2 (Ω − )), where W2 (Ω − ) is the ani2 − 2 − sotropic Sobolev space {v ∈ L (Ω ) : ∂xn v ∈ L (Ω )}. Let χC be the -periodic characteristic function of the set C. It is easy to ∗ see that χC (·/ε) |C| in L∞ (B) as ε → 0, where B = (0, a)n−1 and |C| is the (n − 1)-dimensional Lebesgue measure of C. We recall also that (see Brizzi and Chalot [35]) (0,1)

∗

χΩε− |C| ∗

χΩε ∩Σ |C| ∗

χΓε |C|

(0,1)

in L∞ (Ω − )

as ε → 0,

(14.27)

in L∞ (Σ)

as ε → 0,

(14.28)

∞

in L (Γ0 )

as ε → 0,

(14.29)

Definition 14.6. We say that a sequence {yε ∈ Xμε }ε>0 is weakly convergent to a function y∗ = (y∗+ , y∗− ) (with respect to the space L2 (0, T ; H 1 (Ω + ) × (0,1) W2 (Ω − ))) as ε tends to 0 (in symbols, yε y∗ = (y∗+ , y∗− )) if the following hold: (a) yε+ y∗+ in L2 (0, T ; H 1 (Ω + )); (0,1) (b) y ε− |C| y∗− in L2 (0, T ; W2 (Ω − )). We prove the following compactness property. Proposition 14.7. Let {yε ∈ Xμε }ε>0 be a bounded sequence. Then there exist a subsequence {yε }ε >0 and a function y0 = (y0+ , y0− ) ∈ L2 (0, T ; H 1 (Ω + )) × L2 (0, T ; W2

(0,1)

such that yε y0 = (y0+ , y0− ).

(Ω − ))


525

Proof. In accordance with the initial assumptions, there exists a constant C > 0 independent of ε such that yε Xμε ≤ C. Hence, yε+ L2 (0,T ;H 1 (Ω + )) + y− ε L2 (0,T ;W (0,1) (Ω − )) + yε L2 (0,T ;L2 (Ω − ,dμε )) ≤ C. 2

Therefore, there exist a subsequence {ε } of {ε} (still denoted by ε) and elements y0+ ∈ L2 (0, T ; H 1 (Ω + )), y0− ∈ L2 (0, T ; L2 (Ω − )), and y ∗ ∈ L2 (0, T ; L2 (Ω − )) such that yε+ y0+

in L2 (0, T ; H 1 (Ω + )),

y ε− v =: |C|y0− in L2 (0, T ; L2 (Ω − )), yε y ∗ ∂xn y ε− |C| ∂xn y0−

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

in L2 (0, T ; L2 (Ω − , dμε )), ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ 2 2 − in L (0, T ; L (Ω )).

(14.30)

Remark 14.8. Here, we have used the fact that the bounded sequence {yε }ε>0 is relatively compact with respect to weak convergence in

2 L (0, T ; L2 (Ω − , dμε )) .

T

Indeed, since lim supε→0 0

Ω−

(yε )2 dμε dt < +∞, there exist a subsequence

{ε } of {ε} (still denoted by ε) and an element y∗ ∈ L2 (0, T ; L2 (Ω − )) such that (see Proposition 6.15)

T

lim

ε→0

0

Ω−

ϕ ψ yε dμε dt = 0

T

ϕ ψ y ∗ dx dt Ω−

for any functions ϕ ∈ C0∞ (Rn ) and ψ ∈ C0∞ (R). Note also that the last limit in (14.30) is a consequence of (14.26). Moreover, by analogy with Brizzi and Chalot [35], one can easily prove the relation y0+ = y0−

a.e. on (0, T ) × Σ,

(14.31)

that is, in this case the traces of the limit functions (y0+ , y0− ) coincide on Σ. Obviously, to establish yε (y0+ , y0− ), it suffices to prove that y ∗ = y0−

a.e. on (0, T ) × Ω − .

(14.32)

For this, we introduce some periodic finite Borel measure ν on Rn . Let n = [0, 1)n be the cell of periodicity for ν. Assume that ν is the probability measure, concentrated and uniformly distributed on the set C × [0, 1), so n dν = 1. It is easy to see that for any smooth function, the equality

526


−1

n

−1

f dν = [Ln (C × [0, 1))]

f dx = |C| C×[0,1)

f dx

(14.33)

C×[0,1)

is valid. Now, define a scaling measure νε by the relation νε (A) = εn ν(ε−1 A), where A is an arbitrary Borel set in Rn and ε−1 A = {ε−1 x, x ∈ A}. Then the measure νε is ε-periodic and νε (ε) = εn n dν = εn . Therefore, this measure weakly converges to the Lebesgue measure as ε → 0, that is, ϕ dνε = ϕ dx for all ϕ ∈ C0∞ (Rn ). lim ε→0

Rn

Rn

It means that the weak limits of both sequences

− yε ∈ L2 (0, T ; L2 (Ω − , dμε )) and yε− ∈ L2 (0, T ; L2 (Ω − , dνε )) in the sense of Remark 14.8 have to be the same, namely

T

lim

ε→0

0

Ω−

ϕ ψ yε− dμε dt =

T 0

ϕ ψ y ∗ dx dt

Ω− T

= lim

ε→0

Ω−

0

ϕ ψ yε− dνε dt.

(14.34)

At the same time, for every function yε ∈ Xμε and every fixed ε, the set Ω − can be covered by a system of cubes with edges ε. We denote these cubes by the symbols ε( + kj ). Then in accordance with the definition of the measure νε , we may write 0

T

Ω−

y ε− ϕ ψ dx dt

T

= |C| 0

= |C| 0

j

T

yε− ϕ ψ εn dν(x/ε) dt ε(+kj )

Ω−

yε− ϕ ψ dνε dt,

(14.35)

where ϕ ∈ C0∞ (Rn ) and ψ ∈ C0∞ (R). Now, using (14.35) and (14.34) and taking into account the fact that y ε− |C|y0− in L2 (0, T ; L2 (Ω − )) as ε → 0, we obtain T T − |C|y0 ϕ ψ dx dt = lim y ε− ϕ ψ dx dt 0

Ω−

ε→0

0

Ω− T

= lim |C| ε→0

0

T

= |C| 0

Ω−

yε− ϕ ψ dνε dt

y ∗ ϕ ψ dx dt

Ω−

for all ϕ ∈ C0∞ (Rn ) and ψ ∈ C0∞ (R). Hence, y ∗ = y0− , and we obtain the required result.


527

Definition 14.9. We say that a sequence {(uε , yε ) ∈ Yε } ε>0 is wa -convergent to a triplet (u, y + , y− ) as ε tends to 0 wa in symbols, (uε , yε ) −→ (u, y + , y − ) if uε → u in the sense of Definition 14.4 and yε (y + , y − ) in the sense of Definition 14.6 (here, the space Yε is defined in (14.24)). As follows from the result obtained above and estimate (14.18), the following statement holds. Proposition 14.10. Let {(uε , yε ) ∈ Ξεa } ε>0 be any sequence of admissible a problem. Then there exist a subsequence {(uε , yε )} and pairs for the P ε ε >0 a triplet (u, y + , y − ) ∈ Ya0 :=L2 (0, T ; H 1 (Γ0 )) (0,1)

× L2 (0, T ; H 1 (Ω + )) × L2 (0, T ; W2

(Ω − ))

(14.36)

wa

such that u ∈ Ua and (uε , yε ) −→ (u, y + , y − ), where

Ua = u ∈ L2 (0, T ; H 1 (Γ0 )) | u L2 (0,T ;H 1 (Γ0 )) ≤ C0 .

(14.37)

14.2.2 The convergence concept for the Pbε -problems Let {(uε , yε )}ε>0 be any sequence of admissible pairs for the Pbε problems. Since we cannot assert in this case the existence of extension operators Pε : Uεb → L2 (0, T ; H 1 (Γ0 )) that would be uniformly bounded with respect to ε, it follows that we have to give another convergence concept in the variable space (14.24). Let us denote by u ε the extension by 0 of a function uε ∈ ε ∈ L2 (0, T ; L2 (Γ0 )). L2 (0, T ; H 1 (Γε )) onto Γ0 . Then u Definition 14.11. A sequence

(uε , yε ) ∈ L2 (0, T ; H 1 (Γε )) × Xμε ε>0 is called wb -convergent to a triplet (u, y + , y − ) % & wb in symbols, (uε , yε ) −→ (u, y+ , y− ) as ε tends to 0 if u ε u in L2 (0, T ; L2 (Γ0 )) and yε (y + , y − ) in the sense of Definition 14.9. Then, taking the definition of the sets Uεb , estimate (14.18), and Proposition 14.7 into account, we have the following obvious result.

528


Proposition 14.12. Let (uε , yε ) ∈ Ξεb ε>0 be a sequence of admissible pairs for the Pbε problems such that supε>0 yε Xμε < +∞. Then there exist a subsequence {(uε , yε )} ε >0 and a triplet (u, y + , y − ) ∈ Yb0 :=L2 (0, T ; L2 (Γ0 )) (0,1)

× L2 (0, T ; H 1 (Ω + )) × L2 (0, T ; W2

(Ω − ))

(14.38)

wb

for which (uε , yε ) −→ (u, y + , y− ) as ε → 0. Let us denote by τ a the topology associated with wa -convergence in Ya0 and by τ b the topology associated with wb -convergence in Yb0 . Then, as follows from Propositions 14.10 and 14.12, these topologies can be taken as the most natural ones for limit analysis of the OCPs Paε and Pbε , respectively.

14.3 Definition of the limit problems and their properties As follows from the previous sections, each of the sets of admissible solutions Ξεi (i = a, b) belongs to the corresponding Banach space (Ξεi ⊂ Yε ). We recall the convergence concept of such sets using the wi -sequential version of the set convergence in Kuratowski’s sense (see Sect. 7.4). Hereinafter i = a, b. Definition 14.13. We say that a set Ξi ⊂ Yi0 is the sequential wi -limit in

i i the Kuratowski’s sense (or K(w )-limit) of the sequence Ξε ⊂ Yε ε>0 if the following conditions are satisfied: (i) For every triplet (u, y + , y − ) ∈ Ξi , there exists a sequence {(uε , yε )}ε>0 wi -converging to (u, y + , y− ) and a positive value ε0 > 0 such that (uε , yε ) ∈ Ξεi for every ε ∈ (0, ε0 ).

(ii) For every sequence of admissible pairs (uk , yk ) ∈ Ξεik k∈ N such that wi

εk −→ 0 and (uk , yk ) −→ (u, y + , y − ) as k → ∞, the triplet (u, y + , y − ) belongs to Ξi . Let us show that the sets of admissible pairs for the Paε problems possess the compactness property with respect to the K(wa )-convergence. Theorem 14.14. For the sequence of sets {Ξεa }ε>0 , there exist a subsequence {Ξεa }ε >0 and a set Ξa ⊂ Ya0 such that K(wa )− limε →0 Ξεa = Ξa . Proof. We begin with the obvious fact that the wa -convergence of any sequence of admissible pairs {(uε , yε ) ∈ Ξ aε } ε>0 is equivalent to the weak convergence of its image {(Pε (uε ), yε+ , y − ε )} ε>0 in the space (0,1)

Ya0 = L2 (0, T ; H 1 (Γ0 )) × L2 (0, T ; H 1 (Ω + )) × L2 (0, T ; W2

(Ω − )).

14.3 Definition of the limit problems and their properties

529

Since the space Ya0 is separable and reflexive, there exists a metric d such that for any sequence {pk = (wk , yk , vk )}k∈N in Ya0 , the following conditions are equivalent (see, e.g., Dunford and Schwartz [99]): (j) {pk } −→p = (w, y, v) weakly in Ya0 . (jj) {pk } is bounded in Ya0 and d(pk , p) → 0 as k → ∞. a }ε>0 Let η be the topology associated with the metric d on Ya0 and let {Ξ ε a a be the images sequence of the sets Ξε in Y0 , that is, a = Ξ ε

Pε (uε ), yε+ , y − ε

: (uε , yε ) ∈ Ξεa } .

Since the η topology has a countable base, then by the Kuratowski

acom ε pactness theorem (see Dal Maso [78]), there exists a subsequence of Ξ ε>0

a a still denoted by Ξ that K(η)-converges to a set A ⊂ Y . Now, we prove ε ε>0 0 that the set A coincides with the K(w a )-limit of the family {Ξεa }ε>0 . With this aim, it is enough to show that Ξa ⊆ A, A ⊆ Ξa ,

(14.39) (14.40)

where by Ξa we denoted the K(w a )-limit of the sequence {Ξεa }ε>0 in the sense of Definition 14.13. First, let us verify inclusion (14.39). Let (u, y+ , y − ) be any triplet in Ya0 for which one can find a sequence {(uk , yk )}k ∈N , wa -converging to (u, y + , y − ) and a subsequence {εk }k ∈N such that (uk , yk ) ∈ Ξεak for every k ∈ N. Then (u, y + , y − ) ∈ Ξ0a by Definition 14.13. Let {Pk } be any sequence of the extension operators Pk : L2 (0, T ; H 1 (Γε )) → L2 (0, T ; H 1 (Γ0 )). Then

(Pk uk , yk+ , y k− ) (u, y + , y − ) in Ya0 , a for every k ∈ N. (Pk uk , y + , y − ) ∈ Ξ k

k

εk

Therefore, the equivalence between conditions (j) and (jj) yields η-convergence

of (Pk uk , yk+ , y k− ) to (u, y + , y − ). Hence, (u, y + , y − ) ∈ A by definition of Kuratowski’s limit. So, inclusion (14.39) is proved. Now we verify (14.40). Let (u, y + , y − ) be any triplet of A. Then there exists a sequence {(vε , pε , qε )}ε>0 η-converging to (u, y + , y − ) such that (vε , pε , qε ) ∈ ε for ε small enough. It follows that each pair (pε , qε ) can be represented Ξ as pε = yε+ and qε = y ε− , where yε is a weak solution of the boundary value problem (14.2) with uε = vε |Γε . However, the realization of the condition ε implies that the pair (uε , yε ) is admissible (i.e., (uε , yε ) ∈ Ξ a ). (vε , pε , qε ) ∈ Ξ ε Since the sequence of functions {vε } is bounded in L2 (0, T ; H 1 (Γ0 )) (vε ∈ Ua ε ), we get that the sequence of corresponding soby definition of the sets Ξ (0,1) (Ω − )) lutions {(pε , qε )}ε>0 is bounded in L2 (0, T ; H 1 (Ω + )) × L2 (0, T ; W2 as well. Hence, the equivalence between conditions (j) and (jj) yields the

530


weak convergence of the sequence {(vε , pε , qε )}ε>0 to (u, y + , y − ). However, in view of Definition 14.9, it is equivalent to the wa -convergence of its prototype {(uε , yε )}ε>0 to (u, y + , y− ). Thus, (u, y + , y − ) ∈ Ξa by Definition 14.13. The theorem is proved. Definition 14.15. We say that Pbε satisfies the property (N ) if for any u ∈ Ub = v ∈ L2 ((0, T ) × Γ0 ) : v L2 ((0,T )×Γ0 ) ≤ |C|C0 ,

there exists a sequence uε ∈ Uεb ε>0 such that u ε u in L2 ((0, T ) × Γ0 ) and sup uε L2 (0,T ;H 1 (Γε )) < +∞. ε>0

Then the following compactness property for the sets Ξeb with respect to the K(wb )-convergence takes place. b Theorem

b 14.16. If Pε possesses the (N )-property, then the sequence of the sets Ξε ∈ Yε ε>0 has a subsequence (still denoted by ε) for which there exists

a nonempty set Ξb ⊂ Yb0 that is the K(wb )-limit of Ξεb ε>0 with respect to the space Yb0 .

Remark 14.17. It is well known (see Dal Maso [78]) that the Kuratowski limit A0 of a sequence of subsets {An }n ∈N in a topological space (X, τ ) does not change if we replace the sets An by their τ -closures, that is, K(τ )− lim An = A0 = K(τ )− lim clτ An n→∞

n→∞

(τεb denotes the product of the weak topologies of the spaces L2 (0, T ; L2 (Γε )) b and L2 (0, T ; H 1 (Ωε ))). Thus, {Ξεb }ε>0 coincides with the

the bK(w 2)-limit of b b 2 K(w )-limit of τε -closures clτεb Ξε ⊂ L (0, T ; L (Γε )) × Xμε . Let us turn back to the main object of this section, namely to the sequences of constrained minimization problems (14.23). Using the concept of variational convergence, we give the definition of the “appropriate limits” for these sequences. Definition 14.18. We say that the minimization problem + − Ii (u, y , y ) (i = a, b), inf (u,y+ ,y− )∈ Ξi

(14.41)

where Ξi ⊂ Yi0 , is the variational wi -limit of the sequence (14.23) with respect to the wi -convergence if the following hold:

(i) Ξi is the K(wi )-limit of the sets Ξεi .

14.3 Definition of the limit problems and their properties

(ii) For any triplet (u, y + , y − ) ∈ Ξi and for any sequence

531

(uk , yk ) ∈ Ξεik

wi

such that εk −→ 0 and (uk , yk ) =⇒ (u, y + , y− ) as k → ∞, we have Ii (u, y + , y − ) ≤ lim inf Iεk (uk , yk ). k→∞

(14.42)

(iii) For every triplet (u, y + , y − ) ∈ Ξi , there exist a positive constant ε0 and a sequence {(uε , yε )}ε>0 such that (uε , yε ) ∈ Ξεi

wi

(uε , yε ) =⇒ (u, y + , y− ),

for every ε ≤ ε0 ,

Ii (u, y + , y − ) ≥ lim sup Iε (uε , yε ).

(14.43)

ε→0

Using the same arguments as in the proof of Theorem 8.25 and taking into account the (N )-property, one can establish the following results for the variational w a - and wb -limits. Theorem 14.19. Assume that the constrained minimization problem + − Ia (u, y , y ) (14.44) inf (u,y+ ,y − )∈ Ξa

is the variational w a -limit of the corresponding sequence (14.23) and that this problem has a unique solution (ua , (y a )+ , (y a )− ) in Ξa . Let {(uaε , yεa ) ∈ Ξεa }ε>0 be a sequence of the optimal pairs for the Paε problem. Then wa

(uaε , yεa ) −→ (ua , (y a )+ , (y a )− )

as

ε→0

(14.45)

and, furthermore, inf

(u, y + , y − )∈ Ξa

Ia (u, y + , y − ) = Ia ua , (y a )+ , (y a )− = lim Iε (uaε , yεa ). (14.46) ε→0

Theorem 14.20. Assume that the constrained minimization problem + − Ib (u, y , y ) (14.47) inf + − (u,y ,y )∈ Ξb

is the variational w b -limit of the corresponding sequence (14.23) and that this problem has a unique solution (ub , (y b )+ , (y b )− ) ∈ Ξb ⊂ Yb0 . Let (ubε , yεb ) ∈ Ξεb ε>0 be a sequence of optimal pairs of Pbε problems such wb

that supε>0 yεb Xμε < +∞. Then (ubε , yεb ) −→ (ub , (y b )+ , (y b )− ) as ε → 0 and, furthermore,

inf Ib (u, y + , y − ) = Ib ub , (y b )+ , (y b )− = lim Iε (ubε , yεb ). (u, y + , y − )∈ Ξb

ε→0

Definition 14.21. We say that there exists a limit problem for the family of optimal control problems {Piε }ε>0 (i = a, b) with respect to the wi convergence, if for the corresponding sequence of constrained minimization problems (14.23), there exists a variational limit which can be represented in the form of some optimal control problem.

532


14.4 Analytical representation of the limit sets of admissible solutions The main objects of our consideration in this section are the sequences of the sets of admissible pairs {Ξea ⊂ Yε }ε>0 and Ξeb ⊂ Yε ε>0 and their Kuratowski limits with respect to w a - and w b -convergence, respectively. In view of Theorems 14.14 and 14.16, we may always suppose that for those sequences, there exist sets Ξa and Ξb such that Ξa = K(wa )− lim Ξea , ε→0

Ξb = K(w b )− lim Ξeb . ε→0

To formulate our next results, we introduce the space !

V(Ω) =

y ∈ L2 (Ω) :

∂y ∈ L2 (Ω − ), y ∈ H 1 (Ω + ) ∂xn

and endow it with the scalar product ∇y · ∇v dx + y v dx (y, v)V(Ω) = Ω+ Ω+ ∂xn y∂xn v dx + (|C| + k0 |∂C|H ) + |C| Ω−

y v dx. Ω−

By analogy with Corbo Esposito et al. [74], it can easily be shown that V(Ω) is a Hilbert space and H 1 (Ω) is dense in V(Ω). Moreover, as follows from (14.30) and (14.31), for any function y∗ = (y∗+ , y∗− ), which is a weak limit in the sense of Definition 14.6, we have the following: (i) y∗ = (y∗+ , y∗− ) ∈ L2 (0, T ; V(Ω)); (ii) y∗+ (·, ·) = y∗− (·, ·) almost everywhere on (0, T ) × Σ. Moreover, it should be stressed here that any function of V(Ω) has a trace on any hyperplane L in Ω − such that L = {(x , xn ) ∈ Ω − : xn = const}. 14.4.1 Recovery of the set Ξa The crucial point in the study of the K(wa )-limit properties for the sequence of admissible pairs is the following result. Lemma 14.22. Let {uε }ε>0 be any sequence of admissible controls for the Paε problems which is weakly convergent to a function u0 in L2 (0, T ; H 1 (Γ0 )). Let {yε ∈ Xμε } be the corresponding solutions of the problem (14.2). Then wa

(uε , yε ) −→ (u0 , v0+ , v0− ) as ε → 0, where

14.4 Analytical representation of the limit sets

$ v0 (x) =

v0+ (x) v0− (x)

if x ∈ Ω + , if x ∈ Ω −

533

(14.48)

is the unique weak solution in L2 (0, T ; V(Ω)) of the limit problem ⎫ (v0+ ) − Δx v0+ + v0+ = f0 in (0, T ) × Ω + , ⎪ ⎪ ⎪ ⎪ ⎪ |C| + k |∂C| ⎪ 0 H − − − 2 − ⎪ (v0 ) − ∂xn v0 + v0 = f0 in (0, T ) × Ω , ⎪ ⎪ ⎪ |C| ⎬ + + ∂ν v0 = 0 in (0, T ) × ∂Ω \ Σ, (14.49) ⎪ ⎪ v0− = u0 on (0, T ) × Γ0 , ⎪ ⎪ ⎪ ⎪ ⎪ v0+ = v0− , ∂xn v0+ = |C| ∂xn v0− on (0, T ) × Σ, ⎪ ⎪ ⎪ ⎭ 0 v0 (0, x) = y (x) a.e. x ∈ Ω. Remark 14.23. Here, the weak formulation of the problem (14.49) means that ⎫ v0 ∈ L2 (0, T ; V(Ω)), ⎪ ⎪ ⎪ T T ⎪ ⎪ ⎪ − (χΩ+ + |C|χΩ − ) v0 ϕ ψ dx dt + (v0 , ϕ)V(Ω) ψ dt ⎪ ⎪ ⎪ ⎬ Ω 0 0 T ⎪ (χΩ + + |C|χΩ − ) f0 ϕ ψ dx dt, = ⎪ ⎪ ⎪ 0 Ω ⎪ ∞ ⎪ ⎪ ∀ ϕ ∈ V(Ω; Γ0 ), ∀ ψ ∈ C0 (0, T ), ⎪ ⎪ ⎭ − 0 v0 (0, x) = y (x) a.e. x ∈ Ω, v0 = u0 , on (0, T ) × Γ0 , (14.50) where

V(Ω; Γ0 ) = v ∈ L2 (Ω) :

∂xn v ∈ L2 (Ω − ), v ∈ H 1 (Ω + ), v = 0 a.e. on Γ0 .

Moreover, in this case, we have v0 ∈ L2 (0, T ; (V(Ω)) ) (see Lions [169, p. 107]). Proof. From Proposition 14.7, it follows that there exist a subsequence {ε } of {ε} (still denoted by {ε}) and a triplet (u0 , v0+ , v0− ) ∈ Ya0 such that wa

(uε , yε ) −→ (u0 , v0+ , v0− ) as ε → 0. Similar to the proof of Proposition 14.7 (see relations (14.30) and (14.31)), we can show that ⎫ in L2 (0, T ; H 1 (Ω + )), yε+ v0+ ⎪ ⎪ ⎪ − ⎬ − 2 2 − y ε |C| v0 in L (0, T ; L (Ω )), (14.51) + − + − ⎪ v0 = v0 , ∂xn v0 = |C|∂xn v0 a.e. on (0, T ) × Σ, ⎪ ⎪ ⎭ in {L2 (0, T ; L2 (Ω − , dμε ))}. yε− v0−

534


Moreover, there exist functions γi ∈ L2 (0, T ; L2 (Ω − )) (i = 1, . . . , n − 1) such that

n − − in L2 (0, T ; L2 (Ω − )) . (14.52) ∇ x yε γ1 , . . . , γn−1 , |C|∂v0 /∂xn Let us prove that the function v0− satisfies the boundary condition v0− = u0

almost everywhere on (0, T ) × Γ0 .

(14.53)

We note that any function f ∈ V(Ω) has a trace f |Γ0 ∈ L2 (Γ0 ) (see Brizzi & Chalot [35]), so (14.53) makes sense. It is easy to see that the following statements hold: y ε− = χΓε Pε (uε )

a.e. on (0, T ) × Γ0 , ∀ ε > 0,

χΓε Pε (uε ) |C|u0

in L ((0, T ) × Γ0 ) 2

(14.54) (14.55)

(as a product of strongly and weakly convergent sequences). Then from the integral identity

T

0

y ε− ϕ ψ dx dt Γ0

T

= −

Ω−

0

∂ yε− /∂xn ϕ ψ dx dt −

T

Ω−

0

y ε− ∂ψ/∂xn ϕ dx dt,

∀ϕ ∈ C0∞ (Rn ; ∂Ω − \ Γ0 ), ∀ ψ ∈ C0∞ (R), (14.56) where C0∞ (Rn ; ∂Ω − \ Γ0 ) = {ϕ ∈ C0∞ (Rn ) : ϕ = 0 on ∂Ω − \ Γ0 },

' we immediately get ( yε− )|Γ0 −→ |C| v0− 'Γ0 weakly in L2 ((0, T ) × Γ0 ). Thus, passing to the limit in (14.54) as ε → 0, we obtain the required relation (14.53). Now, let us show that the function v0 is the unique weak solution of problem (14.49). With this aim, we rewrite the integral identity (14.16) as T − yε+ ϕ ψ dx dt 0

Ω+

T

− 0

T

Ω−

+ Ω−

0

T

+ Ω−

0

T

y ε− ϕ ψ

0

− ∇y ε · ∇ϕ ψ dx dt +

Ω+

Ω+ T

y ε− ϕ ψ dx dt + k0 |∂C|H

T 0

∇yε+ · ∇ϕ ψ dx dt yε+ ϕ ψ dx dt

0

fε ϕ ψ dx dt + 0

dx dt +

=

T

Ω−

Ω+ T

0

Ω−

yε− ϕ ψ dμε dt

χΩε− fε ϕ ψ dx dt.

(14.57)


535

Passing to the limit in (14.57) as ε → 0 and using the properties (14.27), and (14.51), and (14.52), we get T T v0+ ϕ ψ dx dt − |C| v0− ϕ ψ dx dt − Ω+

0

Ω+

T

Ω−

0

T

Ω+

T

+ k0 |∂C|H

T

0

Ω−

T

Ω−

0

γi (∂ϕ/∂xi ) ψ dx dt

v0− ϕ ψ dx dt

v0− ϕ ψ dx dt

Ω+

0

Ω − i=1

+ |C|

f0 ϕ ψ dx dt + |C|

=

n−1

(∂v0− /∂xn ) (∂ϕ/∂xn ) ψ dx dt

v0+ ϕ ψ dx dt

+ 0

T

0

+ |C|

∇v0+ · ∇ϕ ψ dx dt +

+ 0

Ω−

0

T

0

T

Ω−

f0 ϕ ψ dx dt,

∀ ϕ ∈ C0∞ (Rn ; Γ0 ), ∀ ψ ∈ C0∞ (0, T ).

(14.58)

Let us fix i ∈ {1, . . . , n − 1} and let wεi be a sequence in W 1,∞ (Ω − ) satisfying the following conditions: strongly in L∞ (Ω − ),

wεi −→ xi D wεi

= 0 a.e. on

(14.59)

Ωε−

(14.60)

for every ε > 0. The existence of such a sequence is proved in Brizzi and Chalot [35] and Corbo Esposito et al. [74]. Let us prove that γi = 0 a.e. on (0, T )×Ω − . Take the test functions ϕ = wεi φ and ϕ = x i φ with φ ∈ C0∞ (Ω − ) in (14.57). Then, by virtue of (14.60), we have T T − i yε− φ wεi ψ dx dt + − ∇y ε · ∇φ wε ψ dx dt Ω−

0

T

y ε− φ wεi ψ dx dt + k0 |∂C|H

+

0 T

Ω−

= Ω−

0

− 0

T

Ω−

T

Ω−

= 0

Ω−

T

Ω−

yε− φ wεi ψ dμε dt

χΩε− fε φ wεi ψ dx dt,

+ 0 T

0

yε− φ x i ψ dx dt +

Ω−

0

T 0

Ω−

(14.61)

− ∇y ε · ∇(φ x i ) ψ dx dt

y ε− φ x i ψ dx dt + k0 |∂C|H

T 0

χΩε− fε φ x i ψ dx dt

Ω−

yε− φ x i ψ dμε dt (14.62)

536


for every ε > 0, φ ∈ C0∞ (Ω − ), and ψ ∈ C0∞ (0, T ). Hence, passing to the limit in (14.61) and (14.62) as ε → 0 and using the properties (14.27), (14.51), (14.52), and (14.59) and Proposition 14.7, we obtain T n−1 T v0− φxi ψ dx dt + γk (∂φ/∂xk ) xi ψ dx dt −|C| Ω−

0

T

+ |C|

Ω−

0 T

T

Ω−

0

T

+ |C|

Ω−

0 T

Ω−

0

T

= |C| 0

v0− φ xi ψ dx dt

(14.63) n−1

Ω − k=1

γk (∂(φ xi )/∂xk ) ψ dx dt

(∂v0− /∂xn ) (∂φ/∂xn ) xi ψ dx dt

+ |C|

T

0

Ω−

f0 φ xi ψ dx dt,

v0− φxi ψ dx dt +

0

Ω−

0

T

ψ dx dt + k0 |∂C|H

T

= |C|

v0− φ xi

Ω−

0

−|C|

(∂v0− /∂xn ) (∂φ/∂xn ) xi ψ dx dt

+ |C|

Ω − k=1

0

v0− φ xi

ψ dx dt + k0 |∂C|H 0

Ω−

T

Ω−

v0− φ xi ψ dx dt

f0 φ xi ψ dx dt.

(14.64)

Comparing (14.63) with (14.64), we conclude that T γk φ ψ dx dt = 0, Ω− ∞ C0 (Ω − ),

0

and ∀ ψ ∈ C0∞ (0, T ). Thus, γi = 0 a.e. ∀ k ∈ {1, . . . , n − 1}, ∀φ ∈ − on (0, T ) × Ω , and we obtain the required result. As for the function v0 , we have the identity T T + v0 ϕ ψ dx dt − |C| v0− ϕ ψ dx dt −

Ω+

0 T

∇v0+

+

Ω+

0 T

Ω+

T

0

T

Ω−

Ω+

0

Ω−

0

Ω−

(∂v0− /∂xn ) (∂ϕ/∂xn ) ψ dx dt

v0− ϕ ψ dx dt

v0− ϕ ψ dx dt

T

f0 ϕ ψ dx dt, +|C|

= 0

T

v0+ ϕ ψ dx dt + |C|

+k0 |∂C|H

· ∇ϕ ψ dx dt + |C|

+ 0

Ω− T

0

0

Ω−

f0 ϕ ψ dx dt

∀ ϕ ∈ C0∞ (Rn ; Γ0 ), ∀ ψ ∈ C0∞ (0, T ).

(14.65)


537

However, using the fact that C0∞ (Rn ; Γ0 ) is dense in H 1 (Ω; Γ0 ) = {ϕ ∈ H 1 (Ω) : ϕ = 0 on Γ0 } and H 1 (Ω; Γ0 ) is dense in

V(Ω; Γ0 ) = v ∈ L2 (Ω) : ∂xn v ∈ L2 (Ω − ), v ∈ H 1 (Ω + ), v = 0 a.e. on Γ0 with the continuous injection H 1 (Ω; Γ0 ) → V(Ω; Γ0 ) (see Corbo Esposito et al. [74]), we observe that the integral identity (14.65) is valid with ϕ ∈ V(Ω; Γ0 ). Hence, it can be rewritten in the form

T

−

T

(χΩ + + |C|χΩ − ) v0 ϕ ψ dx dt + 0

Ω

T

0

(v0 , ϕ)V(Ω) ψ dt

(χΩ + + |C|χΩ − ) f0 ϕ ψ dx dt,

= 0

Ω

∀ ϕ ∈ V(Ω; Γ0 ),

∀ ψ ∈ C0∞ (0, T ).

(14.66)

In addition, taking the initial supposition (14.6) into account and using the approach of De Maio et al. [88], one can prove the relation v0 (0, x) = y 0 (x) a.e. on Ω. Following the standard Hilbert space method and the arguments in [88], we can state that the function v0 is a unique weak solution of the problem (14.49) in the sense of Remark 14.23. However, due to the uniqueness of the solution to the problem (14.49), the above reasoning holds for any subsequence of {ε} chosen at the beginning of the proof. Thus, the lemma is proved. We are now in a position to state the first important result which deals with the recovery problem of the Kuratowski K(wa )-limit set Ξa in the analytical form. Theorem 14.24. For the sequence of the sets of admissible pairs for the Paε problems {Ξεa }, there exists a nonempty K(wa )-limit set Ξa ⊂ Ya0 with the following structure: (u0 , v0+ , v0− ) ∈ Ξa if ⎫ ⎧ u0 ∈ Ua , ⎪ ⎪ ⎪ ⎪ ⎪ + + + ⎪ ⎪ + ⎪ ⎪ ⎪ (v ) − Δ v + v = f in (0, T ) × Ω , x 0 ⎪ ⎪ 0 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ |∂C| |C| + k 0 H ⎪ ⎪ − − − 2 ⎪ ⎪ v (v ) − ∂ v + ⎪ ⎪ xn 0 0 0 ⎪ ⎪ |C| ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ − = f0 in (0, T ) × Ω , . (14.67) Ξa = ⎪ ⎪ + + ⎪ ⎪ ⎪ ⎪ ∂ν v0 = 0 in (0, T ) × ∂Ω \ Σ,⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − ⎪ ⎪ = u on (0, T ) × Γ , v ⎪ ⎪ 0 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + − + − ⎪ ⎪ ⎪ v0 = v0 , ∂xn v0 = |C| ∂xn v0 on (0, T ) × Σ,⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 0 v0 (0, x) = y (x) a.e. x ∈ Ω

Here, Ua = u ∈ L2 (0, T ; H 1 (Γ0 )) : u L2 (0,T ;H 1 (Γ0 )) ≤ C0 .

538


Proof. First, we note that in view of Theorem 14.14, the sequence of sets {Ξεa ⊂ Yaε } is relatively compact with respect to K(wa )-convergence. We show that the K(w a )-limit set exists for the whole sequence and it can be represented in the form (14.67). For this, in accordance with the definition of the K(wa )-limit, we have to verify conditions (i) and (ii) of Definition 14.13. From the previous lemma, we see that the set Ξa is nonempty. Let (u, y + , y − ) be any triplet of the set Ξa . We construct the wa -convergent to (u, y + , y− ) sequence {(uε , yε )}ε>0 as follows: uε ∈ L2 (0, T ; H 1 (Γε )) is the restriction of the control u ∈ Ua on Γε given above and yε is the corresponding to uε weak solution of the boundary value problem (14.2). Then, in view of Definition 14.4, we have uε → u weakly with respect to the space L2 (0, T ; H 1 (Γ0 )). Further, wa

using Lemma 14.22, we obtain (uε , yε ) −→ (u, v0+ , v0− ), where (u, v0+ , v0− ) is a solution in L2 (0, T ; V(Ω)) of the limit problem (14.49). Since this problem has a unique solution, we immediately deduce that (u, v0+ , v0− ) = (u, y + , y − ); thereby, property (i) of Definition 14.13 is valid. The second condition of this definition is an evident consequence of Lemma 14.22 and the lower semicontinuity of the norm in L2 ((0, T ) × Γ0 ) with respect to the weak convergence. This concludes the proof. 14.4.2 Recovery of the set Ξb To establish the structure of the Kuratowski’s K(wb )-limit set Ξb , we make use of the following result (see Sect. 7.2, Proposition 7.12). Proposition 14.25. For every bounded sequence {uε ∈ L2 (0, T ; L2 (Γε ))}ε>0 such that u ε −→ u∗ weakly in L2 ((0, T ) × Γ0 ), the following inequality holds:

T

u2ε

lim inf ε →0

0

−1

T

dx dt ≥ |C|

0

Γε

Γ0

u2∗ dx dt.

(14.68)

The following assertion can be viewed as an analogue of Lemma 14.22 with respect to w b -convergence. Lemma 14.26. Let {uε ∈ L2 (0, T ; H 1 (Γε ))}ε>0 be any sequence of admissible controls for the Pbε problems such that u ε u∗ in L2 ((0, T ) × Γ0 ). Let {yε } be the corresponding solutions of the parabolic problem (14.2) for which wb

supε>0 yε Xμε < +∞. Then (uε , yε ) −→ (u∗ , v0+ , v0− ) as ε → 0, where $ v0+ (x) if x ∈ Ω + , v0 (x) = (14.69) v0− (x) if x ∈ Ω − , is the unique weak solution in L2 (0, T ; V(Ω)) of the limit problem


(v0+ ) − Δx v0+ + v0+ = f0 in (0, T ) × Ω + , |C| + k0 |∂C|H − (v0− ) − ∂x2n v0− + v0 = f0 in (0, T ) × Ω − , |C| ∂ν v0+ = 0 in (0, T ) × ∂Ω + \ Σ, v0− = |C|−1 u∗ on (0, T ) × Γ0 , v0+ = v0− ,

∂xn v0+ = |C| ∂xn v0− on (0, T ) × Σ, v0 (0, x) = y 0 (x) a.e. x ∈ Ω.

539

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(14.70)

Proof. As follows from Proposition 14.12 and Lemma 14.22, we have to show that the relation (14.71) v0− = |C|−1 u∗ on (0, T ) × Γ0 holds true. For this, we note that

− '

' y ε 'Γ |C| v0− 'Γ 0

0

in L2 ((0, T ) × Γ0 )

and the statements y ε− = u ε

a.e. on (0, T ) × Γ0 , ∀ ε > 0,

u ε u0

in L2 ((0, T ) × Γ0 )

(14.72)

are valid. Then, the required relation (14.71) immediately follows from (14.72) after passing to the limit in (14.72) as ε → 0. In order to conclude our proof, we have to follow the arguments in the proof of Lemma 14.22 closely. Now, we are able to prove the theorem concerning the structure of the Kuratowski K(w b )-limit set Ξb .

Theorem 14.27. Let Ξεb ε>0 be the sets of admissible pairs for the Pbε problems possessing the (N )-property. Then for this sequence there exists a nonempty K(wb )-limit set Ξb ⊂ Yb0 which can be represented in the form (u0 , v0+ , v0− ) ∈ Ξb if ⎧ ⎫ u0 ∈ Ub ,⎪ ⎪ ⎪ ⎪ ⎪ + + + ⎪ ⎪ + ⎪ ⎪ ⎪ (v ) − Δ v + v = f in (0, T ) × Ω , x 0 ⎪ ⎪ 0 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ |∂C| |C| + k 0 H ⎪ ⎪ − − − 2 ⎪ ⎪ v (v ) − ∂ v + ⎪ ⎪ xn 0 0 0 ⎪ ⎪ |C| ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ − = f0 in (0, T ) × Ω , . Ξb = ⎪ ⎪ ⎪ ⎪ ⎪ ∂ν v0+ = 0 in (0, T ) × ∂Ω + \ Σ,⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − −1 ⎪ ⎪ v = |C| u on (0, T ) × Γ , ⎪ ⎪ 0 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + − + − ⎪ ⎪ ⎪ v0 = v0 ∂xn v0 = |C| ∂xn v0 on (0, T ) × Σ,⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 0 v0 (0, x) = y (x) a.e. x ∈ Ω

Here, Ub = u ∈ L2 ((0, T ) × Γ0 ) : u L2 ((0,T )×Γ0 ) ≤ |C|C0 .

(14.73)

540


Proof. To obtain the representation (14.73), we have to verify conditions (i) and (ii) of Definition 14.13. Let (u, y+ , y− ) be any triplet of the set Ξb . In accordance with (N )-property of the Pbε problems, there is a wb convergent to (u, y + , y − ) sequence {( uε , yε )}ε>0 such that ( uε , yε ) ∈ Ξεb for wb

every ε > 0. However, due to Lemma 14.26, we have ( uε , yε ) −→ (u, v0+ , v0− ), + − 2 where v = (v0 , v0 ) is a weak solution in L (0, T ; V(Ω)) of the limit problem (14.70). Since this problem has a unique weak solution (see Lions [169]), we immediately deduce that (u, v0+ , v0− ) = (u, y + , y − ). Thus, property (i) of Definition 14.13 holds for any triplet (u, y + , y − ) ∈ Ξ0 . We now verify the second property of Definition 14.13. Let {(uk , yk )}k ∈N be a wb -convergent sequence for which there exists a sequence {εk → 0} such that (uk , yk ) ∈ Ξεbk for all k ∈ N. Let (u, y + , y − ) be its w b -limit. Then by Proposition 14.25, we immediately have C0 ≥ lim inf uk L2 ((0,T )×Γεk ) ≥ |C|−1 · u L2 ((0,T )×Γ0 ) , k→∞

b . In conclusion, it remains only to apply Lemma 14.26. Thus, that is, u ∈ U (u, y + , y − ) ∈ Ξb , and we obtain the required result. This concludes the proof.

14.5 Identification of the cost functionals Ia and Ib In this section, we show that the cost functionals of the limit constrained minimization problems + − + − inf Ia (u, y , y ) and inf Ib (u, y , y ) (14.74) (u,y + ,y − )∈ Ξa

(u,y + ,y − )∈ Ξb

can be recovered in an explicit form and their analytical representations are different. We begin with the following results. Theorem 14.28. For the sequence of the Paε problems (14.23) there exists a variational w a -limit (in the sense of Definition 14.18) + − inf Ia (u, y , y ) , (14.75) (u, y+ , y − )∈ Ξa

where the set Ξa is defined in (14.67) and T + − + 2 Ia (u, y , y ) = (y − q0 ) dx dt + |C| 0

Ω+

0

T

u2 dx dt.

(14.76)

Γ0

Proof. In order to obtain relation (14.76), we verify conditions (ii) and (iii) of Definition 14.18. Let (u, y + , y − ) be any triplet of Ξa and let {(uk , yk )}k∈N be wa

a wa -convergent sequence such that (uk , yk ) −→ (u, y − , y + ), (uk , yk ) ∈ Ξεk

14.5 Identification of the cost functionals Ia and Ib

541

for every k ∈ N, where {εk } is a subsequence of {ε} converging to 0. Then using Lemma 14.22, the definition of the class of admissible controls, and the properties of wa -convergence, we get

T

Ω+

0

0

T

u2k dx dt =

T

(y + )2 dx dt, Ω+

χΓε (Pεk uk ) dx dt 2

0

Γεk

0

T

T

dx dt −→

(yε+ )2

2

T

u2 dx dt

χΓε (Pεk uk ) dx dt −→ |C| 0

for every k ∈ N,

Γ0

0

Γ0

as k → ∞

Γ0

(as the limit of the product of weakly and strongly convergent sequences) and, therefore,

T

(y + − q0 )2 dx

lim inf Iεk (uk , yk ) = k→∞

0

Ω0 T

u2 dx dt = Ia (u, y + , y − ),

+ |C| 0

(14.77)

Γ0

that is, the property (ii) of Definition 14.18 is valid. Similarly, we can show the correctness of the “reverse” inequality (14.43). Indeed, in this case for a given triplet (u, y+ , y − ) ∈ Ξa , it is enough to consider as a “realized sequence” {(uε , yε )} the following one: uε ∈ L2 (0, T ; H 1 (Γε )) is the restriction of u ∈ Ua on Γε and yε is the corresponding to uε solution of the boundary value problem (14.2). Then, by Lemma 14.22, we have wa

(uε , yε ) −→ (u, y + , y − ). For the conclusion of this proof, we repeat the arguments concerning the correctness of the limit passage (14.77).

Theorem 14.29.

inf

(u, y + , y − )∈ Ξb

Ib (u, y , y ) +

−

(14.78)

is the variational wb -limit for the sequence of the Pbε problems (14.23). Here, the set Ξb is defined in (14.73) and Ib (u, y + , y − ) =

0

T

(y + − q0 )2 dx + |C|−1 Ω+

0

T

u2 dx dt.

(14.79)

Γ0

Proof. To obtain representation (14.79), it is enough to repeat the same arguments of the proof of Theorem 14.28 and to

apply Proposition 14.25 and Lemma 14.26. In this case, for any sequence (uk , yk ) ∈ Ξεbk k∈N wb converging to (u, y + , y − ), we have

542


T

lim inf Iεk (uk , yk ) ≥ k→∞

|∇y + |2 dx dt 0

+ |C|−1

Ω+ T 0

u2 dx dt = Ib (u, y + , y− ).

Γ0

To verify the correctness of inequality (14.43), for arbitrary triplets (u, y + , y − ) ∈ Ξb , we have to construct a special “realizing sequence” {(uε , yε )} satisfying conb dition (iii) of Definition

14.18. With this aim, we construct a w 2-convergent to + − b (u, y , y ) sequence (uε , yε ) ∈ Ξε ε>0 as follows. Let {uε ∈ L ((0, T ) × Γ0 )} be any sequence such that uε u in L2 ((0, T ) × Γ0 ), uε L2 ((0,T )×Γ0 ) < |C| · C0 for every ε > 0. Since the weak topology of L2 ((0, T ) × Γ0 ) is metrizable on the set b = u ∈ L2 ((0, T ) × Γ0 ) : u L2 ((0,T )×Γ ) ≤ |C| · C0 , U 0 one can construct a sequence {wε ∈ L2 ((0, T )×Γ0 )}ε>0 satisfying the following condition: Each element wε is a convex envelope of a finite number of the elements {uε }ε>0 and wε −→ u strongly in L2 ((0, T ) × Γ0 ). Note that in this case we have wε L2 ((0,T )×Γ0 ) < |C| · C0 for every ε > 0. Thus, a weakly to u convergent sequence {uε }ε>0 can be taken in the following form: uε ∈ L2 (0, T ; H 1 (Γε )) are elements such that uε − |C|−1 wε L2 ((0,T )×Γε ) < ε2 .

In view of the (N )-property, we may assume that the sequence of norms uε L2 (0,T ;H 1 (Γε )) is uniformly bounded. Since |C|−1 χΓε wε u in L2 ((0, T ) × Γ0 ) (as the limit of the product of weakly and strongly convergent sequences) and ( T wε2 χΓε dx dt |C|−1 wε L2 ((0,T )×Γε ) = |C|−2 Γ0 0 ( T u2 → dx dt < C0 , −1 0 Γ0 |C| it follows that u ε u in L2 ((0, T ) × Γ0 ) and uε ∈ Uεb for ε sufficiently small.


543

We may assume that the elements wε have the representation wε = ε , where the sequence {w ε } is constructed as |C|−1 w L2 ((0, T ) × Γ0 ) w ε → u w ε L2 ((0,T )×Γ0 )

strongly in L2 ((0, T ) × Γ0 ), < |C| · C0 , ∀ ε > 0.

Then lim wε 2L2 ((0,T )×Γε ) = lim |C|−1 χΓε w ε 2L2 ((0,T )×Γ0 ) ε→0 T = |C|−2 lim χΓε w ε2 dx dt

ε→0

ε→0

−1

= |C|

0

Γ0

u 2L2 ((0,T )×Γ0 ) ,

that is, for the realizing sequence of the Dirichlet boundary controls {uε } we have T T u2ε dx dt = |C|−1 u2 dx dt lim ε→0

0

0

Γε

Γ0

and uε ∈ Uεb for all ε > 0. In order to conclude, we take yε as solutions of the boundary value problem (14.2) corresponding to uε . Then by the (N )-property and Lemma 14.26, wb

we have (uε , yε ) −→ (u, y + , y − ) and, therefore, lim supIε (uε , pε , yε ) ε→0

T

= lim

ε→0

T

0

≥ 0

Ω+

ε→0

(y + − q0 )2 dx dt + |C|−1

Ω+

T

u2ε dx dt

(yε+ − q0 )2 dx dt + lim

0 T

0

Γε

u2 dx dt.

Γ0

This concludes the proof. Thus, in accordance with Definition 14.21 and the results obtained above, we may infer the following: Each of the constrained minimization problems (14.74) can be recovered in the form of some OCPs, namely (Pahom ) (see (14.1), (14.2), and (14.4)) and Pbhom (see (14.1), (14.2), and (14.5)). Hence, there exist limiting OCPs to the problems Paε and Pbε as ε tends to 0. However, the corresponding limit problems have different mathematical descriptions and these differences appear not only in the state equation and boundary conditions, but also in the control constrains and limit cost functionals. In fact, the reason for this gap phenomenon is the choice of different topologies for the limit analysis of the original control problem (14.2), (14.4) that were associated with wa - and wb -convergence, respectively. It should be stressed that in our case, this choice was dictated by the characteristic properties of the control constraints.

544


In conclusion, we give some results concerning the variational properties

of the limit problems. As was noted earlier, the problems (Pahom ) and Pbhom have to preserve the well-known variational property, namely both optimal pairs and minimal values of the cost functionals for the the original problems have to converge to the corresponding characteristics of the limit OCPs as ε tends to 0. To establish this result, we begin with the following evident assertion. Proposition 14.30. Each of the limit OCPs (see (14.8)–(14.10) and (14.11)– (14.13)) has a unique solution. Indeed, taking into account the weak lower semicontinuity of the cost functionals Ii : Ξi → R (i = a, b), the topological properties of their domains Ξi (i = a, b), and applying the direct method of Calculus of Variation, we just obtain the required result. a b b + b − b Let us denote by (ua , (y a )+ , (y a ) − ) ∈ Y 0 and (u , (y ) , (y ) ) ∈ Y0 the a b optimal triplets for the (Phom ) and Phom problems, respectively. Lemma 14.31. If the functions fε and yε0 satisfy conditions (14.6)–(14.7), then the sequence of optimal solutions {(uaε , yεa ) ∈ Ξεa } to the Paε problems and the corresponding minimal values of the cost functional (14.1) possess the following properties: lim Iε (uaε , yεa ) = lim

ε→ 0

inf

ε→ 0 (uε ,yε )∈Ξεa

=

inf

(u, y + , y − )∈Ξa

Iε (uε , yε )

Ia (u, y + , y− ) = Ia (ua , (y a )+ , (y a )− ), (14.80) wa

(uaε , yεa ) −→ (ua , (y a )+ , (y a )− ).

(14.81)

Proof. It is clear that for every value of ε, the OCP (14.1)–(14.4) has a unique solution (uaε , yεa ) ∈ Ξεa . Since the sequence {uaε }ε>0 ⊂ Uεa is bounded, there exists a subsequence {ε } of {ε}, which we again denote by {ε}, such that uaε → ua ∈ Ua weakly with respect to the space L2 (0, T ; H 1 (Γ0 )) as ε → 0. wa

Then, in view of Lemma 14.22, we have (uaε , yεa ) −→ (ua , (y ∗ )+ , (y ∗ )− ) as ε → 0, where the triplet (ua , (y ∗ )+ , (y ∗ )− ) is the unique solution of the problem (14.8) with the Dirichlet condition y − = ua on Γ0 . By Theorem 14.19, we immediately conclude that (ua , (y ∗ )− , (y ∗ )+ ) is an optimal solution to the limit problem (14.8)–(14.10) and property (14.80) is valid. Hence, (ua , (y ∗ )− , (y ∗ )+ ) = (ua , (y a )+ , (y a )− ). So, we have obtained the required result. Remark 14.32. It should be noted that the realization of conditions (14.80)– (14.81) does not imply strong convergence of the optimal states yεa .


545

Lemma 14.33. Assume that the functions fε and yε0 satisfy conditions (14.6)– is solvable (14.7), the Pbε problem

for every value ε > 0, and the sequence of optimal solutions (ubε , yεb ) ∈ Ξεb for the Pbε problems is such that sup yεb Xμε < +∞. ε>0

Then lim Iε (ubε , yεb ) = lim

ε→ 0

inf

ε→ 0 (uε ,yε )∈Ξεb

=

inf

(u, y+ , y − )∈Ξb

Iε (uε , yε )

Ib (u, y + , y− ) = Ib (ub , (y b )+ , (y b )− ),

wb

(ubε , yεb ) −→ (ub , (y b )+ , (y b )− ), u bε

−1

− |C|

χΓε u −→ 0 b

(14.82)

(14.83)

strongly in L ((0, T ) × Γ0 ). 2

(14.84)

Proof. Using the arguments of Lemma 14.31 and Theorem 14.27, it can be easily checked that conditions (14.82) and (14.83) hold. To conclude the proof, it remains to verify assertion (14.84). However, in view of (14.29) and the fact that ubε 2L2 (0,T ;L2 (Γε )) −→ |C|−1 ub 2L2 ((0,T )×Γ0 ) (see (14.82)), we get T

b 2 u ε − |C|−1 χΓe ub dx dt 0

Γ0

T

b 2 ue dx dt

= 0

Γe

− 2|C|−1

T

Γ0

0

−→ |C|−1 + |C|−1

T

0

T 0

ub u bε dx dt + |C|−2

0

(ub )2 dx dt − 2|C|−1

(ub )2 dx dt = 0

Γ0 T

0

Γ0

T

χΓe (ub )2 dx dt (ub )2 dx dt

Γ0

as ε → 0,

Γ0

which yields (14.84). This completes the proof.

To conclude this chapter, we would like to point out a possible application of the results obtained above. It concerns the approximation of the optimal solutions to the original problems for ε small enough. Since the computational calculation of the solutions of these problems is very complicated, it is natural to define suboptimal solutions which have to guarantee the closeness of the sub corresponding value of the cost functional Iε (usub ε , yε ) to its minimum for ε small enough (see Definition 11.30).

546


As follows from Proposition 14.30, each of the limit OCPs (14.8)–(14.10) and (14.11)–(14.13) has a unique solution. Then Lemma 14.31 and the a priori estimate (14.18) immediately lead us to the following result. a Theorem 14.34. Let (ua , (y a )+ , (y a )− ) ∈

Ya0 be an optimal solution for the a limit (Phom )-problem. Then the sequence u |Γε ε>0 is asymptotically suboptimal for the Paε problem in the sense of Definition 11.30.

Using the idea of Sect. 13.7, a similar result can be established for the Pbε problem. In conclusion, we would like to note that the asymptotic analysis of OCPs may essentially depend on the differential properties of its solutions. Choosing different topologies on the space of “control-state”, the corresponding limit OCPs may have drastically different mathematical descriptions. So, the choice of such topologies is a very important and nontrivial matter when dealing with the questions of asymptotic behavior of the OCPs. In the homogenization theory of boundary value problems, this fact is called the Lavrentieff phenomenon [263].

15 Boundary Velocity Suboptimal Control of Incompressible Flow in Cylindrically Perforated Domains

Optimal control problems (OCPs) for the Navier–Stokes equations have been the subject of extensive study in recent years. A systematic mathematical and numerical analysis of OCPs of different types (e.g., having Dirichlet, Neumann, and distributed controls) for the steady-state Navier–Stokes system was given by Abergel and Temam [1], Fursikov, Gunzburger, and Hou [111, 112], Hou and Ravindran [123], Gunzburger, Hou, and Svobodny [116], Ivanenko and Mel’nik [126], and Zgurovsky and Mel’nik [253]. Dirichlet controls (i.e., boundary velocity controls or boundary mass flux control) are common in applications [237]. However, as is shown in [124], even though the admissible controls are smooth, the optimality systems for optimal Dirichlet control problems involve a boundary Laplacian or a boundary biharmonic equation. This circumstance makes the numerical resolution of the optimality systems, and hence the numerical calculation of an optimal control for such systems, very complicated. So, much effort has been made for the development of penalty, approximation, and relaxation methods for solving optimal Dirichlet control problems (see [1, 12, 123]). These problems are especially complicated in perforated domains and in domains with quickly oscillating boundaries (see [222]). So, we are concerned with the development of approximate methods for the solution of optimal Dirichlet control problems for Navier–Stokes equations. The approach we propose gives the possibility to replace the original OCP with some limit problem defined in a more simple domain. We show that an optimal control for the limit problem can be taken as a suboptimal control to the original one. We now turn to a more detailed description of the main object of our ⊂ R2 be a bounded connected open domain with a simple study. Let Ω and let ε be a small positive parameter. To connected smooth boundary ∂ Ω, define a cylindrically perforated flow domain Ωε in R3 , we introduce the sets: Y = [−1/2, 1/2)2 ; Q is a compact subset of Y such that 0 ∈ ∂Q,


547

548

15 Suboptimal Control of Incompressible Flow

Θε = {k = (k1 , k2 ) ∈ N2 : ε(rε Q + k) ⊂⊂ Ω}, Tεk = ε(rε Q + k), k ∈ Θε ,

Tε =

Tεk =

k∈Θε

ε = Ω \ Tε , Tε = Tε × [0, ], Ω × {0}, Γ 2 = Ω × {}, Γ1 = Ω

Jε

(15.1) Tjk ,

(15.2)

j=1

× (0, ), Ω=Ω × [0, ], Γ 3 = ∂Ω

(15.3) (15.4)

where the parameter rε denotes the cross-size of the thin cylinders Tεk = Tεk × [0, ] and satisfies the conditions 0 < rε ≤ 1 and limε→0 rε = 0. Then the domain Ωε is defined by removing the cylinders Tεk from Ω, that is (see Fig. 15.1), k Ωε = Ω \ Tε = Ω \ ε(rε Q + k) × [0, ] . (15.5) k∈Θε

k∈Θε

We use the following decomposition for the boundary of this domain: ∂Ωε = Γε1 ∪ Γε2 ∪ Γ 3 ∪ ∂Tε , where ε × {0}, Γε1 = Ω

ε × {}. Γε2 = Ω

(15.6)

Fig. 15.1. An example of a cylindrically perforated domain

To characterize the different possible cross-sizes of thin cylinders which can be considered (“critical”, smaller, and larger cylinders), we define a ratio σε between the current size of the cross-sections and the critical one: σε = ε2 (log 1/rε ) .

(15.7)


549

If the limit of σε , as ε tends to 0, is positive and finite, then the cross-size of the cylinders is called critical. If limε→0 σε = +∞, the cross-size of cylinders is smaller, and if limε→0 σε = 0, the cross-size is larger (see Cioranescu and Murat [65] and Allaire [5]). The OCP we consider is to minimize the vorticity of viscous, incompressible flow by choosing an appropriate boundary velocity on the “vertical” sides of thin cylinders Tεk = {(x1 , x2 , x3 ) : (x1 , x2 ) ∈ ε(rε Q + k), 0 ≤ x3 ≤ } ,

∀ k ∈ Θε .

Precisely, we study the following OCP: Find a boundary velocity field αε = αk1 , αk2 , . . . , αkJε and a corresponding velocity–pressure pair (yε , pε ) such that the functional

ε βε Jε (αε , yε ) = λ |∇ yε | dx + rε j=1 Ωε

J

2

k ∂Tε j

αkj 2 dH2

(15.8)

is minimized subject to the steady-state Navier–Stokes equations −ν yε + (yε · ∇)yε + ∇pε = f ε div yε = 0 in Ωε ,

in Ωε ,

(15.9) (15.10)

yε |Γε1 = y1ε , yε |Γε2 = y2ε , yε |Γ 3 = 0,

(15.11)

yε |∂T kj = αkj ,

(15.12)

ε

∀ j = 1, . . . , Jε .

Here, ν denotes the constant viscosity; yε and pε denote the velocity field and the pressure field, respectively; f ε is a prescribed forcing term; y1ε and y2ε are given boundary velocities on the lower and upper boundaries Γε1 and Γε2 , respectively, and αε is the boundary velocity – the control field. Because of the divergence-free condition on yε , the vector-valued functions y1ε , y2ε , and αkj must necessarily satisfy the relations ⎫ ⎪ y1ε · n dH2 = 0, y2ε · n dH2 = 0, ⎪ ⎬ 1 Γε2 Γε (15.13) ⎪ 2 ⎪ α · n dH = 0, ∀ j = 1, . . . , J . ⎭ k ε j k ∂Tε j

Throughout this chapter, we assume that there are functions y∗ ε ∈ H2 (Ω) ∩ H1sol (Ωε ), f ∈ L2 (Ω), and y∗ ∈ H2 (Ω) such that y∗ε |Γ 1 = y1ε , y∗ε |Γ 2 = y2ε , y∗ε |Γ 3 = 0, and y∗ε y∗ in H2 (Ω), ε ε 2

f ε f in L (Ω).

(15.14) (15.15)

550


We also say that a boundary velocity field αε = αk1 , . . . , αkJε is admissible if there exists a function u ∈ H1sol (Ωε ) ∩ H2 (Ω) (the so-called prototype of the boundary control αε ) such that uH2 (Ω) ≤ γ (for a given value γ > 0) and u|Γ 1 = y1ε , u|Γ 2 = y2ε , u|Γ 3 = 0, u| ε ε

k

∂Tε j

= αkj ,

∀j = 1, . . . , Jε . (15.16)

The constants λ and β in the functional (15.8) are two positive parameters that adjust the relative weights of the two terms in the functional. Such a choice of the cost functional is motivated by the fact that irrotational flows have no local flow recirculations. On the other hand, for both physical and mathematical reasons, the size of the control should be constrained. So, following the representation 2

|curl yε | = curl yε : curl yε

2 2 = |∇yε | + t (∇yε ) − ∇yε : t (∇yε ) − t (∇yε ) : ∇yε ,

(15.17)

we hope that minimizing the functional (15.8) will lead to a reduction in flow recirculations. Our main result is: The boundary velocity field

= Λε (u0 ) ∂T αεsub = αksub , αksub , . . . , αksub 1 2 Jε ε

can be taken as the suboptimal control for the problem (15.8)–(15.12), where Λε : H1sol (Ω) → H1sol (Ωε ) is some linear bounded operator (see (15.103) and (15.104)) and u0 is a solution to one of the following problems: 1. In the case when C0 = limε→0 ε2 (log 1/rε ) = +∞, the functional 2 |u(x)| dx (15.18) Ω

is minimized subject to the constraints ⎧

⎫

uH2 (Ω) ≤ γ, ∇ · u = 0 on Ω, ⎪ ⎪ ⎨

⎬

2 ∗ ∗ u(x) ∈ H (Ω) u|Γ 1 = y |Γ 1 , u|Γ 2 = y |Γ 2 , u|Γ 3 = 0, . (15.19) ⎪

⎪ ⎩ ⎭

y∗ · n = 0 on Γ 1 ∪ Γ 2 2. When 0 < C0 < +∞, the cost functional J0 (u, y) = λ |∇y|2 dx Ω 2πλ 2 2 |y − u| dx + β|∂Q|H |u| dx + C0 Ω Ω

(15.20)

is minimized subject to the boundary value problem for a Brinkman-type law

15.1 Preliminaries and notation

551

2πν (y − u) + (y · ∇)y + ∇p = f in Ω, C0 div y = 0 in Ω, y|∂Ω = u|∂Ω , u|Γ 1 = y∗ |Γ 1 , u|Γ 2 = y∗ |Γ 2 , u|Γ 3 = 0,

(15.22) (15.23)

p ∈ L20 (Ω), u ∈ H2 (Ω), y − u ∈ H10,sol (Ω), uH2 (Ω) ≤ γ.

(15.24)

−ν y +

(15.21)

15.1 Preliminaries and notation is of class C ∞ , so Throughout this chapter, we suppose that the boundary ∂ Ω Ω = Ω ×(0, l) is a measurable set in the sense of Jordan; the small parameter ε varies in a strictly decreasing sequence of positive numbers which converges to 0; Q is a compact subset of Y with Lipschitz boundary ∂Q, int Q is a strongly connected set, Q ⊂ {x = (x1 , . . . , xn ) ∈ Rn : x1 ≥ 0}, and its boundary ∂Q contains the origin; A = B(0, r0 ) is an open ball centered at the origin with a radius r0 < 1/2, so that A ⊂⊂ Y and Q ⊂⊂ A (see Fig. 15.1); C or Ci (where i is any subscript) denotes a constant independent of ε. For any subset E ⊂ Rn , we denote by |E| its n-dimensional Lebesgue measure Ln (E), whereas |∂E|H denotes the (n − 1)-dimensional Hausdorff measure of the manifold ∂E on Rn . For the definition of fractional ordered Sobolev spaces H l (∂Ω) (l noninteger), see [2]. Let L20 (Ω) =

q ∈ L2 (Ω) :

q dx = 0 Ω

with the norm p 2L2 (Ω) = 0

2 p(x) dx dx.

p(x) −

Ω

Ω

Let Mb (Ω) be the space of bounded Borel measures on Ω. Let M+ 0 (Ω) be the cone of all non-negative Borel measures μ on Ω such that μ(B) = 0 for every set B ⊆ Ω with cap(B, Ω) = 0 and μ(B) = inf{μ(U ) : U quasi-open, B ⊆ U } for every Borel set B ⊆ Ω. The space D (Ω) of distributions on Ω is the dual of the space C0∞ (Ω). For m ≥ 0, we introduce the subspaces H0m (Ω) of the Sobolev spaces H m (Ω) as the closure of C0∞ (Ω) in H m (Ω) and the dual spaces H −m (Ω) = (H0m (Ω))∗ . The duality pairing between a Sobolev space H s (Ω) (s > 0) and its dual space is denoted by ·, ·H −s (Ω);H s (Ω) . The trace spaces H l (∂Ω) are the restriction to the boundary of H l+1/2 (Ω) (see [237]). The vector-valued counterparts of these spaces are denoted by bold∞ face symbols (e.g., Lr (Ω), Hm (Ω), Hl (∂Ω), Hm 0 (Ω), and C0 (Ω)). If u is a N N vector-valued function from R to R , then we have the following: 1. The gradient of u is an N × N tensor: ∇u = (∂ui /∂xj )1≤i,j≤N . 2. The inner product of two N ×N tensors A = (aij ) and B = (bij ) is denoted by A : B = tr (t AB) = 1≤i,j≤N aij bij .

552


We will also use the spaces of solenoidal vector fields 3 ∂yi ∞ ∞ C0,sol (Ω) = y ∈ C0 (Ω) : div y = i=1 = 0 in Ω , ∂xi 2 m m y · n dH = 0 , Hsol (Ω) = y ∈ H (Ω) : ∇ · y = 0, ∂Ω

∞ m Hm 0,sol (Ω) = the closure of C0,sol (Ω) in H (Ω)-norm, y · n dH2 is the H −1/2 (∂Ω); H 1/2 (∂Ω) duality where when m = 0, ∂Ω

pairing between the function (y · n) ∈ H −1/2 (∂Ω) and the constant scalar m function 1 ∈ H 1/2 (∂Ω). The norm on Hm sol (Ω) and H0,sol (Ω) is chosen to be m that of H (Ω). We use the standard definition of the divergence operator: y · ∇ϕ dx, ∀ ϕ ∈ H01 (Ω). (15.25) div y, ϕH −1 (Ω);H 1 (Ω) = − 0

Ω

Let H(div, Ω) = y ∈ L (Ω) : div y ∈ L2 (Ω) . We will need the following lemma on integration by parts for functions in the space H(div, Ω) (for the proof, see [237]). 2

Lemma 15.1. Let w ∈ H(div, Ω). Then (w · n)|∂Ω ∈ H −1/2 (∂Ω) and w · n, vH −1/2 (∂Ω);H 1/2 (∂Ω) = v div w dx + w · ∇v dx, Ω

∀ v ∈ H 1 (Ω). (15.26)

Ω

To end this section, we define the standard bilinear and trilinear forms associated with the Navier–Stokes equations: (∇y) : (∇v) dx, ∀ y, v ∈ H1 (Ωε ), aε (y, v) = Ωε q div y dx, ∀ y ∈ H1 (Ωε ), ∀ q ∈ L2 (Ωε ), bε (y, q) = − Ωε cε (y, v, w) = (y · ∇)v · w dx, y, v, w ∈ H1 (Ωε ). Ωε

15.2 Admissible controls and regularity of solutions to the boundary value problem for Navier–Stokes equations We devote this section to the study of the boundary value problem (15.9)– (15.12). For this, we give the definition of a solution for the Navier–Stokes equations with inhomogeneous Dirichlet boundary conditions. Throughout, we assume that f ε ∈ L2 (Ω) and supε>0 f ε L2 (Ω) < +∞. We also assume that there are functions y∗ ε ∈ H2 (Ω) ∩ H1sol (Ωε ) and y∗ ∈ H2 (Ω) satisfying (15.14).

15.2 Admissible controls and regularity solutions

553

Definition 15.2. A pair (yε , pε ) ∈ H1 (Ωε ) × L20 (Ωε ) is said to be a solution of the Navier–Stokes equations (15.9)–(15.12) if and only if f ε · v dx, ∀ v ∈ H10 (Ωε ), (15.27) νaε (yε , v) + cε (yε , yε , v) + bε (v, pε ) = Ωε

bε (yε , q) = 0,

∀ q ∈ L20 (Ωε ),

(15.28)

and yε |Γε1 = y1ε , yε |Γε2 = y2ε , yε |Γε3 = 0, yε |∂T kj = αkj , ∀j = 1, . . . , Jε . ε (15.29) A proof of the existence of a solution in the sense of Definition 15.2 can be found in [237]. field αε = Let γ > 0 be a given value. We say that a boundary velocity αk1 , . . . , αkJε is admissible if there exists a function u ∈ H1sol (Ωε ) ∩ H2 (Ω) (the so-called prototype of the boundary control αε ) such that uH2 (Ω) ≤ γ and conditions (15.16) are satisfied. Such a choice of admissible controls is motivated by the fact that the trace space H1/2 (∂Ωε ) is the restriction to the boundary of H1 (Ωε ). Hence, in view of the initial supposition, for a fixed αε there exists a function u ∈ H1 (Ω) satisfying conditions (15.16). Then, due to (15.13) and the Stokes formula (15.26), we have Ωε div u dx = 0. We denote by Uε the set of all admissible controls for a fixed ε

⎫ ⎧

αkj = u| kj , ∀j = 1, . . . , Jε , ⎪ ⎪ ∂Tε

⎪ ⎪ ⎬ ⎨ u ∈ H1 (Ωε ) ∩ H2 (Ω), sol

αε = αk1 , αk2 , . . . , αkJε

. (15.30) ⎪ ⎪

uH2 (Ω) ≤ γ, ⎪ ⎪ ⎭ ⎩

u| 1 = y1 , u| 2 = y2 , u| 3 = 0 ε ε Γ Γ Γ ε

ε

ε

Definition 15.3. We say that a triplet (αε , yε , pε ) is admissible to the OCP (15.8)–(15.12) if αε ∈ Uε and the pair (yε , pε ) is a corresponding solution of the variational problem (15.27)–(15.29). At the end of this section, we cite the following results (their proof can be found in Duvaut and Lions [102], Fursikov [111], and Temam [237]) that will be useful in the sequel. Theorem 15.4. Let αε be an admissible control (αε ∈ Uε ) and let uε ∈ H1sol (Ωε )∩H2 (Ω) be its prototype. Then there exists a corresponding velocity– pressure pair (yε , pε ) ∈ H1sol (Ωε ) × L20 (Ωε ) satisfying the boundary value problem (15.9)–(15.12) in the following variational sense: yε − uε ∈ H10,sol (Ωε ), νaε (yε , v) + cε (yε , yε , v) = f ε · v dx, ∀ v ∈ H10,sol (Ωε ),

(15.31) (15.32)

Ωε

∇pε = ν yε − (yε · ∇)yε + f ε

in D (Ωε ).

(15.33)

554


Proposition 15.5. Let uε ∈ Uε be a prototype of some admissible control. Assume yε ∈ H1sol (Ωε ) satisfies conditions (15.31)–(15.32). Then there exists a continuous positive function B : R × R × R → R (independent of ε) such that the following estimate holds true: (15.34) yε H1 (Ωε ) ≤ B f ε L2 (Ω) , uε H1 (Ω) , uε 2H1 (Ω) .

15.3 On solvability of the optimal boundary control problem The optimal Dirichlet control problem we consider can be precisely stated as follows: Find a triplet (α0ε , y0ε , p0ε ) ∈ Uε × H1sol (Ωε ) × L20 (Ωε ) such that the cost functional (15.8) is minimized subject to the relations (15.27)–(15.29). We denote this problem by (Pε ). We also define the set of admissible solutions for (Pε ) by ⎫ ⎧

αε = αk1 , αk2 , . . . , αkJ ∈ Uε , ⎪ ⎪ ε ⎪

⎪ ⎪ ⎪ ⎪

⎪ 1 ⎬ ⎨

yε − uε ∈ H0,sol (Ωε ),

Ξε = (αε , yε , pε )

. (15.35) 1 2 ⎪ ⎪ ⎪

(yε , pε ) ∈ Hsol (Ωε ) × L0 (Ωε ), ⎪ ⎪ ⎪ ⎪

⎪ ⎩

(αε , yε , pε ) satisfies (15.27)–(15.29) ⎭ We are now in a position to prove the existence of a solution to (Pε ). Theorem 15.6. The OCP (Pε ) has a solution if and only if this problem is regular, that is, Ξε = ∅ for every fixed ε > 0. Proof. It is clear that we need to prove only sufficient conditions of this statement. Let us fix ε > 0. Since Uε = ∅, it obviously follows from Theorem 15.4 that the set Ξε is nonempty as well. Hence, we may choose a minimizing sequence {(αε,m , yε,m , pε,m ) ∈ Ξε } such that lim Jε (αε,m , yε,m ) =

m→∞

inf

(αε ,yε ,pε )∈Ξε

Jε (αε , yε ).

The boundedness of {Jε (αε,m , yε,m )} and implicit control constraints imply 2 (Ω) 2 the boundedness of control prototypes u and ε,m H ∇ yε,m L (Ωε ) . Then using (15.34), we see that the set yε,m H1 (Ωε ) is also bounded independent of m. Hence, we may extract subsequences (still denoted by uε,m and yε,m , respectively) such that uε,m uε in H2 (Ω) and yε,m yε in H1 (Ω) for some (uε , yε ) ∈ H2 (Ω)×H1 (Ω) as m → ∞. Since uε,m ∈ H1sol (Ωε ) for every m ∈ N, uε,m H2 (Ω) ≤ γ, and uε,m → uε in H1 (Ω), it follows that uε ∈ H2 (Ω) ∩ H1sol (Ωε ), uε H2 (Ω) ≤ γ, that is, uε is an admissible control prototype (uε ∈ Uε ). On the other hand, since the trace of H2 (Ω) equals H3/2 (∂Tε ) and the space H3/2 (∂Tε ) is compactly imbedded into C(∂Tε ), we have that

15.4 Reformulation of the problem (Pε )

555

uε,m uε in H2 (Ω) implies uε,m |∂T kj ≡ αkj ,m → αkj ≡ uε |∂T kj ε

in C(Ω)

ε

as m → ∞ for all j = 1, . . . , Jε . Moreover, due to the imbedding results for Sobolev spaces, we also get the strong convergence yε,m → yε in L4 (Ωε ) as m → ∞. Then, using standard techniques in proving the existence of a solution to the steady-state Navier– Stokes equations, we may pass to the limit in the relation yε,m − uε,m ∈ H10,sol (Ωε ), νaε (yε,m , v) + cε (yε,m , yε,m , v) = f ε · v dx,

∀ v ∈ H10,sol (Ωε ),

Ωε

as m → ∞ to conclude that yε − uε ∈ H10,sol (Ωε ) and yε satisfies (15.32). As a result, applying Theorem 15.4, we get that yε ∈ H1sol (Ωε ) and there exists a corresponding pressure pε ∈ L20 (Ωε ) such that relations (15.27)–(15.28) hold true, that is, the triplet ( uε |∂Tε , yε , pε ) is admissible to the problem (Pε ). Finally, using the compact imbedding results and the sequential weak lower semicontinuity of the cost functional Jε (·, ·) with respect to the product of the weak topologies of H2 (Ω) × H1 (Ωε ), we obtain Jε ( uε |∂Tε , yε ) ≤ lim inf Jε (αε,m , yε,m ) = m→∞

inf (αε ,yε ,pε )∈Ξε

Jε (αε , yε ).

Hence, ( uε |∂Tε , yε ) is an optimal triplet for the problem (Pε ).

Since the boundary value problem (15.9)–(15.12) may have a nonunique solution for a fixed boundary control, in what follows we define the binary ε , pε ) if ε, y relation L; Ξε on each of the sets Ξε by the rule (αε , yε , pε ) L (α ε . It is easily seen that L; Ξε is an equivalence relation. and only if αε = α So, hereinafter we will not distinguish the triplets belonging to the same class of equivalence.

15.4 Reformulation of the problem (Pε) We begin this section with the description of the geometry of the perforated domain Ωε . We describe the class of admissible solutions to the OCP (15.8)– (15.12) in terms of singular periodic Borel measures on R3 , using the approach of Zhikov, and Bouchitté and Fragala (see [26, 256]). Let us denote by Qr the homothetic contraction of the set Q with a factor of r −1 . In what follows, it is assumed that 0 < r ≤ 1. Let η0r be the normalized periodic Borel measure on R2 with the periodicity cell Y such that η0r is concentrated and uniformly distributed on the set ∂Qr , and is proportional to the 1D Hausdorff measure. It is clear that in this case, η0r (Y \ ∂Qr ) = 0.

556


Now, we consider the measure dη r = dη0r × dx3 on Y = [−1/2, 1/2)2 × [0, 1). It is easy to see that this measure concentrated on the set ∂Qr × [0, 1), and for any smooth function g, we have 1 −1 g dη r = g dx3 dη0r = H2 (∂Qr × [0, 1)) g dH2 . 0

Y

e Y

∂Qr ×[0,1)

However, as follows from the properties of the Hausdorff measure, we have H2 (∂Qr × [0, 1)) = H1 (∂Qr ) = rH1 (∂Q) (see [106]). Using the notation |∂Q|H = H1 (∂Q), the previous relation can be rewritten in the form 1 −1 r r g dη = r g dx3 dη0 = |∂Q|H g dH2 . (15.36) r Y

0

e Y

∂Q×(0,1)

Thus, |∂Q|H is the 1D Hausdorff measure of the set ∂Q. We introduce the scaling measure ηεr by setting ηεr (B) = ε3 η r (ε−1 B) for every Borel set B ⊂ R3 . The parameters r and ε are related by the rule r(ε) = rε , where 0 < rε ≤ 1 and limε→0 rε = 0. Then dηεr = ε3 dη r = ε3 . εY

Y

ηεr

weakly converges to the Lebesgue measure: It means that the measure dηεr dx, that is, for every ϕ ∈ C0∞ (R3 ), we have lim ϕ dηεr = ϕ dx. (15.37) ε→0

R3

R3

r By analogy, it can be shown that the scaling measure η0,ε on R2 (by r 2 r −1 2 definition, η0,ε (B) = ε η0 (ε B) for every Borel set B ⊂ R ) weakly converges to the 2D Lebesgue measure.

Remark 15.7. It is easy to see that the scaling measure ηεr belongs to the + 1 class M+ 0 (Ω). Note that if η ∈ M0 (Ω), then the functions of H (Ω) are defined η-almost everywhere and are η-measurable on Ω. Hence, the space H1 (Ω) ∩ L2 (Ω, dηεr ) is well defined (see Sect. 2.6). We note that, in view of relation (15.36), the term ε βε rε j=1

J

k ∂Tε j

can be rewritten in the equivalent form

αk 2 dH2 j

15.4 Reformulation of the problem (Pε ) ε βε rε j=1

J

k ∂Tε j

557

αk 2 dH2 j = βε |∂Q|H 2

Jε 0

j=1

= βε3 |∂Q|H

1

2

e +kj ) ε(Y

Jε j=1

r(ε)

|uε | dη0

(x/ε) dx3

2

|uε | dη r(ε) (x/ε) ε(Y +kj )

2

|uε | dηεr ,

= β|∂Q|H

(15.38)

Ω

where uε is a prototype of the control function αε = αk1 , αk2 , . . . , αkJε . As a result, the original cost functional (15.8) takes the form 2 2 Jε (αε , yε ) = λ χε |∇ y ˘ε | dx + β|∂Q|H |uε | dηεr(ε) . (15.39) Ω

Ω

˘ε ∈ Here, χε is the characteristic function of the perforated domain Ωε , and y H1 (Ω) is “some” extension of the function yε onto the whole Ω. Remark 15.8. Note that any admissible control αε = αk1 , αk2 , . . . , αkJε ∈ r(ε) Uε can be obviously interpreted as an element of the space L2 (Ω, dηε ). 1 2 Indeed, let uε ∈ Hsol (Ωε ) ∩ H (Ω) be a prototype of αε . Then, using the imbedding result H2 (Ω) → C(Ω), we get that uε ∈ C(Ω). Hence, uε is a ηεr (ε)-measurable function such that 2 2 |uε | dηεr ≤ uε 2C(Ω) ηεr (Ω) < +∞. (15.40) uε L2 (Ω,dηr(ε) ) = ε

r(ε)

So, uε ∈ L2 (Ω, dηε

Ω

), and we obtain the required result.

Definition 15.9. Let (αε , yε , pε ) be any admissible solution to the problem (Pε ). Then we say that a triplet (uε , y ˘ε , p˘ε ) ∈ Xε is a prototype to (αε , yε , pε ) if Xε = H1sol (Ωε ) ∩ H2 (Ω) ∩ L2 (Ω, dηεr(ε) ) × H1sol (Ωε ) ∩ H1 (Ω) × L20 (Ω), uε is a control prototype, and (˘ yε , p˘ε ) are some extensions of the functions (yε , pε ) on the whole Ω. Remark 15.10. Let us recall that the perforated domain Ωε considered here satisfies the so-called “condition of strong connectedness” (see [182]). It means that there exist a family {Pε }ε>0 of extension operators Pε : H1 (Ωε ) → H1 (Ω) and a constant C independent of ε and yε such that (Pε yε )H1 (Ω) ≤ C yε H1 (Ωε ) for every yε ∈ H1 (Ωε ).

558


So, we can assume that y ˘ε := Pε yε for some extension operator with the above properties. The main problem is to find an a priori estimate for the pressure pε , which yields the existence of the corresponding prototype p˘ε ∈ L20 (Ω) with a uniformly bounded norm in L20 (Ω). In fact, it is not obvious how to construct such an extension of the pressure pε . For that purpose, following Allaire [5], we introduce an abstract framework of hypotheses on the cylindrical holes. We denote by {ek }k=1,2,3 the canonical basis in R3 and by the extension by 0 onto the cylindrical holes. Hypotheses (H1)–(H5). Let us assume that there exist functions (wεk , qkε , μk ) and a linear map

(1 ≤ k ≤ 3)

Rε ∈ L H10 (Ω); H10 (Ωε )

such that the following hold: (H1) (H2) (H3) (H4)

wεk ∈ H1 (Ω), qkε ∈ L20 (Ω), μk ∈ W−1,∞ (Ω). ∇ · wεk = 0 on Ω and wεk = 0 on Tε . wεk ek in H1 (Ω), and qkε 0 in L20 (Ω). ∀ vε ∈ H1 (Ω) and ∀ v ∈ H1 (Ω) such that vε v in H1 (Ω), vε = 0 on the cylinders Tε , it follows that lim ∇qkε − νΔwεk , ϕ vε H−1 (Ω),H1 (Ω)

ε→0

0

= μk , ϕ vH−1 (Ω),H1 (Ω) , 0

∀ ϕ ∈ C0∞ (Ω).

(H5) If y ∈ H10 (Ωε ), then Rε (χε y) = y on Ωε ; if ∇ · y = 0 on Ω, then ∇ · (Rε y) = 0 on Ωε ; Rε yH10 (Ωε ) ≤ CyH10 (Ω) , and C does not depend on ε. Remark 15.11. In this case, following Tartar’s [236] idea, a linear continuous extension operator Pε ∈ L(L20 (Ωε ); L20 (Ω)) can be constructed as ∇ [Pε qε ] , wH −1 (Ω);H 1 (Ω) 0

= ∇qε , Rε wH −1 (Ωε );H 1 (Ωε ) , 0

∀ w ∈ H10 (Ω). (15.41)

Then it can be easily proved that the following properties hold true (see [5, Theorem 1.1.8]): (i) Pε qε = qε in L20 (Ωε ); (ii) Pε qε L20 (Ω) ≤ Cqε L20 (Ωε ) ; (iii) ∇ [Pε qε ] H−1 (Ω) ≤ C∇qε H−1 (Ωε ) , where the constant C is independent of qε and ε.


559

15.5 Convergence in the variable space X ε ε ) is the fact that for every fixed The characteristic feature of the OCP (P ε belongs to the value of ε, each of the admissible triplets (uε , yε , pε ) ∈ Ξ corresponding functional space X ε . In view of this, we will use the convergence concept in variable spaces following definitions of Sect. 6.2. We observe that, in fact, the characteristic function χε of the perforated domain Ωε is a two-parametric and ε-periodic one, that is, 1 if x ∈ Y \ [Qr × [0, 1)] , r r x r χε (x) = χε (x) = χ , χ (x) = 0 if x ∈ Qr × [0, 1) = rQ × [0, 1). ε It means that the Radon measure χε (x)dx can be viewed as a scaling measure dηεr such that ηεr (B) = ε3 η r (ε−1 B) for every Borel set B ⊂ R3 , where η r is the Y -periodic Borel measure on R3 concentrated on Y \ [Qr × [0, 1)] and proportional there to the Lebesgue measure L3 , that is, dη r = |Y \ Qr |L2 = |Y |L2 − r 2 |Q|L2 = 1 − r2 |Q|L2 . Y

Therefore, dη r |Y |L2 dx = dx as r → 0, and x lim ϕχε dx = lim ϕχr(ε) ϕ dx = ϕ dx dx = |Y | ε→0 Ω ε→0 Ω ε Ω Ω

(15.42)

for every ϕ ∈ C0∞ (Ω). Taking the definition of strong convergence in the variable space L2 (Ω, χε dx) and relation (15.42) into account, we get the following obvious result. Lemma 15.12. χε converges strongly to 1 both in L2 (Ω) and in the variable space L2 (Ω, χε dx) as ε → 0. To introduce the convergence formalism for the sequences {(αε , yε , pε ) ∈ Ξ ε }ε>0 , we begin with the following concepts. Definition 15.13. We say that a sequence of controls {αε ∈ Uε }ε>0 wa converges to a function a0 if some sequence of its prototypes " ! uε ∈ H2 (Ω) ∩ L2 (Ω, dηεr(ε) ) ε>0

converges to a0 weakly in H2 (Ω). Definition 15.14. We say that a sequence {αε ∈ Uε }ε>0 wb -converges to a function b0 ∈ L2 (Ω) if some sequence of its prototypes

560


! " uε ∈ H2 (Ω) ∩ L2 (Ω, dηεr(ε) )

ε>0

converges to b0 weakly in L2 (Ω, dηεr ), that is, lim sup uε L2 (Ω,dηεr ) < +∞ and urε b0 in L2 (Ω, dηεr ).

(15.43)

ε→0

In order to relate the wa - and wb -limits, we prove the following result. Lemma 15.15. Any sequence of admissible controls {αε ∈ Uε }ε>0 contains a subsequence for which the wa - and wb -limits coincide almost everywhere on the domain Ω. r(ε) Proof. Let uε ∈ H2 (Ω) ∩ L2 (Ω, dηε ) ε>0 be a sequence of some control prototypes. Since this sequence is bounded in H2 (Ω), we may suppose that there is an element a0 ∈ H2 (Ω) and a subsequence of {uε }ε>0 (still denoted by the same index) such that uε a0 as ε → 0. By the Sobolev imbedding theorem, we have that a0 ∈ C(Ω) and uε → u0 in C(Ω). Then 2 |uε | dηεr lim sup ε→0 Ω 2 2 ≤ 2 lim sup |uε − a0 | dηεr + 2 lim sup |a0 | dηεr ε→0 ε→0 Ω Ω 2 2 ≤ 2 lim sup uε − a0 C(Ω) ηεr Ω + 2 a0 C(Ω) ηεr Ω ε→0 = 2 a0 2C(Ω) ηεr Ω . Hence, the sequence {uε }ε>0 is bounded in L2 (Ω, dηεr ), and by the compactness criterium of the weak convergence in L2 (Ω, dηεr ), we may suppose that there is an element b0 ∈ L2 (Ω) such that uε b0 in L2 (Ω, dηεr ) (passing to a subsequence, if it is necessary). On the other hand, for any function ϕ ∈ C0∞ (Ω), we have ϕ (a0 − b0 ) dx Ω

r

ϕa0 dηε +

ϕ (a0 − uε ) dηεr

≤ ϕa0 dx − Ω

Ω

Ω

r r

ϕ b0 dx ≤ ϕa0 dx − ϕa0 dηε

+ ϕuε dηε − Ω Ω Ω

Ω

r r

|ϕ| dηε + ϕuε dηε − ϕ b0 dx

+ uε − a0 C(Ω) Ω

= I1 + I2 + I3 .

Ω

Ω

(15.44)

Owing to the weak convergence dηεr dx and to the fact that (ϕ a0 ) ∈ C0 (Ω), we obtain I1 → 0 as ε → 0. By analogy, we also have that I2 → 0 as ε → 0.


561

Then, taking into account (15.43), inequality (15.44) leads us to the following conclusion: ϕ (a0 − b0 ) dx = 0, ∀ ϕ ∈ C0∞ (Ω), Ω

that is, a0 = b0 almost everywhere in Ω. The proof is complete. As a consequence, the following statements are readily true. Lemma 15.16. Let {αε ∈ Uε }ε>0 be a sequence of admissible controls. Then the weak limits in H2 (Ω) of any weakly convergent sequences of prototypes " ! 2 2 r(ε) ∈ H (Ω) ∩ L (Ω, dη ) u(1) ε ε ε>0

"

!

and

2 2 r(ε) ) u(2) ε ∈ H (Ω) ∩ L (Ω, dηε

ε>0

are the same. Lemma 15.17. Any sequence of admissible controls {αε ∈ Uε }ε>0 is relatively compact with respect to the wa -convergence. Moreover, its wa -limit u0 belongs to the set U = u ∈ H2 (Ω) : uH2 (Ω) ≤ γ . (15.45) Using the above results, we are now able to introduce the convergence concept for the admissible triplets. As follows from Remark 15.10, for any uniformly bounded sequence of functions yε ∈ H1 (Ωε ) ε>0 there are extension operators Pε : H1 (Ωε ) → H1 (Ω) and a constant C independent of ε such that ˘ yε H1 (Ω) ≡ (Pε yε )H1 (Ω) ≤ C yε H1 (Ωε ) for every ε. Let us suppose that there are two different bounded sequences of prototypes (1) (2) {˘ y(1) y(2) ε = Pε (yε )}ε>0 and {˘ ε = Pε (yε )}ε>0

such that y ˘ε y∗1 and y ˘ε y∗2 in H1 (Ω). Then using Lemma 15.12 and passing to the limit in the integral identity (1) ˘ ε · ϕ dx = ˘ (2) χε y χε y ∀ ϕ ∈ H1 (Ω), ε · ϕ dx, (1)

(2)

Ω

Ω

as ε tends to 0, we obtain ∗ y1 · ϕ dx = y∗2 · ϕ dx, Ω

∀ ϕ ∈ H1 (Ω).

Ω

Hence, y∗1 = y∗2 . In view of this, we give the following notion (we refer to Sects. 6.1 and 6.6 for the details).

562


Definition 15.18. We say that a bounded sequence {(αε , yε , pε ) ∈ Ξ ε }ε>0 is w-convergent to a triplet (u, y, p) ∈ H2 (Ω) × H1 (Ω) × L20 (Ω) in the variable space X ε as ε tends to 0 w in symbols, (αε , yε , pε ) −→ (u, y, p) ε if some bounded sequence of its prototypes (uε , y ˘ε , p˘ε ) ∈ Ξ converges ε>0 to (u, y, p) in the following sense: w

a u in H2 (Ω); (i) uε −→ (ii) p˘ε p in L20 (Ω); (iii) y ˘ε y in H1 (Ω).

As a consequence, we have the following result. Theorem 15.19. Let {(αε , yε , pε ) ∈ Ξ ε }ε>0 be a sequence of admissible triplets for the Pε -problems. Then there exist a subsequence {(αε , yε , pε )}ε >0 and a triplet (u, y, p) ∈ H2 (Ω) × H1 (Ω) × L20 (Ω) w

such that u ∈ U, (αε , yε , pε ) −→ (u, y, p), and y − u ∈ H10,sol (Ω).

(15.46)

Proof. We set y ˘ε = uε in Ω \ Ωε . Thanks to Lemma 15.17, Proposition 15.5, and Remark 15.10, the sequence {˘ yε }ε>0 is uniformly bounded in H1 (Ω). Let ! " ε ˘ε , p˘ε ) ∈ Ξ (15.47) (uε , y ε>0

2 be a sequence of some prototypes, where p˘ε = Pε (p2ε ) ∈ L0 (Ω) (see Remark 15.11). Then the sequence of pressures p˘ ∈ L0 (Ω) ε>0 is also uniformly bounded. So, the sequence (15.47) is relatively compact with respect to the weak convergence in the space H2 (Ω) × H1 (Ω) × L20 (Ω). Let (u, y, p) ∈ H2 (Ω) × H1 (Ω) × L20 (Ω) be its weak limit. We aim to show that y − u ∈ H10,sol (Ω). Since

y ˘ε − uε ∈ H10,sol (Ωε ) ∩ H10 (Ω) and div : H10 (Ωε ) → L2 (Ωε )/R =

g ∈ L2 (Ωε ) :

g(x) dx = 0 , Ωε

it follows that yε − uε ∈ H(div, Ωε ) for every ε. Therefore,

15.6 Definition of suboptimal controls

ϕ div (yε −uε ) dx = −

0= Ωε

(yε −uε )·∇ϕ dx,

563

∀ ϕ ∈ C0∞ (R3 ), (15.48)

Ωε

∀ ε > 0 (see Lemma 15.1). However, we have ϕ χε div (˘ yε − uε ) dx = − χε (˘ yε − uε ) · ∇ϕ dx, Ω

∀ ϕ ∈ C0∞ (R3 ).

Ω

Hence, yε − uε )] = χε div (˘ yε − uε ), div [χε (˘ and we get yε − uε ) ∈ H(div, Ω). χε (˘ Then, due to Lemma 15.12, we can pass to the limit in (15.48). As a result, we have χε (˘ yε − uε ) · ∇ϕ dx = (y − u)∇ϕ dx, ∀ ϕ ∈ C0∞ (R3 ). 0 = lim ε→0

Ω

Ω

1 Thus, (y − u) ∈ H0,div (Ω), and this concludes the proof.

Corollary 15.20. The supposition of Theorem 15.19 and condition (15.14) imply the inclusions y ∈ H1sol (Ω),

u ∈ U = u ∈ H1sol (Ω) ∩ H2 (Ω) : uH2 (Ω) ≤ γ .

(15.49)

Proof. Since uε ∈ H1sol (Ωε ) ∩ H2 (Ω) for every ε, it follows that v div uε dx = 0 for every v ∈ H 1 (Ω). Ωε

On the other hand, uε u in H2 (Ω) as ε → 0. Hence, taking into account (15.25), we get χε uε · ∇v dx = u · ∇v dx 0 = lim ε→0

Ω

Ω

= div u, vH −1 (Ω);H 1 (Ω) , 0

∀ v ∈ H01 (Ω).

Since div u ∈ L2 (Ω), this yields div u = 0. As a result, we have u ∈ H1sol (Ω). The inclusion y ∈ H1sol (Ω) immediately follows from the previous one and (15.46).

15.6 Definition of suboptimal controls The main question we are going to consider in this section concerns the approximation of the optimal solutions to the original problem (Pε ) for ε small

564


enough. We focus our attention on the possibility of defining suboptimal solutions which have to guarantee the closeness of the corresponding value of the sub cost functional Jε (usub ε , yε ) to its minimum if ε is small enough. In contrast to Definition 11.30, we introduce the following concept. Definition 15.21. We say that a sequence of functions " ! sub sub αεsub = αksub , α , . . . , α k k 1 2 Jε

ε>0

is an asymptotically suboptimal control for the problem (Pε ) if sub 1/2 kj n · αksub dH2 = 0, ∀j = 1, . . . , Jε , αkj ∈ H (∂Tε ), j k

(15.50)

∂Tε j

and for every δ > 0, there is ε0 > 0 such that

sub sub

J (α , y ) − J (α , y ) inf ε ε ε

< δ,

(αε ,yε ,pε )∈ Ξε ε ε ε

∀ ε < ε0 ,

(15.51)

where yεsub = yε (αεsub ) denotes the corresponding solution of the boundary value problem (15.9)–(15.12). To construct such controls, we use the variational convergence of constrained minimization problems (see Sect. 8.4). In view of this, we study the asymptotic behavior of the problem (Pε ) as ε → 0. We represent the Pε problem for various values of ε in the form of the following sequence: $ # Jε (αε , yε ) ; ε > 0 . (15.52) inf (αε ,yε ,pε )∈Ξε

Then the definition of an appropriate limit problem to the family (15.52) can be reduced to the analysis of the limit properties of this sequence as ε → 0. To get this limit in the form of some constrained minimization problem, we introduce the following definition (for comparison, see Definition 8.24). Definition 15.22. We say that a minimization problem # $ inf J0 (u, y) (u,y,p)∈ Ξ0

(15.53)

is the variational w-limit of the sequence (15.52) if the following conditions are satisfied: (i) If a sequence {(αk , yk , pk )}k∈N w-converges to a triplet (u, y, p) as k → ∞ and there exists a subsequence {εk } of {ε} such that εk → 0 as k → ∞ and (αk , yk , pk ) ∈ Ξεk for all k, then (u, y, p) ∈ Ξ0 ;

J0 (u, y) ≤ lim inf Jεk (αk , yk ). k→∞

(15.54)

15.7 Convergence theorem

565

(ii) For every equivalence class Ξ0 /L, there exist a triplet (u, y, p) ∈ Ξ0 /L and a realizing sequence {(αε , yε , pε ) ∈ Ξε }ε>0 such that w

(αε , yε , pε ) −→ (u, y, p), and J0 (u, y) ≥ lim sup Jε (αε , yε ). (15.55) ε→0

The following theorem deals with the main property of the variational w-limit problems (for the proof, we refer to Theorem 9.26). Theorem 15.23. Assume that (15.53) is a weak variational w-limit of the sequence (15.52), and that this problem has a solution. Let (α0ε , y0ε , p0ε ) ∈ Ξε be a sequence of optimal triplets for the Pε -problems. Then there exists a triplet (u0 , y0 , p0 ) ∈ Ξ0 such that w

inf (u,y,p)∈ Ξ0

(α0ε , y0ε , p0ε ) −→ (u0 , y0 , p0 ), J0 (u, y) = J0 u0 , y0 = lim inf

ε→0 (αε ,yε ,pε )∈ Ξε

(15.56) Jε (αε , yε ).

(15.57)

Taking into account this result, we will show below that any solution of the w-limit problem (15.53) can be taken as a prototype for the construction of the suboptimal controls in the sense of Definition 15.21.

15.7 Convergence theorem The main question of this section is the study of asymptotic behavior of the boundary value problem (15.9)–(15.12) as ε tends to 0. To begin, we give some technical lemmas. We suppose that hypotheses (H1)–(H5) are fulfilled. Lemma 15.24. μij ∈ Mb (Ω), ∀ i, j (1 ≤ i, j ≤ 3). Proof. We will prove that for every compact set B ⊂ Ω of zero capacity, we have μij (K) = 0. By standard properties of Radon measures, it follows that μij (D) = 0 for any Borel set D ⊂ Ω of zero capacity. Let K be a compact subset of Ω. Then for any k ∈ N, there exists ϕk ∈ C0∞ (Ω) such that ϕk ≥ χK , 0 ≤ ϕk ≤ 1, ϕk H01 (Ω) ≤ 1/k. In view of hypothesis (H4), we have lim ∇qiε − νΔwεi , vε H−1 (Ω),H1 (Ω) = 0,

ε→0

(15.58)

0

∀ vε ∈ H10 (Ω) : vε 0 in H1 (Ω) and vε = 0 on Tε . Applying this to the sequence vε,k = ϕk wεj , we obtain that for any δ > 0,

∇qiε , vε,k H−1 (Ω),H1 (Ω) + ν −Δwεi , vε,k H−1 (Ω),H1 (Ω) ≤ δ, 0

0

∀ ε < ε0 (δ), k > k0 (δ). Since wεj is divergence-free (see hypothesis (H2)), it follows that

566


∇qiε , vε,k H−1 (Ω),H1 (Ω) 0

=−

qiε wεj · ∇ϕk dx. Ω

Then taking into account the obvious estimates

C

∇qiε , vε,k H−1 (Ω),H10 (Ω) ≤ qiε L20 (Ω) wεj H1 (div,Ω) ϕk H01 (Ω) ≤ , k ε ε ε ε ε −Δwi , vε,k H−1 (Ω),H1 (Ω) = ϕk ∇wi : ∇wj dx + ∇wi : wj ∇ϕk dx, 0

and

Ω

Ω

∇wεi : wεj ∇ϕk dx ≤ wεi H1 (Ω) wεj H1 (Ω) ∇ϕk L2 (Ω) ≤ C ,

k Ω

we obtain

ϕk ∇wε : ∇wε dx ≤ 2δ, i j

∀ ε < ε1 (δ), k > k1 (δ).

(15.59)

Ω

Due to hypothesis (H3), each of the sequences (1 ≤ i, j ≤ 3) ∇wεi : ∇wεj is bounded in L1 (Ω). So, extracting, if necessary, a subsequence, we can suppose the existence of a symmetric matrix M = {μij }1≤i,j≤3 of bounded Radon measures μij such that ∇wεi : ∇wεj converges to μij in the weak-* sense of the space Mb (Ω). Then, passing to the limit in inequality (15.59) as ε → 0, we get ϕk dμij ≤ 2δ, ∀ k > k1 (δ). Ω

Since ϕk ≥ χK , this yields μij (K) ≤ 2δ, ∀ δ > 0, and we obtain the required result. Lemma 15.25. For any ϕ ∈ H01 (Ω) ∩ L∞ (Ω) and each i, j (1 ≤ i, j ≤ 3), we have ϕ ∇wεi : ∇wεj dx = ϕ dμij . (15.60) lim ε→0

Ω

Ω

Proof. Let ϕ ∈ H01 (Ω) ∩ L∞ (Ω). By analogy with Casado-Diaz [55], we consider a sequence of functions {ϕk ∈ C0∞ (Ω)} satisfying the conditions sup ϕk L∞ (Ω) < +∞,

k∈ N

ϕk → ϕ in H01 (Ω) and μij -a.e. on Ω

(the existence of such a sequence has been proved in [267]). From Lebesgue’s dominated convergence theorem, we have ϕ ∈ L∞ (Ω, dμij ) and ϕk converges strongly to ϕ in L1 (Ω, dμij ). Then


567

ε ε

ϕ ∇wi : ∇wj dx − ϕ dμij

Ω Ω ≤ ∇wεi : ∇wεj |ϕ − ϕk | dx

Ω

ϕk dμij

+

ϕk ∇wεi : ∇wεj dx − Ω Ω |ϕk − ϕ| dμij . + Ω

Passing to the limit in this relation for a fixed k and taking into account the weak-* convergence of ∇wεi : ∇wεj to μij in Mb (Ω), we obtain

ε ε

lim sup ϕ ∇wi : ∇wj dx − ϕ dμij

ε→0 Ω Ω

ε ε

≤ lim sup ∇wi : ∇wj |ϕ − ϕk | dx + lim sup |ϕk − ϕ| dμij . ε→0

ε→0

Ω

Ω

Passing, now, to the limit as k → ∞, we find

ε ε

ϕ dμij

lim sup ϕ ∇wi : ∇wj dx − ε→0 Ω Ω

∇wεi : ∇wεj |ϕ − ϕk | dx. ≤ lim lim sup k→∞

ε→0

Ω

Let us show that the limit on the right-hand side is 0. We apply property (15.58) to the sequence vε,k = ±|ϕk − ϕ|wεj . One gets |ϕk − ϕ| ∇wεi : ∇wεj dx lim lim sup ± k→∞ ε→0 Ω ± ∇wεi : wεj ∇|ϕk − ϕ| dx = 0. Ω

Since ∇wεi : wεj ∇|ϕk − ϕ| dx ≤ 2wεi H1 (Ω) wεj H1 (Ω) ∇|ϕk − ϕ|L2 (Ω) Ω

and ϕk tends to ϕ strongly in H01 (Ω), it immediately follows that

ε ε

lim lim sup ∇wi : wj ∇|ϕk − ϕ| dx

= 0. k→∞

ε→0

Thus,

Ω

lim lim sup

k→∞

ε→0

∇wε : ∇wε |ϕ − ϕk | dx = 0, i j

Ω


568


Lemma 15.26. If a sequence vε ∈ H10 (Ω) and v ∈ H10 (Ω) are such that vε = 0 on Tε and vε v in H10 (Ω), then v ∈ L1 (Ω, dμi ) for each i (1 ≤ i ≤ 3). Proof. For every k > 0, we define the truncation function Tk : R → R by Tk (s) = k if s ≥ k, Tk (s) = s if −k ≤ s ≤ k, and Tk (s) = −k if s ≤ −k. We denote the vector-valued counterpart of this function with the bold symbol Tk : R3 → R3 . Let vε ∈ H10 (Ω) and v ∈ H10 (Ω) be the above given functions. We wish to show the fulfillment of the relation

lim lim sup

∇qiε − νΔwεi , vε H−1 (Ω),H1 (Ω) − Tk (v) dμi

= 0, (15.61) k→∞

0

ε→0

Ω

which implies that Ω Tk (v) dμi is bounded independently of k, and hence, by the Beppo Levi’s monotone convergence theorem, v ∈ L1 (Ω, dμi ). For every ε > 0 and k ∈ R, we define the functions vε,k by the rule vε = Tk (vε ) + vε,k and note that

Tk (v) dμi

∇qiε − νΔwεi , vε H−1 (Ω),H10 (Ω) − Ω

≤ ∇qiε − νΔwεi , vε,k H−1 (Ω),H1 (Ω)

0

+

∇qiε − νΔwεi , Tk (vε )H−1 (Ω),H1 (Ω) − Tk (v) dμi

0 Ω

= I1 + I 2 .

(15.62)

Since I2 tends to 0 as ε → 0 for a fixed k by hypothesis (H4) and Lemma 15.24, and I1 tends to zero when k → ∞ and ε → 0 by property (15.58), this concludes the proof. Our next step concerns the structural identification of the matrix M = {μij }, where μij is the weak-∗ limit of ∇wεi : ∇wεj in the the space Mb (Ω). (15.63) Lemma 15.27. Assume the functions ε t ε ε , wj,2 , wj,3 ∈ H1 (Ω) wεj = wj,1

(1 ≤ j ≤ 3)

are such that ε wj,k ∈ H01 (Ω) if j = k.

(15.64)

Then M = diag(μ11 , μ22 , μ33 ). Proof. According to hypothesis (H3) and Rellich–Kondrachov’s compactness ε t ε ε theorem, we conclude that wεj = wj,1 , wj,2 , wj,3 converges to ej strongly 2 in L (Ω). In addition, due to the initial assumptions (15.64), we also have


569

ε ε wj,k

0 in H01 (Ω) and wj,k → 0 in L2 (Ω), ∀ j = k. Then, by Poincaré inequality on Ω, we can deduce ε ε L2 (Ω) ≤ C lim sup wj,k L2 (Ω) = 0, lim sup ∇wj,k ε→0

ε→0

∀ j = k.

(15.65)

Let (i, j) be a pair of indices such that i = j and 1 ≤ i, j ≤ 3. Extracting, if necessary, a subsequence, we can assume the existence of a bounded Radon measure μij such that ∇wεi : ∇wεj converges to μij in the weak-* sense of Mb (Ω). It means that lim ∇wεi : ∇wεj ϕ dx = ϕ dμij , ∀ ϕ ∈ C0 (Ω). ε→0

Observing that

Ω

Ω

∇wεi

:

∇wεj

=

3

ε k=1 ∇wi,k

ε · ∇wj,k , we have the estimate

3 ε ε ∇wεi : ∇wεj ϕ dx ≤ ϕC0 (Ω) ∇wi,k L2 (Ω) ∇wj,k L2 (Ω) . (15.66)

Ω

k=1 ε L2 (Ω) ∇wi,k

ε and ∇wj,k L2 (Ω) contains at least one Since each of the terms multiplier with noncoinciding indices, we can apply property (15.65). As a result, passing to the limit in (15.66) as ε → 0, we obtain the required result: μij = 0.

To check hypotheses (H1)–(H5) and obtain a precise description for the into squares εY with edges ε and denote measures μij , we partition the set Ω these squares with εYj . The corresponding cylindrical cells εYj × (0, ) are denoted by Zjε . Following the ideas of Cioranescu and Murat [65] and Allaire [5], we introduce the functions wεk ∈ H1 (Zjε ) and qkε ∈ L2 (Zjε ) (k = 1, 2, 3) with Z ε qk dx = 0 as follows: j

(i) For each cell Zjε that meets the boundary Γ3 , {wεk = ek ,

qkε = 0 }

on

Zjε ∩ Ω.

(15.67)

(ii) For each cell Zjε entirely included in Ω (precisely, for Zkε where k ∈ Θε ), ε wk = ek on Zkε \ [ε (A + k) × (0, )] , ε q = 0 k −νΔwεk + ∇qkε = 0 on ε (A \ Qrε + k) × (0, ), (15.68) ε = 0 ∇ · w k ε wk = 0 on ε (Qrε + k) × (0, ). qkε = 0 It is now clear that the functions wεk are independent of x3 , that is, wεk (x1 , x2 , x3 ) = wεk (x1 , x2 ),

1 ≤ k ≤ 3.

Following Allaire [5] closely, a quite similar result can be proved.

(15.69)

570


Proposition 15.28. Assume that the size of thin cylinders Tεk satisfies the condition (15.70) lim σε = lim ε2 (log 1/rε ) > 0 ε→0

ε→0

(wk , qkε )

and the functions (k = 1, 2, 3) are defined as in (15.67) and (15.68). Then there exists a symmetric positive matrix M = (μ1 , μ2 , μ3 ) ∈ L(R3 ; R3 ) such that hypotheses (H1)–(H3) are satisfied. Moreover, in this case there exists a linear map Rε satisfying (H5) such that the extension Pε of the pressure pε turns out to be equal to Pε = pε on Ωε and 1 Pε = pε dx on each cylinder Tεk . |Aε \ Qεrε | (Aε \Qεrε +εk)×(0, ) Now, it is clear that, in view of Lemma 15.27 and relations (15.67) and (15.68), the matrix M = {μij }1≤i,j≤3 , which appears in Proposition 15.28, has a diagonal structure M = diag(μ11 , μ22 , μ33 ). In fact, the measure μii ∈ M+ 0 that appeared as the weak limit of ∇wεi : ∇wεi in the space Mb (Ω) can be recovered in an explicit form. For this, we give the following result. Lemma 15.29. Let Q be a compact subset of Y with Lipschitz boundary ∂Q and let int Q be a strongly connected set, Q ⊂ {x = (x1 , . . . , xn ) ∈ Rn : x1 ≥ 0} , and its boundary ∂Q contains the origin. Let A = B(0, r0 ) be an open ball centered at the origin with a radius r0 < 1/2, so that A ⊂⊂ Y and Q ⊂⊂ A (see Fig. 15.1). Then, under condition (15.70) for a sequence wεi ∈ H1 (Ω) (1 ≤ i ≤ 3) defined by (15.67)–(15.68), we have (∇wεi : ∇wεi ) μ∗ii weakly-* in Mb (Ω), where

μ∗ii = 2π lim σε−1 , ε→0

∀ i : 1 ≤ i ≤ 3.

(15.71)

Proof. The proof follows standard techniques (see [106]) and, in some aspects, it is similar to the one given in [73]. To begin, we use the notation |∇wεi |2 = (∇wεi : ∇wεi ) and partition the cylindrical domain Ω into cubes εY with edges ε. We denote these cubes by εYj . It is clear that |εYj | = ε|εYj | = ε3 .

ε 2 ε Let us observe that |∇wεi |2 = 3k=1 ∇wi,k , where wi,i

1 in H 1 (Ω) ε

0 in H01 (Ω) for all k = i. Then and wi,k

2 ϕ |∇wεi |

ε 2 ϕ |∇wii |

dx =

εYj

εYj

dx +

k=i

2

2

ε ϕ |∇wik | dx εYj

ε ϕ |∇wii | dx + Sj (ε)

= εYj


571

for any ϕ ∈ C0 (Ω). Hence, using (15.69), we have the following obvious relation: ε ε 2 ε ε 2 ε 2 ϕ(xj )ε |∇wii | dx = ϕ(xj ) |∇wii | dx ≤ ϕ |∇wii | dx ej εY

εYj

εYj

2

ε |∇wii | dx = ϕ(yjε )ε

≤ ϕ(yjε ) εYj

2

ej εY

ε |∇wii | dx,

where xεj , yjε ∈ εYj . Combining this relation with the previous one, we obtain 2 ε 2 ε |∇wii | dx + Sj (ε) ≤ |∇wεi | ϕ dx e εYj εYj ε 2 |∇wii | dx + Sj (ε). (15.72) ≤ ϕ(yjε )ε ej εY

From the definition of the capacity and its properties, it readily follows that ε 2 |∇wii | dx = cap Qεr(ε) , Aε ej εY

= cap rε Q, A = cap Q, rε−1 A .

(15.73)

Since 0 ∈ ∂Q and using the arguments of [73] (see Lemma 3.3), we can conclude that cap Q, rε−1 A =

2π (1 + cε ) = 2πε2 σε−1 (1 + cε ) , log(1/rε ) where limε→0 cε = 0.

(15.74)

Then, summing up inequalities (15.72) for all admissible indices j and taking into account relations (15.73) and (15.74), we obtain 2πσε−1 (1 + cε ) ε3 ϕ(xεj ) + Sj (ε) j

≤

j

j 2

|∇wεi | ϕ dx

εYj

≤ 2πσε−1 (1 + cε )

ε3 ϕ(yjε ) +

j

Sj (ε).

(15.75)

j

Observing that ε 2 |∇wik |

−ϕC0 (Ω) lim

ε→0

Ω

dx ≤ lim

ε→0

j

2

ε ϕ |∇wik | dx

εYj

2

ε |∇wik | dx

≤ ϕC0 (Ω) lim

ε→0

Ω

572


and taking into account Lemma 15.27 and the construction of the Riemann sum for the integral ϕ dx, we can pass to the limit in (15.75) as ε tends Ω

to 0. As a result, we obtain 2 −1 ϕ dx ≤ lim |∇wεi | ϕ dx 2π lim σε ε→0 ε→0 Ω Ω ≤ 2π lim σε−1 ϕ dx, ε→0

Hence,

|∇wε |2 ϕ dx = 2π lim σε−1 ,

lim

ε→0

∀ ϕ ∈ C0 (Ω).

Ω

ε→0

Ω

and, therefore, Lemma 15.29 is proved. As a consequence of this result, we have the following one. Corollary 15.30. Let wεk ∈ H1 (Zjε ) and qkε ∈ L2 (Zjε ) (k = 1, 2, 3) be the functions defined by (15.67)–(15.68). Then ε ε ε −1 lim ∇qi − νΔwi , ϕwk H−1 (Ω),H1 (Ω) = 2π lim σε δik ϕ dx (15.76) ε→0

ε→0

0

Ω

for all ϕ ∈ C0∞ (Ω) and 1 ≤ k, i ≤ 3, where δik = 0 if i = k, and δii = 1. Proof. Integrating the expression ∇qiε − νΔwεi , ϕwεk H−1 (Ω),H1 (Ω) by parts 0 and using property (H2), we can reduce it to the form ∇qiε − νΔwεi , ϕwεk H−1 (Ω),H1 (Ω) = − qiε (wεk · ∇ϕ) dx 0 Ω ∇wεi : wεk ∇ϕ dx +ν Ω (∇wεi : ∇wεk ) ϕ dx. (15.77) +ν Ω

Then, combining Rellich’s theorem, the conditions wεk ek in H1 (Ω) and qkε 0 in L20 (Ω), and Lemmas 15.27–15.29, we can pass to the limit in the above equation as ε → 0. As a result, we conclude lim ∇qiε − νΔwεi , ϕwεk H−1 (Ω),H1 (Ω) 0 ε ε −1 = ν lim (∇wi : ∇wk ) ϕ dx = 2πν lim σε δik ϕ dx.

ε→0

ε→0

Ω

ε→0

Ω


573

Now, we are able to state and prove the main result of this section concerning the passage to the limit as ε → 0 in the following integral identities: (∇yε : ∇v) dx + (yε · ∇)yε · v dx − pε div v dx ν Ωε Ωε Ωε f ε · v dx, ∀ v ∈ H10 (Ωε ), = (15.78) Ωε q div yε dx = 0, ∀ q ∈ L20 (Ωε ), Ωε

yε |∂Ωε = uε |∂Ωε . The scheme of the proof is rather standard and is based on the energy method, introduced by Tartar [236] and adapted later by Allaire [5] for the Navier–Stokes equations. Theorem 15.31. Let f ε f in L2 (Ω) and let {(αε , yε , pε ) ∈ Ξ ε }ε>0 by a sequence of admissible triplets for the Pε -problems such that w

(αε , yε , pε ) −→ (u, y, p).

(15.79)

Then u ∈ U and the pair (y, p) ∈ H1 (Ω) × L20 (Ω) is a solution of the variational problem y − u ∈ H10,sol (Ω), 2πν (y − u)v dx + (y · ∇)yε · v dx ν (∇y : ∇v) dx + C0 Ω Ω Ω ∇p · v dx = f · v dx, ∀ v ∈ H10 (Ω), + Ω Ω q div y dx = 0, ∀ q ∈ L20 (Ω).

(15.80)

(15.81)

Ω

Remark 15.32. The corresponding limit boundary problem to (15.80) and (15.81) can be formally described as −ν y +

2πν (y − u) + (y · ∇)y + ∇p = f C0 div y = 0 in Ω, y|∂Ω = u|∂Ω .

in Ω,

(15.82) (15.83) (15.84)

These relations correspond to the so-called Brinkman-type law that was introduced in the late 1940s in [34] as a new set of equations, intermediate between the Darcy and Stokes equations. The Brinkman law is obtained from the Stokes equations by adding to the momentum equation a term proportional to the velocity. In our case, the term 2πν/C0 (y − u) takes that role and expresses the presence of the cylindrical holes of critical size ( 0 < C0 < +∞) and Dirichlet controls supported on their boundaries, which disappeared passing to the limit.

574


ε Proof. For a given sequence of admissible solutions, let (uε , y ˘ ε , p˘ε ) ∈ Ξ ε>0 be a sequence of their prototypes. Here, we suppose that {˘ yε } are uniformly bounded in H1 (Ω) and each p˘ε is defined as Pε (pε ) ∈ L20 (Ω) (see Remark 15.11). Due to Theorem 15.19, we may assume that the sequence ˘ε , p˘ε )}ε>0 is uniformly bounded in Xε , and hence there exists a triplet {(uε , y (u, y, p) ∈ H2 (Ω) ∩ H1sol (Ω) × H1sol (Ω) × L2 (Ω) satisfying (15.79). Let (wεk , qkε ) ∈ H1 (Ω) × L2 (Ω) ε>0 be a sequence defined by hypotheses (H1)–(H4). Let ϕ ∈ C0∞ (Ω) be a fixed function. It is clear that ϕwεk ∈ 1 (Ω) and ϕqkε ∈ L20 (Ω) for every ε > 0. Now, we consider the variaH0,sol 1 tional problem (15.78) with the test functions v = ϕwεk ∈ H0,sol (Ωε ) and ε 2 q = ϕqk ∈ L0 (Ωε ). As a result, we obtain ε ∇(yε − uε ) : ∇(ϕwk ) dx + ν ∇uε : ∇(ϕwεk ) dx ν Ωε Ωε + χε (uε · ∇)˘ yε · (ϕwεk ) dx Ω χε ((˘ yε − uε ) · ∇)(˘ yε − uε ) · (ϕwεk ) dx + Ω χε (˘ yε · ∇)uε · (ϕwεk ) dx − χε (uε · ∇)uε · (ϕwεk ) dx + Ω Ω ε pε div (ϕwk ) dx = χε f ε · (ϕwεk ) dx, (15.85) − Ωε

Ω

ϕ qkε div(yε − uε ) dx = 0.

ϕ qkε div yε dx = Ωε

(15.86)

Ωε

Expanding (15.85) and using the fact that wεk is divergence-free, we have ν ϕ∇(yε − uε ) : ∇wεk dx + ν ∇(yε − uε ) : wεk ∇ϕ dx Ωε Ωε ε ϕ(∇uε : ∇wk ) dx + ν ∇uε : wεk ∇ϕ dx +ν Ωε Ωε χε ((˘ yε − uε ) · ∇)(˘ yε − uε ) · (ϕwεk ) dx + χε (uε · ∇)˘ yε · (ϕwεk ) dx + Ω Ω ε χε (˘ yε · ∇)uε · (ϕwk ) dx − χε (uε · ∇)uε · (ϕwεk ) dx + Ω Ω − pε wεk · ∇ϕ dx = χε f ε · ϕwεk dx. (15.87) Ωε

Ω

yε − uε ) ∈ H10 (Ω), after integration of (15.86) by parts, we get Since χε (˘


575

∇qkε , ϕ χε (˘ yε − uε )H−1 (Ω);H1 (Ω) + 0

χε qkε ∇ϕ · (˘ yε − uε ) dx = 0. (15.88) Ω

Then adding two last equations and using the fact that ϕ∇(yε − uε ) : ∇wεk dx Ωε ϕ∇(χε (˘ yε − uε )) : ∇wεk dx = Ω

yε − uε )H−1 (Ω);H1 (Ω) = − Δwεk , ϕχε (˘ 0 − χε (˘ yε − uε )∇ϕ : ∇wεk dx,

(15.89)

Ω

we obtain %

& ∇qkε − νΔwεk , ϕχε (˘ yε − uε ) H−1 (Ω);H1 (Ω) + ν χε ∇uε : wεk ∇ϕ dx 0 Ω ε χε qk ∇ϕ · (˘ yε − uε ) dx − ν χε (˘ yε − uε )∇ϕ : ∇wεk dx + Ω Ω ε χε ∇(˘ yε − uε ) : wk ∇ϕ dx + ν χε ϕ(∇uε : ∇wεk ) dx +ν Ω Ω χε ((˘ yε − uε ) · ∇)(˘ yε − uε ) · (ϕwεk ) dx + Ω + χε (uε · ∇)˘ yε · (ϕwεk ) dx + χε (˘ yε · ∇)uε · (ϕwεk ) dx Ω Ω ε χε (uε · ∇)uε · (ϕwk ) dx − χε p˘ε wεk · ∇ϕ dx − Ω Ω χε f ε · ϕwεk dx. (15.90) = Ω

Now, we can pass to the limit in (15.90) as ε tends to 0. To do so, we recall the following facts: wεk ek in H1 (Ω); ∇wεj converges pointwise and weakly in L2 (Ω) to 0; χε → 0 in L2 (Ω); qkε 0 in L20 (Ω); p˘ε = Pε (pε ) p in yε − uε )} fulfills the conditions of hypothesis (H4); L20 (Ω); the sequence {χε (˘ the nonlinear term ((˘ yε − uε ) · ∇)(˘ yε − uε ) converges strongly to ((y − u) · ∇)(y − u) in H−1 (Ω); and condition (15.79) implies uε → u in H1 (Ω). As a result, we obtain & % ε yε − uε ) H−1 (Ω);H1 (Ω) ∇qk − νΔwεk ,ϕχε (˘ 0

→ ν μk , ϕ(y − u)H−1 (Ω);H1 (Ω) 0 by Lemma 15.29 2πν = ϕ ek · (y − u) dx, C0 Ω and hence, using Rellich’s theorem, from (15.90) we get

576


2πν C0

Ω

ϕ ek ·(y − u) dx ∇y : ek ∇ϕ dx + (y · ∇)y · (ϕek ) dx +ν Ω Ω p ek · ∇ϕ dx = f · ϕek dx, − Ω

Ω

∀ϕ ∈ C0∞ (Ω) and each k = 1, 2, 3.

(15.91)

Integrating the term Ω p ek · ∇ϕ dx by parts and regrouping (15.91) and (15.46), we deduce that the limit triplet (u, y, p) must satisfy the following relations: 2πν ν ∇y : ∇Φ dx + (y − u) · Φ dx + (y · ∇)y · Φ dx C Ω Ω 0 Ω ∇p · Φ dx = f · Φ dx, ∀Φ ∈ C∞ (15.92) + 0 (Ω), Ω

Ω

y − u ∈ H10,sol (Ω),

u ∈ U.

(15.93)

The proof is complete.

15.8 Identification of the limit optimal control problem In this section, we show that for the sequence of constrained minimization problems (15.52), there exists a weak variational limit with respect to the w-convergence, and it can be recovered in an explicit form. We begin with the following result. Lemma 15.33. Let {(αε , yε , pε ) ∈ Ξ ε }ε>0 be a bounded sequence of admissible solutions, assumed to be w-convergent to a triplet (u, y, p) ∈ H2 (Ω) ∩ H1sol (Ω) × H1sol (Ω) × L20 (Ω). Then

2π |∇yε | dx = |∇y| dx + lim ε→0 Ω C0 Ω ε 2

2

|y − u| dx.

2

Proof. We first observe that |∇yε |2 dx = χε |∇˘ yε − ∇uε |2 dx Ωε Ω χε (∇˘ yε : ∇uε ) dx − χε |∇uε |2 dx. +2 Ω

(15.94)

Ω

(15.95)

Ω

˘ε y in Then, taking into account the facts that ∇uε → ∇u in L2 (Ω), y H1 (Ω), and χε → 1 in L2 (Ω), we have


ε→0

χε (∇˘ yε : ∇uε ) dx −→ ∇y : ∇u dx, Ω Ω ε→0 χε |∇uε |2 dx −→ |∇u|2 dx. Ω

577

(15.96) (15.97)

Ω

Since (yε − uε ) ∈ H10,sol (Ωε ) for every ε > 0, it follows that we can take v = yε − uε as a test function in (15.78). Then the following equality is ensured: ν ∇(yε − uε ) : ∇(yε − uε ) dx Ωε χε ∇uε : ∇(˘ yε − uε ) dx =−ν Ω χε ((˘ yε − uε ) · ∇)(˘ yε − uε ) · (˘ yε − uε ) dx − Ω χε (uε · ∇)˘ yε · (˘ yε − uε ) dx − Ω χε (˘ yε · ∇)uε · (˘ yε − uε ) dx + χε (uε · ∇)uε · (˘ yε − uε ) dx − Ω Ω χε p˘ε div (˘ yε − uε ) dx + χε f ε · (˘ yε − uε ) dx. + Ω

Ω

By the arguments of the previous theorem, we obtain lim ν ∇(yε − uε ) : ∇(yε − uε ) dx ε→0 Ωε ∇u : ∇(y − u) dx =−ν Ω − ((y − u) · ∇)(y − u) · (y − u) dx − (u · ∇)y · (y − u) dx Ω Ω − (y · ∇)u · (y − u) dx + (u · ∇)u · (y − u) dx Ω Ω p div (y − u) dx + f · (y − u) dx. (15.98) + Ω

Ω

We now consider the integral identity (15.91) with the test function y − u. By a rearrangement, we have 2πν 2 2 ϕ |y − u| dx + ν |∇(y − u)| dx + ν ∇u : ∇(y − u) dx C0 Ω Ω Ω + ((y − u) · ∇)(y − u) · (y − u) dx + (u · ∇)y · (y − u) dx Ω Ω + (y · ∇)u · (y − u) dx − (u · ∇)u · (y − u) dx Ω Ω = p div (y − u) dx + f · (y − u) dx. (15.99) Ω

Ω

578


The comparison of (15.98) with (15.99) leads to the following equality: 2π 2 2 2 |∇(yε − uε )| dx = |∇(y − u)| dx + ϕ |y − u| dx, lim ε→0 Ω C 0 Ω Ω ε which, together with (15.95)–(15.97), concludes the proof.

Remark 15.34. Using this approach, we can prove a more general result: Under the supposition of Lemma 15.33, the following relation is valid: 2π 2 ∇yε : (∇yε )t dx = ∇y : (∇y)t dx + |y − u| dx. (15.100) lim ε→0 Ω C 0 Ω Ω ε We are now able to establish the identification result of the variational limit for the sequence of constrained minimization problems (15.52). Theorem 15.35. For the sequence (15.52), there exists a unique variational w-limit which can be represented in the form (15.53), where the cost functional J0 and the set of admissible solutions Ξ0 = {(u, y, p )} are defined as follows: |∇y|2 dx J0 (u, y) =λ Ω 2πλ 2 2 |y − u| dx + β|∂Q|H |u| dx, (15.101) + C0 Ω Ω ⎧ ⎫ p ∈ L20 (Ω), u ∈ H2 (Ω), y − u ∈ H10,sol (Ω), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 (Ω) ≤ γ, u| = 0, u H ⎪ ⎪ Γ3 ⎨ ⎬ 2πν Ξ0 = −ν y + C0 (y − u) + (y · ∇)y + ∇p = f in Ω, . (15.102) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ div y = 0 in Ω, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ y| = u| ⎭ ∂Ω

∂Ω

Proof. The proof of this theorem is divided into two steps, each of them concerns the verification of the corresponding item of Definition 15.22. Step 1: Statement (i) of Definition 15.22 is valid. Let {(αε , yε , pε )}ε>0 be a bounded sequence of admissible triplets which is w-convergent to a triplet (u, y, p). Let {εk } be a subsequence of {ε} such that εk → 0 as k → ∞ and (αk , yk , pk ) ∈ Ξεk for all k ∈ N. Then, due to Theorem 15.31, we have that the w-limit triplet (u, y, p) satisfies relations (15.92) and (15.93), and moreover, γ ≥ lim inf uk H2 (Ω) ≥ uH2 (Ω) , k→∞

by the lower semicontinuity of · H 2 (Ω) with respect to the weak convergence in H2 (Ω). So, inclusion (15.54) holds true. The fulfillment of inequality (15.54)


579

immediately follows from Lemma 15.33 and the property of lower semicontinuity of the weak convergence in variable spaces (see Proposition 6.16) ' ( 2 2 r(εk ) |uk | dηεk |u| dx. ≥ β|∂Q|H lim β|∂Q|H k→∞

Ω

Ω

Step 2: Statement (ii) of Definition 15.22 holds true. Let (u, y, p ) ∈ Ξ0 be an admissible triplet for the minimization problem (15.53), (15.101), (15.102). In view of Hypothesis (H5), we may suppose that there exist operators Λε : H1sol (Ω) → H1sol (Ωε ) such that y) = y on Ωε , if y ∈ H1sol (Ωε ), then Λε (˘

(15.103)

Λε (u)H1 (Ωε ) ≤ CuH1 (Ω) , where C > 0 is independent of ε. (15.104) Hence, for a given u ∈ U, we have

∇ · Λε (u) = 0 and

Λε (u) · n dH2 = 0. ∂Ωε

As follows from the structure of the set of admissible controls U, Λε (u) · n dH2 = 0 ⇔ Λε (u) · n dH2 = 0, k

(15.105)

∀ j = 1, . . . , Jε . Let us define the sequence ! " αε∗ = αk∗ 1 , αk∗ 2 , . . . , αk∗ Jε = Λε (u)|∂Tε

(15.106)

∂Tε j

∂Ωε

.

ε>0

Here, we suppose that the sequence {wε = Λε (u)} can be extended to the whole domain Ω with uniformly bounded norms in H1 (Ω) (see Remark 15.10). It is easy to see that each of the functions {αε∗ } satisfy conditions (15.50) and Λε (u) ∈ L2 (Ω, dηεrε ) (see Remark 15.7). Moreover, in view of the estimates uH2 (Ω) ≤ γ and (15.104), the sequence {Λε (u)} is uniformly bounded in the variable space L2 (Ω, dηεrε ). Hence, we may suppose the existence of a function u∗ ∈ L2 (Ω) such that {Λε (u)} → u∗ weakly in L2 (Ω, dηεrε ).

(15.107)

On the other hand, thanks to Lemmas 15.15–15.17, we have u ∈ L2 (Ω, dηεrε ),

∀ ε > 0,

as ε → 0, u u in L 2 2 lim |u| dηεrε = |u| dx, 2

ε→0

w

Ω

(Ω, dηεrε )

Ω

a u. Let us show that u = u∗ . and hence, u|∂Tε −→

(15.108) (15.109) (15.110)

580


To do so, we note that u ∈ C(Ω), by the imbedding results for Sobolev spaces. Hence, for every δ > 0, there exists ε0 > 0 such that

2

j = 1, . . . , Jε and ∀ ε < ε0 . u · n dH (15.111)

< δ, k ∂Tε j

Let us partition the set Ω into cubes εY with edges ε and denote these cubes with εYi . Then, combining (15.105) with (15.111), we obtain that the following estimate holds true ∀ ε < ε0 :

2

kj

(u − Λ (u)) · n dH ε

< δ, ∀ j = 1, . . . , Jε , ∀ i : ∂Tε ∩ εYi = ∅. k ∂Tε j ∩εYi

k

Hence, for every fixed indices (i, j), there exist points xi ∈ ∂Tε j ∩ εYi such that

rε

u(xi ) − 1 Λε (u) dηε

ε3 εYi

1 i 2

Λε (u) dH < δ. (15.112) = u(x ) − 3 kj ε ∂Tε ∩εYi

We are able to estimate the difference

∗

ϕ u dx − ϕ u dx

,

Ω

Ω

where ϕ ∈ C0 (Ω). As a result, we have

∗ rε

ϕ u dx − ϕ u dx ≤ ϕ u dx − ϕ u dηε

Ω Ω Ω

Ω

rε rε

rε ∗

ϕ Λε (u) dηε + ϕ Λε (u) dηε − ϕ u dx

+ ϕ u dηε − Ω

Ω

Ω

Ω

= I1 + I2 + I3 . Taking the weak convergence of the sequences u ∈ L2 (Ω, dηεrε ) into account and Λε (u) ∈ L2 (Ω, dηεrε ) (see (15.107), (15.109), and (15.110)), we observe that I1 → 0 and I3 → 0 as ε tends to 0. Now, we show that I2 → 0. Since u ∈ C(Ω), it follows that there exists a constant C ∗ such that

rε rε

I2 ≤ ϕ u dηε − ϕ Λε (u) dηε

i

εYi

εYi

1 ∗ 3 i i rε

ϕ Λε (u) dηε

≤C ε

ϕ(x ) u(x ) − ε3 εYi i

ϕ(x) rε

u(xi ) − 1 ≤ C ∗ ε3 ϕC(Ω) Λ (u) dη ε ε .

3 i ε εYi ϕ(x ) i


581

Let us suppose the converse, that is,

1 ϕ(x) 3 i rε

Λε (u) dηε > 0, lim ε

u(x ) − ε3 i ε→0 εYi ϕ(x ) i

then there exist a constant D ∗ and a value ε∗ > 0 such that

ϕ(x) rε

u(xi ) − 1 Λε (u) dηε ≥ D∗ , ε < ε∗ .

3 i ε εYi ϕ(x ) Assuming that ϕ(x)/ϕ(xi ) = 1, we just come into conflict with (15.112). So, our supposition was wrong, and we get limε→0 I2 = 0. Thus, u = u∗ . As a result, we have (see Lemma 15.15) wa αε∗ = αk∗ 1 , αk∗ 2 , . . . , αk∗ Jε = Λε (u)|∂Tε −→ u. (15.113) Using the fact that the weak limit of Λε (u) in H1 (Ω) coincides with its weak r(ε) limit in the variable space L2 (Ω, dηε ) (see [86]) and closely following the previous arguments, it can be proved that 2 2 lim |Λε (u)| dηεrε = |u∗ | dx. (15.114) ε→0

Ω

Ω

ε , p ) ∈ Ξε }ε>0 , where Let us consider the sequence of triplets {(αε∗ , y yε (αε∗ ), pε (αε∗ )) are the corresponding solutions of the boundary ( yε , p ) = ( value problem (15.9)–(15.12). Due to Theorem 15.19, this sequence is relatively compact with respect to the w-convergence. Hence, taking into account ε→0 ε , p) −→ (u, y , p ), Lemma 15.33 and property (15.114), we deduce that (αε∗ , y , p ) and (u, y, p ) belong to the same class of equivalence (here we where (u, y suppose that the problem (15.53) may have more than one solution). Thus, to end the proof, it remains only to apply Lemma 15.33 and property (15.114). Remark 15.36. It is now clear that the structure of the limit problem (15.53) essentially depends on the parameter C0 . If C0 = +∞ (which corresponds to the case when the cylinders have smaller cross-size), then the limit minimization problem (15.53), (15.101), (15.102) takes the form of the calculus variation problem (15.18)–(15.19). If 0 < C0 < +∞ (which corresponds to the case when the cylinders are of critical cross-size), then the limit minimization problem (15.53), (15.101), (15.102) can be recovered in the form of the OCP (15.20)–(15.24). Remark 15.37. Following the above approach, we can obtain the explicit mathematical description for the variational w-limit problem (15.53) in the case when the cost functional to the Pε -problem takes the form Ωε

ε βε rε j=1

J

|curl yε |2 dx +

Jε (αε , yε ) = λ

k ∂Tε j

αk 2 dH2 . j

(15.115)

582


Indeed, to do this, we have to use relation (15.17) and to apply identity (15.100). As a result, we obtain the following structure for the limit cost functional: 2 J0 (u, y) = λ |curl y|2 dx + β|∂Q|H |u| dx, Ω

in which the relaxed term 2πλ + C0

Ω

2

|y − u| dx Ω

does not appear. To conclude this section, we combine the results of Theorem 15.23 and Theorem 15.35 and come to the following conclusion concerning the variational properties of the limit problem (15.53), (15.101), (15.102). Theorem 15.38. Let (α0ε , y0ε , p0ε ) ∈ Ξε ε>0 be the optimal solutions of the problems (Pε ). Then each w-cluster triplet (u0 , y0 , p0 ) of this sequence satisfies the conditions inf J0 (u, y) = J0 u0 , y0 = lim Jε (α0ε , y0ε ). (u0 , y0 , p0 ) ∈ Ξ0 , ε→0

(u,y,p)∈ Ξ0

15.9 Suboptimal controls and their approximation properties In this section, we deal with the construction of suboptimal controls to the original optimal boundary control problem (15.8)–(15.12). The main result can be formulated as follows. Theorem 15.39. Let Λε : H1sol (Ω) → H1sol (Ωε ) be the linear continuous map such that properties (15.103) and (15.104) hold true, and let (u0 , y0 , p0 ) ∈ Ξ0 be an optimal solution to the limit problem (15.53). Then the function

= Λε (u0 ) ∂Tε αεsub = αksub , αksub , . . . , αksub (15.116) 1 2 Jε is an asymptotically suboptimal control for the original problem (Pε ) in the sense of Definition 15.21. Proof. To begin, we note that, following the proof of Theorem 15.35, we can immediately establish the following approximation property: If (u0 , y0 , p 0 ) is an optimal solution to the limit minimization problem (15.53), then Λε (u0 ) ∈ L2 (Ω, dηεr(ε) ), ∀ ε > 0,

wa u0 as ε → 0, Λε (u0 ) ∂Tε −→

0 2

Λε (u0 ) 2 dηεr(ε) =

u dx. lim

ε→0

Ω

Ω

(15.117) (15.118) (15.119)

15.9 Suboptimal controls and their approximation properties

Let us consider now the sequence of triplets " !

ε , pε ) ∈ Ξε ( Λε (u0 ) ∂Tε , y

583

,

ε>0

where

ε ( Λε (u0 ) ∂T ), pε ( Λε (u0 ) ∂T ) ( yε , pε ) = y ε ε

are the corresponding solutions of the boundary value problem (15.9)–(15.12). Then following the motivation given in the proof of Theorem 15.35, we conclude that this sequence is relatively compact with respect to the wconvergence, and extracting, if necessary, a subsequence, it satisfies condition

ε→0 0 , p0 ), ε ( Λε (u0 ) ∂T ), pε ( Λε (u0 ) ∂T ) −→ (u 0 , y ( yε , pε ) = y ε

ε

0 , p0 ) and (u 0 , y 0 , p 0 ) belong to the same equivalence class. Conwhere (u 0 , y sequently, 0 ). J0 (u0 , y 0 ) = J0 (u0 , y 0 0 0 Let (αε , yε , pε ) ∈ Ξε ε>0 be the optimal solutions to the Pε -problems. We observe that

ε )

inf Jε (αε , yε ) − Jε ( Λε (u0 ) ∂T , y

ε (αε ,yε ,pε )∈ Ξε

ε ) ≤ Jε (α0ε , y0ε ) − J0 (u0 , y0 )

= Jε (αε0 , yε0 ) − Jε ( Λε (u0 ) ∂Tε , y

0

2

2π

y − u0 dx − ε |2 dx

+ λ

|∇ y 0 |2 dx + |∇ y C0 Ω Ω Ωε

Jε

2

0 2

ε

0

2

dx − u + β |∂Q|H

Λε (u ) ∂T kj dH

kj ε rε j=1 ∂Tε Ω

= J1 + J2 + J3 . To conclude the proof, we note that for a given δ > 0, one can always find the following: (i) ε1 > 0 such that J1 < δ/3 for all ε < ε1 by Theorem 15.23; (ii) ε2 > 0 such that J2 < δ/3 for all ε < ε2 by the Lemma 15.33; and (iii) ε3 > 0 such that J3 < δ/3 for all ε < ε3 by property (15.119). Thus, as expected, the estimate of suboptimality (15.51) is valid for all ε < min{ε1 , ε2 , ε3 }.

16 Optimal Control Problems in Coefficients: Sensitivity Analysis and Approximation

In this chapter, we are mainly interested in optimal L∞ (Ω)-control of the coefficients of an elliptic Dirichlet problem. We prove an existence result for this problem using the direct method of Calculus of Variations, and then we provide a sensitivity analysis of this problem on a reticulated structured with respect to the domain perturbation and discuss possible ways for the approximation of its solutions. Let Ω be a fixed open subset of D which we associate with a thick lattice structure or a network contained in D. As a particular example of a nonlinear optimal control problem (OCP) we consider in this chapter is minimizing the discrepancy between a given distribution yd ∈ L2 (D) and the solution of a Dirichlet problem by choosing an appropriate matrix of coefficients A ∈ L∞ (D; Rn×n ). More precisely, we consider the following minimization problem: |y(x) − yd (x)|2 dx Minimize I Ω (A, y) = Ω |∇y(x)|2Rn dx + J(A) (16.1) + Ω

subject to the constraints A ∈ M, A ≥ αI,

y ∈ H01 (Ω),

(16.2)

α > 0,

(16.3)

−div (A(x)∇y) + a0 (x)y = f

in Ω,

(16.4)

where M is a weakly-∗ closed subset of L∞ (D; Rn×n ), and J : M → R is a lower semicontinuous functional with respect to the weak-∗ topology of L∞ (D; Rn×n ). In (16.3), I is the identity matrix in Rn×n , and the above inequality is in the sense of the quadratic forms defined by (Aξ, ξ)Rn for ξ ∈ Rn . P.I. Kogut, G.R. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains, Systems & Control: Foundations & Applications, DOI 10.1007/978-0-8176-8149-4 16, © Springer Science+Business Media, LLC 2011

585

586

16 Optimal Control Problems in Coefficients

Thus, we seek a matrix of coefficients A ∈ L∞ (D; Rn×n ) such that the corresponding weak solution y = yΩ,A,f of (16.2)–(16.4) is as close as possible to a desired state yd . Note that the range of OCPs in coefficients is very wide, including, for example, optimal shape design problems, topology optimization and material optimization, and many others. This topic has been widely studied by many authors. We mainly mention Allaire [7], Buttazzo and Dal Maso [45, 46], Calvo-Jurado and Casado-D´ıaz [50, 51, 52], Haslinger and Neittaanmäki [119], Lions [169], Litvinov [176], Lurie [181], Murat [193], Murat and Tartar [195], Pironneau [214], Raytum [219], Serova˘iski˘i [226], Sokolowski and Zolesio [233], and Tiba [240]. However, the principal feature of OCPs in coefficients is the fact that an optimal solution for (16.1)–(16.4) does not exist, in general (see, e.g., [45, 50, 193, 219]). We have once again a typical situation of optimal control theory, namely the original control object is described by the well-posed boundary value problem, but the associated OCP is ill-posed and requires relaxation. Taking this fact into account, we will relax the problem (16.1)–(16.4) by introducing the solenoidal controls A ∈ U sol . Notice that this class of admissible controls does not belong to the Sobolev space W 1,∞ (Ω) but is still a uniformly bounded subset of L∞ (Ω). We give a precise definition of such controls and prove that, in this case, the original OCP admits at least one solution. Note that we do not invoke in this process the homogenization method and the relaxation procedure.

16.1 Notation and preliminaries Throughout this chapter, Ω and D are bounded open subsets of Rn , n ≥ 1, such that Ω ⊂⊂ D. We will, as always, associate Ω with some lattice structure. Let χΩ be the characteristic function of the set Ω. For two real numbers 1 < p < +∞ and 1 < q < +∞ such that 1/p + 1/q = 1, the space W01, p (Ω) is the 1, p −1, q (Ω) is the space closure of C0∞ (Ω) in the Sobolev space W (Ω) and W of all distributions of the form f = f0 + j Dj fj , with f0 , f1 , . . . , fn ∈ Lq (Ω) (i.e., W −1, q (Ω) is the dual space of W01, p (Ω)). n We recall that for any vector field v ∈ Lp (Ω) = [Lp (Ω)] , the divergence −1, q is an element of the space W (Ω) defined by the formula div v, ϕ W −1, q (Ω);W 1, p (Ω) = − (v, Dϕ)Rn dx, ∀ ϕ ∈ W01, p (Ω), (16.5) 0

Ω

where ·, · W −1, q (Ω);W 1, p (Ω) denotes the duality pairing between W −1, q (Ω) 0

and W01, p (Ω) and (·, ·)Rn denotes the scalar product of two vectors in Rn . Using the terminology of Sect. 2.3 (see also Sect. 6.4.2 for the periodic case), we say that a vector field v is solenoidal if div v = 0. For any vector

16.1 Notation and preliminaries

587

field v ∈ Lp (Ω), the relations ij

curl v, ϕ W −1, q (Ω);W 1, p (Ω) = − 0

(vi Dj ϕ − vj Di ϕ) dx, Ω

∀ ϕ ∈ W01, p (Ω),

i, j = 1, . . . , n,

define a skew-symmetric matrix curl v, whose elements belong to the space W −1, q (Ω). We recall that a vector field v is said to be vortex-free if curl v = 0. We say that a vector field v ∈ Lp (Ω) is potential if v can be represented in the form v = Du, where u ∈ W 1, p (Ω). Obviously, any potential vector is vortex-free. In order to identify a weakly-∗ closed subset M of L∞ (D; Rn×n ) (see (16.2) and (16.3)), we fix two constants α and β such that 0 < α ≤ β < +∞. We define M as the set Mαβ (D) of all symmetric matrices A = [ai j ] in L∞ (D; Rn×n ) such that ⎫

A(x) Rn×n ≤ β for a.e. x ∈ D, ⎪ ⎪ ⎬ p−2 p n ]ζ, ζ Rn ≥ α|ζ|p a.e. on Ω, ∀ ζ ∈ R , A(x)[ζ ⎪ ⎪ ⎭ A(x)([ζ p−2 ]ζ − [η p−2 ]η), ζ − η Rn ≥ 0 a.e. on Ω, ∀ ζ = η ∈ Rn , (16.6) n 1/p and where |η|p = ( k=1 |η|pk ) [η p−2 ] = diag{|η1 |p−2 , |η2 |p−2 , . . . , |ηn |p−2 },

A(x)[ζ p−2 ]ζ, ζ

Rn

=

n

∀η ∈ Rn ,

aij (x)|ηj |p−2 ηj ηi .

(16.7) (16.8)

i,j=1

Remark 16.1. It is clear that Mαβ (D) is a nonempty subset of L∞ (D; Rn×n ) and its typical representatives are the diagonal matrixes A(x) = diag{δ1 (x), δ2 (x), . . . , δn (x)} such that α ≤ δi (x) ≤ β almost everywhere on Ω, ∀ i ∈ {1, . . . , n}. Indeed, let ζ and η be any two fixed vectors of Rn . Then the estimates (16.6)1 and (16.6)2 are obvious. As for the verification of the monotonicity property (16.6)3 , we make use of the following chain of transformations:

588


A(x)([ζ p−2 ]ζ − [η p−2 ]η), ζ − η Rn = [ζ p−2 ]A(x)ζ − [η p−2 ]A(x)η, ζ − η Rn = [ζ p−2 ]A(x)ζ, ζ Rn − [ζ p−2 ]A(x)ζ, η Rn − [η p−2 ]A(x)η, ζ Rn + [η p−2 ]A(x)η, η Rn =

n

δi (x)|ζi |p−2 ζi2 −

i=1

− =

n i=1 n

n

δi (x)|ζi |p−2 ζi ηi

i=1 n

δi (x)|ηi |p−2 ζi ηi +

δi (x)|ηi |p−2 ηi2

i=1

δi (x)|ζi |p−2 ζi (ζi − ηi ) −

i=1

=

n

n

δi (x)|ηi |p−2 ηi (ζi − ηi )

i=1

δi (x) |ζi |p−2 ζi − |ηi |p−2 ηi (ζi − ηi ) .

(16.9)

i=1

It remains to use the following estimates (see, for instance, [39] and [55]): (|a|p−2 a − |b|p−2 b)(a − b) ≥ 22−p |a − b|p , p ≥ 2, p−2

(|a|p−2 a − |b|p−2 b)(a − b) ≥ (|a| + |b|)

|a − b|2 , 1 < p ≤ 2,

(16.10) (16.11)

for every a, b ∈ R. As a result, the monotonicity property (16.6)3 is a direct consequence of the coercivity assumption (16.6)2 . However, in a general case, these properties should be considered as independent. Remark 16.2. If p = 2 and a matrix A(x) possesses the coerciveness property (16.6)2 , then A(x) satisfies the strong monotonicity property (A(x)(ζ − η), ζ − η)Rn ≥ α|ζ − η|2Rn

a.e. on Ω, ∀ ζ, η ∈ Rn .

Hence, in this case, the strong monotonicity (16.6)3 is a direct consequence of the coerciveness property (16.6)2 . For a fixed matrix A ∈ Mαβ (D) and a real number 1 < p < +∞, we define the mapping a : Rn × Rn → Rn as follows: a(x, ζ) = A(x)[ζ p−2 ]ζ. Then, as follows from the definition of the set Mαβ (D) (see, (16.6) ) for a.e. x ∈ Rn and for every ζ1 = ζ2 ∈ Rn , we have: (a(x, ζ1 ) − a(x, ζ2 ), ζ1 − ζ2 )Rn ≥ 0, (a(x, ζ), ζ)Rn ≥

α|ζ|pp ,

|a(x, ζ)|Rn ≤

β|ζ|pp−1 .

(16.12) (16.13)

For a given function a0 ∈ L∞ (D) such that a0 (x) ≥ 0 for a.e. x ∈ D, we consider the operator Ay = −div (a(x, Dy)) + a0 (x)|y|p−2 y

16.1 Notation and preliminaries

defined via the paring

Ay, v W −1, q (Ω),W 1, p (Ω) = 0

A(x)[Dy p−2 ]Dy, Dv

Ω a0 (x)|y|p−2 yv dx

+

Rn

589

dx

∀ y, v ∈ W01, p (Ω).

Ω

As a result, since A ∈ the following estimate takes place a0 (x) |y|p−2 y(y − v) − |v|p−2 v(y − v) dx Ω a0 (x) |y|p−2 (y 2 − |y||v|) − |v|p−2 (|y||v| − v 2 ) dx ≥ Ω (16.14) a0 (x) |y|p−1 − |v|p−1 (|y| − |v|) dx ≥ 0, = Mαβ (D),

Ω

the operator A turns out to be coercive and monotone from W01, p (Ω) into its dual W −1, q (Ω), and demicontinuous in the following sense: yk → y0 strongly in W01, p (Ω) implies that Ayk → Ay0 weakly in W −1, q (Ω) (see [113, 168]). Moreover, if the element a0 (·) ∈ L∞ (Ω) is strictly separated from 0 in Ω, then inequality (16.14) and estimates (16.10) and (16.11) imply that the monotonicity property for the operator A : W01, p (Ω) → W −1, q (Ω) can be specified as follows: Proposition 16.3. If there exists a positive value β such that a0 (x) ≥ β > 0 a.e. on Ω, then the operator A : W01, p (Ω) → W −1, q (Ω) is strictly monotone, that is, A(y) − A(v), y − v W −1, q (Ω);W 1, p (Ω) > 0, 0

∀ y, v ∈

W01, p (Ω)

such that y = v in L (Ω). p

Then by the well-known existence results for nonlinear elliptic equations with strictly monotone demicontinuous coercive operators (see [113, 126]), one can easily see that for every open set Ω ⊂ D and every f ∈ Lq (D), the nonlinear Dirichlet boundary value problem Ay = f has a weak solution in solution of (16.15) if

in

W01, p (Ω).

y ∈ W01, p (Ω),

Ω,

(16.15)

Let us recall that a function y is the weak

y ∈ W01, p (Ω), p−2 A(x)[Dy ]Dy, Dv Rn dx + a0 (x)|y|p−2 yv dx Ω Ω = f v dx, ∀ v ∈ W01, p (Ω).

(16.16) (16.17)

Ω

Following [39] and [82], we conclude this section by giving a couple results concerning the estimates for the solution of the Dirichlet problem (16.15).

590


Proposition 16.4. For every A ∈ Mαβ (D), a0 ∈ L∞ (D) (a0 (x) ≥ 0 a.e. x ∈ D), and f ∈ W −1, q (Ω), a weak solution yΩ,A,f to the problem (16.15) satisfies the estimate |DyΩ,A,f |pRn dx ≤ C f qW −1, q (Ω) , (16.18) Ω

where C is a constant depending only on p, α, and β. Proposition 16.5. For every open set Ω ⊆ D let yΩ,A,f be a weak solution of (16.15), and let y Ω,A,f denote the zero-extension of yΩ,A,f to D. Then

yΩ,A,f W 1, p (D) ≤ C, 0

where C depends on f , a0 , p, α, and β.

16.2 H-Convergence and a counterexample of Murat Murat showed in 1970 (see [193, 194]) that the problem (16.1)–(16.4) has no solution, in general. It turns out that this feature is typical for the majority of OCPs with respect to coefficients. In addition, this fact is not just a mathematical problem, but it is also very restrictive from the point of view of numerical applications. The initial problem that Murat considered was to minimize the cost functional 1 y(x) − 1 − x2 2 dx, (16.19) I(a, y) = 0

where y is the solution of the boundary value problem dy d a + ay = 0 in (0, 1), − dx dx y(0) = 1,

y(1) = 2,

y ∈ H 1 (0, 1)

(16.20) (16.21)

and the control a belongs to the following admissible control set A = {a | a ∈ L∞ (0, 1), α ≤ a ≤ β a.e. on (0, 1)} .

(16.22)

Trying to apply the direct method of the Calculus of Variations, he noticed ∞ that for a sequence {an }n=1 ⊂ A such that an a+ and

1 1 weakly-∗ in L∞ (0, 1), an a−

(16.23)

the corresponding sequence of solutions yn = y(an ) of (16.20) and (16.21) converges weakly in H 1 (0, 1) to the solution y∞ of the problem

16.2 H-Convergence and a counterexample of Murat

−

d dx

a−

y∞ (0) = 1,

dy∞ dx

591

+ a+ y∞ = 0 in (0, 1),

y∞ (1) = 2,

y∞ ∈ H 1 (0, 1),

(16.24) (16.25)

with the properties dyn dy∞ −→ a− strongly in L2 (0, 1), dx dx 1

I(an , yn ) → I(a− , a+ , y∞ ) = |y∞ (x) − 1 − x2 |2 dx. an

(16.26) (16.27)

0

Indeed, yn is bounded in H 1 (0, 1) and vn = an dyn /dx is bounded in L2 (0, 1), but as its derivative is an yn which is bounded in L2 (0, 1), vn is actually bounded in H 1 (0, 1). Therefore, one can extract a subsequence such that ym weakly converges in H 1 (0, 1) and strongly in L2 (0, 1) to y∞ , and vm strongly converges in L2 (0, 1) to v∞ . Then am ym and dym /dx = (1/am )vm weakly converge in L2 (0, 1), respectively to a+ y∞ and to dy∞ /dx = (1/a− )v∞ , showing (16.26), and therefore (16.24) and (16.25); the fact that y∞ is uniquely determined by (16.24) and (16.25) shows that the extraction of a subsequence is not necessary. Moreover, in this case, it can be shown that all possible pairs (a− , a+ ) which could appear in (16.23) satisfy the inequality α ≤ a− (x) ≤ a+ (x) ≤

a− (x)(α + β) − αβ ≤ β for a.e. x ∈ (0, 1), a− (x)

or equivalently, 1 1 α + β − a+ (x) ≤ ≤ for a.e. x ∈ (0, 1). a+ (x) a− (x) αβ √ √ √ √ 2 − 1 / 2, β = 2 + 1 / 2, and In particular, if α = ⎧ 2 ⎪ ⎪ 1 − 1 − x if x ∈ 2k , 2k + 1 , k = 0, . . . , n − 1, ⎨ 6 2n 2n 2 an (x) = 2 ⎪ 1 x 2k + 1 2k + 2 ⎪ ⎩1 + − if x ∈ , , k = 0, . . . , n − 1, 2 6 2n 2n then Murat has shown in [193] that a− =

1 x2 + ; a+ = 1; y∞ = 1 + x2 on (0, 1), 2 6 lim I(an , yn ) = inf I(a, y) = 0,

n→∞

(16.28) (16.29)

(a,y)∈Ξ

where Ξ denotes the set of all admissible pairs for the problem (16.19)–(16.22). However, it is not possible to have y = 1 + x2 in (16.20) and (16.21) for some

592


a ∈ A. It means that a pair (a, 1 + x2 ) does not belong to the set Ξ for any a ∈ A. In order to clarify this situation, we recall the notion of H-convergence introduced by Tartar in 1977 (see [236]) and later developed by Murat and Tartar (see [196]). Let {Aε }ε>0 be a sequence of matrices in Mαβ (Ω) which are not necessary symmetric. Definition 16.6. A sequence {Aε }ε>0 in Mαβ (Ω) with p = 2 is said to be H

H-convergent to A ∈ Mαβ (Ω) (in symbols, Aε → A) as ε → 0 if the following condition holds. Whenever vector fields dε , d, vε , v ∈ L2 (Ω) satisfy dε = Aε vε ,

∀ ε > 0,

(16.30)

dε → d weakly in L (Ω), 2

(16.31)

vε → v weakly in L (Ω), 2

(16.32) −1,2

(Ω), {div dε }ε>0 is relatively compact in W −1,2 {div vε }ε>0 is relatively compact in W (Ω) ,

(16.33) (16.34)

then d = Av.

(16.35)

It can be proved that the above definition of H-convergence is equivalent to the slightly different definition given in [196] concerning the limit behavior of the elliptic equations governed by matrix Aε (see Definition 16.7). Definition 16.7. A sequence {Aε }ε>0 of matrices in Mαβ (Ω) with p = 2 Hconverges to a matrix A ∈ Mαβ (Ω) if for every f ∈ W −1,2 (Ω), the sequence {uε }ε>0 of the solutions to the Dirichlet problems uε ∈ W01,2 (Ω), (16.36) −div (Aε Duε ) = f in D (Ω), satisfies uε → u weakly in W01,2 (Ω), Aε Duε → A Du weakly in L2 (Ω, Rn ), where u is the solution to the problem u ∈ W01,2 (Ω), −div (A Du) = f in D (Ω).

(16.37)

We list some of the basic properties of H-convergence: (i) The H-limit of an H-converging sequence is unique. (ii) The set Mαβ (Ω) with p = 2 is sequentially compact with respect to the H-convergence.

16.3 Setting of the optimal control problem and existence theorem

593

(iii) H-convergence is stable under transposition of the matrices. H H (iv) If {Aε }ε>0 , {Bε }ε>0 ⊂ Mαβ (Ω), Aε → A, Bε → B, and for some open set U ⊂ Ω, one has Aε = Bε in U for every ε > 0, then A = B in U . Remark 16.8. As follows from the counterexample given above, in general, the H-limit of a bounded sequence {Aε }ε>0 and its weak-∗ limit in L∞ (Ω, Rn×n ) may be drastically different. Moreover, this example indicates that the set of admissible solutions Ξ to the problem (16.19)–(16.22) is not sequentially closed with respect to the product of H-convergence in L∞ (Ω, Rn×n ) and the weak convergence in W01,2 (Ω) and also closedness of Ξ is lacking with respect to the product of weak-∗ convergence in L∞ (Ω, Rn×n ) and the weak convergence in W01,2 (Ω). Indeed, in this case, we have (a− , y∞ ) ∈ Ξ and (a+ , y∞ ) ∈ Ξ. At the same time, Proposition 16.5, the definition of H-convergence, and its properties immediately imply that the set of admissible solutions Ξ ⊂ L∞ (Ω, Rn×n ) × W01,2 (Ω) to the OCP Minimize I(A, y) = |y(x) − yd (x)|2 dx + |Dy(x)|2Rn dx (16.38) Ω

Ω

subject to the constraints A ∈ Mαβ (Ω),

p = 2, y ∈ H01 (Ω), −div (A(x)Dy) + a0 (x)y = f in Ω

(16.39) (16.40)

is sequentially compact with respect to the product of H-convergence in L∞ (Ω, Rn×n ) and the weak convergence in W01,2 (Ω). Moreover, in this case, the cost functional (16.38) is lower semicontinuous in the following sense: H Ak → A∗ in L∞ (Ω; Rn×n ) =⇒ lim inf I(Ak , yk ) ≥ I(A∗ , y∗ ). (16.41) k→∞ yk y∗ in W01,2 (Ω) As a result, we come to the conclusion. Theorem 16.9. The problem (16.38)–(16.40) possesses at least one solution. However, as for the OCP (16.1)–(16.4), the set M = Mαβ (Ω) is not Hclosed in general, and the property of lower semicontinuity for the cost functional IΩ (A, y) like (16.41) cannot be guaranteed. Thus, the existence of optimal solutions for the problem (16.1)–(16.4) is, in general, an open question.

16.3 Setting of the optimal control problem and existence theorem Let us associate with (16.15) the following minimization problem:

594


Minimize

I Ω (A, y) =

|y(x) − yd (x)|p dx |Dy(x)|pRn dx + J(A) + Ω

(16.42)

Ω

subject to the constraints

A ∈ Mαβ (D), y ∈ W01, p (Ω), A(x)[Dy p−2 ]Dy, Dv Rn dx + a0 (x)|y|p−2 yv dx Ω Ω f v dx, ∀ v ∈ W01, p (Ω), =

(16.43) (16.44)

Ω

where f ∈ L (D), yd ∈ L (D), a0 ∈ L∞ (D), a0 (x) ≥ 0 a. e. on D, and J : L∞ (D; Rn×n ) → R is a lower semicontinuous functional with respect to the weak-∗ topology of L∞ (D; Rn×n ) in the following sense: q

p

∗

Ak A∗ in L∞ (D; Rn×n ) =⇒ lim inf J(Ak ) ≥ J(A∗ ). k→∞

It is well known that for both physical and mathematical reasons, the size of the controls should be constrained. Physically, one cannot realize controls of arbitrary size. On the other hand, limits on the size of the control are needed in order to obtain a mathematically meaningful problem (e.g., to guarantee the existence of an optimal solution in the prescribed functional space). Let {Q1 , . . . , Qn } be a collection of nonempty compact subsets of the space W −1, q (D). To define the class of admissible controls, we introduce two sets (see [131]) Ub = A = [ai j ] ∈ Mαβ (D) , (16.45) Usol = A = [a1 , . . . , an ] ∈ Mαβ (D) div ai ∈ Qi , ∀ i = 1, . . . , n , (16.46) assuming that the intersection Ub ∩ Usol ⊂ L∞ (D; Rn×n ) is a nonempty set. Definition 16.10. We say that a matrix A = [ai j ] is an admissible control to the nonlinear Dirichlet problem (16.15) if A ∈ Uad := Ub ∩ Usol . Hereinafter we suppose that the set of admissible controls Uad is sufficiently rich in some sense; otherwise, in view of Proposition 16.4, the OCP |y(x) − yd (x)|p dx I Ω (A, y) = Ω |Dy(x)|pRn dx + J(A) −→ inf, (16.47) + Ω

−div A(x)[Dy p−2 ]Dy + a0 |y|p−2 y = f y∈

W01, p (Ω),

A ∈ Uad

in

Ω,

(16.48) (16.49)


595

becomes trivial. The existence of admissible controls is important both from a theoretical and an application point of view. Usually, this type of controls arises in the optimization of materials (represented by the matrix A). This question is largely open, except for some special cases, and an affirmative answer is usually just put as a hypothesis (see [7, 51, 181]). Let us denote by Ξ ⊂ L∞ (D; Rn×n ) × W01, p (Ω) the set of all admissible pairs to the OCP (16.47)–(16.49). The first question to be answered for the problem (16.47)–(16.49) is about solvability: Does there exists an optimal pair (A0 , y 0 ) in L∞ (D; Rn×n ) × W01, p (Ω) satisfying (16.48) and (16.49)? To begin with, we prove the following result. Proposition 16.11. The set Uad is sequentially compact with respect to the weak-∗ topology of L∞ (D; Rn×n ). Proof. Let {Ak = [a1 k , . . . , an k ]}k∈N ⊂ Uad be an arbitrary sequence of admissible controls. Since Uad ⊂ Ub and Ub is the sequentially weakly-∗ compact subset of L∞ (D; Rn×n ), we may suppose that there exist a matrix A0 = [a1 0 , . . . , an 0 ] ∈ Ub and elements fi ∈ Qi , i = 1, . . . , n, such that k→∞ (ai k , φ)Rn dx −→ (ai 0 , φ)Rn dx, (16.50) D D ∀φ ∈ L1 (D) = [L1 (D)]n , ∀ i = 1, 2, . . . , n, div ai k −→ fi in W −1, q (D), ∀ i = 1, . . . , n. k→∞

(16.51)

It remains to prove that div ai 0 = fi for all i = 1, . . . , n. To do this, we choose φ in (16.50) as a potential vector, that is, φ = Dv, where v ∈ W01, p (D). Then, in view of (16.5), relation (16.51) implies (ai k , Dv)Rn dx = − div a i k , v W −1, q (D),W 1, p (D) 0

D

k→∞

−→ − fi , v W −1, q (D);W 1, p (D) , ∀i = 1, . . . , n. 0

Using this and relation (16.50), we finally get lim (a i k , Dv)Rn dx = (a i 0 , Dv)Rn dx k→∞

D

D

= − div a i 0 , v W −1, q (D);W 1, p (D) 0

= − fi , v W −1, q (D);W 1, p (D) , 0

∀ i = 1, . . . , n.

As a result, we have A0 = [a1 0 , . . . , an 0 ] ∈ Usol . This concludes the proof. In order to discuss the existence of solutions for the problem (16.48)– (16.49), we make use of the following result (for comparison, we refer to the Lemma on Compensated Compactness in [261]).

596


Lemma 16.12. Let {f k }k∈N ⊂ Lq (D), {gk }k∈N ⊂ Lp (D) be a bounded sequences of vector functions such that f k f 0 in Lq (D) and gk g0 in Lp (D). If {div f k }k∈N

is compact with respect to the strong topology of W −1, q (D), and curl gk = 0,

then

lim

k→∞

∀ k ∈ N,

D

φ (f k , gk )Rn dx =

D

φ (f 0 , g0 )Rn dx,

(16.52) (16.53)

∀ φ ∈ C0∞ (D).

(16.54)

Remark 16.13. Equality (16.54) can be interpreted as the weak-∗ convergence in L1 (D) of the sequence {(f k , gk )Rn }k∈N to the element (f 0 , g0 )Rn . Proof. We have (f k , gk )Rn = (f k − f 0 , gk − g0 )Rn − (f 0 , g0 )Rn + (f k , g0 )Rn + (f 0 , gk )Rn . The sum of the last three terms obviously has a weak-∗ limit in L1 (D) which is equal to (f 0 , g0 )Rn . Therefore, it is sufficient to consider only the case f 0 = g0 = 0. Notice that the weak-∗ convergence in L1 (D) has a local property and, thus, for the simplicity, the domain D may be assumed to be a ball. Then, as follows from (16.53), gk is a potential vector field for every k ∈ N (i.e., gk = Dvk ) and one can assume that the potential function vk satisfies the condition v dx = 0, ∀ k ∈ N. By the initial assumptions, we have gk = Dvk 0 in D k Lp (D). Hence, by the Poincaré inequality p p p vk dx ≤ C vk dx + |Dvk |Rn dx = C |Dvk |pRn dx D

D

D

D

and the Sobolev embedding theorem, it follows that vk 0 in W 1, p (D),

vk → 0 in Lp (D).

Notice also that due to condition (16.52), we may suppose that (passing to a subsequence if necessary) div f k → 0 strongly in W −1, q (D). Now, for every fixed function φ ∈ C0∞ (D), we have (f k , gk )Rn φ dx = (f k , D(φ vk ))Rn dx − vk (f k , Dφ)Rn dx D

D

D

= div f k , φ vk W −1, q (D);W 1, p (D) 0 − vk (f k , Dφ)Rn dx. D

(16.55)


597

Since div f k → 0 strongly in W −1, q (D) and the sequence {φ vk }k∈N is bounded in W01, p (D), it follows that k→∞

div f k , φ vk W −1, q (D);W 1, p (D) −→ 0. 0

The last integral converges to 0, since vk → 0 strongly in Lp (D), and the sequence {(f k , Dφ)Rn }k∈N is bounded in Lq (D). We are now in a position to study the topological properties of the set Ξ ⊂ L∞ (D; Rn×n ) × W01, p (Ω) of all admissible pairs to the OCP (16.47)–(16.49). Let τ be the topology on the set L∞ (D; Rn×n )×W01, p (Ω) which we define as the product of the weak-∗ topology of L∞ (D; Rn×n ) and the weak topology of W01, p (Ω). Theorem 16.14 ([139]). For every f ∈ W −1, q (Ω), the set Ξ is sequentially τ -closed. Proof. Let {(Ak , yk )}k∈N ⊂ Ξ be any τ -convergent sequence of admissible pairs to the problem (16.47)–(16.49). Let (A0 , y0 ) be its τ -limit. Our aim is to prove that (A0 , y0 ) ∈ Ξ. Let us set A(A, y) = −div A(x) [D y p−2 ] D y + a0 (x) |y|p−2 y = A1 (A, y) + A2 (y), A1 (A, y) = −div A(x) [D yp−2 ] D y = −div a(A(x), D y). By Proposition 16.11 and the initial assumptions, we have A0 ∈ Uad ; therefore, Ak A0 = [a1 0 , . . . , an 0 ] weakly-∗ in L∞ (D, Rn×n ), div ai k → div ai 0 strongly in W yk y0 in

−1, q

(D), ∀ i = 1, . . . , n,

W01, p (Ω).

(16.56) (16.57) (16.58)

Hence, [Dykp−2 ]yk is bounded in Lq (Ω), q = p/(p − 1), k∈N |yk |p−2 yk k∈N is bounded in Lq (Ω),

yk → y0 in L (Ω), p

yk (x) → y0 (x) for a.e. x ∈ Ω.

(16.59) (16.60)

Then, by (16.60) and the monotonicity of the function g(ζ) = |ζ|p−2 ζ, we have |yk |p−2 yk → |y0 |p−2 y0 almost everywhere on Ω. Using this and (16.59), we conclude (see [168]) |yk |p−2 yk |y0 |p−2 y0

in Lq (Ω).

598


Consider the sequence fk := f − a0 |yk |p−2 yk k∈N . It is clear that fk ∈ W −1, q (Ω) for every k ∈ N and, since the embedding Lq (Ω) → W −1, q (Ω) is compact, it follows that fk → f0 = f − a0 |y0 |p−2 y0

strongly in

W −1, q (Ω).

(16.61)

In view of this and the fact that −div a(Ak (x), D yk ) = fk in Ω, ∀ k ∈ N, we come to the following conclusion (see (16.13)): {a(Ak (x), D yk )}k∈N is a bounded sequence in Lq (Ω). So, passing to a subsequence, we may assume that there exists a vector function ξ ∈ Lq (Ω) such that a(Ak (x), D yk ) = Ak [D ykp−2 ] D yk =: ξk ξ

on Lq (Ω).

(16.62)

Passing to the limit in the relation as k → ∞ (in the sense of the distributions) −div ξ k = f − a0 |yk |p−2 |yk | in D (Ω) yields

− div ξ = f − a0 |y0 |p−2 |y0 | in D (Ω).

It remains to show that ξ = scalar function

p−2 A0 |D y0 |R n

(16.63)

D y0 . To do so, we consider the

v(x) = (z, x)Rn ,

(16.64)

where z is a fixed element of R . Since the operator A1 is monotone, it follows that for every z ∈ Rn and every positive function φ ∈ C0∞ (Ω), we have φ(x) (a(Ak , Dyk ) − a(Ak , Dv), Dyk − Dv)Rn dx ≥ 0, n

Ω

or, taking into account (16.64), this inequality can be rewritten as φ(x) (a(Ak , Dyk ) − a(Ak , z), Dyk − z)Rn dx ≥ 0.

(16.65)

Ω

Our next intention is to pass to the limit in (16.65) as k → ∞ using Lemma 16.12. Since −div a(Ak , Dyk ) → f − a0 |y0 |p−2 |y0 | strongly in W −1, q (Ω), (16.66) curv (D yk − z) = curv D yk = 0, ∀ k ∈ N, it remains to show that the sequence {div a(Ak , z)}k∈N is compact with respect to the strong topology of W −1, q (Ω). Indeed, for every φ ∈ C0∞ (Ω), we have


599

− div a(Ak , z), φ W −1, q (Ω);W 1, p (Ω) 0 Ak (x)[z p−2 ]z, Dφ dx = (a(Ak , z), Dφ)Rn dx = Ω

= Ω

=

=

=

=

Ω

⎞ ⎤ (a1 k (x), [z p−2 ]z)Rn ⎝⎣ ⎦ , Dφ⎠ dx ... (an k (x), [z p−2 ]z)Rn Rn ⎛⎡

n

ai k (x), [z p−2 ]z

Ω i=1 n n

akij (x)

Ω i=1 j=1 n |zj |p−2 zj j=1 Ω n p−2 |zj | zj j=1 D

=−

n

Rn

∂φ dx ∂xi

∂φ |zj |p−2 zj dx ∂xi

(aj k (x), Dφ)Rn dx

Rn dx (aj k (x), Dφ)

' ( |zj |p−2 zj div aj k , φ

j=1

W −1, q (D);W01, p (D)

= Jk .

(16.67)

Then, using (16.57), we get lim Jk =

k→∞

=

n j=1 n

|zj |p−2 zj lim

k→∞

' ( −div aj k , φ

W −1, q (D);W01, p (D)

' ( |zj |p−2 zj −div aj 0 , φ

W −1, q (D);W01, p (D)

j=1

.

(16.68)

Making the reverse transformations with (16.68) as we did it in (16.67), we come to the relation lim −div a(Ak , z), φ W −1, q (Ω);W 1, p (Ω)

k→∞

0

= −div a(A0 , z), φ W −1, q (Ω);W 1, p (Ω) . (16.69) 0

Since, for every i = 1, . . . , n, the sequences {div ai k }k∈N are strongly convergent in W −1, q (D), it follows from (16.67)–(16.69) that lim −div a(Ak , z), φk W −1, q (Ω);W 1, p (Ω)

k→∞

0

= −div a(A0 , z), φ W −1, q (Ω);W 1, p (Ω) 0

(16.70)

600


for each sequence {φk }k∈N ⊂ C0∞ (Ω) such that φk φ in W01, p (Ω). Thus, summing up the above results, we obtain div a(Ak , z) → div a(A0 , z) strongly in W −1, q (Ω), (16.71) a(Ak , z) = Ak [z p−2 ]z A0 [z p−2 ]z weakly-∗ in L∞ (Ω). As a result, combining properties (16.66) and (16.71), it has been shown that all suppositions of Lemma 16.12 are fulfilled. So, taking into account (16.58), (16.66), and (16.71) and passing to the limit in inequality (16.65) as k → ∞, we get φ(x) (ξ − a(A0 , z), D y0 − z)Rn dx ≥ 0, ∀z ∈ Rn , Ω

for all positive φ ∈ C0∞ (Ω). After localization, we have (ξ − a(A0 , z), Dy0 − z)Rn ≥ 0,

∀z ∈ Rn .

Since the operator A1 is monotone, it follows that ξ = a(A0 , Dy0 ) = A0 (x)[Dy0p−2 ]Dy0 . As a result, (16.63) takes the form * ) −div A0 (x)[Dy0p−2 ]Dy0 + a0 |y0 |p−2 |y0 | = f

in

(16.72)

D (Ω).

Hence, the τ -limit pair (A0 , y0 ) is an admissible to the problem (16.47)– (16.49), and this concludes the proof. Now, we can turn to the existence of optimal pairs. Theorem 16.15. Under the control admissibility hypothesis Uad = Ub ∩ Usol = ∅, the optimal control problem (16.47)–(16.49) admits at least one solution (Aopt , y opt ) ∈ Ξ ⊂ L∞ (D; Rn×n ) × W01, p (Ω) for every open set Ω ⊆ D. Proof. Due to Proposition 16.4, the control admissibility condition ensures the existence of a minimizing sequence {(Ak , yk ) ∈ Ξ}k∈N , that is, lim I Ω (Ak , yk ) =

k→∞

inf (A,y)∈ Ξ

I Ω (A, y) < +∞.

(16.73)

Since the controls {Ak ∈ Uad }k∈N are uniformly bounded in L∞ (D; Rn×n ), by the a priori estimate (16.18) it follows that the minimizing sequence is

16.4 Domain perturbations for optimal control problems in coefficients

601

bounded in L∞ (D; Rn×n ) × W01, p (Ω) and, by taking a subsequence, we may assume that there exists a pair (A∗ , y∗ ) such that Ak A∗ weakly-∗ in L∞ (D, Rn×n ),

yk y ∗ in W01, p (Ω).

By Theorem 16.14, the pair (A∗ , y ∗ ) is admissible to the problem (16.47)– (16.49). Moreover, since the cost functional I is lower τ -semicontinuous, we get I Ω (A∗ , y ∗ ) ≤ lim inf I Ω (Ak , yk ) = inf I Ω (A, y). k→∞

∗

(A, y)∈ Ξ

∗

Hence, (A , y ) is an optimal pair. Remark 16.16. The argument used in the proof of Theorem 16.15 is related to the so-called “direct method” in the Calculus of Variations which, roughly speaking, aims to construct a minimizing sequence {(Ak , yk ) ∈ Ξ}k∈N . However, from the practical point of view, it makes sense to use an “indirect method” in the study of control problems, based on the first order necessary optimality conditions (see [169, 219, 226]). Since there is no good topology a priory given on the set of all open subsets of Rn , we study the stability properties of the original control problem imposing some constraints on the domain perturbation. Namely, we consider three types of domain perturbations: the so-called topologically admissible perturbations (following Dancer [83]), perturbations in the Hausdorff complementary topology (following Bucur and Zolesio [40]), and the so-called parametrical perturbations (following Liu, Neittaanm¨ aki, and Tiba [178]). In particular, we show that in this case the sequence of the sets of admissible pairs {Ξε }ε>0 is compact with respect to the Mosco-set convergence in L∞ (D; Rn×n ) × H01 (D). Further, we study the stability properties of the optimal control problem (16.1)–(16.4) under the domain perturbation. Our treatment of this question is based on a new stability concept for optimal control problems. We show that Mosco-stable optimal control problems possess “good” variational properties which allow using the optimal solutions to the perturbed problems in more “simple” domains as a basis for the construction of approximate controls for the original control problem.

16.4 H c- and t-admissible domain perturbations for optimal control problems in coefficients opt The aim of this section is to study how the solutions (Aopt ε , yε ) of the OCP |yε (x) − yd (x)|p dx I Ωε (Aε , yε ) = Ωε + |Dyε (x)|pRn dx + J(Aε ) −→ inf, (16.74) Ωε

602


−div Aε (x)[Dyεp−2 ]Dyε + a0 |yε |p−2 yε = f yε ∈

W01, p (Ωε ),

in

Aε ∈ Uad ,

Ωε ,

(16.75) (16.76)

depend on a perturbation Ωε of a fixed domain Ω ⊆ D. As before, we suppose that f ∈ Lq (D) and yd ∈ Lp (D) are given functions, J : L∞ (D; Rn×n ) → R is a lower semicontinuous functional with respect to the weak-∗ topology of L∞ (D; Rn×n ) and this functional is continuous with respect to the strong topology of L∞ (D; Rn×n ), and the set of admissible controls Uad and, hence, the corresponding set of admissible solutions Ξε ⊂ L∞ (D; Rn×n ) × W01, p (Ωε ) are nonempty for every ε > 0. Before we give the precise definition of the shape stability of the above problem and admissible perturbations for the open set Ω, we remark that neiHc ther the set convergence Ωε −→ Ω in the Hausdorff complementary topology top (see Definition 7.36) nor the topological set convergence Ωε −→ Ω (see Definition 7.43) is a sufficient condition to prove the shape stability of control probopt lem (16.47)–(16.49). In general, a limit pair for the sequence {(Aopt ε , yε )}ε>0 , c under H -perturbations of Ω, can be a nonadmissible pair to the original problem (16.47)–(16.49). We refer to [82] for simple counterexamples. So, we have to impose some constraints on the moving domain. In view of this, we begin with the following concepts. Definition 16.17. Let Ω and {Ωε }ε>0 be open subsets of D. We say that the sets {Ωε }ε>0 form an H c -admissible perturbation of Ω if the following hold: Hc

(i) Ωε −→ Ω as ε → 0; (ii) Ωε ∈ Ww (D) for every ε > 0, where the class of subsets Ww (D) is defined in (7.36). Definition 16.18. Let Ω and {Ωε }ε>0 be open subsets of D. We say that the sets {Ωε }ε>0 form a topologically admissible perturbation of Ω (in short, top

t-admissible), if Ωε −→ Ω in the sense of Definition 7.43. Remark 16.19. As Theorem 7.41 indicates, a subset Ω ⊂ D admits the existence of H c -admissible perturbations if and only if Ω belongs to the family Ww (D). However, this condition is not very restrictive. Indeed, it turns out that the assertion “y ∈ W 1, p (Rn ), Ω ∈ Ww (D), and supp y ⊂ Ω imply y ∈ W01, p (Ω)” is not true, in general. In particular, it fails in the case when the open domain Ω has a crack. So, Ww (D) is a rather general class of open subsets of the domain D. Remark 16.20. The above remark motivates us to say that Ω ⊂ D is p-stable if for any y ∈ W 1, p (Rn ) such that y = 0 almost everywhere on int Ω c , we get y| Ω ∈ W01, p (Ω). Note that this property holds for all reasonably regular domains such as Lipschitz domains. A more precise discussion of this property may be found in [83].


603

We begin with the following result. Proposition 16.21. Let Ω ∈ Ww (D) be a fixed subdomain of D and let {Ωε }ε>0 be an H c -admissible perturbation of Ω. Let {(Aε , y Ωε , Aε ) ∈ ΞΩε }ε>0 be a sequence of admissible pairs for the problems (16.74)–(16.76). Then {(Aε , y Ωε , Aε )}ε>0 is uniformly bounded in L∞ (D; Rn×n ) × W01, p (D), and for every τ -cluster pair (A∗ , y ∗ ) ∈ L∞ (D; Rn×n ) × W01, p (D) of this sequence, we have A∗ ∈ Uad , ∗ A [D(y ∗ )p−2 ]Dy ∗ , Dϕ

Rn dx+ a0 |y ∗ |p−2 y ∗ ϕ

dx D D = fϕ

dx, ∀ ϕ ∈ C0∞ (Ω).

(16.77)

(16.78)

D

Proof. For simplicity, we write yε = y Ωε , Aε . As usual, the zero-extension of yε to Rn is denoted by y ε . Since each of the pairs (Aε , yε ) is admissible to the corresponding problem (16.74)–(16.76), the uniform boundedness of the sequence {(Aε , y ε )}ε>0 with respect to the norm of L∞ (D; Rn×n ) × W01, p (D) is a direct consequence of Proposition 16.5 and Definition 16.10. Therefore, we may assume that there exists a pair (A∗ , y ∗ ) such that (up to passing to τ a subsequence still indexed by ε) (Aε , y ε ) −→ (A∗ , y∗ ) in L∞ (D; Rn×n ) × 1, p W0 (D). Then, in view of Proposition 16.11, we have A∗ ∈ Uad . Hc

∞ Let us take as a test function ϕ 0 (Ω). ∈1,C Since Ωε −→ Ω, then by p Theorem 7.41, the Sobolev spaces W0 (Ωε ) ε>0 converge in the sense of Mosco to W01, p (Ω). the function ϕ ∈ W01, p (Ω) fixed earlier, there Hence,1, for p

ε → ϕ

strongly in W 1, p (D) exists a sequence ϕε ∈ W0 (Ωε ) ε>0 such that ϕ (see property (M1 ) of Definition 7.39). Since (Aε , yε ) is an admissible pair for the corresponding problem in Ωε , we can write p−2 p−2 Aε D (yε ) Dyε , Dϕε Rn dx + a0 |yε | yε ϕε dx = f ϕε dx, Ωε

Ωε

Ωε

and hence, p−2 Aε D ( yε ) D yε , Dϕ

ε Rn dx + a0 | yε |p−2 y ε ϕ

ε dx D

(16.79)

D

fϕ

ε dx, ∀ ε > 0.

=

(16.80)

D

To prove (16.78), we pass to the limit in the integral (16.80) identity as ε → 0. Using the arguments of the proof of Theorem 16.14, we have

604


div ai ε → div a∗i strongly in W −1, q (D), ∀ i = 1, . . . , n, [D( yε )p−2 ]D yε ε>0 is bounded in Lq (D), q = p/(p − 1), p−2 | yε | y ε ε>0 is bounded in Lq (D), y ε → y∗ in Lp (D),

y ε (x) → y ∗ (x) for a.e. x ∈ D,

| yε |p−2 y ε |y ∗ |p−2 y ∗

in Lq (D),

where Aε = [a1 ε , . . . , an ε ] and A∗ = [a∗1 , . . . , a∗n ]. yε |p−2 y ε ε>0 . It is clear that Consider the sequence fε := f − a0 | fε → f0 = f − a0 |y ∗ |p−2 y ∗

strongly in

W −1, q (D).

In view of this and the a priori estimate (16.18), the sequence {a(Aε (x), D y ε )}ε>0 is bounded in Lq (D). So, up to a subsequence, we may suppose that there exists a vector function ξ ∈ Lq (D) such that yε )p−2 ] D y ε ξ ξε := a(Aε (x), D y ε ) = Aε [D (

in Lq (D).

in W 1, p (D) and passing to the limit as Using the strong convergence ϕ

ε → ϕ ε → 0 in relation (16.80), we obtain f − a0 |y ∗ |p−2 y ∗ ϕ

dx. (16.81) (ξ, Dϕ)

Rn dx = D

D

It remains to show that ξ = A∗ D (y ∗ )p−2 Dy∗ .

(16.82)

This, however, can be done as in (16.72). In particular, we should repeat all the arguments of that proof, replacing Ω by D, Ak by Aε , yk by y ε , A0 by

A∗ , y0 by y ∗ , and φ by φ. As a result, we show that representation (16.82) holds true, and hence the integral identity (16.81) takes the form of the desired equality (16.78). The proof is completed. Our next intention is to prove that every τ -cluster pair (A∗ , y ∗ ) ∈ L (D; Rn×n ) × W01, p (D) for the sequence {(Aε , y Ωε , Aε ) ∈ ΞΩε }ε>0 (see Proposition 16.21) is admissible to the original OCP (16.47)–(16.49). As follows from (16.77) and (16.78), it remains to show that y ∗ |Ω ∈ W01, p (Ω) and, hence, (A∗ , y ∗ ) ∈ Ξ. To do so, we recall the following result which is the direct consequence of Theorem 1.1 from [39]. ∞

Hc

Lemma 16.22. Let Ω, {Ωε }ε>0 ∈ Ww (D) and Ωε −→ Ω as ε → 0. Let A0 be any fixed matrix of the set Mαβ (D). Then v Ωε , h → v Ω, h strongly in W01, p (D),

∀ h ∈ W01, p (D),

(16.83)


605

where v Ωε , h and v Ω, h are the unique weak solutions to the boundary value problems −div A0 [Dvp−2 ]Dv + a0 |v|p−2 v = 0 in Ωε , (16.84) v − h ∈ W01, p (Ωε ), and

−div A0 [Dvp−2 ]Dv + a0 |v|p−2 v = 0

in

Ω,

v − h ∈ W01, p (Ω),

(16.85)

respectively. Remark 16.23. Notice that this assertion is not true in the case when top

Ωε −→ Ω. We are now in a position to prove the desired property. Proposition 16.24. Let {(Aε , y Ωε , Aε ) ∈ ΞΩε }ε>0 be a sequence of admissible pairs for the family of the problems (16.74)–(16.76), where {Ωε }ε>0 is some H c -admissible perturbation of the set Ω ∈ Ww (D). If for a subsequence of the given sequence {(Aε , y Ωε , Aε ) ∈ ΞΩε }ε>0 (still denoted by indices ε), we have τ (Aε , y Ωε , Aε ) −→ (A∗ , y∗ ), then

y ∗ = y Ω, A∗ , and hence, (A∗ , y ∗ | Ω ) ∈ Ξ,

where by y Ω, A∗ we denote the weak solution of the boundary value problem (16.48)–(16.49) with A = A∗ . Proof. For simplicity, we use the following notation: yε = y Ωε , Aε , y = y Ω, A∗ . By Propositions 16.4, 16.5, and 16.21, we have (up to passing to a subsequence) Aε A∗ = [a∗1 , . . . , a∗n ] ∈ Uad weakly-∗ in L∞ (D, Rn×n ), y ε y ∗ in W01, p (D), y ∈ W01, p (Ω),

(16.86) (16.87)

y ∈ W01, p (D).

We prove that y ∗ = y . Following Bucur and Trebeschi [39], for every ε > 0 we consider the new boundary value problem −div A∗ [Dϕεp−2 ]Dϕε + a0 |ϕε |p−2 ϕε = 0 in Ωε , (16.88) ϕε = −y ∗ on D \ Ωε . In the weak sense, it means that

606


) *

εp−2 Dϕ

ε , Dψ ε A∗ D ϕ D

Rn

dx +

a 0 |ϕ

ε |p−2 ϕ

ε ψ ε dx = 0,

D

∀ ψ ∈ C0∞ (Ωε ), ∀ ε > 0. (16.89) Taking (16.89) as the test function ψ ε = ϕ

ε + y ∗ − y ε , we have ∗

ε + y ∗ − y ε ) Rn dx A Dϕ

εp−2 Dϕ

ε , D (ϕ D

+

a 0 |ϕ

ε |p−2 ϕ

ε (ϕ

ε + y ∗ − y ε ) dx = 0, ∀ ε > 0. (16.90)

D

Let ϕ ∈ W

1, p

(Ω) be a weak solution of the problem −div A∗ [Dϕp−2 ]Dϕ + a0 |ϕ|p−2 ϕ = 0

in

Ω,

ϕ = −y ∗ on D \ Ω. Then, by Lemma 16.22, we have v Ωε ,h → v Ω,h strongly in W01, p (D). Hence, Dϕ

ε → D ϕ

strongly in Lp (D),

Dϕ

εp−2 Dϕ

ε qLq (D) = D ϕ

qLq (D) ,

ε pLp (D) → D ϕ

pLp (D) = D ϕ

p−2 Dϕ

for a.e. x ∈ D, Dϕ

ε (x) → D ϕ(x) and ϕ

ε → ϕ

strongly in Lp (D), p−2

|ϕ

ε |p−2 ϕ

ε qLq (D) = ϕ

ε pLp (D) → ϕ

pLp (D) = |ϕ|

ϕ

qLq (D) ,

for a.e. x ∈ D. ϕ

ε (x) → ϕ(x) Since the norm convergence together with pointwise convergence implies the strong convergence, it follows that p−2

Rn Dϕ

strongly in Lq (D), Dϕ

εp−2 Dϕ

ε → |D ϕ| p−2 |ϕ

ε |p−2 ϕ

ε → |ϕ|

ϕ

strongly in Lq (D),

D (ϕ

ε + y ∗ − y ε ) Dϕ

weakly in Lp (D) (see (16.87)),

strongly in Lp (D), (ϕ

ε + y ∗ − y ε ) → ϕ Hence, the integral identity (16.90) contains only the products of weakly and strongly convergent sequences. So passing to the limit in (16.90) as ε tends to 0, we get


A∗ D ϕ

p−2 Dϕ,

Dϕ

Rn dx +

D

607

a0 |ϕ|

p dx = 0. D

Taking into account the properties of A∗ and a0 prescribed above, this implies ϕ

= 0 a.e. on D. However, by definition, ϕ

= −y ∗ on D \ Ω. So y ∗ = 0 on D \ Ω, and we obtain the required property y A∗ , Ω = y ∗ |Ω ∈ W01, p (Ω). This completes the proof. The results given above motivate us to study the asymptotic behavior of sequences of admissible pairs {(Aε , y Ωε , Aε ) ∈ ΞΩε }ε>0 for the case of tadmissible perturbations of the set Ω. Proposition 16.25. Let Ω be a p-stable open subset of D. Let {(Aε , y Ωε , Aε ) ∈ ΞΩε }ε>0 be a sequence of admissible pairs for the family of the OCPs (16.74)–(16.76), where {Ωε }ε>0 ⊂ D form a t-admissible perturbation of Ω. Then {(Aε , y Ωε , Aε )}ε>0 is uniformly bounded in L∞ (D; Rn×n ) × W01, p (D), and for every τ -cluster pair (A∗ , y ∗ ) ∈ L∞ (D; Rn×n ) × W01, p (D) of this sequence, we have the following: (j) The pair (A∗ , y ∗ ) satisfies relations (16.77)–(16.78). (jj) The pair (A∗ , y ∗ | Ω ) is an admissible to the problem (16.47)–(16.49), that is, y∗ = y Ω, A∗ , where by y Ω, A∗ we denote the weak solution of the boundary value problem (16.48)–(16.49) with A = A∗ . top

Proof. Since Ωε −→ Ω in the sense of Definition 7.43, it follows that for any ϕ ∈ C0∞ (Ω \ K0 ), we have supp ϕ ⊂ Ωε for all ε > 0 small enough. Moreover, as the set K0 has zero p-capacity, it follows that C0∞ (Ω \ K0 ) is dense in W01, p (Ω). Therefore, the verification of item (j) can be done in a way analogous to theproof of Proposition 16.21, replacing therein the sequence ϕε ∈ W01, p (Ωε ) ε>0 by the function ϕ. As for the remaining assertions, we have to repeat all of the arguments of that proof. To prove assertion (jj), we have to show that y ∗ | Ω ∈ W01, p (Ω). To do so, let B0 be any closed ball not intersecting Ω ∪ K1 . Then it follows from (16.75) and (16.76) that for the pair (Aε , y Ωε , Aε ) ∈ ΞΩε we have y Ωε , Aε = 0 almost everywhere in B0 , whenever the parameter ε is small enough. Since by (j) and the Sobolev embedding theorem, y Ωε , Aε converges to y ∗ strongly in Lp (D), it follows that y ∗ = 0 almost everywhere in B0 . As the ball B0 was chosen arbitrary and K1 is of Lebesgue measure zero, it follows that supp y ∗ ⊂ Ω. Then, by Fubini’s theorem, we have supp y ∗ ⊂ Ω. Hence, using the properties of p-stable domains (see Remark 16.20), we just come to the desired conclusion: y ∗ | Ω ∈ W01, p (Ω). This completes the proof.

608


Corollary 16.26. Let {Aε ≡ A∗ }ε>0 be a constant sequence, where A∗ ∈ Uad is any admissible control. Let y Ωε , A∗ ∈ W01, p (Ωε ) ε>0 be the corresponding solutions of (16.75) and (16.76). Then, under the assumptions of Proposition 16.24 or Proposition 16.25, we have y Ωε , A∗ → y Ω, A∗ strongly in W01, p (D). Proof. As follows from Propositions 16.24 and 16.25, for the sequence of admissible pairs {(A∗ , y Ωε , A∗ ) ∈ ΞΩε }ε>0 , there exists a τ -limit pair (A∗ , y ∗ ) such that y ∗ |Ω = y A∗ , Ω . Denote yε = y Ωε , Aε and y = y Ω, A∗ . We prove strong convergence of y ε to y in W01, p (D) proving the convergence of the norms, that is, we show that

yε W 1, p (D) →

y W 1, p (D) as ε → 0.

(16.91)

Since A∗ ∈ Mαβ (D) and L∞ (D) a0 ≥ 0 a.e. in D, it follows that we can take as an equivalent norm in W01, p (D) the following:

∗

, a0

y A = W 1, p (D) 0

D

∗ A [Dy p−2 ]Dy, Dy Rn dx +

1/p a0 |y|p dx .

D

∗

, a0 As a result, the space W01, p (D), · A endowed with this norm is W 1, p (D) 0

uniformly convex (see [2]). Hence, instead of (16.91), we establish that ∗

∗

, a0 , a0 y A

y ε A W 1, p (D) →

W 1, p (D) as ε → 0.

(16.92)

Using (16.48) and (16.75), we take as the test functions y and y ε , respectively. Then passing to the limit in (16.75), we get ) * lim A∗ [D yεp−2 ]D yε , D yε Rn dx + a0 | yε |p dx ε→0

D

)

∗

, a0 yε A = lim

W 1, p (D) ε→0

= D

∗

p−2

A [D y

*p

D

ε→0

D

]D y , D y Rn dx +

) *p ∗ , a0 . =

y A 1, p W (D)

f y ε dx =

= lim

f y dx D

a0 | y |p dx D

Since (16.92) together with the weak convergence in W01, p (D) implies the strong convergence, we obtain the required conclusion. We a now in a position to prove the main result of this section. Theorem 16.27. Let Ω, {Ωε }ε>0 be open subsets of D. Let ΞΩε ⊂ L∞ (D; Rn×n ) × W01, p (Ωε ) and ΞΩ ⊂ L∞ (D; Rn×n ) × W01, p (Ω)

16.5 p-Perturbation of elliptic optimal control problems in coefficients

609

be the sets of admissible solutions to the OCPs (16.74)–(16.76) and (16.47)– (16.49), respectively. Assume that at least one of the following suppositions holds true: 1. Ω ∈ Ww (D) and {Ωε }ε>0 is an H c -admissible perturbation of Ω. 2. Ω is a p-stable domain and {Ωε }ε>0 is an t-admissible perturbation of Ω. Then the sequence {ΞΩε }ε>0 converges to ΞΩ in the sense of Mosco, that is, the following conditions hold true: (ΞM1 ) For every pair (A, y) ∈ ΞΩ , there exists a sequence {(Aε , yε ) ∈ ΞΩε }ε>0 such that Aε → A strongly in L∞ (D; Rn×n ) and y ε → y strongly in W01, p (D). (ΞM2 ) If {εk }k∈N is a numerical sequence converging to 0, and the sequence {(Ak , yk )}k∈N is such that (Ak , yk ) ∈ ΞΩεk

∀ k ∈ N, and

(Ak , y k ) −→ (A, ψ) in L∞ (D; Rn×n ) × W01, p (D), τ

then there exists a function y ∈ W01, p (Ω) such that y = ψ|Ω and (A, y) ∈ ΞΩ . Proof. To begin, we note that property (ΞM2 ) is a direct consequence of Propositions 16.24 and 16.25. So it remains to check property (ΞM1 ). In view of our initial assumptions, the set of admissible pairs ΞΩ to the problem (16.47)–(16.49) is nonempty. Let (A, y) ∈ ΞΩ be any pair. To construct a sequence {(Aε , yε ) ∈ ΞΩε }ε>0 satisfying property (ΞM1 ), we proceed as follows: Aε = A ∀ ε > 0 and yε = y Ωε ,A is the corresponding solution of the boundary value problem (16.75)–(16.76). Note that such a choice is possible since the matrix A is an admissible control to the problem (16.74)–(16.76) for every ε > 0. Then, by Corollary 16.26, we have y Ωε , A → y Ω, A strongly in W01, p (D). Since y Ω, A∗ is a unique solution to (16.48) and (16.49) and (A, y) ∈ ΞΩ , it follows that y = y Ω, A , and we come to the required conclusion: (A, y ε ) −→ (A, y ) strongly in L∞ (D; Rn×n ) × W01, p (D). This concludes the proof.

16.5 p-Perturbation of elliptic optimal control problems in coefficients In this section, we proceed the study of perturbed elliptic OCPs in coefficients. However, for the sake of simplicity, we restrict our analysis to the linear case,

610


namely the main question is: How the admissible solutions (Aε , yε ) ∈ ΞΩε to the OCP I Ωε (Aε , yε ) = |yε (x) − yd (x)|2 dx Ωε |Dyε (x)|2Rn dx + J(Aε ) −→ inf, (16.93) + Ωε

−div (Aε (x)Dyε ) + a0 yε = f yε ∈

H01 (Ωε ),

Aε ∈ Uad ,

in

Ωε ,

p = 2,

(16.94) (16.95)

depend on a perturbation Ωε of a fixed domain Ω ⊆ D if Ω = p − limε→0 Ωε in the sense of Definition 7.47. Here, the set Uad is defined in Definition 16.10 with the modification that {Q1 , . . . , Qn } is a collection of nonempty compact subsets of H −1 (D). Thus, in contrast to the previous section, we suppose that a perturbation {Ωε }ε>0 of a fixed domain Ω ⊂ D forms a parametrically convergent sequence. In view of this, we begin with the following concepts. Definition 16.28. Let Ω and {Ωε }ε>0 be open subsets of D. We say that the sets {Ωε }ε>0 form a p-admissible perturbation of Ω if the following hold: (i) Ω = p − lim Ωε . ε→0

(ii) Ωε ∈ O for every ε > 0, where the class O is defined in Definition 7.55. older condition, that is, ∂Ωε ∈ (iii) The family {Ωε }ε>0 satisfies the uniform H¨ C 1,α ∀ ε > 0 (see Definition 1.7.1 in [214]). Remark 16.29. Condition (iii) of Definition 16.28 implies that the p-limit domain Ω possesses a similar Hölder condition of the boundary (see [178]). To begin, we prove the following result. Proposition 16.30. Let Ω ∈ O be a fixed subdomain of D and let {Ωε }ε>0 be a p-admissible perturbation of Ω. Let {(Aε , y Ωε , Aε ) ∈ ΞΩε }ε>0 be a sequence of admissible pairs for the problems (16.74) and (16.76). Then {(Aε , y Ωε , Aε )}ε>0 is uniformly bounded in L∞ (D; Rn×n ) × H01 (D) and there is a pair (A∗ , y ∗ ) ∈ ΞΩ ⊂ L∞ (D; Rn×n ) × H01 (Ω) such that (up to a subsequence) the following hold: (i) Aε A∗ weakly-∗ in L∞ (D, Rn×n ). (ii) y ∗ = L−lim y Ωε , Aε as ε → 0 in the sense of Definition 7.52. (iii) y ∗ = y Ω, A∗ is a solution of the boundary value problem (16.48)–(16.49) with A = A∗ .

16.5 p-Perturbation of elliptic optimal control problems in coefficients

611

Proof. As usual, we set yε = y Ωε , Aε . Due to Proposition 16.5 and Definition 16.10, we see that the sequence {(Aε , y ε )}ε>0 of admissible pairs to the problems (16.74) and (16.76) is uniformly bounded in L∞ (D; Rn×n ) × H01 (D). So, in view of Theorem 7.53, we may assume that there exists a pair (A∗ , y ∗ ) such that (up to passing to a subsequence still indexed by ε) Aε A∗ weakly∗ in L∞ (D; Rn×n ) and yε ∈ H01 (Ωε ) locally converge to y ∗ ∈ H 1 (Ω) in the sense of Definition 7.52. Then, by Proposition 16.11, we have A∗ ∈ Uad . Let us show that y ∗ ∈ H01 (Ω) and y ∗ = y Ω, A∗ . We now make use of the Γ -property of parametrical convergence (see Proposition 7.53). Then for any open set G ⊂⊂ Ω, there is a value εG > 0 such that G ⊂ Ωε for all ε ≤ εG . Hence, using the weak formulation of the boundary value problem (16.94)–(16.95), we have that the integral identity (Aε Dyε , Dϕ)Rn dx + a0 yε ϕ dx − χΩε f ϕ

dx G G D ((Aε Dyε , Dϕ)Rn + a0 yε ϕ) , dx, ∀ ϕ ∈ C01 (G) =

(16.96)

Ωε \G

holds true for all ε ∈ (0, εG ]. We can estimate the right-hand side of (16.96): (Aε Dyε , Dϕ)Rn + a0 yε ϕ dx Ωε \G

+ ≤ ϕ C 1 (D) yε H 1 (Ωε ) Ln (Ωε \ G) × max Aε L∞ (D;Rn×n ) , a0 L∞ (D) .

(16.97)

Let y ∗ ∈ H 1 (Ω) be the L-limit of the sequence yε ∈ H01 (Ωε ) ε>0 given by Theorem 7.53. Then, due to Theorem 16.14 and the strong convergence of the characteristic functions χΩε in L2 (D) (see Proposition 7.51), we can pass to the limit as ε → 0 in (16.97). As a result, (16.96) implies the estimate ∗ (A∗ Dy ∗ , Dϕ) n dx + a0 y ϕ dx − fϕ

dx R G

G

Ω

≤ M ϕ C 1 (D)

+

Ln (Ω \ G). (16.98) ∞

We now can take an increasing sequence of open sets {Gj ⊂⊂ Ω}j=1 such that ∪j Gj = Ω. Since the set Ω has the segment property (see Definition 7.55 and Theorem 7.57), that for any fixed function ϕ ∈ H01 (Ω), there it follows 1 exists a sequence ϕj ∈ C0 (Gj ) ε>0 such that ϕ

j → ϕ

strongly in H01 (D) (see [178]). Then (16.98) yields ∗ ∗ ∗ (A Dy , Dϕ)Rn dx + a0 y ϕ dx = fϕ

dx, ∀ ϕ ∈ H01 (Ω). Ω

Ω

Ω

Thus, y ∗ = y Ω, A∗ , and this concludes the proof.

612


We a now in a position to prove the main result of this section. Theorem 16.31. Let Ω, {Ωε }ε>0 be open subsets of D. Let ΞΩε ⊂ L∞ (D; Rn×n ) × H01 (Ωε ) and ΞΩ ⊂ L∞ (D; Rn×n ) × H01 (Ω) be the sets of admissible solutions to the OCPs (16.93)–(16.95) and |y(x) − yd (x)|2 dx I Ω (A, y) = Ω |Dy(x)|2Rn dx + J(A) −→ inf, (16.99) + Ω

−div (A(x)Dy) + a0 y = f y∈

H01 (Ω),

in

Ω,

A ∈ Uad ,

(16.100) (16.101)

respectively. Assume that Ω ∈ O and {Ωε }ε>0 is a p-admissible perturbation of Ω. Then, the sequence {ΞΩε }ε>0 converges to ΞΩ in the following sense: (P1 ) For every pair (A, y) ∈ ΞΩ , there exists a sequence {(Aε , yε ) ∈ ΞΩε }ε>0 such that Aε → A strongly in L∞ (D; Rn×n ) and y = L−lim yε . (P2 ) If {εk }k∈N is a numerical sequence converging to 0 and {(Ak , yk )}k∈N is a sequence satisfying (Ak , yk ) ∈ ΞΩεk ∞

∀ k ∈ N, and

Ak A weakly-∗ in L (D; Rn×n ),

L−lim yk = y,

then (A, y) ∈ ΞΩ . Proof. Since property (P2 ) is a direct consequence of Propositions 16.30, it remains to check property (P1 ). In view of our initial assumptions, the set of admissible pairs ΞΩ to the problem (16.99)–(16.101) is nonempty. Let (A, y) ∈ ΞΩ be any pair. To construct a sequence {(Aε , yε ) ∈ ΞΩε }ε>0 satisfying property (P1 ), we make it as follows: Aε = A, ∀ ε > 0, and yε = y Ωε ,A is the corresponding solution of the boundary value problem (16.94)–(16.95). Note that such a choice is possible since the matrix A is an admissible control to the problem (16.93)–(16.95) for every ε > 0. Then, by Propositions 16.30, we have y ∗ = L−lim yε as ε → 0 in the sense of Definition 7.52. Since y ∗ is a unique solution to (16.100)–(16.101) and (A, y) ∈ ΞΩ , it follows that y = y ∗ , and we come to the required conclusion.

16.6 Mosco-stability of optimal control problems Typically, the mathematical description of an OCP consists of independent mathematical objects such as a state equation, control and state constraints,

16.6 Mosco-stability of optimal control problems

613

and a cost functional. As a rule, each of these ingredients depends on a domain Ω where the control process is studied. Hence, when the domain Ω changes, we obtain, roughly speaking, an absolutely different OCP, probably with another constraints, another cost functional, and so on. Following the classical approach (see, for instance, [38, 39, 80, 83, 84]), if a sequence {Ωε }ε>0 converges to Ω in some sense, then the shape stability of the OCP (16.47)–(16.49) should opt be achieved by proving that any sequence of optimal pairs {(Aopt ε , yε )}ε>0 to the perturbed problems (16.74)–(16.76) is relatively compact with respect to an appropriate topology and each of its limits pairs is an optimal one to the original nonlinear problem (16.47)–(16.49) on the domain Ω. However, the optimal pair may not fully characterize the OCP. As a rule, the full identification of an OCP (including a cost functional, a state equation, and existing restrictions on the controls and state function) is impossible by means of the optimal solution only. Moreover, such an approach to the study of the shape stability is unacceptable when the existence of optimal solutions may fail [38] or when these solutions have nonclassical character (solutions satisfying only the requirements of Tonelli’s partial regularity theorem and they do not have a “variational character”) [17]. Therefore, in order to study the stability of the OCP (16.47)–(16.49) with respect to the variation of the open set Ω ⊂ D, we should be able to compare two control problems defined on two different domains. In general, this question is open now. However, having taken the “asymptotic behavior” of OCPs, when Ωε → Ω in some sense, as a basis to study the stability properties, we come to the following concept (see [139]). Definition 16.32. We say that the OCP (16.47)–(16.49) on Ω is Moscostable in L∞ (D; Rn×n ) × W01, p (D) along the sequence {Ωε }ε>0 which forms some perturbation of Ω if the following conditions are satisfied: (M S1 ) The set of admissible pairs Ξ Ω for (16.47)–(16.49) is the limit in the sense of Mosco of the sequence {ΞΩε }ε>0 of admissible sets for the perturbed problems (16.74)–(16.76). (M S2 ) If {εk }k∈N is a numerical sequence converging to 0 and {(Ak , yk )}k∈N is a sequence such that (Ak , yk ) ∈ ΞΩεk τ

∀ k ∈ N, and ∞

(Ak , y k ) −→ (A, y) in L (D; Rn×n ) × W01, p (D), where (A, y|Ω ) ∈ Ξ Ω , then lim inf I Ωεk (Ak , yk ) ≥ I Ω (A, y|Ω ). k→∞

(M S3 ) For every pair (A, y) ∈ ΞΩ , there exists a sequence {(Aε , yε ) ∈ ΞΩε }ε>0

614


such that Aε → A strongly in L∞ (D; Rn×n ), y ε → y strongly in W01, p (D), and lim sup I Ωε (Aε , yε ) ≤ I Ω (A, y). ε→0

Remark 16.33. Notice that Definition 16.32 can be interpreted as a natural extension of the well-known notion of Γ -convergence of functionals (see Definition 8.9). We do not want here to enter into details of Γ -convergence theory but only emphasize that the Mosco-stable optimal control problem in the sense of Definition 16.32 possesses the fine variational properties that are similar to those of a Γ -limit. We begin with the following result concerning the variational properties of Mosco-stable OCPs. Theorem 16.34 ([139]). Assume that for a given perturbation {Ωε }ε>0 of the domain Ω, the OCP (16.47)–(16.49) on Ω is Mosco-stable in L∞ (D; Rn×n ) × W01, p (D). Let (A0ε , yε0 ) ∈ ΞΩε ε>0 be a sequence of optimal solutions for the corresponding perturbed problems (16.74)–(16.76). Then this sequence is relatively τ -compact in L∞ (D; Rn×n ) × W01, p (D) and every τ -cluster pair is an optimal solution to the original problem (16.47)–(16.49). Moreover, if τ

(A0ε , y ε0 ) −→ (A0 , y 0 ),

(16.102)

then (A0 , y 0 Ω ) ∈ Ξ Ω and inf (A, y)∈ ΞΩ

I Ω (A, y) = I Ω (A0 , y 0 Ω ) = lim

inf

ε→0 (Aε ,yε )∈ Ξ Ωε

I Ωε (Aε , yε ). (16.103)

Proof. Taking the a priori estimate (16.18) into immediately con account, we clude that any sequence of optimal pairs (A0ε , yε0 ) ∈ ΞΩε ε>0 to the perturbed problems (16.74)–(16.76) is relatively τ -compact in L∞ (D; Rn×n )× W01, p (D). So we may suppose that there exist a subsequence (A0εk , yε0k ) k∈ N τ

and a pair (A∗ , y∗ ) such that (A0εk , y ε0k ) −→ (A∗ , y∗ ) as k → ∞. Then by Theorem 16.27 (see property (ΞM2 )), we have (A∗ , y ∗ |Ω ) ∈ Ξ Ω . Hence, in view of property (M S2 ) of Definition 16.32, we get lim inf

IΩεk (A, y) = lim inf IΩεk (A0εk , yε0k )

min

k→∞ (A, y)∈ ΞΩε

k

k→∞

≥ IΩ (A∗ , y ∗ |Ω ) ≥

min (A, y)∈ ΞΩ

= I Ω (Aopt , y opt ). However, condition (M S3 ) implies the existence of a sequence

I Ω (A, y) (16.104)


615

,ε , y,ε ) ∈ Ξ Ω (A ε

ε>0

such that ,ε , y,ε ). ,ε , y,ε ) −→ (Aopt , y opt ) and I Ω (Aopt , y opt ) ≥ lim sup I Ωε (A (A τ

ε→0

Using this fact, we have min (A, y)∈ ΞΩ

,ε , y,ε ) I Ω (A, y) = I Ω (Aopt , y opt ) ≥ lim sup I Ωε (A ε→0

≥ lim sup ε→0

min

(A, y) ∈Ξ Ωε

≥ lim sup

I Ωε (A, y)

min

k→∞ (A, y)∈ ΞΩεk

IΩεk (A, y)

= lim sup IΩεk (A0εk , yε0k ).

(16.105)

k→∞

From this and (16.104), we deduce lim inf IΩεk (A0εk , yε0k ) ≥ lim sup IΩεk (A0εk , yε0k ). k→∞

k→∞

Thus, combining the relations (16.104) and (16.105) and rewriting them in the form of equalities, we obtain IΩ (A∗ , y ∗ |Ω ) = I Ω (Aopt , yopt ) = I Ω (Aopt , y opt ) = lim

min

I Ω (A, y),

(16.106)

IΩεk (A, y).

(16.107)

(A, y)∈ ΞΩ

min

k→∞ (A, y)∈ ΞΩε

k

Since (16.106) and (16.107) hold true for every τ -convergentsubsequence of the original sequence of optimal solutions (A0ε , yε0 ) ∈ ΞΩε ε>0 , it follows that the limits in (16.106) and (16.107) coincide and, therefore, the value I Ω (Aopt , y opt ) is the limit of the whole sequence of minimal values inf I Ωε (A, y) . IΩε (A0ε , yε0 ) = (A,y)∈ Ξ Ωε

ε>0

This concludes the proof. Our next intention is to derive the sufficient conditions for the Moscostability of the OCP (16.47)–(16.49). To do so, we provide the following result. Lemma 16.35. Let Ω be an open subset of D. Assume that a sequence {Ωε }ε>0 forms an admissible perturbation of Ω (in the sense of either Definition 16.17 or 16.18). Let {χ Ωε }ε>0 be the sequence of the associated characteristic functions. Let χ∗ be a weak-∗ limit point of this sequence in L∞ (D; [0, 1]). Then (16.108) χ Ω (1 − χ∗ ) = 0 a.e. on D.

616


Proof. It is clear that for a fixed perturbation {Ωε }ε>0 of the set Ω, up to a subsequence, there exists a function χ∗ such that χ Ωε converges weakly-∗ to χ∗ in L∞ (D; [0, 1]). Let yε ∈ W01, p (Ωε ) ε>0 be any sequence such that y ε y ∗ in W01, p (D) and y ∗ | Ω ∈ W01, p (Ω). Due to Propositions 16.24 and 16.25, such a choice is always possible. Then lim y ε ϕ dx = y ∗ ϕ dx = χ Ω y ∗ ϕ dx, ∀ ϕ ∈ Lq (D). ε→0

D

D

D

On the other hand, using the fact that y ε → y∗ strongly in Lp (D), we have y ε ϕ dx = lim χ Ωε y ε ϕ dx lim ε→0 D ε→0 D χ∗ y ∗ ϕ dx = χ∗ χ Ω y ∗ ϕ dx, ∀ ϕ ∈ Lq (D) = D

D

as a limit of the product of strongly and weakly-∗ convergent sequences. This concludes the proof. We are now in a position to prove the result concerning sufficient conditions for Mosco-stability for the class of the OCPs (16.47)–(16.49). Theorem 16.36. Let Ω be an open subset of D. Assume that the distribution yd ∈ Lp (D) in the cost functional (16.47) is such that yd (x) = yd (x)χ Ω (x)

for a.e. x ∈ D.

(16.109)

Assume also that at least one of the following suppositions holds true: 1. Ω ∈ Ww (D) and {Ωε }ε>0 is an H c -admissible perturbation of Ω. 2. Ω is a p-stable domain and {Ωε }ε>0 is an t-admissible perturbation of Ω. Then the OCP (16.47)–(16.49) is Mosco-stable in L∞ (D; Rn×n ) × W01, p (D). Proof. We prove (M S1 )–(M S3 ) of Definition 16.32. (M S1 ) has been proved in Theorem 16.27. Let {(Ak , yk )}k∈N be a sequence with the properties prescribed in (M S2 ) and let (A, y) be its τ -limit. Then | yεk − yd |p → |y − yd |p 1 strongly in L (D) and yεk pLp (D) ≥ Dy pLp (D) , lim inf D k→∞

by the lower semicontinuity of the norm with respect to the weak convergence. Hence,


617

lim inf I Ωεk (Ak , yk ) k→∞ χΩεk | yεk − yd |p dx + |D yεk |pRn dx + J(Ak ) = lim inf k→∞ D D ≥ χ∗ |y − yd |p dx + |Dy|pRn dx + J(A) = {by (16.109)} D D ∗ p = χ Ω χ |y − yd | dx + |Dy|pRn dx + J(A) = {by (16.108)} Ω D p |y − yd | dx + |Dy|pRn dx + J(A) = I Ω (A, y| Ω ). = Ω

Ω

This completes the verification of (M S2 ). It remains only to verify the last item of Definition 16.32. However, this fact seems to be rather obvious because of the strong convergence (Aε , yε ) → (A, y) in L∞ (D; Rn×n ) × W01, p (D) and properties (16.108) and (16.109). Indeed, in this case we have lim sup I Ωε (Aε , yε ) ε→0 χΩε | yε − yd |p dx + |D yε |pRn dx + J(Aε ) = lim ε→0 D D χ∗ |y − yd |p dx + |Dy|pRn dx + J(A) = D D χ Ω χ∗ |y − yd |p dx + |Dy|pRn dx + J(A) = Ω D p = |y − yd | dx + |Dy|pRn dx + J(A) = I Ω (A, y| Ω ). Ω

Ω

This concludes the proof. To end of this section, we provide more precise information concerning the variational properties of the OCP (16.47)–(16.49) under its Mosco-stable perturbation. Proposition 16.37. Assume that all suppositions of Theorem 16.34 hold true. Let (A0 , y 0 ) be anoptimal pair to the control problem (16.47)–(16.49) and let (A0ε , yε0 ) ∈ ΞΩε ε>0 be a sequence of optimal pairs for the perturbed problems (16.74)–(16.76) such that (A0ε , y ε0 ) −→ (A0 , y 0 ) in L∞ (D; Rn×n ) × W01, p (D). τ

(16.110)

Then condition (16.109) implies that y ε0 → y 0 strongly in W01, p (D), lim

ε→0

and

J(A0ε )

0

= J(A ),

(16.111) (16.112)

618


lim

ε→0

Ωε

Aε0 [D(yε0 )p−2 |Dyε0 , Dyε0 Rn dx 0 A [D(y 0 )p−2 ]Dy 0 , Dy 0 Rn dx. =

(16.113)

Ω

Proof. As follows from Theorem 16.36, for a given perturbation {Ωε }ε>0 of the domain Ω the OCP (16.47)–(16.49) on Ω is Mosco-stable in L∞ (D; Rn×n ) × W01, p (D). Moreover, due to Theorem 16.34, any sequence of optimal pairs for the perturbed problems (16.74)–(16.76) is relatively τ -compact in L∞ (D; Rn×n ) × W01, p (D) and its every τ -cluster pair is an optimal solution to the original problem (16.47)–(16.49). So the supposition (16.110) is not restrictive. To prove (16.111), we make use relation (16.103). Then ε→0

χΩε | yε0

lim

D

|D yε0 |pRn

− yd | dx + dx + D χ Ω | y 0 − yd |p dx + |D y 0 |pRn dx + J(A0 ). (16.114) = p

D

J(A0ε )

D

By the Sobolev embedding theorem, we have y ε0 → y 0 strongly in L0p (D). As a result, using properties (16.108) and (16.109), we obtain lim χΩε | yε0 − yd |p dx = χ∗ χ Ω | y 0 − yd |p dx ε→0 D D χ Ω | y 0 − yd |p dx. (16.115) = D

Combining this with (16.114), we come to the relation lim |D yε0 |pRn dx + J(A0ε ) = |D y 0 |pRn dx + J(A0 ). ε→0

D

D

Taking into account the lower semicontinuity property of the functional J, we have lim

ε→0

D

lim J(A0ε ) = J(A0 ), 0 p |D yε |Rn dx = |D y 0 |pRn dx. ε→0

D

Then the last equality together with the weak convergence in W01, p (D) implies (16.111). It remains only to prove the convergence of energies (16.113). To do this, we use (16.48) and (16.75) replacing therein y by y 0 and yε by yε0 , respectively. Then for the corresponding integral identities, we take as the test functions y 0 and y ε0 , respectively. As a result, passing to the limit in (16.75), we get


)

lim

ε→0

D

619

* A0ε [D( yε0 )p−2 ]D yε0 , D yε0 Rn dx + a0 | yε0 |p dx D 0 f y ε dx = f y 0 dx = lim ε→0 D D 0 0 p−2 y ) ]D y 0 , D y 0 Rn dx = A [D( D + a0 | y 0 |p dx. D

To this end, it remains only to use the equality a0 | yε0 |p dx = a0 | y 0 |p dx lim ε→0

D

(see (16.115)). The proof is completed.

D

In conclusion, we note that the characteristic feature of the above problem is the fact that the “real” lattice structure (junction structure, thing structure) Ω is never perfectly smooth but contains microscopic asperities of the size significantly smaller than characteristic length scale of the domain. So a direct numerical computation of the solutions of OCPs in such domains is extremely difficult. Usually, a very fine discretization mesh is needed, which means significant computation time, and such a computation is often irrelevant. In view of the variational properties of Mosco-stable problems (see Theorem 16.34 and Proposition 16.37), the above approach allows us to replace the “rough” domain Ω by a family of more “regular” domains {Ωε }ε>0 forming some admissible perturbation and to approximate the original problem by the corresponding perturbed problems.

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Index

G-convergence, 152 H-convergence, 592 H c -admissible perturbation of a domain, 602 K-lower limit of sets, 234 K-upper limit of sets, 234 ε-approximate solution, 114 ε-suboptimal solution, 116 γ-convergence of Borel measures, 260 γp -convergence of sets, 258 D-weak compactness, 59 D-weak convergence, 59 μ-Laplacian, 55 σ(X∗ , X)-locally convex topology, 58 ε-periodic graphlike structures, 410 p-admissible perturbation of a domain, 610 p-connected measure, 182 p-stable domain, 602 t-admissible perturbation of a domain, 602

compactness condition, 65 constrained Γ -limit in a Banach space, 271 constrained Γ -limit in a metric space, 269 convergence in the sense Kuratowski, 234 cylindrically perforated domain, 547 distributed singular system, 110 divergence, 25 divergence with respect to the measure, 54 Ekeland metric in L∞ (D), 218 epigraph of a cost functional, 157 finely Borel subset, 46 Friedrichs inequality, 23, 33 Gˆ ateaux derivative, 102 Green formula, 23

approximation of singular measure, 189 Banach–Alaoglu theorem, 58 Borel measure, 15 Borel measure space, 15 Borel subsets, 15 Brinkman-type law, 550 by-component homogenization, 134 capacity of sets, 44 Clarkson inequalities, 178 coerciveness condition, 65

H¨ older inequality, 71 Hausdorff distance, 236 Hausdorff outer measure, 18 ill-posed control object, 83 Lagrange function, 101 Lagrange multiplier principle, 101 Lebesgue points, 17 Lemma on Compensated Compactness, 595

P.I. Kogut, G.R. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains, Systems & Control: Foundations & Applications, DOI 10.1007/978-0-8176-8149-4, © Springer Science+Business Media, LLC 2011

635

636

Index

mean value of periodic function, 33 mean value property, 33 Mosco-convergence of Sobolev spaces, 238 Mosco-stable optimal control problem, 613 nondegenerate measure, 184 parametrical convergence of open sets, 243 Poincaré inequality, 24 Pompeiu–Hausdorff distance, 237 potential vector field, 25 problem of concentration, 38 problem of wild oscillation, 38 quasi-continuous functions, 44 quasi-open subsets, 44

starting control, 91 state functional, 159 steady-state Navier–Stokes equation, 549 strong χ-convergence, 219 strong two-scale convergence, 174 structure of critical thickness, 6 sufficiently thick structure, 5 sufficiently thin structure, 5 tangential gradient, 53 tangential space, 53 the weak approximation property, 491 thick multistructure, 478 thin structure, 4 Tikhonov minimizing sequence, 115 topological set convergence, 240 total variation of a measure, 16 trace operator, 22

Radon positive measure, 16 Radon–Nikodym decomposition, 17 Radon–Nikodym Theorem, 17 reduced defect measure, 38 regular set of the measure, 46 regularity condition, 65 rigid optimal control, 86

uniform set convergence, 236

scheme of homogenization of OCP, 158 set convergence in the Hausdorff complementary topology, 238 set convergence in the Hausdorff metric, 236 singular set of the measure, 46 smoothing operator, 190 solenoidal periodic field, 34 solenoidal vector field, 25

weak ε-approximate solution, 115 weak approximation property with respect to the scale of spaces, 284 weak minimizing sequence, 115 weak two-scale convergence, 174 weakened ε-approximate solution, 116 weakened minimizing sequence, 116 weakly convergent sequence, 35 weakly-∗ convergent sequence, 36

variational limit of constrained minimization problems, 276 variational limit of constrained minimization problems in variable spaces, 284

Optimal Control Problems for Partial Differential Equations on Reticulated Domains: Approximation and Asymptotic Analysis

Partial Differential Equations V: Asymptotic Methods for Partial Differential Equations

Partial Differential Equations V: Asymptotic Methods for Partial Differential Equations

Partial Differential Equations V: Asymptotic Methods for Partial Differential Equations

Constrained Optimization and Optimal Control for Partial Differential Equations

Constrained optimization and optimal control for partial differential equations

Asymptotic analysis and numerical solution of partial differential equations

Inverse Problems for Partial Differential Equations

Inverse problems for partial differential equations

Inverse Problems for Partial Differential Equations

Control of partial differential equations

Problems in Distributions and Partial Differential Equations

Analysis of Singularities for Partial Differential Equations

Analysis of singularities for partial differential equations

Numerical approximation of partial differential equations

Numerical approximation of partial differential equations

Boundary value problems and partial differential equations

Boundary Value Problems: and Partial Differential Equations

Control of partial differential equations and applications

Partial Differential Equations V: Asymptotic Methods for Partial Differential Equations (Encyclopaedia of Mathematical Sciences) (v. 5)

Lectures on Partial Differential Equations

Lectures on Partial Differential Equations

Lectures on Partial Differential Equations

Lectures on Partial Differential Equations

Lectures on Partial Differential Equations

Lectures on partial differential equations

Lectures on Partial Differential Equations

Notes on Partial Differential Equations

Lectures On Partial Differential Equations

Harmonic analysis and partial differential equations

Fourier Analysis and Nonlinear Partial Differential Equations

Optimal Control Problems for Partial Differential Equations on Reticulated Domains: Approximation and Asymptotic Analysis

Partial Differential Equations V: Asymptotic Methods for Partial Differential Equations

Partial Differential Equations V: Asymptotic Methods for Partial Differential Equations

Partial Differential Equations V: Asymptotic Methods for Partial Differential Equations

Constrained Optimization and Optimal Control for Partial Differential Equations

Constrained optimization and optimal control for partial differential equations

Asymptotic analysis and numerical solution of partial differential equations

Inverse Problems for Partial Differential Equations

Inverse problems for partial differential equations

Inverse Problems for Partial Differential Equations

Control of partial differential equations

Problems in Distributions and Partial Differential Equations

Analysis of Singularities for Partial Differential Equations

Analysis of singularities for partial differential equations

Numerical approximation of partial differential equations

Numerical approximation of partial differential equations

Boundary value problems and partial differential equations

Boundary Value Problems: and Partial Differential Equations

Control of partial differential equations and applications

Partial Differential Equations V: Asymptotic Methods for Partial Differential Equations (Encyclopaedia of Mathematical Sciences) (v. 5)

Lectures on Partial Differential Equations

Lectures on Partial Differential Equations

Lectures on Partial Differential Equations

Lectures on Partial Differential Equations

Lectures on Partial Differential Equations

Lectures on partial differential equations

Lectures on Partial Differential Equations

Notes on Partial Differential Equations

Lectures On Partial Differential Equations

Harmonic analysis and partial differential equations

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