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Control and Boundary Analysis

edited by

John Cagnol Pole Universitaire Leonard de Vinci Paris, France

Jean-Paul Zolésio CNRS/INRIA, Projet OPALE Valbonne, France

Boca Raton London New York Singapore

A CRC title, part of the Taylor & Francis imprint, a member of the Taylor & Francis Group, the academic division of T&F Informa plc.

Copyright © 2005 Marcel Dekker, Inc.

DK3828 disclaimer.fm Page 1 Tuesday, January 4, 2005 1:16 PM

Library of Congress Cataloging-in-Publication Data Control and boundary analysis / edited by John Cagnol, Jean-Paul Zolésio. p. cm. — (Lecture notes in pure and applied mathematics ; v. 239) Includes bibliographical references and index. ISBN 1-57444-594-4 (alk. paper) 1. Control theory—Congresses. 2. Boundary element methods—Congresses. 3. Differential equations, Partial—Congresses. I. Cagnol, John. II. Zolésio, J. P. III. Series. QA402.3.C6175 2005 003'.5--dc22

2004061831

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. All rights reserved. Authorization to photocopy items for internal or personal use, or the personal or internal use of specific clients, may be granted by CRC Press, provided that $1.50 per page photocopied is paid directly to Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923 USA. The fee code for users of the Transactional Reporting Service is ISBN 1-57444-594-4/05/$0.00+$1.50. The fee is subject to change without notice. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. The consent of Marcel Dekker and CRC Press does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from Marcel Dekker/CRC Press for such copying. Direct all inquiries to CRC Press, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe.

Visit the CRC Press Web site at www.crcpress.com © 2005 by Marcel Dekker/CRC Press No claim to original U.S. Government works International Standard Book Number 1-57444-594-4 Library of Congress Card Number 2004061831 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper

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Foreword

This volume comprises selected papers from the 21st Conference on System Modeling and Optimization that took place from July 21 to July 25, 2003, in Sophia Antipolis, France. These papers pertain to moving boundary and boundary control in systems described by partial diﬀerential systems and are speciﬁcally relevant to Working Group 7.2 of the International Federation for Information Processing (IFIP). In the 1970s, the study of partial diﬀerential equations focused on the modeling of continuous mechanics involving ﬁxed boundary, that is, in abstract form, autonomous systems. At the same time, control theory, which was born in the ﬁeld of ﬁnite dimensional systems, emerged in the domains of active and passive control of partial diﬀerential equations at the boundary. In the 1980s, shape optimization became a mathematical area that used control theory extended to moving boundaries. In that sense, it included free boundary and phase transition problems. More recently, topological shape optimization has been developed as an alternative approach to the homogenization process in material sciences. Currently, the focus is on moving boundary analysis, which combines and extends all of the previous mathematical and numerical backgrounds. In this book, all of these aspects are covered by the selected papers, including the important stochastic approach. New existence results are derived from ﬁne analysis of boundary behavior; the ﬁrst variation of the boundary is taken into account by a transverse backward vector ﬁeld. Intrinsic geometry is developed for shell modeling, avoiding local representations, such as the Christoﬀel symbol. Non-diﬀerentiabilities are involved when the moving domain is a contact area related to some unilateral problems. New algorithms associated with current computing power allow impressive simulations for particle ﬂow. This leads to numerical treatment of the mathematical speed method, or Eulerian parametrization tool. John Cagnol and Jean-Paul Zol´esio

Copyright © 2005 Marcel Dekker, Inc.

Contributing Authors

Viorel Barbu Alexandru Ioan Cuza University, Romania Youssef Belhamadia Universit´e Laval, Canada John Cagnol Pˆ ole Universitaire L´eonard de Vinci, France Giuseppe Da Prato Scuola Normale Superiore di Pisa, Italy Edward J. Dean University of Houston, USA Michel C. Delfour Universit´e de Montr´eal, Canada Zdzislaw Denkowski Jagiellonian University, Poland Nicolas Doyon Universit´e Laval, Canada Karsten Eppler Weierstraß Institut f¨ ur Angewandte Analysis und Stochastik, Germany Lorella Fatone Universit`a di Modena e Reggio Emilia, Italy Andr´ e Fortin Universit´e Laval, Canada H´ el` ene Frankowska CNRS, CREA, Ecole Polytechnique, France

Copyright © 2005 Marcel Dekker, Inc.

Leszek Gasi´ nski Jagiellonian University, Poland Roland Glowinski University of Houston, USA Martin Gugat University of Erlangen-Nuremberg, Germany Scott W. Hansen Iowa State University, USA Helmut Harbrecht Christian–Albrechts–Universit¨at zu Kiel, Germany Mary Ann Horn Vanderbilt University, USA Michal Koˇ cvara University of Erlangen, Germany Irena Lasiecka University of Virginia, USA Catherine Lebiedzik Wayne State University, USA Kazimierz Malanowski Polish Academy of Sciences, Poland Helmut Maurer Westf¨alische Wilhelms Universit¨at, Germany Stanislaw Mig´ orski Jagiellonian University, Poland Kirsten Morris University of Waterloo, Canada Marwan Moubachir INRIA, France Carmeliza Navasca University of California, USA

Copyright © 2005 Marcel Dekker, Inc.

Arian Novruzi University of Ottawa, Canada Jiˇ ri V. Outrata Academy of Sciences of the Czech Republic, Czech Republic Tsorng-Whay Pan University of Houston, USA Sabine Pickenhain Brandenburgische Technische Universit¨at, Germany Michael P. Polis Oakland University, USA Vicentiu Radulescu University of Craiova, Romania Mohammad A. Rammaha University of Nebraska-Lincoln, USA Maria Cristina Recchioni Universit`a Politecnica delle Marche, Italy Irina F. Sivergina Kettering University, USA Jan Sokolowski Universit´e Henri Poincar´e Nancy I, France Xudong Yao Texas A&M University, USA Jianxin Zhou Texas A&M University, USA Francesco Zirilli Universit`a di Roma La Sapienza, Italy ˙ Antoni Zochowski Polish Academy of Sciences, Poland Jean-Paul Zol´ esio CNRS, INRIA, France

Copyright © 2005 Marcel Dekker, Inc.

Contents

I. Operator-Splitting Methods and Applications to the Direct Numerical Simulation of Particulate Flow and to the Solution of the Elliptic Monge-Amp` ere Equation 1 Edward J. Dean, Roland Glowinski, Tsorng-Whay Pan 1 Operator-Splitting Schemes for the Time-Discretization of Initial Value Problems 2 Operator-Splitting Methods for the Direct Numerical Simulation of Particulate Flow 3 An Operator-Splitting Method for the Elliptic Monge-Amp`ere Equations in Two-Dimension

14

II. Dynamical Shape Sensitivity

29

Marwan 1 2 3 4 5

30 30 31 33 36

Moubachir, Jean-Paul Zol´esio Introduction Mechanical Problem Inverse Problem Dynamical Shape Sensitivity of the Fluid Conclusion

2 6

III. Optimal Control of a Structural Acoustic Model with Flexible Curved Walls

37

John Cagnol, Catherine Lebiedzik 1 Introduction 2 Background and Literature 3 Statement of Main Results 4 Intrinsic Geometry 5 The Unforced, Conservative Shell Model 6 Structurally Damped, Forced Shell Wall of an Acoustic Chamber 7 Optimal Control Problem 8 Proof of Lemma 3

37 38 39 41 42 45 47 48

IV. Nonlinear Wave Equations with Degenerate Damping and Source Terms

53

Viorel Barbu, Irena Lasiecka, Mohammad A. Rammaha 1 Deﬁnitions and Main Results 2 Outline of the Proof of Theorem 3

55 58

Copyright © 2005 Marcel Dekker, Inc.

V. Numerical Modeling of Phase Change Problems

63

Andr´e Fortin, Youssef Belhamadia 1 Stefan Problem and Semi-Phase-Field Formulation 2 Finite Element Discretization 3 Adaptive Strategy 4 Numerical Results 5 Conclusions

64 65 65 66 67

VI. Shape Optimization of Free Air-Porous Media Transmission Coeﬃcient

73

Arian Novruzi 1 Computation of the Porous Shape 2 Optimal Porous Shape Domain

76 81

VII. The Uniform Fat Segment and Uniform Cusp Properties in Shape Optimization

85

Michel C. Delfour, Nicolas Doyon, Jean-Paul Zol´esio 1 Preliminaries: Topologies on Families of Sets 2 The New Uniform Fat Segment Property 3 Equivalence of the Uniform Cusp and Fat Segment Properties

86 87 91

VIII. Topology Optimization for Unilateral Problems

97

˙ Jan Sokolowski, Antoni Zochowski 1 Transformations of the Energy Functional for the Laplace Equation 2 Signorini Problem

98 102

IX. Second Order Lagrange Multiplier Approximation for Constrained Shape Optimization Problems

107

Karsten Eppler, Helmut Harbrecht 1 Shape Optimization 2 Optimization of Constrained Problems 3 Numerical Results

X. Mathematical Models of “Active” Obstacles in Acoustic Scattering Lorella Fatone, Maria Cristina Recchioni, Francesco Zirilli 1 The Mathematical Models of the Masking and Ghost Obstacle Problems 2 The Optimality Conditions for the Ghost Obstacle Problem and Some Numerical Experience

XI. Local Null Controllability in a State Constrained Thermoelastic Contact Problem Irina F. Sivergina, Michael P. Polis 1 Control of the Linear System Copyright © 2005 Marcel Dekker, Inc.

108 112 114

119 122 125

131 135

2 3 4

Nonlinear Case: Application of a Fixed Point Theorem The State Constrained Null Control Problem Conclusion

XII. On Sensitivity of Optimal Solutions to Control Problems for Hyperbolic Hemivariational Inequalities Zdzislaw 1 2 3 4

136 143 143

145

Denkowski, Stanislaw Mig´ orski Class of Problems Preliminaries Sensitivity of Solution Sets for Second Order HVIs Complementary Γ-convergence of Cost Functionals

146 147 151 153

XIII. Evolution Hemivariational Inequality with Hysteresis and Optimal Control Problem

157

Leszek Gasi´ nski 1 Preliminaries 2 Hemivariational Inequality with Hysteresis 3 Estimates on the Solutions of (HVI) 4 Optimal Control Problem

XIV. On the Modeling and Control of Delamination Processes Michal Koˇcvara, Jiˇri V. Outrata 1 Introduction 2 Delamination Process 3 Optimality Conditions 4 Optimization of Delamination Processes 5 Conclusion

XV. On a Spectral Variational Problem Arising in the Study of Earthquakes Vicentiu Radulescu 1 Main Results and Physical Motivation 2 Proofs

XVI. Nodal Control of Conservation Laws on Networks Martin Gugat 1 Notation 2 The Linearized Problem 3 Existence of Solutions 4 Directional Diﬀerentiability 5 Evaluation of Directional Derivatives 6 Example 7 Conclusion Copyright © 2005 Marcel Dekker, Inc.

158 161 163 164

169 169 171 177 182 185

189 190 194

201 202 204 205 206 206 211 214

XVII. Invariance of Closed Sets under Stochastic Control Systems Giuseppe 1 2 3

Da Prato, H´el`ene Frankowska Preliminaries Necessary and Suﬃcient Conditions for the Invariance Invariance of Stochastic Control Systems

XVIII. Uniform Stabilization of an Anisotropic System of Thermoelasticity Mary Ann Horn 1 Statement of the Problem 2 Proof of Theorem 3: Uniform Stabilization 3 Discussion

217 219 220 224

231 232 235 241

XIX. Semigroup Well-Posedness of a Multilayer Mead-Markus Plate with Shear Damping

243

Scott W. Hansen 1 Multilayer Mead-Markus Model 2 State Variable Formulation 3 Semigroup Formulation of Homogeneous Problem

244 248 251

XX. Solution of Algebraic Riccati Equations Arising in Control of Partial Diﬀerential Equations Kirsten Morris, Carmeliza Navasca 1 Solution of Lyapunov Equation 2 Benchmark Examples 3 Euler-Bernoulli Beam

XXI. Stabilization in Computing Saddle Points Jianxin Zhou, Xudong Yao 1 The Local Minimax Method 2 Results on the Order of Saddle Points 3 Reduce Instability by Using Symmetries 4 Selected Numerical Examples

XXII. Second Order Suﬃcient Conditions for Optimal Control Subject to First Order State Constraints Kazimierz Malanowski, Helmut Maurer, Sabine Pickenhain 1 Preliminaries 2 Second Order Suﬃcient Conditions via Hamilton-Jacobi Inequality 3 Checking Positive Deﬁniteness with Riccati Equation 4 Example

Copyright © 2005 Marcel Dekker, Inc.

257 260 263 268

281 282 283 286 289

293 294 298 300 302

CHAPTER I OPERATOR-SPLITTING METHODS AND APPLICATIONS TO THE DIRECT NUMERICAL SIMULATION OF PARTICULATE FLOW AND TO THE SOLUTION OF THE ELLIPTIC ` MONGE-AMPERE EQUATION Edward J. Dean Department of Mathematics University of Houston Houston, TX, USA

Roland Glowinski Department of Mathematics University of Houston Houston, TX, USA

Tsorng-Whay Pan Department of Mathematics University of Houston Houston, TX, USA

Dedicated to A.V. Balakrishnan, J.-L. Lions, and G.I. Marchuk for their outstanding contributions to computational and applied mathematics and for making this IFIP conference possible. Abstract

The main goal of this article is to investigate some new applications of operator-splitting methods. We will show that these methods (almost) trivialize the numerical solution of outstanding problems from various areas in mechanics, physics, and diﬀerential geometry, such as the direct simulation of particulate ﬂow and the real elliptic Monge-Amp`ere equations. The results of numerical experiments will be given to illustrate the capabilities of these methods.

Keywords: Operator-splitting methods, particulate ﬂow, Monge-Amp`ere equations

Copyright © 2005 Marcel Dekker, Inc.

2

CONTROL AND BOUNDARY ANALYSIS

Introduction It has been known for decades that operator-splitting methods provide eﬃcient tools for the numerical solution of complicated problems from various areas in science and engineering. New applications appear almost daily and we know of many instances where the only available practical solution methods are of the operator-splitting type. Our goal, here, is to show that operator-splitting makes almost trivial the numerical solution of outstanding problems such as: (i) The simulation of particulate ﬂow when the number of particles exceeds 102 . (ii) Fully nonlinear elliptic equations of the real Monge-Amp`ere type. The content of this article is as follows: In Section 1 we will brieﬂy discuss the time-discretization of initial value problems by various operator-splitting methods and show that wellknown iterative methods are indeed disguised operator-splitting schemes. In Section 2 we will address the direct numerical simulation of particulate ﬂow via a methodology combining operator-splitting and ﬁctitious domain methods. Finally, in Section 3 we will discuss the numerical solution of the two-dimensional Dirichlet problem for the elliptic MongeAmp`ere equation. The results of numerical experiments will be given in Sections 2 and 3; they will conﬁrm the capabilities of operator-splitting methods for solving problems still considered complicated by today’s standards. Operator-splitting methods have generated a huge literature; let us mention among many others, refs.: [3], [10](Chapters 2 and 6) (see also the references therein), [17], and [11].

1.

Operator-Splitting Schemes for the Time-Discretization of Initial Value Problems

1.1

Generalities

Let us consider the following autonomous initial value problem: dϕ + A(ϕ) = 0 on (0, T ) (with 0 < T ≤ +∞), dt

ϕ(0) = ϕ0 .

(IVP)

Operator A maps a vector space V into itself and we suppose that ϕ0 ∈ V. We suppose also that A has a non-trivial decomposition such as A=

J j=1

Copyright © 2005 Marcel Dekker, Inc.

Aj ,

(1)

Operator-Splitting Methods and Applications

3

with J ≥ 2 (by non-trivial we mean that the operators Aj are individually simpler than A). A question which arises naturally is clearly: Can we take advantage of decomposition (1) for the solution of (IVP)? It has been known for a long time that the answer to the above question is yes. Many schemes have been designed to take advantage of (1) when solving (IVP); two of them will be brieﬂy discussed in the following paragraphs, namely the Lie scheme and the Strang scheme.

1.2

Time-Discretization of (IVP) by Lie’s Scheme

Let τ (> 0) be a time-discretization step (we suppose τ uniform, for simplicity); we denote nτ by tn . With ϕn an approximation of ϕ(tn ), Lie’s scheme reads as follows (for its derivation see, e.g.,[10](Chapter 6)): For n ≥ 0, assuming that ϕn is known, we compute ϕn+1 via dϕ + Aj (ϕ) = 0 on (tn , tn+1 ), (2) dt ϕ(tn ) = ϕn+(j−1)/J ; ϕn+j/J = ϕ(tn+1 ), for j = 1, ..., J, with ϕ0 = ϕ0 . If (IVP) is taking place in a ﬁnite dimensional space and if operators Aj are smooth enough, then ϕ(tn ) − ϕn = O(τ ), the function ϕ being the solution of (IVP).

Remark 1 The above scheme applies also to multivalued operators (such as the subgradient of a proper l.s.c. convex functional) but in such a case ﬁrst order accuracy is not guaranteed anymore. Remark 2 The above scheme is easy to generalize to non-autonomous problems by observing that dϕ dϕ dt + A(ϕ, θ) = 0, + A(ϕ, t) = 0, dθ ⇔ dt − 1 = 0, ϕ(0) = ϕ 0 dt ϕ(0) = ϕ0 , θ(0) = 0. Remark 3 Scheme ( 2) is semi-constructive in the sense that we still have to solve the sub-initial value problem in ( 2) for each j. Suppose Copyright © 2005 Marcel Dekker, Inc.

4

CONTROL AND BOUNDARY ANALYSIS

that we discretize this sub-problem using just one step of the backward Euler scheme. The resulting scheme reads as follows: For n ≥ 0, assuming that ϕn is known, we compute ϕn+1 via ϕn+j/J − ϕn+(j−1)/J + Aj (ϕn+j/J ) = 0, τ

(3)

for j = 1, ..., J, with ϕ0 = ϕ0 . Scheme ( 3) is known as the MarchukYanenko scheme (see, e.g., refs. [10](Chapter 6) and [17] for more details).

1.3

Time-Discretization of (IVP) by Strang’s Scheme

In order to improve the accuracy of Lie’s scheme, G. Strang suggested a symmetrized variant of scheme (2) (ref. [22]). When applied to nonautonomous problems, in the case where J = 2, we obtain (with tn+1/2 = tn + τ /2) : (4) ϕ0 = ϕ0 ; then, for n ≥ 0, assuming that ϕn is known, we compute ϕn+1 via dϕ + A1 (ϕ, t) = 0 on (tn , tn+1/2 ), dt ϕ(tn ) = ϕn ; ϕn+1/2 = ϕ(tn+1/2 ), dϕ + A2 (ϕ, tn+1/2 ) = 0 on (0, τ ), dt ϕ(0) = ϕn+1/2 ; ϕˆn+1/2 = ϕ(τ ), dϕ + A1 (ϕ, t) = 0 on (tn+1/2 , tn+1 ), dt ϕ(tn+1/2 ) = ϕˆn+1/2 ; ϕn+1 = ϕ(tn+1 ).

(5)

(6)

(7)

If (IVP) is taking place in a ﬁnite dimensional space and if operators A1 and A2 are smooth enough, then ϕ(tn ) − ϕn = O(τ 2 ), the function ϕ being the solution of (IVP).

Remark 4 In order to preserve the second order accuracy of scheme ( 4)-( 7) (assuming it holds) we still have to discretize the initial value problems in ( 5), ( 6), and ( 7) by schemes which are themselves second order accurate (at least); examples of such schemes can be found in, e.g., ref. [10](Chapter 6) (which contains also a discussion of the case J > 2). Copyright © 2005 Marcel Dekker, Inc.

5

Operator-Splitting Methods and Applications

1.4

Application

Applications of operator-splitting are everywhere; indeed, some wellknown methods or algorithms are disguised operator-splitting schemes. Our favorite example in that direction is the following: Suppose that A is a real d×d matrix, symmetric and positive deﬁnite. Ordering the eigenvalues of A as follows: 0 < λ1 ≤ λ2 ≤ ... ≤ λd , our goal is to compute λ1 . We have (with obvious notation) λ1 = minv∈S Av · v, with S = {v|v ∈ IRd , v = 1}.

(8)

The minimization problem in (8) is equivalent to 1 minv∈IRd { Av · v + IS (v)}, 2

(9)

where, in (9), functional IS : IRd → IR ∪ {+∞} is deﬁned as follows: 0, if v ∈ S, IS (v) = +∞, otherwise, implying that IS is the indicator functional of sphere S. Suppose that u is a solution of problem (9); we have then Au + ∂IS (u) = 0,

(10)

∂IS (u) being in (10) a (kind of) generalized gradient of functional IS at u. Next, we associate to the (necessary) optimality system (10) the following initial value problem (ﬂow in the dynamical system terminology): du + Au + ∂IS (u) = 0 on (0, +∞), (11) dt u(0) = u . 0

If we apply the Marchuk-Yanenko scheme (3) to the solution of problem (11) we obtain (12) u0 = u0 , and for n ≥ 0, un being known, un+1/2 − un + Aun+1/2 = 0, τ

(13)

un+1 − un+1/2 + ∂IS (un+1 ) = 0. τ

(14)

Copyright © 2005 Marcel Dekker, Inc.

6

CONTROL AND BOUNDARY ANALYSIS

Relation (13) implies un+1/2 = (I + τ A)−1 un .

(15)

On the other hand, relation (14) can be interpreted as a necessary optimality condition for the following minimization problem: 1 minv∈S { v2 − un+1/2 · v}. 2

(16)

Since v = 1 over S, the solution of problem (16) is given by un+1 =

un+1/2 . un+1/2

(17)

Algorithm (12)-(14) reduces then to (12), (15), and (17), which is nothing but the inverse power method with shift, a well-known algorithm from Numerical Linear Algebra. Clearly, numerical analysts have not been waiting for operator-splitting to compute matrix eigenvalues and eigenvectors; on the other hand, operator-splitting has provided eﬃcient algorithms for the solution of complicated problems from diﬀerential geometry, mechanics, physics, physico-chemistry, etc. Some of these applications will be discussed in Sections 2 and 3.

2. 2.1

Operator-Splitting Methods for the Direct Numerical Simulation of Particulate Flow Generalities. Problem Formulation

It is the (very likely biased) opinion of these authors that the direct numerical simulation of particulate ﬂow has been one of the success stories of operator-splitting methods. Albeit this “story” has been told in several publications (see, e.g., [10](Chapter 8) and [12], and the references therein); owing to its importance, we decided to return to it again (without too many details, due to page limitation). For simplicity, we only consider the one-particle case (see the two above references for the many-particle case). Let Ω be a bounded, connected, and open region of IRd (d = 2 or 3 in applications); the boundary of Ω is denoted by Γ. We suppose that Ω contains: (i) A Newtonian incompressible viscous ﬂuid of density ρf and viscosity µf ; ρf and µf are both positive constants. (ii) A rigid body B of boundary ∂B, mass M, center of mass G, and inertia I at the center of mass (see Figure 1 for additional details). Copyright © 2005 Marcel Dekker, Inc.

7

Operator-Splitting Methods and Applications

G B

B

Ω n Figure 1.

Γ

Visualization of the ﬂow region and of the rigid body.

We suppose that the ﬂuid occupies the region Ω \ B and the distance (∂B(0), Γ) > 0. From now on, x = {xi }di=1 will denote the generic point of IRd , dx = dx1 ...dxd , and ϕ(t) will denote the function x → ϕ(x, t). Assuming that the only external force is gravity, the ﬂuid ﬂow-rigid body motion coupling is modeled by ∂u + (u · ∇)u] − µf ∆u + ∇p = ρf g in ∪t∈(0,T ) Ω \ B(t), (18) ∂t ∇ · u(t) = 0 in Ω \ B(t), ∀t ∈ (0, T ), (19) u(t) = uΓ (t) on Γ, ∀t ∈ (0, T ), with uΓ (t) · ndΓ = 0, (20)

ρf [

Γ

u(0) = u0 in Ω \ B(0) with ∇ · u0 = 0,

(21)

and dG = V, dt dV = M g + RH , M dt d(Iω)/dt = TH , G(0) = G0 , V(0) = V0 , ω(0) = ω0 , B(0) = B0 .

(22) (23) (24) (25)

In relations (18)-(21) and (22)-(25), vector u = {ui }di=1 is the ﬂuid (ﬂow) velocity and p is the pressure; n is the unit normal vector at Γ ∪ ∂B, outward to Ω \ B; u0 and uΓ are given functions; RH and TH denote, respectively, the resultant and the torque of the hydrodynamical forces, namely the forces the ﬂuid exerts on B; we have, actually, −→ σndγ and TH = − RH = − Gx × σndγ; (26) ∂B

Copyright © 2005 Marcel Dekker, Inc.

∂B

8

CONTROL AND BOUNDARY ANALYSIS

V (resp., ω) is the translation (resp., angular) velocity of B. In (26) the stress-tensor σ is deﬁned by σ = 2µf D(u) − pId , with D(v) = 1 t 2 [∇v + (∇v) ]. Concerning the compatibility conditions on ∂B we shall assume that: (i) The forces exerted by the ﬂuid on the solid body cancel those exerted by the solid body on the ﬂuid. (ii) On ∂B the no-slip boundary condition holds, namely −−−→ u(x, t) = V(t) + ω(t) × G(t)x, ∀x ∈ ∂B(t).

(27)

Remark 5 System ( 18)–( 21) (resp., ( 22)–( 25)) is of the incompressible Navier-Stokes (resp., Euler-Newton) type. Also, the above model can be generalized to multiple-particle situations and/or non-Newtonian incompressible viscous ﬂuids. 2 The (local in time) existence of weak solutions for problems such as (18)–(21), and (22)–(25) has been proved in ref. [7], assuming that at t = 0 the particles do not touch each other and do not touch Γ (see also [13], [21]). Concerning the numerical solution of (18)–(21), and (22)– (25), completed by interface conditions, we can divide them, roughly, in two classes, namely: (i) The Arbitrary Lagrange-Euler (ALE) methods; these methods relying on moving meshes are discussed in, e.g., refs. [14], [15], and [18]. (ii) The non-boundary ﬁtted ﬁctitious domain methods; these methods rely on ﬁxed meshes and are discussed in, e.g., [10](Chapter 8) and [12](see also the references therein). These methods seem to enjoy a growing popularity, justifying thus the (brief) discussion hereafter.

Remark 6 Even if theory suggests that collisions never take place in ﬁnite time, near-collisions take place, and, after discretization, “real” collisions may occur. These phenomena can be avoided by introducing well-chosen repulsion potentials reminiscent of those encountered in Molecular Dynamics (see [10](Chapter 8) and [12] for details).

2.2

A Fictitious Domain Formulation

Considering the ﬂuid-rigid body mixture as a unique medium, we are going to derive a ﬁctitious domain-based variational formulation. The Copyright © 2005 Marcel Dekker, Inc.

Operator-Splitting Methods and Applications

9

principle of this derivation is pretty simple; it relies on the following steps (see, e.g., [10] and [12] for details): a. Start from the following global weak formulation (of the virtual power type): ∂u + (u · ∇)u · vdx + 2µf D(u) : D(v)dx ρf ∂t Ω\B(t) Ω\B(t) d(Iω) dV ·Y+ ·θ p∇ · vdx + M − dt dt Ω\B(t) (28) g · vdx + M g · Y, ∀{Y, θ} ∈ IRd × Θ, = ρf Ω\B(t) and ∀v ∈ (H 1 (Ω \ B(t)))d and verif ying v = 0 on Γ, −−−→ v(x) = Y + θ × G(t)x, ∀x ∈ ∂B(t), t ∈ (0, T ) with Θ = IR3 if d = 3, Θ = {{0, 0, θ}|θ ∈ IR} if d = 2, q∇ · u(t)dx = 0, ∀q ∈ L2 (Ω \ B(t)), t ∈ (0, T ), (29) Ω\B(t)

u(t) = uΓ (t) on Γ, t ∈ (0, T ), −−−→ u(x, t) = V(t) + ω(t) × G(t)x, ∀x ∈ ∂B(t), t ∈ (0, T ), dG = V, dt u(x, 0) = u0 (x), ∀x ∈ Ω \ B(0), G(0) = G0 , V(0) = V0 , ω(0) = ω0 , B(0) = B0 .

(30) (31) (32) (33)

b. Fill B with the surrounding ﬂuid.

c. Impose a rigid body motion to the ﬂuid inside B.

d. Modify the global weak formulation (28)–(33) accordingly, taking advantage of the fact that if v is a rigid body motion velocity ﬁeld, then ∇ · v = 0 and D(v) = 0. e. Use a Lagrange multiplier deﬁned over B to force the rigid body motion inside B. Copyright © 2005 Marcel Dekker, Inc.

10

CONTROL AND BOUNDARY ANALYSIS

Assuming that B is made of a homogeneous material of density ρs , the above “program” leads to: ∂u + (u · ∇)u · vdx + 2µf D(u) : D(v)dx ρf Ω ∂t Ω dV d(Iω) − p∇ · vdx + (1 − ρf /ρs ) M ·Y+ ·θ Ω dt dt −−−→ (34) + < λ, v − Y − θ × G(t)x >B(t) = ρf g · vdx Ω ρf 1 (Ω))d × IRd × Θ, t ∈ (0, T ), )M g · Y, ∀{v, Y, θ} ∈ (H +(1 − 0 ρs 3 with Θ = IR if d = 3, Θ = {{0, 0, θ}|θ ∈ IR} if d = 2, q∇ · u(t)dx = 0, ∀q ∈ L2 (Ω), t ∈ (0, T ),

(35)

Ω

(36) u(t) = uΓ (t) on Γ, t ∈ (0, T ), −−−→ < µ, u(t) − V(t) − ω(t) × G(t)x >B(t) = 0, ∀µ ∈ Λ(t), t ∈ (0, T ), (37) dG = V, (38) dt G(0) = G0 , V(0) = V0 , ω(0) = ω0 , B(0) = B0 , u0 (x), ∀x ∈ Ω \ B 0 , (39) u(x, 0) = −−→ u(x, 0) = V0 + ω0 × G0 x, ∀x ∈ B 0 with Λ(t) = (H 1 (B(t)))d . From a theoretical point of view,

a natural choice for < ·, · >B(t) is provided by, e.g., < µ, v >B(t) = B(t) [µ · v + δ2 D(µ) : D(v)]dx, with δ as a characteristic length, the diameter of B, for example. From a practical point of view, when it comes to space discretization, a simple and eﬃcient strategy is the following one (cf. [10](Chapter 8) and [12]): “approximate” Λ(t) by Λh (t) = {µ|µ =

N

µj δ(x − xj ), with µj ∈ IRd , ∀j = 1, ..., N }, (40)

j=1

N and the above pairing by < µ, v >B(t)h = j=1 µj · v(xj ). In (40), x → δ(x − xj ) is the Dirac measure at xj , and the set {xj }N j=1 is the union of two subsets, namely: (i) The set of the points of the velocity grid contained in B(t) and whose distance to ∂B(t) is ≥ ch, h being a space discretization step and c a constant 1. (ii) A set of control points located on ∂B(t) and forming a mesh whose step size is of the order of h. It is clear that, using the above approach, one forces the rigid body motion inside the particle by collocation. Copyright © 2005 Marcel Dekker, Inc.

Operator-Splitting Methods and Applications

2.3

11

Solving Problem (34)–(39) by Operator-Splitting

We do not consider collisions; after (formal) elimination of p and λ, problem (34)–(39) reduces to a dynamical system of the following form dX + Aj (X, t) = 0 on (0, T ), X(0) = X0 , dt J

(41)

j=1

where X = {u, V, ω, G} (or {u, V, Iω, G}). A typical situation will be the one where, with J = 4, operator A1 will be associated to incompressibility, A2 to advection, A3 to diﬀusion, A4 to ﬁctitious domain and body motion; other decompositions are possible as shown in, e.g., [10](Chapter 8) and [12]. Lie’s scheme (2) applies “beautifully” to the solution of the formulation (41) of problem (34)–(39). The resulting method is quite modular implying that diﬀerent space and time approximations can be used to treat the various steps; the only constraint is that two successive steps have to communicate (by projection in general).

2.4

Numerical Experiments

Generalities. The methods described in the above paragraphs have been validated by numerous experiments (see, e.g., [10](Chapters 8 and 9), [12], and [20]). In this article, we shall focus on two test problems, the ﬁrst one involving only one particle, while the second test problem concerns a channel ﬂow with 300 particles. The ﬁctitious domain/operatorsplitting approach has made the solution of both problems (almost) routine, but no later than the mid-nineties, solving such problems was considered a Grand Challenge (actually, “our” ﬁrst systematic attack of such problems was part of a National Science Foundation supported Grand Challenge project). First Test Problem: Lifting of a Ball by a Pressure Driven Flow. Let us denote by Ω a truncated circular cylinder of length L and diameter D; we denote by Γ the boundary of Ω and suppose that: (a) The axis of the cylinder is parallel to the horizontal axis Ox2 . (b) The cylinder has been truncated by the vertical planes x2 = 0 and x2 = L; we denote by Γ1 and Γ2 the two vertical disks limiting Ω. (c) The cylinder is ﬁlled with an incompressible Newtonian viscous ﬂuid of density ρf and viscosity µf ; it contains also a rigid solid ball B of diameter d and density ρs . (d) The only external force is gravity. (e) At t = 0, the ﬂuid and the solid are at rest, the ball lying on the bottom of the cylinder. (f) There exists a pressure drop ∆P between Γ1 and Copyright © 2005 Marcel Dekker, Inc.

12

CONTROL AND BOUNDARY ANALYSIS 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

t=0

−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1

Figure 2.

0.5

0

t=40

229.2 229.4 229.6

229.8 230 230.2

230.4 230.6 230.8

231

1

0.5

0

Position of the ball at t = 0 and 40 (from left to right). The ratio between the distance to the central axis and the radius of the pipe 1 0.9 0.8 0.7 Ratio

0.6 0.5 0.4 0.3 0.2 0.1 0 0

Figure 3.

5

10

15

20 25 time

30

35

40

45

Time variation of the distance of the ball center to the cylinder axis.

Γ2 . (g) The following boundary conditions prevail on Γ : u = 0 on Γ \ (Γ1 ∪ Γ2 ), and u(x, t) = u(x + Le2 , t), ∀x ∈ Γ1 , e2 being the unit vector of axis Ox2 (periodic boundary conditions). If ∆P is large enough, the lift acting on B will be suﬃciently large for the ball to take oﬀ and possibly reach an equilibrium height. This is indeed the phenomenon we intend to simulate. In order to do so, we are going to apply the ﬁctitious domain methodology discussed in Sections 2.2 and 2.3. Actually the above methodology is used twice: (i) To treat the ﬂuid-rigid ball coupling as done in Section 2.2. (ii) To solve all the ˜ containing Ω; we impose u = 0 partial diﬀerential equations in a “box” Ω ˜ in Ω \ Ω via a distributed Lagrange multiplier method very close to the one discussed in [10](Section 41). In the particular problem that we are considering: L = 2, D = ˜ is approximated by Ω ˜h = 1, d = 0.2, ρf = 1, µf = 1, ρs = 1.001, Ω (0, 1 + 4h) × (0, 2) × (0, 1 + 4h), (h being a uniform space-discretization step), the cylinder axis is on the line x1 = 12 + 2h, x3 = 12 + 2h, and ∆P = 320, so that, without the ball, the maximal steady ﬂow speed would have been 10. The mesh size used to compute the velocity ﬁeld (resp., the pressure ) is hv = h = 1/96 (resp., hp = 2h = 1/48), while Copyright © 2005 Marcel Dekker, Inc.

Operator-Splitting Methods and Applications

13

we took 1/1000 for the time-discretization step; with these values the velocity mesh contains approximately 2 × 106 grid-points. The lifting phenomenon has been simulated for t ∈ [0, 40], leading to the following results: (1) The translation speed of the ball averaged during the last ﬁve time units is 6.126. (2) The Reynolds number averaged during the last ﬁve time units is 1.2252. (3) The distance of the ball center to the cylinder axis, also averaged during the last ﬁve time units, is 0.30129. The following have been visualized in Figures 2 to 3, respectively: (1) The position of the ball at time t = 0. (2) The position of the ball at t = 40; it has, essentially, reached its equilibrium height (height of the ball center .2). (3) The variation versus time of the ratio distance of the ball center to the cylinder axis /D; the equilibrium height has been reached around t = 27.5.

Second Test Problem: Motion of 300 Neutrally Buoyant Disks in a Two-Dimensional Horizontal Channel. The second test problems involving 300 particles and a solid volume/ﬂuid volume ratio of the order of 37.86% collisions (or near-collisions) have to be accounted for in the simulations; to do so we have used the methods discussed in, e.g., refs. [10](Chapter 8) and [12]. Another peculiarity of this second test problem is that ρs = ρf for all particles (a neutrally buoyant situation). Indeed, neutrally buoyant models are more delicate to handle than those in the general case, since 1 − ρf /ρs = 0 in (34); however, this diﬃculty can be overcome, as shown in ref. [20]. For this test problem, we have: (a) Ω = (0, 42) × (0, 12). (b) Ω contains the mixture of a Newtonian incompressible viscous ﬂuid of density 1 and viscosity 1 with 300 rigid solid disks of density 1 and radius .45. (c) At time t = 0, ﬂuid and particles are at rest, the particle centers being located at the points of a regular lattice. (d) The mixture is put into motion by a uniform pressure drop of 25/18 per unit length (without the particles the steady ﬂow would be of the Poiseuille type with 25 as maximal ﬂow speed). (e) The boundary conditions are given by u = 0 on (0, 42) × {0, 12} (no-slip boundary condition on the horizontal parts of the boundary), and u(0, x2 , t) = u(42, x2 , t), 0 < x2 < 12, 0 ≤ t ≤ 700 (spaceperiodic boundary conditions). (f) hv = 1/10, hp = 1/5, and the timediscretization step is 1/1000. The particle distribution at t = 70, 120, and 700 has been visualized on Figure 4. These ﬁgures show that, initially, we have the sliding motion of horizontal particle layers (see Figure 4), then after some time, a chaotic ﬂow-motion takes place, the higher particle concentration being along the axis of the channel. The maximal speed is 10.04 at t = 700, implying that the corresponding particle Reynolds number is very close to 9. For more details, and further comCopyright © 2005 Marcel Dekker, Inc.

14

Figure 4.

CONTROL AND BOUNDARY ANALYSIS

Particle distribution at t = 70, 120, and 700 (from top to bottom).

ments and results on pressure driven neutrally buoyant particulate ﬂow in two-dimensional channels, see [10](Chapter 9) and [20].

3. 3.1

An Operator-Splitting Method for the Elliptic Monge-Amp` ere Equations in Two-Dimension Generalities

These last years have been witnessing a surge of interest in MongeAmp`ere equations and their numerical solution. Indeed, beside the important role they play in diﬀerential geometry, these equations occur in the modeling of various phenomena in mechanics, physics, etc. (see [2], [19]) and their “full nonlinearity” (in the sense of Caﬀarelli and Cabr´e; see ref. [1]) presents an interesting challenge to numerical analysts. Our goal, here, is to show that via operator-splitting based methods, and an appropriate reformulation, we can make the solution of these equations almost trivial (at least for two-dimensional Dirichlet problems in the elliptic case). In this article, following [5], [6], we will discuss an augmented Lagrangian approach to the solution of the elliptic DirichletMonge-Amp`ere problem; a least-squares based method is also discussed in ref. [5]. Copyright © 2005 Marcel Dekker, Inc.

15

Operator-Splitting Methods and Applications

3.2

Formulation of the Problem

Let Ω be a bounded domain of IR2 ; we denote by Γ the boundary of Ω. The two-dimensional Dirichlet problem for the Monge-Amp`ere equation reads as follows: detD2 ψ = f in Ω, ψ = g on Γ,

(42)

∂2ψ ∂xi ∂xj 1≤i,j≤2 and where f and g are two given functions, with f > 0. Unlike the (closely related) Dirichlet problem for the Laplace operator, problem (42) may have multiple solutions (actually, two at most; cf., e.g., [4](Chapter 4)), and the smoothness of the data does not imply the existence of a smooth solution. Concerning the last property, suppose that Ω = (0, 1) × (0, 1) and consider the special case where (42) is deﬁned by 2 ∂ 2 ψ ∂ 2 ψ ∂ 2 ψ − = 1 in Ω, ψ = 0 on Γ. (43) ∂x21 ∂x22 ∂x1 ∂x2

where, in (42), D2 ψ is the Hessian of ψ, i.e., D2 ψ =

Problem (43) cannot have smooth solutions since, for those solutions, the ∂2ψ ∂2ψ and boundary condition ψ = 0 on Γ implies that the product ∂x21 ∂x22 ∂2ψ the cross-derivative vanish at the boundary, implying in turn ∂x1 ∂x2 that detD2 ψ is strictly less than one in some neighborhood of Γ. The above (non-existence) result is not a consequence of the non-smoothness of Γ, since a similar non-existence property holds if in (43) one replaces the above Ω by the ovoid-shaped domain whose C ∞ -boundary is deﬁned by Γ = ∪4i=1 Γi , with Γ1 = {x|x = {x1 , x2 }, x2 = 0, 0 ≤ x1 ≤ 1}, Γ2 = {x|x = {x1 , x2 }, x1 = 1 + e4−1/x2 (1−x2 ) , 0 ≤ x2 ≤ 1}, Γ3 = {x|x = {x1 , x2 }, x2 = 1, 0 ≤ x1 ≤ 1}, Γ4 = {x|x = {x1 , x2 }, x1 = −e4−1/x2 (1−x2 ) , 0 ≤ x2 ≤ 1}. Actually, for the above two Ωs the nonexistence of solutions for problem (43) follows from the non-strict convexity of these domains.

Remark 7 Suppose that Ω is simply connected; let us deﬁne a vector ∂ψ ∂ψ ,− (= {u1 , u2 }); problem ( 42) valued function u by u = ∂x2 ∂x1 takes then the equivalent formulation det∇u = f in Ω, ∇ · u = 0 in Ω, (44) dg u · n = on Γ, ds Copyright © 2005 Marcel Dekker, Inc.

16

CONTROL AND BOUNDARY ANALYSIS

where, in ( 44), n denotes the outward unit vector normal at Γ, and s a counter-clockwise curvilinear abscissa. Once u is known, one obtains ψ via the solution of the following Poisson-Dirichlet problem: −∆ψ =

∂u2 ∂u1 − in Ω, ψ = g on Γ. ∂x1 ∂x2

Problem ( 44) has clearly an incompressible ﬂuid ﬂow ﬂavor, ψ playing here the role of a stream function. Relations (44) can be used to solve problem (42) but this approach will not be further investigated here.

3.3

An Augmented Lagrangian Approach for the Solution of Problem (42)

Suppose that in (42) f ∈ L1 (Ω) and g ∈ H 3/2 (Γ); it makes sense then to attempt solving problem (42) in H 2 (Ω) by considering it as a (kind of) nonlinear bi-harmonic problem. A way to do so is to consider the following problem from Calculus of Variations: 1 |∆ϕ|2 dx, (45) min ϕ∈Ef g 2 Ω with Ef g = {ϕ|ϕ ∈ Vg , detD2 ϕ = f } and Vg = {ϕ|ϕ ∈ H 2 (Ω), ϕ = g on Γ}. Problem (45) is in turn equivalent to 1 |∆ϕ|2 dx, (46) min {ϕ,q}∈Ef g 2 Ω with Ef g = {{ϕ, q}|ϕ ∈ Vg , q ∈ Q, detq = f, D2 ϕ = q} and Q = {q = {qij }1≤i,j≤2 |q ∈ (L2 (Ω))4 , q12 = q21 }. Following, e.g., refs. [9], [11], and [16], we associate to problem (46): (i) The augmented Lagrangian functional Lr : (H 2 (Ω) × Q) × Q → IR deﬁned, with r > 0, by 1 r 2 2 2 |∆ϕ| dx + |D ϕ − q| dx + µ : (D2 ϕ − q)dx, Lr (ϕ, q; µ) = 2 Ω 2 Ω Ω (47) with S : T = 1≤i,j≤2 sij tij , if S = {sij } and T = {tij }, and (ii) The saddle-point problem F ind {ψ, p; λ} ∈ (Vg × Qf ) × Q such that (48) Lr (ψ, p; µ) ≤ Lr (ψ, p; λ) ≤ Lr (ϕ, q; λ), ∀{ϕ, q; µ} ∈ (Vg × Qf ) × Q, Copyright © 2005 Marcel Dekker, Inc.

Operator-Splitting Methods and Applications

17

with Qf = {q|q ∈ Q, detq = f }. One can easily show that if {ψ, p; λ} is a solution of the saddle-point problem (48), then ψ is solution of the Monge-Amp`ere problem (42), p = D2 ψ, and λ is a Lagrange multiplier associated to relation p − D2 ψ = 0. Concerning the solution of problem (48) we advocate (following, e.g., refs. [9], [11]) the (relatively simple) Douglas-Rachford-Uzawa algorithm below: {ψ −1 , λ0 } is given in Vg × Q;

(49)

for n ≥ 0, assuming that {ψ n−1 , ψ n } is known, solve pn ∈ Qf ; Lr (ψ n−1 , pn ; λn ) ≤ Lr (ψ n−1 , q; λn ), ∀q ∈ Qf , ψ n ∈ Vg ; Lr (ψ n , pn ; λn ) ≤ Lr (ϕ, pn ; λn ), ∀ϕ ∈ Vg , λn+1 = λn + r(D2 ψ n − pn ).

(50) (51) (52)

Algorithm (49)-(52) deserves many comments, among them: Concerning the initialization of algorithm (49)–(52), we advocate λ0 = 0 and ψ −1 as the solution of the following Poisson-Dirichlet problem (53) −∆ψ −1 = f 1/2 in Ω, ψ −1 = g on Γ; the rational for such a choice is given in ref. [5]. Problem (50) can be solved point-wise (in practice at the grid points of a ﬁnite element or ﬁnite diﬀerence mesh). Indeed, (50) reduces, a.e. on Ω, to the solution of a ﬁnite dimensional problem of the following type: r minz { (z12 + z22 + 2z32 ) − bn (x) · z} 2

(54)

with z (= {zi }3i=1 ) ∈ {z|z ∈ IR3 , z1 z2 − z32 = f (x)}. The solution of problem (54) (a generalized eigenvalue problem) is discussed in [5]. Problem (51) reduces to a linear variational problem of the following type: ∆ψ n ∆ϕdx + r D2 ψ n : D2 ϕdx = L (ϕ), n (55) Ω Ω 2 1 n ∀ϕ ∈ V0 (= H (Ω) ∩ H0 ); ψ ∈ Vg , the functional Ln (·) being linear and continuous over V0 . The unique solution of problem (55) can be computed by a conjugate gradient algorithm operating in Vg and V0 equipped with the scalar Copyright © 2005 Marcel Dekker, Inc.

18

CONTROL AND BOUNDARY ANALYSIS

product {v, w} →

∆v∆wdx and the corresponding norm. Such Ω

an algorithm is described in [5]; its most important feature is that for well-chosen ﬁnite element approximations, its discrete variants require no more than the solution of two discrete Poisson-Dirichlet problems at each iteration (see [5] for details). Suppose that problem (42) has no solution in H 2 (Ω), but that neither Vg nor Qf are empty (as it is the case for problem (43)). In that case, we expect the arithmetic divergence of sequence {λn }n and the geometric convergence of sequence {{ψ n , pn }}n to a pair {ψ, p} ∈ Vg × Qf such as {ψ, p} minimizes over Vg × Qf (locally or globally) (56) the f unctional{ϕ, q} → D2 ϕ − q(L2 (Ω))4 . The rational for this prediction is discussed at length in [5], and, indeed, when applying algorithm (49)–(52) to the solution of problem (43), the numerical results show that the above algorithm behaves as expected; these numerical results are reported in Section 3.4 (see also [5], [6]). On the basis of these results it seems that for parameter r well chosen, algorithm (49)–(52) produces either a solution of problem (42) if such a solution exists in H 2 (Ω), or a generalized solution in the sense of (56), i.e., a least-squares solution, if (42) is without solution in H 2 (Ω), while Vg and Qf are non-empty. This result justiﬁes the least-squares methodology discussed in [5]. From a geometrical point of view, problem (42) has solutions in H 2 (Ω) if D2 Vg and Qf (both subsets of space Q) intersect; such a situation has been visualized on Figure 5(a). Figure 5(b) corresponds to a situation where (42) has no solution in H 2 (Ω), but neither Vg nor Qf are empty; a generalized solution, in the sense of (56) has been visualized on this ﬁgure. The saddle-point formulation (48) of problem (42) is a mixed variational formulation where the “nonlinearity burden” has been transferred from ψ to p, making it purely algebraic. Actually, this approach (this is even truer for the discrete analogues of problems (42) and (48); see [5] for details) provides a solution method where instead of solving (42) directly (i.e., without introducing additional functions, like p), we solve it via its associated Pfaﬀ system (see, e.g., [8](Chapter A)), namely: dψ − u1 dx1 − u2 dx2 = 0 in Ω, Copyright © 2005 Marcel Dekker, Inc.

(57)

19

Operator-Splitting Methods and Applications

D2Vg

Qf

D2 Vg

Qf D2ψ

p= D 2 ψ

Q

Qf

Qf p

Q

Figure 5. (a). Problem (42) has a solution in H 2 (Ω) (left). (b). Problem (42) has no solution in H 2 (Ω) (right).

du1 − p11 dx1 − p12 dx2 = 0 in Ω, du2 − p12 dx1 − p22 dx2 = 0 in Ω, p11 p22 − p212 = f,

(58) (59) (60)

completed by the boundary condition ψ = g on Γ. System (57)– (60) provides clearly a mixed formulation of problem (42). Let us return to problem (IVP) of Section 1.1 and suppose that A = A1 + A2 . An alternative to Lie’s and Strang’s schemes is provided by the following (ﬁrst order accurate) Douglas-Rachford scheme: Given ϕ0 = ϕ0 ; then, for n ≥ 0, ϕn being known, solve (ϕn+1/2 − ϕn )/τ + A1 (ϕn+1/2 ) + A2 (ϕn ) = 0, (ϕn+1 − ϕn )/τ + A1 (ϕn+1/2 ) + A2 (ϕn+1 ) = 0.

(61) (62)

The basic properties of scheme (61)–(62), can be found in, e.g., [10](Chapter 2) (see also the references therein). It is shown in, e.g., ref. [11](see also [9]) that algorithms such as (49)–(52) are in fact “disguised” Douglas-Rachford algorithms, with parameter r the reciprocal of a time step (see the two above references for details). The ﬁnite element implementation of algorithm (49)–(52) is discussed in ref. [5]; it relies on a mixed ﬁnite element approximation of problems (42) and (48) where ψ and the four components of the tensor-valued functions p and λ are approximated by functions belonging to the ﬁnite dimensional space Vh deﬁned by Vh = {ϕ|ϕ ∈ C 0 (Ω), ϕ|T ∈ P1 , ∀T ∈ Th }, Copyright © 2005 Marcel Dekker, Inc.

(63)

20

CONTROL AND BOUNDARY ANALYSIS

x2

x1 Figure 6.

A uniform triangulation of Ω (h = 1/4).

where in (63): P1 is the space of the two-variable polynomials of degree ≤ 1, Th is a triangulation of Ω, and h is a space discretization step. The numerical results shown in Section 3.4 have been obtained using the above approximation.

3.4

Numerical Results

We are going to apply the methodology (brieﬂy) discussed in Section 3.3 to the solution of three test problems. For all these test problems, we shall assume that Ω = (0, 1)×(0, 1) and that Th is a uniform triangulation like the one in Figure 6, with h the length of the edges of Th adjacent to the right angles. The ﬁrst test problem is deﬁned as follows: (i) f (x) =

√ R2 , ∀x ∈ Ω, with R ≥ 2 and |x| = (x21 + x22 )1/2 . (R2 − |x|2 )2

(ii) g(x) = (R2 − |x|2 )1/2 , ∀x ∈ Γ (= ∂Ω). If the above data prevail, function ψ given by ψ(x) = (R2 − |x|2 )1/2

(64)

is a solution to the Monge-Amp`ere problem (42). The graph of function ψ is clearly a piece of the sphere of radius R centered at √ {0, 0, 0}. If √ R > 2 we have ψ ∈ C ∞ (Ω); on the other hand, if R = 2 we have no better than ψ ∈ W 1,p (Ω) with p ∈ [1, 4), implying that in that particular case, ψ does not have the H 2 -regularity. When applying the computational methods discussed in Section 3.3 to the solution of the above problem (with r = 1 in algorithm (49)–(52)), we obtain if R = 2, Copyright © 2005 Marcel Dekker, Inc.

21

Operator-Splitting Methods and Applications Table 1.

Results for the First Test Problem (R = 2) h 1/32 1/64 1/128

Table 2.

ψhc − ψL2 (Ω) 4.45 × 10−6 1.14 × 10−6 2.97 × 10−7

D2h ψhc − pch (L2 (Ω))4 9.48 × 10−7 1.35 × 10−6 1.58 × 10−6

Results for the First Test Problem (R = h 1/32 1/64 1/128

ψhc − ψL2 (Ω) 2.20 × 10−5 5.51 × 10−6 1.37 × 10−6

√

2 + 10−1 )

D2h ψhc − pch (L2 (Ω))4 9.68 × 10−7 1.54 × 10−6 2.04 × 10−6

and after 78 iterations of the discrete variant of the above algorithm, the results displayed in Table 1. In Table 1, ψhc is the computed approximate solution, D2h ψhc is the corresponding discrete Hessian, and pch is the computed approximation of tensor p. The above results strongly suggest second order accuracy (a textbook one, indeed), which is in some sense optimal considering √ the type of ﬁnite element approximations we are using. If we take R = 2, our methodology which has been designed to solve the Monge-Amp`ere problem (42) in H 2 (Ω) is unable to capture any solution of the above problem, the corresponding algorithm (49)–(52) being divergent for any √ value of r. The same troubles persist if one takes R = 2 + 10−2 ; on √ the other hand, if one takes R = 2 + 10−1 , things are back to normal since, using again r = 1, we obtain after 117 iterations of algorithm (49)–(52) the results summarized in Table 2. The above results show that second order accuracy√holds. However, the second order derivatives of ψ being larger for R = 2 + 10−1 than for R = 2, the corresponding approximations errors are also larger. On Figures 7 and 8 we have visualized, respectively: √ (i) The graph of ψ when R = 2; the singularity of ∇ψ at {1, 1} appears clearly on Figure 7(a). (ii) The graph of ψhc corresponding to h = 1/128 and R = 2. √ (iii) The graph of ψhc corresponding to h = 1/128 and R = 2 + 10−1 . √ (iv) The graph of f when R = 2 + 10−1 .

Copyright © 2005 Marcel Dekker, Inc.

22

CONTROL AND BOUNDARY ANALYSIS Computed spherical surface, R=2

Spherical surface solution, R=sqrt(2) 1.5 1 0.5 0 0

0.2

0.4

0.6

0.8

10

1 0.6 0.8 0.2 0.4

2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0

0.2

0.4

0.6

0.8

Figure 7. (a). First test problem: graph of ψ when R = problem: graph of ψhc when R = 2 and h = 1/128 (right).

Computed spherical solution, R=sqrt(2)+.1 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 1 0 0.2 0.6 0.8 0.4 0.6 0.8 1 0 0.2 0.4

1 0

√

0.2

0.4

0.6

0.8

1

2 (left). (b). First test

f, R=sqrt(2)+.1 30 25 20 15 10 5 0 0

0.2

0.4

0.6

0.8

1 0

0.2

0.4

0.6

0.8

1

√ Figure 8. (a). First test problem: graph of ψhc when√R = 2 + 10−1 and h = 1/128 (left). (b). First test problem: graph of f when R = 2 + 10−1 (right).

Remark 8 When computing the approximate solutions for h = 1/32, we stopped the iterations of algorithm ( 49)–( 52) as soon as D2h ψhn − pnh (L2 (Ω))4 ≤ 10−6 .

(65)

The √ corresponding number of iterations is 78 for R = 2, and 117 for R = 2 + 10−1 (we did not try to ﬁnd the optimal value of r, or to use a variable r strategy). Next, when computing the approximate solutions for h = 1/64 and 1/128, we stopped iterating once the iteration numbers associated to h = 1/32 were reached (actually, using ( 65) as stopping criteria for h = 1/64 and 1/ 128 did not change much the approximation errors shown in Tables 1 and 2). 2 The second test problem is deﬁned as follows: (i) f (x) =

(2|x|)3/2 1 , ∀x ∈ Ω. (ii) g(x) = , ∀x ∈ Γ. |x| 3

Copyright © 2005 Marcel Dekker, Inc.

23

Operator-Splitting Methods and Applications Table 3.

Results for the Second Test Problem h 1/32 1/64 1/128

D2h ψhc − pch (L2 (Ω))4 9.91 × 10−7 1.60 × 10−6 2.02 × 10−6

ψhc − ψL2 (Ω) 5.56 × 10−5 1.50 × 10−5 3.94 × 10−5

Test Problem 2

1000 900 800 700 600 500 400 300 200 100 0 1 0.8 0.6 0.4

1.5 1 0.5 0 1 0.2 0 0

1 0.6 0.8 0.2 0.4

0.8

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

Figure 9. (a). Second test problem: graph of f (left). (b). Second test problem: graph of ψhc (h = 1/128) (right).

With these data, a solution to the Monge-Amp`ere problem (42) is the function ψ deﬁned by ψ(x) =

(2|x|)3/2 , ∀x ∈ Ω. 3

(66)

One can easily check that ψ does not belong to C 2 (Ω); however, since ψ ∈ W 2,p (Ω) for all p ∈ [1, 4), it has, in principle, enough regularity so that we can apply algorithm (49)–(52) to the solution of the corresponding problem (42). Indeed, despite the singularity of function f at {0, 0} (see Figure 9(a)), algorithm (49)–(52), with r = 1, provides after 160 iterations the results summarized in Table 3. From the results in Table 3 we can infer that second order accuracy still holds. On Figure 9(a) (resp., 9(b)) we have visualized function f (resp., the computed approximate solution obtained with h = 1/128). The third test problem is—in some sense—the more interesting since we consider this time the solution of problem (43), namely: 2 ∂ 2 ψ ∂ 2 ψ ∂ 2 ψ − = 1 in Ω, ψ = 0 on Γ. ∂x21 ∂x22 ∂x1 ∂x2 Despite the smoothness of its data, the above problem has no smooth solution, the troubles coming from the non-strict convexity of Ω = Copyright © 2005 Marcel Dekker, Inc.

24

CONTROL AND BOUNDARY ANALYSIS

(0, 1) × (0, 1). When applying algorithm (49)–(52) (in fact a discrete variant of it) to the solution of problem (43) we observe the following phenomena: (i) For r suﬃciently small (r = 1/10 here) sequence {ψhn , pnh }n≥0 converges geometrically (albeit slowly) to a limit {ψhc , pch }, while sequence {λnh }n≥0 diverges arithmetically. (ii) A close inspection of the numerical results shows that the curvature of the graph of ψhc becomes negative close to the corners, in violation of the Monge-Amp`ere equation; actually, as expected, it is violated also along the boundary, since D2h ψhc − pch (L2 (Ω))4 = 1.8 × 10−2 if h = 1/32, 3.3 × 10−2 if h = 1/64, 4.2 × 10−2 if h = 1/128, while D2h ψhc − pch (L2 (Ω1 ))4 = 2.7 × 10−4 if h = 1/32, 4.1 × 10−4 if h = 1/64, 4.9 × 10−4 if h = 1/128, and D2h ψhc − pch (L2 (Ω2 ))4 = 4.4 × 10−5 if h = 1/32, 4.9 × 10−5 if h = 1/64, 5.1 × 10−5 if h = 1/128, where Ω1 = (1/8, 7/8) × (1/8, 7/8) and Ω2 = (1/4, 3/4) × (1/4, 3/4). These results suggest that detD2 ψ = 1 is “almost” veriﬁed in Ω2 . The graph of ψhc obtained with h = 1/64 has been shown in Figure 10, while the intersections of this graph with the planes x1 = 12 and x1 = x2 have been shown in Figures 11(a) and 11(b), respectively, for h = 1/32, 1/64, and 1/128. Since ψhc does not vary much with h, we suspect that, according to Section 3.3, what we have here is a (good) approximation of one of those functions of H 2 (Ω) ∩ H01 (Ω) whose Hessian is at a minimal L2 -distance (global or local) from the set Qf deﬁned in Section 3.3. Assuming that the above is true we can claim that the solutionless problem (43) has been solved in a least-squares sense in the functional space H 2 (Ω), leading to a (not so novel in general, but possibly new in the Monge-Amp`ere “environment”) concept of generalized solution.

Remark 9 When applying algorithm ( 49)–( 52) to the solution of problem ( 42) we have to solve at each iteration a discrete analogue of the linear variational problem ( 55); to solve this ﬁnite dimensional problem, we have used a preconditioned conjugate gradient algorithm, converging typically in 6 to 8 iterations, the preconditioning requiring the solution of two discrete Poisson-Dirichlet problems per conjugate gradient iteration. The mesh being uniform we have used Fast Poisson Solvers to achieve preconditioning. Copyright © 2005 Marcel Dekker, Inc.

25

Operator-Splitting Methods and Applications Test Problem 3 0.25 0.2 0.15 0.1 0.05 0 1

Figure 10.

0.8

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

Third test problem: graph of the computed solution (h = 1/64).

cross sections

0.25 0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

0

diagonal cross sections

0.25

0 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.5

1

1.5

Figure 11. (a). Third test problem: graph of the computed solution restricted to the line x1 = 1/2 (h=1/32, 1/64, and 1/128) (left). (b). Third test problem: graph of the computed solution restricted to the line x1 = x2 (h = 1/32, 1/64, and 1/128) (right).

Acknowledgments The authors would like to thank J.D. Benamou, Y. Brennier, L.A. Caffarelli, B. Dacorogna, J. Cagnol, G.P. Galdi, D.D. Joseph, P.-L. Lions, P. Muscarello, and J.-P. Zol´esio for assistance and helpful comments and suggestions. The support of NSF (grants ECS-9527123, CTS-9873236, DMS-9902035, DMS-0209066, DMS-0412267) and DOE / LACSI (grant R71700K-292-000-99) is also acknowledged.

References ´ Fully Nonlinear Elliptic Equations. Amer[1] L. A. CAFFARELLI and X. CABRE. ican Math. Society, Providence, RI, 1995.

Copyright © 2005 Marcel Dekker, Inc.

26

CONTROL AND BOUNDARY ANALYSIS

[2] L.A. CAFFARELLI and eds. MILMAN, M. Monge-Amp`ere Equation: Application to Geometry and Optimization. American Math. Society, Providence, RI, 1999. [3] A.J. CHORIN, T.J.R. HUGHES, M.F. McCRACKEN, and J.E. MARSDEN. Product formulas and numerical algorithms. Comm. Pure and Appl. Math., 31:205–256, 1978. [4] R. COURANT and D. HILBERT. Methods of Mathematical Physics, Vol. II. Wiley Interscience, New York, NY, 1989. [5] E.J. DEAN and R. GLOWINSKI. Numerical methods for fully nonlinear elliptic equations of the Monge-Amp`ere type. (in preparation). [6] E.J. DEAN and R. GLOWINSKI. Numerical solution of the two-dimensional elliptic Monge-Amp`ere equation with Dirichlet boundary conditions: an augmented Lagrangian approach. C. R. Acad. Sci. Paris, Mathematiques, S´ er. I, 336:779–784, 2003. [7] B. DESJARDINS and M.J. ESTEBAN. On weak solution for ﬂuid-rigid structure interaction: compressible and incompressible models. Arch. Rat. Mech. Anal., 146:59–71, 1999. [8] J. DIEUDONNE. Panorama des Math´ematiques Pures: Le Choix Bourbachique. Editions Jacques Gabay, Paris, 2003. [9] M. FORTIN and R. GLOWINSKI. Augmented Lagrangians Methods: Application to the Numerical Solution of Boundary-Value Problems. North-Holland, Amsterdam, 1983. [10] R. GLOWINSKI. Finite Element Methods for Incompressible Viscous Flow, volume IX of Handbook of Numerical Analysis, P.G. Ciarlet and J.L. Lions eds. North-Holland, Amsterdam, 2003. [11] R. GLOWINSKI and P. LE TALLEC. Augmented Lagrangians and Operator Splitting Methods in Nonlinear Mechanics. SIAM, Philadelphia, PA, 1989. [12] R. GLOWINSKI, T.W. PAN, T.I. HESLA, D.D. JOSEPH, and J. PERIAUX. A ﬁctitious domain approach to the direct numerical simulation of incompressible viscous ﬂuid ﬂow past moving rigid bodies: Application to particulate ﬂow. J. Comput. Phys., 169:363–426, 2001. [13] C. GRANDMONT and Y. MADAY. Existence for an unsteady ﬂuid-structure interaction problem. Math. Model. Num. Anal., 34:609–636, 2000. [14] H.H. HU, N.A. PATANKAR, and M.Y. ZHU. Direct numerical simulation of ﬂuid-solid systems using Arbitrary Lagrangian-Eulerian techniques. J. Comput. Phys., 169:427–462, 2001. [15] A. JOHNSON and T. TEZDUYAR. 3-d simulations of ﬂuid-particle interactions with the number of particles reaching 100. Comput. Methods Mech. Engrg., 145:301–321, 1997. [16] P. LE TALLEC. Numerical Methods for Nonlinear Three-Dimensional Elasticity, volume III of Handbook of Numerical Analysis, P.G. Ciarlet and J.L. Lions eds. North-Holland, Amsterdam, 1994. [17] G.I. MARCHUK. Splitting and alternating direction methods, volume I of Handbook of Numerical Analysis, P.G. Ciarlet and J.L. Lions eds. North-Holland, Amsterdam, 1990.

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Operator-Splitting Methods and Applications

27

[18] B. MAURY. Direct simulation of 2-d ﬂuid-particle ﬂows in biperiodic domains. J. Comp. Phys., 156:325–351, 1999. [19] J.R. OCKENDON, S. HOWISON, A. LACEY, and A. MOVCHAN. Applied Partial Diﬀerential Equations. Oxford University Press, Oxford, UK, 1999. [20] T.W. PAN and R. GLOWINSKI. Direct simulation of the motion of neutrally buoyant circular cylinders in plane Poiseuille ﬂow. J. Comp.Phys., 181:260–279, 2002. [21] J.A. SAN MARTIN, V. STAROVOITOV, and M. TUCSNAK. Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous ﬂuid. Arch. Ration. Mech. Anal., 161:113–147, 2002. [22] G. STRANG. On the construction and comparison of diﬀerence schemes. SIAM J. Num. Anal., 5:506–517, 1968.

Copyright © 2005 Marcel Dekker, Inc.

CHAPTER II DYNAMICAL SHAPE SENSITIVITY Application to a Coupled Fluid-Structure System Marwan Moubachir Institut National de Recherche en Informatique et Automatique Projet OPALE Sophia Antipolis, France

Jean-Paul Zol´esio CNRS and INRIA Projet OPALE Sophia Antipolis, France

Abstract

Nowadays the design of systems involving the interaction of a ﬂuid with a moving interface is part of many industrial processes, from aeronautics to biomechanics. This interface is usually the boundary between the ﬂuid and a material with diﬀerent physical and rheological properties, such as an elastic solid, a gas, or a diﬀerent ﬂuid phase. Up to now, the associated control and inverse problems have been treated by neglecting the motion of the interface, resulting in the analysis of classical optimization problems for partial diﬀerential equations in cylindrical time-space domain endowed with transpiration boundary conditions on the ﬁxed interface. However, the search for performances and robustness and the need to take into account extreme running conditions make this approximation too restrictive. This article introduces several tools that may be useful while studying the optimization problem of industrial devices involving large interface displacements. The keystone of this analysis is the generalization of cylindrical shape optimization tools to the non-cylindrical case. These tools are illustrated by showing how to compute the cost function gradient involved in an inverse problem for a coupled system consisting of an elastic solid in rigid motion surrounded by a Navier-Stokes ﬂow.

Keywords: Shape optimization, moving domain, Navier-Stokes, ﬂuid-solid interaction, optimal control, inverse problem

Copyright © 2005 Marcel Dekker, Inc.

30

1.

CONTROL AND BOUNDARY ANALYSIS

Introduction

This article deals with the analysis of an inverse boundary problem for a coupled ﬂuid-structure system which enters as a generic tool in the design and the control of many industrial devices, such as aircraft wings, cable-stayed bridges, automobile shapes, satellite reservoir tanks, and more generally, systems involving ﬂuid-solid interactions. This article is focused on the derivation of optimality conditions associated with the minimization problem involved in a typical open-loop control strategy. Our aeroelastic system consists of an elastically supported rigid solid moving inside an incompressible ﬂuid ﬂow in 2-D. It is described by a non-cylindrical system of partial diﬀerential equations, where the evolution of the moving boundary is unknown [5]. Using a new methodology to handle non-cylindrical sensitivity analysis introduced in [6, 3], we are able to perform the sensitivity analysis of this coupled system where the evolution of the moving boundary is governed by an ordinary diﬀerential equation. In the ﬁrst section, we introduce the reader to the coupled ﬂuidstructure mechanical system. Then, we set up the inverse problem framework and state the main result of this article, which consists of furnishing the expression of the gradient of a speciﬁc cost function depending on the dynamical evolution of the solid, with respect to the far-ﬁeld boundary condition considered as a control variable. The keystone of its proof lies in the Navier-Stokes dynamical shape sensitivity analysis introduced in the last section.

2.

Mechanical Problem

We consider a two-dimensional ﬂexible solid in rigid motion. For the sake of simplicity, we only consider one degree of freedom for the structural motion: the vertical displacement d(t)e2 where e2 is the element of the cartesian basis (e1 , e2 , e3 ) in R3 . The structure is surrounded by a viscous ﬂuid in the plane (e1 , e2 ). We consider a control volume Ω = Ωft Ωst ⊂ R2 with boundary Γ∞ , containing the solid Ωst with boundary Γst for every time t ∈ (0, T ). We introduce a diﬀeomorphic map Tt sending a ﬁxed reference domain Ω0 = Ωf0 Ωs0 into the physical conﬁguration Ω at time t ≥ 0. This map is built thanks to the Lagrangian ﬂow of Eulerian velocity ﬁeld V deﬁned as follows: ˙ x ∈ Ωst V (x, t) = d(t)e 2, f ˙ V (x, t) = Ext(d(t)e 2 ), x ∈ Ωt V (x, t) · n = 0, x ∈ Γ∞ Copyright © 2005 Marcel Dekker, Inc.

(1)

31

Dynamical Shape Sensitivity

with Ext is an arbitrary extension operator from Γst into Ωft . The solid is described by the evolution of its displacement d e2 and its ˙ is solution of the following ordinary velocity d˙ e2 and the couple (d, d) second order diﬀerential equation: ¨ k d = Ff , m d + (2) d, d˙ (t = 0) = [d0 , d1 ] where (m, k) stand for the structural mass and stiﬀness. The right-hand term Ff is the projection of the ﬂuid loads on Γst along the motion direction e2 . We consider a viscous incompressible Newtonian ﬂuid. Its evolution is described by its velocity u and its pressure p. The couple (u, p) satisﬁes the classical Navier-Stokes equations written in non-conservative form: ∂t u + D u · u − ν∆u + ∇p = 0, Qf (V ) div(u) = 0, Qf (V ) (3) u = u∞ , Σ∞ u(t = 0) = u0 , Ωf0 where ν stands for the kinematic viscosity, u∞ is the far-ﬁeld velocity ﬁeld and def def Qf (V ) = {t} × Ωft , Σ∞ = (0, T ) × Γ∞ 0 0 are constant. (ii) The thickness h of the shell is small enough to accomodate the curvatures H and K, i.e., the product of the thickness by the curvatures is small as compared to 1. As a consequence we shall drop terms of order equal to or greater than 2 in the series expansions with respect to the radial variable. We also suppose that j(z) = det(DTz ) = det(I − zD 2 b) = 1. (iii) (Kirchhoﬀ Hypothesis) Let T be a transformation of the shell Sh , and let e = (eΓ , w) be the corresponding transformation of the midsurface. In the classical thin plate theory named after Kirchhoﬀ, the displacement vectors T and e ◦ p are related by the hypothesis that the ﬁlaments of the plate initially perpendicular to the middle surface remain straight and perpendicular to the deformed surface, and undergo neither contraction nor extension. We may generalize this hypothesis to the case of a shell using the intrinsic geometry, yielding T = e ◦ p − b ( ∗ DΓ e ∇b) ◦ p (12) The equations given below correspond to the behavior of the midsurface Γ0 . On Γ0 , w is the normal displacement of the shell, and eΓ is a vector describing the tangential displacements. Both of these quantities are speciﬁed in local coordinates. Let ρsh be the density of the shell, and λ and µ be the Lam´e coeﬃcients. Let the function b, and the matrices built on that function, depend on the geometry of the shell. The curvature is given by K and H, and it is also taken into account in the matrix D2 b. Deﬁne the following operator C acting on a matrix A: C(A) = λ tr (A) I + 2µ A

(13)

χ ˜ = CΓ eΓ − εΓ (D 2 b eΓ )

(14)

the expression χ ˜ and β = (λ + 2µ)−1 . Next, deﬁne CΓ and SΓ as CΓ u = SΓ w = Copyright © 2005 Marcel Dekker, Inc.

1 2 ∗ (D b DΓ u + DΓ u D 2 b) 2 1 2 (D w + ∗ DΓ2 w) 2 Γ

(15a) (15b)

44

CONTROL AND BOUNDARY ANALYSIS

CΓ is a ﬁrst-order operator, that in practice operates on a tangential vector u. SΓ is the symmetrization of the Hessian matrix of a scalar function w (the Hessian matrix is not symmetric in the tangential calculus [9]). Then, given appropriate initial conditions, the displacement (w, eΓ ) of the shell satisﬁes the following system of shell equations: ∂tt w − γ∆Γ ∂tt w + ∆Γ 2 w +

γ divΓ (D 2 b ∂tt eΓ ) 2

+P1 (eΓ ) + Q1 (w) = 0

(16a)

(I + γ(D 2 b)2 )∂tt eΓ − β[γ −1 divΓ C(εΓ (eΓ )) + D2 b divΓ C(χ) ˜ γ ˜ − D2 b∇Γ ∂tt w + P2 (w) + Q2 (eΓ ) = 0 (16b) −divΓ (D 2 b C(χ))] 2 where P1 denotes coupling terms and Q1 denotes lower order terms in the plate equation; and P2 , Q2 in the wave equation: (17a) P1 (eΓ ) = β[2λHγ −1 divΓ eΓ + 2µγ −1 tr (D 2 b εΓ (eΓ )) 3 2 +2µdivΓ divΓ (χ) ˜ + 4µH tr (D beΓ D b) −λ∆Γ 2∇Γ H, eΓ − λ(4H 2 − 2K) 2∇Γ H, eΓ −2µ (D 2 b)2 ..D3 b, eΓ Q1 (w) = β[kγ w + 4µdivΓ ((D 2 b)2 ∇Γ w) + λ∆Γ ((4H 2 − 2K)w) (17b) +2µdivΓ divΓ ((D2 b)2 w) + 2µdivΓ (K∇Γ w) +λ(4H 2 − 2K)∆Γ w] + 2µ tr (SΓ w(D 2 b)2 ) +4µH tr ((D 2 b)3 w) P2 (w) = β[−2λγ −1 ∇Γ (Hw) + 2µγ −1 divΓ (w D2 b) (17c) 2 2 2 3 +λ2∇Γ H(∆Γ w − (4H − 2K)w) − 2µ(D b) ..D bw] −2µdivΓ (D 2 b SΓ w) + 2µD2 b divΓ (SΓ w) Q2 (eΓ ) = −2βµγ −1 (K eΓ + 2(D2 b)2 eΓ )

(17d)

On ∂Γ0 × (0, T ), which is clamped, one has the boundary conditions: ∂w =0 (18) ∂ν For ease of computation, whenever possible we replace equations (16)– (18) with the following abstract form: eΓ = 0,

w = 0,

M∂tt e + Ae = 0 Copyright © 2005 Marcel Dekker, Inc.

on

Γ0 × (0, T )

(19a)

45

Optimal Control of a Structural Acoustic Model

eΓ = 0,

w=

∂w = 0 on ∂ν

∂Γ0 × (0, T )

(19b)

where the vector e = (eΓ , w). The deﬁnition of the operators M and A can be seen by inspection of (16). The operators M and A are positive, self-adjoint, densely deﬁned, and closable. A is bounded and 2 coercive on H01 (Γ0 ) × H02 (Γ0 ) and M is bounded and coercive on [L2 (Γ0 )]2 × H01 (Γ0 ) [5].

6.

Structurally Damped, Forced Shell Wall of an Acoustic Chamber

Next, we use the basic operators deﬁned above in analysis of the control problem. In order to model damping in the shell, we account for KelvinVoight type structural damping. In other words, we add the potential energy operator A acting on the time derivative of displacement, ∂t e. It is necessary to take into account the action of the piezoceramic actuators on the shell. This will take the form of a speciﬁc operator B with a control input u. We consider N patches perfectly bonded to the shell Γ0 as shown on Figure 1. The vector u will be composed of the N scalar inputs ui where ui is the input voltage at patch i. Following [12] and references therein, when a voltage is applied along the polarization of a piezoelectric patch, this element develops a compressive strain proportional to the voltage applied. We here consider a generic unconstrained 2D-actuator, polarized along the normal direction, numered i0 . We will denote as Ξ the area included on the mid-surface, above which the patches are located, while ∂Ξ denotes the closed curves which delineate this area. The operator B is given by (Bu, e)Γ0 = (B1 u, eΓ0 ) + (B2 u, w) N 1 uj [αj eΓ0 + βj ∇Γ w, ∇b] dΞj , for e ∈ D(A 2 ) (20) = j=1

Ξj

where the coeﬃcients αj and βj are based on the material properties of the patches. Then B : [L2 (Ξi )]N i=1 ≡ U → D(A) , and therefore A−r B ∈ L(U; [L2 (Γ0 )]3 ) for

3 1 0: |eAt B|L(U ;H) ≤ with

3 8

CT tr

for

0 0, we have that | |T (e)|L

1/2 2 (0,T ;D(Mγ ))

≤ C|M−1/2 (∂t z|Γ0 ∇b)|L2 (0,T ;L2 (Γ0 )) γ

≤ C|(∂t z|Γ0 )|L2 (0,T ;H −1/3 (Γ0 )) ≤ C|∂t w|L2 (0,T ;H 1/3 (Γ0 )) ≤ C|e|L2 (0,T ;He ) (37) Since −Ash generates an analytic semigroup on He , for f ∈ L2 (0, T ; He ) we immediately have the estimate t −A (t−s) sh e f (s) ds ≤ C|f |L2 (0,T ;He ) (38) 1/2− 0

C([0,T ];D(Ash

))

Combining (37) and (38) gives, for arbitrary v ∈ L2 (0, T ; He ), that |Y (e)|C([0,T ];D(A1/2− )) ≤ C|T (e)|L2 (0,T ;He ) ≤ C|e|L2 (0,T ;He ) sh

(39)

which proves the ﬁrst part of Proposition 6. For the second, note that the estimate (39) implies that |Y (e)|L2 (0,T ;He ) ≤ CT |e|L2 (0,T ;He )

(40)

where CT → 0 as T → 0. This allows for the bounded invertibility of I −Y on a small interval (0, T0 ), and repeatedly using this estimate gives the estimate for arbitrary T > 0 after ﬁnitely many steps. Step 2: From the regularity shown in (21) we have

We will show that

|Bu|D(Ar ) = |Ar Bu|U ≤ C|u|U

(41)

0 M−1 Bu

(42)

D(Arsh )

≤ C|u|U

We take φ = (φ1 , φ2 ) ∈ D(Arsh ) and compute with ﬁxed u: 0 ,φ = (M−1 Bu, φ2 ) 1 = (Bu, φ2 )[L2 (Γ )]3 0 D(M 2 ) M−1 Bu H e

= (A−r Bu, Ar φ2 )[L2 (Γ0 )]3 = C|Ar φ2 |[L2 (Γ0 )]3 |u|U ≤ C|Arsh φ|He |u|U (43) Copyright © 2005 Marcel Dekker, Inc.

50

CONTROL AND BOUNDARY ANALYSIS

where we have used (37) and the regularity (21). This implies the estimate (42). Additionally, we can use (42), analyticity of e−Ash t and Lemma 4 to give, for a ﬁxed u, 2 T −A t 0 e sh dt ≤ C|u|2U (44a) −1 M Bu D(A1/2−r ) 0 sh −A t 0 ≤ C |u|U , for 0 < t ≤ T e sh (44b) −1 M Bu H tr e Step 3: Denote the vector

z ∂t z At e = e Bu ∂t e

(45)

Then, (z, ∂t z) satisﬁes the wave equation (35) and e is given by 0 −Ash (t) e(t) = e + Y (e)(t) M−1 Bu Thus, −Ash (t)

[I − Y ]e(t) = e

(46)

0

(47)

M−1 Bu

Applying Proposition 6 (ii) and (44) leads to

T 0

|e(t)|2He

≤ C|[I − Y ≤C 0

T

]−1 |2L(L2 (0,T ;He ) e−Ash (t)

−A (t) e sh

2 −1 M Bu L 0

2 (0,T ;He )

2 0 dt ≤ C|u|2U M−1 Bu D(A1/2−r )

(48)

sh

1 if we take 38 < r ≤ where we note that 12 − r ≥ 12 following a priori regularity of the variable e:

5 12 .

Thus we have the

|e|L2 (0,T ;He ) ≤ C|u|U

e ∈ L2 (0, T ; He ),

(49)

Step 4: We can now use the regularity (49) and Lemma 5 which gives the following regularity of the wave component z = (z, ∂t z) in (45): |z|C([0,T ];Hz ) ≤ C|u|U

(50)

Now, using Proposition 6 gives 1/2−ε

Y e ∈ C([0, T ]; D(Ash Copyright © 2005 Marcel Dekker, Inc.

)) ⊂ C([0, T ]; He )

(51)

Optimal Control of a Structural Acoustic Model

|Y e|C([0,T ];He ) ≤ C|u|U

51 (52)

Since we have the representation (46) for all 0 ≤ t ≤ T , (38) gives

1 1 (53) |e(t)|He ≤ C|e|L2 (0,T ;He ) + C r |u|U ≤ C 1 + r |u|U t t Combining (50) and (53) gives the result.

References [1] G. Avalos. The exponential stability of a coupled hyperbolic/parabolic system arising in structural acoustics. Abstract and Applied Analysis, 1(2):203–217, 1996. [2] G. Avalos and I. Lasiecka. Diﬀerential Riccati equation for the active control of a problem in structural acoustics. Journal of Optimization Theory and Applications, 91(3):695–728, 1996. [3] H. T. Banks, R. J. Silcox, and R. C. Smith. The modeling and control of acoustic/structure interaction problems via piezoceramic actuators: 2-d numerical examples. ASME Journal of Vibration and Acoustics, 116(3):386–396, 1994. [4] H. T. Banks, R. C. Smith, and Y. Y. Wang. The modeling of piezoceramic patch interactions with shells, plates, and beams. Quarterly of Applied Mathematics, 53(2):353–381, 1995. [5] J. Cagnol, I. Lasiecka, C. Lebiedzik, and J.-P. Zol´esio. Uniform stability in structural acoustic models with ﬂexible curved walls. Journal of Diﬀerential Equations, 186(1):88–121, 2002. [6] J. Cagnol and C. Lebiedzik. Free boundary conditions for intrinsic shell model. In V. Barbu, I. Lasiecka, D. Tiba, and C. Varsan, editors, Analysis and optimization of diﬀerential systems, pages 77–88. Kluwer Academic Publishers, Boston, MA, 2003. [7] M. Camurdan and G. Ji. A noise reduction problem arising in structural acoustics: a three-dimensional solution. Contemporary Mathematics, 267, 2000. [8] S. Chen and R. Triggiani. Proof of extensions of two conjectures on structural damping for elastic systems. Paciﬁc Journal of Mathematics, 136(1):15–55, 1989. [9] M. C. Delfour and J.-P. Zol´esio. Intrinsic diﬀerential geometry and theory of thin shells. To appear. [10] M. C. Delfour and J.-P. Zol´esio. A boundary diﬀerential equation for thin shells. Journal of Diﬀerential Equations, 119(2):426–449, 1995. [11] M. C. Delfour and J.-P. Zol´esio. Tangential diﬀerential equations for dynamical thin/shallow shells. Journal of Diﬀerential Equations, 128(1):125–167, 1996. [12] E. K. Dimitradis, C. R. Fuller, and C. A. Rogers. Piezoelectric actuators for distributed vibration excitation of thin plates. Journal of Vibration and Acoustics, 113:100–107, 1991. [13] I. Lasiecka. Mathematical Control Theory of Coupled PDE Systems: NSF-CMBS Lecture Notes. SIAM, 2000.

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[14] I. Lasiecka and C. Lebiedzik. Uniform stability in structural acoustic systems with thermal eﬀects and nonlinear boundary damping. Control and Cybernetics, 28(3):557–581, 1999. [15] I. Lasiecka and R. Marchand. Control and stabilization in nonlinear structural acoustic problems. In Proceedings of SPIE’s 4th Annual Symposium on Smart Structures and Materials, Mathematics and Control in Smart Structures, 1997. [16] I. Lasiecka and R. Marchand. Riccati equations arising in acoustic structure interactions with curved walls. Dynamics and Control, 8:269–292, 1998. [17] I. Lasiecka and R. Triggiani. Sharp regularity theory for second order hyperbolic equations of Neumann type. Part I. nonhomogeneous data. Annali di Matematica Pura ed Applicata IV, CLVII:285–367, 1990.

Copyright © 2005 Marcel Dekker, Inc.

CHAPTER IV NONLINEAR WAVE EQUATIONS WITH DEGENERATE DAMPING AND SOURCE TERMS Viorel Barbu Department of Mathematics Alexandru Ioan Cuza University Iasi, Romania

Irena Lasiecka Department of Mathematics University of Virginia Charlottesville, VA, USA

Mohammad A. Rammaha Department of Mathematics and Statistics University of Nebraska-Lincoln Lincoln, NE, USA

Abstract

In this article we present some results on the global well-posedness of the wave equation utt − ∆u + |u|k ∂j(ut ) = |u|p−1 u in Ω × (0, T ), where ∂j is a sub-diﬀerential of a continuous convex function j, and Ω is a bounded smooth domain in Rn . Under some conditions on j and the parameters in the equations we obtain several results on the existence of global solutions, uniqueness, nonexistence of global solutions, and propagation of regularity.

Keywords: Wave equation, damping and source terms, weak solutions, blow-up of solutions, sub-diﬀerentials, energy estimates

Introduction Let Ω be a bounded domain in Rn with a smooth boundary Γ. In this article we study the solvability and long-time behavior of the following initial-boundary value problem: utt − ∆u + |u|k ∂j(ut ) = |u|p−1 u, Copyright © 2005 Marcel Dekker, Inc.

in Ω × (0, T ) ≡ QT ,

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CONTROL AND BOUNDARY ANALYSIS

u(x, 0) = u0 (x) ∈ H01 (Ω),

ut (x, 0) = u1 (x) ∈ L2 (Ω),

(1)

u = 0, on Γ × (0, T ), where j(s) is a continuous, convex real valued function deﬁned on R and ∂j is its sub-diﬀerential (see [4]). The initial-boundary value problem (1) is studied under the following assumption: Assumption A: k, m, p ≥ 0. In addition, k and p

0. Strict monotonicity: (∂j(s) − ∂j(v))(s − v) ≥ c1 |s − v|m+1 , where c1 > 0. Continuity: ∂j(s) is single valued and |∂j(s)| ≤ c0 |s|m + c2 , for some constants c0 > 0, c2 ≥ 0. We remark here that although ∂j(s) is single valued, we choose to use the sub-diﬀerential notation in this paper for the sake of convenience. The main goal of the paper is to study the long-time behavior of solutions to the initial-boundary value problem (1). Of particular interest is the relationship of the source and damping terms to the behavior of solutions. It is important to note that the partial diﬀerential equation in (1) is a special case of the prototype evolution equation utt − ∆u + R(x, t, u, ut ) = F(x, u),

(2)

where in (2) the nonlinearities satisfy: vR(x, t, u, v) ≥ 0, R(x, t, u, 0) = F(x, 0) = 0, and F(x, u) ∼ |u|p−1 u for large |u|. Various special cases of (2) arise in quantum ﬁeld theory and some important mechanical applications. See for example J¨ orgens [11]and Segal [28]. Also, one of the important special cases of (1) is the following well known polynomially damped wave equation studied extensively in the literature (see for instance [24, 26]). utt − ∆u + |u|k |ut |m−1 ut = |u|p−1 u, u(x, 0) = u0 (x),

in Ω × (0, T ),

ut (x, 0) = u1 (x),

u(x, t) = 0 on Γ × (0, T ), Copyright © 2005 Marcel Dekker, Inc.

in Ω,

(3)

Nonlinear Wave Equations with Degenerate Damping and Source Terms

55

1 It is easy to see that by taking j(s) = m+1 |s|m+1 , then ∂j(s) = |s|m−1 s, and therefore Assumption A is satisﬁed. Moreover, in this case, the initial-boundary value problem (1) is equivalent to (3). It is worth noting here that when the damping term |u|k |ut |m−1 ut is absent, the source term |u|p−1 u drives the solution of (3) to blow-up in ﬁnite time [8, 17, 23, 31]. In addition, if the source term |u|p−1 u is removed from the equation, then damping terms of various forms are known to yield existence of global solutions, (cf. [2–4]). However, the interaction between the damping and source terms is often diﬃcult to analyze, as one can see from the work in [7, 20, 18, 24, 26, 29]. It should be noted that if k = 0 and p = 0 then equation (1) can be studied via the theory of monotone operators and the full well-posedness of strong solutions (in the terminology of monotone operator theory) is classical [4]. In addition, with k = 0 the presence of a source term, which is locally Lipschitz in H 1 (Ω) does not aﬀect the arguments for establishing the existence of local solutions via the theory of monotone operators. Moreover, if p ≤ k + m then one can derive the necessary a priori bounds that guarantee that, in this case, every local solution is indeed global in time. The situation is, however, diﬀerent when the damping term is degenerate, leading to the degeneracy of the monotonicity argument. In fact, when k > 0, (3) is no longer locally Lipschitz perturbation of a monotone problem. Thus, standard monotone operator theory does not apply. This fact combined with a potential strong growth of the damping term (the case when m > 1 ), makes the problem interesting and the analysis is more subtle. Indeed, one needs to be careful about the meaning of the solution and its relation to the equation.

1.

Deﬁnitions and Main Results

In this section we provide results on the existence and uniqueness of various types of solutions such as generalized solutions, weak solutions, and strong solutions. In order to proceed with the presentation of our results we shall introduce the appropriate deﬁnitions. First, we give the deﬁnition of a generalized solution, which satisﬁes a given variational inequality.

Definition 1 A function u ∈ Cw ([0, T ], H01 (Ω))∩Cw1 ([0, T ], L2 (Ω)) with |u|k j(ut ) ∈ L1 (Ω × (0, T )) is said to be a generalized solution to ( 1) if and only if for all 0 < t ≤ T the following inequality holds: t (ut vt − ∇u∇v)dxdt + 1/2 [u2t (t) + |∇u(t)|2 ]dx 0

Ω

Copyright © 2005 Marcel Dekker, Inc.

Ω

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CONTROL AND BOUNDARY ANALYSIS

t

+ |u|k [j(ut ) − j(v)]dxdt 0 Ω t |u|p−1 u(ut − v)dxdt + 1/2 [u21 + |∇u0 |2 + 2u1 v(0)]dx ≤ 0

Ω

(4)

Ω

for all test functions v satisfying v ∈ H 1 (0, t; L2 (Ω)) ∩ L2 (0, t; H01 (Ω)) ∩ L∞ (Ω × (0, T )), v(t) = vt (t) = 0. It should be noted here that Deﬁnition 1 is a proper extension of the classical deﬁnition of weak solutions, which sometimes is called variational.

Definition 2 A function u ∈ Cw ([0, T ],H01 (Ω))∩Cw1 ([0, T ],L2 (Ω)) which satisﬁes ∆u − utt and |u|k j(ut ) ∈ L2 (Ω × (0, T )) is said to be a weak solution to ( 1) if and only if for all 0 < t ≤ T the following variational equality holds: t t (utt − ∆u)vdxdt + |u|k ∂j(ut )vdxdt 0 0 Ω Ω t |u|p−1 uvdxdt = 0

u(0) = u0 , ut (0) = u1

(5)

for all test functions v ∈ L2 (Ω × (0, T )). Equivalently, we may write t t (−ut vt + ∇u∇v)dxdt + u1 v(0)dx + |u|k ∂j(ut )vdxdt 0 0 Ω Ω Ω t |u|p−1 uvdxdt, (6) = 0

for all test functions v satisfying v ∈ H 1 (0, T ; L2 (Ω)) ∩ L2 (0, T ; H01 (Ω)) ∩ L2 (Ω × (0, T )), v(t) = 0 It should be noted here that Deﬁnition 2 implies that utt − ∆u = −|u|k ∂j(ut ) + |u|p−1 u, a.e in x, t ∈ Ω × (0, T ). At this end, we remark that the following notation will be used in the sequel: |u|s,Ω ≡ |u|H s (Ω) and |u|p ≡ |u|Lp (Ω) , Copyright © 2005 Marcel Dekker, Inc.

Nonlinear Wave Equations with Degenerate Damping and Source Terms

57

where H s (Ω) and Lp (Ω) stands for the classical Sobolev spaces and the Lebesgue spaces, respectively. Also, we let A : L2 (Ω) → L2 (Ω), where A = −∆ with its domain D(A) = H 2 (Ω) ∩ H01 (Ω). Our main results read as follows:

Theorem 3 Generalized solutions. Under Assumption A and the condition p ≤ k + m, there exists a global generalized solution to ( 1). By assuming more restrictions on the parameters k, m, we obtain global existence of weak solutions. Speciﬁcally, we have:

Theorem 4 Weak solutions. Part 1 In addition to Assumption A and the condition p ≤ k + m, we also assume that m < 1 if n = 1, 2; (7) k m 1 2n ∗ p∗ + 2 ≤ 2 , if n ≥ 3; where p = n−2 . Then, there exists a global weak solution to ( 1) which is deﬁned on (0, T ), for any T > 0. Moreover, the solution is unique and depends continuously on the initial data. The same conclusion holds when instead of ( 7) we assume k = 0. Part 2 In addition to Assumption A, assume that m = 1, k + 1 ≤ p∗ . Then, there exists a global solution u to ( 1) such that u ∈ 2 Cw ([0, T ], H01 (Ω))∩ Cw ([0, T ], L2 (Ω)) where u satisﬁes ( 4) with the test functions v ∈ L∞ (Ω×(0, T )), and T is arbitrary . In addition, ∆u − utt ∈ L1 (Ω × (0, T )) and such a solution u is unique, but it may not be continuously dependent on the initial data in the ﬁnite energy norm. Our next result addresses the issue of a strong source (large values of p) which may lead to a ﬁnite-time blow-up of solutions.

Theorem 5 Blow-up of weak solutions. Assume the validity of Assumption A with c2 = 0 and that p > k + m. In addition, assume that E(0) < 0, where E(0) is the initial energy given by ( 1 1' |u1 |20,Ω + |A1/2 u0 |20,Ω − |u0 |p+1 . E(0) = Lp+1 (Ω) 2 p+1 Then, the weak solution to ( 1) blows up in a ﬁnite time. Our next Theorem addresses the issue of propagation of regularity. This means that more regular data produce more regular solutions. In fact, Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

the result below states that this is always the case locally (i.e., for sufﬁciently small times). However, in the special case when the parameter p is below the critical value k + m, then the propagation of regularity is a global phenomena.

Theorem 6 Strong (regular) solutions. Assume that k > 1, p > 2, ∗ n and that j is deﬁned almost everywhere on R. Then, k + m ≤ p2 = n−2 for every initial data satisfying u0 ∈ H 2 (Ω) ∩ H01 (Ω), u1 ∈ H01 (Ω), there exists T0 > 0 such that ( 1) has a local solution u with the regularity that u ∈ C([0, T ], H 2 (Ω)) ∩ C 1 ([0, T ], H 1 (Ω)), for some T ≤ T0 where T0 may be ﬁnite. In addition, if we assume that p ≤ k + m, and either k = 0 1 or pk∗ + m 2 ≤ 2 , then regular solutions are global and T0 can be taken arbitrarily. As we have already mentioned in the Introduction, the particular feature of the problem studied in this paper is the potential degeneracy of the damping term. Thus, the beneﬁcial eﬀects of damping may not be present when the displacement has strong oscillations. In fact, this problem was studied ﬁrst in [24] in the special case when the damping 1 term is sub-linear in the velocity, i.e., m < 1 and subject to pk∗ + m 2 ≤ 2. In this case and with the assumption that p ≤ k + m, the authors in [24] established the existence of a unique global weak solution. When m > 1, the situation is more complex. This is mostly due to the notorious diﬃculties in passing to the limit in a super-linear term in ut with a weakly convergent subsequence. The lack of good structure, namely the lack of monotonicity, and super-linearity of ut in a degenerate damping term render standard tools nonapplicable. Nevertheless, our results establish the existence of global generalized solutions with no restrictions on m. Under additional restrictions on the parameters (see Theorem 4), one obtains weak solutions which propagate additional regularity as stated in Theorem 6. The ﬁrst statement in Part 1 in Theorem 4 was shown in [24]. However, our proof is diﬀerent and, we believe, it is simpler. The result stated in Part 2 of Theorem 4 is new. Here, the main issue was the uniqueness of the corresponding solution. While it is shown that the solution is indeed unique, it may not depend continuously on initial data.

2.

Outline of the Proof of Theorem 3

In this section we only present an outline of the proof of Theorem 3. Full proofs of our results are in [6] and will appear elsewhere. Copyright © 2005 Marcel Dekker, Inc.

Nonlinear Wave Equations with Degenerate Damping and Source Terms

59

The following a priori bound for generalized solutions in the topology speciﬁed by the Deﬁnition 1 is crucial in our proofs. Speciﬁcally, we have the following Lemma.

Lemma 7 Let u be a generalized solution of problem ( 1) with the assumption p ≤ k + m. Then for all initial data u0 ∈ H01 (Ω), u1 ∈ L2 (Ω) and all T > 0, we have the inequality |u(t)|1,Ω + |ut (t)|0,Ω +

t 0

|u|k j(ut )dxdτ ≤ CT (|u0 |1,Ω , |u1 |0,Ω ),

Ω

for all t ∈ [0, T ]. The a priori bound obtained in Lemma 7 above allows us to construct a multi-valued ﬁxed point argument that completes the proof of Theorem 3. However, in order to show the existence of a ﬁxed point, one must establish several facts: First, for a ﬁxed function w ∈ C(0, T ; Lq (Ω)) establish the solvability of the corresponding problem of the following equation (this is accomplished by applying an appropriate Faedo-Galerkin method): utt − ∆u + |w|k ∂j(ut ) = |w|p−1 w,

(8)

For a given argument w ∈ C(0, T ; Lq (Ω)) we consider the multivalued mapping F : C(0, T ; Lq (Ω)) → C(0, T ; Lq (Ω)), where the multi-valued action of F is deﬁned by u ∈ F w if and only if u is a solution to the following variational inequality: t 1 (ut vt − ∇u∇v)dxdt + [u2 (t) + |∇u(t)|2 ]dx 2 Ω t 0 Ω t |w|k [j(ut ) − j(v)]dxdt + 0 Ω t |w|p−1 w(ut − v)dxdt ≤ 0 Ω 1 [u2 + |∇u0 |2 + 2u1 v(0)]dx (9) + 2 Ω 1 for all test functions v ∈ H 1 (0, t; L2 (Ω)) ∩ L2 (0, t; H01 (Ω)) ∩ L∞ (Qt ), Copyright © 2005 Marcel Dekker, Inc.

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v(t) = vt (t) = 0. We remark here, by the previous bullet F (w) = ∅ for every ﬁxed w ∈ C(0, T ; Lq (Ω)) Next, we show that: If K is a ball in C(0, T ; Lq (Ω)), then F (K) is compact and F (w) is convex in C(0, T ; Lq (Ω)), for every w ∈ K. Our next step is to show the upper semi-continuity of the nonlinear map F . For this part, our argument is based on subtle approximations by weakly lower semi-continuous functions. Finally, we apply Kakutani’s ﬁxed point theorem to establish a ﬁxed point for F in an appropriate ball in C(0, T ; Lq (Ω)).

References [1] R. A. Adams. Sobolev spaces. Academic Press [A subsidiary of Harcourt Brace Jovanovich Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. [2] K. Agre and M. A. Rammaha. Global solutions to boundary value problems for a nonlinear wave equation in high space dimensions. Diﬀ. Integral Eq., 14(11):1315–1331, 2001. ´ [3] ¯D. ¯D. Ang and A. Pham Ngoc Dinh. Mixed problem for some semilinear wave equation with a nonhomogeneous condition. Nonlinear Anal., 12(6):581–592, 1988. [4] V. Barbu. Nonlinear Diﬀerential Equations in Banach spaces. Nordhoﬀ, Leyden, The Netherlands, 1976. [5] V. Barbu and Th. Precupanu. Convexity and optimization in Banach spaces. Editura Academiei, Bucharest, revised edition, 1978. Translated from the Romanian. [6] V. Barbu, I. Lasiecka, and M. A. Rammaha. On nonlinear wave equations with degenerate damping and source terms. Trans. Amer. Math. Soc. To appear. [7] V. Georgiev and G. Todorova. Existence of a solution of the wave equation with nonlinear damping and source terms. J. Diﬀerential Equations, 109(2):295–308, 1994. [8] R. T. Glassey. Blow-up theorems for nonlinear wave equations. Math. Z., 132:183–203, 1973. [9] P. Grisvard. Caract´erisation de quelques espaces d’interpolation. Arch. Rational Mech. Anal., 25:40–63, 1967. ´ ´ [10] P. Grisvard. Equations diﬀ´erentielles abstraites. Ann. Sci. Ecole Norm. Sup. (4), 2:311–395, 1969. [11] K. J¨ orgens. Das Anfangswertproblem im Grossen f¨ ur eine Klasse nichtlinearer Wellengleichungen. Math. Z., 77:295–308, 1961.

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61

[12] H. Koch and I. Lasiecka. Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity: full von Karman systems. In Evolution equations, semigroups and functional analysis (Milano, 2000), volume 50 of Progr. Nonlinear Diﬀerential Equations Appl., pages 197–216. Birkh¨ auser, Basel, 2002. [13] I. Lasiecka, J.-L. Lions, and R. Triggiani. Nonhomogeneous boundary value problems for second order hyperbolic operators. J. Math. Pures Appl. (9), 65(2):149–192, 1986. [14] I. Lasiecka and R. Triggiani. A cosine operator approach to modeling L2 (0, T ; L2 (Γ))—boundary input hyperbolic equations. Appl. Math. Optim., 7(1):35–93, 1981. [15] I. Lasiecka and R. Triggiani. Regularity theory of hyperbolic equations with nonhomogeneous Neumann boundary conditions. II. General boundary data. J. Diﬀerential Equations, 94(1):112–164, 1991. [16] I. Lasiecka and J. Ong. Global solvability and uniform decays of solutions to quasilinear equation with nonlinear boundary dissipation. Comm. Partial Diﬀerential Equations, 24(11–12):2069–2107, 1999. [17] H. A. Levine. Instability and nonexistence of global solutions to nonlinear wave equations of the form P utt = −Au + F(u). Trans. Amer. Math. Soc., 192:1–21, 1974. [18] H. A. Levine, S. R. Park, and J. Serrin. Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation. J. Math. Anal. Appl., 228(1):181–205, 1998. [19] H. A. Levine, S. R. Park, and J. Serrin. Global existence and nonexistence theorems for quasilinear evolution equations of formally parabolic type. J. Differential Equations, 142(1):212–229, 1998. [20] H. A. Levine and J. Serrin. Global nonexistence theorems for quasilinear evolution equations with dissipation. Arch. Rational Mech. Anal., 137(4):341–361, 1997. [21] J.-L. Lions and E. Magenes. Non-homogeneous boundary value problems and applications. Vol. II. Springer-Verlag, New York, 1972. Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 182. [22] J.-L. Lions and W. A. Strauss. Some non-linear evolution equations. Bull. Soc. Math. France, 93:43–96, 1965. [23] L. E. Payne and D. H. Sattinger. Saddle points and instability of nonlinear hyperbolic equations. Israel J. Math., 22(3–4):273–303, 1975. [24] D. R. Pitts and M. A. Rammaha. Global existence and non-existence theorems for nonlinear wave equations. Indiana Univ. Math. J., 51(6):1479–1509, 2002. [25] P. Pucci and J. Serrin. Global nonexistence for abstract evolution equations with positive initial energy. J. Diﬀerential Equations, 150(1):203–214, 1998. [26] M. A. Rammaha and T. A. Strei. Global existence and nonexistence for nonlinear wave equations with damping and source terms. Trans. Amer. Math. Soc., 354(9):3621–3637 (electronic), 2002. [27] R. Seeley. Interpolation in Lp with boundary conditions. Studia Math., 44:47– 60, 1972. Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientiﬁc activity, I.

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[28] I. Segal. Non-linear semi-groups. Ann. of Math. (2), 78:339–364, 1963. [29] G. Todorova. Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms. C. R. Acad. Sci. Paris S´er. I Math., 326(2):191– 196, 1998. [30] G. Todorova. Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms. Nonlinear Anal., 41(7-8, Ser. A: Theory Methods):891–905, 2000. [31] M. Tsutsumi. On solutions of semilinear diﬀerential equations in a Hilbert space. Math. Japon., 17:173–193, 1972. [32] G. F. Webb. Existence and asymptotic behavior for a strongly damped nonlinear wave equation. Canad. J. Math., 32(3):631–643, 1980.

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CHAPTER V NUMERICAL MODELING OF PHASE CHANGE PROBLEMS Andr´e Fortin GIREF, D´epartement de Math´ematiques et de Statistiques Universit´e Laval Quebec, Canada

Youssef Belhamadia GIREF, D´epartement de Math´ematiques et de Statistiques Universit´e Laval Qu´ebec, Canada

Abstract

Recent developments for the numerical simulation of phase change problems are presented. A semi-phase ﬁeld formulation is described which requires the discretization of both the phase-ﬁeld function φ and the temperature T . Numerical results using anisotropic mesh adaptation are presented for two- and three-dimensional problems.

Keywords: Stefan problem, phase change, semi-phase-ﬁeld formulation, ﬁnite element method, error estimators, anisotropic mesh adaptation

Introduction Phase change problems are extremely important in many applications. The main objective of this work is the development of numerical techniques for the simulation of cryosurgery problems. In this surgical technique a cryoprobe is inserted inside a tumor and cancerous cells are destroyed by the freezing-thawing process (see Keanini and Rubinski [6]). The objective of this work is an accurate prediction of freezing fronts in order to destroy completely the tumor while preserving healthy tissues. The mathematical formulation of such problems is known as the Stefan problem where the heat conduction equation has to be solved for the temperature T in a domain typically consisting of a solid and a liquid part, and separated by a moving interface (the freezing or melting front). On this interface, a heat balance condition, the Stefan condition, Copyright © 2005 Marcel Dekker, Inc.

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must be imposed. The precise location and form of the interface is thus extremely important. Standard ﬁnite element methods often result in inaccurate prediction of this interface. Enthalpy formulations greatly alleviate the diﬃculties since the interface is computed implicitly and the Stefan condition is satisﬁed automatically. The reader is referred to Fortin and Belhamadia [4] and to Nochetto et al. [8] for a complete discussion. Phase-ﬁeld formulations have been introduced for the numerical simulation of physical problems such as dendritic growth. In these formulations, a phase-ﬁeld function is introduced to determine the interface position which is the solution of a partial diﬀerential equation (see Mackenzie and Robertson [7]). In the present paper, a semi-phase-ﬁeld is introduced as an alternative to the enthalpy formulation of the Stefan problem. In this new formulation, an algebraic equation allows the determination of the phase-ﬁeld function, thus the name semi-phase-ﬁeld. An anisotropic mesh adaptation procedure is also introduced and applied to two- and three-dimensional problems. As shall be seen, the adaptive method concentrates the elements in the vicinity of the freezing front allowing a better prediction of the interface position and form.

1.

Stefan Problem and Semi-Phase-Field Formulation

The Stefan problem can be formulated as: ∂T ρi ci − ∇ · (K i ∇T ) = fi ∂t

T

= Tf

(K s ∇T ) · ns + (K l ∇T ) · nl = ρl L VΓ

in Ωi i = s, l, on Γ,

(1)

on Γ.

Subscripts s and l refer to the solid and liquid phases and K i is the thermal conductivity tensor, ρi is the density, ci is the speciﬁc heat, fi is a possible heat source, L is the latent heat of fusion, and VΓ is the interface normal velocity. The semi-phase-ﬁeld requires the introduction of the phase-ﬁeld variable φ deﬁned as: 0 in Ωs i.e. for T ≤ Tf , φ= 1 in Ωl i.e. for T ≥ Tf . and consequently, the phase-ﬁeld function is completely determined by this non-linear algebraic equation. As described in Belhamadia et al. [2], Copyright © 2005 Marcel Dekker, Inc.

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the semi-phase-ﬁeld formulation takes the form of a non-linear system: ∂φ ∂T + ρL − ∇ · (K(φ)∇T ) = f (φ), α(φ) (2) ∂t ∂t φ = F (T ). where:

α(φ) = ρs cs + φ (ρl cl − ρs cs ) , K(φ) = K s + φ (K l − K s ) , f (φ) = fs + φ (fl − fs ) .

This last relation underlines the fact that physical properties such as thermal conductivity may vary according to the phase and thus depend on the phase-ﬁeld function φ.

2.

Finite Element Discretization

The weak formulation of system (2) is straightforward. An implicit Euler scheme is used for the time discretization and the variational formulation becomes: (n+1) − T (n) (n+1) − φ(n) T φ α(φ(n+1) ) vT dΩ + ρl L vT dΩ ∆t ∆t Ω Ω f (φ(n+1) )vT dΩ, + K(φ(n+1) )∇T (n+1) · ∇vT dΩ = Ω Ω ' ( φ(n+1) − F (T (n+1) ) v dΩ = 0.

φ

Ω

(3) Newton’s method is used to solve the non-linear system above at each time step. In all numerical simulations, a quadratic discretization of the temperature T and a linear approximation of φ were used. For twodimensional problems, linear systems resulting from Newton’s method are solved by a direct method (LU factorization). For three-dimensional problems, however, an iterative method is employed since memory requirements for direct methods would rapidly exceed the capacity of available computers. In the speciﬁc case of system (3), an ILU preconditioned GMRES solver [9] from the PetSc librairy [1] was used.

3.

Adaptive Strategy

As already mentioned, the accurate prediction of the interface which is determined by the discontinuous phase-ﬁeld function φ is crucial. It is well known that approximating a discontinuous function by standard Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

ﬁnite element methods may lead to oscillations and thus inaccurate prediction of the interface form. To avoid oscillations, the mesh has to be reﬁned near the interface location. This can be done with appropriate mesh adaptation techniques. A brief description of adaptive methods for time dependent problems will now be presented. Depending on the dimension of the problem, two diﬀerent methods for estimating the error have been used. Both strategies result in anisotropic meshes. For two-dimensional problems, a hierarchical error estimator described in Belhamadia et al. [2] was used. This error estimator is based on a hierarchical decomposition of the ﬁnite element space Vk+1 of polynomials of degree k + 1. Starting from a numerical solution in Vk , a hierarchical correction of this solution is computed and used as an approximation of the error. The extension to the three-dimensional case of the hierarchical error estimator is not yet fully implemented in our code. This is why, for three-dimensional problems, an error estimator based on a deﬁnition of edge length using a solution-dependent metric was used (see Habashi et al. [5]). Equirepartition of the error is then achieved when all the element edges are of the same length in this metric. For a complete presentation the reader is referred to Belhamadia et al. [3]. The overall adaptive strategy is the following: 1 Starting from a solution (T (n) , φ(n) ) and a mesh M(n) at time t(n) ; 2 Solve system (3) on mesh M(n) to obtain a ﬁrst approximation (T˜(n+1) , φ˜(n+1) ) of the solution at time t(n+1) ; 3 Adapt the mesh on the new solution (T˜(n+1) , φ˜(n+1)) and (T (n) ,φ(n)) to obtain M(n+1) ; 4 Reinterpolate (T (n) , φ(n) ) on M(n+1) ; 5 Solve system (3) on mesh M(n+1) for (T (n+1) , φ(n+1) ). The mesh is thus adapted at each time step in order to preserve the accuracy of the solution. They are adapted to ﬁt the actual and proceeding solutions.

4.

Numerical Results

A test problem will now be solved in the two- and three-dimensional cases. The chosen test case is the oscillating sphere (circle in two dimensions) since it possesses an analytical solution. The problem will be described in the three-dimensional case but its two-dimensional restriction is straightforward. Copyright © 2005 Marcel Dekker, Inc.

Numerical Modeling of Phase Change Problems

67

The computational domain is the box Ω = [0, 5] × [−1, 4] × [−1.5, 1.5]. The interface has the form of a half-sphere of radius 1 moving up and down along the plane x = 0. Its position at all times is (0,α(t),0), where α(t) = 0.5 + sin(1.25t) The half-period of oscillation is thus 4π/5. If a proper function f is chosen, then the solution of the Stefan problem (1) is given by: r < 1, 0.75(r 2 − 1), (4) T (x, y, z, t) = ˙ (1.5 − α(t) sin φ)(r − 1), r ≥ 1, ˙ = where r = (x2 + (y − α(t))2 + z 2 )1/2 , sin φ = ((y − α(t))/r) and α(t) dα (t). The other parameters are given in the following table: dt Oscillating sphere ρs = ρl = 1 L=1 cs = cl = 1 Tf = 0 Ks = Kl = I A homogeneous Neumann boundary condition is imposed on the plane x = 0 (where the center of the sphere is located) while Dirichlet boundary conditions compatible with the analytical solution are given on all the other sides. The initial condition is also compatible with the analytical solution (4). The time step was ﬁxed to 4π/500 and to 0.0125. Figure 1 shows the numerical solution (in blue) and analytical one (in red) over one period of oscillation. The two colors are superimposed proving that the form and position of the interface are accurately predicted. Figure 2 shows clearly how the mesh is concentrated and strongly anisotropic. In the three-dimensional case the initial mesh was also structured ( 180 000 elements) but at the ﬁrst time step, the number of elements is greatly reduced since they are concentrated in the vicinity of the sphere (see Figure 3). Cross sections of the mesh in diﬀerent planes passing through the sphere clearly show the quality and anisotropy of the mesh.

5.

Conclusions

An anisotropic mesh adaptation strategy was introduced for the numerical simulation of phase change problems. The numerical examples presented clearly show the advantages of mesh adaptation over classical ﬁnite element methods.

Acknowledgments The authors wish to acknowledge the ﬁnancial support of NSERC project SKALPEL-ITC. Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

a) t = 0

b) t = 4π/25

c) t = 8π/25

d) t = 12π/25

e) t = 16π/25

f) t = 4π/5

Figure 1.

Copyright © 2005 Marcel Dekker, Inc.

Numerical and analytical solutions.

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Numerical Modeling of Phase Change Problems

Figure 2.

a) t = 0

b) t = 4π/25

c) t = 8π/25

d) t = 12π/25

e) t = 16π/25

f) t = 20π/25

Mesh evolution for the oscillating circle adapting on T and φ.

Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

a) Numerical and analytical solutions

b) External envelope of the mesh

c) Mesh on plane x = 0

d) Mesh on plane y = 1.5

b) Mesh on plane z = 0

e) Close up view on plane x = 0

Figure 3.

Oscillating sphere: solution and cross sections of the mesh at t = 10π/25.

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71

References [1] S. Balay, K. Buschelman, V. Eijkhout, W. Gropp, D. Kaushik, M. Knepley, L. C. McInnes, B. Smith, and H. Zhang. PETSc Users Manual. Technical Report ANL-95/11-Revision 2.1.6, Argonne National Laboratory, Argonne, Illinois, 2003. http://www.mcs.anl.gov/petsc/. ´ Chamberland. Anisotropic Mesh Adaptation [2] Y. Belhamadia, A. Fortin, and E. for the Solution of the Stefan Problem. J. of Comp. Phys. 194(1):233–255, 2004. ´ Chamberland. Three-Dimensional Anisotropic [3] Y. Belhamadia, A. Fortin, and E. Mesh Adaptation for Phase Change Problems. J. of Comp. Phys., 2004. In press. [4] A. Fortin and Y. Belhamadia. A New Enthalpy Finite Element Method for Phase Change Problems. Comm. Num. Meth. Eng., 2003. Submitted. [5] W.G. Habashi, J. Dompierre, Y. Bourgault, D. Ait Ali Yahia, M. Fortin, and M.-G. Vallet. Anisotropic Mesh Adaptation: Towards User-Independent, MeshIndependent and Solver-Independent CFD. Part I: General Principles. Int. J. Numer. Meth. Fluids, 32:725–744, 2000. [6] R. G. Keanini and B. Rubinski. Optimization of Multiprobe Cryosurgery. ASME Journal of Heat Transfer, 114:796–801, 1992. [7] J. A. Mackenzie and M. L. Robertson. A Moving Mesh Method for the Solution of the One-Dimensional Phase-Field Equations. J. of Comp. Phys., 181:526–544, 2002. [8] R. H. Nochetto, M. Paolini, and C. Verdi. An Adaptive Finite Element Method for Two-Phase Stefan Problems in Two Space Dimensions. Part II: Implementation and Numerical Experiments. SIAM J. Sci. Stat. Comp., 12(5):1207–1244, 1991. [9] Saad Y. Iterative Methods for Sparse Linear Systems. PWS Publishing Company, Massachusetts, 1996.

Copyright © 2005 Marcel Dekker, Inc.

CHAPTER VI SHAPE OPTIMIZATION OF FREE AIR-POROUS MEDIA TRANSMISSION COEFFICIENT Arian Novruzi∗ Department of Mathematics and Statistics University of Ottawa Ottawa, Canada

Abstract

In this work we consider the shape analysis of the free air-porous domain transmission coeﬃcient as a function of the pore shapes. Shape derivatives of the transmission coeﬃcient and of the permeability are computed. To a given number we have also computed the porous domain which transmission coeﬃcient is equal to. Moreover, among all the porous domains having a given transmission coeﬃcient, we prove that under the constraint that the measure is given, there exists one optimal domain maximizing the velocity in porous domain in a given direction. The computation of this domain is performed as well.

Keywords: Shape optimization, free air-porous media interface, Stokes equation

Introduction The interaction of free air-porous media appears in many physical systems and has been studied intensively these last 30 years. The interaction is characterized by the so-called transmission coeﬃcient. The transmission coeﬃcient is very important because it deﬁnes the coupling of the equations in free air and porous domain. The transmission coeﬃcient on the free air-porous domain interface has been reported for the ﬁrst time in [1]. A proof at the physical level of rigor has been presented in [2]. The interface boundary condition has been proved rigorously in [4], [9], where it is given also a formula for the transmission coeﬃcient. In [5] it has been considered the computation of the transmission coeﬃcient appearing on the interface boundary condition.

∗ This

work was funded partially from NSERC and MITACS of Canada.

Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

The main objective of this paper is the construction of the porous domain having a given transmission coeﬃcient and the shape optimization with respect to the pore shape of a cost functional deﬁned later. Our study is motivated by the numerical computation of ﬂuid dynamics in hydrogen fuel cells, where the properties of the free air (channels)-porous media (graphite diﬀusive layers) interface plays an important role in the hydrogen fuel cell performance. As it is standard, we will consider a 2d viscous ﬂow over a periodic porous domain with period size , separated from the air by a horizontal interface, like in [4]-[5] (we will use almost the same notations as in these papers). The solution (u , p ) of this problem can be approached by a Poiseuille ﬂow in the free air domain, while in the porous domain one can approach the velocity by zero. It has been reported, see [1], [2], that in many cases this approximation is not suitable. The approximation to a further order involves the solution of the following problem. Let us set Q := (0, 1)2 and let Ω, Z ⊂ Q, Ω open such that Ω∪Z = Q, Z ∩ Ω = ∂Q ∩ ∂Z = φ. The set Z represents the solid part, Ω represents the ﬂuid part. We will assume that Ω is a smooth set. Let us also set + + Ω+ − = Ω ∪ S ∪ Ω− where Ω = (0, 1) × (0, +∞), Ω− = {Ω − (0, n), n = 1, 2, . . .} and Z− = {Z − (0, n), n = 1, 2, . . .}, S = (0, 1) × {0}. The solution (β, ω), β = (β1 , β2 ) of the problem −∆β + ∇ω = 0 in Ω+ ∪ Ω− , ∇ · β = 0 in Ω+ ∪ Ω− , β = 0 on S, (1) (∇β − ωI)e 2 = e1 on S, β = 0 on Z− , β, ω x1 periodic, where β = lim β(t) − β(−t), {e1 , e2 } is the base of R2 , plays a crucial t→0

role on approaching (u , p ). Moreover, in [4] it has been found that the eﬀective velocity is given by eﬀ ∂u eﬀ on S, u1 = −C 1 ∂x2 where ueﬀ is the average over the period of u and 1 C= β1 (x1 , x2 )dx1 , x2 ≥ 0.

(2)

(3)

0

In [5] it has been studied in depth (β, ω) and C, and for a given domain Ω the computation of C has been done. In fact, as Ω+ − is unbounded, Copyright © 2005 Marcel Dekker, Inc.

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an approximation of C is computed. More precisely, for k, l ∈ N, let us set Ωkl = Ωk ∪ S ∪ Ωl , where Ωk = (0, 1) × (0, k), Ωl = {Ω − (0, n), n = 1, 2, . . . , l}, and Zl = {Z − (0, n), n = 1, 2, . . . , l}. Now let (β kl , ω kl ) be the solution of −∆β kl + ∇ω kl = 0 in Ωk ∪ Ωl , ∇ · β kl = 0 in Ωk ∪ Ωl , β kl = 0 on S, (∇β kl − ω kl I)e = e on S, 2 1 (4) kl = (0, 0) on ∂Z ∪ {(·, −l)} β l kl kl = 0 on {(·, k)}, 2 β1

β2 = ∂kl Ωk ∪Ωl ω dx = 0, β, ω x1 periodic. Diﬀerent estimations for (β −β kl , ω −ω kl ) are proved in [5]. In particular, if we deﬁne 1

C kl =

β1kl (x1 , 0)dx1 ,

(5)

0

then C kl approaches C as follows |C − C kl | ≤ M (e−2πk − e−γ|l| ),

(6)

where M and γ are constants. In this paper we will deal with the following problems. We consider the coeﬃcient C as a shape functional Ω → C(Ω) given by (3) and we look for: a) If c is given, does there exist a domain Ω such that C(Ω) = c? If the answer is yes, ﬁnd an algorithm to compute it. b) If for a given c there exist many domains Ω such that C(Ω) = c, how one can choose reasonably one of them? Let O = {Ω ⊂ Q, Ω open, Z = Q\Ω ⊃ Bδ ( 12 , 12 ), d(∂Z, ∂Q) ≥ δ, #(Z) = 1 },

(7)

where #(Z) is the number of connected components of Z and δ > 0 is an arbitrary small number. Then, the problem a) has a solution if c ∈ R(C) := {C(Ω), Ω ∈ O}. Finding R(C) is a quite diﬃcult problem. In this paper, for a given c we will compute a domain Ω such that C kl (Ω) = c (of course when c ∈ R(C kl )). It will be shown numerically that such a domain Ω is not unique. One can look for one of these domains that optimizes the permeability on a given direction, under the Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

constraint m(Ω) = m0 . Indeed, if the permeabilty is deﬁned by the matrix (8) K(Ω) = [α1 , α2 ], Ω

where

α1 ,

α2

are solutions of

−∆αi + ∇σ i = ei in i in ∇·α = 0 i = (0, 0) on α x1 , x2 αi , σ i

Ω, i = 1, 2, Ω, ∂Z, periodic,

(9)

and the pressure gradient ∇p in the porous media is considered known, then − µ1 K(Ω) · ∇p gives the velocity. Often it is required to maximize the velocity on a given direction d. Thus, we look for Ω∗ ∈ O solution of K(Ω∗ ) = min{−d · K(Ω) · ∇p, C(Ω) = c, m(Ω) = m0 , Ω ∈ O}, where d ∈ R2 is a given vector.

1.

Computation of the Porous Shape

For a given number c, one may look for Ω ∈ O ∩C 2 solution of C(Ω) = c. As Ω+ and Ω− are unbounded, we will assume that for k, l ∈ N large ˜ ∗ ) = c then Ω∗ and Ω ˜ ∗ are close enough (for enough, if C(Ω∗ ) = c, C kl (Ω any topology, for example that of Hausdorﬀ). So, we will deal with the solution of C kl (Ω) = c instead of C(Ω) = c. To solve this equation we will use a Newton method. As C kl is with values in R, starting from an initial domain Ω0 , one parametric family of domains (assumed to contain ˜ ∗ ), for example of the form {x + tξ(x), x ∈ Ω0 }, where ξ is the solution Ω a regular vector ﬁeld, is suﬃcient. Assuming that the shape derivative in direction ξ of β kl (Ωn ) is β kl (Ωn ; ξ), where Ωn ∈ O, Zn = Q\Ωn , the Newton algorithm is: Ω0 given,

C kl (Ωn ) (10) (∂Ωn ), n = 0, 1, . . . I − kl ∂Ωn+1 = C (Ωn ; ξ)

where kl

β1kl (Ωn ; ξ)(x1 , 0)dx1 .

C (Ωn ; ξ) = S

We have the following shape derivative result.

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77

Proposition 1 For Ω ∈ O ∩ C 2 ﬁxed let us set 02 (Ω) = D(Ωl ∪ Zl ); R2 ) · C 2 (R2 ;R2 ) . C 2 (Ω) let set Ωθ = (I +θ)(Ω) and (β kl (Ωθ ), ω kl (Ωθ )) the solution For θ ∈ C 0 of ( 4) for Ωθ instead of Ω. Then the map θ → β kl (Ωθ ) ◦ (I + θ) is C 1 2 (Ω) into H 1 (Ωkl ; R2 ) in a neighborhood of θ = 0. Moreover, from C 0 if β kl (Ω; ξ) is the shape derivative of β kl (Ωθ ) at θ = 0 in the direction 2 (Ω), then there exists ω kl (Ω; ξ) ∈ L2 (Ωkl ) such that ξ∈C 0 −∆β kl (Ω; ξ) + ∇ω kl (Ω; ξ) ∇ · β kl (Ω; ξ) β kl (Ω; ξ) (∇β kl (Ω; ξ) − ω kl (Ω; ξ) · I)e2 β kl (Ω; ξ) β kl (Ω; ξ) β2kl (Ω; ξ) = ∂2 β1kl (Ω; ξ) ω kl (Ω; ξ)dx

= = = = = = =

0 in Ωk ∪ Ωl , 0 in Ωk ∪ Ωl , 0 on S, 0 on S, −∇β kl (Ω) · ξ on ∂Zl , (0, 0) on {(·, −l)}, 0 on {(·, k)},

= 0,

(11) (12) (13) (14) (15) (16) (17) (18)

Ωkl

x1 periodic,

β(Ω; ξ), ω(Ω; ξ)

(19)

and C (Ωn ; ξ) = − kl

kl ∂Zn

|∇β1kl (Ωn )|2 (ξ · νnkl ),

(20)

where νnkl is the unit exterior normal vector to Ωkl n. Proof. As usual in shape derivative results, we will use implicit function theorem. Let us introduce the spaces 1 (Ω) H p uH 1 (Ω) p

L2p (Ω) −1 (Ω) H p

= H 1 (Ωkl ; R2 ) ∩ {(u1 , u2 ) = 0 on ∂Zl ∪ {(·, −l)}, u1 = ∂1 u2 = 0 on {·, k)}, x1 periodic}, = ∇uH 1 (Ωkl ;R2 ) ,

= L2 (Ωkl ) ∩ { Ωkl p(x)dx = 0, x1 periodic}, 1 (Ω). = dual space of H p

p1 (Ω). For θ ∈ C 2 (Ω) small, Let us point out that cot H 1 (Ω) is norm in H 0 p kl kl 1 2 let (β (Ωθ ), ω (Ωθ )) ∈ Hp (Ωθ ) × Lp (Ωθ ) be the unique solution of the problem (4) for Ω = Ωθ , see [5]. If u = β(Ωθ ) ◦ (I + θ), p = ω(Ωθ ) ◦ (I + θ) 1 (Ω) × L2 (Ω) and satisﬁes then (v, p) ∈ H p p Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

t t kl ( i=1,2 ∇ ui · M (θ) · M (θ)

·ϕ Ω +M (θ) · ∇p)Jac(I + θ) − S ϕ1 = 0, i,j=1,2 Mj,i (θ)∂j ui = 0,

(21)

1 (Ω). where M (θ) = t [∇(I + θ)]−1 and ϕ = (ϕ1 , ϕ2 ) ∈ H p Let us introduce the function f as follows 2 (Ω) × H p1 (Ω) × L2p (Ω) → H p−1 (Ω) × L2p (Ω), : C 0

f (θ, u, p) = ( Ω ( i=1,2 ∇t ui · t M (θ) · M (θ) · ϕ+ M (θ) · ∇p)Jac(I + θ) − S ϕ1 , i,j=1,2 Mj,i (θ)∂j ui ). f

(22)

It is clear that the function f is of class C 1 near a neighborhood of (0, β kl (Ω), ω kl (Ω)). Its (u, p) derivative at (0, β kl (Ω), ω kl (Ω)) in the direction (v, q) is ∂(u,p) f ((0, β kl (Ω), ω kl (Ω))(v, q) =

∇v · ∇ϕ + ∇q · ϕ − ϕ1 , ∇ · v . Ωkl

(23)

S

p1 (Ω) × L2p (Ω) into The derivative (23) deﬁnes an isomorphism from H −1 (Ω) × L2 (Ω) let us prove that −1 (Ω) × L2 (Ω). Indeed, for (h, g) ∈ H H p p p p

∇v · ∇ϕ + ∇q · ϕ − ϕ1 , ∇ · v = (h, g) (24) Ωkl

S

has a unique solution.

p1 (Ω) ∩ {∇ · a) g = 0. It is known that Ωkl ∇v · ∇ϕ − S ϕ1 = h(ϕ), ϕ ∈ H

1 (Ω) because kl |∇ϕ|2 is a norm u = 0} has a unique solution v ∈ H p Ω p1 (Ω), see [5]. Therefore, from Proposition 1.1, 1.2, [7], it follows the in H existence of p ∈ L2 (Ω) such that −∆v + ∇p = h(·) + (·)1 . S

in distribution sense. From h(ϕ) = i,j=1,2 Ωkl ∂i ϕj phij , for some phi,j ∈

L2p (Ω) and the periodicity of v it follows that ∂(0,1)×(−l.k) (ϕ · ν)p = p1 (Ω) and ν is the exterior unitary normal vector to 0, where ϕ ∈ H

(0, 1) × (−l, k). Thus p ∈ L2p (Ω).

1 (Ω) such that b) g = 0. It is enough to prove that there exists vg ∈ H p Copyright © 2005 Marcel Dekker, Inc.

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79

1 (Ω) is continuous. ∇ · vg = g and that the map g ∈ L2p (Ω) → vg ∈ H p Indeed, in such a case one can look for v = w + vg and apply a) with hg (ϕ) = h(ϕ) − Ωkl ∇vg · ϕ instead of h(ϕ). But the existence of vg follows from Lemma 2.4, Proposition 1.2, [7] (the only modiﬁcation is to 1 (Ω) instead of H −1 (Ω)). take H p 1 (Ω) × L2 (Ω) → (h, g) ∈ H −1 (Ω) × c) It is clear that the map (v, p) ∈ H p p p L2p (Ωkl ) it is continuous. Thus, f satisﬁes the conditions of implicit function theorem. It follows 2 (Ω) → (u(θ), p(θ)) ∈ H p1 (Ω) × L2p (Ωkl ) that there exists a C 1 map θ ∈ C satisfying f (θ, u(θ), p(θ)) = 0. From the uniqueness of (4) it follows that u(θ) = β kl (Ωθ ) ◦ (I + θ) which proves the diﬀerentiability of β kl . The proof of (11)–(12) follows from Theorem 3.1, [3], while (13)–(17) follow from Theorem 3.2, [3] (let us point out that β kl (Ω) is H 2 close to the boundary ∂Zl ). The equation (20) follows from the fact that

C kl (Ω) = − Ωkl |∇β kl (Ω)|2 and Theorem 3.3, [3] after using the simple

fact that Ωkl ∇β kl (Ω) · ∇β kl (Ω; ξ) = 0.

1.1

Computation of the Pore Shape Corresponding to a Given Transmission Coeﬃcient

As it has been also reported in [5] for k, l ∈ N large enough, C is well approached by C kl . We have made numerical computations with k = l = 2. First, we have solved the direct problem as in [5], i.e., for a given domain Ω compute C kl . In the examples of Figure 1 we present the velocity ﬁeld and the transmission coeﬃcient C kl . The domain Ω is deﬁned by the nodes: square domain: U domain:

triangle domain:

{(0.4, −0.4), (0.6, −0.4), (0, 4, −0.6), (0.6, −0.6)} + {(0, 0), (0, −1)}, {(0.2, −0.25), (0.4, −0.25), (0.6, −0.25), (0.8, −0.25), (0.4, −0.6), (0.6, −0.6), (0.2, −0.75), (0.8 − 0.75)}+ {(0, 0), (0, −1)}, {(0.4, −0.4), (0.7, −0.3), (0.6, −0.7)} + {(0, 0), (0, −1)}.

Taking as initial domain Ω0 one of the three domains of Figure 1, we have solved the equation C kl (Ω) = −0.15 using the algorithm (10) with ξ = ν0 (of course, we have been ensured that −0.15 is in the range of C kl (Ω) after some numerical experiences). The results are summarized in Figure 2 where the initial and ﬁnal domains are shown, as well as the velocity ﬁeld. As we see, the problem C kl (Ω) = c has not a unique Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

S= -0.437631 S=-0.245949 S=-0.348377 Figure 1. For three diﬀerent domains: the velocity ﬁeld and C bl .

S(Ω) = −0.43, S(Ω) = −0.15

Figure 2. ﬁeld.

S(Ω) = −0.24, S(Ω) = −0.15

∗

S(Ω) = −0.43, S(Ω) = −0.15

22

For three diﬀerent cases: Ω0 , Ω such that C (Ω∗ ) = −0.15 and velocity

solution Ω. To deﬁne Ω in a unique way we need to add another criteria. A good criteria is to maximize the velocity in the porous domain (in a given direction) which is closely related to the permeability. Copyright © 2005 Marcel Dekker, Inc.

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2.

Optimal Porous Shape Domain

In the previous section we saw numerically that to a given transmission coeﬃcient c correspond many porous domains. In this section we discuss the criteria to impose in order to deﬁne the optimal domain among them. In practice, often it is required to maximize the velocity on a given direction. Assuming that the pressure gradient ∇p in the porous media is given, the velocity in porous media is given by − K(Ω) µ ∇p, where K(Ω) is the permeability matrix deﬁned by (8) and µ is a constant. If d ∈ R2 is a vector then one can look for solution of the following problem: Find Ω∗ ∈ O solution of K(Ω∗ ) = max{−d · K(Ω) · ∇p, C kl (Ω) = c, m(Ω) = m0 , Ω ∈ O} = min{d · K(Ω) · ∇p, C kl (Ω) = c, m(Ω) = m0 , Ω ∈ O},

(25)

It is diﬃcult to prove any existence result for the problem (25). However, we can prove an existence result if the problem is modiﬁed by adding a perimeter term. More precisely, let us consider Ω∗ ∈ O solution of K(Ω∗ ) = min{d · K(Ω) · ∇p + P (Ω), C(Ω) = c, m(Ω) = m0 } (26) where > 0 arbitrary small and P (Ω) is the perimeter of Ω. Then we have

Theorem 2 The problem ( 26) has a solution in Ω∗ ∈ O. Proof. Let Ωn ∈ O be any minimizing sequence of (26) and (β kl,n , ω kl,n ) (αi,n , σ i,n ), i = 1, 2 solutions of (4), (9) with Ω = Ωn . It is easy to prove that there exist a constant M1 (k) and M2 (δ) such that ∇β kl,n H 1 (Ω) ≤ M1 (k), p

∇α1,n H 1 (Ω) + ∇α2,n H 1 (Ω) ≤ M2 (δ).

Thus, if Ωδ = Q\B δ ( 12 , 12 ) then β kl,n , resp. α1,n , α2,n , are bounded in 1 (Ωδ ), resp. H 1 (Ωδ ; R2 ). It follows that there exists a subsequence of H p β kl,n , resp. α1,n , α2,n , again denoted with the same letters, converging p1 (Ωδ ), resp. α1 , α2 ∈ H 1 (Ωδ ). H 1 weakly to β kl ∈ H On the other side, knowing that K(Ωn ) = [Kij (Ωn )] = Ωn

Copyright © 2005 Marcel Dekker, Inc.

∇αi,n · ∇αj,n

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CONTROL AND BOUNDARY ANALYSIS

it follows that {P (Ωn ), n = 1, 2, . . .} is bounded. Then, there exists a subsequence of {Ωn , n ∈ N}, again denoted by Ωn converging weakly in BV sense to a Ω∗ . As the set O is closed and compact for the complimentary Hausdorﬀ topology, see [6], we may consider that Ωn → Ω∗ , for weak BV and complementary Hausdorﬀ topologies. (27) Therefore Ω∗ ∈ O, C kl (Ω∗ ) = c and m(Ω∗ ) = m0 . On the other side, from Theorem 5.3, [8], it follows that convergence of β kl,n , and α1,n , p1 (Ω∗ ), resp. α1,n , α2,n ∈ H 1 (Ω∗ ) ∩ {u = α2,n is strong and β kl ∈ H ∗ 0 on ∂(Q\Ω )} , which proves that Ω∗ is a solution of (26). 2 (Ω) = C 2 (Ω; R2 )∩ Proposition 3 Let Ω ∈ O∩C 2 be ﬁxed and let set C 0 2 (Ω) let set Ωθ = (I + θ)(Ω) and {ϕ = (0, 0) on ∂Q}. For θ ∈ C 0 (αi (Ωθ ), σ i (Ωθ )), i = 1, 2, the solution of ( 9) for Ωθ instead of Ω. Then 2 (Ω) into H 2 (Ω) in a neighthe map θ → αi (Ωθ ) ◦ (I + θ) is C 1 from C 0 i borhood of θ = 0. Moreover, α (Ω; ξ), the shape derivative of αi (Ωθ ) at 2 (Ω) is given by θ = 0 in the direction ξ ∈ C 0 −∆αi (Ω; ξ) + ∇σ i (Ω; ξ) = 0 in Ω, ∇ · αi (Ω; ξ) = 0 in Ω, αi (Ω; ξ) = −∇αi · ξ on ∂Ω, x1 , x2 periodic. αi (Ω; ξ), σ i (Ω; ξ)

(28) (29) (30) (31)

Proof. The proof of this result is very similar to the proof of Proposition 1 and we will not present it here.

2.1

Numerical Results

To solve (26) we have used a gradient method with penalization. Precisely, let λ, µ > 0 be two big numbers. Then we look for solution of min{E(Ω) := d · K(Ω) · ∇p + P (Ω) + λ(C kl (Ω) − c)2 + µ(m(Ω) − m0 )2 }.

(32)

Based on Propositions 1, 3 and Theorem 3.1, [3], we have

Proposition 4 The shape derivative of E(Ω) in the direction ξ is given by i j ∇α (Ω) · ∇α (Ω)(ξ · ν) · ∇p + H(ξ · ν) dE(Ω; ξ) = d · ∂Z ∂Z 2 |∇β(Ω)| (ξ · ν) + µ (ξ · ν), (33) + λ ∂Z kl

Copyright © 2005 Marcel Dekker, Inc.

∂Z

Shape Optimization of Free Air-Porous Media Transmission Coeﬃcient

83

where H is the mean curvature of ∂Z. We will present two numerical results for solving (26). The initial √ domain is Ω0 = Br ( 12 , 12 ), the circle with center ( 12 , 12 ) and radius r = 0.02. We have taken = 10−2 , λ = µ = 102 , c = −0.3, m0 = 0.9. We have considered two following cases. The results are shown in Figure 3. The algorithm continues as long as E and dE decrease. The stop test used is when the gradient dE does not decrease anymore (it oscillates). 1) d = −∇p = (1, 1). 2) d = (1, −1), −∇p = (0, −1).

Initial domain

Figure 3. ﬁelds.

d = −∇p = (1, 1)

d = (1, −1), −∇p = (0, −11).

Initial domain and two ﬁnal domains solution of (26) as well as velocity

References [1] Beavers G. S. and Joseph D. D. Boundary condition at a naturally permeable wall. J. Fluid Mech, 30:197–207, 1967. [2] Saﬀman P. G. On the boundary condition at the surface of a porous medium. Studies in Appl. Math., L(2), 1971.

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[3] Simon J. Diﬀerentiation with respect to the domain in boundary value problems. Numer. Func. Anal. and Optimiz., 2 (7&8):649–687, 1980. [4] Jager W. and Mikeli´c A. On the interface boundary condition of beavers, joseph and saﬀman. SIAM, J. Appl. Math., 60(4):1111–1127, 2000. [5] Mikeli´c A., Jager W. and Neuss N. Asymptotic analysis of the laminar viscous ﬂow over a porous bed. SIAM J. Sci. Comp., 22(6):2006–2028, 2001. [6] Kuratoski K. Topolgy, volume I. Academic Press, New York, 1966. [7] Temam R. Navier-Stokes Equations. Theory and Numerical Analysis. NorthHolland Publishing Company, Amsterdam-New York-Oxford, 1977. [8] Sverak V. On optimal shape design. J. Math. Pure Appl., 72:537–551, 1993. [9] Jager W. and Mikeli´c A. On the boundary condition at the interface between a porous medium and a free ﬂuid. Annali Scuala Norm. Super. Pisa Cl. Sci., 23:403–465, 1996.

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CHAPTER VII THE UNIFORM FAT SEGMENT AND CUSP PROPERTIES IN SHAPE OPTIMIZATION Michel C. Delfour∗ Centre de Recherches Math´ematiques et D´epartement de Math´ematiques et de Statistique Universit´e de Montr´eal Montr´eal, Canada

Nicolas Doyon D´epartement de Math´ematiques et de Statistique Universit´e Laval Qu´ebec, Canada

Jean-Paul Zol´esio CNRS and INRIA Sophia Antipolis, France

Abstract

The object of this paper is to ﬁrst introduce the new uniform fat segment property along with a compactness theorem for a family of subsets of a bounded holdall verifying that property. In a second part the uniform cusp property introduced in [1], its extension in [2] to the larger class of dominating cusp functions which are only continuous at the origin, and the uniform fat segment property are shown to be equivalent.

Keywords: Segment property, cusp, compactness, oriented distance function, signed distance function, fat segment property, uniform cusp property, intrinsic shell model

∗ The

research of the ﬁrst author was supported by National Sciences and Engineering Research Council of Canada discovery grant A–8730, by a FQRNT grant from the Minist`ere de ´ l’Education du Qu´ebec, and by a Fellowship from the John Simon Memorial Foundation.

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CONTROL AND BOUNDARY ANALYSIS

Introduction A large class of domains Ω in RN can be characterized by a local geometric property, the segment property, which is equivalent to the property that Ω be locally a C 0 -epigraph. This property is suﬃcient to get the density of the function space C k (Ω) in the Sobolev space W m,p (Ω) for any 1 ≤ m ≤ k and it plays a key role in establishing the continuity of the solution of the Neumann problem for the Laplace operator with respect to the underlying domain Ω. Yet, even the stronger uniform segment property is not suﬃcient to get the compactness of subsets of a bounded open holdall verifying that property in any one of the metric topologies introduced in [1] via distance or characteristic functions (cf. [1], Chapter 5, Example 11.1, p. 256). The compactness can be recovered by going to the stronger uniform cusp property ﬁrst introduced in [1] (Chapter 5, § 11, p. 257) and it remains true for the larger class of dominating cusp functions which are only continuous at the origin [2]. The object of this paper is to ﬁrst introduce the new uniform fat segment property that does not a priori assume that the open region that fattens the segment be a cone or a cusp. Then we show that the uniform cusp property introduced in [1], its extension in [2] to the larger class of dominating cusp functions which are only continuous at the origin, and the uniform fat segment property are all equivalent. This emphasizes the fact that it is only the modulus of continuity of the cusp function at the origin that matters (see also [2] for the local epigraph viewpoint).

1.

Preliminaries: Topologies on Families of Sets

We ﬁrst introduce some notation. Given an integer N ≥ 1, mN and HN −1 will denote the N -dimensional Lebesgue and (N − 1)-dimensional Hausdorﬀ measures. The inner product and the norm in RN will be written x · y and |x|. The complement {x ∈ RN : x ∈ / Ω} and the N boundary Ω ∩ Ω of a subset Ω of R will be respectively denoted by Ω or RN \Ω and by ∂Ω or Γ. The distance function dA (x) from a point x to a subset A = ∅ of RN is deﬁned as inf{|y − x| : y ∈ A}. We recall a few results on metric topologies deﬁned on spaces of equivalence classes of sets constructed from the characteristic function, the distance or the oriented distance functions to a set. Given Ω ⊂ RN , Γ = ∅, the oriented distance function is deﬁned as def

bΩ (x) = dΩ (x) − dΩ (x).

(1)

It is Lipschitz continuous of constant 1, and ∇bΩ exists and |∇bΩ | ≤ 1 1,p (RN ) for all p, 1 ≤ p ≤ ∞. almost everywhere in RN . Thus bΩ ∈ Wloc Copyright © 2005 Marcel Dekker, Inc.

The Uniform Fat Segment and Uniform Cusp Properties

87

− Recall that b+ Ω = dΩ , bΩ = dΩ , and |bΩ | = dΓ , and that χintΩ = |∇dΩ |, χintΩ = |∇dΩ |, and χΓ = 1 − |∇dΓ | a.e. in RN , where χA denotes the characteristic function of a subset A of RN . Given a nonempty subset D of RN , the family Cb (D) = {bΩ : Ω ⊂ D and Γ = ∅} is closed in W 1,p (D). The following theorem is central. It shows that convergence and compactness in the metric on Cb (D) associated with W 1,p (D) will imply the same properties in the other topologies introduced in [1].

Theorem 1 Let D ⊂ RN be bounded open and 1 ≤ p < ∞. The maps − 1,p (D) → W 1,p (D)3 (2) bΩ → (b+ Ω , bΩ , |bΩ |) = (dΩ , dΩ , d∂Ω ) : Cb (D) ⊂ W (3) bΩ → (χ∂Ω , χintΩ , χintΩ ) : W 1,p (D) → Lp (D)3

are continuous. Proof. – They are well deﬁned from [1] (Chapter 5, Theorem 2.1 (iii), p. 207) for the map (2) and [1] (Chapter 5, Thm 2.2 (iv)-(v), p. 210) for the map (3). They are continuous from [1] (Chapter 5, Thm 5.1).

2. 2.1

The New Uniform Fat Segment Property Deﬁnitions

An open segment between two points x and y of RN will be denoted def

(x, y) = {x + t(y − x) : ∀t, 0 < t < 1}.

Definition 2 Let Ω be a subset of RN such that ∂Ω = ∅. (i) Ω is said to satisfy the segment property if ∀x ∈ ∂Ω, ∃r > 0, ∃λ > 0, ∃d ∈ RN , |d| = 1 such that

∀y ∈ B(x, r) ∩ Ω,

(y, y + λd) ⊂ intΩ.

(ii) Ω is said to satisfy the uniform segment property if ∃r > 0, ∃λ > 0 such that ∀x ∈ ∂Ω, ∃d ∈ RN , |d| = 1 such that

∀y ∈ B(x, r) ∩ Ω,

(y, y + λd) ⊂ intΩ.

Theorem 3 ([1], Chapter 2, Thm 7.2) If ∂Ω is compact, then the segment and uniform segment properties of Deﬁnition 2 coincide. The uniform segment property is too weak to yield the compactness of the corresponding family of sets which are contained in a bounded holdall D of RN (cf. [1], Chapter 5, Example 11.1, p. 256). This Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

suggests to fatten the open segment (0, λeN ) by replacing it with an open region containing (0, λd). It is natural to introduce the following assumption ∃O open in RN and λ > 0 such that (0, λeN ) ⊂ O and 0 ∈ / O,

(4)

where eN is the unit vector (0, . . . , 0, 1) ∈ RN . Recall the deﬁnition of the orthogonal subgroup of N × N matrices O(N) = {A : ∗A A = A ∗A = I} , def

(5)

where ∗A is the transposed matrix of A. A direction can be speciﬁed either by a matrix (of rotation) A ∈ O(N) or the corresponding unit vector d = AeN ∈ RN .

Definition 4 A set Ω ⊂ RN , ∂Ω = ∅, is said to satisfy the uniform fat segment property if there exist r > 0, λ > 0, and an open region O of RN containing (0, λeN ) and not 0 such that for all x ∈ ∂Ω, ∃A(x) ∈ O(N) such that ∀y ∈ Ω ∩ B(x, r),

y + A(x)O ⊂ intΩ.

(6)

It is clear that there are open regions O for which no set Ω ⊂ RN , ∂Ω = ∅, will satisfy the conditions of Deﬁnition 4 and that the class of signiﬁcant open regions can be considerably reduced.

Lemma 5 If Ω satisﬁes the uniform fat segment property with r, λ, and O, it also does with r, λ, and def

O+ = {y ∈ Oc : 0 < y · eN < λ} ,

(7)

where Oc is the connected open component of O which contains (0, λeN ). / O+ , and (0, λeN ) ⊂ O+ . Proof. – ∅ = O+ ⊂ Oc ⊂ O is open, 0 ∈

2.2

Compactness and Uniform Fat Segment Property

Given a bounded open subset D of RN consider the family Ω satisﬁes the uniform fat def L(D, r, O, λ) = Ω ⊂ D : segment property for (r, O, λ)

(8)

in the sense of Deﬁnition 4.

Theorem 6 Let D be a nonempty bounded open subset of RN and 1 ≤ p < ∞. Assume that L(D, r, O, λ) is not empty for r > 0, λ > 0, and Copyright © 2005 Marcel Dekker, Inc.

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The Uniform Fat Segment and Uniform Cusp Properties

an open subset O of RN such that (0, λeN ) ⊂ O and 0 ∈ / O. Then the family def

B(D, r, O, λ) = {bΩ : ∀Ω ∈ L(D, r, O, λ)} is compact in C(D) and W 1,p (D). As a consequence the families def

Bd (D, r, O, λ) = {dΩ : ∀Ω ∈ L(D, r, O, λ)} , def Bdc (D, r, O, λ) = {dΩ : ∀Ω ∈ L(D, r, O, λ)} , def Bd∂ (D, r, O, λ) = {d∂Ω : ∀Ω ∈ L(D, r, O, λ)} are compact in C(D) and W 1,p (D), and the following families are compact in Lp (D) def

X(D, r, O, λ) = {χΩ : ∀Ω ∈ L(D, r, O, λ)} , def X c (D, r, O, λ) = {χΩ : ∀Ω ∈ L(D, r, O, λ)}.

Remark 10 The compactness Theorem 6 is no longer true when the uniform fat segment property is replaced by the weaker uniform segment property: that is, when the region O containing (0, λeN ) is reduced to the open segment (0, λeN ) (cf. Example 11.1, Chapter 5 of [1]). The proof of Theorem 6 will require the following two lemmas.

Lemma 7 Given a bounded open subset D of RN , let {Ωn } be a sequence of subsets of D such that ∂Ωn = ∅ and m(∂Ωn ) = 0. Further assume ¯ such that ∂Ω = ∅ and m(∂Ω) = 0. Then for all that there exists Ω ⊂ D p, 1 ≤ p < ∞, bΩn bΩ in W 1,2 (D)-weak

⇒ bΩn → bΩ in W 1,p (D)-strong.

Proof. – From [2] or part (ii) of the proof of Theorem 10.1 and Theorem 5.1 (i) in Chapter 5 of [1] for the equivalence of the W 1,p -topologies.

Lemma 8 ( [1], Chapter 5, § 10, Lemma 10.1 ) Given a sequence {bΩn } ⊂ Cb (D) such that bΩn → bΩ in C(D) for some bΩ ∈ Cb (D), we have the following properties: ∀x ∈ Ω, ∀R > 0,

∃N (x, R) > 0, ∀n ≥ N (x, R),

B(x, R) ∩ Ωn = ∅,

and for all x ∈ Ω, ∀R > 0,

∃N (x, R) > 0,

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∀n ≥ N (x, R),

B(x, R) ∩ Ωn = ∅. (9)

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Moreover, ∀x ∈ ∂Ω, ∀R > 0, ∃N (x, R) > 0, ∀n ≥ N (x, R), B(x, R) ∩ Ωn = ∅ and B(x, R) ∩ Ωn ) = ∅, and B(x, R) ∩ ∂Ωn = ∅. Proof. – [Proof of Theorem 6] (i) Compactness in C(D). Since for all Ω in L(D, r, O, λ), Ω is locally a C 0 -epigraph, ∂Ω = ∅, bΩ ∈ Cb (D), and m(∂Ω) = 0. Consider an arbitrary sequence {Ωn } in L(D, r, O, λ). For D compact, Cb (D) is compact in C(D) and there exists Ω ⊂ D and a subsequence {Ωnk } such that bΩnk → bΩ in C(D). It remains to prove that Ω ∈ L(D, r, O, λ), that is ∀x ∈ ∂Ω, ∃A ∈ O(N) such that ∀y ∈ Ω ∩ B(x, r),

y + AO ⊂ intΩ.

From Lemma 8, for each x ∈ ∂Ω, ∀k ≥ 1, ∃nk ≥ k such that ' r( B x, k ∩ ∂Ωnk = ∅. 2 Denote by xk an element of that intersection: ' r( ∀k ≥ 1, xk ∈ B x, k ∩ ∂Ωnk . 2 By construction xk → x. Next consider y ∈ B(x, r) ∩ Ω. From the ﬁrst part of the lemma, there exists a subsequence of {Ωnk }, still denoted {Ωnk }, such that for all k ≥ 1, B(y, r/2k ) ∩ Ωnk = ∅. For each k ≥ 1 denote by yk a point of that intersection. By construction yk ∈ Ωnk → y ∈ Ω ∩ B(x, r). There exists K > 0 large enough such that for all k ≥ K, yk ∈ B(xk , r). To see this, note that y ∈ B(x, r) and that ∃ρ > 0,

B(y, ρ) ⊂ B(x, r) and

|y − x| +

ρ < r. 2

Now |yk − xk | ≤ |yk − y| + |y − x| + |x − xk | r r ρ ρ r < r. ≤ k + r − + k ≤ r + k−1 − 2 2 2 2 2 Since r/ρ > 1 the result is true for r ρ − < 0 ⇒ k > 2 + log(r/ρ). k−1 2 2 Copyright © 2005 Marcel Dekker, Inc.

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So we have constructed a subsequence {Ωnk } such that for k ≥ K xk ∈ ∂Ωnk → x ∈ ∂Ω and yk ∈ Ωnk ∩ B(xk , r) → y ∈ Ω ∩ B(x, r). ∀k, ∃Ak ∈ O(N), Ak ∗Ak = ∗Ak Ak = I, such that yk + Ak O ⊂ intΩnk . Pick another subsequence of {Ωnk }, still denoted {Ωnk }, such that ∃A ∈ O(N), A ∗A = ∗A A = I,

Ak → A.

Now consider z ∈ y + AO. Since y + AO is open there exists ρ > 0 such that B(z, ρ) ⊂ y + AO, and there exists K ≥ K such that ∀k ≥ K ,

B(z, ρ/2) ⊂ yk + Ak O ⊂ intΩnk = Ωnk ,

⇒ B(z, ρ/2) ⊃ Ωnk ⇒ 0 < ρ/2 = dB(z,ρ/2) (z) ≤ dΩn (z) → dΩ (z) k

⇒ 0 < ρ/2 ≤ dΩ (z)

⇒ z ∈ Ω = intΩ

⇒ y + AO ⊂ intΩ.

Hence Ω ⊂ D satisﬁes the uniform fat segment property. (ii) W 1,p (D)-Compactness. From Theorem 1 (i) it is suﬃcient to prove the result for p = 2. Consider the subsequence {Ωnk } ⊂ L(D, λ, O, r) and let Ω ∈ L(D, λ, O, r) be the set previously constructed such as bΩnk → bΩ in C(D). Hence bΩnk → bΩ in L2 (D). Since D is compact 2 |bΩnk | dx ≤ diam(D)2 dx ≤ diam(D)2 m(D), ∀Ω ⊂ D, D D |∇bΩnk |2 dx ≤ dx = m(D), D

D

and there exists a subsequence, still denoted {bΩnk }, which converges weakly to bΩ . Since all the sets are locally C 0 -epigraphs, m(∂Ωnk ) = 0 = m(∂Ω) and, by Lemma 7, bΩnk → bΩ in W 1,p (D)-strong, 1 ≤ p < ∞. (iii) The other compactnesses follow by continuity of the maps (2) and (3) in Theorem 1 and the fact that m(∂Ω) = 0 implies χintΩ = χΩ and χintΩ = χΩ almost everywhere for Ω ∈ L(D, λ, O, r).

3. 3.1

Equivalence of the Uniform Cusp and Fat Segment Properties Extended Cusp Property

The uniform cusp property introduced in [1] (Chapter 5, § 11) was speciﬁed by a continuous function h : [0, ρ[ → R such that h(0) = 0,

h(ρ) = λ,

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∀θ, 0 < θ < ρ,

0 < h(θ) < λ.

(10)

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Recall that with h of the form h(θ) = λ (θ/ρ)α , 0 < α ≤ 1, we recover the uniform cusp property for 0 < α < 1 and the uniform cone property for α = 1, ρ = λ tan ω and h(θ) = θ/ tan ω which corresponds to an open cone in 0 of aperture ω, height λ, and axis eN . This property can be viewed as a special case of condition (6) in Deﬁnition 4 by choosing the open set (11) O = (ζ , ζN ) ∈ RN : |ζ | < ρ and h(|ζ |) < ζN < λ . In [2] the uniform cusp property was extended to the following larger space of cusp functions h def

H = {h : [0, ∞[ → R : h(0) = 0 and h is continuous in 0}

(12)

by associating with h ∈ H, ρ > 0, and λ > 0 the axi-symmetrical region ) def

C(λ, h, ρ) =

(ζ , ζN ) ∈ RN : |ζ | < ρ and lim sup h(|ξ |) < ζN < λ ξ →ζ

(13)

around the axis eN = (0, . . . , 0, 1) in RN .

Lemma 9 Given ρ > 0, λ > 0, and h ∈ H, the region C(λ, h, ρ) contains the segment (0, λeN ), does not contain 0, and is open. Proof. – By deﬁnition N h(|ξ |) < ζN ⊂ C(λ, h, ρ), (0, λeN ) ⊂ (0, ζN ) ∈ R : 0 = lim ξ →0

since h(0) = 0 and h is continuous in 0. Also 0 = (0, 0) ∈ / C(λ, h, ρ) since it would yield the contradiction 0 = limξ →0 h(|ξ |) < ζN = 0. To show that C(λ, h, ρ) is open we ﬁx a point ζ = (ζ , ζN ) ∈ C(λ, h, ρ) and construct a neighborhood of ζ contained in C(λ, h, ρ). Let ε = ζN − ¯lζ > 0,

¯lζ def = lim sup h(|ξ |). ξ →ζ

By deﬁnition of the lim sup there exists ρ > 0 such that sup

ξ ∈B(ζ ,ρ)

h(|ξ |) < ¯lζ + ε/2. def

For all ξ = ζ in B(ζ , ρ) and ρξ = min{|ξ − ζ |, ρ − |ξ − ζ |} > 0 B(ξ , ρξ ) ⊂ B(ζ , ρ)

Copyright © 2005 Marcel Dekker, Inc.

⇒ ¯lξ < ¯lζ + ε/2,

¯lξ def = lim sup h(|η |). η →ξ

The Uniform Fat Segment and Uniform Cusp Properties

93

For all ξN ∈ R such that |ξN − ζN | < ε/2, we get ¯lξ − ξN = ¯lζ − ζN + ¯lξ − ¯lζ + ζN − ξN = −ε + ¯lξ − ¯lζ + |ζN − ξN | < −ε + ε/2 + ε/2 = 0 ⇒ (ξ , ξN ) ∈ C(λ, h, ρ), B(ζ, min{ρ, ε/2}) ⊂ C(λ, h, ρ) and C(λ, h, ρ) is open. Given λ > 0, ρ > 0, h ∈ H, and a direction d ∈ RN , |d| = 1, the rotated region from the direction eN to d is deﬁned as ) |PHd (y)| < ρ and def C(λ, h, ρ, d) = y ∈ RN : lim sup h(|P (z)|) < y · d < λ , (14) Hd z→y

where Hd = {d}⊥ is the hyperplane through 0 orthogonal to the direction d. The orientation can also be speciﬁed by the matrix of rotation A ∈ O(N) such that d = AeN and hence C(λ, h, ρ, d) = AC(λ, h, ρ). Finally, the translation of C(λ, h, ρ, d) to the point x will be denoted def

Cx (λ, h, ρ, d) = x + C(λ, h, ρ, d).

Lemma 10 For all λ > 0, ρ > 0, h ∈ H, and x ∈ RN , the regions C(λ, h, ρ) and Cx (λ, h, ρ, d) are nonempty and open. Moreover the segment (x, x + λd) is contained in Cx (λ, h, ρ, d). This led in [2] to the extension of the uniform cusp property in [1] (Chapter 5, § 11) to the larger class H. The function h is referred to as a cusp function and the space H as the space of cusp functions.

Definition 11 Let Ω be a subset of RN such that ∂Ω = ∅. (i) Ω satisﬁes the local uniform cusp property if ∀x ∈ ∂Ω, ∃h ∈ H, ∃λ > 0, ∃ρ > 0, ∃r > 0, ∃d ∈ RN , |d| = 1, such that ∀y ∈ B(x, r) ∩ Ω, Cy (λ, h, ρ, d) ⊂ intΩ. (ii) Given h ∈ H, Ω satisﬁes the h-local uniform cusp property if ∀x ∈ ∂Ω, such that

∃λ > 0, ∃ρ > 0, ∃r > 0, ∃d ∈ RN , |d| = 1, ∀y ∈ B(x, r) ∩ Ω, Cy (λ, h, ρ, d) ⊂ intΩ.

(iii) Ω satisﬁes the uniform cusp property for (r, λ, h, ρ) if ∃h ∈ H, ∃λ > 0, ∃ρ > 0, ∃r > 0, ∀x ∈ ∂Ω, ∃d ∈ RN , |d| = 1, such that ∀y ∈ B(x, r) ∩ Ω, Cy (λ, h, ρ, d) ⊂ intΩ. Copyright © 2005 Marcel Dekker, Inc.

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Equivalence

It turns out that the uniform fat segment and cusp properties are equivalent either with a cusp function which is only continuous at the origin or with a cusp function which is continuous, non-negative, and monotone strictly increasing.

Theorem 12 Given a subset Ω of RN with nonempty boundary Γ, the following three conditions are equivalent. (i) Ω satisﬁes the uniform cusp property of Deﬁnition 11 (iii) for some r, λ, ρ, and a cusp function h ∈ H that is continuous, nonnegative, monotone strictly increasing in [0, ρ], and λ = h(ρ). (ii) Ω satisﬁes the uniform cusp property of Deﬁnition 11 (iii) for some r, λ, ρ, and a cusp function h ∈ H (that is continuous at the origin). (iii) Ω satisﬁes the uniform fat segment property of Deﬁnition 4 for some r, λ, and an open set O. Proof. – (i) ⇔ (ii). (i) ⇒ (ii) is obvious since h ∈ H. Conversely, let r > 0 and C(λ, h, ρ) be associated with the uniform cusp property of Deﬁnition 11 (iii). By continuity of h ∈ H in 0, ∃0 < θ0 ≤ ρ, ∀n ≥ 1,

∀0 ≤ θ ≤ θ0 ,

∃0 < θn < θn−1 /2,

|h(θ)| ≤ λ/2,

∀0 ≤ θ ≤ θn ,

|h(θ)| ≤ λ/2n+1 .

At each step n ≥ 0 construct the continuous monotone strictly increasing and non-negative function kn : [0, θ0 ] → R deﬁned as follows θj −θ θ−θ λ + 2λj θj −θj+1 , if θj+1 < θ ≤ θj , 0 ≤ j ≤ n − 1 def j+1 j+1 j+1 kn (θ) = 2 λ θθj −θ n −θ + 2λn θθn , if 0 ≤ θ ≤ θn . 2n+1 θn By continuity of h at the origin and the fact that h(0) = 0, θn → 0 and kn (0) → 0. By construction, 0 ≤ |h(θ)| ≤ kn+1 (θ) ≤ kn (θ) in [0, θ0 ], kn+1 (θ) = kn (θ) in [θn+1 , θ0 ], and kn+1 − kn C[0,θn+1 ] ≤ λ/2n+1 . Therefore there exists a continuous non-negative and monotone strictly increasing function k ∈ C[0, θ0 ] such that kn → k in C[0, θ0 ], k(0) = 0, and |h(θ)| ≤ k(θ) ≤ λ in [0, θ0 ]. Finally, if k(θ0 ) = λ, choose ρ such that k(ρ ) = λ, λ = λ, and h = k. If k(θ0 ) < λ, choose ρ = θ0 , λ = k(θ0 ), and h = k. From the construction, ρ ≤ ρ, λ ≤ λ, h ≥ h, and hence C(λ , h , ρ ) ⊂ C(λ, h, ρ). Therefore the local uniform cusp property of Deﬁnition 11 is veriﬁed with a non-negative, continuous, and monotone strictly increasing cusp function of the form (10). Copyright © 2005 Marcel Dekker, Inc.

The Uniform Fat Segment and Uniform Cusp Properties

95

(ii) ⇔ (iii). (ii) ⇒ (iii) from Lemma 9 by choosing O = C(λ, h, ρ). Conversely, assume that there exist r > 0, λ > 0, and O such that the uniform fat segment property is veriﬁed: for each x ∈ ∂Ω, there exists A(x) ∈ O(N) such that for all y ∈ Ω ∩ B(x, r), y + A(x)O ⊂ intΩ. From Lemma 5 the uniform fat segment property is also veriﬁed for r and λ and the subset O+ of O deﬁned in (7). Denote by H = {eN }⊥ the hyperplane through 0 orthogonal to eN . The projection PH (O+ ∩ B(0, r/2)) of the open set O+ ∩ B(0, r/2) onto the hyperplane H is an open set which contains 0 since the segment min{λ, r/2)}(0, eN ) is contained in O+ . Hence there exists an open ball BH (0, ρ) in H of radius ρ > 0 such that BH (0, ρ) ⊂ PH (O+ ∩ B(0, r/2)). Deﬁne the function a : BH (0, ρ) → R a(y ) =

def

inf

z∈O + ∩B(0,r/2) PH (z)=y

z · eN .

It is well deﬁned since the region O+ ∩ B(0, r/2) is bounded below by H, a(y ) ≥ 0, and a(0) = 0. It is continuous in 0. For any ε > 0, there exists 0 < δ ≤ min{ρ, ε/4} such that B((0, ε/2), δ) ⊂ O+ ∩ B(0, r) since the segment (0, λeN ) is a subset of the open set O+ . Hence for all |y | < δ, 0 ≤ a(y ) < ε. By construction of a, for all |ζ | ≤ ρ, (ζ , a(ζ )) ∈ B(0, r/2) ∩ O+ ⊂ B(0, r/2) ∩ O+ and, by the uniform fat segment property, for all y ∈ Ω ∩ B(x, r/2) ∀|ζ | ≤ ρ,

y + A(x)(ζ , a(ζ )) ∈ B(x, r)

∀|ζ | ≤ ρ,

y + A(x)(ζ , a(ζ )) ∈ y + A(x)O ⊂ intΩ ⊂ Ω

⇒ ∀|ζ | ≤ ρ, ⇒ ∀|ζ | ≤ ρ,

y + A(x)(ζ , a(ζ )) ∈ Ω ∩ B(x, r) y + A(x)(ζ , a(ζ )) + A(x)O ⊂ intΩ

and the fat segment property is veriﬁed for r/2, λ, and the new open region O =

def

(ζ , a(ζ )) + O : ∀|ζ | ≤ ρ .

To complete the proof we now construct a function h ∈ H from the function a such that C(λ, h, ρ) ⊂ O , where C(λ, h, ρ) is as deﬁned in (13). Consider the function sup|ζ |=1 a(θζ ), 0 ≤ θ < ρ def h(θ) = λ, otherwise is well deﬁned, non-negative, and h(0) = 0. By continuity of a in 0, ∀ε > 0, ∃0 < δ ≤ ρ such that |y − 0| < δ ⇒ |a(y )| = |a(y ) − a(0)| < ε Copyright © 2005 Marcel Dekker, Inc.

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⇒ ∀θ, 0 ≤ θ < δ, ∀|ζ | = 1, |a(θζ )| = |a(θζ ) − a(0)| < ε ⇒ ∀θ, 0 ≤ θ < δ, |h(θ) − h(0)| = |h(θ)| = sup |a(θζ )| < ε |ζ |=1

and h is continuous in 0. Thus h ∈ H. By deﬁnition for all |ξ | < ρ, a(ξ ) ≤ h(|ξ |), and C(λ, h, ρ) ⊂ C(λ, a, ρ), where ) def

C(λ, a, ρ) =

(ζ , ζN ) ∈ RN : |ζ | < ρ and lim sup a(ξ ) < ζN < λ . ξ →ζ

To conclude we show that C(λ, a, ρ) ⊂ O . Since a(ζ ) ≥ 0, def ∀(ζ , ζN ) ∈ C(λ, a, ρ), 0 ≤ ¯lζ = lim sup a(ξ ) < ζN < λ ξ →ζ

⇒ 0 < ζN − ¯lζ < λ. By assumption the segment (0, λeN ) is contained in the open set O. Hence there exists a ball B((0, ζN − ¯lζ ), η) of radius η at the point (0, ζN − ¯lζ ) that is contained in O. By deﬁnition of the lim sup, there exists a sequence {ξn } ⊂ B(0, ρ) such that ξn → ζ and a(ξn ) → ¯lζ . Therefore there exists n ¯ such that for all n ≥ n ¯, (ζ , ζN ) = (ζ , ¯lζ ) + (0, ζN − ¯lζ ) ∈ (ξn , a(ξn )) + B((0, ζN − ¯lζ ), η) ⇒ ∀n ≥ n ¯ , (ζ , ζN ) ∈ (ξn , a(ξn )) + O ⊂ O ⇒ C(λ, a, ρ) ⊂ O ⇒ C(λ, h, ρ) ⊂ C(λ, a, ρ) ⊂ O . Hence the uniform cusp property is veriﬁed with r = r/2, λ, and C(λ, h, ρ).

References [1] M.C. Delfour and J.-P. Zol´esio, Shapes and Geometries: Analysis, Diﬀerential Calculus and Optimization, SIAM series on Advances in Design and Control, Society for Industrial and Applied Mathematics, Philadelphia, USA 2001. [2] M.C. Delfour, N. Doyon, and J.-P. Zol´esio, Extension of the uniform cusp property in shape optimization, in Control of Partial Diﬀerential Equations, G. Leugering, O. Imanuvilov, R. Triggiani, and B. Zhang, eds. Lectures Notes in Pure and Applied Mathematics, Marcel Dekker, New York, 2004. [3] M.C. Delfour, N. Doyon, and J.-P. Zol´esio, Uniform cusp property, boundary integral, and compactness for shape optimization, in System Modeling and Optimization, J. Cagnol and J.-P. Zol´esio, eds., pp. 25–40, Kluwer Academic Publishers, Norwell, MA, 2004.

Copyright © 2005 Marcel Dekker, Inc.

CHAPTER VIII TOPOLOGY OPTIMIZATION FOR UNILATERAL PROBLEMS Jan Sokolowski Institut Elie Cartan, Laboratoire de Math´ematiques Universit´e Henri Poincar´e Nancy I Vandoeuvre-L`es-Nancy, France

∗ ˙ Antoni Zochowski

Systems Research Institute of the Polish Academy of Sciences Warszawa, Poland

Abstract

The shape sensitivity analysis of unilateral problems is performed in [18]. The analysis is combined with outer asymptotic expansions of solutions to variational inequalities in order to derive topological derivatives. In particular, the problem of topology optimization is considered for free boundary problems of thin obstacle types. The formulae for the ﬁrst term of asymptotics for energy functionals are derived. The precision of obtained terms is veriﬁed numerically. The topological diﬀerentiability of solutions to variational inequalities is established.

Keywords: Shape optimization, topology optimization, asymptotic analysis, singular perturbations, unilateral problem

Introduction The boundary variations technique for unilateral problems is applied in [18] in the framework of nonsmooth analysis combined with the speed method. Nonsmooth analysis is necessary since the shape diﬀerentiability of solutions to unilateral problems is obtained only in the framework of Hadamard diﬀerentiability of metric projections onto polyhedric sets in the appropriate Sobolev spaces. However, to our best knowledge, there are no numerical methods for simultaneous shape and topology optimization [16] of unilateral problems. The main diﬃculty in analysis of

∗ Funding

provided by grant no. 4 T11A 01524 of the State Committee for the Scientiﬁc Research of the Republic of Poland (KBN).

Copyright © 2005 Marcel Dekker, Inc.

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unilateral problems, e.g., the contact problems in elasticity, is associated with the nonlinearity of the nonpenetration condition over the contact zone which makes the boundary value problem nonsmooth. In the paper we propose a method for numerical evaluation of topological derivatives for such problems. The notion of topological derivative of a shape functional is introduced in [14]. The knowledge of topological derivatives is required for the optimality conditions of simultaneous shape and topology optimization. The topological derivative of a given shape functional can be determined from the variations of the shape functional created by the variations of the topology of geometrical domains. The topology variations are deﬁned by nucleation of small holes or cavities or more generally small defects in geometrical domains. The main idea we use to derive the topological derivatives for unilateral problems is the modiﬁcation of the energy functional by an appropriate correction term and subsequent minimization of the resulting energy functional over the cone of admissible displacements. The asymptotic analysis of elliptic boundary value problems is performed in [4], [8], [10]. The Signorini problem is considered in [1], [2], [3]. Conical diﬀerentiability of solutions to variational inequalities is established, e.g., in [6], [12] in the framework used in the paper. The topological derivatives of shape functionals and optimality conditions for topology optimization are derived in [7], [11], [14], [15], [16] for elliptic problems. Applications to inverse problems are given in [5]. In the paper we derive useful formulae for the correction terms of the energy functionals. We restrict ourselves to two-dimensional problems and to singular perturbations of geometrical domains in the form of small discs. The correction terms are derived in such a form, that the numerical veriﬁcation of its precision is straightforward. On the other hand, the terms are directly used to establish the topological diﬀerentiability of solutions to variational inequalities.

1. 1.1

Transformations of the Energy Functional for the Laplace Equation Fundamental Formulae

Let us consider the bulk energy functional of the form 1 ∇u2 dx, E(u) = 2 Ω

(1)

where u satisﬁes inside the domain Ω ⊂ R2 the Laplace equation ∆u = 0 Copyright © 2005 Marcel Dekker, Inc.

(2)

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Topology Optimization for Unilateral Problems

with suitable boundary conditions. Our goal is to study the inﬂuence of the small circular hole of the variable radius ρ contained in the domain; for simplicity we put its center at x = 0. We do not want to study variable domains, so we isolate this hole inside the ring C(ρ, R) = { x | ρ < x < R } and replace E(u) by the equivalent expression over ΩR = Ω \ B(R), with an additional boundary integral term over ΓR = ∂B(R). On Γρ = ∂B(ρ) we assume homogeneous Neumann conditions. In the ﬁrst step we modify E(u). Since 2 2 ∇u dx = ∇u dx + ∇u2 dx = (3) Ω ΩR B(R) ∂u ds, ∇u2 dx + u ΩR ΓR ∂n where n is the outward normal vector on the boundary of B(R), we may concentrate on the expression ∂u ds. (4) ER (u) = u ΓR ∂n Here the values of ∂u/∂n on ΓR are given by the Dirichlet to Neumann mapping, namely the solution of ∆v = 0 in B(R), v = u

on ΓR ,

(5)

by means of the relation ∂u/∂n = ∂v/∂n. The function v may be constructed with the help of the Poisson kernel, i.e., expressed as its integral over the circle ΓR . Using this representation and asymptotic expansions of solutions to the Laplace equation around the small hole it is possible [17] to calculate the ﬁrst term in the perturbation of ER (u) caused by the introduction of B(ρ): *

2

2 + ρ2 ux1 ds + ux2 ds . (6) δER (u) = − 6 πR ΓR ΓR This allows us to consider the following approximation. Let uρ satisfy (2) inside Ωρ = Ω \ B(ρ) and take 1 ∇uρ 2 dx. (7) E(ρ ; uρ ) = 2 Ωρ Then we may consider uρ in a ﬁxed domain ΩR and add boundary terms 1 1 1 ∇uρ 2 dx + ER (uρ ) + δER (uρ ) + o(ρ2 ). (8) E(ρ ; uρ ) = 2 ΩR 2 2 Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

The representation (6) allows us to transform the task from solving the boundary value problem for the Laplace equation in variable domain Ωρ to the same problem in the ﬁxed domain ΩR , but with energy parametrized by ρ. In fact, since the ﬁrst terms represent energy for the whole of Ω, we may even avoid using ΩR (and the remeshing of Ω in order to get discretization of ΩR ). This results from observation that outside B(R) the function wR coincides with w given by the solution of the boundary value problem with bulk energy given by

(9) E0 (ρ ; w) = * +

2 2 ρ2 1 ∇w2 dx − wx1 ds + wx2 ds + o(ρ2 ) 2 Ω 2πR6 ΓR ΓR The weak form of this problem is easily obtained by taking variation of E0 (ρ ; w), adding external work and using boundary conditions on ∂Ω. As we see, modiﬁcation of bulk energy accounts to introducing source term concentrated on the ﬁxed circle ΓR and parametrized by ρ. The last formula may be expresed also in another form. Let us denote by eu (x) the energy density at the point x, eu (x0 ) = ∇u(x0 )2 . If the function u is harmonic in B(R), then, as may be proved [17], the expressions for derivatives 1 u · (x1 − x1,0 ) ds, u/1 (x0 ) = πR3 ΓR (x0 ) 1 u/2 (x0 ) = u · (x2 − x2,0 ) ds. πR3 ΓR (x0 ) are exact. In view of this, formula (9) can be rewritten in the equivalent form 1 E0 (ρ ; w) = 2

Copyright © 2005 Marcel Dekker, Inc.

1 ∇w2 dx − πρ2 ew (0) + o(ρ2 ). 2 Ω

(10)

Topology Optimization for Unilateral Problems

1.2

101

Numerical Tests of the Transformed Energy Approach

We consider the test problem ∆uρ = 0 in 0 1 uρ = x21 √ x1 ∂uρ = 0 on ∂n

Ωρ = [0, 1] × [0, 1] \ B(x0 , ρ), for for for for

x1 x1 x2 x2

= 0, = 1, = 0, = 1.

(11)

∂B(x0 , ρ).

Three types of approximations are used. Here by u we denote the solution in the domain without void. 1 Double correction: the function uρ is represented in the form uρ = u + sρ (u) + pρ + sρ (pρ ), where sρ (u) is the ﬁrst correction for u. However, sρ (u) disturbs the boundary conditions on ∂Ω and we introduce second corrections. The function pρ solves the Laplace equation with the boundary condition pρ = −sρ (u) on ∂Ω, and then again sρ (pρ ) nulliﬁes the Neumann condition on the boundary of the void. We consider this solution as nearly exact in Ωρ . 2 “Exact” solution: here the right-hand side is augmented by the expression containing derivatives of Dirac’s delta, ∆uρ = 2πρ2 (∇δ(x − x0 ) · ∇u). In theory this solution is also exact in Ωρ up to the higher than 2 powers of ρ, see [9], but of course there are diﬃculties with numerical approximation. 3 Solution obtained by the modiﬁcation of the energy term given by (10). This should be exact up to the higher than 2 powers of ρ outside the ring of the radius R > ρ. The hole was positioned at x0 = [0.5, 0.7] and had the radius ρ = 0.05. The ﬁgures show the results of computations for diﬀerent positions of the void and ratios R/ρ, in the form of sections through the surface u(x1 ) = u(x1 , x02 ) along the line x2 = x02 , i.e., going through the middle of the hole. Other sections look very similar. We may conclude that the Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS 1

flat area - position at the hole

0.8

with double correction

0.6

solution obtained via correction term in energy

0.4 solution in the intact domain

0.2 0

exact - with singular source term

–0.2 boundary of B(R)

–0.4 –0.6 –0.8 0

10

20

30

40

50

60

70

80

90 100

Figure 1. Comparison of solutions for R = 1.2ρ. Visible slight loss of accuracy near boundary of the ring, due to the small ratio of the radii.

modiﬁed energy allows, in concordance with the theory developed in the last subsection, to compute accurately the solution outside the ring ΓR . Moreover we have seen a very good agreement of the solution with the singular right-hand side with the nearly exact solution produced by double correction.

2.

Signorini Problem

We establish the conical diﬀerentiability of solutions to the Signorini problem with respect to the small parameter. The obtained expansion of solutions to the Signorini problem can be interpreted as the ﬁrst order outer asymptotic expansion in the spirit, e.g., of [11]. So, this way, we can deﬁne the topological derivatives of some shape functionals, including the energy functional, for variational inequalities. To our best knowledge this is the ﬁrst result in this direction derived without any assumptions on the strict complementarity conditions for the unknown solutions to the obstacle problems. In order to introduce the Signorini problem we need the bilinear form a(·, ·), the convex set K, the linear form L(·), and the convex cone , ⊂ V . We are going to deﬁne the variational in the Sobolev space K inequality over the space H 1 (Ωρ ), where Ωρ = Ω \ B(ρ), with the small ball B(ρ) = {x ∈ R2 |x − O < ρ}, here O is the center of the ball, and we assume that the center is the origin. So, the variational inequality reads: Copyright © 2005 Marcel Dekker, Inc.

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Topology Optimization for Unilateral Problems

, ρ ) = {v ∈ H 1 (Ωρ )|v ≥ 0 on Γs } such that a(u, v − u) ≥ Find u ∈ K(Ω Γ0 , L(v − u) for all v ∈ K(Ωρ ). The unique solution u(Ωρ ) of the variational inequality depends on the small parameter ρ. In order to analyze the dependence of the solution u(Ωρ ) on ρ we proceed in the following way. The bilinear form ∇u · ∇vdx a(u, v) = Ωρ

deﬁned on the variable geometrical domain Ωρ is linearized using the asymptotic expansions technique for the energy functionals. The linearized bilinear form is denoted by a(ρ; ·, ·) and contains in two dimensions the term of order zero with respect to ρ and the ﬁrst order perturbation depending on ρ2 . In Section VIII.1.1 the following bilinear form is identiﬁed as the ﬁrst order perturbation 1 1 wx1 ds vx1 ds − wx ds vx2 ds . b(w, v) = − 6 2 πR ΓR πR6 ΓR ΓR ΓR The bilinear form b(w, v) is continuous on the space H 1 (Ω), where ΓR = ∂B(R), and the ball B(R) ⊂ Ω. Thus, the bilinear form a(·, ·) is replaced in the variational inequality by the integral expression, depending only on ρ as a small parameter, ∇w · ∇vdx + ρ2 b(w, v) , (12) a(ρ; w, v) = Ω

a(ρ; w, v) is deﬁned over the domain of integration Ω ⊂ R2 with the smooth boundary ∂Ω = Γ0 ∪ Γs , with w, v in the energy space HΓ10 (Ω) = {v ∈ H 1 (Ω)|v = 0 on Γ0 } . Thus, the singular perturbation of the geometrical domain Ωρ is replaced in the variational inequality under considerations by the regular perturbation of the bilinear form. For such regular perturbation the standard sensitivity analysis of variational inequalities over polyhedric sets in Dirichlet spaces applies (see, e.g., [13], [6]) and the ﬁrst order expansion of the solution with respect to the small parameter is obtained. We refer the reader to [3] for the direct asymptotic analysis of the energy functional for the Signorini problem. For ρ suﬃciently small we can consider the variational inequality with the bilinear form a(ρ; w, v) over the convex set K = {v ∈ H 1 (Ω)|v ≥ 0 Copyright © 2005 Marcel Dekker, Inc.

on Γs ,

v=g

on Γ0 } .

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CONTROL AND BOUNDARY ANALYSIS

Let uρ denote the unique solution of the following variational inequality uρ ∈ K :

a(ρ; w, v − w) ≥ 0 ∀v ∈ K .

(13)

The following theorem may be proved [17].

Theorem 1 For ρ suﬃciently small we have the following expansion of the solution uρ , with respect to the parameter ρ, at 0+, uρ = u + ρ2 q + o(ρ2 ) ,

(14)

where the outer topological derivative q of the solution u = u(Ω) to the Signorini problem is given by the unique solution of the following variational inequality q ∈ SK (u) = {v ∈ HΓ10 (Ω)|v ≥ 0 on Ξ(u) , a(u, v) = 0} a(q, v − q) + b(u, v − q) ≥ 0 ∀v ∈ SK (u) .

(15) (16)

The coincidence set Ξ(u) = {x ∈ Γs |u(x) = 0} is well deﬁned [13] for any function u ∈ H 1 (Ω), and u ∈ K ⊂ V = HΓ10 (Ω) is the solution of the variational inequality for ρ = 0.

Concluding Remarks We presented the topological derivatives of solutions for variational inequalities. The same approach is applied in [17] to the frictionless contact problems in elasticity. Since the analysis becomes more involved for contact problems, we restricted our presentation to a scalar problem. The case of evolution problems seems to be open, although a result on conical diﬀerentiability of solutions to parabolic variational inequalities is established in [6].

References [1] I.I. Argatov and S.A. Nazarov. Asymptotic solution to the Signorini problem with small parts of the free boundary. Siberian Mat. Zh., 35:258–277, 1994. [2] I.I. Argatov and S.A. Nazarov. Asymptotic solution of the Signorini problem with an obstacle on a thin elongated set. Mat. sbornik, 187:3–32, 1996 (English translation in: Math. Sbornik. 187:1411–1442, 1996). [3] I.I. Argatov and J. Sokolowski. On asymptotic behaviour of the energy functional for the Signorini problem under small singular perturbation of the domain. Journal of Computational Mathematics and Mathematical Physics, 43:742–756, 2003. [4] A.M. Il’in. Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, volume 102 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1992.

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˙ [5] L. Jackowska-Strumillo, J. Sokolowski, A. Zochowski, and A. Henrot. On numerical solution of shape inverse problems. Computational Optimization and Applications, 23:231–255, 2002. [6] J. Jaruˇsek, M. Krbec, M. Rao, and J. Sokolowski. Conical diﬀerentiability for evolution variational inequalities. Journal of Diﬀerential Equations, 193:131– 146, 2003. [7] T. Lewinski and J. Sokolowski. Energy change due to appearing of cavities in elastic solids. International Journal of Solids and Structures, 40:1765–1803, 2003. [8] V.G. Maz’ya, S.A. Nazarov, and B.A. Plamenevskii. Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains. Birkhuser Verlag, Basel, Vol. 1, 2, 2000. [9] S.A. Nazarov. Asymptotic conditions at a point, self-adjoint extensions of operators, and the method of matched asymptotic expansions. American Mathematical Society Translations, 198:77–125, 1999. [10] S.A. Nazarov and B.A. Plamenevsky. Elliptic Problems in Domains with Piecewise Smooth Boundaries. Walter de Gruyter, De Gruyter Exposition in Mathematics 13, New York, 1994. [11] S.A. Nazarov and J. Sokolowski. Asymptotic analysis of shape functionals. Journal de Math´ematiques pures et appliqu´ ees, 82:125–196, 2003. [12] M. Rao and J. Sokolowski. Non-linear balayage and applications. Illinois Journal of Mathematics, 44:310–328, 2000. [13] M. Rao and J. Sokolowski. Tangent sets in banach spaces and applications to variational inequalities. Technical Report 42, Institute Elie Cartan, http://www.iecn.u-nancy.fr/ sokolows/, 2000. Submitted for publication. ˙ [14] J. Sokolowski and A. Zochowski. On topological derivative in shape optimization. SIAM Journal on Control and Optimization, 37:1251–1272, 1999. ˙ [15] J. Sokolowski and A. Zochowski. Topological derivatives of shape functionals for elasticity systems. Mechanics of Structures and Machines, 29:333–351, 2001. ˙ [16] J. Sokolowski and A. Zochowski. Optimality conditions for simultaneous topology and shape optimization. SIAM Journal on Control and Optimization, 42:1198–1221, 2003. ˙ [17] J. Sokolowski and A. Zochowski. Topological optimization for contact problems. Technical Report 25, Institute Elie Cartan, http://www.iecn.unancy.fr/ sokolows/, 2003. Submitted for publication. [18] J. Sokolowski and J.-P. Zol´esio. Introduction to Shape Optimization, volume 14 of SSCM. Springer-Verlag, New York, 1992.

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CHAPTER IX SECOND ORDER LAGRANGE MULTIPLIER APPROXIMATION FOR CONSTRAINED SHAPE OPTIMIZATION PROBLEMS M˚ artensson’s Approach for Shape Problems Karsten Eppler∗ Weierstraß Institut f¨ ur Angewandte Analysis und Stochastik Berlin, Germany

Helmut Harbrecht Institut f¨ ur Informatik und Praktische Mathematik Christian–Albrechts–Universit¨ at zu Kiel Kiel, Germany

Abstract

The present paper is dedicated to the solution of constrained shape optimization problems by second order algorithms with respect to both the primal and dual variables. This goal is realized by combining a Newton scheme for the primal variables with M˚ artensson’s concept of Lagrange multiplier functions for augmented Lagrangians.

Keywords: Shape optimization, elliptic PDE, Newton method, wavelet BEM, augmented Lagrangian approach

Introduction Shape optimization is quite indispensable for designing and constructing industrial components. Many problems that arise in application, particularly in structural mechanics and in the optimal control of distributed parameter systems, can be formulated as the minimization of functionals deﬁned over a class of admissible domains. Therefore, such problems have been intensively studied in the literature throughout the last 25–30 years (see [16–17], [21, 23], and the references therein). From ∗ Funding

provided by DFG project no. TR 302/6-1.

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the practical as well as from the theoretical point of view, to prevent optimal shapes from degeneration, several constraints have to be taken into account. More recently, the computation of the related dual variables, i.e. the Lagrange multipliers, becomes of increasing interest due to applicational and theoretical reasons. On the one hand, there are several applications where these multipliers have an important physical meaning like in the electromagnetic shaping of liquid metals (see, e.g., [19–20]). On the other hand, their computation is important by itself for the investigation of suﬃcient optimality conditions in shape optimization [6]. In [9, 11]we developed ﬁrst and second order optimization algorithms for a class of elliptic shape optimization problems, where a powerful wavelet BEM is proposed for the computation of the state and related quantities. Additional equality and/or inequality constraints of functional type are treated by an augmented Lagrangian technique, which turns out to be more stable and eﬃcient than a penalty method. In particular, by the standard update procedure, a linear convergence is provided for the Lagrange multipliers. However, due to the eﬃciency of the Newton method with respect to the primal variables, that is, the shape respective to its ﬁnite dimensional approximation, this slow convergence of the Lagrange multipliers becomes, in our opinion, the bottleneck of the overall algorithm. Furthermore, the question of a faster approximation of the dual variables is important as we explained above. Following an idea of M˚ artensson [18], second order convergence is realizable if only active equality constraints are present. Due to known degeneration tendencies in shape problems, exactly this situation occurs very often in practical and theoretical shape optimization problems. Consequently, the goal of the present paper is to demonstrate this approach for the class of shape problems considered in [9]. Of course, both the standard method and its modiﬁcation are independent of the underlying shape calculus and of the numerical method for solving the state equation. The outline of the present paper is as follows. In the ﬁrst section we brieﬂy introduce our setup for solving the considered shape optimization problems developed in [9–11]. The second section is dedicated to the augmented Lagrangian approach and the improvement of M˚ artensson by introducing so-called Lagrange multiplier functions [18]. The last section presents the numerical results.

1. 1.1

Shape Optimization The Model Problem

Let Υ denote the set of all bounded domains Ω ∈ C 2,α , Ω ⊆ D ⊆ R2 , starshaped with respect to Bδ (0). The outer security set D ⊆ R2 (the Copyright © 2005 Marcel Dekker, Inc.

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Second Order Lagrange Multiplier Approximation

hold all set) is simply connected and closed, but not necessarily bounded. Setting Γ := ∂Ω, we consider the following shape problem: h(x)u(x) + h0 (x)dx = min (1) J(Ω) = Ω∈Υ

Ω

subject to

J1 (Ω) =

h1 (x)dx = c1 , Ω

J2 (Ω) = .. . Jm (Ω) =

h2 (x)dx = c2 , Ω

(2) hm (x)dx = cm ,

Ω

where the state function u solves the Dirichlet boundary value problem −∆u = f u=g

in Ω, on Γ.

(3)

In order to conceive a well-posed problem the functions f , g, h and h0 , . . . , hm are assumed suﬃciently regular on the whole set D. Of course, constraints of the type hi (x)dσx Ji (Ω) = Γ

may be considered in our setup as well. Constraints

of practical interest in shape problems are the volume of the domain Ω 1 dx or its perimeter

1 dσ , for example. x Γ Let us remark that (1) includes also the important case of energy functionals. Consider the state equation having homogeneous boundary conditions (g ≡ 0), then integration by parts yields 2 ∇u(x) dx = f udx, J(Ω) = Ω

Ω

that is h ≡ f and h0 ≡ 0.

1.2

Shape Calculus

Clearly, the starshaped domain Ω ∈ Υ can be identiﬁed with a function which describes its boundary, i.e., in polar coordinates we have φ : φ ∈ [0, 2π] , Γ := γ(φ) = r(φ) cos sin φ Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

2,α where r ∈ Cper [0, 2π] is a positive function with r > δ and 2,α [0, 2π] = {r ∈ C 2,α [0, 2π] : r (i) (0) = r (i) (2π), i = 0, 1, 2}. Cper

As a standard variation for perturbed domains Ωε and boundaries Γε , 2,α [0, 2π] respectively, we introduce a function dr ∈ Cper rε (φ) = r(φ) + εdr(φ), where γε (φ) = rε (φ)er (φ) is always a Jordan curve. Herein, er (φ) = (cos φ, sin φ)T denotes the unit vector in the outer radial direction. The main advantage of this simple approach is a complete embedding of the shape problem into a Banach space setting. That is, both the shapes 2,α [0, 2π]. and its increments, can be viewed as elements of Cper Let us remark that boundary variation is an appropriate ansatz since ﬁrst and second order directional derivatives can be represented by boundary integrals, cf. [7–8].

1.3

BIE Formulation and Fast Solvers

It turns out that introducing the adjoint state p by −∆p = h p=0

in Ω, on Γ,

the directional derivative ∇J(Ω)[dr] requires the knowledge of ∂u/∂n and ∂p/∂n, where the state function u satisﬁes (3), cf. [9]. In addition, deﬁning the local shape derivative du[dr] and the adjoint local shape derivative dp[dr] via ∆du[dr] = 0 du[dr] = drer , n

in Ω, ∂(g − u) ∂n

on Γ,

and ∆dp[dr] = 0 ∂p dp[dr] = −drer , n ∂n

in Ω, on Γ,

respectively, the computation of ∇2 J(Ω)[dr1 , dr2 ] requires the quantities ∂ 2 u/(∂n∂t), ∂ 2 p/(∂n∂t), ∂du[dr]/∂n, and ∂dp[dr]/∂n, see [9]for details. As shown in [9, 11], employing suitable Newton potentials, the computation of the derivatives of u, du, and the associated adjoints can be performed eﬃciently by boundary integral equations (BIEs). Boundary element methods provide a common tool for the solution of such Copyright © 2005 Marcel Dekker, Inc.

Second Order Lagrange Multiplier Approximation

111

equations. In general, cardinal B-splines are used as ansatz functions in the Galerkin formulation. However, these traditional discretizations yield densely populated and in general ill-conditioned system matrices. Hence, a numerical scheme is of at least quadratic complexity. The crucial idea of the wavelet Galerkin scheme is a change of bases, i.e., applying appropriate wavelet bases instead of the traditional singlescale bases. On the one hand, based on the well-known norm equivalences of wavelet bases, the diagonals of the system matrices deﬁne optimal preconditioners, cf. [3, 22]. On the other hand, the resulting quasi-sparse system matrices can be compressed without loss of accuracy such that the complexity for the solution of the boundary integral equations becomes linear , cf. [5, 13, 14, 15].

1.4

Boundary Approximation

Since the inﬁnite dimensional optimization problem cannot be solved directly, we replace it by a ﬁnite dimensional problem. Based on the polar coordinate approach, we can express the smooth function r ∈ 2,α ([0, 2π]) by the Fourier series Cper r(φ) = a0 +

∞

an cos nφ + a−n sin nφ.

n=1

Hence, it is reasonable to take the truncated Fourier series rN (φ) = a0 +

N

an cos nφ + a−n sin nφ.

(4)

n=1

as approximation of r. We mention that other boundary representations like B-splines can be considered as well. The advantage of our approach is an exponential convergence rN → r if the shape is analytical. To this end, since rN has the 2N + 1 degrees of freedom a−N , a1−N , . . ., aN , we replace the inﬁnite dimensional set of admissible domains Υ by an 2N + 1 dimensional set ΥN deﬁned by ΥN := {a−N , a1−N , . . . , aN ∈ R : rN (φ) > 0, φ ∈ [0, 2π]} ⊂ R2N +1 . Consequently, according to this ﬁnite dimensional setting, we conclude ∇J(Ω)[cos N φ] ∇J(Ω)[cos(N −1)φ]

∇J(Ω) =

.. .

∈ R2N +1

∇J(Ω)[sin N φ],

and likewise for the shape hessian ∇2 J(Ω) ∈ R(2N +1)×(2N +1) . Copyright © 2005 Marcel Dekker, Inc.

112

CONTROL AND BOUNDARY ANALYSIS

2.

Optimization of Constrained Problems

2.1

Augmented Lagrangian Functionals

Setting λ = (λ1 , λ2 . . . , λm )T ∈ Rm , the minimization problem deﬁned by (1) and (2) implies to ﬁnd the stationary point (Ω , λ ) ∈ ΥN × Rm of the following modiﬁed augmented Lagrangian functional - - C1 (Ω)−c1 -2 C1 (Ω)−c1 2 2 α- C2 (Ω)−c C2 (Ω)−c .. .. Lα (Ω, λ) = J(Ω) + λT (5) + - - . . . 2Cm (Ω)−cm Cm (Ω)−cm Of course, the choice α = 0 yields the pure Lagrangian while λ = 0 and α → ∞ is known as standard quadratic penalty method. However, both methods have some drawbacks from the numerical point of view, cf. [12, 18], for example. In order to avoid these diﬃculties, we consider α > 0 ﬁxed but appropriately chosen and perform the following standard augmented Lagrangian algorithm: initialization: choose initial guesses λ(0) for λ and Ω(0) for Ω inner iteration: solve Lα (Ω, λ(n) ) = min with initial guess Ω(n) outer iteration: update

λ(n+1) := λ(n) − α

C1 (Ω)−c1 C2 (Ω)−c2

.. .

(6)

Cm (Ω)−cm

and choose Ω(n+1) as the solution from the inner iteration. As we have observed in [9], a Newton scheme for minimizing Lα (Ω, λ(n) ) makes a line-search nearly obsolete in comparison to ﬁrst order methods. As a consequence, the bottleneck during the iteration process is now the slow ﬁrst order approach of the Lagrange multipliers according to the following theorem, cf. [2, 12].

Theorem 1 Assuming that the minimization problem (1) and (2) has a solution Ω ∈ ΥN with λ ∈ Rm satisfying the KKT-conditions as well as the suﬃcient second order conditions, then the update rule (6) yields a linear convergence λ(n) → λ , provided that the initial guesses are properly chosen. We like to mention that augmented Lagrangian techniques are also available for inequality constraints, but due to the lack of smoothness second order algorithms would not work. Copyright © 2005 Marcel Dekker, Inc.

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Second Order Lagrange Multiplier Approximation

The gradient of the functional (5) reads as ∇Lα (Ω, λ) = ∇J(Ω) +

m

λi + α(Ci (Ω) − ci ) ∇Ci (Ω),

i=1

and its hessian as ∇ Lα (Ω, λ) = ∇ J(Ω) + 2

2

+α

m i=1

∇C1 (Ω) ∇C2 (Ω)

.. .

T

∇Cm (Ω)

2.2

λi + α(Ci (Ω) − ci ) ∇2 Ci (Ω) ∇C1 (Ω) ∇C2 (Ω)

.. .

.

∇Cm (Ω)

Second Order Lagrange Multiplier Update

M˚ artensson [18] proposed a second order method for the Lagrange multiplier. More precisely, the multiplier is seen as a function depending on the primal variable, i.e., λ = λ(Ω), namely T ∇ C1 (Ω)∇J(Ω) T C (Ω)∇J(Ω) 2

∇ λ(Ω) = −A(Ω)−1

.. .

m ∈R ,

(7)

∇T Cm (Ω)∇J(Ω)

where the matrix A(Ω) is given by T A(Ω) =

∇ C1 (Ω)∇C1 (Ω) ... ∇T C1 (Ω)∇Cm (Ω) ∇T C2 (Ω)∇C1 (Ω) ... ∇T C2 (Ω)∇Cm (Ω)

.. .

.. .

m×m . ∈R

∇T Cm (Ω)∇C1 (Ω) ... ∇T Cm (Ω)∇Cm (Ω)

Instead of the constant multiplier in the shape functional (5), a ﬁrst order Taylor expansion of the above multiplier function, developed at the given initial guess Ω(n) , is inserted into the augmented Lagrangian functional. Consequently, using T ∇ C1 (Ω)∇2 J(Ω)+∇T J(Ω)∇2 C1 (Ω) ∇T C2 (Ω)∇2 J(Ω)+∇T J(Ω)∇2 C2 (Ω) .. λ (Ω) = −A(Ω)−1 . T ∇ Cm (Ω)∇2 J(Ω)+∇T J(Ω)∇2 Cn (Ω) T ∇ C1 (Ω)∇2 Ci (Ω)+∇T Ci (Ω)∇2 C1 (Ω) m ∇T C2 (Ω)∇2 Ci (Ω)+∇T Ci (Ω)∇2 C2 (Ω) m×(2N +1) .. − λi , ∈R . i=1 T 2 T 2 ∇ Cm (Ω)∇ Ci (Ω)+∇ Ci (Ω)∇ Cm (Ω)

Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

we consider the functional (n) ) + λ (Ω(n) )(Ω − Ω(n) ) . L(n) α (Ω) = Lα Ω, λ(Ω

(8)

Herein, the diﬀerence Ω − Ω(n) has to be understood as the diﬀerence of the associated Fourier coeﬃcients. Of course, the gradient and the hessian of (8) are slightly diﬀerent compared to those of (5), but the computation is straightforward and all ingredients appear also in the ﬁrst and second order derivatives of (5): the gradient is given by (n) ∇L(n) ) + λ (Ω(n) )(Ω − Ω(n) ) α (Ω) = ∇Lα Ω, λ(Ω + λ (Ω(n) )T

C1 (Ω)−c1 C2 (Ω)−c2

.. .

Cm (Ω)−cm

and the hessian by

2 (n) ∇2 L(n) ) + λ (Ω(n) )(Ω − Ω(n) ) α (Ω) = ∇ Lα Ω, λ(Ω T T T + λ (Ω(n) )T

∇ C1 (Ω) ∇T C2 (Ω)

.. .

∇T Cm (Ω)

+

∇ C1 (Ω) ∇T C2 (Ω)

.. .

(n) λ (Ω ).

∇T Cm (Ω)

M˚ artensson’s approach yields the following optimization algorithm: initialization: choose an initial guess Ω(0) for Ω (n)

inner iteration: solve Lα (Ω) = min by the Newton scheme with initial guess Ω(n) outer iteration: choose Ω(n+1) as the solution from the inner iteration

Theorem 2 (M˚ artensson [18]) Assume α suﬃciently large and (Ω , λ ) ∈ ΥN × Rm satisfying the assertion of Theorem 1, then the above algorithm converges quadratically to (Ω , λ ), i.e., this algorithm has one step convergence for linear-quadratic problems.

3. 3.1

Numerical Results A Shape Problem from Planar Elasticity

For comparison reasons we employ ﬁrst a model problem where the optimal shape and the dual variables are known analytically. We consider Copyright © 2005 Marcel Dekker, Inc.

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Second Order Lagrange Multiplier Approximation

a cylindric circular bar which is homogeneous and isotropic with a planar, simply connected cross section Ω ∈ R2 . We follow Banichuk and Karihaloo [1] but normalize the shear modulus G = 1 and the elastic modulus E = 1. We want to solve the problem of maximizing the torsional rigidity of the bar subject to given equality constraints on the stiﬀness rigidity and the volume. First, we brieﬂy recall the mathematical formulation of the quantities. The torsional rigidity is calculated by u(x)dx. J(Ω) = 2 Ω

Thus, since we want to maximize the torsional rigidity, we have h ≡ −2 and h0 ≡ 0 in (1). The stress function u = u(Ω) satisﬁes −∆u = 2 u=0

in Ω, on Γ.

The bending rigidity with respect to a ﬁxed barycenter in the origin is given by J1 (Ω) = y 2 dx. Ω

The volume of the domain and its (simpliﬁed) barycenter coordinates read as J2 (Ω) = dx, J3 (Ω) = xdx, J4 (Ω) = ydx. Ω

Ω

Ω

√

Choosing c1 = 2π/4, c2 = π, and c3 = c4 = 0, then the optimal shape 1/4 . The associated is the ellipse with semiaxes hx = 2−1/4 and h√ y = 2 Lagrange multipliers are λ1 = −4/9, λ2 = 8 2/9, and λ3 = λ4 = 0, cf. [1]. The numerical setting is as follows. We approximate the boundary via 33 Fourier coeﬃcients, that is the choice N = 16 according to (4). The boundary integral equations are discretized using 256 boundary elements. The penalty parameter α is set to 10. In the inner iteration we use a Newton scheme, see [9] for the details. The outer iteration is started with the unit circle. In Figure 1 we compare produced errors of M˚ artensson’s method (solid line) with that of the traditional update rule (6) using diﬀerent initial guesses for the Lagrange multipliers, namely (0, 1, 0, 0)T produced from M˚ artensson’s formula (7) at the unit circle (dashed line) Copyright © 2005 Marcel Dekker, Inc.

116

CONTROL AND BOUNDARY ANALYSIS (0)

(−1, 1, 0, 0)T , that is λi

= sgn λi

(1, 1, 1, 1)T (indicated by circles) λ (dash-dotted line) As one ﬁgures out, M˚ artensson’s method yields the best convergence while the best traditional scheme uses the initial guess computed by (7).

3.2

Exterior Electromagnetic Shaping

We consider a cylindric vertical column of molten liquid metal with planar, simply connected cross section falling down in an electromagnetic ﬁeld generated by vertical conductors of radii ε > 0. The frequency of the imposed current is very high, such that electromagnetic forces are reduced to the magnetic pressure acting on the interface. In the equilibrium case, a stationary horizontal cross section Ωc = R2 \ Ω of ﬁxed volume c1 arises and the three-dimensional problem reduces to the Error of λ1

102

Martensson method Martensson start initial guess (1,1,1,1) initial guess (–1,1,0,0) exact start

101

100

10–1

10–1

–2

10

10–2

10–3

10–3

10–4

10–4

–5

10–5

10–6

0

5

10 15 20 25 30 35 40 45

Error of |(λ3,λ4)|

102 10–2

10–6

0

5

10 15 20 25 30 35 40 45

Shape Error

100 Martensson method Martensson start initial guess (1,1,1,1) initial guess (–1,1,0,0) exact start

100

Martensson method Martensson start initial guess (1,1,1,1) initial guess (–1,1,0,0) exact start

101

100

10

Error of λ2

102

Martensson method Martensson start initial guess (1,1,1,1) initial guess (–1,1,0,0) exact start

–1

10

10–2

10–4 –6

10

10–3

10–8

10–4

10–10 10–12

10–5

10–14 10–16 0

5

Figure 1.

10 15 20 25 30 35 40 45

10–6 0

5

10 15 20 25 30 35 40 45

Errors of the dual variables and the shape during the iteration.

Copyright © 2005 Marcel Dekker, Inc.

117

Second Order Lagrange Multiplier Approximation Electromagnetic Shaping 3

–1

2

+1

1 0

–1

–1

–1

+1

–1

+1

+1

–2

+1 –1

–3 –3

Figure 2.

+1

–1

–2

–1

0

1

2

3

Conﬁguration of electromagnets and liquid metal.

following two-dimensional shape problem: Seek Ω such that J(Ω) = − ∇u2 dx + A 1dσ = min, Ω

Γ

where

αi in Ω, −∆u = j := M i=1 πε2 χBε (xi ) u=0 on Γ, u = O(1) as x → ∞, as x → ∞. ∇u = O(x−2 )

subject to J1 (Ω) = Ωc 1 dx = c1 . We emphasize that the Lagrange multiplier λ1 corresponds to the magnetic pressure on the surface of the liquid metal. We choose a A = 0.001, c1 = 3π/4, and twelve conductors in the positions and with amperage αi in accordance with Figure 2. The choice of ε > 0 does not inﬂuence the solution, cf. [10]. We use 65 Fourier coeﬃcients for the boundary, i.e. N = 32, 512 boundary elements, and α = 5. We use the circle of volume c1 as initial guess for the Newton scheme. M˚ artensson’s method ﬁnishes the iteration in 110.38 sec. after 10 Newton steps while the traditional scheme (with (0) initial guess λ1 = 0) requires 192.55 sec. and 25 Newton steps. Both schemes produce the same λ1 = −0.0101458.

References [1] N.V. Banichuk and B.L. Karihaloo. Minimum-weight design of multi-purpose cylindrical bars. Intl. J. of Solids and Structures, 12:267–273, 1976. [2] D.P. Bertsekas. Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York, 1982. [3] W. Dahmen. Wavelet and multiscale methods for operator equations. Acta Numerica, 6:55–228, 1997. [4] W. Dahmen and A. Kunoth. Multilevel preconditioning. Numer. Math., 63:315– 344, 1992.

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[5] W. Dahmen, H. Harbrecht, and R. Schneider. Compression techniques for boundary integral equations – optimal complexity estimates. Technical Report SFB 393/02-06, Institute of Mathematics, TU Chemnitz, 2002. Submitted. [6] M. Dambrine and M. Pierre. About stability of equilibrium shapes. M2AN, 34:811–834, 2000. [7] K. Eppler. Boundary integral representations of second derivatives in shape optimization. Discussiones Mathematicae (Diﬀerential Inclusion Control and Optimization), 20:63–78, 2000. [8] K. Eppler. Optimal shape design for elliptic equations via BIE-methods. J. of Applied Mathematics and Computer Science, 10:487–516, 2000. [9] K. Eppler and H. Harbrecht. 2nd order shape optimization using wavelet BEM. Technical Report 06-2003, Institute of Mathematics, TU Berlin, 2003. Submitted. [10] K. Eppler and H. Harbrecht. Exterior electromagnetic shaping using wavelet BEM. Technical Report 13-2003, Institute of Mathematics, TU Berlin, 2003. To appear in Math. Meth. Appl. Sci. [11] K. Eppler and H. Harbrecht. Numerical solution of elliptic shape optimization problems using wavelet-based BEM. Optim. Methods Softw., 18:105–123, 2003. [12] Ch. Grossmann and J. Terno. Numerik der Optimierung. B.G. Teubner, Stuttgart, 1993. [13] H. Harbrecht, F. Paiva, C. P´erez, and R. Schneider. Biorthogonal wavelet approximation for the coupling of FEM-BEM. Numer. Math., 92:325–356, 2002. [14] H. Harbrecht, F. Paiva, C. P´erez, and R. Schneider. Wavelet preconditioning for the coupling of FEM-BEM. Numerical Linear Algebra with Applications, 10:197–222, 2003. [15] H. Harbrecht and R. Schneider. Wavelet Galerkin schemes for 2D-BEM. In Operator Theory: Advances and Applications 121. Birkh¨ auser, Basel, 2001. [16] J. Haslinger and P. Neitaanm¨ aki. Finite element approximation for optimal shape, material and topology design, 2nd edition. Wiley, Chichester, 1996. [17] A.M. Khludnev and J. Sokolowski. Modelling and control in solid mechanics. Birkh¨ auser, Basel, 1997. [18] K. M˚ artensson. A new approach to constrained function optimization. J. of Optimization Theory and Application, 12:531–554, 1973. [19] A. Novruzi and J.-R. Roche. Second derivatives, newton method, application to shape optimization. Technical Report No. 2555, INRIA, Nancy, France, 1995. [20] M. Pierre and J.-R. Roche. Computation of free surfaces in the electromagnetic shaping of liquid metals by optimization algorithms. Eur. J. Mech, B/Fluids, 10:489–500, 1991. [21] O. Pironneau. Optimal shape design for elliptic systems. Springer-Verlag, New York, 1983. [22] R. Schneider. Multiskalen- und Wavelet-Matrixkompression: Analysisbasierte Methoden zur L¨ osung großer vollbesetzter Gleichungssysteme. B.G. Teubner, Stuttgart, 1998. [23] J. Sokolowski and J.-P. Zolesio. Introduction to shape optimization. SpringerVerlag, Berlin, 1992.

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CHAPTER X MATHEMATICAL MODELS OF “ACTIVE” OBSTACLES IN ACOUSTIC SCATTERING Lorella Fatone Dipartimento di Matematica Pura e Applicata Universit` a di Modena e Reggio Emilia Modena, Italy

Maria Cristina Recchioni Istituto di Teoria delle Decisioni e Finanza Innovativa Universit` a Politecnica delle Marche Ancona, Italy

Francesco Zirilli Dipartimento di Matematica “G. Castelnuovo” Universit` a di Roma “La Sapienza” Roma, Italy

Abstract

We consider the following problem: given an acoustic incident ﬁeld that hits a bounded obstacle Ω characterized by a constant acoustic boundary impedance χ and immersed in an isotropic and homogeneous medium that ﬁlls R3 \ Ω choose a pressure current, acting on the boundary of Ω, in order to pursue a given goal that involves the ﬁeld scattered by Ω when hit by the incident ﬁeld. Obstacles of this kind are called active obstacles. Let us give two examples of goals that can be pursued. Let D and DG be bounded sets contained in R3 and characterized by a nonnegative constant boundary acoustic impedance χ and χG respectively, such that D ⊆ Ω and DG ∩ Ω = ∅, DG = ∅. The problems considered consist in choosing a pressure current deﬁned on the boundary of Ω in order to make the wave scattered by Ω when hit by the incident acoustic ﬁeld to appear like the wave scattered by D or by DG in the same circumstances outside a bounded region that contains Ω and D or Ω and DG respectively. The problem involving D is called “masking problem”, the problem involving DG is called “ghost obstacle problem.” In this paper the two problems proposed are modeled as optimal control problems for the wave equation. Using the Pontryagin maximum

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CONTROL AND BOUNDARY ANALYSIS principle the ﬁrst order optimality conditions associated to these two problems are formulated as exterior problems deﬁned outside Ω for a system of two coupled wave equations. These exterior problems are well suited to be solved with an adapted version of the operator expansion method proposed in [5]. Finally the numerical results obtained on some test cases are discussed.

Keywords: Time dependent acoustic obstacle scattering, optimal control problems for the wave equation

Introduction Let R be the set of real numbers, R3 be the three-dimensional real Euclidean space, and x = (x1 , x2 , x3 )T ∈ R3 be a generic vector, where the superscript T denotes transposed. We denote with (·, ·) the Euclidean scalar product in R3 and with · the corresponding vector norm. Let R3 be ﬁlled with a homogeneous isotropic medium in equilibrium at rest with no source terms present. Let Ω, D, and DG be bounded simply connected open sets contained in R3 with locally Lipschitz boundaries ∂Ω, ∂D, ∂DG and let Ω, D, D G be their closures respectively. We denote with n(x) = (n1 (x), n2 (x), n3 (x))T ∈ R3 , x ∈ ∂Ω the outward unit normal vector to ∂Ω in x ∈ ∂Ω. Similarly n(x) = (n1 (x), n2 (x), n3 (x))T ∈ R3 , x ∈ ∂D or x ∈ ∂DG denotes the outward unit normal to ∂D in x ∈ ∂D or to ∂DG in x ∈ ∂DG . Since Ω has a locally Lipschitz boundary n(x), x ∈ ∂Ω exists almost everywhere (see [7] Lemma 2.42 p. 88) and similar statements hold for D and DG . Furthermore let D ⊆ Ω and let DG be such that Ω ∩ DG = ∅ and DG = ∅. We assume that Ω, D, and DG are characterized by non-negative constant boundary acoustic impedances χ, χ , and χG respectively, and we refer to (Ω; χ) as the obstacle, to (D; χ ) as the mask, and to (DG ; χG ) as the ghost obstacle. We consider (Ω; χ), (D; χ ), (DG ; χG ) immersed in the medium mentioned previously that ﬁlls R3 \Ω, R3 \ D, R3 \ DG respectively. Without loss of generality we can assume that the origin of the coordinate system is a point in the interior of Ω and in the interior of D when D = ∅. We always assume Ω = ∅. We note that the last assumption on Ω implies that there exists a > 0 such that the closed sphere of center of the origin and radius a is contained in Ω. Finally let t ∈ R denote the time variable and let ui (x, t), (x, t) ∈ R3 ×R, be an acoustic incident wave propagating in the medium satisfying the wave equation in R3 ×R with wave propagation velocity c > 0. When the incident wave ui hits the obstacle (Ω; χ) a scattered wave is generated. The same happens when ui hits the mask (D; χ ) or the ghost obstacle (DG ; χG ). We denote with us (x, t), (x, t) ∈ (R3 \ Ω) × R, with usD (x, t), Copyright © 2005 Marcel Dekker, Inc.

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(x, t) ∈ (R3 \ D) × R and with usG (x, t), (x, t) ∈ (R3 \ DG ) × R the wave scattered by the obstacle (Ω; χ), by the mask (D; χ ) and by the ghost obstacle (DG ; χG ) respectively, when hit by the incoming ﬁeld ui . We consider the following problems: Problem 1 “Masking” Problem: Given the incoming acoustic wave ui , the obstacle (Ω; χ), the mask (D; χ ), such that D ⊆ Ω, choose a pressure current circulating on ∂Ω such that the wave scattered by (Ω; χ) when hit by ui is, outside Ω, as similar as possible to the wave scattered by the mask (D; χ ) in the same circumstances. Problem 2 “Ghost Obstacle” Problem: Given the incoming acoustic wave ui , the obstacle (Ω; χ), and the ghost obstacle (DG ; χG ), such that Ω ∩ DG = ∅, DG = ∅, choose a pressure current circulating on ∂Ω such that the wave scattered by (Ω; χ) when hit by ui appears, outside a given set containing Ω and DG , “as similar as possible” to the wave scattered by the ghost obstacle (DG ; χG ) in the same circumstances. Note that the obstacle (Ω; χ) considered in Problem 1 and in Problem 2 is an active obstacle that reacts to the presence of the incoming acoustic ﬁeld circulating a pressure current on its boundary ∂Ω for t ∈ R to pursue a given goal. The obstacles that do not react to the presence of an incoming ﬁeld are called passive. Indeed, a masking problem can be interpreted as a ghost obstacle problem if we allow the choice DG ⊆ Ω. Finally, when D ⊂ Ω and D = ∅ we have usD (x, t) = 0, (x, t) ∈ R3 × R. In this last case we call the masking problem a furtivity problem. In fact, in this case, the goal pursued will be that of making the obstacle undetectable. The mathematical model of the ghost obstacle problem proposed is similar to the mathematical model of the masking problem, so that we give in detail only the formulation of the model for the ghost obstacle problem. In [2] a detailed treatment of the masking problem can be found. Note that Problem 1 and Problem 2 have great relevance in many practical situations. In fact, they are natural direct scattering problems whose solution ﬁnds applications in many ﬁelds such as stealth technologies, noise control, and acoustic compatibility in medical echo-graphy (see the website: http://www.jhuapl.edu/programs/specialapp/). The mathematical formulation of the ghost obstacle problem given here is an optimal control problem for the wave equation. The most usual methods to solve optimal control problems in wave scattering involve the iterative solution of the direct scattering problem in the attempt of minimizing the cost functional. This approach is computationally very expensive and, sometimes, not really of practical value. We avoid this iterative approach thanks to the use of the Pontryagin maximum Copyright © 2005 Marcel Dekker, Inc.

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principle (see [4] for a survey of optimal control problems for partial diﬀerential equations). In fact, the ﬁrst order optimality conditions for the ghost obstacle problem can be formulated as the exterior problem (22)–(28) of Section X.2 for two coupled wave equations deﬁned in R3 \ Ω, and consequently the optimal control problem considered, that is, problem (8), (1), (6), (4), (5), can be solved approximately at the same computational cost as the cost necessary to solve the direct scattering problem for the passive obstacle (1), (2), (4), (5). This is a relevant result that has been obtained also in the study of several other control problems in acoustic and electromagnetic scattering, see [2], [3], [5]. To solve problem (22)–(28), we propose the use of the operator expansion method in the formulation developed in [1], [5], [6]. In Section X.1 we formulate the masking problem and the ghost obstacle problem as optimal control problems. In Section X.2 we derive the ﬁrst order optimality conditions for the control problems formulated in Section X.1 and we show some numerical results obtained solving the “ghost obstacle” optimal control problem in a test case.

1.

The Mathematical Models of the Masking and Ghost Obstacle Problems

Let us begin formulating the direct scattering problem associated to the (passive) obstacle (Ω; χ). That is, in the hypotheses described above the obstacle (Ω; χ) when hit by an acoustic incident wave ui (x, t), (x, t) ∈ R3 × R, generates a scattered wave us (x, t), (x, t) ∈ (R3 \ Ω) × R that is the unique solution of the following problem (see [6]): 1 ∂ 2 us (x, t) = 0, (x, t) ∈ (R3 \ Ω) × R, c2 ∂t2 with the boundary condition: us (x, t) −

−

∂us ∂us (x, t) + cχ = g(x, t), (x, t) ∈ ∂Ω × R, ∂t ∂n(x)

(1)

(2)

where g(x, t) is given by: g(x, t) =

∂ui ∂ui (x, t) − cχ (x, t), (x, t) ∈ ∂Ω × R, ∂t ∂n(x)

the boundary condition at inﬁnity: 1 us (x, t) = O( ), r → +∞, t ∈ R, r and the radiation condition: 1 ∂us 1 ∂us (x, t) + (x, t) = o( ), r → +∞, t ∈ R, ∂r c ∂t r Copyright © 2005 Marcel Dekker, Inc.

(3)

(4)

(5)

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∂2 where r = x, x ∈ R3 , = 3i=1 ∂x 2 is the Laplace operator, and O(·) i and o(·) are the Landau symbols. We note that g(x, t), (x, t) ∈ ∂Ω × R is deﬁned almost everywhere and that the boundary condition (2) can be adapted to deal with the limit case of the acoustically hard obstacles, i.e., χ = +∞ (see [5], [6]). Note that similar problems are satisﬁed by usD and usG . Let us model the masking problem and the ghost obstacle problem as optimal control problems. We introduce a control variable ψ, a pressure current acting on the boundary of the obstacle ∂Ω for t ∈ R. That is, in problem (1), (2), (4), (5) we replace the boundary condition (2), with the following boundary condition: −

∂us ∂us (x, t) + cχ = g(x, t) + (1 + χ)ψ(x, t), (x, t) ∈ ∂Ω × R. (6) ∂t ∂n(x)

We distinguish the obstacle that generates a scattered ﬁeld satisfying problem (1), (2), (4), (5) from the obstacle that generates a scattered ﬁeld satisfying problem (1), (6), (4), (5); we refer to them as passive obstacle and active obstacle respectively. Let us formulate Problem 1 and Problem 2 as optimal control problems for the wave equation. That is: Problem 1a) “Masking” Optimal Control Problem: Given the incoming acoustic wave ui , the obstacle (Ω; χ), and the mask (D; χ ), such that D ⊆ Ω, choose a pressure current (i.e., a control function in a suitable class of admissible controls) deﬁned on ∂Ω for t ∈ R in order to minimize a cost functional that measures the “magnitude” of the pressure current used and the “magnitude” of the “diﬀerence” between the wave scattered by the active obstacle (Ω; χ) when hit by ui and usD . Problem 2a) “Ghost Obstacle” Optimal Control Problem: Given the incoming acoustic wave ui , the obstacle (Ω; χ), and the ghost obstacle (DG ; χG ), such that DG ∩ Ω = ∅ and DG = ∅, choose a pressure current deﬁned on ∂Ω for t ∈ R in order to minimize a cost functional that measures the “magnitude” of the pressure current used and the “magnitude” of the diﬀerence outside a given set containing Ω and DG between the wave scattered by the active obstacle (Ω; χ) when hit by ui and usG . We concentrate our attention to the mathematical formulation of Problem 2a). The mathematical formulation of Problem 1a) is very similar to that of Problem 2a). Let Ω be a bounded simply connected open set, containing Ω and DG , with locally Lipschitz boundary ∂Ω and let ds∂Ω , ds∂Ω be the surface measures on ∂Ω and ∂Ω respectively, λ ≥ 0, µ ≥ 0 be adimensional constants such that λ + µ = 1, and ς be a Copyright © 2005 Marcel Dekker, Inc.

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positive dimensional constant. We consider the following functional: s s 2 2 dt (1 + χ)λ(u − uG ) ds∂Ω + (1 + χ)µςψ ds∂Ω Fλ,µ, (ψ) = R

∂Ω

∂Ω

(7) and we model the ghost obstacle problem via the following optimal control problem: min Fλ,µ, (ψ), ψ∈C

(8)

subject to conditions (1), (6), (4), (5) where C is the set of the admissible controls. Note that the solution of (1), (6), (4), (5) depends on ψ through the boundary condition (6). The cases µ = 0 and µ = 1 are trivial so that we restrict our analysis to the case 0 < µ < 1. We choose the cost functional (7) since when 0 < µ < 1 we have λ > 0, that is, minimizing (7) we try to make small the diﬀerence between the wave us scattered by the active obstacle (Ω; χ) and the wave usG scattered by the ghost (DG ; χG ) on ∂Ω and to make small the magnitude of the pressure current ψ used. When us − usG is small for (x, t) ∈ ∂Ω × R an observer located in R3 \ Ω observes a ﬁeld scattered by (Ω; χ) that resembles the ﬁeld scattered by the ghost (DG ; χG ), that is, an observer located in R3 \ Ω is induced to believe that the obstacle (Ω; χ) is indeed the ghost obstacle (DG ; χG ). That is, the optimal control problem (8), (1), (6), (4), (5) is a legitimate mathematical model for Problem 2. When we consider the masking problem (i.e., Problem 1) we have D ⊆ Ω and we can choose Ω = Ω. With this choice the optimal control problem for the masking problem is given by problem (8), (1), (6), (4), (5) where in (8) usD replaces usG . Let (r, θ, φ) be the usual spherical coordinate system in R3 , let B be the sphere of center of the origin and radius one, and let ∂B be its boundary. We assume that: (a) the boundary of the obstacle Ω is a starlike surface with respect to the origin, and that Ω can be represented as follows: x ∈ R3 | 0 ≤ r < ξ(ˆ x), x ˆ ∈ ∂B}, Ω = {x = rˆ

(9)

where ξ(ˆ x) > 0, x ˆ ∈ ∂B is a single valued function deﬁned on ∂B that is assumed suﬃciently regular for the manipulations that follow; (b) we deﬁne: x ∈ R3 | 0 ≤ r < ξ(ˆ x) + η, x ˆ ∈ ∂B}, η ≥ 0, Ωη = {x = rˆ then there exists > 0 such that DG ⊂ Ω . Copyright © 2005 Marcel Dekker, Inc.

(10)

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Let vη be the following function: 1 2 ∂ξ ∂ξ 2 2 vη (ˆ x(θ, φ)) = (ξ + η) sin θ + + (ξ + η)2 sin2 θ , ∂θ ∂φ x ∈ ∂B, (11) η ≥ 0, 0 ≤ θ ≤ π, 0 ≤ φ < 2π, x ˆ= x an easy computation gives: ds∂Ω = v0 (ˆ x(θ, φ))dθ dφ, ds∂Ωη = vη (ˆ x(θ, φ))dθ dφ, η ≥ 0, 0 ≤ θ ≤ π, 0 ≤ φ < 2π.

(12)

Let fη (x), x ∈ ∂Ω be the function deﬁned by: fη (x) =

vη (ˆ x(θ, φ) x(θ, φ)) , x ˆ (θ, φ) = ∈ ∂B, x ∈ ∂Ω, v0 (ˆ x(θ, φ)) x(θ, φ) η ≥ 0, 0 ≤ θ ≤ π, 0 ≤ φ < 2π, (13)

thank to assumptions (a) and (b), we can express the measure on ∂Ω via the measure on ∂Ω as follows: x ) = f (x)ds∂Ω (x), x ∈ ∂Ω . ds∂Ω (x + (14) x Using formula (14) in (7) we can rewrite the cost functional Fλ,µ, as follows: dt ds∂Ω (x)(1 + χ) f (x)λ· (15) Fλ,µ, (ψ) = R ∂Ω )

2 x x s s 2 u (x + , t) − uG (x + , t) + µςψ (x, t) . x x Hence, thanks to assumptions (a) and (b), the cost functional Fλ,µ, relative to the ghost obstacle problem can be treated as the cost functionals of the furtivity and masking problem considered in [2], [5].

2.

The Optimality Conditions for the Ghost Obstacle Problem and Some Numerical Experience

To solve the ghost obstacle problem, that is, the optimal control problem (8), (1), (6), (4), (5) we use the Pontryagin maximum principle, see [5] and [8] for more details. Let us introduce the vector space C of admissible controls. Let us denote for example with us |∂Ω the restriction of us to ∂Ω. With a similar Copyright © 2005 Marcel Dekker, Inc.

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notation we denote the restriction of a generic function to a set and let L∞ (∂Ω) be the space of the real functions deﬁned on ∂Ω essentially bounded on ∂Ω. Let L2 (∂Ω×R), L2 (∂Ω ×R), > 0 be the usual spaces of the real functions deﬁned on ∂Ω × R and ∂Ω × R square integrable with respect to the measures ds∂Ω dt and ds∂Ω dt, > 0 respectively. Let us deﬁne the space of admissible controls to be: C = v ∈ L2 (∂Ω × R), v(x, t) ∈ L∞ (∂Ω). t ∈ R Moreover, there exists F : (R3 \ Ω) × R → R s.t. F |∂Ω×R = v, lim F (x, t) = 0, t→+∞

x ∈ R3 \ Ω, and F |∂Ω×R ,

∂F ∂F |∂Ω×R ∈ L2 (∂Ω × R), |∂Ω×R , ∂n ∂t

F satisﬁes equations (1), (4) and the following condition at inﬁnity: 1 ∂F 1 ∂F , r → +∞, t ∈ R . (16) (x, t) − (x, t) = o ∂r c ∂t r The vector space that is supposed to contain the restriction of us to ∂Ω × R is deﬁned similarly (see [1] for further details). We note that us and usG must belong to L2 (∂Ω × R), > 0. In Section X.1 we have shown that the cost functional (7) under the assumptions (a) and (b) can be reduced to the cost functional (15). Using (15), (1), (6), (4), (5) we obtain the following Hamiltonian associated to the ghost obstacle problem (see [8]): ds∂Ω (x) · (17) H(u , ψ, ϕ, t) = −(1 + χ) ∂Ω (2 ' x x , t) − usG (x + , t) + µςψ(x, t)2 + λf (x) us (x + x x s ∂u ds∂Ω (x) ϕ(x, t) cχ (x, t) − (1 + χ)ψ(x, t) − g(x, t) , t ∈ R, ∂n ∂Ω s

where ϕ(x, t), (x, t) ∈ ∂Ω × R is the adjoint variable of us (x, t), (x, t) ∈ ∂Ω × R. We underline that the second adjoint variable, that is,the one that should appear inside the ﬁrst integral in (17), has been chosen equal to −1 identically (see [5] for further details). Note that the Hamiltonian (17) can be adapted to deal with the limit case χ = +∞. Let δH δϕ δH and δu s be the functional derivatives of the Hamiltonian H with respect to ϕ and us respectively, ψ2 ∈ C be the optimal control function associated to the control problem (8), (1), (6), (4), (5), and u ˆs be the corresponding optimal solution of problem (1), (6), (4), (5). Thanks Copyright © 2005 Marcel Dekker, Inc.

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to the Pontryagin maximum principle we can assert that there exists a function ϕ(x, ˆ t), (x, t) ∈ ∂Ω × R that satisﬁes the following conditions:

δH ∂u ˆs 2 ϕ, (x, t) = (ˆ us , ψ, ˆ t), (x, t) ∈ ∂Ω × R, (18) ∂t δϕ

δH ∂ ϕˆ 2 ϕ, (x, t) = − (ˆ us , ψ, ˆ t), (x, t) ∈ ∂Ω × R, (19) ∂t δus 2 ϕ, ˆ t) ≥ H(ˆ us , ψ, ϕ, ˆ t), H(ˆ us , ψ,

ψ ∈ C, t ∈ R,

(20)

2 That where u ˆs is the ﬁeld scattered by the active obstacle when ψ = ψ. s 2 t), ˆ t) takes its maximum value in ψ(x, t) = ψ(x, is, for t ∈ R, H(ˆ u , ·, ϕ, x ∈ ∂Ω, i.e., 2 ϕ, ˆ t) = max H(ˆ us , ψ, ϕ, ˆ t), t ∈ R. H(ˆ us , ψ, ψ∈C

(21)

Extending the adjoint variable ϕˆ from (x, t) ∈ ∂Ω × R to (x, t) ∈ (R3 \ Ω) × R, assuming ϕ| ˆ ∂Ω ∈ C and arguing as in [1], [5], in particular applying the Green’s formula, under the assumptions (a) and (b) and the assumption that the incident wave belongs to a class of functions that guarantees conditions (28), we can show that the necessary ﬁrst order optimality conditions associated to problem (8), (1), (6), (4), (5) are given by the following exterior value problem for a system of two coupled wave equations: ˆ us (x, t) −

1 ∂2u ˆs (x, t) = 0, (x, t) ∈ (R3 \ Ω) × R, c2 ∂t2

(22)

ˆs 1 ∂u 1 ∂u 1 ˆs (x, t) + (x, t) = o( ), r → +∞, t ∈ R, (23) u ˆs (x, t) = O( ), r ∂r c ∂t r s s ∂u ˆ ∂u ˆ (1 + χ) − (x, t) + cχ (x) = g(x, t) − ϕ(x, ˆ t), (x, t) ∈ ∂Ω × R, ∂t ∂n(x) 2µς (24) 2 1 ∂ ϕˆ (25) ϕ(x, ˆ t) − 2 2 (x, t) = 0, (x, t) ∈ (R3 \ Ω) × R, c ∂t 1 ∂ ϕˆ 1 ∂ ϕˆ 1 ϕ(x, ˆ t) = O( ), (x, t) − (x, t) = o( ), r → +∞, t ∈ R, (26) r ∂r c ∂t r ∂ ϕˆ ∂ ϕˆ (x, t) − cχ (x) = −2λ(1 + χ)f (x) · ∂t ∂n(x) x x , t) − usG (x + , t)), (x, t) ∈ ∂Ω × R, (ˆ us (x + x x −

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(27)

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lim u ˆs (x, t) = 0, lim ϕ(x, ˆ t) = 0, x ∈ R3 \ Ω.

t→−∞

t→+∞

(28)

The relation between the adjoint variable ϕ ˆ solution of (22)–(28) and 2 the optimal control ψ of problem (8), (1), (4), (5), (6) is the following one: 2 t) = − 1 ϕ(x, ˆ t), (x, t) ∈ ∂Ω × R, 0 < µ < 1. (29) ψ(x, 2µς Note that equations (24) and (27) correspond to equations (18) and (19) respectively, after some algebraic manipulations and notational changes. Finally, note that in the masking problem when we choose D ⊆ Ω and Ω = Ω, that is, when we choose η = 0 in (10) we have f0 (x) = 1, x ∈ ∂Ω in equation (27) so that, in this case, equations (22)–(28) reduce to the ﬁrst order optimality conditions for the masking problem (see [2]).

Figure 1. Acoustic ﬁeld scattered by the ghost obstacle, by the active obstacle, and by the passive obstacle plotted on the sphere BR2

We conclude this section with a numerical experiment relative to the ghost obstacle problem where the obstacle involved is a cube with χ = 0. The cube has its center of mass in the origin, the largest sphere contained in the cube has radius equal to 0.65, and the facets of the cube are orthogonal to the cartesian coordinate axis that intersects them. Copyright © 2005 Marcel Dekker, Inc.

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The ghost obstacle has the same shape and the same impedance of the obstacle, that is, DG is a cube obtained translating Ω of a vector x∗ = (1.5, 0, 0)T ∈ R3 and χG = 0. We have chosen c = 1, ς = 1, λ = 0.999, µ = 0.001, = 1.5, and ui (x, t) = exp(−(x1 − t)2 ), (x, t) ∈ R3 × R. We have computed the wave scattered by the active obstacle usa when the optimal “pressure current” is used, the wave scattered by the passive obstacle usp , and the wave scattered by the ghost usG on a sphere of center of the origin and radius R2 = 4 such that Ω ⊂ BR2 . Figure 1 shows in a gray map the ﬁeld scattered by the ghost obstacle, the active obstacle, and the passive obstacle on BR2 . Three values of the time variable t are represented; that is: t = 0, t = 2 and t = 3.5. Note that the ﬁeld scattered by the ghost obstacle and by the active obstacle are similar. We note that the incident acoustic wave packet comes from the negative x1 -axis; hence it hits ﬁrst Ω then the ghost obstacle whose center of mass is located in x∗ = (1.5, 0, 0)T . In fact, when t = 0 the passive obstacle generates a measurable scattered wave while the ghost obstacle substantially does not irradiate energy. When t = 3.5 the situation is reversed. In fact, when t = 3.5 the wave packet has already left Ω but is still touching the ghost obstacle.

References [1] L. Fatone, G. Pacelli, M.C. Recchioni, and F. Zirilli. The use of optimal control methods to study two new classes of smart obstacles in time dependent acoustic scattering. Submitted to Journal of Computational Acoustics. [2] L. Fatone, M.C. Recchioni, and F. Zirilli. A masking problem in time dependent acoustic obstacle scattering ARLO (Acoustics Research Letters Online) 5 (2):2530, 2004. [3] L. Fatone, M. C. Recchioni, and F. Zirilli. Some control problems for the Maxwell equations related to furtivity and masking problems in electromagnetic obstacle scattering. In P. Joly, G.C. Cohen, E. Heikkola, and P. Neittaanmaki, editors, Mathematical and Numerical Aspects of Wave Propagation. Waves 2003, pages 189–194, Berlin, 2003. Springer-Verlag. [4] J.-L. Lions. Optimal Control of Systems Governed by Partial Diﬀerential Equations. Springer-Verlag, Berlin, 1969. [5] F. Mariani, M. C. Recchioni, and F. Zirilli. The use of the Pontryagin maximum principle in a furtivity problem in time-dependent acoustic obstacle scattering. Waves in Random Media, 11:549–575, 2001. [6] E. Mecocci, L. Misici, M. C. Recchioni, and F. Zirilli. A new formalism for time dependent wave scattering from a bounded obstacle. Journal of the Acoustical Society of America, 107:1825–1840, 2000. ´ [7] J. Ne˘cas. Les M´ethodes Directes en Th´ eorie des Equations Elliptiques. Masson & Cie., Paris, 1967. [8] L. S. Pontryagin, V. G. Boltiamskii, R. V. Gamkrelidze, and F. Mischenico. Th´eorie Math´ematique des Processus Optimaux. Editions Mir, Moscow, 1974.

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CHAPTER XI LOCAL NULL CONTROLLABILITY IN A STATE CONSTRAINED THERMOELASTIC CONTACT PROBLEM Irina F. Sivergina Department of Science and Mathematics Kettering University Flint, MI, USA

Michael P. Polis School of Engineering and Computer Science Oakland University Rochester, MI, USA

Abstract

We study the controllability properties of a nonlinear parabolic problem that models the temperature evolution of a one-dimensional thermoelastic rod that may come into contact with a rigid obstacle. Basically the system dynamics is described by a one-dimensional nonlocal heat equation with a nonlinear and nonlocal boundary condition of Newmann type. We focus on the control problem and treat the case when the control is distributed over the whole space domain. In this case, the system is proved to be locally null controllable. This result is then used to prove the existence of a control bringing the system to zero without contact with the obstacle, provided the initial state is small enough. The proofs are based on a transposition technique and on using a ﬁxed point theorem.

Keywords: Null controllability, state constrained control problem, thermoelastic system, quasistatic approximation, nonlocal parabolic equation, nonlinear boundary conditions

Introduction The system to be analyzed is an isotropic rod 0 ≤ x ≤ l that undergoes a motion in which the displacement vector is parallel to the x-axis. The displacement u(x, t) and the temperature θ(x, t) are functions of the coordinate x and the time t ≥ 0 only. Consider the equations of dynamic

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CONTROL AND BOUNDARY ANALYSIS

linear thermoelasticity [5, 18] κθxx = cθt + (3λ + 2µ)αθ0 uxt , (λ + 2µ)uxx = (3λ + 2µ)αθx + utt , where θ0 is the uniform reference temperature, is the mass density, λ and µ are the elastic moduli, c is the speciﬁc heat per unit mass, α is the coeﬃcient of thermal expansion, and κ is the thermal conductivity. The subscripts containing t or x denote partial derivatives. Following the usual practice, we change variables to simplify the notation x → 3

λ+2µ θ−θ0 kt u t → cl 2, θ → θ0 , u → l cθ0 , and get the system above in a nondimensional form: θxx = θt + auxt , butt − uxx = −aθx where 2 (3λ+2µ)2 k2 , b = c2 (λ+2µ)l a2 = θ0 αc(λ+2µ) 2 . These equations are considered in the domain ΩT = (0, 1) × (0, T ). To establish the boundary conditions, we consider the case when the rod is situated between two walls that are kept at diﬀerent temperatures. One end of the rod is permanently attached to a wall, while the other end is free to expand or contract. The expansion of the rod resulting from the evolution of the temperature and the stresses is limited by the existence of the other wall. For the displacement at the free edge, we choose the Signorini boundary conditions [7]: u(1, t) ≤ g, ux (1, t) ≤ aθ(1, t), [u(1, t) − g][ux (1, t) − aθ(1, t)] = 0 where the constant g is the nominal gap size between the wall and the free edge in the reference conﬁguration. These conditions are completed by θ(0, t) = 0, −θx (1, t) = kθ(1, t), u(0, t) = 0, and all boundary conditions hold for 0 < t < T. When it is reasonable to expect small acceleration in the process, the term butt is treated as negligible. This assumption allows the problem to be decoupled and formulated in terms of the temperature only; for the details, see [3, 6]. This results in the following initial-boundary value problem considered in this paper: 1 d (1 + a2 )θt − θxx = a max a θ(ξ, t)dξ − g, 0 +v(x, t), in ΩT , (1) dt x l,

0

θ(0, t) = 0, in (0, T ), (2)

1

−θx (1, t) = k[g − a

θ(ξ, t) dξ]θ(1, t), in (0, T ), (3) 0

θ(x, 0) = θ0 (x), in Ω. (4) Here k = k[s] is a function of the scalar argument s ∈ R. We need to clarify a sense, in which the solution is treated for the system (1)– (4). Let W22,1 (ΩT ) be the Sobolev space, which consists of all L2 (ΩT )Copyright © 2005 Marcel Dekker, Inc.

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Local Null Controllability

summable functions that possess generalized L2 (ΩT )-summable secondorder space and ﬁrst-order time derivatives, a norm for which is given

2 ) dxdt. A strong solution of (1)– by θ2 2,1 = ΩT (θ 2 + θt2 + θx2 + θxx W2 (ΩT )

(4) is deﬁned as a function from W22,1 (ΩT ), which satisﬁes (1) almost everywhere in ΩT , (2) and (3) almost everywhere in (0, T ), and for which (4) holds for a trace θ(·, 0) ∈ L2 (Ω). Referring to [2, Theorem 2.1] and [3, Theorem 3.1], we have the following result for a homogeneous system: given v = 0, θ0 ∈ H 1 (0, 1) 1 (R), there exists a unique with θ0 (0) = 0, and a nonnegative k ∈ W∞ strong solution to (1)–(4). The nonhomogeneous case was considered in [1], but the existence and uniqueness theorems were formulated for v(x, t) being suﬃciently smooth. The latter result, based on a special transformation of the original equation, is of no use for the problem considered here, since v(·, t) will be treated as a control function in the system (1)–(4). We are interested in the local null controllability of this system.

Definition 1 The system (1)–(4) is said to be locally null controllable on the time interval [0, T ], if there is γ > 0 such that for each θ0 ∈ H 1 (Ω), θ0 H 1 (Ω) < γ, θ0 (0) = 0, there exists a control v ∈ L2 (ΩT ), for which (i) there exists a unique solution θ ∈ W22,1 (ΩT ) to the system (1)–(4), and (ii) θ(·, T ) = 0. Definition 2 The system (1)–(4) is said to be globally null controllable on the time interval [0, T ], if for any θ0 ∈ H 1 (Ω) with θ0 (0) = 0, there exists a control v ∈ L2 (ΩT ) such that (i) and (ii) from Deﬁnition 1 hold. Note that if for a v ∈ L2 (ΩT ), there exists a strong solution of system (1)–(4), it is unique. This statement follows from the uniqueness theorem proved in [2, Theorem 2.1]. In this paper, we prove the following results. 1 (R) be nonnegative. Then for any T > 0, Theorem 3 Let k ∈ W∞ system (1)–(4) is locally null controllable on [0, T ]. 1 (R) be nonnegative and the constant η in Theorem 4 Let k ∈ W∞ ess sup |k (s)| ≤ η be small enough. Then for any T > 0, the system s

(1)–(4) is globally null controllable on [0, T ]. Next we focus on the behavior of the system when the free end of the rod is constrained to avoid contact with the obstacle, that is, when the following state constraint holds 1 θ(ξ, t)dξ − g < 0, t ∈ [0, T ].

a 0

Copyright © 2005 Marcel Dekker, Inc.

(5)

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CONTROL AND BOUNDARY ANALYSIS

Definition 5 The system (1)–(5) is said to be locally state constrained null controllable on the time interval [0, T ] if there exists a constant γ1 > 0 such that for any θ0 (x) yielding θ0 H 1 (Ω) < γ1 and θ0 (0) = 0, there exists a control v ∈ L2 (ΩT ) such that in addition to (i) and (ii) in Deﬁnition 1, (5) holds. The following theorem gives our result on handling the local state constrained null controllability for system (1)–(5).

Theorem 6 Under the assumption of Theorem 3, for any T > 0, the system (1)–(5) is locally state constrained null controllable on the time interval [0, T ]. This theorem can be interpreted in the following way: if the free end of the rod is not in contact with the obstacle at the initial time instant, and if θ0 H 1 (Ω) is small enough, then the temperature of the rod may be brought to zero by the desired time instant T without contact with the obstacle at all. There is a large body of literature devoted to controllability problems for linear and semilinear parabolic systems. In [4], [8], [10], [13], [14], [15], [19], the approximate null controllability and null controllability were studied. The typical assumption for establishing the controllability property for semilinear equations is a Lipschitz nonlinearity. For example, in [12], the boundary controllablity is proved under this condition. In [19], the local null controllability is established for semilinear parabolic systems. The system (1)–(4) diﬀers from those treated previously since, ﬁrst, there is a nonlinear boundary condition, and, second, equation (1) contains a nonlocal term describing a stress in the system. Thus, existing techniques do not cover the system considered here. Nevertheless, as stated in Theorem 3 and proved below, the system (1)–(4) is locally null controllable. We note also that from equation (1), it follows that if the initial condition is small, then the diﬀerential equation governing the system is linear on an initial portion of the time interval. The system as a whole however, is not linear because of the boundary condition (3) and the possibility 1

that a state fails to satisfy the inequality max a θ(ξ, t)dξ − g, 0 > 0 0

beginning at some time instant t. This observation raises the question of the state constrained null controllability when the contact of the free end of the rod, being permitted, nevertheless does not happen at all. Copyright © 2005 Marcel Dekker, Inc.

135

Local Null Controllability

1.

Control of the Linear System

First of all we observe that it suﬃces for our purpose to prove null controllability of the system (1 + a2 )θ˜t − θ˜xx = v˜(x, t), in ΩT , ˜ t) = 0, in (0, T ), θ(0, 1 ˜ t) dξ]θ(1, ˜ t), θ(ξ, in (0, T ), −θ˜x (1, t) = k[g − a

(6) (7) (8)

0

˜ 0) = θ0 (x), θ(x,

in Ω.

(9)

˜ t) of If there exists a control v˜ ∈ L2 (ΩT ) such that the solution θ(x, 2,1 ˜ T ) = 0, then the system (6)–(9) belongs to W2 (ΩT ) and yields θ(x, control 1 d ˜ t)dξ − g, 0 (10) v(x, t) = v˜(x, t) − a max a θ(ξ, dt 0

is also in L2 (ΩT ) and solves the null controllability problem for the original system (1)–(4). With this observation, we will now focus on studying the null controllability of (6)–(9). For simplicity we use θ and v below instead of θ˜ and v˜, respectively, in (6)–(9). We start by proving the following auxiliary result.

Theorem 7 Let in the system (1 + a2 )θt = θxx + v, in ΩT , θ(0, t) = 0, in (0, T ), −θx (1, t) = k[t]θ(1, t), in (0, T ), θ(x, 0) = θ0 (x), in Ω,

(11)

k[·] ∈ H 1 (0, T ) be nonnegative and ﬁxed. Then for any θ0 ∈ L2 (Ω), there exists the control v ∈ L2 (ΩT ) such that θ(·, T ) = 0. To prove this theorem, we need one inequality which we formulate without proof. This inequality is a generalization of a well-known estimate proved in [17, p. 81].

Lemma 8 Let in (11), k[·] ∈ H 1 (0, T ) be nonnegative and v = 0. Then there exists a constant R > 0, which is independent of k[·], such that for any θ0 ∈ L2 (Ω), solution of system (11) yields the inequality θ(·, T )L2 (Ω) ≤ RθL2 (ΩT ) . Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

Proof of Theorem 7. Consider a set Z in the space L2 (ΩT ) that consists of the solutions of the system (1 + a2 )zt + zxx = 0, in ΩT , z(0, t) = 0, in (0, T ), −zx (1, t) = k[t] z(1, t), in (0, T ), z(x, T ) = z0 , in Ω

(12)

for a z0 ∈ H 1 (Ω) with z0 (0) = 0. We deﬁne a linear functional on

1 Z by setting l(z) = − 0 z(x, 0)θ0 (x) dx. According to Lemma 8, this functional is continuous on Z and clearly sup lZ ≤ z(x, 0)θ0 (x) dx ≤ Rθ0 L2 (Ω) . z(x,0) L2 (Ω) ≤R

Ω

In the last estimate, R is the constant found in Lemma 8. Applying the Hahn-Banach theorem, we conclude that l can be extended over 2 2 all

L (ΩT ), and there exists a function v˜ ∈ L (ΩT ) such that l(z) = v (x, t) dxdt and lZ = lL2 (ΩT ) = ˜ v L2 (ΩT ) . Hence, ΩT z(x, t)˜ ˜ v L2 (ΩT ) ≤ Rθ0 L2 (Ω) .

(13)

Let θ(x, t) be the solution to (11) with v = v˜, and let z satisfy (12). We have ((1 + a2 )θt − θxx − v)z + ((1 + a2 )zt + zxx )θ dxdt 0= ΩT 1 2 θ(x, T )z0 (x) dx = (1 + a ) 0

for any z0 ∈ H 1 (0, 1), and, hence, θ(·, T ) = 0.

2.

Nonlinear Case: Application of a Fixed Point Theorem

The next step in proving our controllability theorems is in appropriately applying a version of the Kakutani ﬁxed point theorem for topological spaces, which is formulated as follows [9, Theorem 1]: Let L be a locally convex topological linear space and K a compact convex subset in L. Let (K) be the family of all closed convex (nonempty) subsets of K. Then for any upper semicontinuous point-to-set transformation f from K into (K), there exists a point x0 ∈ K such that x0 ∈ F[x0 ]. Here upper semicontinuity means that lim xn = x0 , yn ∈ F[xn ], lim yn = y0 imply y0 ∈ F[x0 ]. Copyright © 2005 Marcel Dekker, Inc.

137

Local Null Controllability

In the next subsection we deﬁne a set in W22,1 (ΩT ) and a point-to-set mapping F in an appropriate way. In subsection 2.2 we prove that this map is upper semicontinuous. And, ﬁnally, in subsection 2.3 we give some details of applying the Katutani theorem to our problem.

2.1

The Imbedding Result

Consider the set K deﬁned as K = {θ ∈ W22,1 (ΩT ) : θ(0, t) = 0, θ(x, 0) = θ0 }

(14)

where θ0 is the same as in the initial condition in (11). For ﬁxed θ ∈ K and v ∈ L2 (ΩT ), we consider a solution w to the system (1 + a2 )wt − wxx = v(x, t),

in ΩT ,

w(0, t) = 0,

in (0, T ),

−wx (1, t) = k[g − a θ(ξ, t)dξ] w(1, t),

in (0, T ),

Ω

w(x, 0) = θ0 (x), in Ω. It follows from the standard theory of linear parabolic equations, see, e.g. [17, p. 33], that this system has a unique solution from W22,1 (ΩT ) and thereby w ∈ K. As proved in section 1, there is a control v, vL2 (ΩT ) ≤ Rθ0 L2 (Ω) , such that w(·, T ) = 0. Having ﬁxed ρ > 0, we denote V = {v ∈ L2 (Ω) : vL2 (ΩT ) ≤ Rθ0 L2 (Ω) + ρ, w(·, T ; v) = 0}. By w(x, t; v), we denote the solution of system (15) corresponding to the control v. V is a bounded closed convex subset in L2 (ΩT ). Now we deﬁne the point-to-set mapping F : K θ → F[θ] ⊂ K by setting F[θ] = {w(·, ·; v), v ∈ V }.

Lemma 9 F[θ] is nonempty weakly closed in W22,1 (ΩT ) for any θ ∈ K. Proof. Suppose that {wn } is a sequence in F[θ] that converges weakly in W22,1 (ΩT ) to a function w. We have to prove that w ∈ F[θ]. By the deﬁnition of the mapping F, there are the controls vn ∈ V such that (1 + a2 )wnt − wnxx = vn (x, t),

in ΩT ,

wn (0, t) = 0, in (0, T ), −wnx (1, t) = k[g − a θ(ξ, t)dξ] wn (1, t), Ω

wn (x, 0) = θ0 (x), wn (x, T ) = 0, in Ω. Copyright © 2005 Marcel Dekker, Inc.

in (0, T ),

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CONTROL AND BOUNDARY ANALYSIS

Since V is a weak compact in L2 (ΩT ), we may assume that the sequence vn is weakly convergent to a v ∈ V . Taking the limit as n → ∞ in (15), from the Sobolev embedding theorem and the weak continuity of the corresponding trace operators in W22,1 (ΩT ), we obtain that (1 + a2 )wt − wxx = v(x, t),

in ΩT , (15)

w(0, t) = 0, in (0, T ), w(x, 0) = θ0 (x), w(x, T ) = 0, in Ω and that wnx (1, t) → wx (1, t), wn (1, t) → w(1, t) weakly in L2 (0, T ). Since k[g − a

1

θ(ξ, t)dξ] is continuous with respect to t, we conclude

1 that −wx (1, t) = k[g − a 0 θ(ξ, t)dξ] w(1, t). The proof of Lemma 9 is complete. The next lemma is to state an imbedding result for the map F. We want to prove that there is a subset in K, denoted by Kδ , such that F[θ] ⊆ Kδ for any θ ∈ Kδ . An existence of such a set Kδ is proved under two conditions, that will bring us respectively either to local or to globall null controllability for the system (6)–(9). 0

Lemma 10 (i) There exist γ > 0 and δ > 0 such that θW 2,1 (ΩT ) ≤ δ implies 2

sup wW 2,1 (ΩT ) ≤ δ

w∈F [θ]

2

(16)

whenever θ0 H 1 (Ω) < γ. (ii) There exist η > 0 and δ > 0 such (16) holds if kW2∞ (R) < η. Proof. We square both sides of the ﬁrst equation in system (15) and integrate by parts over Ωτ = Ω × (0, τ ) for τ ∈ (0, T ] to obtain 2 2 2 2 2 (1 + a ) wt + wxx dxdt = 2(1 + a ) wt wxx dxdt + v 2 dxdt Ωτ Ωτ Ωτ 2 2 = −(1 + a )k[g − a θ(ξ, τ )dξ] w (1, τ ) Ω 2 +(1 + a )k[g − a θ0 (ξ) dξ] w2 (1, 0) Ω τ 2 +(1 + a ) k [g − a θ(ξ, t)dξ] θt (ξ, t) dξ w2 (1, t) dt 0 Ω Ω 2 2 2 2 wx (x, τ ) dx + (1 + a ) wx (x, 0) dx + v 2 dxdt. −(1 + a ) Ω

Ω

Ωτ

(17) Copyright © 2005 Marcel Dekker, Inc.

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Local Null Controllability

1 (R), there exists a constant k such that |k [s]| ≤ k for Since k ∈ W∞ 1 1 almost all s. Then, 2 2 2 2 2 (1 + a ) wt + wxx dxdt + (1 + a ) wx2 (x, τ ) dx Ωτ Ω 2 2 +(1 + a )k[g − a θ(ξ, τ )dξ] w (1, τ ) Ω

≤ (1 + a )k[g − a 2

τ

θ0 (ξ)

dξ] θ02 (1)

|θt (ξ, t)| dξ

+(1 + a )k1

(θ0 )2x dx

+ (1 + a ) Ω

2

Ω

2

0

2

2

v 2 (x, t) dxdt.

w (1, t) dt +

Ω

Ωτ

(18) Using the trace theorems for the space W22,1 (Ωτ ) (see, e.g., [16, p. 21]) ¯ for Ω ⊂ R1 [16, and the continuity of the injection H 1 (Ω) ⊂ C 0 (Ω) p. 45], we can estimate the third term on the right-hand side of (18) 2

τ

θ2 2,1 . as follows 0 Ω |θt (ξ, t)| dξ w2 (1, t) dt ≤ C12 w2 2,1 W2 (Ωτ )

Returning to (18) and using (13), we obtain 2 (1 + a2 )2 wt2 + wxx dxdt + (1 + a2 ) wx2 (x, τ ) dx Ωτ

Ω

1

+(1 + a2 )k[g − a

W2 (Ωτ )

θ(ξ, τ )dξ] w2 (1, τ ) 0

(19)

≤ (1 + a2 )k[g − a

θ0 (ξ) dξ] θ02 (1) + (1 + a2 )

(θ0 )2x dx Ω

Ω

+(1 + a2 )k1 C12 w2W 2,1 (Ω ) θ2W 2,1 (Ω 2

τ

2

τ)

+ (Rθ0 L2 (Ω) + ρ)2 .

In particular, for τ = T , we have 2 (1 + a2 )2 wt2 + wxx dxdt + (1 + a2 ) wx2 (x, T ) dx ΩT Ω 2 2 +(1 + a )k[(g − a θ(ξ, T )dξ] w (1, T ) Ω

(20)

≤ (1 + a2 )k[g − a

θ0 (ξ) dξ] θ02 (1) + (1 + a2 ) Ω

+(1 + a2 )k1 C12 w2W 2,1 (Ω ) θ2W 2,1 (Ω 2

T

2

1(θ0 )2x dx Ω

T

+ (Rθ0 L2 (Ω) + ρ)2 . )

After integrating both sides of the inequality (19) with respect to τ over (0, T ), we obtain the following inequality that holds for any , Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

0 < < T: T − 2 (1 + a2 )2 wt2 + wxx dxdt + Ω

0

ΩT

1

≤ T (1 + a )k[g − a 2

2

+T (1 + a

wx2 (x, t) dxdt

dξ] θ02 (1)

θ0 (ξ)

(θ0 )2x dx

+ T (1 + a )

0 )k1 C12 w2W 2,1 (Ω ) θ2W 2,1 (Ω ) T T 2

2

2

Ω

+ T (Rθ0 L2 (Ω) + ρ)2 .

Adding respective sides of (20) and (21), we get 2 (1 + a2 )2 wt2 + wx2 + wxx dxdt ΩT 1 ≤ (T + 1)(1 + a2 ) k[g − a θ0 (ξ) dξ] θ02 (1) + (θ0 )2x dx Ω

0

+(T + 1)(1 + a2 )k1 C12 w2W 2,1 (Ω ) θ2W 2,1 (Ω 2

2

T

(21)

(22)

T)

+(T + 1)(Rθ0 L2 (Ω) + ρ) . 2

Since for w(0, t) ≡ 0, we have 2 w (x, t) dxdt ≤

ΩT

wx2 (x, t) dxdt, ΩT

2 and, hence, ΩT wt2 + wx2 + wxx dxdt is equivalent to the standard norm

in W22,1 (ΩT ). This will result in the following inequality obtained from (22): 2 1 2 2 wW 2,1 (Ω ) ≤ wt + wx2 + wxx dxdt T 2 2 Ω T 1 ≤ (T + 1)(1 + a2 ) k[g − a θ0 (ξ) dξ] θ02 (1) + (θ0 )2x dx (23) Ω 0 2 (wt2 + wx2 + wxx ) dxdt +2(T + 1)(1 + a2 )k1 C12 θ2W 2,1 (Ω ) 2

T

ΩT

+(T + 1)(Rθ0 L2 (Ω) + ρ) ≤ αθ2W 2,1 (Ω ) w2W 2,1 (Ω 2

2

T

2

T)

+β

where the positive constants α, β are α = 2(T + 1)(1 + a2 )k1 C12 ,

β = (T + 1)(1 + a2 )k[g − a θ0 (ξ) dξ]θ02 (1) Ω 2 2 +(T + 1)(1 + a ) (θ0 )x dx + (T + 1)(Rθ0 L2 (Ω) + ρ)2 . Ω

Copyright © 2005 Marcel Dekker, Inc.

141

Local Null Controllability

Now, we assume that θW 2,1 (ΩT ) ≤ δ. Then, (23) yields 2

1 ( − αδ2 )w2W 2,1 (Ω ) ≤ β. T 2 2

(24)

To prove (i), we choose δ to be small enough to guarantee that 12 − αδ2 > 0. Since β is basically determined by the norm of θ0 in H 1 (Ω) and by ρ, we can take γ > 0 and ρ > 0 so small that for any θ0 yielding ≤ θ0 2H 1 (Ω) ≤ γ, we will have β ≤ 12 − αδ2 δ2 , and, hence, w2 2,1 W2 (ΩT )

δ2 . For proving (ii), we see that the value α basically depends on k1 . Then if δ is big enough so that δ2 ≥ 4β, and, hence, there are θ yielding 1 θW 2,1 (ΩT ) ≤ δ, while k1 is so small that k1 ≤ η ≡ 8(T +1)(1+a 2 )C 2 δ 2 , we 2

will have β/( 12 − αδ2 ) ≤ δ2 , and this again leads to w2

1

W22,1 (ΩT )

≤ δ2 .

The proof of Lemma 10 is complete. Now, take δ > 0 found from either the conditions (i) or (ii) if we focus respectively on the local null controllability, or on the global null controllability. Consider the subset Kδ = {w ∈ K : wW 2,1 (ΩT ) ≤ δ}. 2 What is proved in Lemma 10 means that F[θ] ⊆ Kδ for all θ ∈ Kδ . Our next goal is to prove that F is upper semicontinuous in the weak topology of W22,1 (ΩT ) and this is done in the next subsection.

2.2

The Upper Semicontinuity of F

Lemma 11 The operator F is upper semicontinuous in the weak topology of W22,1 (ΩT ). Proof. Suppose that {θn } is a sequence in K that converges weakly in W22,1 (ΩT ) to a function θ. Consider an arbitrary sequence {wn }, wn ∈ F[θn ], that converges weakly in W22,1 (ΩT ) to a function w. We want to prove that w ∈ F[θ], i.e. that there exists v ∈ V such that w yields (15) and thereby w(x, T ) = 0. The proof is basically the same as that used in the proof of Lemma 9. We start by deﬁning the sequence {vn }, vn ∈ V such that (1 + a2 )wnt − wnxx = vn (x, t),

in ΩT ,

wn (0, t) = 0, in (0, T ), −wnx (1, t) = k[g − a θn (ξ, t)dξ] wn (1, t), Ω

wn (x, 0) = θ0 (x), wn (x, T ) = 0, in Ω. Denote its weak limit by v and observe that v ∈ V . Copyright © 2005 Marcel Dekker, Inc.

in (0, T ),

142

CONTROL AND BOUNDARY ANALYSIS

Taking the limit in (25) as n tends

to inﬁnity, we get equations (15). The equation −wx (1, t) = k[g − a θ(ξ, t)dξ] w(1, t) in L2 (0, T ) follows Ω

from Lemma 12, whose proof we omit.

Lemma 12 Suppose {fn }, {gn } are sequences of functions in L2 (0, T ), such that fn → f in L∞ (0, T ) and gn → g weakly in L2 (0, T ). Then fn gn → f g weakly in L2 (0, T ). Summarizing what we have proved, we see that for a v ∈ V , the equations yield (1 + a2 )wt − wxx = v(x, t), in ΩT , w(0, t) = 0, in (0,T ), −wx (1, t) = k[g − a

θ(ξ, t) dξ]w(1, t), Ω

w(x, 0) = θ0 (x), w(·, T ) = 0,

in (0, T ),

in Ω.

This means that w ∈ F[θ] is what we need. The proof of Lemma 12 is complete.

2.3

The Application of the Kakutani Theorem

Now, we can apply the Kakutani ﬁxed point theorem. Let us choose δ according to either the conditions (i) or (ii) in Lemma 10. The set Kδ is nonempty, convex, and weakly compact in W22,1 (ΩT ). The sets F[θ] are convex and weakly closed in W22,1 (ΩT ) for any θ ∈ Kδ . And, ﬁnally, the mapping F is upper semicontinuous in the weak topology of W22,1 (ΩT ). According to the Kakutani theorem, there exists θ ∈ Kδ such that θ ∈ F[θ]. This means that for this θ, there exists v ∈ V such that (1 + a2 )θt − θxx = v(x, t), in ΩT , θ(0, t) = 0,

in (0, T ), −θx (1, t) = k[g − a θ(ξ, t) dξ] θ(1, t),

in (0, T ),

Ω

θ(x, 0) = θ0 (x), θ(·, T ) = 0,

in Ω.

If δ has been chosen according to the conditions (i) in Lemma 10, this means the local null controllability of system (6)–(9). The choice of δ according to the conditions (ii) in Lemma 10 leads to the global null controllability when the derivative of k is small enough. So, the problems of local and global null controllability for system (6)–(9) along with the same problems for system (1)–(4) are solved. Copyright © 2005 Marcel Dekker, Inc.

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Local Null Controllability

3.

The State Constrained Null Control Problem

In this section, we prove Theorem 6. Our proof is based on an estimate for the solution of the parabolic equations. According to Theorem 3, there exists γ > 0 so small that for θ0 yielding θ0 H 1 (Ω) < γ, there is a control v˜ ∈ L2 (ΩT ) such that for ˜ T ) = 0. Hence, the the solution of the system (6)–(9) we have θ(x, control (10) solves the null controllability problem for the system (1)–(4). Applying the results of section 3, we may assume that ˜ v L2 (ΩT ) ≤ 2Rγ. Now we will prove that there is γ1 ≤ γ such that for all T ∈ [0, T ], 1 ˜ t) dξ − g < 0 θ(ξ,

a 0

if θ0 H 1 (Ω) < γ1 . This will mean that the controls v˜ and (10) are identical and so are the solutions θ˜ and θ for (6)–(9) and (1)–(4), respectively. We multiply (6) by θ˜ and integrate the resulting equation with respect to x over Ω: 1 1 1 (1 + a2 ) v˜2 (ξ, τ ) dξ. θ˜t2 (x, τ ) dx ≤ θ˜2 (ξ, τ ) dξ + 0

0

0

Applying Gronwall’s lemma to the last inequality yields 1 4R2 θ0 2L2 (Ω) T . θ˜2 (ξ, t) dξ ≤ e 1+a2 θ0 2L2 (Ω) + 1 + a2 0 1 ( ' T 4R2 γ 2 2 < g. Then for We choose 0 < γ1 < γ so that ae 2(1+a2 ) × γ12 + 1+a21

1 ˜ t) dξ − g < 0 holds. This means all t ∈ [0, T ], the inequality a 0 θ(ξ, that v˜(x, t) and v(x, t) are equal, and so are θ˜ and θ, and this completes the proof of Theorem 6.

4.

Conclusion

In this paper, we have obtained local and global null controllability results for a thermoelastic contact problem described by a nonlinear, nonlocal parabolic equation. The proofs are based on applying the Kakutani ﬁxed point theorem to a point-to-set map in a topological space. The local null controllability result is then used to prove the local null controllability of the state constrained contact problem, where no contact is permitted.

Acknowledgments The authors thank Professor I. Lasiecka for useful discussions. Copyright © 2005 Marcel Dekker, Inc.

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References [1] W. Allegretto, J.R. Cannon, and Y. Lin. A parabolic integro-diﬀerential equation arising from thermoelastic contact. Discrete and Continuous Dynamical Systems, 3:217–234, 1997. [2] K.T. Andrews, A. Mikelich, P. Shi, M. Shillor, and S. Wright. One-dimensional thermoelastic contact with a stress-dependent radiation condition. SIAM J. Math. Anal., 23:1393–1416, 1992. [3] K.T. Andrews, P. Shi, M. Shillor, and S. Wright. Thermoelastic contact with Barber’s heat exchange condition. J. Appl. Math. Optim., 28:11–48, 1993. [4] O. Bodart and C. Farbe. Controls insensitizing the norm of the solution of a semilinear heat equation. J. Appl. Math. Appl., 195:658–683, 1995. [5] B.A. Boley and J.H. Weiner. Theory of Thermal Stress. Wiley, New York, 1960. [6] W.A. Day. Justiﬁcation of the uncoupled and quasistatic approximation in a problem of dynamic thermoelasticity. Arch. Rat. Mech. Anal., 80:135–158, 1988. [7] G. Duvant and J.-L. Lions. Inequalities in Mechanics and Physics. SpringerVerlag, Berlin, 1976. [8] M. Eller, I. Lasiecka, and R. Triggiani. Exact boundary controllability of thermoelastic plates with variable coeﬃcients. Semigroups of operators: theory and applications, vol. 42 of Progress in Nonlinear Diﬀerential Equations and Applications. Birkh¨ auser, Basel, 2000. [9] B.K. Fan. Fixed point and minimax theorems in locally convex topological linear spaces. Proc NSA, 38:121–126, 1952. [10] C. Farbe, J.P. Puel, and E. Zuazua. Approximate controllability for the semilinear heat equation. Proc. Royal Soc Edinburgh, Sect A, 125:31–61, 1995. [11] V.P. Fonf, J. Lindenstrauss, and R.R. Phelps. Inﬁnite dimensional convexity, 2001. In: Jonson, W.B. and Lindenstrauss, J. (ed), Handbook of the Geometry of Bahach Spaces, vol. 1, Elsevier, Amsterdam, 2001. [12] A.V. Fursikov and O.Y. Imanuvilov. Controllability of evolution equations. Lectures at the Seoul National University, Seoul, 1996. [13] S.W. Hansen. Boundary control of a one-dimensional linear thermoelastic rod. SIAM J. Control Optim., 32:1052–1074, 1994. [14] G. Lebeau and L. Robbiano. Controle exact de l’equation de la chaleur. Commun. Partial Diﬀ. Equations, 20:335–356, 1995. [15] G. Lebeau and E. Zuazua. Null-controllability of a system of linear thermoelasticity. Arch. Rational Mech. Anal., 141:297–329, 1998. [16] J.-L. Lions and E. Magenes. Non-Homogeneous Boundary Value Problems and Applications, Vol. I. Springer-Verlag, New York, 1972. [17] J.-L. Lions and E. Magenes. Non-Homogeneous Boundary Value Problems and Applications, Vol. II. Springer-Verlag, New York, 1972. [18] M. Slemrod. Global existence, uniqueness and asymptotic stability of classical smoothy solutions in one-dimensional nonlinear thermoelasticity. Arch. Rational Mech. Anal., 76:97–133, 1981. [19] X. Zhang. Exact controllability of semilinear evolution systems and its applications. J. Optim. Theory Appl., 107:415–432, 2000.

Copyright © 2005 Marcel Dekker, Inc.

CHAPTER XII ON SENSITIVITY OF OPTIMAL SOLUTIONS TO CONTROL PROBLEMS FOR HYPERBOLIC HEMIVARIATIONAL INEQUALITIES Zdzislaw Denkowski∗ Faculty of Mathematics and Computer Science Institute of Computer Science Jagiellonian University Krakow, Poland

Stanislaw Mig´ orski Faculty of Mathematics and Computer Science Institute of Computer Science Jagiellonian University Krakow, Poland

Abstract

The results on sensitivity of solution sets to special classes of second order hemivariational inequalities (HVIs) are formulated and then applied to get variational convergence of optimal solutions of control problems for systems described by such HVIs.

Keywords: Control problems, hemivariational inequality, sensitivity, multivalued G, P G, and Γ-convergences

Introduction This note is, in a sense, a continuation of the papers of Denkowski [5] and Denkowski and Mig´ orski [7] concerning the sensitivity of optimal solutions to control problems for systems governed by the stationary (elliptic) and the ﬁrst order evolution (parabolic) hemivariational inequalities (HVIs). Here our main results provide suﬃcient conditions for the variational convergence of optimal solutions of perturbed control ∗ Supported

in part by the State Committee for Scientiﬁc Research of the Republic of Poland (KBN) under research grants no. 2 P03A 003 25 and 4 T07A 027 26.

Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

problems for systems governed by second order evolution (hyperbolic) HVIs. We consider two special classes of such HVIs: with multivalued and time independent, and single-valued and time dependent damping. Variational convergence means the convergence of optimal values, as well as minimizers. In our approach, it follows from general theory of the sequential Γ-convergence (see, e.g., [4, 1, 3, 10]), which is presented in detail in [7] and is important for our purpose. So here we omit the relevant deﬁnitions and theorems. Two basic properties needed in this approach are (i) Kuratowski convergence of solution sets of state relations, here it means the sensitivity results (K-usc and K-lsc) for solution sets of hyperbolic HVIs (ii) Some complementary Γ-convergence of cost functionals (i.e. their uniform convergence with respect to state variable and Γ-convergence with respect to control variable) The nonemptiness of solution sets for HVIs considered in this note follows from the general result on hyperbolic HVIs presented at IFIP conference (see [6]). The paper is organized as follows. In Section 1 we present the class of problems we deal with. Next in Section 3, after preliminaries of Section 2, we formulate theorems concerning the Kuratowski upper and lower semicontinuity (at k = +∞) of the sequence of solution sets of the controlled HVIs. In Section 4 ﬁrst we give some regularity and convergence results for cost functionals and ﬁnally the main sensitivity theorem is presented.

1.

Class of Problems

We are interested in the “sensitivity property” for the following class of control problems (CP ): control

8 9: ; minimize F(u, y 0 , y 1 , y) := F (1) (y) + F (2) (u) + F (3) (y 0 ) + F (4) (y 1 ) , where control u ˜ = (u, y 0 , y 1 ) ∈ U × V × H and state y ∈ Y are subject to the second order diﬀerential inclusion (called also HVI) y (t) + A(t, y (t)) + ∂J(y(t)) f (t) + (Cu)(t) a.e. in Q (HV I) y(0) = y0 , y (0) = y1 in Ω. Above y(t), y (t), y (t) denote, respectively, function y(t, x) and its distributional derivatives with respect to time, ∂J stands for the Clarke Copyright © 2005 Marcel Dekker, Inc.

On Sensitivity in Control Problems for Hyperbolic HVIs

147

subdiﬀerential of superpotential J, Q = (0, T ) × Ω, Ω ⊂ RN and C is a controller operator. Two classes of damping operator A will be considered: A(t, y (t)) = −diva(t, x, Dy (t, x)) (1) with single-valued, time dependent a ∈ M (the Svanstedt class of monotone functions) and A(t, y (t)) = A(y (t)) = {−divη : η(t, x) ∈ a(x, Dy (t, x))}

(2)

with multivalued, time independent a ∈ MΩ (RN ) (the Chiado’Piat, Dal Maso and Defranceschi class of maximal monotone mappings). The above diﬀerential inclusion is equivalent to corresponding inequality (hence the name HVI): y (t)+A(t, y (t)), v −y(t)+J 0 (y(t); v −y(t)) ≥ f (t)+(Cu)(t), v −y(t) for all v ∈ V . Notice that both HVIs (with single-valued and multivalued damping) are particular cases of the more general hyperbolic HVI: y + A(t, y (t)) + By + ∂J(y) f + Cu (HV Ih ) y(0) = y 0 , y (0) = y 1 , y ∈ Y, where, in the abstract setting, we are given the following spaces and operators: V ⊂ Z ⊆ H ⊆ Z ∗ ⊂ V ∗ is the Gelfand ﬁvefold of spaces, ∗ A : (0, T ) × V → 2V is multivalued and pseudomonotone, (Ay)(t) := A(t)y(t), t ∈ (0, T ) is the Nemitsky operator, J : Z → R is a locally Lipschitz superpotential (linked to a nonmonotone, possibly multivalued law between, e.g., stress-strain, reaction-displacement, etc.), C : U → V ∗ is a controller (linear or not), U is the space of controls, ∂J : Z → ∗ 2Z denotes the Clarke subdiﬀerential of J, and J 0 (u; v) stands for its generalized directional derivative. By means of such an (HV Ih ) one can model, for instance, short memory materials, the movement of elastic beam on gluing support, etc. The nonemptiness of the solution set for (HV Ih ) can be proved for large classes of operators (see [6]). In order to get sensitivity results we have to restrict ourselves to special classes of operators for which the notion of G or P G-convergence can be deﬁned.

2.

Preliminaries

First, in an abstract way, we formulate the problems of sensitivity and then we recall the deﬁnition and some properties of multivalued G-convergence. Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

2.1

Abstract Setting

We are given a state relation R linking the state y ∈ YR to the control u ∈ U (e.g., R = ODE, PDE, DI, VI, HVI) and a cost functional F : (Λ := graph R) (u, y) −→ F(u, y) ∈ R. We look for an optimal solution, i.e., an element (u∗ , y ∗ ) ∈ Λ which minimizes the cost functional m := min {F(u, y) : (u, y) ∈ Λ}

(CP )R

The set Λ of admissible pairs can be represented as: Λ = graph SR = {(u, y) : y ∈ SR (u), u ∈ U } =

(= F(u∗ , y ∗ )) . u∈U

SR (u),

where the solution map is given by SR : U u −→ SR (u) := {y ∈ YR : (u, y) ∈ R} ⊂ YR . For the well posed problem the set SR (u) consists of one element but it is not the case for R = HV I. Let us denote the set of optimal solutions by ∗ := {(u∗ , y ∗ ) ∈ Λ : F(u∗ , y ∗ ) = m} . SR The “sensitivity” is understood as a “nice” asymptotic behavior of optimal solutions to the perturbed problems (i.e., perturbed state relations Rk as well as perturbed cost functionals Fk ). Here index k indicates “perturbation”, k = ∞ denotes unperturbed original problem. So we consider the whole sequence (k ∈ N = N ∪ {+∞}) of problems: (CP )Rk

mk := min {Fk (u, y) : (u, y) ∈ Λk }

(= Fk (u∗k , yk∗ ))

and we look for conditions which assure that (i)

mk → m∞

(ii)

∗ K– lim sup Sk∗ ⊂ S∞

∗ The condition (ii) is equivalent to saying that if (u∗k , yk∗ ) ∈ SR and k ∗ ∗ ∗ ∗ ∗ ∗ ∗ (ukn , ykn ) → (u∞ , y∞ ), then (u∞ , y∞ ) ∈ SR∞ . Setting 0 x∈A χA (x) = +∞ x ∈, /

we can reformulate the problem to unconstrained optimization: (CP )Rk

mk := min {Fk (u, y) + χΛk (u, y) : (u, y) ∈ U × YRk } ,

and then apply the Γ-convergence theory (see [4, 1, 3, 10]) to get the required sensitivity results. Copyright © 2005 Marcel Dekker, Inc.

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On Sensitivity in Control Problems for Hyperbolic HVIs

2.2

Functional Spaces and Operators

Let Ω ⊂ RN be an open bounded set with regular boundary. We ﬁx p and q such that 1 < p < +∞ and 1/p + 1/q = 1, and we admit:

Definition 1 (Chiado’Piat, Dal Maso, Defranceschi [2]) For ﬁxed mi ∈ L1 (Ω), ci > 0, i = 1, 2, we denote MΩ (RN ) = {a : Ω × RN → 2R

N

such that (i) - (iii) below hold}

(i) a(x, ξ) is maximal monotone with respect to ξ for all x ∈ Ω (ii) a is L(Ω) ⊗ B(RN ) ⊗ B(RN ) measurable (i.e., a−1 (C) ∈ L(Ω) ⊗ B(RN ) for all closed set C ⊂ RN ) (iii) for every (ξ, η) with η ∈ a(x, ξ) we have |η|q ≤ m1 (x) + c1 (η, ξ)RN and |ξ|p ≤ m2 (x) + c2 (η, ξ)RN This class is large enough, e.g., it contains the mappings of the form a(x, ξ) = ∂ξ ψ(x, ξ) for some ψ : Ω × RN → [0, +∞) which is measurable in both variables, convex in ξ and satisﬁes c1 |ξ|p ≤ ψ(x, ξ) ≤ c2 |ξ|p . In Lq (Ω; RN ) we introduce the topology σ accordingly to the following σ

Definition 2 We say that ηk −→ η if and only if ηk → η in w − Lq (Ω; RN ) and divηk → divη in s − (W 1,p (Ω))∗ . For any function a ∈ MΩ (RN ) we deﬁne multivalued operators: A : W 1,p (Ω) y → Ay := {η ∈ Lq (Ω; RN ) : η(x) ∈ a(x, Dy(x)) a.e.}, A : W 1,p (Ω) y → Ay := {−divη : η ∈ Ay} ⊂ (W 1,p (Ω))∗ and we admit:

Definition 3 (“multivalued G-convergence”) For {ak }, a ∈ MΩ (RN ) G

we say that ak is G-convergent to a (written as ak −→ a) if and only if K(w, σ)– lim sup GrAk ⊂ GrA. We quote the result of Chiado’Piat, Dal Maso, and Defranceschi [2]:

Theorem 4 For ak , a ∈ MΩ (RN ) we have G

ak −→ a

=⇒

K(w, s)– lim GrAk = GrA. k→∞

K(w,s)

The latter convergence is denoted by GrAk −→ GrA. In the above theorem the implication ”⇐=” does not hold (see [2]). Copyright © 2005 Marcel Dekker, Inc.

150

CONTROL AND BOUNDARY ANALYSIS

The multivalued G-convergence appeared to be very useful in proving the convergence of solution sets and sensitivity analysis for stationary (elliptic) HVIs (see [7]). Now, it will be applied to evolution HVIs. Let V = W01,p (Ω), Z = Lp (Ω), H = L2 (Ω), Z ∗ = Lq (Ω), V ∗ = = Lp (Ω, RN ) Z N , Z ∗ = Lq (Ω, RN ) Z ∗ N . Notice W −1,q (Ω), and Z that for p ≥ 2 we have V ⊂ Z ⊆ H ⊆ Z∗ ⊂ V ∗ and the ﬁrst inclusion is a compact embedding. For evolution HVIs we need also time dependent spaces. To this end let Q = (0, T )×Ω and let us deﬁne the spaces V = Lp (0, T ; V ), Z = Lp (0, T ; Z), H = L2 (0, T ; H)

= Lp (0, T ; Z), Z ∗ = L2 (Q), Z ∗ = Lq (0, T ; Z ∗ ), V ∗ = Lq (0, T ; V ∗ ), Z ∗ ), and Lq (0, T ; Z Wpq = {y ∈ V : y ∈ V },

Y = {y ∈ V : y ∈ Wpq }.

So Wpq ⊂ V ⊂ Z ⊆ H ⊆ Z ∗ ⊂ V ∗ and the embedding Wpq into Z is compact. Below we quote a result on convergence of solutions of ﬁrst order (parabolic) diﬀerential inclusions which will be used to prove the Kuratowski convergence of solution sets of the second order (hyperbolic) HVIs with multivalued damping. Let ak ∈ MΩ (RN ), gk ∈ V ∗ , zk0 ∈ H for k ∈ N = N ∪ {+∞} be given. For operators Ak corresponding (see formula (2)) to ak we denote by zk the unique solution in Wpq to diﬀerential inclusion z (t) + Ak z(t) gk (t) for a.e. t ∈ (0, T ) (DI k ) z(0) = zk0 . Recall that zk ∈ Wpq is a solution to (DI k ) if ∗, zk (t) − divζk (t) = gk (t) for some ζk ∈ Z ζk (t, x) ∈ ak (x, Dzk (t, x)) a.e. in Q, zk (0) = zk0 . G

Theorem 5 (Denkowski, Mig´ orski, and Papageorgiou [9]) If ak −→ a∞ , s−V ∗

w−H

w−Wpq

∗ w−Z

0 , then z −→ z gk −→ g∞ , zk0 −→ z∞ ∞ and ζk −→ ζ∞ . k

Copyright © 2005 Marcel Dekker, Inc.

151

On Sensitivity in Control Problems for Hyperbolic HVIs

3.

Sensitivity of Solution Sets for Second Order HVIs

3.1

HVIs with Multivalued Damping

We start with a “K-usc” result for sequence (k ∈ N = N ∪ {+∞}) of problems: y (t) + Ak (y (t)) + ∂Jk (y(t)) fk (t) a.e. in (0, T ) × Ω (HV I)k y(0) = yk0 , y (0) = yk1 in Ω. We admit the hypotheses below which are satisﬁed uniformly in k ∈ N: ∗

H(A) : Ak : V → 2V \ ∅ correspond to functions ak ∈ MΩ (RN ) (i.e. Ak v = −divak (x, Dv) =: {−divη ∈ V ∗ : η(x) ∈ a(x, Dv(x))}) and H(J) :

G

ak −→ a∞ Jk : (0, T ) × Z → R satisﬁes the conditions:

(i) Jk (·, z) is measurable for all z ∈ Z (ii) Jk (t, ·) is locally Lipschitz for a.e. t ∈ (0, T ) 2/q

(iii) ||∂Jk (t, z)||Z ∗ ≤ cJ (1 + ||z||Z ) for all z ∈ Z, t ∈ (0, T ) with some cJ > 0 (iv) K(s − Z, w − Z ∗ )– lim sup Gr∂Jk (t, ·) ⊂ Gr∂J∞ (t, ·) for a.e. t s−V ∗

w−Z ∗

(H0 ) : either fk , f∞ ∈ V ∗ , fk −→ f∞ or fk , f∞ ∈ Z ∗ , fk −→ f∞ and 0 1 s−H 1 0 ∈ V , y 0 w−V 1 1 yk0 , y∞ k −→ y∞ , and yk , y∞ ∈ H, yk −→ y∞ (H1 ) : if p = 2, then α > cJ β 2 T , where α > 0 is the coercivity constant for Ak and β > 0 is an embedding constant of V into Z Now, (HV I)k reads as follows ∗ and ξk ∈ Z ∗ such that Find yk ∈ Y, ηk ∈ Z y (t, x) − divη (t, x) + ξ (t, x) = f (t, x) a.e. in Q k k k k ηk (t, x) ∈ ak (x, Dyk (t, x)), ξk (t, x) ∈ ∂Jk (t, yk (t, x)) a.e. in Q yk (0, x) = yk0 (x), yk (0, x) = yk1 (x) in Ω and we have

Theorem 6 Under hypotheses H(A), H(J), (H0 ), and (H1 ) it holds: Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

(j) The solution set S(Ak , Jk , fk , yk0 , yk1 ) of (HV Ik ) is nonempty for every k ∈ N. (jj) For every sequence yk ∈ S(Ak , Jk , fk , yk0 , yk1 ) there exists a subsew−Wpq

w−V

as n → ∞ and quence {ykn } such that ykn −→ y∞ , yk n −→ y∞ 0 , y 1 ). y∞ ∈ S(A∞ , J∞ , f∞ , y∞ ∞

Proof. Condition (j) follows from a general result on surjectivity of “Lpseudomonotone, coercive and bounded” operators (cf. Theorem 1.3.73 in [8]), while for (jj) we use in calculations Theorem 5 on convergence of solutions for (DI)k . Because of the lack of space here the complete proof will be published elsewhere. Next, we apply this theorem to our control systems y (t) + A(t, y (t)) + ∂J(y(t)) f (t) + (Cu)(t) a.e. in Q (HV Ic )k y(0) = y 0 , y (0) = y 1 in Ω, where the controller operators (providing us with distributed control) satisfy: c s−V ∗ H(C) : Ck : U → V ∗ , Ck −→ C∞ (continuously, i.e., Ck uk −→ C∞ u∞ U

for all uk −→ u∞ ).

Corollary 7 Under hypotheses H(A), H(J), H(C), (H0 ), and (H1 ) if we assume u ˜k = (uk , yk0 , yk1 )

U ×(w−V )×(w−H)

−→

then

0 1 (u∞ , y∞ , y∞ )=u ˜∞ ,

˜ ∈ U × V × H there exists at least one (j) for every k ∈ N and u u) := S(Ak , Jk , Ck , u ˜) = ∅ for solution to problem (HV Ic )k i.e. Sk (˜ all u ˜ ∈ U × V × H. uk ), where (jj) for any sequence {yk } ∈ Sk (˜ u ˜k = (uk , yk0 , yk1 )

U ×(w−V )×(w−H)

−→

u ˜∞ ,

there exists a subsequence which converges to a solution of the limit w−Y u∞ ). So problem, i.e., ykn −→ y∞ and y∞ ∈ S∞ (˜ uk ) ⊂ S∞ (˜ u∞ ). K(w − Y)– lim sup Sk (˜ k→∞

(jjj) Moreover, if the unperturbed (limit) problem (HV Ih )∞ has the uni˜ then u) = {y(˜ u)} for all u ˜ ∈ U), city of solution property (i.e., S∞ (˜ also we have S∞ (˜ u) ⊂ K(w − Y)– lim inf Sk (˜ uk ), so k→∞

uk ) Sk (˜ Copyright © 2005 Marcel Dekker, Inc.

K(w−Y)

−→

S∞ (˜ u∞ ), as k → ∞.

On Sensitivity in Control Problems for Hyperbolic HVIs

3.2

153

HVIs with Single-Valued Damping

An analogous result can be obtained for HVIs with single-valued time dependent damping of the form A(t, y (t)) = −diva(t, x, Dy (t, x)) with a ∈ M, where M is deﬁned by

Definition 8 (Svanstedt class of monotone functions, see [11]) Given nonnegative constants m0 , m1 , m2 , and 0 < α ≤ 1 we set M = M(m0 , m1 , m2 , α) := = {a : Q × RN → RN such that (i) - (iv) below hold} (i) |a(t, x, 0)| ≤ m0 a.e. in Q (ii) a(·, ·, ξ) is Lebesgue measurable on Q, ∀ ξ ∈ RN (iii) |a(t, x, ξ) − a(t, x, η)| ≤ m1 (1 + |ξ| + |η|)p−1−α |ξ − η|α a.e. (iv) (a(t, x, ξ) − a(t, x, η), ξ − η)RN ≥ m2 |ξ − η|α a.e. in Q, ∀ ξ, η In this case the assumption of G-convergence should be replaced by P G−convergence deﬁned as follows:

Definition 9 Let ak , a∞ ∈ M. We say that ak is P G-convergent to PG a∞ , written as ak −→ a∞ if and only if for every g ∈ V ∗ we have w−Lq (Q,RN )

w−Wpq

−→ a(t, x, Dy∞ ), where yk , for yk −→ y∞ and ak (t, x, Dyk ) k ∈ N, is the unique solution to the problem y − divak (t, x, Dy) = g,

y(0) = 0.

We admit the following hypothesis H(A)1 :

Ak : (0, T ) × V → V ∗ correspond to functions ak ∈ M PG

(i.e., Ak v = −divak (t, x, Dv(x))) and ak −→ a∞ . We have

Theorem 10 Assume in sequences of (HV I)k and (HV Ic )k we replace the multivalued damping by a single-valued one satisfying the hypothesis H(A)1 . Then for these sequences Theorem 6 and Corollary 7 (with hypothesis H(A) being replaced by H(A)1 ) remain true.

4.

Complementary Γ-convergence of Cost Functionals

We consider a sequence of “perturbed” cost functionals: u ˜

8 9: ; (1) (2) (3) (4) Fk (u, y 0 , y 1 , y) := Fk (y) + Fk (u) + Fk (y 0 ) + Fk (y 1 ), Copyright © 2005 Marcel Dekker, Inc.

(3)

154 where

CONTROL AND BOUNDARY ANALYSIS (1)

Fk (y) =

(2)

Fk (u) =

(1) Q Fk (t, x, y(t, x)) dtdx,

(2) Q Fk (t, x, (Ck u)(t, x)) dtdx,

(3)

(4)

Fk (y 0 ) = Fk (y 1 ) =

(3) 0 0 Ω Fk (x, y (x), Dy (x)) dx, (4) 1 Ω Fk (x, y (x)) dx.

We would like to establish the regularity and convergence conditions which assure: (i)

1o for every ﬁxed k ∈ N, Fk (·), i = 1, . . . , 4 are lsc in suitable topologies; Γseq ((U˜)− ,Y ± )

2o Fk (˜ u, y) −→ ˜ U × V × H =: U.

F∞ (˜ u, y), as k → ∞, where u ˜ = (u, y 0 , y 1 ) ∈

The lsc property 1o is needed for the direct method when proving existence of optimal solutions. The “complementary” Γ-convergence in 2o , which means “continuous” convergence in y and Γ-convergence in u, together with K-convergence of solution sets of HVIs imply variational convergence of optimal solutions of (CP )HV I . To establish properties 1o and 2o , we admit the regularity and convergence assumptions for integrands (below conditions (i) and (ii) are uniform with respect to k ∈ N). H(F (1) ) :

(1)

(i) Fk : Q × R → R is measurable in (t, x) ∈ Q, (1)

Fk (t, x, 0) ∈ Lp (Q) (1)

(1)

(1)

w−L1 (Q)

(ii) |Fk (t, x, z1 ) − Fk (t, x, z2 )| ≤ c(1 + |z1 |)|z1 − z2 | (iii) Fk (·, ·, z) H(F (2) ) :

−→

(1) F∞ (·, ·, z), ∀ z ∈ R

(2)

(i) Fk : Q × R → R is measurable in (t, x), convex in z (2)

(ii) 0 ≤ Fk (t, x, z) ≤ λ|z|q , a.e. in Q (2)

(iii) Fk (·, ·, z) H(F (3) ) :

w−L1 (Q)

−→

(2) F∞ (·, ·, z), ∀ z ∈ R

(3)

(i) Fk : Ω × RN +1 → R is measurable in x ∈ Ω convex in z ∈ RN +1 (3) (ii) λ|z|p ≤ Fk (t, x, z), a.e. in Q (3)∗

(iii) Fk

(·, ·, z)

Copyright © 2005 Marcel Dekker, Inc.

w−L1 (Q)

(3)∗ −→ F∞ (·, ·, z), ∀ z ∈ RN +1 (convergence of the Fenchel conjugates)

155

On Sensitivity in Control Problems for Hyperbolic HVIs (4)

(i) Fk : Ω × R → R is measurable in x ∈ Ω, convex in z

H(F (4) ) :

(4)

(ii) 0 ≤ Fk (x, z) ≤ λ|z|2 , a.e. in Ω w−L1 (Ω)

(4)

−→

(iii) Fk (·, z)

(4) F∞ (·, z), ∀ z ∈ R

Lemma 11 Let U = Lq (0, T ; Lq (Ω)) Lq (Q), Ck = id : U → Z ∗

Lq (Q) ⊂ V ∗ . Then: 10 For every ﬁxed k ∈ N under regularity assumptions (i), (ii) of H(F (1) ) − H(F (4) ), respectively, we obtain: (1)

(j) Fk (·) is s − Lp (Q) so also (w − Y)-continuous (2)

(jj) Fk (·) is s − Lq (Q) continuous (3)

(jjj) Fk (·) is w − W 1,p (Ω) continuous (4)

(jv) Fk (·) is w − L2 (Ω) continuous 20 Under convergence conditions (iii) of H(F (1) ) − H(F (4) ), respectively, we have (1)

c−Y

(1) (y), Fk (y) −→ F∞ (3)

Fk (y 0 )

Γseq (w−W 1,p (Ω)− )

−→

(2)

Fk (u)

(3) 0 F∞ (y ),

Γseq (s−Lq (Q)− )

−→

(4)

Fk (y 1 ) (1)

(2) F∞ (u)

Γseq (w−L2 (Ω)− )

−→

(2)

(4) 1 F∞ (y ) (3)

so for the functional Fk (u, y 0 , y 1 , y) = Fk (y) + Fk (u) + Fk (y 1 ) + (4) Fk (y 1 ), we obtain Γseq s − Lq (Q)− , w − W 1,p (Ω)− , w − L2 (Ω), w − Y ± lim Fk = F∞ . k→∞

The proof of (j) − (jv) follows from classical theorems on continuity on Lp spaces of integral functionals with Carath´eodory integrands with respect to the strong topology (or weak topology provided the integrands are convex). The proof of the ﬁrst convergence follows from direct calculations, while those of the next three follow from theorems on Γconvergence (cf. [3]); that it is equivalent to the pointwise convergence for sequences of locally equicontinuous functions (e.g., convex and locally bounded).

Theorem 12 We admit the hypothesis H(A), H(J), (H0 ) for (HV Ic )k with U = Lq (0, T ; Lq (Ω)), Ck = id : U → Z ∗ Lq (Q) ⊂ V ∗ . We assume the hypotheses H(F (j) ), j = 1, . . . , 4, for cost functional Fk (u, y 0 , y 1 , y) given by ( 3) hold. Then: Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

(i) For every k ∈ N the problem (CP )(HV Ic )k has at least one optimal solution (u∗k , yk0∗ , yk1∗ , yk∗ ) ∈ Sk∗ , mk := Fk (u∗k , yk0∗ , yk1∗ , yk∗ ) being its “minimal value”. (ii) If the unperturbed (original) problem (CP )(HV Ic )∞ has the “unicity ˜ S(HV I ) (˜ u) = {y∞ (˜ u)}), of solution property” (for all u ˜ ∈ U, c ∞ ˜∞ , S(HV Ic )k (˜ uk ) are “equicoercive”, so there exists then for u ˜k → u convergent subsequence of optimal solutions such that 0∗ 1∗ ∗ 0∗ 1∗ ∗ ∗ , y∞ , y∞ ) and (u∗∞ , y∞ , y∞ , y∞ ) ∈ S∞ . (u∗kn , yk0∗n , yk1∗n , yk∗n )to(u∗∞ , y∞

(iii) We also have mk → m∞ , as k → ∞. Finally, we remark that an analogous theorem holds for (CP )HV I with single-valued damping, provided we replace hypothesis H(A) by H(A)1 .

References [1] G. Buttazzo and G. Dal Maso. Gamma-convergence and optimal control problems. J. Optim. Theory Appl., 38:385–407, 1982. [2] V. ChiadoPiat, G. Dal Maso, and A. Defranceschi. G-convergence of monotone operators. Annales de l’Institut Henri Poincar´e, 7:124–160, 1990. [3] G. Dal Maso. An Introduction to Γ-convergence. Birkh¨ auser, Boston, 1993. [4] E. De Giorgi and T. Franzoni. Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Natur., 58:842–850, 1975. [5] Z. Denkowski. Control problems for systems described by hemivariational inequalities. Control and Cybernetics, 31:713–738, 2002. [6] Z. Denkowski and S. Mig´ orski. Existence of solutions to evolution second order hemivariational inequalities with multivalued damping, in System Modeling and Optimization, J. Cagnol and J.-P. Zol´esio (Eds.), pp. 203–215, Kluwer Academic Publishers, Norwell, MA, 2004. [7] Z. Denkowski and S. Mig´ orski. Sensitivity of optimal solutions to control problems for systems described by hemivariational inequalities, Control and Cybernetics, 33(2), 2004. In press. [8] Z. Denkowski, S. Mig´ orski, and N.S. Papageorgiou. An Introduction to Nonlinear Analysis: Applications. Kluwer Academic/Plenum Publishers, Boston, 2003. [9] Z. Denkowski, S. Mig´ orski, and N.S. Papageorgiou. On the convergence of solutions of multivalued parabolic equations and applications. Nonlinear Analysis, 54:667–682, 2003. [10] Z. Denkowski and S. Mortola. Asymptotic behaviour of optimal solutions to control problems for systems described by diﬀerential inclusions corresponding to partial diﬀerential equations. J. Optim. Theory Appl., 78:365–391, 1993. [11] N. Svanstedt. G-convergence of parabolic operators. Nonlinear Analysis, 36:807– 842, 1999.

Copyright © 2005 Marcel Dekker, Inc.

CHAPTER XIII EVOLUTION HEMIVARIATIONAL INEQUALITY WITH HYSTERESIS AND OPTIMAL CONTROL PROBLEM Leszek Gasi´ nski∗ Institute of Computer Science Jagiellonian University Cracow, Poland

Abstract

We study an optimal control problem of Bolza type for evolution hemivariational inequality with hysteresis operator.

Keywords: Hemivariational inequality, locally Lipschitz function, Clarke’s subdifferential, hysteresis operator, control problem of Bolza type

Introduction In this paper we study an optimal control problem of the form: min min J(y, u), y∈U u∈S(y)

where J is a Bolza-type functional and S(y) is the set of solutions of the following evolution hemivariational inequality with hysteresis: u (t) + w(t) + A(u (t)) + B(u(t)) + χ(t) = f (t) + D(t)y(t) in V . In this statement, χ is a selection of Clarke’s subdiﬀerential of a locally Lipschitz potential function and w is an output function of a hysteresis operator with the input u. The notion of hemivariational inequalities was introduced in the 1980s by P.D. Panagiotopoulos (see [13–15]). These new expressions are a natural extension of variational inequalities. They arise in physical problems when we deal with nonsmooth and nonconvex potential functionals and thus with nonmonotone and possibly multivalued laws. Such functionals appear quite often in mechanics and engineering if one wants ∗ This

paper was partially supported by the State Committee for Scientiﬁc Research of the Republic of Poland (KBN) under research grants no. 2 P03A 003 25 and no. 4 T07A 027 26.

Copyright © 2005 Marcel Dekker, Inc.

158

CONTROL AND BOUNDARY ANALYSIS

to consider more realistic mechanical laws of nonmonotone, multivalued nature. Hemivariational inequalities use the notion of Clarke’s subdiﬀerential of locally Lipschitz functions (see F.H. Clarke [4]), which is a generalization of classical derivative of smooth functions as well as subdiﬀerential of convex functions in the sense of convex analysis. For particular applications of hemivariational inequalities to problems in mechanics and engineering we refer to the books of P.D. Panagiotopoulos [13, 16], Z. Naniewicz and P.D. Panagiotopoulos [12], and references therein. Some existence results for hemivariational inequalities of second order were obtained, e.g., by P.D. Panagiotopoulos [17], W. Bian [2], L. Gasi´ nski [6, 8]. In the ﬁrst of the above mentioned papers we can also ﬁnd some applications of this type of inequalities in mechanics, namely for the description of plane linear elastic body with nonmonotone skin eﬀects. Parallel to the theory of hemivariational inequalities, the theory of hysteresis has been developed. The latter is based on the notion of the hysteresis operator, introduced by M.A. Krasnoselskii and A.V. Pokrovskii [10] and developed, e.g., by M. Brokate and J. Sprekels [3], A. Visintin [18], and others. Our starting point in this paper is the existence result for hemivariational inequality with hysteresis shown in [7].

1.

Preliminaries

Let X be a Banach space with a norm · X and X its topological dual. By ·, ·X ×X we shall denote the duality brackets for the pair (X, X ). If X and Y are two Banach spaces such that X ⊆ Y and the embedding X ⊆ Y is continuous, then by cX Y we denote the embedding constant, i.e. the smallest number such that xY ≤ cX Y xX for all x ∈ X. If X is in addition a Hilbert space, then by (·, ·)X we shall denote the scalar product in X. Given a locally Lipschitz function ϕ : X −→ R, we deﬁne generalized directional derivative ϕ0 (x; h) of ϕ at x ∈ X in the direction h ∈ X, by df

ϕ0 (x; h) = lim sup y→x t0

ϕ(y + th) − ϕ(y) . t

It is easy to see that the function X h −→ ϕ0 (x; h) ∈ R is sublinear, continuous, and so by the Hahn-Banach theorem, ϕ0 (x; ·) is the support function of a nonempty, w∗ -compact and convex set ∂ϕ(x) deﬁned by df

∂ϕ(x) = {x∗ ∈ X : x∗ , hX ≤ ϕ0 (x; h) Copyright © 2005 Marcel Dekker, Inc.

∀h ∈ X}.

Evolution Hemivariational Inequality with Hysteresis and OCP

159

The multifunction ∂ϕ : X −→ 2X \ {∅} is known as the Clarke’s or generalized subdiﬀerential of ϕ. If ϕ, ψ : X −→ R are both locally Lipschitz functionals, then ∂(ϕ + ψ)(x) ⊆ ∂ϕ(x) + ∂ψ(x)

∀x ∈ X

and ∂(tϕ)(x) = tϕ(x)

∀x ∈ X, t ∈ R.

If in addition ϕ is convex, then the subdiﬀerential ∂ϕ coincides with the subdiﬀerential in the sense of convex analysis (see S. Hu and N.S. Papageorgiou [9, p. 267]). Finally, if ϕ is continuously Gateaux diﬀerentiable at x ∈ X (i.e., ϕ ∈ C 1 (X)), then ∂ϕ(x) = {ϕ (x)}. Now we recall the basic facts and notions from the theory of hysteresis operators. Let T > 0. A function v : [0, T ] −→ R is called piecewise monotone if there exists a partition 0 ≤ t0 < t1 < . . . < tn = T of interval [0, T ] (called monotonicity partition) such that v|[ti−1 ,ti ] is monotone, for all i = 1, . . . , n. Such a partition is called standard monotonicity partition if the number of subintervals n is minimal. By Cpm ([0, T ]), we denote the space of all continuous and piecewise monotone functions on [0, T ]. Functional H : Cpm ([0, T ]) −→ R is called rate independent if and only if H[v ◦ ϕ] = H[v], (1) for all v ∈ Cpm ([0, T ]) and all ϕ : [0, T ] −→ R, continuous increasing functions satisfying ϕ(0) = 0 and ϕ(T ) = T . It can be easily seen that only the local extremal values of v have an inﬂuence on H[v]. By S we denote the set of all ﬁnite strings of real numbers and by SA , the set of all alternating strings, i.e., df

SA = {(s0 , . . . , sn ) ∈ S : (si+1 −si )(si −si−1 ) < 0, 1 ≤ i ≤ n−1}. (2) We deﬁne the restriction operator ρA : Cpm ([0, T ]) −→ SA , as follows: df

ρA (v) = (v(t0 ), v(t1 ), . . . , v(tn )),

(3)

for all v ∈ Cpm ([0, T ]), where 0 = t0 < t1 < . . . < tn = T is the standard monotonicity partition of v. By πA : SA −→ Cpm ([0, T ]), we denote the so-called prolongation operator, which maps any string s = (s0 , s1 , . .. , sn ) into the linear interpolate function v : [0, T ] −→ R of the points iT n , si for i = 0, 1 . . . , n. Using these operators, we obtain , : SA −→ R and rate the bijective correspondence between functionals H independent functionals H : Cpm ([0, T ]) −→ R, as follows , ◦ ρA , H = H Copyright © 2005 Marcel Dekker, Inc.

, = H ◦ πA . H

(4)

160

CONTROL AND BOUNDARY ANALYSIS

An operator W : Cpm ([0, T ]) −→ C [0, T ] is said to be a hysteresis operator on Cpm ([0, T ]) if there exists a rate independent functional H called a generating functional of W such that W[v](t) = H[vt ] ∀t ∈ [0, T ], ∀v ∈ Cpm ([0, T ]), where df

vt (ξ) =

v(ξ), v(t),

for 0 ≤ ξ ≤ t, for t < ξ ≤ T.

(5)

(6)

, : SA −→ S is called a hysteresis operator on SA , if An operator W , , 0 ), H(s , 0 , s1 ), . . . , H(s)) , W(s) = (H(s

∀s = (s0 , s1 , . . . , sn ) ∈ SA , (7)

, and H is a rate , = H ◦ πA is called a generating functional of W where H independent functional on Cpm ([0, T ]). , are called ﬁnal value mapThe unique generating functionals H and H , f , respectively. Due to (4), we also pings and are denoted by Wf and W have a bijective correspondence between hysteresis operators W deﬁned , deﬁned on SA . We will use the same notation W on Cpm ([0, T ]) and W , f . Finally, we can , and also Wf for both Wf and W for both W and W extend the hysteresis operators deﬁned on Cpm ([0, T ]) to the set of all continuous functions C [0, T ] , by the use of the density of the embedding Cpm ([0, T ]) ⊆ C [0, T ] . For more details on hysteresis operators we refer to M. Brokate and J. Sprekels [3, Chapter 2], M.A. Krasnoselskii and A.V. Pokrovskii [10] and A. Visintin [18, Chapter I]. As for the optimal control problem, our main result will be based on the following theorem (see E.J. Balder [1, Theorem 2.1, p.1400]).

Theorem 1 Let X be a separable Banach space and Z a separable reﬂexive Banach space. Let F : [0, T ] × X × Z −→ R be a measurable function such that (i) for almost all t ∈ [0, T ], the function F (t, ·, ·) is sequentially lower semicontinuous on X × Z (ii) for all x ∈ X and for almost all t ∈ [0, T ], the function F (t, x, ·) is convex on Z (iii) there exists M > 0 and ψ ∈ L1 0, T ; R such that F (t, x, z) ≥ ψ(t) − M (xX + zZ )

∀x ∈ X, z ∈ Z, a.a. t ∈ [0, T ],

then the functional F : L1 0, T ; X × L1 0, T ; Z −→ [−∞, +∞], Copyright © 2005 Marcel Dekker, Inc.

Evolution Hemivariational Inequality with Hysteresis and OCP

deﬁned by df

161

T

F(x, z) =

F (t, x(t), z(t))dt is sequentially lower semicontinuous in L1 0, T ; X × (w–L1 0, T ; Z )topology. Moreover, the conditions (i)-(iii) are also necessary provided that there 1 1 exists (x, z) ∈ L 0, T ; X × L 0, T ; Z such that F(x, z) < +∞. 0

We will also make use of the following fact from the multivalued analysis (see S. Hu and N.S. Papageorgiou [9, Proposition 3.9, p.694]).

Theorem 2 Let Z be a separable Banach space, 1 ≤ p < ∞, and for almost all t ∈ [0, T ], let the set U (t) be a weakly compact subset of Z. p p p If {fn } ⊆ L 0, T ; Z , f ∈ L 0, T ; Z , fn −→ f weakly in L 0, T ; Z and fn (t) ∈ U (t) for all n and almost all t ∈ [0, T ], then f (t) ∈ convK(w–Z)– lim sup fn (t) n→+∞

2.

for a.a. t ∈ [0, T ].

Hemivariational Inequality with Hysteresis

Let Ω ∈ RN be an open, bounded set with Lipschitz boundary ∂Ω and 2 T > 0. Let H = L Ω . By V we denote a Hilbert space such that the embedding V ⊆ H 1 (Ω) is dense and continuous. We introduce the following spaces: H = L2 0, T ; H = L2 (0, T ) × Ω , V = L2 0, T ; V , W = {v : v ∈ V, v ∈ V }, Y = H 1 (0, T ; H) ∩ L∞ 0, T ; V , X = {v : v ∈ C [0, T ]; V , v ∈ Y} X, = {v : v ∈ C [0, T ]; V , v ∈ W}. The following embeddings are continuous: X ⊆ X, ⊆ Y ⊆ W. We consider the following evolution hemivariational inequality: Find u ∈ X, and χ ∈ H, such that u (t) + w(t) + A(u (t)) + B(u(t)) + χ(t) = f (t) + D(t)y(t) in V (HV I) u(0) = ψ0 , u (0) = ψ1 in Ω w(t, x) = W[u(·, x); x](t) for a.a. (t, x) ∈ (0, T ) × Ω χ(t, x) ∈ ∂j(x, g(u(t, x), u (t, x))) ∀t ∈ [0, T ], for a.a. x ∈ Ω. Copyright © 2005 Marcel Dekker, Inc.

162

CONTROL AND BOUNDARY ANALYSIS

Our hypotheses on the right-hand side f , operator D, initial conditions ψ0 , ψ1 , the hysteresis operator W, the operators A and B, the nonsmooth potential function j, and the function g are the following: H(f, ψ) f ∈ H, ψ0 , ψ1 ∈ V . H(W) W[·, x] is a family of hysteresis operators (indexed by x ∈ Ω) such that (i) for all x ∈ Ω, operator W[·; x] : C [0, T ] −→ C [0, T ] is continuous (ii) the parameterized ﬁnal value mapping Ω x −→ Wf (s; x) ∈ R is measurable for all n ∈ N and all s = (s0 , s1 , . . . , sn ) ∈ S; we also have |Wf (s; x)| ≤ aW(x) + cW max |sk | ∀x ∈ Ω, s ∈ S, n ∈ N, 0≤k≤n

where aW ∈ H and cW > 0 (iii) Wf (ψ1 (·); ·) ∈ H H(g) g : R × R −→ R is a function such that (i) g is continuous (ii) we have |g(ξ, ζ)| ≤ αg |ξ| + βg |ζ| ∀ξ, ζ ∈ R, with αg , βg ≥ 0 H(j) j : Ω × R −→ R is a function such that (i) for all ξ ∈ R, the function Ω x −→ j(x, ξ) ∈ R is measurable (ii) for almost all x ∈ Ω, the function R ξ −→ j(x, ξ) ∈ R is locally Lipschitz (iii) for almost all x ∈ Ω, all ξ ∈ R, and all η ∈ ∂j(x, ξ), we have that |η| ≤ cj (1 + |ξ|), where cj > 0 H(A) A : V −→ V is a linear operator such that (i) for all v, w ∈ V , we have that Av, wV ≤ αA vV wV , with some αA > 0 (ii) for all v, w ∈ V , we have Av, wV = Aw, vV (iii) for all v ∈ V , we have Av, vV ≥ βA v2V − γA v2H , with 2 some βA > 2 cVH cj βg and γA ≥ 0 Copyright © 2005 Marcel Dekker, Inc.

Evolution Hemivariational Inequality with Hysteresis and OCP

163

H(B) B : V −→ V is a linear operator such that (i) for all v, w ∈ V , we have Bv, wV ≤ αB vV wV , with some αB > 0 (ii) for all v, w ∈ V , we have Bv, wV = Bw, vV (iii) for all v ∈ V , we have Bv, vV ≥ 0 H(D) D ∈ L∞ 0, T ; L(Y, V ) and Y is a separable reﬂexive Banach space. Taking g(ξ, ζ) = ξ, we obtain the law governed by Clarke’s subdiﬀerential depending only on function u. Such a relation for elliptic hemivariational inequality was studied by Z. Naniewicz and P.D. Panagiotopoulos [12], and for parabolic hemivariational inequality by M. Miettinen [11] and L. Gasi´ nski [5]. On the other hand, taking g(ξ, ζ) = ζ, we obtain the law governed by Clarke’s subdiﬀerential depending only on the derivative u of the function u. Such a relation for evolution hemivariational inequality was studied by P.D. Panagiotopoulos [17]. Now we can formulate an existence result for (HV I).

Theorem 3 If hypotheses H(f, ψ), H(W), H(g), H(j), H(A), H(B), and H(D) hold, then for any y ∈ L2 0, T ; Y , problem (HV I) admits a solution u ∈ X ⊆ X,. Proof: For a ﬁxed y ∈ L2 0, T ; Y , we have D(·)y(·)2V

T

= 0

D(t)y(t)2V dt ≤

T 0

D(t)2L(Y,V ) y(t)2Y dt

≤ D2L∞ (0,T ;L(Y,V )) y2L2 (0,T ;Y )

(8)

and so for the right-hand side of the equation in (HV I), we have that f (·) + D(·)y(·) ∈ V .

(9)

Now the existence theorem follows from Theorem 3.1 of L. Gasi´ nski [7]. 2

3.

Estimates on the Solutions of (HVI)

The crucial role in the proof of the main result will be played by a priori estimates on the solutions of (HV I). Let us start with two simple lemmas on the estimates of selections of ∂j x, · and the output of the hysteresis operator.

Copyright © 2005 Marcel Dekker, Inc.

164

CONTROL AND BOUNDARY ANALYSIS

Lemma 4 If hypotheses hold, u ∈ W, u(0) = ψ0 , η ∈ H H(j) and H(g) and η(t, x) ∈ ∂j x, g u(t, x), u (t, x) for almost all (t, x) ∈ (0, T ) × Ω, then for all t ∈ [0, T ], we have

t t t s 2 2 2 η(s)H ds ≤ c 1 + u (s)H ds + u (τ )H dτ ds , 0

0 df

where c =

3c2j

0

0

(10)

max T |Ω| + 2T α2g ψ0 2H , βg2 , 2T α2g .

Lemma 5 If hypotheses H(W) hold, u ∈ Y, u(0) = ψ0 , and w ∈ H are such that w(t, x) = W[u(·, x); x](t) then t

w(s)2H

ds ≤ 2 c 1+

0

for a.a. (t, x) ∈ (0, T ) × Ω,

t 0

s

u

(τ )2H

dτ ds

∀t ∈ [0, T ],

(11)

0

and in particular

c 1 + T u 2H w2H ≤ 2 df with 2 c = 2T max aW2H + 2c2Wψ0 2H , 2c2W .

(12)

The next lemma gives some estimates on the solutions of (HV I). In its proof both Lemmas 4 and 5 are needed.

Lemma 6 If hypotheses H(f, ψ), H(W), H(g), H(j), H(A), H(B), and H(D) hold and u ∈ X, is a solution of (HV I) for some y ∈ L2 0, T ; Y , then uX, = uC([0,T ];V ) + u W ≤ c 1 + yL2 (0,T ;Y ) , where c > 0 is a constant not depending on y.

4.

Optimal Control Problem

In this section we consider an optimal control problem governed by our hemivariational inequality with hysteresis. Let J be the Bolza-type functional deﬁned by: T df J(y, u) = l(u(T ), u (T )) + L(t, u(t), u (t), y(t))dt, 0

where y is any admissible control and u ∈ S(y), where by S(y) ⊆ X, we denote the set of solutions for (HV I) with the control y at the right-hand side. Copyright © 2005 Marcel Dekker, Inc.

Evolution Hemivariational Inequality with Hysteresis and OCP

165

By optimal control problem, we mean the following problem: (CP )

Find y ∗ ∈ U and u∗ ∈ S(y ∗ ) such that: J(y ∗ , u∗ ) = min min J(y, u). y∈U u∈S(y)

Due to the lack of convexity of j (or some additional assumptions) we are not able to guarantee the uniqueness of the solution of (HV I), so S(y) contains, in general, more than one element. To obtain our existence result we will need the following assumptions on the set of admissible controls U and the cost functional J: df H(U) U = y ∈ L2 0, T ; Y : y(t) ∈ U (t) , where U : [0, T ] −→ 2Y \ {∅} is a multifunction such that for all t ∈ [0, T ], the set U (t) is a nonempty closed convex subset of Y and the function df

t → |U (t)| = sup{yY : y ∈ U (t)} is in L∞ + (0, T ). H(J) l : V × H −→ R is weakly lower semicontinuous and L : [0, T ] × V × H × Y −→ R ∪ {+∞} is a measurable function such that (i) L(t, ·, ·, ·) is sequentially lower semicontinuous on V × H × Y for almost all t ∈ [0, T ] (ii) L(t, v, w, ·) is convex on Y for all v ∈ V , w ∈ H and almost all t ∈ [0, T ] (iii) there exists M > 0 and ψ ∈ L1 (0, T ) such that for all v ∈ V , w ∈ H, y ∈ Y and almost all t ∈ [0, T ], we have L(t, v, w, y) ≥ ψ(t) − M (vV + wH + yY ) The following lemma says that the map ,

U y −→ S(y) ∈ 2X \ {∅}, which for every y ∈ U admits the value S(y), the set of solutions of (HV I) with the control y has a closed graph in (w–U )×(w–X,)-topology.

Lemma 7 Let hypotheses H(f, ψ), H(W), H(g), H(j), H(A), H(B), H(D) and H(U) hold. If {yn } ⊆ U and {un } ⊆ X, are two sequences such that un ∈ S(yn ) and yn −→ y ∗ Copyright © 2005 Marcel Dekker, Inc.

weakly in U ,

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CONTROL AND BOUNDARY ANALYSIS

weakly in X,,

un −→ u∗

with some y ∗ ∈ U and u∗ ∈ X,, then u∗ ∈ S(y ∗ ). Now we can formulate our main result:

Theorem 8 If hypotheses H(f, ψ), H(W), H(g), H(j), H(A), H(B), H(D), H(U) and H(J) hold, then problem (CP ) admits a solution. Proof: We apply the direct method of the calculus of variations. Let (yn , un ) be a minimizing sequence for (CP ), i.e., yn ∈ U,

un ∈ S(yn )

∀n ≥ 1

and lim J(yn , un ) = min min J(y, u).

n→+∞

y∈U u∈S(y)

From the hypotheses H(U), we obtain that T yn (t)2Y dt ≤ M1 , yn 2L2 (0,T ;Y ) = 0

with some M1 > 0 not dependent on n and so we may assume, passing to a subsequence if necessary, that yn −→ y ∗ weakly in L2 0, T ; Y . (13) We will show that for almost all t ∈ [0, T ], sets U (t) are weakly compact subsets of Y . Indeed, for a ﬁxed t ∈ [0, T ], the set U (t) is closed and convex, so also weakly closed. Moreover, for any sequence {zn } ⊆ U (t), we have that zn Y ≤ M2 , with some M2 > 0. Since Y is a reﬂexive Banach space, we get, at least for a subsequence, that zn −→ z

weakly in Y.

In view of the weak closedness of U (t), we obtain that z(t) ∈ U (t), which proves that U (t) is weakly compact for almost all t ∈ [0, T ]. Now, applying Theorem 2, for almost all t ∈ [0, T ], we get that y ∗ (t) ∈ convK(w–Y )– lim sup yn (t) ⊆ convU (t), n→+∞

what, together with the fact that U (t) is a closed and convex subset of Y , implies y ∗ (t) ∈ U (t) for a.a. t ∈ [0, T ]. (14) Copyright © 2005 Marcel Dekker, Inc.

Evolution Hemivariational Inequality with Hysteresis and OCP

167

Recalling that the sequence {yn }n≥1 is bounded in L2 0, T ; Y , from the a priori estimate of Lemma 6, we immediately get that un V ≤ M3

and

un W ≤ M3 ,

with some M3 > 0. So, for a subsequence, we have un −→ u∗ un −→ u∗

weakly in V weakly in W.

(15) (16)

Hence, and from (13), applying Lemma 7, we obtain that u∗ ∈ S(y ∗ ). Thus from (14), we get that (y ∗ , u∗ ) is an admissible state-control pair. To conclude, it is enough to show that it is also an optimal solution. First, from the continuity of the embedding W ⊆ C [0, T ]; H we have that un (T ) −→ u∗ (T ) weakly in H un (T ) −→ u∗ (T ) weakly in H. Using the compactness of the embedding W ⊆ H, from (15) and (16), for a next subsequence, we have un −→ u∗ in H un −→ u∗ in H.

(17) (18)

Invoking Theorem 1 and the fact that l is weakly lower semicontinuous (see hypotheses H(J)), we obtain that the cost functional J is sequen tially lower semicontinuous on L2 0, T ; H × H × (w–L2 0, T ; Y ). So in consequence, from (13), (17), and (18), we get J(y ∗ , u∗ ) ≤ lim inf J(yn , un ) = min min J(y, u), n→+∞

which proves the theorem.

y∈U u∈S(y)

2

References [1] E.J. Balder. Necessary and suﬃcient conditions for l1 -strong-weak lower semicontinuity of integral functional. Nonlin. Anal., 11:1399–1404, 1987. [2] W. Bian. Existence results for second order nonlinear evolution inclusions. Indian J. Pure Appl. Math., 11:1177–1193, 1998. [3] M. Brokate and J. Sprekels. Hysteresis and Phase Transitions. Springer-Verlag, Berlin/New York, 1996.

Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

[4] F.H. Clarke. Optimization and Nonsmooth Analysis. John Wiley & Sons, New York, 1983. [5] L. Gasi´ nski. Optimal shape design problems for a class of systems described by parabolic hemivariational inequality. J. Global Optim., 12:299–317, 1998. [6] L. Gasi´ nski. Existence result for hyperbolic hemivariational inequalities. Nonlinear Anal., 47:681–686, 2001. [7] L. Gasi´ nski. Evolution hemivariational inequalities with hysteresis. Nonlinear Anal., 57:323-340, 2004. [8] L. Gasi´ nski. Existence of solutions for hyperbolic hemivariational inequalities. J. Math. Anal. Appl., 276:723–746, 2002. [9] S. Hu and N.S. Papageorgiou. Handbook of Multivalued Analysis, Volume I: Theory. Kluwer, Dordrecht, 1997. [10] M.A. Krasnoselskii and A.V. Pokrowski. Systems with Hysteresis. SpringerVerlag, Heidelberg, 1989. [11] M. Miettinen. A parabolic hemivariational inequality. Nonlin. Anal., 26:725– 734, 1996. [12] Z. Naniewicz and P.D. Panagiotopoulos. Mathematical Theory of Hemivariational Inequalities and Applications. Dekker, New York, 1995. [13] P.D. Panagiotopoulos. Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions. Birkh¨ auser, Basel, 1985. [14] P.D. Panagiotopoulos. Nonconvex problems of semipermeable media and related topics. Z. Angew. Math. Mech., 65:29–36, 1985. [15] P.D. Panagiotopoulos. Coercive and semicoercive hemivariational inequalities. Nonlin. Anal., 16:209–231, 1991. [16] P.D. Panagiotopoulos. Hemivariational Inequalities, Applications in Mechanics and Engineering. Springer-Verlag, Berlin, 1993. [17] P.D. Panagiotopoulos. Modelling of nonconvex nonsmooth energy problems. Dynamic hemivariational inequalities with impact eﬀects. J. Comp. Appl. Math., 63:123–138, 1995. [18] A. Visintin. Diﬀerential Models of Hysteresis. Springer-Verlag, Berlin/New York, 1994.

Copyright © 2005 Marcel Dekker, Inc.

CHAPTER XIV ON THE MODELING AND CONTROL OF DELAMINATION PROCESSES Michal Koˇcvara∗ Institute of Applied Mathematics University of Erlangen Erlangen, Germany

Jiˇri V. Outrata Institute of Information Theory and Automation Academy of Sciences of the Czech Republic Prague, Czech Republic

Abstract

This paper is motivated by problem of optimal shape design of laminated elastic bodies. We use a recently introduced model of delamination, based on minimization of potential energy which includes the free (Gibbs-type) energy and (pseudo)potential of dissipative forces, to introduce and analyze a special mathematical program with equilibrium constraints. The equilibrium is governed by a ﬁnite sequence of coupled mathematical programs that have to be solved one after another in the direction of increasing time. We derive optimality conditions for the control problem and illustrate them on an academic example.

Keywords: Inelastic damage, mathematical program with equilibrium constraints, evolution equilibrium, hemivariational inequality

1.

Introduction

The design of crashworthy vehicles depends upon developing structures capable of absorbing large amounts of crash energy. Composite materials have been shown to have energy absorbing properties superior to conventional metallic structures. With the goal of designing crash elements (structural elements with high energy absorption by crash), we consider laminated structures. The energy is absorbed by delamination of such structures. Mathematical simulation of vehicle crash requires,

∗ On

leave from the Academy of Sciences of the Czech Republic.

Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

ﬁrst of all, ﬁnding a reliable mathematical model. In this paper, we use a recently introduced model of delamination, based on minimization of potential energy which includes the free (Gibbs-type) energy and (pseudo)potential of dissipative forces ([9]). Based on this model, we introduce and analyze an evolution control problem with the ﬁnal goal of designing optimal shape of crash elements. Delamination is a progressive separation of bonded laminate and, simultaneously, degradation of the used adhesive. The mechanism of delamination is very complex and involves phenomena like debonding and unilateral contact with nonmonotone friction. In the recent literature, the modeling of the delamination problem is generally approached either by using fracture mechanics (see, e.g., [3, 20]) or by introducing special constitutive laws for the interface material in the spirit of damage mechanics, or simply quasistatically. In the second approach, delamination is described by a damage variable reﬂecting the destruction of the bonds in the a priori known delamination surface (see, e.g., [5, 17]). Another, static approach was proposed by Panagiotopoulos [16] who formulated the problem of equilibrium positions as a hemivariational inequality (HVI), a generalization of variational inequality for nonmonotone operators; see also [2]. Thus, this method has limited applications only in processes with simple time-dependent loadings. After discretization by the ﬁnite elements method, this approach results in a nonsmooth and nonconvex optimization problem. The variables of this problem are the elements of the discretized displacement vector. In the model introduced in [9], delamination is considered as a fracture-like process that can run along a priori known surfaces between homogeneous isotropic elastic bodies that are in frictionless unilateral contact. It is as an activated and rate-independent process, based on the philosophy that a speciﬁc energy is needed to cut the macromolecular structure of the adhesive, no matter how fast or slow this process is. This model is supported by a rigorous analysis based on the apparatus recently developed for rate-independent process in, e.g., [10–11]. In this paper, we will refer this model as RI model. After discretization in time and space, the RI model results in a sequence of smooth nonconvex optimization problems. Compared to the static HVI model, the dimension of one problem increases by the number of damage parameters. Assume that the discretized RI model contains parameters, the values of which should be optimized with respect to an (upper-level) objective. In this way, we arrive at a special mathematical program with equilibrium constraints (MPEC) in which the underlying equilibrium has evolution nature. From the viewpoint of optimality conditions, similar problems were investigated in [21]. There, however, the equilibrium is Copyright © 2005 Marcel Dekker, Inc.

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171

governed by a standard optimal control problem (i.e., the optimization is performed over the whole time interval). In our case, we have to do with a ﬁnite sequence of coupled mathematical programs that have to be solved one after another in the direction of increasing time. This difference is naturally reﬂected in the character of the resulting optimality conditions. The paper is organized as follows: In the next section, we introduce the RI model of [9] and compare it to the static VI model. After that, we present a conceptual MPEC with the discretized RI delamination process as equilibrium constraint. Section 3 is devoted to necessary optimality conditions for this MPEC. As a workhorse we use the generalized diﬀerential calculus of B. Mordukhovich ([13, 12]) which proved to be an eﬃcient tool in the treatment of various equilibria. The resulting conditions are illustrated by means of an academic example introduced in Section 2. The following notation is employed: If f is a diﬀerentiable function of two variables, then ∇1 f and ∇2 f denote the partial derivatives with respect to the ﬁrst and the second variable, respectively. B is the unit ball and for a multifunction Q[Rn ; Rm ], Gph Q := {(x, y) ∈ Rn × Rm | y ∈ Q(x)}. To describe the local properties of sets, multifunctions, and realvalued functions, we make use of suitable concepts from the generalized diﬀerential calculus of B. Mordukhovich ([13],[12]). So, NΩ (x) denotes the limiting normal cone to the set Ω at x ∈ clΩ, D ∗ Q(x, y)(·) denotes the coderivative of the multifunction Q at (x, y) ∈ cl Gph Q, and ∂f (x) is the limiting subdiﬀerential of a real-valued function f at x ∈ domf . For the reader’s convenience, we close this section with a useful stability concept for multifunctions which plays an important role in Section 3.

Definition 1 ([19]) A multifunction Q[Rn ; Rm ] is calm at (¯ x, y¯) ∈ Gph Q provided there is a constant κ ∈ R+ along with neighborhoods U of x ¯ and V of y¯ such that Q(x) ∩ V ⊂ Q(¯ x) + κx − x ¯B for all x ∈ U.

2.

Delamination Process

In this section we recall the rate-independent model of the delamination process, as recently introduced in [9]. In this model, after a (time and space) discretization, one has to solve in each time step a smooth nonconvex optimization problem. We further propose a conceptual optimization problem in which the discretized delamination process arises as a constraint. Copyright © 2005 Marcel Dekker, Inc.

172

2.1

CONTROL AND BOUNDARY ANALYSIS

Modeling

For simplicity of notation, we only consider laminates consisting of two elastic bodies Ω1 and Ω2 with an interface boundary denoted by Γ12 ; see Figure 1. The two bodies are in unilateral contact along Γ12 ; further they are glued along Γ12 by an adhesive. The matrix b : Γ12 → R2×2 reﬂects the elastic properties of the adhesive. We adopt, in fact, a dimensional reduction of the two-dimensional adhesive layer to a one-dimensional surface Γ12 . The state of the system will be considered as y = (u1 , u2 , ζ) where ui : Ωi → R2 is the (small) displacement in the domain Ωi , i = 1, 2, and ζ : Γ12 → [0, 1] is a damage parameter indicating how much of the adhesive is eﬀective: 1 means 100% of the adhesive glues at x ∈ Γ12 , 0 means that the surface is completely delaminated at the current point x ∈ Γ12 , and 0 < ζ(x) < 1 means that some portion of macromolecules of the adhesive is already cut while the rest is still eﬀective. A rate-independent delamination model has recently been introduced in [9]. This model is based on the minimization of the elastic stored energy and the dissipation potential subject to several constraints. The constraints reﬂect – the unilateral contact of the two elastic bodies – the time-dependent Dirichlet loading by a “hard device” – the irreversibility (in time) of the delamination of the adhesive The resulting minimization problem is ﬁrst discretized in the time variable and further in the space variable by the standard ﬁnite element

Γn

Ω1

&

Γt

Ω2

Figure 1.

Copyright © 2005 Marcel Dekker, Inc.

Γt Γ12

Γt

On the Modeling and Control of Delamination Processes

173

method. Below we present the fully discretized problem that will be further studied in the subsequent sections. We use the same ﬁnite element mesh for both variables, u and ζ. In order to allow for the separation (delamination) of the elastic bodies, the joint boundary Γ12 is discretized by N pairs of nodes, say (n1,j , n2,j ), j = 1, . . . , N , with ni,j ∈ Ωi , i = 1, 2. At the beginning of the delamination process the nodes n1,j , n2,j have the same positions for all j = 1, . . . , N . Let A1 , A2 be the stiﬀness matrices of the elastic bodies Ω1 , Ω2 , respectively. The discretized elastic stored energy becomes V (u, ζ) =

2

uTi Ai ui +

i=1

N

ωj ζj (u1,j −u2,j ) b(nj )(u1,j −u2,j )

j=1

where ωj are integration weights. Similarly, the discretized dissipation potential is R(ζ) =

N

−ωj d(nj )ζj .

j=1

Here d(x) is a phenomenological parameter having the meaning of a speciﬁc energy needed to delaminate the surface Γ12 at a point x ∈ Γ12 , i.e., the energy needed to switch ζ(x) from 1 to 0. This energy is irreversibly dissipated to the structural change of the adhesive. The case R(ζ) = +∞ will be respected by a corresponding linear constraint. We assume that the laminate is loaded by time-dependent unilateral Dirichlet loading (“hard-device”) on parts of the boundaries of Ω1 , Ω2 . The load vector (prescribed displacements) at a time step i is denoted by ui . Let L1 be a rectangular matrix selecting the “loaded” boundary components from the whole vector u. Finally, denote by L2 a rectangular matrix that takes care of the nonpenetration of Ω1 and Ω2 on Γ12 . The discrete version of the delamination problem in time step i is then (we omit the current time step index for simplicity): Minimize V (u, ζ) + R(ζ−ζ i−1 ) Subject to

(1)

L 1 u ≥ ui L2 u ≤ 0 ζ i−1 ≥ ζ ≥ 0 Copyright © 2005 Marcel Dekker, Inc.

componentwise.

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CONTROL AND BOUNDARY ANALYSIS

Here the index i−1 refers to the previous time step. The irreversibility of the dissipation process is guaranteed by the left-hand side of the last constraint. As we are only interested in the components of the displacement vector lying on the boundary Γ12 , we can eliminate all components corresponding to the interior nodes. This reduces the number of variables in (1) to the number of interface boundary nodes times ﬁve (two times two components of the displacement vector plus the components of ζ) plus the number of boundary nodes with prescribed nonzero Dirichlet condition (loaded nodes) times two. In [9] it was shown that (1) can be eﬃciently solved by state-of-the-art optimization software and that the proposed modeling of the delamination process delivers reasonable results.

2.2

Relation to the HVI Model

Let us brieﬂy show the relation of the RI model described above to the HVI model, introduced by Panagiotopoulos and numerically solved in [2]. The hemivariational inequality model of the static delamination problem amounts to ﬁrst-order optimality condition of the following nonsmooth nonconvex optimization problem (we use the same notation as in the previous section): Minimize 2

uTi Ai ui +

i=1

N

ωj min{(u1,j −u2,j ) b(nj )(u1,j −u2,j ), d(nj )}

j=1

Subject to L 1 u ≥ ui L2 u ≤ 0 . Recall the simple fact that, for smooth convex functions f and g, the set of (Clarke) stationary points of the (nonsmooth) pointwise minimum function min{f (x), g(x)} is equal to the set of x-components of stationary points of the (smooth) function αf (x)+(1−α)g(x), α ∈ [0, 1]. Hence the (Clarke) stationary points of problem (1) are just the u-components of stationary points of the smooth problem (1), assuming that ζ i−1 ≡ 1. In other words, solving the static HVI model “is the same” as solving the RI model for the ﬁrst time step, starting from the non-delaminated state (of course, one has to take into account that diﬀerent algorithms may ﬁnd diﬀerent stationary points of these two nonconvex problems). Copyright © 2005 Marcel Dekker, Inc.

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175

The static model tries to “jump” directly into the solution at the terminal time—the time discretization is reduced to only one interval. The HVI results may then only be reliable for monotone (in time) loadings and simple geometries, contrary to the RI model that allows for general nonmonotone loading (and unloading). The diﬀerence between the two models, even for monotone loading and simple problems, is clearly seen when solving the optimal control problems; see, in particular, Example 2 in Section 4.

2.3

Control

Our second goal is to control the (discretized) RI delamination process by certain parameters. The aim is to ﬁnd such parameters that an objective function depending on the terminal state is minimized. For instance, having in mind our motivation from the Introduction, we want to ﬁnd such a shape of the boundaries of Ω1 , Ω2 that, at the terminal time, as much energy is dissipated as possible. This optimal control problem can thus be written as the following “conceptual” MPEC: Minimize ϕ(x, y) Subject to x ∈ Uad , y is the terminal state of the delamination process depending on parameter x,

(2)

where Uad ⊂ Rn is the set of admissible controls. The detailed formulation of the problem is given in the next section. Before going on, let us introduce a simpliﬁed example that demonstrates the diﬃculties connected with the control of the delamination process. Example 1 Consider the four-string example as shown in Figure 2. In reality, the strings are at the same horizontal position; here they are plotted with some gap for presentation reasons. The elasticity modulae of the strings are e1 , . . . , e4 . Assume that e3 = e4 and denote it by e. The end-nodes of the strings are denoted n0 , n1 , n2 . The vertical displacements at the nodes are u0 , u1 , u2 . Node n0 is ﬁxed (u0 = 0), node n2 is subjected to nonzero Dirichlet boundary condition (prescribed displacements) u2 = u. Copyright © 2005 Marcel Dekker, Inc.

176

CONTROL AND BOUNDARY ANALYSIS n0 e1

e3 n1

e2

e4 n2

Figure 2.

The left-hand strings are elastic and do “never” break—they simulate the elastic bodies. The right-hand strings simulate the adhesive: they are also elastic but can break when the relative displacement reaches a certain value. The Dirichlet condition u depends on time. The equilibrium state in the i-th time interval is obtained by solving the optimization problem min

u1 ,u2 ,ζ1 ,ζ2

2 ' ( 2 2 i−1 ej (uj −uj−1 ) + ζj e(uj −uj−1 ) + (ζj −ζj )ed E := j=1

Subject to

(3)

u 2 ≥ ui uj − uj−1 ≥ 0, ζ

i−1

j = 1, 2

≥ζ ≥0

The ﬁrst term under the sum is the strain energy of the elastic strings 1 and 2. The second term is the strain energy of the breakable strings 3 and 4 (the “adhesive”). The third term is the dissipation energy. Number d is the energy dissipated by breaking one string (this is given in our case). In the MPEC, the control variables are e1 and e2 . The goal is to ﬁnd such design that, at the terminal time k, as much energy is dissipated as possible: ϕ := ζ1k + ζ2k max e1 ,e2 ,u1 ,u2 ,ζ1 ,ζ2

Subject to e1 + e2 = 2 (u1 , u2 , ζ1 , ζ2 ) solves (3) at time k. Assume the following problem data: d = 1 · 10−6 (dissipation parameter) Copyright © 2005 Marcel Dekker, Inc.

(4)

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177

uk = 0.007 (ﬁnal prescribed displacement) e = 10. The optimal solution is e1 = 1,

e2 = 1,

ζ1k = ζ2k = 0, ϕ = 2,

uk1 = 0.0035,

uk2 = 0.007,

E = 4.45 · 10−5 ,

when both strings 3 and 4 break. However, there may be more solutions to (4) within some neighborhood of 1 that also lead to the break of both strings. One such solution is e1 = 1.05, ζ1k = ζ2k = 0, ϕ = 2,

e2 = 0.95, uk1 = 0.003325,

uk2 = 0.007,

E = 4.44388 · 10−5 .

Obviously, everything within the interval e1 ∈ (0.95, 1.05), e2 = 2 − e1 is a solution to (4). Note that, for instance, (e1 = 1.1, e2 = 0.9) is not an optimal control of the MPEC (4) anymore, because it gives ζ1k = 1, ζ2k = 0 as a solution of the state problem (3). That means only one string breaks and the corresponding upper-level criterium is only ϕ = 1. The next two sections are devoted to necessary optimality conditions for MPECs of the type (2).

3.

Optimality Conditions

The discretized delamination model, introduced and discussed in the previous section, can be generally written down in the form of a ﬁnite sequence of coupled optimization problems Minimize f i (y i−1 , y i ) Subject to

(5) y ∈ Γ (y i

i

i−1

), i = 1, 2, . . . , k

0

y given, where y i−1 ∈ Rm is the parameter, y i ∈ Rm is the unknown variable, f i (y i−1 , ·) is the objective, and the multifunction Γi [Rm ; Rm ] speciﬁes the feasible set. This sequence of optimization problems has to be Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

solved starting from the initial “state” y 0 , in the direction of the increasing time index. In this way, one obtains eventually the whole trajectory y 1 , y 2 , . . . , y k . The aim of this section is to analyze MPECs with equilibria governed by such a sequence of coupled optimization problems. In each of them, additionally, a control variable x ∈ Rn arises so that the ith problem attains the “controlled” form Minimize f i (x, y i−1 , y i ) Subject to

(6)

y i ∈ Γi (x, y i−1 ). Because our attention is paid to necessary optimality conditions for MPECs with such equilibria, we can replace the single problems (6) by the respective ﬁrst-order necessary optimality conditions. Assuming that all objectives f i (x, y i−1 , ·) are continuously diﬀerentiable for each pair (x, y i−1 ), these conditions can be written down in the form 0 ∈ ∇3 f i (x, y i−1 , y i ) + NΓ(x,yi−1 ) (y i ), i = 1, 2, . . . , k.

(7)

Instead of the sequence of relations (6), we can now consider the sequence of coupled generalized equations (GEs) 0 ∈ F i (x, y i−1 , y i ) + Qi (x, y i−1 , y i ), i = 1, 2, . . . , k,

(8)

where F i [Rn × Rm × Rm → Rm ], Qi [Rn × Rm × Rm ; Rm ] represent the single-valued and the multivalued part, respectively. This generalization enlarges our applicability area also to equilibria governed, e.g., by a ﬁnite sequence of coupled complementarity problems or variational inequalities. In this way, we have arrived at a hopefully useful paradigm describing a speciﬁc time evolution of a fairly large class of equilibrium problems. On the respective MPEC we impose the following simplifying assumptions: (i) The (upper level) objective depends only on x and y k . (ii) The state variables y 1 , y 2 , . . . , y k are not subject to any constraints. (iii) All function F i are continuously diﬀerentiable and all maps Qi have closed graphs. The ﬁrst two assumptions are not essential and can be removed. In the MPECs associated with the delamination process, however, they are fulﬁlled and so we decided to simplify the statement of the next theorem Copyright © 2005 Marcel Dekker, Inc.

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On the Modeling and Control of Delamination Processes

by imposing them from the very beginning. We thus have to deal with the following MPEC: Minimize ϕ(x, y k ) Subject to 0 ∈ F i (x, y i−1 , y i ) + Qi (x, y i−1 , y i ), i = 1, 2, . . . , k

(9)

y 0 given x ∈ Uad , where ϕ[Rn × Rm → R] is an (upper level) objective and Uad ⊂ Rn is as before the set of admissible controls. Throughout the rest of the paper it is assumed that ϕ is locally Lipschitz and Uad is nonempty and closed. By y we denote the trajectory (y 1 , y 2 , . . . , y k ) and m := km.

Theorem 2 Let (2 x, y2) be a (local) solution of MPEC ( 9). Denote z2i = −F i (2 x, y2i−1 , y2i ), i = 1, 2, . . . , k, and deﬁne the multifunction Ξ[Rm ; n R × Rm ] by Ξ(ξ 1 , ξ 2 , . . . , ξ k ) := (x, y) ∈ Uad × Rm |ξ i ∈ F i (x, y i−1 , y i ) + Qi (x, y i−1 , y i ), i = 1, 2, . . . , k} . Assume that Ξ is calm at (0, x 2, y2). Then there exist four sequences of adjoint vectors p1 , p2 , . . . , pk , q 1 , q 2 , . . . , q k , v 1 , v 2 , . . . , v k , w1 , w2 , . . . , wk and subgradients (κ, η) ∈ ∂ϕ(2 x, y2k ) such that (with yˆ0 = y 0 ) x, y2i−1 , y2i , z2i )(wi ), i = 1, 2, . . . , k (pi , q i , v i ) ∈ D∗ Qi (2 and the adjoint equation system x, y2k−1 , y2k ))T wk + v k 0 = η + (∇3 F k (2 0 = (∇3 F k−1 (2 x, y2k−2 , y2k−1 ))T wk−1 + v k−1 + (∇2 F k (2 x, y2k−1 , y2k ))T wk + q k

(10)

1 0 1 T 1 1 2 1 2 T 2 2 x, y , y2 )) w + v + (∇2 F (2 x, y2 , y2 )) w + q 0 = (∇3 F (2

(11)

.. .

is fulﬁlled. Moreover, one has 0 ∈κ+

k T x, y2i−1 , y2i ) wi + pi + NUad (2 x). ∇1 F i (2 i=1

Copyright © 2005 Marcel Dekker, Inc.

(12)

180

CONTROL AND BOUNDARY ANALYSIS

Proof. The constraints in (9) can be written down in the form 0 ∈ Φ(x, y) + Λ, x ∈ Uad , where

Φ(x, y) = −

x y0 y1 1 −F (x, y 0 , y 1 ) x y1 y2 2 −F (x, y 1 , y 2 ) ······ x y k−1 yk k −F (x, y k−1 , y k )

k and Λ = Xi=1 GphQi

Due to the imposed calmness assumption, one can invoke [14, Thm.2.4] which yields the existence of a Karush-Kuhn-Tucker (KKT) vector x, y2)) b = (p1 , q 1 , v 1 , −w1 , p2 , q 2 , v 2 , −w2 , . . . , pk , q k , v k , −wk ) ∈ NΛ (−Φ(2 and subgradients (κ, η) ∈ ∂ϕ(2 x, y2k ) such that κ NUad (2 x) 0 0 x, y2))T b + 0 ∈ ... − (∇Φ(2 .. . 0 0 η

.

(13)

k N x, y2)) = Xi=1 x, y2i−1 , y2i , z2i ) see [13, Prop.1.6], Because NΛ (−Φ(2 GphQi (2 x, y2i−1 , y2i , z2i ), i = 1, 2, . . . , k. In it follows that (pi , q i , v i , −wi ) ∈ NGphQi (2 this way relations (10) have been established. The ﬁrst line of (13) leads now directly to relation (12), whereas the remaining k lines generate the adjoint system (11).

The calmness assumption is automatically fulﬁlled provided Uad is convex polyhedral, all function F i are aﬃne, and all sets GphQi are unions of ﬁnitely many convex polyhedral sets. Indeed, in such a case, GphΞ is also a union of ﬁnitely many convex polyhedral sets and Ξ is locally upper Lipschitz around 0, cf. [18]. This is, however, a stronger Copyright © 2005 Marcel Dekker, Inc.

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181

property than the required calmness at (0, x 2, y2). Another possibility is to ensure the Aubin property of Ξ around (0, x 2, y2), which is also stronger than the required calmness condition. This can be done by the following Mangasarian-Fromowitz constraint qualiﬁcation.

Theorem 3 Assume that the system 0 = (∇3 F k (2 x, y2k−1 , y2k ))T wk + v k 0 = (∇3 F k−1 (2 x, y2k−2 , y2k−1 ))Twk−1 +v k−1 +(∇2 F k (2 x, y2k−1 , y2k ))T wk +q k .. . x, y 0 , y21 ))T w1 + v 1 + (∇2 F 2 (2 x, y21 , y22 ))T w2 + q 2 0 = (∇3 F 1 (2 k T ∇1 F i (2 0∈ x, y2i−1 , y2i ) wi + pi + NUad (2 x) i=1

x, y2i−1 , y2i , z2i )(wi ), i = 1, 2, . . . , k, possesses only with (pi , q i , v i ) ∈ D∗ Qi (2 the trivial solution p1 = p2 = . . . = pk = 0, q 2 = q 3 = . . . = q k = 0, v 1 = v 2 = . . . = v k = 0 and w1 = w2 = . . . wk = 0. Then Ξ has the Aubin property around (0, x 2, y2). The statement follows from the Mordukhovich characterization of the Aubin property ([19, Thm.9.40]) and standard rules of the coderivative calculus [12]. As shown in [14], the above condition is automatically fulﬁlled, provided the multifunctions Qi do not depend on the control and the multifunction ∆[Rm ; Rm ], deﬁned by x, y2i−1 , y2i ) ∆(ξ) = { y ∈ Rm | ξ i ∈ F i (2 x, y2i−1 , y2i )(y i−1 − y2i−1 ) + ∇2 F i (2 + ∇3 F i (2 x, y2i−1 , y2i )(y i − y2i ) x, y2i−1 , y2i ), i = 1, 2, . . . , k } , + Qi (2 is locally single-valued and Lipschitz around (0, y2) (Robinson’s strong regularity). This is, however, not the case if we deal with the delamination model of Section 2. The optimality conditions of Theorem 2 can be substantially simpliﬁed provided the maps Qi do not depend on x or y i−1 . This is expressed in the following corollaries.

Corollary 4 Let all assumptions of Theorem 2 be fulﬁlled and assume that the maps Qi , i = 1, 2, . . . , k, do not depend on x. Then there exist three sequences of adjoint vectors q 1 , q 2 , . . . , q k , v 1 , v 2 , . . . , v k , x, y2k ) such that w1 , w2 , . . . , wk and subgradients (κ, η) ∈ ∂ϕ(2 y i−1 , y2i , z2i )(wi ), i = 1, 2, . . . , k, (q i , v i ) ∈ D∗ Qi (2 Copyright © 2005 Marcel Dekker, Inc.

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the adjoint equation system ( 11) is satisﬁed, and 0∈κ+

k

T ∇1 F i (2 x, y2i−1 , y2i ) wi + NUad (2 x).

(14)

i=1

Corollary 5 Let all assumptions of Theorem 2 be fulﬁlled and assume that for i = 1, 2, . . . , k the maps Qi depend exclusively on variables y i , respectively. Then there exist two sequences of adjoint vectors v 1 , v 2 , . . . , v k , w1 , w2 , . . . , wk and subgradients (κ, η) ∈ ∂ϕ(2 x, y2k ) such that y i , z2i )(wi ), i = 1, 2, . . . , k v i ∈ D∗ Qi (2 and the adjoint equation system x, y2k−1 , y2k ))T wk + v k 0 = η + (∇3 F k (2 0 = (∇3 F k−1 (2 x, y2k−2 , y2k−1 ))T wk−1 + v k−1 + (∇2 F k (2 x, y2k−1 , y2k ))T wk ······ x, y 0 , y21 ))T w1 + v 1 + (∇2 F 2 (2 x, y21 , y22 ))T w2 0 = (∇3 F 1 (2

(15)

is fulﬁlled. Moreover, relation ( 14) holds true. Remark As in the classic discrete-time optimal control problems [8], the adjoint systems (11),(15) have to be solved backwards starting from the terminal condition. Together with the GEs (8) they represent a special two-point boundary value problem.

4.

Optimization of Delamination Processes

The aim of this section is to apply the preceding theory to an MPEC, where the equilibrium is governed by a sequence of optimization problems (6) with the data (functions f i and multifunctions Γi ) speciﬁed in Section 2. Without giving the structure of f i in detail, the ith problem attains the form Minimize f (x, ζ i−1 , ui , ζ i ) Subject to

¯i L1 ui ≥ u L2 ui ≤ 0

(16)

ζi ≥ 0 ζ i ≤ ζ i−1 , where the control x arises only in the objective, all other variables were described in Section 2, and also the problem data f, L1 L2 were deﬁned Copyright © 2005 Marcel Dekker, Inc.

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183

there. The next step consists in the construction of such optimality conditions for (16) that will facilitate a subsequent application of Theorem 2 as much as possible. To this purpose, we introduce the polyhedral sets Ωi := (ui , ζ i ) | L1 ui ≥ u ¯i , ζ i ≥ 0 , consisting (due to the structure of L1 ) only of lower bounds for some variables.

Theorem 6 Let x = x ˜ and ζ i−1 = ζ˜i−1 be given and (˜ u, ζ˜i ) be a solution ˜ i such of the respective problem ( 16). Then there exists a KKT vector λ that ˜ i + NΩi (˜ x, ζ˜i−1 , u ˜i , ζ˜i ) + (∇2,3 G(ζ˜i−1 , u˜i , ζ˜i ))T λ ui , ζ˜i ) 0 ∈ ∇3,4 f (˜ (17) ˜ i ), 0 ∈ −G(ζ˜i−1 , u ˜i , ζ˜i ) + N l (λ R+

where G(ζ

i−1

i

i

,u ,ζ ) : =

L 2 ui i ζ − ζ i−1

and l is the dimension of the image space of G. Note that in the above optimality conditions we do not need any constraint qualiﬁcation, because all functions arising in the constraints are aﬃne. The GE (17) is already in the required form (8); it suﬃces to put y i : = (ui , ζ i , λi ), ∇3,4 f (x, ζ i−1 , ui , ζ i ) + (∇2,3 G(ζ i−1 , ui , ζ i ))T λi i−1 i F (x, y , y ) : = −G(ζ i−1 , ui , ζ i ) and

i

i

Q (y ) : =

NΩi (ui , ζ i ) NRl (λi )

.

+

Since the sets Ωi are translated nonnegative orthants, the coderivatives of the maps Qi can easily be computed on the basis of [15, Lemma 2.1]. Moreover, F is aﬃne and all sets GphQi are unions of a ﬁnite number of convex polyhedral sets ([18]). Hence, we do not need to take care about the calmness condition in Theorem 2, whenever Uad is convex polyhedral. We now illustrate the application of Corollary 5 and the form of the resulting optimality conditions by means of an MPEC generated on the basis of the academic equilibrium from Example 1. Copyright © 2005 Marcel Dekker, Inc.

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Example 2 Consider the four-string problem from Example 1, where we set xj = ej for j = 1, 2. The respective problem (16) thus attains the form 2 Minimize (x1 +eζ1i )(ui1 )2 + (x2 +eζ2i )(ui2 −ui1 )2 − ed (ζji −ζji−1 ) j=1

Subject to

ui1 − ui2 ≤ 0 ζ1i − ζ1i−1 ≤ 0 ζ2i − ζ2i−1 ≤ 0 (ui1 , ui2 , ζ1i , ζ2i )

(18)

∈ Ωi ,

where Ωi = ui1 , ui2 , ζ1i , ζ2i | ui1 ≥ 0, ui2 ≥ u ¯i , ζ1i ≥ 0, ζ2i ≥ 0 . The aim is to maximize the energy dissipated until the terminal time k so that ϕ(x, y k ) = ζ1k + ζ2k . Finally, Uad = {x ∈ R2 | x1 + x2 = 2} and ζ10 = ζ20 = 1 (the remaining initial values are not needed). The application of Theorem 6 to (18) yields the following GE: 2x1 ui1 − 2x2 (ui2 − ui1 ) + 2ζ1i eui1 − 2ζ2i e(ui2 − ui1 ) + λi1 2x2 (ui − ui ) + 2ζ i e(ui − ui ) − λi 2 1 2 2 1 1 0∈ e(ui1 )2 − de + λi2 i i 2 i e(u2 − u1 ) − de + λ3 + NΩi (ui1 , ui2 , ζ1i , ζ2i ) 0∈

(19)

− ui1 ui2 ζ1i−1 − ζ1i + NR3+ (λi1 , λi2 , λi3 ). ζ2i−1 − ζ2i

Hence, even in this simple academic example, one has m = 7, n = 2, 2k , ζ21 , . . . , ζ2k ), where and l = 3. It is clear that each vector (2 x, u 21 , . . . , u i i 2 2 u , ζ2 ) ∈ R × R is a solution of (18) with x = x 2, ζ i−1 = ζ2i−1 x 2 ∈ Uad , (2 for i = 1, 2, . . . , k and ζ21k = ζ22k = 0 generates a global solution of the above MPEC. As explained in Example 1, for the data given there, it is possible to construct such a vector on the basis of physical consideration; 22 = 1, u 211 = 0.0032, u 212 = in the case k = 2 we obtain, e.g., x 21 = x 2 2 1 1 2 2 22 = 0.007, ζ21 = 0, ζ22 = 1, ζ21 = ζ22 = 0. The 0.0035, u 21 = 0.0035, u 21 = 0, λ 21 = 10−6 , λ 22 = 21 = λ optimality conditions (19) are fulﬁlled with λ 1 2 3 1 Copyright © 2005 Marcel Dekker, Inc.

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On the Modeling and Control of Delamination Processes

22 = 0, and the vectors z2i ∈ NΩ1 (2 2i , λ 2i 2i 22 = λ ui1 , u 2i2 , ζ21i , ζ22i ) × NR3+ (λ λ 2 3 1 2 , λ3 ), i = 1, 2, equal to z21 = (0, −0.0064, −10−5 , 0, −0.0003, −1, 0)T z22 = (0, −0.007, −0.0001, −0.0001, −0.0035, 0, −1)T . To evaluate the coderivatives of the multivalued part in (19), we invoke, as already mentioned, [15, Lemma 2.1]. It follows that v11 = v41 = v71 = 0, w21 = w31 = w51 = w61 = 0 and v12 = 0, w22 = w32 = w42 = w52 = w72 = 0. Only the adjoint variables v62 and w62 are related in a more complicated way, because the respective constraint violates the strict complementarity. One has either v62 w62 = 0 or v62 < 0, w62 < 0. The respective adjoint equation system (15) can thus be substantially simpliﬁed and attains the form 0 4 0 0 0 −2 v2 0 2 1 0.07 v32 −1 w2 2 1 + −0, 07 0 v 1 0= + 2 w 42 6 0 1 v5 0 2 0 0 v6 0 0 0 0 v72 0=

24 −0.006 0 −22 0.006 0 1 0.064 0 0 w1 1 −0.006 0 −1 w41 + 1 0 0 w7 0 0 0 0 1 0

0 v21 v31 0 v51 v61 0

+

0 0 w62 0 0 0 0

.

This equation system in variables (v 1 , v 2 , w1 , w2 ) possesses a solution in which (w1 , w2 ) = 0. Because κ = 0, relation (14) holds true and the optimality conditions of Corollary 5 have been veriﬁed.

5.

Conclusion

The optimality condition derived in Section 3 can be used to test the stationarity of approximate solutions to considered MPECs computed by Copyright © 2005 Marcel Dekker, Inc.

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available numerical methods [1],[4]. The complexity of these MPECs and the respective optimality conditions increases naturally with the number of time levels (k). On the other hand, as explained in Section 2.1, in modeling of the delamination processes, a ﬁne time discretization is really needed to compute a physically acceptable equilibrium. In MPECs associated with such equilibria, the need of a large k may be even stronger, because the (upper level) objective may force the equilibrium to attain physically unacceptable values. For instance, in the illustrative four-string problem, the choice k = 2 is deﬁnitely too small to solve the associated MPEC (Example 2) by an SQP code. As soon as the starting point is not extremely close to a solution, the procedure terminates at a physically unacceptable equilibrium. The choice of k is thus a certain trade-oﬀ between the complexity of the problem (and the associated optimality conditions) and our eﬀort to arrive at a physically acceptable equilibrium.

Acknowledgments This research was supported by grant A1075005 of the Grant Agency of the Academy of Sciences of the Czech Republic and by BMBF project 03ZOM3ER.

References [1] M. Anitescu. On solving mathematical programs with complementarity constraints as nonlinear programs. Preprint ANL/MCS-P864-1200, Argonne National Laboratory, Argonne, IL, 2000. [2] C. C. Baniotopoulos, J. Haslinger, and Z. Mor´ avkov´ a. Mathematical modeling of delamination and nonmonotone friction problems by hemivariational inequalities. Technical report, Charles University, Prague, 2002. [3] B. D. Davidson. An analytical investigation of delamination front curvature in double cantilever beam specimens. J. Compos. Mater., 24:1124–1137, 1990. [4] R. Fletcher and S. Leyﬀer. Numerical experience with solving MPECs as NLPs. Report NA\210, University of Dundee, 2002. [5] M. Fr´emond. Dissipation dans l’adhrence des solides. C. R. Acad. Sci., Paris, Sr. II, 300:709–714, 1985. [6] M. Fr´emond. Contact with adhesion. In J. J. Moreau, G. Panagiotopoulos, and G. Strang, editors, Topics in Nonsmooth Mechanics. Birkh¨ auser, Basel, 1988. [7] H. Guidouche and N. Point. Unilateral contact with adherence. In A. Bossavit, A. Damlamian, and M. Fr´emond, editors, Free Boundary Problems: Applications and Theory IV, pages 340–346. Pitman Res. Notes Math. 121, Longman Science and Technology, UK, 1985. [8] H. Halkin. Optimal Control for Systems Described by Diﬀerence Equations. Academic Press, New York, 1964.

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[9] M. Koˇcvara, A. Mielke, and T. Roub´ıˇcek. Rate-independent approach to the delamination problem. Preprint 2003/29, SFB 404 Mehrfeldprobleme in der Kontinuumsmechanik, Universit¨ at Stuttgart, 2003. [10] A. Mielke and F. Theil. A mathematical model for rate-independent phase transformations with hysteresis. In H.-D. Alder, R. Balean, and R. Farwig, editors, Models of Continuum Mechanics in Analysis and Engineering, pages 117–129. Shaker Ver., Aachen, 1999. [11] A. Mielke, F. Theil, and V. I. Levitas. A variational formulation of rateindependent phase transformations using an extremum principle. Archive Rat. Mech. Anal., 162:137–177, 2002. [12] B. S. Mordukhovich. Generalized diﬀerential calculus for nonsmooth and setvalued mappings. J. Math. Anal. Appl., 183:250–288, 1994. [13] B.S. Mordukhovich. Approximation Methods in Problems of Optimization and Control. Nauka, Moscow, 1988. In Russian. [14] J. Outrata. A generalized mathematical program with equilibrium constraints. SIAM J. Control. Optim., 38:1623–1638, 2000. [15] J. V. Outrata. Optimality conditions for a class of mathematical programs with equilibrium constraints. Math. Oper. Res., 24:627–644, 1999. [16] G. Panagiotopoulos. Hemivariational Inequalities. Springer-Verlag, Heidelberg, 1993. [17] N. Point and E. Sacco. A delamination model for laminated composites. Int. J. Solids Structures, 33:483–509, 1996. [18] S. M. Robinson. Some continuity properties of polyhedral multifunctions. Mathematical Programming Study, 14:206–214, 1981. [19] R. T. Rockafellar and R. Wets. Variational Analysis. Springer-Verlag, Berlin, 1998. [20] H. L. Schreyer, D. L. Sulsky, and S.-J. Zhou. Modelling delamination as strong discontinuity with the material point method. Comput. Methods Appl. Mech. Engrg., 191:2483–2507, 2002. [21] J. J. Ye. Necessary optimality conditions for bilevel dynamic problems. In Proc. of the 36th Conference on Decision and Control, San Diego, pages 1405–1410. IEEE, 1997.

Copyright © 2005 Marcel Dekker, Inc.

CHAPTER XV ON A SPECTRAL VARIATIONAL PROBLEM ARISING IN THE STUDY OF EARTHQUAKES Multiplicity and Perturbation from Symmetry Vicentiu Radulescu Department of Mathematics University of Craiova Craiova, Romania

Abstract

We consider an eigenvalue variational inequality problem arising in the earthquake initiation. Our purpose is twofold. Firstly, in the symmetric case, we establish the existence of inﬁnitely many distinct solutions. Next, in the case where the problem is aﬀected by a non-symmetric perturbation, we prove that the number of solutions of the perturbed problem becomes larger and larger if the perturbation “tends” to zero with respect to a suitable topology. Since the canonical energy functionals are included neither in the theory of monotone operators, nor in their Lipschitz perturbations, the proofs of the main results rely on nonsmooth critical point theories in the sense of De Giorgi and Degiovanni combined with methods from algebraic topology.

Keywords: Earthquake initiation, variational inequality, non-smooth critical point theory, perturbation

Introduction The purpose of this paper is twofold: ﬁrst, we establish a multiplicity result for a nonlinear symmetric variational inequality; next, we study the eﬀect of an arbitrary perturbation. For related results and further comments we refer to our recent papers [11] (for an appropriate variational inequality) and [2, 5, 14] for the hemivariational framework. We have been inspired in this work by the following simple phenomenon which occurs in elementary mathematics. Usually, an equation with a certain symmetry admits inﬁnitely many solutions. For instance, the equation sin x = 1/2 (x ∈ R) has an inﬁnite number of solutions. In this case, the “symmetry” is given by periodicity. If the above equation Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

is aﬀected in an arbitrary non-symmetric way by a certain perturbation, then the number of solutions of the new equation becomes larger and larger if the perturbation “tends to zero” in a suitable sense. For instance, the equation

sin x =

1 + ε x2 , 2

x∈R

(1)

has ﬁnitely many solutions, for any ε > 0. However, the number of solutions of (1) tends to inﬁnity if ε tends to 0. Our purpose in this work is to illustrate the above elementary phenomenon to the study of a nonlinear eigenvalue variational inequality arising in earthquake initiation. More precisely, using a multiplicity result of Lusternik-Schnirelmann type combined with the fact that an adequate function space is inﬁnitely dimensional, we ﬁrst establish the existence of inﬁnitely many solutions. Next, we are concerned with the study of the eﬀect of a small non-symmetric perturbation and we prove that the number of solutions of the perturbed problem tends to inﬁnity if the perturbation tends to zero with respect to an appropriate topology. The main novelty in our framework is the presence of the convex cone of functions with non-negative jump across an internal boundary which is composed of a ﬁnite number of bounded connected arcs.

1.

Main Results and Physical Motivation

Let Ω ⊂ RN be a domain, not necessarily bounded, containing a ﬁnite number of cuts. Its boundary ∂Ω is supposed to be smooth and divided ¯ and the internal into two disjoint parts: the exterior boundary Γd = ∂ Ω i one Γ composed by Nf bounded connected arcs Γf , i = 1, .., Nf , called cracks or faults. On Γ we denote by [ ] the jump across Γ, (i.e., [w] = w+ − w− ) and by ∂n = ∇ · n the corresponding normal derivative with the unit normal n outwards the positive side. Set V = {v ∈ H 1 (Ω); v = 0 on Γd }. Denote by · the norm in the space V , and by Λ0 : L2 (Ω) → L2 (Ω)∗ and Λ1 : V → V ∗ the duality 2

isomorphisms Λ0 u(v) = Ω uvdx, for any u, v ∈ L (Ω), and Λ1 u(v) = Ω ∇u · ∇vdx, for any u, v ∈ V. Consider the Lipschitz map γ = i ◦ η : V → L2 (Γ), where η : V → H 1/2 (Γ) is the trace operator, η(v) = [v] on Γ and i : H 1/2 (Γ) → L2 (Γ) is the embedding operator. Let K be the convex closed cone deﬁned by K = {v ∈ V ; [v] ≥ 0 on Γ}. Copyright © 2005 Marcel Dekker, Inc.

On a Spectral Variational Problem Arising in the Study of Earthquakes

Consider the nonlinear eigenvalue problem ﬁnd u ∈ K and λ ∈ Rsuch that ∇u · ∇(v − u)dx + j (γ(u(x)); γ(v(x)) − γ(u(x))) dσ+ Ω Γ λ u(v − u)dx ≥ 0, ∀v ∈ K,

191

(2)

Ω

where j(t) = − β2 t2 , t ∈ R, for some real constant β. If N = 2, then any solution of problem (2) can be viewed as a stationary solution of an appropriate evolution variational inequality that describes the slip-dependent friction law introduced in the geophysical context of earthquake modeling (see [1], [3]). We look for w : R+ × Ω → R solution of the wave equation ∂tt w(t) = c2 ∆w(t) in Ω, with the boundary condition w(t) = 0 on Γd . On the contact zone Γ we have [∂n w] = 0 and the following slip-dependent friction law (introduced in the geophysical context of earthquakes modeling) is assumed (see [1], [3]): G∂n w(t) = −µ(|[w(t)]|))Ssign([∂t w(t)]) − q, if [∂t w(t)] = 0, |G∂n w(t) + q| ≤ µ(|[w(t)]|)S if ∂t [w(t)] = 0, where G < denotes the elastic medium shear rigidity, ρ is the density, and c = G/ρ is the shear velocity. The non-vanishing shear stress components are σzx = τx∞ + G∂x w, σzy = τy∞ + G∂y w, σxx = σyy = −S (S > 0 is the normal stress on the fault plane), and q = τx∞ nx + τy∞ ny . The initial conditions are w(0) = w0 ,

∂t w(0) = w1 in Ω.

Any solution of the above problem satisﬁes the following nonlinear eigenvalue problem: ﬁnd w : [0, T ] → V such that 1 ∂tt w(t)(v − ∂t w(t) dx + ∇w(t) · ∇(v − ∂t w(t)) dx + (3) c2 Ω Ω S 1 µ(|[w(t)]|)(|[v]| − |[∂t w(t)]|) dσ ≥ q([v] − [∂t w(t)]) dσ, Γ G Γ G for all v ∈ V . Copyright © 2005 Marcel Dekker, Inc.

192

CONTROL AND BOUNDARY ANALYSIS

The main diﬃculty in the study of problem (3) is the non-monotone dependence of µ with respect to the slip |[w]|. However, in modeling unstable phenomena, as earthquakes, we have to expect “bad” mathematical properties of the operators involved in the abstract problem. The existence of a solution w ∈ W 1,∞ (0, T, V ) ∩ W 2,∞ (0, T, L2 (Ω)) (if N = 2) was recently proved by Ionescu et al. [10]. The uniqueness was obtained only in the one-dimensional case. Since our intention is to study the evolution of the elastic system near an unstable equilibrium position, we shall suppose that q = µ(0)S. We remark that w ≡ 0 is an equilibrium solution of (3), and w0 , w1 may be considered as small perturbations of it. For simplicity, let us assume in the following that the friction law is homogeneous on the fault plane having the form of a piecewise linear function (see [13]):

µ(x, u) = µs −

µs − µd u if u ≤ 2Dc , µ(x, u) = µd if u > 2Dc , 2Dc

(4)

where u is the relative slip, µs and µd (µs > µd ) are the static and dynamic friction coeﬃcients, and Dc is the critical slip. Since the initial perturbation (w0 , w1 ) of the equilibrium (w ≡ 0) is small we have [w(t, x))] ≤ 2Dc for t ∈ [0, Tc ] for all x ∈ Γ, where Tc is a critical time for which the slip on the fault reaches the critical value 2Dc at least at one point. Hence for a ﬁrst period [0, Tc ], called the initiation phase, we deal with a linear function µ. Our aim is to analyze the evolution of the perturbation during this initial phase. That is why we are interested in the existence of solutions of the type

w(t, x) = sinh(|λ|ct)u(x),

w(t, x) = sin(|λ|ct)u(x)

(5)

during the initiation phase t ∈ [0, Tc ]. If we put the above expression in (3) and we have in mind that from (4) we have µ(s) = µs − (µs − µd )/(2Dc )s then we deduce that (u, λ) is solution of the problem (2), where β = (µs − µd )S/(2Dc G) > 0. The ﬁrst type of solution described by (5) has an exponential growth in time and corresponds to λ > 0. The second one has the same amplitude during the initiation phase and corresponds to λ < 0. Returning to problem (2), we observe that, due to its homogeneity, we can reformulate this problem in terms of a constrained inequality problem as follows. For any ﬁxed r > 0, consider the smooth manifold Copyright © 2005 Marcel Dekker, Inc.

On a Spectral Variational Problem Arising in the Study of Earthquakes

M = u ∈ V ; Ω u2 dx = r 2 . We shall study the problem ﬁnd u ∈ K ∩ M and λ∈ R such that ∇u · ∇(v − u)dx + j (γ(u(x)); γ(v(x)) − γ(u(x))) dσ+ Ω Γ λ u(v − u)dx ≥ 0, ∀v ∈ K.

193

(6)

Ω

The multiplicity of solutions to problem (6) is described in

Theorem 1 Problem ( 6) has inﬁnitely many solutions (u, λ) and the set of eigenvalues {λ} is bounded from above and its inﬁmum equals to −∞. Let λ0 = sup{λ}. Then there exists u0 such that (u0 , λ0 ) is a solution of ( 6). Moreover the function β −→ λ0 (β) is convex and the following inequality holds |∇v|2 dx + λ0 (β) v 2 dx ≥ β [v]2 dσ, ∀v ∈ K. (7) Ω

Ω

Γ

Our next purpose is to describe the eﬀect of an arbitrary perturbation in problem (2). More precisely, we consider the problem ﬁnd uε ∈ K and λε ∈ R such that ∇uε · ∇(v − uε )dx+ Ω (8) j + εg (γ(uε (x)); γ(v(x)) − γ(uε (x))) dσ+ Γ λε uε (v − uε )dx ≥ 0, ∀v ∈ K, Ω

where ε > 0 and g : R → R is a continuous function with no symmetry hypothesis, but satisfying the growth assumption 2(N − 1) : |g(t)| ≤ a(1 + |t|p ) , if N ≥ 3; N −2 , if N = 2. ∃ a > 0, ∃ 2 ≤ p < +∞ : |g(t)| ≤ a(1 + |t|p ) ∃ a > 0, ∃ 2 ≤ p ≤

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This hypothesis is motivated by the following embedding inequality of Ionescu [9] that will be used in an essential manner in the proof.

Lemma 2 (Lemma 5.1 in [9]). Let 2 ≤ α ≤ 2(N − 1)/(N − 2) if N ≥ 3 and 2 ≤ α < +∞ if N = 2. Then for β = [(α − 2)N + 2]/(2α) if N ≥ 3 or if N = 2 and α = 2 and for all (α − 1)/α < β < 1 if N = 2 and α > 2, there exists C = C(β) such that

1/α

(1−β)/2

β/2 |[u]|α dσ ≤C u2 dx |∇u|2 dx , (10) Γ

Copyright © 2005 Marcel Dekker, Inc.

Ω

Ω

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for any u ∈ V . Our perturbation result is

Theorem 3 For every positive integer n, there exists εn > 0 such that problem ( 8) has at least n distinct solutions (uε , λε ) if ε < εn . There exists and is ﬁnite λ0ε = sup{λε } and there exists u0ε such that (u0ε , λ0ε ) is a solution of ( 8). Moreover, λ0ε converges to λ0 as ε tends to 0, where λ0 was deﬁned in Theorem 1.

2.

Proofs

We ﬁrst recall some of the notions used in the proofs of Theorems 1 and 3. An important role in our arguments in order to locate the solution of problem (6) is played by the indicator function of M , that is, 0 , if u ∈ M IM (u) = +∞ , if u ∈ V \ M. Then IM is lower semicontinuous. However, since the natural energy functional associated to problem (6) is neither smooth nor convex, it is necessary to introduce a more general concept of gradient. We shall employ the following notion of lower subdiﬀerential which is due to De Giorgi, Marino, and Tosques [8].

Definition 4 Let X be a Banach space and let f : X → R ∪ {+∞} be an arbitrary proper functional. Let x ∈ D(f ). The Fr´echet (regular) subdiﬀerential of f at x is the (possibly empty) set f (y) − f (x) − ξ(y − x) − ∗ ≥0 . ∂ f (x) = ξ ∈ X ; lim inf y→x y − x An element ξ ∈ ∂ − f (x) is called a lower subgradient of f at x. Accordingly, we say that x ∈ D(f ) is a critical (lower stationary) point of f if 0 ∈ ∂ − f (x). Then ∂ − f (x) is a convex set. If ∂ − f (x) = ∅ we denote by grad− f (x) the element of minimal norm of ∂ − f (x), that is, grad− f (x) = min{ξX ∗ ; ξ ∈ ∂ − f (x)}. This notion plays a central role in the statement of the following basic compactness condition.

Definition 5 Let f : X → R ∪ {+∞} be an arbitrary functional. We say that (xn ) ⊂ D(f ) is a Palais-Smale sequence if supn |f (xn )| < +∞ Copyright © 2005 Marcel Dekker, Inc.

On a Spectral Variational Problem Arising in the Study of Earthquakes

195

and limn→∞ grad− f (xn ) = 0. The functional f is said to satisfy the Palais-Smale condition provided that any Palais-Smale sequence is relatively compact. Deﬁnition 4 implies that if g : X → R is Fr´echet diﬀerentiable and f : X → R ∪ {+∞} is an arbitrary proper function then ∂ − (f + g)(x) = ξ + g (x); ξ ∈ ∂ − f (x) , for any x ∈ D(f ). We also point out that in [4] it is proved the formula ∂ − IM (u) = {λΛ0 u; λ ∈ R} ⊂ L2 (Ω)∗ ⊂ V ∗ ,

(11)

for any u ∈ M , where Λ0 : L2 (Ω) → L2 (Ω)∗ denotes the canonical duality isomorphism.

2.1

Proof of Theorem 1

Deﬁne E = F + G : V → R ∪ {+∞}, where 1 |∇u|2 dx , F (u) = 2 Ω +∞ , and

β G(u) = − 2

if u ∈ K if u ∈ K

[γ(u(x))]2 dσ. Γ

Then E +IM is lower semicontinuous. Moreover, E +IM is the canonical energy functional associated to problem (6). This assertion is reﬁned in the following auxiliary result.

Lemma 6 Let (u, λ) be an arbitrary solution of problem ( 6). Then 0 ∈ ∂ − (E + IM )(u). Conversely, let u be a critical point of E + IM and denote λ = −2E(u)r −2 . Then (u, λ) is a solution of problem ( 6). Proof of Lemma 6. If (u, λ) is a solution of problem (6) then, by the deﬁnition of the lower subdiﬀerential, −λu ∈ ∂ − E(u).

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On the other hand, ∂ − (E + IM )(u) = ∂ − E(u) + ∂ − IM (u), So, by (11) and (12), 0 ∈ ∂ − (E + IM )(u). Copyright © 2005 Marcel Dekker, Inc.

for any u ∈ K ∩ M. (13)

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Conversely, let 0 ∈ ∂ − (E + IM )(u). Thus, by (11) and (13), there exists λ ∈ R such that (u, λ) is a solution of problem (6). If we put v = 0 in (6) then we deduce λr 2 ≤ −2E(u) and for v = 2u we get λr 2 ≥ −2E(u), that is λ = −2E(u)r −2 . The next step in our proof consists in showing that the functional E + IM satisﬁes the Palais-Smale condition. This is done by using standard arguments, but applied in the framework of the non-smooth critical point theory in the sense of De Giorgi, Marino, and Tosques. Due to the symmetry of problem (6), we can extend our study to the symmetric cone (−K). More precisely, if (u, λ) is a solution of (6) then u0 := −u ∈ (−K) ∩ M satisﬁes ∇u0 · ∇(v − u0 )dx + j (γ(u0 (x)); γ(v(x)) − γ(u0 (x))) dσ+ Ω Γ λ u0 (v − u0 )dx ≥ 0, for all v ∈ (−K). Ω

This means that we can extend the energy functional associated to prob, := K ∪ (−K). We put, by deﬁnition, lem (6) to the symmetric set K , if u ∈ K E(u) , E(−u) , if u ∈ (−K) E(u) = +∞ , otherwise. We are interested in ﬁnding the lower stationary points of the extended , + IM . energy functional J := E , ∩ M with the graph metric of E , deﬁned by We endow the set K , , d(u, v) = u − v + |E(u) − E(v)|,

, ∩ M. for any u, v ∈ K

, ∩ M, d). Denote by X the metric space (K The next step in the proof of Theorem 1 consists in showing , ∩ M ) = +∞. Lemma 7 We have CatX (K The proof is straightforward and is accomplished by using adequate tools from Algebraic Topology. The above results enable us to apply the Lusternik-Schnirelmann theorem in the sense established by Marino and Scolozzi [12]. This implies that problem (6) admits inﬁnitely many solutions (u, λ). Next, we observe that the set of eigenvalues is bounded from above. Indeed, if (u, λ) is a solution of our problem, then choosing v = 0 in (6) and using (10), it follows that λr 2 ≤ −2u2 + β2 u2L2 (Γ) ≤ C, where C does not depend on u. Copyright © 2005 Marcel Dekker, Inc.

On a Spectral Variational Problem Arising in the Study of Earthquakes

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It remains to prove that inf{λ; λ is an eigenvalue of problem (6)} = −∞. For this purpose, it is suﬃcient to show that , ∩ M } = +∞. sup{J(u); u ∈ K But this follows directly from (10) after observing that |∇u|2 dx = +∞. sup , u∈K∩M

Ω

In order to prove the last part of the theorem we remark that −λ0 , as a function of β, is the upper bound of a family of aﬃne functions 1 2 2 |∇v| dx − β [v] dσ , (14) −λ0 (β) = inf v∈K∩M r 2 Ω Γ hence it is a concave function. Thus β −→ λ0 (β) is convex and (7) yields. This completes the proof of Theorem 1.

2.2

Proof of Theorem 3

The main idea is to establish the multiplicity result with respect to a prescribed level of energy. let us ﬁx r > 0. Consider the

More precisely, manifold N = u ∈ V ; Γ [u]p dσ = r p , where p is as in (9). We reformulate problem (8) as follows: ﬁnd uε ∈ K ∩ N and λε ∈ R such that ∇uε · ∇(v − uε )dx+ Ω (15) j (γ(uε (x)); γ(v(x)) − γ(uε (x))) dσ+ + εg Γ λε uε (v − uε )dx ≥ 0, ∀v ∈ K. Ω

We start with the preliminary result

Lemma 8 There exists a sequence (bn ) of essential values of E such that bn → +∞ as n → ∞. Proof of Lemma 8. For any n ≥ 1, set an = inf S∈Γn supu∈S E(u), where Γn is the family of compact subsets of K ∩ N of the form φ(S n−1 ), with φ : S n−1 → K ∩ N continuous and odd. The function E restricted to K ∩ N is continuous, even, and bounded from below. So, by Theorem 2.12 in [7], it is suﬃcient to prove that an → +∞ as n → ∞. But, as in Copyright © 2005 Marcel Dekker, Inc.

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the preceding section, the functional E restricted to K ∩ N satisﬁes the Palais-Smale condition. So, taking into account Theorem 3.5 in [6] and Theorem 3.9 in [7], we deduce that the set E c has ﬁnite genus for any c ∈ R. Using now the deﬁnition of the genus combined with the fact that K ∩ N is a weakly locally contractible metric space, we deduce that an → +∞. This completes our proof. The canonical energy associated to problem (15) is the functional J restricted to K ∩ N , where J = E + Φ and Φ is deﬁned by Φ(u) = ε g(γ(u(x)))dσ. Γ

A straightforward computation with the same arguments as in the proof of Lemma 6 shows that if u is a lower stationary point of J then there exists λ ∈ R such that (u, λ) is a solution of problem (15). In virtue of this result, it is suﬃcient for concluding the proof of Theorem 3 to show that the functional J has at least n distinct critical values, provided that ε > 0 is suﬃciently small. We ﬁrst prove that J is a small perturbation of E. More precisely, we have

Lemma 9 For every η > 0, there exists δ = δη > 0 such that for any ε ≤ δ, supu∈K∩N |J(u) − E(u)| ≤ η. Proof of Lemma 9. We have |J(u) − E(u)| = |Φ(u)| ≤ ε

|g(γ(u(x)))| dσ. Γ

So, by (9) and Lemma 2, |J(u) − E(u)| ≤ ε a

(1 + [u(x)]p ) dσ ≤ Cε ≤ η, Γ

if ε is suﬃciently small. By Lemma 8, there exists a sequence (bn ) of essential values of E|K∩N such that bn → +∞. Without loss of generality we can assume that bi < bj if i < j. Fix an integer n ≥ 1 and choose ε0 > 0 such that ε0 < 1/2 min2≤i≤n (bi − bi−1 ). Applying now [7, Theorem 2.6], we obtain that for any 1 ≤ j ≤ n, there exists ηj > 0 such that if supK∩N |J(u) − E(u)| < ηj then J|K∩N has an essential value cj ∈ (bj − ε0 , bj + ε0 ). So, by Lemma 9 applied for η = min{η1 , . . . , ηn }, there exists δn > 0 such that supK∩N |J(u) − E(u)| < η, provided that ε ≤ δn . This shows that the energy functional J has at least n distinct essential values c1 , . . . , cn in (b1 − ε0 , bn + ε0 ). Copyright © 2005 Marcel Dekker, Inc.

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The next step consists in showing that c1 , . . . , cn are critical values of J|K∩N . Arguing by contradiction, let us suppose that cj is not a critical value of J|K∩N . We show in what follows that: (A1 ) There exists δ¯ > 0 such that J|K∩N has no critical value in (cj − ¯ ¯ cj + δ). δ, a ¯ cj + δ) ¯ with a < b, the pair (J b (A2 ) For every a, b ∈ (cj − δ, |K∩N , J|K∩N ) is trivial. Suppose, by contradiction, that (A1 ) is not valid. Then there exists a sequence (dk ) of critical values of J|K∩N with dk → cj as k → ∞. Since dk is a critical value, it follows that there exists uk ∈ K ∩ N such that J(uk ) = dk and 0 ∈ ∂ − J(uk ). Using now the fact that J satisﬁes the Palais-Smale condition at the level cj , it follows that, up to a subsequence, (uk ) converges to some u ∈ K ∩ N as k → ∞. So, by the continuity of J and the lower semicontinuity of grad J( · ), we obtain J(u) = cj and 0 ∈ ∂ − J(u), which contradicts the initial assumption on cj . Let us now prove assertion (A2 ). For this purpose we apply the Noncritical Point Theorem (see [6], Theorem 2.15]). So, there exists a continuous map χ : (K ∩ N ) × [0, 1] → K ∩ N such that χ(u, 0) = u, J(χ(u, t)) ≤ J(u), J(u) ≤ b ⇒ J(χ(u, 1)) ≤ a, J(u) ≤ a ⇒ χ(u, t) = u.

(16)

b a → J|K∩N by ρ(u) = χ(u, 1). From (16) we Deﬁne the map ρ : J|K∩N obtain that ρ is well deﬁned and it is a retraction. Set b b × [0, 1] → J|K∩N , J : J|K∩N

J (u, t) = χ(u, t).

b The deﬁnition of J implies that, for every u ∈ J|K∩N ,

J (u, 0) = u

and

J (u, 1) = ρ(u)

(17)

a × [0, 1], and, for any (u, t) ∈ J|K∩N

J (u, t) = J (u, 0).

(18)

a -homotopic to the identity From (17) and (18) it follows that J is J|K∩N a of J|K∩N ; that is, J is a strong deformation retraction, so the pair b a , J|K∩N ) is trivial. Assertions (A1 ), and (A2 ) show that cj is not (J|K∩N an essential value of J|K∩N . This contradiction concludes the proof of Theorem 3.

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References [1] J.-P. Ampuero, J-P. Vilotte, and F. J. Sanchez-Sesma. Nucleation of rupture under slip dependent friction law: simple models of fault zone. J. Geophys. Res., 107:1–15, 2002. [2] M. Bocea, P. D. Panagiotopoulos, and V. R˘ adulescu. A perturbation result for a double eigenvalue hemivariational inequality and applications. J. Global Optimiz., 14:137–156, 1999. [3] M. Campillo and I.-R. Ionescu. Initiation of antiplane shear instability under slip dependent friction. J. Geophys. Res., 122:20363–20371, 1997. [4] G. Chobanov, A. Marino, and D. Scolozzi. Multiplicity of eigenvalues for the laplace operator with respect to an obstacle and non-tangency conditions. Nonlinear Analysis, 15:199–215, 1990. [5] F. Cˆırstea and V. R˘ adulescu. Multiplicity of solutions for a class of nonsymmetric eigenvalue hemivariational inequalities. J. Global Optimiz., 17:43–54, 2000. [6] J. N. Corvellec, M. Degiovanni, and M. Marzocchi. Deformation properties for continuous functionals and critical point theory. Topol. Methods Nonlinear Anal., 1:151–171, 1993. [7] M. Degiovanni and S. Lancelotti. Perturbations of even nonsmooth functionals. Diﬀerential Integral Equations, 8:981–992, 1995. [8] E. De Giorgi, A. Marino, and M. Tosques. Problemi di evoluzione in spazi metrici e curve di massima pendenza. Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat., 68:180–187, 1980. [9] I.-R. Ionescu. Viscosity solutions for dynamic problems with slip-rate dependent friction. Quart. Appl. Math., 60:461–476, 2002. [10] I.-R. Ionescu, Q.-L. Nguyen, and S. Wolf. Slip dependent friction in dynamic elasticity. Nonlinear Analysis, 53:375–390, 2003. [11] I.-R. Ionescu and V. R˘ adulescu. Nonlinear eigenvalue problems arising in earthquake initiation. Adv. Diﬀer. Equations, 8:769–786, 2003. [12] A. Marino and D. Scolozzi. Geodetiche con ostacolo. Boll. Un. Mat. Ital., 2B:1–31, 1983. [13] A. C. Palmer and J. R. Rice. The growth of slip surfaces in the progressive failure of overconsolidated clay slopes. Proc. Roy. Soc. London A, 332:527–548, 1973. [14] P. D. Panagiotopoulos and V. R˘ adulescu. Perturbations of hemivariational inequalities with constraints and applications. J. Global Optimiz., 12:285–297, 1998.

Copyright © 2005 Marcel Dekker, Inc.

CHAPTER XVI NODAL CONTROL OF CONSERVATION LAWS ON NETWORKS Sensitivity Calculations for the Control of Systems of Conservation Laws with Source Terms on Networks Martin Gugat∗ Lehrstuhl f¨ ur Angewandte Mathematik II Erlangen, Germany

Abstract

We consider a network where a dynamical process governed by hyperbolic conservation laws takes place. At the vertices, the conservation laws are coupled by algebraic node conditions. The system is also controlled at the vertices through these conditions. To solve problems of optimal control for such a system, an adjoint sensitivity calculus is useful since it allows the eﬃcient evaluation of the gradient of the objective function. We present such a calculus in the framework of classical solutions.

Keywords: Conservation law, network, node conditions, adjoint system

Introduction Consider a network of open channels and pipes with free surface ﬂow as in a sewage system. The dynamic behavior of the water ﬂow in each single channel or pipe of this system can be modeled by the de St. Venant equations, which form a system of hyperbolic conservation laws. The channels and pipes correspond to the edges of a graph, where the nodes correspond to the junctions. The ﬂow through these points where the channels/pipes are connected is governed by a set of algebraic node conditions that guarantee for example that in the node mass is neither generated nor destroyed. At certain points in the network, the ﬂow is controlled by devices such as pumps or underﬂow gates. What is the best way to operate these control devices? This question gives rise to a problem of optimal control of the type that we consider in this paper. ∗ This

work was supported by DFG–research cluster: real-time optimization of complex systems; grant number Le595/13-1.

Copyright © 2005 Marcel Dekker, Inc.

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In this paper we consider systems of this type, where a process governed by hyperbolic conservation laws takes place on a network. On each edge of the corresponding graph, the ﬂow is converned by a system of hyperbolic conservation laws with source term. (There are numerous studies of hyperbolic conservation laws, see for example [12, 4, 11, 7]. Classical solutions are considered in [14].) At the vertices of the graph, these systems are coupled by a system of algebraic equations. Since the system is controlled at these vertices, also the control functions appear in these node conditions, therefore we speak of nodal control as a generalization of boundary control. We consider problems of optimal control for systems of this type, where the system is controlled at the nodes of the graph. We state results about directional diﬀerentiability with respect to the control function of objective functions that are deﬁned as integrals that depend on the state of the system. We introduce an adjoint backwards problem that is useful for the formulation of optimality conditions and allows the eﬃcient evaluation of the derivatives in an optimize–then–discretize approach, where ﬁrst the equations for the gradients are determined on the PDE level and then discretized. As stated in [6], this approach yields ﬂexibility since it allows the use of diﬀerent methods in the state and adjoint discretizations. Related studies can be found in [3, 2, 17, 16] but no networks are considered there. Similar problems have also been studied in [6] for problems with space dimension two, but without networks; the application considered there is the optimal boundary control of aeroaccoustic noise governed by the two-dimensional unsteady compressible Euler equations. Up to now, an adjoint sensitivity calculus for the type of networks with dynamics governed by conservation laws that we consider here cannot be found in the literature. A related study with dynamics governed by hyperbolic quasilinear systems in diagonal form is presented in [10], where also the proofs of some of the results that we use in this paper are given.

1.

Notation

Let a ﬁnite directed graph (V, E) with vertices (nodes) V and edges E be given. Each edge e ∈ E corresponds to a space interval [0, Le ]. We consider the system on the time interval [0, T ]. The dynamic behavior on the edge is governed by a hyperbolic system of conservation laws of the form vte + F e (v e )x + S e (v e ) = 0. Copyright © 2005 Marcel Dekker, Inc.

(1)

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203

Here v e = (v1e , ..., vne )T is a vector-valued function from [0, Le ] × [0, T ] into IRn , and S e = (S1e , ..., Sne )T and F e = (F1e , ..., Fne )T are twice continuously diﬀerentiable functions from IRn into IRn . The initial state of the system is given by an initial condition of the form v e (x, 0) = v0e (x), e ∈ E, x ∈ [0, Le ] (2) where for all edges e ∈ E, the function v0e is continuously diﬀerentiable. At the nodes ω ∈ V of the graph, the values of the functions v e are coupled by algebraic node conditions. For a node ω ∈ V , let E0 (ω) denote the set of edges that are connected with ω. For e ∈ E0 (ω), let xe (ω) ∈ {0, Le } denote the end-point of the interval [0, Le ] that corresponds to the vertex. There are two types of nodes: boundary nodes, where the set E0 (ω) has only one element and interior nodes, where E0 (ω) has at least two elements. Let VB ⊂ V denote the set of boundary nodes. For ω ∈ VB , let e(ω) denote the adjacent edge. Let V 0 (V L respectively) denote the set of boundary nodes that are at the end zero (the end Le ) of the adjacent edge e. For e ∈ E0 (ω), let se (ω) = −1 if xe (ω) = 0 and se (ω) = 1 if xe (ω) = Le .

1.1

Node Conditions

To state the node conditions, we need the following notation: For all e ∈ E, we assume that for the numbers ne+ = the number of positive eigenvalues of (F e ) (v0e (x)), x ∈ [0, Le ] ne− = the number of negative eigenvalues of (F e ) (v0e (x)), x ∈ [0, Le ] are well deﬁned, that is, the signs of the eigenvalues are independent of the space variable x and that we have ne+ + ne− = n, that is there are no zero eigenvalues. We consider controls that generate states v e where for all x ∈ [0, Le ] and all t ∈ (0, T ] ne+ = the number of positive eigenvalues of (F e ) (v e (x, t)), (3) ne− = the number of negative eigenvalues of (F e ) (v e (x, t)) (4) that is the signs of the eigenvalues are also independent of the time t. e , I e ⊂ {1, ..., n} be given where I e has Let two disjoint index sets I+ − + e e e e ∪ I e = {1, ..., n}. n+ elements and I− has n− elements. Then I+ − Copyright © 2005 Marcel Dekker, Inc.

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For e ∈ E0 (ω), deﬁne the index sets e e e (ω) = {I− if xe (ω) = 0 and I+ if xe (ω) = Le }, Iin e e e Iout (ω) = {I+ if xe (ω) = 0 and I− if xe (ω) = Le }.

For ω ∈ V , B0 (ω, ·) denotes a continuously diﬀerentiable function and for e ∈ E0 (ω), ue (ω, ·) is a continuously diﬀerentiable control function. The numbers of components of B0e (ω, ·) is equal to the number of components of ue (ω, ·) and depends on the numbers of positive and negative eigenvalues of the system matrix. A discussion of boundary conditions for hyperbolic systems is given in [1]. We consider node conditions of e (ω): the following form. For e ∈ E0 (ω) and i ∈ Iout f (ω)) + uei (ω, t) vie (xe (ω), t) = (B0 )ei (ω, t, vjf (xf (ω), t) : f ∈ E0 (ω), j ∈ Iin (5)

2.

The Linearized Problem

For an edge e ∈ E, the linearized problem is Hte + (F e ) (v e )H e x + (S e ) (v e )H e = 0. Here (S e ) (w) = ( and

∂Sie ∂vj

(6)

(w))nij=1 denotes the derivative of the source term (F e ) (w) = (

∂Fie (w))nij=1 ∂vj

is the system matrix. The initial state v0e is independent of the control, hence the initial condition for H e is H e (x, 0) = 0, x ∈ [0, Le ].

(7)

If the solution v e of (1) is continuously diﬀerentiable, in general we cannot expect that the linearized problem has also a continuously differentiable solution, since the linearization causes a loss of regularity.

2.1

Node Conditions for the Linearized Problem

Let a node ω ∈ V be given. The node conditions of the linearized problem have the following form. e (ω): For e ∈ E0 (ω) and i ∈ Iout f e bef (8) Hie (xe (ω), t) = ij (ω, t) Hj (0, t) + di (ω, t). f ∈ E0 (ω) : f (ω) j ∈ Iin Copyright © 2005 Marcel Dekker, Inc.

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205

The functions bef ij are given by the corresponding partial derivatives of B0 , ∂ ef Bij = f (B0 )ei . ∂vj The functions d(ω, ·) give the direction in which the control function is linearized and are continuously diﬀerentiable with de (ω, 0) = (de ) (ω, 0) = 0.

3.

(9)

Existence of Solutions

We assume that a control function u∗ is given that generates a continuously diﬀerentiable state. Then all control functions u in a certain C 1 neighborhood of u∗ that satisfy the compatibility conditions u(ω, 0) = u∗ (ω, 0), u (ω, 0) = (u∗ ) (ω, 0), ω ∈ V also generate continuously diﬀerentiable states. This follows from results of [5]. The following results state that if the nonlinear initial boundary value problem has a continuously diﬀerentiable solution, the linearized initial boundary value problem has a continuous solution in the characteristic sense, that is a solution that satisﬁes the corresponding integral relations along the characteristic curves. Since the linearized system has the same system matrix as (1) and the characteristic curves are determined by the eigenvalues of this matrix, the linearized system has the same characteristic curves as the original system. If the original system has a continuously diﬀerentiable solution, the characteristic curves corresponding to the same eigenvalue do not intersect, so we can also transform the linearized system to the same characteristic coordinates.

Theorem 1 Let a control function u be given that generates a continuously diﬀerentiable solution of equation ( 1) on the time interval [0, T ] that satisﬁes the initial condition ( 2) and the node conditions ( 5). Assume that the function d satisﬁes the compatibility conditions ( 9). Then on the time interval [0, T ] there exists a unique continuous solution of the initial boundary value problem deﬁned by the initial condition H e (·, 0) = 0 for all e ∈ E, the node conditions ( 8), and the linearized equation ( 6) in the characteristic sense. The theorem is proved by a ﬁxed-point iteration of Picard-Lindel¨ of type along the characteristic curves (see Theorem 1.1 in [15], p. 47). Due to the linearity of the problem, we need not assume a bound for the size of the norm of the direction d or its derivatives. Copyright © 2005 Marcel Dekker, Inc.

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Directional Diﬀerentiability

The following result states that the directional derivative of the solution of the nonlinear initial boundary value problem with respect to the boundary control functions is given by the solution of the linearized initial boundary value problem.

Theorem 2 Let a control function u be given that generates a continuously diﬀerentiable solution of equation ( 1) on the time interval [0, T ] that satisﬁes the initial condition ( 2) and the node conditions ( 5). Assume that the function d satisﬁes the compatibility conditions ( 9). Then for all real numbers h that are suﬃciently small, the control functions u + hd also generate continuously diﬀerentiable solutions denoted by v e (x, t, u + hd). Moreover, with the unique continuous solution H e (x, t, u, d) of the linearized problem ( 7), ( 8), and ( 6) in the characteristic sense the following equation holds for all e ∈ E, x ∈ [0, Le ], t ∈ [0, T ]: v e (x, t, u + hd) − v e (x, t, u) = H(x, t, u, d). h→0+ h lim

5.

(10)

Evaluation of Directional Derivatives

We consider objective or constraint functions of the form J(u) =

ω∈V e∈E0 (ω)

+

e∈E

T 0

Le

T

f ω,e (v e (xe (ω), t), t) dt 0

f e (v e (x, t), x, t) dx dt.

0

The functions f ω,e and f e are assumed to be continuously diﬀerentiable, and v e (the state of the system) is the solution of the initial boundary value problem (1), (2), (5) generated by the control function u. For e ∈ E, the adjoint system equation is µet + µex (F e ) (v e ) − µe (S e ) (v e ) = (∇v f e )T

(11)

where µe = (µe1 , ..., µen ) is a row vector. The end conditions for µe are µe (x, T ) = 0 for all e ∈ E, x ∈ [0, Le ]. Copyright © 2005 Marcel Dekker, Inc.

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Nodal Control of Conservation Laws on Networks

207

Using the solution of a backwards terminal boundary value problem, we give a representation of the directional derivative J(u + hd) − J(u) h→0 h

Dd J(u) = lim

where the solution of the original nonlinear forward problem and the adjoint solution appear, but not the solution of the linearized problem. To deﬁne the backwards terminal boundary value problem, we need the corresponding adjoint node conditions. For ω ∈ V and t ∈ [0, T ] deﬁne the column vectors ω e win (t) = (Hie (xe (ω), t) : e ∈ E0 (ω), i ∈ Iin (ω) )T ω e wall (t) = (Hie (xe (ω), t) : e ∈ E0 (ω), i ∈ Iin (ω); e e Hi (xe (ω), t) : e ∈ E0 (ω), i ∈ Iout (ω))T e dω (t) = (dei (ω, t) : e ∈ E0 (ω), i ∈ Iout (ω))T .

In what follows let the order of the components in the vectors be ﬁxed and compatible in all three vectors. We can write the node condition (8) in the form

I 0 ω ω wall (t) = win (t) + (13) B ω (t) dω (t) ω (t) and where I denotes the identity matrix in the space containing win ω ω ω (t) B (t) is the appropriate matrix. Let min denote the length of win and mωout denote the length of dω (t). Then B ω (t) is a mωout × mωin matrix. Deﬁne the row vectors e µωin (t) = (µei (xe (ω), t) : e ∈ E0 (ω), i ∈ Iout (ω))T e µωall (t) = (µei (xe (ω), t) : e ∈ E0 (ω), i ∈ Iin (ω); e e µi (xe (ω), t) : e ∈ E0 (ω), i ∈ Iout (ω))T .

For the adjoint node conditions we make the ansatz µωall (t) = µωin (t)(Aω (t), I) + (αω (t), 0)

(14)

where I denotes the identity matrix in the mωout -dimensional space and Aω (t) is a mωout × mωin matrix that we have to determine as well as the row vector αω (t) that has mωin components. Deﬁne the matrices ∂Fie e f ω e e M1 (t) = s (ω) f (v (xe (ω), t) : e, f ∈ E0 (ω), i ∈ Iin (ω), j ∈ Iin (ω) ∂vj Copyright © 2005 Marcel Dekker, Inc.

208

CONTROL AND BOUNDARY ANALYSIS

M2ω (t)

e

= s (ω)

M3ω (t)

e

= s (ω)

M4ω (t) = se (ω)

∂Fie ∂vjf ∂Fie ∂vjf

(v (xe (ω), t) : e, f ∈ E0 (ω), i ∈

e Iin (ω), j

(v (xe (ω), t) : e, f ∈ E0 (ω), i ∈

e Iout (ω), j

e

∈

f Iout (ω)

∈

f Iin (ω)

e (v e (xe (ω), t) : e, f ∈ E0 (ω), i ∈ Iout (ω), j ∈

f Iout (ω)

e

∂Fie ∂vjf

ω

M (t) =

M1ω (t) M2ω (t) M3ω (t) M4ω (t)

.

(15)

The matrix M1ω (t) is a mωin × mωin matrix and the matrix M4ω (t) is a mωout × mωout matrix. We deﬁne the real-valued function ψ ω (t) by the following equation: ω (t) (16) ψ ω (t) := µωall (t) M ω (t) wall e e e e e = s (ω) µ (xe (ω), t) (F ) (v (xe (ω), t)) H (xe (ω), t) e∈E0 (ω)

Then we have ψ ω (t) = [ µωin (t) (Aω (t), I) + (αω (t), 0)]

0 I M1ω (t) M2ω (t) ω win (t) + dω (t) B ω (t) M3ω (t) M4ω (t) ω (t) = µωin (t)[Aω (t){M1ω (t) + M2ω (t)B ω (t)} + M3ω (t) + M4ω (t)B ω (t)]win ω ω ω ω ω +µin (t) [A (t)M2 (t) + M4 (t)]d (t) ω (t) +αω (t) [M1ω (t) + M2ω (t)B ω (t)] win ω ω ω +α (t) M2 (t) d (t).

Assume that for all ω ∈ V , t ∈ [0, T ] the matrix [M1ω (t)+M2ω (t)B ω (t)] is invertible. We deﬁne the matrix Aω (t) by the equation Aω (t) = − [M3ω (t) + M4ω (t)B ω (t)] [M1ω (t) + M2ω (t)B ω (t)]−1 . Then the expression for ψ ω (t) is much simpler, namely ψ ω (t) = µωin (t) [Aω (t) M2ω (t) + M4ω (t)]dω (t) Copyright © 2005 Marcel Dekker, Inc.

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Nodal Control of Conservation Laws on Networks

209

ω + αω (t) [M1ω (t) + M2ω (t) B ω (t)] win (t) ω ω ω + α (t) M2 (t) d (t).

Deﬁne the column vectors ω e Dfin (t) = (∂vi f e,ω (xe (ω), t), e ∈ E0 (ω), i ∈ Iin (ω))T , ω e (t) = (∂vi f e,ω (xe (ω), t), e ∈ E0 (ω), i ∈ Iout (ω))T , Dfout

ω (t) Dfin ω . Dfall (t) = ω (t) Dfout

We deﬁne the row vector αω (t) by the equation ω ω αω (t) = −Dfin (t)T − Dfout (t)T B ω (t) [M1ω (t) + M2ω (t) B ω (t)]−1 . (18) Then we have ψ ω (t) = µωin (t) [Aω (t) M2ω (t) + M4ω (t)] dω (t) + αω (t) M2ω (t) dω (t) ω (t) − [Df1ω (t)T + Df2ω (t)T B ω (t)] win = {µωin (t) [Aω (t) M2ω (t) + M4ω (t)] + αω (t) M2ω (t)} dω (t)

0 ω ω (t)T wall (t) − . −Dfall dω (t)

(19)

Now we can give a representation of the directional derivative.

Theorem 3 Let the assumptions of Theorem 2 hold. Let µ denote the solution of the terminal value problem that satisﬁes for all e ∈ E the end condition ( 12) and for all (x, t) ∈ (0, Le ) × (0, T ) the adjoint equation ( 11) and the adjoint node conditions deﬁned by ( 14), ( 17), ( 18). Assume that for all ω ∈ V , t ∈ [0, T ] the matrix [M1ω (t)+M2ω (t)B ω (t)] is invertible, with B ω deﬁned in ( 13) and M ω deﬁned in ( 15). Again we consider solutions that satisfy the corresponding integral equations along the characteristic curves. Then the directional derivative of J is given by the equation Dd J(u) =

(20)

ω [µωin (t)(Aω (t)M2ω (t) + M4ω (t)) + αω (t) M2ω (t) + Dfout (t)T ] dω (t) dt.

ω∈V

Proof of Theorem 3. Consider the directional derivative of J. By Theorem 1 a continuous solution H of the linearized problem exists, and Copyright © 2005 Marcel Dekker, Inc.

210

CONTROL AND BOUNDARY ANALYSIS

by Theorem 2 and integration by parts the deﬁnitions of the linearized and the adjoint problem imply the equation T (∇v f ω,e (v e (xe (ω), t), t))T H e (xe (ω), t) dt Dd J(u) = 0

ω∈V e∈E0 (ω)

+

e∈E e

T

0

Le

(∇v f e (v e (x, t), x, t))T H e (x, t)

0

+ µ Hte + (F e ) (v e )H e x + (S e ) (v e )H e |(x,t) dx dt T (∇v f ω,e (v e (xe (ω), t), t))T H e (xe (ω), t) dt = 0

ω∈V e∈E0 (ω)

+

e∈E

T

0

Le

(∇v f e (v e (x, t), x, t))T H e (x, t)

0

− µet + µex (F e ) (v e ) − µe (S e ) (v e ) H e |(x,t) dx dt T e µe (x, t) (F e ) (v e (x, t)) H e (x, t)|L + x=0 dt 0

=

ω∈V e∈E0 (ω)

+

−

T

∇v f ω,e (v e (xe (ω), t), t) H e (xe (ω), t) dt

0

T

e ∈ E0 (ω) : xe (ω) = Le

0

e ∈ E0 (ω) : xe (ω) = 0

0

T

µe (Le , t) (F e ) (v e (Le , t)) H e (Le , t) dt

µe (0, t) (F e ) (v e (0, t)) H e (0, t) dt.

Now we have represented Dd J(u) in a form that requires only the values of µ and H at the nodes of the network. Up to now, for µ only the adjoint equation (11) and the end condition (12) have been used. It remains to be shown that due to the deﬁnition of the adjoint node conditions only the values of the solution µ of the adjoint backwards problem are necessary and the values of the linearized problem H are not needed. The deﬁnition (16) of ψ ω (t) and equation (19) imply that T (∇v f ω,e (v e (xe (ω), t), t))T H e (xe (ω), t) dt Dd J(u) = ω∈V e∈E0 (ω)

Copyright © 2005 Marcel Dekker, Inc.

0

211

Nodal Control of Conservation Laws on Networks

+

ω∈V

=

T

ψ ω (t) dt

0

T ω ω Dfall (t)T wall (t)

+ +

ω∈V 0 µωin (t) (Aω (t) M2ω (t) + αω (t) M2ω (t) dω (t) dt

−

ω Dfall (t)T

M4ω (t)) dω (t)

ω (t) wall

−

0 dω (t)

dt

T ω [Dfout (t)T + µωin (t) (Aω (t)M2ω (t) + M4ω (t)) + αω (t)M2ω (t)] dω (t) dt

= 0

and the assertion follows.

6.

Example

We consider a network of rectangular water channels where on each channel, the ﬂow is governed by the de St. Venant equations [8]. The network is described by a graph (V, E).

6.1

Shallow Water Equations

For e ∈ E, let Qe (x, t) denote the ﬂow rate on channel e at the point x and time t and let Ae (x, t) denote the corresponding wetted cross section. The conservation of mass yields the equation Aet + Qex = 0. The balance of momentum yields the second equation

g (Qe )2 e e 2 Qt + (A ) + + Ae S2e = 0. 2be Ae x Here g is the gravitational constant, be denotes the width of channel e and S2e = g(γ e + β e (Ae , Qe )), where γ e is the slope of the channel. If γ e < 0, the channel proceeds downwards. The function β e is the friction slope. Let e e v1 A = . (21) ve = v2e Qe Copyright © 2005 Marcel Dekker, Inc.

212

CONTROL AND BOUNDARY ANALYSIS

@f @ Rx @

g

ω

h

?

Figure 1.

6.2

A junction of three channels.

Node Conditions

Consider a node ω ∈ V with three adjacent channels f , g, h and xf (ω) = Lf , xg (ω) = Lg , xh (ω) = 0 (see Figure XVI.6.2). We consider the node conditions Qf + Qg = Qh , Af = (bf /bh )Ah , Ag = (bg /bh ) Ah . We can write these node conditions in the form f f Q 0 0 bf /bh A Ag = 0 0 bg /bh Qg Qh 1 1 0 Ah

(22)

and the matrix B ω from (13) is given by the matrix in equation (22). f g We have Iin (ω) = Iin (ω) = {2} and for the matrices M1 , M2 , M3 , M4 we have the equations ∂F2f 2Qf 0 0 f 0 0 ∂Q f A g ∂F g M1ω (t) = 0 0 , 0 2Q 0 ∂Q2g = Ag ∂F1h 0 0 0 0 0 ∂Ah

M2ω (t) = =

gAf bf

Copyright © 2005 Marcel Dekker, Inc.

−

∂F2f ∂Af

0

0

∂F2g ∂Ag

0

0

(Qf )2 (Af )2

0 0

0

0 h

∂F1 ∂Qh

0 0

gAg bg

−

0 , 0 1

(Qg )2 (Ag )2

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Nodal Control of Conservation Laws on Networks

M3ω (t) = M4ω (t) =

∂F1f ∂Qf

0

0

∂F1g ∂Qg

0

0

∂F1f ∂Af

0

0

∂F1g ∂Ag

0

0

0

1 0 0 1 0 = 0 0 ∂F2h ∂Ah 0 0 0 0 0 0 = ∂F2h 0 0 ∂Qh

gAh bh

−

0 0 , 2

(Qh ) (Ah )2

0 0 .

2Qh Ah

We have

f g 2Qg 2Qf f f Q 2 g g Q 2 − b ( ) − b ( ) gA − gA bh Ag Af bh Af Ag g Qf 2 Q f 3 f f 2 g 3 g g 2 g(A ) − b (Q ) + g g(A ) − b (Q ) =− h f g . b A A Af A

det[M1ω + M2ω B ω ] = −

If the ﬂow is subcritical, that is, if be (Ae )2 [ghe − (U e )2 ] = g(Ae )3 − be (Qe )2 > 0, e ∈ {f, g, h} (where he denotes the water height and U e the velocity) and if Qf Qg > 0, this determinant does not vanish. If Qf = Qg = 0, the determinant is zero, hence in this case Theorem 3 is not applicable. Deﬁne the number α = −Ag Qg [g(Af )3 − bf (Qf )2 ] − Af Qf [g(Ag )3 − bg (Qg )2 ] bh (Af Ag )2 det[M1ω + M2ω B ω ]. 2 Then if Qf Qg > 0, we have α [M1ω + M2ω B ω ]−1 = =

(Af )2 [bg (Qg )2 −g(Ag )3 ] 2 (Af )2 [g(Ag )3 −bg (Qg )2 ] 2 −bh (Af )2 Ag Qg

(Ag )2 [g(Af )3 −bf (Qf )2 ] 2 (Ag )2 [bf (Qf )2 −g(Af )3 ] 2 −bh Af (Ag )2 Qf

Ag Qg [bf (Qf )2 − g(Af )3 ]

Af Qf [bg (Qg )2 − g(Ag )3 ] . 2bh Af Ag Qf Qg

For the matrix Aω deﬁned in (17) the ﬁrst two rows of the matrix −α Aω are the same as in α [M1ω + M2ω B ω ]−1 and the third row of the matrix −α Aω has the entries h h 2 f 2 g g b (Q ) h (A ) A Q − gA , (Ah )2 h h 2 b (Q ) h − gA , Af (Ag )2 Qf (Ah )2 Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

2Qh g g f f 2 f 3 f f g g 2 g 3 A Q [b (Q ) − g(A ) ] + A Q [b (Q ) − g(A ) ] Ah +2Af Ag Qf Qg [g Ah − bh

(Qh )2 ]. (Ah )2

The adjoint node conditions from (14) are (µf2 , µg2 , µh1 ) = (µf1 , µg1 , µh2 ) Aω + αω

(23)

with the row vector αω as deﬁned in (18).

7.

Conclusion

We have presented an adjoint sensitivity calculus for hyperbolic systems of conservation laws that are deﬁned on networks. We have stated the adjoint problem with terminal conditions and adjoint node conditions. With the adjoint solution of the corresponding linear backwards problem, the directional derivatives for real-valued integral functions of the state can be computed relatively cheaply in several directions, which is useful to apply gradient-based optimization algorithms. Our theory has been stated within the framework of continuously diﬀerentiable solutions. This is an interesting case, since often it is desirable to steer the system in such a way that the state is regular and, in particular, no shocks occur. The question whether this is indeed possible has been studied in [13, 9] for ﬂows in networks of open channels. Still extensions to more general classes of solutions that also allow for the formation of shocks would be interesting, since if the optimal control generates a regular state, during the iterations of an optimization algorithm controls that generate shocks can occur.

Acknowledgments The author wants to thank G. Leugering for helpful discussions.

References [1] C. Bardos, A.Y. Leroux, and J.C. Nedelec. First order quasilinear equations with boundary conditions. Comm. in Partial Diﬀerential Equations, 4:1017– 1034, 1979. [2] C. Bardos and O. Pironneau. A formalism for the diﬀerentiation of conservation laws. Comptes Rendus Mathematique, 335:839–845, 2002. [3] J. Borggaard, J. Burns, E. Cliﬀ, and M. Gunzburger. Sensitivity calculations for a 2d, inviscid, supersonic forebody problem. In Identiﬁcation and Control in Systems Governed by Partial Diﬀerential Equations, pages 14–25. SIAM, Philadelphia, 1993.

Copyright © 2005 Marcel Dekker, Inc.

Nodal Control of Conservation Laws on Networks

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[4] A. Bressan. Hyperbolic Systems of Conservation Laws. The One-dimensional Cauchy Problem. Oxford University Press, 2000. [5] M. Cirina. Nonlinear hyperbolic problems with solutions on preassigned sets. Michigan Math. J., 17:193–209, 1970. [6] S. S. Collis, K. Ghayour, and M. Heinkenschloss. Optimal transpiration boundary control for aeroaccoustics. Preprint, pages 1–37, 2002. [7] C. M. Dafermos. Hyperbolic Conservation Laws in Continuum Physics. Springer, Berlin, 2000. [8] B. de Saint-Venant. Theorie du mouvement non–permanent des eaux avec application aux crues des rivi`eres et ` a l‘introduction des marees dans leur lit. Comptes Rendus Academie des Sciences, 73:148–154,237–240, 1871. [9] M. Gugat. Boundary controllability between sub- and supercritical ﬂow. SIAM Journal on Control and Optimization, 42:1056–1070, 2003. [10] M. Gugat. Nodal control of networked hyperbolic systems. Submitted to ESAIM:COCV, 2003. [11] H. Holden and N. H. Risebro. Front tracking for hyperbolic conservation laws. Springer, Berlin, 2002. [12] P. G. LeFloch. Hyperbolic Systems of Conservation Laws. The Theory of Classical and Nonclassical Shock Waves. Birkhaeuser, Basel, 2002. [13] G. Leugering and E.J.P. Georg Schmidt. On the modelling and stabilisation of ﬂows in networks of open canals. SIAM J. on Control and Optimization, 41:164–180, 2002. [14] L. Ta-tsien. Global Classical solutions for quasilinear hyperbolic systems. Masson, Paris, 1994. [15] L. Ta-tsien and Y. Wen-ci. Boundary value problems for quasilinear hyperbolic systems. Duke University Mathematics Series V, 1985. [16] S. Ulbrich. A sensitivity and adjoint calculus for discontinuous solutions of hyperbolic conservation laws with source terms. SIAM J. Control Optim., 41:740– 797, 2002. [17] S. Ulbrich. Adjoint-based derivative computations for the optimal control of discontinuous solutions of hyperbolic conservation laws. Systems and Control Letters, 48:309–324, 2003.

Copyright © 2005 Marcel Dekker, Inc.

CHAPTER XVII INVARIANCE OF CLOSED SETS UNDER STOCHASTIC CONTROL SYSTEMS Giuseppe Da Prato∗ Scuola Normale Superiore di Pisa Pisa, Italy

H´el`ene Frankowska∗ CNRS, CREA, Ecole Polytechnique Paris, France

Abstract

We prove that a closed set of a ﬁnite dimensional space is invariant under the stochastic control system if and only if it is invariant under a deterministic control system with two controls.

Keywords: Stochastic invariance, Stratonovitch drift, stochastic control system, state-constraints

Introduction We are given two ﬁnite dimensional spaces H and H1 , a complete probability space (Ω, F, P), an increasing family of σ−sub-algebras Ft ⊂ F augmented by P−null sets of F and an H1 -valued Ft -Brownian motion W (t), t ≥ 0 such that W (0) = 0. This paper is devoted to the problem of invariance of the stochastic control system dX = b(X, v(t))dt + σ(X, v(t))dW (t), v(t) ∈ U,

(1)

where U is a complete separable metric space, b : H × U → H, σ : H × U → L(H1 , H) are bounded continuous mappings which are Lipschitz with respect to the ﬁrst variable, and controls v(t) are U −valued mappings which are progressively measurable with respect to the family Ft , called admissible controls. ∗ Work

supported in part by the European Community’s Human Potential Programme under contract HPRN-CT-2002-00281, Evolution Equations.

Copyright © 2005 Marcel Dekker, Inc.

218

CONTROL AND BOUNDARY ANALYSIS

A set K ⊂ H is called invariant under the control system (1) if for every F0 -random variable X0 such that X0 ∈ K almost surely and every admissible control v(·), the strong solution X to (1) starting at X0 satisﬁes for all t ≥ 0, X(t) ∈ K almost surely. When b and σ are control independent, the above system reduces to the stochastic diﬀerential equation dX = b(X)dt + σ(X)dW (t).

(2)

Recently a number of papers were written on stochastic viability and invariance of closed sets. In the case of stochastic equation (2) conditions for the invariance were expressed using the Stratonovitch drift [9] or stochastic contingent sets [2]. For stochastic control systems and diﬀerential inclusions, diﬀerent authors used stochastic contingent sets [3], viscosity solutions of second order partial diﬀerential equations [5–6], and derivatives of the distance function [7], see also [11–12]for several other approaches. The method based on the second order partial diﬀerential equations deals with value functions of some associated optimal control problems. These tools use the second order sets of continuous solutions to PDE’s. So the second order normal cones to K arise naturally in characterizations of invariance. However, there is no good calculus available for the second order normal cones. In contrast, the results of [9, Doss] obtained in the context of stochastic equations use only ﬁrst order normals to K. This approach is based on an equivalence between invariance of stochastic equation (2) and that of an associated deterministic control system. Namely, it was shown in [9] that if σ ∈ C 3 and has bounded derivatives up to the order three, then K is invariant under the stochastic equation (2) if and only if K is invariant under the (well understood) deterministic control system x = b(x) −

1 Dσj (x)σj (x) + σ(x)u(t), u(t) ∈ H1 , 2 m

(3)

j=1

where σj (x) denotes the column j of the matrix σ(x) and Dσj the jacobian of σj . Theory of [9] needs, however, more regularity of the diﬀusion term σ (Cb3 instead of bounded and Lipschitz continuous) and is based on the Stroock-Varhadan support theorem (see for instance [11]) which is not applicable in the presence of time dependent controls. In this paper we prove a similar ﬁrst order characterization of invariance of stochastic control systems, when σ ∈ Cb1,1 . That is, we extend the Doss theorem into two directions: to control systems and less regular σ. Copyright © 2005 Marcel Dekker, Inc.

Invariance of Closed Sets under Stochastic Control Systems

219

Recall that in the deterministic case a necessary and suﬃcient condition for invariance of K under (3) can be expressed by using tangents to K. Thus K is invariant under the stochastic system (2) if and only if the so-called Stratonovitch drift is tangent to K: 1 Dσj (x)σj (x) ∈ TK (x), ∀ x ∈ K b(x) − 2 m

(4)

j=1

and the image of the diﬀusion σ is tangent to K: σ(x)u ∈ TK (x), ∀ x ∈ K, ∀ u ∈ H1 .

(5)

Condition (5) in turn is equivalent to the invariance of the boundary of K under the deterministic control system x = σ(x)u(t), u(t) ∈ H1 .

(6)

Instead of the support theorem, we use (6) to show that K is invariant under the stochastic control system (1) if and only if it is invariant under the deterministic control system with two controls x = b(x, v(t)) −

1 Dσj (x, v(t))σj (x, v(t)) + σ(x, v(t))u(t), 2 m

(7)

j=1

where u(t) ∈ H1 , v(t) ∈ U. In Section 3 the characterization of the invariance is also stated in terms of proximal normals and tangent cones. Several results below are provided without detailed proofs. They will appear elsewhere.

1.

Preliminaries

Consider a closed non-empty subset K of H. We denote by ∂K the boundary of K and by dK the distance of x ∈ H from K : dK (x) = inf y∈K |x − y|. The contingent cone TK (x) to K at x ∈ K is the set of all vectors v ∈ H such that lim inf h→0+ dK (x + hv)/h = 0. A vector p ∈ H is called a proximal normal to K at x ∈ K if |p| = dK (x + p). Clearly p = 0 is a proximal normal and it is the only proximal normal when x is in the interior of K. It is well known that if p is a proximal normal to K at x, then for some c > 0 ∀ z ∈ K, p, z − x ≤ c|z − x|2 . The following result can be easily deduced from [1].

Copyright © 2005 Marcel Dekker, Inc.

(8)

220

CONTROL AND BOUNDARY ANALYSIS

Lemma 1 Assume that σ : H → L(H1 , H) is locally Lipschitz. Then K and ∂K are invariant under the deterministic control system ( 6) if and only if and for all x ∈ K and any proximal normal p at x we have σ(x)∗ p = 0. For a function ψ : R+ → R the lower directional derivative in the . Using the direction one is deﬁned by dψ(t) := lim inf h→0+ ψ(t+h)−ψ(t) h viability theorem from [1] we obtain the following result.

Lemma 2 Consider T > 0 and a continuous function ψ : [0, T ] → R+ with ψ(0) = 0. Assume that for some L ≥ 0 and every t ∈ [0, T [ such that ψ(t) > 0 we have dψ(t) ≤ Lψ(t). Then ψ = 0. We next recall a necessary condition for the stochastic viability. The result below may be applied to any weak solution of (2). See for instance [13] for the deﬁnition of weak solution. A mapping X : R+ → L2 (Ω, H) is called an adapted process if for every t ≥ 0, X(t) is Ft −measurable. Let b, σ be bounded and continuous. Assume that an adapted process X(·) is continuous and for some x ∈ H, t t b(X(s))ds + σ(X(s))dW (s) a.s. (9) ∀ t ≥ 0, X(t) = x + 0

0

The process X(·) is called viable in K, if for all t ≥ 0, X(t) ∈ K a.s.

Theorem 3 Assume that b and σ are bounded and continuous. If an adapted process X(·) is continuous, satisﬁes (9) with x ∈ K, and for some hi → 0+, X(hi ) ∈ K a.s., then for any proximal normal p to K at x we have σ(x)∗ p = 0. In particular, if ( 2) has a weak solution starting at some x ∈ K which is viable in K, then for any proximal normal p to K at x we have σ(x)∗ p = 0. The proof is similar to the one provided in the appendix of [8].

2.

Necessary and Suﬃcient Conditions for the Invariance

We denote by Cb1,1 (H; L(H1 , H)) the set of all functions σ : H → L(H1 , H) such that σ and its derivative σ are bounded and σ (·) is Lipschitz.

Theorem 4 Assume that K is closed, b is Lipschitz and bounded, and that σ ∈ Cb1,1 . The set K is invariant under the system (2) if and only Copyright © 2005 Marcel Dekker, Inc.

Invariance of Closed Sets under Stochastic Control Systems

221

if for every x ∈ ∂K and every proximal normal p to K at x we have @ ? m 1 Dσj (x)σj (x) ≤ 0, σ(x)∗ p = 0. (10) p, b(x) − 2 j=1

Theorem 4 allows to extend to a less regular σ a result from [9]:

Corollary 5 Assume that K is closed, b is Lipschitz and bounded, and that σ ∈ Cb1,1 . Then K is invariant under the stochastic system (2) if and only if K is invariant under the deterministic control system ( 3) or, equivalently, if and only if ( 4) and ( 5) hold true. Proof of Theorem 4 — Necessity. Assume that K is invariant. Then, by Theorem 3, σ ∗ (y)py = 0 for all y ∈ K and any proximal normal py to K at y. Fix x ∈ K and a proximal normal p to K at x. Then for some c > 0 A (t) in the the inequality (8) holds true. Fix 0 < t ≤ 1, an element W , class of functions equivalent to W (t) ∈ L2 (Ω; H1 ), and an element X(t) 2 in the class of functions equivalent to X(t) ∈ L (Ω; H). For every ω ∈ Ω deﬁne Aω (t) = 0, if W 0 Aω (t) uω := W otherwise. − A |Wω (t)| For all ω ∈ Ω, let zω (·) be the solution to the deterministic system ,ω (t). z (s) = σ(z(s))uω , z(0) = X , Since X(t) ∈ K almost surely, by Lemma 1 we know that for almost all ω ∈ Ω and for all s ≥ 0, zω (s) ∈ K. Let σij denote elements of σ. For all ξ ∈ H deﬁne F (t, ξ) : Ω → L(H1 , H) by Aω (t). F (t, ξ) := (fij (t, ξ)), fij (t, ξ)ω = ∇σij (ξ), σ(ξ)W Then Aω (t)|) = X ,ω (t) − σ(X ,ω (t))W Aω (t) + 1 F (t, X ,ω (t))W Aω (t) + Φω (t), zω (|W 2 Aω (t)|3 . Set yω (t) = zω (|W Aω (t)|). Using the projecwhere Φω (t) ≤ M |W tion theorem we check that y(t) is Ft −measurable and y(t) ∈ L2 (Ω, H). Since y(t) ∈ K almost surely, Ep, y(t) − x ≤ c E|y(t) − x|2 . Copyright © 2005 Marcel Dekker, Inc.

(11)

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CONTROL AND BOUNDARY ANALYSIS

This implies that for all 0 ≤ t ≤ 1 E p, y(t) − x = O(t2 ).

(12)

By the Lipschitz continuity of σ and ∇σij , for some c1 > 0 and all s ∈ [0, 1] E|σ ∗ (X(s))∇σij (X(s)) − σ ∗ (x)∇σ(x)|2 ≤ c1 s.

(13)

This and the H¨ older inequality imply that , E|F (t, X(t))W (t) − F (t, x)W (t)| = O(t3/2 ).

(14)

Furthermore, by the H¨ older inequality and (13), for a constant c1 > 0 t E|p, ( (σ ∗ ∇σij )(X(s)) − (σ ∗ ∇σij )(x), dW (s))W (t)| ≤ 0 t √ (15) c1 t max(E| (σ ∗ ∇σij )(X(s)) − (σ ∗ ∇σij )(x), dW (s)|2 )1/2 i,j

= O(t3/2 ).

0

Thus it follows from (12), (14) and (15) that for all 0 < t ≤ 1 1 Ep, tb(x) + E p, F (t, x)W (t) − 2 B t C E p, ∇σij (x), σ(x)dW (s) W (t) = O(t3/2 ).

(16)

0

Finally observe that t Ep, ( ∇σij (x), σ(x)dW (s))W (t) = 0 Ep, ( ∇σij (x), σ(x)W (t)W j (t)) = Ep, F (t, x)W (t) = j

∂σij pi (x)σkr (x)W r (t)W j (t)) = E( ∂xk i j,k,r ∂σij pi (x)σkj (x)E(W j (t))2 = tp, Dσj (x)σj (x). ∂xk i

j,k

(17)

j

E D From (16), (17) we obtain t p, b(x) − 12 j Dσj (x)σj (x) = O(t3/2 ) for every 0 < t ≤ 1. Dividing by t and taking the limit completes the proof of necessary conditions. Suﬃciency. Fix an F0 - random variable X0 ∈ K a.s. and consider the strong solution X(t) to (2). Set ψ(t) = Ed2K (X(t)). To prove that Copyright © 2005 Marcel Dekker, Inc.

Invariance of Closed Sets under Stochastic Control Systems

223

X(t) ∈ K almost surely, we have to show that ψ(t) = 0. Since ψ(0) = 0, by Lemma 2 it is enough to prove that for some L > 0 and all t > 0 such that ψ(t) > 0 we have dψ(t) ≤ Lψ(t). Fix t > 0 such that ψ(t) > 0, , and also an element X(t) in the class of functions equivalent to X(t) ∈ 2 2 ,ω (h)). L (Ω; H). Set ϕω (h) = dK (X By the measurable selection theorem there exists an Ft - measurable ,ω (t) − ζω |. In particular, X ,ω (t) − map ω → ζω ∈ K such that ϕω (t) = |X ζω is a proximal normal to K at ζω . A (t) in the class of functions Fix h ∈ (0, 1), and also an element W A (t + h) in the class equivalent to W (t) ∈ L2 (Ω; H1 ), and an element W 2 of functions equivalent to W (t + h) ∈ L (Ω; H1 ). For all ω ∈ Ω set Aω (t) Aω (t + h) = W if W 0 Aω (t) Aω (t + h) − W uω (h) := W otherwise. A Aω (t)| |Wω (t + h) − W For every ω ∈ Ω consider the solution zω (·) to the deterministic system z (s) = σ(z(s))uω (h), zω (0) = ζω . By (10) and Lemma 1, zω (s) ∈ K for all s ≥ 0. On the other hand, for all ω ∈ Ω Aω (t + h) − W Aω (t)|) = ζω + σ(ζω )(W Aω (t + h) − W Aω (t)) + Φω (t)+ (|W zω D E 1 Aω (t + h) − W Aω (t)) (W A j (t + h) − W A j (t)) ∇σij (ζω ), σ(ζω )(W ω ω 2 j

Aω (t + h) − W Aω (t)|3 . Set yω (h) = zω (|W Aω (t + h) − where Φω (t) ≤ M |W Aω (t)|). It is not diﬃcult to verify that y(h) is Ft+h −measurable and W y(h) ∈ L2 (Ω; H). Observe next that d2K (X(t + h)) ≤ |X(t + h) − y(h)|2 . The above inequality, Lipschitz continuity of b, σ, and the H¨ older inequality imply that for some L > 0 independent from t and for all 0 < h ≤ 1, ψ(t + h) ≤

D E ψ(t) + Lhψ(t) + 2hE X(t) − ζ, b(ζ) − 12 j Dσj (ζ)σj (ζ) + O(h3/2 ).

Thus, by (10), ψ(t + h) ≤ ψ(t) + Lhψ(t) + O(h3/2 ) and therefore dψ(t) ≤ Lψ(t). By Lemma 2, ψ ≡ 0 implying that dK (X(t)) = 0 a.s. 2 Copyright © 2005 Marcel Dekker, Inc.

224

3.

CONTROL AND BOUNDARY ANALYSIS

Invariance of Stochastic Control Systems

Let U be a complete separable metric space and b : H × U → H, and σ : H × U → L(H1 , H) be bounded continuous mappings. Assume that there exists a constant C > 0 such that for all x, y ∈ H and v ∈ U |b(x, v) − b(y, v)| + σ(x, v) − σ(y, v) + σx (x, v) − σx (y, v) ≤ C|x − y|.

(18)

Denote by A the set of all U −valued mappings v(·) deﬁned on R+ which are progressively measurable with respect to the family Ft , i.e. for every t ≥ 0, v(t) ∈ U a.s. and the mapping [0, t] × Ω (s, ω) → vω (s) is B1 × Ft −measurable. Elements of A are called admissible controls. We associate to the above data the stochastic control system (1). Let X0 be an F0 - random variable, v(·) ∈ A and consider the diﬀerential stochastic equation dX = b(X, v(t))dt + σ(X, v(t))dW (t), (19) X(0) = X0 . Under the above assumptions, system (19) has a unique strong solution X(·), i.e. for all t ≥ 0, t t b(X(s), v(s))ds + σ(X(s), v(s))dW (s) a.s.. X(t) = X0 + 0

0

Definition 6 A set K ⊂ H is called invariant under the stochastic control system ( 1) if for every F0 - random variable X0 such that X0 ∈ K almost surely and every admissible control v(·) ∈ A, the strong solution X to ( 19) satisﬁes for all t ≥ 0, X(t) ∈ K almost surely. Theorem 7 Assume that K is closed, b, σ are bounded and continuous, that there exists a constant C > 0 such that ( 18) holds true. Then K is invariant under the stochastic control system ( 1) if and only if for every x ∈ ∂K, v ∈ U and for every proximal normal p to K at x we have @ ? m 1 Dx σj (x, v)σj (x, v) ≤ 0, σ(x, v)∗ p = 0, (20) p, b(x, v) − 2 j=1

where σj (x, v) denotes the column j of the matrix σ(x, v) and Dx σj (x, v) the jacobian of σj (·, v) at x.

Corollary 8 If all the assumptions of Theorem 7 hold true, then K is invariant under the stochastic control system ( 1) if and only if K is Copyright © 2005 Marcel Dekker, Inc.

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Invariance of Closed Sets under Stochastic Control Systems

invariant under the deterministic control system with two controls ( 7) or, equivalently, if and only if for all v ∈ U and x ∈ K 1 Dσj (x, v)σj (x, v) ∈ TK (x), Im(σ(x, v)) ⊂ TK (x). 2 m

b(x, v) −

j=1

Proof of Theorem 7 — If the set K is invariant under the system (1), then for every v0 ∈ U the mapping v ≡ v0 belongs to A. Thus, for every F0 -measurable random variable X0 ∈ K a.s., the solution X to (19) satisﬁes X(t) ∈ K a.s. This and Theorem 4 imply that for every x ∈ ∂K and every proximal normal p to K at x relations (20) hold true. Assume next that (20) is satisﬁed. Case 1 of constant controls. The proof of suﬃciency is essentially the same as the one of the invariance for stochastic diﬀerential equations provided in Section 2. Case 2 of piecewise constant controls. Let v ∈ A be such that for some 0 = s0 < s1 < ... < sk < ... and for all k ≥ 0, v is time independent on the time interval [sk , sk+1 ). Then, Case 1 and an induction argument imply that X(s) ∈ K a.s. for all s ≥ 0. The general case. Consider v ∈ A, F0 -random variable X0 ∈ K a.s. and the strong solution X(t) to (19). Fix t > 0. We have to show that X(t) ∈ K a.s. For this end let us v(s)) ∈ ﬁx 0 < ε < 1 and deﬁne the mapping R+ s → g(s) := b(X(s),

s L2 (Ω, H). Then for almost all ω ∈ Ω, the mapping s → 0 gω (τ )dτ ∈ H is absolutely continuous on bounded intervals. Deﬁne f : [0, t] ×

s+h (R+ \{0}) → L2 (Ω, H) by f (s, h) = h1 s g(τ )dτ. Then, for almost

t all ω ∈ Ω, limh→0+ 0 |fω (s, h) − gω (s)|ds = 0. Hence, by the dominated

t convergence theorem, limh→0+ E 0 |f (s, h)−g(s)|ds = 0. Next, applying the Fubini theorem, we obtain that lim

h→0+ 0

t

E|f (s, h) − g(s)|ds = 0.

(21)

Let hi → 0+ be such for all i ≥ 1, hi ≤ ε2 . Claim. We claim that for all i large enough, there exist δi → 0+ i ≤ t such that for all and 0 = τ0i ≤ si1 < τ1i ≤ si2 ... ≤ simi < τm i i i 1 ≤ j ≤ mi , τj = sj + hi and 0 ≤ t − mi hi ≤ δi + hi , Copyright © 2005 Marcel Dekker, Inc.

E|

sij +hi

sij

g(τ )dτ − hi g(sij )| ≤ εhi .

(22)

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The proof of this claim follows from (21). It is quite technical and is omitted. Consider piecewise constant controls if for some 1 ≤ j ≤ mi − 1, s ∈ [sij , τji ), v(sij ) v(τji ) if for some 0 ≤ j ≤ mi − 1, s ∈ [τji , sij+1 ), ui (s) := i i , v(τmi ) if s ≥ τm i and piecewise constant functions if for some 1 ≤ j ≤ mi − 1, s ∈ [sij , τji ), X(sij ) if for some 0 ≤ j ≤ mi − 1, s ∈ [τji , sij+1 ) X(τji ) Xi (s) := i i . X(τmi ) if s ≥ τm i Then, by the very deﬁnition of the Itˆ o integral, for all s ∈ [0, t], s lim E| (σ(X(ρ), u(ρ)) − σ(Xi (ρ), ui (ρ)))dW (ρ)|2 = 0. i→∞

(23)

0

Consider solutions Yi to dY = b(Y, ui )dt + σ(Y, ui )dW (t), X(0) = X0 . Then by Case 2, Yi (s) ∈ K a.s. for all s ≥ 0. Set ψiε (s) := E|X(s) − Yi (s)|2 . Notice that Ed2K (X(t)) ≤ ψiε (t), (24) and that for some α > 1 and all s ∈ [0, t], s 1 ε E|b(X(ρ), ui (ρ)) − b(Yi (ρ), ui (ρ))|2 dρ+ ψ (s) ≤ α i 0 s (σ(X(ρ), ui (ρ)) − σ(Yi (ρ), ui (ρ)))dW (ρ)|2 + E| 0 s E| (σ(X(ρ), ui (ρ)) − σ(Xi (ρ), ui (ρ)))dW (ρ)|2 + 0 s (g(ρ) − b(X(ρ), ui (ρ))dρ|2 + E| 0 s E| (σ(X(ρ), u(ρ)) − σ(Xi (ρ), ui (ρ)))dW (ρ)|2 =

(25)

0

I1i (s) + I2i (s) + I3i (s) + I4i (s) + I5i (s). Since b and σ are C−Lipschitz in the ﬁrst variable, s i i 2 E|X(ρ) − Yi (ρ)| dρ = 2C I1 (s) + I2 (s) ≤ 2C 0

Copyright © 2005 Marcel Dekker, Inc.

0

s

ψiε (ρ)dρ.

(26)

Invariance of Closed Sets under Stochastic Control Systems

By the Lipschitz continuity of σ with respect to x, we get s I3i (s) ≤ C E|X(ρ) − Xi (ρ)|2 dρ ≤ 0 mi τ i j C (ρ − sij )dρ + 2M Ct(δi + hi ) ≤ C

i j=1 sj mi (τji − sij )2 j=1

sij

2

2b∞ (ε + δi ) +

mi j=1

By (22), E that

mi

τji

j=1 | si j

mi | E j=1

(27)

+ 2M Ct(δi + hi ) ≤ Cε2 t + 2M Ct(δi + ε2 ).

On the other hand, s (g(ρ) − b(X(ρ), ui (ρ)))dρ ≤ 0 i m i τ j | (g(ρ) − b(X(sij ), v(sij )))dρ|+ j=1

227

τji sij

τji

sij

(28)

|b(X(ρ), ui (ρ)) − b(X(sij ), v(sij )|dρ.

(g(ρ) −

b(X(sij ), v(sij ))dρ|

≤ εt, which implies

2 (g(ρ) − b(X(sij ), v(sij ))dρ| ≤ 2εt2 b∞ .

(29)

Furthermore, by the Lipschitz continuity of b with respect to x, for a constant c1 > 0 independent from i, ε, 2 mi τ i j |b(X(ρ), ui (ρ)) − b(X(sij ), v(sij )|dρ ≤ E sij mi τ i j

j=1

2tb∞

sij

E|b(X(ρ), ui (ρ)) − b(X(sij ), v(sij ))|dρ ≤

j=1 mi τ i j

2Ctb∞ c1

mi j=1

j=1 τji 3

sij

sij

(30) E|X(ρ) − X(sij )|dρ ≤

ρ − sij dρ ≤ c1

mi (τji − sij )3/2 ≤ c1 εt. j=1

By (28) - (30) for constant c2 > 0 independent from i, ε, s, I4i (s) ≤ c2 (ε + δi ). This and (25) - (27) imply for a constant c3 > 0 independent Copyright © 2005 Marcel Dekker, Inc.

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from i, ε, and for all s ∈ [0, t]

s ψiε (s) ≤ c3 0 ψiε (ρ)dρ + c3 (ε + δi ) + I5i (s). Then it follows from the Gronwall inequality that for a constant c4 > 0 independent from ε and for all i,

t (31) ψiε (t) ≤ c4 (ε + δi + I5i (t)) + c4 0 I5i (s)ds. From (23) we know that for every s ∈ [0, t], limi→∞ I5i (s) = 0. On the other hand, s i Eσ(X(ρ), u(ρ)) − σ(Xi (ρ), ui (ρ))2 dρ ≤ 2sσ2∞ . I5 (s) = 0

From the

t Lebesgue dominated convergence theorem we deduce that limi→∞ 0 I5i (s)ds = 0. This and (24), (31) imply that E(dK (X(t)2 ) ≤ lim supi→∞ ψiε (t) ≤ c4 ε. Since ε > 0 is arbitrary and c4 does not depend on ε, we get X(t) ∈ K a.s. 2

References [1] AUBIN J.-P. (1991) Viability Theory, Birkh¨ auser, Boston, Basel, Berlin. [2] AUBIN J.-P. and DA PRATO G. (1990) Stochastic viability and invariance, Annali Scuola Normale di Pisa, 27, 595–694. [3] AUBIN J.-P., DA PRATO G., and FRANKOWSKA H. (2001) Stochastic invariance for diﬀerential inclusions, Set-Valued Analysis, 8, 181–201. [4] AUBIN J.-P. and DOSS H. (to appear) Characterization of stochastic viability of any nonsmooth set involving its generalized contingent curvature, Stochastic Analysis and Applications. [5] BARDI M. and GOATIN P. (1999) Invariant sets for controlled degenerate diffusions: a viscosity solutions approach, W. M. M. Mc Eneaney, G. G. Yin, Q. Zhang Eds. LNiM 1660, Birkh¨ auser, 191–208. [6] BUCKDAHN R., PENG S., QUINCAMPOIX M., and RAINER C. (1998) Existence of stochastic control under state constraints, Comptes-Rendus de l’Acad´emie des Sciences, 327, 17–22. [7] DA PRATO G. and FRANKOWSKA H. (2001) Stochastic viability for compact sets in terms of the distance function, Dynamic Systems and Applications, vol. 10, 177–184. [8] DA PRATO G. and FRANKOWSKA H. (to appear) Invariant measure for a class of parabolic degenerate equations, NODEA. [9] DOSS H. (1977) Liens entre ´equations diﬀ´ erentielles stochastiques et ordinaires, Ann. Inst. Henri Poincar´e, Calcul des Probabilit´es et Statistique, 23, 99–125. [10] DOSS H. and LENGLART E. (1977) Sur le comportement asymptotique des solutions d’´equations diﬀ´ erentielles stochastiques, CRAS, 284, 971–974.

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229

[11] IKEDA N. and WATANABE S. (1981) Stochastic diﬀerential equations and diffusion processes, North-Holland, Amsterdam-New York. [12] MILIAN A. (1997) Invariance for stochastic equations with regular coeﬃcients, Stochastic Anal. Appl., 15, 91–101. [13] STROOCK D.V. and VARHADAN S.R.S (1979) Multidimensional Diﬀusion Processes, Springer-Verlag, Berlin.

Copyright © 2005 Marcel Dekker, Inc.

CHAPTER XVIII UNIFORM STABILIZATION OF AN ANISOTROPIC SYSTEM OF THERMOELASTICITY Achieving Stability with Minimal Geometric Assumptions Mary Ann Horn∗ Department of Mathematics Vanderbilt University Nashville, TN, USA

Abstract

Stabilization of thermoelasticity is inherently easier than stabilization of the corresponding elastic model due to the additional smoothing that occurs through the coupling of thermal eﬀects. However, in asking if an anisotropic thermoelastic system may be uniformly stabilized through the use of boundary control when the domain is assumed to be nonconvex, open questions still exist. In this article, the question of stabilization for a three-dimensional thermoelastic system is addressed. Generalizations to nonconvex domains require unique continuation for the system, a question that remains largely unanswered for anisotropic systems.

Keywords: Boundary stabilization, thermoelasticity, anisotropic elasticity, nonlinear control

Introduction In the past decade, numerous results have demonstrated that uniform decay of the energy can be obtained for elastic systems through the use of boundary dissipation. Early results focused on the case of a homogeneous, isotropic elastic body, beginning with the work of Lagnese [8–9]. In the latter, nonlinear boundary feedback, dependent on both the velocity and the tangential component of the position, was shown to uniformly stabilize the system under the assumption that the domain

∗ Research

partially supported by National Science Foundation Grant DMS-9803547.

Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

was star-shaped. Extending this result to more general domains, Horn used only velocity feedback acting as a traction force along the boundary, thus eliminating the additional feedback involving the tangential derivative [5]. Initial results on anisotropic elasticity were achieved by Alabau and Komornik with the underlying assumption that the domain was spherical, thus imposing strict geometric conditions even if control was assumed to be active on the entire boundary [2]. More recently, the question of uniform stability for thermoelastic systems with boundary dissipation has received considerable attention. While there is now a considerable amount of literature on boundary stabilization of thermoelastic Kirchhoﬀ plates (see [3, 12] and references therein) and the full von K´ arm´an system of dynamic elasticity [4, 10], literature most closely related to the problem of interest in this paper focuses on boundary stabilization results for isotropic and anisotropic thermoelasticity. Rivera and Olivera established uniform stability of an anisotropic thermoelastic system with linear velocity feedback acting through natural boundary conditions [15]. Application of the divergence theorem within their proof necessitates the assumption that the domain be spherical, however, even with this restrictive assumption, this result was among the ﬁrst to establish positive results on uniform stability for anisotropic systems. An extension of this result to include nonlinear boundary feedback control was seen in the work of Aassila [1], again with strict geometric assumptions. The goal of this paper is to extend the results of Rivera and Olivera to more general domains while retaining the nonlinear boundary feedback control seen in the work of Aassila. To do so necessitates the use of sharp trace regularity results and requires unique continuation results for the corresponding anisotropic elastic system. Because few unique continuation results are available for anisotropic systems, to facilitate the proof, a light interior damping term will be included. While this requirement is solely due to a technical challenge in the proof, this assumption is reasonable, given that all elastic bodies have some inherent internal damping.

1.

Statement of the Problem

To formulate the system of thermoelasticity, let n be a positive integer and let (aijkl ) be a tensor satisfying the following symmetries, aijkl = aklij = ajikl , 1 ≤ i, j, k, l ≤ n.

(1)

For some α > 0, the above tensor satisﬁes a strong ellipticity condition, aijkl εij εkl ≥ αεij εkl , Copyright © 2005 Marcel Dekker, Inc.

(2)

Uniform Stabilization of an Anisotropic System of Thermoelasticity

233

for every symmetric tensor εij . Assume Ω is a nonempty bounded domain in IRn with boundary ∂Ω. Let u = (u1 , . . . , un ) : Ω → IRn denote the displacement vector. Then with the strain tensor εij deﬁned by 1 εij (u) = 2

∂uj ∂ui + ∂xj ∂xi

,

(3)

the stress-strain relation can be expressed by σij (u) = aijkl εkl (u).

(4)

By letting θ denote the temperature, consider the system of anisotropic thermoelasticity with light internal damping deﬁned in the domain Ω with suﬃciently smooth boundary ∂Ω = ∂Ω0 ∪ ∂Ω1 : in Ω × (0, T ) utt − ∇ · σ(u) + ∇θ + but = 0 θt = ∆θ − divut in Ω × (0, T ) u = 0, θ = 0 on ∂Ω0 × (0, T ) σ(u)ν − θν = −g(ut ) on ∂Ω1 × (0, T ) ∂θ + λθ = 0 on ∂Ω1 × (0, T ) ∂ν

(5) (6) (7) (8) (9)

with corresponding initial conditions u(·, 0) = u0 (·), ut (·, 0) = u1 (·), θ(·, 0) = θ0 (·)

in Ω.

(10)

In the above equations, ν represents the unit outward normal to ∂Ω and b(x) > 0 for all x ∈ Ω. To avoid the necessity of compatibility conditions at the junction of ∂Ω0 and ∂Ω1 , the assumption ∂Ω0 ∩ ∂Ω1 = ∅ is imposed. The control function, g, which is acting via traction forces on the boundary, is a continuous vector-valued function, with each component assumed to be monotone increasing and zero at the origin. Additionally, g ∈ C 1 (IRn ) is subject to the following constraints: g(s) · s > 0, for s = 0, m|s| ≤ |g(s)| ≤ M |s|, for |s| > 1,

(11)

for two positive constants, m and M . Notice that while growth conditions are imposed for large values of |s|, none are required near the origin. Copyright © 2005 Marcel Dekker, Inc.

234

1.1

CONTROL AND BOUNDARY ANALYSIS

Wellposedness of the System

To state wellposedness of this anisotropic thermoelastic system, the following function space deﬁnitions are required. Let 1 (Ω) ≡ {v ∈ H 1 (Ω) : v = 0 on ∂Ω0 } H∂Ω 0

(12)

1 (Ω) to be and deﬁne H∂Ω 0 1 (Ω) ≡ {v ∈ [H 1 (Ω)]n : v = 0 on ∂Ω0 }. H∂Ω 0

(13)

Theorem 1 (Existence of strong solutions.) Let (u0 (x), u1 (x), θ0 (x)) ∈ 1 (Ω) × H1 (Ω) × H 1 (Ω) satisfy the following compat[H 2 (Ω)]n ∩ H∂Ω ∂Ω0 ∂Ω0 0 ibility condition: σ(u0 )ν − θ0 ν = −g(u1 ) on ∂Ω1 . Then there exists a unique solution, 1 1 (Ω) × H∂Ω (Ω)), (u, ut ) ∈ C([0, T ]; [H 2 (Ω)]n ∩ H∂Ω 0 0 1 θ ∈ C([0, T ]; H∂Ω (Ω)), 0

satisfying system (5-10).

Theorem 2 (Existence of weak solutions.) Let (u0 (x), u1 (x), θ0 (x)) ∈ 1 (Ω) × [L2 (Ω)]n × L2 (Ω). Then there exists a unique solution (in the H∂Ω 0 sense of distributions), 1 (Ω) × [L2 (Ω)]n ), (u, ut ) ∈ C([0, T ]; H∂Ω 0

θ ∈ C([0, T ]; L2 (Ω)), satisfying system (5-10). Proof of the existence and uniqueness of solutions can be established via semigroup theory [14]. The reader is referred to discussion in [1, 15] for further details on the proofs for anisotropic thermoelasticity. In order to concentrate on the question of uniform stability, the details of wellposedness will not be discussed in this paper.

1.2

Uniform Decay of the Energy of the System

The natural energy for the system of anisotropic thermoelasticity shown in (5-10) is deﬁned by 1 (14) ut 2[L2 (Ω)]n + (σ(u), ε(u))Ω + θ2L2 (Ω) . E(t) ≡ 2 Copyright © 2005 Marcel Dekker, Inc.

Uniform Stabilization of an Anisotropic System of Thermoelasticity

235

Note that the inner product appearing in the deﬁnition, (σ(u), ε(u))Ω is topologically equivalent to the standard norm on [H 1 (Ω)]n . In order to show that the energy of the solution to the system of anisotropic thermoelasticity decays uniformly to zero with respect to the initial energy, we begin by focusing on the nonlinear control. To state the stability result, accounting for the nonlinearities, the outline of Lasiecka and Tataru is followed [11]. Let G(x) be a concave, strictly increasing function which is zero at the origin and satisﬁes G(s · g(s)) ≥ |s|2 + |g(s)|2 for |s| ≤ 1

and deﬁne ˜ G(x) ≡G

x meas(∂Ω × (0, T ))

(15)

.

(16)

Since G˜ is monotone increasing, the operator cI + G˜ is invertible for any constant c ≥ 0. Therefore, the function ˜ −1 (kx), p(x) ≡ (cI + G)

k > 0,

(17)

is positive, continuous, and strictly increasing with p(0) = 0.

Theorem 3 (Uniform stability of the system.) Let (u, θ) be a solution of the anisotropic thermoelastic system (5-10). Assume h(x) · ν ≤ 0 on ∂Ω0 , where h(x) = x − x0 for some x0 ∈ IRn . Then for some T0 > 0,

t E(t) ≤ S −1 ∀t > T0 , T0

(18)

(19)

where S(t) → 0 as t → 0 and is the contraction semigroup satisfying the diﬀerential equation St (t) + q(S(t)) = 0,

S(0) = E(0),

(20)

where q(x) is deﬁned by q(x) ≡ x − (I − p)−1 (x) for x > 0.

2.

Proof of Theorem 3: Uniform Stabilization

The system of anisotropic thermoelasticity can be reformulated as a variational problem for equation (5), taking into account the boundary conditions in (7-8), as follows: (utt , φ)Ω + (σ(u), ε(φ))Ω + (but , φ)Ω + (∇θ, φ)Ω +(g(ut ) − θν, φ)∂Ω1 − (σ(u)ν, φ)∂Ω0 = 0, Copyright © 2005 Marcel Dekker, Inc.

(21)

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CONTROL AND BOUNDARY ANALYSIS

plus the variational form of equation (6), including boundary condition (9): (θt , ψ)Ω + (∇θ, ∇ψ)Ω + (divut , ψ)Ω + (λθ, ψ)∂Ω1 = 0.

(22)

As a ﬁrst step in the proof of Theorem 3, the energy of the system is shown to satisfy a dissipativity inequality. Note that the corresponding anisotropic system of elasticity with b(x) ≡ 0 and with homogeneous boundary conditions is a conservative system. For the thermoelastic system, the following dissipativity result can be shown to hold.

Lemma 4 (Dissipativity inequality.) Let (u, θ) be a solution of system (5-10). Then for any s < t, the following inequality holds: t s

t

g(ut ) · ut dxdt +

E(t) +

s

∂Ω1

t

|∇θ| dxdt +

s

Ω

Ω

t

2

+

b|ut |2 dxdt

λθ 2 dxdt = E(s). s

(23)

∂Ω1

Proof: Let φ ≡ ut and ψ ≡ θ in the variational formulation, integrate with respect to time, apply the divergence theorem, and add the resulting equations.

2.1

Trace Regularity

Sharp trace regularity estimates play a critical role in eliminating many of the strict geometric assumptions frequently seen in the literature. In particular, the following result for a general elastic system will be used within the proof of stabilization. Note that standard trace theory does not allow the tangential component of the derivative of u on the boundary to be bounded by the energy norm.

Theorem 5 (Trace regularity [6].) Let u be a solution to the following general elastic system: in Ω × (0, T ) utt − ∇ · σ(u) = F u = 0 on ∂Ω0 × (0, T ) σ(u)ν = G on ∂Ω1 × (0, T )

(24) (25) (26)

with corresponding initial conditions u(·, 0) = u0 (·), ut (·, 0) = u1 (·) Copyright © 2005 Marcel Dekker, Inc.

in Ω.

(27)

Uniform Stabilization of an Anisotropic System of Thermoelasticity

and let 0 < α < T /2. Then

T −α

2 ∂Ω |∇uτ α | dxdt ≤ C ut 2[L2 (∂Ω)]n + u2[L2 (0,T ;H 1/2+ (Ω))]n +F 2[H −1/2 (0,T ;Ω)]n

+

G2[L2 (0,T ;∂Ω1 )]n

237

(28)

dt.

Remark: Proof of this theorem is based on microlocal analysis, under the assumption that the spatial operator, ∇ · σ(u), is uniformly elliptic. While the result stated in [6] is in the context of isotropic elasticity, the estimate is also valid for anisotropic elasticity. This is due to the fact that ellipticity is preserved under the partition of unity and the ﬂattening of the boundary procedure used in the proof. For the thermoelastic system under consideration here, application of Theorem 5 yields the inequality,

T −α

2 ∂Ω |∇uτ α | dxdt ≤ C ut 2[L2 (∂Ω)]n + g(ut )2[L2 (∂Ω)]n + θ2L2 (0,T ;∂Ω) (29) +θ2H 1/2+ (0,T ;Ω) + u2[L2 (0,T ;H 1− (Ω))]n dt.

2.2

Stability Estimates

The critical step in proving the stability result stated in Theorem 3 is to establish the following estimate.

Lemma 6 (Stability estimate.) Let (u, θ) be a strong solution to the system of anisotropic thermoelasticity deﬁned in (5-10). Assume h(x) · ν ≤ 0

on ∂Ω0 .

Then there exists a suﬃciently large time T and a constant CT (E(0)), which is dependent upon T and, possibly, upon the initial energy E(0), such that the following estimate holds: T 2 |ut | + |g(ut )|2 dxdt E(T ) ≤ CT (E(0)) 0

T

∂Ω1

b|ut | dxdt +

T

+ 0

Ω

|∇θ| dxdt

2

2

0

(30)

Ω

Once Lemma 6 has been established, the proof of uniform stabilization follows from the results of Lasiecka and Tataru [11] as it did for the case of anisotropic elasticity [5], where details as to the application of their work can be found. Copyright © 2005 Marcel Dekker, Inc.

238

CONTROL AND BOUNDARY ANALYSIS

The proof of Lemma 6 is comprised of three basic steps. First, an appropriate choice of test functions will lead to a preliminary bound on the energy. Such test functions, or multipliers, depend upon the problem at hand. Among the functions chosen are those that have been seen in anisotropic elasticity, φ1 ≡ ∇uh and φ2 ≡ u. Secondly, sharp trace regularity estimates for the system of anisotropic elasticity are applied to remove traces of the solution which cannot be bounded by the energy using standard trace theory. Finally, a compactness-uniqueness argument allows the ﬁnal estimate to be obtained by removing terms involving u which are lower order with respect to the energy norm.

Preliminary Estimate. tablished.

To begin, the following lemma will be es-

Lemma 7 Let (u, θ) be a solution of the system of anisotropic thermoelasticity deﬁned in (5-10). Assume h(x) · ν ≤ 0

on ∂Ω0 .

Then there exists a constant, C, such that for any ∈ (0, 14 ), the following estimate holds: T T 2 |ut | + |g(ut )|2 + |∇u|2 dxdt E(t)dt ≤ C E(T ) + 0

0

∂Ω1

T

+

θ21,Ω dt + l.o.t.(u) ,

(31)

0

where

l.o.t.(u) ≡ 0

T

u2[H 1− (Ω)]n dt

which includes terms involving the norms of u which are strictly lower order with respect to the energy norm. Proof: Analogously to the system of anisotropic elasticity, the proof follows the series of steps outlined below, taking into account the thermal eﬀects now included. For further technical details, the reader is referred to [7]. Let φ = ∇uh be the test function in the variational formulation (21). Integrating over time, the resulting inequality is

n T n−2 T 2 ε(u)) dt 0 (σ(u), 2 0 Ω |ut | dxdt − 2

T Ω ≤ C(E(0) + E(T )) + 12 0 ∂Ω1 |ut |2h · νdxdt

T (32)

T − 0 (but , ∇uh)Ω dt − 0 (g(ut ) − θν, ∇uh)∂Ω1 dt

T − 0 (σ(u), ε(u)ν)∂Ω dt. Copyright © 2005 Marcel Dekker, Inc.

Uniform Stabilization of an Anisotropic System of Thermoelasticity

239

With a second multiplier, φ = u, the following identity can be obtained:

T dt 0 (σ(u), ε(u))

T

T Ω = 0 ut 2[L2 (Ω)]n dt − (ut , u)T0 − 0 (but , u)Ω dt (33)

T

T + 0 (σ(u)ν, u)∂Ω0 dt − 0 (g(ut ) − θν, u)∂Ω1 dt Multiplying the second identity by n−1 2 , adding it to the ﬁrst and applying trace theory to the terms involving θ yields

1 T 1 T 2 2 0 ut [L2 (Ω)]n dxdt + 2 0 (σ(u), ε(u))Ω dt

T

≤ C E(0) + E(T ) + 0 ∂Ω1 |ut |2 + |g(ut )|2 dxdt (34)

T

T

|∇u|2 dxdt + 0 Ω b|ut |2 dxdt + 0 ∂Ω1

T

− 0 Ω |∇θ|2 dxdt + l.o.t.(u) .

Elimination of Higher Order Traces. At this stage, the higher order traces of u on the boundary must be bounded. To do so, Theorem 5 is applied, resulting in the following estimate. Lemma 8 Let (u, θ) be a solution of the anisotropic thermoelastic system in (5-10). Assuming h · ν ≤ 0 on ∂Ω0 , then for some suﬃciently large time T , there exists a constant C(T ) such that for any 0 < < 14 , the solution satisﬁes T E(T ) ≤ C(T ) 0 Ω b|ut |2 + |∇θ|2 dxdt (35)

T

+ 0 ∂Ω1 |ut |2 + |g(ut )|2 dxdt + l.o.t.(u) . Proof follows by application of Theorem 5 combined with a careful analysis of the tangential and normal traces of the solution on the boundary.

Removal of Lower Order Terms. To reach the conclusion of Lemma 6, the ﬁnal step is to remove the lower order terms, i.e., the following estimate must be established: Lemma 9 Let (u, θ) be a solution to the system of thermoelasticity deﬁned in (5-10). Then there exists a constant, CT (E(0)), dependent upon T and, possibly, dependent upon the initial energy E(0), such that

T l.o.t.(u) ≤ CT (E(0)) 0 Ω |∇θ|2 + b|ut |2 dx

(36) + ∂Ω1 |ut |2 + |g(ut )|2 dx dt. Remark: It is precisely at this stage that the assumption b(x) > 0 for all x ∈ Ω plays a key role. Proof of Lemma 9 is via a compactnessCopyright © 2005 Marcel Dekker, Inc.

240

CONTROL AND BOUNDARY ANALYSIS

uniqueness argument and with the inclusion of this light internal damping, the question of uniqueness for the full dynamic system can be reduced to that of a corresponding stationary problem. Proof: Assume (36) does not hold. Then there exists a sequence of solutions {(um , θm )}∞ m=1 satisfying system (5-10) with initial data 1 (Ω) × [L2 (Ω)]n × L2 (Ω) (um,0 , um,1 , θm,0 ) ∈ H∂Ω 0

such that l.o.t.(u) →∞

2 + b|u |2 ) dx + 2 + |g(u )|2 ) dx dt (|∇θ| (|u | t t t Ω ∂Ω1

T

0

(37)

as m → ∞. Let Em (t) denote the energy of the system satisfying by (um , θm ) at time t. Since (um , θm ) satisﬁes system (5-10), Lemma 8 can be applied and, therefore, Em (t) ≤ C(T ) uniformly in m. Thus, the sequence of solutions has the following convergence properties: in [L∞ (0, T ; H 1 (Ω)]n weakly star, um → u um,t → ut in [L∞ (0, T ; L2 (Ω))]n weakly star. Therefore, l.o.t.(um ) → l.o.t.(u) as m → ∞, since the lower order terms are compact with respect to the energy norm. Case 1: Assume l.o.t.(u) = 0. Then, if (37) holds, θm → 0 um,t → 0 um,t → 0 g(um,t ) → 0

in in in in

L2 (0, T ; H 1 (Ω)), [L∞ (0, T ; Ω]n , [L∞ (0, T ; ∂Ω1 )]n , [L∞ (0, T ; ∂Ω1 )]n .

Passing the limit on the original system in (5-10), the limit function u must satisfy ∇ · σ(u) = 0 in Ω × (0, T ), u = 0 on ∂Ω0 × (0, T ), σ(u)ν = 0 on ∂Ω1 × (0, T ), and, therefore, Korn’s inequality implies u ≡ 0 [13], which contradicts the assumption that l.o.t.(u) = 0. Case 2: Assume l.o.t.(u) = 0. Deﬁne cm ≡ l.o.t.(um ) and u ˜m ≡ um /cm , θ˜m ≡ θm /cm . Then l.o.t.(˜ um ) ≡ 1, Copyright © 2005 Marcel Dekker, Inc.

Uniform Stabilization of an Anisotropic System of Thermoelasticity

1 c2m and 1 c2m

T

0

T

241

b|um,t |2 + |∇θm |2 dxdt → 0,

Ω

0

|um,t |2 + |g(ut )|2 dxdt → 0,

∂Ω1

implying, θ˜m → 0 u ˜m,t → 0 u ˜m,t → 0 1 cm g(um,t ) → 0

in in in in

L2 (0, T ; H 1 (Ω)), [L∞ (0, T ; Ω]n , [L∞ (0, T ; ∂Ω1 )]n , [L∞ (0, T ; ∂Ω1 )]n .

Lemma 8 and the above convergence yields the inequality T 1 1 2 2 E (T ) ≤ 2 m cm c2m 0 Ω b|um,t | + |∇θm | dxdt

T

1 2 + |g(u 2 dxdt + l.o.t.(˜ |u u | )| ) m,t m,t m 2 0 ∂Ω1 c m

Therefore, the right-hand side is bounded uniformly for all m, implying, as in the ﬁrst case, that u) = 1 l.o.t.(˜ um ) → l.o.t.(˜ and u ˜ satisﬁes the corresponding homogeneous limiting equation. Hence, as before, u ˜ ≡ 0, a contradiction.

Conclusion of the Proof of Lemma 6. Combining the results of Lemmas 8 and 9 yields the desired stability estimate of Lemma 6.

3.

Discussion

The proof of stabilization demonstrates that the addition of thermal eﬀects to the anisotropic system of elasticity creates no signiﬁcant diﬃculties. Of course, this is to be expected since it is well known that thermoelastic systems have inherent stability properties not seen in purely elastic systems, which are typically conservative unless additional damping is assumed. However, the same challenge that appeared in extending uniform stability results to anisotropic elastic systems when working in nonconvex domains is again seen here. Unique continuation properties play a critical role when considering general domains, but in the case of anisotropic systems, many open questions remain. It is clear that this diﬃculty can be avoided by the inclusion of the internal damping term, represented by b(x)ut , however, whether this term can be easily eliminated remains to be seen. Copyright © 2005 Marcel Dekker, Inc.

242

CONTROL AND BOUNDARY ANALYSIS

References [1] M. Aassila. Nonlinear boundary stabilization of an inhomogeneous and anisotropic thermoelasticity system. Appl. Math. Letters, 13:71–76, 2000. [2] F. Alabau and V. Komornik. Boundary observability, controllability and stabilization of linear elastodynamic systems. SIAM J. Control Optim., 37(2):521– 542, 1999. [3] G. Avalos and I. Lasiecka. Exponential stability of a thermoelastic system with free boundary conditions without mechanical dissipation. SIAM J. Math. Anal., 29(1):155–182, 1998. [4] A. Benabdallah and I. Lasiecka. Exponential decay rates for a full von k´ arm´ an system of dynamic thermoelasticity. J. Diﬀ. Eq., 160:51–93, 2000. [5] M. A. Horn. Implications of sharp trace regularity results on boundary stabilization of the system of linear elasticity. J. Math. Anal. Appl., 223:126–150, 1998. [6] M. A. Horn. Sharp trace regularity for the solutions of the equations of dynamic elasticity. J. Math. Sys. Estim. Cont., 8(2):217–219, 1998. [7] M. A. Horn. Nonlinear boundary stabilization of a system of anisotropic elasticity with light internal damping. Contemporary Mathematics, 268:177–189, 2000. [8] J. E. Lagnese. Boundary stabilization of linear elastodynamic systems. SIAM J. Control Optim., 21(6):968–984, 1983. [9] J. E. Lagnese. Uniform asymptotic energy estimates for solutions of the equations of dynamic plane elasticity with nonlinear dissipation at the boundary. Nonlinear Analysis, 16(1):35–54, 1991. [10] I. Lasiecka. Uniform stabilizability for a full von k´ arm´ an system with nonlinear boundary dissipation. SIAM J. Cont. Optim., 36(4):1376–1422, 1998. [11] I. Lasiecka and D. Tataru. Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Diﬀ. Int. Equations, 6:507–533, 1993. [12] Z. Liu and S. Zheng. Semigroups Associated with Dissipative Systems. Chapman & Hall, Boca Raton, Florida, 1999. [13] A. S. Shamaev, O. A. Oleinik, and G. A. Yosiﬁan. Mathematical Problems in Elasticity and Homogenization. North-Holland, Amsterdam, 1992. [14] A. Pazy. Semigroups of Linear Operators and Applications to Partial Diﬀerential Equations. Springer-Verlag, New York, 1993. [15] J. E. M. Rivera and M. L. Olivera. Stability in inhomogeneous and anisotropic thermoelasticity. Boll. U. M. I., 11-A(7):115–127, 1997.

Copyright © 2005 Marcel Dekker, Inc.

CHAPTER XIX SEMIGROUP WELL-POSEDNESS OF A MULTILAYER MEAD-MARKUS PLATE WITH SHEAR DAMPING Scott W. Hansen∗ Department of Mathematics Iowa State University Ames, IA, USA

Abstract

A multilayer plate model consisting of m + 1 relatively stiﬀ plate layers bonded together by m compliant plate layers is described. In the case of only three layers, the model reduces to a Mead-Markus sandwich plate with viscous damping in the central layer. Existence and uniqueness of solutions are established using semigroup theory. In particular, the homogeneous problem may be written in the form x (t) = Ax(t) where A is the generator of a C0 contraction semigroup.

Keywords: Multilayer plate, Mead-Markus plate, sandwich plate, shear damping

Introduction The classical sandwich plate is a three-layer plate model consisting of two relatively stiﬀ outer layers and a more compliant inner layer. Various sandwich plate and constrained layer models have been proposed and analyzed, see e.g., DiTaranto [2], Mead and Markus [9], Rao and Nakra [10]. For a review and some comparisons of these models see e.g., Sun and Lu [11]or Mead [8]. In Hansen [3], a multilayer generalization of the Mead-Markus model (and Rao-Nakra model) was derived. The multilayer models consist of alternating “stiﬀ” and “compliant” layers. The stiﬀ layers do not allow shear and are modeled as Kirchhoﬀ plates, while the compliant layers allow shear and may be modeled in various ways. For the multilayer Mead-Markus model, the in-plane inertia is ignored in each layer and bending stiﬀness is ignored in the compliant layers. In the undamped ∗ This

research was supported in part by the National Science Foundation under grant DMS0205148.

Copyright © 2005 Marcel Dekker, Inc.

244

CONTROL AND BOUNDARY ANALYSIS

case, existence and uniqueness of solutions for the multilayer systems in [3] was established on the natural energy spaces using the standard variational theory (e.g., Lions and Dautray [1]). However, in the case when shear damping is included in the multilayer Mead-Markus model, the variational approach in ([3]) could not be applied (at least directly) due to a lack of coerciveness in the bilinear form associated with the highest order time-derivatives. We overcome this problem in this article for the special case in which all of the stiﬀ layers have the same Poisson’s ratio. In this case it is possible to solve for the shear in each compliant layer and write the homogeneous problem in the standard form y = Ay. We then show that A is the generator of a strongly continuous semigroup of contractions on the natural energy space. Analogous models to the one of this article have been investigated in the case of a layered beam. For example, an analogous three-layer beam model was investigated in [4], where it was found (for either clamped or simply supported boundary conditions) that the associated semigroup is in fact exponentially stable (moreover is analytic, if rotational moment of inertia is neglected). Similar results were found in the multilayer beam model in [5] for the case of simply supported boundary conditions.

1.

Multilayer Mead-Markus Model

We ﬁrst describe the “multilayer Mead-Markus model” of [3]. (For a more careful derivation see [3].) The multilayer sandwich plate is assumed to consist of n = 2m + 1 layers of alternating “stiﬀ” and “compliant” plates that occupy the region Ω × (0, h) at equilibrium, where Ω is a smooth bounded domain in the plane. The layers are indexed from 1 to n, from bottom to top, with odd indices for stiﬀ layers and even indices for compliant layers. We use coordinates x = {x1 , x2 } to denote points in Ω and use x3 as the transverse independent variable. As is typical in plate theories, it is assumed that the transverse displacement is independent of x3 , i.e., we may use the scalar w(x) to denote the transverse displacement at the point x ∈ Ω. We let v i = {v1i , v2i } i = 1, 2, . . . n denote the in-plane displacements along the midplane of the ith layer. It is assumed that all the layers are bonded to one another so that no slip occurs. The Kirchhoﬀ hypothesis applies to the stiﬀ layers (i.e., no shear), while the compliant layers allow shear and deform linearly with respect to the transverse variable. Under these assumptions any displacement is completely determined by speciﬁcation of the state variables v i , i odd, and w.

Copyright © 2005 Marcel Dekker, Inc.

245

Semigroup Well-Posedness of a Multilayer Mead-Markus Plate

If θ and ξ are matrices in Rlm , by θ : ξ we mean the scalar product in Rlm . We also denote (θ, ξ)Ω = θ : ξ dx, (θ, ξ)Γ = θ : ξ dΓ. Ω

Γ

Deﬁne the form for functions θ(x) = {θ1 (x), θ2 (x)} by ( ( ( ' ' ' ∂θ1 ∂ θˆ1 ∂θ2 ∂ θˆ2 ∂θ2 ∂ θˆ1 ˆ = , + , + ν , (θ; θ) ∂x1 ∂x1 Ω ∂x2 ∂x1 Ω 2 ∂x2 Ω ( ∂x' ( ' ' 1−ν ' ∂θ1 ∂θ1 ∂ θˆ2 ∂θ2 ∂ θˆ1 + ν ∂x1 , ∂x2 + + 2 ∂x2 ∂x1 , ∂x2 + Ω

∂ θˆ2 ∂x1

(( Ω

.

where ν is the Poisson’s ratio (0 < ν < 1/2). It is assumed that the in-plane kinetic energy is negligilble and bending potential energy of the even layers are negligible in comparison to those of the surrounding odd layers. The energy is the sum of the kinetic and potential energies of each layer Ei = Ki + Pi , where 1

2 2 i odd 2 Ω ρi hi (w ) + αi |∇w | dx Ki = 1 2 ρi hi (w ) dx i even 21 Ω 3 i i i odd 2 Ω K(hi Di ∇w; ∇w) + 12(hi Di v ; v ) dx Pi = 1 2 G h |ϕ | dx i even 2 Ω i i i In the above primes (e.g., w ) denote diﬀerentiation with respect to time, Di = Ei /12(1 − ν 2 ), αi = ρi h3i /12 where Ei > 0 denotes the in-plane Young’s modulus, hi the thickness, ρi the volume density, all for the ith layer. In addition, Gi is the transverse shear modulus. The variable ϕi is the shear of the ith layer deﬁned in (1) below. Deﬁne the following n by n matrices: h = diag (h1 , h2 , . . . , hn ) p = diag (ρ1 , ρ2 , . . . , ρn )

D = diag (D1 , D2 , . . . , Dn ) G = diag (G1 , G2 , . . . , Gn ).

In addition, we let e.g., hO and hE denote the diagonal matrices of oddindexed and even-indexed thicknesses hi , respectively. Actually, we will only need to refer to hO , hE , DO , GE , pO . Also deﬁne 1E and 1O as the column vectors of m and m + 1 ones, respectively. Let vO denote the (m+1)×2 matrix with rows v i , i = 1, 3, 5, . . . 2m+1. Likewise, let ϕO and ϕE denote the matrix with rows ϕi , i odd and even, respectively. Since no shear occurs in the odd layers, ϕO = 0. The shear ϕE can be expressed in terms of vO , ∇w as follows: hE ϕE

= BvO + hE N ∇w;

N = h−1 E AhO 1O + 1E

(1)

where A = (aij ), B = (bij ) are the m × (m + 1) matrices deﬁned by 1/2 if j = i or i + 1 (−1)i+j+1 if j = i or i + 1 aij = bij = 0 otherwise 0 otherwise. Copyright © 2005 Marcel Dekker, Inc.

246

CONTROL AND BOUNDARY ANALYSIS

Also deﬁne O (vO , vˆO ) =

n

(v i ; vˆi ).

i odd

Collecting the energies one ﬁnds that the total potential and kinetic energy may be expressed as , w ; vO , w )/2 K(t) = c(vO

P(t) = a(vO , w; vO , w)/2

where c(· ; ·) and a(· ; ·) denote the bilinear forms ˆ = (mw, w) ˆ Ω + (α∇w, ∇w) ˆ Ω c(vO , w; vˆO , w) a(vO , w; vˆO , w) ˆ = O (K1O ∇w; 1O ∇w) ˆ + 12O (hO DO vO ; vˆO ) +(GE hE ϕE , ϕˆE )Ω where ϕ and ϕˆ satisfy (1), and m=

n

hi ρi ,

i=1

n 1 α= ρi h3i , 12 i odd

K=

Di h3i .

i odd

Let us assume the plate is clamped on a portion Γ0 of the boundary Γ and subject to applied forces on the complementary portion Γ1 and distributed forces on Ω. The equations of motion are determined by Hamilton’s principle from the energy. The variational diﬀerential equation one obtains is the following: ˆ Ω + (α∇w , ∇w) ˆ Ω + (K∇O w, ∇w) ˆ (mw , w) −1 vO + N ∇w) ˆ +12O (hO DO vO ; vˆO ) + (GE hE ϕE , hE Bˆ

ˆ = Ω wf ˆ 3 + vˆO fO dx + Γ1 wg ˆ 3 + vˆO gO − w ˆn Mn ds. = W ({ˆ vO , w}) In the above, f3 is the (scalar) transverse applied force in Ω and fO is the net in-plane force acting on the odd layers in Ω. (fO has rows f i = {f1i , f2i }, i = 1, 3, 5, . . . 2m + 1.) The boundary forces acting on Γ1 are: the transverse force g3 (scalar valued), the in-plane force (dimensions matching fO ), and the bending moment Mn (scalar valued). (For precise description of forces see [3].) The test functions w, ˆ vˆO satisfy the clamped boundary conditions on Γ0 and are suﬃciently regular.

Inclusion of Shear Damping. Damping may be introduced into any of the plate layers by replacing the standard stress-strain relation for transversely isotropic materials by an appropriate dissipative constitutive law. In the case of strain-rate shear damping, the stress-strain relation for transverse shear: σ13 = 2G13 is replaced by a dissipative Copyright © 2005 Marcel Dekker, Inc.

Semigroup Well-Posedness of a Multilayer Mead-Markus Plate

247

˜ d )13 . The equations stress-strain relation of the form: σ13 = 2(G + G dt ˜ ∂ ). of motion are then modiﬁed by the correspondence G → (G + G ∂t ˜E In our situation, shear damping occurs only in the even layers. Let G ˜ be the diagonal matrix with diagonal elements Gi , i even. We obtain the following variational diﬀerential equation when shear damping is included in the even layers: ˆ Ω + (α∇w , ∇w) ˆ Ω + (K∇w, ∇w) ˆ + 12O (hO DO vO ; vˆO ) (mw , w) −1 ˜ +(GE hE ϕE + GE hE ϕE , hE Bˆ vO + N ∇w) ˆ = W ({ˆ vO , w}). ˆ (2)

1.1

Boundary Value Problem

We ﬁrst need to deﬁne some operators and boundary operators. Deﬁne the operator L by Lφ = L{φ1 , φ2 } = {L1 (φ), L2 (φ)} where L1 φ = L2 φ =

∂φ1 ∂ ∂x1 [( ∂x1 ∂φ2 ∂ ∂x2 [( ∂x2

2 + ν ∂φ ∂x2 )] + ∂φ1 + ν ∂x1 )] +

∂ 1−ν ∂φ1 ∂x2 [( 2 )( ∂x2 ∂ 1−ν ∂φ2 ∂x1 [( 2 )( ∂x1

+ +

∂φ2 ∂x1 )] ∂φ1 ∂x2 )].

The associated boundary operator Bφ = {B1 (φ1 , φ2 ), (B2 (φ1 , φ2 )} is deﬁned by ( ' ' ∂φ1 ∂φ2 ( ∂φ2 1−ν 1 + n + ν n B1 (φ1 , φ2 ) = ∂φ ∂x1 1 ∂x2 1 2 ∂x2 + ∂x1 n2 ' ' ( ∂φ2 ∂φ1 ( ∂φ1 1−ν 2 n + ν n B2 (φ1 , φ2 ) = ∂φ + 2 2 ∂x2 ∂x1 2 ∂x1 + ∂x2 n1 . where n = (n1 , n2 ) denotes the outward unit normal to Γ. (For later reference: also deﬁne τ = (−n2 , n1 ) as the unit tangent vector to Γ.) ˆ φ: The following Green’s formula is valid for all suﬃciently smooth φ, ˆ Ω. ˆ = (Bφ, φ) ˆ Γ − (Lφ, φ) (φ, φ)

(3)

For ξ = (ξji ) (i = 1, 2, . . . , n, j = 1, 2) deﬁne the matrices Lξ and Bξ by (Lξ)ij = (Lj ξ i ),

(Bξ)ij = (Bj ξ i ),

i = 1, 2, . . . , n, j = 1, 2.

Furthermore we deﬁne the operators LO , LE , BO , BE from L and B based upon the convention that O and E subscripts refer to the parts of the operators that act upon odd and even rows respectively. Similar to (3) the following Green’s formula is valid for all suﬃciently ˆ smooth ξ, ξ: ˆ = (BO ξ, ξ) ˆ Γ − (LO ξ, ξ) ˆ Ω. O (ξ, ξ) (4) Copyright © 2005 Marcel Dekker, Inc.

248

CONTROL AND BOUNDARY ANALYSIS

The equations of motion can be found from (2) using the Green’s formulas (3), (4) and further integrations by parts: T hE (GE ϕE + G ˜ E ϕ ) = f3 mw − α∆w + K∆2 w − div N E (5) ˜ ) = fO −12hO DO LO vO + B T (GE ϕE + Gϕ E where ϕE = h−1 E BvO + N ∇w ∂ ¯ αwn − K( ∂τ (B∇w) · τ ) − K(∆w)n T hE (GE ϕE + G ˜ E ϕE ) · n = g3 +N (6) K(B∇w) · n = −Mn 12DO BO hO vO = gO w = wn = 0 (7) vO = 0 where (5) holds on Q = Ω × (0, ∞), (6) holds on Σ1 = Γ1 × (0, ∞), and (7) holds on Σ0 = Γ0 × (0, ∞). Appropriate initial conditions are of the form w(0) = w0 ,

2.

w (0) = w1 ,

ϕE (0) = ϕ0E

(8)

State Variable Formulation

There is a diﬃculty in proving existence and uniqueness in the formulation above since it is not an easy task to solve for ϕE in (5), which is needed in order to solve for the generator of a semigroup. However, this can be accomplished in the present situation (when all the Poisson ratios in the stiﬀ layers are the same) due to the decomposition described in this section. Let W = span 1O , and V = span DO hO1O . Clearly these spaces are not orthogonal. Therefore there is a unique decomposition of Rm+1 (or Cm+1 ) into a part in W ⊥ and a part in V . Since the kernel of B is easily seen to be W , any vector in W ⊥ can be written as an element in the image of B T . Thus at each point in Ω and Γ1 , respectively, we have the decomposition fO = B T fE + f˜O ,

gO = B T gE + g˜O

where f˜O and g˜O at each point belong to V × V and fE and gE at each point belong to Rm . We return to (5)–(7) but focus on the terms coupled to vO . The relevant part of the system is ˜ E ϕ ) = f˜O + B T fE in Ω (9) −12hO DO LO vO + B T (GE ϕE + G E 12DO hO BO vO = g˜O + B T gE on Γ1 (10) Copyright © 2005 Marcel Dekker, Inc.

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Semigroup Well-Posedness of a Multilayer Mead-Markus Plate

v = 0 on Γ0 .

(11)

We multiply (9) and (10) on the left by 1TO and obtain Di hi v i ) = −12L( f˜Oi in Ω i odd

12B(

(12)

i odd i

Di hi v ) =

i odd

i g˜O

in Γ1 .

(13)

i odd

If we deﬁne T vO = v¯O = C

Di hi v i ,

f¯ =

i odd

f˜Oi ,

g¯ =

i odd

i g˜O

(14)

i odd

then (12)–(13) can be written as −12L¯ vO = f¯ 12B¯ vO = g¯ v¯O = 0

in Ω on Γ1 on Γ0 .

(15)

When the Poisson’s ratios are constant we can write ∇w). BLO vO = LE BvO = LE hE (ϕE − N

(16)

Deﬁne the matrix P by P = We multiply (9) and (10) by

1 −1 T BD−1 O hO B . 12 −1 −1 1 12 BhO DO

−BLO vO + P GE ϕE

(17)

and obtain

= P fE

in Ω.

(18)

Thus, after substitution of (16) into the above, we obtain the system ˜ E ϕ ) = P fE in Ω (19) ∇w) + P (GE ϕE + G −LE hE (ϕE − N E ∇w) = P gE on Γ1 (20) BE hE (ϕE − N vO = 0 on ΓO . (21)

Lemma 1 The matrix P in ( 17) is positive, i.e., z T P z > 0 for all z ∈ Rm . Moreover, every element pij of P −1 is positive. Proof: We can write P = BΛB T where Λ = diag (λ1 , λ2 , . . . , λm+1 ), with λi > 0, all i. Moreover B can be written as the sum of partitioned . . matrices B = (0 .. Im ) − (Im .. 0), where Im is the identity on Rm , and 0 is Copyright © 2005 Marcel Dekker, Inc.

250

CONTROL AND BOUNDARY ANALYSIS

a column vector of zeros. Performing the block multiplications in (17), one obtains P = Λ11 + Λm+1,m+1 − Λ1,m+1 − Λm+1,1 where Λij is the minor for the ij-th spot in Λ. Thus the diagonal of P has the elements pii = λi + λi+1 , for i = 1, 2, . . . m. The superdiagonal and subdiagonal sequence (beginning in the upper left) are each −λ2 , −λ3 , . . . , −λm . All other elements are zero. We therefore have p11 = λ1 + λ2 >

pmm = λm + λm+1 >

m

|p1,k | = λ2

k=2 m−1

|pm,k | = λm

k=1

pkk = λk + λk+1 ≥

|p1,j | = λk + λk+1 .

j=k

Thus P is diagonally dominant (with strict inequality in the ﬁrst and last rows). Furthermore it is easily veriﬁed that P is irreducible. It follows from the theory of M -matrices (see e.g., [7],Theorem 3, p. 531) that P −1 is a nonnegative matrix (every element nonnegative). This completes the proof. ˜ E ϕ in (19)–(21) Since P is invertible, one can solve for GE ϕE + G E and substitute into (5)–(7) to obtain the following system: mw − α∆w + K∆2 w T −1 T in Q (22) −div N {hE P LE BvO } = f3 − div N hE fE ˜ E ϕ E + GE ϕE − P −1 LE BvO = fE G ∂ α(wn ) − ∂τ (τ · KB∇w) + K(∆w)n T hE (GE ϕE + G ˜ E ϕ˙ E ) · n = g3 +N on Σ1(23) −KB∇w · n = Mn P −1 BE BvO = gE ϕE = 0, w = 0, wn = 0 on Σ0(24) Where ∇w), BvO = hE (ϕE − N

T vO = v¯O , C

(25)

where v¯O is the solution to the stationary Lam´e system (15). Initial conditions are speciﬁed for w, w and ϕE . Suppose for the moment that the system (22)-(24) above together with appropriate initial conditions uniquely determines ϕE and v¯O . Then vO Copyright © 2005 Marcel Dekker, Inc.

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Semigroup Well-Posedness of a Multilayer Mead-Markus Plate

is also determined: Indeed, deﬁne BC to be the partitioned (square) . is deﬁned in (14)). Then (since BC matrix deﬁned by BCT = (B T .. C), (C is invertible from our earlier discussion) vO can be obtained from the following:

∇w hE ϕE − hE N . (26) BC vO = v¯O

3.

Semigroup Formulation of Homogeneous Problem

We consider the homogeneous problem T {hE P −1 LE BvO } = 0 in Q mw − α∆w + K∆2 w − div N ˜ E ϕ + GE ϕE − P −1 LE BvO = 0 in Q G E

(27)

For simplicity, we restrict our interest here to the case of case of hinged boundary conditions: B∇w · n = 0 on Σ (28) BE BvO = 0 w=0 on Σ. (29) Here Σ = ∂Ω × (0, ∞). T vO we assume ﬁxed boundary conditions, so For the variable v¯O = C that v¯O is the solution of −12L¯ vO = 0 v¯O = 0

in Ω on Γ0 .

(30)

The system (30) has the unique solution v¯O = 0 since the form is known to be coercive on H01 (Ω) ([6]). Therefore we eliminate vO and T vO = 0. henceforth may assume C Deﬁne the spaces L2O (Ω) 1 HO,Γ 2 LE (Ω) HE1

= = = =

{vO = (vji ), i = 1, 3, 5, . . . n, j = 1, 2 : vji ∈ L2 (Ω)} {vO ∈ L2O (Ω) : vji ∈ H01 (Ω)} {φE = (φij ), i = 2, 4, . . . 2m, j = 1, 2 : φij ∈ L2 (Ω)} {φE ∈ L2E (Ω) : φij ∈ H 1 (Ω)}.

T vO = 0} where The energy space is {w, w , vO ∈ V : C 1 . V = H 2 (Ω) ∩ H01 (Ω) × H01 (Ω) × HO,Γ

Let φ = Ry denote the solution to the following elliptic problem mφ − α∆φ = y, Copyright © 2005 Marcel Dekker, Inc.

φ = 0 on Γ.

(31)

252

CONTROL AND BOUNDARY ANALYSIS

Then R maps H −1 (Ω) into H01 (Ω) continuously. Let y T = (y1 , y2 , y3 ) = (w, w , ϕE ) then y2 T {hE P −1 LE hE (y3 − N ∇y1 )} . y = Ay = R(−K∆2 y1 + div N −1 −1 −1 ˜ GE y3 + G ˜ P LE hE (y3 − N ∇y1 ) −G E E (32) Deﬁne matrix S by a (33) BC y = · · · ⇔ y = Sa 0 It is simple to verify that BSy = y

∀ y ∈ Rm

SBu = u

∀u ∈ R

m+1

(34) Tu = 0 :C

(35)

∇y1 ). Along with the state variables y, deﬁne vy = ShE (y3 − N The energy inner product is deﬁned by (y, z)E

= (my2 , z2 )Ω + α(∇y2 , ∇z2 )Ω + KO (10 ∇y1 , 10 ∇z1 ) +(GE hE y3 , z3 )Ω + 120 (h0 DO vy , vz ).

Deﬁne H by H = {w, w , ϕE : w ∈ H 2 (Ω) ∩ H01 (Ω), w ∈ H01 (Ω), ϕE ∈ HE1 }. (36) Then one can verify that A : D(A) → H where D(A) = {y ∈ H : y1 ∈ H 3 (Ω), y2 ∈ H 2 (Ω), y3 ∈ HE3 , + BC’s}

(37)

where “+BC’s” means the boundary conditions (B∇y1 ) · n = 0 and BE (Bvy ) = 0 are imposed on Γ.

Theorem 2 The operator A in ( 32) is the generator of a strongly continuous semigroup of contractions on H. Moreover for any y ∈ D(A), ˜ −1 hE U, U )Ω ; Re (y, Ay)E = −(G E

U = P −1 BLO vy − GE y3 .

(38)

Proof: To show that A is the generator of a contraction semigroup, by the Lumer-Phillips theorem it is enough to demonstrate that A is dissipative and satisﬁes the range condition: i.e., (I − A)−1 maps D(A) onto H. Copyright © 2005 Marcel Dekker, Inc.

Semigroup Well-Posedness of a Multilayer Mead-Markus Plate

253

First we show dissipativity. Let z = Ay, then (y, Ay)E = (my2 , z2 )Ω + α(∇y2 , ∇z2 )Ω + K(∇y1 , ∇z1 ) +(GE hE y3 , z3 )Ω + 120 (h0 DO vy , vz ) T {hE P −1 LE hE (y3 − N ∇y1 )}])Ω = (my2 , R[−K∆2 y1 + div N 2 T −1 ∇y1 )}])Ω +α(∇y2 , ∇R[−K∆ y1 + div N {hE P LE hE (y3 − N +K(∇y1 , ∇y2 ) ˜ −1 GE y3 + G ˜ −1 P −1 LE hE (y3 − N ∇y1 ))Ω +(GE hE y3 , −G E E −1 ˜ GE y3 +12O (hO DO vy , ShE (−G E ˜ −1 P −1 LE hE (y3 − N ∇y1 ) − N ∇y2 )) +G E T hE P −1 LE Bvy )Ω = {(y2 , −K∆2 y1 )Ω + K(∇y1 , ∇y2 )} + {(y2 , div N ∇y2 )} + {terms with G ˜ −1 } +12O (hO DO vy , −She N E =: {T1 } + {T2 } + {T3 }. The real part of the ﬁrst bracketed term T1 is easily shown to vanish using integrations by parts. For the second term T2 we integrate by parts, apply (16), and use the boundary conditions to obtain T hE P −1 LE Bvy )Ω + 12(hO DO BO vy , −ShE N ∇y2 )Γ T2 = (y2 , div N T hE P −1 BLO vy )Ω ∇y2 )Ω = (y2 , div N +12(hO DO LO vy , ShE N T hE S T LO hO DO vy , y2 )Ω −12(div N y2 )Γ +12(S T LO hO DO vy · n, hE N T hE P −1 BLO vy )Ω = (y2 , div N T hE P −1 BD−1 h−1 B T S T hO DO LO vy , y2 )Ω −(div N O

O

One has that −1 T T −1 −1 T T (B − BD−1 O hO B S hO DO ) = BDO hO (I − B S )hO DO = 0

where we have applied the transposition of formula (35) to obtain the last equality. Thus the real part of T2 vanishes. Hence T3

˜ −1 GE y3 )Ω + (GE hE y3 , G ˜ −1 P −1 LE Bvy )Ω = (GE hE y3 , −G E E ˜ −1 GE y3 + G ˜ −1 P −1 LE Bvy )) +12O (hO DO vy , ShE (−G E

E

˜ −1 GE y3 )Ω + (GE hE y3 , G ˜ −1 P −1 BLO vy )Ω = −(GE hE y3 , G E E ˜ −1 S T hO DO LO vy , hE GE y3 )Ω +12(G E

˜ −1 P −1 BLO vy ))Ω . −12(hO DO LO vy , ShE G E One can easily show using (35) and the deﬁnition of P that 12S T hO DO = P −1 B. Copyright © 2005 Marcel Dekker, Inc.

(39)

254

CONTROL AND BOUNDARY ANALYSIS

Applying (39) to the present calculation one ﬁnds that Re (y, Ay)E ˜ −1 hE GE y3 , GE y3 )Ω + Re {(G ˜ −1 hE GE y3 , P −1 BLO vy )Ω = −(G E E ˜ −1 hE P −1 BLO vy , GE y3 )Ω } +(G E

˜ −1 hE P −1 BLO vy , P −1 BLO vy )Ω −(G E −1 ˜ = −(G−1 BLO vy − GE y3 ), (P −1 BLO vy − GE y3 ))Ω . E hE (P ˜ −1 hE is nonnegative, A is dissipative. Since G Next we check the range condition. In what follows we use the notation |w|s = wH s (Ω) ,

|ϕE |E,s =

m

|ϕij |s .

i even j=1,2

Assume that −Ay + y = z where z ∈ H. This is the same as −y2 + y1 = z1 (40) −1

∇y1 )) + y2 = z2 (41) −R(−K∆ y1 + div N hE P LE hE (y3 − N −1 −1 −1 ˜ P LE hE (y3 − N ∇y1 ) + y3 = z3 (42) ˜ GE y3 − G G E E 2

T

Solving for ∆2 y1 and LE y3 in (41) and (42) gives K∆2 y1 − α∆y1 + my1 ˜ E hE (y3 − z3 ) + (m − α∆)(z1 + z2 ). TG = div N

(43)

˜ E y3 GE y3 − P −1 LE hE y3 + G ˜ E z3 . ∇y1 + G = −P −1 LE hE N

(44)

The left-hand side of (43) is associated with a coercive bilinear symmetric form on H 2 (Ω) ∩ H01 (Ω) (see [3]) and hence for functions y ∈ D(A) one can show that |y1 |4 ≤ C|K∆2 y1 − α∆y1 + my1 |0 . Thus, considering the right hand side of (43), |y1 |4 ≤ C1 (|z3 |E,1 + |y3 |E,1 + |z1 |2 + |z2 |2 ).

(45)

Estimation of (43) with R1/2 applied to both sides leads to |y1 |3 ≤ C2 (|z3 |E,1 + |y3 |E,1 + |z1 |1 + |z2 |1 ). Copyright © 2005 Marcel Dekker, Inc.

(46)

Semigroup Well-Posedness of a Multilayer Mead-Markus Plate

255

Using that P −1 is symmetric, positive deﬁnite, we also have that ˜ E y3 |0 ≤ C1 (|z3 |E,0 + |y1 |3 ). (47) |y3 |E,2 ≤ C|GE y3 − P −1 LE hE y3 + G Hence from (46) |y3 |E,2 ≤ C|z3 |E,1 + |y3 |E,1 + |z1 |1 + |z2 |1 )

(48)

From (40) we also have |y2 |2 ≤ C(|y1 |2 + |z1 |2 ).

(49)

Since A has is dissipative, I − A is injective and hence any solutions to (40)–(42) are unique. Applying the usual compactness/uniqueness argument to the estimates (46), (48), (49) yields |y1 |3 + |y2 |2 + |y3 |E,2 ≤ C(|z2 |1 + |z3 |E,1 + |z1 |2 ).

(50)

Thus y ∈ D(A) and I − A is surjective from D(A) to H. This completes the proof.

References [1] R. Dautray and J.-L.. Lions (with collaboration of M. Artola, M. Cessenat, and H. Lanchon). Mathematical Analysis and Numerical Methods for Science and Technology; Evolution Problems I, volume 5. Springer-Verlag, Berlin, 1992. [2] R.A. DiTaranto. Theory of vibratory bending for elastic and viscoelastic layered ﬁnite-length beams. J. Appl. Mech., 32:881–886, 1965. [3] S.W. Hansen. Several related models for multilayer sandwich plates, 2003. Submitted. [4] S.W. Hansen and I. Lasiecka. Analyticity, hyperbolicity and uniform stability of semigroups arising in models of composite beams. Math. Models Methods Appl. Sci., 10:555–580, 2000. [5] S.W. Hansen and Z. Liu. Analyticity of semigroup associated with laminated composite beam, 1999. in: Control of Distributed Parameter and Sochastic Systems; eds. S. Chen, X. Li, J. Yong, X.Y. Zhou, pp 467–54, (Kluwer Academic Publisher, MA). [6] J.E. Lagnese and J.-L. Lions. Modelling, Analysis and Control of Thin Plates (collection: “Recherches en Math´ematiques Appliqu´ees”), RMA 6. Masson, Paris, 1988. [7] P. Lancaster and M. Tismenetsky. The Theory of Matrices. series: Computer Science and Applied Mathematics: Academic Press, San Diego, 1985. [8] D.J. Mead. A comparison of some equations for the ﬂexural vibration of damped sandwich beams. J. Sound Vib., 83:363–377, 1982. [9] D.J. Mead and S. Markus. The forced vibration of a three-layer, damped sandwich beam with arbitrary boundary conditions. J. Sound Vibr., 10:163–175, 1969.

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[10] Y.V.K.S. Rao and B.C. Nakra. Vibrations of unsymmetrical sandwich beams and plates with viscoelastic cores. J. Sound Vibr., 34:309–326, 1974. [11] C.T. Sun and Y.P. Lu. Vibration Damping of Structural Elements. Prentice Hall, NJ, 1995.

Copyright © 2005 Marcel Dekker, Inc.

CHAPTER XX SOLUTION OF ALGEBRAIC RICCATI EQUATIONS ARISING IN CONTROL OF PARTIAL DIFFERENTIAL EQUATIONS Kirsten Morris Department of Applied Mathematics University of Waterloo Waterloo, Canada

Carmeliza Navasca Department of Mathematics University of California Los Angeles, CA, USA

Abstract

Algebraic Riccati equations of large dimension arise when using approximations to design controllers for systems modeled by partial diﬀerential equations. For large model order direct solution methods based on eigenvector calculation fail. In this paper we describe an iterative method that takes advantage of several special features of these problems: (1) sparsity of the matrices, (2) much fewer controls than approximation order, and (3) convergence of the control with increasing model order. The algorithm is straightforward to code. Performance is illustrated with a number of standard examples.

Introduction We consider the problem of calculating feedback controls for systems modeled by partial diﬀerential or delay diﬀerential equations. In these systems the state x(t) lies in an inﬁnite-dimensional space. A classical controller design objective is to ﬁnd a control u(t) so that the objective function ∞

Cx(t), Cx(t) + u∗ (t)Ru(t)dt

(1)

0

is minimized where R is a positive deﬁnite matrix and the observation C ∈ L(X, Rp ). The theoretical solution to this problem for many inﬁnite-dimensional systems parallels the theory for ﬁnite-dimensional Copyright © 2005 Marcel Dekker, Inc.

258

CONTROL AND BOUNDARY ANALYSIS

systems [10, 17, 18, e.g.]. In practice, the control is calculated through approximation. This leads to solving an algebraic Riccati equation A∗ P + P A − P BR−1 B ∗ P = −C ∗ C

(2)

for a feedback operator K = −R−1 B P.

(3)

The matrices A, B, C arise in a ﬁnite-dimensional approximation of the inﬁnite-dimensional system. Let n indicate the order of the approximation, m the number of control inputs, and p the number of observations. Thus, A is n × n, B is n × m, and C is p × n. There have been many papers written describing conditions under which approximations lead to approximating controls that converge to the control for the original inﬁnite-dimensional system [3, 11, 14, 17, 18, e.g.]. In this paper we will assume that an approximation has been chosen so that a solution to the Riccati equation (2) exists for suﬃciently large n and also that the approximating feedback operators converge. For problems where the model order is small, n < 50, a direct method based on calculating the eigenvectors of the associated Hamiltonian works well [19]. Due the limitations of the calculation of eigenvectors for large non-symmetric matrices, this method is not suitable for problems where n becomes large. Unfortunately, many inﬁnite-dimensional control problems lead to Riccati equations of large order. This is particularly evident in control of systems modeled by partial diﬀerential equations with more than one space dimension. For such problems, iterative methods are more appropriate. There are two methods that may be used: Chandrasekhar and Newton-Kleinman iterations. In Chandrasekhar iterations, the Riccati equation is not itself solved directly [2, 7]. A system of 2 diﬀerential equations ˙ K(t) = −B ∗ L∗ (t)L(t), ˙ L(t) = L(t)(A − BK(t)),

K(0) = 0, L(0) = C,

is solved for K ∈ Rm×n , L ∈ Rp×n . The feedback operator is obtained as limt→−∞ K(t). The advantage to this approach is that the number of controls m and number of observations p is typically much less than the approximation model order n. This leads to signiﬁcant savings in storage. Furthermore, the matrices arising in approximation are typically sparse and this can be used in implementation of this algorithm. Unfortunately, the convergence of K(t) can be very slow and a very accurate Copyright © 2005 Marcel Dekker, Inc.

Solution of Algebraic Riccati Equations Arising in Control of PDEs

259

algorithm suitable for stiﬀ systems must be used. This can lead to very large computation times. Another approach to solving large Riccati equations is the NewtonKleinman method [16]. The Riccati equation (2) can be rewritten as (A − BK)∗ P + P (A − BK) = −C ∗ C − K ∗ RK.

(4)

We say a matrix Ao is Hurwitz if σ(Ao ) ⊂ C− . If A − BK is Hurwitz, then the above equation is a Lyapunov equation. An initial feedback K0 must be chosen so A − BK0 is Hurwitz. Deﬁne Si = A − BKi , and solve the Lyapunov equation Si∗ Xi + Xi Si = −C ∗ C − Ki∗ RKi

(5)

for Xi and then update the feedback as Ki+1 = −R−1 B ∗ Xi . If A − BK0 is Hurwitz, then Xi converges quadratically to P [16]. For an arbitrary large Riccati equation, this condition may be diﬃcult to satisfy. However, this condition is not restrictive for Riccati equations arising in control of inﬁnite-dimensional systems. First, many of these systems are stable even when uncontrolled and so the initial iterate K0 may be chosen as zero. Second, if the approximation procedure is valid then convergence of the feedback gains is obtained with increasing model order. Thus, a gain obtained from a lower order approximation, perhaps using a direct solution, may be used as an initial estimate, or ansatz, for a higher order approximation. This technique was used successfully in [13, 26] and later in this paper. In this paper we use a modiﬁed Newton-Kleinman iteration ﬁrst proposed by Banks and Ito [2] as a reﬁnement for a partial solution to the Chandrasekhar equation. In that paper, they partially solve the Chandrasekhar equations and then use the resulting feedback K as a stabilizing initial guess for a modiﬁed Newton-Kleinman method. Instead of the standard Newton-Kleinman form (5) above, Banks and Ito rewrote the Riccati equation in the form (A − BKi )∗ Xi + Xi (A − BKi ) = −Di∗ Di

(6)

where Xi = Pi−1 − Pi , Ki+1 = Ki − B T Xi , and Di = Ki − Ki−1 . The resulting Lyapunov equation is solved for Xi . Equation (6) has fewer inhomogeneous terms than the equation in the standard NewtonKleinman method (5). Also, the non-homogeneous term D depends on m inputs, not the observation C. In [2] a Smith’s method was used to solve the Lyapunov equations. Although convergent, this method is slow. Solution of the Lyapunov equation is a key step in implementing either modiﬁed or standard Newton-Kleinman. The Lyapunov equations Copyright © 2005 Marcel Dekker, Inc.

260

CONTROL AND BOUNDARY ANALYSIS

arising in the Newton-Kleinman method have several special features: (1) the model order n is generally much larger than m or p and (2) the matrices are often sparse. We use a recently developed method [20, 24] that uses these features. In the next section we describe this Lyapunov solver. Incorporating this algorithm with the Newton-Kleinman method is described in [6, 25]. We use this Lyapunov solver with both standard and modiﬁed NewtonKleinman to solve a number of standard control examples, including one with several space variables. Our results indicate that modiﬁed NewtonKleinman achieves considerable savings in computation time over standard Newton-Kleinman. We also found that using the solution from a lower-order approximation as an ansatz for a higher-order approximation signiﬁcantly reduced the computation time.

1.

Solution of Lyapunov Equation

Solution of a Lyapunov equation is a key step in each iteration of the Newton-Kleinman method. Thus, it is imperative to use a good Lyapunov algorithm. As for the Riccati equation, direct methods such as Bartels-Stewart [4] are only appropriate for low model order and do not take advantage of sparsity in the matrices. The Alternating Direction Implicit (ADI) and Smith methods are two well-known iterative schemes. These will be brieﬂy described before describing a modiﬁcation that leads to reduced memory requirements and faster computation. Consider the Lyapunov equation XAo + A∗o X = −DD∗

(7)

where Ao ∈ Rn×n and D ∈ Rn×r . In the case of standard NewtonKleinman, r = m + p while for modiﬁed Newton-Kleinman, r is only m. If Ao is Hurwitz, then the Lyapunov equation has a symmetric positive semideﬁnite solution X. For p < 0, deﬁne U = (Ao − pI)(Ao + pI)−1 and V = −2p(A∗o + pI)−1 DD∗ (Ao + pI)−1 . In Smith’s method [28], equation (7) is rewritten. The solution X is found by using successive substitutions: X = limi→∞ Xi where Xi = U ∗ Xi−1 U + V

(8)

with X0 = 0. Convergence of the iterations can be improved by careful choice of the parameter p e.g. [27, pg. 197]. This method of successive substitution is unconditionally convergent, but has only linear convergence. The ADI method [21, 30] improves Smith’s method by using a diﬀerent parameter pi at each step. Two Copyright © 2005 Marcel Dekker, Inc.

Solution of Algebraic Riccati Equations Arising in Control of PDEs

261

alternating linear systems, (A∗o + pi I)Xi− 1 (A∗o

2

+

pi I)Xi∗

= −DD∗ − Xi−1 (Ao − pi I) ∗

= −DD −

∗ Xi− 1 (Ao 2

− pi I)

(9) (10)

are solved recursively starting with X0 = 0 ∈ Rn×n and parameters pi < 0. If all parameters pi = p, then equations (9,10) reduce to Smith’s method. If the ADI parameters pi are chosen appropriately, then convergence is obtained in J iterations where J ! n. Choice of the ADI parameters is discussed below. If Ao is sparse, then the linear systems (9,10) can be solved eﬃciently. However, full calculation of the dense iterates Xi is required at each step. Setting X0 = 0, it can be easily shown that Xi is symmetric and positive semideﬁnite for all i, and so we can write X = ZZ ∗ where Z is a Cholesky factor of X [20, 24]. (A Cholesky factor does not need to be square or be lower triangular.) Substituting Zi Zi∗ for Xi in (9,10) and setting X0 = 0, we obtain the following iterates for < Z1 = −2p1 (A∗o + p1 I)−1 D < Zi = [ −2pi (A∗o + pi I)−1 D, (A∗o + pi I)−1 (A∗o − pi I)Zi−1 ]. (11) Note that Z1 ∈ Rn×r , Z2 ∈ Rn×2r , and Zi ∈ Rn×ir . Recall that for standard Newton-Kleinman, r is the sum of observations p and controls m. For modiﬁed Newton-Kleinman r is equal to m. In practice, the number of controls m (and often the number of observations p) is much less than the model order, n. This form of solution results in considerable savings in computation time and memory. The algorithm is stopped when the Cholesky iterations converge within some tolerance. In [20] these iterates are reformulated in a more eﬃcient form, using the observation that the order in which the ADI parameters are used is irrelevant. This leads to the algorithm shown in Table 1.

1.1

ADI Parameter Selection

As mentioned above, choice of the ADI parameters signiﬁcantly aﬀects the convergence of the ADI method. The parameter selection problem has been studied extensively [9, 21, 29, e.g.]. Optimal ADI parameters are the solution to the min-max problem J F pj − λ . {p1 , p2 , . . . , pJ } = arg min max pi λj ∈σ(Ao ) pj + λ j It is not feasible to solve this problem. First, solution of the Lyapunov equation arises as a step in the iterative solution of the Riccati equation Copyright © 2005 Marcel Dekker, Inc.

262 Table 1.

CONTROL AND BOUNDARY ANALYSIS Cholesky-ADI Method

Given Ao and D Choose ADI√ parameters {p1 , . . . , pJ } with (pi ) < 0 Deﬁne z1 = −2p1 (A∗o + p1 I)−1 D and Z1 = [z1 ] For i = 2, . . . , J √ −2pi+1

Deﬁne Wi = ( √−2p )[I − (pi+1 − pi )(A∗o + pi+1 I)−1 ] i (1) zi = Wi zi−1 (2) If z > tol Zi = [Zi−1 zi ] Else, stop.

and Ao = A−BKi where Ki is the feedback calculated at the it h iterate. Thus, the matrix Ao and its spectrum change at each iterate. Second, when Ao is large, solving this eigenvalue problem is computationally diﬃcult. The optimal ADI parameters are approximated in several respects. First, the spectrum of the original matrix A − BK0 is used and the resulting parameters used for each subsequent Lyapunov solution. In most applications, A is Hurwitz and so we can use the spectrum of the original matrix A. If these eigenvalues are real and contained in the interval [−b, −a] then the solution to (11) is known in closed form [21]. For more general problems, the selection procedure in [9] yields parameters that are approximately optimal. Let λi indicate the eigenvalues of A − BK0 . Deﬁne a = min("λi ),

(12)

b = max("λi ),

(13)

i

i

α = tan−1 max | i

λi |. "λi

(14)

These parameters determine an elliptic domain Ω that contains the spectrum and is used to determine the ADI parameters pi . The closeness of Ω to the smallest domain containing the spectrum aﬀects the number of iterations required for convergence of the Cholesky-ADI. The parameter α is the maximum angle between the eigenvalues and the real axis. When the spectrum contains lightly damped complex eigenvalues, α is close to π/2. In this case, Ω is a poor estimate of this domain. This point is investigated in the third example below. Copyright © 2005 Marcel Dekker, Inc.

Solution of Algebraic Riccati Equations Arising in Control of PDEs

2.

263

Benchmark Examples

In this section we test the algorithm with a number of standard examples: a one-dimensional heat equation, a two-dimensional partial diﬀerential equation, and a beam equation. All computations were done within MATLAB on a computer with two 1.2 GHz AMD processors. (Shorter computation time would be obtained by running optimized code outside of a package such as MATLAB. The CPU times are given only for comparison purposes.) The relative error for the Cholesky iterates was set to 10−8 .

2.1

Heat Equation

Consider the linear quadratic regulator problem of minimizing a cost functional [2, 8] ∞ (|Cz(t)|2 + |u(t)|2 )dt J(u) = 0

subject to ∂z(t, x) ∂ 2 z(t, x) , x ∈ (0, 1), = ∂t ∂x2 z(0, x) = ψ(x)

(15)

with boundary conditions ∂z(t, 0) ∂x ∂z(t, 1) ∂x

= u(t) = 0.

(16)

z(t, x)dx,

(17)

Setting Cz(t) =

1

0

and R = 1, the solution to the inﬁnite-dimensional Riccati equation is 1 Kz = k(x)z(x)dx 0

where k = 1 [3]. Thus, for this problem we have an exact solution to which we can compare the approximations. The equations (14–17) are discretized using the standard Galerkin approximation with linear spline ﬁnite element basis on a uniform partition of [0, 1]. The resulting A matrix is symmetric and tridiagonal while Copyright © 2005 Marcel Dekker, Inc.

264 Table 2.

CONTROL AND BOUNDARY ANALYSIS Heat Equation: Feedback Gain at Each Newton-Kleinman Iteration Newton-Kleinman Iteration 1 2 3 4 5 6 7 8 9 10 11

Optimal Feedback Gain 50.005 25.0125 12.5262 6.303 3.2308 1.7702 1.1675 1.012 1.0001 1 1

B is a column vector with only one non-zero entry. Denote each basis element by li , i = 1..n. For an approximation with n elements, the approximating optimal feedback operator K is

1

kn (x)z(x)dx,

Kz =

(18)

0

where kn (x) = ni=1 ki li (x). The solutions to the approximating Riccati equations converge [3, 14] and so do the feedback operators. Table 2 shows the approximated optimal feedback gain at each Newton-Kleinman iteration. The data in Table 2 is identical for n = 25, 50, 100, 200 and for both Newton-Kleinman methods. The error in K versus Newton-Kleinman iteration in shown in Figure 1 for standard Newton-Kleinman and in Figure 2 for the modiﬁed algorithm. In Tables 3 and 4 we compare the number of Newton-Kleinman and Lyapunov iterations as well as the CPU time per order n. We use the ansatz k0 (x) = 100 for all n. With the modiﬁed algorithm, there are 1 to 2 fewer Riccati loops than with the original Newton-Kleinman iteration. Also, the modiﬁed Newton-Kleinman method requires fewer Lyapunov iterations within the last few Newton-Kleinman loops. The computation time with the modiﬁed Newton-Kleinman algorithm is signiﬁcantly less than that of the original algorithm. Copyright © 2005 Marcel Dekker, Inc.

Solution of Algebraic Riccati Equations Arising in Control of PDEs

Table 3.

Table 4.

Heat Equation: Standard Newton-Kleinman Iterations n 25

Newton-Kleinman Itn’s 11

50

11

100

11

200

11

Lyapunov Itn’s 19,22,23,26,27,29 30,31,31,31,31 24,26,28,30,32,34 35,35,35,35,35 28,31,32,35,36,38 39,40,40,40,40 33,35,37,39,41,43 44,44,44,44,44

CPU time 0.83 1.2 3.49 23.1

Heat Equation: Modiﬁed Newton-Kleinman Iterations n 25

Newton-Kleinman Itn’s 10

50

10

100

9

200

9

Copyright © 2005 Marcel Dekker, Inc.

Lyapunov Itn’s 19,22,23,26,27,29 30,31,29,1 24,26,28,30,32,34 35,35,33,1 28,31,32,35,36,38 39,40,1 33,35,37,39,41,43 44,44,1

CPU time 0.66 0.94 2.32 14.2

265

266

2.2

CONTROL AND BOUNDARY ANALYSIS

Two-Dimensional Example

Deﬁne the rectangle Ω = [0, 1] × [0, 1] with boundary ∂Ω. Consider the two-dimensional partial diﬀerential equation [8] ∂z ∂t

2

∂ z = ∂x 2 + z(x, y, t) = 0,

∂2z ∂y 2

∂z + 20 ∂y + 100z = f (x, y)u(t),

(x, y) ∈ Ω (x, y) ∈ ∂Ω (19)

where z is a function of x, y and t. Let f (x, y) =

100, if .1 < x < .3 and .4 < y < .6 . 0, else

Central diﬀerence approximations are used to discretize (19) on a grid of N × M points. The resulting approximation has dimension n = N × M : A ∈ Rn×n and B ∈ Rn×1 . The A matrix is sparse with at most 5 nonzero entries in any row. The B matrix is a sparse column vector. We chose C = B ∗ and R = 1. We solved the Riccati equation on a number of grids, using both standard and modiﬁed Newton-Kleinman methods. The data is shown in Tables 5 and 6. Modiﬁed Newton-Kleinman is clearly much more eﬃcient. Fewer Lyapunov iterations are required for convergence and this leads to a reduction in computation time of nearly 50%, see Figure 3. We also investigated the use of non-zero initial estimates for K in reducing computation time. We ﬁrst solve the Riccati equation on a 12×12 grid. Since σ(A) ⊂ C− , K0144 = 0 is a possible ansatz. It required 13 Newton-Kleinman iterations and a total of 419 Lyapunov iterations to obtain a relative error in K of 10−11 . Linear interpolation was used to project this solution to a function on a ﬁner grid, 23×12, where n = 276. 12×12 . On the ﬁner grid 23 × 12 where Indicate this projection by Kproj 12×12 n = 276, we used both zero and Kproj as initial estimates. As indicated in Table 6, the error of K was 10−12 after only 150 Lyapunov and 54 Newton-Kleinman iterations. The same procedure is applied to generate a guess K0529 where the mesh is 23 × 23 and n = 529. Neglecting the computation time to perform the projection, use of a previous solution led to a total computation time over both grids of only 6.3 seconds versus 11.9 seconds for n = 276 and a total computation time of only 27.8 versus 79.5 for n = 529. Similar improvements in computation time were obtained with standard Newton-Kleinman. Copyright © 2005 Marcel Dekker, Inc.

267

Solution of Algebraic Riccati Equations Arising in Control of PDEs

Table 5.

Iterations for 2-d Equation (Newton-Kleinman)

grid

K0

n

12 × 12

0

144

Newton Kleinman Itn’s 14

23 × 12

0

276

15

23 × 12 23 × 23

12x12 Kproj 0

276 529

6 16

23 × 23

23x12 Kproj

529

5

Table 6.

Lyapunov Itn’s

CPU time

12,44,41,39,36,34,32 30,28,27,27,27,27,27 16,47,45,42,40,38,35,33 31,30,30,30,30,30,30 29,30,30,30,30,30 20,51,48,46,44,41,39,37 35,34,33,33,32,32,32,32 33,32,32,32,32

4.98 22.2 10.8 139. 65.7

Iterations for 2-d Equation (Modiﬁed Newton-Kleinman)

grid

K0

n

12 × 12

0

144

NewtonKleinman Itn’s 13

23 × 12

0

276

13

23 × 12 23 × 23

12x12 Kproj 0

276 529

4 14

23 × 23

23x12 Kproj

529

4

Copyright © 2005 Marcel Dekker, Inc.

Lyapunov Itn’s

CPU time

12,44,41,39,36,34,32 30,28,27,27,27,1 16,47,45,42,40,38,35 33,31,30,30,30,29 29,30,30,29 20,50,48,46,44,41,39 37,35,34,33,33,33,31 33,32,32,1

2.94 11.9 3.36 79.5 21.5

268

CONTROL AND BOUNDARY ANALYSIS

3.

Euler-Bernoulli Beam

Consider a Euler-Bernoulli beam clamped at one end (r = 0) and free to vibrate at the other end (r = 1). Let w(r, t) denote the deﬂection of the beam from its rigid body motion at time t and position r. The deﬂection is controlled by applying a a torque u(t) at the clamped end (r = 0). We assume that the hub inertia Ih is much larger than the beam inertia Ib so that Ih θ¨ ≈ u(t). The partial diﬀerential equation model with Kelvin-Voigt and viscous damping is ∂2 EIr ρr wrr (r, t) = u(t), wtt (r, t) + Cv wt (r, t) + 2 Cd Ib wrrt (x, t) + ∂r ρA Ih with boundary conditions w(0, t) = 0 wr (1, t) = 0. EIwrr (1, t) + Cd Ib wrrt (1, t) = 0 ∂ [EI(1)wrr (r, t) + Cd Ib wrrt (r, t)]r=1 = 0. ∂r The values of the physical parameters in Table 7 are as in [1]. Deﬁne H to be the closed linear subspace of the Sobolev space H 2 (0, 1) dw 2 H = w ∈ H (0, 1) : w(0) = (0) = 0 dr and deﬁne the state-space to be X = H × L2 (0, 1) with state z(t) = ∂ (w(·, t), ∂t w(·, t)). A state-space formulation of the above partial diﬀerential equation problem is d x(t) = Ax(t) + Bu(t), dt where A=

0

I

d4 − EI ρ dr 4

4 − Cρd I ddr4

, −

B=

Cv ρ

0

,

r Ih

with domain dom (A) = {(φ, ψ) ∈ X : ψ ∈ H and 2

2

d d 2 M = EI dr 2 φ + Cd I dr 2 ψ ∈ H (0, 1) with M (L) =

Copyright © 2005 Marcel Dekker, Inc.

d dr M (L)

= 0} .

Solution of Algebraic Riccati Equations Arising in Control of PDEs Table 7.

269

Table of Physical Parameters E Ib ρ Cv Cd L Ih d

2.68 × 1010 N/m2 1.64 × 10−9 m4 1.02087 kg/m 1.8039 N s/m 1.99 × 105 N s/m 1m 121.9748 kg m2 .041 kg −1

We use R = 1 and deﬁne C by the tip position: w(1, t) = C[w(x, t) w(x, ˙ t)]. Let H N ⊂ H be a sequence of ﬁnite-dimensional subspaces spanned by the standard cubic B-splines with a uniform partition of [0, 1] into N subintervals. This yields an approximation in H N × H N [15, e.g.] of dimension n = 2N. This approximation method yields a sequence of solutions to the algebraic Riccati equation that converge strongly to the solution to the inﬁnite-dimensonal Riccati equation corresponding to the original partial diﬀerential equation description [22]. The spectrum of A for various n is shown in Figure 4. For small values of n, the spectrum of An only contains complex eigenvalues with fairly constant angle. As n increases, the spectrum curves into the real axis. For large values of n, the spectrum shows behavior like that of the original diﬀerential operator, and contains two branches on the real axis. For these large values of n, the ADI parameters are complex numbers. We calculated the complex ADI parameters as in [9]. Although there are methods to eﬃciently calculate with complex parameters by splitting the calculation into 2 real parts [20] their presence increases computation time. Figure 5 shows the total Lyapunov iterations for various values of n. Although the number of Newton-Kleinman iterations remained at 2 for all n, the number of Lyapunov iterates increases as n → ∞. Figure 6 shows the change in the spectrum of A as Cd is varied. Essentially, increasing Cd increases the angle that the spectrum makes with the imaginary axis. Recall that the spectral bounds α, a, and b deﬁne the elliptic function domain that contains the spectrum of A [21]. The ADI parameters depend entirely on these bounds. The quantity π 2 − α is the angle between the spectrum and the imaginary axis and so α → π/2 as Cd is decreased. Figure 7 shows the eﬀect of varying Cd Copyright © 2005 Marcel Dekker, Inc.

270

CONTROL AND BOUNDARY ANALYSIS

on the number of iterations required for convergence. Larger values of Cd (i.e., smaller values of α) leads to a decreasing number of iterations. Small values of Cd lead to a large number of required iterations in each solution of a Lyapunov equation. There are several possible reasons for this. As the spectrum of A ﬂattens with increasing Cd , the spectral bounds (12–14) give sharper estimates for the elliptic function domain Ω and thus the ADI parameters are closer to optimal. Improvement in calculation of ADI parameters for problems where the spectrum is nearly vertical in the complex plane is an open problem. Another explanation lies in the nature of the mathematical problem being solved. If Cd > 0 the semigroup for the original partial diﬀerential equation is parabolic and the solution to the Riccati equation converges uniformly in operator norm [17, Chap.4]. However, if Cd = 0, the partial diﬀerential equation is hyperbolic and only strong convergence of the solution is obtained [18]. Thus, one might expect a greater number of iterations in the Lyapunov loop to be required as Cd is decreased. Any X ∈ Rn×n to the matrix Lyapunov equation is symmetric and positive semi-deﬁnite and so we can order its eigenvalues λ1 ≥ λ2 ≥ ...λn ≥ 0. ˆ is determined The ability to approximate X by a matrix of lower rank X by the following relation [12, Thm. 2.5.2]: ˜ λk (X) X − X = . ˜ X λ1 (X) rank X≤k−1 min

This ratio is plotted for several values of Cd in Figures 8 and 9. For larger values of Cd the solution X is closer to a low rank matrix than it is for smaller values of Cd . Recall that the CF-ADI algorithm used here starts with a rank 1 initial estimate of the Cholesky factor and the rank of the solution is increased at each step. The fact that the solution X is closer to a low rank matrix for larger values of Cd implies a smaller number of iterations are required for convergence. If the fundamental reason for the slow convergence with small Cd is the “hyperbolic-like” behavior of the problem, then this convergence will not be improved by better ADI parameter selection. This may have consequences for control of coupled acoustic-structure problems where the spectra are closer to those of hyperbolic systems than those of parabolic systems.

Copyright © 2005 Marcel Dekker, Inc.

271

Solution of Algebraic Riccati Equations Arising in Control of PDEs

Beam: Eﬀect of Changing Cd (Standard Newton-Kleinman)

Table 8. Cv 2

Cd 1 × 104 3 × 105 4 × 105 1 × 107 1 × 108 5 × 108

Newton-Kleinman Itn’s – – 3 3 3 3

Lyapunov Itn’s – – 1620;1620;1620 1316;1316;1316 744;744;744 301;301;301

CPU time – – 63.14 42.91 18.01 5.32

Beam: Eﬀect of Changing Cd (Modiﬁed Newton-Kleinman)

Table 9. Cv 2

α 1.5699 1.5661 1.5654 1.5370 1.4852 1.3102

Cd 1 × 104 3 × 105 4 × 105 1 × 107 1 × 108 5 × 108

α 1.5699 1.5661 1.5654 1.5370 1.4852 1.3102

Newton-Kleinman It’s 2 2 2 2 2 2

Lyapunov It’s – – 1620;1 1316;1 744;1 301;1

CPU time – – 24.83 16.79 7.49 2.32

0

10

−1

10

−2

10

−3

10

−4

Error

10

−5

10

−6

10

−7

10

N=25 N=50 N=100 N=200

−8

10

−9

10

Figure 1.

1

2

3

4

5

6 Iterations

7

8

9

10

11

Heat equation: error versus atandard Newton-Kleinman iterations.

Copyright © 2005 Marcel Dekker, Inc.

272

CONTROL AND BOUNDARY ANALYSIS

0

10

−2

10

−4

10

−6

Error

10

−8

10

−10

10

N=25 N=50 N=100 N=200

−12

10

−14

10

1

2

3

4

5

6

7

8

9

10

Iterations

Figure 2.

Heat equation: error versus modiﬁed Newton-Kleinman iterations.

Copyright © 2005 Marcel Dekker, Inc.

Solution of Algebraic Riccati Equations Arising in Control of PDEs

273

0

10

N=144 N=276 N=529 −2

10

−4

10

−6

Error

10

−8

10

−10

10

−12

10

−14

10

0

2

4

6

8

10

12

14

Iterations

Figure 3. Convergence rate for 2-d problem (modiﬁed Newton-Kleinman). Initial 144 276 , K 529 = Kproj . estimates: K 144 = 0, K 276 = Kproj

Copyright © 2005 Marcel Dekker, Inc.

274

CONTROL AND BOUNDARY ANALYSIS

4

4

x 10

2 0 −2 −4 −6 4 x 10 4

−5

−4

−3

−2

−1

0 5

x 10

2 0 −2 −4 −6 4 x 10 4

−5

−4

−3

−2

−1

0 5

x 10

2 0 −2 −4 −6 4 x 10 4

−5

−4

−3

−2

−1

0 5

x 10

2 0 −2 −4 −6

−5

−4

−3

−2

−1

0 5

x 10

Figure 4.

Beam: spectrum of An for n = 48, 56, 64, 88 (Cd = 9 × 105 ).

Copyright © 2005 Marcel Dekker, Inc.

275

Solution of Algebraic Riccati Equations Arising in Control of PDEs 5000

4500

Lyapunov Iterations

4000

3500

3000

2500

2000

1500

1000 45

50

55

60

65

70

75

80

85

90

n

Figure 5.

Beam: n versus Lyapunov iterations (Cd = 9 × 105 ).

4

x 10 5 0 −5

−5 −4.5 4 x 10

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0 4

x 10

5 0 −5 −5 −4.5 4 x 10

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0 4

x 10

5 0 −5 −5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0 4

x 10

Figure 6.

Beam: spectrum of A for Cd = 1x104 , 4x105 , 1x108 (n = 80).

Copyright © 2005 Marcel Dekker, Inc.

276

CONTROL AND BOUNDARY ANALYSIS

Lyapunov Iterations

3500 3000 2500 2000 1500 1000 500

0

0.5

1

1.5

2

2.5 Cd

3

3.5

4

4.5

5 8

x 10

7

10

x 10

8

Cd

6 4 2 0 1.48

Figure 7.

1.5

1.52

α

1.54

1.56

1.58

Beam: eﬀect of Cd on Lyapunov iterations and α (n = 80).

Copyright © 2005 Marcel Dekker, Inc.

277

Solution of Algebraic Riccati Equations Arising in Control of PDEs

Eigenvalue Ratio for N=64

0

10

5x108 1x108 1x107 −5

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−10

λk / λ1

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−15

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−20

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−25

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−30

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Figure 8.

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25 k

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Beam: eﬀect of varying Cd on low rank approximation.

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Eigenvalue Ratio for N=80

0

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5x108 1x108 1x107 −5

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−10

λk / λ1

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−15

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−20

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−25

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−30

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0

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Figure 9.

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45

Beam: eﬀect of varying Cd on low rank approximation.

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279

References [1] H.T. Banks and D. J. Inman, “On Damping Mechanisms in Beams,” ICASE Report No. 89-64, NASA, Langley, 1989. [2] H. T. Banks and K. Ito, A Numerical Algorithm for Optimal Feedback Gains in High Dimensional Linear Quadratic Regulator Problems, SIAM J. Control Optim., vol. 29, no. 3, pp. 499–515, 1991. [3] H. T. Banks and K. Kunisch, The linear regulator problem for parabolic systems, SIAM J. Control Optim., vol. 22, no. 5, pp. 684–698, 1984. [4] R.H. Bartels and W. Stewart, “Solution of the matrix equation AX+XB=C,” Comm. of ACM, vol. 15, pp. 820–826, 1972. [5] D. Bau, and L. Trefethen, Numerical Linear Algebra, SIAM, Philadephia, 1997. [6] P. Benner, “Eﬃcient Algorithms for Large-Scale Quadratic Matrix Equations,” Proc. in Applied Mathematics and Mechanics, vol. 1, no. 1, pp. 492–495, 2002. [7] J.A. Burns and K. P. Hulsing, “Numerical Methods for Approximating Functional Gains in LQR Boundary Control Problems,” Mathematical and Computer Modelling, vol. 33, no. 1, pp. 89–100, 2001. [8] Y. Chahlaoui and P. Van Dooren, A collection of Benchmark examples for model reduction of linear time invariant dynamical systems, SLICOT Working Note 2002-2, http://www.win.tue.nl/niconet/ [9] N. Ellner and E. L. Wachpress, Alternating Direction Implicit Iteration for Systems with Complex Spectra, SIAM J. Numer. Anal., vol. 28, no. 3, pp. 859–870, 1991. [10] J.S. Gibson, “The Riccati Integral Equations for Optimal Control Problems on Hilbert Spaces”, SIAM J. Control and Optim., vol. 17, pp. 637–565, 1979. [11] J.S. Gibson, “Linear-quadratic optimal control of hereditary diﬀerential systems: inﬁnite dimensional Riccati equations and numerical approximations”, SIAM J. Control and Optim., vol. 21, pp. 95–139, 1983. [12] G.H. Golub and C.F. van Loan, Matrix Computations, John Hopkins, 1989. [13] J.R. Grad and K. A. Morris, “Solving the Linear Quadratic Control Problem for Inﬁnite-Dimensional Systems”, Computers and Mathematics with Applications, vol. 32, no. 9, pp. 99–119, 1996. [14] K. Ito, Strong convergence and convergence rates of approximating solutions for algebraic Riccati equations in Hilbert spaces, Distributed Parameter Systems, eds. F. Kappel, K. Kunisch, W. Schappacher, Springer-Verlag, 1987, pp. 151– 166. [15] K. Ito and K. A. Morris, An approximation theory of solutions to operator Riccati equations for H ∞ control, SIAM J. Control Optim., vol. 36, pp. 82–99, 1998. [16] D. Kleinman, On an Iterative Technique for Riccati Equation Computations, IEEE Transactions on Automat. Control, vol. 13, pp. 114–115, 1968. [17] I. Lasiecka and R. Triggiani, Control Theory for Partial Diﬀerential Equations: Continuous and Approximation Theories, Part 1, Cambridge University Press, 2000.

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[18] I. Lasiecka and R. Triggiani, Control Theory for Partial Diﬀerential Equations: Continuous and Approximation Theories, Part 2, Cambridge University Press, 2000. [19] A.J. Laub, “A Schur method for solving algebraic Riccati equations”, IEEE Trans. Auto. Control, vol. 24, pp. 913–921, 1979. [20] J. R. Li and J White, Low rank solution of Lyapunov equations, SIAM J. Matrix Anal. Appl., vol. 24, pp. 260–280, 2002. [21] A. Lu and E. L. Wachpress, Solution of Lyapunov Equations by Alternating Direction Implicit Iteration, Computers Math. Applic., vol. 21, no. 9, pp. 43–58, 1991. [22] K.A. Morris, “Design of Finite-Dimensional Controllers for Inﬁnite-Dimensional Systems by Approximation,” J. Mathematical Systems, Estimation and Control, vol. 4, pp. 1–30, 1994. [23] K.A. Morris, Introduction to Feedback Control, Harcourt-Brace, 2001. [24] T. Penzl, “A Cyclic Low-Rank Smith Method for Large Sparse Lyapunov Equations”, SIAM J. Sci. Comput., vol. 21, no. 4, pp. 1401–1418, 2000. [25] T. Penzl, LYAPACK User’s Guide, http://www.win.tue.nl/niconet/ [26] I. G. Rosen and C. Wang, “A multilevel technique for the approximate solution of operator Lyapunov and algebraic Riccati equations”, SIAM J. Numer. Anal. vol. 32, pp. 514–541, 1994. [27] Russell, D.L., Mathematics of Finite Dimensional Control Systems: Theory and Design, Marcel Dekker, New York, 1979. [28] R.A. Smith, “Matrix Equation XA + BX = C”, SIAM Jour. of Applied Math., vol. 16, pp. 198–201, 1968. [29] E. L. Wachpress, The ADI Model Problem, preprint, 1995. [30] E. Wachpress, “Iterative Solution of the Lyapunov Matrix Equation,” Appl. Math. Lett., vol. 1, pp. 87–90, 1988.

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CHAPTER XXI STABILIZATION IN COMPUTING SADDLE POINTS Jianxin Zhou∗ Texas A&M University College Station, TX, USA

Xudong Yao∗ Texas A&M University College Station, TX, USA

Abstract

Due to unstable nature, saddle points are very elusive to capture numerically. In this paper, two stabilization techniques in numerical computation of saddle points are developed. The ﬁrst is to deﬁne a solution set and then minimize the functional restricted to the solution set. The second is to use symmetry to deﬁne an invariant subspace. Mathematical justiﬁcations of the two techniques are established in a Banach space setting.

Keywords: Saddle points and their orders, local minimax method, invariance

Introduction Many nonlinear boundary value problems can be reduced to solve their Euler-Lagrange equation ∇J(u) = 0 (1) where ∇J is the gradient of a C 1 energy function J on a Banach space B. A solution u∗ ∈ B to (1) is called a critical point of J. u∗ is said to be nondegenerate if J (u∗ ) is invertible. The ﬁrst candidates for a critical point are the local extrema to which the classical critical point theory was devoted in calculus of variation. Most conventional numerical algorithms focus on ﬁnding such stable solutions. Critical points that are not local extrema are unstable and called saddle points. In physical systems, saddle points appear as unstable equilibria or transient excited ∗ This

research was supported in part by NSF grant DMS-0311905.

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states. For example, a common and important problem in computational chemistry and condensed matter physics is the calculation of the rate of transition states. Due to unstable nature, saddle points are very elusive to capture numerically. Most results in critical point theory [2], [1], [3], [4], [5], [10], [11], [12], [14] focus on the existence issue and are not for algorithm implementation. For example, minimax method is the most popular approach in critical point theory, such as the mountain pass, various linking and saddle point theorems. It characterizes a critical point as a solution to a two-level minimax problem min max J(u)

A∈A u∈A

where A is usually compact, max is global on A, min is global among all A’s. It becomes a two-level global optimization problem and is not for algorithm implementation. Newton’s method depends on an initial guess and is not for degenerate cases. However degeneracy exists in every multiple saddle point problem. A local minimax method (LMM) is developed in [7],[8],[15] and successfully applied to numerically solve several semi and quasi linear elliptic boundary value problems for multiple unstable solutions. The computations are quite stable. The purpose of this paper is to provide such mathematical evidences in a Banach space.

1.

The Local Minimax Method

Let L and L be closed subspaces of B such that B = L ⊕ L . Let P : B → L be the corresponding linear projection operator. Let SL be the unit sphere in L . For each v ∈ SL denote [L, v] the subspace spanned by L and v.

Definition 1 A mapping P : SL → 2B is the peak mapping of J w.r.t. L if ∀v ∈ SL , P (v) is the set of all local maximum points of J in [L, v]. A single-valued mapping p : SL → B is a peak selection of J w.r.t. L if p(v) ∈ P (v) ∀v ∈ SL . For a given u ∈ SL , J is said to have a local peak selection at u if a peak selection p is locally deﬁned near u. Definition 2 For given 0 < θ ≤ 1, if u ∈ B with ∇J(u) = 0, a point Ψ(u) ∈ B is called a pseudo-gradient of J at u if Ψ(u) ≤ 1 and ∇J(u), Ψ(u) ≥ θ∇J(u). Lemma 3 [15] For v0 ∈ SL , if J has a local peak selection p w.r.t. L at v0 satisfying (1) p is continuous at v0 , (2) d(p(v0 ), L) > 0, and (3) Copyright © 2005 Marcel Dekker, Inc.

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283

∇J(p(v0 )) ≥ δ > 0. Denote p(v0 ) = t0 v0 + w0 for some t0 = 0 and w0 ∈ L. Then there exists s0 > 0 such that for 0 < s < s0 J(p(v(s))) − J(p(v0 )) < −

θs d(p(v0 ), L)∇J(p(v0 )) 4

(2)

v0 − sign(t0 )sG(p(v0 )) , G(p(v0 )) = P(Ψ(p(v0 ))) and v0 − sign(t0 )sG(p(v0 )) Ψ(p(v0 )) is a pseudo-gradient of J at u0 = p(v0 ) with constant θ ∈ (0, 1).

where v(s) =

Then the following local minimax characterization of saddle points can be established.

Theorem 4 [15] Assume that a local peak selection function p is continuous at v ∗ ∈ SL . If J(p(v ∗ )) = minv∈SL J(p(v)) and d(p(v ∗ ), L) > 0, then u∗ = p(v ∗ ) is a critical point of J. Deﬁne the solution set (manifold) M = {p(v) : v ∈ SL }. Theorem 4 states that a local minimum point of J on M yields a saddle point u∗ = p(v ∗ ). It leads to the local minimax algorithm [7], [8], [15]. As stated in Lemma 3, for v0 ∈ SL , v1 = v0 (s1 ) where s1 is the maximum s > 0 such that (2) is satisﬁed. When B is a Hilbert space, we can simply take L = L⊥ and G(p(v)) = ∇J(p(v)).

2.

Results on the Order of Saddle Points

To establish an order or instability indices for saddle points is an important issue in saddle point analysis and application. When B = H is a Hilbert space, the Morse index is widely used to deﬁne an order (instability index) for nondegenerate saddle points [6]. Let H + , H 0 , H − be respectively, the maximum positive deﬁnite, null, maximum negative deﬁnite subspaces of J (u∗ ). The number MI(u∗ ) = dim(H − ) is called the Morse index of u∗ .

Theorem 5 [16] Let v ∗ ∈ SL⊥ . Assume that J has a local peak selection p w.r.t. L at v ∗ s.t. (a) p is continuous at v ∗ , (b) u∗ ≡ p(v ∗ ) ∈ L and (c) v ∗ = arg minv∈SL⊥ J(p(v)). If p is diﬀerentiable at v ∗ , then u∗ is a critical point with dim(L) + 1 = MI (u∗ ) + dim(H 0 ∩ [L, v ∗ ]).

(3)

Theorem 5 implies that in order for our local minimax algorithm to have a stable convergence in computing a nondegenerate saddle point of Copyright © 2005 Marcel Dekker, Inc.

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MI= n, the support L must contain at least n − 1 critical points at the lower critical level. Since many diﬀerent cases may happen in H 0 , the Morse index is basically for nondegenerate cases and is not deﬁned in a Banach space. For allowing degenerate cases in a Banach space to happen, an other approach has to be used.

Definition 6 Let J ∈ C 1 (B, R) and u∗ ∈ B be a critical point of J. If B = B I ⊕ B D for some subspaces B I , B D in B and for each u1 ∈ B I and u2 ∈ B D with u1 = 1 and u2 = 1 there exist constants r1 > 0 and r2 > 0 s.t. J(u∗ + tu1 ) ≥ J(u∗ ), ∀ 0 < |t| ≤ r1 , J(u∗ + tu2 ) < J(u∗ ), ∀ 0 < |t| ≤ r2 .

(4) (5)

It is clear that u∗ is a saddle point of J and the integer dim(B D ) is called the order of u∗ . Such a saddle point is called a local linking in the literature [9] and is of particular interests in computational chemistry and condensed matter physics, since the subspace B D corresponds to the reaction coordinates. Since (4) and (5) lack of characterization and robustness, they are difﬁcult to apply in discussion, in particular in a Banach space. Thus we modify the deﬁnition by replacing (5) with J(u∗ + tu2 + o(t)) < J(u∗ ), ∀ 0 < |t| ≤ r2 ,

(6)

where o(t) represents a higher order term. If u∗ is a nondegenerate saddle point in a Hilbert space H, then we have J(u∗ + tu1 + o(t)) − J(u∗ ) = J(u∗ + tu2 + o(t)) − J(u∗ ) =

t2 ∗ J (u )u1 , u1 + o(t2 ) > 0 2 t2 ∗ J (u )u2 , u2 + o(t2 ) < 0 2

for any u1 ∈ H + , u2 ∈ H − and |t| small, i.e., B D = H − and B I = H + or the order of u∗ is the Morse index of u∗ . The deﬁnitions in (4) and (6) do not require nondegeneracy and are much more general than the Morse index. Next we assume that (4) and (6) are satisﬁed and prove that for a saddle point u∗ = p(v ∗ ) found by our local minimax algorithm, its order is equal to k = dim(L) + 1.

Theorem 7 Let v ∗ ∈ SL . Assume that J has a local peak selection p w.r.t. L at v ∗ s.t. p is diﬀerentiable at v ∗ , u∗ ≡ p(v ∗ ) ∈ L, and Copyright © 2005 Marcel Dekker, Inc.

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v ∗ = arg minv∈SL J(p(v)). If we assume that · exists in B \ {0} and u∗ is a strict local maximum point of J restricted to [L, v ∗ ], then u∗ is a saddle point of order equal to k = dim(L) + 1. Proof. Let B D and B I be deﬁned as in (4) and (6). Assume p(v ∗ ) = t0 v ∗ + vL∗ for some t0 = 0 and vL∗ ∈ L. Since k > dim(B D ) implies [L, v ∗ ] ∩ B I = ∅, it contradicts our assumption that u∗ is a strict local maximum of J restricted to [L, v ∗ ]. Thus we have k ≤ dim(B D ). To prove k = dim(B D ), recall we have B = L ⊕ L , v ∗ ∈ SL . Denote Lv∗ = {u ∈ L : v ∗ , u = 0}. Then B = [L, v ∗ ] ⊕ Lv∗ . We ﬁrst prove that p (v ∗ )(Lv∗ ) ⊕ [L, v ∗ ] = B. (7) For any w ∈ Lv∗ with w = 1, denote v ∗ (s) = d ∗ ds |s=0 v (s) = N (v ∗ ) ∩ SL .

v∗ +sw v∗ +sw .

Then

w and there exists s0 > 0 s.t. when |s| < s0 , v ∗ (s) ∈

Let α(s) ≡ P(p(v ∗ (s))) = t(s)v ∗ (s), where P is the linear projection operator onto L . Since p is diﬀerentiable at v ∗ and v ∗ (s) smoothly depends on s, α is diﬀerentiable at 0 and α (0) = P(p (v ∗ )(w)) = (t(s)v ∗ (s))|s=0 = t (0)v ∗ + t0 w.

(8)

Since t0 = 0, it leads to w ∈ [p (v ∗ )(w), L, v ∗ ], ∀w ∈ Lv∗ .

(9)

Now if w ∈ Lv∗ and p (v ∗ )(w) ∈ [L, v ∗ ], (9) implies w ∈ [L, v ∗ ] and then w = 0. Thus p (v ∗ )(w) ∈ [L, v ∗ ], ∀w ∈ Lv∗ , w = 0, and we conclude (7). Now we suppose k < dim(B D ). Then (7) implies p (v ∗ )(Lv∗ ) ∩ B D contains nonzero points. Let w ∈ Lv∗ with w = 1 and p (v ∗ )(w) ∈ ∗ ∗ ∗ (B D \ {0}). Denote v ∗ (s) = vv∗ +sw +sw . We have v (s) ∈ N (v ) ∩ SL for |s| small and

d ∗ ds |s=0 v (s)

= w. It follows

p(v ∗ (s)) = p(v ∗ ) + sp (v ∗ )(w) + o(s) = u∗ + sp (v ∗ )(w) + o(s). Since p (v ∗ )(w) ∈ (B D \ {0}), for |s| small, we have J(p(v ∗ (s)) = J(u∗ + sp (v ∗ )(w) + o(s)) < J(u∗ ) = J(p(v ∗ )). On the other hand, by our assumption on v ∗ = arg minv∈SL J(p(v)), we have J(p(v ∗ (s)) ≥ J(p(v ∗ )). Copyright © 2005 Marcel Dekker, Inc.

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It leads to a contradiction. Thus we conclude that k = dim(B D ).2 The above result implies that in our local minimax algorithm, the computation of a saddle point of order n will be stable if the support L contains n − 1 linearly independent critical points at the lower critical level. A method to check if p is diﬀerentiable is developed in [17].

3.

Reduce Instability by Using Symmetries

Symmetries exist in many natural phenomena, such as in crystals, elementary particle physics, etc. Symmetries are usually handled by “compact group actions” in mathematics. Here we show that symmetry can be used to reduce instability in computing saddle points.

Definition 8 ([13]) Let B be a Banach space and J ∈ C 1 (B, R). A closed subspace BI of B is said to be a J-invariant space if u = p(v) ∈ BI and ∇J(u) = 0 imply that there exists a pseudo-gradient Ψ(u) of J at u such that G(u) = PΨ(u) ∈ BI . We have the following modiﬁed Principle of Symmetric Criticality (PSC).

Theorem 9 Let BI be a J-invariant space of B. If u∗ = p(v ∗ ) ∈ BI is a critical point of J restricted to BI , then u∗ is a critical point of J in B. Proof. Note ∇J(u∗ ), v = 0, ∀v ∈ BI . If ∇J(u∗ ) = 0, then we have G(u∗ ) = PΨ(u∗ ) ∈ BI . By the deﬁnition of a peak selection, we have Ψ(u∗ ) − G(u∗ ) ∈ L and ∇J(p(v ∗ )), Ψ(u∗ ) − G(u∗ ) = 0. Thus 0 = ∇J(p(v ∗ )), G(p(v ∗ )) = ∇J(p(v ∗ )), Ψ(p(v ∗ )) ≥ θ∇J(p(v ∗ )). It leads to a contradiction. To compute a saddle point u∗ , we identify its symmetries and deﬁne the smallest invariant subspace BI containing u∗ , then replace B by BI and L by L ∩ BI in the local minimax algorithm. We have v0 ∈ BI =⇒ u0 = p(v0 ) ∈ BI =⇒ G(u0 ) ∈ BI =⇒ v1 ∈ BI .... v0 − sign(t0 )s1 G(p(v0 )) and s1 is the maximum v0 − sign(t0 )s1 G(p(v0 )) s > 0 such that (2) is satisﬁed.

where v1 = v0 (s1 ) =

Copyright © 2005 Marcel Dekker, Inc.

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287

Theorem 10 The local minimax method is closed in an invariant subspace, i.e., if v0 ∈ BI then vn , un = p(vn ) ∈ BI . The order (instability index) of u∗ in BI is dim(L∩BI )+1 and is greatly reduced. Within BI , LMM becomes much more eﬃcient and stable. But in numerical computation of the modiﬁed pseudo-gradient dk = −G(p(v k )), it involves discretization, approximation, round-oﬀ error, etc. The numerical error will break the invariance of dk and leads to dk ∈ BI . Due to the unstable nature of u∗ outside BI , it will ﬁnd a slider and bypass u∗ . To preserve the invariance, we project dk back to BI . With this projected dk ∈ BI , the updated point v k+1 is in BI and the invariance is preserved. For symmetries that can be described by a compact group of linear isomorphisms, there is a much easier way to ﬁnd the projection to BI . Let G be a compact group of linear isomorphisms of B. A set A ⊂ B is G-invariant if g(A) = A for every g ∈ G. The subspace BG = {u ∈ B : gu = u, ∀g ∈ G} is called the invariant subspace of B under G. J is said to be G-invariant if J(gu) = J(u) for every (g, u) ∈ G × B. A map F : B → B is G-equivariant if g ◦ F = F ◦ g for every g ∈ G. Since J ∈ C 1 (B, R) is G-invariant, it implies that ∇J is G-equivariant, i.e., ∇J(gu) = ∇J(u) ∀(g, u) ∈ G × B. Then one can ﬁnd a Gequivariant pseudo-gradient ﬂow Ψ(u) ∈ BG (see page 86 in [12]). Since L is chosen to be a subspace of BG , and BG = L ⊕ L , G(p(v)) = PΨ(p(v)) ∈ BG where P is the linear projection onto L . Thus our deﬁnition of an invariant subspace is more general. To ﬁnd the projection from B to BG , let us cite the Haar integral.

Theorem 11 (Haar, 1933) Let G be a compact group and C(G) be the space of real-valued continuous functions on G. Then there exists a unique

positive integral (the Haar integral) s.t. the map : C(G) → R by f → G f (g) dg is

(a) linear, monotone, and normalized ( G 1 dg = 1);

2 (b) left-invariant, i.e., G f (hg) dg = G f (g) dg, ∀h ∈ G. It leads to the Haar integral operator H from B to B deﬁned by Hu = gu dg ∈ BG , ∀u ∈ B where BG = {u ∈ B : Hu = u}. G

This operator was used in the literature to preserve an invariance. Copyright © 2005 Marcel Dekker, Inc.

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Example 4.1. Let G = {g(1), ..., g(m)} be a ﬁnite group of linear isomorphisms of B where g(m) = id. Then for each u ∈ B, 1 g(k)u ∈ BG . m m

Hu = uG =

k=1

≥ 1) be a bounded open set and B = Example 4.2. Let Ω ⊂ W 1,p (Ω) where p > 1 be the Sobolev space. Let G be the set of all orthogonal matrices g ∈ Rn×n s.t. g(Ω) = Ω. For each g ∈ G and u ∈ B, if we deﬁne gu(x) = u(gx), ∀x ∈ Ω, then G is a compact group of linear isomorphisms of B. In particular, if g ∈ Rn×n is an orthogonal matrix s.t. g(Ω) = Ω and m > 0 is the least integer s.t. gm = I. For each u ∈ H, we deﬁne g(u)(x) = u(gx), ∀x ∈ Ω. Then we have a ﬁnite cyclic group G = {g, g2 , ..., gm } ∼ = ZZm of linear isomorphisms of B. Example 4.3. Let Ω ⊂ Rn be bounded and open. Assume that Ω is symmetric about the ﬁrst n − 1 axes. Let B = W 1,p (Ω). Deﬁne g : B → B by (gu)(x1 , ..., , xn−1 , xn ) = −u(−x1 , ..., −xn−1 , xn ). Then G = {id, g} ∼ = ZZ2 . There are symmetries that can not be deﬁned by a compact group of linear isomorphisms, e.g., a composite symmetry involving partially deﬁned symmetry. We may identify those symmetries by using several projections. Example 4.4. Let Ω = [−a, a] × [−a, a] in R2 and B = W01,p (Ω). We are interested in ﬁnding a critical point u∗ with the following symmetries. The proﬁle of u∗ is even symmetric about the y = −x axis, even symmetric about the x-axis for points (x, y) with 0 ≥ y ≥ −x ≥ −a, and even symmetric about the y-axis for points (x, y) with 0 ≥ −x ≥ y ≥ −a. The invariant space BT is deﬁned by two projections T = T2 · T1 , where Rn , (n

1 (T1 u)(x, y) = (u(x, y) + u(−y, −x)), (x, y) ∈ Ω 2 is the projection from B to BT1 with which computational error is minimized and T2 is a projection from BT1 to BT deﬁned by u(x, −y), 0 ≥ y ≥ −x; u(−x, y), 0 ≥ −x ≥ y; (T2 u)(x, y) = u(x, y), otherwise. Note that T2 can map B to outside of B. However T2 projects BT1 into B. Thus T = T2 · T1 projects B into BT ⊂ B. In [13], such an example for the Lane-Emden equation has been successfully carried out with B = H01 (Ω). We have numerically veriﬁed that BT is an invariant space in the sense of Deﬁnition 8 Asymm of ∇JH 1 = 2.·10−4 ∇JH01 . Copyright © 2005 Marcel Dekker, Inc.

Stabilization in Computing Saddle Points

4.

289

Selected Numerical Examples

Consider ﬁnding the eigenpairs (u, λ) ∈ W 1,p (Ω)×R+ of the p-Laplacian operator on Ω = [0, 2] × [0, 2] ∆p u(x) = λ|u(x)|p−2 u(x), x ∈ Ω, (10) u(x) = 0, x ∈ ∂Ω ∂u 2 12 ) ) and where u = 0 and ∆p u = div(|∇u|p−2 ∇u), |∇u| = ( ni=1 ( ∂x i n = 2. The p-Laplacian operator has a variety of applications in physical ﬁelds, such as in ﬂuid dynamics when the shear stress and the velocity gradient are related in a certain manner where p = 2, p < 2, p > 2 if the ﬂuid is Newtonian, pseudo-plastic, dilatant, respectively. The pLaplacian operator also appears in the study of ﬂow in a porous media (p = 32 ), nonlinear elasticity (p > 2), and glaciology (p ∈ (1, 43 )). To solve the eigenpair problem (10), we deﬁne the Rayleigh quotient

|∇u(x)|p dx J(u) = Ω . p Ω |u(x)| dx Then critical points u∗ of J correspond to eigenfunctions of the pLaplacian with respect to the eigenvalue λ = J(u∗ ). In our numerical computation, we have utilized the symmetry of a saddle point to be found to reduce its instability. In all our numerical examples here, we use 1000 × 1000 piecewise linear ﬁnite elements and set ε < 10−4 . For illustration, we present the ﬁrst six eigenfunctions for the p-Laplacian with p = 2.75 and p = 1.7. An interesting phenomenon is the change of sequential order of the eigenvalues. When p = 2, λ2 = λ3 and λ5 = λ6 . When p = 2, the two pairs have diﬀerent values and change the sequential orders when p crosses 2.

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1.4

1.5

1.2

1

1 0.5

0.8 0

0.6 −0.5

0.4

0.2

−1

0 2

−1.5 2

2

1.5

2

1.5

1.5

1

1.5

1

1 0.5

1 0.5

0.5 0

0.5 0

0

0

p = 2.75, Eigenfunctions for λ1 = 7.0907 (left) and for λ2 = 25.6218

Figure 1. (right).

1.5

1.5

1

1

0.5

0.5

0

0

−0.5

−0.5

−1

−1

−1.5 2

−1.5 2

2

1.5

2

1.5

1.5

1

1.5

1

1 0.5

1 0.5

0.5 0

0.5 0

0

0

p = 2.75, Eigenfunctions for λ3 = 26.6583 (left) and for λ4 = 47.6999

Figure 2. (right).

1.5

0.15

0.1

1

0.05

0.5

0

0 −0.05

−0.5 −0.1

−1

−0.15

−1.5 2

−0.2 2

2

1.5

1.5

1

2

1.5

1.5

1

1 0.5

0.5 0

Figure 3. (right).

0

1 0.5

0.5 0

0

p = 2.75, Eigenfunctions for λ5 = 66.4530 (left) and for λ6 = 68.6633

Copyright © 2005 Marcel Dekker, Inc.

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Stabilization in Computing Saddle Points

0.9

1

0.8

0.7

0.5

0.6

0.5 0

0.4

0.3 −0.5

0.2

0.1

0 2

−1 2

2

1.5

2

1.5

1.5

1

1.5

1

1 0.5

1 0.5

0.5 0

0.5 0

0

0

p = 1.7, Eigenfunctions for λ1 = 4.1091 (left) and for λ2 = 8.7889 (right).

Figure 4.

1

1

0.5

0.5

0

0

−0.5

−0.5

−1 2

−1 2

2

1.5

2

1.5

1.5

1

1.5

1

1 0.5

1 0.5

0.5 0

0.5 0

0

0

p = 1.7, Eigenfunctions for λ3 = 8.8895 (left) and for λ4 = 13.3503

Figure 5. (right).

0.15

1

0.1

0.05

0.5

0

−0.05 0

−0.1

−0.15 −0.5

−0.2

−0.25

−0.3 2

−1 2

2

1.5

1.5

1

2

1.5

1.5

1

1 0.5

0.5 0

Figure 6. (right).

0

1 0.5

0.5 0

0

p = 1.7, Eigenfunctions for λ5 = 15.6864 (left) and for λ6 = 16.0231

Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

References [1] On the existence of sign-changing solutions for semilinear dirichlet problems. Topol. Methods Nonlinear Anal., 7:115–131, 1996. [2] A. Ambrosetti and P. Rabinowitz. Dual variational methods in critical point theory and applications. J. Funct. Anal., 14:349–381, 1973. [3] H. Brezis and L. Nirenberg. Remarks on ﬁnding critical points. Communications on Pure and Applied Mathematics, XLIV:939–963, 1991. [4] K.C. Chang. Inﬁnite Dimensional Morse Theory and Multiple Solution Problems. Birkh¨ auser, Boston, 1993. [5] I. Ekeland and N. Ghoussoub. Selected new aspects of the calculus of variations in the large. Bull. Amer. Math. Soc. (N.S.), 39:207–265, 2002. [6] J. Smoller Shock Waves and Reaction-Diﬀusion Equations. Springer-Verlag, New York, 1983. [7] Y. Li and J. Zhou. A minimax method for ﬁnding multiple critical points and its applications to semilinear pde. SIAM J. Sci. Comp., 23:840–865, 2001. [8] Y. Li and J. Zhou. Convergence results of a local minimax method for ﬁnding multiple critical points. SIAM J. Sci. Comp., 24:840–865, 2002. [9] J. Q. Liu and S. J. Li. Some existence theorems on multiple critical points and their applications. Kexue Tongbao, 17, 1984. [10] J. Mawhin and M. Willem. Critical Point Theory and Hamiltonian Systems. Springer-Verlag, New York, 1989. [11] P. Rabinowitz. Minimax Method in Critical Point Theory with Applications to Diﬀerential Equations, volume 65 of CBMS Regional Conf. Series in Math. AMS, Providence, 1986. [12] M. Struwe. Variational Methods. Springer-Verlag, Berlin, 1996. [13] Z. Q. Wang and J. Zhou. An eﬃcient and stable method for computing multiple saddle points with symmetries. To appear. [14] M. Willem. Minimax Theorems. Birkh¨ auser, Boston, 1996. [15] X. Yao and J. Zhou. A minimax method for ﬁnding multiple critical points in Banach spaces and its application to quasilinear elliptic pde. To appear. [16] J. Zhou. Local instability analysis of saddle points by a local minimax method. Math. Comp. to appear. [17] J. Zhou. A min-orthogonal method for ﬁnding multiple saddle points. J. Math. Anal. Appl., 291:66–81, 2004.

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CHAPTER XXII SECOND ORDER SUFFICIENT CONDITIONS FOR OPTIMAL CONTROL SUBJECT TO FIRST ORDER STATE CONSTRAINTS Kazimierz Malanowski Systems Research Institute Polish Academy of Sciences Warszawa, Poland

Helmut Maurer Institut f¨ ur Numerische Mathematik Westf¨ alische Wilhelms Universit¨ at M¨ unster, Germany

Sabine Pickenhain Institut f¨ ur Mathematik Brandenburgische Technische Universit¨ at Cottbus, Germany

Abstract

Second order suﬃcient optimality conditions (SSC) are derived, for an optimal control problem subject to pure state constraints of order one. The proof is based on a Hamilton-Jacobi inequality and it exploits regularity of the control function, as well as the associated Lagrange multipliers. The obtained SSC involve Legendre-Clebsch conditions and solvability of an auxiliary Riccati equation. The latter condition is weakened by taking into account the strongly active constraints.

Keywords: Nonlinear optimal control, ﬁrst order state constraints, second order suﬃcient optimality conditions, solvability of a Riccati equation, constraint qualiﬁcations, Legendre-Clebsch condition

Introduction We shall derive second order suﬃcient optimality conditions (SSC) for an optimal control problem for nonlinear ODEs, subject to pure state Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

constraints of order one. We follow the approach based on the HamiltonJacobi inequality which was used in [8] for multidimensional control problems. For one-dimensional problems we are able to obtain results more constructive than those in [8]. We continue the line of [6], where mixed control-state constraints were considered. As in [6], we try to obtain a SSC in a possibly weak form which takes into account the strongly active constraints. In the whole analysis, the regularity of the control as well as of the associated Lagrange multipliers plays a crucial role. We consider the case where the control is a continuous function, and we impose constraint qualiﬁcations which ensure existence, uniqueness and regularity of the normal Lagrange multipliers. By these constraints qualiﬁcations, the analysis is conﬁned to the so-called ﬁrst order state constraints [9]. The relevant regularity results obtained in [4] are recalled in Section 1. Note that in the regularity analysis a Lagrangian in the socalled indirect form is more convenient, while in the analysis of suﬃciency in Sections 2 and 3 a Lagrangian in the direct form is used (see [9] for the deﬁnitions). Following [8] and [6], we formulate in Section 3 a second order suﬃcient optimality condition in terms of a solution to a Hamilton-Jacobi inequality. A quadratic form is chosen as a candidate for a solution to the Hamilton-Jacobi inequality, and conditions are derived under which this quadratic form is the needed solution. These conditions involve the previous constraint qualiﬁcations, as well as some pointwise coercivity conditions with respect to control (Legendre-Clebsch condition) and with respect to state. The latter is expressed in terms of solvability of a modiﬁed matrix Riccati equation which is weakened by taking into account strongly active constraints. We believe that the obtained Riccati equation is original and that it constitutes the main contribution of this paper to suﬃciency analysis for optimal control problems. A simple example in Section 4 shows that the obtained SSC are really weaker than the classical one.

1.

Preliminaries

We consider the following optimal control problem: (O)

Find (x0 , u0 ) ∈ W 1,∞ (0, 1; Rn ) × L∞ (0, 1; Rm ) such that 1 ϕ(x(t), u(t))dt + ψ(x(0), x(1))} F (x0 , u0 ) = min{F (x, u) := subject to x(t) ˙ − f (x(t), u(t)) = 0 ξ(x(0), x(1)) = 0, ϑ(x(t)) ≤ 0

Copyright © 2005 Marcel Dekker, Inc.

0

for a.a. t ∈ [0, 1], for all t ∈ [0, 1],

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Second Order Suﬃcient Conditions

where ϕ : Rn × Rm → R, ξ : Rn × Rn → Rd ,

ψ : Rn × Rn → R, ϑ : Rn → Rl .

f : Rn × Rm → Rn ,

Denote J = {1, ..., l} the set of indices of constraints. Throughout the paper we assume that the data satisfy the following regularity conditions: the functions ϕ, φ, f , ξ, and ϑ are twice Fr´echet diﬀerentiable in all arguments, and all diﬀerentials are Lipschitz continuous. Note that control or mixed control-state constraints can be easily included in our suﬃciency analysis, in a way similar to that in [6] and [10]. For p ∈ [1, ∞], let us introduce the spaces X p := W 1,p (0, 1; Rn ) × Lp (0, 1; Rm ), Y p := W 1,p (0, 1; Rn ) × Rd × W 1,p (0, 1; Rl ), Z p := W 1,p (0, 1; Rn ) × Rd × Lp (0, 1; Rl ) × Rd × Rd . Deﬁne the following Lagrangian and Hamiltonians associated with (O): L, : X ∞ × Y ∞ → R, H : Rn × Rm × Rn × Rk × Rl → R , u, q, ρ, µ) = F (x, u) − (q, x˙ − f (x, u)) + ρ, ξ(x(0), x(1)) L(x, +µ(0), ϑ(x(0)) + (µ, ˙ Dx ϑ(x)f (x, u)), H(x, u, q) = ϕ(x, u) + q, f (x, u) , H(x, u, q, µ) ˙ = H(x, u, q) + µ, ˙ Dx ϑ(x)f (x, u).

(1) (2)

Remark. The Lagrangian (1) is in the normal form, i.e., the Lagrange multiplier corresponding to the functional F (x, u) is equal to one. The Lagrangian is written in the so-called indirect or Pontryagin form with an absolutely continuous adjoint function q (see Section 7 in [9] as well as [1] and [7]). The state constraints are considered in W 1,∞ (0, T ; Rl ) rather than in C 0 (0, 1; Rl ). This form is convenient in the analysis of existence and regularity of Lagrange multipliers [4]. In the sequel, we introduce still another form of the Lagrangian. Let (x0 , u0 ) with u0 ∈ C 0 (0, 1; Rm ) be a ﬁxed pair admissible for (O). We are going to analyze conditions under which (x0 , u0 ) is a locally isolated local solution to (O). For the sake of simplicity, the functions evaluated at (x0 , u0 ) will be denoted by subscript “0”, e.g., f0 (t) = f (x0 (t), u0 (t)), ϑ0 (t) = ϑ(x0 (t)). First we will consider the stationarity conditions of the Lagrangian (1) at (x0 , u0 ). To get existence and uniqueness of a Lagrange multiplier associated with (x0 , u0 ) we need some constraints qualiﬁcations. To this end, for t ∈ [0, 1] introduce the sets of active constraints J0 (t) := {j ∈ J | ϑj0 (t) = 0}, and deﬁne the matrices Υ0 (t) := [Dx ϑj0 (t)]j∈J0 (t) . Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

We assume: (A1) (Linear Independence) There exists β > 0 such that |[Du f0 (t)∗ Υ∗0 ] χ| ≥ β|χ| for all χ of appropriate dimension and all t ∈ [0, 1]. (A2) (Controllability) For any g ∈ Rd there exists a solution (y, v) ∈ X ∞ of the following equations: y(t) ˙ − Dx f0 (t)y(t) − Du f0 (t)v(t) = 0, D0 ξ0 y(0) + D1 ξ0 y(1) = g, Υ0 (t)y(t) = 0. Here and in the sequel, we denote Di := Dx(i) , i = 0, 1. Similarly 2 := D 2 Dij x(i)x(j) , i, j = 0, 1. Note that (A1) restricts the analysis to problems with so-called ﬁrst order state constraints (see [9]). Condition (A1) implies in particular that |[Υ0 (t)∗ ] η| ≥ β|η| for all η of appropriate dimension and all t ∈ [0, 1]. This means that, pointwisely along the trajectory (x0 , u0 ), the gradients of all active constraints are linearly independent, uniformly on [0, 1]. Theorem 4.3 in [4] implies:

Lemma 1 If assumptions (A1) and (A2) hold, then there exists a unique normal Lagrange multiplier λ0 := (q0 , ρ0 , µ0 ) ∈ Y ∞ for (O) associated with (x0 , u0 ), i.e., the following ﬁrst order optimality conditions are satisﬁed: , 0 , u0 , q0 , ρ0 , µ0 ) = 0, Dx L(x , 0 , u0 , q0 , ρ0 , µ0 ) = 0, Du L(x ,+, µ0 (0), ϑ(x(0)) + (µ˙ 0 , Dx ϑ(x0 )f (x0 , u0 )) = 0, µ0 ∈ L where ,+ = {µ ∈ W 1,∞ (0, 1; Rl )|0 ≤ µ˙ j (t) ≤ µj (t), and µ˙ j (t) is L nonincreasing, for all j ∈ J and a.a. t ∈ [0, 1]}. In addition to (A1) and (A2), we assume: (A3) (Legendre-Clebsch condition) There exists γ > 0 such that 2 , H0 (t)v ≥ γ|v|2 v, Duu

Copyright © 2005 Marcel Dekker, Inc.

for all v ∈ Rm and a.a. t ∈ [0, 1].

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Second Order Suﬃcient Conditions

In the same way as in Theorem 5.2 in [4], we obtain:

Corollary 2 If (A1) – (A3) hold, then x˙ 0 , u0 , q˙0 , µ˙ 0 are Lipschitz continuous on (0, 1).

(3)

Taking advantage of the regularity (3), we can deﬁne a Lagrangian for (O) in the so-called direct form [9]: L : X ∞ × Z ∞ → R, L(x, u, p, ρ, ν, σ 0 , σ 1 ) = F (x, u) − (p, x˙ − f (x, u)) + ρ, ξ(x(0), x(1)) +(ν, ϑ(x)) + σ 0 , ϑ(x(0)) + σ 1 , ϑ(x(1). (4) We also introduce still another Hamiltonian: 2 : Rn × Rm × Rn × Rl → R, H

2 u, p, ν) = H(x, u, p) + ν, ϑ(x). (5) H(x,

The stationarity of the Lagrangian (4) yields the following ﬁrst order optimality conditions: 2 0 , u0 , p0 , ν0 ) p˙ 0 + Dx H(x (6) = p˙ 0 + Dx f (x0 , u0 )∗ p0 + Dx ϕ(x0 , u0 ) + Dx ϑ(x0 )∗ ν0 = 0, p0 (0) + D0 ψ(x0 (0), x0 (1)) + D0 ξ(x0 (0), x0 (1))∗ ρ0 +Dx ϑ(x0 (0))∗ σ00 = 0,

(7)

−p0 (1) + D1 ψ(x0 (0), x0 (1)) + D1 ξ(x0 (0), x0 (1))∗ ρ0 +Dx ϑ(x0 (1))∗ σ01 = 0,

(8)

Du H(x0 (t), u0 (t), p0 (t), ν0 ) := Du ϕ(x0 , u0 ) + Du f (x0 (t), u0 (t))∗ p0 = 0, (ν0 , ϑ(x0 )) = 0,

σ00 , ϑ(x0 (0)) = 0,

(9)

σ01 , ϑ(x0 (1)) = 0,

where σ00 ∈ Rd+ ,

σ01 ∈ Rd+ ,

ν0 ∈ L+ := {ν ∈ L∞ (0, 1; Rl ) | ν i (t) ≥ 0, for all j ∈ J and a.a. t ∈ [0, 1]}. In view of (3), simple calculations show that µ0 ∈ L∞ (0, 1; Rl ), p0 = Dx ϑ(x0 )∗ µ˙ 0 + q0 ∈ W 1,∞ (0, 1; Rn ), ν0 = −¨ σ01 = µ˙ 0 (1). σ00 = µ0 (0) − µ˙ 0 (0), (10) Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

In particular, 2 0 , u0 , p0 , ν0 ) − ν0 , ϑ(x0 ) , 0 , u0 , q0 , µ˙ 0 ) = H(x H(x , 0 , u0 , q0 , µ˙ 0 ) = Du H(x 2 0 , u0 , p0 , ν0 ), Du H(x 2 H(x 2 H(x , 0 , u0 , q0 , µ˙ 0 ) = Duu 2 0 , u0 , p0 , ν0 ). Duu

i.e., (11)

For a ﬁxed α ≥ 0, let us deﬁne: Jα+ (t) = {j ∈ J0 (t) | ν0 (t) > α}, 2 α (t) = Dx ϑj (t) . Υ 0

α (t) = cardJα+ (t),

+ j∈Jα (t)

Note that Jα+ (t) are the sets of those active constraints for which pointwise strict complementarity is satisﬁed with the margin α.

2.

Second Order Suﬃcient Conditions via Hamilton-Jacobi Inequality

To derive second order suﬃcient optimality conditions for (O), we follow the approach used in [8] and [6] which is based on the Hamilton-Jacobi inequality. By B X (x) := {y ∈ X |y − xX ≤ } we denote a closed ball of radius in a normed space X, centered at a point x. The following theorem can be proved in the same way as Theorem 3.1 in [6] (see also Assertion 2 in [8]).

Theorem 3 Suppose there exists a function V : [0, 1] × Rn → R that is of class C 1 with respect to x and Lipschitz continuous with respect to t, such that the following conditions hold for suitable > 0 and c > 0: Dt V (t, x) + H(x, u, Dx V (t, x)) ≥ c{|x − x0 (t)|2 + |u − u0 |2 } Rn+m

for all (x, u) ∈ B

(12)

((x0 (t), u0 (t))), s.t. ϑ(x) ≤ 0

and for a.a. t ∈ [0, 1], Dt V (t, x0 (t)) + H(x0 (t), u0 (t), Dx V (t, x0 (t))) = 0 for a.a. t ∈ [0, 1],

(13) [V (1, x0 (1)) − V (0, x0 (0))] − V (1, x ) − V (0, x ) +ψ(x0 , x1 ) − ψ(x0 (0), x0 (1)) ≥ c |x0 − x0 (0)|2 + |x1 − x0 (1)|2 (14)

1

0

for all xi such that ϑ(xi ) ≤ 0, i = 0, 1 and ξ(x0 , x1 ) = 0. Then the estimate F (x, u) ≥ F (x0 , u0 ) + c(x, u) − (x0 , u0 )2X 2 Copyright © 2005 Marcel Dekker, Inc.

(15)

299

Second Order Suﬃcient Conditions ∞

holds for all admissible pairs (x, u) ∈ B X (x0 , u0 ), i.e., (x0 , u0 ) is a locally isolated weak local minimizer. As in [8] and [6], the function V : [0, 1] × Rn → R used in Theorem 3 is deﬁned by the following quadratic expression: 1 V (t, x) = e(t) + p0 (t)∗ (x − x0 (t)) + (x − x0 (t))∗ Q(t)(x − x0 (t)), (16) 2 1,∞ 1,∞ where e ∈ W (0, 1; R), p0 ∈ W (0, 1; Rn ) is the adjoint variable given in (6), while Q(t) is a symmetric matrix and Q ∈ W 1,∞ (0, 1; Rn×n ). The function e is chosen in such a way that the HJ equality (13) holds, while the appropriate choice of Q ensures that inequalities (12) and (14) are satisﬁed. To this end, sensitivity results for parametric mathematical programs are used, in the same way as in [6]. We obtain the following results (see Theorem 4.1 in [6]):

Lemma 4 Suppose that (C) There exist constants α > 0 and δ > 0, independent of t ∈ [0, 1] such that y ∗ ∗ ≥ δ(|y|2 + |v|2 ) [y , u ]K(t) v for all (y, v) ∈ Rn+m such that Dx ϑi0 , y = 0 for all j ∈ Jα+ (t), and for a.a. t ∈ [0, 1], where + * 2 H 2 H + QD f 2 0 Dxu Q˙ + QDx f0 + Dx f0∗ Q + Dxx 0 u 0 (t). K(t) = 2 H + D f ∗Q 2 H Dux Duu 0 u 0 0 Then inequality (12) holds.

Lemma 5 Suppose that: (A4)

[y0∗ , y1∗ ]

L00 L01 L∗01 L11

y0 y1

>0

for all (y 0 , y 1 ) ∈ R2n such that: 2 0 (1)y 1 = 0, 2 0 (0)y 0 = 0, Υ Υ D0 ξ(x0 (0), x0 (1))y 0 + D1 ξ(x0 (0), x0 (1))y 1 = 0, where 2 (ψ(x (0), x (1)) + ξ(x (0), x (1))∗ ρ L00 = Q(0) + D00 0 0 0 0 0 ∗ 0 +ϑ(x0 (0)) σ0 ), 2 (ψ(x (0), x (1)) + ξ(x (0), x (1))∗ ρ ), L01 = D01 0 0 0 0 0 2 (ψ(x (0), x (1)) + ξ(x (0), x (1))∗ ρ L11 = −Q(1) + D11 0 0 0 0 0 +ϑ(x0 (1))∗ σ01 ). Copyright © 2005 Marcel Dekker, Inc.

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Then inequality (14) holds. By Theorem 3 and Lemmas 4 and 5 we obtain:

Theorem 6 Suppose that (A1) – (A3) hold and there exists a symmetric matrix function Q ∈ W 1,∞ (0, 1; Rn×n ) such that conditions (C) and (A4) are satisﬁed. Then there exist c > 0 and > 0 such that F (x, u) − F (x0 , u0 ) ≥ c(x, u) − (x0 , u0 )2X 2 ∞ for all admissible x, u ∈ B X (x0 , u0 ), i.e., (x0 , u0 ) is a locally isolated strong local minimizer of (O).

3.

Checking Positive Deﬁniteness with Riccati Equation

Theorem 6 is not constructive, in the sense that it provides no information on the existence and the form of a matrix function Q satisfying (C) and (A4). In this section we are going to show that conditions (C) and (A4) of positive deﬁniteness can be expressed in terms of solvability of an auxiliary matrix Riccati diﬀerential equation, satisfying appropriate boundary conditions. To this end, let us ﬁx α > 0 and denote by P(t) : Rn → Rn the projection map onto the (t)-dimensional subspace 2 α (t)y = 0}. Condition (C) can be expressed in the form: {y ∈ Rn | Υ y ∗ ∗ ≥ δ (|P(t)y|2 + |v|2 ) (17) [y , u ]M (t) v for all (y, v) ∈ Rn+m , and a.a. t ∈ [0, 1], where M (t) = * 2 + 2 H 20 P P ∗ Dxu P ∗ Q˙ + QDx f0 + Dx f0∗ Q + Dxx H0 + QDu f0 (t). 2 2 H Duu Dux H0 + Du f0∗ Q P 0 (18) Let π(S) denote the number of positive eigenvalues of a symmetric matrix S. Condition (17) takes the form π(M (t) − δI) = n + m − (t), where I is the unit matrix of the appropriate dimension. We assume: Copyright © 2005 Marcel Dekker, Inc.

(19)

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Second Order Suﬃcient Conditions

(A5) The following matrix Riccati equation 2 H 20 Q˙ + QDx f0 + Dx f0∗ Q + Dxx 2 −1 2 2 Dux H0 + Du f0∗ Q = S − Dxu H0 + QDu f0 Duu H0 (20) has a solution uniformly bounded on (0, 1). Here S ∈ L∞ (0, 1; Rn×n ) is any symmetric matrix function such that P ∗ (t)S(t)P(t) = 0 for a.a. t ∈ (0, 1). (21)

Remark If S(t) ≡ 0, then (20) reduces to the classical Riccati equation, used in second order suﬃcient optimality conditions for optimal control problems (see e.g., [5]). Clearly, (A5) is weaker than the latter condition, since S can be any arbitrary matrix function satisfying (21). Thus, in (A5) strongly active state constraints are taken into account.

Lemma 7 If (A1) - (A3) and (A5) hold, then (C) is satisﬁed. Proof Let π(S) denote the number of positive eigenvalues of a symmetric matrix S. We will need the following result from linear algebra (see Theorem 1 in [2]):

Lemma 8 Let A, B, C be n×n, m×n, and m×m dimensional matrices, respectively, where A and C are symmetric and C is invertible. Deﬁne A B∗ D= . B C Then π(D) = π(C) + π(A − B ∗ C −1 B), where A − B ∗ C −1 B is the Schur complement of C. In view of (A3) we have 2 H0 − δI) = m π(Duu

for δ ∈ (0, γ).

(22)

Hence, by (A5) and by well known stability results for ODEs, the following perturbed Riccati equation 2 H 20 Q˙ + QDx f0 + Dx f0∗ Q + Dxx 2 −1 2 2 − Dxu H0 + QDu f0 Duu H0 − δI Dux H0 + Du f0∗ Q = S + 2δI

(23)

has a bounded solution for any δ ∈ (0, γ) suﬃciently small. Multiplying (23) from the left and right by P(t)∗ and P(t), respectively, and taking Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

into account (21), we get 2 H 20 R := P ∗ Q˙ + QDx f0 + Dx f0∗ Q + Dxx 2 2 −1 2 ( ∗ Dux H0 + Du f0 Q P − Dxu H0 + QDu f0 Duu H0 − δI −δP ∗ P = δP ∗ P. (24) Simple calculations show that R(t) is the Schur complement of the sub2 H (t) − δI of the matrix M (t) − δI, where M (t) is deﬁned matrix Duu 0 in (18). Clearly, (24) yields that π(R(t)) = n − (t). Hence, by Lemma 8 and by (22) we get π(M (t) − δI) = n + m − (t). It implies that (C) holds. Using Theorem 4 and Lemma 7 we arrive at the principal result of this paper:

Theorem 9 (Suﬃcient Optimality Condition) Suppose that conditions (A1) – (A3) are satisﬁed, and there exists a symmetric matrix function Q ∈ W 1,∞ (0, 1; Rn×n ) such that (A4) and (A5) hold. Then there exist c > 0 and > 0, such that F (x, u) − F (x0 , u0 ) ≥ c(x, u) − (x0 , u0 )X 2 ∞ for all admissible (x, u) ∈ B X ((x0 , u0 )), i.e., (x0 , u0 ) is a locally isolated strong local minimizer of (O).

4.

Example

Consider the following variational problem in a time interval [0, T ] with ﬁxed endtime T > 0 but free endpoint x(T ): T (u(t)2 − x(t)2 ) dt (25) Minimize F (x, u) = 0.5 0

subject to

x(t) ˙ = u(t), x(0) = x0 , t ∈ [0, T ], −a ≤ x(t) ≤ b for t ∈ [0, T ], 0 < a < b .

(26) (27)

The initial value x0 is supposed to satisfy −a < x0 < b. It is the same problem as Example 7.1 in Kawasaki, Zeidan [3] except that we consider a free endpoint. Consider the Hamiltonian (5) 2 u, p) = 0.5(u2 − x2 ) + pu + νa (−x − a) + νb (x − b) H(x,

(28)

with two multipliers νa , νb associated with the state constraints −x−a ≤ 0 and x − b ≤ 0. The adjoint equation (6) is p˙ = x + νa − νb . Copyright © 2005 Marcel Dekker, Inc.

(29)

Second Order Suﬃcient Conditions

303

2 u = u + p = 0, the optimal control is given by In view of H u = −p .

(30)

It is obvious that the strengthened Legendre-Clebsch condition (A3) holds and that the state constraint is of order one. Depending on the initial value x0 ∈ (−a, b), the optimal solution is composed by an interior arc with −a < x(t) < b in [0, t1 ) and a boundary arc either on the lower boundary, x(t) ≡ −a, or on the upper boundary, x(t) ≡ b, for t1 ≤ t ≤ T . The solutions on the boundary arcs are as follows: x(t) ≡ −a : u = 0, p = 0, νa = a > 0, x(t) ≡ b : u = 0, p = 0, νb = b > 0.

(31)

To determine the unknown entry time t1 , we use the continuity of the control and obtain the entry conditions x(t1 ) ∈ (−a, b), p(t1 ) = 0. Thus, the following boundary value problem has to be solved on the interior arc: (32) x˙ = −p, p˙ = x, x(0) = x0 , x(t1 ) ∈ (−a, b), p(t1 ) = 0. The solution is x(t) = α sin(t) + β cos(t) with β = x0 , α = x0 tan(t1 ). The entry time is determined by the condition cos(t1 ) = x0 /b on the upper boundary, resp., cos(t1 ) = −x0 /a on the lower boundary. Thus, for initial values with −a < x0 < a the last equation is solvable with 0 < t1 < π both for the upper and the lower boundary. Hence, we have found two extremal solutions. For x0 = a, we also get two extremals with x(t1 ) = −a for t1 = π and x(t1 ) = b with t1 < π. Finally, for all initial values a < x0 < b there exists only one extremal joining the upper boundary with x(t1 ) = b for t1 < π. We are going to use the SSC derived in Section 3 to check optimality of all these extremals. On the interior arc [0, t1 ), the function S in the Riccati equation (20) becomes zero and the equation takes the form (33) Q˙ − Q2 − 1 = 0. This equation has a bounded solution Q(t) = tan(t − t1 /2) for t1 < π. However, there is no bounded solution of (33) in the whole interval [0, T ] for an endtime T ≥ π. So, in particular, the classical SSC condition with S ≡ 0 fails. Here, we can take advantage of the freedom in choosing any S satisfying (21). In our case, S ∈ W 1,∞ (t1 , T ) can be arbitrary. Setting S(t) ≡ (−Q(t1 ) − 1) we ﬁnd that (20) reduces to Q˙ = 0. The boundary condition for Q(0) and Q(T ) are both vacuous since the only admissible variations in (A4) are y 0 = y 1 = 0. Hence, the solution of (20) needed in Theorem 9 takes the form: for t ∈ [0, t1 ], tan(t − t1 /2) Q(t) = for t ∈ [t1 , T ], tan(t1 /2) and by that theorem the extremal is the solution of the optimal control problem. Copyright © 2005 Marcel Dekker, Inc.

304

CONTROL AND BOUNDARY ANALYSIS

References [1] W.W. Hager. Lipschitz continuity for constrained processes. SIAM J. Control and Optimization, 17:321–338, 1979. [2] E.V. Haynsworth. Determination of the inertia of a partitioned hermitian matrix. Linear Algebra Appl., 1:73–81, 1968. [3] H. Kawasaki and V. Zeidan. Conjugate points for variational problems with equality and inequality state constraints. SIAM J. Control and Optimization, 39:433–456, 2000. [4] K. Malanowski. On normality of Lagrange multipliers for state constrained optimal control problems. Optimization, 52:75–91, 2003. [5] H. Maurer. First and second order suﬃcient optimality conditions in mathematical programming and optimal control. Mathematical Programming Study, 14:163–177, 1981. [6] H. Maurer and S. Pickenhain. Second-order suﬃcient conditions for control problems with mixed control-state constraints. J. Optim. Theory Appl., 86:649– 667, 1995. [7] L.W. Neustadt. Optimization: a theory of necessary conditions. Princeton University Press, Princeton, NJ, 1976. [8] S. Pickenhain and K. Tammer. Suﬃcient conditions for local optimality in multidimensional control problems with state restrictions. Zeitschrift f¨ ur Analysis und ihre Anwendungen, 11:397–405, 1994. [9] S.P. Sethi, R.F. Hartl, and R.G. Vickson. A survey of the maximum principle for optimal control problems with state constraints. SIAM Review, 37:181–218, 1995. [10] V. Zeidan. The Riccati equation for optimal control problems with mixed state control problems: necessity and suﬃciency. SIAM J. Control and Optimization, 32:1297–1321, 1994.

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1/5/05

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Control and Boundary Analysis

edited by

John Cagnol Pole Universitaire Leonard de Vinci Paris, France

Jean-Paul Zolésio CNRS/INRIA, Projet OPALE Valbonne, France

Boca Raton London New York Singapore

A CRC title, part of the Taylor & Francis imprint, a member of the Taylor & Francis Group, the academic division of T&F Informa plc.

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DK3828 disclaimer.fm Page 1 Tuesday, January 4, 2005 1:16 PM

Library of Congress Cataloging-in-Publication Data Control and boundary analysis / edited by John Cagnol, Jean-Paul Zolésio. p. cm. — (Lecture notes in pure and applied mathematics ; v. 239) Includes bibliographical references and index. ISBN 1-57444-594-4 (alk. paper) 1. Control theory—Congresses. 2. Boundary element methods—Congresses. 3. Differential equations, Partial—Congresses. I. Cagnol, John. II. Zolésio, J. P. III. Series. QA402.3.C6175 2005 003'.5--dc22

2004061831

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. All rights reserved. Authorization to photocopy items for internal or personal use, or the personal or internal use of specific clients, may be granted by CRC Press, provided that $1.50 per page photocopied is paid directly to Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923 USA. The fee code for users of the Transactional Reporting Service is ISBN 1-57444-594-4/05/$0.00+$1.50. The fee is subject to change without notice. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. The consent of Marcel Dekker and CRC Press does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from Marcel Dekker/CRC Press for such copying. Direct all inquiries to CRC Press, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe.

Visit the CRC Press Web site at www.crcpress.com © 2005 by Marcel Dekker/CRC Press No claim to original U.S. Government works International Standard Book Number 1-57444-594-4 Library of Congress Card Number 2004061831 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper

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Foreword

This volume comprises selected papers from the 21st Conference on System Modeling and Optimization that took place from July 21 to July 25, 2003, in Sophia Antipolis, France. These papers pertain to moving boundary and boundary control in systems described by partial diﬀerential systems and are speciﬁcally relevant to Working Group 7.2 of the International Federation for Information Processing (IFIP). In the 1970s, the study of partial diﬀerential equations focused on the modeling of continuous mechanics involving ﬁxed boundary, that is, in abstract form, autonomous systems. At the same time, control theory, which was born in the ﬁeld of ﬁnite dimensional systems, emerged in the domains of active and passive control of partial diﬀerential equations at the boundary. In the 1980s, shape optimization became a mathematical area that used control theory extended to moving boundaries. In that sense, it included free boundary and phase transition problems. More recently, topological shape optimization has been developed as an alternative approach to the homogenization process in material sciences. Currently, the focus is on moving boundary analysis, which combines and extends all of the previous mathematical and numerical backgrounds. In this book, all of these aspects are covered by the selected papers, including the important stochastic approach. New existence results are derived from ﬁne analysis of boundary behavior; the ﬁrst variation of the boundary is taken into account by a transverse backward vector ﬁeld. Intrinsic geometry is developed for shell modeling, avoiding local representations, such as the Christoﬀel symbol. Non-diﬀerentiabilities are involved when the moving domain is a contact area related to some unilateral problems. New algorithms associated with current computing power allow impressive simulations for particle ﬂow. This leads to numerical treatment of the mathematical speed method, or Eulerian parametrization tool. John Cagnol and Jean-Paul Zol´esio

Copyright © 2005 Marcel Dekker, Inc.

Contributing Authors

Viorel Barbu Alexandru Ioan Cuza University, Romania Youssef Belhamadia Universit´e Laval, Canada John Cagnol Pˆ ole Universitaire L´eonard de Vinci, France Giuseppe Da Prato Scuola Normale Superiore di Pisa, Italy Edward J. Dean University of Houston, USA Michel C. Delfour Universit´e de Montr´eal, Canada Zdzislaw Denkowski Jagiellonian University, Poland Nicolas Doyon Universit´e Laval, Canada Karsten Eppler Weierstraß Institut f¨ ur Angewandte Analysis und Stochastik, Germany Lorella Fatone Universit`a di Modena e Reggio Emilia, Italy Andr´ e Fortin Universit´e Laval, Canada H´ el` ene Frankowska CNRS, CREA, Ecole Polytechnique, France

Copyright © 2005 Marcel Dekker, Inc.

Leszek Gasi´ nski Jagiellonian University, Poland Roland Glowinski University of Houston, USA Martin Gugat University of Erlangen-Nuremberg, Germany Scott W. Hansen Iowa State University, USA Helmut Harbrecht Christian–Albrechts–Universit¨at zu Kiel, Germany Mary Ann Horn Vanderbilt University, USA Michal Koˇ cvara University of Erlangen, Germany Irena Lasiecka University of Virginia, USA Catherine Lebiedzik Wayne State University, USA Kazimierz Malanowski Polish Academy of Sciences, Poland Helmut Maurer Westf¨alische Wilhelms Universit¨at, Germany Stanislaw Mig´ orski Jagiellonian University, Poland Kirsten Morris University of Waterloo, Canada Marwan Moubachir INRIA, France Carmeliza Navasca University of California, USA

Copyright © 2005 Marcel Dekker, Inc.

Arian Novruzi University of Ottawa, Canada Jiˇ ri V. Outrata Academy of Sciences of the Czech Republic, Czech Republic Tsorng-Whay Pan University of Houston, USA Sabine Pickenhain Brandenburgische Technische Universit¨at, Germany Michael P. Polis Oakland University, USA Vicentiu Radulescu University of Craiova, Romania Mohammad A. Rammaha University of Nebraska-Lincoln, USA Maria Cristina Recchioni Universit`a Politecnica delle Marche, Italy Irina F. Sivergina Kettering University, USA Jan Sokolowski Universit´e Henri Poincar´e Nancy I, France Xudong Yao Texas A&M University, USA Jianxin Zhou Texas A&M University, USA Francesco Zirilli Universit`a di Roma La Sapienza, Italy ˙ Antoni Zochowski Polish Academy of Sciences, Poland Jean-Paul Zol´ esio CNRS, INRIA, France

Copyright © 2005 Marcel Dekker, Inc.

Contents

I. Operator-Splitting Methods and Applications to the Direct Numerical Simulation of Particulate Flow and to the Solution of the Elliptic Monge-Amp` ere Equation 1 Edward J. Dean, Roland Glowinski, Tsorng-Whay Pan 1 Operator-Splitting Schemes for the Time-Discretization of Initial Value Problems 2 Operator-Splitting Methods for the Direct Numerical Simulation of Particulate Flow 3 An Operator-Splitting Method for the Elliptic Monge-Amp`ere Equations in Two-Dimension

14

II. Dynamical Shape Sensitivity

29

Marwan 1 2 3 4 5

30 30 31 33 36

Moubachir, Jean-Paul Zol´esio Introduction Mechanical Problem Inverse Problem Dynamical Shape Sensitivity of the Fluid Conclusion

2 6

III. Optimal Control of a Structural Acoustic Model with Flexible Curved Walls

37

John Cagnol, Catherine Lebiedzik 1 Introduction 2 Background and Literature 3 Statement of Main Results 4 Intrinsic Geometry 5 The Unforced, Conservative Shell Model 6 Structurally Damped, Forced Shell Wall of an Acoustic Chamber 7 Optimal Control Problem 8 Proof of Lemma 3

37 38 39 41 42 45 47 48

IV. Nonlinear Wave Equations with Degenerate Damping and Source Terms

53

Viorel Barbu, Irena Lasiecka, Mohammad A. Rammaha 1 Deﬁnitions and Main Results 2 Outline of the Proof of Theorem 3

55 58

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V. Numerical Modeling of Phase Change Problems

63

Andr´e Fortin, Youssef Belhamadia 1 Stefan Problem and Semi-Phase-Field Formulation 2 Finite Element Discretization 3 Adaptive Strategy 4 Numerical Results 5 Conclusions

64 65 65 66 67

VI. Shape Optimization of Free Air-Porous Media Transmission Coeﬃcient

73

Arian Novruzi 1 Computation of the Porous Shape 2 Optimal Porous Shape Domain

76 81

VII. The Uniform Fat Segment and Uniform Cusp Properties in Shape Optimization

85

Michel C. Delfour, Nicolas Doyon, Jean-Paul Zol´esio 1 Preliminaries: Topologies on Families of Sets 2 The New Uniform Fat Segment Property 3 Equivalence of the Uniform Cusp and Fat Segment Properties

86 87 91

VIII. Topology Optimization for Unilateral Problems

97

˙ Jan Sokolowski, Antoni Zochowski 1 Transformations of the Energy Functional for the Laplace Equation 2 Signorini Problem

98 102

IX. Second Order Lagrange Multiplier Approximation for Constrained Shape Optimization Problems

107

Karsten Eppler, Helmut Harbrecht 1 Shape Optimization 2 Optimization of Constrained Problems 3 Numerical Results

X. Mathematical Models of “Active” Obstacles in Acoustic Scattering Lorella Fatone, Maria Cristina Recchioni, Francesco Zirilli 1 The Mathematical Models of the Masking and Ghost Obstacle Problems 2 The Optimality Conditions for the Ghost Obstacle Problem and Some Numerical Experience

XI. Local Null Controllability in a State Constrained Thermoelastic Contact Problem Irina F. Sivergina, Michael P. Polis 1 Control of the Linear System Copyright © 2005 Marcel Dekker, Inc.

108 112 114

119 122 125

131 135

2 3 4

Nonlinear Case: Application of a Fixed Point Theorem The State Constrained Null Control Problem Conclusion

XII. On Sensitivity of Optimal Solutions to Control Problems for Hyperbolic Hemivariational Inequalities Zdzislaw 1 2 3 4

136 143 143

145

Denkowski, Stanislaw Mig´ orski Class of Problems Preliminaries Sensitivity of Solution Sets for Second Order HVIs Complementary Γ-convergence of Cost Functionals

146 147 151 153

XIII. Evolution Hemivariational Inequality with Hysteresis and Optimal Control Problem

157

Leszek Gasi´ nski 1 Preliminaries 2 Hemivariational Inequality with Hysteresis 3 Estimates on the Solutions of (HVI) 4 Optimal Control Problem

XIV. On the Modeling and Control of Delamination Processes Michal Koˇcvara, Jiˇri V. Outrata 1 Introduction 2 Delamination Process 3 Optimality Conditions 4 Optimization of Delamination Processes 5 Conclusion

XV. On a Spectral Variational Problem Arising in the Study of Earthquakes Vicentiu Radulescu 1 Main Results and Physical Motivation 2 Proofs

XVI. Nodal Control of Conservation Laws on Networks Martin Gugat 1 Notation 2 The Linearized Problem 3 Existence of Solutions 4 Directional Diﬀerentiability 5 Evaluation of Directional Derivatives 6 Example 7 Conclusion Copyright © 2005 Marcel Dekker, Inc.

158 161 163 164

169 169 171 177 182 185

189 190 194

201 202 204 205 206 206 211 214

XVII. Invariance of Closed Sets under Stochastic Control Systems Giuseppe 1 2 3

Da Prato, H´el`ene Frankowska Preliminaries Necessary and Suﬃcient Conditions for the Invariance Invariance of Stochastic Control Systems

XVIII. Uniform Stabilization of an Anisotropic System of Thermoelasticity Mary Ann Horn 1 Statement of the Problem 2 Proof of Theorem 3: Uniform Stabilization 3 Discussion

217 219 220 224

231 232 235 241

XIX. Semigroup Well-Posedness of a Multilayer Mead-Markus Plate with Shear Damping

243

Scott W. Hansen 1 Multilayer Mead-Markus Model 2 State Variable Formulation 3 Semigroup Formulation of Homogeneous Problem

244 248 251

XX. Solution of Algebraic Riccati Equations Arising in Control of Partial Diﬀerential Equations Kirsten Morris, Carmeliza Navasca 1 Solution of Lyapunov Equation 2 Benchmark Examples 3 Euler-Bernoulli Beam

XXI. Stabilization in Computing Saddle Points Jianxin Zhou, Xudong Yao 1 The Local Minimax Method 2 Results on the Order of Saddle Points 3 Reduce Instability by Using Symmetries 4 Selected Numerical Examples

XXII. Second Order Suﬃcient Conditions for Optimal Control Subject to First Order State Constraints Kazimierz Malanowski, Helmut Maurer, Sabine Pickenhain 1 Preliminaries 2 Second Order Suﬃcient Conditions via Hamilton-Jacobi Inequality 3 Checking Positive Deﬁniteness with Riccati Equation 4 Example

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257 260 263 268

281 282 283 286 289

293 294 298 300 302

CHAPTER I OPERATOR-SPLITTING METHODS AND APPLICATIONS TO THE DIRECT NUMERICAL SIMULATION OF PARTICULATE FLOW AND TO THE SOLUTION OF THE ELLIPTIC ` MONGE-AMPERE EQUATION Edward J. Dean Department of Mathematics University of Houston Houston, TX, USA

Roland Glowinski Department of Mathematics University of Houston Houston, TX, USA

Tsorng-Whay Pan Department of Mathematics University of Houston Houston, TX, USA

Dedicated to A.V. Balakrishnan, J.-L. Lions, and G.I. Marchuk for their outstanding contributions to computational and applied mathematics and for making this IFIP conference possible. Abstract

The main goal of this article is to investigate some new applications of operator-splitting methods. We will show that these methods (almost) trivialize the numerical solution of outstanding problems from various areas in mechanics, physics, and diﬀerential geometry, such as the direct simulation of particulate ﬂow and the real elliptic Monge-Amp`ere equations. The results of numerical experiments will be given to illustrate the capabilities of these methods.

Keywords: Operator-splitting methods, particulate ﬂow, Monge-Amp`ere equations

Copyright © 2005 Marcel Dekker, Inc.

2

CONTROL AND BOUNDARY ANALYSIS

Introduction It has been known for decades that operator-splitting methods provide eﬃcient tools for the numerical solution of complicated problems from various areas in science and engineering. New applications appear almost daily and we know of many instances where the only available practical solution methods are of the operator-splitting type. Our goal, here, is to show that operator-splitting makes almost trivial the numerical solution of outstanding problems such as: (i) The simulation of particulate ﬂow when the number of particles exceeds 102 . (ii) Fully nonlinear elliptic equations of the real Monge-Amp`ere type. The content of this article is as follows: In Section 1 we will brieﬂy discuss the time-discretization of initial value problems by various operator-splitting methods and show that wellknown iterative methods are indeed disguised operator-splitting schemes. In Section 2 we will address the direct numerical simulation of particulate ﬂow via a methodology combining operator-splitting and ﬁctitious domain methods. Finally, in Section 3 we will discuss the numerical solution of the two-dimensional Dirichlet problem for the elliptic MongeAmp`ere equation. The results of numerical experiments will be given in Sections 2 and 3; they will conﬁrm the capabilities of operator-splitting methods for solving problems still considered complicated by today’s standards. Operator-splitting methods have generated a huge literature; let us mention among many others, refs.: [3], [10](Chapters 2 and 6) (see also the references therein), [17], and [11].

1.

Operator-Splitting Schemes for the Time-Discretization of Initial Value Problems

1.1

Generalities

Let us consider the following autonomous initial value problem: dϕ + A(ϕ) = 0 on (0, T ) (with 0 < T ≤ +∞), dt

ϕ(0) = ϕ0 .

(IVP)

Operator A maps a vector space V into itself and we suppose that ϕ0 ∈ V. We suppose also that A has a non-trivial decomposition such as A=

J j=1

Copyright © 2005 Marcel Dekker, Inc.

Aj ,

(1)

Operator-Splitting Methods and Applications

3

with J ≥ 2 (by non-trivial we mean that the operators Aj are individually simpler than A). A question which arises naturally is clearly: Can we take advantage of decomposition (1) for the solution of (IVP)? It has been known for a long time that the answer to the above question is yes. Many schemes have been designed to take advantage of (1) when solving (IVP); two of them will be brieﬂy discussed in the following paragraphs, namely the Lie scheme and the Strang scheme.

1.2

Time-Discretization of (IVP) by Lie’s Scheme

Let τ (> 0) be a time-discretization step (we suppose τ uniform, for simplicity); we denote nτ by tn . With ϕn an approximation of ϕ(tn ), Lie’s scheme reads as follows (for its derivation see, e.g.,[10](Chapter 6)): For n ≥ 0, assuming that ϕn is known, we compute ϕn+1 via dϕ + Aj (ϕ) = 0 on (tn , tn+1 ), (2) dt ϕ(tn ) = ϕn+(j−1)/J ; ϕn+j/J = ϕ(tn+1 ), for j = 1, ..., J, with ϕ0 = ϕ0 . If (IVP) is taking place in a ﬁnite dimensional space and if operators Aj are smooth enough, then ϕ(tn ) − ϕn = O(τ ), the function ϕ being the solution of (IVP).

Remark 1 The above scheme applies also to multivalued operators (such as the subgradient of a proper l.s.c. convex functional) but in such a case ﬁrst order accuracy is not guaranteed anymore. Remark 2 The above scheme is easy to generalize to non-autonomous problems by observing that dϕ dϕ dt + A(ϕ, θ) = 0, + A(ϕ, t) = 0, dθ ⇔ dt − 1 = 0, ϕ(0) = ϕ 0 dt ϕ(0) = ϕ0 , θ(0) = 0. Remark 3 Scheme ( 2) is semi-constructive in the sense that we still have to solve the sub-initial value problem in ( 2) for each j. Suppose Copyright © 2005 Marcel Dekker, Inc.

4

CONTROL AND BOUNDARY ANALYSIS

that we discretize this sub-problem using just one step of the backward Euler scheme. The resulting scheme reads as follows: For n ≥ 0, assuming that ϕn is known, we compute ϕn+1 via ϕn+j/J − ϕn+(j−1)/J + Aj (ϕn+j/J ) = 0, τ

(3)

for j = 1, ..., J, with ϕ0 = ϕ0 . Scheme ( 3) is known as the MarchukYanenko scheme (see, e.g., refs. [10](Chapter 6) and [17] for more details).

1.3

Time-Discretization of (IVP) by Strang’s Scheme

In order to improve the accuracy of Lie’s scheme, G. Strang suggested a symmetrized variant of scheme (2) (ref. [22]). When applied to nonautonomous problems, in the case where J = 2, we obtain (with tn+1/2 = tn + τ /2) : (4) ϕ0 = ϕ0 ; then, for n ≥ 0, assuming that ϕn is known, we compute ϕn+1 via dϕ + A1 (ϕ, t) = 0 on (tn , tn+1/2 ), dt ϕ(tn ) = ϕn ; ϕn+1/2 = ϕ(tn+1/2 ), dϕ + A2 (ϕ, tn+1/2 ) = 0 on (0, τ ), dt ϕ(0) = ϕn+1/2 ; ϕˆn+1/2 = ϕ(τ ), dϕ + A1 (ϕ, t) = 0 on (tn+1/2 , tn+1 ), dt ϕ(tn+1/2 ) = ϕˆn+1/2 ; ϕn+1 = ϕ(tn+1 ).

(5)

(6)

(7)

If (IVP) is taking place in a ﬁnite dimensional space and if operators A1 and A2 are smooth enough, then ϕ(tn ) − ϕn = O(τ 2 ), the function ϕ being the solution of (IVP).

Remark 4 In order to preserve the second order accuracy of scheme ( 4)-( 7) (assuming it holds) we still have to discretize the initial value problems in ( 5), ( 6), and ( 7) by schemes which are themselves second order accurate (at least); examples of such schemes can be found in, e.g., ref. [10](Chapter 6) (which contains also a discussion of the case J > 2). Copyright © 2005 Marcel Dekker, Inc.

5

Operator-Splitting Methods and Applications

1.4

Application

Applications of operator-splitting are everywhere; indeed, some wellknown methods or algorithms are disguised operator-splitting schemes. Our favorite example in that direction is the following: Suppose that A is a real d×d matrix, symmetric and positive deﬁnite. Ordering the eigenvalues of A as follows: 0 < λ1 ≤ λ2 ≤ ... ≤ λd , our goal is to compute λ1 . We have (with obvious notation) λ1 = minv∈S Av · v, with S = {v|v ∈ IRd , v = 1}.

(8)

The minimization problem in (8) is equivalent to 1 minv∈IRd { Av · v + IS (v)}, 2

(9)

where, in (9), functional IS : IRd → IR ∪ {+∞} is deﬁned as follows: 0, if v ∈ S, IS (v) = +∞, otherwise, implying that IS is the indicator functional of sphere S. Suppose that u is a solution of problem (9); we have then Au + ∂IS (u) = 0,

(10)

∂IS (u) being in (10) a (kind of) generalized gradient of functional IS at u. Next, we associate to the (necessary) optimality system (10) the following initial value problem (ﬂow in the dynamical system terminology): du + Au + ∂IS (u) = 0 on (0, +∞), (11) dt u(0) = u . 0

If we apply the Marchuk-Yanenko scheme (3) to the solution of problem (11) we obtain (12) u0 = u0 , and for n ≥ 0, un being known, un+1/2 − un + Aun+1/2 = 0, τ

(13)

un+1 − un+1/2 + ∂IS (un+1 ) = 0. τ

(14)

Copyright © 2005 Marcel Dekker, Inc.

6

CONTROL AND BOUNDARY ANALYSIS

Relation (13) implies un+1/2 = (I + τ A)−1 un .

(15)

On the other hand, relation (14) can be interpreted as a necessary optimality condition for the following minimization problem: 1 minv∈S { v2 − un+1/2 · v}. 2

(16)

Since v = 1 over S, the solution of problem (16) is given by un+1 =

un+1/2 . un+1/2

(17)

Algorithm (12)-(14) reduces then to (12), (15), and (17), which is nothing but the inverse power method with shift, a well-known algorithm from Numerical Linear Algebra. Clearly, numerical analysts have not been waiting for operator-splitting to compute matrix eigenvalues and eigenvectors; on the other hand, operator-splitting has provided eﬃcient algorithms for the solution of complicated problems from diﬀerential geometry, mechanics, physics, physico-chemistry, etc. Some of these applications will be discussed in Sections 2 and 3.

2. 2.1

Operator-Splitting Methods for the Direct Numerical Simulation of Particulate Flow Generalities. Problem Formulation

It is the (very likely biased) opinion of these authors that the direct numerical simulation of particulate ﬂow has been one of the success stories of operator-splitting methods. Albeit this “story” has been told in several publications (see, e.g., [10](Chapter 8) and [12], and the references therein); owing to its importance, we decided to return to it again (without too many details, due to page limitation). For simplicity, we only consider the one-particle case (see the two above references for the many-particle case). Let Ω be a bounded, connected, and open region of IRd (d = 2 or 3 in applications); the boundary of Ω is denoted by Γ. We suppose that Ω contains: (i) A Newtonian incompressible viscous ﬂuid of density ρf and viscosity µf ; ρf and µf are both positive constants. (ii) A rigid body B of boundary ∂B, mass M, center of mass G, and inertia I at the center of mass (see Figure 1 for additional details). Copyright © 2005 Marcel Dekker, Inc.

7

Operator-Splitting Methods and Applications

G B

B

Ω n Figure 1.

Γ

Visualization of the ﬂow region and of the rigid body.

We suppose that the ﬂuid occupies the region Ω \ B and the distance (∂B(0), Γ) > 0. From now on, x = {xi }di=1 will denote the generic point of IRd , dx = dx1 ...dxd , and ϕ(t) will denote the function x → ϕ(x, t). Assuming that the only external force is gravity, the ﬂuid ﬂow-rigid body motion coupling is modeled by ∂u + (u · ∇)u] − µf ∆u + ∇p = ρf g in ∪t∈(0,T ) Ω \ B(t), (18) ∂t ∇ · u(t) = 0 in Ω \ B(t), ∀t ∈ (0, T ), (19) u(t) = uΓ (t) on Γ, ∀t ∈ (0, T ), with uΓ (t) · ndΓ = 0, (20)

ρf [

Γ

u(0) = u0 in Ω \ B(0) with ∇ · u0 = 0,

(21)

and dG = V, dt dV = M g + RH , M dt d(Iω)/dt = TH , G(0) = G0 , V(0) = V0 , ω(0) = ω0 , B(0) = B0 .

(22) (23) (24) (25)

In relations (18)-(21) and (22)-(25), vector u = {ui }di=1 is the ﬂuid (ﬂow) velocity and p is the pressure; n is the unit normal vector at Γ ∪ ∂B, outward to Ω \ B; u0 and uΓ are given functions; RH and TH denote, respectively, the resultant and the torque of the hydrodynamical forces, namely the forces the ﬂuid exerts on B; we have, actually, −→ σndγ and TH = − RH = − Gx × σndγ; (26) ∂B

Copyright © 2005 Marcel Dekker, Inc.

∂B

8

CONTROL AND BOUNDARY ANALYSIS

V (resp., ω) is the translation (resp., angular) velocity of B. In (26) the stress-tensor σ is deﬁned by σ = 2µf D(u) − pId , with D(v) = 1 t 2 [∇v + (∇v) ]. Concerning the compatibility conditions on ∂B we shall assume that: (i) The forces exerted by the ﬂuid on the solid body cancel those exerted by the solid body on the ﬂuid. (ii) On ∂B the no-slip boundary condition holds, namely −−−→ u(x, t) = V(t) + ω(t) × G(t)x, ∀x ∈ ∂B(t).

(27)

Remark 5 System ( 18)–( 21) (resp., ( 22)–( 25)) is of the incompressible Navier-Stokes (resp., Euler-Newton) type. Also, the above model can be generalized to multiple-particle situations and/or non-Newtonian incompressible viscous ﬂuids. 2 The (local in time) existence of weak solutions for problems such as (18)–(21), and (22)–(25) has been proved in ref. [7], assuming that at t = 0 the particles do not touch each other and do not touch Γ (see also [13], [21]). Concerning the numerical solution of (18)–(21), and (22)– (25), completed by interface conditions, we can divide them, roughly, in two classes, namely: (i) The Arbitrary Lagrange-Euler (ALE) methods; these methods relying on moving meshes are discussed in, e.g., refs. [14], [15], and [18]. (ii) The non-boundary ﬁtted ﬁctitious domain methods; these methods rely on ﬁxed meshes and are discussed in, e.g., [10](Chapter 8) and [12](see also the references therein). These methods seem to enjoy a growing popularity, justifying thus the (brief) discussion hereafter.

Remark 6 Even if theory suggests that collisions never take place in ﬁnite time, near-collisions take place, and, after discretization, “real” collisions may occur. These phenomena can be avoided by introducing well-chosen repulsion potentials reminiscent of those encountered in Molecular Dynamics (see [10](Chapter 8) and [12] for details).

2.2

A Fictitious Domain Formulation

Considering the ﬂuid-rigid body mixture as a unique medium, we are going to derive a ﬁctitious domain-based variational formulation. The Copyright © 2005 Marcel Dekker, Inc.

Operator-Splitting Methods and Applications

9

principle of this derivation is pretty simple; it relies on the following steps (see, e.g., [10] and [12] for details): a. Start from the following global weak formulation (of the virtual power type): ∂u + (u · ∇)u · vdx + 2µf D(u) : D(v)dx ρf ∂t Ω\B(t) Ω\B(t) d(Iω) dV ·Y+ ·θ p∇ · vdx + M − dt dt Ω\B(t) (28) g · vdx + M g · Y, ∀{Y, θ} ∈ IRd × Θ, = ρf Ω\B(t) and ∀v ∈ (H 1 (Ω \ B(t)))d and verif ying v = 0 on Γ, −−−→ v(x) = Y + θ × G(t)x, ∀x ∈ ∂B(t), t ∈ (0, T ) with Θ = IR3 if d = 3, Θ = {{0, 0, θ}|θ ∈ IR} if d = 2, q∇ · u(t)dx = 0, ∀q ∈ L2 (Ω \ B(t)), t ∈ (0, T ), (29) Ω\B(t)

u(t) = uΓ (t) on Γ, t ∈ (0, T ), −−−→ u(x, t) = V(t) + ω(t) × G(t)x, ∀x ∈ ∂B(t), t ∈ (0, T ), dG = V, dt u(x, 0) = u0 (x), ∀x ∈ Ω \ B(0), G(0) = G0 , V(0) = V0 , ω(0) = ω0 , B(0) = B0 .

(30) (31) (32) (33)

b. Fill B with the surrounding ﬂuid.

c. Impose a rigid body motion to the ﬂuid inside B.

d. Modify the global weak formulation (28)–(33) accordingly, taking advantage of the fact that if v is a rigid body motion velocity ﬁeld, then ∇ · v = 0 and D(v) = 0. e. Use a Lagrange multiplier deﬁned over B to force the rigid body motion inside B. Copyright © 2005 Marcel Dekker, Inc.

10

CONTROL AND BOUNDARY ANALYSIS

Assuming that B is made of a homogeneous material of density ρs , the above “program” leads to: ∂u + (u · ∇)u · vdx + 2µf D(u) : D(v)dx ρf Ω ∂t Ω dV d(Iω) − p∇ · vdx + (1 − ρf /ρs ) M ·Y+ ·θ Ω dt dt −−−→ (34) + < λ, v − Y − θ × G(t)x >B(t) = ρf g · vdx Ω ρf 1 (Ω))d × IRd × Θ, t ∈ (0, T ), )M g · Y, ∀{v, Y, θ} ∈ (H +(1 − 0 ρs 3 with Θ = IR if d = 3, Θ = {{0, 0, θ}|θ ∈ IR} if d = 2, q∇ · u(t)dx = 0, ∀q ∈ L2 (Ω), t ∈ (0, T ),

(35)

Ω

(36) u(t) = uΓ (t) on Γ, t ∈ (0, T ), −−−→ < µ, u(t) − V(t) − ω(t) × G(t)x >B(t) = 0, ∀µ ∈ Λ(t), t ∈ (0, T ), (37) dG = V, (38) dt G(0) = G0 , V(0) = V0 , ω(0) = ω0 , B(0) = B0 , u0 (x), ∀x ∈ Ω \ B 0 , (39) u(x, 0) = −−→ u(x, 0) = V0 + ω0 × G0 x, ∀x ∈ B 0 with Λ(t) = (H 1 (B(t)))d . From a theoretical point of view,

a natural choice for < ·, · >B(t) is provided by, e.g., < µ, v >B(t) = B(t) [µ · v + δ2 D(µ) : D(v)]dx, with δ as a characteristic length, the diameter of B, for example. From a practical point of view, when it comes to space discretization, a simple and eﬃcient strategy is the following one (cf. [10](Chapter 8) and [12]): “approximate” Λ(t) by Λh (t) = {µ|µ =

N

µj δ(x − xj ), with µj ∈ IRd , ∀j = 1, ..., N }, (40)

j=1

N and the above pairing by < µ, v >B(t)h = j=1 µj · v(xj ). In (40), x → δ(x − xj ) is the Dirac measure at xj , and the set {xj }N j=1 is the union of two subsets, namely: (i) The set of the points of the velocity grid contained in B(t) and whose distance to ∂B(t) is ≥ ch, h being a space discretization step and c a constant 1. (ii) A set of control points located on ∂B(t) and forming a mesh whose step size is of the order of h. It is clear that, using the above approach, one forces the rigid body motion inside the particle by collocation. Copyright © 2005 Marcel Dekker, Inc.

Operator-Splitting Methods and Applications

2.3

11

Solving Problem (34)–(39) by Operator-Splitting

We do not consider collisions; after (formal) elimination of p and λ, problem (34)–(39) reduces to a dynamical system of the following form dX + Aj (X, t) = 0 on (0, T ), X(0) = X0 , dt J

(41)

j=1

where X = {u, V, ω, G} (or {u, V, Iω, G}). A typical situation will be the one where, with J = 4, operator A1 will be associated to incompressibility, A2 to advection, A3 to diﬀusion, A4 to ﬁctitious domain and body motion; other decompositions are possible as shown in, e.g., [10](Chapter 8) and [12]. Lie’s scheme (2) applies “beautifully” to the solution of the formulation (41) of problem (34)–(39). The resulting method is quite modular implying that diﬀerent space and time approximations can be used to treat the various steps; the only constraint is that two successive steps have to communicate (by projection in general).

2.4

Numerical Experiments

Generalities. The methods described in the above paragraphs have been validated by numerous experiments (see, e.g., [10](Chapters 8 and 9), [12], and [20]). In this article, we shall focus on two test problems, the ﬁrst one involving only one particle, while the second test problem concerns a channel ﬂow with 300 particles. The ﬁctitious domain/operatorsplitting approach has made the solution of both problems (almost) routine, but no later than the mid-nineties, solving such problems was considered a Grand Challenge (actually, “our” ﬁrst systematic attack of such problems was part of a National Science Foundation supported Grand Challenge project). First Test Problem: Lifting of a Ball by a Pressure Driven Flow. Let us denote by Ω a truncated circular cylinder of length L and diameter D; we denote by Γ the boundary of Ω and suppose that: (a) The axis of the cylinder is parallel to the horizontal axis Ox2 . (b) The cylinder has been truncated by the vertical planes x2 = 0 and x2 = L; we denote by Γ1 and Γ2 the two vertical disks limiting Ω. (c) The cylinder is ﬁlled with an incompressible Newtonian viscous ﬂuid of density ρf and viscosity µf ; it contains also a rigid solid ball B of diameter d and density ρs . (d) The only external force is gravity. (e) At t = 0, the ﬂuid and the solid are at rest, the ball lying on the bottom of the cylinder. (f) There exists a pressure drop ∆P between Γ1 and Copyright © 2005 Marcel Dekker, Inc.

12

CONTROL AND BOUNDARY ANALYSIS 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

t=0

−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1

Figure 2.

0.5

0

t=40

229.2 229.4 229.6

229.8 230 230.2

230.4 230.6 230.8

231

1

0.5

0

Position of the ball at t = 0 and 40 (from left to right). The ratio between the distance to the central axis and the radius of the pipe 1 0.9 0.8 0.7 Ratio

0.6 0.5 0.4 0.3 0.2 0.1 0 0

Figure 3.

5

10

15

20 25 time

30

35

40

45

Time variation of the distance of the ball center to the cylinder axis.

Γ2 . (g) The following boundary conditions prevail on Γ : u = 0 on Γ \ (Γ1 ∪ Γ2 ), and u(x, t) = u(x + Le2 , t), ∀x ∈ Γ1 , e2 being the unit vector of axis Ox2 (periodic boundary conditions). If ∆P is large enough, the lift acting on B will be suﬃciently large for the ball to take oﬀ and possibly reach an equilibrium height. This is indeed the phenomenon we intend to simulate. In order to do so, we are going to apply the ﬁctitious domain methodology discussed in Sections 2.2 and 2.3. Actually the above methodology is used twice: (i) To treat the ﬂuid-rigid ball coupling as done in Section 2.2. (ii) To solve all the ˜ containing Ω; we impose u = 0 partial diﬀerential equations in a “box” Ω ˜ in Ω \ Ω via a distributed Lagrange multiplier method very close to the one discussed in [10](Section 41). In the particular problem that we are considering: L = 2, D = ˜ is approximated by Ω ˜h = 1, d = 0.2, ρf = 1, µf = 1, ρs = 1.001, Ω (0, 1 + 4h) × (0, 2) × (0, 1 + 4h), (h being a uniform space-discretization step), the cylinder axis is on the line x1 = 12 + 2h, x3 = 12 + 2h, and ∆P = 320, so that, without the ball, the maximal steady ﬂow speed would have been 10. The mesh size used to compute the velocity ﬁeld (resp., the pressure ) is hv = h = 1/96 (resp., hp = 2h = 1/48), while Copyright © 2005 Marcel Dekker, Inc.

Operator-Splitting Methods and Applications

13

we took 1/1000 for the time-discretization step; with these values the velocity mesh contains approximately 2 × 106 grid-points. The lifting phenomenon has been simulated for t ∈ [0, 40], leading to the following results: (1) The translation speed of the ball averaged during the last ﬁve time units is 6.126. (2) The Reynolds number averaged during the last ﬁve time units is 1.2252. (3) The distance of the ball center to the cylinder axis, also averaged during the last ﬁve time units, is 0.30129. The following have been visualized in Figures 2 to 3, respectively: (1) The position of the ball at time t = 0. (2) The position of the ball at t = 40; it has, essentially, reached its equilibrium height (height of the ball center .2). (3) The variation versus time of the ratio distance of the ball center to the cylinder axis /D; the equilibrium height has been reached around t = 27.5.

Second Test Problem: Motion of 300 Neutrally Buoyant Disks in a Two-Dimensional Horizontal Channel. The second test problems involving 300 particles and a solid volume/ﬂuid volume ratio of the order of 37.86% collisions (or near-collisions) have to be accounted for in the simulations; to do so we have used the methods discussed in, e.g., refs. [10](Chapter 8) and [12]. Another peculiarity of this second test problem is that ρs = ρf for all particles (a neutrally buoyant situation). Indeed, neutrally buoyant models are more delicate to handle than those in the general case, since 1 − ρf /ρs = 0 in (34); however, this diﬃculty can be overcome, as shown in ref. [20]. For this test problem, we have: (a) Ω = (0, 42) × (0, 12). (b) Ω contains the mixture of a Newtonian incompressible viscous ﬂuid of density 1 and viscosity 1 with 300 rigid solid disks of density 1 and radius .45. (c) At time t = 0, ﬂuid and particles are at rest, the particle centers being located at the points of a regular lattice. (d) The mixture is put into motion by a uniform pressure drop of 25/18 per unit length (without the particles the steady ﬂow would be of the Poiseuille type with 25 as maximal ﬂow speed). (e) The boundary conditions are given by u = 0 on (0, 42) × {0, 12} (no-slip boundary condition on the horizontal parts of the boundary), and u(0, x2 , t) = u(42, x2 , t), 0 < x2 < 12, 0 ≤ t ≤ 700 (spaceperiodic boundary conditions). (f) hv = 1/10, hp = 1/5, and the timediscretization step is 1/1000. The particle distribution at t = 70, 120, and 700 has been visualized on Figure 4. These ﬁgures show that, initially, we have the sliding motion of horizontal particle layers (see Figure 4), then after some time, a chaotic ﬂow-motion takes place, the higher particle concentration being along the axis of the channel. The maximal speed is 10.04 at t = 700, implying that the corresponding particle Reynolds number is very close to 9. For more details, and further comCopyright © 2005 Marcel Dekker, Inc.

14

Figure 4.

CONTROL AND BOUNDARY ANALYSIS

Particle distribution at t = 70, 120, and 700 (from top to bottom).

ments and results on pressure driven neutrally buoyant particulate ﬂow in two-dimensional channels, see [10](Chapter 9) and [20].

3. 3.1

An Operator-Splitting Method for the Elliptic Monge-Amp` ere Equations in Two-Dimension Generalities

These last years have been witnessing a surge of interest in MongeAmp`ere equations and their numerical solution. Indeed, beside the important role they play in diﬀerential geometry, these equations occur in the modeling of various phenomena in mechanics, physics, etc. (see [2], [19]) and their “full nonlinearity” (in the sense of Caﬀarelli and Cabr´e; see ref. [1]) presents an interesting challenge to numerical analysts. Our goal, here, is to show that via operator-splitting based methods, and an appropriate reformulation, we can make the solution of these equations almost trivial (at least for two-dimensional Dirichlet problems in the elliptic case). In this article, following [5], [6], we will discuss an augmented Lagrangian approach to the solution of the elliptic DirichletMonge-Amp`ere problem; a least-squares based method is also discussed in ref. [5]. Copyright © 2005 Marcel Dekker, Inc.

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Operator-Splitting Methods and Applications

3.2

Formulation of the Problem

Let Ω be a bounded domain of IR2 ; we denote by Γ the boundary of Ω. The two-dimensional Dirichlet problem for the Monge-Amp`ere equation reads as follows: detD2 ψ = f in Ω, ψ = g on Γ,

(42)

∂2ψ ∂xi ∂xj 1≤i,j≤2 and where f and g are two given functions, with f > 0. Unlike the (closely related) Dirichlet problem for the Laplace operator, problem (42) may have multiple solutions (actually, two at most; cf., e.g., [4](Chapter 4)), and the smoothness of the data does not imply the existence of a smooth solution. Concerning the last property, suppose that Ω = (0, 1) × (0, 1) and consider the special case where (42) is deﬁned by 2 ∂ 2 ψ ∂ 2 ψ ∂ 2 ψ − = 1 in Ω, ψ = 0 on Γ. (43) ∂x21 ∂x22 ∂x1 ∂x2

where, in (42), D2 ψ is the Hessian of ψ, i.e., D2 ψ =

Problem (43) cannot have smooth solutions since, for those solutions, the ∂2ψ ∂2ψ and boundary condition ψ = 0 on Γ implies that the product ∂x21 ∂x22 ∂2ψ the cross-derivative vanish at the boundary, implying in turn ∂x1 ∂x2 that detD2 ψ is strictly less than one in some neighborhood of Γ. The above (non-existence) result is not a consequence of the non-smoothness of Γ, since a similar non-existence property holds if in (43) one replaces the above Ω by the ovoid-shaped domain whose C ∞ -boundary is deﬁned by Γ = ∪4i=1 Γi , with Γ1 = {x|x = {x1 , x2 }, x2 = 0, 0 ≤ x1 ≤ 1}, Γ2 = {x|x = {x1 , x2 }, x1 = 1 + e4−1/x2 (1−x2 ) , 0 ≤ x2 ≤ 1}, Γ3 = {x|x = {x1 , x2 }, x2 = 1, 0 ≤ x1 ≤ 1}, Γ4 = {x|x = {x1 , x2 }, x1 = −e4−1/x2 (1−x2 ) , 0 ≤ x2 ≤ 1}. Actually, for the above two Ωs the nonexistence of solutions for problem (43) follows from the non-strict convexity of these domains.

Remark 7 Suppose that Ω is simply connected; let us deﬁne a vector ∂ψ ∂ψ ,− (= {u1 , u2 }); problem ( 42) valued function u by u = ∂x2 ∂x1 takes then the equivalent formulation det∇u = f in Ω, ∇ · u = 0 in Ω, (44) dg u · n = on Γ, ds Copyright © 2005 Marcel Dekker, Inc.

16

CONTROL AND BOUNDARY ANALYSIS

where, in ( 44), n denotes the outward unit vector normal at Γ, and s a counter-clockwise curvilinear abscissa. Once u is known, one obtains ψ via the solution of the following Poisson-Dirichlet problem: −∆ψ =

∂u2 ∂u1 − in Ω, ψ = g on Γ. ∂x1 ∂x2

Problem ( 44) has clearly an incompressible ﬂuid ﬂow ﬂavor, ψ playing here the role of a stream function. Relations (44) can be used to solve problem (42) but this approach will not be further investigated here.

3.3

An Augmented Lagrangian Approach for the Solution of Problem (42)

Suppose that in (42) f ∈ L1 (Ω) and g ∈ H 3/2 (Γ); it makes sense then to attempt solving problem (42) in H 2 (Ω) by considering it as a (kind of) nonlinear bi-harmonic problem. A way to do so is to consider the following problem from Calculus of Variations: 1 |∆ϕ|2 dx, (45) min ϕ∈Ef g 2 Ω with Ef g = {ϕ|ϕ ∈ Vg , detD2 ϕ = f } and Vg = {ϕ|ϕ ∈ H 2 (Ω), ϕ = g on Γ}. Problem (45) is in turn equivalent to 1 |∆ϕ|2 dx, (46) min {ϕ,q}∈Ef g 2 Ω with Ef g = {{ϕ, q}|ϕ ∈ Vg , q ∈ Q, detq = f, D2 ϕ = q} and Q = {q = {qij }1≤i,j≤2 |q ∈ (L2 (Ω))4 , q12 = q21 }. Following, e.g., refs. [9], [11], and [16], we associate to problem (46): (i) The augmented Lagrangian functional Lr : (H 2 (Ω) × Q) × Q → IR deﬁned, with r > 0, by 1 r 2 2 2 |∆ϕ| dx + |D ϕ − q| dx + µ : (D2 ϕ − q)dx, Lr (ϕ, q; µ) = 2 Ω 2 Ω Ω (47) with S : T = 1≤i,j≤2 sij tij , if S = {sij } and T = {tij }, and (ii) The saddle-point problem F ind {ψ, p; λ} ∈ (Vg × Qf ) × Q such that (48) Lr (ψ, p; µ) ≤ Lr (ψ, p; λ) ≤ Lr (ϕ, q; λ), ∀{ϕ, q; µ} ∈ (Vg × Qf ) × Q, Copyright © 2005 Marcel Dekker, Inc.

Operator-Splitting Methods and Applications

17

with Qf = {q|q ∈ Q, detq = f }. One can easily show that if {ψ, p; λ} is a solution of the saddle-point problem (48), then ψ is solution of the Monge-Amp`ere problem (42), p = D2 ψ, and λ is a Lagrange multiplier associated to relation p − D2 ψ = 0. Concerning the solution of problem (48) we advocate (following, e.g., refs. [9], [11]) the (relatively simple) Douglas-Rachford-Uzawa algorithm below: {ψ −1 , λ0 } is given in Vg × Q;

(49)

for n ≥ 0, assuming that {ψ n−1 , ψ n } is known, solve pn ∈ Qf ; Lr (ψ n−1 , pn ; λn ) ≤ Lr (ψ n−1 , q; λn ), ∀q ∈ Qf , ψ n ∈ Vg ; Lr (ψ n , pn ; λn ) ≤ Lr (ϕ, pn ; λn ), ∀ϕ ∈ Vg , λn+1 = λn + r(D2 ψ n − pn ).

(50) (51) (52)

Algorithm (49)-(52) deserves many comments, among them: Concerning the initialization of algorithm (49)–(52), we advocate λ0 = 0 and ψ −1 as the solution of the following Poisson-Dirichlet problem (53) −∆ψ −1 = f 1/2 in Ω, ψ −1 = g on Γ; the rational for such a choice is given in ref. [5]. Problem (50) can be solved point-wise (in practice at the grid points of a ﬁnite element or ﬁnite diﬀerence mesh). Indeed, (50) reduces, a.e. on Ω, to the solution of a ﬁnite dimensional problem of the following type: r minz { (z12 + z22 + 2z32 ) − bn (x) · z} 2

(54)

with z (= {zi }3i=1 ) ∈ {z|z ∈ IR3 , z1 z2 − z32 = f (x)}. The solution of problem (54) (a generalized eigenvalue problem) is discussed in [5]. Problem (51) reduces to a linear variational problem of the following type: ∆ψ n ∆ϕdx + r D2 ψ n : D2 ϕdx = L (ϕ), n (55) Ω Ω 2 1 n ∀ϕ ∈ V0 (= H (Ω) ∩ H0 ); ψ ∈ Vg , the functional Ln (·) being linear and continuous over V0 . The unique solution of problem (55) can be computed by a conjugate gradient algorithm operating in Vg and V0 equipped with the scalar Copyright © 2005 Marcel Dekker, Inc.

18

CONTROL AND BOUNDARY ANALYSIS

product {v, w} →

∆v∆wdx and the corresponding norm. Such Ω

an algorithm is described in [5]; its most important feature is that for well-chosen ﬁnite element approximations, its discrete variants require no more than the solution of two discrete Poisson-Dirichlet problems at each iteration (see [5] for details). Suppose that problem (42) has no solution in H 2 (Ω), but that neither Vg nor Qf are empty (as it is the case for problem (43)). In that case, we expect the arithmetic divergence of sequence {λn }n and the geometric convergence of sequence {{ψ n , pn }}n to a pair {ψ, p} ∈ Vg × Qf such as {ψ, p} minimizes over Vg × Qf (locally or globally) (56) the f unctional{ϕ, q} → D2 ϕ − q(L2 (Ω))4 . The rational for this prediction is discussed at length in [5], and, indeed, when applying algorithm (49)–(52) to the solution of problem (43), the numerical results show that the above algorithm behaves as expected; these numerical results are reported in Section 3.4 (see also [5], [6]). On the basis of these results it seems that for parameter r well chosen, algorithm (49)–(52) produces either a solution of problem (42) if such a solution exists in H 2 (Ω), or a generalized solution in the sense of (56), i.e., a least-squares solution, if (42) is without solution in H 2 (Ω), while Vg and Qf are non-empty. This result justiﬁes the least-squares methodology discussed in [5]. From a geometrical point of view, problem (42) has solutions in H 2 (Ω) if D2 Vg and Qf (both subsets of space Q) intersect; such a situation has been visualized on Figure 5(a). Figure 5(b) corresponds to a situation where (42) has no solution in H 2 (Ω), but neither Vg nor Qf are empty; a generalized solution, in the sense of (56) has been visualized on this ﬁgure. The saddle-point formulation (48) of problem (42) is a mixed variational formulation where the “nonlinearity burden” has been transferred from ψ to p, making it purely algebraic. Actually, this approach (this is even truer for the discrete analogues of problems (42) and (48); see [5] for details) provides a solution method where instead of solving (42) directly (i.e., without introducing additional functions, like p), we solve it via its associated Pfaﬀ system (see, e.g., [8](Chapter A)), namely: dψ − u1 dx1 − u2 dx2 = 0 in Ω, Copyright © 2005 Marcel Dekker, Inc.

(57)

19

Operator-Splitting Methods and Applications

D2Vg

Qf

D2 Vg

Qf D2ψ

p= D 2 ψ

Q

Qf

Qf p

Q

Figure 5. (a). Problem (42) has a solution in H 2 (Ω) (left). (b). Problem (42) has no solution in H 2 (Ω) (right).

du1 − p11 dx1 − p12 dx2 = 0 in Ω, du2 − p12 dx1 − p22 dx2 = 0 in Ω, p11 p22 − p212 = f,

(58) (59) (60)

completed by the boundary condition ψ = g on Γ. System (57)– (60) provides clearly a mixed formulation of problem (42). Let us return to problem (IVP) of Section 1.1 and suppose that A = A1 + A2 . An alternative to Lie’s and Strang’s schemes is provided by the following (ﬁrst order accurate) Douglas-Rachford scheme: Given ϕ0 = ϕ0 ; then, for n ≥ 0, ϕn being known, solve (ϕn+1/2 − ϕn )/τ + A1 (ϕn+1/2 ) + A2 (ϕn ) = 0, (ϕn+1 − ϕn )/τ + A1 (ϕn+1/2 ) + A2 (ϕn+1 ) = 0.

(61) (62)

The basic properties of scheme (61)–(62), can be found in, e.g., [10](Chapter 2) (see also the references therein). It is shown in, e.g., ref. [11](see also [9]) that algorithms such as (49)–(52) are in fact “disguised” Douglas-Rachford algorithms, with parameter r the reciprocal of a time step (see the two above references for details). The ﬁnite element implementation of algorithm (49)–(52) is discussed in ref. [5]; it relies on a mixed ﬁnite element approximation of problems (42) and (48) where ψ and the four components of the tensor-valued functions p and λ are approximated by functions belonging to the ﬁnite dimensional space Vh deﬁned by Vh = {ϕ|ϕ ∈ C 0 (Ω), ϕ|T ∈ P1 , ∀T ∈ Th }, Copyright © 2005 Marcel Dekker, Inc.

(63)

20

CONTROL AND BOUNDARY ANALYSIS

x2

x1 Figure 6.

A uniform triangulation of Ω (h = 1/4).

where in (63): P1 is the space of the two-variable polynomials of degree ≤ 1, Th is a triangulation of Ω, and h is a space discretization step. The numerical results shown in Section 3.4 have been obtained using the above approximation.

3.4

Numerical Results

We are going to apply the methodology (brieﬂy) discussed in Section 3.3 to the solution of three test problems. For all these test problems, we shall assume that Ω = (0, 1)×(0, 1) and that Th is a uniform triangulation like the one in Figure 6, with h the length of the edges of Th adjacent to the right angles. The ﬁrst test problem is deﬁned as follows: (i) f (x) =

√ R2 , ∀x ∈ Ω, with R ≥ 2 and |x| = (x21 + x22 )1/2 . (R2 − |x|2 )2

(ii) g(x) = (R2 − |x|2 )1/2 , ∀x ∈ Γ (= ∂Ω). If the above data prevail, function ψ given by ψ(x) = (R2 − |x|2 )1/2

(64)

is a solution to the Monge-Amp`ere problem (42). The graph of function ψ is clearly a piece of the sphere of radius R centered at √ {0, 0, 0}. If √ R > 2 we have ψ ∈ C ∞ (Ω); on the other hand, if R = 2 we have no better than ψ ∈ W 1,p (Ω) with p ∈ [1, 4), implying that in that particular case, ψ does not have the H 2 -regularity. When applying the computational methods discussed in Section 3.3 to the solution of the above problem (with r = 1 in algorithm (49)–(52)), we obtain if R = 2, Copyright © 2005 Marcel Dekker, Inc.

21

Operator-Splitting Methods and Applications Table 1.

Results for the First Test Problem (R = 2) h 1/32 1/64 1/128

Table 2.

ψhc − ψL2 (Ω) 4.45 × 10−6 1.14 × 10−6 2.97 × 10−7

D2h ψhc − pch (L2 (Ω))4 9.48 × 10−7 1.35 × 10−6 1.58 × 10−6

Results for the First Test Problem (R = h 1/32 1/64 1/128

ψhc − ψL2 (Ω) 2.20 × 10−5 5.51 × 10−6 1.37 × 10−6

√

2 + 10−1 )

D2h ψhc − pch (L2 (Ω))4 9.68 × 10−7 1.54 × 10−6 2.04 × 10−6

and after 78 iterations of the discrete variant of the above algorithm, the results displayed in Table 1. In Table 1, ψhc is the computed approximate solution, D2h ψhc is the corresponding discrete Hessian, and pch is the computed approximation of tensor p. The above results strongly suggest second order accuracy (a textbook one, indeed), which is in some sense optimal considering √ the type of ﬁnite element approximations we are using. If we take R = 2, our methodology which has been designed to solve the Monge-Amp`ere problem (42) in H 2 (Ω) is unable to capture any solution of the above problem, the corresponding algorithm (49)–(52) being divergent for any √ value of r. The same troubles persist if one takes R = 2 + 10−2 ; on √ the other hand, if one takes R = 2 + 10−1 , things are back to normal since, using again r = 1, we obtain after 117 iterations of algorithm (49)–(52) the results summarized in Table 2. The above results show that second order accuracy√holds. However, the second order derivatives of ψ being larger for R = 2 + 10−1 than for R = 2, the corresponding approximations errors are also larger. On Figures 7 and 8 we have visualized, respectively: √ (i) The graph of ψ when R = 2; the singularity of ∇ψ at {1, 1} appears clearly on Figure 7(a). (ii) The graph of ψhc corresponding to h = 1/128 and R = 2. √ (iii) The graph of ψhc corresponding to h = 1/128 and R = 2 + 10−1 . √ (iv) The graph of f when R = 2 + 10−1 .

Copyright © 2005 Marcel Dekker, Inc.

22

CONTROL AND BOUNDARY ANALYSIS Computed spherical surface, R=2

Spherical surface solution, R=sqrt(2) 1.5 1 0.5 0 0

0.2

0.4

0.6

0.8

10

1 0.6 0.8 0.2 0.4

2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0

0.2

0.4

0.6

0.8

Figure 7. (a). First test problem: graph of ψ when R = problem: graph of ψhc when R = 2 and h = 1/128 (right).

Computed spherical solution, R=sqrt(2)+.1 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 1 0 0.2 0.6 0.8 0.4 0.6 0.8 1 0 0.2 0.4

1 0

√

0.2

0.4

0.6

0.8

1

2 (left). (b). First test

f, R=sqrt(2)+.1 30 25 20 15 10 5 0 0

0.2

0.4

0.6

0.8

1 0

0.2

0.4

0.6

0.8

1

√ Figure 8. (a). First test problem: graph of ψhc when√R = 2 + 10−1 and h = 1/128 (left). (b). First test problem: graph of f when R = 2 + 10−1 (right).

Remark 8 When computing the approximate solutions for h = 1/32, we stopped the iterations of algorithm ( 49)–( 52) as soon as D2h ψhn − pnh (L2 (Ω))4 ≤ 10−6 .

(65)

The √ corresponding number of iterations is 78 for R = 2, and 117 for R = 2 + 10−1 (we did not try to ﬁnd the optimal value of r, or to use a variable r strategy). Next, when computing the approximate solutions for h = 1/64 and 1/128, we stopped iterating once the iteration numbers associated to h = 1/32 were reached (actually, using ( 65) as stopping criteria for h = 1/64 and 1/ 128 did not change much the approximation errors shown in Tables 1 and 2). 2 The second test problem is deﬁned as follows: (i) f (x) =

(2|x|)3/2 1 , ∀x ∈ Ω. (ii) g(x) = , ∀x ∈ Γ. |x| 3

Copyright © 2005 Marcel Dekker, Inc.

23

Operator-Splitting Methods and Applications Table 3.

Results for the Second Test Problem h 1/32 1/64 1/128

D2h ψhc − pch (L2 (Ω))4 9.91 × 10−7 1.60 × 10−6 2.02 × 10−6

ψhc − ψL2 (Ω) 5.56 × 10−5 1.50 × 10−5 3.94 × 10−5

Test Problem 2

1000 900 800 700 600 500 400 300 200 100 0 1 0.8 0.6 0.4

1.5 1 0.5 0 1 0.2 0 0

1 0.6 0.8 0.2 0.4

0.8

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

Figure 9. (a). Second test problem: graph of f (left). (b). Second test problem: graph of ψhc (h = 1/128) (right).

With these data, a solution to the Monge-Amp`ere problem (42) is the function ψ deﬁned by ψ(x) =

(2|x|)3/2 , ∀x ∈ Ω. 3

(66)

One can easily check that ψ does not belong to C 2 (Ω); however, since ψ ∈ W 2,p (Ω) for all p ∈ [1, 4), it has, in principle, enough regularity so that we can apply algorithm (49)–(52) to the solution of the corresponding problem (42). Indeed, despite the singularity of function f at {0, 0} (see Figure 9(a)), algorithm (49)–(52), with r = 1, provides after 160 iterations the results summarized in Table 3. From the results in Table 3 we can infer that second order accuracy still holds. On Figure 9(a) (resp., 9(b)) we have visualized function f (resp., the computed approximate solution obtained with h = 1/128). The third test problem is—in some sense—the more interesting since we consider this time the solution of problem (43), namely: 2 ∂ 2 ψ ∂ 2 ψ ∂ 2 ψ − = 1 in Ω, ψ = 0 on Γ. ∂x21 ∂x22 ∂x1 ∂x2 Despite the smoothness of its data, the above problem has no smooth solution, the troubles coming from the non-strict convexity of Ω = Copyright © 2005 Marcel Dekker, Inc.

24

CONTROL AND BOUNDARY ANALYSIS

(0, 1) × (0, 1). When applying algorithm (49)–(52) (in fact a discrete variant of it) to the solution of problem (43) we observe the following phenomena: (i) For r suﬃciently small (r = 1/10 here) sequence {ψhn , pnh }n≥0 converges geometrically (albeit slowly) to a limit {ψhc , pch }, while sequence {λnh }n≥0 diverges arithmetically. (ii) A close inspection of the numerical results shows that the curvature of the graph of ψhc becomes negative close to the corners, in violation of the Monge-Amp`ere equation; actually, as expected, it is violated also along the boundary, since D2h ψhc − pch (L2 (Ω))4 = 1.8 × 10−2 if h = 1/32, 3.3 × 10−2 if h = 1/64, 4.2 × 10−2 if h = 1/128, while D2h ψhc − pch (L2 (Ω1 ))4 = 2.7 × 10−4 if h = 1/32, 4.1 × 10−4 if h = 1/64, 4.9 × 10−4 if h = 1/128, and D2h ψhc − pch (L2 (Ω2 ))4 = 4.4 × 10−5 if h = 1/32, 4.9 × 10−5 if h = 1/64, 5.1 × 10−5 if h = 1/128, where Ω1 = (1/8, 7/8) × (1/8, 7/8) and Ω2 = (1/4, 3/4) × (1/4, 3/4). These results suggest that detD2 ψ = 1 is “almost” veriﬁed in Ω2 . The graph of ψhc obtained with h = 1/64 has been shown in Figure 10, while the intersections of this graph with the planes x1 = 12 and x1 = x2 have been shown in Figures 11(a) and 11(b), respectively, for h = 1/32, 1/64, and 1/128. Since ψhc does not vary much with h, we suspect that, according to Section 3.3, what we have here is a (good) approximation of one of those functions of H 2 (Ω) ∩ H01 (Ω) whose Hessian is at a minimal L2 -distance (global or local) from the set Qf deﬁned in Section 3.3. Assuming that the above is true we can claim that the solutionless problem (43) has been solved in a least-squares sense in the functional space H 2 (Ω), leading to a (not so novel in general, but possibly new in the Monge-Amp`ere “environment”) concept of generalized solution.

Remark 9 When applying algorithm ( 49)–( 52) to the solution of problem ( 42) we have to solve at each iteration a discrete analogue of the linear variational problem ( 55); to solve this ﬁnite dimensional problem, we have used a preconditioned conjugate gradient algorithm, converging typically in 6 to 8 iterations, the preconditioning requiring the solution of two discrete Poisson-Dirichlet problems per conjugate gradient iteration. The mesh being uniform we have used Fast Poisson Solvers to achieve preconditioning. Copyright © 2005 Marcel Dekker, Inc.

25

Operator-Splitting Methods and Applications Test Problem 3 0.25 0.2 0.15 0.1 0.05 0 1

Figure 10.

0.8

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

Third test problem: graph of the computed solution (h = 1/64).

cross sections

0.25 0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

0

diagonal cross sections

0.25

0 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.5

1

1.5

Figure 11. (a). Third test problem: graph of the computed solution restricted to the line x1 = 1/2 (h=1/32, 1/64, and 1/128) (left). (b). Third test problem: graph of the computed solution restricted to the line x1 = x2 (h = 1/32, 1/64, and 1/128) (right).

Acknowledgments The authors would like to thank J.D. Benamou, Y. Brennier, L.A. Caffarelli, B. Dacorogna, J. Cagnol, G.P. Galdi, D.D. Joseph, P.-L. Lions, P. Muscarello, and J.-P. Zol´esio for assistance and helpful comments and suggestions. The support of NSF (grants ECS-9527123, CTS-9873236, DMS-9902035, DMS-0209066, DMS-0412267) and DOE / LACSI (grant R71700K-292-000-99) is also acknowledged.

References ´ Fully Nonlinear Elliptic Equations. Amer[1] L. A. CAFFARELLI and X. CABRE. ican Math. Society, Providence, RI, 1995.

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CONTROL AND BOUNDARY ANALYSIS

[2] L.A. CAFFARELLI and eds. MILMAN, M. Monge-Amp`ere Equation: Application to Geometry and Optimization. American Math. Society, Providence, RI, 1999. [3] A.J. CHORIN, T.J.R. HUGHES, M.F. McCRACKEN, and J.E. MARSDEN. Product formulas and numerical algorithms. Comm. Pure and Appl. Math., 31:205–256, 1978. [4] R. COURANT and D. HILBERT. Methods of Mathematical Physics, Vol. II. Wiley Interscience, New York, NY, 1989. [5] E.J. DEAN and R. GLOWINSKI. Numerical methods for fully nonlinear elliptic equations of the Monge-Amp`ere type. (in preparation). [6] E.J. DEAN and R. GLOWINSKI. Numerical solution of the two-dimensional elliptic Monge-Amp`ere equation with Dirichlet boundary conditions: an augmented Lagrangian approach. C. R. Acad. Sci. Paris, Mathematiques, S´ er. I, 336:779–784, 2003. [7] B. DESJARDINS and M.J. ESTEBAN. On weak solution for ﬂuid-rigid structure interaction: compressible and incompressible models. Arch. Rat. Mech. Anal., 146:59–71, 1999. [8] J. DIEUDONNE. Panorama des Math´ematiques Pures: Le Choix Bourbachique. Editions Jacques Gabay, Paris, 2003. [9] M. FORTIN and R. GLOWINSKI. Augmented Lagrangians Methods: Application to the Numerical Solution of Boundary-Value Problems. North-Holland, Amsterdam, 1983. [10] R. GLOWINSKI. Finite Element Methods for Incompressible Viscous Flow, volume IX of Handbook of Numerical Analysis, P.G. Ciarlet and J.L. Lions eds. North-Holland, Amsterdam, 2003. [11] R. GLOWINSKI and P. LE TALLEC. Augmented Lagrangians and Operator Splitting Methods in Nonlinear Mechanics. SIAM, Philadelphia, PA, 1989. [12] R. GLOWINSKI, T.W. PAN, T.I. HESLA, D.D. JOSEPH, and J. PERIAUX. A ﬁctitious domain approach to the direct numerical simulation of incompressible viscous ﬂuid ﬂow past moving rigid bodies: Application to particulate ﬂow. J. Comput. Phys., 169:363–426, 2001. [13] C. GRANDMONT and Y. MADAY. Existence for an unsteady ﬂuid-structure interaction problem. Math. Model. Num. Anal., 34:609–636, 2000. [14] H.H. HU, N.A. PATANKAR, and M.Y. ZHU. Direct numerical simulation of ﬂuid-solid systems using Arbitrary Lagrangian-Eulerian techniques. J. Comput. Phys., 169:427–462, 2001. [15] A. JOHNSON and T. TEZDUYAR. 3-d simulations of ﬂuid-particle interactions with the number of particles reaching 100. Comput. Methods Mech. Engrg., 145:301–321, 1997. [16] P. LE TALLEC. Numerical Methods for Nonlinear Three-Dimensional Elasticity, volume III of Handbook of Numerical Analysis, P.G. Ciarlet and J.L. Lions eds. North-Holland, Amsterdam, 1994. [17] G.I. MARCHUK. Splitting and alternating direction methods, volume I of Handbook of Numerical Analysis, P.G. Ciarlet and J.L. Lions eds. North-Holland, Amsterdam, 1990.

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Operator-Splitting Methods and Applications

27

[18] B. MAURY. Direct simulation of 2-d ﬂuid-particle ﬂows in biperiodic domains. J. Comp. Phys., 156:325–351, 1999. [19] J.R. OCKENDON, S. HOWISON, A. LACEY, and A. MOVCHAN. Applied Partial Diﬀerential Equations. Oxford University Press, Oxford, UK, 1999. [20] T.W. PAN and R. GLOWINSKI. Direct simulation of the motion of neutrally buoyant circular cylinders in plane Poiseuille ﬂow. J. Comp.Phys., 181:260–279, 2002. [21] J.A. SAN MARTIN, V. STAROVOITOV, and M. TUCSNAK. Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous ﬂuid. Arch. Ration. Mech. Anal., 161:113–147, 2002. [22] G. STRANG. On the construction and comparison of diﬀerence schemes. SIAM J. Num. Anal., 5:506–517, 1968.

Copyright © 2005 Marcel Dekker, Inc.

CHAPTER II DYNAMICAL SHAPE SENSITIVITY Application to a Coupled Fluid-Structure System Marwan Moubachir Institut National de Recherche en Informatique et Automatique Projet OPALE Sophia Antipolis, France

Jean-Paul Zol´esio CNRS and INRIA Projet OPALE Sophia Antipolis, France

Abstract

Nowadays the design of systems involving the interaction of a ﬂuid with a moving interface is part of many industrial processes, from aeronautics to biomechanics. This interface is usually the boundary between the ﬂuid and a material with diﬀerent physical and rheological properties, such as an elastic solid, a gas, or a diﬀerent ﬂuid phase. Up to now, the associated control and inverse problems have been treated by neglecting the motion of the interface, resulting in the analysis of classical optimization problems for partial diﬀerential equations in cylindrical time-space domain endowed with transpiration boundary conditions on the ﬁxed interface. However, the search for performances and robustness and the need to take into account extreme running conditions make this approximation too restrictive. This article introduces several tools that may be useful while studying the optimization problem of industrial devices involving large interface displacements. The keystone of this analysis is the generalization of cylindrical shape optimization tools to the non-cylindrical case. These tools are illustrated by showing how to compute the cost function gradient involved in an inverse problem for a coupled system consisting of an elastic solid in rigid motion surrounded by a Navier-Stokes ﬂow.

Keywords: Shape optimization, moving domain, Navier-Stokes, ﬂuid-solid interaction, optimal control, inverse problem

Copyright © 2005 Marcel Dekker, Inc.

30

1.

CONTROL AND BOUNDARY ANALYSIS

Introduction

This article deals with the analysis of an inverse boundary problem for a coupled ﬂuid-structure system which enters as a generic tool in the design and the control of many industrial devices, such as aircraft wings, cable-stayed bridges, automobile shapes, satellite reservoir tanks, and more generally, systems involving ﬂuid-solid interactions. This article is focused on the derivation of optimality conditions associated with the minimization problem involved in a typical open-loop control strategy. Our aeroelastic system consists of an elastically supported rigid solid moving inside an incompressible ﬂuid ﬂow in 2-D. It is described by a non-cylindrical system of partial diﬀerential equations, where the evolution of the moving boundary is unknown [5]. Using a new methodology to handle non-cylindrical sensitivity analysis introduced in [6, 3], we are able to perform the sensitivity analysis of this coupled system where the evolution of the moving boundary is governed by an ordinary diﬀerential equation. In the ﬁrst section, we introduce the reader to the coupled ﬂuidstructure mechanical system. Then, we set up the inverse problem framework and state the main result of this article, which consists of furnishing the expression of the gradient of a speciﬁc cost function depending on the dynamical evolution of the solid, with respect to the far-ﬁeld boundary condition considered as a control variable. The keystone of its proof lies in the Navier-Stokes dynamical shape sensitivity analysis introduced in the last section.

2.

Mechanical Problem

We consider a two-dimensional ﬂexible solid in rigid motion. For the sake of simplicity, we only consider one degree of freedom for the structural motion: the vertical displacement d(t)e2 where e2 is the element of the cartesian basis (e1 , e2 , e3 ) in R3 . The structure is surrounded by a viscous ﬂuid in the plane (e1 , e2 ). We consider a control volume Ω = Ωft Ωst ⊂ R2 with boundary Γ∞ , containing the solid Ωst with boundary Γst for every time t ∈ (0, T ). We introduce a diﬀeomorphic map Tt sending a ﬁxed reference domain Ω0 = Ωf0 Ωs0 into the physical conﬁguration Ω at time t ≥ 0. This map is built thanks to the Lagrangian ﬂow of Eulerian velocity ﬁeld V deﬁned as follows: ˙ x ∈ Ωst V (x, t) = d(t)e 2, f ˙ V (x, t) = Ext(d(t)e 2 ), x ∈ Ωt V (x, t) · n = 0, x ∈ Γ∞ Copyright © 2005 Marcel Dekker, Inc.

(1)

31

Dynamical Shape Sensitivity

with Ext is an arbitrary extension operator from Γst into Ωft . The solid is described by the evolution of its displacement d e2 and its ˙ is solution of the following ordinary velocity d˙ e2 and the couple (d, d) second order diﬀerential equation: ¨ k d = Ff , m d + (2) d, d˙ (t = 0) = [d0 , d1 ] where (m, k) stand for the structural mass and stiﬀness. The right-hand term Ff is the projection of the ﬂuid loads on Γst along the motion direction e2 . We consider a viscous incompressible Newtonian ﬂuid. Its evolution is described by its velocity u and its pressure p. The couple (u, p) satisﬁes the classical Navier-Stokes equations written in non-conservative form: ∂t u + D u · u − ν∆u + ∇p = 0, Qf (V ) div(u) = 0, Qf (V ) (3) u = u∞ , Σ∞ u(t = 0) = u0 , Ωf0 where ν stands for the kinematic viscosity, u∞ is the far-ﬁeld velocity ﬁeld and def def Qf (V ) = {t} × Ωft , Σ∞ = (0, T ) × Γ∞ 0 0 are constant. (ii) The thickness h of the shell is small enough to accomodate the curvatures H and K, i.e., the product of the thickness by the curvatures is small as compared to 1. As a consequence we shall drop terms of order equal to or greater than 2 in the series expansions with respect to the radial variable. We also suppose that j(z) = det(DTz ) = det(I − zD 2 b) = 1. (iii) (Kirchhoﬀ Hypothesis) Let T be a transformation of the shell Sh , and let e = (eΓ , w) be the corresponding transformation of the midsurface. In the classical thin plate theory named after Kirchhoﬀ, the displacement vectors T and e ◦ p are related by the hypothesis that the ﬁlaments of the plate initially perpendicular to the middle surface remain straight and perpendicular to the deformed surface, and undergo neither contraction nor extension. We may generalize this hypothesis to the case of a shell using the intrinsic geometry, yielding T = e ◦ p − b ( ∗ DΓ e ∇b) ◦ p (12) The equations given below correspond to the behavior of the midsurface Γ0 . On Γ0 , w is the normal displacement of the shell, and eΓ is a vector describing the tangential displacements. Both of these quantities are speciﬁed in local coordinates. Let ρsh be the density of the shell, and λ and µ be the Lam´e coeﬃcients. Let the function b, and the matrices built on that function, depend on the geometry of the shell. The curvature is given by K and H, and it is also taken into account in the matrix D2 b. Deﬁne the following operator C acting on a matrix A: C(A) = λ tr (A) I + 2µ A

(13)

χ ˜ = CΓ eΓ − εΓ (D 2 b eΓ )

(14)

the expression χ ˜ and β = (λ + 2µ)−1 . Next, deﬁne CΓ and SΓ as CΓ u = SΓ w = Copyright © 2005 Marcel Dekker, Inc.

1 2 ∗ (D b DΓ u + DΓ u D 2 b) 2 1 2 (D w + ∗ DΓ2 w) 2 Γ

(15a) (15b)

44

CONTROL AND BOUNDARY ANALYSIS

CΓ is a ﬁrst-order operator, that in practice operates on a tangential vector u. SΓ is the symmetrization of the Hessian matrix of a scalar function w (the Hessian matrix is not symmetric in the tangential calculus [9]). Then, given appropriate initial conditions, the displacement (w, eΓ ) of the shell satisﬁes the following system of shell equations: ∂tt w − γ∆Γ ∂tt w + ∆Γ 2 w +

γ divΓ (D 2 b ∂tt eΓ ) 2

+P1 (eΓ ) + Q1 (w) = 0

(16a)

(I + γ(D 2 b)2 )∂tt eΓ − β[γ −1 divΓ C(εΓ (eΓ )) + D2 b divΓ C(χ) ˜ γ ˜ − D2 b∇Γ ∂tt w + P2 (w) + Q2 (eΓ ) = 0 (16b) −divΓ (D 2 b C(χ))] 2 where P1 denotes coupling terms and Q1 denotes lower order terms in the plate equation; and P2 , Q2 in the wave equation: (17a) P1 (eΓ ) = β[2λHγ −1 divΓ eΓ + 2µγ −1 tr (D 2 b εΓ (eΓ )) 3 2 +2µdivΓ divΓ (χ) ˜ + 4µH tr (D beΓ D b) −λ∆Γ 2∇Γ H, eΓ − λ(4H 2 − 2K) 2∇Γ H, eΓ −2µ (D 2 b)2 ..D3 b, eΓ Q1 (w) = β[kγ w + 4µdivΓ ((D 2 b)2 ∇Γ w) + λ∆Γ ((4H 2 − 2K)w) (17b) +2µdivΓ divΓ ((D2 b)2 w) + 2µdivΓ (K∇Γ w) +λ(4H 2 − 2K)∆Γ w] + 2µ tr (SΓ w(D 2 b)2 ) +4µH tr ((D 2 b)3 w) P2 (w) = β[−2λγ −1 ∇Γ (Hw) + 2µγ −1 divΓ (w D2 b) (17c) 2 2 2 3 +λ2∇Γ H(∆Γ w − (4H − 2K)w) − 2µ(D b) ..D bw] −2µdivΓ (D 2 b SΓ w) + 2µD2 b divΓ (SΓ w) Q2 (eΓ ) = −2βµγ −1 (K eΓ + 2(D2 b)2 eΓ )

(17d)

On ∂Γ0 × (0, T ), which is clamped, one has the boundary conditions: ∂w =0 (18) ∂ν For ease of computation, whenever possible we replace equations (16)– (18) with the following abstract form: eΓ = 0,

w = 0,

M∂tt e + Ae = 0 Copyright © 2005 Marcel Dekker, Inc.

on

Γ0 × (0, T )

(19a)

45

Optimal Control of a Structural Acoustic Model

eΓ = 0,

w=

∂w = 0 on ∂ν

∂Γ0 × (0, T )

(19b)

where the vector e = (eΓ , w). The deﬁnition of the operators M and A can be seen by inspection of (16). The operators M and A are positive, self-adjoint, densely deﬁned, and closable. A is bounded and 2 coercive on H01 (Γ0 ) × H02 (Γ0 ) and M is bounded and coercive on [L2 (Γ0 )]2 × H01 (Γ0 ) [5].

6.

Structurally Damped, Forced Shell Wall of an Acoustic Chamber

Next, we use the basic operators deﬁned above in analysis of the control problem. In order to model damping in the shell, we account for KelvinVoight type structural damping. In other words, we add the potential energy operator A acting on the time derivative of displacement, ∂t e. It is necessary to take into account the action of the piezoceramic actuators on the shell. This will take the form of a speciﬁc operator B with a control input u. We consider N patches perfectly bonded to the shell Γ0 as shown on Figure 1. The vector u will be composed of the N scalar inputs ui where ui is the input voltage at patch i. Following [12] and references therein, when a voltage is applied along the polarization of a piezoelectric patch, this element develops a compressive strain proportional to the voltage applied. We here consider a generic unconstrained 2D-actuator, polarized along the normal direction, numered i0 . We will denote as Ξ the area included on the mid-surface, above which the patches are located, while ∂Ξ denotes the closed curves which delineate this area. The operator B is given by (Bu, e)Γ0 = (B1 u, eΓ0 ) + (B2 u, w) N 1 uj [αj eΓ0 + βj ∇Γ w, ∇b] dΞj , for e ∈ D(A 2 ) (20) = j=1

Ξj

where the coeﬃcients αj and βj are based on the material properties of the patches. Then B : [L2 (Ξi )]N i=1 ≡ U → D(A) , and therefore A−r B ∈ L(U; [L2 (Γ0 )]3 ) for

3 1 0: |eAt B|L(U ;H) ≤ with

3 8

CT tr

for

0 0, we have that | |T (e)|L

1/2 2 (0,T ;D(Mγ ))

≤ C|M−1/2 (∂t z|Γ0 ∇b)|L2 (0,T ;L2 (Γ0 )) γ

≤ C|(∂t z|Γ0 )|L2 (0,T ;H −1/3 (Γ0 )) ≤ C|∂t w|L2 (0,T ;H 1/3 (Γ0 )) ≤ C|e|L2 (0,T ;He ) (37) Since −Ash generates an analytic semigroup on He , for f ∈ L2 (0, T ; He ) we immediately have the estimate t −A (t−s) sh e f (s) ds ≤ C|f |L2 (0,T ;He ) (38) 1/2− 0

C([0,T ];D(Ash

))

Combining (37) and (38) gives, for arbitrary v ∈ L2 (0, T ; He ), that |Y (e)|C([0,T ];D(A1/2− )) ≤ C|T (e)|L2 (0,T ;He ) ≤ C|e|L2 (0,T ;He ) sh

(39)

which proves the ﬁrst part of Proposition 6. For the second, note that the estimate (39) implies that |Y (e)|L2 (0,T ;He ) ≤ CT |e|L2 (0,T ;He )

(40)

where CT → 0 as T → 0. This allows for the bounded invertibility of I −Y on a small interval (0, T0 ), and repeatedly using this estimate gives the estimate for arbitrary T > 0 after ﬁnitely many steps. Step 2: From the regularity shown in (21) we have

We will show that

|Bu|D(Ar ) = |Ar Bu|U ≤ C|u|U

(41)

0 M−1 Bu

(42)

D(Arsh )

≤ C|u|U

We take φ = (φ1 , φ2 ) ∈ D(Arsh ) and compute with ﬁxed u: 0 ,φ = (M−1 Bu, φ2 ) 1 = (Bu, φ2 )[L2 (Γ )]3 0 D(M 2 ) M−1 Bu H e

= (A−r Bu, Ar φ2 )[L2 (Γ0 )]3 = C|Ar φ2 |[L2 (Γ0 )]3 |u|U ≤ C|Arsh φ|He |u|U (43) Copyright © 2005 Marcel Dekker, Inc.

50

CONTROL AND BOUNDARY ANALYSIS

where we have used (37) and the regularity (21). This implies the estimate (42). Additionally, we can use (42), analyticity of e−Ash t and Lemma 4 to give, for a ﬁxed u, 2 T −A t 0 e sh dt ≤ C|u|2U (44a) −1 M Bu D(A1/2−r ) 0 sh −A t 0 ≤ C |u|U , for 0 < t ≤ T e sh (44b) −1 M Bu H tr e Step 3: Denote the vector

z ∂t z At e = e Bu ∂t e

(45)

Then, (z, ∂t z) satisﬁes the wave equation (35) and e is given by 0 −Ash (t) e(t) = e + Y (e)(t) M−1 Bu Thus, −Ash (t)

[I − Y ]e(t) = e

(46)

0

(47)

M−1 Bu

Applying Proposition 6 (ii) and (44) leads to

T 0

|e(t)|2He

≤ C|[I − Y ≤C 0

T

]−1 |2L(L2 (0,T ;He ) e−Ash (t)

−A (t) e sh

2 −1 M Bu L 0

2 (0,T ;He )

2 0 dt ≤ C|u|2U M−1 Bu D(A1/2−r )

(48)

sh

1 if we take 38 < r ≤ where we note that 12 − r ≥ 12 following a priori regularity of the variable e:

5 12 .

Thus we have the

|e|L2 (0,T ;He ) ≤ C|u|U

e ∈ L2 (0, T ; He ),

(49)

Step 4: We can now use the regularity (49) and Lemma 5 which gives the following regularity of the wave component z = (z, ∂t z) in (45): |z|C([0,T ];Hz ) ≤ C|u|U

(50)

Now, using Proposition 6 gives 1/2−ε

Y e ∈ C([0, T ]; D(Ash Copyright © 2005 Marcel Dekker, Inc.

)) ⊂ C([0, T ]; He )

(51)

Optimal Control of a Structural Acoustic Model

|Y e|C([0,T ];He ) ≤ C|u|U

51 (52)

Since we have the representation (46) for all 0 ≤ t ≤ T , (38) gives

1 1 (53) |e(t)|He ≤ C|e|L2 (0,T ;He ) + C r |u|U ≤ C 1 + r |u|U t t Combining (50) and (53) gives the result.

References [1] G. Avalos. The exponential stability of a coupled hyperbolic/parabolic system arising in structural acoustics. Abstract and Applied Analysis, 1(2):203–217, 1996. [2] G. Avalos and I. Lasiecka. Diﬀerential Riccati equation for the active control of a problem in structural acoustics. Journal of Optimization Theory and Applications, 91(3):695–728, 1996. [3] H. T. Banks, R. J. Silcox, and R. C. Smith. The modeling and control of acoustic/structure interaction problems via piezoceramic actuators: 2-d numerical examples. ASME Journal of Vibration and Acoustics, 116(3):386–396, 1994. [4] H. T. Banks, R. C. Smith, and Y. Y. Wang. The modeling of piezoceramic patch interactions with shells, plates, and beams. Quarterly of Applied Mathematics, 53(2):353–381, 1995. [5] J. Cagnol, I. Lasiecka, C. Lebiedzik, and J.-P. Zol´esio. Uniform stability in structural acoustic models with ﬂexible curved walls. Journal of Diﬀerential Equations, 186(1):88–121, 2002. [6] J. Cagnol and C. Lebiedzik. Free boundary conditions for intrinsic shell model. In V. Barbu, I. Lasiecka, D. Tiba, and C. Varsan, editors, Analysis and optimization of diﬀerential systems, pages 77–88. Kluwer Academic Publishers, Boston, MA, 2003. [7] M. Camurdan and G. Ji. A noise reduction problem arising in structural acoustics: a three-dimensional solution. Contemporary Mathematics, 267, 2000. [8] S. Chen and R. Triggiani. Proof of extensions of two conjectures on structural damping for elastic systems. Paciﬁc Journal of Mathematics, 136(1):15–55, 1989. [9] M. C. Delfour and J.-P. Zol´esio. Intrinsic diﬀerential geometry and theory of thin shells. To appear. [10] M. C. Delfour and J.-P. Zol´esio. A boundary diﬀerential equation for thin shells. Journal of Diﬀerential Equations, 119(2):426–449, 1995. [11] M. C. Delfour and J.-P. Zol´esio. Tangential diﬀerential equations for dynamical thin/shallow shells. Journal of Diﬀerential Equations, 128(1):125–167, 1996. [12] E. K. Dimitradis, C. R. Fuller, and C. A. Rogers. Piezoelectric actuators for distributed vibration excitation of thin plates. Journal of Vibration and Acoustics, 113:100–107, 1991. [13] I. Lasiecka. Mathematical Control Theory of Coupled PDE Systems: NSF-CMBS Lecture Notes. SIAM, 2000.

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[14] I. Lasiecka and C. Lebiedzik. Uniform stability in structural acoustic systems with thermal eﬀects and nonlinear boundary damping. Control and Cybernetics, 28(3):557–581, 1999. [15] I. Lasiecka and R. Marchand. Control and stabilization in nonlinear structural acoustic problems. In Proceedings of SPIE’s 4th Annual Symposium on Smart Structures and Materials, Mathematics and Control in Smart Structures, 1997. [16] I. Lasiecka and R. Marchand. Riccati equations arising in acoustic structure interactions with curved walls. Dynamics and Control, 8:269–292, 1998. [17] I. Lasiecka and R. Triggiani. Sharp regularity theory for second order hyperbolic equations of Neumann type. Part I. nonhomogeneous data. Annali di Matematica Pura ed Applicata IV, CLVII:285–367, 1990.

Copyright © 2005 Marcel Dekker, Inc.

CHAPTER IV NONLINEAR WAVE EQUATIONS WITH DEGENERATE DAMPING AND SOURCE TERMS Viorel Barbu Department of Mathematics Alexandru Ioan Cuza University Iasi, Romania

Irena Lasiecka Department of Mathematics University of Virginia Charlottesville, VA, USA

Mohammad A. Rammaha Department of Mathematics and Statistics University of Nebraska-Lincoln Lincoln, NE, USA

Abstract

In this article we present some results on the global well-posedness of the wave equation utt − ∆u + |u|k ∂j(ut ) = |u|p−1 u in Ω × (0, T ), where ∂j is a sub-diﬀerential of a continuous convex function j, and Ω is a bounded smooth domain in Rn . Under some conditions on j and the parameters in the equations we obtain several results on the existence of global solutions, uniqueness, nonexistence of global solutions, and propagation of regularity.

Keywords: Wave equation, damping and source terms, weak solutions, blow-up of solutions, sub-diﬀerentials, energy estimates

Introduction Let Ω be a bounded domain in Rn with a smooth boundary Γ. In this article we study the solvability and long-time behavior of the following initial-boundary value problem: utt − ∆u + |u|k ∂j(ut ) = |u|p−1 u, Copyright © 2005 Marcel Dekker, Inc.

in Ω × (0, T ) ≡ QT ,

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CONTROL AND BOUNDARY ANALYSIS

u(x, 0) = u0 (x) ∈ H01 (Ω),

ut (x, 0) = u1 (x) ∈ L2 (Ω),

(1)

u = 0, on Γ × (0, T ), where j(s) is a continuous, convex real valued function deﬁned on R and ∂j is its sub-diﬀerential (see [4]). The initial-boundary value problem (1) is studied under the following assumption: Assumption A: k, m, p ≥ 0. In addition, k and p

0. Strict monotonicity: (∂j(s) − ∂j(v))(s − v) ≥ c1 |s − v|m+1 , where c1 > 0. Continuity: ∂j(s) is single valued and |∂j(s)| ≤ c0 |s|m + c2 , for some constants c0 > 0, c2 ≥ 0. We remark here that although ∂j(s) is single valued, we choose to use the sub-diﬀerential notation in this paper for the sake of convenience. The main goal of the paper is to study the long-time behavior of solutions to the initial-boundary value problem (1). Of particular interest is the relationship of the source and damping terms to the behavior of solutions. It is important to note that the partial diﬀerential equation in (1) is a special case of the prototype evolution equation utt − ∆u + R(x, t, u, ut ) = F(x, u),

(2)

where in (2) the nonlinearities satisfy: vR(x, t, u, v) ≥ 0, R(x, t, u, 0) = F(x, 0) = 0, and F(x, u) ∼ |u|p−1 u for large |u|. Various special cases of (2) arise in quantum ﬁeld theory and some important mechanical applications. See for example J¨ orgens [11]and Segal [28]. Also, one of the important special cases of (1) is the following well known polynomially damped wave equation studied extensively in the literature (see for instance [24, 26]). utt − ∆u + |u|k |ut |m−1 ut = |u|p−1 u, u(x, 0) = u0 (x),

in Ω × (0, T ),

ut (x, 0) = u1 (x),

u(x, t) = 0 on Γ × (0, T ), Copyright © 2005 Marcel Dekker, Inc.

in Ω,

(3)

Nonlinear Wave Equations with Degenerate Damping and Source Terms

55

1 It is easy to see that by taking j(s) = m+1 |s|m+1 , then ∂j(s) = |s|m−1 s, and therefore Assumption A is satisﬁed. Moreover, in this case, the initial-boundary value problem (1) is equivalent to (3). It is worth noting here that when the damping term |u|k |ut |m−1 ut is absent, the source term |u|p−1 u drives the solution of (3) to blow-up in ﬁnite time [8, 17, 23, 31]. In addition, if the source term |u|p−1 u is removed from the equation, then damping terms of various forms are known to yield existence of global solutions, (cf. [2–4]). However, the interaction between the damping and source terms is often diﬃcult to analyze, as one can see from the work in [7, 20, 18, 24, 26, 29]. It should be noted that if k = 0 and p = 0 then equation (1) can be studied via the theory of monotone operators and the full well-posedness of strong solutions (in the terminology of monotone operator theory) is classical [4]. In addition, with k = 0 the presence of a source term, which is locally Lipschitz in H 1 (Ω) does not aﬀect the arguments for establishing the existence of local solutions via the theory of monotone operators. Moreover, if p ≤ k + m then one can derive the necessary a priori bounds that guarantee that, in this case, every local solution is indeed global in time. The situation is, however, diﬀerent when the damping term is degenerate, leading to the degeneracy of the monotonicity argument. In fact, when k > 0, (3) is no longer locally Lipschitz perturbation of a monotone problem. Thus, standard monotone operator theory does not apply. This fact combined with a potential strong growth of the damping term (the case when m > 1 ), makes the problem interesting and the analysis is more subtle. Indeed, one needs to be careful about the meaning of the solution and its relation to the equation.

1.

Deﬁnitions and Main Results

In this section we provide results on the existence and uniqueness of various types of solutions such as generalized solutions, weak solutions, and strong solutions. In order to proceed with the presentation of our results we shall introduce the appropriate deﬁnitions. First, we give the deﬁnition of a generalized solution, which satisﬁes a given variational inequality.

Definition 1 A function u ∈ Cw ([0, T ], H01 (Ω))∩Cw1 ([0, T ], L2 (Ω)) with |u|k j(ut ) ∈ L1 (Ω × (0, T )) is said to be a generalized solution to ( 1) if and only if for all 0 < t ≤ T the following inequality holds: t (ut vt − ∇u∇v)dxdt + 1/2 [u2t (t) + |∇u(t)|2 ]dx 0

Ω

Copyright © 2005 Marcel Dekker, Inc.

Ω

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CONTROL AND BOUNDARY ANALYSIS

t

+ |u|k [j(ut ) − j(v)]dxdt 0 Ω t |u|p−1 u(ut − v)dxdt + 1/2 [u21 + |∇u0 |2 + 2u1 v(0)]dx ≤ 0

Ω

(4)

Ω

for all test functions v satisfying v ∈ H 1 (0, t; L2 (Ω)) ∩ L2 (0, t; H01 (Ω)) ∩ L∞ (Ω × (0, T )), v(t) = vt (t) = 0. It should be noted here that Deﬁnition 1 is a proper extension of the classical deﬁnition of weak solutions, which sometimes is called variational.

Definition 2 A function u ∈ Cw ([0, T ],H01 (Ω))∩Cw1 ([0, T ],L2 (Ω)) which satisﬁes ∆u − utt and |u|k j(ut ) ∈ L2 (Ω × (0, T )) is said to be a weak solution to ( 1) if and only if for all 0 < t ≤ T the following variational equality holds: t t (utt − ∆u)vdxdt + |u|k ∂j(ut )vdxdt 0 0 Ω Ω t |u|p−1 uvdxdt = 0

u(0) = u0 , ut (0) = u1

(5)

for all test functions v ∈ L2 (Ω × (0, T )). Equivalently, we may write t t (−ut vt + ∇u∇v)dxdt + u1 v(0)dx + |u|k ∂j(ut )vdxdt 0 0 Ω Ω Ω t |u|p−1 uvdxdt, (6) = 0

for all test functions v satisfying v ∈ H 1 (0, T ; L2 (Ω)) ∩ L2 (0, T ; H01 (Ω)) ∩ L2 (Ω × (0, T )), v(t) = 0 It should be noted here that Deﬁnition 2 implies that utt − ∆u = −|u|k ∂j(ut ) + |u|p−1 u, a.e in x, t ∈ Ω × (0, T ). At this end, we remark that the following notation will be used in the sequel: |u|s,Ω ≡ |u|H s (Ω) and |u|p ≡ |u|Lp (Ω) , Copyright © 2005 Marcel Dekker, Inc.

Nonlinear Wave Equations with Degenerate Damping and Source Terms

57

where H s (Ω) and Lp (Ω) stands for the classical Sobolev spaces and the Lebesgue spaces, respectively. Also, we let A : L2 (Ω) → L2 (Ω), where A = −∆ with its domain D(A) = H 2 (Ω) ∩ H01 (Ω). Our main results read as follows:

Theorem 3 Generalized solutions. Under Assumption A and the condition p ≤ k + m, there exists a global generalized solution to ( 1). By assuming more restrictions on the parameters k, m, we obtain global existence of weak solutions. Speciﬁcally, we have:

Theorem 4 Weak solutions. Part 1 In addition to Assumption A and the condition p ≤ k + m, we also assume that m < 1 if n = 1, 2; (7) k m 1 2n ∗ p∗ + 2 ≤ 2 , if n ≥ 3; where p = n−2 . Then, there exists a global weak solution to ( 1) which is deﬁned on (0, T ), for any T > 0. Moreover, the solution is unique and depends continuously on the initial data. The same conclusion holds when instead of ( 7) we assume k = 0. Part 2 In addition to Assumption A, assume that m = 1, k + 1 ≤ p∗ . Then, there exists a global solution u to ( 1) such that u ∈ 2 Cw ([0, T ], H01 (Ω))∩ Cw ([0, T ], L2 (Ω)) where u satisﬁes ( 4) with the test functions v ∈ L∞ (Ω×(0, T )), and T is arbitrary . In addition, ∆u − utt ∈ L1 (Ω × (0, T )) and such a solution u is unique, but it may not be continuously dependent on the initial data in the ﬁnite energy norm. Our next result addresses the issue of a strong source (large values of p) which may lead to a ﬁnite-time blow-up of solutions.

Theorem 5 Blow-up of weak solutions. Assume the validity of Assumption A with c2 = 0 and that p > k + m. In addition, assume that E(0) < 0, where E(0) is the initial energy given by ( 1 1' |u1 |20,Ω + |A1/2 u0 |20,Ω − |u0 |p+1 . E(0) = Lp+1 (Ω) 2 p+1 Then, the weak solution to ( 1) blows up in a ﬁnite time. Our next Theorem addresses the issue of propagation of regularity. This means that more regular data produce more regular solutions. In fact, Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

the result below states that this is always the case locally (i.e., for sufﬁciently small times). However, in the special case when the parameter p is below the critical value k + m, then the propagation of regularity is a global phenomena.

Theorem 6 Strong (regular) solutions. Assume that k > 1, p > 2, ∗ n and that j is deﬁned almost everywhere on R. Then, k + m ≤ p2 = n−2 for every initial data satisfying u0 ∈ H 2 (Ω) ∩ H01 (Ω), u1 ∈ H01 (Ω), there exists T0 > 0 such that ( 1) has a local solution u with the regularity that u ∈ C([0, T ], H 2 (Ω)) ∩ C 1 ([0, T ], H 1 (Ω)), for some T ≤ T0 where T0 may be ﬁnite. In addition, if we assume that p ≤ k + m, and either k = 0 1 or pk∗ + m 2 ≤ 2 , then regular solutions are global and T0 can be taken arbitrarily. As we have already mentioned in the Introduction, the particular feature of the problem studied in this paper is the potential degeneracy of the damping term. Thus, the beneﬁcial eﬀects of damping may not be present when the displacement has strong oscillations. In fact, this problem was studied ﬁrst in [24] in the special case when the damping 1 term is sub-linear in the velocity, i.e., m < 1 and subject to pk∗ + m 2 ≤ 2. In this case and with the assumption that p ≤ k + m, the authors in [24] established the existence of a unique global weak solution. When m > 1, the situation is more complex. This is mostly due to the notorious diﬃculties in passing to the limit in a super-linear term in ut with a weakly convergent subsequence. The lack of good structure, namely the lack of monotonicity, and super-linearity of ut in a degenerate damping term render standard tools nonapplicable. Nevertheless, our results establish the existence of global generalized solutions with no restrictions on m. Under additional restrictions on the parameters (see Theorem 4), one obtains weak solutions which propagate additional regularity as stated in Theorem 6. The ﬁrst statement in Part 1 in Theorem 4 was shown in [24]. However, our proof is diﬀerent and, we believe, it is simpler. The result stated in Part 2 of Theorem 4 is new. Here, the main issue was the uniqueness of the corresponding solution. While it is shown that the solution is indeed unique, it may not depend continuously on initial data.

2.

Outline of the Proof of Theorem 3

In this section we only present an outline of the proof of Theorem 3. Full proofs of our results are in [6] and will appear elsewhere. Copyright © 2005 Marcel Dekker, Inc.

Nonlinear Wave Equations with Degenerate Damping and Source Terms

59

The following a priori bound for generalized solutions in the topology speciﬁed by the Deﬁnition 1 is crucial in our proofs. Speciﬁcally, we have the following Lemma.

Lemma 7 Let u be a generalized solution of problem ( 1) with the assumption p ≤ k + m. Then for all initial data u0 ∈ H01 (Ω), u1 ∈ L2 (Ω) and all T > 0, we have the inequality |u(t)|1,Ω + |ut (t)|0,Ω +

t 0

|u|k j(ut )dxdτ ≤ CT (|u0 |1,Ω , |u1 |0,Ω ),

Ω

for all t ∈ [0, T ]. The a priori bound obtained in Lemma 7 above allows us to construct a multi-valued ﬁxed point argument that completes the proof of Theorem 3. However, in order to show the existence of a ﬁxed point, one must establish several facts: First, for a ﬁxed function w ∈ C(0, T ; Lq (Ω)) establish the solvability of the corresponding problem of the following equation (this is accomplished by applying an appropriate Faedo-Galerkin method): utt − ∆u + |w|k ∂j(ut ) = |w|p−1 w,

(8)

For a given argument w ∈ C(0, T ; Lq (Ω)) we consider the multivalued mapping F : C(0, T ; Lq (Ω)) → C(0, T ; Lq (Ω)), where the multi-valued action of F is deﬁned by u ∈ F w if and only if u is a solution to the following variational inequality: t 1 (ut vt − ∇u∇v)dxdt + [u2 (t) + |∇u(t)|2 ]dx 2 Ω t 0 Ω t |w|k [j(ut ) − j(v)]dxdt + 0 Ω t |w|p−1 w(ut − v)dxdt ≤ 0 Ω 1 [u2 + |∇u0 |2 + 2u1 v(0)]dx (9) + 2 Ω 1 for all test functions v ∈ H 1 (0, t; L2 (Ω)) ∩ L2 (0, t; H01 (Ω)) ∩ L∞ (Qt ), Copyright © 2005 Marcel Dekker, Inc.

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v(t) = vt (t) = 0. We remark here, by the previous bullet F (w) = ∅ for every ﬁxed w ∈ C(0, T ; Lq (Ω)) Next, we show that: If K is a ball in C(0, T ; Lq (Ω)), then F (K) is compact and F (w) is convex in C(0, T ; Lq (Ω)), for every w ∈ K. Our next step is to show the upper semi-continuity of the nonlinear map F . For this part, our argument is based on subtle approximations by weakly lower semi-continuous functions. Finally, we apply Kakutani’s ﬁxed point theorem to establish a ﬁxed point for F in an appropriate ball in C(0, T ; Lq (Ω)).

References [1] R. A. Adams. Sobolev spaces. Academic Press [A subsidiary of Harcourt Brace Jovanovich Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. [2] K. Agre and M. A. Rammaha. Global solutions to boundary value problems for a nonlinear wave equation in high space dimensions. Diﬀ. Integral Eq., 14(11):1315–1331, 2001. ´ [3] ¯D. ¯D. Ang and A. Pham Ngoc Dinh. Mixed problem for some semilinear wave equation with a nonhomogeneous condition. Nonlinear Anal., 12(6):581–592, 1988. [4] V. Barbu. Nonlinear Diﬀerential Equations in Banach spaces. Nordhoﬀ, Leyden, The Netherlands, 1976. [5] V. Barbu and Th. Precupanu. Convexity and optimization in Banach spaces. Editura Academiei, Bucharest, revised edition, 1978. Translated from the Romanian. [6] V. Barbu, I. Lasiecka, and M. A. Rammaha. On nonlinear wave equations with degenerate damping and source terms. Trans. Amer. Math. Soc. To appear. [7] V. Georgiev and G. Todorova. Existence of a solution of the wave equation with nonlinear damping and source terms. J. Diﬀerential Equations, 109(2):295–308, 1994. [8] R. T. Glassey. Blow-up theorems for nonlinear wave equations. Math. Z., 132:183–203, 1973. [9] P. Grisvard. Caract´erisation de quelques espaces d’interpolation. Arch. Rational Mech. Anal., 25:40–63, 1967. ´ ´ [10] P. Grisvard. Equations diﬀ´erentielles abstraites. Ann. Sci. Ecole Norm. Sup. (4), 2:311–395, 1969. [11] K. J¨ orgens. Das Anfangswertproblem im Grossen f¨ ur eine Klasse nichtlinearer Wellengleichungen. Math. Z., 77:295–308, 1961.

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[12] H. Koch and I. Lasiecka. Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity: full von Karman systems. In Evolution equations, semigroups and functional analysis (Milano, 2000), volume 50 of Progr. Nonlinear Diﬀerential Equations Appl., pages 197–216. Birkh¨ auser, Basel, 2002. [13] I. Lasiecka, J.-L. Lions, and R. Triggiani. Nonhomogeneous boundary value problems for second order hyperbolic operators. J. Math. Pures Appl. (9), 65(2):149–192, 1986. [14] I. Lasiecka and R. Triggiani. A cosine operator approach to modeling L2 (0, T ; L2 (Γ))—boundary input hyperbolic equations. Appl. Math. Optim., 7(1):35–93, 1981. [15] I. Lasiecka and R. Triggiani. Regularity theory of hyperbolic equations with nonhomogeneous Neumann boundary conditions. II. General boundary data. J. Diﬀerential Equations, 94(1):112–164, 1991. [16] I. Lasiecka and J. Ong. Global solvability and uniform decays of solutions to quasilinear equation with nonlinear boundary dissipation. Comm. Partial Diﬀerential Equations, 24(11–12):2069–2107, 1999. [17] H. A. Levine. Instability and nonexistence of global solutions to nonlinear wave equations of the form P utt = −Au + F(u). Trans. Amer. Math. Soc., 192:1–21, 1974. [18] H. A. Levine, S. R. Park, and J. Serrin. Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation. J. Math. Anal. Appl., 228(1):181–205, 1998. [19] H. A. Levine, S. R. Park, and J. Serrin. Global existence and nonexistence theorems for quasilinear evolution equations of formally parabolic type. J. Differential Equations, 142(1):212–229, 1998. [20] H. A. Levine and J. Serrin. Global nonexistence theorems for quasilinear evolution equations with dissipation. Arch. Rational Mech. Anal., 137(4):341–361, 1997. [21] J.-L. Lions and E. Magenes. Non-homogeneous boundary value problems and applications. Vol. II. Springer-Verlag, New York, 1972. Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 182. [22] J.-L. Lions and W. A. Strauss. Some non-linear evolution equations. Bull. Soc. Math. France, 93:43–96, 1965. [23] L. E. Payne and D. H. Sattinger. Saddle points and instability of nonlinear hyperbolic equations. Israel J. Math., 22(3–4):273–303, 1975. [24] D. R. Pitts and M. A. Rammaha. Global existence and non-existence theorems for nonlinear wave equations. Indiana Univ. Math. J., 51(6):1479–1509, 2002. [25] P. Pucci and J. Serrin. Global nonexistence for abstract evolution equations with positive initial energy. J. Diﬀerential Equations, 150(1):203–214, 1998. [26] M. A. Rammaha and T. A. Strei. Global existence and nonexistence for nonlinear wave equations with damping and source terms. Trans. Amer. Math. Soc., 354(9):3621–3637 (electronic), 2002. [27] R. Seeley. Interpolation in Lp with boundary conditions. Studia Math., 44:47– 60, 1972. Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientiﬁc activity, I.

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[28] I. Segal. Non-linear semi-groups. Ann. of Math. (2), 78:339–364, 1963. [29] G. Todorova. Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms. C. R. Acad. Sci. Paris S´er. I Math., 326(2):191– 196, 1998. [30] G. Todorova. Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms. Nonlinear Anal., 41(7-8, Ser. A: Theory Methods):891–905, 2000. [31] M. Tsutsumi. On solutions of semilinear diﬀerential equations in a Hilbert space. Math. Japon., 17:173–193, 1972. [32] G. F. Webb. Existence and asymptotic behavior for a strongly damped nonlinear wave equation. Canad. J. Math., 32(3):631–643, 1980.

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CHAPTER V NUMERICAL MODELING OF PHASE CHANGE PROBLEMS Andr´e Fortin GIREF, D´epartement de Math´ematiques et de Statistiques Universit´e Laval Quebec, Canada

Youssef Belhamadia GIREF, D´epartement de Math´ematiques et de Statistiques Universit´e Laval Qu´ebec, Canada

Abstract

Recent developments for the numerical simulation of phase change problems are presented. A semi-phase ﬁeld formulation is described which requires the discretization of both the phase-ﬁeld function φ and the temperature T . Numerical results using anisotropic mesh adaptation are presented for two- and three-dimensional problems.

Keywords: Stefan problem, phase change, semi-phase-ﬁeld formulation, ﬁnite element method, error estimators, anisotropic mesh adaptation

Introduction Phase change problems are extremely important in many applications. The main objective of this work is the development of numerical techniques for the simulation of cryosurgery problems. In this surgical technique a cryoprobe is inserted inside a tumor and cancerous cells are destroyed by the freezing-thawing process (see Keanini and Rubinski [6]). The objective of this work is an accurate prediction of freezing fronts in order to destroy completely the tumor while preserving healthy tissues. The mathematical formulation of such problems is known as the Stefan problem where the heat conduction equation has to be solved for the temperature T in a domain typically consisting of a solid and a liquid part, and separated by a moving interface (the freezing or melting front). On this interface, a heat balance condition, the Stefan condition, Copyright © 2005 Marcel Dekker, Inc.

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must be imposed. The precise location and form of the interface is thus extremely important. Standard ﬁnite element methods often result in inaccurate prediction of this interface. Enthalpy formulations greatly alleviate the diﬃculties since the interface is computed implicitly and the Stefan condition is satisﬁed automatically. The reader is referred to Fortin and Belhamadia [4] and to Nochetto et al. [8] for a complete discussion. Phase-ﬁeld formulations have been introduced for the numerical simulation of physical problems such as dendritic growth. In these formulations, a phase-ﬁeld function is introduced to determine the interface position which is the solution of a partial diﬀerential equation (see Mackenzie and Robertson [7]). In the present paper, a semi-phase-ﬁeld is introduced as an alternative to the enthalpy formulation of the Stefan problem. In this new formulation, an algebraic equation allows the determination of the phase-ﬁeld function, thus the name semi-phase-ﬁeld. An anisotropic mesh adaptation procedure is also introduced and applied to two- and three-dimensional problems. As shall be seen, the adaptive method concentrates the elements in the vicinity of the freezing front allowing a better prediction of the interface position and form.

1.

Stefan Problem and Semi-Phase-Field Formulation

The Stefan problem can be formulated as: ∂T ρi ci − ∇ · (K i ∇T ) = fi ∂t

T

= Tf

(K s ∇T ) · ns + (K l ∇T ) · nl = ρl L VΓ

in Ωi i = s, l, on Γ,

(1)

on Γ.

Subscripts s and l refer to the solid and liquid phases and K i is the thermal conductivity tensor, ρi is the density, ci is the speciﬁc heat, fi is a possible heat source, L is the latent heat of fusion, and VΓ is the interface normal velocity. The semi-phase-ﬁeld requires the introduction of the phase-ﬁeld variable φ deﬁned as: 0 in Ωs i.e. for T ≤ Tf , φ= 1 in Ωl i.e. for T ≥ Tf . and consequently, the phase-ﬁeld function is completely determined by this non-linear algebraic equation. As described in Belhamadia et al. [2], Copyright © 2005 Marcel Dekker, Inc.

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Numerical Modeling of Phase Change Problems

the semi-phase-ﬁeld formulation takes the form of a non-linear system: ∂φ ∂T + ρL − ∇ · (K(φ)∇T ) = f (φ), α(φ) (2) ∂t ∂t φ = F (T ). where:

α(φ) = ρs cs + φ (ρl cl − ρs cs ) , K(φ) = K s + φ (K l − K s ) , f (φ) = fs + φ (fl − fs ) .

This last relation underlines the fact that physical properties such as thermal conductivity may vary according to the phase and thus depend on the phase-ﬁeld function φ.

2.

Finite Element Discretization

The weak formulation of system (2) is straightforward. An implicit Euler scheme is used for the time discretization and the variational formulation becomes: (n+1) − T (n) (n+1) − φ(n) T φ α(φ(n+1) ) vT dΩ + ρl L vT dΩ ∆t ∆t Ω Ω f (φ(n+1) )vT dΩ, + K(φ(n+1) )∇T (n+1) · ∇vT dΩ = Ω Ω ' ( φ(n+1) − F (T (n+1) ) v dΩ = 0.

φ

Ω

(3) Newton’s method is used to solve the non-linear system above at each time step. In all numerical simulations, a quadratic discretization of the temperature T and a linear approximation of φ were used. For twodimensional problems, linear systems resulting from Newton’s method are solved by a direct method (LU factorization). For three-dimensional problems, however, an iterative method is employed since memory requirements for direct methods would rapidly exceed the capacity of available computers. In the speciﬁc case of system (3), an ILU preconditioned GMRES solver [9] from the PetSc librairy [1] was used.

3.

Adaptive Strategy

As already mentioned, the accurate prediction of the interface which is determined by the discontinuous phase-ﬁeld function φ is crucial. It is well known that approximating a discontinuous function by standard Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

ﬁnite element methods may lead to oscillations and thus inaccurate prediction of the interface form. To avoid oscillations, the mesh has to be reﬁned near the interface location. This can be done with appropriate mesh adaptation techniques. A brief description of adaptive methods for time dependent problems will now be presented. Depending on the dimension of the problem, two diﬀerent methods for estimating the error have been used. Both strategies result in anisotropic meshes. For two-dimensional problems, a hierarchical error estimator described in Belhamadia et al. [2] was used. This error estimator is based on a hierarchical decomposition of the ﬁnite element space Vk+1 of polynomials of degree k + 1. Starting from a numerical solution in Vk , a hierarchical correction of this solution is computed and used as an approximation of the error. The extension to the three-dimensional case of the hierarchical error estimator is not yet fully implemented in our code. This is why, for three-dimensional problems, an error estimator based on a deﬁnition of edge length using a solution-dependent metric was used (see Habashi et al. [5]). Equirepartition of the error is then achieved when all the element edges are of the same length in this metric. For a complete presentation the reader is referred to Belhamadia et al. [3]. The overall adaptive strategy is the following: 1 Starting from a solution (T (n) , φ(n) ) and a mesh M(n) at time t(n) ; 2 Solve system (3) on mesh M(n) to obtain a ﬁrst approximation (T˜(n+1) , φ˜(n+1) ) of the solution at time t(n+1) ; 3 Adapt the mesh on the new solution (T˜(n+1) , φ˜(n+1)) and (T (n) ,φ(n)) to obtain M(n+1) ; 4 Reinterpolate (T (n) , φ(n) ) on M(n+1) ; 5 Solve system (3) on mesh M(n+1) for (T (n+1) , φ(n+1) ). The mesh is thus adapted at each time step in order to preserve the accuracy of the solution. They are adapted to ﬁt the actual and proceeding solutions.

4.

Numerical Results

A test problem will now be solved in the two- and three-dimensional cases. The chosen test case is the oscillating sphere (circle in two dimensions) since it possesses an analytical solution. The problem will be described in the three-dimensional case but its two-dimensional restriction is straightforward. Copyright © 2005 Marcel Dekker, Inc.

Numerical Modeling of Phase Change Problems

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The computational domain is the box Ω = [0, 5] × [−1, 4] × [−1.5, 1.5]. The interface has the form of a half-sphere of radius 1 moving up and down along the plane x = 0. Its position at all times is (0,α(t),0), where α(t) = 0.5 + sin(1.25t) The half-period of oscillation is thus 4π/5. If a proper function f is chosen, then the solution of the Stefan problem (1) is given by: r < 1, 0.75(r 2 − 1), (4) T (x, y, z, t) = ˙ (1.5 − α(t) sin φ)(r − 1), r ≥ 1, ˙ = where r = (x2 + (y − α(t))2 + z 2 )1/2 , sin φ = ((y − α(t))/r) and α(t) dα (t). The other parameters are given in the following table: dt Oscillating sphere ρs = ρl = 1 L=1 cs = cl = 1 Tf = 0 Ks = Kl = I A homogeneous Neumann boundary condition is imposed on the plane x = 0 (where the center of the sphere is located) while Dirichlet boundary conditions compatible with the analytical solution are given on all the other sides. The initial condition is also compatible with the analytical solution (4). The time step was ﬁxed to 4π/500 and to 0.0125. Figure 1 shows the numerical solution (in blue) and analytical one (in red) over one period of oscillation. The two colors are superimposed proving that the form and position of the interface are accurately predicted. Figure 2 shows clearly how the mesh is concentrated and strongly anisotropic. In the three-dimensional case the initial mesh was also structured ( 180 000 elements) but at the ﬁrst time step, the number of elements is greatly reduced since they are concentrated in the vicinity of the sphere (see Figure 3). Cross sections of the mesh in diﬀerent planes passing through the sphere clearly show the quality and anisotropy of the mesh.

5.

Conclusions

An anisotropic mesh adaptation strategy was introduced for the numerical simulation of phase change problems. The numerical examples presented clearly show the advantages of mesh adaptation over classical ﬁnite element methods.

Acknowledgments The authors wish to acknowledge the ﬁnancial support of NSERC project SKALPEL-ITC. Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

a) t = 0

b) t = 4π/25

c) t = 8π/25

d) t = 12π/25

e) t = 16π/25

f) t = 4π/5

Figure 1.

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Numerical and analytical solutions.

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Numerical Modeling of Phase Change Problems

Figure 2.

a) t = 0

b) t = 4π/25

c) t = 8π/25

d) t = 12π/25

e) t = 16π/25

f) t = 20π/25

Mesh evolution for the oscillating circle adapting on T and φ.

Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

a) Numerical and analytical solutions

b) External envelope of the mesh

c) Mesh on plane x = 0

d) Mesh on plane y = 1.5

b) Mesh on plane z = 0

e) Close up view on plane x = 0

Figure 3.

Oscillating sphere: solution and cross sections of the mesh at t = 10π/25.

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Numerical Modeling of Phase Change Problems

71

References [1] S. Balay, K. Buschelman, V. Eijkhout, W. Gropp, D. Kaushik, M. Knepley, L. C. McInnes, B. Smith, and H. Zhang. PETSc Users Manual. Technical Report ANL-95/11-Revision 2.1.6, Argonne National Laboratory, Argonne, Illinois, 2003. http://www.mcs.anl.gov/petsc/. ´ Chamberland. Anisotropic Mesh Adaptation [2] Y. Belhamadia, A. Fortin, and E. for the Solution of the Stefan Problem. J. of Comp. Phys. 194(1):233–255, 2004. ´ Chamberland. Three-Dimensional Anisotropic [3] Y. Belhamadia, A. Fortin, and E. Mesh Adaptation for Phase Change Problems. J. of Comp. Phys., 2004. In press. [4] A. Fortin and Y. Belhamadia. A New Enthalpy Finite Element Method for Phase Change Problems. Comm. Num. Meth. Eng., 2003. Submitted. [5] W.G. Habashi, J. Dompierre, Y. Bourgault, D. Ait Ali Yahia, M. Fortin, and M.-G. Vallet. Anisotropic Mesh Adaptation: Towards User-Independent, MeshIndependent and Solver-Independent CFD. Part I: General Principles. Int. J. Numer. Meth. Fluids, 32:725–744, 2000. [6] R. G. Keanini and B. Rubinski. Optimization of Multiprobe Cryosurgery. ASME Journal of Heat Transfer, 114:796–801, 1992. [7] J. A. Mackenzie and M. L. Robertson. A Moving Mesh Method for the Solution of the One-Dimensional Phase-Field Equations. J. of Comp. Phys., 181:526–544, 2002. [8] R. H. Nochetto, M. Paolini, and C. Verdi. An Adaptive Finite Element Method for Two-Phase Stefan Problems in Two Space Dimensions. Part II: Implementation and Numerical Experiments. SIAM J. Sci. Stat. Comp., 12(5):1207–1244, 1991. [9] Saad Y. Iterative Methods for Sparse Linear Systems. PWS Publishing Company, Massachusetts, 1996.

Copyright © 2005 Marcel Dekker, Inc.

CHAPTER VI SHAPE OPTIMIZATION OF FREE AIR-POROUS MEDIA TRANSMISSION COEFFICIENT Arian Novruzi∗ Department of Mathematics and Statistics University of Ottawa Ottawa, Canada

Abstract

In this work we consider the shape analysis of the free air-porous domain transmission coeﬃcient as a function of the pore shapes. Shape derivatives of the transmission coeﬃcient and of the permeability are computed. To a given number we have also computed the porous domain which transmission coeﬃcient is equal to. Moreover, among all the porous domains having a given transmission coeﬃcient, we prove that under the constraint that the measure is given, there exists one optimal domain maximizing the velocity in porous domain in a given direction. The computation of this domain is performed as well.

Keywords: Shape optimization, free air-porous media interface, Stokes equation

Introduction The interaction of free air-porous media appears in many physical systems and has been studied intensively these last 30 years. The interaction is characterized by the so-called transmission coeﬃcient. The transmission coeﬃcient is very important because it deﬁnes the coupling of the equations in free air and porous domain. The transmission coeﬃcient on the free air-porous domain interface has been reported for the ﬁrst time in [1]. A proof at the physical level of rigor has been presented in [2]. The interface boundary condition has been proved rigorously in [4], [9], where it is given also a formula for the transmission coeﬃcient. In [5] it has been considered the computation of the transmission coeﬃcient appearing on the interface boundary condition.

∗ This

work was funded partially from NSERC and MITACS of Canada.

Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

The main objective of this paper is the construction of the porous domain having a given transmission coeﬃcient and the shape optimization with respect to the pore shape of a cost functional deﬁned later. Our study is motivated by the numerical computation of ﬂuid dynamics in hydrogen fuel cells, where the properties of the free air (channels)-porous media (graphite diﬀusive layers) interface plays an important role in the hydrogen fuel cell performance. As it is standard, we will consider a 2d viscous ﬂow over a periodic porous domain with period size , separated from the air by a horizontal interface, like in [4]-[5] (we will use almost the same notations as in these papers). The solution (u , p ) of this problem can be approached by a Poiseuille ﬂow in the free air domain, while in the porous domain one can approach the velocity by zero. It has been reported, see [1], [2], that in many cases this approximation is not suitable. The approximation to a further order involves the solution of the following problem. Let us set Q := (0, 1)2 and let Ω, Z ⊂ Q, Ω open such that Ω∪Z = Q, Z ∩ Ω = ∂Q ∩ ∂Z = φ. The set Z represents the solid part, Ω represents the ﬂuid part. We will assume that Ω is a smooth set. Let us also set + + Ω+ − = Ω ∪ S ∪ Ω− where Ω = (0, 1) × (0, +∞), Ω− = {Ω − (0, n), n = 1, 2, . . .} and Z− = {Z − (0, n), n = 1, 2, . . .}, S = (0, 1) × {0}. The solution (β, ω), β = (β1 , β2 ) of the problem −∆β + ∇ω = 0 in Ω+ ∪ Ω− , ∇ · β = 0 in Ω+ ∪ Ω− , β = 0 on S, (1) (∇β − ωI)e 2 = e1 on S, β = 0 on Z− , β, ω x1 periodic, where β = lim β(t) − β(−t), {e1 , e2 } is the base of R2 , plays a crucial t→0

role on approaching (u , p ). Moreover, in [4] it has been found that the eﬀective velocity is given by eﬀ ∂u eﬀ on S, u1 = −C 1 ∂x2 where ueﬀ is the average over the period of u and 1 C= β1 (x1 , x2 )dx1 , x2 ≥ 0.

(2)

(3)

0

In [5] it has been studied in depth (β, ω) and C, and for a given domain Ω the computation of C has been done. In fact, as Ω+ − is unbounded, Copyright © 2005 Marcel Dekker, Inc.

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an approximation of C is computed. More precisely, for k, l ∈ N, let us set Ωkl = Ωk ∪ S ∪ Ωl , where Ωk = (0, 1) × (0, k), Ωl = {Ω − (0, n), n = 1, 2, . . . , l}, and Zl = {Z − (0, n), n = 1, 2, . . . , l}. Now let (β kl , ω kl ) be the solution of −∆β kl + ∇ω kl = 0 in Ωk ∪ Ωl , ∇ · β kl = 0 in Ωk ∪ Ωl , β kl = 0 on S, (∇β kl − ω kl I)e = e on S, 2 1 (4) kl = (0, 0) on ∂Z ∪ {(·, −l)} β l kl kl = 0 on {(·, k)}, 2 β1

β2 = ∂kl Ωk ∪Ωl ω dx = 0, β, ω x1 periodic. Diﬀerent estimations for (β −β kl , ω −ω kl ) are proved in [5]. In particular, if we deﬁne 1

C kl =

β1kl (x1 , 0)dx1 ,

(5)

0

then C kl approaches C as follows |C − C kl | ≤ M (e−2πk − e−γ|l| ),

(6)

where M and γ are constants. In this paper we will deal with the following problems. We consider the coeﬃcient C as a shape functional Ω → C(Ω) given by (3) and we look for: a) If c is given, does there exist a domain Ω such that C(Ω) = c? If the answer is yes, ﬁnd an algorithm to compute it. b) If for a given c there exist many domains Ω such that C(Ω) = c, how one can choose reasonably one of them? Let O = {Ω ⊂ Q, Ω open, Z = Q\Ω ⊃ Bδ ( 12 , 12 ), d(∂Z, ∂Q) ≥ δ, #(Z) = 1 },

(7)

where #(Z) is the number of connected components of Z and δ > 0 is an arbitrary small number. Then, the problem a) has a solution if c ∈ R(C) := {C(Ω), Ω ∈ O}. Finding R(C) is a quite diﬃcult problem. In this paper, for a given c we will compute a domain Ω such that C kl (Ω) = c (of course when c ∈ R(C kl )). It will be shown numerically that such a domain Ω is not unique. One can look for one of these domains that optimizes the permeability on a given direction, under the Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

constraint m(Ω) = m0 . Indeed, if the permeabilty is deﬁned by the matrix (8) K(Ω) = [α1 , α2 ], Ω

where

α1 ,

α2

are solutions of

−∆αi + ∇σ i = ei in i in ∇·α = 0 i = (0, 0) on α x1 , x2 αi , σ i

Ω, i = 1, 2, Ω, ∂Z, periodic,

(9)

and the pressure gradient ∇p in the porous media is considered known, then − µ1 K(Ω) · ∇p gives the velocity. Often it is required to maximize the velocity on a given direction d. Thus, we look for Ω∗ ∈ O solution of K(Ω∗ ) = min{−d · K(Ω) · ∇p, C(Ω) = c, m(Ω) = m0 , Ω ∈ O}, where d ∈ R2 is a given vector.

1.

Computation of the Porous Shape

For a given number c, one may look for Ω ∈ O ∩C 2 solution of C(Ω) = c. As Ω+ and Ω− are unbounded, we will assume that for k, l ∈ N large ˜ ∗ ) = c then Ω∗ and Ω ˜ ∗ are close enough (for enough, if C(Ω∗ ) = c, C kl (Ω any topology, for example that of Hausdorﬀ). So, we will deal with the solution of C kl (Ω) = c instead of C(Ω) = c. To solve this equation we will use a Newton method. As C kl is with values in R, starting from an initial domain Ω0 , one parametric family of domains (assumed to contain ˜ ∗ ), for example of the form {x + tξ(x), x ∈ Ω0 }, where ξ is the solution Ω a regular vector ﬁeld, is suﬃcient. Assuming that the shape derivative in direction ξ of β kl (Ωn ) is β kl (Ωn ; ξ), where Ωn ∈ O, Zn = Q\Ωn , the Newton algorithm is: Ω0 given,

C kl (Ωn ) (10) (∂Ωn ), n = 0, 1, . . . I − kl ∂Ωn+1 = C (Ωn ; ξ)

where kl

β1kl (Ωn ; ξ)(x1 , 0)dx1 .

C (Ωn ; ξ) = S

We have the following shape derivative result.

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Proposition 1 For Ω ∈ O ∩ C 2 ﬁxed let us set 02 (Ω) = D(Ωl ∪ Zl ); R2 ) · C 2 (R2 ;R2 ) . C 2 (Ω) let set Ωθ = (I +θ)(Ω) and (β kl (Ωθ ), ω kl (Ωθ )) the solution For θ ∈ C 0 of ( 4) for Ωθ instead of Ω. Then the map θ → β kl (Ωθ ) ◦ (I + θ) is C 1 2 (Ω) into H 1 (Ωkl ; R2 ) in a neighborhood of θ = 0. Moreover, from C 0 if β kl (Ω; ξ) is the shape derivative of β kl (Ωθ ) at θ = 0 in the direction 2 (Ω), then there exists ω kl (Ω; ξ) ∈ L2 (Ωkl ) such that ξ∈C 0 −∆β kl (Ω; ξ) + ∇ω kl (Ω; ξ) ∇ · β kl (Ω; ξ) β kl (Ω; ξ) (∇β kl (Ω; ξ) − ω kl (Ω; ξ) · I)e2 β kl (Ω; ξ) β kl (Ω; ξ) β2kl (Ω; ξ) = ∂2 β1kl (Ω; ξ) ω kl (Ω; ξ)dx

= = = = = = =

0 in Ωk ∪ Ωl , 0 in Ωk ∪ Ωl , 0 on S, 0 on S, −∇β kl (Ω) · ξ on ∂Zl , (0, 0) on {(·, −l)}, 0 on {(·, k)},

= 0,

(11) (12) (13) (14) (15) (16) (17) (18)

Ωkl

x1 periodic,

β(Ω; ξ), ω(Ω; ξ)

(19)

and C (Ωn ; ξ) = − kl

kl ∂Zn

|∇β1kl (Ωn )|2 (ξ · νnkl ),

(20)

where νnkl is the unit exterior normal vector to Ωkl n. Proof. As usual in shape derivative results, we will use implicit function theorem. Let us introduce the spaces 1 (Ω) H p uH 1 (Ω) p

L2p (Ω) −1 (Ω) H p

= H 1 (Ωkl ; R2 ) ∩ {(u1 , u2 ) = 0 on ∂Zl ∪ {(·, −l)}, u1 = ∂1 u2 = 0 on {·, k)}, x1 periodic}, = ∇uH 1 (Ωkl ;R2 ) ,

= L2 (Ωkl ) ∩ { Ωkl p(x)dx = 0, x1 periodic}, 1 (Ω). = dual space of H p

p1 (Ω). For θ ∈ C 2 (Ω) small, Let us point out that cot H 1 (Ω) is norm in H 0 p kl kl 1 2 let (β (Ωθ ), ω (Ωθ )) ∈ Hp (Ωθ ) × Lp (Ωθ ) be the unique solution of the problem (4) for Ω = Ωθ , see [5]. If u = β(Ωθ ) ◦ (I + θ), p = ω(Ωθ ) ◦ (I + θ) 1 (Ω) × L2 (Ω) and satisﬁes then (v, p) ∈ H p p Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

t t kl ( i=1,2 ∇ ui · M (θ) · M (θ)

·ϕ Ω +M (θ) · ∇p)Jac(I + θ) − S ϕ1 = 0, i,j=1,2 Mj,i (θ)∂j ui = 0,

(21)

1 (Ω). where M (θ) = t [∇(I + θ)]−1 and ϕ = (ϕ1 , ϕ2 ) ∈ H p Let us introduce the function f as follows 2 (Ω) × H p1 (Ω) × L2p (Ω) → H p−1 (Ω) × L2p (Ω), : C 0

f (θ, u, p) = ( Ω ( i=1,2 ∇t ui · t M (θ) · M (θ) · ϕ+ M (θ) · ∇p)Jac(I + θ) − S ϕ1 , i,j=1,2 Mj,i (θ)∂j ui ). f

(22)

It is clear that the function f is of class C 1 near a neighborhood of (0, β kl (Ω), ω kl (Ω)). Its (u, p) derivative at (0, β kl (Ω), ω kl (Ω)) in the direction (v, q) is ∂(u,p) f ((0, β kl (Ω), ω kl (Ω))(v, q) =

∇v · ∇ϕ + ∇q · ϕ − ϕ1 , ∇ · v . Ωkl

(23)

S

p1 (Ω) × L2p (Ω) into The derivative (23) deﬁnes an isomorphism from H −1 (Ω) × L2 (Ω) let us prove that −1 (Ω) × L2 (Ω). Indeed, for (h, g) ∈ H H p p p p

∇v · ∇ϕ + ∇q · ϕ − ϕ1 , ∇ · v = (h, g) (24) Ωkl

S

has a unique solution.

p1 (Ω) ∩ {∇ · a) g = 0. It is known that Ωkl ∇v · ∇ϕ − S ϕ1 = h(ϕ), ϕ ∈ H

1 (Ω) because kl |∇ϕ|2 is a norm u = 0} has a unique solution v ∈ H p Ω p1 (Ω), see [5]. Therefore, from Proposition 1.1, 1.2, [7], it follows the in H existence of p ∈ L2 (Ω) such that −∆v + ∇p = h(·) + (·)1 . S

in distribution sense. From h(ϕ) = i,j=1,2 Ωkl ∂i ϕj phij , for some phi,j ∈

L2p (Ω) and the periodicity of v it follows that ∂(0,1)×(−l.k) (ϕ · ν)p = p1 (Ω) and ν is the exterior unitary normal vector to 0, where ϕ ∈ H

(0, 1) × (−l, k). Thus p ∈ L2p (Ω).

1 (Ω) such that b) g = 0. It is enough to prove that there exists vg ∈ H p Copyright © 2005 Marcel Dekker, Inc.

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1 (Ω) is continuous. ∇ · vg = g and that the map g ∈ L2p (Ω) → vg ∈ H p Indeed, in such a case one can look for v = w + vg and apply a) with hg (ϕ) = h(ϕ) − Ωkl ∇vg · ϕ instead of h(ϕ). But the existence of vg follows from Lemma 2.4, Proposition 1.2, [7] (the only modiﬁcation is to 1 (Ω) instead of H −1 (Ω)). take H p 1 (Ω) × L2 (Ω) → (h, g) ∈ H −1 (Ω) × c) It is clear that the map (v, p) ∈ H p p p L2p (Ωkl ) it is continuous. Thus, f satisﬁes the conditions of implicit function theorem. It follows 2 (Ω) → (u(θ), p(θ)) ∈ H p1 (Ω) × L2p (Ωkl ) that there exists a C 1 map θ ∈ C satisfying f (θ, u(θ), p(θ)) = 0. From the uniqueness of (4) it follows that u(θ) = β kl (Ωθ ) ◦ (I + θ) which proves the diﬀerentiability of β kl . The proof of (11)–(12) follows from Theorem 3.1, [3], while (13)–(17) follow from Theorem 3.2, [3] (let us point out that β kl (Ω) is H 2 close to the boundary ∂Zl ). The equation (20) follows from the fact that

C kl (Ω) = − Ωkl |∇β kl (Ω)|2 and Theorem 3.3, [3] after using the simple

fact that Ωkl ∇β kl (Ω) · ∇β kl (Ω; ξ) = 0.

1.1

Computation of the Pore Shape Corresponding to a Given Transmission Coeﬃcient

As it has been also reported in [5] for k, l ∈ N large enough, C is well approached by C kl . We have made numerical computations with k = l = 2. First, we have solved the direct problem as in [5], i.e., for a given domain Ω compute C kl . In the examples of Figure 1 we present the velocity ﬁeld and the transmission coeﬃcient C kl . The domain Ω is deﬁned by the nodes: square domain: U domain:

triangle domain:

{(0.4, −0.4), (0.6, −0.4), (0, 4, −0.6), (0.6, −0.6)} + {(0, 0), (0, −1)}, {(0.2, −0.25), (0.4, −0.25), (0.6, −0.25), (0.8, −0.25), (0.4, −0.6), (0.6, −0.6), (0.2, −0.75), (0.8 − 0.75)}+ {(0, 0), (0, −1)}, {(0.4, −0.4), (0.7, −0.3), (0.6, −0.7)} + {(0, 0), (0, −1)}.

Taking as initial domain Ω0 one of the three domains of Figure 1, we have solved the equation C kl (Ω) = −0.15 using the algorithm (10) with ξ = ν0 (of course, we have been ensured that −0.15 is in the range of C kl (Ω) after some numerical experiences). The results are summarized in Figure 2 where the initial and ﬁnal domains are shown, as well as the velocity ﬁeld. As we see, the problem C kl (Ω) = c has not a unique Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

S= -0.437631 S=-0.245949 S=-0.348377 Figure 1. For three diﬀerent domains: the velocity ﬁeld and C bl .

S(Ω) = −0.43, S(Ω) = −0.15

Figure 2. ﬁeld.

S(Ω) = −0.24, S(Ω) = −0.15

∗

S(Ω) = −0.43, S(Ω) = −0.15

22

For three diﬀerent cases: Ω0 , Ω such that C (Ω∗ ) = −0.15 and velocity

solution Ω. To deﬁne Ω in a unique way we need to add another criteria. A good criteria is to maximize the velocity in the porous domain (in a given direction) which is closely related to the permeability. Copyright © 2005 Marcel Dekker, Inc.

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Shape Optimization of Free Air-Porous Media Transmission Coeﬃcient

2.

Optimal Porous Shape Domain

In the previous section we saw numerically that to a given transmission coeﬃcient c correspond many porous domains. In this section we discuss the criteria to impose in order to deﬁne the optimal domain among them. In practice, often it is required to maximize the velocity on a given direction. Assuming that the pressure gradient ∇p in the porous media is given, the velocity in porous media is given by − K(Ω) µ ∇p, where K(Ω) is the permeability matrix deﬁned by (8) and µ is a constant. If d ∈ R2 is a vector then one can look for solution of the following problem: Find Ω∗ ∈ O solution of K(Ω∗ ) = max{−d · K(Ω) · ∇p, C kl (Ω) = c, m(Ω) = m0 , Ω ∈ O} = min{d · K(Ω) · ∇p, C kl (Ω) = c, m(Ω) = m0 , Ω ∈ O},

(25)

It is diﬃcult to prove any existence result for the problem (25). However, we can prove an existence result if the problem is modiﬁed by adding a perimeter term. More precisely, let us consider Ω∗ ∈ O solution of K(Ω∗ ) = min{d · K(Ω) · ∇p + P (Ω), C(Ω) = c, m(Ω) = m0 } (26) where > 0 arbitrary small and P (Ω) is the perimeter of Ω. Then we have

Theorem 2 The problem ( 26) has a solution in Ω∗ ∈ O. Proof. Let Ωn ∈ O be any minimizing sequence of (26) and (β kl,n , ω kl,n ) (αi,n , σ i,n ), i = 1, 2 solutions of (4), (9) with Ω = Ωn . It is easy to prove that there exist a constant M1 (k) and M2 (δ) such that ∇β kl,n H 1 (Ω) ≤ M1 (k), p

∇α1,n H 1 (Ω) + ∇α2,n H 1 (Ω) ≤ M2 (δ).

Thus, if Ωδ = Q\B δ ( 12 , 12 ) then β kl,n , resp. α1,n , α2,n , are bounded in 1 (Ωδ ), resp. H 1 (Ωδ ; R2 ). It follows that there exists a subsequence of H p β kl,n , resp. α1,n , α2,n , again denoted with the same letters, converging p1 (Ωδ ), resp. α1 , α2 ∈ H 1 (Ωδ ). H 1 weakly to β kl ∈ H On the other side, knowing that K(Ωn ) = [Kij (Ωn )] = Ωn

Copyright © 2005 Marcel Dekker, Inc.

∇αi,n · ∇αj,n

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CONTROL AND BOUNDARY ANALYSIS

it follows that {P (Ωn ), n = 1, 2, . . .} is bounded. Then, there exists a subsequence of {Ωn , n ∈ N}, again denoted by Ωn converging weakly in BV sense to a Ω∗ . As the set O is closed and compact for the complimentary Hausdorﬀ topology, see [6], we may consider that Ωn → Ω∗ , for weak BV and complementary Hausdorﬀ topologies. (27) Therefore Ω∗ ∈ O, C kl (Ω∗ ) = c and m(Ω∗ ) = m0 . On the other side, from Theorem 5.3, [8], it follows that convergence of β kl,n , and α1,n , p1 (Ω∗ ), resp. α1,n , α2,n ∈ H 1 (Ω∗ ) ∩ {u = α2,n is strong and β kl ∈ H ∗ 0 on ∂(Q\Ω )} , which proves that Ω∗ is a solution of (26). 2 (Ω) = C 2 (Ω; R2 )∩ Proposition 3 Let Ω ∈ O∩C 2 be ﬁxed and let set C 0 2 (Ω) let set Ωθ = (I + θ)(Ω) and {ϕ = (0, 0) on ∂Q}. For θ ∈ C 0 (αi (Ωθ ), σ i (Ωθ )), i = 1, 2, the solution of ( 9) for Ωθ instead of Ω. Then 2 (Ω) into H 2 (Ω) in a neighthe map θ → αi (Ωθ ) ◦ (I + θ) is C 1 from C 0 i borhood of θ = 0. Moreover, α (Ω; ξ), the shape derivative of αi (Ωθ ) at 2 (Ω) is given by θ = 0 in the direction ξ ∈ C 0 −∆αi (Ω; ξ) + ∇σ i (Ω; ξ) = 0 in Ω, ∇ · αi (Ω; ξ) = 0 in Ω, αi (Ω; ξ) = −∇αi · ξ on ∂Ω, x1 , x2 periodic. αi (Ω; ξ), σ i (Ω; ξ)

(28) (29) (30) (31)

Proof. The proof of this result is very similar to the proof of Proposition 1 and we will not present it here.

2.1

Numerical Results

To solve (26) we have used a gradient method with penalization. Precisely, let λ, µ > 0 be two big numbers. Then we look for solution of min{E(Ω) := d · K(Ω) · ∇p + P (Ω) + λ(C kl (Ω) − c)2 + µ(m(Ω) − m0 )2 }.

(32)

Based on Propositions 1, 3 and Theorem 3.1, [3], we have

Proposition 4 The shape derivative of E(Ω) in the direction ξ is given by i j ∇α (Ω) · ∇α (Ω)(ξ · ν) · ∇p + H(ξ · ν) dE(Ω; ξ) = d · ∂Z ∂Z 2 |∇β(Ω)| (ξ · ν) + µ (ξ · ν), (33) + λ ∂Z kl

Copyright © 2005 Marcel Dekker, Inc.

∂Z

Shape Optimization of Free Air-Porous Media Transmission Coeﬃcient

83

where H is the mean curvature of ∂Z. We will present two numerical results for solving (26). The initial √ domain is Ω0 = Br ( 12 , 12 ), the circle with center ( 12 , 12 ) and radius r = 0.02. We have taken = 10−2 , λ = µ = 102 , c = −0.3, m0 = 0.9. We have considered two following cases. The results are shown in Figure 3. The algorithm continues as long as E and dE decrease. The stop test used is when the gradient dE does not decrease anymore (it oscillates). 1) d = −∇p = (1, 1). 2) d = (1, −1), −∇p = (0, −1).

Initial domain

Figure 3. ﬁelds.

d = −∇p = (1, 1)

d = (1, −1), −∇p = (0, −11).

Initial domain and two ﬁnal domains solution of (26) as well as velocity

References [1] Beavers G. S. and Joseph D. D. Boundary condition at a naturally permeable wall. J. Fluid Mech, 30:197–207, 1967. [2] Saﬀman P. G. On the boundary condition at the surface of a porous medium. Studies in Appl. Math., L(2), 1971.

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[3] Simon J. Diﬀerentiation with respect to the domain in boundary value problems. Numer. Func. Anal. and Optimiz., 2 (7&8):649–687, 1980. [4] Jager W. and Mikeli´c A. On the interface boundary condition of beavers, joseph and saﬀman. SIAM, J. Appl. Math., 60(4):1111–1127, 2000. [5] Mikeli´c A., Jager W. and Neuss N. Asymptotic analysis of the laminar viscous ﬂow over a porous bed. SIAM J. Sci. Comp., 22(6):2006–2028, 2001. [6] Kuratoski K. Topolgy, volume I. Academic Press, New York, 1966. [7] Temam R. Navier-Stokes Equations. Theory and Numerical Analysis. NorthHolland Publishing Company, Amsterdam-New York-Oxford, 1977. [8] Sverak V. On optimal shape design. J. Math. Pure Appl., 72:537–551, 1993. [9] Jager W. and Mikeli´c A. On the boundary condition at the interface between a porous medium and a free ﬂuid. Annali Scuala Norm. Super. Pisa Cl. Sci., 23:403–465, 1996.

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CHAPTER VII THE UNIFORM FAT SEGMENT AND CUSP PROPERTIES IN SHAPE OPTIMIZATION Michel C. Delfour∗ Centre de Recherches Math´ematiques et D´epartement de Math´ematiques et de Statistique Universit´e de Montr´eal Montr´eal, Canada

Nicolas Doyon D´epartement de Math´ematiques et de Statistique Universit´e Laval Qu´ebec, Canada

Jean-Paul Zol´esio CNRS and INRIA Sophia Antipolis, France

Abstract

The object of this paper is to ﬁrst introduce the new uniform fat segment property along with a compactness theorem for a family of subsets of a bounded holdall verifying that property. In a second part the uniform cusp property introduced in [1], its extension in [2] to the larger class of dominating cusp functions which are only continuous at the origin, and the uniform fat segment property are shown to be equivalent.

Keywords: Segment property, cusp, compactness, oriented distance function, signed distance function, fat segment property, uniform cusp property, intrinsic shell model

∗ The

research of the ﬁrst author was supported by National Sciences and Engineering Research Council of Canada discovery grant A–8730, by a FQRNT grant from the Minist`ere de ´ l’Education du Qu´ebec, and by a Fellowship from the John Simon Memorial Foundation.

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CONTROL AND BOUNDARY ANALYSIS

Introduction A large class of domains Ω in RN can be characterized by a local geometric property, the segment property, which is equivalent to the property that Ω be locally a C 0 -epigraph. This property is suﬃcient to get the density of the function space C k (Ω) in the Sobolev space W m,p (Ω) for any 1 ≤ m ≤ k and it plays a key role in establishing the continuity of the solution of the Neumann problem for the Laplace operator with respect to the underlying domain Ω. Yet, even the stronger uniform segment property is not suﬃcient to get the compactness of subsets of a bounded open holdall verifying that property in any one of the metric topologies introduced in [1] via distance or characteristic functions (cf. [1], Chapter 5, Example 11.1, p. 256). The compactness can be recovered by going to the stronger uniform cusp property ﬁrst introduced in [1] (Chapter 5, § 11, p. 257) and it remains true for the larger class of dominating cusp functions which are only continuous at the origin [2]. The object of this paper is to ﬁrst introduce the new uniform fat segment property that does not a priori assume that the open region that fattens the segment be a cone or a cusp. Then we show that the uniform cusp property introduced in [1], its extension in [2] to the larger class of dominating cusp functions which are only continuous at the origin, and the uniform fat segment property are all equivalent. This emphasizes the fact that it is only the modulus of continuity of the cusp function at the origin that matters (see also [2] for the local epigraph viewpoint).

1.

Preliminaries: Topologies on Families of Sets

We ﬁrst introduce some notation. Given an integer N ≥ 1, mN and HN −1 will denote the N -dimensional Lebesgue and (N − 1)-dimensional Hausdorﬀ measures. The inner product and the norm in RN will be written x · y and |x|. The complement {x ∈ RN : x ∈ / Ω} and the N boundary Ω ∩ Ω of a subset Ω of R will be respectively denoted by Ω or RN \Ω and by ∂Ω or Γ. The distance function dA (x) from a point x to a subset A = ∅ of RN is deﬁned as inf{|y − x| : y ∈ A}. We recall a few results on metric topologies deﬁned on spaces of equivalence classes of sets constructed from the characteristic function, the distance or the oriented distance functions to a set. Given Ω ⊂ RN , Γ = ∅, the oriented distance function is deﬁned as def

bΩ (x) = dΩ (x) − dΩ (x).

(1)

It is Lipschitz continuous of constant 1, and ∇bΩ exists and |∇bΩ | ≤ 1 1,p (RN ) for all p, 1 ≤ p ≤ ∞. almost everywhere in RN . Thus bΩ ∈ Wloc Copyright © 2005 Marcel Dekker, Inc.

The Uniform Fat Segment and Uniform Cusp Properties

87

− Recall that b+ Ω = dΩ , bΩ = dΩ , and |bΩ | = dΓ , and that χintΩ = |∇dΩ |, χintΩ = |∇dΩ |, and χΓ = 1 − |∇dΓ | a.e. in RN , where χA denotes the characteristic function of a subset A of RN . Given a nonempty subset D of RN , the family Cb (D) = {bΩ : Ω ⊂ D and Γ = ∅} is closed in W 1,p (D). The following theorem is central. It shows that convergence and compactness in the metric on Cb (D) associated with W 1,p (D) will imply the same properties in the other topologies introduced in [1].

Theorem 1 Let D ⊂ RN be bounded open and 1 ≤ p < ∞. The maps − 1,p (D) → W 1,p (D)3 (2) bΩ → (b+ Ω , bΩ , |bΩ |) = (dΩ , dΩ , d∂Ω ) : Cb (D) ⊂ W (3) bΩ → (χ∂Ω , χintΩ , χintΩ ) : W 1,p (D) → Lp (D)3

are continuous. Proof. – They are well deﬁned from [1] (Chapter 5, Theorem 2.1 (iii), p. 207) for the map (2) and [1] (Chapter 5, Thm 2.2 (iv)-(v), p. 210) for the map (3). They are continuous from [1] (Chapter 5, Thm 5.1).

2. 2.1

The New Uniform Fat Segment Property Deﬁnitions

An open segment between two points x and y of RN will be denoted def

(x, y) = {x + t(y − x) : ∀t, 0 < t < 1}.

Definition 2 Let Ω be a subset of RN such that ∂Ω = ∅. (i) Ω is said to satisfy the segment property if ∀x ∈ ∂Ω, ∃r > 0, ∃λ > 0, ∃d ∈ RN , |d| = 1 such that

∀y ∈ B(x, r) ∩ Ω,

(y, y + λd) ⊂ intΩ.

(ii) Ω is said to satisfy the uniform segment property if ∃r > 0, ∃λ > 0 such that ∀x ∈ ∂Ω, ∃d ∈ RN , |d| = 1 such that

∀y ∈ B(x, r) ∩ Ω,

(y, y + λd) ⊂ intΩ.

Theorem 3 ([1], Chapter 2, Thm 7.2) If ∂Ω is compact, then the segment and uniform segment properties of Deﬁnition 2 coincide. The uniform segment property is too weak to yield the compactness of the corresponding family of sets which are contained in a bounded holdall D of RN (cf. [1], Chapter 5, Example 11.1, p. 256). This Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

suggests to fatten the open segment (0, λeN ) by replacing it with an open region containing (0, λd). It is natural to introduce the following assumption ∃O open in RN and λ > 0 such that (0, λeN ) ⊂ O and 0 ∈ / O,

(4)

where eN is the unit vector (0, . . . , 0, 1) ∈ RN . Recall the deﬁnition of the orthogonal subgroup of N × N matrices O(N) = {A : ∗A A = A ∗A = I} , def

(5)

where ∗A is the transposed matrix of A. A direction can be speciﬁed either by a matrix (of rotation) A ∈ O(N) or the corresponding unit vector d = AeN ∈ RN .

Definition 4 A set Ω ⊂ RN , ∂Ω = ∅, is said to satisfy the uniform fat segment property if there exist r > 0, λ > 0, and an open region O of RN containing (0, λeN ) and not 0 such that for all x ∈ ∂Ω, ∃A(x) ∈ O(N) such that ∀y ∈ Ω ∩ B(x, r),

y + A(x)O ⊂ intΩ.

(6)

It is clear that there are open regions O for which no set Ω ⊂ RN , ∂Ω = ∅, will satisfy the conditions of Deﬁnition 4 and that the class of signiﬁcant open regions can be considerably reduced.

Lemma 5 If Ω satisﬁes the uniform fat segment property with r, λ, and O, it also does with r, λ, and def

O+ = {y ∈ Oc : 0 < y · eN < λ} ,

(7)

where Oc is the connected open component of O which contains (0, λeN ). / O+ , and (0, λeN ) ⊂ O+ . Proof. – ∅ = O+ ⊂ Oc ⊂ O is open, 0 ∈

2.2

Compactness and Uniform Fat Segment Property

Given a bounded open subset D of RN consider the family Ω satisﬁes the uniform fat def L(D, r, O, λ) = Ω ⊂ D : segment property for (r, O, λ)

(8)

in the sense of Deﬁnition 4.

Theorem 6 Let D be a nonempty bounded open subset of RN and 1 ≤ p < ∞. Assume that L(D, r, O, λ) is not empty for r > 0, λ > 0, and Copyright © 2005 Marcel Dekker, Inc.

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The Uniform Fat Segment and Uniform Cusp Properties

an open subset O of RN such that (0, λeN ) ⊂ O and 0 ∈ / O. Then the family def

B(D, r, O, λ) = {bΩ : ∀Ω ∈ L(D, r, O, λ)} is compact in C(D) and W 1,p (D). As a consequence the families def

Bd (D, r, O, λ) = {dΩ : ∀Ω ∈ L(D, r, O, λ)} , def Bdc (D, r, O, λ) = {dΩ : ∀Ω ∈ L(D, r, O, λ)} , def Bd∂ (D, r, O, λ) = {d∂Ω : ∀Ω ∈ L(D, r, O, λ)} are compact in C(D) and W 1,p (D), and the following families are compact in Lp (D) def

X(D, r, O, λ) = {χΩ : ∀Ω ∈ L(D, r, O, λ)} , def X c (D, r, O, λ) = {χΩ : ∀Ω ∈ L(D, r, O, λ)}.

Remark 10 The compactness Theorem 6 is no longer true when the uniform fat segment property is replaced by the weaker uniform segment property: that is, when the region O containing (0, λeN ) is reduced to the open segment (0, λeN ) (cf. Example 11.1, Chapter 5 of [1]). The proof of Theorem 6 will require the following two lemmas.

Lemma 7 Given a bounded open subset D of RN , let {Ωn } be a sequence of subsets of D such that ∂Ωn = ∅ and m(∂Ωn ) = 0. Further assume ¯ such that ∂Ω = ∅ and m(∂Ω) = 0. Then for all that there exists Ω ⊂ D p, 1 ≤ p < ∞, bΩn bΩ in W 1,2 (D)-weak

⇒ bΩn → bΩ in W 1,p (D)-strong.

Proof. – From [2] or part (ii) of the proof of Theorem 10.1 and Theorem 5.1 (i) in Chapter 5 of [1] for the equivalence of the W 1,p -topologies.

Lemma 8 ( [1], Chapter 5, § 10, Lemma 10.1 ) Given a sequence {bΩn } ⊂ Cb (D) such that bΩn → bΩ in C(D) for some bΩ ∈ Cb (D), we have the following properties: ∀x ∈ Ω, ∀R > 0,

∃N (x, R) > 0, ∀n ≥ N (x, R),

B(x, R) ∩ Ωn = ∅,

and for all x ∈ Ω, ∀R > 0,

∃N (x, R) > 0,

Copyright © 2005 Marcel Dekker, Inc.

∀n ≥ N (x, R),

B(x, R) ∩ Ωn = ∅. (9)

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Moreover, ∀x ∈ ∂Ω, ∀R > 0, ∃N (x, R) > 0, ∀n ≥ N (x, R), B(x, R) ∩ Ωn = ∅ and B(x, R) ∩ Ωn ) = ∅, and B(x, R) ∩ ∂Ωn = ∅. Proof. – [Proof of Theorem 6] (i) Compactness in C(D). Since for all Ω in L(D, r, O, λ), Ω is locally a C 0 -epigraph, ∂Ω = ∅, bΩ ∈ Cb (D), and m(∂Ω) = 0. Consider an arbitrary sequence {Ωn } in L(D, r, O, λ). For D compact, Cb (D) is compact in C(D) and there exists Ω ⊂ D and a subsequence {Ωnk } such that bΩnk → bΩ in C(D). It remains to prove that Ω ∈ L(D, r, O, λ), that is ∀x ∈ ∂Ω, ∃A ∈ O(N) such that ∀y ∈ Ω ∩ B(x, r),

y + AO ⊂ intΩ.

From Lemma 8, for each x ∈ ∂Ω, ∀k ≥ 1, ∃nk ≥ k such that ' r( B x, k ∩ ∂Ωnk = ∅. 2 Denote by xk an element of that intersection: ' r( ∀k ≥ 1, xk ∈ B x, k ∩ ∂Ωnk . 2 By construction xk → x. Next consider y ∈ B(x, r) ∩ Ω. From the ﬁrst part of the lemma, there exists a subsequence of {Ωnk }, still denoted {Ωnk }, such that for all k ≥ 1, B(y, r/2k ) ∩ Ωnk = ∅. For each k ≥ 1 denote by yk a point of that intersection. By construction yk ∈ Ωnk → y ∈ Ω ∩ B(x, r). There exists K > 0 large enough such that for all k ≥ K, yk ∈ B(xk , r). To see this, note that y ∈ B(x, r) and that ∃ρ > 0,

B(y, ρ) ⊂ B(x, r) and

|y − x| +

ρ < r. 2

Now |yk − xk | ≤ |yk − y| + |y − x| + |x − xk | r r ρ ρ r < r. ≤ k + r − + k ≤ r + k−1 − 2 2 2 2 2 Since r/ρ > 1 the result is true for r ρ − < 0 ⇒ k > 2 + log(r/ρ). k−1 2 2 Copyright © 2005 Marcel Dekker, Inc.

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So we have constructed a subsequence {Ωnk } such that for k ≥ K xk ∈ ∂Ωnk → x ∈ ∂Ω and yk ∈ Ωnk ∩ B(xk , r) → y ∈ Ω ∩ B(x, r). ∀k, ∃Ak ∈ O(N), Ak ∗Ak = ∗Ak Ak = I, such that yk + Ak O ⊂ intΩnk . Pick another subsequence of {Ωnk }, still denoted {Ωnk }, such that ∃A ∈ O(N), A ∗A = ∗A A = I,

Ak → A.

Now consider z ∈ y + AO. Since y + AO is open there exists ρ > 0 such that B(z, ρ) ⊂ y + AO, and there exists K ≥ K such that ∀k ≥ K ,

B(z, ρ/2) ⊂ yk + Ak O ⊂ intΩnk = Ωnk ,

⇒ B(z, ρ/2) ⊃ Ωnk ⇒ 0 < ρ/2 = dB(z,ρ/2) (z) ≤ dΩn (z) → dΩ (z) k

⇒ 0 < ρ/2 ≤ dΩ (z)

⇒ z ∈ Ω = intΩ

⇒ y + AO ⊂ intΩ.

Hence Ω ⊂ D satisﬁes the uniform fat segment property. (ii) W 1,p (D)-Compactness. From Theorem 1 (i) it is suﬃcient to prove the result for p = 2. Consider the subsequence {Ωnk } ⊂ L(D, λ, O, r) and let Ω ∈ L(D, λ, O, r) be the set previously constructed such as bΩnk → bΩ in C(D). Hence bΩnk → bΩ in L2 (D). Since D is compact 2 |bΩnk | dx ≤ diam(D)2 dx ≤ diam(D)2 m(D), ∀Ω ⊂ D, D D |∇bΩnk |2 dx ≤ dx = m(D), D

D

and there exists a subsequence, still denoted {bΩnk }, which converges weakly to bΩ . Since all the sets are locally C 0 -epigraphs, m(∂Ωnk ) = 0 = m(∂Ω) and, by Lemma 7, bΩnk → bΩ in W 1,p (D)-strong, 1 ≤ p < ∞. (iii) The other compactnesses follow by continuity of the maps (2) and (3) in Theorem 1 and the fact that m(∂Ω) = 0 implies χintΩ = χΩ and χintΩ = χΩ almost everywhere for Ω ∈ L(D, λ, O, r).

3. 3.1

Equivalence of the Uniform Cusp and Fat Segment Properties Extended Cusp Property

The uniform cusp property introduced in [1] (Chapter 5, § 11) was speciﬁed by a continuous function h : [0, ρ[ → R such that h(0) = 0,

h(ρ) = λ,

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∀θ, 0 < θ < ρ,

0 < h(θ) < λ.

(10)

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Recall that with h of the form h(θ) = λ (θ/ρ)α , 0 < α ≤ 1, we recover the uniform cusp property for 0 < α < 1 and the uniform cone property for α = 1, ρ = λ tan ω and h(θ) = θ/ tan ω which corresponds to an open cone in 0 of aperture ω, height λ, and axis eN . This property can be viewed as a special case of condition (6) in Deﬁnition 4 by choosing the open set (11) O = (ζ , ζN ) ∈ RN : |ζ | < ρ and h(|ζ |) < ζN < λ . In [2] the uniform cusp property was extended to the following larger space of cusp functions h def

H = {h : [0, ∞[ → R : h(0) = 0 and h is continuous in 0}

(12)

by associating with h ∈ H, ρ > 0, and λ > 0 the axi-symmetrical region ) def

C(λ, h, ρ) =

(ζ , ζN ) ∈ RN : |ζ | < ρ and lim sup h(|ξ |) < ζN < λ ξ →ζ

(13)

around the axis eN = (0, . . . , 0, 1) in RN .

Lemma 9 Given ρ > 0, λ > 0, and h ∈ H, the region C(λ, h, ρ) contains the segment (0, λeN ), does not contain 0, and is open. Proof. – By deﬁnition N h(|ξ |) < ζN ⊂ C(λ, h, ρ), (0, λeN ) ⊂ (0, ζN ) ∈ R : 0 = lim ξ →0

since h(0) = 0 and h is continuous in 0. Also 0 = (0, 0) ∈ / C(λ, h, ρ) since it would yield the contradiction 0 = limξ →0 h(|ξ |) < ζN = 0. To show that C(λ, h, ρ) is open we ﬁx a point ζ = (ζ , ζN ) ∈ C(λ, h, ρ) and construct a neighborhood of ζ contained in C(λ, h, ρ). Let ε = ζN − ¯lζ > 0,

¯lζ def = lim sup h(|ξ |). ξ →ζ

By deﬁnition of the lim sup there exists ρ > 0 such that sup

ξ ∈B(ζ ,ρ)

h(|ξ |) < ¯lζ + ε/2. def

For all ξ = ζ in B(ζ , ρ) and ρξ = min{|ξ − ζ |, ρ − |ξ − ζ |} > 0 B(ξ , ρξ ) ⊂ B(ζ , ρ)

Copyright © 2005 Marcel Dekker, Inc.

⇒ ¯lξ < ¯lζ + ε/2,

¯lξ def = lim sup h(|η |). η →ξ

The Uniform Fat Segment and Uniform Cusp Properties

93

For all ξN ∈ R such that |ξN − ζN | < ε/2, we get ¯lξ − ξN = ¯lζ − ζN + ¯lξ − ¯lζ + ζN − ξN = −ε + ¯lξ − ¯lζ + |ζN − ξN | < −ε + ε/2 + ε/2 = 0 ⇒ (ξ , ξN ) ∈ C(λ, h, ρ), B(ζ, min{ρ, ε/2}) ⊂ C(λ, h, ρ) and C(λ, h, ρ) is open. Given λ > 0, ρ > 0, h ∈ H, and a direction d ∈ RN , |d| = 1, the rotated region from the direction eN to d is deﬁned as ) |PHd (y)| < ρ and def C(λ, h, ρ, d) = y ∈ RN : lim sup h(|P (z)|) < y · d < λ , (14) Hd z→y

where Hd = {d}⊥ is the hyperplane through 0 orthogonal to the direction d. The orientation can also be speciﬁed by the matrix of rotation A ∈ O(N) such that d = AeN and hence C(λ, h, ρ, d) = AC(λ, h, ρ). Finally, the translation of C(λ, h, ρ, d) to the point x will be denoted def

Cx (λ, h, ρ, d) = x + C(λ, h, ρ, d).

Lemma 10 For all λ > 0, ρ > 0, h ∈ H, and x ∈ RN , the regions C(λ, h, ρ) and Cx (λ, h, ρ, d) are nonempty and open. Moreover the segment (x, x + λd) is contained in Cx (λ, h, ρ, d). This led in [2] to the extension of the uniform cusp property in [1] (Chapter 5, § 11) to the larger class H. The function h is referred to as a cusp function and the space H as the space of cusp functions.

Definition 11 Let Ω be a subset of RN such that ∂Ω = ∅. (i) Ω satisﬁes the local uniform cusp property if ∀x ∈ ∂Ω, ∃h ∈ H, ∃λ > 0, ∃ρ > 0, ∃r > 0, ∃d ∈ RN , |d| = 1, such that ∀y ∈ B(x, r) ∩ Ω, Cy (λ, h, ρ, d) ⊂ intΩ. (ii) Given h ∈ H, Ω satisﬁes the h-local uniform cusp property if ∀x ∈ ∂Ω, such that

∃λ > 0, ∃ρ > 0, ∃r > 0, ∃d ∈ RN , |d| = 1, ∀y ∈ B(x, r) ∩ Ω, Cy (λ, h, ρ, d) ⊂ intΩ.

(iii) Ω satisﬁes the uniform cusp property for (r, λ, h, ρ) if ∃h ∈ H, ∃λ > 0, ∃ρ > 0, ∃r > 0, ∀x ∈ ∂Ω, ∃d ∈ RN , |d| = 1, such that ∀y ∈ B(x, r) ∩ Ω, Cy (λ, h, ρ, d) ⊂ intΩ. Copyright © 2005 Marcel Dekker, Inc.

94

3.2

CONTROL AND BOUNDARY ANALYSIS

Equivalence

It turns out that the uniform fat segment and cusp properties are equivalent either with a cusp function which is only continuous at the origin or with a cusp function which is continuous, non-negative, and monotone strictly increasing.

Theorem 12 Given a subset Ω of RN with nonempty boundary Γ, the following three conditions are equivalent. (i) Ω satisﬁes the uniform cusp property of Deﬁnition 11 (iii) for some r, λ, ρ, and a cusp function h ∈ H that is continuous, nonnegative, monotone strictly increasing in [0, ρ], and λ = h(ρ). (ii) Ω satisﬁes the uniform cusp property of Deﬁnition 11 (iii) for some r, λ, ρ, and a cusp function h ∈ H (that is continuous at the origin). (iii) Ω satisﬁes the uniform fat segment property of Deﬁnition 4 for some r, λ, and an open set O. Proof. – (i) ⇔ (ii). (i) ⇒ (ii) is obvious since h ∈ H. Conversely, let r > 0 and C(λ, h, ρ) be associated with the uniform cusp property of Deﬁnition 11 (iii). By continuity of h ∈ H in 0, ∃0 < θ0 ≤ ρ, ∀n ≥ 1,

∀0 ≤ θ ≤ θ0 ,

∃0 < θn < θn−1 /2,

|h(θ)| ≤ λ/2,

∀0 ≤ θ ≤ θn ,

|h(θ)| ≤ λ/2n+1 .

At each step n ≥ 0 construct the continuous monotone strictly increasing and non-negative function kn : [0, θ0 ] → R deﬁned as follows θj −θ θ−θ λ + 2λj θj −θj+1 , if θj+1 < θ ≤ θj , 0 ≤ j ≤ n − 1 def j+1 j+1 j+1 kn (θ) = 2 λ θθj −θ n −θ + 2λn θθn , if 0 ≤ θ ≤ θn . 2n+1 θn By continuity of h at the origin and the fact that h(0) = 0, θn → 0 and kn (0) → 0. By construction, 0 ≤ |h(θ)| ≤ kn+1 (θ) ≤ kn (θ) in [0, θ0 ], kn+1 (θ) = kn (θ) in [θn+1 , θ0 ], and kn+1 − kn C[0,θn+1 ] ≤ λ/2n+1 . Therefore there exists a continuous non-negative and monotone strictly increasing function k ∈ C[0, θ0 ] such that kn → k in C[0, θ0 ], k(0) = 0, and |h(θ)| ≤ k(θ) ≤ λ in [0, θ0 ]. Finally, if k(θ0 ) = λ, choose ρ such that k(ρ ) = λ, λ = λ, and h = k. If k(θ0 ) < λ, choose ρ = θ0 , λ = k(θ0 ), and h = k. From the construction, ρ ≤ ρ, λ ≤ λ, h ≥ h, and hence C(λ , h , ρ ) ⊂ C(λ, h, ρ). Therefore the local uniform cusp property of Deﬁnition 11 is veriﬁed with a non-negative, continuous, and monotone strictly increasing cusp function of the form (10). Copyright © 2005 Marcel Dekker, Inc.

The Uniform Fat Segment and Uniform Cusp Properties

95

(ii) ⇔ (iii). (ii) ⇒ (iii) from Lemma 9 by choosing O = C(λ, h, ρ). Conversely, assume that there exist r > 0, λ > 0, and O such that the uniform fat segment property is veriﬁed: for each x ∈ ∂Ω, there exists A(x) ∈ O(N) such that for all y ∈ Ω ∩ B(x, r), y + A(x)O ⊂ intΩ. From Lemma 5 the uniform fat segment property is also veriﬁed for r and λ and the subset O+ of O deﬁned in (7). Denote by H = {eN }⊥ the hyperplane through 0 orthogonal to eN . The projection PH (O+ ∩ B(0, r/2)) of the open set O+ ∩ B(0, r/2) onto the hyperplane H is an open set which contains 0 since the segment min{λ, r/2)}(0, eN ) is contained in O+ . Hence there exists an open ball BH (0, ρ) in H of radius ρ > 0 such that BH (0, ρ) ⊂ PH (O+ ∩ B(0, r/2)). Deﬁne the function a : BH (0, ρ) → R a(y ) =

def

inf

z∈O + ∩B(0,r/2) PH (z)=y

z · eN .

It is well deﬁned since the region O+ ∩ B(0, r/2) is bounded below by H, a(y ) ≥ 0, and a(0) = 0. It is continuous in 0. For any ε > 0, there exists 0 < δ ≤ min{ρ, ε/4} such that B((0, ε/2), δ) ⊂ O+ ∩ B(0, r) since the segment (0, λeN ) is a subset of the open set O+ . Hence for all |y | < δ, 0 ≤ a(y ) < ε. By construction of a, for all |ζ | ≤ ρ, (ζ , a(ζ )) ∈ B(0, r/2) ∩ O+ ⊂ B(0, r/2) ∩ O+ and, by the uniform fat segment property, for all y ∈ Ω ∩ B(x, r/2) ∀|ζ | ≤ ρ,

y + A(x)(ζ , a(ζ )) ∈ B(x, r)

∀|ζ | ≤ ρ,

y + A(x)(ζ , a(ζ )) ∈ y + A(x)O ⊂ intΩ ⊂ Ω

⇒ ∀|ζ | ≤ ρ, ⇒ ∀|ζ | ≤ ρ,

y + A(x)(ζ , a(ζ )) ∈ Ω ∩ B(x, r) y + A(x)(ζ , a(ζ )) + A(x)O ⊂ intΩ

and the fat segment property is veriﬁed for r/2, λ, and the new open region O =

def

(ζ , a(ζ )) + O : ∀|ζ | ≤ ρ .

To complete the proof we now construct a function h ∈ H from the function a such that C(λ, h, ρ) ⊂ O , where C(λ, h, ρ) is as deﬁned in (13). Consider the function sup|ζ |=1 a(θζ ), 0 ≤ θ < ρ def h(θ) = λ, otherwise is well deﬁned, non-negative, and h(0) = 0. By continuity of a in 0, ∀ε > 0, ∃0 < δ ≤ ρ such that |y − 0| < δ ⇒ |a(y )| = |a(y ) − a(0)| < ε Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

⇒ ∀θ, 0 ≤ θ < δ, ∀|ζ | = 1, |a(θζ )| = |a(θζ ) − a(0)| < ε ⇒ ∀θ, 0 ≤ θ < δ, |h(θ) − h(0)| = |h(θ)| = sup |a(θζ )| < ε |ζ |=1

and h is continuous in 0. Thus h ∈ H. By deﬁnition for all |ξ | < ρ, a(ξ ) ≤ h(|ξ |), and C(λ, h, ρ) ⊂ C(λ, a, ρ), where ) def

C(λ, a, ρ) =

(ζ , ζN ) ∈ RN : |ζ | < ρ and lim sup a(ξ ) < ζN < λ . ξ →ζ

To conclude we show that C(λ, a, ρ) ⊂ O . Since a(ζ ) ≥ 0, def ∀(ζ , ζN ) ∈ C(λ, a, ρ), 0 ≤ ¯lζ = lim sup a(ξ ) < ζN < λ ξ →ζ

⇒ 0 < ζN − ¯lζ < λ. By assumption the segment (0, λeN ) is contained in the open set O. Hence there exists a ball B((0, ζN − ¯lζ ), η) of radius η at the point (0, ζN − ¯lζ ) that is contained in O. By deﬁnition of the lim sup, there exists a sequence {ξn } ⊂ B(0, ρ) such that ξn → ζ and a(ξn ) → ¯lζ . Therefore there exists n ¯ such that for all n ≥ n ¯, (ζ , ζN ) = (ζ , ¯lζ ) + (0, ζN − ¯lζ ) ∈ (ξn , a(ξn )) + B((0, ζN − ¯lζ ), η) ⇒ ∀n ≥ n ¯ , (ζ , ζN ) ∈ (ξn , a(ξn )) + O ⊂ O ⇒ C(λ, a, ρ) ⊂ O ⇒ C(λ, h, ρ) ⊂ C(λ, a, ρ) ⊂ O . Hence the uniform cusp property is veriﬁed with r = r/2, λ, and C(λ, h, ρ).

References [1] M.C. Delfour and J.-P. Zol´esio, Shapes and Geometries: Analysis, Diﬀerential Calculus and Optimization, SIAM series on Advances in Design and Control, Society for Industrial and Applied Mathematics, Philadelphia, USA 2001. [2] M.C. Delfour, N. Doyon, and J.-P. Zol´esio, Extension of the uniform cusp property in shape optimization, in Control of Partial Diﬀerential Equations, G. Leugering, O. Imanuvilov, R. Triggiani, and B. Zhang, eds. Lectures Notes in Pure and Applied Mathematics, Marcel Dekker, New York, 2004. [3] M.C. Delfour, N. Doyon, and J.-P. Zol´esio, Uniform cusp property, boundary integral, and compactness for shape optimization, in System Modeling and Optimization, J. Cagnol and J.-P. Zol´esio, eds., pp. 25–40, Kluwer Academic Publishers, Norwell, MA, 2004.

Copyright © 2005 Marcel Dekker, Inc.

CHAPTER VIII TOPOLOGY OPTIMIZATION FOR UNILATERAL PROBLEMS Jan Sokolowski Institut Elie Cartan, Laboratoire de Math´ematiques Universit´e Henri Poincar´e Nancy I Vandoeuvre-L`es-Nancy, France

∗ ˙ Antoni Zochowski

Systems Research Institute of the Polish Academy of Sciences Warszawa, Poland

Abstract

The shape sensitivity analysis of unilateral problems is performed in [18]. The analysis is combined with outer asymptotic expansions of solutions to variational inequalities in order to derive topological derivatives. In particular, the problem of topology optimization is considered for free boundary problems of thin obstacle types. The formulae for the ﬁrst term of asymptotics for energy functionals are derived. The precision of obtained terms is veriﬁed numerically. The topological diﬀerentiability of solutions to variational inequalities is established.

Keywords: Shape optimization, topology optimization, asymptotic analysis, singular perturbations, unilateral problem

Introduction The boundary variations technique for unilateral problems is applied in [18] in the framework of nonsmooth analysis combined with the speed method. Nonsmooth analysis is necessary since the shape diﬀerentiability of solutions to unilateral problems is obtained only in the framework of Hadamard diﬀerentiability of metric projections onto polyhedric sets in the appropriate Sobolev spaces. However, to our best knowledge, there are no numerical methods for simultaneous shape and topology optimization [16] of unilateral problems. The main diﬃculty in analysis of

∗ Funding

provided by grant no. 4 T11A 01524 of the State Committee for the Scientiﬁc Research of the Republic of Poland (KBN).

Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

unilateral problems, e.g., the contact problems in elasticity, is associated with the nonlinearity of the nonpenetration condition over the contact zone which makes the boundary value problem nonsmooth. In the paper we propose a method for numerical evaluation of topological derivatives for such problems. The notion of topological derivative of a shape functional is introduced in [14]. The knowledge of topological derivatives is required for the optimality conditions of simultaneous shape and topology optimization. The topological derivative of a given shape functional can be determined from the variations of the shape functional created by the variations of the topology of geometrical domains. The topology variations are deﬁned by nucleation of small holes or cavities or more generally small defects in geometrical domains. The main idea we use to derive the topological derivatives for unilateral problems is the modiﬁcation of the energy functional by an appropriate correction term and subsequent minimization of the resulting energy functional over the cone of admissible displacements. The asymptotic analysis of elliptic boundary value problems is performed in [4], [8], [10]. The Signorini problem is considered in [1], [2], [3]. Conical diﬀerentiability of solutions to variational inequalities is established, e.g., in [6], [12] in the framework used in the paper. The topological derivatives of shape functionals and optimality conditions for topology optimization are derived in [7], [11], [14], [15], [16] for elliptic problems. Applications to inverse problems are given in [5]. In the paper we derive useful formulae for the correction terms of the energy functionals. We restrict ourselves to two-dimensional problems and to singular perturbations of geometrical domains in the form of small discs. The correction terms are derived in such a form, that the numerical veriﬁcation of its precision is straightforward. On the other hand, the terms are directly used to establish the topological diﬀerentiability of solutions to variational inequalities.

1. 1.1

Transformations of the Energy Functional for the Laplace Equation Fundamental Formulae

Let us consider the bulk energy functional of the form 1 ∇u2 dx, E(u) = 2 Ω

(1)

where u satisﬁes inside the domain Ω ⊂ R2 the Laplace equation ∆u = 0 Copyright © 2005 Marcel Dekker, Inc.

(2)

99

Topology Optimization for Unilateral Problems

with suitable boundary conditions. Our goal is to study the inﬂuence of the small circular hole of the variable radius ρ contained in the domain; for simplicity we put its center at x = 0. We do not want to study variable domains, so we isolate this hole inside the ring C(ρ, R) = { x | ρ < x < R } and replace E(u) by the equivalent expression over ΩR = Ω \ B(R), with an additional boundary integral term over ΓR = ∂B(R). On Γρ = ∂B(ρ) we assume homogeneous Neumann conditions. In the ﬁrst step we modify E(u). Since 2 2 ∇u dx = ∇u dx + ∇u2 dx = (3) Ω ΩR B(R) ∂u ds, ∇u2 dx + u ΩR ΓR ∂n where n is the outward normal vector on the boundary of B(R), we may concentrate on the expression ∂u ds. (4) ER (u) = u ΓR ∂n Here the values of ∂u/∂n on ΓR are given by the Dirichlet to Neumann mapping, namely the solution of ∆v = 0 in B(R), v = u

on ΓR ,

(5)

by means of the relation ∂u/∂n = ∂v/∂n. The function v may be constructed with the help of the Poisson kernel, i.e., expressed as its integral over the circle ΓR . Using this representation and asymptotic expansions of solutions to the Laplace equation around the small hole it is possible [17] to calculate the ﬁrst term in the perturbation of ER (u) caused by the introduction of B(ρ): *

2

2 + ρ2 ux1 ds + ux2 ds . (6) δER (u) = − 6 πR ΓR ΓR This allows us to consider the following approximation. Let uρ satisfy (2) inside Ωρ = Ω \ B(ρ) and take 1 ∇uρ 2 dx. (7) E(ρ ; uρ ) = 2 Ωρ Then we may consider uρ in a ﬁxed domain ΩR and add boundary terms 1 1 1 ∇uρ 2 dx + ER (uρ ) + δER (uρ ) + o(ρ2 ). (8) E(ρ ; uρ ) = 2 ΩR 2 2 Copyright © 2005 Marcel Dekker, Inc.

100

CONTROL AND BOUNDARY ANALYSIS

The representation (6) allows us to transform the task from solving the boundary value problem for the Laplace equation in variable domain Ωρ to the same problem in the ﬁxed domain ΩR , but with energy parametrized by ρ. In fact, since the ﬁrst terms represent energy for the whole of Ω, we may even avoid using ΩR (and the remeshing of Ω in order to get discretization of ΩR ). This results from observation that outside B(R) the function wR coincides with w given by the solution of the boundary value problem with bulk energy given by

(9) E0 (ρ ; w) = * +

2 2 ρ2 1 ∇w2 dx − wx1 ds + wx2 ds + o(ρ2 ) 2 Ω 2πR6 ΓR ΓR The weak form of this problem is easily obtained by taking variation of E0 (ρ ; w), adding external work and using boundary conditions on ∂Ω. As we see, modiﬁcation of bulk energy accounts to introducing source term concentrated on the ﬁxed circle ΓR and parametrized by ρ. The last formula may be expresed also in another form. Let us denote by eu (x) the energy density at the point x, eu (x0 ) = ∇u(x0 )2 . If the function u is harmonic in B(R), then, as may be proved [17], the expressions for derivatives 1 u · (x1 − x1,0 ) ds, u/1 (x0 ) = πR3 ΓR (x0 ) 1 u/2 (x0 ) = u · (x2 − x2,0 ) ds. πR3 ΓR (x0 ) are exact. In view of this, formula (9) can be rewritten in the equivalent form 1 E0 (ρ ; w) = 2

Copyright © 2005 Marcel Dekker, Inc.

1 ∇w2 dx − πρ2 ew (0) + o(ρ2 ). 2 Ω

(10)

Topology Optimization for Unilateral Problems

1.2

101

Numerical Tests of the Transformed Energy Approach

We consider the test problem ∆uρ = 0 in 0 1 uρ = x21 √ x1 ∂uρ = 0 on ∂n

Ωρ = [0, 1] × [0, 1] \ B(x0 , ρ), for for for for

x1 x1 x2 x2

= 0, = 1, = 0, = 1.

(11)

∂B(x0 , ρ).

Three types of approximations are used. Here by u we denote the solution in the domain without void. 1 Double correction: the function uρ is represented in the form uρ = u + sρ (u) + pρ + sρ (pρ ), where sρ (u) is the ﬁrst correction for u. However, sρ (u) disturbs the boundary conditions on ∂Ω and we introduce second corrections. The function pρ solves the Laplace equation with the boundary condition pρ = −sρ (u) on ∂Ω, and then again sρ (pρ ) nulliﬁes the Neumann condition on the boundary of the void. We consider this solution as nearly exact in Ωρ . 2 “Exact” solution: here the right-hand side is augmented by the expression containing derivatives of Dirac’s delta, ∆uρ = 2πρ2 (∇δ(x − x0 ) · ∇u). In theory this solution is also exact in Ωρ up to the higher than 2 powers of ρ, see [9], but of course there are diﬃculties with numerical approximation. 3 Solution obtained by the modiﬁcation of the energy term given by (10). This should be exact up to the higher than 2 powers of ρ outside the ring of the radius R > ρ. The hole was positioned at x0 = [0.5, 0.7] and had the radius ρ = 0.05. The ﬁgures show the results of computations for diﬀerent positions of the void and ratios R/ρ, in the form of sections through the surface u(x1 ) = u(x1 , x02 ) along the line x2 = x02 , i.e., going through the middle of the hole. Other sections look very similar. We may conclude that the Copyright © 2005 Marcel Dekker, Inc.

102

CONTROL AND BOUNDARY ANALYSIS 1

flat area - position at the hole

0.8

with double correction

0.6

solution obtained via correction term in energy

0.4 solution in the intact domain

0.2 0

exact - with singular source term

–0.2 boundary of B(R)

–0.4 –0.6 –0.8 0

10

20

30

40

50

60

70

80

90 100

Figure 1. Comparison of solutions for R = 1.2ρ. Visible slight loss of accuracy near boundary of the ring, due to the small ratio of the radii.

modiﬁed energy allows, in concordance with the theory developed in the last subsection, to compute accurately the solution outside the ring ΓR . Moreover we have seen a very good agreement of the solution with the singular right-hand side with the nearly exact solution produced by double correction.

2.

Signorini Problem

We establish the conical diﬀerentiability of solutions to the Signorini problem with respect to the small parameter. The obtained expansion of solutions to the Signorini problem can be interpreted as the ﬁrst order outer asymptotic expansion in the spirit, e.g., of [11]. So, this way, we can deﬁne the topological derivatives of some shape functionals, including the energy functional, for variational inequalities. To our best knowledge this is the ﬁrst result in this direction derived without any assumptions on the strict complementarity conditions for the unknown solutions to the obstacle problems. In order to introduce the Signorini problem we need the bilinear form a(·, ·), the convex set K, the linear form L(·), and the convex cone , ⊂ V . We are going to deﬁne the variational in the Sobolev space K inequality over the space H 1 (Ωρ ), where Ωρ = Ω \ B(ρ), with the small ball B(ρ) = {x ∈ R2 |x − O < ρ}, here O is the center of the ball, and we assume that the center is the origin. So, the variational inequality reads: Copyright © 2005 Marcel Dekker, Inc.

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Topology Optimization for Unilateral Problems

, ρ ) = {v ∈ H 1 (Ωρ )|v ≥ 0 on Γs } such that a(u, v − u) ≥ Find u ∈ K(Ω Γ0 , L(v − u) for all v ∈ K(Ωρ ). The unique solution u(Ωρ ) of the variational inequality depends on the small parameter ρ. In order to analyze the dependence of the solution u(Ωρ ) on ρ we proceed in the following way. The bilinear form ∇u · ∇vdx a(u, v) = Ωρ

deﬁned on the variable geometrical domain Ωρ is linearized using the asymptotic expansions technique for the energy functionals. The linearized bilinear form is denoted by a(ρ; ·, ·) and contains in two dimensions the term of order zero with respect to ρ and the ﬁrst order perturbation depending on ρ2 . In Section VIII.1.1 the following bilinear form is identiﬁed as the ﬁrst order perturbation 1 1 wx1 ds vx1 ds − wx ds vx2 ds . b(w, v) = − 6 2 πR ΓR πR6 ΓR ΓR ΓR The bilinear form b(w, v) is continuous on the space H 1 (Ω), where ΓR = ∂B(R), and the ball B(R) ⊂ Ω. Thus, the bilinear form a(·, ·) is replaced in the variational inequality by the integral expression, depending only on ρ as a small parameter, ∇w · ∇vdx + ρ2 b(w, v) , (12) a(ρ; w, v) = Ω

a(ρ; w, v) is deﬁned over the domain of integration Ω ⊂ R2 with the smooth boundary ∂Ω = Γ0 ∪ Γs , with w, v in the energy space HΓ10 (Ω) = {v ∈ H 1 (Ω)|v = 0 on Γ0 } . Thus, the singular perturbation of the geometrical domain Ωρ is replaced in the variational inequality under considerations by the regular perturbation of the bilinear form. For such regular perturbation the standard sensitivity analysis of variational inequalities over polyhedric sets in Dirichlet spaces applies (see, e.g., [13], [6]) and the ﬁrst order expansion of the solution with respect to the small parameter is obtained. We refer the reader to [3] for the direct asymptotic analysis of the energy functional for the Signorini problem. For ρ suﬃciently small we can consider the variational inequality with the bilinear form a(ρ; w, v) over the convex set K = {v ∈ H 1 (Ω)|v ≥ 0 Copyright © 2005 Marcel Dekker, Inc.

on Γs ,

v=g

on Γ0 } .

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CONTROL AND BOUNDARY ANALYSIS

Let uρ denote the unique solution of the following variational inequality uρ ∈ K :

a(ρ; w, v − w) ≥ 0 ∀v ∈ K .

(13)

The following theorem may be proved [17].

Theorem 1 For ρ suﬃciently small we have the following expansion of the solution uρ , with respect to the parameter ρ, at 0+, uρ = u + ρ2 q + o(ρ2 ) ,

(14)

where the outer topological derivative q of the solution u = u(Ω) to the Signorini problem is given by the unique solution of the following variational inequality q ∈ SK (u) = {v ∈ HΓ10 (Ω)|v ≥ 0 on Ξ(u) , a(u, v) = 0} a(q, v − q) + b(u, v − q) ≥ 0 ∀v ∈ SK (u) .

(15) (16)

The coincidence set Ξ(u) = {x ∈ Γs |u(x) = 0} is well deﬁned [13] for any function u ∈ H 1 (Ω), and u ∈ K ⊂ V = HΓ10 (Ω) is the solution of the variational inequality for ρ = 0.

Concluding Remarks We presented the topological derivatives of solutions for variational inequalities. The same approach is applied in [17] to the frictionless contact problems in elasticity. Since the analysis becomes more involved for contact problems, we restricted our presentation to a scalar problem. The case of evolution problems seems to be open, although a result on conical diﬀerentiability of solutions to parabolic variational inequalities is established in [6].

References [1] I.I. Argatov and S.A. Nazarov. Asymptotic solution to the Signorini problem with small parts of the free boundary. Siberian Mat. Zh., 35:258–277, 1994. [2] I.I. Argatov and S.A. Nazarov. Asymptotic solution of the Signorini problem with an obstacle on a thin elongated set. Mat. sbornik, 187:3–32, 1996 (English translation in: Math. Sbornik. 187:1411–1442, 1996). [3] I.I. Argatov and J. Sokolowski. On asymptotic behaviour of the energy functional for the Signorini problem under small singular perturbation of the domain. Journal of Computational Mathematics and Mathematical Physics, 43:742–756, 2003. [4] A.M. Il’in. Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, volume 102 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1992.

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˙ [5] L. Jackowska-Strumillo, J. Sokolowski, A. Zochowski, and A. Henrot. On numerical solution of shape inverse problems. Computational Optimization and Applications, 23:231–255, 2002. [6] J. Jaruˇsek, M. Krbec, M. Rao, and J. Sokolowski. Conical diﬀerentiability for evolution variational inequalities. Journal of Diﬀerential Equations, 193:131– 146, 2003. [7] T. Lewinski and J. Sokolowski. Energy change due to appearing of cavities in elastic solids. International Journal of Solids and Structures, 40:1765–1803, 2003. [8] V.G. Maz’ya, S.A. Nazarov, and B.A. Plamenevskii. Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains. Birkhuser Verlag, Basel, Vol. 1, 2, 2000. [9] S.A. Nazarov. Asymptotic conditions at a point, self-adjoint extensions of operators, and the method of matched asymptotic expansions. American Mathematical Society Translations, 198:77–125, 1999. [10] S.A. Nazarov and B.A. Plamenevsky. Elliptic Problems in Domains with Piecewise Smooth Boundaries. Walter de Gruyter, De Gruyter Exposition in Mathematics 13, New York, 1994. [11] S.A. Nazarov and J. Sokolowski. Asymptotic analysis of shape functionals. Journal de Math´ematiques pures et appliqu´ ees, 82:125–196, 2003. [12] M. Rao and J. Sokolowski. Non-linear balayage and applications. Illinois Journal of Mathematics, 44:310–328, 2000. [13] M. Rao and J. Sokolowski. Tangent sets in banach spaces and applications to variational inequalities. Technical Report 42, Institute Elie Cartan, http://www.iecn.u-nancy.fr/ sokolows/, 2000. Submitted for publication. ˙ [14] J. Sokolowski and A. Zochowski. On topological derivative in shape optimization. SIAM Journal on Control and Optimization, 37:1251–1272, 1999. ˙ [15] J. Sokolowski and A. Zochowski. Topological derivatives of shape functionals for elasticity systems. Mechanics of Structures and Machines, 29:333–351, 2001. ˙ [16] J. Sokolowski and A. Zochowski. Optimality conditions for simultaneous topology and shape optimization. SIAM Journal on Control and Optimization, 42:1198–1221, 2003. ˙ [17] J. Sokolowski and A. Zochowski. Topological optimization for contact problems. Technical Report 25, Institute Elie Cartan, http://www.iecn.unancy.fr/ sokolows/, 2003. Submitted for publication. [18] J. Sokolowski and J.-P. Zol´esio. Introduction to Shape Optimization, volume 14 of SSCM. Springer-Verlag, New York, 1992.

Copyright © 2005 Marcel Dekker, Inc.

CHAPTER IX SECOND ORDER LAGRANGE MULTIPLIER APPROXIMATION FOR CONSTRAINED SHAPE OPTIMIZATION PROBLEMS M˚ artensson’s Approach for Shape Problems Karsten Eppler∗ Weierstraß Institut f¨ ur Angewandte Analysis und Stochastik Berlin, Germany

Helmut Harbrecht Institut f¨ ur Informatik und Praktische Mathematik Christian–Albrechts–Universit¨ at zu Kiel Kiel, Germany

Abstract

The present paper is dedicated to the solution of constrained shape optimization problems by second order algorithms with respect to both the primal and dual variables. This goal is realized by combining a Newton scheme for the primal variables with M˚ artensson’s concept of Lagrange multiplier functions for augmented Lagrangians.

Keywords: Shape optimization, elliptic PDE, Newton method, wavelet BEM, augmented Lagrangian approach

Introduction Shape optimization is quite indispensable for designing and constructing industrial components. Many problems that arise in application, particularly in structural mechanics and in the optimal control of distributed parameter systems, can be formulated as the minimization of functionals deﬁned over a class of admissible domains. Therefore, such problems have been intensively studied in the literature throughout the last 25–30 years (see [16–17], [21, 23], and the references therein). From ∗ Funding

provided by DFG project no. TR 302/6-1.

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the practical as well as from the theoretical point of view, to prevent optimal shapes from degeneration, several constraints have to be taken into account. More recently, the computation of the related dual variables, i.e. the Lagrange multipliers, becomes of increasing interest due to applicational and theoretical reasons. On the one hand, there are several applications where these multipliers have an important physical meaning like in the electromagnetic shaping of liquid metals (see, e.g., [19–20]). On the other hand, their computation is important by itself for the investigation of suﬃcient optimality conditions in shape optimization [6]. In [9, 11]we developed ﬁrst and second order optimization algorithms for a class of elliptic shape optimization problems, where a powerful wavelet BEM is proposed for the computation of the state and related quantities. Additional equality and/or inequality constraints of functional type are treated by an augmented Lagrangian technique, which turns out to be more stable and eﬃcient than a penalty method. In particular, by the standard update procedure, a linear convergence is provided for the Lagrange multipliers. However, due to the eﬃciency of the Newton method with respect to the primal variables, that is, the shape respective to its ﬁnite dimensional approximation, this slow convergence of the Lagrange multipliers becomes, in our opinion, the bottleneck of the overall algorithm. Furthermore, the question of a faster approximation of the dual variables is important as we explained above. Following an idea of M˚ artensson [18], second order convergence is realizable if only active equality constraints are present. Due to known degeneration tendencies in shape problems, exactly this situation occurs very often in practical and theoretical shape optimization problems. Consequently, the goal of the present paper is to demonstrate this approach for the class of shape problems considered in [9]. Of course, both the standard method and its modiﬁcation are independent of the underlying shape calculus and of the numerical method for solving the state equation. The outline of the present paper is as follows. In the ﬁrst section we brieﬂy introduce our setup for solving the considered shape optimization problems developed in [9–11]. The second section is dedicated to the augmented Lagrangian approach and the improvement of M˚ artensson by introducing so-called Lagrange multiplier functions [18]. The last section presents the numerical results.

1. 1.1

Shape Optimization The Model Problem

Let Υ denote the set of all bounded domains Ω ∈ C 2,α , Ω ⊆ D ⊆ R2 , starshaped with respect to Bδ (0). The outer security set D ⊆ R2 (the Copyright © 2005 Marcel Dekker, Inc.

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Second Order Lagrange Multiplier Approximation

hold all set) is simply connected and closed, but not necessarily bounded. Setting Γ := ∂Ω, we consider the following shape problem: h(x)u(x) + h0 (x)dx = min (1) J(Ω) = Ω∈Υ

Ω

subject to

J1 (Ω) =

h1 (x)dx = c1 , Ω

J2 (Ω) = .. . Jm (Ω) =

h2 (x)dx = c2 , Ω

(2) hm (x)dx = cm ,

Ω

where the state function u solves the Dirichlet boundary value problem −∆u = f u=g

in Ω, on Γ.

(3)

In order to conceive a well-posed problem the functions f , g, h and h0 , . . . , hm are assumed suﬃciently regular on the whole set D. Of course, constraints of the type hi (x)dσx Ji (Ω) = Γ

may be considered in our setup as well. Constraints

of practical interest in shape problems are the volume of the domain Ω 1 dx or its perimeter

1 dσ , for example. x Γ Let us remark that (1) includes also the important case of energy functionals. Consider the state equation having homogeneous boundary conditions (g ≡ 0), then integration by parts yields 2 ∇u(x) dx = f udx, J(Ω) = Ω

Ω

that is h ≡ f and h0 ≡ 0.

1.2

Shape Calculus

Clearly, the starshaped domain Ω ∈ Υ can be identiﬁed with a function which describes its boundary, i.e., in polar coordinates we have φ : φ ∈ [0, 2π] , Γ := γ(φ) = r(φ) cos sin φ Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

2,α where r ∈ Cper [0, 2π] is a positive function with r > δ and 2,α [0, 2π] = {r ∈ C 2,α [0, 2π] : r (i) (0) = r (i) (2π), i = 0, 1, 2}. Cper

As a standard variation for perturbed domains Ωε and boundaries Γε , 2,α [0, 2π] respectively, we introduce a function dr ∈ Cper rε (φ) = r(φ) + εdr(φ), where γε (φ) = rε (φ)er (φ) is always a Jordan curve. Herein, er (φ) = (cos φ, sin φ)T denotes the unit vector in the outer radial direction. The main advantage of this simple approach is a complete embedding of the shape problem into a Banach space setting. That is, both the shapes 2,α [0, 2π]. and its increments, can be viewed as elements of Cper Let us remark that boundary variation is an appropriate ansatz since ﬁrst and second order directional derivatives can be represented by boundary integrals, cf. [7–8].

1.3

BIE Formulation and Fast Solvers

It turns out that introducing the adjoint state p by −∆p = h p=0

in Ω, on Γ,

the directional derivative ∇J(Ω)[dr] requires the knowledge of ∂u/∂n and ∂p/∂n, where the state function u satisﬁes (3), cf. [9]. In addition, deﬁning the local shape derivative du[dr] and the adjoint local shape derivative dp[dr] via ∆du[dr] = 0 du[dr] = drer , n

in Ω, ∂(g − u) ∂n

on Γ,

and ∆dp[dr] = 0 ∂p dp[dr] = −drer , n ∂n

in Ω, on Γ,

respectively, the computation of ∇2 J(Ω)[dr1 , dr2 ] requires the quantities ∂ 2 u/(∂n∂t), ∂ 2 p/(∂n∂t), ∂du[dr]/∂n, and ∂dp[dr]/∂n, see [9]for details. As shown in [9, 11], employing suitable Newton potentials, the computation of the derivatives of u, du, and the associated adjoints can be performed eﬃciently by boundary integral equations (BIEs). Boundary element methods provide a common tool for the solution of such Copyright © 2005 Marcel Dekker, Inc.

Second Order Lagrange Multiplier Approximation

111

equations. In general, cardinal B-splines are used as ansatz functions in the Galerkin formulation. However, these traditional discretizations yield densely populated and in general ill-conditioned system matrices. Hence, a numerical scheme is of at least quadratic complexity. The crucial idea of the wavelet Galerkin scheme is a change of bases, i.e., applying appropriate wavelet bases instead of the traditional singlescale bases. On the one hand, based on the well-known norm equivalences of wavelet bases, the diagonals of the system matrices deﬁne optimal preconditioners, cf. [3, 22]. On the other hand, the resulting quasi-sparse system matrices can be compressed without loss of accuracy such that the complexity for the solution of the boundary integral equations becomes linear , cf. [5, 13, 14, 15].

1.4

Boundary Approximation

Since the inﬁnite dimensional optimization problem cannot be solved directly, we replace it by a ﬁnite dimensional problem. Based on the polar coordinate approach, we can express the smooth function r ∈ 2,α ([0, 2π]) by the Fourier series Cper r(φ) = a0 +

∞

an cos nφ + a−n sin nφ.

n=1

Hence, it is reasonable to take the truncated Fourier series rN (φ) = a0 +

N

an cos nφ + a−n sin nφ.

(4)

n=1

as approximation of r. We mention that other boundary representations like B-splines can be considered as well. The advantage of our approach is an exponential convergence rN → r if the shape is analytical. To this end, since rN has the 2N + 1 degrees of freedom a−N , a1−N , . . ., aN , we replace the inﬁnite dimensional set of admissible domains Υ by an 2N + 1 dimensional set ΥN deﬁned by ΥN := {a−N , a1−N , . . . , aN ∈ R : rN (φ) > 0, φ ∈ [0, 2π]} ⊂ R2N +1 . Consequently, according to this ﬁnite dimensional setting, we conclude ∇J(Ω)[cos N φ] ∇J(Ω)[cos(N −1)φ]

∇J(Ω) =

.. .

∈ R2N +1

∇J(Ω)[sin N φ],

and likewise for the shape hessian ∇2 J(Ω) ∈ R(2N +1)×(2N +1) . Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

2.

Optimization of Constrained Problems

2.1

Augmented Lagrangian Functionals

Setting λ = (λ1 , λ2 . . . , λm )T ∈ Rm , the minimization problem deﬁned by (1) and (2) implies to ﬁnd the stationary point (Ω , λ ) ∈ ΥN × Rm of the following modiﬁed augmented Lagrangian functional - - C1 (Ω)−c1 -2 C1 (Ω)−c1 2 2 α- C2 (Ω)−c C2 (Ω)−c .. .. Lα (Ω, λ) = J(Ω) + λT (5) + - - . . . 2Cm (Ω)−cm Cm (Ω)−cm Of course, the choice α = 0 yields the pure Lagrangian while λ = 0 and α → ∞ is known as standard quadratic penalty method. However, both methods have some drawbacks from the numerical point of view, cf. [12, 18], for example. In order to avoid these diﬃculties, we consider α > 0 ﬁxed but appropriately chosen and perform the following standard augmented Lagrangian algorithm: initialization: choose initial guesses λ(0) for λ and Ω(0) for Ω inner iteration: solve Lα (Ω, λ(n) ) = min with initial guess Ω(n) outer iteration: update

λ(n+1) := λ(n) − α

C1 (Ω)−c1 C2 (Ω)−c2

.. .

(6)

Cm (Ω)−cm

and choose Ω(n+1) as the solution from the inner iteration. As we have observed in [9], a Newton scheme for minimizing Lα (Ω, λ(n) ) makes a line-search nearly obsolete in comparison to ﬁrst order methods. As a consequence, the bottleneck during the iteration process is now the slow ﬁrst order approach of the Lagrange multipliers according to the following theorem, cf. [2, 12].

Theorem 1 Assuming that the minimization problem (1) and (2) has a solution Ω ∈ ΥN with λ ∈ Rm satisfying the KKT-conditions as well as the suﬃcient second order conditions, then the update rule (6) yields a linear convergence λ(n) → λ , provided that the initial guesses are properly chosen. We like to mention that augmented Lagrangian techniques are also available for inequality constraints, but due to the lack of smoothness second order algorithms would not work. Copyright © 2005 Marcel Dekker, Inc.

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Second Order Lagrange Multiplier Approximation

The gradient of the functional (5) reads as ∇Lα (Ω, λ) = ∇J(Ω) +

m

λi + α(Ci (Ω) − ci ) ∇Ci (Ω),

i=1

and its hessian as ∇ Lα (Ω, λ) = ∇ J(Ω) + 2

2

+α

m i=1

∇C1 (Ω) ∇C2 (Ω)

.. .

T

∇Cm (Ω)

2.2

λi + α(Ci (Ω) − ci ) ∇2 Ci (Ω) ∇C1 (Ω) ∇C2 (Ω)

.. .

.

∇Cm (Ω)

Second Order Lagrange Multiplier Update

M˚ artensson [18] proposed a second order method for the Lagrange multiplier. More precisely, the multiplier is seen as a function depending on the primal variable, i.e., λ = λ(Ω), namely T ∇ C1 (Ω)∇J(Ω) T C (Ω)∇J(Ω) 2

∇ λ(Ω) = −A(Ω)−1

.. .

m ∈R ,

(7)

∇T Cm (Ω)∇J(Ω)

where the matrix A(Ω) is given by T A(Ω) =

∇ C1 (Ω)∇C1 (Ω) ... ∇T C1 (Ω)∇Cm (Ω) ∇T C2 (Ω)∇C1 (Ω) ... ∇T C2 (Ω)∇Cm (Ω)

.. .

.. .

m×m . ∈R

∇T Cm (Ω)∇C1 (Ω) ... ∇T Cm (Ω)∇Cm (Ω)

Instead of the constant multiplier in the shape functional (5), a ﬁrst order Taylor expansion of the above multiplier function, developed at the given initial guess Ω(n) , is inserted into the augmented Lagrangian functional. Consequently, using T ∇ C1 (Ω)∇2 J(Ω)+∇T J(Ω)∇2 C1 (Ω) ∇T C2 (Ω)∇2 J(Ω)+∇T J(Ω)∇2 C2 (Ω) .. λ (Ω) = −A(Ω)−1 . T ∇ Cm (Ω)∇2 J(Ω)+∇T J(Ω)∇2 Cn (Ω) T ∇ C1 (Ω)∇2 Ci (Ω)+∇T Ci (Ω)∇2 C1 (Ω) m ∇T C2 (Ω)∇2 Ci (Ω)+∇T Ci (Ω)∇2 C2 (Ω) m×(2N +1) .. − λi , ∈R . i=1 T 2 T 2 ∇ Cm (Ω)∇ Ci (Ω)+∇ Ci (Ω)∇ Cm (Ω)

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CONTROL AND BOUNDARY ANALYSIS

we consider the functional (n) ) + λ (Ω(n) )(Ω − Ω(n) ) . L(n) α (Ω) = Lα Ω, λ(Ω

(8)

Herein, the diﬀerence Ω − Ω(n) has to be understood as the diﬀerence of the associated Fourier coeﬃcients. Of course, the gradient and the hessian of (8) are slightly diﬀerent compared to those of (5), but the computation is straightforward and all ingredients appear also in the ﬁrst and second order derivatives of (5): the gradient is given by (n) ∇L(n) ) + λ (Ω(n) )(Ω − Ω(n) ) α (Ω) = ∇Lα Ω, λ(Ω + λ (Ω(n) )T

C1 (Ω)−c1 C2 (Ω)−c2

.. .

Cm (Ω)−cm

and the hessian by

2 (n) ∇2 L(n) ) + λ (Ω(n) )(Ω − Ω(n) ) α (Ω) = ∇ Lα Ω, λ(Ω T T T + λ (Ω(n) )T

∇ C1 (Ω) ∇T C2 (Ω)

.. .

∇T Cm (Ω)

+

∇ C1 (Ω) ∇T C2 (Ω)

.. .

(n) λ (Ω ).

∇T Cm (Ω)

M˚ artensson’s approach yields the following optimization algorithm: initialization: choose an initial guess Ω(0) for Ω (n)

inner iteration: solve Lα (Ω) = min by the Newton scheme with initial guess Ω(n) outer iteration: choose Ω(n+1) as the solution from the inner iteration

Theorem 2 (M˚ artensson [18]) Assume α suﬃciently large and (Ω , λ ) ∈ ΥN × Rm satisfying the assertion of Theorem 1, then the above algorithm converges quadratically to (Ω , λ ), i.e., this algorithm has one step convergence for linear-quadratic problems.

3. 3.1

Numerical Results A Shape Problem from Planar Elasticity

For comparison reasons we employ ﬁrst a model problem where the optimal shape and the dual variables are known analytically. We consider Copyright © 2005 Marcel Dekker, Inc.

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Second Order Lagrange Multiplier Approximation

a cylindric circular bar which is homogeneous and isotropic with a planar, simply connected cross section Ω ∈ R2 . We follow Banichuk and Karihaloo [1] but normalize the shear modulus G = 1 and the elastic modulus E = 1. We want to solve the problem of maximizing the torsional rigidity of the bar subject to given equality constraints on the stiﬀness rigidity and the volume. First, we brieﬂy recall the mathematical formulation of the quantities. The torsional rigidity is calculated by u(x)dx. J(Ω) = 2 Ω

Thus, since we want to maximize the torsional rigidity, we have h ≡ −2 and h0 ≡ 0 in (1). The stress function u = u(Ω) satisﬁes −∆u = 2 u=0

in Ω, on Γ.

The bending rigidity with respect to a ﬁxed barycenter in the origin is given by J1 (Ω) = y 2 dx. Ω

The volume of the domain and its (simpliﬁed) barycenter coordinates read as J2 (Ω) = dx, J3 (Ω) = xdx, J4 (Ω) = ydx. Ω

Ω

Ω

√

Choosing c1 = 2π/4, c2 = π, and c3 = c4 = 0, then the optimal shape 1/4 . The associated is the ellipse with semiaxes hx = 2−1/4 and h√ y = 2 Lagrange multipliers are λ1 = −4/9, λ2 = 8 2/9, and λ3 = λ4 = 0, cf. [1]. The numerical setting is as follows. We approximate the boundary via 33 Fourier coeﬃcients, that is the choice N = 16 according to (4). The boundary integral equations are discretized using 256 boundary elements. The penalty parameter α is set to 10. In the inner iteration we use a Newton scheme, see [9] for the details. The outer iteration is started with the unit circle. In Figure 1 we compare produced errors of M˚ artensson’s method (solid line) with that of the traditional update rule (6) using diﬀerent initial guesses for the Lagrange multipliers, namely (0, 1, 0, 0)T produced from M˚ artensson’s formula (7) at the unit circle (dashed line) Copyright © 2005 Marcel Dekker, Inc.

116

CONTROL AND BOUNDARY ANALYSIS (0)

(−1, 1, 0, 0)T , that is λi

= sgn λi

(1, 1, 1, 1)T (indicated by circles) λ (dash-dotted line) As one ﬁgures out, M˚ artensson’s method yields the best convergence while the best traditional scheme uses the initial guess computed by (7).

3.2

Exterior Electromagnetic Shaping

We consider a cylindric vertical column of molten liquid metal with planar, simply connected cross section falling down in an electromagnetic ﬁeld generated by vertical conductors of radii ε > 0. The frequency of the imposed current is very high, such that electromagnetic forces are reduced to the magnetic pressure acting on the interface. In the equilibrium case, a stationary horizontal cross section Ωc = R2 \ Ω of ﬁxed volume c1 arises and the three-dimensional problem reduces to the Error of λ1

102

Martensson method Martensson start initial guess (1,1,1,1) initial guess (–1,1,0,0) exact start

101

100

10–1

10–1

–2

10

10–2

10–3

10–3

10–4

10–4

–5

10–5

10–6

0

5

10 15 20 25 30 35 40 45

Error of |(λ3,λ4)|

102 10–2

10–6

0

5

10 15 20 25 30 35 40 45

Shape Error

100 Martensson method Martensson start initial guess (1,1,1,1) initial guess (–1,1,0,0) exact start

100

Martensson method Martensson start initial guess (1,1,1,1) initial guess (–1,1,0,0) exact start

101

100

10

Error of λ2

102

Martensson method Martensson start initial guess (1,1,1,1) initial guess (–1,1,0,0) exact start

–1

10

10–2

10–4 –6

10

10–3

10–8

10–4

10–10 10–12

10–5

10–14 10–16 0

5

Figure 1.

10 15 20 25 30 35 40 45

10–6 0

5

10 15 20 25 30 35 40 45

Errors of the dual variables and the shape during the iteration.

Copyright © 2005 Marcel Dekker, Inc.

117

Second Order Lagrange Multiplier Approximation Electromagnetic Shaping 3

–1

2

+1

1 0

–1

–1

–1

+1

–1

+1

+1

–2

+1 –1

–3 –3

Figure 2.

+1

–1

–2

–1

0

1

2

3

Conﬁguration of electromagnets and liquid metal.

following two-dimensional shape problem: Seek Ω such that J(Ω) = − ∇u2 dx + A 1dσ = min, Ω

Γ

where

αi in Ω, −∆u = j := M i=1 πε2 χBε (xi ) u=0 on Γ, u = O(1) as x → ∞, as x → ∞. ∇u = O(x−2 )

subject to J1 (Ω) = Ωc 1 dx = c1 . We emphasize that the Lagrange multiplier λ1 corresponds to the magnetic pressure on the surface of the liquid metal. We choose a A = 0.001, c1 = 3π/4, and twelve conductors in the positions and with amperage αi in accordance with Figure 2. The choice of ε > 0 does not inﬂuence the solution, cf. [10]. We use 65 Fourier coeﬃcients for the boundary, i.e. N = 32, 512 boundary elements, and α = 5. We use the circle of volume c1 as initial guess for the Newton scheme. M˚ artensson’s method ﬁnishes the iteration in 110.38 sec. after 10 Newton steps while the traditional scheme (with (0) initial guess λ1 = 0) requires 192.55 sec. and 25 Newton steps. Both schemes produce the same λ1 = −0.0101458.

References [1] N.V. Banichuk and B.L. Karihaloo. Minimum-weight design of multi-purpose cylindrical bars. Intl. J. of Solids and Structures, 12:267–273, 1976. [2] D.P. Bertsekas. Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York, 1982. [3] W. Dahmen. Wavelet and multiscale methods for operator equations. Acta Numerica, 6:55–228, 1997. [4] W. Dahmen and A. Kunoth. Multilevel preconditioning. Numer. Math., 63:315– 344, 1992.

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[5] W. Dahmen, H. Harbrecht, and R. Schneider. Compression techniques for boundary integral equations – optimal complexity estimates. Technical Report SFB 393/02-06, Institute of Mathematics, TU Chemnitz, 2002. Submitted. [6] M. Dambrine and M. Pierre. About stability of equilibrium shapes. M2AN, 34:811–834, 2000. [7] K. Eppler. Boundary integral representations of second derivatives in shape optimization. Discussiones Mathematicae (Diﬀerential Inclusion Control and Optimization), 20:63–78, 2000. [8] K. Eppler. Optimal shape design for elliptic equations via BIE-methods. J. of Applied Mathematics and Computer Science, 10:487–516, 2000. [9] K. Eppler and H. Harbrecht. 2nd order shape optimization using wavelet BEM. Technical Report 06-2003, Institute of Mathematics, TU Berlin, 2003. Submitted. [10] K. Eppler and H. Harbrecht. Exterior electromagnetic shaping using wavelet BEM. Technical Report 13-2003, Institute of Mathematics, TU Berlin, 2003. To appear in Math. Meth. Appl. Sci. [11] K. Eppler and H. Harbrecht. Numerical solution of elliptic shape optimization problems using wavelet-based BEM. Optim. Methods Softw., 18:105–123, 2003. [12] Ch. Grossmann and J. Terno. Numerik der Optimierung. B.G. Teubner, Stuttgart, 1993. [13] H. Harbrecht, F. Paiva, C. P´erez, and R. Schneider. Biorthogonal wavelet approximation for the coupling of FEM-BEM. Numer. Math., 92:325–356, 2002. [14] H. Harbrecht, F. Paiva, C. P´erez, and R. Schneider. Wavelet preconditioning for the coupling of FEM-BEM. Numerical Linear Algebra with Applications, 10:197–222, 2003. [15] H. Harbrecht and R. Schneider. Wavelet Galerkin schemes for 2D-BEM. In Operator Theory: Advances and Applications 121. Birkh¨ auser, Basel, 2001. [16] J. Haslinger and P. Neitaanm¨ aki. Finite element approximation for optimal shape, material and topology design, 2nd edition. Wiley, Chichester, 1996. [17] A.M. Khludnev and J. Sokolowski. Modelling and control in solid mechanics. Birkh¨ auser, Basel, 1997. [18] K. M˚ artensson. A new approach to constrained function optimization. J. of Optimization Theory and Application, 12:531–554, 1973. [19] A. Novruzi and J.-R. Roche. Second derivatives, newton method, application to shape optimization. Technical Report No. 2555, INRIA, Nancy, France, 1995. [20] M. Pierre and J.-R. Roche. Computation of free surfaces in the electromagnetic shaping of liquid metals by optimization algorithms. Eur. J. Mech, B/Fluids, 10:489–500, 1991. [21] O. Pironneau. Optimal shape design for elliptic systems. Springer-Verlag, New York, 1983. [22] R. Schneider. Multiskalen- und Wavelet-Matrixkompression: Analysisbasierte Methoden zur L¨ osung großer vollbesetzter Gleichungssysteme. B.G. Teubner, Stuttgart, 1998. [23] J. Sokolowski and J.-P. Zolesio. Introduction to shape optimization. SpringerVerlag, Berlin, 1992.

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CHAPTER X MATHEMATICAL MODELS OF “ACTIVE” OBSTACLES IN ACOUSTIC SCATTERING Lorella Fatone Dipartimento di Matematica Pura e Applicata Universit` a di Modena e Reggio Emilia Modena, Italy

Maria Cristina Recchioni Istituto di Teoria delle Decisioni e Finanza Innovativa Universit` a Politecnica delle Marche Ancona, Italy

Francesco Zirilli Dipartimento di Matematica “G. Castelnuovo” Universit` a di Roma “La Sapienza” Roma, Italy

Abstract

We consider the following problem: given an acoustic incident ﬁeld that hits a bounded obstacle Ω characterized by a constant acoustic boundary impedance χ and immersed in an isotropic and homogeneous medium that ﬁlls R3 \ Ω choose a pressure current, acting on the boundary of Ω, in order to pursue a given goal that involves the ﬁeld scattered by Ω when hit by the incident ﬁeld. Obstacles of this kind are called active obstacles. Let us give two examples of goals that can be pursued. Let D and DG be bounded sets contained in R3 and characterized by a nonnegative constant boundary acoustic impedance χ and χG respectively, such that D ⊆ Ω and DG ∩ Ω = ∅, DG = ∅. The problems considered consist in choosing a pressure current deﬁned on the boundary of Ω in order to make the wave scattered by Ω when hit by the incident acoustic ﬁeld to appear like the wave scattered by D or by DG in the same circumstances outside a bounded region that contains Ω and D or Ω and DG respectively. The problem involving D is called “masking problem”, the problem involving DG is called “ghost obstacle problem.” In this paper the two problems proposed are modeled as optimal control problems for the wave equation. Using the Pontryagin maximum

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CONTROL AND BOUNDARY ANALYSIS principle the ﬁrst order optimality conditions associated to these two problems are formulated as exterior problems deﬁned outside Ω for a system of two coupled wave equations. These exterior problems are well suited to be solved with an adapted version of the operator expansion method proposed in [5]. Finally the numerical results obtained on some test cases are discussed.

Keywords: Time dependent acoustic obstacle scattering, optimal control problems for the wave equation

Introduction Let R be the set of real numbers, R3 be the three-dimensional real Euclidean space, and x = (x1 , x2 , x3 )T ∈ R3 be a generic vector, where the superscript T denotes transposed. We denote with (·, ·) the Euclidean scalar product in R3 and with · the corresponding vector norm. Let R3 be ﬁlled with a homogeneous isotropic medium in equilibrium at rest with no source terms present. Let Ω, D, and DG be bounded simply connected open sets contained in R3 with locally Lipschitz boundaries ∂Ω, ∂D, ∂DG and let Ω, D, D G be their closures respectively. We denote with n(x) = (n1 (x), n2 (x), n3 (x))T ∈ R3 , x ∈ ∂Ω the outward unit normal vector to ∂Ω in x ∈ ∂Ω. Similarly n(x) = (n1 (x), n2 (x), n3 (x))T ∈ R3 , x ∈ ∂D or x ∈ ∂DG denotes the outward unit normal to ∂D in x ∈ ∂D or to ∂DG in x ∈ ∂DG . Since Ω has a locally Lipschitz boundary n(x), x ∈ ∂Ω exists almost everywhere (see [7] Lemma 2.42 p. 88) and similar statements hold for D and DG . Furthermore let D ⊆ Ω and let DG be such that Ω ∩ DG = ∅ and DG = ∅. We assume that Ω, D, and DG are characterized by non-negative constant boundary acoustic impedances χ, χ , and χG respectively, and we refer to (Ω; χ) as the obstacle, to (D; χ ) as the mask, and to (DG ; χG ) as the ghost obstacle. We consider (Ω; χ), (D; χ ), (DG ; χG ) immersed in the medium mentioned previously that ﬁlls R3 \Ω, R3 \ D, R3 \ DG respectively. Without loss of generality we can assume that the origin of the coordinate system is a point in the interior of Ω and in the interior of D when D = ∅. We always assume Ω = ∅. We note that the last assumption on Ω implies that there exists a > 0 such that the closed sphere of center of the origin and radius a is contained in Ω. Finally let t ∈ R denote the time variable and let ui (x, t), (x, t) ∈ R3 ×R, be an acoustic incident wave propagating in the medium satisfying the wave equation in R3 ×R with wave propagation velocity c > 0. When the incident wave ui hits the obstacle (Ω; χ) a scattered wave is generated. The same happens when ui hits the mask (D; χ ) or the ghost obstacle (DG ; χG ). We denote with us (x, t), (x, t) ∈ (R3 \ Ω) × R, with usD (x, t), Copyright © 2005 Marcel Dekker, Inc.

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(x, t) ∈ (R3 \ D) × R and with usG (x, t), (x, t) ∈ (R3 \ DG ) × R the wave scattered by the obstacle (Ω; χ), by the mask (D; χ ) and by the ghost obstacle (DG ; χG ) respectively, when hit by the incoming ﬁeld ui . We consider the following problems: Problem 1 “Masking” Problem: Given the incoming acoustic wave ui , the obstacle (Ω; χ), the mask (D; χ ), such that D ⊆ Ω, choose a pressure current circulating on ∂Ω such that the wave scattered by (Ω; χ) when hit by ui is, outside Ω, as similar as possible to the wave scattered by the mask (D; χ ) in the same circumstances. Problem 2 “Ghost Obstacle” Problem: Given the incoming acoustic wave ui , the obstacle (Ω; χ), and the ghost obstacle (DG ; χG ), such that Ω ∩ DG = ∅, DG = ∅, choose a pressure current circulating on ∂Ω such that the wave scattered by (Ω; χ) when hit by ui appears, outside a given set containing Ω and DG , “as similar as possible” to the wave scattered by the ghost obstacle (DG ; χG ) in the same circumstances. Note that the obstacle (Ω; χ) considered in Problem 1 and in Problem 2 is an active obstacle that reacts to the presence of the incoming acoustic ﬁeld circulating a pressure current on its boundary ∂Ω for t ∈ R to pursue a given goal. The obstacles that do not react to the presence of an incoming ﬁeld are called passive. Indeed, a masking problem can be interpreted as a ghost obstacle problem if we allow the choice DG ⊆ Ω. Finally, when D ⊂ Ω and D = ∅ we have usD (x, t) = 0, (x, t) ∈ R3 × R. In this last case we call the masking problem a furtivity problem. In fact, in this case, the goal pursued will be that of making the obstacle undetectable. The mathematical model of the ghost obstacle problem proposed is similar to the mathematical model of the masking problem, so that we give in detail only the formulation of the model for the ghost obstacle problem. In [2] a detailed treatment of the masking problem can be found. Note that Problem 1 and Problem 2 have great relevance in many practical situations. In fact, they are natural direct scattering problems whose solution ﬁnds applications in many ﬁelds such as stealth technologies, noise control, and acoustic compatibility in medical echo-graphy (see the website: http://www.jhuapl.edu/programs/specialapp/). The mathematical formulation of the ghost obstacle problem given here is an optimal control problem for the wave equation. The most usual methods to solve optimal control problems in wave scattering involve the iterative solution of the direct scattering problem in the attempt of minimizing the cost functional. This approach is computationally very expensive and, sometimes, not really of practical value. We avoid this iterative approach thanks to the use of the Pontryagin maximum Copyright © 2005 Marcel Dekker, Inc.

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principle (see [4] for a survey of optimal control problems for partial diﬀerential equations). In fact, the ﬁrst order optimality conditions for the ghost obstacle problem can be formulated as the exterior problem (22)–(28) of Section X.2 for two coupled wave equations deﬁned in R3 \ Ω, and consequently the optimal control problem considered, that is, problem (8), (1), (6), (4), (5), can be solved approximately at the same computational cost as the cost necessary to solve the direct scattering problem for the passive obstacle (1), (2), (4), (5). This is a relevant result that has been obtained also in the study of several other control problems in acoustic and electromagnetic scattering, see [2], [3], [5]. To solve problem (22)–(28), we propose the use of the operator expansion method in the formulation developed in [1], [5], [6]. In Section X.1 we formulate the masking problem and the ghost obstacle problem as optimal control problems. In Section X.2 we derive the ﬁrst order optimality conditions for the control problems formulated in Section X.1 and we show some numerical results obtained solving the “ghost obstacle” optimal control problem in a test case.

1.

The Mathematical Models of the Masking and Ghost Obstacle Problems

Let us begin formulating the direct scattering problem associated to the (passive) obstacle (Ω; χ). That is, in the hypotheses described above the obstacle (Ω; χ) when hit by an acoustic incident wave ui (x, t), (x, t) ∈ R3 × R, generates a scattered wave us (x, t), (x, t) ∈ (R3 \ Ω) × R that is the unique solution of the following problem (see [6]): 1 ∂ 2 us (x, t) = 0, (x, t) ∈ (R3 \ Ω) × R, c2 ∂t2 with the boundary condition: us (x, t) −

−

∂us ∂us (x, t) + cχ = g(x, t), (x, t) ∈ ∂Ω × R, ∂t ∂n(x)

(1)

(2)

where g(x, t) is given by: g(x, t) =

∂ui ∂ui (x, t) − cχ (x, t), (x, t) ∈ ∂Ω × R, ∂t ∂n(x)

the boundary condition at inﬁnity: 1 us (x, t) = O( ), r → +∞, t ∈ R, r and the radiation condition: 1 ∂us 1 ∂us (x, t) + (x, t) = o( ), r → +∞, t ∈ R, ∂r c ∂t r Copyright © 2005 Marcel Dekker, Inc.

(3)

(4)

(5)

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∂2 where r = x, x ∈ R3 , = 3i=1 ∂x 2 is the Laplace operator, and O(·) i and o(·) are the Landau symbols. We note that g(x, t), (x, t) ∈ ∂Ω × R is deﬁned almost everywhere and that the boundary condition (2) can be adapted to deal with the limit case of the acoustically hard obstacles, i.e., χ = +∞ (see [5], [6]). Note that similar problems are satisﬁed by usD and usG . Let us model the masking problem and the ghost obstacle problem as optimal control problems. We introduce a control variable ψ, a pressure current acting on the boundary of the obstacle ∂Ω for t ∈ R. That is, in problem (1), (2), (4), (5) we replace the boundary condition (2), with the following boundary condition: −

∂us ∂us (x, t) + cχ = g(x, t) + (1 + χ)ψ(x, t), (x, t) ∈ ∂Ω × R. (6) ∂t ∂n(x)

We distinguish the obstacle that generates a scattered ﬁeld satisfying problem (1), (2), (4), (5) from the obstacle that generates a scattered ﬁeld satisfying problem (1), (6), (4), (5); we refer to them as passive obstacle and active obstacle respectively. Let us formulate Problem 1 and Problem 2 as optimal control problems for the wave equation. That is: Problem 1a) “Masking” Optimal Control Problem: Given the incoming acoustic wave ui , the obstacle (Ω; χ), and the mask (D; χ ), such that D ⊆ Ω, choose a pressure current (i.e., a control function in a suitable class of admissible controls) deﬁned on ∂Ω for t ∈ R in order to minimize a cost functional that measures the “magnitude” of the pressure current used and the “magnitude” of the “diﬀerence” between the wave scattered by the active obstacle (Ω; χ) when hit by ui and usD . Problem 2a) “Ghost Obstacle” Optimal Control Problem: Given the incoming acoustic wave ui , the obstacle (Ω; χ), and the ghost obstacle (DG ; χG ), such that DG ∩ Ω = ∅ and DG = ∅, choose a pressure current deﬁned on ∂Ω for t ∈ R in order to minimize a cost functional that measures the “magnitude” of the pressure current used and the “magnitude” of the diﬀerence outside a given set containing Ω and DG between the wave scattered by the active obstacle (Ω; χ) when hit by ui and usG . We concentrate our attention to the mathematical formulation of Problem 2a). The mathematical formulation of Problem 1a) is very similar to that of Problem 2a). Let Ω be a bounded simply connected open set, containing Ω and DG , with locally Lipschitz boundary ∂Ω and let ds∂Ω , ds∂Ω be the surface measures on ∂Ω and ∂Ω respectively, λ ≥ 0, µ ≥ 0 be adimensional constants such that λ + µ = 1, and ς be a Copyright © 2005 Marcel Dekker, Inc.

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positive dimensional constant. We consider the following functional: s s 2 2 dt (1 + χ)λ(u − uG ) ds∂Ω + (1 + χ)µςψ ds∂Ω Fλ,µ, (ψ) = R

∂Ω

∂Ω

(7) and we model the ghost obstacle problem via the following optimal control problem: min Fλ,µ, (ψ), ψ∈C

(8)

subject to conditions (1), (6), (4), (5) where C is the set of the admissible controls. Note that the solution of (1), (6), (4), (5) depends on ψ through the boundary condition (6). The cases µ = 0 and µ = 1 are trivial so that we restrict our analysis to the case 0 < µ < 1. We choose the cost functional (7) since when 0 < µ < 1 we have λ > 0, that is, minimizing (7) we try to make small the diﬀerence between the wave us scattered by the active obstacle (Ω; χ) and the wave usG scattered by the ghost (DG ; χG ) on ∂Ω and to make small the magnitude of the pressure current ψ used. When us − usG is small for (x, t) ∈ ∂Ω × R an observer located in R3 \ Ω observes a ﬁeld scattered by (Ω; χ) that resembles the ﬁeld scattered by the ghost (DG ; χG ), that is, an observer located in R3 \ Ω is induced to believe that the obstacle (Ω; χ) is indeed the ghost obstacle (DG ; χG ). That is, the optimal control problem (8), (1), (6), (4), (5) is a legitimate mathematical model for Problem 2. When we consider the masking problem (i.e., Problem 1) we have D ⊆ Ω and we can choose Ω = Ω. With this choice the optimal control problem for the masking problem is given by problem (8), (1), (6), (4), (5) where in (8) usD replaces usG . Let (r, θ, φ) be the usual spherical coordinate system in R3 , let B be the sphere of center of the origin and radius one, and let ∂B be its boundary. We assume that: (a) the boundary of the obstacle Ω is a starlike surface with respect to the origin, and that Ω can be represented as follows: x ∈ R3 | 0 ≤ r < ξ(ˆ x), x ˆ ∈ ∂B}, Ω = {x = rˆ

(9)

where ξ(ˆ x) > 0, x ˆ ∈ ∂B is a single valued function deﬁned on ∂B that is assumed suﬃciently regular for the manipulations that follow; (b) we deﬁne: x ∈ R3 | 0 ≤ r < ξ(ˆ x) + η, x ˆ ∈ ∂B}, η ≥ 0, Ωη = {x = rˆ then there exists > 0 such that DG ⊂ Ω . Copyright © 2005 Marcel Dekker, Inc.

(10)

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Let vη be the following function: 1 2 ∂ξ ∂ξ 2 2 vη (ˆ x(θ, φ)) = (ξ + η) sin θ + + (ξ + η)2 sin2 θ , ∂θ ∂φ x ∈ ∂B, (11) η ≥ 0, 0 ≤ θ ≤ π, 0 ≤ φ < 2π, x ˆ= x an easy computation gives: ds∂Ω = v0 (ˆ x(θ, φ))dθ dφ, ds∂Ωη = vη (ˆ x(θ, φ))dθ dφ, η ≥ 0, 0 ≤ θ ≤ π, 0 ≤ φ < 2π.

(12)

Let fη (x), x ∈ ∂Ω be the function deﬁned by: fη (x) =

vη (ˆ x(θ, φ) x(θ, φ)) , x ˆ (θ, φ) = ∈ ∂B, x ∈ ∂Ω, v0 (ˆ x(θ, φ)) x(θ, φ) η ≥ 0, 0 ≤ θ ≤ π, 0 ≤ φ < 2π, (13)

thank to assumptions (a) and (b), we can express the measure on ∂Ω via the measure on ∂Ω as follows: x ) = f (x)ds∂Ω (x), x ∈ ∂Ω . ds∂Ω (x + (14) x Using formula (14) in (7) we can rewrite the cost functional Fλ,µ, as follows: dt ds∂Ω (x)(1 + χ) f (x)λ· (15) Fλ,µ, (ψ) = R ∂Ω )

2 x x s s 2 u (x + , t) − uG (x + , t) + µςψ (x, t) . x x Hence, thanks to assumptions (a) and (b), the cost functional Fλ,µ, relative to the ghost obstacle problem can be treated as the cost functionals of the furtivity and masking problem considered in [2], [5].

2.

The Optimality Conditions for the Ghost Obstacle Problem and Some Numerical Experience

To solve the ghost obstacle problem, that is, the optimal control problem (8), (1), (6), (4), (5) we use the Pontryagin maximum principle, see [5] and [8] for more details. Let us introduce the vector space C of admissible controls. Let us denote for example with us |∂Ω the restriction of us to ∂Ω. With a similar Copyright © 2005 Marcel Dekker, Inc.

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notation we denote the restriction of a generic function to a set and let L∞ (∂Ω) be the space of the real functions deﬁned on ∂Ω essentially bounded on ∂Ω. Let L2 (∂Ω×R), L2 (∂Ω ×R), > 0 be the usual spaces of the real functions deﬁned on ∂Ω × R and ∂Ω × R square integrable with respect to the measures ds∂Ω dt and ds∂Ω dt, > 0 respectively. Let us deﬁne the space of admissible controls to be: C = v ∈ L2 (∂Ω × R), v(x, t) ∈ L∞ (∂Ω). t ∈ R Moreover, there exists F : (R3 \ Ω) × R → R s.t. F |∂Ω×R = v, lim F (x, t) = 0, t→+∞

x ∈ R3 \ Ω, and F |∂Ω×R ,

∂F ∂F |∂Ω×R ∈ L2 (∂Ω × R), |∂Ω×R , ∂n ∂t

F satisﬁes equations (1), (4) and the following condition at inﬁnity: 1 ∂F 1 ∂F , r → +∞, t ∈ R . (16) (x, t) − (x, t) = o ∂r c ∂t r The vector space that is supposed to contain the restriction of us to ∂Ω × R is deﬁned similarly (see [1] for further details). We note that us and usG must belong to L2 (∂Ω × R), > 0. In Section X.1 we have shown that the cost functional (7) under the assumptions (a) and (b) can be reduced to the cost functional (15). Using (15), (1), (6), (4), (5) we obtain the following Hamiltonian associated to the ghost obstacle problem (see [8]): ds∂Ω (x) · (17) H(u , ψ, ϕ, t) = −(1 + χ) ∂Ω (2 ' x x , t) − usG (x + , t) + µςψ(x, t)2 + λf (x) us (x + x x s ∂u ds∂Ω (x) ϕ(x, t) cχ (x, t) − (1 + χ)ψ(x, t) − g(x, t) , t ∈ R, ∂n ∂Ω s

where ϕ(x, t), (x, t) ∈ ∂Ω × R is the adjoint variable of us (x, t), (x, t) ∈ ∂Ω × R. We underline that the second adjoint variable, that is,the one that should appear inside the ﬁrst integral in (17), has been chosen equal to −1 identically (see [5] for further details). Note that the Hamiltonian (17) can be adapted to deal with the limit case χ = +∞. Let δH δϕ δH and δu s be the functional derivatives of the Hamiltonian H with respect to ϕ and us respectively, ψ2 ∈ C be the optimal control function associated to the control problem (8), (1), (6), (4), (5), and u ˆs be the corresponding optimal solution of problem (1), (6), (4), (5). Thanks Copyright © 2005 Marcel Dekker, Inc.

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to the Pontryagin maximum principle we can assert that there exists a function ϕ(x, ˆ t), (x, t) ∈ ∂Ω × R that satisﬁes the following conditions:

δH ∂u ˆs 2 ϕ, (x, t) = (ˆ us , ψ, ˆ t), (x, t) ∈ ∂Ω × R, (18) ∂t δϕ

δH ∂ ϕˆ 2 ϕ, (x, t) = − (ˆ us , ψ, ˆ t), (x, t) ∈ ∂Ω × R, (19) ∂t δus 2 ϕ, ˆ t) ≥ H(ˆ us , ψ, ϕ, ˆ t), H(ˆ us , ψ,

ψ ∈ C, t ∈ R,

(20)

2 That where u ˆs is the ﬁeld scattered by the active obstacle when ψ = ψ. s 2 t), ˆ t) takes its maximum value in ψ(x, t) = ψ(x, is, for t ∈ R, H(ˆ u , ·, ϕ, x ∈ ∂Ω, i.e., 2 ϕ, ˆ t) = max H(ˆ us , ψ, ϕ, ˆ t), t ∈ R. H(ˆ us , ψ, ψ∈C

(21)

Extending the adjoint variable ϕˆ from (x, t) ∈ ∂Ω × R to (x, t) ∈ (R3 \ Ω) × R, assuming ϕ| ˆ ∂Ω ∈ C and arguing as in [1], [5], in particular applying the Green’s formula, under the assumptions (a) and (b) and the assumption that the incident wave belongs to a class of functions that guarantees conditions (28), we can show that the necessary ﬁrst order optimality conditions associated to problem (8), (1), (6), (4), (5) are given by the following exterior value problem for a system of two coupled wave equations: ˆ us (x, t) −

1 ∂2u ˆs (x, t) = 0, (x, t) ∈ (R3 \ Ω) × R, c2 ∂t2

(22)

ˆs 1 ∂u 1 ∂u 1 ˆs (x, t) + (x, t) = o( ), r → +∞, t ∈ R, (23) u ˆs (x, t) = O( ), r ∂r c ∂t r s s ∂u ˆ ∂u ˆ (1 + χ) − (x, t) + cχ (x) = g(x, t) − ϕ(x, ˆ t), (x, t) ∈ ∂Ω × R, ∂t ∂n(x) 2µς (24) 2 1 ∂ ϕˆ (25) ϕ(x, ˆ t) − 2 2 (x, t) = 0, (x, t) ∈ (R3 \ Ω) × R, c ∂t 1 ∂ ϕˆ 1 ∂ ϕˆ 1 ϕ(x, ˆ t) = O( ), (x, t) − (x, t) = o( ), r → +∞, t ∈ R, (26) r ∂r c ∂t r ∂ ϕˆ ∂ ϕˆ (x, t) − cχ (x) = −2λ(1 + χ)f (x) · ∂t ∂n(x) x x , t) − usG (x + , t)), (x, t) ∈ ∂Ω × R, (ˆ us (x + x x −

Copyright © 2005 Marcel Dekker, Inc.

(27)

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lim u ˆs (x, t) = 0, lim ϕ(x, ˆ t) = 0, x ∈ R3 \ Ω.

t→−∞

t→+∞

(28)

The relation between the adjoint variable ϕ ˆ solution of (22)–(28) and 2 the optimal control ψ of problem (8), (1), (4), (5), (6) is the following one: 2 t) = − 1 ϕ(x, ˆ t), (x, t) ∈ ∂Ω × R, 0 < µ < 1. (29) ψ(x, 2µς Note that equations (24) and (27) correspond to equations (18) and (19) respectively, after some algebraic manipulations and notational changes. Finally, note that in the masking problem when we choose D ⊆ Ω and Ω = Ω, that is, when we choose η = 0 in (10) we have f0 (x) = 1, x ∈ ∂Ω in equation (27) so that, in this case, equations (22)–(28) reduce to the ﬁrst order optimality conditions for the masking problem (see [2]).

Figure 1. Acoustic ﬁeld scattered by the ghost obstacle, by the active obstacle, and by the passive obstacle plotted on the sphere BR2

We conclude this section with a numerical experiment relative to the ghost obstacle problem where the obstacle involved is a cube with χ = 0. The cube has its center of mass in the origin, the largest sphere contained in the cube has radius equal to 0.65, and the facets of the cube are orthogonal to the cartesian coordinate axis that intersects them. Copyright © 2005 Marcel Dekker, Inc.

Mathematical Models of “Active” Obstacles in Acoustic Scattering

129

The ghost obstacle has the same shape and the same impedance of the obstacle, that is, DG is a cube obtained translating Ω of a vector x∗ = (1.5, 0, 0)T ∈ R3 and χG = 0. We have chosen c = 1, ς = 1, λ = 0.999, µ = 0.001, = 1.5, and ui (x, t) = exp(−(x1 − t)2 ), (x, t) ∈ R3 × R. We have computed the wave scattered by the active obstacle usa when the optimal “pressure current” is used, the wave scattered by the passive obstacle usp , and the wave scattered by the ghost usG on a sphere of center of the origin and radius R2 = 4 such that Ω ⊂ BR2 . Figure 1 shows in a gray map the ﬁeld scattered by the ghost obstacle, the active obstacle, and the passive obstacle on BR2 . Three values of the time variable t are represented; that is: t = 0, t = 2 and t = 3.5. Note that the ﬁeld scattered by the ghost obstacle and by the active obstacle are similar. We note that the incident acoustic wave packet comes from the negative x1 -axis; hence it hits ﬁrst Ω then the ghost obstacle whose center of mass is located in x∗ = (1.5, 0, 0)T . In fact, when t = 0 the passive obstacle generates a measurable scattered wave while the ghost obstacle substantially does not irradiate energy. When t = 3.5 the situation is reversed. In fact, when t = 3.5 the wave packet has already left Ω but is still touching the ghost obstacle.

References [1] L. Fatone, G. Pacelli, M.C. Recchioni, and F. Zirilli. The use of optimal control methods to study two new classes of smart obstacles in time dependent acoustic scattering. Submitted to Journal of Computational Acoustics. [2] L. Fatone, M.C. Recchioni, and F. Zirilli. A masking problem in time dependent acoustic obstacle scattering ARLO (Acoustics Research Letters Online) 5 (2):2530, 2004. [3] L. Fatone, M. C. Recchioni, and F. Zirilli. Some control problems for the Maxwell equations related to furtivity and masking problems in electromagnetic obstacle scattering. In P. Joly, G.C. Cohen, E. Heikkola, and P. Neittaanmaki, editors, Mathematical and Numerical Aspects of Wave Propagation. Waves 2003, pages 189–194, Berlin, 2003. Springer-Verlag. [4] J.-L. Lions. Optimal Control of Systems Governed by Partial Diﬀerential Equations. Springer-Verlag, Berlin, 1969. [5] F. Mariani, M. C. Recchioni, and F. Zirilli. The use of the Pontryagin maximum principle in a furtivity problem in time-dependent acoustic obstacle scattering. Waves in Random Media, 11:549–575, 2001. [6] E. Mecocci, L. Misici, M. C. Recchioni, and F. Zirilli. A new formalism for time dependent wave scattering from a bounded obstacle. Journal of the Acoustical Society of America, 107:1825–1840, 2000. ´ [7] J. Ne˘cas. Les M´ethodes Directes en Th´ eorie des Equations Elliptiques. Masson & Cie., Paris, 1967. [8] L. S. Pontryagin, V. G. Boltiamskii, R. V. Gamkrelidze, and F. Mischenico. Th´eorie Math´ematique des Processus Optimaux. Editions Mir, Moscow, 1974.

Copyright © 2005 Marcel Dekker, Inc.

CHAPTER XI LOCAL NULL CONTROLLABILITY IN A STATE CONSTRAINED THERMOELASTIC CONTACT PROBLEM Irina F. Sivergina Department of Science and Mathematics Kettering University Flint, MI, USA

Michael P. Polis School of Engineering and Computer Science Oakland University Rochester, MI, USA

Abstract

We study the controllability properties of a nonlinear parabolic problem that models the temperature evolution of a one-dimensional thermoelastic rod that may come into contact with a rigid obstacle. Basically the system dynamics is described by a one-dimensional nonlocal heat equation with a nonlinear and nonlocal boundary condition of Newmann type. We focus on the control problem and treat the case when the control is distributed over the whole space domain. In this case, the system is proved to be locally null controllable. This result is then used to prove the existence of a control bringing the system to zero without contact with the obstacle, provided the initial state is small enough. The proofs are based on a transposition technique and on using a ﬁxed point theorem.

Keywords: Null controllability, state constrained control problem, thermoelastic system, quasistatic approximation, nonlocal parabolic equation, nonlinear boundary conditions

Introduction The system to be analyzed is an isotropic rod 0 ≤ x ≤ l that undergoes a motion in which the displacement vector is parallel to the x-axis. The displacement u(x, t) and the temperature θ(x, t) are functions of the coordinate x and the time t ≥ 0 only. Consider the equations of dynamic

Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

linear thermoelasticity [5, 18] κθxx = cθt + (3λ + 2µ)αθ0 uxt , (λ + 2µ)uxx = (3λ + 2µ)αθx + utt , where θ0 is the uniform reference temperature, is the mass density, λ and µ are the elastic moduli, c is the speciﬁc heat per unit mass, α is the coeﬃcient of thermal expansion, and κ is the thermal conductivity. The subscripts containing t or x denote partial derivatives. Following the usual practice, we change variables to simplify the notation x → 3

λ+2µ θ−θ0 kt u t → cl 2, θ → θ0 , u → l cθ0 , and get the system above in a nondimensional form: θxx = θt + auxt , butt − uxx = −aθx where 2 (3λ+2µ)2 k2 , b = c2 (λ+2µ)l a2 = θ0 αc(λ+2µ) 2 . These equations are considered in the domain ΩT = (0, 1) × (0, T ). To establish the boundary conditions, we consider the case when the rod is situated between two walls that are kept at diﬀerent temperatures. One end of the rod is permanently attached to a wall, while the other end is free to expand or contract. The expansion of the rod resulting from the evolution of the temperature and the stresses is limited by the existence of the other wall. For the displacement at the free edge, we choose the Signorini boundary conditions [7]: u(1, t) ≤ g, ux (1, t) ≤ aθ(1, t), [u(1, t) − g][ux (1, t) − aθ(1, t)] = 0 where the constant g is the nominal gap size between the wall and the free edge in the reference conﬁguration. These conditions are completed by θ(0, t) = 0, −θx (1, t) = kθ(1, t), u(0, t) = 0, and all boundary conditions hold for 0 < t < T. When it is reasonable to expect small acceleration in the process, the term butt is treated as negligible. This assumption allows the problem to be decoupled and formulated in terms of the temperature only; for the details, see [3, 6]. This results in the following initial-boundary value problem considered in this paper: 1 d (1 + a2 )θt − θxx = a max a θ(ξ, t)dξ − g, 0 +v(x, t), in ΩT , (1) dt x l,

0

θ(0, t) = 0, in (0, T ), (2)

1

−θx (1, t) = k[g − a

θ(ξ, t) dξ]θ(1, t), in (0, T ), (3) 0

θ(x, 0) = θ0 (x), in Ω. (4) Here k = k[s] is a function of the scalar argument s ∈ R. We need to clarify a sense, in which the solution is treated for the system (1)– (4). Let W22,1 (ΩT ) be the Sobolev space, which consists of all L2 (ΩT )Copyright © 2005 Marcel Dekker, Inc.

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Local Null Controllability

summable functions that possess generalized L2 (ΩT )-summable secondorder space and ﬁrst-order time derivatives, a norm for which is given

2 ) dxdt. A strong solution of (1)– by θ2 2,1 = ΩT (θ 2 + θt2 + θx2 + θxx W2 (ΩT )

(4) is deﬁned as a function from W22,1 (ΩT ), which satisﬁes (1) almost everywhere in ΩT , (2) and (3) almost everywhere in (0, T ), and for which (4) holds for a trace θ(·, 0) ∈ L2 (Ω). Referring to [2, Theorem 2.1] and [3, Theorem 3.1], we have the following result for a homogeneous system: given v = 0, θ0 ∈ H 1 (0, 1) 1 (R), there exists a unique with θ0 (0) = 0, and a nonnegative k ∈ W∞ strong solution to (1)–(4). The nonhomogeneous case was considered in [1], but the existence and uniqueness theorems were formulated for v(x, t) being suﬃciently smooth. The latter result, based on a special transformation of the original equation, is of no use for the problem considered here, since v(·, t) will be treated as a control function in the system (1)–(4). We are interested in the local null controllability of this system.

Definition 1 The system (1)–(4) is said to be locally null controllable on the time interval [0, T ], if there is γ > 0 such that for each θ0 ∈ H 1 (Ω), θ0 H 1 (Ω) < γ, θ0 (0) = 0, there exists a control v ∈ L2 (ΩT ), for which (i) there exists a unique solution θ ∈ W22,1 (ΩT ) to the system (1)–(4), and (ii) θ(·, T ) = 0. Definition 2 The system (1)–(4) is said to be globally null controllable on the time interval [0, T ], if for any θ0 ∈ H 1 (Ω) with θ0 (0) = 0, there exists a control v ∈ L2 (ΩT ) such that (i) and (ii) from Deﬁnition 1 hold. Note that if for a v ∈ L2 (ΩT ), there exists a strong solution of system (1)–(4), it is unique. This statement follows from the uniqueness theorem proved in [2, Theorem 2.1]. In this paper, we prove the following results. 1 (R) be nonnegative. Then for any T > 0, Theorem 3 Let k ∈ W∞ system (1)–(4) is locally null controllable on [0, T ]. 1 (R) be nonnegative and the constant η in Theorem 4 Let k ∈ W∞ ess sup |k (s)| ≤ η be small enough. Then for any T > 0, the system s

(1)–(4) is globally null controllable on [0, T ]. Next we focus on the behavior of the system when the free end of the rod is constrained to avoid contact with the obstacle, that is, when the following state constraint holds 1 θ(ξ, t)dξ − g < 0, t ∈ [0, T ].

a 0

Copyright © 2005 Marcel Dekker, Inc.

(5)

134

CONTROL AND BOUNDARY ANALYSIS

Definition 5 The system (1)–(5) is said to be locally state constrained null controllable on the time interval [0, T ] if there exists a constant γ1 > 0 such that for any θ0 (x) yielding θ0 H 1 (Ω) < γ1 and θ0 (0) = 0, there exists a control v ∈ L2 (ΩT ) such that in addition to (i) and (ii) in Deﬁnition 1, (5) holds. The following theorem gives our result on handling the local state constrained null controllability for system (1)–(5).

Theorem 6 Under the assumption of Theorem 3, for any T > 0, the system (1)–(5) is locally state constrained null controllable on the time interval [0, T ]. This theorem can be interpreted in the following way: if the free end of the rod is not in contact with the obstacle at the initial time instant, and if θ0 H 1 (Ω) is small enough, then the temperature of the rod may be brought to zero by the desired time instant T without contact with the obstacle at all. There is a large body of literature devoted to controllability problems for linear and semilinear parabolic systems. In [4], [8], [10], [13], [14], [15], [19], the approximate null controllability and null controllability were studied. The typical assumption for establishing the controllability property for semilinear equations is a Lipschitz nonlinearity. For example, in [12], the boundary controllablity is proved under this condition. In [19], the local null controllability is established for semilinear parabolic systems. The system (1)–(4) diﬀers from those treated previously since, ﬁrst, there is a nonlinear boundary condition, and, second, equation (1) contains a nonlocal term describing a stress in the system. Thus, existing techniques do not cover the system considered here. Nevertheless, as stated in Theorem 3 and proved below, the system (1)–(4) is locally null controllable. We note also that from equation (1), it follows that if the initial condition is small, then the diﬀerential equation governing the system is linear on an initial portion of the time interval. The system as a whole however, is not linear because of the boundary condition (3) and the possibility 1

that a state fails to satisfy the inequality max a θ(ξ, t)dξ − g, 0 > 0 0

beginning at some time instant t. This observation raises the question of the state constrained null controllability when the contact of the free end of the rod, being permitted, nevertheless does not happen at all. Copyright © 2005 Marcel Dekker, Inc.

135

Local Null Controllability

1.

Control of the Linear System

First of all we observe that it suﬃces for our purpose to prove null controllability of the system (1 + a2 )θ˜t − θ˜xx = v˜(x, t), in ΩT , ˜ t) = 0, in (0, T ), θ(0, 1 ˜ t) dξ]θ(1, ˜ t), θ(ξ, in (0, T ), −θ˜x (1, t) = k[g − a

(6) (7) (8)

0

˜ 0) = θ0 (x), θ(x,

in Ω.

(9)

˜ t) of If there exists a control v˜ ∈ L2 (ΩT ) such that the solution θ(x, 2,1 ˜ T ) = 0, then the system (6)–(9) belongs to W2 (ΩT ) and yields θ(x, control 1 d ˜ t)dξ − g, 0 (10) v(x, t) = v˜(x, t) − a max a θ(ξ, dt 0

is also in L2 (ΩT ) and solves the null controllability problem for the original system (1)–(4). With this observation, we will now focus on studying the null controllability of (6)–(9). For simplicity we use θ and v below instead of θ˜ and v˜, respectively, in (6)–(9). We start by proving the following auxiliary result.

Theorem 7 Let in the system (1 + a2 )θt = θxx + v, in ΩT , θ(0, t) = 0, in (0, T ), −θx (1, t) = k[t]θ(1, t), in (0, T ), θ(x, 0) = θ0 (x), in Ω,

(11)

k[·] ∈ H 1 (0, T ) be nonnegative and ﬁxed. Then for any θ0 ∈ L2 (Ω), there exists the control v ∈ L2 (ΩT ) such that θ(·, T ) = 0. To prove this theorem, we need one inequality which we formulate without proof. This inequality is a generalization of a well-known estimate proved in [17, p. 81].

Lemma 8 Let in (11), k[·] ∈ H 1 (0, T ) be nonnegative and v = 0. Then there exists a constant R > 0, which is independent of k[·], such that for any θ0 ∈ L2 (Ω), solution of system (11) yields the inequality θ(·, T )L2 (Ω) ≤ RθL2 (ΩT ) . Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

Proof of Theorem 7. Consider a set Z in the space L2 (ΩT ) that consists of the solutions of the system (1 + a2 )zt + zxx = 0, in ΩT , z(0, t) = 0, in (0, T ), −zx (1, t) = k[t] z(1, t), in (0, T ), z(x, T ) = z0 , in Ω

(12)

for a z0 ∈ H 1 (Ω) with z0 (0) = 0. We deﬁne a linear functional on

1 Z by setting l(z) = − 0 z(x, 0)θ0 (x) dx. According to Lemma 8, this functional is continuous on Z and clearly sup lZ ≤ z(x, 0)θ0 (x) dx ≤ Rθ0 L2 (Ω) . z(x,0) L2 (Ω) ≤R

Ω

In the last estimate, R is the constant found in Lemma 8. Applying the Hahn-Banach theorem, we conclude that l can be extended over 2 2 all

L (ΩT ), and there exists a function v˜ ∈ L (ΩT ) such that l(z) = v (x, t) dxdt and lZ = lL2 (ΩT ) = ˜ v L2 (ΩT ) . Hence, ΩT z(x, t)˜ ˜ v L2 (ΩT ) ≤ Rθ0 L2 (Ω) .

(13)

Let θ(x, t) be the solution to (11) with v = v˜, and let z satisfy (12). We have ((1 + a2 )θt − θxx − v)z + ((1 + a2 )zt + zxx )θ dxdt 0= ΩT 1 2 θ(x, T )z0 (x) dx = (1 + a ) 0

for any z0 ∈ H 1 (0, 1), and, hence, θ(·, T ) = 0.

2.

Nonlinear Case: Application of a Fixed Point Theorem

The next step in proving our controllability theorems is in appropriately applying a version of the Kakutani ﬁxed point theorem for topological spaces, which is formulated as follows [9, Theorem 1]: Let L be a locally convex topological linear space and K a compact convex subset in L. Let (K) be the family of all closed convex (nonempty) subsets of K. Then for any upper semicontinuous point-to-set transformation f from K into (K), there exists a point x0 ∈ K such that x0 ∈ F[x0 ]. Here upper semicontinuity means that lim xn = x0 , yn ∈ F[xn ], lim yn = y0 imply y0 ∈ F[x0 ]. Copyright © 2005 Marcel Dekker, Inc.

137

Local Null Controllability

In the next subsection we deﬁne a set in W22,1 (ΩT ) and a point-to-set mapping F in an appropriate way. In subsection 2.2 we prove that this map is upper semicontinuous. And, ﬁnally, in subsection 2.3 we give some details of applying the Katutani theorem to our problem.

2.1

The Imbedding Result

Consider the set K deﬁned as K = {θ ∈ W22,1 (ΩT ) : θ(0, t) = 0, θ(x, 0) = θ0 }

(14)

where θ0 is the same as in the initial condition in (11). For ﬁxed θ ∈ K and v ∈ L2 (ΩT ), we consider a solution w to the system (1 + a2 )wt − wxx = v(x, t),

in ΩT ,

w(0, t) = 0,

in (0, T ),

−wx (1, t) = k[g − a θ(ξ, t)dξ] w(1, t),

in (0, T ),

Ω

w(x, 0) = θ0 (x), in Ω. It follows from the standard theory of linear parabolic equations, see, e.g. [17, p. 33], that this system has a unique solution from W22,1 (ΩT ) and thereby w ∈ K. As proved in section 1, there is a control v, vL2 (ΩT ) ≤ Rθ0 L2 (Ω) , such that w(·, T ) = 0. Having ﬁxed ρ > 0, we denote V = {v ∈ L2 (Ω) : vL2 (ΩT ) ≤ Rθ0 L2 (Ω) + ρ, w(·, T ; v) = 0}. By w(x, t; v), we denote the solution of system (15) corresponding to the control v. V is a bounded closed convex subset in L2 (ΩT ). Now we deﬁne the point-to-set mapping F : K θ → F[θ] ⊂ K by setting F[θ] = {w(·, ·; v), v ∈ V }.

Lemma 9 F[θ] is nonempty weakly closed in W22,1 (ΩT ) for any θ ∈ K. Proof. Suppose that {wn } is a sequence in F[θ] that converges weakly in W22,1 (ΩT ) to a function w. We have to prove that w ∈ F[θ]. By the deﬁnition of the mapping F, there are the controls vn ∈ V such that (1 + a2 )wnt − wnxx = vn (x, t),

in ΩT ,

wn (0, t) = 0, in (0, T ), −wnx (1, t) = k[g − a θ(ξ, t)dξ] wn (1, t), Ω

wn (x, 0) = θ0 (x), wn (x, T ) = 0, in Ω. Copyright © 2005 Marcel Dekker, Inc.

in (0, T ),

138

CONTROL AND BOUNDARY ANALYSIS

Since V is a weak compact in L2 (ΩT ), we may assume that the sequence vn is weakly convergent to a v ∈ V . Taking the limit as n → ∞ in (15), from the Sobolev embedding theorem and the weak continuity of the corresponding trace operators in W22,1 (ΩT ), we obtain that (1 + a2 )wt − wxx = v(x, t),

in ΩT , (15)

w(0, t) = 0, in (0, T ), w(x, 0) = θ0 (x), w(x, T ) = 0, in Ω and that wnx (1, t) → wx (1, t), wn (1, t) → w(1, t) weakly in L2 (0, T ). Since k[g − a

1

θ(ξ, t)dξ] is continuous with respect to t, we conclude

1 that −wx (1, t) = k[g − a 0 θ(ξ, t)dξ] w(1, t). The proof of Lemma 9 is complete. The next lemma is to state an imbedding result for the map F. We want to prove that there is a subset in K, denoted by Kδ , such that F[θ] ⊆ Kδ for any θ ∈ Kδ . An existence of such a set Kδ is proved under two conditions, that will bring us respectively either to local or to globall null controllability for the system (6)–(9). 0

Lemma 10 (i) There exist γ > 0 and δ > 0 such that θW 2,1 (ΩT ) ≤ δ implies 2

sup wW 2,1 (ΩT ) ≤ δ

w∈F [θ]

2

(16)

whenever θ0 H 1 (Ω) < γ. (ii) There exist η > 0 and δ > 0 such (16) holds if kW2∞ (R) < η. Proof. We square both sides of the ﬁrst equation in system (15) and integrate by parts over Ωτ = Ω × (0, τ ) for τ ∈ (0, T ] to obtain 2 2 2 2 2 (1 + a ) wt + wxx dxdt = 2(1 + a ) wt wxx dxdt + v 2 dxdt Ωτ Ωτ Ωτ 2 2 = −(1 + a )k[g − a θ(ξ, τ )dξ] w (1, τ ) Ω 2 +(1 + a )k[g − a θ0 (ξ) dξ] w2 (1, 0) Ω τ 2 +(1 + a ) k [g − a θ(ξ, t)dξ] θt (ξ, t) dξ w2 (1, t) dt 0 Ω Ω 2 2 2 2 wx (x, τ ) dx + (1 + a ) wx (x, 0) dx + v 2 dxdt. −(1 + a ) Ω

Ω

Ωτ

(17) Copyright © 2005 Marcel Dekker, Inc.

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Local Null Controllability

1 (R), there exists a constant k such that |k [s]| ≤ k for Since k ∈ W∞ 1 1 almost all s. Then, 2 2 2 2 2 (1 + a ) wt + wxx dxdt + (1 + a ) wx2 (x, τ ) dx Ωτ Ω 2 2 +(1 + a )k[g − a θ(ξ, τ )dξ] w (1, τ ) Ω

≤ (1 + a )k[g − a 2

τ

θ0 (ξ)

dξ] θ02 (1)

|θt (ξ, t)| dξ

+(1 + a )k1

(θ0 )2x dx

+ (1 + a ) Ω

2

Ω

2

0

2

2

v 2 (x, t) dxdt.

w (1, t) dt +

Ω

Ωτ

(18) Using the trace theorems for the space W22,1 (Ωτ ) (see, e.g., [16, p. 21]) ¯ for Ω ⊂ R1 [16, and the continuity of the injection H 1 (Ω) ⊂ C 0 (Ω) p. 45], we can estimate the third term on the right-hand side of (18) 2

τ

θ2 2,1 . as follows 0 Ω |θt (ξ, t)| dξ w2 (1, t) dt ≤ C12 w2 2,1 W2 (Ωτ )

Returning to (18) and using (13), we obtain 2 (1 + a2 )2 wt2 + wxx dxdt + (1 + a2 ) wx2 (x, τ ) dx Ωτ

Ω

1

+(1 + a2 )k[g − a

W2 (Ωτ )

θ(ξ, τ )dξ] w2 (1, τ ) 0

(19)

≤ (1 + a2 )k[g − a

θ0 (ξ) dξ] θ02 (1) + (1 + a2 )

(θ0 )2x dx Ω

Ω

+(1 + a2 )k1 C12 w2W 2,1 (Ω ) θ2W 2,1 (Ω 2

τ

2

τ)

+ (Rθ0 L2 (Ω) + ρ)2 .

In particular, for τ = T , we have 2 (1 + a2 )2 wt2 + wxx dxdt + (1 + a2 ) wx2 (x, T ) dx ΩT Ω 2 2 +(1 + a )k[(g − a θ(ξ, T )dξ] w (1, T ) Ω

(20)

≤ (1 + a2 )k[g − a

θ0 (ξ) dξ] θ02 (1) + (1 + a2 ) Ω

+(1 + a2 )k1 C12 w2W 2,1 (Ω ) θ2W 2,1 (Ω 2

T

2

1(θ0 )2x dx Ω

T

+ (Rθ0 L2 (Ω) + ρ)2 . )

After integrating both sides of the inequality (19) with respect to τ over (0, T ), we obtain the following inequality that holds for any , Copyright © 2005 Marcel Dekker, Inc.

140

CONTROL AND BOUNDARY ANALYSIS

0 < < T: T − 2 (1 + a2 )2 wt2 + wxx dxdt + Ω

0

ΩT

1

≤ T (1 + a )k[g − a 2

2

+T (1 + a

wx2 (x, t) dxdt

dξ] θ02 (1)

θ0 (ξ)

(θ0 )2x dx

+ T (1 + a )

0 )k1 C12 w2W 2,1 (Ω ) θ2W 2,1 (Ω ) T T 2

2

2

Ω

+ T (Rθ0 L2 (Ω) + ρ)2 .

Adding respective sides of (20) and (21), we get 2 (1 + a2 )2 wt2 + wx2 + wxx dxdt ΩT 1 ≤ (T + 1)(1 + a2 ) k[g − a θ0 (ξ) dξ] θ02 (1) + (θ0 )2x dx Ω

0

+(T + 1)(1 + a2 )k1 C12 w2W 2,1 (Ω ) θ2W 2,1 (Ω 2

2

T

(21)

(22)

T)

+(T + 1)(Rθ0 L2 (Ω) + ρ) . 2

Since for w(0, t) ≡ 0, we have 2 w (x, t) dxdt ≤

ΩT

wx2 (x, t) dxdt, ΩT

2 and, hence, ΩT wt2 + wx2 + wxx dxdt is equivalent to the standard norm

in W22,1 (ΩT ). This will result in the following inequality obtained from (22): 2 1 2 2 wW 2,1 (Ω ) ≤ wt + wx2 + wxx dxdt T 2 2 Ω T 1 ≤ (T + 1)(1 + a2 ) k[g − a θ0 (ξ) dξ] θ02 (1) + (θ0 )2x dx (23) Ω 0 2 (wt2 + wx2 + wxx ) dxdt +2(T + 1)(1 + a2 )k1 C12 θ2W 2,1 (Ω ) 2

T

ΩT

+(T + 1)(Rθ0 L2 (Ω) + ρ) ≤ αθ2W 2,1 (Ω ) w2W 2,1 (Ω 2

2

T

2

T)

+β

where the positive constants α, β are α = 2(T + 1)(1 + a2 )k1 C12 ,

β = (T + 1)(1 + a2 )k[g − a θ0 (ξ) dξ]θ02 (1) Ω 2 2 +(T + 1)(1 + a ) (θ0 )x dx + (T + 1)(Rθ0 L2 (Ω) + ρ)2 . Ω

Copyright © 2005 Marcel Dekker, Inc.

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Local Null Controllability

Now, we assume that θW 2,1 (ΩT ) ≤ δ. Then, (23) yields 2

1 ( − αδ2 )w2W 2,1 (Ω ) ≤ β. T 2 2

(24)

To prove (i), we choose δ to be small enough to guarantee that 12 − αδ2 > 0. Since β is basically determined by the norm of θ0 in H 1 (Ω) and by ρ, we can take γ > 0 and ρ > 0 so small that for any θ0 yielding ≤ θ0 2H 1 (Ω) ≤ γ, we will have β ≤ 12 − αδ2 δ2 , and, hence, w2 2,1 W2 (ΩT )

δ2 . For proving (ii), we see that the value α basically depends on k1 . Then if δ is big enough so that δ2 ≥ 4β, and, hence, there are θ yielding 1 θW 2,1 (ΩT ) ≤ δ, while k1 is so small that k1 ≤ η ≡ 8(T +1)(1+a 2 )C 2 δ 2 , we 2

will have β/( 12 − αδ2 ) ≤ δ2 , and this again leads to w2

1

W22,1 (ΩT )

≤ δ2 .

The proof of Lemma 10 is complete. Now, take δ > 0 found from either the conditions (i) or (ii) if we focus respectively on the local null controllability, or on the global null controllability. Consider the subset Kδ = {w ∈ K : wW 2,1 (ΩT ) ≤ δ}. 2 What is proved in Lemma 10 means that F[θ] ⊆ Kδ for all θ ∈ Kδ . Our next goal is to prove that F is upper semicontinuous in the weak topology of W22,1 (ΩT ) and this is done in the next subsection.

2.2

The Upper Semicontinuity of F

Lemma 11 The operator F is upper semicontinuous in the weak topology of W22,1 (ΩT ). Proof. Suppose that {θn } is a sequence in K that converges weakly in W22,1 (ΩT ) to a function θ. Consider an arbitrary sequence {wn }, wn ∈ F[θn ], that converges weakly in W22,1 (ΩT ) to a function w. We want to prove that w ∈ F[θ], i.e. that there exists v ∈ V such that w yields (15) and thereby w(x, T ) = 0. The proof is basically the same as that used in the proof of Lemma 9. We start by deﬁning the sequence {vn }, vn ∈ V such that (1 + a2 )wnt − wnxx = vn (x, t),

in ΩT ,

wn (0, t) = 0, in (0, T ), −wnx (1, t) = k[g − a θn (ξ, t)dξ] wn (1, t), Ω

wn (x, 0) = θ0 (x), wn (x, T ) = 0, in Ω. Denote its weak limit by v and observe that v ∈ V . Copyright © 2005 Marcel Dekker, Inc.

in (0, T ),

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CONTROL AND BOUNDARY ANALYSIS

Taking the limit in (25) as n tends

to inﬁnity, we get equations (15). The equation −wx (1, t) = k[g − a θ(ξ, t)dξ] w(1, t) in L2 (0, T ) follows Ω

from Lemma 12, whose proof we omit.

Lemma 12 Suppose {fn }, {gn } are sequences of functions in L2 (0, T ), such that fn → f in L∞ (0, T ) and gn → g weakly in L2 (0, T ). Then fn gn → f g weakly in L2 (0, T ). Summarizing what we have proved, we see that for a v ∈ V , the equations yield (1 + a2 )wt − wxx = v(x, t), in ΩT , w(0, t) = 0, in (0,T ), −wx (1, t) = k[g − a

θ(ξ, t) dξ]w(1, t), Ω

w(x, 0) = θ0 (x), w(·, T ) = 0,

in (0, T ),

in Ω.

This means that w ∈ F[θ] is what we need. The proof of Lemma 12 is complete.

2.3

The Application of the Kakutani Theorem

Now, we can apply the Kakutani ﬁxed point theorem. Let us choose δ according to either the conditions (i) or (ii) in Lemma 10. The set Kδ is nonempty, convex, and weakly compact in W22,1 (ΩT ). The sets F[θ] are convex and weakly closed in W22,1 (ΩT ) for any θ ∈ Kδ . And, ﬁnally, the mapping F is upper semicontinuous in the weak topology of W22,1 (ΩT ). According to the Kakutani theorem, there exists θ ∈ Kδ such that θ ∈ F[θ]. This means that for this θ, there exists v ∈ V such that (1 + a2 )θt − θxx = v(x, t), in ΩT , θ(0, t) = 0,

in (0, T ), −θx (1, t) = k[g − a θ(ξ, t) dξ] θ(1, t),

in (0, T ),

Ω

θ(x, 0) = θ0 (x), θ(·, T ) = 0,

in Ω.

If δ has been chosen according to the conditions (i) in Lemma 10, this means the local null controllability of system (6)–(9). The choice of δ according to the conditions (ii) in Lemma 10 leads to the global null controllability when the derivative of k is small enough. So, the problems of local and global null controllability for system (6)–(9) along with the same problems for system (1)–(4) are solved. Copyright © 2005 Marcel Dekker, Inc.

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Local Null Controllability

3.

The State Constrained Null Control Problem

In this section, we prove Theorem 6. Our proof is based on an estimate for the solution of the parabolic equations. According to Theorem 3, there exists γ > 0 so small that for θ0 yielding θ0 H 1 (Ω) < γ, there is a control v˜ ∈ L2 (ΩT ) such that for ˜ T ) = 0. Hence, the the solution of the system (6)–(9) we have θ(x, control (10) solves the null controllability problem for the system (1)–(4). Applying the results of section 3, we may assume that ˜ v L2 (ΩT ) ≤ 2Rγ. Now we will prove that there is γ1 ≤ γ such that for all T ∈ [0, T ], 1 ˜ t) dξ − g < 0 θ(ξ,

a 0

if θ0 H 1 (Ω) < γ1 . This will mean that the controls v˜ and (10) are identical and so are the solutions θ˜ and θ for (6)–(9) and (1)–(4), respectively. We multiply (6) by θ˜ and integrate the resulting equation with respect to x over Ω: 1 1 1 (1 + a2 ) v˜2 (ξ, τ ) dξ. θ˜t2 (x, τ ) dx ≤ θ˜2 (ξ, τ ) dξ + 0

0

0

Applying Gronwall’s lemma to the last inequality yields 1 4R2 θ0 2L2 (Ω) T . θ˜2 (ξ, t) dξ ≤ e 1+a2 θ0 2L2 (Ω) + 1 + a2 0 1 ( ' T 4R2 γ 2 2 < g. Then for We choose 0 < γ1 < γ so that ae 2(1+a2 ) × γ12 + 1+a21

1 ˜ t) dξ − g < 0 holds. This means all t ∈ [0, T ], the inequality a 0 θ(ξ, that v˜(x, t) and v(x, t) are equal, and so are θ˜ and θ, and this completes the proof of Theorem 6.

4.

Conclusion

In this paper, we have obtained local and global null controllability results for a thermoelastic contact problem described by a nonlinear, nonlocal parabolic equation. The proofs are based on applying the Kakutani ﬁxed point theorem to a point-to-set map in a topological space. The local null controllability result is then used to prove the local null controllability of the state constrained contact problem, where no contact is permitted.

Acknowledgments The authors thank Professor I. Lasiecka for useful discussions. Copyright © 2005 Marcel Dekker, Inc.

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References [1] W. Allegretto, J.R. Cannon, and Y. Lin. A parabolic integro-diﬀerential equation arising from thermoelastic contact. Discrete and Continuous Dynamical Systems, 3:217–234, 1997. [2] K.T. Andrews, A. Mikelich, P. Shi, M. Shillor, and S. Wright. One-dimensional thermoelastic contact with a stress-dependent radiation condition. SIAM J. Math. Anal., 23:1393–1416, 1992. [3] K.T. Andrews, P. Shi, M. Shillor, and S. Wright. Thermoelastic contact with Barber’s heat exchange condition. J. Appl. Math. Optim., 28:11–48, 1993. [4] O. Bodart and C. Farbe. Controls insensitizing the norm of the solution of a semilinear heat equation. J. Appl. Math. Appl., 195:658–683, 1995. [5] B.A. Boley and J.H. Weiner. Theory of Thermal Stress. Wiley, New York, 1960. [6] W.A. Day. Justiﬁcation of the uncoupled and quasistatic approximation in a problem of dynamic thermoelasticity. Arch. Rat. Mech. Anal., 80:135–158, 1988. [7] G. Duvant and J.-L. Lions. Inequalities in Mechanics and Physics. SpringerVerlag, Berlin, 1976. [8] M. Eller, I. Lasiecka, and R. Triggiani. Exact boundary controllability of thermoelastic plates with variable coeﬃcients. Semigroups of operators: theory and applications, vol. 42 of Progress in Nonlinear Diﬀerential Equations and Applications. Birkh¨ auser, Basel, 2000. [9] B.K. Fan. Fixed point and minimax theorems in locally convex topological linear spaces. Proc NSA, 38:121–126, 1952. [10] C. Farbe, J.P. Puel, and E. Zuazua. Approximate controllability for the semilinear heat equation. Proc. Royal Soc Edinburgh, Sect A, 125:31–61, 1995. [11] V.P. Fonf, J. Lindenstrauss, and R.R. Phelps. Inﬁnite dimensional convexity, 2001. In: Jonson, W.B. and Lindenstrauss, J. (ed), Handbook of the Geometry of Bahach Spaces, vol. 1, Elsevier, Amsterdam, 2001. [12] A.V. Fursikov and O.Y. Imanuvilov. Controllability of evolution equations. Lectures at the Seoul National University, Seoul, 1996. [13] S.W. Hansen. Boundary control of a one-dimensional linear thermoelastic rod. SIAM J. Control Optim., 32:1052–1074, 1994. [14] G. Lebeau and L. Robbiano. Controle exact de l’equation de la chaleur. Commun. Partial Diﬀ. Equations, 20:335–356, 1995. [15] G. Lebeau and E. Zuazua. Null-controllability of a system of linear thermoelasticity. Arch. Rational Mech. Anal., 141:297–329, 1998. [16] J.-L. Lions and E. Magenes. Non-Homogeneous Boundary Value Problems and Applications, Vol. I. Springer-Verlag, New York, 1972. [17] J.-L. Lions and E. Magenes. Non-Homogeneous Boundary Value Problems and Applications, Vol. II. Springer-Verlag, New York, 1972. [18] M. Slemrod. Global existence, uniqueness and asymptotic stability of classical smoothy solutions in one-dimensional nonlinear thermoelasticity. Arch. Rational Mech. Anal., 76:97–133, 1981. [19] X. Zhang. Exact controllability of semilinear evolution systems and its applications. J. Optim. Theory Appl., 107:415–432, 2000.

Copyright © 2005 Marcel Dekker, Inc.

CHAPTER XII ON SENSITIVITY OF OPTIMAL SOLUTIONS TO CONTROL PROBLEMS FOR HYPERBOLIC HEMIVARIATIONAL INEQUALITIES Zdzislaw Denkowski∗ Faculty of Mathematics and Computer Science Institute of Computer Science Jagiellonian University Krakow, Poland

Stanislaw Mig´ orski Faculty of Mathematics and Computer Science Institute of Computer Science Jagiellonian University Krakow, Poland

Abstract

The results on sensitivity of solution sets to special classes of second order hemivariational inequalities (HVIs) are formulated and then applied to get variational convergence of optimal solutions of control problems for systems described by such HVIs.

Keywords: Control problems, hemivariational inequality, sensitivity, multivalued G, P G, and Γ-convergences

Introduction This note is, in a sense, a continuation of the papers of Denkowski [5] and Denkowski and Mig´ orski [7] concerning the sensitivity of optimal solutions to control problems for systems governed by the stationary (elliptic) and the ﬁrst order evolution (parabolic) hemivariational inequalities (HVIs). Here our main results provide suﬃcient conditions for the variational convergence of optimal solutions of perturbed control ∗ Supported

in part by the State Committee for Scientiﬁc Research of the Republic of Poland (KBN) under research grants no. 2 P03A 003 25 and 4 T07A 027 26.

Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

problems for systems governed by second order evolution (hyperbolic) HVIs. We consider two special classes of such HVIs: with multivalued and time independent, and single-valued and time dependent damping. Variational convergence means the convergence of optimal values, as well as minimizers. In our approach, it follows from general theory of the sequential Γ-convergence (see, e.g., [4, 1, 3, 10]), which is presented in detail in [7] and is important for our purpose. So here we omit the relevant deﬁnitions and theorems. Two basic properties needed in this approach are (i) Kuratowski convergence of solution sets of state relations, here it means the sensitivity results (K-usc and K-lsc) for solution sets of hyperbolic HVIs (ii) Some complementary Γ-convergence of cost functionals (i.e. their uniform convergence with respect to state variable and Γ-convergence with respect to control variable) The nonemptiness of solution sets for HVIs considered in this note follows from the general result on hyperbolic HVIs presented at IFIP conference (see [6]). The paper is organized as follows. In Section 1 we present the class of problems we deal with. Next in Section 3, after preliminaries of Section 2, we formulate theorems concerning the Kuratowski upper and lower semicontinuity (at k = +∞) of the sequence of solution sets of the controlled HVIs. In Section 4 ﬁrst we give some regularity and convergence results for cost functionals and ﬁnally the main sensitivity theorem is presented.

1.

Class of Problems

We are interested in the “sensitivity property” for the following class of control problems (CP ): control

8 9: ; minimize F(u, y 0 , y 1 , y) := F (1) (y) + F (2) (u) + F (3) (y 0 ) + F (4) (y 1 ) , where control u ˜ = (u, y 0 , y 1 ) ∈ U × V × H and state y ∈ Y are subject to the second order diﬀerential inclusion (called also HVI) y (t) + A(t, y (t)) + ∂J(y(t)) f (t) + (Cu)(t) a.e. in Q (HV I) y(0) = y0 , y (0) = y1 in Ω. Above y(t), y (t), y (t) denote, respectively, function y(t, x) and its distributional derivatives with respect to time, ∂J stands for the Clarke Copyright © 2005 Marcel Dekker, Inc.

On Sensitivity in Control Problems for Hyperbolic HVIs

147

subdiﬀerential of superpotential J, Q = (0, T ) × Ω, Ω ⊂ RN and C is a controller operator. Two classes of damping operator A will be considered: A(t, y (t)) = −diva(t, x, Dy (t, x)) (1) with single-valued, time dependent a ∈ M (the Svanstedt class of monotone functions) and A(t, y (t)) = A(y (t)) = {−divη : η(t, x) ∈ a(x, Dy (t, x))}

(2)

with multivalued, time independent a ∈ MΩ (RN ) (the Chiado’Piat, Dal Maso and Defranceschi class of maximal monotone mappings). The above diﬀerential inclusion is equivalent to corresponding inequality (hence the name HVI): y (t)+A(t, y (t)), v −y(t)+J 0 (y(t); v −y(t)) ≥ f (t)+(Cu)(t), v −y(t) for all v ∈ V . Notice that both HVIs (with single-valued and multivalued damping) are particular cases of the more general hyperbolic HVI: y + A(t, y (t)) + By + ∂J(y) f + Cu (HV Ih ) y(0) = y 0 , y (0) = y 1 , y ∈ Y, where, in the abstract setting, we are given the following spaces and operators: V ⊂ Z ⊆ H ⊆ Z ∗ ⊂ V ∗ is the Gelfand ﬁvefold of spaces, ∗ A : (0, T ) × V → 2V is multivalued and pseudomonotone, (Ay)(t) := A(t)y(t), t ∈ (0, T ) is the Nemitsky operator, J : Z → R is a locally Lipschitz superpotential (linked to a nonmonotone, possibly multivalued law between, e.g., stress-strain, reaction-displacement, etc.), C : U → V ∗ is a controller (linear or not), U is the space of controls, ∂J : Z → ∗ 2Z denotes the Clarke subdiﬀerential of J, and J 0 (u; v) stands for its generalized directional derivative. By means of such an (HV Ih ) one can model, for instance, short memory materials, the movement of elastic beam on gluing support, etc. The nonemptiness of the solution set for (HV Ih ) can be proved for large classes of operators (see [6]). In order to get sensitivity results we have to restrict ourselves to special classes of operators for which the notion of G or P G-convergence can be deﬁned.

2.

Preliminaries

First, in an abstract way, we formulate the problems of sensitivity and then we recall the deﬁnition and some properties of multivalued G-convergence. Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

2.1

Abstract Setting

We are given a state relation R linking the state y ∈ YR to the control u ∈ U (e.g., R = ODE, PDE, DI, VI, HVI) and a cost functional F : (Λ := graph R) (u, y) −→ F(u, y) ∈ R. We look for an optimal solution, i.e., an element (u∗ , y ∗ ) ∈ Λ which minimizes the cost functional m := min {F(u, y) : (u, y) ∈ Λ}

(CP )R

The set Λ of admissible pairs can be represented as: Λ = graph SR = {(u, y) : y ∈ SR (u), u ∈ U } =

(= F(u∗ , y ∗ )) . u∈U

SR (u),

where the solution map is given by SR : U u −→ SR (u) := {y ∈ YR : (u, y) ∈ R} ⊂ YR . For the well posed problem the set SR (u) consists of one element but it is not the case for R = HV I. Let us denote the set of optimal solutions by ∗ := {(u∗ , y ∗ ) ∈ Λ : F(u∗ , y ∗ ) = m} . SR The “sensitivity” is understood as a “nice” asymptotic behavior of optimal solutions to the perturbed problems (i.e., perturbed state relations Rk as well as perturbed cost functionals Fk ). Here index k indicates “perturbation”, k = ∞ denotes unperturbed original problem. So we consider the whole sequence (k ∈ N = N ∪ {+∞}) of problems: (CP )Rk

mk := min {Fk (u, y) : (u, y) ∈ Λk }

(= Fk (u∗k , yk∗ ))

and we look for conditions which assure that (i)

mk → m∞

(ii)

∗ K– lim sup Sk∗ ⊂ S∞

∗ The condition (ii) is equivalent to saying that if (u∗k , yk∗ ) ∈ SR and k ∗ ∗ ∗ ∗ ∗ ∗ ∗ (ukn , ykn ) → (u∞ , y∞ ), then (u∞ , y∞ ) ∈ SR∞ . Setting 0 x∈A χA (x) = +∞ x ∈, /

we can reformulate the problem to unconstrained optimization: (CP )Rk

mk := min {Fk (u, y) + χΛk (u, y) : (u, y) ∈ U × YRk } ,

and then apply the Γ-convergence theory (see [4, 1, 3, 10]) to get the required sensitivity results. Copyright © 2005 Marcel Dekker, Inc.

149

On Sensitivity in Control Problems for Hyperbolic HVIs

2.2

Functional Spaces and Operators

Let Ω ⊂ RN be an open bounded set with regular boundary. We ﬁx p and q such that 1 < p < +∞ and 1/p + 1/q = 1, and we admit:

Definition 1 (Chiado’Piat, Dal Maso, Defranceschi [2]) For ﬁxed mi ∈ L1 (Ω), ci > 0, i = 1, 2, we denote MΩ (RN ) = {a : Ω × RN → 2R

N

such that (i) - (iii) below hold}

(i) a(x, ξ) is maximal monotone with respect to ξ for all x ∈ Ω (ii) a is L(Ω) ⊗ B(RN ) ⊗ B(RN ) measurable (i.e., a−1 (C) ∈ L(Ω) ⊗ B(RN ) for all closed set C ⊂ RN ) (iii) for every (ξ, η) with η ∈ a(x, ξ) we have |η|q ≤ m1 (x) + c1 (η, ξ)RN and |ξ|p ≤ m2 (x) + c2 (η, ξ)RN This class is large enough, e.g., it contains the mappings of the form a(x, ξ) = ∂ξ ψ(x, ξ) for some ψ : Ω × RN → [0, +∞) which is measurable in both variables, convex in ξ and satisﬁes c1 |ξ|p ≤ ψ(x, ξ) ≤ c2 |ξ|p . In Lq (Ω; RN ) we introduce the topology σ accordingly to the following σ

Definition 2 We say that ηk −→ η if and only if ηk → η in w − Lq (Ω; RN ) and divηk → divη in s − (W 1,p (Ω))∗ . For any function a ∈ MΩ (RN ) we deﬁne multivalued operators: A : W 1,p (Ω) y → Ay := {η ∈ Lq (Ω; RN ) : η(x) ∈ a(x, Dy(x)) a.e.}, A : W 1,p (Ω) y → Ay := {−divη : η ∈ Ay} ⊂ (W 1,p (Ω))∗ and we admit:

Definition 3 (“multivalued G-convergence”) For {ak }, a ∈ MΩ (RN ) G

we say that ak is G-convergent to a (written as ak −→ a) if and only if K(w, σ)– lim sup GrAk ⊂ GrA. We quote the result of Chiado’Piat, Dal Maso, and Defranceschi [2]:

Theorem 4 For ak , a ∈ MΩ (RN ) we have G

ak −→ a

=⇒

K(w, s)– lim GrAk = GrA. k→∞

K(w,s)

The latter convergence is denoted by GrAk −→ GrA. In the above theorem the implication ”⇐=” does not hold (see [2]). Copyright © 2005 Marcel Dekker, Inc.

150

CONTROL AND BOUNDARY ANALYSIS

The multivalued G-convergence appeared to be very useful in proving the convergence of solution sets and sensitivity analysis for stationary (elliptic) HVIs (see [7]). Now, it will be applied to evolution HVIs. Let V = W01,p (Ω), Z = Lp (Ω), H = L2 (Ω), Z ∗ = Lq (Ω), V ∗ = = Lp (Ω, RN ) Z N , Z ∗ = Lq (Ω, RN ) Z ∗ N . Notice W −1,q (Ω), and Z that for p ≥ 2 we have V ⊂ Z ⊆ H ⊆ Z∗ ⊂ V ∗ and the ﬁrst inclusion is a compact embedding. For evolution HVIs we need also time dependent spaces. To this end let Q = (0, T )×Ω and let us deﬁne the spaces V = Lp (0, T ; V ), Z = Lp (0, T ; Z), H = L2 (0, T ; H)

= Lp (0, T ; Z), Z ∗ = L2 (Q), Z ∗ = Lq (0, T ; Z ∗ ), V ∗ = Lq (0, T ; V ∗ ), Z ∗ ), and Lq (0, T ; Z Wpq = {y ∈ V : y ∈ V },

Y = {y ∈ V : y ∈ Wpq }.

So Wpq ⊂ V ⊂ Z ⊆ H ⊆ Z ∗ ⊂ V ∗ and the embedding Wpq into Z is compact. Below we quote a result on convergence of solutions of ﬁrst order (parabolic) diﬀerential inclusions which will be used to prove the Kuratowski convergence of solution sets of the second order (hyperbolic) HVIs with multivalued damping. Let ak ∈ MΩ (RN ), gk ∈ V ∗ , zk0 ∈ H for k ∈ N = N ∪ {+∞} be given. For operators Ak corresponding (see formula (2)) to ak we denote by zk the unique solution in Wpq to diﬀerential inclusion z (t) + Ak z(t) gk (t) for a.e. t ∈ (0, T ) (DI k ) z(0) = zk0 . Recall that zk ∈ Wpq is a solution to (DI k ) if ∗, zk (t) − divζk (t) = gk (t) for some ζk ∈ Z ζk (t, x) ∈ ak (x, Dzk (t, x)) a.e. in Q, zk (0) = zk0 . G

Theorem 5 (Denkowski, Mig´ orski, and Papageorgiou [9]) If ak −→ a∞ , s−V ∗

w−H

w−Wpq

∗ w−Z

0 , then z −→ z gk −→ g∞ , zk0 −→ z∞ ∞ and ζk −→ ζ∞ . k

Copyright © 2005 Marcel Dekker, Inc.

151

On Sensitivity in Control Problems for Hyperbolic HVIs

3.

Sensitivity of Solution Sets for Second Order HVIs

3.1

HVIs with Multivalued Damping

We start with a “K-usc” result for sequence (k ∈ N = N ∪ {+∞}) of problems: y (t) + Ak (y (t)) + ∂Jk (y(t)) fk (t) a.e. in (0, T ) × Ω (HV I)k y(0) = yk0 , y (0) = yk1 in Ω. We admit the hypotheses below which are satisﬁed uniformly in k ∈ N: ∗

H(A) : Ak : V → 2V \ ∅ correspond to functions ak ∈ MΩ (RN ) (i.e. Ak v = −divak (x, Dv) =: {−divη ∈ V ∗ : η(x) ∈ a(x, Dv(x))}) and H(J) :

G

ak −→ a∞ Jk : (0, T ) × Z → R satisﬁes the conditions:

(i) Jk (·, z) is measurable for all z ∈ Z (ii) Jk (t, ·) is locally Lipschitz for a.e. t ∈ (0, T ) 2/q

(iii) ||∂Jk (t, z)||Z ∗ ≤ cJ (1 + ||z||Z ) for all z ∈ Z, t ∈ (0, T ) with some cJ > 0 (iv) K(s − Z, w − Z ∗ )– lim sup Gr∂Jk (t, ·) ⊂ Gr∂J∞ (t, ·) for a.e. t s−V ∗

w−Z ∗

(H0 ) : either fk , f∞ ∈ V ∗ , fk −→ f∞ or fk , f∞ ∈ Z ∗ , fk −→ f∞ and 0 1 s−H 1 0 ∈ V , y 0 w−V 1 1 yk0 , y∞ k −→ y∞ , and yk , y∞ ∈ H, yk −→ y∞ (H1 ) : if p = 2, then α > cJ β 2 T , where α > 0 is the coercivity constant for Ak and β > 0 is an embedding constant of V into Z Now, (HV I)k reads as follows ∗ and ξk ∈ Z ∗ such that Find yk ∈ Y, ηk ∈ Z y (t, x) − divη (t, x) + ξ (t, x) = f (t, x) a.e. in Q k k k k ηk (t, x) ∈ ak (x, Dyk (t, x)), ξk (t, x) ∈ ∂Jk (t, yk (t, x)) a.e. in Q yk (0, x) = yk0 (x), yk (0, x) = yk1 (x) in Ω and we have

Theorem 6 Under hypotheses H(A), H(J), (H0 ), and (H1 ) it holds: Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

(j) The solution set S(Ak , Jk , fk , yk0 , yk1 ) of (HV Ik ) is nonempty for every k ∈ N. (jj) For every sequence yk ∈ S(Ak , Jk , fk , yk0 , yk1 ) there exists a subsew−Wpq

w−V

as n → ∞ and quence {ykn } such that ykn −→ y∞ , yk n −→ y∞ 0 , y 1 ). y∞ ∈ S(A∞ , J∞ , f∞ , y∞ ∞

Proof. Condition (j) follows from a general result on surjectivity of “Lpseudomonotone, coercive and bounded” operators (cf. Theorem 1.3.73 in [8]), while for (jj) we use in calculations Theorem 5 on convergence of solutions for (DI)k . Because of the lack of space here the complete proof will be published elsewhere. Next, we apply this theorem to our control systems y (t) + A(t, y (t)) + ∂J(y(t)) f (t) + (Cu)(t) a.e. in Q (HV Ic )k y(0) = y 0 , y (0) = y 1 in Ω, where the controller operators (providing us with distributed control) satisfy: c s−V ∗ H(C) : Ck : U → V ∗ , Ck −→ C∞ (continuously, i.e., Ck uk −→ C∞ u∞ U

for all uk −→ u∞ ).

Corollary 7 Under hypotheses H(A), H(J), H(C), (H0 ), and (H1 ) if we assume u ˜k = (uk , yk0 , yk1 )

U ×(w−V )×(w−H)

−→

then

0 1 (u∞ , y∞ , y∞ )=u ˜∞ ,

˜ ∈ U × V × H there exists at least one (j) for every k ∈ N and u u) := S(Ak , Jk , Ck , u ˜) = ∅ for solution to problem (HV Ic )k i.e. Sk (˜ all u ˜ ∈ U × V × H. uk ), where (jj) for any sequence {yk } ∈ Sk (˜ u ˜k = (uk , yk0 , yk1 )

U ×(w−V )×(w−H)

−→

u ˜∞ ,

there exists a subsequence which converges to a solution of the limit w−Y u∞ ). So problem, i.e., ykn −→ y∞ and y∞ ∈ S∞ (˜ uk ) ⊂ S∞ (˜ u∞ ). K(w − Y)– lim sup Sk (˜ k→∞

(jjj) Moreover, if the unperturbed (limit) problem (HV Ih )∞ has the uni˜ then u) = {y(˜ u)} for all u ˜ ∈ U), city of solution property (i.e., S∞ (˜ also we have S∞ (˜ u) ⊂ K(w − Y)– lim inf Sk (˜ uk ), so k→∞

uk ) Sk (˜ Copyright © 2005 Marcel Dekker, Inc.

K(w−Y)

−→

S∞ (˜ u∞ ), as k → ∞.

On Sensitivity in Control Problems for Hyperbolic HVIs

3.2

153

HVIs with Single-Valued Damping

An analogous result can be obtained for HVIs with single-valued time dependent damping of the form A(t, y (t)) = −diva(t, x, Dy (t, x)) with a ∈ M, where M is deﬁned by

Definition 8 (Svanstedt class of monotone functions, see [11]) Given nonnegative constants m0 , m1 , m2 , and 0 < α ≤ 1 we set M = M(m0 , m1 , m2 , α) := = {a : Q × RN → RN such that (i) - (iv) below hold} (i) |a(t, x, 0)| ≤ m0 a.e. in Q (ii) a(·, ·, ξ) is Lebesgue measurable on Q, ∀ ξ ∈ RN (iii) |a(t, x, ξ) − a(t, x, η)| ≤ m1 (1 + |ξ| + |η|)p−1−α |ξ − η|α a.e. (iv) (a(t, x, ξ) − a(t, x, η), ξ − η)RN ≥ m2 |ξ − η|α a.e. in Q, ∀ ξ, η In this case the assumption of G-convergence should be replaced by P G−convergence deﬁned as follows:

Definition 9 Let ak , a∞ ∈ M. We say that ak is P G-convergent to PG a∞ , written as ak −→ a∞ if and only if for every g ∈ V ∗ we have w−Lq (Q,RN )

w−Wpq

−→ a(t, x, Dy∞ ), where yk , for yk −→ y∞ and ak (t, x, Dyk ) k ∈ N, is the unique solution to the problem y − divak (t, x, Dy) = g,

y(0) = 0.

We admit the following hypothesis H(A)1 :

Ak : (0, T ) × V → V ∗ correspond to functions ak ∈ M PG

(i.e., Ak v = −divak (t, x, Dv(x))) and ak −→ a∞ . We have

Theorem 10 Assume in sequences of (HV I)k and (HV Ic )k we replace the multivalued damping by a single-valued one satisfying the hypothesis H(A)1 . Then for these sequences Theorem 6 and Corollary 7 (with hypothesis H(A) being replaced by H(A)1 ) remain true.

4.

Complementary Γ-convergence of Cost Functionals

We consider a sequence of “perturbed” cost functionals: u ˜

8 9: ; (1) (2) (3) (4) Fk (u, y 0 , y 1 , y) := Fk (y) + Fk (u) + Fk (y 0 ) + Fk (y 1 ), Copyright © 2005 Marcel Dekker, Inc.

(3)

154 where

CONTROL AND BOUNDARY ANALYSIS (1)

Fk (y) =

(2)

Fk (u) =

(1) Q Fk (t, x, y(t, x)) dtdx,

(2) Q Fk (t, x, (Ck u)(t, x)) dtdx,

(3)

(4)

Fk (y 0 ) = Fk (y 1 ) =

(3) 0 0 Ω Fk (x, y (x), Dy (x)) dx, (4) 1 Ω Fk (x, y (x)) dx.

We would like to establish the regularity and convergence conditions which assure: (i)

1o for every ﬁxed k ∈ N, Fk (·), i = 1, . . . , 4 are lsc in suitable topologies; Γseq ((U˜)− ,Y ± )

2o Fk (˜ u, y) −→ ˜ U × V × H =: U.

F∞ (˜ u, y), as k → ∞, where u ˜ = (u, y 0 , y 1 ) ∈

The lsc property 1o is needed for the direct method when proving existence of optimal solutions. The “complementary” Γ-convergence in 2o , which means “continuous” convergence in y and Γ-convergence in u, together with K-convergence of solution sets of HVIs imply variational convergence of optimal solutions of (CP )HV I . To establish properties 1o and 2o , we admit the regularity and convergence assumptions for integrands (below conditions (i) and (ii) are uniform with respect to k ∈ N). H(F (1) ) :

(1)

(i) Fk : Q × R → R is measurable in (t, x) ∈ Q, (1)

Fk (t, x, 0) ∈ Lp (Q) (1)

(1)

(1)

w−L1 (Q)

(ii) |Fk (t, x, z1 ) − Fk (t, x, z2 )| ≤ c(1 + |z1 |)|z1 − z2 | (iii) Fk (·, ·, z) H(F (2) ) :

−→

(1) F∞ (·, ·, z), ∀ z ∈ R

(2)

(i) Fk : Q × R → R is measurable in (t, x), convex in z (2)

(ii) 0 ≤ Fk (t, x, z) ≤ λ|z|q , a.e. in Q (2)

(iii) Fk (·, ·, z) H(F (3) ) :

w−L1 (Q)

−→

(2) F∞ (·, ·, z), ∀ z ∈ R

(3)

(i) Fk : Ω × RN +1 → R is measurable in x ∈ Ω convex in z ∈ RN +1 (3) (ii) λ|z|p ≤ Fk (t, x, z), a.e. in Q (3)∗

(iii) Fk

(·, ·, z)

Copyright © 2005 Marcel Dekker, Inc.

w−L1 (Q)

(3)∗ −→ F∞ (·, ·, z), ∀ z ∈ RN +1 (convergence of the Fenchel conjugates)

155

On Sensitivity in Control Problems for Hyperbolic HVIs (4)

(i) Fk : Ω × R → R is measurable in x ∈ Ω, convex in z

H(F (4) ) :

(4)

(ii) 0 ≤ Fk (x, z) ≤ λ|z|2 , a.e. in Ω w−L1 (Ω)

(4)

−→

(iii) Fk (·, z)

(4) F∞ (·, z), ∀ z ∈ R

Lemma 11 Let U = Lq (0, T ; Lq (Ω)) Lq (Q), Ck = id : U → Z ∗

Lq (Q) ⊂ V ∗ . Then: 10 For every ﬁxed k ∈ N under regularity assumptions (i), (ii) of H(F (1) ) − H(F (4) ), respectively, we obtain: (1)

(j) Fk (·) is s − Lp (Q) so also (w − Y)-continuous (2)

(jj) Fk (·) is s − Lq (Q) continuous (3)

(jjj) Fk (·) is w − W 1,p (Ω) continuous (4)

(jv) Fk (·) is w − L2 (Ω) continuous 20 Under convergence conditions (iii) of H(F (1) ) − H(F (4) ), respectively, we have (1)

c−Y

(1) (y), Fk (y) −→ F∞ (3)

Fk (y 0 )

Γseq (w−W 1,p (Ω)− )

−→

(2)

Fk (u)

(3) 0 F∞ (y ),

Γseq (s−Lq (Q)− )

−→

(4)

Fk (y 1 ) (1)

(2) F∞ (u)

Γseq (w−L2 (Ω)− )

−→

(2)

(4) 1 F∞ (y ) (3)

so for the functional Fk (u, y 0 , y 1 , y) = Fk (y) + Fk (u) + Fk (y 1 ) + (4) Fk (y 1 ), we obtain Γseq s − Lq (Q)− , w − W 1,p (Ω)− , w − L2 (Ω), w − Y ± lim Fk = F∞ . k→∞

The proof of (j) − (jv) follows from classical theorems on continuity on Lp spaces of integral functionals with Carath´eodory integrands with respect to the strong topology (or weak topology provided the integrands are convex). The proof of the ﬁrst convergence follows from direct calculations, while those of the next three follow from theorems on Γconvergence (cf. [3]); that it is equivalent to the pointwise convergence for sequences of locally equicontinuous functions (e.g., convex and locally bounded).

Theorem 12 We admit the hypothesis H(A), H(J), (H0 ) for (HV Ic )k with U = Lq (0, T ; Lq (Ω)), Ck = id : U → Z ∗ Lq (Q) ⊂ V ∗ . We assume the hypotheses H(F (j) ), j = 1, . . . , 4, for cost functional Fk (u, y 0 , y 1 , y) given by ( 3) hold. Then: Copyright © 2005 Marcel Dekker, Inc.

156

CONTROL AND BOUNDARY ANALYSIS

(i) For every k ∈ N the problem (CP )(HV Ic )k has at least one optimal solution (u∗k , yk0∗ , yk1∗ , yk∗ ) ∈ Sk∗ , mk := Fk (u∗k , yk0∗ , yk1∗ , yk∗ ) being its “minimal value”. (ii) If the unperturbed (original) problem (CP )(HV Ic )∞ has the “unicity ˜ S(HV I ) (˜ u) = {y∞ (˜ u)}), of solution property” (for all u ˜ ∈ U, c ∞ ˜∞ , S(HV Ic )k (˜ uk ) are “equicoercive”, so there exists then for u ˜k → u convergent subsequence of optimal solutions such that 0∗ 1∗ ∗ 0∗ 1∗ ∗ ∗ , y∞ , y∞ ) and (u∗∞ , y∞ , y∞ , y∞ ) ∈ S∞ . (u∗kn , yk0∗n , yk1∗n , yk∗n )to(u∗∞ , y∞

(iii) We also have mk → m∞ , as k → ∞. Finally, we remark that an analogous theorem holds for (CP )HV I with single-valued damping, provided we replace hypothesis H(A) by H(A)1 .

References [1] G. Buttazzo and G. Dal Maso. Gamma-convergence and optimal control problems. J. Optim. Theory Appl., 38:385–407, 1982. [2] V. ChiadoPiat, G. Dal Maso, and A. Defranceschi. G-convergence of monotone operators. Annales de l’Institut Henri Poincar´e, 7:124–160, 1990. [3] G. Dal Maso. An Introduction to Γ-convergence. Birkh¨ auser, Boston, 1993. [4] E. De Giorgi and T. Franzoni. Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Natur., 58:842–850, 1975. [5] Z. Denkowski. Control problems for systems described by hemivariational inequalities. Control and Cybernetics, 31:713–738, 2002. [6] Z. Denkowski and S. Mig´ orski. Existence of solutions to evolution second order hemivariational inequalities with multivalued damping, in System Modeling and Optimization, J. Cagnol and J.-P. Zol´esio (Eds.), pp. 203–215, Kluwer Academic Publishers, Norwell, MA, 2004. [7] Z. Denkowski and S. Mig´ orski. Sensitivity of optimal solutions to control problems for systems described by hemivariational inequalities, Control and Cybernetics, 33(2), 2004. In press. [8] Z. Denkowski, S. Mig´ orski, and N.S. Papageorgiou. An Introduction to Nonlinear Analysis: Applications. Kluwer Academic/Plenum Publishers, Boston, 2003. [9] Z. Denkowski, S. Mig´ orski, and N.S. Papageorgiou. On the convergence of solutions of multivalued parabolic equations and applications. Nonlinear Analysis, 54:667–682, 2003. [10] Z. Denkowski and S. Mortola. Asymptotic behaviour of optimal solutions to control problems for systems described by diﬀerential inclusions corresponding to partial diﬀerential equations. J. Optim. Theory Appl., 78:365–391, 1993. [11] N. Svanstedt. G-convergence of parabolic operators. Nonlinear Analysis, 36:807– 842, 1999.

Copyright © 2005 Marcel Dekker, Inc.

CHAPTER XIII EVOLUTION HEMIVARIATIONAL INEQUALITY WITH HYSTERESIS AND OPTIMAL CONTROL PROBLEM Leszek Gasi´ nski∗ Institute of Computer Science Jagiellonian University Cracow, Poland

Abstract

We study an optimal control problem of Bolza type for evolution hemivariational inequality with hysteresis operator.

Keywords: Hemivariational inequality, locally Lipschitz function, Clarke’s subdifferential, hysteresis operator, control problem of Bolza type

Introduction In this paper we study an optimal control problem of the form: min min J(y, u), y∈U u∈S(y)

where J is a Bolza-type functional and S(y) is the set of solutions of the following evolution hemivariational inequality with hysteresis: u (t) + w(t) + A(u (t)) + B(u(t)) + χ(t) = f (t) + D(t)y(t) in V . In this statement, χ is a selection of Clarke’s subdiﬀerential of a locally Lipschitz potential function and w is an output function of a hysteresis operator with the input u. The notion of hemivariational inequalities was introduced in the 1980s by P.D. Panagiotopoulos (see [13–15]). These new expressions are a natural extension of variational inequalities. They arise in physical problems when we deal with nonsmooth and nonconvex potential functionals and thus with nonmonotone and possibly multivalued laws. Such functionals appear quite often in mechanics and engineering if one wants ∗ This

paper was partially supported by the State Committee for Scientiﬁc Research of the Republic of Poland (KBN) under research grants no. 2 P03A 003 25 and no. 4 T07A 027 26.

Copyright © 2005 Marcel Dekker, Inc.

158

CONTROL AND BOUNDARY ANALYSIS

to consider more realistic mechanical laws of nonmonotone, multivalued nature. Hemivariational inequalities use the notion of Clarke’s subdiﬀerential of locally Lipschitz functions (see F.H. Clarke [4]), which is a generalization of classical derivative of smooth functions as well as subdiﬀerential of convex functions in the sense of convex analysis. For particular applications of hemivariational inequalities to problems in mechanics and engineering we refer to the books of P.D. Panagiotopoulos [13, 16], Z. Naniewicz and P.D. Panagiotopoulos [12], and references therein. Some existence results for hemivariational inequalities of second order were obtained, e.g., by P.D. Panagiotopoulos [17], W. Bian [2], L. Gasi´ nski [6, 8]. In the ﬁrst of the above mentioned papers we can also ﬁnd some applications of this type of inequalities in mechanics, namely for the description of plane linear elastic body with nonmonotone skin eﬀects. Parallel to the theory of hemivariational inequalities, the theory of hysteresis has been developed. The latter is based on the notion of the hysteresis operator, introduced by M.A. Krasnoselskii and A.V. Pokrovskii [10] and developed, e.g., by M. Brokate and J. Sprekels [3], A. Visintin [18], and others. Our starting point in this paper is the existence result for hemivariational inequality with hysteresis shown in [7].

1.

Preliminaries

Let X be a Banach space with a norm · X and X its topological dual. By ·, ·X ×X we shall denote the duality brackets for the pair (X, X ). If X and Y are two Banach spaces such that X ⊆ Y and the embedding X ⊆ Y is continuous, then by cX Y we denote the embedding constant, i.e. the smallest number such that xY ≤ cX Y xX for all x ∈ X. If X is in addition a Hilbert space, then by (·, ·)X we shall denote the scalar product in X. Given a locally Lipschitz function ϕ : X −→ R, we deﬁne generalized directional derivative ϕ0 (x; h) of ϕ at x ∈ X in the direction h ∈ X, by df

ϕ0 (x; h) = lim sup y→x t0

ϕ(y + th) − ϕ(y) . t

It is easy to see that the function X h −→ ϕ0 (x; h) ∈ R is sublinear, continuous, and so by the Hahn-Banach theorem, ϕ0 (x; ·) is the support function of a nonempty, w∗ -compact and convex set ∂ϕ(x) deﬁned by df

∂ϕ(x) = {x∗ ∈ X : x∗ , hX ≤ ϕ0 (x; h) Copyright © 2005 Marcel Dekker, Inc.

∀h ∈ X}.

Evolution Hemivariational Inequality with Hysteresis and OCP

159

The multifunction ∂ϕ : X −→ 2X \ {∅} is known as the Clarke’s or generalized subdiﬀerential of ϕ. If ϕ, ψ : X −→ R are both locally Lipschitz functionals, then ∂(ϕ + ψ)(x) ⊆ ∂ϕ(x) + ∂ψ(x)

∀x ∈ X

and ∂(tϕ)(x) = tϕ(x)

∀x ∈ X, t ∈ R.

If in addition ϕ is convex, then the subdiﬀerential ∂ϕ coincides with the subdiﬀerential in the sense of convex analysis (see S. Hu and N.S. Papageorgiou [9, p. 267]). Finally, if ϕ is continuously Gateaux diﬀerentiable at x ∈ X (i.e., ϕ ∈ C 1 (X)), then ∂ϕ(x) = {ϕ (x)}. Now we recall the basic facts and notions from the theory of hysteresis operators. Let T > 0. A function v : [0, T ] −→ R is called piecewise monotone if there exists a partition 0 ≤ t0 < t1 < . . . < tn = T of interval [0, T ] (called monotonicity partition) such that v|[ti−1 ,ti ] is monotone, for all i = 1, . . . , n. Such a partition is called standard monotonicity partition if the number of subintervals n is minimal. By Cpm ([0, T ]), we denote the space of all continuous and piecewise monotone functions on [0, T ]. Functional H : Cpm ([0, T ]) −→ R is called rate independent if and only if H[v ◦ ϕ] = H[v], (1) for all v ∈ Cpm ([0, T ]) and all ϕ : [0, T ] −→ R, continuous increasing functions satisfying ϕ(0) = 0 and ϕ(T ) = T . It can be easily seen that only the local extremal values of v have an inﬂuence on H[v]. By S we denote the set of all ﬁnite strings of real numbers and by SA , the set of all alternating strings, i.e., df

SA = {(s0 , . . . , sn ) ∈ S : (si+1 −si )(si −si−1 ) < 0, 1 ≤ i ≤ n−1}. (2) We deﬁne the restriction operator ρA : Cpm ([0, T ]) −→ SA , as follows: df

ρA (v) = (v(t0 ), v(t1 ), . . . , v(tn )),

(3)

for all v ∈ Cpm ([0, T ]), where 0 = t0 < t1 < . . . < tn = T is the standard monotonicity partition of v. By πA : SA −→ Cpm ([0, T ]), we denote the so-called prolongation operator, which maps any string s = (s0 , s1 , . .. , sn ) into the linear interpolate function v : [0, T ] −→ R of the points iT n , si for i = 0, 1 . . . , n. Using these operators, we obtain , : SA −→ R and rate the bijective correspondence between functionals H independent functionals H : Cpm ([0, T ]) −→ R, as follows , ◦ ρA , H = H Copyright © 2005 Marcel Dekker, Inc.

, = H ◦ πA . H

(4)

160

CONTROL AND BOUNDARY ANALYSIS

An operator W : Cpm ([0, T ]) −→ C [0, T ] is said to be a hysteresis operator on Cpm ([0, T ]) if there exists a rate independent functional H called a generating functional of W such that W[v](t) = H[vt ] ∀t ∈ [0, T ], ∀v ∈ Cpm ([0, T ]), where df

vt (ξ) =

v(ξ), v(t),

for 0 ≤ ξ ≤ t, for t < ξ ≤ T.

(5)

(6)

, : SA −→ S is called a hysteresis operator on SA , if An operator W , , 0 ), H(s , 0 , s1 ), . . . , H(s)) , W(s) = (H(s

∀s = (s0 , s1 , . . . , sn ) ∈ SA , (7)

, and H is a rate , = H ◦ πA is called a generating functional of W where H independent functional on Cpm ([0, T ]). , are called ﬁnal value mapThe unique generating functionals H and H , f , respectively. Due to (4), we also pings and are denoted by Wf and W have a bijective correspondence between hysteresis operators W deﬁned , deﬁned on SA . We will use the same notation W on Cpm ([0, T ]) and W , f . Finally, we can , and also Wf for both Wf and W for both W and W extend the hysteresis operators deﬁned on Cpm ([0, T ]) to the set of all continuous functions C [0, T ] , by the use of the density of the embedding Cpm ([0, T ]) ⊆ C [0, T ] . For more details on hysteresis operators we refer to M. Brokate and J. Sprekels [3, Chapter 2], M.A. Krasnoselskii and A.V. Pokrovskii [10] and A. Visintin [18, Chapter I]. As for the optimal control problem, our main result will be based on the following theorem (see E.J. Balder [1, Theorem 2.1, p.1400]).

Theorem 1 Let X be a separable Banach space and Z a separable reﬂexive Banach space. Let F : [0, T ] × X × Z −→ R be a measurable function such that (i) for almost all t ∈ [0, T ], the function F (t, ·, ·) is sequentially lower semicontinuous on X × Z (ii) for all x ∈ X and for almost all t ∈ [0, T ], the function F (t, x, ·) is convex on Z (iii) there exists M > 0 and ψ ∈ L1 0, T ; R such that F (t, x, z) ≥ ψ(t) − M (xX + zZ )

∀x ∈ X, z ∈ Z, a.a. t ∈ [0, T ],

then the functional F : L1 0, T ; X × L1 0, T ; Z −→ [−∞, +∞], Copyright © 2005 Marcel Dekker, Inc.

Evolution Hemivariational Inequality with Hysteresis and OCP

deﬁned by df

161

T

F(x, z) =

F (t, x(t), z(t))dt is sequentially lower semicontinuous in L1 0, T ; X × (w–L1 0, T ; Z )topology. Moreover, the conditions (i)-(iii) are also necessary provided that there 1 1 exists (x, z) ∈ L 0, T ; X × L 0, T ; Z such that F(x, z) < +∞. 0

We will also make use of the following fact from the multivalued analysis (see S. Hu and N.S. Papageorgiou [9, Proposition 3.9, p.694]).

Theorem 2 Let Z be a separable Banach space, 1 ≤ p < ∞, and for almost all t ∈ [0, T ], let the set U (t) be a weakly compact subset of Z. p p p If {fn } ⊆ L 0, T ; Z , f ∈ L 0, T ; Z , fn −→ f weakly in L 0, T ; Z and fn (t) ∈ U (t) for all n and almost all t ∈ [0, T ], then f (t) ∈ convK(w–Z)– lim sup fn (t) n→+∞

2.

for a.a. t ∈ [0, T ].

Hemivariational Inequality with Hysteresis

Let Ω ∈ RN be an open, bounded set with Lipschitz boundary ∂Ω and 2 T > 0. Let H = L Ω . By V we denote a Hilbert space such that the embedding V ⊆ H 1 (Ω) is dense and continuous. We introduce the following spaces: H = L2 0, T ; H = L2 (0, T ) × Ω , V = L2 0, T ; V , W = {v : v ∈ V, v ∈ V }, Y = H 1 (0, T ; H) ∩ L∞ 0, T ; V , X = {v : v ∈ C [0, T ]; V , v ∈ Y} X, = {v : v ∈ C [0, T ]; V , v ∈ W}. The following embeddings are continuous: X ⊆ X, ⊆ Y ⊆ W. We consider the following evolution hemivariational inequality: Find u ∈ X, and χ ∈ H, such that u (t) + w(t) + A(u (t)) + B(u(t)) + χ(t) = f (t) + D(t)y(t) in V (HV I) u(0) = ψ0 , u (0) = ψ1 in Ω w(t, x) = W[u(·, x); x](t) for a.a. (t, x) ∈ (0, T ) × Ω χ(t, x) ∈ ∂j(x, g(u(t, x), u (t, x))) ∀t ∈ [0, T ], for a.a. x ∈ Ω. Copyright © 2005 Marcel Dekker, Inc.

162

CONTROL AND BOUNDARY ANALYSIS

Our hypotheses on the right-hand side f , operator D, initial conditions ψ0 , ψ1 , the hysteresis operator W, the operators A and B, the nonsmooth potential function j, and the function g are the following: H(f, ψ) f ∈ H, ψ0 , ψ1 ∈ V . H(W) W[·, x] is a family of hysteresis operators (indexed by x ∈ Ω) such that (i) for all x ∈ Ω, operator W[·; x] : C [0, T ] −→ C [0, T ] is continuous (ii) the parameterized ﬁnal value mapping Ω x −→ Wf (s; x) ∈ R is measurable for all n ∈ N and all s = (s0 , s1 , . . . , sn ) ∈ S; we also have |Wf (s; x)| ≤ aW(x) + cW max |sk | ∀x ∈ Ω, s ∈ S, n ∈ N, 0≤k≤n

where aW ∈ H and cW > 0 (iii) Wf (ψ1 (·); ·) ∈ H H(g) g : R × R −→ R is a function such that (i) g is continuous (ii) we have |g(ξ, ζ)| ≤ αg |ξ| + βg |ζ| ∀ξ, ζ ∈ R, with αg , βg ≥ 0 H(j) j : Ω × R −→ R is a function such that (i) for all ξ ∈ R, the function Ω x −→ j(x, ξ) ∈ R is measurable (ii) for almost all x ∈ Ω, the function R ξ −→ j(x, ξ) ∈ R is locally Lipschitz (iii) for almost all x ∈ Ω, all ξ ∈ R, and all η ∈ ∂j(x, ξ), we have that |η| ≤ cj (1 + |ξ|), where cj > 0 H(A) A : V −→ V is a linear operator such that (i) for all v, w ∈ V , we have that Av, wV ≤ αA vV wV , with some αA > 0 (ii) for all v, w ∈ V , we have Av, wV = Aw, vV (iii) for all v ∈ V , we have Av, vV ≥ βA v2V − γA v2H , with 2 some βA > 2 cVH cj βg and γA ≥ 0 Copyright © 2005 Marcel Dekker, Inc.

Evolution Hemivariational Inequality with Hysteresis and OCP

163

H(B) B : V −→ V is a linear operator such that (i) for all v, w ∈ V , we have Bv, wV ≤ αB vV wV , with some αB > 0 (ii) for all v, w ∈ V , we have Bv, wV = Bw, vV (iii) for all v ∈ V , we have Bv, vV ≥ 0 H(D) D ∈ L∞ 0, T ; L(Y, V ) and Y is a separable reﬂexive Banach space. Taking g(ξ, ζ) = ξ, we obtain the law governed by Clarke’s subdiﬀerential depending only on function u. Such a relation for elliptic hemivariational inequality was studied by Z. Naniewicz and P.D. Panagiotopoulos [12], and for parabolic hemivariational inequality by M. Miettinen [11] and L. Gasi´ nski [5]. On the other hand, taking g(ξ, ζ) = ζ, we obtain the law governed by Clarke’s subdiﬀerential depending only on the derivative u of the function u. Such a relation for evolution hemivariational inequality was studied by P.D. Panagiotopoulos [17]. Now we can formulate an existence result for (HV I).

Theorem 3 If hypotheses H(f, ψ), H(W), H(g), H(j), H(A), H(B), and H(D) hold, then for any y ∈ L2 0, T ; Y , problem (HV I) admits a solution u ∈ X ⊆ X,. Proof: For a ﬁxed y ∈ L2 0, T ; Y , we have D(·)y(·)2V

T

= 0

D(t)y(t)2V dt ≤

T 0

D(t)2L(Y,V ) y(t)2Y dt

≤ D2L∞ (0,T ;L(Y,V )) y2L2 (0,T ;Y )

(8)

and so for the right-hand side of the equation in (HV I), we have that f (·) + D(·)y(·) ∈ V .

(9)

Now the existence theorem follows from Theorem 3.1 of L. Gasi´ nski [7]. 2

3.

Estimates on the Solutions of (HVI)

The crucial role in the proof of the main result will be played by a priori estimates on the solutions of (HV I). Let us start with two simple lemmas on the estimates of selections of ∂j x, · and the output of the hysteresis operator.

Copyright © 2005 Marcel Dekker, Inc.

164

CONTROL AND BOUNDARY ANALYSIS

Lemma 4 If hypotheses hold, u ∈ W, u(0) = ψ0 , η ∈ H H(j) and H(g) and η(t, x) ∈ ∂j x, g u(t, x), u (t, x) for almost all (t, x) ∈ (0, T ) × Ω, then for all t ∈ [0, T ], we have

t t t s 2 2 2 η(s)H ds ≤ c 1 + u (s)H ds + u (τ )H dτ ds , 0

0 df

where c =

3c2j

0

0

(10)

max T |Ω| + 2T α2g ψ0 2H , βg2 , 2T α2g .

Lemma 5 If hypotheses H(W) hold, u ∈ Y, u(0) = ψ0 , and w ∈ H are such that w(t, x) = W[u(·, x); x](t) then t

w(s)2H

ds ≤ 2 c 1+

0

for a.a. (t, x) ∈ (0, T ) × Ω,

t 0

s

u

(τ )2H

dτ ds

∀t ∈ [0, T ],

(11)

0

and in particular

c 1 + T u 2H w2H ≤ 2 df with 2 c = 2T max aW2H + 2c2Wψ0 2H , 2c2W .

(12)

The next lemma gives some estimates on the solutions of (HV I). In its proof both Lemmas 4 and 5 are needed.

Lemma 6 If hypotheses H(f, ψ), H(W), H(g), H(j), H(A), H(B), and H(D) hold and u ∈ X, is a solution of (HV I) for some y ∈ L2 0, T ; Y , then uX, = uC([0,T ];V ) + u W ≤ c 1 + yL2 (0,T ;Y ) , where c > 0 is a constant not depending on y.

4.

Optimal Control Problem

In this section we consider an optimal control problem governed by our hemivariational inequality with hysteresis. Let J be the Bolza-type functional deﬁned by: T df J(y, u) = l(u(T ), u (T )) + L(t, u(t), u (t), y(t))dt, 0

where y is any admissible control and u ∈ S(y), where by S(y) ⊆ X, we denote the set of solutions for (HV I) with the control y at the right-hand side. Copyright © 2005 Marcel Dekker, Inc.

Evolution Hemivariational Inequality with Hysteresis and OCP

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By optimal control problem, we mean the following problem: (CP )

Find y ∗ ∈ U and u∗ ∈ S(y ∗ ) such that: J(y ∗ , u∗ ) = min min J(y, u). y∈U u∈S(y)

Due to the lack of convexity of j (or some additional assumptions) we are not able to guarantee the uniqueness of the solution of (HV I), so S(y) contains, in general, more than one element. To obtain our existence result we will need the following assumptions on the set of admissible controls U and the cost functional J: df H(U) U = y ∈ L2 0, T ; Y : y(t) ∈ U (t) , where U : [0, T ] −→ 2Y \ {∅} is a multifunction such that for all t ∈ [0, T ], the set U (t) is a nonempty closed convex subset of Y and the function df

t → |U (t)| = sup{yY : y ∈ U (t)} is in L∞ + (0, T ). H(J) l : V × H −→ R is weakly lower semicontinuous and L : [0, T ] × V × H × Y −→ R ∪ {+∞} is a measurable function such that (i) L(t, ·, ·, ·) is sequentially lower semicontinuous on V × H × Y for almost all t ∈ [0, T ] (ii) L(t, v, w, ·) is convex on Y for all v ∈ V , w ∈ H and almost all t ∈ [0, T ] (iii) there exists M > 0 and ψ ∈ L1 (0, T ) such that for all v ∈ V , w ∈ H, y ∈ Y and almost all t ∈ [0, T ], we have L(t, v, w, y) ≥ ψ(t) − M (vV + wH + yY ) The following lemma says that the map ,

U y −→ S(y) ∈ 2X \ {∅}, which for every y ∈ U admits the value S(y), the set of solutions of (HV I) with the control y has a closed graph in (w–U )×(w–X,)-topology.

Lemma 7 Let hypotheses H(f, ψ), H(W), H(g), H(j), H(A), H(B), H(D) and H(U) hold. If {yn } ⊆ U and {un } ⊆ X, are two sequences such that un ∈ S(yn ) and yn −→ y ∗ Copyright © 2005 Marcel Dekker, Inc.

weakly in U ,

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CONTROL AND BOUNDARY ANALYSIS

weakly in X,,

un −→ u∗

with some y ∗ ∈ U and u∗ ∈ X,, then u∗ ∈ S(y ∗ ). Now we can formulate our main result:

Theorem 8 If hypotheses H(f, ψ), H(W), H(g), H(j), H(A), H(B), H(D), H(U) and H(J) hold, then problem (CP ) admits a solution. Proof: We apply the direct method of the calculus of variations. Let (yn , un ) be a minimizing sequence for (CP ), i.e., yn ∈ U,

un ∈ S(yn )

∀n ≥ 1

and lim J(yn , un ) = min min J(y, u).

n→+∞

y∈U u∈S(y)

From the hypotheses H(U), we obtain that T yn (t)2Y dt ≤ M1 , yn 2L2 (0,T ;Y ) = 0

with some M1 > 0 not dependent on n and so we may assume, passing to a subsequence if necessary, that yn −→ y ∗ weakly in L2 0, T ; Y . (13) We will show that for almost all t ∈ [0, T ], sets U (t) are weakly compact subsets of Y . Indeed, for a ﬁxed t ∈ [0, T ], the set U (t) is closed and convex, so also weakly closed. Moreover, for any sequence {zn } ⊆ U (t), we have that zn Y ≤ M2 , with some M2 > 0. Since Y is a reﬂexive Banach space, we get, at least for a subsequence, that zn −→ z

weakly in Y.

In view of the weak closedness of U (t), we obtain that z(t) ∈ U (t), which proves that U (t) is weakly compact for almost all t ∈ [0, T ]. Now, applying Theorem 2, for almost all t ∈ [0, T ], we get that y ∗ (t) ∈ convK(w–Y )– lim sup yn (t) ⊆ convU (t), n→+∞

what, together with the fact that U (t) is a closed and convex subset of Y , implies y ∗ (t) ∈ U (t) for a.a. t ∈ [0, T ]. (14) Copyright © 2005 Marcel Dekker, Inc.

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Recalling that the sequence {yn }n≥1 is bounded in L2 0, T ; Y , from the a priori estimate of Lemma 6, we immediately get that un V ≤ M3

and

un W ≤ M3 ,

with some M3 > 0. So, for a subsequence, we have un −→ u∗ un −→ u∗

weakly in V weakly in W.

(15) (16)

Hence, and from (13), applying Lemma 7, we obtain that u∗ ∈ S(y ∗ ). Thus from (14), we get that (y ∗ , u∗ ) is an admissible state-control pair. To conclude, it is enough to show that it is also an optimal solution. First, from the continuity of the embedding W ⊆ C [0, T ]; H we have that un (T ) −→ u∗ (T ) weakly in H un (T ) −→ u∗ (T ) weakly in H. Using the compactness of the embedding W ⊆ H, from (15) and (16), for a next subsequence, we have un −→ u∗ in H un −→ u∗ in H.

(17) (18)

Invoking Theorem 1 and the fact that l is weakly lower semicontinuous (see hypotheses H(J)), we obtain that the cost functional J is sequen tially lower semicontinuous on L2 0, T ; H × H × (w–L2 0, T ; Y ). So in consequence, from (13), (17), and (18), we get J(y ∗ , u∗ ) ≤ lim inf J(yn , un ) = min min J(y, u), n→+∞

which proves the theorem.

y∈U u∈S(y)

2

References [1] E.J. Balder. Necessary and suﬃcient conditions for l1 -strong-weak lower semicontinuity of integral functional. Nonlin. Anal., 11:1399–1404, 1987. [2] W. Bian. Existence results for second order nonlinear evolution inclusions. Indian J. Pure Appl. Math., 11:1177–1193, 1998. [3] M. Brokate and J. Sprekels. Hysteresis and Phase Transitions. Springer-Verlag, Berlin/New York, 1996.

Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

[4] F.H. Clarke. Optimization and Nonsmooth Analysis. John Wiley & Sons, New York, 1983. [5] L. Gasi´ nski. Optimal shape design problems for a class of systems described by parabolic hemivariational inequality. J. Global Optim., 12:299–317, 1998. [6] L. Gasi´ nski. Existence result for hyperbolic hemivariational inequalities. Nonlinear Anal., 47:681–686, 2001. [7] L. Gasi´ nski. Evolution hemivariational inequalities with hysteresis. Nonlinear Anal., 57:323-340, 2004. [8] L. Gasi´ nski. Existence of solutions for hyperbolic hemivariational inequalities. J. Math. Anal. Appl., 276:723–746, 2002. [9] S. Hu and N.S. Papageorgiou. Handbook of Multivalued Analysis, Volume I: Theory. Kluwer, Dordrecht, 1997. [10] M.A. Krasnoselskii and A.V. Pokrowski. Systems with Hysteresis. SpringerVerlag, Heidelberg, 1989. [11] M. Miettinen. A parabolic hemivariational inequality. Nonlin. Anal., 26:725– 734, 1996. [12] Z. Naniewicz and P.D. Panagiotopoulos. Mathematical Theory of Hemivariational Inequalities and Applications. Dekker, New York, 1995. [13] P.D. Panagiotopoulos. Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions. Birkh¨ auser, Basel, 1985. [14] P.D. Panagiotopoulos. Nonconvex problems of semipermeable media and related topics. Z. Angew. Math. Mech., 65:29–36, 1985. [15] P.D. Panagiotopoulos. Coercive and semicoercive hemivariational inequalities. Nonlin. Anal., 16:209–231, 1991. [16] P.D. Panagiotopoulos. Hemivariational Inequalities, Applications in Mechanics and Engineering. Springer-Verlag, Berlin, 1993. [17] P.D. Panagiotopoulos. Modelling of nonconvex nonsmooth energy problems. Dynamic hemivariational inequalities with impact eﬀects. J. Comp. Appl. Math., 63:123–138, 1995. [18] A. Visintin. Diﬀerential Models of Hysteresis. Springer-Verlag, Berlin/New York, 1994.

Copyright © 2005 Marcel Dekker, Inc.

CHAPTER XIV ON THE MODELING AND CONTROL OF DELAMINATION PROCESSES Michal Koˇcvara∗ Institute of Applied Mathematics University of Erlangen Erlangen, Germany

Jiˇri V. Outrata Institute of Information Theory and Automation Academy of Sciences of the Czech Republic Prague, Czech Republic

Abstract

This paper is motivated by problem of optimal shape design of laminated elastic bodies. We use a recently introduced model of delamination, based on minimization of potential energy which includes the free (Gibbs-type) energy and (pseudo)potential of dissipative forces, to introduce and analyze a special mathematical program with equilibrium constraints. The equilibrium is governed by a ﬁnite sequence of coupled mathematical programs that have to be solved one after another in the direction of increasing time. We derive optimality conditions for the control problem and illustrate them on an academic example.

Keywords: Inelastic damage, mathematical program with equilibrium constraints, evolution equilibrium, hemivariational inequality

1.

Introduction

The design of crashworthy vehicles depends upon developing structures capable of absorbing large amounts of crash energy. Composite materials have been shown to have energy absorbing properties superior to conventional metallic structures. With the goal of designing crash elements (structural elements with high energy absorption by crash), we consider laminated structures. The energy is absorbed by delamination of such structures. Mathematical simulation of vehicle crash requires,

∗ On

leave from the Academy of Sciences of the Czech Republic.

Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

ﬁrst of all, ﬁnding a reliable mathematical model. In this paper, we use a recently introduced model of delamination, based on minimization of potential energy which includes the free (Gibbs-type) energy and (pseudo)potential of dissipative forces ([9]). Based on this model, we introduce and analyze an evolution control problem with the ﬁnal goal of designing optimal shape of crash elements. Delamination is a progressive separation of bonded laminate and, simultaneously, degradation of the used adhesive. The mechanism of delamination is very complex and involves phenomena like debonding and unilateral contact with nonmonotone friction. In the recent literature, the modeling of the delamination problem is generally approached either by using fracture mechanics (see, e.g., [3, 20]) or by introducing special constitutive laws for the interface material in the spirit of damage mechanics, or simply quasistatically. In the second approach, delamination is described by a damage variable reﬂecting the destruction of the bonds in the a priori known delamination surface (see, e.g., [5, 17]). Another, static approach was proposed by Panagiotopoulos [16] who formulated the problem of equilibrium positions as a hemivariational inequality (HVI), a generalization of variational inequality for nonmonotone operators; see also [2]. Thus, this method has limited applications only in processes with simple time-dependent loadings. After discretization by the ﬁnite elements method, this approach results in a nonsmooth and nonconvex optimization problem. The variables of this problem are the elements of the discretized displacement vector. In the model introduced in [9], delamination is considered as a fracture-like process that can run along a priori known surfaces between homogeneous isotropic elastic bodies that are in frictionless unilateral contact. It is as an activated and rate-independent process, based on the philosophy that a speciﬁc energy is needed to cut the macromolecular structure of the adhesive, no matter how fast or slow this process is. This model is supported by a rigorous analysis based on the apparatus recently developed for rate-independent process in, e.g., [10–11]. In this paper, we will refer this model as RI model. After discretization in time and space, the RI model results in a sequence of smooth nonconvex optimization problems. Compared to the static HVI model, the dimension of one problem increases by the number of damage parameters. Assume that the discretized RI model contains parameters, the values of which should be optimized with respect to an (upper-level) objective. In this way, we arrive at a special mathematical program with equilibrium constraints (MPEC) in which the underlying equilibrium has evolution nature. From the viewpoint of optimality conditions, similar problems were investigated in [21]. There, however, the equilibrium is Copyright © 2005 Marcel Dekker, Inc.

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171

governed by a standard optimal control problem (i.e., the optimization is performed over the whole time interval). In our case, we have to do with a ﬁnite sequence of coupled mathematical programs that have to be solved one after another in the direction of increasing time. This difference is naturally reﬂected in the character of the resulting optimality conditions. The paper is organized as follows: In the next section, we introduce the RI model of [9] and compare it to the static VI model. After that, we present a conceptual MPEC with the discretized RI delamination process as equilibrium constraint. Section 3 is devoted to necessary optimality conditions for this MPEC. As a workhorse we use the generalized diﬀerential calculus of B. Mordukhovich ([13, 12]) which proved to be an eﬃcient tool in the treatment of various equilibria. The resulting conditions are illustrated by means of an academic example introduced in Section 2. The following notation is employed: If f is a diﬀerentiable function of two variables, then ∇1 f and ∇2 f denote the partial derivatives with respect to the ﬁrst and the second variable, respectively. B is the unit ball and for a multifunction Q[Rn ; Rm ], Gph Q := {(x, y) ∈ Rn × Rm | y ∈ Q(x)}. To describe the local properties of sets, multifunctions, and realvalued functions, we make use of suitable concepts from the generalized diﬀerential calculus of B. Mordukhovich ([13],[12]). So, NΩ (x) denotes the limiting normal cone to the set Ω at x ∈ clΩ, D ∗ Q(x, y)(·) denotes the coderivative of the multifunction Q at (x, y) ∈ cl Gph Q, and ∂f (x) is the limiting subdiﬀerential of a real-valued function f at x ∈ domf . For the reader’s convenience, we close this section with a useful stability concept for multifunctions which plays an important role in Section 3.

Definition 1 ([19]) A multifunction Q[Rn ; Rm ] is calm at (¯ x, y¯) ∈ Gph Q provided there is a constant κ ∈ R+ along with neighborhoods U of x ¯ and V of y¯ such that Q(x) ∩ V ⊂ Q(¯ x) + κx − x ¯B for all x ∈ U.

2.

Delamination Process

In this section we recall the rate-independent model of the delamination process, as recently introduced in [9]. In this model, after a (time and space) discretization, one has to solve in each time step a smooth nonconvex optimization problem. We further propose a conceptual optimization problem in which the discretized delamination process arises as a constraint. Copyright © 2005 Marcel Dekker, Inc.

172

2.1

CONTROL AND BOUNDARY ANALYSIS

Modeling

For simplicity of notation, we only consider laminates consisting of two elastic bodies Ω1 and Ω2 with an interface boundary denoted by Γ12 ; see Figure 1. The two bodies are in unilateral contact along Γ12 ; further they are glued along Γ12 by an adhesive. The matrix b : Γ12 → R2×2 reﬂects the elastic properties of the adhesive. We adopt, in fact, a dimensional reduction of the two-dimensional adhesive layer to a one-dimensional surface Γ12 . The state of the system will be considered as y = (u1 , u2 , ζ) where ui : Ωi → R2 is the (small) displacement in the domain Ωi , i = 1, 2, and ζ : Γ12 → [0, 1] is a damage parameter indicating how much of the adhesive is eﬀective: 1 means 100% of the adhesive glues at x ∈ Γ12 , 0 means that the surface is completely delaminated at the current point x ∈ Γ12 , and 0 < ζ(x) < 1 means that some portion of macromolecules of the adhesive is already cut while the rest is still eﬀective. A rate-independent delamination model has recently been introduced in [9]. This model is based on the minimization of the elastic stored energy and the dissipation potential subject to several constraints. The constraints reﬂect – the unilateral contact of the two elastic bodies – the time-dependent Dirichlet loading by a “hard device” – the irreversibility (in time) of the delamination of the adhesive The resulting minimization problem is ﬁrst discretized in the time variable and further in the space variable by the standard ﬁnite element

Γn

Ω1

&

Γt

Ω2

Figure 1.

Copyright © 2005 Marcel Dekker, Inc.

Γt Γ12

Γt

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173

method. Below we present the fully discretized problem that will be further studied in the subsequent sections. We use the same ﬁnite element mesh for both variables, u and ζ. In order to allow for the separation (delamination) of the elastic bodies, the joint boundary Γ12 is discretized by N pairs of nodes, say (n1,j , n2,j ), j = 1, . . . , N , with ni,j ∈ Ωi , i = 1, 2. At the beginning of the delamination process the nodes n1,j , n2,j have the same positions for all j = 1, . . . , N . Let A1 , A2 be the stiﬀness matrices of the elastic bodies Ω1 , Ω2 , respectively. The discretized elastic stored energy becomes V (u, ζ) =

2

uTi Ai ui +

i=1

N

ωj ζj (u1,j −u2,j ) b(nj )(u1,j −u2,j )

j=1

where ωj are integration weights. Similarly, the discretized dissipation potential is R(ζ) =

N

−ωj d(nj )ζj .

j=1

Here d(x) is a phenomenological parameter having the meaning of a speciﬁc energy needed to delaminate the surface Γ12 at a point x ∈ Γ12 , i.e., the energy needed to switch ζ(x) from 1 to 0. This energy is irreversibly dissipated to the structural change of the adhesive. The case R(ζ) = +∞ will be respected by a corresponding linear constraint. We assume that the laminate is loaded by time-dependent unilateral Dirichlet loading (“hard-device”) on parts of the boundaries of Ω1 , Ω2 . The load vector (prescribed displacements) at a time step i is denoted by ui . Let L1 be a rectangular matrix selecting the “loaded” boundary components from the whole vector u. Finally, denote by L2 a rectangular matrix that takes care of the nonpenetration of Ω1 and Ω2 on Γ12 . The discrete version of the delamination problem in time step i is then (we omit the current time step index for simplicity): Minimize V (u, ζ) + R(ζ−ζ i−1 ) Subject to

(1)

L 1 u ≥ ui L2 u ≤ 0 ζ i−1 ≥ ζ ≥ 0 Copyright © 2005 Marcel Dekker, Inc.

componentwise.

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CONTROL AND BOUNDARY ANALYSIS

Here the index i−1 refers to the previous time step. The irreversibility of the dissipation process is guaranteed by the left-hand side of the last constraint. As we are only interested in the components of the displacement vector lying on the boundary Γ12 , we can eliminate all components corresponding to the interior nodes. This reduces the number of variables in (1) to the number of interface boundary nodes times ﬁve (two times two components of the displacement vector plus the components of ζ) plus the number of boundary nodes with prescribed nonzero Dirichlet condition (loaded nodes) times two. In [9] it was shown that (1) can be eﬃciently solved by state-of-the-art optimization software and that the proposed modeling of the delamination process delivers reasonable results.

2.2

Relation to the HVI Model

Let us brieﬂy show the relation of the RI model described above to the HVI model, introduced by Panagiotopoulos and numerically solved in [2]. The hemivariational inequality model of the static delamination problem amounts to ﬁrst-order optimality condition of the following nonsmooth nonconvex optimization problem (we use the same notation as in the previous section): Minimize 2

uTi Ai ui +

i=1

N

ωj min{(u1,j −u2,j ) b(nj )(u1,j −u2,j ), d(nj )}

j=1

Subject to L 1 u ≥ ui L2 u ≤ 0 . Recall the simple fact that, for smooth convex functions f and g, the set of (Clarke) stationary points of the (nonsmooth) pointwise minimum function min{f (x), g(x)} is equal to the set of x-components of stationary points of the (smooth) function αf (x)+(1−α)g(x), α ∈ [0, 1]. Hence the (Clarke) stationary points of problem (1) are just the u-components of stationary points of the smooth problem (1), assuming that ζ i−1 ≡ 1. In other words, solving the static HVI model “is the same” as solving the RI model for the ﬁrst time step, starting from the non-delaminated state (of course, one has to take into account that diﬀerent algorithms may ﬁnd diﬀerent stationary points of these two nonconvex problems). Copyright © 2005 Marcel Dekker, Inc.

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175

The static model tries to “jump” directly into the solution at the terminal time—the time discretization is reduced to only one interval. The HVI results may then only be reliable for monotone (in time) loadings and simple geometries, contrary to the RI model that allows for general nonmonotone loading (and unloading). The diﬀerence between the two models, even for monotone loading and simple problems, is clearly seen when solving the optimal control problems; see, in particular, Example 2 in Section 4.

2.3

Control

Our second goal is to control the (discretized) RI delamination process by certain parameters. The aim is to ﬁnd such parameters that an objective function depending on the terminal state is minimized. For instance, having in mind our motivation from the Introduction, we want to ﬁnd such a shape of the boundaries of Ω1 , Ω2 that, at the terminal time, as much energy is dissipated as possible. This optimal control problem can thus be written as the following “conceptual” MPEC: Minimize ϕ(x, y) Subject to x ∈ Uad , y is the terminal state of the delamination process depending on parameter x,

(2)

where Uad ⊂ Rn is the set of admissible controls. The detailed formulation of the problem is given in the next section. Before going on, let us introduce a simpliﬁed example that demonstrates the diﬃculties connected with the control of the delamination process. Example 1 Consider the four-string example as shown in Figure 2. In reality, the strings are at the same horizontal position; here they are plotted with some gap for presentation reasons. The elasticity modulae of the strings are e1 , . . . , e4 . Assume that e3 = e4 and denote it by e. The end-nodes of the strings are denoted n0 , n1 , n2 . The vertical displacements at the nodes are u0 , u1 , u2 . Node n0 is ﬁxed (u0 = 0), node n2 is subjected to nonzero Dirichlet boundary condition (prescribed displacements) u2 = u. Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS n0 e1

e3 n1

e2

e4 n2

Figure 2.

The left-hand strings are elastic and do “never” break—they simulate the elastic bodies. The right-hand strings simulate the adhesive: they are also elastic but can break when the relative displacement reaches a certain value. The Dirichlet condition u depends on time. The equilibrium state in the i-th time interval is obtained by solving the optimization problem min

u1 ,u2 ,ζ1 ,ζ2

2 ' ( 2 2 i−1 ej (uj −uj−1 ) + ζj e(uj −uj−1 ) + (ζj −ζj )ed E := j=1

Subject to

(3)

u 2 ≥ ui uj − uj−1 ≥ 0, ζ

i−1

j = 1, 2

≥ζ ≥0

The ﬁrst term under the sum is the strain energy of the elastic strings 1 and 2. The second term is the strain energy of the breakable strings 3 and 4 (the “adhesive”). The third term is the dissipation energy. Number d is the energy dissipated by breaking one string (this is given in our case). In the MPEC, the control variables are e1 and e2 . The goal is to ﬁnd such design that, at the terminal time k, as much energy is dissipated as possible: ϕ := ζ1k + ζ2k max e1 ,e2 ,u1 ,u2 ,ζ1 ,ζ2

Subject to e1 + e2 = 2 (u1 , u2 , ζ1 , ζ2 ) solves (3) at time k. Assume the following problem data: d = 1 · 10−6 (dissipation parameter) Copyright © 2005 Marcel Dekker, Inc.

(4)

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177

uk = 0.007 (ﬁnal prescribed displacement) e = 10. The optimal solution is e1 = 1,

e2 = 1,

ζ1k = ζ2k = 0, ϕ = 2,

uk1 = 0.0035,

uk2 = 0.007,

E = 4.45 · 10−5 ,

when both strings 3 and 4 break. However, there may be more solutions to (4) within some neighborhood of 1 that also lead to the break of both strings. One such solution is e1 = 1.05, ζ1k = ζ2k = 0, ϕ = 2,

e2 = 0.95, uk1 = 0.003325,

uk2 = 0.007,

E = 4.44388 · 10−5 .

Obviously, everything within the interval e1 ∈ (0.95, 1.05), e2 = 2 − e1 is a solution to (4). Note that, for instance, (e1 = 1.1, e2 = 0.9) is not an optimal control of the MPEC (4) anymore, because it gives ζ1k = 1, ζ2k = 0 as a solution of the state problem (3). That means only one string breaks and the corresponding upper-level criterium is only ϕ = 1. The next two sections are devoted to necessary optimality conditions for MPECs of the type (2).

3.

Optimality Conditions

The discretized delamination model, introduced and discussed in the previous section, can be generally written down in the form of a ﬁnite sequence of coupled optimization problems Minimize f i (y i−1 , y i ) Subject to

(5) y ∈ Γ (y i

i

i−1

), i = 1, 2, . . . , k

0

y given, where y i−1 ∈ Rm is the parameter, y i ∈ Rm is the unknown variable, f i (y i−1 , ·) is the objective, and the multifunction Γi [Rm ; Rm ] speciﬁes the feasible set. This sequence of optimization problems has to be Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

solved starting from the initial “state” y 0 , in the direction of the increasing time index. In this way, one obtains eventually the whole trajectory y 1 , y 2 , . . . , y k . The aim of this section is to analyze MPECs with equilibria governed by such a sequence of coupled optimization problems. In each of them, additionally, a control variable x ∈ Rn arises so that the ith problem attains the “controlled” form Minimize f i (x, y i−1 , y i ) Subject to

(6)

y i ∈ Γi (x, y i−1 ). Because our attention is paid to necessary optimality conditions for MPECs with such equilibria, we can replace the single problems (6) by the respective ﬁrst-order necessary optimality conditions. Assuming that all objectives f i (x, y i−1 , ·) are continuously diﬀerentiable for each pair (x, y i−1 ), these conditions can be written down in the form 0 ∈ ∇3 f i (x, y i−1 , y i ) + NΓ(x,yi−1 ) (y i ), i = 1, 2, . . . , k.

(7)

Instead of the sequence of relations (6), we can now consider the sequence of coupled generalized equations (GEs) 0 ∈ F i (x, y i−1 , y i ) + Qi (x, y i−1 , y i ), i = 1, 2, . . . , k,

(8)

where F i [Rn × Rm × Rm → Rm ], Qi [Rn × Rm × Rm ; Rm ] represent the single-valued and the multivalued part, respectively. This generalization enlarges our applicability area also to equilibria governed, e.g., by a ﬁnite sequence of coupled complementarity problems or variational inequalities. In this way, we have arrived at a hopefully useful paradigm describing a speciﬁc time evolution of a fairly large class of equilibrium problems. On the respective MPEC we impose the following simplifying assumptions: (i) The (upper level) objective depends only on x and y k . (ii) The state variables y 1 , y 2 , . . . , y k are not subject to any constraints. (iii) All function F i are continuously diﬀerentiable and all maps Qi have closed graphs. The ﬁrst two assumptions are not essential and can be removed. In the MPECs associated with the delamination process, however, they are fulﬁlled and so we decided to simplify the statement of the next theorem Copyright © 2005 Marcel Dekker, Inc.

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by imposing them from the very beginning. We thus have to deal with the following MPEC: Minimize ϕ(x, y k ) Subject to 0 ∈ F i (x, y i−1 , y i ) + Qi (x, y i−1 , y i ), i = 1, 2, . . . , k

(9)

y 0 given x ∈ Uad , where ϕ[Rn × Rm → R] is an (upper level) objective and Uad ⊂ Rn is as before the set of admissible controls. Throughout the rest of the paper it is assumed that ϕ is locally Lipschitz and Uad is nonempty and closed. By y we denote the trajectory (y 1 , y 2 , . . . , y k ) and m := km.

Theorem 2 Let (2 x, y2) be a (local) solution of MPEC ( 9). Denote z2i = −F i (2 x, y2i−1 , y2i ), i = 1, 2, . . . , k, and deﬁne the multifunction Ξ[Rm ; n R × Rm ] by Ξ(ξ 1 , ξ 2 , . . . , ξ k ) := (x, y) ∈ Uad × Rm |ξ i ∈ F i (x, y i−1 , y i ) + Qi (x, y i−1 , y i ), i = 1, 2, . . . , k} . Assume that Ξ is calm at (0, x 2, y2). Then there exist four sequences of adjoint vectors p1 , p2 , . . . , pk , q 1 , q 2 , . . . , q k , v 1 , v 2 , . . . , v k , w1 , w2 , . . . , wk and subgradients (κ, η) ∈ ∂ϕ(2 x, y2k ) such that (with yˆ0 = y 0 ) x, y2i−1 , y2i , z2i )(wi ), i = 1, 2, . . . , k (pi , q i , v i ) ∈ D∗ Qi (2 and the adjoint equation system x, y2k−1 , y2k ))T wk + v k 0 = η + (∇3 F k (2 0 = (∇3 F k−1 (2 x, y2k−2 , y2k−1 ))T wk−1 + v k−1 + (∇2 F k (2 x, y2k−1 , y2k ))T wk + q k

(10)

1 0 1 T 1 1 2 1 2 T 2 2 x, y , y2 )) w + v + (∇2 F (2 x, y2 , y2 )) w + q 0 = (∇3 F (2

(11)

.. .

is fulﬁlled. Moreover, one has 0 ∈κ+

k T x, y2i−1 , y2i ) wi + pi + NUad (2 x). ∇1 F i (2 i=1

Copyright © 2005 Marcel Dekker, Inc.

(12)

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CONTROL AND BOUNDARY ANALYSIS

Proof. The constraints in (9) can be written down in the form 0 ∈ Φ(x, y) + Λ, x ∈ Uad , where

Φ(x, y) = −

x y0 y1 1 −F (x, y 0 , y 1 ) x y1 y2 2 −F (x, y 1 , y 2 ) ······ x y k−1 yk k −F (x, y k−1 , y k )

k and Λ = Xi=1 GphQi

Due to the imposed calmness assumption, one can invoke [14, Thm.2.4] which yields the existence of a Karush-Kuhn-Tucker (KKT) vector x, y2)) b = (p1 , q 1 , v 1 , −w1 , p2 , q 2 , v 2 , −w2 , . . . , pk , q k , v k , −wk ) ∈ NΛ (−Φ(2 and subgradients (κ, η) ∈ ∂ϕ(2 x, y2k ) such that κ NUad (2 x) 0 0 x, y2))T b + 0 ∈ ... − (∇Φ(2 .. . 0 0 η

.

(13)

k N x, y2)) = Xi=1 x, y2i−1 , y2i , z2i ) see [13, Prop.1.6], Because NΛ (−Φ(2 GphQi (2 x, y2i−1 , y2i , z2i ), i = 1, 2, . . . , k. In it follows that (pi , q i , v i , −wi ) ∈ NGphQi (2 this way relations (10) have been established. The ﬁrst line of (13) leads now directly to relation (12), whereas the remaining k lines generate the adjoint system (11).

The calmness assumption is automatically fulﬁlled provided Uad is convex polyhedral, all function F i are aﬃne, and all sets GphQi are unions of ﬁnitely many convex polyhedral sets. Indeed, in such a case, GphΞ is also a union of ﬁnitely many convex polyhedral sets and Ξ is locally upper Lipschitz around 0, cf. [18]. This is, however, a stronger Copyright © 2005 Marcel Dekker, Inc.

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181

property than the required calmness at (0, x 2, y2). Another possibility is to ensure the Aubin property of Ξ around (0, x 2, y2), which is also stronger than the required calmness condition. This can be done by the following Mangasarian-Fromowitz constraint qualiﬁcation.

Theorem 3 Assume that the system 0 = (∇3 F k (2 x, y2k−1 , y2k ))T wk + v k 0 = (∇3 F k−1 (2 x, y2k−2 , y2k−1 ))Twk−1 +v k−1 +(∇2 F k (2 x, y2k−1 , y2k ))T wk +q k .. . x, y 0 , y21 ))T w1 + v 1 + (∇2 F 2 (2 x, y21 , y22 ))T w2 + q 2 0 = (∇3 F 1 (2 k T ∇1 F i (2 0∈ x, y2i−1 , y2i ) wi + pi + NUad (2 x) i=1

x, y2i−1 , y2i , z2i )(wi ), i = 1, 2, . . . , k, possesses only with (pi , q i , v i ) ∈ D∗ Qi (2 the trivial solution p1 = p2 = . . . = pk = 0, q 2 = q 3 = . . . = q k = 0, v 1 = v 2 = . . . = v k = 0 and w1 = w2 = . . . wk = 0. Then Ξ has the Aubin property around (0, x 2, y2). The statement follows from the Mordukhovich characterization of the Aubin property ([19, Thm.9.40]) and standard rules of the coderivative calculus [12]. As shown in [14], the above condition is automatically fulﬁlled, provided the multifunctions Qi do not depend on the control and the multifunction ∆[Rm ; Rm ], deﬁned by x, y2i−1 , y2i ) ∆(ξ) = { y ∈ Rm | ξ i ∈ F i (2 x, y2i−1 , y2i )(y i−1 − y2i−1 ) + ∇2 F i (2 + ∇3 F i (2 x, y2i−1 , y2i )(y i − y2i ) x, y2i−1 , y2i ), i = 1, 2, . . . , k } , + Qi (2 is locally single-valued and Lipschitz around (0, y2) (Robinson’s strong regularity). This is, however, not the case if we deal with the delamination model of Section 2. The optimality conditions of Theorem 2 can be substantially simpliﬁed provided the maps Qi do not depend on x or y i−1 . This is expressed in the following corollaries.

Corollary 4 Let all assumptions of Theorem 2 be fulﬁlled and assume that the maps Qi , i = 1, 2, . . . , k, do not depend on x. Then there exist three sequences of adjoint vectors q 1 , q 2 , . . . , q k , v 1 , v 2 , . . . , v k , x, y2k ) such that w1 , w2 , . . . , wk and subgradients (κ, η) ∈ ∂ϕ(2 y i−1 , y2i , z2i )(wi ), i = 1, 2, . . . , k, (q i , v i ) ∈ D∗ Qi (2 Copyright © 2005 Marcel Dekker, Inc.

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the adjoint equation system ( 11) is satisﬁed, and 0∈κ+

k

T ∇1 F i (2 x, y2i−1 , y2i ) wi + NUad (2 x).

(14)

i=1

Corollary 5 Let all assumptions of Theorem 2 be fulﬁlled and assume that for i = 1, 2, . . . , k the maps Qi depend exclusively on variables y i , respectively. Then there exist two sequences of adjoint vectors v 1 , v 2 , . . . , v k , w1 , w2 , . . . , wk and subgradients (κ, η) ∈ ∂ϕ(2 x, y2k ) such that y i , z2i )(wi ), i = 1, 2, . . . , k v i ∈ D∗ Qi (2 and the adjoint equation system x, y2k−1 , y2k ))T wk + v k 0 = η + (∇3 F k (2 0 = (∇3 F k−1 (2 x, y2k−2 , y2k−1 ))T wk−1 + v k−1 + (∇2 F k (2 x, y2k−1 , y2k ))T wk ······ x, y 0 , y21 ))T w1 + v 1 + (∇2 F 2 (2 x, y21 , y22 ))T w2 0 = (∇3 F 1 (2

(15)

is fulﬁlled. Moreover, relation ( 14) holds true. Remark As in the classic discrete-time optimal control problems [8], the adjoint systems (11),(15) have to be solved backwards starting from the terminal condition. Together with the GEs (8) they represent a special two-point boundary value problem.

4.

Optimization of Delamination Processes

The aim of this section is to apply the preceding theory to an MPEC, where the equilibrium is governed by a sequence of optimization problems (6) with the data (functions f i and multifunctions Γi ) speciﬁed in Section 2. Without giving the structure of f i in detail, the ith problem attains the form Minimize f (x, ζ i−1 , ui , ζ i ) Subject to

¯i L1 ui ≥ u L2 ui ≤ 0

(16)

ζi ≥ 0 ζ i ≤ ζ i−1 , where the control x arises only in the objective, all other variables were described in Section 2, and also the problem data f, L1 L2 were deﬁned Copyright © 2005 Marcel Dekker, Inc.

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183

there. The next step consists in the construction of such optimality conditions for (16) that will facilitate a subsequent application of Theorem 2 as much as possible. To this purpose, we introduce the polyhedral sets Ωi := (ui , ζ i ) | L1 ui ≥ u ¯i , ζ i ≥ 0 , consisting (due to the structure of L1 ) only of lower bounds for some variables.

Theorem 6 Let x = x ˜ and ζ i−1 = ζ˜i−1 be given and (˜ u, ζ˜i ) be a solution ˜ i such of the respective problem ( 16). Then there exists a KKT vector λ that ˜ i + NΩi (˜ x, ζ˜i−1 , u ˜i , ζ˜i ) + (∇2,3 G(ζ˜i−1 , u˜i , ζ˜i ))T λ ui , ζ˜i ) 0 ∈ ∇3,4 f (˜ (17) ˜ i ), 0 ∈ −G(ζ˜i−1 , u ˜i , ζ˜i ) + N l (λ R+

where G(ζ

i−1

i

i

,u ,ζ ) : =

L 2 ui i ζ − ζ i−1

and l is the dimension of the image space of G. Note that in the above optimality conditions we do not need any constraint qualiﬁcation, because all functions arising in the constraints are aﬃne. The GE (17) is already in the required form (8); it suﬃces to put y i : = (ui , ζ i , λi ), ∇3,4 f (x, ζ i−1 , ui , ζ i ) + (∇2,3 G(ζ i−1 , ui , ζ i ))T λi i−1 i F (x, y , y ) : = −G(ζ i−1 , ui , ζ i ) and

i

i

Q (y ) : =

NΩi (ui , ζ i ) NRl (λi )

.

+

Since the sets Ωi are translated nonnegative orthants, the coderivatives of the maps Qi can easily be computed on the basis of [15, Lemma 2.1]. Moreover, F is aﬃne and all sets GphQi are unions of a ﬁnite number of convex polyhedral sets ([18]). Hence, we do not need to take care about the calmness condition in Theorem 2, whenever Uad is convex polyhedral. We now illustrate the application of Corollary 5 and the form of the resulting optimality conditions by means of an MPEC generated on the basis of the academic equilibrium from Example 1. Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

Example 2 Consider the four-string problem from Example 1, where we set xj = ej for j = 1, 2. The respective problem (16) thus attains the form 2 Minimize (x1 +eζ1i )(ui1 )2 + (x2 +eζ2i )(ui2 −ui1 )2 − ed (ζji −ζji−1 ) j=1

Subject to

ui1 − ui2 ≤ 0 ζ1i − ζ1i−1 ≤ 0 ζ2i − ζ2i−1 ≤ 0 (ui1 , ui2 , ζ1i , ζ2i )

(18)

∈ Ωi ,

where Ωi = ui1 , ui2 , ζ1i , ζ2i | ui1 ≥ 0, ui2 ≥ u ¯i , ζ1i ≥ 0, ζ2i ≥ 0 . The aim is to maximize the energy dissipated until the terminal time k so that ϕ(x, y k ) = ζ1k + ζ2k . Finally, Uad = {x ∈ R2 | x1 + x2 = 2} and ζ10 = ζ20 = 1 (the remaining initial values are not needed). The application of Theorem 6 to (18) yields the following GE: 2x1 ui1 − 2x2 (ui2 − ui1 ) + 2ζ1i eui1 − 2ζ2i e(ui2 − ui1 ) + λi1 2x2 (ui − ui ) + 2ζ i e(ui − ui ) − λi 2 1 2 2 1 1 0∈ e(ui1 )2 − de + λi2 i i 2 i e(u2 − u1 ) − de + λ3 + NΩi (ui1 , ui2 , ζ1i , ζ2i ) 0∈

(19)

− ui1 ui2 ζ1i−1 − ζ1i + NR3+ (λi1 , λi2 , λi3 ). ζ2i−1 − ζ2i

Hence, even in this simple academic example, one has m = 7, n = 2, 2k , ζ21 , . . . , ζ2k ), where and l = 3. It is clear that each vector (2 x, u 21 , . . . , u i i 2 2 u , ζ2 ) ∈ R × R is a solution of (18) with x = x 2, ζ i−1 = ζ2i−1 x 2 ∈ Uad , (2 for i = 1, 2, . . . , k and ζ21k = ζ22k = 0 generates a global solution of the above MPEC. As explained in Example 1, for the data given there, it is possible to construct such a vector on the basis of physical consideration; 22 = 1, u 211 = 0.0032, u 212 = in the case k = 2 we obtain, e.g., x 21 = x 2 2 1 1 2 2 22 = 0.007, ζ21 = 0, ζ22 = 1, ζ21 = ζ22 = 0. The 0.0035, u 21 = 0.0035, u 21 = 0, λ 21 = 10−6 , λ 22 = 21 = λ optimality conditions (19) are fulﬁlled with λ 1 2 3 1 Copyright © 2005 Marcel Dekker, Inc.

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On the Modeling and Control of Delamination Processes

22 = 0, and the vectors z2i ∈ NΩ1 (2 2i , λ 2i 2i 22 = λ ui1 , u 2i2 , ζ21i , ζ22i ) × NR3+ (λ λ 2 3 1 2 , λ3 ), i = 1, 2, equal to z21 = (0, −0.0064, −10−5 , 0, −0.0003, −1, 0)T z22 = (0, −0.007, −0.0001, −0.0001, −0.0035, 0, −1)T . To evaluate the coderivatives of the multivalued part in (19), we invoke, as already mentioned, [15, Lemma 2.1]. It follows that v11 = v41 = v71 = 0, w21 = w31 = w51 = w61 = 0 and v12 = 0, w22 = w32 = w42 = w52 = w72 = 0. Only the adjoint variables v62 and w62 are related in a more complicated way, because the respective constraint violates the strict complementarity. One has either v62 w62 = 0 or v62 < 0, w62 < 0. The respective adjoint equation system (15) can thus be substantially simpliﬁed and attains the form 0 4 0 0 0 −2 v2 0 2 1 0.07 v32 −1 w2 2 1 + −0, 07 0 v 1 0= + 2 w 42 6 0 1 v5 0 2 0 0 v6 0 0 0 0 v72 0=

24 −0.006 0 −22 0.006 0 1 0.064 0 0 w1 1 −0.006 0 −1 w41 + 1 0 0 w7 0 0 0 0 1 0

0 v21 v31 0 v51 v61 0

+

0 0 w62 0 0 0 0

.

This equation system in variables (v 1 , v 2 , w1 , w2 ) possesses a solution in which (w1 , w2 ) = 0. Because κ = 0, relation (14) holds true and the optimality conditions of Corollary 5 have been veriﬁed.

5.

Conclusion

The optimality condition derived in Section 3 can be used to test the stationarity of approximate solutions to considered MPECs computed by Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

available numerical methods [1],[4]. The complexity of these MPECs and the respective optimality conditions increases naturally with the number of time levels (k). On the other hand, as explained in Section 2.1, in modeling of the delamination processes, a ﬁne time discretization is really needed to compute a physically acceptable equilibrium. In MPECs associated with such equilibria, the need of a large k may be even stronger, because the (upper level) objective may force the equilibrium to attain physically unacceptable values. For instance, in the illustrative four-string problem, the choice k = 2 is deﬁnitely too small to solve the associated MPEC (Example 2) by an SQP code. As soon as the starting point is not extremely close to a solution, the procedure terminates at a physically unacceptable equilibrium. The choice of k is thus a certain trade-oﬀ between the complexity of the problem (and the associated optimality conditions) and our eﬀort to arrive at a physically acceptable equilibrium.

Acknowledgments This research was supported by grant A1075005 of the Grant Agency of the Academy of Sciences of the Czech Republic and by BMBF project 03ZOM3ER.

References [1] M. Anitescu. On solving mathematical programs with complementarity constraints as nonlinear programs. Preprint ANL/MCS-P864-1200, Argonne National Laboratory, Argonne, IL, 2000. [2] C. C. Baniotopoulos, J. Haslinger, and Z. Mor´ avkov´ a. Mathematical modeling of delamination and nonmonotone friction problems by hemivariational inequalities. Technical report, Charles University, Prague, 2002. [3] B. D. Davidson. An analytical investigation of delamination front curvature in double cantilever beam specimens. J. Compos. Mater., 24:1124–1137, 1990. [4] R. Fletcher and S. Leyﬀer. Numerical experience with solving MPECs as NLPs. Report NA\210, University of Dundee, 2002. [5] M. Fr´emond. Dissipation dans l’adhrence des solides. C. R. Acad. Sci., Paris, Sr. II, 300:709–714, 1985. [6] M. Fr´emond. Contact with adhesion. In J. J. Moreau, G. Panagiotopoulos, and G. Strang, editors, Topics in Nonsmooth Mechanics. Birkh¨ auser, Basel, 1988. [7] H. Guidouche and N. Point. Unilateral contact with adherence. In A. Bossavit, A. Damlamian, and M. Fr´emond, editors, Free Boundary Problems: Applications and Theory IV, pages 340–346. Pitman Res. Notes Math. 121, Longman Science and Technology, UK, 1985. [8] H. Halkin. Optimal Control for Systems Described by Diﬀerence Equations. Academic Press, New York, 1964.

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[9] M. Koˇcvara, A. Mielke, and T. Roub´ıˇcek. Rate-independent approach to the delamination problem. Preprint 2003/29, SFB 404 Mehrfeldprobleme in der Kontinuumsmechanik, Universit¨ at Stuttgart, 2003. [10] A. Mielke and F. Theil. A mathematical model for rate-independent phase transformations with hysteresis. In H.-D. Alder, R. Balean, and R. Farwig, editors, Models of Continuum Mechanics in Analysis and Engineering, pages 117–129. Shaker Ver., Aachen, 1999. [11] A. Mielke, F. Theil, and V. I. Levitas. A variational formulation of rateindependent phase transformations using an extremum principle. Archive Rat. Mech. Anal., 162:137–177, 2002. [12] B. S. Mordukhovich. Generalized diﬀerential calculus for nonsmooth and setvalued mappings. J. Math. Anal. Appl., 183:250–288, 1994. [13] B.S. Mordukhovich. Approximation Methods in Problems of Optimization and Control. Nauka, Moscow, 1988. In Russian. [14] J. Outrata. A generalized mathematical program with equilibrium constraints. SIAM J. Control. Optim., 38:1623–1638, 2000. [15] J. V. Outrata. Optimality conditions for a class of mathematical programs with equilibrium constraints. Math. Oper. Res., 24:627–644, 1999. [16] G. Panagiotopoulos. Hemivariational Inequalities. Springer-Verlag, Heidelberg, 1993. [17] N. Point and E. Sacco. A delamination model for laminated composites. Int. J. Solids Structures, 33:483–509, 1996. [18] S. M. Robinson. Some continuity properties of polyhedral multifunctions. Mathematical Programming Study, 14:206–214, 1981. [19] R. T. Rockafellar and R. Wets. Variational Analysis. Springer-Verlag, Berlin, 1998. [20] H. L. Schreyer, D. L. Sulsky, and S.-J. Zhou. Modelling delamination as strong discontinuity with the material point method. Comput. Methods Appl. Mech. Engrg., 191:2483–2507, 2002. [21] J. J. Ye. Necessary optimality conditions for bilevel dynamic problems. In Proc. of the 36th Conference on Decision and Control, San Diego, pages 1405–1410. IEEE, 1997.

Copyright © 2005 Marcel Dekker, Inc.

CHAPTER XV ON A SPECTRAL VARIATIONAL PROBLEM ARISING IN THE STUDY OF EARTHQUAKES Multiplicity and Perturbation from Symmetry Vicentiu Radulescu Department of Mathematics University of Craiova Craiova, Romania

Abstract

We consider an eigenvalue variational inequality problem arising in the earthquake initiation. Our purpose is twofold. Firstly, in the symmetric case, we establish the existence of inﬁnitely many distinct solutions. Next, in the case where the problem is aﬀected by a non-symmetric perturbation, we prove that the number of solutions of the perturbed problem becomes larger and larger if the perturbation “tends” to zero with respect to a suitable topology. Since the canonical energy functionals are included neither in the theory of monotone operators, nor in their Lipschitz perturbations, the proofs of the main results rely on nonsmooth critical point theories in the sense of De Giorgi and Degiovanni combined with methods from algebraic topology.

Keywords: Earthquake initiation, variational inequality, non-smooth critical point theory, perturbation

Introduction The purpose of this paper is twofold: ﬁrst, we establish a multiplicity result for a nonlinear symmetric variational inequality; next, we study the eﬀect of an arbitrary perturbation. For related results and further comments we refer to our recent papers [11] (for an appropriate variational inequality) and [2, 5, 14] for the hemivariational framework. We have been inspired in this work by the following simple phenomenon which occurs in elementary mathematics. Usually, an equation with a certain symmetry admits inﬁnitely many solutions. For instance, the equation sin x = 1/2 (x ∈ R) has an inﬁnite number of solutions. In this case, the “symmetry” is given by periodicity. If the above equation Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

is aﬀected in an arbitrary non-symmetric way by a certain perturbation, then the number of solutions of the new equation becomes larger and larger if the perturbation “tends to zero” in a suitable sense. For instance, the equation

sin x =

1 + ε x2 , 2

x∈R

(1)

has ﬁnitely many solutions, for any ε > 0. However, the number of solutions of (1) tends to inﬁnity if ε tends to 0. Our purpose in this work is to illustrate the above elementary phenomenon to the study of a nonlinear eigenvalue variational inequality arising in earthquake initiation. More precisely, using a multiplicity result of Lusternik-Schnirelmann type combined with the fact that an adequate function space is inﬁnitely dimensional, we ﬁrst establish the existence of inﬁnitely many solutions. Next, we are concerned with the study of the eﬀect of a small non-symmetric perturbation and we prove that the number of solutions of the perturbed problem tends to inﬁnity if the perturbation tends to zero with respect to an appropriate topology. The main novelty in our framework is the presence of the convex cone of functions with non-negative jump across an internal boundary which is composed of a ﬁnite number of bounded connected arcs.

1.

Main Results and Physical Motivation

Let Ω ⊂ RN be a domain, not necessarily bounded, containing a ﬁnite number of cuts. Its boundary ∂Ω is supposed to be smooth and divided ¯ and the internal into two disjoint parts: the exterior boundary Γd = ∂ Ω i one Γ composed by Nf bounded connected arcs Γf , i = 1, .., Nf , called cracks or faults. On Γ we denote by [ ] the jump across Γ, (i.e., [w] = w+ − w− ) and by ∂n = ∇ · n the corresponding normal derivative with the unit normal n outwards the positive side. Set V = {v ∈ H 1 (Ω); v = 0 on Γd }. Denote by · the norm in the space V , and by Λ0 : L2 (Ω) → L2 (Ω)∗ and Λ1 : V → V ∗ the duality 2

isomorphisms Λ0 u(v) = Ω uvdx, for any u, v ∈ L (Ω), and Λ1 u(v) = Ω ∇u · ∇vdx, for any u, v ∈ V. Consider the Lipschitz map γ = i ◦ η : V → L2 (Γ), where η : V → H 1/2 (Γ) is the trace operator, η(v) = [v] on Γ and i : H 1/2 (Γ) → L2 (Γ) is the embedding operator. Let K be the convex closed cone deﬁned by K = {v ∈ V ; [v] ≥ 0 on Γ}. Copyright © 2005 Marcel Dekker, Inc.

On a Spectral Variational Problem Arising in the Study of Earthquakes

Consider the nonlinear eigenvalue problem ﬁnd u ∈ K and λ ∈ Rsuch that ∇u · ∇(v − u)dx + j (γ(u(x)); γ(v(x)) − γ(u(x))) dσ+ Ω Γ λ u(v − u)dx ≥ 0, ∀v ∈ K,

191

(2)

Ω

where j(t) = − β2 t2 , t ∈ R, for some real constant β. If N = 2, then any solution of problem (2) can be viewed as a stationary solution of an appropriate evolution variational inequality that describes the slip-dependent friction law introduced in the geophysical context of earthquake modeling (see [1], [3]). We look for w : R+ × Ω → R solution of the wave equation ∂tt w(t) = c2 ∆w(t) in Ω, with the boundary condition w(t) = 0 on Γd . On the contact zone Γ we have [∂n w] = 0 and the following slip-dependent friction law (introduced in the geophysical context of earthquakes modeling) is assumed (see [1], [3]): G∂n w(t) = −µ(|[w(t)]|))Ssign([∂t w(t)]) − q, if [∂t w(t)] = 0, |G∂n w(t) + q| ≤ µ(|[w(t)]|)S if ∂t [w(t)] = 0, where G < denotes the elastic medium shear rigidity, ρ is the density, and c = G/ρ is the shear velocity. The non-vanishing shear stress components are σzx = τx∞ + G∂x w, σzy = τy∞ + G∂y w, σxx = σyy = −S (S > 0 is the normal stress on the fault plane), and q = τx∞ nx + τy∞ ny . The initial conditions are w(0) = w0 ,

∂t w(0) = w1 in Ω.

Any solution of the above problem satisﬁes the following nonlinear eigenvalue problem: ﬁnd w : [0, T ] → V such that 1 ∂tt w(t)(v − ∂t w(t) dx + ∇w(t) · ∇(v − ∂t w(t)) dx + (3) c2 Ω Ω S 1 µ(|[w(t)]|)(|[v]| − |[∂t w(t)]|) dσ ≥ q([v] − [∂t w(t)]) dσ, Γ G Γ G for all v ∈ V . Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

The main diﬃculty in the study of problem (3) is the non-monotone dependence of µ with respect to the slip |[w]|. However, in modeling unstable phenomena, as earthquakes, we have to expect “bad” mathematical properties of the operators involved in the abstract problem. The existence of a solution w ∈ W 1,∞ (0, T, V ) ∩ W 2,∞ (0, T, L2 (Ω)) (if N = 2) was recently proved by Ionescu et al. [10]. The uniqueness was obtained only in the one-dimensional case. Since our intention is to study the evolution of the elastic system near an unstable equilibrium position, we shall suppose that q = µ(0)S. We remark that w ≡ 0 is an equilibrium solution of (3), and w0 , w1 may be considered as small perturbations of it. For simplicity, let us assume in the following that the friction law is homogeneous on the fault plane having the form of a piecewise linear function (see [13]):

µ(x, u) = µs −

µs − µd u if u ≤ 2Dc , µ(x, u) = µd if u > 2Dc , 2Dc

(4)

where u is the relative slip, µs and µd (µs > µd ) are the static and dynamic friction coeﬃcients, and Dc is the critical slip. Since the initial perturbation (w0 , w1 ) of the equilibrium (w ≡ 0) is small we have [w(t, x))] ≤ 2Dc for t ∈ [0, Tc ] for all x ∈ Γ, where Tc is a critical time for which the slip on the fault reaches the critical value 2Dc at least at one point. Hence for a ﬁrst period [0, Tc ], called the initiation phase, we deal with a linear function µ. Our aim is to analyze the evolution of the perturbation during this initial phase. That is why we are interested in the existence of solutions of the type

w(t, x) = sinh(|λ|ct)u(x),

w(t, x) = sin(|λ|ct)u(x)

(5)

during the initiation phase t ∈ [0, Tc ]. If we put the above expression in (3) and we have in mind that from (4) we have µ(s) = µs − (µs − µd )/(2Dc )s then we deduce that (u, λ) is solution of the problem (2), where β = (µs − µd )S/(2Dc G) > 0. The ﬁrst type of solution described by (5) has an exponential growth in time and corresponds to λ > 0. The second one has the same amplitude during the initiation phase and corresponds to λ < 0. Returning to problem (2), we observe that, due to its homogeneity, we can reformulate this problem in terms of a constrained inequality problem as follows. For any ﬁxed r > 0, consider the smooth manifold Copyright © 2005 Marcel Dekker, Inc.

On a Spectral Variational Problem Arising in the Study of Earthquakes

M = u ∈ V ; Ω u2 dx = r 2 . We shall study the problem ﬁnd u ∈ K ∩ M and λ∈ R such that ∇u · ∇(v − u)dx + j (γ(u(x)); γ(v(x)) − γ(u(x))) dσ+ Ω Γ λ u(v − u)dx ≥ 0, ∀v ∈ K.

193

(6)

Ω

The multiplicity of solutions to problem (6) is described in

Theorem 1 Problem ( 6) has inﬁnitely many solutions (u, λ) and the set of eigenvalues {λ} is bounded from above and its inﬁmum equals to −∞. Let λ0 = sup{λ}. Then there exists u0 such that (u0 , λ0 ) is a solution of ( 6). Moreover the function β −→ λ0 (β) is convex and the following inequality holds |∇v|2 dx + λ0 (β) v 2 dx ≥ β [v]2 dσ, ∀v ∈ K. (7) Ω

Ω

Γ

Our next purpose is to describe the eﬀect of an arbitrary perturbation in problem (2). More precisely, we consider the problem ﬁnd uε ∈ K and λε ∈ R such that ∇uε · ∇(v − uε )dx+ Ω (8) j + εg (γ(uε (x)); γ(v(x)) − γ(uε (x))) dσ+ Γ λε uε (v − uε )dx ≥ 0, ∀v ∈ K, Ω

where ε > 0 and g : R → R is a continuous function with no symmetry hypothesis, but satisfying the growth assumption 2(N − 1) : |g(t)| ≤ a(1 + |t|p ) , if N ≥ 3; N −2 , if N = 2. ∃ a > 0, ∃ 2 ≤ p < +∞ : |g(t)| ≤ a(1 + |t|p ) ∃ a > 0, ∃ 2 ≤ p ≤

(9)

This hypothesis is motivated by the following embedding inequality of Ionescu [9] that will be used in an essential manner in the proof.

Lemma 2 (Lemma 5.1 in [9]). Let 2 ≤ α ≤ 2(N − 1)/(N − 2) if N ≥ 3 and 2 ≤ α < +∞ if N = 2. Then for β = [(α − 2)N + 2]/(2α) if N ≥ 3 or if N = 2 and α = 2 and for all (α − 1)/α < β < 1 if N = 2 and α > 2, there exists C = C(β) such that

1/α

(1−β)/2

β/2 |[u]|α dσ ≤C u2 dx |∇u|2 dx , (10) Γ

Copyright © 2005 Marcel Dekker, Inc.

Ω

Ω

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for any u ∈ V . Our perturbation result is

Theorem 3 For every positive integer n, there exists εn > 0 such that problem ( 8) has at least n distinct solutions (uε , λε ) if ε < εn . There exists and is ﬁnite λ0ε = sup{λε } and there exists u0ε such that (u0ε , λ0ε ) is a solution of ( 8). Moreover, λ0ε converges to λ0 as ε tends to 0, where λ0 was deﬁned in Theorem 1.

2.

Proofs

We ﬁrst recall some of the notions used in the proofs of Theorems 1 and 3. An important role in our arguments in order to locate the solution of problem (6) is played by the indicator function of M , that is, 0 , if u ∈ M IM (u) = +∞ , if u ∈ V \ M. Then IM is lower semicontinuous. However, since the natural energy functional associated to problem (6) is neither smooth nor convex, it is necessary to introduce a more general concept of gradient. We shall employ the following notion of lower subdiﬀerential which is due to De Giorgi, Marino, and Tosques [8].

Definition 4 Let X be a Banach space and let f : X → R ∪ {+∞} be an arbitrary proper functional. Let x ∈ D(f ). The Fr´echet (regular) subdiﬀerential of f at x is the (possibly empty) set f (y) − f (x) − ξ(y − x) − ∗ ≥0 . ∂ f (x) = ξ ∈ X ; lim inf y→x y − x An element ξ ∈ ∂ − f (x) is called a lower subgradient of f at x. Accordingly, we say that x ∈ D(f ) is a critical (lower stationary) point of f if 0 ∈ ∂ − f (x). Then ∂ − f (x) is a convex set. If ∂ − f (x) = ∅ we denote by grad− f (x) the element of minimal norm of ∂ − f (x), that is, grad− f (x) = min{ξX ∗ ; ξ ∈ ∂ − f (x)}. This notion plays a central role in the statement of the following basic compactness condition.

Definition 5 Let f : X → R ∪ {+∞} be an arbitrary functional. We say that (xn ) ⊂ D(f ) is a Palais-Smale sequence if supn |f (xn )| < +∞ Copyright © 2005 Marcel Dekker, Inc.

On a Spectral Variational Problem Arising in the Study of Earthquakes

195

and limn→∞ grad− f (xn ) = 0. The functional f is said to satisfy the Palais-Smale condition provided that any Palais-Smale sequence is relatively compact. Deﬁnition 4 implies that if g : X → R is Fr´echet diﬀerentiable and f : X → R ∪ {+∞} is an arbitrary proper function then ∂ − (f + g)(x) = ξ + g (x); ξ ∈ ∂ − f (x) , for any x ∈ D(f ). We also point out that in [4] it is proved the formula ∂ − IM (u) = {λΛ0 u; λ ∈ R} ⊂ L2 (Ω)∗ ⊂ V ∗ ,

(11)

for any u ∈ M , where Λ0 : L2 (Ω) → L2 (Ω)∗ denotes the canonical duality isomorphism.

2.1

Proof of Theorem 1

Deﬁne E = F + G : V → R ∪ {+∞}, where 1 |∇u|2 dx , F (u) = 2 Ω +∞ , and

β G(u) = − 2

if u ∈ K if u ∈ K

[γ(u(x))]2 dσ. Γ

Then E +IM is lower semicontinuous. Moreover, E +IM is the canonical energy functional associated to problem (6). This assertion is reﬁned in the following auxiliary result.

Lemma 6 Let (u, λ) be an arbitrary solution of problem ( 6). Then 0 ∈ ∂ − (E + IM )(u). Conversely, let u be a critical point of E + IM and denote λ = −2E(u)r −2 . Then (u, λ) is a solution of problem ( 6). Proof of Lemma 6. If (u, λ) is a solution of problem (6) then, by the deﬁnition of the lower subdiﬀerential, −λu ∈ ∂ − E(u).

(12)

On the other hand, ∂ − (E + IM )(u) = ∂ − E(u) + ∂ − IM (u), So, by (11) and (12), 0 ∈ ∂ − (E + IM )(u). Copyright © 2005 Marcel Dekker, Inc.

for any u ∈ K ∩ M. (13)

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Conversely, let 0 ∈ ∂ − (E + IM )(u). Thus, by (11) and (13), there exists λ ∈ R such that (u, λ) is a solution of problem (6). If we put v = 0 in (6) then we deduce λr 2 ≤ −2E(u) and for v = 2u we get λr 2 ≥ −2E(u), that is λ = −2E(u)r −2 . The next step in our proof consists in showing that the functional E + IM satisﬁes the Palais-Smale condition. This is done by using standard arguments, but applied in the framework of the non-smooth critical point theory in the sense of De Giorgi, Marino, and Tosques. Due to the symmetry of problem (6), we can extend our study to the symmetric cone (−K). More precisely, if (u, λ) is a solution of (6) then u0 := −u ∈ (−K) ∩ M satisﬁes ∇u0 · ∇(v − u0 )dx + j (γ(u0 (x)); γ(v(x)) − γ(u0 (x))) dσ+ Ω Γ λ u0 (v − u0 )dx ≥ 0, for all v ∈ (−K). Ω

This means that we can extend the energy functional associated to prob, := K ∪ (−K). We put, by deﬁnition, lem (6) to the symmetric set K , if u ∈ K E(u) , E(−u) , if u ∈ (−K) E(u) = +∞ , otherwise. We are interested in ﬁnding the lower stationary points of the extended , + IM . energy functional J := E , ∩ M with the graph metric of E , deﬁned by We endow the set K , , d(u, v) = u − v + |E(u) − E(v)|,

, ∩ M. for any u, v ∈ K

, ∩ M, d). Denote by X the metric space (K The next step in the proof of Theorem 1 consists in showing , ∩ M ) = +∞. Lemma 7 We have CatX (K The proof is straightforward and is accomplished by using adequate tools from Algebraic Topology. The above results enable us to apply the Lusternik-Schnirelmann theorem in the sense established by Marino and Scolozzi [12]. This implies that problem (6) admits inﬁnitely many solutions (u, λ). Next, we observe that the set of eigenvalues is bounded from above. Indeed, if (u, λ) is a solution of our problem, then choosing v = 0 in (6) and using (10), it follows that λr 2 ≤ −2u2 + β2 u2L2 (Γ) ≤ C, where C does not depend on u. Copyright © 2005 Marcel Dekker, Inc.

On a Spectral Variational Problem Arising in the Study of Earthquakes

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It remains to prove that inf{λ; λ is an eigenvalue of problem (6)} = −∞. For this purpose, it is suﬃcient to show that , ∩ M } = +∞. sup{J(u); u ∈ K But this follows directly from (10) after observing that |∇u|2 dx = +∞. sup , u∈K∩M

Ω

In order to prove the last part of the theorem we remark that −λ0 , as a function of β, is the upper bound of a family of aﬃne functions 1 2 2 |∇v| dx − β [v] dσ , (14) −λ0 (β) = inf v∈K∩M r 2 Ω Γ hence it is a concave function. Thus β −→ λ0 (β) is convex and (7) yields. This completes the proof of Theorem 1.

2.2

Proof of Theorem 3

The main idea is to establish the multiplicity result with respect to a prescribed level of energy. let us ﬁx r > 0. Consider the

More precisely, manifold N = u ∈ V ; Γ [u]p dσ = r p , where p is as in (9). We reformulate problem (8) as follows: ﬁnd uε ∈ K ∩ N and λε ∈ R such that ∇uε · ∇(v − uε )dx+ Ω (15) j (γ(uε (x)); γ(v(x)) − γ(uε (x))) dσ+ + εg Γ λε uε (v − uε )dx ≥ 0, ∀v ∈ K. Ω

We start with the preliminary result

Lemma 8 There exists a sequence (bn ) of essential values of E such that bn → +∞ as n → ∞. Proof of Lemma 8. For any n ≥ 1, set an = inf S∈Γn supu∈S E(u), where Γn is the family of compact subsets of K ∩ N of the form φ(S n−1 ), with φ : S n−1 → K ∩ N continuous and odd. The function E restricted to K ∩ N is continuous, even, and bounded from below. So, by Theorem 2.12 in [7], it is suﬃcient to prove that an → +∞ as n → ∞. But, as in Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

the preceding section, the functional E restricted to K ∩ N satisﬁes the Palais-Smale condition. So, taking into account Theorem 3.5 in [6] and Theorem 3.9 in [7], we deduce that the set E c has ﬁnite genus for any c ∈ R. Using now the deﬁnition of the genus combined with the fact that K ∩ N is a weakly locally contractible metric space, we deduce that an → +∞. This completes our proof. The canonical energy associated to problem (15) is the functional J restricted to K ∩ N , where J = E + Φ and Φ is deﬁned by Φ(u) = ε g(γ(u(x)))dσ. Γ

A straightforward computation with the same arguments as in the proof of Lemma 6 shows that if u is a lower stationary point of J then there exists λ ∈ R such that (u, λ) is a solution of problem (15). In virtue of this result, it is suﬃcient for concluding the proof of Theorem 3 to show that the functional J has at least n distinct critical values, provided that ε > 0 is suﬃciently small. We ﬁrst prove that J is a small perturbation of E. More precisely, we have

Lemma 9 For every η > 0, there exists δ = δη > 0 such that for any ε ≤ δ, supu∈K∩N |J(u) − E(u)| ≤ η. Proof of Lemma 9. We have |J(u) − E(u)| = |Φ(u)| ≤ ε

|g(γ(u(x)))| dσ. Γ

So, by (9) and Lemma 2, |J(u) − E(u)| ≤ ε a

(1 + [u(x)]p ) dσ ≤ Cε ≤ η, Γ

if ε is suﬃciently small. By Lemma 8, there exists a sequence (bn ) of essential values of E|K∩N such that bn → +∞. Without loss of generality we can assume that bi < bj if i < j. Fix an integer n ≥ 1 and choose ε0 > 0 such that ε0 < 1/2 min2≤i≤n (bi − bi−1 ). Applying now [7, Theorem 2.6], we obtain that for any 1 ≤ j ≤ n, there exists ηj > 0 such that if supK∩N |J(u) − E(u)| < ηj then J|K∩N has an essential value cj ∈ (bj − ε0 , bj + ε0 ). So, by Lemma 9 applied for η = min{η1 , . . . , ηn }, there exists δn > 0 such that supK∩N |J(u) − E(u)| < η, provided that ε ≤ δn . This shows that the energy functional J has at least n distinct essential values c1 , . . . , cn in (b1 − ε0 , bn + ε0 ). Copyright © 2005 Marcel Dekker, Inc.

On a Spectral Variational Problem Arising in the Study of Earthquakes

199

The next step consists in showing that c1 , . . . , cn are critical values of J|K∩N . Arguing by contradiction, let us suppose that cj is not a critical value of J|K∩N . We show in what follows that: (A1 ) There exists δ¯ > 0 such that J|K∩N has no critical value in (cj − ¯ ¯ cj + δ). δ, a ¯ cj + δ) ¯ with a < b, the pair (J b (A2 ) For every a, b ∈ (cj − δ, |K∩N , J|K∩N ) is trivial. Suppose, by contradiction, that (A1 ) is not valid. Then there exists a sequence (dk ) of critical values of J|K∩N with dk → cj as k → ∞. Since dk is a critical value, it follows that there exists uk ∈ K ∩ N such that J(uk ) = dk and 0 ∈ ∂ − J(uk ). Using now the fact that J satisﬁes the Palais-Smale condition at the level cj , it follows that, up to a subsequence, (uk ) converges to some u ∈ K ∩ N as k → ∞. So, by the continuity of J and the lower semicontinuity of grad J( · ), we obtain J(u) = cj and 0 ∈ ∂ − J(u), which contradicts the initial assumption on cj . Let us now prove assertion (A2 ). For this purpose we apply the Noncritical Point Theorem (see [6], Theorem 2.15]). So, there exists a continuous map χ : (K ∩ N ) × [0, 1] → K ∩ N such that χ(u, 0) = u, J(χ(u, t)) ≤ J(u), J(u) ≤ b ⇒ J(χ(u, 1)) ≤ a, J(u) ≤ a ⇒ χ(u, t) = u.

(16)

b a → J|K∩N by ρ(u) = χ(u, 1). From (16) we Deﬁne the map ρ : J|K∩N obtain that ρ is well deﬁned and it is a retraction. Set b b × [0, 1] → J|K∩N , J : J|K∩N

J (u, t) = χ(u, t).

b The deﬁnition of J implies that, for every u ∈ J|K∩N ,

J (u, 0) = u

and

J (u, 1) = ρ(u)

(17)

a × [0, 1], and, for any (u, t) ∈ J|K∩N

J (u, t) = J (u, 0).

(18)

a -homotopic to the identity From (17) and (18) it follows that J is J|K∩N a of J|K∩N ; that is, J is a strong deformation retraction, so the pair b a , J|K∩N ) is trivial. Assertions (A1 ), and (A2 ) show that cj is not (J|K∩N an essential value of J|K∩N . This contradiction concludes the proof of Theorem 3.

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References [1] J.-P. Ampuero, J-P. Vilotte, and F. J. Sanchez-Sesma. Nucleation of rupture under slip dependent friction law: simple models of fault zone. J. Geophys. Res., 107:1–15, 2002. [2] M. Bocea, P. D. Panagiotopoulos, and V. R˘ adulescu. A perturbation result for a double eigenvalue hemivariational inequality and applications. J. Global Optimiz., 14:137–156, 1999. [3] M. Campillo and I.-R. Ionescu. Initiation of antiplane shear instability under slip dependent friction. J. Geophys. Res., 122:20363–20371, 1997. [4] G. Chobanov, A. Marino, and D. Scolozzi. Multiplicity of eigenvalues for the laplace operator with respect to an obstacle and non-tangency conditions. Nonlinear Analysis, 15:199–215, 1990. [5] F. Cˆırstea and V. R˘ adulescu. Multiplicity of solutions for a class of nonsymmetric eigenvalue hemivariational inequalities. J. Global Optimiz., 17:43–54, 2000. [6] J. N. Corvellec, M. Degiovanni, and M. Marzocchi. Deformation properties for continuous functionals and critical point theory. Topol. Methods Nonlinear Anal., 1:151–171, 1993. [7] M. Degiovanni and S. Lancelotti. Perturbations of even nonsmooth functionals. Diﬀerential Integral Equations, 8:981–992, 1995. [8] E. De Giorgi, A. Marino, and M. Tosques. Problemi di evoluzione in spazi metrici e curve di massima pendenza. Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat., 68:180–187, 1980. [9] I.-R. Ionescu. Viscosity solutions for dynamic problems with slip-rate dependent friction. Quart. Appl. Math., 60:461–476, 2002. [10] I.-R. Ionescu, Q.-L. Nguyen, and S. Wolf. Slip dependent friction in dynamic elasticity. Nonlinear Analysis, 53:375–390, 2003. [11] I.-R. Ionescu and V. R˘ adulescu. Nonlinear eigenvalue problems arising in earthquake initiation. Adv. Diﬀer. Equations, 8:769–786, 2003. [12] A. Marino and D. Scolozzi. Geodetiche con ostacolo. Boll. Un. Mat. Ital., 2B:1–31, 1983. [13] A. C. Palmer and J. R. Rice. The growth of slip surfaces in the progressive failure of overconsolidated clay slopes. Proc. Roy. Soc. London A, 332:527–548, 1973. [14] P. D. Panagiotopoulos and V. R˘ adulescu. Perturbations of hemivariational inequalities with constraints and applications. J. Global Optimiz., 12:285–297, 1998.

Copyright © 2005 Marcel Dekker, Inc.

CHAPTER XVI NODAL CONTROL OF CONSERVATION LAWS ON NETWORKS Sensitivity Calculations for the Control of Systems of Conservation Laws with Source Terms on Networks Martin Gugat∗ Lehrstuhl f¨ ur Angewandte Mathematik II Erlangen, Germany

Abstract

We consider a network where a dynamical process governed by hyperbolic conservation laws takes place. At the vertices, the conservation laws are coupled by algebraic node conditions. The system is also controlled at the vertices through these conditions. To solve problems of optimal control for such a system, an adjoint sensitivity calculus is useful since it allows the eﬃcient evaluation of the gradient of the objective function. We present such a calculus in the framework of classical solutions.

Keywords: Conservation law, network, node conditions, adjoint system

Introduction Consider a network of open channels and pipes with free surface ﬂow as in a sewage system. The dynamic behavior of the water ﬂow in each single channel or pipe of this system can be modeled by the de St. Venant equations, which form a system of hyperbolic conservation laws. The channels and pipes correspond to the edges of a graph, where the nodes correspond to the junctions. The ﬂow through these points where the channels/pipes are connected is governed by a set of algebraic node conditions that guarantee for example that in the node mass is neither generated nor destroyed. At certain points in the network, the ﬂow is controlled by devices such as pumps or underﬂow gates. What is the best way to operate these control devices? This question gives rise to a problem of optimal control of the type that we consider in this paper. ∗ This

work was supported by DFG–research cluster: real-time optimization of complex systems; grant number Le595/13-1.

Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

In this paper we consider systems of this type, where a process governed by hyperbolic conservation laws takes place on a network. On each edge of the corresponding graph, the ﬂow is converned by a system of hyperbolic conservation laws with source term. (There are numerous studies of hyperbolic conservation laws, see for example [12, 4, 11, 7]. Classical solutions are considered in [14].) At the vertices of the graph, these systems are coupled by a system of algebraic equations. Since the system is controlled at these vertices, also the control functions appear in these node conditions, therefore we speak of nodal control as a generalization of boundary control. We consider problems of optimal control for systems of this type, where the system is controlled at the nodes of the graph. We state results about directional diﬀerentiability with respect to the control function of objective functions that are deﬁned as integrals that depend on the state of the system. We introduce an adjoint backwards problem that is useful for the formulation of optimality conditions and allows the eﬃcient evaluation of the derivatives in an optimize–then–discretize approach, where ﬁrst the equations for the gradients are determined on the PDE level and then discretized. As stated in [6], this approach yields ﬂexibility since it allows the use of diﬀerent methods in the state and adjoint discretizations. Related studies can be found in [3, 2, 17, 16] but no networks are considered there. Similar problems have also been studied in [6] for problems with space dimension two, but without networks; the application considered there is the optimal boundary control of aeroaccoustic noise governed by the two-dimensional unsteady compressible Euler equations. Up to now, an adjoint sensitivity calculus for the type of networks with dynamics governed by conservation laws that we consider here cannot be found in the literature. A related study with dynamics governed by hyperbolic quasilinear systems in diagonal form is presented in [10], where also the proofs of some of the results that we use in this paper are given.

1.

Notation

Let a ﬁnite directed graph (V, E) with vertices (nodes) V and edges E be given. Each edge e ∈ E corresponds to a space interval [0, Le ]. We consider the system on the time interval [0, T ]. The dynamic behavior on the edge is governed by a hyperbolic system of conservation laws of the form vte + F e (v e )x + S e (v e ) = 0. Copyright © 2005 Marcel Dekker, Inc.

(1)

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203

Here v e = (v1e , ..., vne )T is a vector-valued function from [0, Le ] × [0, T ] into IRn , and S e = (S1e , ..., Sne )T and F e = (F1e , ..., Fne )T are twice continuously diﬀerentiable functions from IRn into IRn . The initial state of the system is given by an initial condition of the form v e (x, 0) = v0e (x), e ∈ E, x ∈ [0, Le ] (2) where for all edges e ∈ E, the function v0e is continuously diﬀerentiable. At the nodes ω ∈ V of the graph, the values of the functions v e are coupled by algebraic node conditions. For a node ω ∈ V , let E0 (ω) denote the set of edges that are connected with ω. For e ∈ E0 (ω), let xe (ω) ∈ {0, Le } denote the end-point of the interval [0, Le ] that corresponds to the vertex. There are two types of nodes: boundary nodes, where the set E0 (ω) has only one element and interior nodes, where E0 (ω) has at least two elements. Let VB ⊂ V denote the set of boundary nodes. For ω ∈ VB , let e(ω) denote the adjacent edge. Let V 0 (V L respectively) denote the set of boundary nodes that are at the end zero (the end Le ) of the adjacent edge e. For e ∈ E0 (ω), let se (ω) = −1 if xe (ω) = 0 and se (ω) = 1 if xe (ω) = Le .

1.1

Node Conditions

To state the node conditions, we need the following notation: For all e ∈ E, we assume that for the numbers ne+ = the number of positive eigenvalues of (F e ) (v0e (x)), x ∈ [0, Le ] ne− = the number of negative eigenvalues of (F e ) (v0e (x)), x ∈ [0, Le ] are well deﬁned, that is, the signs of the eigenvalues are independent of the space variable x and that we have ne+ + ne− = n, that is there are no zero eigenvalues. We consider controls that generate states v e where for all x ∈ [0, Le ] and all t ∈ (0, T ] ne+ = the number of positive eigenvalues of (F e ) (v e (x, t)), (3) ne− = the number of negative eigenvalues of (F e ) (v e (x, t)) (4) that is the signs of the eigenvalues are also independent of the time t. e , I e ⊂ {1, ..., n} be given where I e has Let two disjoint index sets I+ − + e e e e ∪ I e = {1, ..., n}. n+ elements and I− has n− elements. Then I+ − Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

For e ∈ E0 (ω), deﬁne the index sets e e e (ω) = {I− if xe (ω) = 0 and I+ if xe (ω) = Le }, Iin e e e Iout (ω) = {I+ if xe (ω) = 0 and I− if xe (ω) = Le }.

For ω ∈ V , B0 (ω, ·) denotes a continuously diﬀerentiable function and for e ∈ E0 (ω), ue (ω, ·) is a continuously diﬀerentiable control function. The numbers of components of B0e (ω, ·) is equal to the number of components of ue (ω, ·) and depends on the numbers of positive and negative eigenvalues of the system matrix. A discussion of boundary conditions for hyperbolic systems is given in [1]. We consider node conditions of e (ω): the following form. For e ∈ E0 (ω) and i ∈ Iout f (ω)) + uei (ω, t) vie (xe (ω), t) = (B0 )ei (ω, t, vjf (xf (ω), t) : f ∈ E0 (ω), j ∈ Iin (5)

2.

The Linearized Problem

For an edge e ∈ E, the linearized problem is Hte + (F e ) (v e )H e x + (S e ) (v e )H e = 0. Here (S e ) (w) = ( and

∂Sie ∂vj

(6)

(w))nij=1 denotes the derivative of the source term (F e ) (w) = (

∂Fie (w))nij=1 ∂vj

is the system matrix. The initial state v0e is independent of the control, hence the initial condition for H e is H e (x, 0) = 0, x ∈ [0, Le ].

(7)

If the solution v e of (1) is continuously diﬀerentiable, in general we cannot expect that the linearized problem has also a continuously differentiable solution, since the linearization causes a loss of regularity.

2.1

Node Conditions for the Linearized Problem

Let a node ω ∈ V be given. The node conditions of the linearized problem have the following form. e (ω): For e ∈ E0 (ω) and i ∈ Iout f e bef (8) Hie (xe (ω), t) = ij (ω, t) Hj (0, t) + di (ω, t). f ∈ E0 (ω) : f (ω) j ∈ Iin Copyright © 2005 Marcel Dekker, Inc.

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205

The functions bef ij are given by the corresponding partial derivatives of B0 , ∂ ef Bij = f (B0 )ei . ∂vj The functions d(ω, ·) give the direction in which the control function is linearized and are continuously diﬀerentiable with de (ω, 0) = (de ) (ω, 0) = 0.

3.

(9)

Existence of Solutions

We assume that a control function u∗ is given that generates a continuously diﬀerentiable state. Then all control functions u in a certain C 1 neighborhood of u∗ that satisfy the compatibility conditions u(ω, 0) = u∗ (ω, 0), u (ω, 0) = (u∗ ) (ω, 0), ω ∈ V also generate continuously diﬀerentiable states. This follows from results of [5]. The following results state that if the nonlinear initial boundary value problem has a continuously diﬀerentiable solution, the linearized initial boundary value problem has a continuous solution in the characteristic sense, that is a solution that satisﬁes the corresponding integral relations along the characteristic curves. Since the linearized system has the same system matrix as (1) and the characteristic curves are determined by the eigenvalues of this matrix, the linearized system has the same characteristic curves as the original system. If the original system has a continuously diﬀerentiable solution, the characteristic curves corresponding to the same eigenvalue do not intersect, so we can also transform the linearized system to the same characteristic coordinates.

Theorem 1 Let a control function u be given that generates a continuously diﬀerentiable solution of equation ( 1) on the time interval [0, T ] that satisﬁes the initial condition ( 2) and the node conditions ( 5). Assume that the function d satisﬁes the compatibility conditions ( 9). Then on the time interval [0, T ] there exists a unique continuous solution of the initial boundary value problem deﬁned by the initial condition H e (·, 0) = 0 for all e ∈ E, the node conditions ( 8), and the linearized equation ( 6) in the characteristic sense. The theorem is proved by a ﬁxed-point iteration of Picard-Lindel¨ of type along the characteristic curves (see Theorem 1.1 in [15], p. 47). Due to the linearity of the problem, we need not assume a bound for the size of the norm of the direction d or its derivatives. Copyright © 2005 Marcel Dekker, Inc.

206

4.

CONTROL AND BOUNDARY ANALYSIS

Directional Diﬀerentiability

The following result states that the directional derivative of the solution of the nonlinear initial boundary value problem with respect to the boundary control functions is given by the solution of the linearized initial boundary value problem.

Theorem 2 Let a control function u be given that generates a continuously diﬀerentiable solution of equation ( 1) on the time interval [0, T ] that satisﬁes the initial condition ( 2) and the node conditions ( 5). Assume that the function d satisﬁes the compatibility conditions ( 9). Then for all real numbers h that are suﬃciently small, the control functions u + hd also generate continuously diﬀerentiable solutions denoted by v e (x, t, u + hd). Moreover, with the unique continuous solution H e (x, t, u, d) of the linearized problem ( 7), ( 8), and ( 6) in the characteristic sense the following equation holds for all e ∈ E, x ∈ [0, Le ], t ∈ [0, T ]: v e (x, t, u + hd) − v e (x, t, u) = H(x, t, u, d). h→0+ h lim

5.

(10)

Evaluation of Directional Derivatives

We consider objective or constraint functions of the form J(u) =

ω∈V e∈E0 (ω)

+

e∈E

T 0

Le

T

f ω,e (v e (xe (ω), t), t) dt 0

f e (v e (x, t), x, t) dx dt.

0

The functions f ω,e and f e are assumed to be continuously diﬀerentiable, and v e (the state of the system) is the solution of the initial boundary value problem (1), (2), (5) generated by the control function u. For e ∈ E, the adjoint system equation is µet + µex (F e ) (v e ) − µe (S e ) (v e ) = (∇v f e )T

(11)

where µe = (µe1 , ..., µen ) is a row vector. The end conditions for µe are µe (x, T ) = 0 for all e ∈ E, x ∈ [0, Le ]. Copyright © 2005 Marcel Dekker, Inc.

(12)

Nodal Control of Conservation Laws on Networks

207

Using the solution of a backwards terminal boundary value problem, we give a representation of the directional derivative J(u + hd) − J(u) h→0 h

Dd J(u) = lim

where the solution of the original nonlinear forward problem and the adjoint solution appear, but not the solution of the linearized problem. To deﬁne the backwards terminal boundary value problem, we need the corresponding adjoint node conditions. For ω ∈ V and t ∈ [0, T ] deﬁne the column vectors ω e win (t) = (Hie (xe (ω), t) : e ∈ E0 (ω), i ∈ Iin (ω) )T ω e wall (t) = (Hie (xe (ω), t) : e ∈ E0 (ω), i ∈ Iin (ω); e e Hi (xe (ω), t) : e ∈ E0 (ω), i ∈ Iout (ω))T e dω (t) = (dei (ω, t) : e ∈ E0 (ω), i ∈ Iout (ω))T .

In what follows let the order of the components in the vectors be ﬁxed and compatible in all three vectors. We can write the node condition (8) in the form

I 0 ω ω wall (t) = win (t) + (13) B ω (t) dω (t) ω (t) and where I denotes the identity matrix in the space containing win ω ω ω (t) B (t) is the appropriate matrix. Let min denote the length of win and mωout denote the length of dω (t). Then B ω (t) is a mωout × mωin matrix. Deﬁne the row vectors e µωin (t) = (µei (xe (ω), t) : e ∈ E0 (ω), i ∈ Iout (ω))T e µωall (t) = (µei (xe (ω), t) : e ∈ E0 (ω), i ∈ Iin (ω); e e µi (xe (ω), t) : e ∈ E0 (ω), i ∈ Iout (ω))T .

For the adjoint node conditions we make the ansatz µωall (t) = µωin (t)(Aω (t), I) + (αω (t), 0)

(14)

where I denotes the identity matrix in the mωout -dimensional space and Aω (t) is a mωout × mωin matrix that we have to determine as well as the row vector αω (t) that has mωin components. Deﬁne the matrices ∂Fie e f ω e e M1 (t) = s (ω) f (v (xe (ω), t) : e, f ∈ E0 (ω), i ∈ Iin (ω), j ∈ Iin (ω) ∂vj Copyright © 2005 Marcel Dekker, Inc.

208

CONTROL AND BOUNDARY ANALYSIS

M2ω (t)

e

= s (ω)

M3ω (t)

e

= s (ω)

M4ω (t) = se (ω)

∂Fie ∂vjf ∂Fie ∂vjf

(v (xe (ω), t) : e, f ∈ E0 (ω), i ∈

e Iin (ω), j

(v (xe (ω), t) : e, f ∈ E0 (ω), i ∈

e Iout (ω), j

e

∈

f Iout (ω)

∈

f Iin (ω)

e (v e (xe (ω), t) : e, f ∈ E0 (ω), i ∈ Iout (ω), j ∈

f Iout (ω)

e

∂Fie ∂vjf

ω

M (t) =

M1ω (t) M2ω (t) M3ω (t) M4ω (t)

.

(15)

The matrix M1ω (t) is a mωin × mωin matrix and the matrix M4ω (t) is a mωout × mωout matrix. We deﬁne the real-valued function ψ ω (t) by the following equation: ω (t) (16) ψ ω (t) := µωall (t) M ω (t) wall e e e e e = s (ω) µ (xe (ω), t) (F ) (v (xe (ω), t)) H (xe (ω), t) e∈E0 (ω)

Then we have ψ ω (t) = [ µωin (t) (Aω (t), I) + (αω (t), 0)]

0 I M1ω (t) M2ω (t) ω win (t) + dω (t) B ω (t) M3ω (t) M4ω (t) ω (t) = µωin (t)[Aω (t){M1ω (t) + M2ω (t)B ω (t)} + M3ω (t) + M4ω (t)B ω (t)]win ω ω ω ω ω +µin (t) [A (t)M2 (t) + M4 (t)]d (t) ω (t) +αω (t) [M1ω (t) + M2ω (t)B ω (t)] win ω ω ω +α (t) M2 (t) d (t).

Assume that for all ω ∈ V , t ∈ [0, T ] the matrix [M1ω (t)+M2ω (t)B ω (t)] is invertible. We deﬁne the matrix Aω (t) by the equation Aω (t) = − [M3ω (t) + M4ω (t)B ω (t)] [M1ω (t) + M2ω (t)B ω (t)]−1 . Then the expression for ψ ω (t) is much simpler, namely ψ ω (t) = µωin (t) [Aω (t) M2ω (t) + M4ω (t)]dω (t) Copyright © 2005 Marcel Dekker, Inc.

(17)

Nodal Control of Conservation Laws on Networks

209

ω + αω (t) [M1ω (t) + M2ω (t) B ω (t)] win (t) ω ω ω + α (t) M2 (t) d (t).

Deﬁne the column vectors ω e Dfin (t) = (∂vi f e,ω (xe (ω), t), e ∈ E0 (ω), i ∈ Iin (ω))T , ω e (t) = (∂vi f e,ω (xe (ω), t), e ∈ E0 (ω), i ∈ Iout (ω))T , Dfout

ω (t) Dfin ω . Dfall (t) = ω (t) Dfout

We deﬁne the row vector αω (t) by the equation ω ω αω (t) = −Dfin (t)T − Dfout (t)T B ω (t) [M1ω (t) + M2ω (t) B ω (t)]−1 . (18) Then we have ψ ω (t) = µωin (t) [Aω (t) M2ω (t) + M4ω (t)] dω (t) + αω (t) M2ω (t) dω (t) ω (t) − [Df1ω (t)T + Df2ω (t)T B ω (t)] win = {µωin (t) [Aω (t) M2ω (t) + M4ω (t)] + αω (t) M2ω (t)} dω (t)

0 ω ω (t)T wall (t) − . −Dfall dω (t)

(19)

Now we can give a representation of the directional derivative.

Theorem 3 Let the assumptions of Theorem 2 hold. Let µ denote the solution of the terminal value problem that satisﬁes for all e ∈ E the end condition ( 12) and for all (x, t) ∈ (0, Le ) × (0, T ) the adjoint equation ( 11) and the adjoint node conditions deﬁned by ( 14), ( 17), ( 18). Assume that for all ω ∈ V , t ∈ [0, T ] the matrix [M1ω (t)+M2ω (t)B ω (t)] is invertible, with B ω deﬁned in ( 13) and M ω deﬁned in ( 15). Again we consider solutions that satisfy the corresponding integral equations along the characteristic curves. Then the directional derivative of J is given by the equation Dd J(u) =

(20)

ω [µωin (t)(Aω (t)M2ω (t) + M4ω (t)) + αω (t) M2ω (t) + Dfout (t)T ] dω (t) dt.

ω∈V

Proof of Theorem 3. Consider the directional derivative of J. By Theorem 1 a continuous solution H of the linearized problem exists, and Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

by Theorem 2 and integration by parts the deﬁnitions of the linearized and the adjoint problem imply the equation T (∇v f ω,e (v e (xe (ω), t), t))T H e (xe (ω), t) dt Dd J(u) = 0

ω∈V e∈E0 (ω)

+

e∈E e

T

0

Le

(∇v f e (v e (x, t), x, t))T H e (x, t)

0

+ µ Hte + (F e ) (v e )H e x + (S e ) (v e )H e |(x,t) dx dt T (∇v f ω,e (v e (xe (ω), t), t))T H e (xe (ω), t) dt = 0

ω∈V e∈E0 (ω)

+

e∈E

T

0

Le

(∇v f e (v e (x, t), x, t))T H e (x, t)

0

− µet + µex (F e ) (v e ) − µe (S e ) (v e ) H e |(x,t) dx dt T e µe (x, t) (F e ) (v e (x, t)) H e (x, t)|L + x=0 dt 0

=

ω∈V e∈E0 (ω)

+

−

T

∇v f ω,e (v e (xe (ω), t), t) H e (xe (ω), t) dt

0

T

e ∈ E0 (ω) : xe (ω) = Le

0

e ∈ E0 (ω) : xe (ω) = 0

0

T

µe (Le , t) (F e ) (v e (Le , t)) H e (Le , t) dt

µe (0, t) (F e ) (v e (0, t)) H e (0, t) dt.

Now we have represented Dd J(u) in a form that requires only the values of µ and H at the nodes of the network. Up to now, for µ only the adjoint equation (11) and the end condition (12) have been used. It remains to be shown that due to the deﬁnition of the adjoint node conditions only the values of the solution µ of the adjoint backwards problem are necessary and the values of the linearized problem H are not needed. The deﬁnition (16) of ψ ω (t) and equation (19) imply that T (∇v f ω,e (v e (xe (ω), t), t))T H e (xe (ω), t) dt Dd J(u) = ω∈V e∈E0 (ω)

Copyright © 2005 Marcel Dekker, Inc.

0

211

Nodal Control of Conservation Laws on Networks

+

ω∈V

=

T

ψ ω (t) dt

0

T ω ω Dfall (t)T wall (t)

+ +

ω∈V 0 µωin (t) (Aω (t) M2ω (t) + αω (t) M2ω (t) dω (t) dt

−

ω Dfall (t)T

M4ω (t)) dω (t)

ω (t) wall

−

0 dω (t)

dt

T ω [Dfout (t)T + µωin (t) (Aω (t)M2ω (t) + M4ω (t)) + αω (t)M2ω (t)] dω (t) dt

= 0

and the assertion follows.

6.

Example

We consider a network of rectangular water channels where on each channel, the ﬂow is governed by the de St. Venant equations [8]. The network is described by a graph (V, E).

6.1

Shallow Water Equations

For e ∈ E, let Qe (x, t) denote the ﬂow rate on channel e at the point x and time t and let Ae (x, t) denote the corresponding wetted cross section. The conservation of mass yields the equation Aet + Qex = 0. The balance of momentum yields the second equation

g (Qe )2 e e 2 Qt + (A ) + + Ae S2e = 0. 2be Ae x Here g is the gravitational constant, be denotes the width of channel e and S2e = g(γ e + β e (Ae , Qe )), where γ e is the slope of the channel. If γ e < 0, the channel proceeds downwards. The function β e is the friction slope. Let e e v1 A = . (21) ve = v2e Qe Copyright © 2005 Marcel Dekker, Inc.

212

CONTROL AND BOUNDARY ANALYSIS

@f @ Rx @

g

ω

h

?

Figure 1.

6.2

A junction of three channels.

Node Conditions

Consider a node ω ∈ V with three adjacent channels f , g, h and xf (ω) = Lf , xg (ω) = Lg , xh (ω) = 0 (see Figure XVI.6.2). We consider the node conditions Qf + Qg = Qh , Af = (bf /bh )Ah , Ag = (bg /bh ) Ah . We can write these node conditions in the form f f Q 0 0 bf /bh A Ag = 0 0 bg /bh Qg Qh 1 1 0 Ah

(22)

and the matrix B ω from (13) is given by the matrix in equation (22). f g We have Iin (ω) = Iin (ω) = {2} and for the matrices M1 , M2 , M3 , M4 we have the equations ∂F2f 2Qf 0 0 f 0 0 ∂Q f A g ∂F g M1ω (t) = 0 0 , 0 2Q 0 ∂Q2g = Ag ∂F1h 0 0 0 0 0 ∂Ah

M2ω (t) = =

gAf bf

Copyright © 2005 Marcel Dekker, Inc.

−

∂F2f ∂Af

0

0

∂F2g ∂Ag

0

0

(Qf )2 (Af )2

0 0

0

0 h

∂F1 ∂Qh

0 0

gAg bg

−

0 , 0 1

(Qg )2 (Ag )2

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Nodal Control of Conservation Laws on Networks

M3ω (t) = M4ω (t) =

∂F1f ∂Qf

0

0

∂F1g ∂Qg

0

0

∂F1f ∂Af

0

0

∂F1g ∂Ag

0

0

0

1 0 0 1 0 = 0 0 ∂F2h ∂Ah 0 0 0 0 0 0 = ∂F2h 0 0 ∂Qh

gAh bh

−

0 0 , 2

(Qh ) (Ah )2

0 0 .

2Qh Ah

We have

f g 2Qg 2Qf f f Q 2 g g Q 2 − b ( ) − b ( ) gA − gA bh Ag Af bh Af Ag g Qf 2 Q f 3 f f 2 g 3 g g 2 g(A ) − b (Q ) + g g(A ) − b (Q ) =− h f g . b A A Af A

det[M1ω + M2ω B ω ] = −

If the ﬂow is subcritical, that is, if be (Ae )2 [ghe − (U e )2 ] = g(Ae )3 − be (Qe )2 > 0, e ∈ {f, g, h} (where he denotes the water height and U e the velocity) and if Qf Qg > 0, this determinant does not vanish. If Qf = Qg = 0, the determinant is zero, hence in this case Theorem 3 is not applicable. Deﬁne the number α = −Ag Qg [g(Af )3 − bf (Qf )2 ] − Af Qf [g(Ag )3 − bg (Qg )2 ] bh (Af Ag )2 det[M1ω + M2ω B ω ]. 2 Then if Qf Qg > 0, we have α [M1ω + M2ω B ω ]−1 = =

(Af )2 [bg (Qg )2 −g(Ag )3 ] 2 (Af )2 [g(Ag )3 −bg (Qg )2 ] 2 −bh (Af )2 Ag Qg

(Ag )2 [g(Af )3 −bf (Qf )2 ] 2 (Ag )2 [bf (Qf )2 −g(Af )3 ] 2 −bh Af (Ag )2 Qf

Ag Qg [bf (Qf )2 − g(Af )3 ]

Af Qf [bg (Qg )2 − g(Ag )3 ] . 2bh Af Ag Qf Qg

For the matrix Aω deﬁned in (17) the ﬁrst two rows of the matrix −α Aω are the same as in α [M1ω + M2ω B ω ]−1 and the third row of the matrix −α Aω has the entries h h 2 f 2 g g b (Q ) h (A ) A Q − gA , (Ah )2 h h 2 b (Q ) h − gA , Af (Ag )2 Qf (Ah )2 Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

2Qh g g f f 2 f 3 f f g g 2 g 3 A Q [b (Q ) − g(A ) ] + A Q [b (Q ) − g(A ) ] Ah +2Af Ag Qf Qg [g Ah − bh

(Qh )2 ]. (Ah )2

The adjoint node conditions from (14) are (µf2 , µg2 , µh1 ) = (µf1 , µg1 , µh2 ) Aω + αω

(23)

with the row vector αω as deﬁned in (18).

7.

Conclusion

We have presented an adjoint sensitivity calculus for hyperbolic systems of conservation laws that are deﬁned on networks. We have stated the adjoint problem with terminal conditions and adjoint node conditions. With the adjoint solution of the corresponding linear backwards problem, the directional derivatives for real-valued integral functions of the state can be computed relatively cheaply in several directions, which is useful to apply gradient-based optimization algorithms. Our theory has been stated within the framework of continuously diﬀerentiable solutions. This is an interesting case, since often it is desirable to steer the system in such a way that the state is regular and, in particular, no shocks occur. The question whether this is indeed possible has been studied in [13, 9] for ﬂows in networks of open channels. Still extensions to more general classes of solutions that also allow for the formation of shocks would be interesting, since if the optimal control generates a regular state, during the iterations of an optimization algorithm controls that generate shocks can occur.

Acknowledgments The author wants to thank G. Leugering for helpful discussions.

References [1] C. Bardos, A.Y. Leroux, and J.C. Nedelec. First order quasilinear equations with boundary conditions. Comm. in Partial Diﬀerential Equations, 4:1017– 1034, 1979. [2] C. Bardos and O. Pironneau. A formalism for the diﬀerentiation of conservation laws. Comptes Rendus Mathematique, 335:839–845, 2002. [3] J. Borggaard, J. Burns, E. Cliﬀ, and M. Gunzburger. Sensitivity calculations for a 2d, inviscid, supersonic forebody problem. In Identiﬁcation and Control in Systems Governed by Partial Diﬀerential Equations, pages 14–25. SIAM, Philadelphia, 1993.

Copyright © 2005 Marcel Dekker, Inc.

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[4] A. Bressan. Hyperbolic Systems of Conservation Laws. The One-dimensional Cauchy Problem. Oxford University Press, 2000. [5] M. Cirina. Nonlinear hyperbolic problems with solutions on preassigned sets. Michigan Math. J., 17:193–209, 1970. [6] S. S. Collis, K. Ghayour, and M. Heinkenschloss. Optimal transpiration boundary control for aeroaccoustics. Preprint, pages 1–37, 2002. [7] C. M. Dafermos. Hyperbolic Conservation Laws in Continuum Physics. Springer, Berlin, 2000. [8] B. de Saint-Venant. Theorie du mouvement non–permanent des eaux avec application aux crues des rivi`eres et ` a l‘introduction des marees dans leur lit. Comptes Rendus Academie des Sciences, 73:148–154,237–240, 1871. [9] M. Gugat. Boundary controllability between sub- and supercritical ﬂow. SIAM Journal on Control and Optimization, 42:1056–1070, 2003. [10] M. Gugat. Nodal control of networked hyperbolic systems. Submitted to ESAIM:COCV, 2003. [11] H. Holden and N. H. Risebro. Front tracking for hyperbolic conservation laws. Springer, Berlin, 2002. [12] P. G. LeFloch. Hyperbolic Systems of Conservation Laws. The Theory of Classical and Nonclassical Shock Waves. Birkhaeuser, Basel, 2002. [13] G. Leugering and E.J.P. Georg Schmidt. On the modelling and stabilisation of ﬂows in networks of open canals. SIAM J. on Control and Optimization, 41:164–180, 2002. [14] L. Ta-tsien. Global Classical solutions for quasilinear hyperbolic systems. Masson, Paris, 1994. [15] L. Ta-tsien and Y. Wen-ci. Boundary value problems for quasilinear hyperbolic systems. Duke University Mathematics Series V, 1985. [16] S. Ulbrich. A sensitivity and adjoint calculus for discontinuous solutions of hyperbolic conservation laws with source terms. SIAM J. Control Optim., 41:740– 797, 2002. [17] S. Ulbrich. Adjoint-based derivative computations for the optimal control of discontinuous solutions of hyperbolic conservation laws. Systems and Control Letters, 48:309–324, 2003.

Copyright © 2005 Marcel Dekker, Inc.

CHAPTER XVII INVARIANCE OF CLOSED SETS UNDER STOCHASTIC CONTROL SYSTEMS Giuseppe Da Prato∗ Scuola Normale Superiore di Pisa Pisa, Italy

H´el`ene Frankowska∗ CNRS, CREA, Ecole Polytechnique Paris, France

Abstract

We prove that a closed set of a ﬁnite dimensional space is invariant under the stochastic control system if and only if it is invariant under a deterministic control system with two controls.

Keywords: Stochastic invariance, Stratonovitch drift, stochastic control system, state-constraints

Introduction We are given two ﬁnite dimensional spaces H and H1 , a complete probability space (Ω, F, P), an increasing family of σ−sub-algebras Ft ⊂ F augmented by P−null sets of F and an H1 -valued Ft -Brownian motion W (t), t ≥ 0 such that W (0) = 0. This paper is devoted to the problem of invariance of the stochastic control system dX = b(X, v(t))dt + σ(X, v(t))dW (t), v(t) ∈ U,

(1)

where U is a complete separable metric space, b : H × U → H, σ : H × U → L(H1 , H) are bounded continuous mappings which are Lipschitz with respect to the ﬁrst variable, and controls v(t) are U −valued mappings which are progressively measurable with respect to the family Ft , called admissible controls. ∗ Work

supported in part by the European Community’s Human Potential Programme under contract HPRN-CT-2002-00281, Evolution Equations.

Copyright © 2005 Marcel Dekker, Inc.

218

CONTROL AND BOUNDARY ANALYSIS

A set K ⊂ H is called invariant under the control system (1) if for every F0 -random variable X0 such that X0 ∈ K almost surely and every admissible control v(·), the strong solution X to (1) starting at X0 satisﬁes for all t ≥ 0, X(t) ∈ K almost surely. When b and σ are control independent, the above system reduces to the stochastic diﬀerential equation dX = b(X)dt + σ(X)dW (t).

(2)

Recently a number of papers were written on stochastic viability and invariance of closed sets. In the case of stochastic equation (2) conditions for the invariance were expressed using the Stratonovitch drift [9] or stochastic contingent sets [2]. For stochastic control systems and diﬀerential inclusions, diﬀerent authors used stochastic contingent sets [3], viscosity solutions of second order partial diﬀerential equations [5–6], and derivatives of the distance function [7], see also [11–12]for several other approaches. The method based on the second order partial diﬀerential equations deals with value functions of some associated optimal control problems. These tools use the second order sets of continuous solutions to PDE’s. So the second order normal cones to K arise naturally in characterizations of invariance. However, there is no good calculus available for the second order normal cones. In contrast, the results of [9, Doss] obtained in the context of stochastic equations use only ﬁrst order normals to K. This approach is based on an equivalence between invariance of stochastic equation (2) and that of an associated deterministic control system. Namely, it was shown in [9] that if σ ∈ C 3 and has bounded derivatives up to the order three, then K is invariant under the stochastic equation (2) if and only if K is invariant under the (well understood) deterministic control system x = b(x) −

1 Dσj (x)σj (x) + σ(x)u(t), u(t) ∈ H1 , 2 m

(3)

j=1

where σj (x) denotes the column j of the matrix σ(x) and Dσj the jacobian of σj . Theory of [9] needs, however, more regularity of the diﬀusion term σ (Cb3 instead of bounded and Lipschitz continuous) and is based on the Stroock-Varhadan support theorem (see for instance [11]) which is not applicable in the presence of time dependent controls. In this paper we prove a similar ﬁrst order characterization of invariance of stochastic control systems, when σ ∈ Cb1,1 . That is, we extend the Doss theorem into two directions: to control systems and less regular σ. Copyright © 2005 Marcel Dekker, Inc.

Invariance of Closed Sets under Stochastic Control Systems

219

Recall that in the deterministic case a necessary and suﬃcient condition for invariance of K under (3) can be expressed by using tangents to K. Thus K is invariant under the stochastic system (2) if and only if the so-called Stratonovitch drift is tangent to K: 1 Dσj (x)σj (x) ∈ TK (x), ∀ x ∈ K b(x) − 2 m

(4)

j=1

and the image of the diﬀusion σ is tangent to K: σ(x)u ∈ TK (x), ∀ x ∈ K, ∀ u ∈ H1 .

(5)

Condition (5) in turn is equivalent to the invariance of the boundary of K under the deterministic control system x = σ(x)u(t), u(t) ∈ H1 .

(6)

Instead of the support theorem, we use (6) to show that K is invariant under the stochastic control system (1) if and only if it is invariant under the deterministic control system with two controls x = b(x, v(t)) −

1 Dσj (x, v(t))σj (x, v(t)) + σ(x, v(t))u(t), 2 m

(7)

j=1

where u(t) ∈ H1 , v(t) ∈ U. In Section 3 the characterization of the invariance is also stated in terms of proximal normals and tangent cones. Several results below are provided without detailed proofs. They will appear elsewhere.

1.

Preliminaries

Consider a closed non-empty subset K of H. We denote by ∂K the boundary of K and by dK the distance of x ∈ H from K : dK (x) = inf y∈K |x − y|. The contingent cone TK (x) to K at x ∈ K is the set of all vectors v ∈ H such that lim inf h→0+ dK (x + hv)/h = 0. A vector p ∈ H is called a proximal normal to K at x ∈ K if |p| = dK (x + p). Clearly p = 0 is a proximal normal and it is the only proximal normal when x is in the interior of K. It is well known that if p is a proximal normal to K at x, then for some c > 0 ∀ z ∈ K, p, z − x ≤ c|z − x|2 . The following result can be easily deduced from [1].

Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

Lemma 1 Assume that σ : H → L(H1 , H) is locally Lipschitz. Then K and ∂K are invariant under the deterministic control system ( 6) if and only if and for all x ∈ K and any proximal normal p at x we have σ(x)∗ p = 0. For a function ψ : R+ → R the lower directional derivative in the . Using the direction one is deﬁned by dψ(t) := lim inf h→0+ ψ(t+h)−ψ(t) h viability theorem from [1] we obtain the following result.

Lemma 2 Consider T > 0 and a continuous function ψ : [0, T ] → R+ with ψ(0) = 0. Assume that for some L ≥ 0 and every t ∈ [0, T [ such that ψ(t) > 0 we have dψ(t) ≤ Lψ(t). Then ψ = 0. We next recall a necessary condition for the stochastic viability. The result below may be applied to any weak solution of (2). See for instance [13] for the deﬁnition of weak solution. A mapping X : R+ → L2 (Ω, H) is called an adapted process if for every t ≥ 0, X(t) is Ft −measurable. Let b, σ be bounded and continuous. Assume that an adapted process X(·) is continuous and for some x ∈ H, t t b(X(s))ds + σ(X(s))dW (s) a.s. (9) ∀ t ≥ 0, X(t) = x + 0

0

The process X(·) is called viable in K, if for all t ≥ 0, X(t) ∈ K a.s.

Theorem 3 Assume that b and σ are bounded and continuous. If an adapted process X(·) is continuous, satisﬁes (9) with x ∈ K, and for some hi → 0+, X(hi ) ∈ K a.s., then for any proximal normal p to K at x we have σ(x)∗ p = 0. In particular, if ( 2) has a weak solution starting at some x ∈ K which is viable in K, then for any proximal normal p to K at x we have σ(x)∗ p = 0. The proof is similar to the one provided in the appendix of [8].

2.

Necessary and Suﬃcient Conditions for the Invariance

We denote by Cb1,1 (H; L(H1 , H)) the set of all functions σ : H → L(H1 , H) such that σ and its derivative σ are bounded and σ (·) is Lipschitz.

Theorem 4 Assume that K is closed, b is Lipschitz and bounded, and that σ ∈ Cb1,1 . The set K is invariant under the system (2) if and only Copyright © 2005 Marcel Dekker, Inc.

Invariance of Closed Sets under Stochastic Control Systems

221

if for every x ∈ ∂K and every proximal normal p to K at x we have @ ? m 1 Dσj (x)σj (x) ≤ 0, σ(x)∗ p = 0. (10) p, b(x) − 2 j=1

Theorem 4 allows to extend to a less regular σ a result from [9]:

Corollary 5 Assume that K is closed, b is Lipschitz and bounded, and that σ ∈ Cb1,1 . Then K is invariant under the stochastic system (2) if and only if K is invariant under the deterministic control system ( 3) or, equivalently, if and only if ( 4) and ( 5) hold true. Proof of Theorem 4 — Necessity. Assume that K is invariant. Then, by Theorem 3, σ ∗ (y)py = 0 for all y ∈ K and any proximal normal py to K at y. Fix x ∈ K and a proximal normal p to K at x. Then for some c > 0 A (t) in the the inequality (8) holds true. Fix 0 < t ≤ 1, an element W , class of functions equivalent to W (t) ∈ L2 (Ω; H1 ), and an element X(t) 2 in the class of functions equivalent to X(t) ∈ L (Ω; H). For every ω ∈ Ω deﬁne Aω (t) = 0, if W 0 Aω (t) uω := W otherwise. − A |Wω (t)| For all ω ∈ Ω, let zω (·) be the solution to the deterministic system ,ω (t). z (s) = σ(z(s))uω , z(0) = X , Since X(t) ∈ K almost surely, by Lemma 1 we know that for almost all ω ∈ Ω and for all s ≥ 0, zω (s) ∈ K. Let σij denote elements of σ. For all ξ ∈ H deﬁne F (t, ξ) : Ω → L(H1 , H) by Aω (t). F (t, ξ) := (fij (t, ξ)), fij (t, ξ)ω = ∇σij (ξ), σ(ξ)W Then Aω (t)|) = X ,ω (t) − σ(X ,ω (t))W Aω (t) + 1 F (t, X ,ω (t))W Aω (t) + Φω (t), zω (|W 2 Aω (t)|3 . Set yω (t) = zω (|W Aω (t)|). Using the projecwhere Φω (t) ≤ M |W tion theorem we check that y(t) is Ft −measurable and y(t) ∈ L2 (Ω, H). Since y(t) ∈ K almost surely, Ep, y(t) − x ≤ c E|y(t) − x|2 . Copyright © 2005 Marcel Dekker, Inc.

(11)

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CONTROL AND BOUNDARY ANALYSIS

This implies that for all 0 ≤ t ≤ 1 E p, y(t) − x = O(t2 ).

(12)

By the Lipschitz continuity of σ and ∇σij , for some c1 > 0 and all s ∈ [0, 1] E|σ ∗ (X(s))∇σij (X(s)) − σ ∗ (x)∇σ(x)|2 ≤ c1 s.

(13)

This and the H¨ older inequality imply that , E|F (t, X(t))W (t) − F (t, x)W (t)| = O(t3/2 ).

(14)

Furthermore, by the H¨ older inequality and (13), for a constant c1 > 0 t E|p, ( (σ ∗ ∇σij )(X(s)) − (σ ∗ ∇σij )(x), dW (s))W (t)| ≤ 0 t √ (15) c1 t max(E| (σ ∗ ∇σij )(X(s)) − (σ ∗ ∇σij )(x), dW (s)|2 )1/2 i,j

= O(t3/2 ).

0

Thus it follows from (12), (14) and (15) that for all 0 < t ≤ 1 1 Ep, tb(x) + E p, F (t, x)W (t) − 2 B t C E p, ∇σij (x), σ(x)dW (s) W (t) = O(t3/2 ).

(16)

0

Finally observe that t Ep, ( ∇σij (x), σ(x)dW (s))W (t) = 0 Ep, ( ∇σij (x), σ(x)W (t)W j (t)) = Ep, F (t, x)W (t) = j

∂σij pi (x)σkr (x)W r (t)W j (t)) = E( ∂xk i j,k,r ∂σij pi (x)σkj (x)E(W j (t))2 = tp, Dσj (x)σj (x). ∂xk i

j,k

(17)

j

E D From (16), (17) we obtain t p, b(x) − 12 j Dσj (x)σj (x) = O(t3/2 ) for every 0 < t ≤ 1. Dividing by t and taking the limit completes the proof of necessary conditions. Suﬃciency. Fix an F0 - random variable X0 ∈ K a.s. and consider the strong solution X(t) to (2). Set ψ(t) = Ed2K (X(t)). To prove that Copyright © 2005 Marcel Dekker, Inc.

Invariance of Closed Sets under Stochastic Control Systems

223

X(t) ∈ K almost surely, we have to show that ψ(t) = 0. Since ψ(0) = 0, by Lemma 2 it is enough to prove that for some L > 0 and all t > 0 such that ψ(t) > 0 we have dψ(t) ≤ Lψ(t). Fix t > 0 such that ψ(t) > 0, , and also an element X(t) in the class of functions equivalent to X(t) ∈ 2 2 ,ω (h)). L (Ω; H). Set ϕω (h) = dK (X By the measurable selection theorem there exists an Ft - measurable ,ω (t) − ζω |. In particular, X ,ω (t) − map ω → ζω ∈ K such that ϕω (t) = |X ζω is a proximal normal to K at ζω . A (t) in the class of functions Fix h ∈ (0, 1), and also an element W A (t + h) in the class equivalent to W (t) ∈ L2 (Ω; H1 ), and an element W 2 of functions equivalent to W (t + h) ∈ L (Ω; H1 ). For all ω ∈ Ω set Aω (t) Aω (t + h) = W if W 0 Aω (t) Aω (t + h) − W uω (h) := W otherwise. A Aω (t)| |Wω (t + h) − W For every ω ∈ Ω consider the solution zω (·) to the deterministic system z (s) = σ(z(s))uω (h), zω (0) = ζω . By (10) and Lemma 1, zω (s) ∈ K for all s ≥ 0. On the other hand, for all ω ∈ Ω Aω (t + h) − W Aω (t)|) = ζω + σ(ζω )(W Aω (t + h) − W Aω (t)) + Φω (t)+ (|W zω D E 1 Aω (t + h) − W Aω (t)) (W A j (t + h) − W A j (t)) ∇σij (ζω ), σ(ζω )(W ω ω 2 j

Aω (t + h) − W Aω (t)|3 . Set yω (h) = zω (|W Aω (t + h) − where Φω (t) ≤ M |W Aω (t)|). It is not diﬃcult to verify that y(h) is Ft+h −measurable and W y(h) ∈ L2 (Ω; H). Observe next that d2K (X(t + h)) ≤ |X(t + h) − y(h)|2 . The above inequality, Lipschitz continuity of b, σ, and the H¨ older inequality imply that for some L > 0 independent from t and for all 0 < h ≤ 1, ψ(t + h) ≤

D E ψ(t) + Lhψ(t) + 2hE X(t) − ζ, b(ζ) − 12 j Dσj (ζ)σj (ζ) + O(h3/2 ).

Thus, by (10), ψ(t + h) ≤ ψ(t) + Lhψ(t) + O(h3/2 ) and therefore dψ(t) ≤ Lψ(t). By Lemma 2, ψ ≡ 0 implying that dK (X(t)) = 0 a.s. 2 Copyright © 2005 Marcel Dekker, Inc.

224

3.

CONTROL AND BOUNDARY ANALYSIS

Invariance of Stochastic Control Systems

Let U be a complete separable metric space and b : H × U → H, and σ : H × U → L(H1 , H) be bounded continuous mappings. Assume that there exists a constant C > 0 such that for all x, y ∈ H and v ∈ U |b(x, v) − b(y, v)| + σ(x, v) − σ(y, v) + σx (x, v) − σx (y, v) ≤ C|x − y|.

(18)

Denote by A the set of all U −valued mappings v(·) deﬁned on R+ which are progressively measurable with respect to the family Ft , i.e. for every t ≥ 0, v(t) ∈ U a.s. and the mapping [0, t] × Ω (s, ω) → vω (s) is B1 × Ft −measurable. Elements of A are called admissible controls. We associate to the above data the stochastic control system (1). Let X0 be an F0 - random variable, v(·) ∈ A and consider the diﬀerential stochastic equation dX = b(X, v(t))dt + σ(X, v(t))dW (t), (19) X(0) = X0 . Under the above assumptions, system (19) has a unique strong solution X(·), i.e. for all t ≥ 0, t t b(X(s), v(s))ds + σ(X(s), v(s))dW (s) a.s.. X(t) = X0 + 0

0

Definition 6 A set K ⊂ H is called invariant under the stochastic control system ( 1) if for every F0 - random variable X0 such that X0 ∈ K almost surely and every admissible control v(·) ∈ A, the strong solution X to ( 19) satisﬁes for all t ≥ 0, X(t) ∈ K almost surely. Theorem 7 Assume that K is closed, b, σ are bounded and continuous, that there exists a constant C > 0 such that ( 18) holds true. Then K is invariant under the stochastic control system ( 1) if and only if for every x ∈ ∂K, v ∈ U and for every proximal normal p to K at x we have @ ? m 1 Dx σj (x, v)σj (x, v) ≤ 0, σ(x, v)∗ p = 0, (20) p, b(x, v) − 2 j=1

where σj (x, v) denotes the column j of the matrix σ(x, v) and Dx σj (x, v) the jacobian of σj (·, v) at x.

Corollary 8 If all the assumptions of Theorem 7 hold true, then K is invariant under the stochastic control system ( 1) if and only if K is Copyright © 2005 Marcel Dekker, Inc.

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Invariance of Closed Sets under Stochastic Control Systems

invariant under the deterministic control system with two controls ( 7) or, equivalently, if and only if for all v ∈ U and x ∈ K 1 Dσj (x, v)σj (x, v) ∈ TK (x), Im(σ(x, v)) ⊂ TK (x). 2 m

b(x, v) −

j=1

Proof of Theorem 7 — If the set K is invariant under the system (1), then for every v0 ∈ U the mapping v ≡ v0 belongs to A. Thus, for every F0 -measurable random variable X0 ∈ K a.s., the solution X to (19) satisﬁes X(t) ∈ K a.s. This and Theorem 4 imply that for every x ∈ ∂K and every proximal normal p to K at x relations (20) hold true. Assume next that (20) is satisﬁed. Case 1 of constant controls. The proof of suﬃciency is essentially the same as the one of the invariance for stochastic diﬀerential equations provided in Section 2. Case 2 of piecewise constant controls. Let v ∈ A be such that for some 0 = s0 < s1 < ... < sk < ... and for all k ≥ 0, v is time independent on the time interval [sk , sk+1 ). Then, Case 1 and an induction argument imply that X(s) ∈ K a.s. for all s ≥ 0. The general case. Consider v ∈ A, F0 -random variable X0 ∈ K a.s. and the strong solution X(t) to (19). Fix t > 0. We have to show that X(t) ∈ K a.s. For this end let us v(s)) ∈ ﬁx 0 < ε < 1 and deﬁne the mapping R+ s → g(s) := b(X(s),

s L2 (Ω, H). Then for almost all ω ∈ Ω, the mapping s → 0 gω (τ )dτ ∈ H is absolutely continuous on bounded intervals. Deﬁne f : [0, t] ×

s+h (R+ \{0}) → L2 (Ω, H) by f (s, h) = h1 s g(τ )dτ. Then, for almost

t all ω ∈ Ω, limh→0+ 0 |fω (s, h) − gω (s)|ds = 0. Hence, by the dominated

t convergence theorem, limh→0+ E 0 |f (s, h)−g(s)|ds = 0. Next, applying the Fubini theorem, we obtain that lim

h→0+ 0

t

E|f (s, h) − g(s)|ds = 0.

(21)

Let hi → 0+ be such for all i ≥ 1, hi ≤ ε2 . Claim. We claim that for all i large enough, there exist δi → 0+ i ≤ t such that for all and 0 = τ0i ≤ si1 < τ1i ≤ si2 ... ≤ simi < τm i i i 1 ≤ j ≤ mi , τj = sj + hi and 0 ≤ t − mi hi ≤ δi + hi , Copyright © 2005 Marcel Dekker, Inc.

E|

sij +hi

sij

g(τ )dτ − hi g(sij )| ≤ εhi .

(22)

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CONTROL AND BOUNDARY ANALYSIS

The proof of this claim follows from (21). It is quite technical and is omitted. Consider piecewise constant controls if for some 1 ≤ j ≤ mi − 1, s ∈ [sij , τji ), v(sij ) v(τji ) if for some 0 ≤ j ≤ mi − 1, s ∈ [τji , sij+1 ), ui (s) := i i , v(τmi ) if s ≥ τm i and piecewise constant functions if for some 1 ≤ j ≤ mi − 1, s ∈ [sij , τji ), X(sij ) if for some 0 ≤ j ≤ mi − 1, s ∈ [τji , sij+1 ) X(τji ) Xi (s) := i i . X(τmi ) if s ≥ τm i Then, by the very deﬁnition of the Itˆ o integral, for all s ∈ [0, t], s lim E| (σ(X(ρ), u(ρ)) − σ(Xi (ρ), ui (ρ)))dW (ρ)|2 = 0. i→∞

(23)

0

Consider solutions Yi to dY = b(Y, ui )dt + σ(Y, ui )dW (t), X(0) = X0 . Then by Case 2, Yi (s) ∈ K a.s. for all s ≥ 0. Set ψiε (s) := E|X(s) − Yi (s)|2 . Notice that Ed2K (X(t)) ≤ ψiε (t), (24) and that for some α > 1 and all s ∈ [0, t], s 1 ε E|b(X(ρ), ui (ρ)) − b(Yi (ρ), ui (ρ))|2 dρ+ ψ (s) ≤ α i 0 s (σ(X(ρ), ui (ρ)) − σ(Yi (ρ), ui (ρ)))dW (ρ)|2 + E| 0 s E| (σ(X(ρ), ui (ρ)) − σ(Xi (ρ), ui (ρ)))dW (ρ)|2 + 0 s (g(ρ) − b(X(ρ), ui (ρ))dρ|2 + E| 0 s E| (σ(X(ρ), u(ρ)) − σ(Xi (ρ), ui (ρ)))dW (ρ)|2 =

(25)

0

I1i (s) + I2i (s) + I3i (s) + I4i (s) + I5i (s). Since b and σ are C−Lipschitz in the ﬁrst variable, s i i 2 E|X(ρ) − Yi (ρ)| dρ = 2C I1 (s) + I2 (s) ≤ 2C 0

Copyright © 2005 Marcel Dekker, Inc.

0

s

ψiε (ρ)dρ.

(26)

Invariance of Closed Sets under Stochastic Control Systems

By the Lipschitz continuity of σ with respect to x, we get s I3i (s) ≤ C E|X(ρ) − Xi (ρ)|2 dρ ≤ 0 mi τ i j C (ρ − sij )dρ + 2M Ct(δi + hi ) ≤ C

i j=1 sj mi (τji − sij )2 j=1

sij

2

2b∞ (ε + δi ) +

mi j=1

By (22), E that

mi

τji

j=1 | si j

mi | E j=1

(27)

+ 2M Ct(δi + hi ) ≤ Cε2 t + 2M Ct(δi + ε2 ).

On the other hand, s (g(ρ) − b(X(ρ), ui (ρ)))dρ ≤ 0 i m i τ j | (g(ρ) − b(X(sij ), v(sij )))dρ|+ j=1

227

τji sij

τji

sij

(28)

|b(X(ρ), ui (ρ)) − b(X(sij ), v(sij )|dρ.

(g(ρ) −

b(X(sij ), v(sij ))dρ|

≤ εt, which implies

2 (g(ρ) − b(X(sij ), v(sij ))dρ| ≤ 2εt2 b∞ .

(29)

Furthermore, by the Lipschitz continuity of b with respect to x, for a constant c1 > 0 independent from i, ε, 2 mi τ i j |b(X(ρ), ui (ρ)) − b(X(sij ), v(sij )|dρ ≤ E sij mi τ i j

j=1

2tb∞

sij

E|b(X(ρ), ui (ρ)) − b(X(sij ), v(sij ))|dρ ≤

j=1 mi τ i j

2Ctb∞ c1

mi j=1

j=1 τji 3

sij

sij

(30) E|X(ρ) − X(sij )|dρ ≤

ρ − sij dρ ≤ c1

mi (τji − sij )3/2 ≤ c1 εt. j=1

By (28) - (30) for constant c2 > 0 independent from i, ε, s, I4i (s) ≤ c2 (ε + δi ). This and (25) - (27) imply for a constant c3 > 0 independent Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

from i, ε, and for all s ∈ [0, t]

s ψiε (s) ≤ c3 0 ψiε (ρ)dρ + c3 (ε + δi ) + I5i (s). Then it follows from the Gronwall inequality that for a constant c4 > 0 independent from ε and for all i,

t (31) ψiε (t) ≤ c4 (ε + δi + I5i (t)) + c4 0 I5i (s)ds. From (23) we know that for every s ∈ [0, t], limi→∞ I5i (s) = 0. On the other hand, s i Eσ(X(ρ), u(ρ)) − σ(Xi (ρ), ui (ρ))2 dρ ≤ 2sσ2∞ . I5 (s) = 0

From the

t Lebesgue dominated convergence theorem we deduce that limi→∞ 0 I5i (s)ds = 0. This and (24), (31) imply that E(dK (X(t)2 ) ≤ lim supi→∞ ψiε (t) ≤ c4 ε. Since ε > 0 is arbitrary and c4 does not depend on ε, we get X(t) ∈ K a.s. 2

References [1] AUBIN J.-P. (1991) Viability Theory, Birkh¨ auser, Boston, Basel, Berlin. [2] AUBIN J.-P. and DA PRATO G. (1990) Stochastic viability and invariance, Annali Scuola Normale di Pisa, 27, 595–694. [3] AUBIN J.-P., DA PRATO G., and FRANKOWSKA H. (2001) Stochastic invariance for diﬀerential inclusions, Set-Valued Analysis, 8, 181–201. [4] AUBIN J.-P. and DOSS H. (to appear) Characterization of stochastic viability of any nonsmooth set involving its generalized contingent curvature, Stochastic Analysis and Applications. [5] BARDI M. and GOATIN P. (1999) Invariant sets for controlled degenerate diffusions: a viscosity solutions approach, W. M. M. Mc Eneaney, G. G. Yin, Q. Zhang Eds. LNiM 1660, Birkh¨ auser, 191–208. [6] BUCKDAHN R., PENG S., QUINCAMPOIX M., and RAINER C. (1998) Existence of stochastic control under state constraints, Comptes-Rendus de l’Acad´emie des Sciences, 327, 17–22. [7] DA PRATO G. and FRANKOWSKA H. (2001) Stochastic viability for compact sets in terms of the distance function, Dynamic Systems and Applications, vol. 10, 177–184. [8] DA PRATO G. and FRANKOWSKA H. (to appear) Invariant measure for a class of parabolic degenerate equations, NODEA. [9] DOSS H. (1977) Liens entre ´equations diﬀ´ erentielles stochastiques et ordinaires, Ann. Inst. Henri Poincar´e, Calcul des Probabilit´es et Statistique, 23, 99–125. [10] DOSS H. and LENGLART E. (1977) Sur le comportement asymptotique des solutions d’´equations diﬀ´ erentielles stochastiques, CRAS, 284, 971–974.

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[11] IKEDA N. and WATANABE S. (1981) Stochastic diﬀerential equations and diffusion processes, North-Holland, Amsterdam-New York. [12] MILIAN A. (1997) Invariance for stochastic equations with regular coeﬃcients, Stochastic Anal. Appl., 15, 91–101. [13] STROOCK D.V. and VARHADAN S.R.S (1979) Multidimensional Diﬀusion Processes, Springer-Verlag, Berlin.

Copyright © 2005 Marcel Dekker, Inc.

CHAPTER XVIII UNIFORM STABILIZATION OF AN ANISOTROPIC SYSTEM OF THERMOELASTICITY Achieving Stability with Minimal Geometric Assumptions Mary Ann Horn∗ Department of Mathematics Vanderbilt University Nashville, TN, USA

Abstract

Stabilization of thermoelasticity is inherently easier than stabilization of the corresponding elastic model due to the additional smoothing that occurs through the coupling of thermal eﬀects. However, in asking if an anisotropic thermoelastic system may be uniformly stabilized through the use of boundary control when the domain is assumed to be nonconvex, open questions still exist. In this article, the question of stabilization for a three-dimensional thermoelastic system is addressed. Generalizations to nonconvex domains require unique continuation for the system, a question that remains largely unanswered for anisotropic systems.

Keywords: Boundary stabilization, thermoelasticity, anisotropic elasticity, nonlinear control

Introduction In the past decade, numerous results have demonstrated that uniform decay of the energy can be obtained for elastic systems through the use of boundary dissipation. Early results focused on the case of a homogeneous, isotropic elastic body, beginning with the work of Lagnese [8–9]. In the latter, nonlinear boundary feedback, dependent on both the velocity and the tangential component of the position, was shown to uniformly stabilize the system under the assumption that the domain

∗ Research

partially supported by National Science Foundation Grant DMS-9803547.

Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

was star-shaped. Extending this result to more general domains, Horn used only velocity feedback acting as a traction force along the boundary, thus eliminating the additional feedback involving the tangential derivative [5]. Initial results on anisotropic elasticity were achieved by Alabau and Komornik with the underlying assumption that the domain was spherical, thus imposing strict geometric conditions even if control was assumed to be active on the entire boundary [2]. More recently, the question of uniform stability for thermoelastic systems with boundary dissipation has received considerable attention. While there is now a considerable amount of literature on boundary stabilization of thermoelastic Kirchhoﬀ plates (see [3, 12] and references therein) and the full von K´ arm´an system of dynamic elasticity [4, 10], literature most closely related to the problem of interest in this paper focuses on boundary stabilization results for isotropic and anisotropic thermoelasticity. Rivera and Olivera established uniform stability of an anisotropic thermoelastic system with linear velocity feedback acting through natural boundary conditions [15]. Application of the divergence theorem within their proof necessitates the assumption that the domain be spherical, however, even with this restrictive assumption, this result was among the ﬁrst to establish positive results on uniform stability for anisotropic systems. An extension of this result to include nonlinear boundary feedback control was seen in the work of Aassila [1], again with strict geometric assumptions. The goal of this paper is to extend the results of Rivera and Olivera to more general domains while retaining the nonlinear boundary feedback control seen in the work of Aassila. To do so necessitates the use of sharp trace regularity results and requires unique continuation results for the corresponding anisotropic elastic system. Because few unique continuation results are available for anisotropic systems, to facilitate the proof, a light interior damping term will be included. While this requirement is solely due to a technical challenge in the proof, this assumption is reasonable, given that all elastic bodies have some inherent internal damping.

1.

Statement of the Problem

To formulate the system of thermoelasticity, let n be a positive integer and let (aijkl ) be a tensor satisfying the following symmetries, aijkl = aklij = ajikl , 1 ≤ i, j, k, l ≤ n.

(1)

For some α > 0, the above tensor satisﬁes a strong ellipticity condition, aijkl εij εkl ≥ αεij εkl , Copyright © 2005 Marcel Dekker, Inc.

(2)

Uniform Stabilization of an Anisotropic System of Thermoelasticity

233

for every symmetric tensor εij . Assume Ω is a nonempty bounded domain in IRn with boundary ∂Ω. Let u = (u1 , . . . , un ) : Ω → IRn denote the displacement vector. Then with the strain tensor εij deﬁned by 1 εij (u) = 2

∂uj ∂ui + ∂xj ∂xi

,

(3)

the stress-strain relation can be expressed by σij (u) = aijkl εkl (u).

(4)

By letting θ denote the temperature, consider the system of anisotropic thermoelasticity with light internal damping deﬁned in the domain Ω with suﬃciently smooth boundary ∂Ω = ∂Ω0 ∪ ∂Ω1 : in Ω × (0, T ) utt − ∇ · σ(u) + ∇θ + but = 0 θt = ∆θ − divut in Ω × (0, T ) u = 0, θ = 0 on ∂Ω0 × (0, T ) σ(u)ν − θν = −g(ut ) on ∂Ω1 × (0, T ) ∂θ + λθ = 0 on ∂Ω1 × (0, T ) ∂ν

(5) (6) (7) (8) (9)

with corresponding initial conditions u(·, 0) = u0 (·), ut (·, 0) = u1 (·), θ(·, 0) = θ0 (·)

in Ω.

(10)

In the above equations, ν represents the unit outward normal to ∂Ω and b(x) > 0 for all x ∈ Ω. To avoid the necessity of compatibility conditions at the junction of ∂Ω0 and ∂Ω1 , the assumption ∂Ω0 ∩ ∂Ω1 = ∅ is imposed. The control function, g, which is acting via traction forces on the boundary, is a continuous vector-valued function, with each component assumed to be monotone increasing and zero at the origin. Additionally, g ∈ C 1 (IRn ) is subject to the following constraints: g(s) · s > 0, for s = 0, m|s| ≤ |g(s)| ≤ M |s|, for |s| > 1,

(11)

for two positive constants, m and M . Notice that while growth conditions are imposed for large values of |s|, none are required near the origin. Copyright © 2005 Marcel Dekker, Inc.

234

1.1

CONTROL AND BOUNDARY ANALYSIS

Wellposedness of the System

To state wellposedness of this anisotropic thermoelastic system, the following function space deﬁnitions are required. Let 1 (Ω) ≡ {v ∈ H 1 (Ω) : v = 0 on ∂Ω0 } H∂Ω 0

(12)

1 (Ω) to be and deﬁne H∂Ω 0 1 (Ω) ≡ {v ∈ [H 1 (Ω)]n : v = 0 on ∂Ω0 }. H∂Ω 0

(13)

Theorem 1 (Existence of strong solutions.) Let (u0 (x), u1 (x), θ0 (x)) ∈ 1 (Ω) × H1 (Ω) × H 1 (Ω) satisfy the following compat[H 2 (Ω)]n ∩ H∂Ω ∂Ω0 ∂Ω0 0 ibility condition: σ(u0 )ν − θ0 ν = −g(u1 ) on ∂Ω1 . Then there exists a unique solution, 1 1 (Ω) × H∂Ω (Ω)), (u, ut ) ∈ C([0, T ]; [H 2 (Ω)]n ∩ H∂Ω 0 0 1 θ ∈ C([0, T ]; H∂Ω (Ω)), 0

satisfying system (5-10).

Theorem 2 (Existence of weak solutions.) Let (u0 (x), u1 (x), θ0 (x)) ∈ 1 (Ω) × [L2 (Ω)]n × L2 (Ω). Then there exists a unique solution (in the H∂Ω 0 sense of distributions), 1 (Ω) × [L2 (Ω)]n ), (u, ut ) ∈ C([0, T ]; H∂Ω 0

θ ∈ C([0, T ]; L2 (Ω)), satisfying system (5-10). Proof of the existence and uniqueness of solutions can be established via semigroup theory [14]. The reader is referred to discussion in [1, 15] for further details on the proofs for anisotropic thermoelasticity. In order to concentrate on the question of uniform stability, the details of wellposedness will not be discussed in this paper.

1.2

Uniform Decay of the Energy of the System

The natural energy for the system of anisotropic thermoelasticity shown in (5-10) is deﬁned by 1 (14) ut 2[L2 (Ω)]n + (σ(u), ε(u))Ω + θ2L2 (Ω) . E(t) ≡ 2 Copyright © 2005 Marcel Dekker, Inc.

Uniform Stabilization of an Anisotropic System of Thermoelasticity

235

Note that the inner product appearing in the deﬁnition, (σ(u), ε(u))Ω is topologically equivalent to the standard norm on [H 1 (Ω)]n . In order to show that the energy of the solution to the system of anisotropic thermoelasticity decays uniformly to zero with respect to the initial energy, we begin by focusing on the nonlinear control. To state the stability result, accounting for the nonlinearities, the outline of Lasiecka and Tataru is followed [11]. Let G(x) be a concave, strictly increasing function which is zero at the origin and satisﬁes G(s · g(s)) ≥ |s|2 + |g(s)|2 for |s| ≤ 1

and deﬁne ˜ G(x) ≡G

x meas(∂Ω × (0, T ))

(15)

.

(16)

Since G˜ is monotone increasing, the operator cI + G˜ is invertible for any constant c ≥ 0. Therefore, the function ˜ −1 (kx), p(x) ≡ (cI + G)

k > 0,

(17)

is positive, continuous, and strictly increasing with p(0) = 0.

Theorem 3 (Uniform stability of the system.) Let (u, θ) be a solution of the anisotropic thermoelastic system (5-10). Assume h(x) · ν ≤ 0 on ∂Ω0 , where h(x) = x − x0 for some x0 ∈ IRn . Then for some T0 > 0,

t E(t) ≤ S −1 ∀t > T0 , T0

(18)

(19)

where S(t) → 0 as t → 0 and is the contraction semigroup satisfying the diﬀerential equation St (t) + q(S(t)) = 0,

S(0) = E(0),

(20)

where q(x) is deﬁned by q(x) ≡ x − (I − p)−1 (x) for x > 0.

2.

Proof of Theorem 3: Uniform Stabilization

The system of anisotropic thermoelasticity can be reformulated as a variational problem for equation (5), taking into account the boundary conditions in (7-8), as follows: (utt , φ)Ω + (σ(u), ε(φ))Ω + (but , φ)Ω + (∇θ, φ)Ω +(g(ut ) − θν, φ)∂Ω1 − (σ(u)ν, φ)∂Ω0 = 0, Copyright © 2005 Marcel Dekker, Inc.

(21)

236

CONTROL AND BOUNDARY ANALYSIS

plus the variational form of equation (6), including boundary condition (9): (θt , ψ)Ω + (∇θ, ∇ψ)Ω + (divut , ψ)Ω + (λθ, ψ)∂Ω1 = 0.

(22)

As a ﬁrst step in the proof of Theorem 3, the energy of the system is shown to satisfy a dissipativity inequality. Note that the corresponding anisotropic system of elasticity with b(x) ≡ 0 and with homogeneous boundary conditions is a conservative system. For the thermoelastic system, the following dissipativity result can be shown to hold.

Lemma 4 (Dissipativity inequality.) Let (u, θ) be a solution of system (5-10). Then for any s < t, the following inequality holds: t s

t

g(ut ) · ut dxdt +

E(t) +

s

∂Ω1

t

|∇θ| dxdt +

s

Ω

Ω

t

2

+

b|ut |2 dxdt

λθ 2 dxdt = E(s). s

(23)

∂Ω1

Proof: Let φ ≡ ut and ψ ≡ θ in the variational formulation, integrate with respect to time, apply the divergence theorem, and add the resulting equations.

2.1

Trace Regularity

Sharp trace regularity estimates play a critical role in eliminating many of the strict geometric assumptions frequently seen in the literature. In particular, the following result for a general elastic system will be used within the proof of stabilization. Note that standard trace theory does not allow the tangential component of the derivative of u on the boundary to be bounded by the energy norm.

Theorem 5 (Trace regularity [6].) Let u be a solution to the following general elastic system: in Ω × (0, T ) utt − ∇ · σ(u) = F u = 0 on ∂Ω0 × (0, T ) σ(u)ν = G on ∂Ω1 × (0, T )

(24) (25) (26)

with corresponding initial conditions u(·, 0) = u0 (·), ut (·, 0) = u1 (·) Copyright © 2005 Marcel Dekker, Inc.

in Ω.

(27)

Uniform Stabilization of an Anisotropic System of Thermoelasticity

and let 0 < α < T /2. Then

T −α

2 ∂Ω |∇uτ α | dxdt ≤ C ut 2[L2 (∂Ω)]n + u2[L2 (0,T ;H 1/2+ (Ω))]n +F 2[H −1/2 (0,T ;Ω)]n

+

G2[L2 (0,T ;∂Ω1 )]n

237

(28)

dt.

Remark: Proof of this theorem is based on microlocal analysis, under the assumption that the spatial operator, ∇ · σ(u), is uniformly elliptic. While the result stated in [6] is in the context of isotropic elasticity, the estimate is also valid for anisotropic elasticity. This is due to the fact that ellipticity is preserved under the partition of unity and the ﬂattening of the boundary procedure used in the proof. For the thermoelastic system under consideration here, application of Theorem 5 yields the inequality,

T −α

2 ∂Ω |∇uτ α | dxdt ≤ C ut 2[L2 (∂Ω)]n + g(ut )2[L2 (∂Ω)]n + θ2L2 (0,T ;∂Ω) (29) +θ2H 1/2+ (0,T ;Ω) + u2[L2 (0,T ;H 1− (Ω))]n dt.

2.2

Stability Estimates

The critical step in proving the stability result stated in Theorem 3 is to establish the following estimate.

Lemma 6 (Stability estimate.) Let (u, θ) be a strong solution to the system of anisotropic thermoelasticity deﬁned in (5-10). Assume h(x) · ν ≤ 0

on ∂Ω0 .

Then there exists a suﬃciently large time T and a constant CT (E(0)), which is dependent upon T and, possibly, upon the initial energy E(0), such that the following estimate holds: T 2 |ut | + |g(ut )|2 dxdt E(T ) ≤ CT (E(0)) 0

T

∂Ω1

b|ut | dxdt +

T

+ 0

Ω

|∇θ| dxdt

2

2

0

(30)

Ω

Once Lemma 6 has been established, the proof of uniform stabilization follows from the results of Lasiecka and Tataru [11] as it did for the case of anisotropic elasticity [5], where details as to the application of their work can be found. Copyright © 2005 Marcel Dekker, Inc.

238

CONTROL AND BOUNDARY ANALYSIS

The proof of Lemma 6 is comprised of three basic steps. First, an appropriate choice of test functions will lead to a preliminary bound on the energy. Such test functions, or multipliers, depend upon the problem at hand. Among the functions chosen are those that have been seen in anisotropic elasticity, φ1 ≡ ∇uh and φ2 ≡ u. Secondly, sharp trace regularity estimates for the system of anisotropic elasticity are applied to remove traces of the solution which cannot be bounded by the energy using standard trace theory. Finally, a compactness-uniqueness argument allows the ﬁnal estimate to be obtained by removing terms involving u which are lower order with respect to the energy norm.

Preliminary Estimate. tablished.

To begin, the following lemma will be es-

Lemma 7 Let (u, θ) be a solution of the system of anisotropic thermoelasticity deﬁned in (5-10). Assume h(x) · ν ≤ 0

on ∂Ω0 .

Then there exists a constant, C, such that for any ∈ (0, 14 ), the following estimate holds: T T 2 |ut | + |g(ut )|2 + |∇u|2 dxdt E(t)dt ≤ C E(T ) + 0

0

∂Ω1

T

+

θ21,Ω dt + l.o.t.(u) ,

(31)

0

where

l.o.t.(u) ≡ 0

T

u2[H 1− (Ω)]n dt

which includes terms involving the norms of u which are strictly lower order with respect to the energy norm. Proof: Analogously to the system of anisotropic elasticity, the proof follows the series of steps outlined below, taking into account the thermal eﬀects now included. For further technical details, the reader is referred to [7]. Let φ = ∇uh be the test function in the variational formulation (21). Integrating over time, the resulting inequality is

n T n−2 T 2 ε(u)) dt 0 (σ(u), 2 0 Ω |ut | dxdt − 2

T Ω ≤ C(E(0) + E(T )) + 12 0 ∂Ω1 |ut |2h · νdxdt

T (32)

T − 0 (but , ∇uh)Ω dt − 0 (g(ut ) − θν, ∇uh)∂Ω1 dt

T − 0 (σ(u), ε(u)ν)∂Ω dt. Copyright © 2005 Marcel Dekker, Inc.

Uniform Stabilization of an Anisotropic System of Thermoelasticity

239

With a second multiplier, φ = u, the following identity can be obtained:

T dt 0 (σ(u), ε(u))

T

T Ω = 0 ut 2[L2 (Ω)]n dt − (ut , u)T0 − 0 (but , u)Ω dt (33)

T

T + 0 (σ(u)ν, u)∂Ω0 dt − 0 (g(ut ) − θν, u)∂Ω1 dt Multiplying the second identity by n−1 2 , adding it to the ﬁrst and applying trace theory to the terms involving θ yields

1 T 1 T 2 2 0 ut [L2 (Ω)]n dxdt + 2 0 (σ(u), ε(u))Ω dt

T

≤ C E(0) + E(T ) + 0 ∂Ω1 |ut |2 + |g(ut )|2 dxdt (34)

T

T

|∇u|2 dxdt + 0 Ω b|ut |2 dxdt + 0 ∂Ω1

T

− 0 Ω |∇θ|2 dxdt + l.o.t.(u) .

Elimination of Higher Order Traces. At this stage, the higher order traces of u on the boundary must be bounded. To do so, Theorem 5 is applied, resulting in the following estimate. Lemma 8 Let (u, θ) be a solution of the anisotropic thermoelastic system in (5-10). Assuming h · ν ≤ 0 on ∂Ω0 , then for some suﬃciently large time T , there exists a constant C(T ) such that for any 0 < < 14 , the solution satisﬁes T E(T ) ≤ C(T ) 0 Ω b|ut |2 + |∇θ|2 dxdt (35)

T

+ 0 ∂Ω1 |ut |2 + |g(ut )|2 dxdt + l.o.t.(u) . Proof follows by application of Theorem 5 combined with a careful analysis of the tangential and normal traces of the solution on the boundary.

Removal of Lower Order Terms. To reach the conclusion of Lemma 6, the ﬁnal step is to remove the lower order terms, i.e., the following estimate must be established: Lemma 9 Let (u, θ) be a solution to the system of thermoelasticity deﬁned in (5-10). Then there exists a constant, CT (E(0)), dependent upon T and, possibly, dependent upon the initial energy E(0), such that

T l.o.t.(u) ≤ CT (E(0)) 0 Ω |∇θ|2 + b|ut |2 dx

(36) + ∂Ω1 |ut |2 + |g(ut )|2 dx dt. Remark: It is precisely at this stage that the assumption b(x) > 0 for all x ∈ Ω plays a key role. Proof of Lemma 9 is via a compactnessCopyright © 2005 Marcel Dekker, Inc.

240

CONTROL AND BOUNDARY ANALYSIS

uniqueness argument and with the inclusion of this light internal damping, the question of uniqueness for the full dynamic system can be reduced to that of a corresponding stationary problem. Proof: Assume (36) does not hold. Then there exists a sequence of solutions {(um , θm )}∞ m=1 satisfying system (5-10) with initial data 1 (Ω) × [L2 (Ω)]n × L2 (Ω) (um,0 , um,1 , θm,0 ) ∈ H∂Ω 0

such that l.o.t.(u) →∞

2 + b|u |2 ) dx + 2 + |g(u )|2 ) dx dt (|∇θ| (|u | t t t Ω ∂Ω1

T

0

(37)

as m → ∞. Let Em (t) denote the energy of the system satisfying by (um , θm ) at time t. Since (um , θm ) satisﬁes system (5-10), Lemma 8 can be applied and, therefore, Em (t) ≤ C(T ) uniformly in m. Thus, the sequence of solutions has the following convergence properties: in [L∞ (0, T ; H 1 (Ω)]n weakly star, um → u um,t → ut in [L∞ (0, T ; L2 (Ω))]n weakly star. Therefore, l.o.t.(um ) → l.o.t.(u) as m → ∞, since the lower order terms are compact with respect to the energy norm. Case 1: Assume l.o.t.(u) = 0. Then, if (37) holds, θm → 0 um,t → 0 um,t → 0 g(um,t ) → 0

in in in in

L2 (0, T ; H 1 (Ω)), [L∞ (0, T ; Ω]n , [L∞ (0, T ; ∂Ω1 )]n , [L∞ (0, T ; ∂Ω1 )]n .

Passing the limit on the original system in (5-10), the limit function u must satisfy ∇ · σ(u) = 0 in Ω × (0, T ), u = 0 on ∂Ω0 × (0, T ), σ(u)ν = 0 on ∂Ω1 × (0, T ), and, therefore, Korn’s inequality implies u ≡ 0 [13], which contradicts the assumption that l.o.t.(u) = 0. Case 2: Assume l.o.t.(u) = 0. Deﬁne cm ≡ l.o.t.(um ) and u ˜m ≡ um /cm , θ˜m ≡ θm /cm . Then l.o.t.(˜ um ) ≡ 1, Copyright © 2005 Marcel Dekker, Inc.

Uniform Stabilization of an Anisotropic System of Thermoelasticity

1 c2m and 1 c2m

T

0

T

241

b|um,t |2 + |∇θm |2 dxdt → 0,

Ω

0

|um,t |2 + |g(ut )|2 dxdt → 0,

∂Ω1

implying, θ˜m → 0 u ˜m,t → 0 u ˜m,t → 0 1 cm g(um,t ) → 0

in in in in

L2 (0, T ; H 1 (Ω)), [L∞ (0, T ; Ω]n , [L∞ (0, T ; ∂Ω1 )]n , [L∞ (0, T ; ∂Ω1 )]n .

Lemma 8 and the above convergence yields the inequality T 1 1 2 2 E (T ) ≤ 2 m cm c2m 0 Ω b|um,t | + |∇θm | dxdt

T

1 2 + |g(u 2 dxdt + l.o.t.(˜ |u u | )| ) m,t m,t m 2 0 ∂Ω1 c m

Therefore, the right-hand side is bounded uniformly for all m, implying, as in the ﬁrst case, that u) = 1 l.o.t.(˜ um ) → l.o.t.(˜ and u ˜ satisﬁes the corresponding homogeneous limiting equation. Hence, as before, u ˜ ≡ 0, a contradiction.

Conclusion of the Proof of Lemma 6. Combining the results of Lemmas 8 and 9 yields the desired stability estimate of Lemma 6.

3.

Discussion

The proof of stabilization demonstrates that the addition of thermal eﬀects to the anisotropic system of elasticity creates no signiﬁcant diﬃculties. Of course, this is to be expected since it is well known that thermoelastic systems have inherent stability properties not seen in purely elastic systems, which are typically conservative unless additional damping is assumed. However, the same challenge that appeared in extending uniform stability results to anisotropic elastic systems when working in nonconvex domains is again seen here. Unique continuation properties play a critical role when considering general domains, but in the case of anisotropic systems, many open questions remain. It is clear that this diﬃculty can be avoided by the inclusion of the internal damping term, represented by b(x)ut , however, whether this term can be easily eliminated remains to be seen. Copyright © 2005 Marcel Dekker, Inc.

242

CONTROL AND BOUNDARY ANALYSIS

References [1] M. Aassila. Nonlinear boundary stabilization of an inhomogeneous and anisotropic thermoelasticity system. Appl. Math. Letters, 13:71–76, 2000. [2] F. Alabau and V. Komornik. Boundary observability, controllability and stabilization of linear elastodynamic systems. SIAM J. Control Optim., 37(2):521– 542, 1999. [3] G. Avalos and I. Lasiecka. Exponential stability of a thermoelastic system with free boundary conditions without mechanical dissipation. SIAM J. Math. Anal., 29(1):155–182, 1998. [4] A. Benabdallah and I. Lasiecka. Exponential decay rates for a full von k´ arm´ an system of dynamic thermoelasticity. J. Diﬀ. Eq., 160:51–93, 2000. [5] M. A. Horn. Implications of sharp trace regularity results on boundary stabilization of the system of linear elasticity. J. Math. Anal. Appl., 223:126–150, 1998. [6] M. A. Horn. Sharp trace regularity for the solutions of the equations of dynamic elasticity. J. Math. Sys. Estim. Cont., 8(2):217–219, 1998. [7] M. A. Horn. Nonlinear boundary stabilization of a system of anisotropic elasticity with light internal damping. Contemporary Mathematics, 268:177–189, 2000. [8] J. E. Lagnese. Boundary stabilization of linear elastodynamic systems. SIAM J. Control Optim., 21(6):968–984, 1983. [9] J. E. Lagnese. Uniform asymptotic energy estimates for solutions of the equations of dynamic plane elasticity with nonlinear dissipation at the boundary. Nonlinear Analysis, 16(1):35–54, 1991. [10] I. Lasiecka. Uniform stabilizability for a full von k´ arm´ an system with nonlinear boundary dissipation. SIAM J. Cont. Optim., 36(4):1376–1422, 1998. [11] I. Lasiecka and D. Tataru. Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Diﬀ. Int. Equations, 6:507–533, 1993. [12] Z. Liu and S. Zheng. Semigroups Associated with Dissipative Systems. Chapman & Hall, Boca Raton, Florida, 1999. [13] A. S. Shamaev, O. A. Oleinik, and G. A. Yosiﬁan. Mathematical Problems in Elasticity and Homogenization. North-Holland, Amsterdam, 1992. [14] A. Pazy. Semigroups of Linear Operators and Applications to Partial Diﬀerential Equations. Springer-Verlag, New York, 1993. [15] J. E. M. Rivera and M. L. Olivera. Stability in inhomogeneous and anisotropic thermoelasticity. Boll. U. M. I., 11-A(7):115–127, 1997.

Copyright © 2005 Marcel Dekker, Inc.

CHAPTER XIX SEMIGROUP WELL-POSEDNESS OF A MULTILAYER MEAD-MARKUS PLATE WITH SHEAR DAMPING Scott W. Hansen∗ Department of Mathematics Iowa State University Ames, IA, USA

Abstract

A multilayer plate model consisting of m + 1 relatively stiﬀ plate layers bonded together by m compliant plate layers is described. In the case of only three layers, the model reduces to a Mead-Markus sandwich plate with viscous damping in the central layer. Existence and uniqueness of solutions are established using semigroup theory. In particular, the homogeneous problem may be written in the form x (t) = Ax(t) where A is the generator of a C0 contraction semigroup.

Keywords: Multilayer plate, Mead-Markus plate, sandwich plate, shear damping

Introduction The classical sandwich plate is a three-layer plate model consisting of two relatively stiﬀ outer layers and a more compliant inner layer. Various sandwich plate and constrained layer models have been proposed and analyzed, see e.g., DiTaranto [2], Mead and Markus [9], Rao and Nakra [10]. For a review and some comparisons of these models see e.g., Sun and Lu [11]or Mead [8]. In Hansen [3], a multilayer generalization of the Mead-Markus model (and Rao-Nakra model) was derived. The multilayer models consist of alternating “stiﬀ” and “compliant” layers. The stiﬀ layers do not allow shear and are modeled as Kirchhoﬀ plates, while the compliant layers allow shear and may be modeled in various ways. For the multilayer Mead-Markus model, the in-plane inertia is ignored in each layer and bending stiﬀness is ignored in the compliant layers. In the undamped ∗ This

research was supported in part by the National Science Foundation under grant DMS0205148.

Copyright © 2005 Marcel Dekker, Inc.

244

CONTROL AND BOUNDARY ANALYSIS

case, existence and uniqueness of solutions for the multilayer systems in [3] was established on the natural energy spaces using the standard variational theory (e.g., Lions and Dautray [1]). However, in the case when shear damping is included in the multilayer Mead-Markus model, the variational approach in ([3]) could not be applied (at least directly) due to a lack of coerciveness in the bilinear form associated with the highest order time-derivatives. We overcome this problem in this article for the special case in which all of the stiﬀ layers have the same Poisson’s ratio. In this case it is possible to solve for the shear in each compliant layer and write the homogeneous problem in the standard form y = Ay. We then show that A is the generator of a strongly continuous semigroup of contractions on the natural energy space. Analogous models to the one of this article have been investigated in the case of a layered beam. For example, an analogous three-layer beam model was investigated in [4], where it was found (for either clamped or simply supported boundary conditions) that the associated semigroup is in fact exponentially stable (moreover is analytic, if rotational moment of inertia is neglected). Similar results were found in the multilayer beam model in [5] for the case of simply supported boundary conditions.

1.

Multilayer Mead-Markus Model

We ﬁrst describe the “multilayer Mead-Markus model” of [3]. (For a more careful derivation see [3].) The multilayer sandwich plate is assumed to consist of n = 2m + 1 layers of alternating “stiﬀ” and “compliant” plates that occupy the region Ω × (0, h) at equilibrium, where Ω is a smooth bounded domain in the plane. The layers are indexed from 1 to n, from bottom to top, with odd indices for stiﬀ layers and even indices for compliant layers. We use coordinates x = {x1 , x2 } to denote points in Ω and use x3 as the transverse independent variable. As is typical in plate theories, it is assumed that the transverse displacement is independent of x3 , i.e., we may use the scalar w(x) to denote the transverse displacement at the point x ∈ Ω. We let v i = {v1i , v2i } i = 1, 2, . . . n denote the in-plane displacements along the midplane of the ith layer. It is assumed that all the layers are bonded to one another so that no slip occurs. The Kirchhoﬀ hypothesis applies to the stiﬀ layers (i.e., no shear), while the compliant layers allow shear and deform linearly with respect to the transverse variable. Under these assumptions any displacement is completely determined by speciﬁcation of the state variables v i , i odd, and w.

Copyright © 2005 Marcel Dekker, Inc.

245

Semigroup Well-Posedness of a Multilayer Mead-Markus Plate

If θ and ξ are matrices in Rlm , by θ : ξ we mean the scalar product in Rlm . We also denote (θ, ξ)Ω = θ : ξ dx, (θ, ξ)Γ = θ : ξ dΓ. Ω

Γ

Deﬁne the form for functions θ(x) = {θ1 (x), θ2 (x)} by ( ( ( ' ' ' ∂θ1 ∂ θˆ1 ∂θ2 ∂ θˆ2 ∂θ2 ∂ θˆ1 ˆ = , + , + ν , (θ; θ) ∂x1 ∂x1 Ω ∂x2 ∂x1 Ω 2 ∂x2 Ω ( ∂x' ( ' ' 1−ν ' ∂θ1 ∂θ1 ∂ θˆ2 ∂θ2 ∂ θˆ1 + ν ∂x1 , ∂x2 + + 2 ∂x2 ∂x1 , ∂x2 + Ω

∂ θˆ2 ∂x1

(( Ω

.

where ν is the Poisson’s ratio (0 < ν < 1/2). It is assumed that the in-plane kinetic energy is negligilble and bending potential energy of the even layers are negligible in comparison to those of the surrounding odd layers. The energy is the sum of the kinetic and potential energies of each layer Ei = Ki + Pi , where 1

2 2 i odd 2 Ω ρi hi (w ) + αi |∇w | dx Ki = 1 2 ρi hi (w ) dx i even 21 Ω 3 i i i odd 2 Ω K(hi Di ∇w; ∇w) + 12(hi Di v ; v ) dx Pi = 1 2 G h |ϕ | dx i even 2 Ω i i i In the above primes (e.g., w ) denote diﬀerentiation with respect to time, Di = Ei /12(1 − ν 2 ), αi = ρi h3i /12 where Ei > 0 denotes the in-plane Young’s modulus, hi the thickness, ρi the volume density, all for the ith layer. In addition, Gi is the transverse shear modulus. The variable ϕi is the shear of the ith layer deﬁned in (1) below. Deﬁne the following n by n matrices: h = diag (h1 , h2 , . . . , hn ) p = diag (ρ1 , ρ2 , . . . , ρn )

D = diag (D1 , D2 , . . . , Dn ) G = diag (G1 , G2 , . . . , Gn ).

In addition, we let e.g., hO and hE denote the diagonal matrices of oddindexed and even-indexed thicknesses hi , respectively. Actually, we will only need to refer to hO , hE , DO , GE , pO . Also deﬁne 1E and 1O as the column vectors of m and m + 1 ones, respectively. Let vO denote the (m+1)×2 matrix with rows v i , i = 1, 3, 5, . . . 2m+1. Likewise, let ϕO and ϕE denote the matrix with rows ϕi , i odd and even, respectively. Since no shear occurs in the odd layers, ϕO = 0. The shear ϕE can be expressed in terms of vO , ∇w as follows: hE ϕE

= BvO + hE N ∇w;

N = h−1 E AhO 1O + 1E

(1)

where A = (aij ), B = (bij ) are the m × (m + 1) matrices deﬁned by 1/2 if j = i or i + 1 (−1)i+j+1 if j = i or i + 1 aij = bij = 0 otherwise 0 otherwise. Copyright © 2005 Marcel Dekker, Inc.

246

CONTROL AND BOUNDARY ANALYSIS

Also deﬁne O (vO , vˆO ) =

n

(v i ; vˆi ).

i odd

Collecting the energies one ﬁnds that the total potential and kinetic energy may be expressed as , w ; vO , w )/2 K(t) = c(vO

P(t) = a(vO , w; vO , w)/2

where c(· ; ·) and a(· ; ·) denote the bilinear forms ˆ = (mw, w) ˆ Ω + (α∇w, ∇w) ˆ Ω c(vO , w; vˆO , w) a(vO , w; vˆO , w) ˆ = O (K1O ∇w; 1O ∇w) ˆ + 12O (hO DO vO ; vˆO ) +(GE hE ϕE , ϕˆE )Ω where ϕ and ϕˆ satisfy (1), and m=

n

hi ρi ,

i=1

n 1 α= ρi h3i , 12 i odd

K=

Di h3i .

i odd

Let us assume the plate is clamped on a portion Γ0 of the boundary Γ and subject to applied forces on the complementary portion Γ1 and distributed forces on Ω. The equations of motion are determined by Hamilton’s principle from the energy. The variational diﬀerential equation one obtains is the following: ˆ Ω + (α∇w , ∇w) ˆ Ω + (K∇O w, ∇w) ˆ (mw , w) −1 vO + N ∇w) ˆ +12O (hO DO vO ; vˆO ) + (GE hE ϕE , hE Bˆ

ˆ = Ω wf ˆ 3 + vˆO fO dx + Γ1 wg ˆ 3 + vˆO gO − w ˆn Mn ds. = W ({ˆ vO , w}) In the above, f3 is the (scalar) transverse applied force in Ω and fO is the net in-plane force acting on the odd layers in Ω. (fO has rows f i = {f1i , f2i }, i = 1, 3, 5, . . . 2m + 1.) The boundary forces acting on Γ1 are: the transverse force g3 (scalar valued), the in-plane force (dimensions matching fO ), and the bending moment Mn (scalar valued). (For precise description of forces see [3].) The test functions w, ˆ vˆO satisfy the clamped boundary conditions on Γ0 and are suﬃciently regular.

Inclusion of Shear Damping. Damping may be introduced into any of the plate layers by replacing the standard stress-strain relation for transversely isotropic materials by an appropriate dissipative constitutive law. In the case of strain-rate shear damping, the stress-strain relation for transverse shear: σ13 = 2G13 is replaced by a dissipative Copyright © 2005 Marcel Dekker, Inc.

Semigroup Well-Posedness of a Multilayer Mead-Markus Plate

247

˜ d )13 . The equations stress-strain relation of the form: σ13 = 2(G + G dt ˜ ∂ ). of motion are then modiﬁed by the correspondence G → (G + G ∂t ˜E In our situation, shear damping occurs only in the even layers. Let G ˜ be the diagonal matrix with diagonal elements Gi , i even. We obtain the following variational diﬀerential equation when shear damping is included in the even layers: ˆ Ω + (α∇w , ∇w) ˆ Ω + (K∇w, ∇w) ˆ + 12O (hO DO vO ; vˆO ) (mw , w) −1 ˜ +(GE hE ϕE + GE hE ϕE , hE Bˆ vO + N ∇w) ˆ = W ({ˆ vO , w}). ˆ (2)

1.1

Boundary Value Problem

We ﬁrst need to deﬁne some operators and boundary operators. Deﬁne the operator L by Lφ = L{φ1 , φ2 } = {L1 (φ), L2 (φ)} where L1 φ = L2 φ =

∂φ1 ∂ ∂x1 [( ∂x1 ∂φ2 ∂ ∂x2 [( ∂x2

2 + ν ∂φ ∂x2 )] + ∂φ1 + ν ∂x1 )] +

∂ 1−ν ∂φ1 ∂x2 [( 2 )( ∂x2 ∂ 1−ν ∂φ2 ∂x1 [( 2 )( ∂x1

+ +

∂φ2 ∂x1 )] ∂φ1 ∂x2 )].

The associated boundary operator Bφ = {B1 (φ1 , φ2 ), (B2 (φ1 , φ2 )} is deﬁned by ( ' ' ∂φ1 ∂φ2 ( ∂φ2 1−ν 1 + n + ν n B1 (φ1 , φ2 ) = ∂φ ∂x1 1 ∂x2 1 2 ∂x2 + ∂x1 n2 ' ' ( ∂φ2 ∂φ1 ( ∂φ1 1−ν 2 n + ν n B2 (φ1 , φ2 ) = ∂φ + 2 2 ∂x2 ∂x1 2 ∂x1 + ∂x2 n1 . where n = (n1 , n2 ) denotes the outward unit normal to Γ. (For later reference: also deﬁne τ = (−n2 , n1 ) as the unit tangent vector to Γ.) ˆ φ: The following Green’s formula is valid for all suﬃciently smooth φ, ˆ Ω. ˆ = (Bφ, φ) ˆ Γ − (Lφ, φ) (φ, φ)

(3)

For ξ = (ξji ) (i = 1, 2, . . . , n, j = 1, 2) deﬁne the matrices Lξ and Bξ by (Lξ)ij = (Lj ξ i ),

(Bξ)ij = (Bj ξ i ),

i = 1, 2, . . . , n, j = 1, 2.

Furthermore we deﬁne the operators LO , LE , BO , BE from L and B based upon the convention that O and E subscripts refer to the parts of the operators that act upon odd and even rows respectively. Similar to (3) the following Green’s formula is valid for all suﬃciently ˆ smooth ξ, ξ: ˆ = (BO ξ, ξ) ˆ Γ − (LO ξ, ξ) ˆ Ω. O (ξ, ξ) (4) Copyright © 2005 Marcel Dekker, Inc.

248

CONTROL AND BOUNDARY ANALYSIS

The equations of motion can be found from (2) using the Green’s formulas (3), (4) and further integrations by parts: T hE (GE ϕE + G ˜ E ϕ ) = f3 mw − α∆w + K∆2 w − div N E (5) ˜ ) = fO −12hO DO LO vO + B T (GE ϕE + Gϕ E where ϕE = h−1 E BvO + N ∇w ∂ ¯ αwn − K( ∂τ (B∇w) · τ ) − K(∆w)n T hE (GE ϕE + G ˜ E ϕE ) · n = g3 +N (6) K(B∇w) · n = −Mn 12DO BO hO vO = gO w = wn = 0 (7) vO = 0 where (5) holds on Q = Ω × (0, ∞), (6) holds on Σ1 = Γ1 × (0, ∞), and (7) holds on Σ0 = Γ0 × (0, ∞). Appropriate initial conditions are of the form w(0) = w0 ,

2.

w (0) = w1 ,

ϕE (0) = ϕ0E

(8)

State Variable Formulation

There is a diﬃculty in proving existence and uniqueness in the formulation above since it is not an easy task to solve for ϕE in (5), which is needed in order to solve for the generator of a semigroup. However, this can be accomplished in the present situation (when all the Poisson ratios in the stiﬀ layers are the same) due to the decomposition described in this section. Let W = span 1O , and V = span DO hO1O . Clearly these spaces are not orthogonal. Therefore there is a unique decomposition of Rm+1 (or Cm+1 ) into a part in W ⊥ and a part in V . Since the kernel of B is easily seen to be W , any vector in W ⊥ can be written as an element in the image of B T . Thus at each point in Ω and Γ1 , respectively, we have the decomposition fO = B T fE + f˜O ,

gO = B T gE + g˜O

where f˜O and g˜O at each point belong to V × V and fE and gE at each point belong to Rm . We return to (5)–(7) but focus on the terms coupled to vO . The relevant part of the system is ˜ E ϕ ) = f˜O + B T fE in Ω (9) −12hO DO LO vO + B T (GE ϕE + G E 12DO hO BO vO = g˜O + B T gE on Γ1 (10) Copyright © 2005 Marcel Dekker, Inc.

249

Semigroup Well-Posedness of a Multilayer Mead-Markus Plate

v = 0 on Γ0 .

(11)

We multiply (9) and (10) on the left by 1TO and obtain Di hi v i ) = −12L( f˜Oi in Ω i odd

12B(

(12)

i odd i

Di hi v ) =

i odd

i g˜O

in Γ1 .

(13)

i odd

If we deﬁne T vO = v¯O = C

Di hi v i ,

f¯ =

i odd

f˜Oi ,

g¯ =

i odd

i g˜O

(14)

i odd

then (12)–(13) can be written as −12L¯ vO = f¯ 12B¯ vO = g¯ v¯O = 0

in Ω on Γ1 on Γ0 .

(15)

When the Poisson’s ratios are constant we can write ∇w). BLO vO = LE BvO = LE hE (ϕE − N

(16)

Deﬁne the matrix P by P = We multiply (9) and (10) by

1 −1 T BD−1 O hO B . 12 −1 −1 1 12 BhO DO

−BLO vO + P GE ϕE

(17)

and obtain

= P fE

in Ω.

(18)

Thus, after substitution of (16) into the above, we obtain the system ˜ E ϕ ) = P fE in Ω (19) ∇w) + P (GE ϕE + G −LE hE (ϕE − N E ∇w) = P gE on Γ1 (20) BE hE (ϕE − N vO = 0 on ΓO . (21)

Lemma 1 The matrix P in ( 17) is positive, i.e., z T P z > 0 for all z ∈ Rm . Moreover, every element pij of P −1 is positive. Proof: We can write P = BΛB T where Λ = diag (λ1 , λ2 , . . . , λm+1 ), with λi > 0, all i. Moreover B can be written as the sum of partitioned . . matrices B = (0 .. Im ) − (Im .. 0), where Im is the identity on Rm , and 0 is Copyright © 2005 Marcel Dekker, Inc.

250

CONTROL AND BOUNDARY ANALYSIS

a column vector of zeros. Performing the block multiplications in (17), one obtains P = Λ11 + Λm+1,m+1 − Λ1,m+1 − Λm+1,1 where Λij is the minor for the ij-th spot in Λ. Thus the diagonal of P has the elements pii = λi + λi+1 , for i = 1, 2, . . . m. The superdiagonal and subdiagonal sequence (beginning in the upper left) are each −λ2 , −λ3 , . . . , −λm . All other elements are zero. We therefore have p11 = λ1 + λ2 >

pmm = λm + λm+1 >

m

|p1,k | = λ2

k=2 m−1

|pm,k | = λm

k=1

pkk = λk + λk+1 ≥

|p1,j | = λk + λk+1 .

j=k

Thus P is diagonally dominant (with strict inequality in the ﬁrst and last rows). Furthermore it is easily veriﬁed that P is irreducible. It follows from the theory of M -matrices (see e.g., [7],Theorem 3, p. 531) that P −1 is a nonnegative matrix (every element nonnegative). This completes the proof. ˜ E ϕ in (19)–(21) Since P is invertible, one can solve for GE ϕE + G E and substitute into (5)–(7) to obtain the following system: mw − α∆w + K∆2 w T −1 T in Q (22) −div N {hE P LE BvO } = f3 − div N hE fE ˜ E ϕ E + GE ϕE − P −1 LE BvO = fE G ∂ α(wn ) − ∂τ (τ · KB∇w) + K(∆w)n T hE (GE ϕE + G ˜ E ϕ˙ E ) · n = g3 +N on Σ1(23) −KB∇w · n = Mn P −1 BE BvO = gE ϕE = 0, w = 0, wn = 0 on Σ0(24) Where ∇w), BvO = hE (ϕE − N

T vO = v¯O , C

(25)

where v¯O is the solution to the stationary Lam´e system (15). Initial conditions are speciﬁed for w, w and ϕE . Suppose for the moment that the system (22)-(24) above together with appropriate initial conditions uniquely determines ϕE and v¯O . Then vO Copyright © 2005 Marcel Dekker, Inc.

251

Semigroup Well-Posedness of a Multilayer Mead-Markus Plate

is also determined: Indeed, deﬁne BC to be the partitioned (square) . is deﬁned in (14)). Then (since BC matrix deﬁned by BCT = (B T .. C), (C is invertible from our earlier discussion) vO can be obtained from the following:

∇w hE ϕE − hE N . (26) BC vO = v¯O

3.

Semigroup Formulation of Homogeneous Problem

We consider the homogeneous problem T {hE P −1 LE BvO } = 0 in Q mw − α∆w + K∆2 w − div N ˜ E ϕ + GE ϕE − P −1 LE BvO = 0 in Q G E

(27)

For simplicity, we restrict our interest here to the case of case of hinged boundary conditions: B∇w · n = 0 on Σ (28) BE BvO = 0 w=0 on Σ. (29) Here Σ = ∂Ω × (0, ∞). T vO we assume ﬁxed boundary conditions, so For the variable v¯O = C that v¯O is the solution of −12L¯ vO = 0 v¯O = 0

in Ω on Γ0 .

(30)

The system (30) has the unique solution v¯O = 0 since the form is known to be coercive on H01 (Ω) ([6]). Therefore we eliminate vO and T vO = 0. henceforth may assume C Deﬁne the spaces L2O (Ω) 1 HO,Γ 2 LE (Ω) HE1

= = = =

{vO = (vji ), i = 1, 3, 5, . . . n, j = 1, 2 : vji ∈ L2 (Ω)} {vO ∈ L2O (Ω) : vji ∈ H01 (Ω)} {φE = (φij ), i = 2, 4, . . . 2m, j = 1, 2 : φij ∈ L2 (Ω)} {φE ∈ L2E (Ω) : φij ∈ H 1 (Ω)}.

T vO = 0} where The energy space is {w, w , vO ∈ V : C 1 . V = H 2 (Ω) ∩ H01 (Ω) × H01 (Ω) × HO,Γ

Let φ = Ry denote the solution to the following elliptic problem mφ − α∆φ = y, Copyright © 2005 Marcel Dekker, Inc.

φ = 0 on Γ.

(31)

252

CONTROL AND BOUNDARY ANALYSIS

Then R maps H −1 (Ω) into H01 (Ω) continuously. Let y T = (y1 , y2 , y3 ) = (w, w , ϕE ) then y2 T {hE P −1 LE hE (y3 − N ∇y1 )} . y = Ay = R(−K∆2 y1 + div N −1 −1 −1 ˜ GE y3 + G ˜ P LE hE (y3 − N ∇y1 ) −G E E (32) Deﬁne matrix S by a (33) BC y = · · · ⇔ y = Sa 0 It is simple to verify that BSy = y

∀ y ∈ Rm

SBu = u

∀u ∈ R

m+1

(34) Tu = 0 :C

(35)

∇y1 ). Along with the state variables y, deﬁne vy = ShE (y3 − N The energy inner product is deﬁned by (y, z)E

= (my2 , z2 )Ω + α(∇y2 , ∇z2 )Ω + KO (10 ∇y1 , 10 ∇z1 ) +(GE hE y3 , z3 )Ω + 120 (h0 DO vy , vz ).

Deﬁne H by H = {w, w , ϕE : w ∈ H 2 (Ω) ∩ H01 (Ω), w ∈ H01 (Ω), ϕE ∈ HE1 }. (36) Then one can verify that A : D(A) → H where D(A) = {y ∈ H : y1 ∈ H 3 (Ω), y2 ∈ H 2 (Ω), y3 ∈ HE3 , + BC’s}

(37)

where “+BC’s” means the boundary conditions (B∇y1 ) · n = 0 and BE (Bvy ) = 0 are imposed on Γ.

Theorem 2 The operator A in ( 32) is the generator of a strongly continuous semigroup of contractions on H. Moreover for any y ∈ D(A), ˜ −1 hE U, U )Ω ; Re (y, Ay)E = −(G E

U = P −1 BLO vy − GE y3 .

(38)

Proof: To show that A is the generator of a contraction semigroup, by the Lumer-Phillips theorem it is enough to demonstrate that A is dissipative and satisﬁes the range condition: i.e., (I − A)−1 maps D(A) onto H. Copyright © 2005 Marcel Dekker, Inc.

Semigroup Well-Posedness of a Multilayer Mead-Markus Plate

253

First we show dissipativity. Let z = Ay, then (y, Ay)E = (my2 , z2 )Ω + α(∇y2 , ∇z2 )Ω + K(∇y1 , ∇z1 ) +(GE hE y3 , z3 )Ω + 120 (h0 DO vy , vz ) T {hE P −1 LE hE (y3 − N ∇y1 )}])Ω = (my2 , R[−K∆2 y1 + div N 2 T −1 ∇y1 )}])Ω +α(∇y2 , ∇R[−K∆ y1 + div N {hE P LE hE (y3 − N +K(∇y1 , ∇y2 ) ˜ −1 GE y3 + G ˜ −1 P −1 LE hE (y3 − N ∇y1 ))Ω +(GE hE y3 , −G E E −1 ˜ GE y3 +12O (hO DO vy , ShE (−G E ˜ −1 P −1 LE hE (y3 − N ∇y1 ) − N ∇y2 )) +G E T hE P −1 LE Bvy )Ω = {(y2 , −K∆2 y1 )Ω + K(∇y1 , ∇y2 )} + {(y2 , div N ∇y2 )} + {terms with G ˜ −1 } +12O (hO DO vy , −She N E =: {T1 } + {T2 } + {T3 }. The real part of the ﬁrst bracketed term T1 is easily shown to vanish using integrations by parts. For the second term T2 we integrate by parts, apply (16), and use the boundary conditions to obtain T hE P −1 LE Bvy )Ω + 12(hO DO BO vy , −ShE N ∇y2 )Γ T2 = (y2 , div N T hE P −1 BLO vy )Ω ∇y2 )Ω = (y2 , div N +12(hO DO LO vy , ShE N T hE S T LO hO DO vy , y2 )Ω −12(div N y2 )Γ +12(S T LO hO DO vy · n, hE N T hE P −1 BLO vy )Ω = (y2 , div N T hE P −1 BD−1 h−1 B T S T hO DO LO vy , y2 )Ω −(div N O

O

One has that −1 T T −1 −1 T T (B − BD−1 O hO B S hO DO ) = BDO hO (I − B S )hO DO = 0

where we have applied the transposition of formula (35) to obtain the last equality. Thus the real part of T2 vanishes. Hence T3

˜ −1 GE y3 )Ω + (GE hE y3 , G ˜ −1 P −1 LE Bvy )Ω = (GE hE y3 , −G E E ˜ −1 GE y3 + G ˜ −1 P −1 LE Bvy )) +12O (hO DO vy , ShE (−G E

E

˜ −1 GE y3 )Ω + (GE hE y3 , G ˜ −1 P −1 BLO vy )Ω = −(GE hE y3 , G E E ˜ −1 S T hO DO LO vy , hE GE y3 )Ω +12(G E

˜ −1 P −1 BLO vy ))Ω . −12(hO DO LO vy , ShE G E One can easily show using (35) and the deﬁnition of P that 12S T hO DO = P −1 B. Copyright © 2005 Marcel Dekker, Inc.

(39)

254

CONTROL AND BOUNDARY ANALYSIS

Applying (39) to the present calculation one ﬁnds that Re (y, Ay)E ˜ −1 hE GE y3 , GE y3 )Ω + Re {(G ˜ −1 hE GE y3 , P −1 BLO vy )Ω = −(G E E ˜ −1 hE P −1 BLO vy , GE y3 )Ω } +(G E

˜ −1 hE P −1 BLO vy , P −1 BLO vy )Ω −(G E −1 ˜ = −(G−1 BLO vy − GE y3 ), (P −1 BLO vy − GE y3 ))Ω . E hE (P ˜ −1 hE is nonnegative, A is dissipative. Since G Next we check the range condition. In what follows we use the notation |w|s = wH s (Ω) ,

|ϕE |E,s =

m

|ϕij |s .

i even j=1,2

Assume that −Ay + y = z where z ∈ H. This is the same as −y2 + y1 = z1 (40) −1

∇y1 )) + y2 = z2 (41) −R(−K∆ y1 + div N hE P LE hE (y3 − N −1 −1 −1 ˜ P LE hE (y3 − N ∇y1 ) + y3 = z3 (42) ˜ GE y3 − G G E E 2

T

Solving for ∆2 y1 and LE y3 in (41) and (42) gives K∆2 y1 − α∆y1 + my1 ˜ E hE (y3 − z3 ) + (m − α∆)(z1 + z2 ). TG = div N

(43)

˜ E y3 GE y3 − P −1 LE hE y3 + G ˜ E z3 . ∇y1 + G = −P −1 LE hE N

(44)

The left-hand side of (43) is associated with a coercive bilinear symmetric form on H 2 (Ω) ∩ H01 (Ω) (see [3]) and hence for functions y ∈ D(A) one can show that |y1 |4 ≤ C|K∆2 y1 − α∆y1 + my1 |0 . Thus, considering the right hand side of (43), |y1 |4 ≤ C1 (|z3 |E,1 + |y3 |E,1 + |z1 |2 + |z2 |2 ).

(45)

Estimation of (43) with R1/2 applied to both sides leads to |y1 |3 ≤ C2 (|z3 |E,1 + |y3 |E,1 + |z1 |1 + |z2 |1 ). Copyright © 2005 Marcel Dekker, Inc.

(46)

Semigroup Well-Posedness of a Multilayer Mead-Markus Plate

255

Using that P −1 is symmetric, positive deﬁnite, we also have that ˜ E y3 |0 ≤ C1 (|z3 |E,0 + |y1 |3 ). (47) |y3 |E,2 ≤ C|GE y3 − P −1 LE hE y3 + G Hence from (46) |y3 |E,2 ≤ C|z3 |E,1 + |y3 |E,1 + |z1 |1 + |z2 |1 )

(48)

From (40) we also have |y2 |2 ≤ C(|y1 |2 + |z1 |2 ).

(49)

Since A has is dissipative, I − A is injective and hence any solutions to (40)–(42) are unique. Applying the usual compactness/uniqueness argument to the estimates (46), (48), (49) yields |y1 |3 + |y2 |2 + |y3 |E,2 ≤ C(|z2 |1 + |z3 |E,1 + |z1 |2 ).

(50)

Thus y ∈ D(A) and I − A is surjective from D(A) to H. This completes the proof.

References [1] R. Dautray and J.-L.. Lions (with collaboration of M. Artola, M. Cessenat, and H. Lanchon). Mathematical Analysis and Numerical Methods for Science and Technology; Evolution Problems I, volume 5. Springer-Verlag, Berlin, 1992. [2] R.A. DiTaranto. Theory of vibratory bending for elastic and viscoelastic layered ﬁnite-length beams. J. Appl. Mech., 32:881–886, 1965. [3] S.W. Hansen. Several related models for multilayer sandwich plates, 2003. Submitted. [4] S.W. Hansen and I. Lasiecka. Analyticity, hyperbolicity and uniform stability of semigroups arising in models of composite beams. Math. Models Methods Appl. Sci., 10:555–580, 2000. [5] S.W. Hansen and Z. Liu. Analyticity of semigroup associated with laminated composite beam, 1999. in: Control of Distributed Parameter and Sochastic Systems; eds. S. Chen, X. Li, J. Yong, X.Y. Zhou, pp 467–54, (Kluwer Academic Publisher, MA). [6] J.E. Lagnese and J.-L. Lions. Modelling, Analysis and Control of Thin Plates (collection: “Recherches en Math´ematiques Appliqu´ees”), RMA 6. Masson, Paris, 1988. [7] P. Lancaster and M. Tismenetsky. The Theory of Matrices. series: Computer Science and Applied Mathematics: Academic Press, San Diego, 1985. [8] D.J. Mead. A comparison of some equations for the ﬂexural vibration of damped sandwich beams. J. Sound Vib., 83:363–377, 1982. [9] D.J. Mead and S. Markus. The forced vibration of a three-layer, damped sandwich beam with arbitrary boundary conditions. J. Sound Vibr., 10:163–175, 1969.

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[10] Y.V.K.S. Rao and B.C. Nakra. Vibrations of unsymmetrical sandwich beams and plates with viscoelastic cores. J. Sound Vibr., 34:309–326, 1974. [11] C.T. Sun and Y.P. Lu. Vibration Damping of Structural Elements. Prentice Hall, NJ, 1995.

Copyright © 2005 Marcel Dekker, Inc.

CHAPTER XX SOLUTION OF ALGEBRAIC RICCATI EQUATIONS ARISING IN CONTROL OF PARTIAL DIFFERENTIAL EQUATIONS Kirsten Morris Department of Applied Mathematics University of Waterloo Waterloo, Canada

Carmeliza Navasca Department of Mathematics University of California Los Angeles, CA, USA

Abstract

Algebraic Riccati equations of large dimension arise when using approximations to design controllers for systems modeled by partial diﬀerential equations. For large model order direct solution methods based on eigenvector calculation fail. In this paper we describe an iterative method that takes advantage of several special features of these problems: (1) sparsity of the matrices, (2) much fewer controls than approximation order, and (3) convergence of the control with increasing model order. The algorithm is straightforward to code. Performance is illustrated with a number of standard examples.

Introduction We consider the problem of calculating feedback controls for systems modeled by partial diﬀerential or delay diﬀerential equations. In these systems the state x(t) lies in an inﬁnite-dimensional space. A classical controller design objective is to ﬁnd a control u(t) so that the objective function ∞

Cx(t), Cx(t) + u∗ (t)Ru(t)dt

(1)

0

is minimized where R is a positive deﬁnite matrix and the observation C ∈ L(X, Rp ). The theoretical solution to this problem for many inﬁnite-dimensional systems parallels the theory for ﬁnite-dimensional Copyright © 2005 Marcel Dekker, Inc.

258

CONTROL AND BOUNDARY ANALYSIS

systems [10, 17, 18, e.g.]. In practice, the control is calculated through approximation. This leads to solving an algebraic Riccati equation A∗ P + P A − P BR−1 B ∗ P = −C ∗ C

(2)

for a feedback operator K = −R−1 B P.

(3)

The matrices A, B, C arise in a ﬁnite-dimensional approximation of the inﬁnite-dimensional system. Let n indicate the order of the approximation, m the number of control inputs, and p the number of observations. Thus, A is n × n, B is n × m, and C is p × n. There have been many papers written describing conditions under which approximations lead to approximating controls that converge to the control for the original inﬁnite-dimensional system [3, 11, 14, 17, 18, e.g.]. In this paper we will assume that an approximation has been chosen so that a solution to the Riccati equation (2) exists for suﬃciently large n and also that the approximating feedback operators converge. For problems where the model order is small, n < 50, a direct method based on calculating the eigenvectors of the associated Hamiltonian works well [19]. Due the limitations of the calculation of eigenvectors for large non-symmetric matrices, this method is not suitable for problems where n becomes large. Unfortunately, many inﬁnite-dimensional control problems lead to Riccati equations of large order. This is particularly evident in control of systems modeled by partial diﬀerential equations with more than one space dimension. For such problems, iterative methods are more appropriate. There are two methods that may be used: Chandrasekhar and Newton-Kleinman iterations. In Chandrasekhar iterations, the Riccati equation is not itself solved directly [2, 7]. A system of 2 diﬀerential equations ˙ K(t) = −B ∗ L∗ (t)L(t), ˙ L(t) = L(t)(A − BK(t)),

K(0) = 0, L(0) = C,

is solved for K ∈ Rm×n , L ∈ Rp×n . The feedback operator is obtained as limt→−∞ K(t). The advantage to this approach is that the number of controls m and number of observations p is typically much less than the approximation model order n. This leads to signiﬁcant savings in storage. Furthermore, the matrices arising in approximation are typically sparse and this can be used in implementation of this algorithm. Unfortunately, the convergence of K(t) can be very slow and a very accurate Copyright © 2005 Marcel Dekker, Inc.

Solution of Algebraic Riccati Equations Arising in Control of PDEs

259

algorithm suitable for stiﬀ systems must be used. This can lead to very large computation times. Another approach to solving large Riccati equations is the NewtonKleinman method [16]. The Riccati equation (2) can be rewritten as (A − BK)∗ P + P (A − BK) = −C ∗ C − K ∗ RK.

(4)

We say a matrix Ao is Hurwitz if σ(Ao ) ⊂ C− . If A − BK is Hurwitz, then the above equation is a Lyapunov equation. An initial feedback K0 must be chosen so A − BK0 is Hurwitz. Deﬁne Si = A − BKi , and solve the Lyapunov equation Si∗ Xi + Xi Si = −C ∗ C − Ki∗ RKi

(5)

for Xi and then update the feedback as Ki+1 = −R−1 B ∗ Xi . If A − BK0 is Hurwitz, then Xi converges quadratically to P [16]. For an arbitrary large Riccati equation, this condition may be diﬃcult to satisfy. However, this condition is not restrictive for Riccati equations arising in control of inﬁnite-dimensional systems. First, many of these systems are stable even when uncontrolled and so the initial iterate K0 may be chosen as zero. Second, if the approximation procedure is valid then convergence of the feedback gains is obtained with increasing model order. Thus, a gain obtained from a lower order approximation, perhaps using a direct solution, may be used as an initial estimate, or ansatz, for a higher order approximation. This technique was used successfully in [13, 26] and later in this paper. In this paper we use a modiﬁed Newton-Kleinman iteration ﬁrst proposed by Banks and Ito [2] as a reﬁnement for a partial solution to the Chandrasekhar equation. In that paper, they partially solve the Chandrasekhar equations and then use the resulting feedback K as a stabilizing initial guess for a modiﬁed Newton-Kleinman method. Instead of the standard Newton-Kleinman form (5) above, Banks and Ito rewrote the Riccati equation in the form (A − BKi )∗ Xi + Xi (A − BKi ) = −Di∗ Di

(6)

where Xi = Pi−1 − Pi , Ki+1 = Ki − B T Xi , and Di = Ki − Ki−1 . The resulting Lyapunov equation is solved for Xi . Equation (6) has fewer inhomogeneous terms than the equation in the standard NewtonKleinman method (5). Also, the non-homogeneous term D depends on m inputs, not the observation C. In [2] a Smith’s method was used to solve the Lyapunov equations. Although convergent, this method is slow. Solution of the Lyapunov equation is a key step in implementing either modiﬁed or standard Newton-Kleinman. The Lyapunov equations Copyright © 2005 Marcel Dekker, Inc.

260

CONTROL AND BOUNDARY ANALYSIS

arising in the Newton-Kleinman method have several special features: (1) the model order n is generally much larger than m or p and (2) the matrices are often sparse. We use a recently developed method [20, 24] that uses these features. In the next section we describe this Lyapunov solver. Incorporating this algorithm with the Newton-Kleinman method is described in [6, 25]. We use this Lyapunov solver with both standard and modiﬁed NewtonKleinman to solve a number of standard control examples, including one with several space variables. Our results indicate that modiﬁed NewtonKleinman achieves considerable savings in computation time over standard Newton-Kleinman. We also found that using the solution from a lower-order approximation as an ansatz for a higher-order approximation signiﬁcantly reduced the computation time.

1.

Solution of Lyapunov Equation

Solution of a Lyapunov equation is a key step in each iteration of the Newton-Kleinman method. Thus, it is imperative to use a good Lyapunov algorithm. As for the Riccati equation, direct methods such as Bartels-Stewart [4] are only appropriate for low model order and do not take advantage of sparsity in the matrices. The Alternating Direction Implicit (ADI) and Smith methods are two well-known iterative schemes. These will be brieﬂy described before describing a modiﬁcation that leads to reduced memory requirements and faster computation. Consider the Lyapunov equation XAo + A∗o X = −DD∗

(7)

where Ao ∈ Rn×n and D ∈ Rn×r . In the case of standard NewtonKleinman, r = m + p while for modiﬁed Newton-Kleinman, r is only m. If Ao is Hurwitz, then the Lyapunov equation has a symmetric positive semideﬁnite solution X. For p < 0, deﬁne U = (Ao − pI)(Ao + pI)−1 and V = −2p(A∗o + pI)−1 DD∗ (Ao + pI)−1 . In Smith’s method [28], equation (7) is rewritten. The solution X is found by using successive substitutions: X = limi→∞ Xi where Xi = U ∗ Xi−1 U + V

(8)

with X0 = 0. Convergence of the iterations can be improved by careful choice of the parameter p e.g. [27, pg. 197]. This method of successive substitution is unconditionally convergent, but has only linear convergence. The ADI method [21, 30] improves Smith’s method by using a diﬀerent parameter pi at each step. Two Copyright © 2005 Marcel Dekker, Inc.

Solution of Algebraic Riccati Equations Arising in Control of PDEs

261

alternating linear systems, (A∗o + pi I)Xi− 1 (A∗o

2

+

pi I)Xi∗

= −DD∗ − Xi−1 (Ao − pi I) ∗

= −DD −

∗ Xi− 1 (Ao 2

− pi I)

(9) (10)

are solved recursively starting with X0 = 0 ∈ Rn×n and parameters pi < 0. If all parameters pi = p, then equations (9,10) reduce to Smith’s method. If the ADI parameters pi are chosen appropriately, then convergence is obtained in J iterations where J ! n. Choice of the ADI parameters is discussed below. If Ao is sparse, then the linear systems (9,10) can be solved eﬃciently. However, full calculation of the dense iterates Xi is required at each step. Setting X0 = 0, it can be easily shown that Xi is symmetric and positive semideﬁnite for all i, and so we can write X = ZZ ∗ where Z is a Cholesky factor of X [20, 24]. (A Cholesky factor does not need to be square or be lower triangular.) Substituting Zi Zi∗ for Xi in (9,10) and setting X0 = 0, we obtain the following iterates for < Z1 = −2p1 (A∗o + p1 I)−1 D < Zi = [ −2pi (A∗o + pi I)−1 D, (A∗o + pi I)−1 (A∗o − pi I)Zi−1 ]. (11) Note that Z1 ∈ Rn×r , Z2 ∈ Rn×2r , and Zi ∈ Rn×ir . Recall that for standard Newton-Kleinman, r is the sum of observations p and controls m. For modiﬁed Newton-Kleinman r is equal to m. In practice, the number of controls m (and often the number of observations p) is much less than the model order, n. This form of solution results in considerable savings in computation time and memory. The algorithm is stopped when the Cholesky iterations converge within some tolerance. In [20] these iterates are reformulated in a more eﬃcient form, using the observation that the order in which the ADI parameters are used is irrelevant. This leads to the algorithm shown in Table 1.

1.1

ADI Parameter Selection

As mentioned above, choice of the ADI parameters signiﬁcantly aﬀects the convergence of the ADI method. The parameter selection problem has been studied extensively [9, 21, 29, e.g.]. Optimal ADI parameters are the solution to the min-max problem J F pj − λ . {p1 , p2 , . . . , pJ } = arg min max pi λj ∈σ(Ao ) pj + λ j It is not feasible to solve this problem. First, solution of the Lyapunov equation arises as a step in the iterative solution of the Riccati equation Copyright © 2005 Marcel Dekker, Inc.

262 Table 1.

CONTROL AND BOUNDARY ANALYSIS Cholesky-ADI Method

Given Ao and D Choose ADI√ parameters {p1 , . . . , pJ } with (pi ) < 0 Deﬁne z1 = −2p1 (A∗o + p1 I)−1 D and Z1 = [z1 ] For i = 2, . . . , J √ −2pi+1

Deﬁne Wi = ( √−2p )[I − (pi+1 − pi )(A∗o + pi+1 I)−1 ] i (1) zi = Wi zi−1 (2) If z > tol Zi = [Zi−1 zi ] Else, stop.

and Ao = A−BKi where Ki is the feedback calculated at the it h iterate. Thus, the matrix Ao and its spectrum change at each iterate. Second, when Ao is large, solving this eigenvalue problem is computationally diﬃcult. The optimal ADI parameters are approximated in several respects. First, the spectrum of the original matrix A − BK0 is used and the resulting parameters used for each subsequent Lyapunov solution. In most applications, A is Hurwitz and so we can use the spectrum of the original matrix A. If these eigenvalues are real and contained in the interval [−b, −a] then the solution to (11) is known in closed form [21]. For more general problems, the selection procedure in [9] yields parameters that are approximately optimal. Let λi indicate the eigenvalues of A − BK0 . Deﬁne a = min("λi ),

(12)

b = max("λi ),

(13)

i

i

α = tan−1 max | i

λi |. "λi

(14)

These parameters determine an elliptic domain Ω that contains the spectrum and is used to determine the ADI parameters pi . The closeness of Ω to the smallest domain containing the spectrum aﬀects the number of iterations required for convergence of the Cholesky-ADI. The parameter α is the maximum angle between the eigenvalues and the real axis. When the spectrum contains lightly damped complex eigenvalues, α is close to π/2. In this case, Ω is a poor estimate of this domain. This point is investigated in the third example below. Copyright © 2005 Marcel Dekker, Inc.

Solution of Algebraic Riccati Equations Arising in Control of PDEs

2.

263

Benchmark Examples

In this section we test the algorithm with a number of standard examples: a one-dimensional heat equation, a two-dimensional partial diﬀerential equation, and a beam equation. All computations were done within MATLAB on a computer with two 1.2 GHz AMD processors. (Shorter computation time would be obtained by running optimized code outside of a package such as MATLAB. The CPU times are given only for comparison purposes.) The relative error for the Cholesky iterates was set to 10−8 .

2.1

Heat Equation

Consider the linear quadratic regulator problem of minimizing a cost functional [2, 8] ∞ (|Cz(t)|2 + |u(t)|2 )dt J(u) = 0

subject to ∂z(t, x) ∂ 2 z(t, x) , x ∈ (0, 1), = ∂t ∂x2 z(0, x) = ψ(x)

(15)

with boundary conditions ∂z(t, 0) ∂x ∂z(t, 1) ∂x

= u(t) = 0.

(16)

z(t, x)dx,

(17)

Setting Cz(t) =

1

0

and R = 1, the solution to the inﬁnite-dimensional Riccati equation is 1 Kz = k(x)z(x)dx 0

where k = 1 [3]. Thus, for this problem we have an exact solution to which we can compare the approximations. The equations (14–17) are discretized using the standard Galerkin approximation with linear spline ﬁnite element basis on a uniform partition of [0, 1]. The resulting A matrix is symmetric and tridiagonal while Copyright © 2005 Marcel Dekker, Inc.

264 Table 2.

CONTROL AND BOUNDARY ANALYSIS Heat Equation: Feedback Gain at Each Newton-Kleinman Iteration Newton-Kleinman Iteration 1 2 3 4 5 6 7 8 9 10 11

Optimal Feedback Gain 50.005 25.0125 12.5262 6.303 3.2308 1.7702 1.1675 1.012 1.0001 1 1

B is a column vector with only one non-zero entry. Denote each basis element by li , i = 1..n. For an approximation with n elements, the approximating optimal feedback operator K is

1

kn (x)z(x)dx,

Kz =

(18)

0

where kn (x) = ni=1 ki li (x). The solutions to the approximating Riccati equations converge [3, 14] and so do the feedback operators. Table 2 shows the approximated optimal feedback gain at each Newton-Kleinman iteration. The data in Table 2 is identical for n = 25, 50, 100, 200 and for both Newton-Kleinman methods. The error in K versus Newton-Kleinman iteration in shown in Figure 1 for standard Newton-Kleinman and in Figure 2 for the modiﬁed algorithm. In Tables 3 and 4 we compare the number of Newton-Kleinman and Lyapunov iterations as well as the CPU time per order n. We use the ansatz k0 (x) = 100 for all n. With the modiﬁed algorithm, there are 1 to 2 fewer Riccati loops than with the original Newton-Kleinman iteration. Also, the modiﬁed Newton-Kleinman method requires fewer Lyapunov iterations within the last few Newton-Kleinman loops. The computation time with the modiﬁed Newton-Kleinman algorithm is signiﬁcantly less than that of the original algorithm. Copyright © 2005 Marcel Dekker, Inc.

Solution of Algebraic Riccati Equations Arising in Control of PDEs

Table 3.

Table 4.

Heat Equation: Standard Newton-Kleinman Iterations n 25

Newton-Kleinman Itn’s 11

50

11

100

11

200

11

Lyapunov Itn’s 19,22,23,26,27,29 30,31,31,31,31 24,26,28,30,32,34 35,35,35,35,35 28,31,32,35,36,38 39,40,40,40,40 33,35,37,39,41,43 44,44,44,44,44

CPU time 0.83 1.2 3.49 23.1

Heat Equation: Modiﬁed Newton-Kleinman Iterations n 25

Newton-Kleinman Itn’s 10

50

10

100

9

200

9

Copyright © 2005 Marcel Dekker, Inc.

Lyapunov Itn’s 19,22,23,26,27,29 30,31,29,1 24,26,28,30,32,34 35,35,33,1 28,31,32,35,36,38 39,40,1 33,35,37,39,41,43 44,44,1

CPU time 0.66 0.94 2.32 14.2

265

266

2.2

CONTROL AND BOUNDARY ANALYSIS

Two-Dimensional Example

Deﬁne the rectangle Ω = [0, 1] × [0, 1] with boundary ∂Ω. Consider the two-dimensional partial diﬀerential equation [8] ∂z ∂t

2

∂ z = ∂x 2 + z(x, y, t) = 0,

∂2z ∂y 2

∂z + 20 ∂y + 100z = f (x, y)u(t),

(x, y) ∈ Ω (x, y) ∈ ∂Ω (19)

where z is a function of x, y and t. Let f (x, y) =

100, if .1 < x < .3 and .4 < y < .6 . 0, else

Central diﬀerence approximations are used to discretize (19) on a grid of N × M points. The resulting approximation has dimension n = N × M : A ∈ Rn×n and B ∈ Rn×1 . The A matrix is sparse with at most 5 nonzero entries in any row. The B matrix is a sparse column vector. We chose C = B ∗ and R = 1. We solved the Riccati equation on a number of grids, using both standard and modiﬁed Newton-Kleinman methods. The data is shown in Tables 5 and 6. Modiﬁed Newton-Kleinman is clearly much more eﬃcient. Fewer Lyapunov iterations are required for convergence and this leads to a reduction in computation time of nearly 50%, see Figure 3. We also investigated the use of non-zero initial estimates for K in reducing computation time. We ﬁrst solve the Riccati equation on a 12×12 grid. Since σ(A) ⊂ C− , K0144 = 0 is a possible ansatz. It required 13 Newton-Kleinman iterations and a total of 419 Lyapunov iterations to obtain a relative error in K of 10−11 . Linear interpolation was used to project this solution to a function on a ﬁner grid, 23×12, where n = 276. 12×12 . On the ﬁner grid 23 × 12 where Indicate this projection by Kproj 12×12 n = 276, we used both zero and Kproj as initial estimates. As indicated in Table 6, the error of K was 10−12 after only 150 Lyapunov and 54 Newton-Kleinman iterations. The same procedure is applied to generate a guess K0529 where the mesh is 23 × 23 and n = 529. Neglecting the computation time to perform the projection, use of a previous solution led to a total computation time over both grids of only 6.3 seconds versus 11.9 seconds for n = 276 and a total computation time of only 27.8 versus 79.5 for n = 529. Similar improvements in computation time were obtained with standard Newton-Kleinman. Copyright © 2005 Marcel Dekker, Inc.

267

Solution of Algebraic Riccati Equations Arising in Control of PDEs

Table 5.

Iterations for 2-d Equation (Newton-Kleinman)

grid

K0

n

12 × 12

0

144

Newton Kleinman Itn’s 14

23 × 12

0

276

15

23 × 12 23 × 23

12x12 Kproj 0

276 529

6 16

23 × 23

23x12 Kproj

529

5

Table 6.

Lyapunov Itn’s

CPU time

12,44,41,39,36,34,32 30,28,27,27,27,27,27 16,47,45,42,40,38,35,33 31,30,30,30,30,30,30 29,30,30,30,30,30 20,51,48,46,44,41,39,37 35,34,33,33,32,32,32,32 33,32,32,32,32

4.98 22.2 10.8 139. 65.7

Iterations for 2-d Equation (Modiﬁed Newton-Kleinman)

grid

K0

n

12 × 12

0

144

NewtonKleinman Itn’s 13

23 × 12

0

276

13

23 × 12 23 × 23

12x12 Kproj 0

276 529

4 14

23 × 23

23x12 Kproj

529

4

Copyright © 2005 Marcel Dekker, Inc.

Lyapunov Itn’s

CPU time

12,44,41,39,36,34,32 30,28,27,27,27,1 16,47,45,42,40,38,35 33,31,30,30,30,29 29,30,30,29 20,50,48,46,44,41,39 37,35,34,33,33,33,31 33,32,32,1

2.94 11.9 3.36 79.5 21.5

268

CONTROL AND BOUNDARY ANALYSIS

3.

Euler-Bernoulli Beam

Consider a Euler-Bernoulli beam clamped at one end (r = 0) and free to vibrate at the other end (r = 1). Let w(r, t) denote the deﬂection of the beam from its rigid body motion at time t and position r. The deﬂection is controlled by applying a a torque u(t) at the clamped end (r = 0). We assume that the hub inertia Ih is much larger than the beam inertia Ib so that Ih θ¨ ≈ u(t). The partial diﬀerential equation model with Kelvin-Voigt and viscous damping is ∂2 EIr ρr wrr (r, t) = u(t), wtt (r, t) + Cv wt (r, t) + 2 Cd Ib wrrt (x, t) + ∂r ρA Ih with boundary conditions w(0, t) = 0 wr (1, t) = 0. EIwrr (1, t) + Cd Ib wrrt (1, t) = 0 ∂ [EI(1)wrr (r, t) + Cd Ib wrrt (r, t)]r=1 = 0. ∂r The values of the physical parameters in Table 7 are as in [1]. Deﬁne H to be the closed linear subspace of the Sobolev space H 2 (0, 1) dw 2 H = w ∈ H (0, 1) : w(0) = (0) = 0 dr and deﬁne the state-space to be X = H × L2 (0, 1) with state z(t) = ∂ (w(·, t), ∂t w(·, t)). A state-space formulation of the above partial diﬀerential equation problem is d x(t) = Ax(t) + Bu(t), dt where A=

0

I

d4 − EI ρ dr 4

4 − Cρd I ddr4

, −

B=

Cv ρ

0

,

r Ih

with domain dom (A) = {(φ, ψ) ∈ X : ψ ∈ H and 2

2

d d 2 M = EI dr 2 φ + Cd I dr 2 ψ ∈ H (0, 1) with M (L) =

Copyright © 2005 Marcel Dekker, Inc.

d dr M (L)

= 0} .

Solution of Algebraic Riccati Equations Arising in Control of PDEs Table 7.

269

Table of Physical Parameters E Ib ρ Cv Cd L Ih d

2.68 × 1010 N/m2 1.64 × 10−9 m4 1.02087 kg/m 1.8039 N s/m 1.99 × 105 N s/m 1m 121.9748 kg m2 .041 kg −1

We use R = 1 and deﬁne C by the tip position: w(1, t) = C[w(x, t) w(x, ˙ t)]. Let H N ⊂ H be a sequence of ﬁnite-dimensional subspaces spanned by the standard cubic B-splines with a uniform partition of [0, 1] into N subintervals. This yields an approximation in H N × H N [15, e.g.] of dimension n = 2N. This approximation method yields a sequence of solutions to the algebraic Riccati equation that converge strongly to the solution to the inﬁnite-dimensonal Riccati equation corresponding to the original partial diﬀerential equation description [22]. The spectrum of A for various n is shown in Figure 4. For small values of n, the spectrum of An only contains complex eigenvalues with fairly constant angle. As n increases, the spectrum curves into the real axis. For large values of n, the spectrum shows behavior like that of the original diﬀerential operator, and contains two branches on the real axis. For these large values of n, the ADI parameters are complex numbers. We calculated the complex ADI parameters as in [9]. Although there are methods to eﬃciently calculate with complex parameters by splitting the calculation into 2 real parts [20] their presence increases computation time. Figure 5 shows the total Lyapunov iterations for various values of n. Although the number of Newton-Kleinman iterations remained at 2 for all n, the number of Lyapunov iterates increases as n → ∞. Figure 6 shows the change in the spectrum of A as Cd is varied. Essentially, increasing Cd increases the angle that the spectrum makes with the imaginary axis. Recall that the spectral bounds α, a, and b deﬁne the elliptic function domain that contains the spectrum of A [21]. The ADI parameters depend entirely on these bounds. The quantity π 2 − α is the angle between the spectrum and the imaginary axis and so α → π/2 as Cd is decreased. Figure 7 shows the eﬀect of varying Cd Copyright © 2005 Marcel Dekker, Inc.

270

CONTROL AND BOUNDARY ANALYSIS

on the number of iterations required for convergence. Larger values of Cd (i.e., smaller values of α) leads to a decreasing number of iterations. Small values of Cd lead to a large number of required iterations in each solution of a Lyapunov equation. There are several possible reasons for this. As the spectrum of A ﬂattens with increasing Cd , the spectral bounds (12–14) give sharper estimates for the elliptic function domain Ω and thus the ADI parameters are closer to optimal. Improvement in calculation of ADI parameters for problems where the spectrum is nearly vertical in the complex plane is an open problem. Another explanation lies in the nature of the mathematical problem being solved. If Cd > 0 the semigroup for the original partial diﬀerential equation is parabolic and the solution to the Riccati equation converges uniformly in operator norm [17, Chap.4]. However, if Cd = 0, the partial diﬀerential equation is hyperbolic and only strong convergence of the solution is obtained [18]. Thus, one might expect a greater number of iterations in the Lyapunov loop to be required as Cd is decreased. Any X ∈ Rn×n to the matrix Lyapunov equation is symmetric and positive semi-deﬁnite and so we can order its eigenvalues λ1 ≥ λ2 ≥ ...λn ≥ 0. ˆ is determined The ability to approximate X by a matrix of lower rank X by the following relation [12, Thm. 2.5.2]: ˜ λk (X) X − X = . ˜ X λ1 (X) rank X≤k−1 min

This ratio is plotted for several values of Cd in Figures 8 and 9. For larger values of Cd the solution X is closer to a low rank matrix than it is for smaller values of Cd . Recall that the CF-ADI algorithm used here starts with a rank 1 initial estimate of the Cholesky factor and the rank of the solution is increased at each step. The fact that the solution X is closer to a low rank matrix for larger values of Cd implies a smaller number of iterations are required for convergence. If the fundamental reason for the slow convergence with small Cd is the “hyperbolic-like” behavior of the problem, then this convergence will not be improved by better ADI parameter selection. This may have consequences for control of coupled acoustic-structure problems where the spectra are closer to those of hyperbolic systems than those of parabolic systems.

Copyright © 2005 Marcel Dekker, Inc.

271

Solution of Algebraic Riccati Equations Arising in Control of PDEs

Beam: Eﬀect of Changing Cd (Standard Newton-Kleinman)

Table 8. Cv 2

Cd 1 × 104 3 × 105 4 × 105 1 × 107 1 × 108 5 × 108

Newton-Kleinman Itn’s – – 3 3 3 3

Lyapunov Itn’s – – 1620;1620;1620 1316;1316;1316 744;744;744 301;301;301

CPU time – – 63.14 42.91 18.01 5.32

Beam: Eﬀect of Changing Cd (Modiﬁed Newton-Kleinman)

Table 9. Cv 2

α 1.5699 1.5661 1.5654 1.5370 1.4852 1.3102

Cd 1 × 104 3 × 105 4 × 105 1 × 107 1 × 108 5 × 108

α 1.5699 1.5661 1.5654 1.5370 1.4852 1.3102

Newton-Kleinman It’s 2 2 2 2 2 2

Lyapunov It’s – – 1620;1 1316;1 744;1 301;1

CPU time – – 24.83 16.79 7.49 2.32

0

10

−1

10

−2

10

−3

10

−4

Error

10

−5

10

−6

10

−7

10

N=25 N=50 N=100 N=200

−8

10

−9

10

Figure 1.

1

2

3

4

5

6 Iterations

7

8

9

10

11

Heat equation: error versus atandard Newton-Kleinman iterations.

Copyright © 2005 Marcel Dekker, Inc.

272

CONTROL AND BOUNDARY ANALYSIS

0

10

−2

10

−4

10

−6

Error

10

−8

10

−10

10

N=25 N=50 N=100 N=200

−12

10

−14

10

1

2

3

4

5

6

7

8

9

10

Iterations

Figure 2.

Heat equation: error versus modiﬁed Newton-Kleinman iterations.

Copyright © 2005 Marcel Dekker, Inc.

Solution of Algebraic Riccati Equations Arising in Control of PDEs

273

0

10

N=144 N=276 N=529 −2

10

−4

10

−6

Error

10

−8

10

−10

10

−12

10

−14

10

0

2

4

6

8

10

12

14

Iterations

Figure 3. Convergence rate for 2-d problem (modiﬁed Newton-Kleinman). Initial 144 276 , K 529 = Kproj . estimates: K 144 = 0, K 276 = Kproj

Copyright © 2005 Marcel Dekker, Inc.

274

CONTROL AND BOUNDARY ANALYSIS

4

4

x 10

2 0 −2 −4 −6 4 x 10 4

−5

−4

−3

−2

−1

0 5

x 10

2 0 −2 −4 −6 4 x 10 4

−5

−4

−3

−2

−1

0 5

x 10

2 0 −2 −4 −6 4 x 10 4

−5

−4

−3

−2

−1

0 5

x 10

2 0 −2 −4 −6

−5

−4

−3

−2

−1

0 5

x 10

Figure 4.

Beam: spectrum of An for n = 48, 56, 64, 88 (Cd = 9 × 105 ).

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4500

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4000

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Figure 5.

Beam: n versus Lyapunov iterations (Cd = 9 × 105 ).

4

x 10 5 0 −5

−5 −4.5 4 x 10

−4

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x 10

Figure 6.

Beam: spectrum of A for Cd = 1x104 , 4x105 , 1x108 (n = 80).

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Lyapunov Iterations

3500 3000 2500 2000 1500 1000 500

0

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6 4 2 0 1.48

Figure 7.

1.5

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1.54

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Beam: eﬀect of Cd on Lyapunov iterations and α (n = 80).

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Eigenvalue Ratio for N=64

0

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5x108 1x108 1x107 −5

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Figure 8.

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Eigenvalue Ratio for N=80

0

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5x108 1x108 1x107 −5

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Figure 9.

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Beam: eﬀect of varying Cd on low rank approximation.

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References [1] H.T. Banks and D. J. Inman, “On Damping Mechanisms in Beams,” ICASE Report No. 89-64, NASA, Langley, 1989. [2] H. T. Banks and K. Ito, A Numerical Algorithm for Optimal Feedback Gains in High Dimensional Linear Quadratic Regulator Problems, SIAM J. Control Optim., vol. 29, no. 3, pp. 499–515, 1991. [3] H. T. Banks and K. Kunisch, The linear regulator problem for parabolic systems, SIAM J. Control Optim., vol. 22, no. 5, pp. 684–698, 1984. [4] R.H. Bartels and W. Stewart, “Solution of the matrix equation AX+XB=C,” Comm. of ACM, vol. 15, pp. 820–826, 1972. [5] D. Bau, and L. Trefethen, Numerical Linear Algebra, SIAM, Philadephia, 1997. [6] P. Benner, “Eﬃcient Algorithms for Large-Scale Quadratic Matrix Equations,” Proc. in Applied Mathematics and Mechanics, vol. 1, no. 1, pp. 492–495, 2002. [7] J.A. Burns and K. P. Hulsing, “Numerical Methods for Approximating Functional Gains in LQR Boundary Control Problems,” Mathematical and Computer Modelling, vol. 33, no. 1, pp. 89–100, 2001. [8] Y. Chahlaoui and P. Van Dooren, A collection of Benchmark examples for model reduction of linear time invariant dynamical systems, SLICOT Working Note 2002-2, http://www.win.tue.nl/niconet/ [9] N. Ellner and E. L. Wachpress, Alternating Direction Implicit Iteration for Systems with Complex Spectra, SIAM J. Numer. Anal., vol. 28, no. 3, pp. 859–870, 1991. [10] J.S. Gibson, “The Riccati Integral Equations for Optimal Control Problems on Hilbert Spaces”, SIAM J. Control and Optim., vol. 17, pp. 637–565, 1979. [11] J.S. Gibson, “Linear-quadratic optimal control of hereditary diﬀerential systems: inﬁnite dimensional Riccati equations and numerical approximations”, SIAM J. Control and Optim., vol. 21, pp. 95–139, 1983. [12] G.H. Golub and C.F. van Loan, Matrix Computations, John Hopkins, 1989. [13] J.R. Grad and K. A. Morris, “Solving the Linear Quadratic Control Problem for Inﬁnite-Dimensional Systems”, Computers and Mathematics with Applications, vol. 32, no. 9, pp. 99–119, 1996. [14] K. Ito, Strong convergence and convergence rates of approximating solutions for algebraic Riccati equations in Hilbert spaces, Distributed Parameter Systems, eds. F. Kappel, K. Kunisch, W. Schappacher, Springer-Verlag, 1987, pp. 151– 166. [15] K. Ito and K. A. Morris, An approximation theory of solutions to operator Riccati equations for H ∞ control, SIAM J. Control Optim., vol. 36, pp. 82–99, 1998. [16] D. Kleinman, On an Iterative Technique for Riccati Equation Computations, IEEE Transactions on Automat. Control, vol. 13, pp. 114–115, 1968. [17] I. Lasiecka and R. Triggiani, Control Theory for Partial Diﬀerential Equations: Continuous and Approximation Theories, Part 1, Cambridge University Press, 2000.

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[18] I. Lasiecka and R. Triggiani, Control Theory for Partial Diﬀerential Equations: Continuous and Approximation Theories, Part 2, Cambridge University Press, 2000. [19] A.J. Laub, “A Schur method for solving algebraic Riccati equations”, IEEE Trans. Auto. Control, vol. 24, pp. 913–921, 1979. [20] J. R. Li and J White, Low rank solution of Lyapunov equations, SIAM J. Matrix Anal. Appl., vol. 24, pp. 260–280, 2002. [21] A. Lu and E. L. Wachpress, Solution of Lyapunov Equations by Alternating Direction Implicit Iteration, Computers Math. Applic., vol. 21, no. 9, pp. 43–58, 1991. [22] K.A. Morris, “Design of Finite-Dimensional Controllers for Inﬁnite-Dimensional Systems by Approximation,” J. Mathematical Systems, Estimation and Control, vol. 4, pp. 1–30, 1994. [23] K.A. Morris, Introduction to Feedback Control, Harcourt-Brace, 2001. [24] T. Penzl, “A Cyclic Low-Rank Smith Method for Large Sparse Lyapunov Equations”, SIAM J. Sci. Comput., vol. 21, no. 4, pp. 1401–1418, 2000. [25] T. Penzl, LYAPACK User’s Guide, http://www.win.tue.nl/niconet/ [26] I. G. Rosen and C. Wang, “A multilevel technique for the approximate solution of operator Lyapunov and algebraic Riccati equations”, SIAM J. Numer. Anal. vol. 32, pp. 514–541, 1994. [27] Russell, D.L., Mathematics of Finite Dimensional Control Systems: Theory and Design, Marcel Dekker, New York, 1979. [28] R.A. Smith, “Matrix Equation XA + BX = C”, SIAM Jour. of Applied Math., vol. 16, pp. 198–201, 1968. [29] E. L. Wachpress, The ADI Model Problem, preprint, 1995. [30] E. Wachpress, “Iterative Solution of the Lyapunov Matrix Equation,” Appl. Math. Lett., vol. 1, pp. 87–90, 1988.

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CHAPTER XXI STABILIZATION IN COMPUTING SADDLE POINTS Jianxin Zhou∗ Texas A&M University College Station, TX, USA

Xudong Yao∗ Texas A&M University College Station, TX, USA

Abstract

Due to unstable nature, saddle points are very elusive to capture numerically. In this paper, two stabilization techniques in numerical computation of saddle points are developed. The ﬁrst is to deﬁne a solution set and then minimize the functional restricted to the solution set. The second is to use symmetry to deﬁne an invariant subspace. Mathematical justiﬁcations of the two techniques are established in a Banach space setting.

Keywords: Saddle points and their orders, local minimax method, invariance

Introduction Many nonlinear boundary value problems can be reduced to solve their Euler-Lagrange equation ∇J(u) = 0 (1) where ∇J is the gradient of a C 1 energy function J on a Banach space B. A solution u∗ ∈ B to (1) is called a critical point of J. u∗ is said to be nondegenerate if J (u∗ ) is invertible. The ﬁrst candidates for a critical point are the local extrema to which the classical critical point theory was devoted in calculus of variation. Most conventional numerical algorithms focus on ﬁnding such stable solutions. Critical points that are not local extrema are unstable and called saddle points. In physical systems, saddle points appear as unstable equilibria or transient excited ∗ This

research was supported in part by NSF grant DMS-0311905.

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states. For example, a common and important problem in computational chemistry and condensed matter physics is the calculation of the rate of transition states. Due to unstable nature, saddle points are very elusive to capture numerically. Most results in critical point theory [2], [1], [3], [4], [5], [10], [11], [12], [14] focus on the existence issue and are not for algorithm implementation. For example, minimax method is the most popular approach in critical point theory, such as the mountain pass, various linking and saddle point theorems. It characterizes a critical point as a solution to a two-level minimax problem min max J(u)

A∈A u∈A

where A is usually compact, max is global on A, min is global among all A’s. It becomes a two-level global optimization problem and is not for algorithm implementation. Newton’s method depends on an initial guess and is not for degenerate cases. However degeneracy exists in every multiple saddle point problem. A local minimax method (LMM) is developed in [7],[8],[15] and successfully applied to numerically solve several semi and quasi linear elliptic boundary value problems for multiple unstable solutions. The computations are quite stable. The purpose of this paper is to provide such mathematical evidences in a Banach space.

1.

The Local Minimax Method

Let L and L be closed subspaces of B such that B = L ⊕ L . Let P : B → L be the corresponding linear projection operator. Let SL be the unit sphere in L . For each v ∈ SL denote [L, v] the subspace spanned by L and v.

Definition 1 A mapping P : SL → 2B is the peak mapping of J w.r.t. L if ∀v ∈ SL , P (v) is the set of all local maximum points of J in [L, v]. A single-valued mapping p : SL → B is a peak selection of J w.r.t. L if p(v) ∈ P (v) ∀v ∈ SL . For a given u ∈ SL , J is said to have a local peak selection at u if a peak selection p is locally deﬁned near u. Definition 2 For given 0 < θ ≤ 1, if u ∈ B with ∇J(u) = 0, a point Ψ(u) ∈ B is called a pseudo-gradient of J at u if Ψ(u) ≤ 1 and ∇J(u), Ψ(u) ≥ θ∇J(u). Lemma 3 [15] For v0 ∈ SL , if J has a local peak selection p w.r.t. L at v0 satisfying (1) p is continuous at v0 , (2) d(p(v0 ), L) > 0, and (3) Copyright © 2005 Marcel Dekker, Inc.

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∇J(p(v0 )) ≥ δ > 0. Denote p(v0 ) = t0 v0 + w0 for some t0 = 0 and w0 ∈ L. Then there exists s0 > 0 such that for 0 < s < s0 J(p(v(s))) − J(p(v0 )) < −

θs d(p(v0 ), L)∇J(p(v0 )) 4

(2)

v0 − sign(t0 )sG(p(v0 )) , G(p(v0 )) = P(Ψ(p(v0 ))) and v0 − sign(t0 )sG(p(v0 )) Ψ(p(v0 )) is a pseudo-gradient of J at u0 = p(v0 ) with constant θ ∈ (0, 1).

where v(s) =

Then the following local minimax characterization of saddle points can be established.

Theorem 4 [15] Assume that a local peak selection function p is continuous at v ∗ ∈ SL . If J(p(v ∗ )) = minv∈SL J(p(v)) and d(p(v ∗ ), L) > 0, then u∗ = p(v ∗ ) is a critical point of J. Deﬁne the solution set (manifold) M = {p(v) : v ∈ SL }. Theorem 4 states that a local minimum point of J on M yields a saddle point u∗ = p(v ∗ ). It leads to the local minimax algorithm [7], [8], [15]. As stated in Lemma 3, for v0 ∈ SL , v1 = v0 (s1 ) where s1 is the maximum s > 0 such that (2) is satisﬁed. When B is a Hilbert space, we can simply take L = L⊥ and G(p(v)) = ∇J(p(v)).

2.

Results on the Order of Saddle Points

To establish an order or instability indices for saddle points is an important issue in saddle point analysis and application. When B = H is a Hilbert space, the Morse index is widely used to deﬁne an order (instability index) for nondegenerate saddle points [6]. Let H + , H 0 , H − be respectively, the maximum positive deﬁnite, null, maximum negative deﬁnite subspaces of J (u∗ ). The number MI(u∗ ) = dim(H − ) is called the Morse index of u∗ .

Theorem 5 [16] Let v ∗ ∈ SL⊥ . Assume that J has a local peak selection p w.r.t. L at v ∗ s.t. (a) p is continuous at v ∗ , (b) u∗ ≡ p(v ∗ ) ∈ L and (c) v ∗ = arg minv∈SL⊥ J(p(v)). If p is diﬀerentiable at v ∗ , then u∗ is a critical point with dim(L) + 1 = MI (u∗ ) + dim(H 0 ∩ [L, v ∗ ]).

(3)

Theorem 5 implies that in order for our local minimax algorithm to have a stable convergence in computing a nondegenerate saddle point of Copyright © 2005 Marcel Dekker, Inc.

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MI= n, the support L must contain at least n − 1 critical points at the lower critical level. Since many diﬀerent cases may happen in H 0 , the Morse index is basically for nondegenerate cases and is not deﬁned in a Banach space. For allowing degenerate cases in a Banach space to happen, an other approach has to be used.

Definition 6 Let J ∈ C 1 (B, R) and u∗ ∈ B be a critical point of J. If B = B I ⊕ B D for some subspaces B I , B D in B and for each u1 ∈ B I and u2 ∈ B D with u1 = 1 and u2 = 1 there exist constants r1 > 0 and r2 > 0 s.t. J(u∗ + tu1 ) ≥ J(u∗ ), ∀ 0 < |t| ≤ r1 , J(u∗ + tu2 ) < J(u∗ ), ∀ 0 < |t| ≤ r2 .

(4) (5)

It is clear that u∗ is a saddle point of J and the integer dim(B D ) is called the order of u∗ . Such a saddle point is called a local linking in the literature [9] and is of particular interests in computational chemistry and condensed matter physics, since the subspace B D corresponds to the reaction coordinates. Since (4) and (5) lack of characterization and robustness, they are difﬁcult to apply in discussion, in particular in a Banach space. Thus we modify the deﬁnition by replacing (5) with J(u∗ + tu2 + o(t)) < J(u∗ ), ∀ 0 < |t| ≤ r2 ,

(6)

where o(t) represents a higher order term. If u∗ is a nondegenerate saddle point in a Hilbert space H, then we have J(u∗ + tu1 + o(t)) − J(u∗ ) = J(u∗ + tu2 + o(t)) − J(u∗ ) =

t2 ∗ J (u )u1 , u1 + o(t2 ) > 0 2 t2 ∗ J (u )u2 , u2 + o(t2 ) < 0 2

for any u1 ∈ H + , u2 ∈ H − and |t| small, i.e., B D = H − and B I = H + or the order of u∗ is the Morse index of u∗ . The deﬁnitions in (4) and (6) do not require nondegeneracy and are much more general than the Morse index. Next we assume that (4) and (6) are satisﬁed and prove that for a saddle point u∗ = p(v ∗ ) found by our local minimax algorithm, its order is equal to k = dim(L) + 1.

Theorem 7 Let v ∗ ∈ SL . Assume that J has a local peak selection p w.r.t. L at v ∗ s.t. p is diﬀerentiable at v ∗ , u∗ ≡ p(v ∗ ) ∈ L, and Copyright © 2005 Marcel Dekker, Inc.

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v ∗ = arg minv∈SL J(p(v)). If we assume that · exists in B \ {0} and u∗ is a strict local maximum point of J restricted to [L, v ∗ ], then u∗ is a saddle point of order equal to k = dim(L) + 1. Proof. Let B D and B I be deﬁned as in (4) and (6). Assume p(v ∗ ) = t0 v ∗ + vL∗ for some t0 = 0 and vL∗ ∈ L. Since k > dim(B D ) implies [L, v ∗ ] ∩ B I = ∅, it contradicts our assumption that u∗ is a strict local maximum of J restricted to [L, v ∗ ]. Thus we have k ≤ dim(B D ). To prove k = dim(B D ), recall we have B = L ⊕ L , v ∗ ∈ SL . Denote Lv∗ = {u ∈ L : v ∗ , u = 0}. Then B = [L, v ∗ ] ⊕ Lv∗ . We ﬁrst prove that p (v ∗ )(Lv∗ ) ⊕ [L, v ∗ ] = B. (7) For any w ∈ Lv∗ with w = 1, denote v ∗ (s) = d ∗ ds |s=0 v (s) = N (v ∗ ) ∩ SL .

v∗ +sw v∗ +sw .

Then

w and there exists s0 > 0 s.t. when |s| < s0 , v ∗ (s) ∈

Let α(s) ≡ P(p(v ∗ (s))) = t(s)v ∗ (s), where P is the linear projection operator onto L . Since p is diﬀerentiable at v ∗ and v ∗ (s) smoothly depends on s, α is diﬀerentiable at 0 and α (0) = P(p (v ∗ )(w)) = (t(s)v ∗ (s))|s=0 = t (0)v ∗ + t0 w.

(8)

Since t0 = 0, it leads to w ∈ [p (v ∗ )(w), L, v ∗ ], ∀w ∈ Lv∗ .

(9)

Now if w ∈ Lv∗ and p (v ∗ )(w) ∈ [L, v ∗ ], (9) implies w ∈ [L, v ∗ ] and then w = 0. Thus p (v ∗ )(w) ∈ [L, v ∗ ], ∀w ∈ Lv∗ , w = 0, and we conclude (7). Now we suppose k < dim(B D ). Then (7) implies p (v ∗ )(Lv∗ ) ∩ B D contains nonzero points. Let w ∈ Lv∗ with w = 1 and p (v ∗ )(w) ∈ ∗ ∗ ∗ (B D \ {0}). Denote v ∗ (s) = vv∗ +sw +sw . We have v (s) ∈ N (v ) ∩ SL for |s| small and

d ∗ ds |s=0 v (s)

= w. It follows

p(v ∗ (s)) = p(v ∗ ) + sp (v ∗ )(w) + o(s) = u∗ + sp (v ∗ )(w) + o(s). Since p (v ∗ )(w) ∈ (B D \ {0}), for |s| small, we have J(p(v ∗ (s)) = J(u∗ + sp (v ∗ )(w) + o(s)) < J(u∗ ) = J(p(v ∗ )). On the other hand, by our assumption on v ∗ = arg minv∈SL J(p(v)), we have J(p(v ∗ (s)) ≥ J(p(v ∗ )). Copyright © 2005 Marcel Dekker, Inc.

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It leads to a contradiction. Thus we conclude that k = dim(B D ).2 The above result implies that in our local minimax algorithm, the computation of a saddle point of order n will be stable if the support L contains n − 1 linearly independent critical points at the lower critical level. A method to check if p is diﬀerentiable is developed in [17].

3.

Reduce Instability by Using Symmetries

Symmetries exist in many natural phenomena, such as in crystals, elementary particle physics, etc. Symmetries are usually handled by “compact group actions” in mathematics. Here we show that symmetry can be used to reduce instability in computing saddle points.

Definition 8 ([13]) Let B be a Banach space and J ∈ C 1 (B, R). A closed subspace BI of B is said to be a J-invariant space if u = p(v) ∈ BI and ∇J(u) = 0 imply that there exists a pseudo-gradient Ψ(u) of J at u such that G(u) = PΨ(u) ∈ BI . We have the following modiﬁed Principle of Symmetric Criticality (PSC).

Theorem 9 Let BI be a J-invariant space of B. If u∗ = p(v ∗ ) ∈ BI is a critical point of J restricted to BI , then u∗ is a critical point of J in B. Proof. Note ∇J(u∗ ), v = 0, ∀v ∈ BI . If ∇J(u∗ ) = 0, then we have G(u∗ ) = PΨ(u∗ ) ∈ BI . By the deﬁnition of a peak selection, we have Ψ(u∗ ) − G(u∗ ) ∈ L and ∇J(p(v ∗ )), Ψ(u∗ ) − G(u∗ ) = 0. Thus 0 = ∇J(p(v ∗ )), G(p(v ∗ )) = ∇J(p(v ∗ )), Ψ(p(v ∗ )) ≥ θ∇J(p(v ∗ )). It leads to a contradiction. To compute a saddle point u∗ , we identify its symmetries and deﬁne the smallest invariant subspace BI containing u∗ , then replace B by BI and L by L ∩ BI in the local minimax algorithm. We have v0 ∈ BI =⇒ u0 = p(v0 ) ∈ BI =⇒ G(u0 ) ∈ BI =⇒ v1 ∈ BI .... v0 − sign(t0 )s1 G(p(v0 )) and s1 is the maximum v0 − sign(t0 )s1 G(p(v0 )) s > 0 such that (2) is satisﬁed.

where v1 = v0 (s1 ) =

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287

Theorem 10 The local minimax method is closed in an invariant subspace, i.e., if v0 ∈ BI then vn , un = p(vn ) ∈ BI . The order (instability index) of u∗ in BI is dim(L∩BI )+1 and is greatly reduced. Within BI , LMM becomes much more eﬃcient and stable. But in numerical computation of the modiﬁed pseudo-gradient dk = −G(p(v k )), it involves discretization, approximation, round-oﬀ error, etc. The numerical error will break the invariance of dk and leads to dk ∈ BI . Due to the unstable nature of u∗ outside BI , it will ﬁnd a slider and bypass u∗ . To preserve the invariance, we project dk back to BI . With this projected dk ∈ BI , the updated point v k+1 is in BI and the invariance is preserved. For symmetries that can be described by a compact group of linear isomorphisms, there is a much easier way to ﬁnd the projection to BI . Let G be a compact group of linear isomorphisms of B. A set A ⊂ B is G-invariant if g(A) = A for every g ∈ G. The subspace BG = {u ∈ B : gu = u, ∀g ∈ G} is called the invariant subspace of B under G. J is said to be G-invariant if J(gu) = J(u) for every (g, u) ∈ G × B. A map F : B → B is G-equivariant if g ◦ F = F ◦ g for every g ∈ G. Since J ∈ C 1 (B, R) is G-invariant, it implies that ∇J is G-equivariant, i.e., ∇J(gu) = ∇J(u) ∀(g, u) ∈ G × B. Then one can ﬁnd a Gequivariant pseudo-gradient ﬂow Ψ(u) ∈ BG (see page 86 in [12]). Since L is chosen to be a subspace of BG , and BG = L ⊕ L , G(p(v)) = PΨ(p(v)) ∈ BG where P is the linear projection onto L . Thus our deﬁnition of an invariant subspace is more general. To ﬁnd the projection from B to BG , let us cite the Haar integral.

Theorem 11 (Haar, 1933) Let G be a compact group and C(G) be the space of real-valued continuous functions on G. Then there exists a unique

positive integral (the Haar integral) s.t. the map : C(G) → R by f → G f (g) dg is

(a) linear, monotone, and normalized ( G 1 dg = 1);

2 (b) left-invariant, i.e., G f (hg) dg = G f (g) dg, ∀h ∈ G. It leads to the Haar integral operator H from B to B deﬁned by Hu = gu dg ∈ BG , ∀u ∈ B where BG = {u ∈ B : Hu = u}. G

This operator was used in the literature to preserve an invariance. Copyright © 2005 Marcel Dekker, Inc.

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Example 4.1. Let G = {g(1), ..., g(m)} be a ﬁnite group of linear isomorphisms of B where g(m) = id. Then for each u ∈ B, 1 g(k)u ∈ BG . m m

Hu = uG =

k=1

≥ 1) be a bounded open set and B = Example 4.2. Let Ω ⊂ W 1,p (Ω) where p > 1 be the Sobolev space. Let G be the set of all orthogonal matrices g ∈ Rn×n s.t. g(Ω) = Ω. For each g ∈ G and u ∈ B, if we deﬁne gu(x) = u(gx), ∀x ∈ Ω, then G is a compact group of linear isomorphisms of B. In particular, if g ∈ Rn×n is an orthogonal matrix s.t. g(Ω) = Ω and m > 0 is the least integer s.t. gm = I. For each u ∈ H, we deﬁne g(u)(x) = u(gx), ∀x ∈ Ω. Then we have a ﬁnite cyclic group G = {g, g2 , ..., gm } ∼ = ZZm of linear isomorphisms of B. Example 4.3. Let Ω ⊂ Rn be bounded and open. Assume that Ω is symmetric about the ﬁrst n − 1 axes. Let B = W 1,p (Ω). Deﬁne g : B → B by (gu)(x1 , ..., , xn−1 , xn ) = −u(−x1 , ..., −xn−1 , xn ). Then G = {id, g} ∼ = ZZ2 . There are symmetries that can not be deﬁned by a compact group of linear isomorphisms, e.g., a composite symmetry involving partially deﬁned symmetry. We may identify those symmetries by using several projections. Example 4.4. Let Ω = [−a, a] × [−a, a] in R2 and B = W01,p (Ω). We are interested in ﬁnding a critical point u∗ with the following symmetries. The proﬁle of u∗ is even symmetric about the y = −x axis, even symmetric about the x-axis for points (x, y) with 0 ≥ y ≥ −x ≥ −a, and even symmetric about the y-axis for points (x, y) with 0 ≥ −x ≥ y ≥ −a. The invariant space BT is deﬁned by two projections T = T2 · T1 , where Rn , (n

1 (T1 u)(x, y) = (u(x, y) + u(−y, −x)), (x, y) ∈ Ω 2 is the projection from B to BT1 with which computational error is minimized and T2 is a projection from BT1 to BT deﬁned by u(x, −y), 0 ≥ y ≥ −x; u(−x, y), 0 ≥ −x ≥ y; (T2 u)(x, y) = u(x, y), otherwise. Note that T2 can map B to outside of B. However T2 projects BT1 into B. Thus T = T2 · T1 projects B into BT ⊂ B. In [13], such an example for the Lane-Emden equation has been successfully carried out with B = H01 (Ω). We have numerically veriﬁed that BT is an invariant space in the sense of Deﬁnition 8 Asymm of ∇JH 1 = 2.·10−4 ∇JH01 . Copyright © 2005 Marcel Dekker, Inc.

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4.

289

Selected Numerical Examples

Consider ﬁnding the eigenpairs (u, λ) ∈ W 1,p (Ω)×R+ of the p-Laplacian operator on Ω = [0, 2] × [0, 2] ∆p u(x) = λ|u(x)|p−2 u(x), x ∈ Ω, (10) u(x) = 0, x ∈ ∂Ω ∂u 2 12 ) ) and where u = 0 and ∆p u = div(|∇u|p−2 ∇u), |∇u| = ( ni=1 ( ∂x i n = 2. The p-Laplacian operator has a variety of applications in physical ﬁelds, such as in ﬂuid dynamics when the shear stress and the velocity gradient are related in a certain manner where p = 2, p < 2, p > 2 if the ﬂuid is Newtonian, pseudo-plastic, dilatant, respectively. The pLaplacian operator also appears in the study of ﬂow in a porous media (p = 32 ), nonlinear elasticity (p > 2), and glaciology (p ∈ (1, 43 )). To solve the eigenpair problem (10), we deﬁne the Rayleigh quotient

|∇u(x)|p dx J(u) = Ω . p Ω |u(x)| dx Then critical points u∗ of J correspond to eigenfunctions of the pLaplacian with respect to the eigenvalue λ = J(u∗ ). In our numerical computation, we have utilized the symmetry of a saddle point to be found to reduce its instability. In all our numerical examples here, we use 1000 × 1000 piecewise linear ﬁnite elements and set ε < 10−4 . For illustration, we present the ﬁrst six eigenfunctions for the p-Laplacian with p = 2.75 and p = 1.7. An interesting phenomenon is the change of sequential order of the eigenvalues. When p = 2, λ2 = λ3 and λ5 = λ6 . When p = 2, the two pairs have diﬀerent values and change the sequential orders when p crosses 2.

Copyright © 2005 Marcel Dekker, Inc.

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1.4

1.5

1.2

1

1 0.5

0.8 0

0.6 −0.5

0.4

0.2

−1

0 2

−1.5 2

2

1.5

2

1.5

1.5

1

1.5

1

1 0.5

1 0.5

0.5 0

0.5 0

0

0

p = 2.75, Eigenfunctions for λ1 = 7.0907 (left) and for λ2 = 25.6218

Figure 1. (right).

1.5

1.5

1

1

0.5

0.5

0

0

−0.5

−0.5

−1

−1

−1.5 2

−1.5 2

2

1.5

2

1.5

1.5

1

1.5

1

1 0.5

1 0.5

0.5 0

0.5 0

0

0

p = 2.75, Eigenfunctions for λ3 = 26.6583 (left) and for λ4 = 47.6999

Figure 2. (right).

1.5

0.15

0.1

1

0.05

0.5

0

0 −0.05

−0.5 −0.1

−1

−0.15

−1.5 2

−0.2 2

2

1.5

1.5

1

2

1.5

1.5

1

1 0.5

0.5 0

Figure 3. (right).

0

1 0.5

0.5 0

0

p = 2.75, Eigenfunctions for λ5 = 66.4530 (left) and for λ6 = 68.6633

Copyright © 2005 Marcel Dekker, Inc.

291

Stabilization in Computing Saddle Points

0.9

1

0.8

0.7

0.5

0.6

0.5 0

0.4

0.3 −0.5

0.2

0.1

0 2

−1 2

2

1.5

2

1.5

1.5

1

1.5

1

1 0.5

1 0.5

0.5 0

0.5 0

0

0

p = 1.7, Eigenfunctions for λ1 = 4.1091 (left) and for λ2 = 8.7889 (right).

Figure 4.

1

1

0.5

0.5

0

0

−0.5

−0.5

−1 2

−1 2

2

1.5

2

1.5

1.5

1

1.5

1

1 0.5

1 0.5

0.5 0

0.5 0

0

0

p = 1.7, Eigenfunctions for λ3 = 8.8895 (left) and for λ4 = 13.3503

Figure 5. (right).

0.15

1

0.1

0.05

0.5

0

−0.05 0

−0.1

−0.15 −0.5

−0.2

−0.25

−0.3 2

−1 2

2

1.5

1.5

1

2

1.5

1.5

1

1 0.5

0.5 0

Figure 6. (right).

0

1 0.5

0.5 0

0

p = 1.7, Eigenfunctions for λ5 = 15.6864 (left) and for λ6 = 16.0231

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CONTROL AND BOUNDARY ANALYSIS

References [1] On the existence of sign-changing solutions for semilinear dirichlet problems. Topol. Methods Nonlinear Anal., 7:115–131, 1996. [2] A. Ambrosetti and P. Rabinowitz. Dual variational methods in critical point theory and applications. J. Funct. Anal., 14:349–381, 1973. [3] H. Brezis and L. Nirenberg. Remarks on ﬁnding critical points. Communications on Pure and Applied Mathematics, XLIV:939–963, 1991. [4] K.C. Chang. Inﬁnite Dimensional Morse Theory and Multiple Solution Problems. Birkh¨ auser, Boston, 1993. [5] I. Ekeland and N. Ghoussoub. Selected new aspects of the calculus of variations in the large. Bull. Amer. Math. Soc. (N.S.), 39:207–265, 2002. [6] J. Smoller Shock Waves and Reaction-Diﬀusion Equations. Springer-Verlag, New York, 1983. [7] Y. Li and J. Zhou. A minimax method for ﬁnding multiple critical points and its applications to semilinear pde. SIAM J. Sci. Comp., 23:840–865, 2001. [8] Y. Li and J. Zhou. Convergence results of a local minimax method for ﬁnding multiple critical points. SIAM J. Sci. Comp., 24:840–865, 2002. [9] J. Q. Liu and S. J. Li. Some existence theorems on multiple critical points and their applications. Kexue Tongbao, 17, 1984. [10] J. Mawhin and M. Willem. Critical Point Theory and Hamiltonian Systems. Springer-Verlag, New York, 1989. [11] P. Rabinowitz. Minimax Method in Critical Point Theory with Applications to Diﬀerential Equations, volume 65 of CBMS Regional Conf. Series in Math. AMS, Providence, 1986. [12] M. Struwe. Variational Methods. Springer-Verlag, Berlin, 1996. [13] Z. Q. Wang and J. Zhou. An eﬃcient and stable method for computing multiple saddle points with symmetries. To appear. [14] M. Willem. Minimax Theorems. Birkh¨ auser, Boston, 1996. [15] X. Yao and J. Zhou. A minimax method for ﬁnding multiple critical points in Banach spaces and its application to quasilinear elliptic pde. To appear. [16] J. Zhou. Local instability analysis of saddle points by a local minimax method. Math. Comp. to appear. [17] J. Zhou. A min-orthogonal method for ﬁnding multiple saddle points. J. Math. Anal. Appl., 291:66–81, 2004.

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CHAPTER XXII SECOND ORDER SUFFICIENT CONDITIONS FOR OPTIMAL CONTROL SUBJECT TO FIRST ORDER STATE CONSTRAINTS Kazimierz Malanowski Systems Research Institute Polish Academy of Sciences Warszawa, Poland

Helmut Maurer Institut f¨ ur Numerische Mathematik Westf¨ alische Wilhelms Universit¨ at M¨ unster, Germany

Sabine Pickenhain Institut f¨ ur Mathematik Brandenburgische Technische Universit¨ at Cottbus, Germany

Abstract

Second order suﬃcient optimality conditions (SSC) are derived, for an optimal control problem subject to pure state constraints of order one. The proof is based on a Hamilton-Jacobi inequality and it exploits regularity of the control function, as well as the associated Lagrange multipliers. The obtained SSC involve Legendre-Clebsch conditions and solvability of an auxiliary Riccati equation. The latter condition is weakened by taking into account the strongly active constraints.

Keywords: Nonlinear optimal control, ﬁrst order state constraints, second order suﬃcient optimality conditions, solvability of a Riccati equation, constraint qualiﬁcations, Legendre-Clebsch condition

Introduction We shall derive second order suﬃcient optimality conditions (SSC) for an optimal control problem for nonlinear ODEs, subject to pure state Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

constraints of order one. We follow the approach based on the HamiltonJacobi inequality which was used in [8] for multidimensional control problems. For one-dimensional problems we are able to obtain results more constructive than those in [8]. We continue the line of [6], where mixed control-state constraints were considered. As in [6], we try to obtain a SSC in a possibly weak form which takes into account the strongly active constraints. In the whole analysis, the regularity of the control as well as of the associated Lagrange multipliers plays a crucial role. We consider the case where the control is a continuous function, and we impose constraint qualiﬁcations which ensure existence, uniqueness and regularity of the normal Lagrange multipliers. By these constraints qualiﬁcations, the analysis is conﬁned to the so-called ﬁrst order state constraints [9]. The relevant regularity results obtained in [4] are recalled in Section 1. Note that in the regularity analysis a Lagrangian in the socalled indirect form is more convenient, while in the analysis of suﬃciency in Sections 2 and 3 a Lagrangian in the direct form is used (see [9] for the deﬁnitions). Following [8] and [6], we formulate in Section 3 a second order suﬃcient optimality condition in terms of a solution to a Hamilton-Jacobi inequality. A quadratic form is chosen as a candidate for a solution to the Hamilton-Jacobi inequality, and conditions are derived under which this quadratic form is the needed solution. These conditions involve the previous constraint qualiﬁcations, as well as some pointwise coercivity conditions with respect to control (Legendre-Clebsch condition) and with respect to state. The latter is expressed in terms of solvability of a modiﬁed matrix Riccati equation which is weakened by taking into account strongly active constraints. We believe that the obtained Riccati equation is original and that it constitutes the main contribution of this paper to suﬃciency analysis for optimal control problems. A simple example in Section 4 shows that the obtained SSC are really weaker than the classical one.

1.

Preliminaries

We consider the following optimal control problem: (O)

Find (x0 , u0 ) ∈ W 1,∞ (0, 1; Rn ) × L∞ (0, 1; Rm ) such that 1 ϕ(x(t), u(t))dt + ψ(x(0), x(1))} F (x0 , u0 ) = min{F (x, u) := subject to x(t) ˙ − f (x(t), u(t)) = 0 ξ(x(0), x(1)) = 0, ϑ(x(t)) ≤ 0

Copyright © 2005 Marcel Dekker, Inc.

0

for a.a. t ∈ [0, 1], for all t ∈ [0, 1],

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Second Order Suﬃcient Conditions

where ϕ : Rn × Rm → R, ξ : Rn × Rn → Rd ,

ψ : Rn × Rn → R, ϑ : Rn → Rl .

f : Rn × Rm → Rn ,

Denote J = {1, ..., l} the set of indices of constraints. Throughout the paper we assume that the data satisfy the following regularity conditions: the functions ϕ, φ, f , ξ, and ϑ are twice Fr´echet diﬀerentiable in all arguments, and all diﬀerentials are Lipschitz continuous. Note that control or mixed control-state constraints can be easily included in our suﬃciency analysis, in a way similar to that in [6] and [10]. For p ∈ [1, ∞], let us introduce the spaces X p := W 1,p (0, 1; Rn ) × Lp (0, 1; Rm ), Y p := W 1,p (0, 1; Rn ) × Rd × W 1,p (0, 1; Rl ), Z p := W 1,p (0, 1; Rn ) × Rd × Lp (0, 1; Rl ) × Rd × Rd . Deﬁne the following Lagrangian and Hamiltonians associated with (O): L, : X ∞ × Y ∞ → R, H : Rn × Rm × Rn × Rk × Rl → R , u, q, ρ, µ) = F (x, u) − (q, x˙ − f (x, u)) + ρ, ξ(x(0), x(1)) L(x, +µ(0), ϑ(x(0)) + (µ, ˙ Dx ϑ(x)f (x, u)), H(x, u, q) = ϕ(x, u) + q, f (x, u) , H(x, u, q, µ) ˙ = H(x, u, q) + µ, ˙ Dx ϑ(x)f (x, u).

(1) (2)

Remark. The Lagrangian (1) is in the normal form, i.e., the Lagrange multiplier corresponding to the functional F (x, u) is equal to one. The Lagrangian is written in the so-called indirect or Pontryagin form with an absolutely continuous adjoint function q (see Section 7 in [9] as well as [1] and [7]). The state constraints are considered in W 1,∞ (0, T ; Rl ) rather than in C 0 (0, 1; Rl ). This form is convenient in the analysis of existence and regularity of Lagrange multipliers [4]. In the sequel, we introduce still another form of the Lagrangian. Let (x0 , u0 ) with u0 ∈ C 0 (0, 1; Rm ) be a ﬁxed pair admissible for (O). We are going to analyze conditions under which (x0 , u0 ) is a locally isolated local solution to (O). For the sake of simplicity, the functions evaluated at (x0 , u0 ) will be denoted by subscript “0”, e.g., f0 (t) = f (x0 (t), u0 (t)), ϑ0 (t) = ϑ(x0 (t)). First we will consider the stationarity conditions of the Lagrangian (1) at (x0 , u0 ). To get existence and uniqueness of a Lagrange multiplier associated with (x0 , u0 ) we need some constraints qualiﬁcations. To this end, for t ∈ [0, 1] introduce the sets of active constraints J0 (t) := {j ∈ J | ϑj0 (t) = 0}, and deﬁne the matrices Υ0 (t) := [Dx ϑj0 (t)]j∈J0 (t) . Copyright © 2005 Marcel Dekker, Inc.

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CONTROL AND BOUNDARY ANALYSIS

We assume: (A1) (Linear Independence) There exists β > 0 such that |[Du f0 (t)∗ Υ∗0 ] χ| ≥ β|χ| for all χ of appropriate dimension and all t ∈ [0, 1]. (A2) (Controllability) For any g ∈ Rd there exists a solution (y, v) ∈ X ∞ of the following equations: y(t) ˙ − Dx f0 (t)y(t) − Du f0 (t)v(t) = 0, D0 ξ0 y(0) + D1 ξ0 y(1) = g, Υ0 (t)y(t) = 0. Here and in the sequel, we denote Di := Dx(i) , i = 0, 1. Similarly 2 := D 2 Dij x(i)x(j) , i, j = 0, 1. Note that (A1) restricts the analysis to problems with so-called ﬁrst order state constraints (see [9]). Condition (A1) implies in particular that |[Υ0 (t)∗ ] η| ≥ β|η| for all η of appropriate dimension and all t ∈ [0, 1]. This means that, pointwisely along the trajectory (x0 , u0 ), the gradients of all active constraints are linearly independent, uniformly on [0, 1]. Theorem 4.3 in [4] implies:

Lemma 1 If assumptions (A1) and (A2) hold, then there exists a unique normal Lagrange multiplier λ0 := (q0 , ρ0 , µ0 ) ∈ Y ∞ for (O) associated with (x0 , u0 ), i.e., the following ﬁrst order optimality conditions are satisﬁed: , 0 , u0 , q0 , ρ0 , µ0 ) = 0, Dx L(x , 0 , u0 , q0 , ρ0 , µ0 ) = 0, Du L(x ,+, µ0 (0), ϑ(x(0)) + (µ˙ 0 , Dx ϑ(x0 )f (x0 , u0 )) = 0, µ0 ∈ L where ,+ = {µ ∈ W 1,∞ (0, 1; Rl )|0 ≤ µ˙ j (t) ≤ µj (t), and µ˙ j (t) is L nonincreasing, for all j ∈ J and a.a. t ∈ [0, 1]}. In addition to (A1) and (A2), we assume: (A3) (Legendre-Clebsch condition) There exists γ > 0 such that 2 , H0 (t)v ≥ γ|v|2 v, Duu

Copyright © 2005 Marcel Dekker, Inc.

for all v ∈ Rm and a.a. t ∈ [0, 1].

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Second Order Suﬃcient Conditions

In the same way as in Theorem 5.2 in [4], we obtain:

Corollary 2 If (A1) – (A3) hold, then x˙ 0 , u0 , q˙0 , µ˙ 0 are Lipschitz continuous on (0, 1).

(3)

Taking advantage of the regularity (3), we can deﬁne a Lagrangian for (O) in the so-called direct form [9]: L : X ∞ × Z ∞ → R, L(x, u, p, ρ, ν, σ 0 , σ 1 ) = F (x, u) − (p, x˙ − f (x, u)) + ρ, ξ(x(0), x(1)) +(ν, ϑ(x)) + σ 0 , ϑ(x(0)) + σ 1 , ϑ(x(1). (4) We also introduce still another Hamiltonian: 2 : Rn × Rm × Rn × Rl → R, H

2 u, p, ν) = H(x, u, p) + ν, ϑ(x). (5) H(x,

The stationarity of the Lagrangian (4) yields the following ﬁrst order optimality conditions: 2 0 , u0 , p0 , ν0 ) p˙ 0 + Dx H(x (6) = p˙ 0 + Dx f (x0 , u0 )∗ p0 + Dx ϕ(x0 , u0 ) + Dx ϑ(x0 )∗ ν0 = 0, p0 (0) + D0 ψ(x0 (0), x0 (1)) + D0 ξ(x0 (0), x0 (1))∗ ρ0 +Dx ϑ(x0 (0))∗ σ00 = 0,

(7)

−p0 (1) + D1 ψ(x0 (0), x0 (1)) + D1 ξ(x0 (0), x0 (1))∗ ρ0 +Dx ϑ(x0 (1))∗ σ01 = 0,

(8)

Du H(x0 (t), u0 (t), p0 (t), ν0 ) := Du ϕ(x0 , u0 ) + Du f (x0 (t), u0 (t))∗ p0 = 0, (ν0 , ϑ(x0 )) = 0,

σ00 , ϑ(x0 (0)) = 0,

(9)

σ01 , ϑ(x0 (1)) = 0,

where σ00 ∈ Rd+ ,

σ01 ∈ Rd+ ,

ν0 ∈ L+ := {ν ∈ L∞ (0, 1; Rl ) | ν i (t) ≥ 0, for all j ∈ J and a.a. t ∈ [0, 1]}. In view of (3), simple calculations show that µ0 ∈ L∞ (0, 1; Rl ), p0 = Dx ϑ(x0 )∗ µ˙ 0 + q0 ∈ W 1,∞ (0, 1; Rn ), ν0 = −¨ σ01 = µ˙ 0 (1). σ00 = µ0 (0) − µ˙ 0 (0), (10) Copyright © 2005 Marcel Dekker, Inc.

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In particular, 2 0 , u0 , p0 , ν0 ) − ν0 , ϑ(x0 ) , 0 , u0 , q0 , µ˙ 0 ) = H(x H(x , 0 , u0 , q0 , µ˙ 0 ) = Du H(x 2 0 , u0 , p0 , ν0 ), Du H(x 2 H(x 2 H(x , 0 , u0 , q0 , µ˙ 0 ) = Duu 2 0 , u0 , p0 , ν0 ). Duu

i.e., (11)

For a ﬁxed α ≥ 0, let us deﬁne: Jα+ (t) = {j ∈ J0 (t) | ν0 (t) > α}, 2 α (t) = Dx ϑj (t) . Υ 0

α (t) = cardJα+ (t),

+ j∈Jα (t)

Note that Jα+ (t) are the sets of those active constraints for which pointwise strict complementarity is satisﬁed with the margin α.

2.

Second Order Suﬃcient Conditions via Hamilton-Jacobi Inequality

To derive second order suﬃcient optimality conditions for (O), we follow the approach used in [8] and [6] which is based on the Hamilton-Jacobi inequality. By B X (x) := {y ∈ X |y − xX ≤ } we denote a closed ball of radius in a normed space X, centered at a point x. The following theorem can be proved in the same way as Theorem 3.1 in [6] (see also Assertion 2 in [8]).

Theorem 3 Suppose there exists a function V : [0, 1] × Rn → R that is of class C 1 with respect to x and Lipschitz continuous with respect to t, such that the following conditions hold for suitable > 0 and c > 0: Dt V (t, x) + H(x, u, Dx V (t, x)) ≥ c{|x − x0 (t)|2 + |u − u0 |2 } Rn+m

for all (x, u) ∈ B

(12)

((x0 (t), u0 (t))), s.t. ϑ(x) ≤ 0

and for a.a. t ∈ [0, 1], Dt V (t, x0 (t)) + H(x0 (t), u0 (t), Dx V (t, x0 (t))) = 0 for a.a. t ∈ [0, 1],

(13) [V (1, x0 (1)) − V (0, x0 (0))] − V (1, x ) − V (0, x ) +ψ(x0 , x1 ) − ψ(x0 (0), x0 (1)) ≥ c |x0 − x0 (0)|2 + |x1 − x0 (1)|2 (14)

1

0

for all xi such that ϑ(xi ) ≤ 0, i = 0, 1 and ξ(x0 , x1 ) = 0. Then the estimate F (x, u) ≥ F (x0 , u0 ) + c(x, u) − (x0 , u0 )2X 2 Copyright © 2005 Marcel Dekker, Inc.

(15)

299

Second Order Suﬃcient Conditions ∞

holds for all admissible pairs (x, u) ∈ B X (x0 , u0 ), i.e., (x0 , u0 ) is a locally isolated weak local minimizer. As in [8] and [6], the function V : [0, 1] × Rn → R used in Theorem 3 is deﬁned by the following quadratic expression: 1 V (t, x) = e(t) + p0 (t)∗ (x − x0 (t)) + (x − x0 (t))∗ Q(t)(x − x0 (t)), (16) 2 1,∞ 1,∞ where e ∈ W (0, 1; R), p0 ∈ W (0, 1; Rn ) is the adjoint variable given in (6), while Q(t) is a symmetric matrix and Q ∈ W 1,∞ (0, 1; Rn×n ). The function e is chosen in such a way that the HJ equality (13) holds, while the appropriate choice of Q ensures that inequalities (12) and (14) are satisﬁed. To this end, sensitivity results for parametric mathematical programs are used, in the same way as in [6]. We obtain the following results (see Theorem 4.1 in [6]):

Lemma 4 Suppose that (C) There exist constants α > 0 and δ > 0, independent of t ∈ [0, 1] such that y ∗ ∗ ≥ δ(|y|2 + |v|2 ) [y , u ]K(t) v for all (y, v) ∈ Rn+m such that Dx ϑi0 , y = 0 for all j ∈ Jα+ (t), and for a.a. t ∈ [0, 1], where + * 2 H 2 H + QD f 2 0 Dxu Q˙ + QDx f0 + Dx f0∗ Q + Dxx 0 u 0 (t). K(t) = 2 H + D f ∗Q 2 H Dux Duu 0 u 0 0 Then inequality (12) holds.

Lemma 5 Suppose that: (A4)

[y0∗ , y1∗ ]

L00 L01 L∗01 L11

y0 y1

>0

for all (y 0 , y 1 ) ∈ R2n such that: 2 0 (1)y 1 = 0, 2 0 (0)y 0 = 0, Υ Υ D0 ξ(x0 (0), x0 (1))y 0 + D1 ξ(x0 (0), x0 (1))y 1 = 0, where 2 (ψ(x (0), x (1)) + ξ(x (0), x (1))∗ ρ L00 = Q(0) + D00 0 0 0 0 0 ∗ 0 +ϑ(x0 (0)) σ0 ), 2 (ψ(x (0), x (1)) + ξ(x (0), x (1))∗ ρ ), L01 = D01 0 0 0 0 0 2 (ψ(x (0), x (1)) + ξ(x (0), x (1))∗ ρ L11 = −Q(1) + D11 0 0 0 0 0 +ϑ(x0 (1))∗ σ01 ). Copyright © 2005 Marcel Dekker, Inc.

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Then inequality (14) holds. By Theorem 3 and Lemmas 4 and 5 we obtain:

Theorem 6 Suppose that (A1) – (A3) hold and there exists a symmetric matrix function Q ∈ W 1,∞ (0, 1; Rn×n ) such that conditions (C) and (A4) are satisﬁed. Then there exist c > 0 and > 0 such that F (x, u) − F (x0 , u0 ) ≥ c(x, u) − (x0 , u0 )2X 2 ∞ for all admissible x, u ∈ B X (x0 , u0 ), i.e., (x0 , u0 ) is a locally isolated strong local minimizer of (O).

3.

Checking Positive Deﬁniteness with Riccati Equation

Theorem 6 is not constructive, in the sense that it provides no information on the existence and the form of a matrix function Q satisfying (C) and (A4). In this section we are going to show that conditions (C) and (A4) of positive deﬁniteness can be expressed in terms of solvability of an auxiliary matrix Riccati diﬀerential equation, satisfying appropriate boundary conditions. To this end, let us ﬁx α > 0 and denote by P(t) : Rn → Rn the projection map onto the (t)-dimensional subspace 2 α (t)y = 0}. Condition (C) can be expressed in the form: {y ∈ Rn | Υ y ∗ ∗ ≥ δ (|P(t)y|2 + |v|2 ) (17) [y , u ]M (t) v for all (y, v) ∈ Rn+m , and a.a. t ∈ [0, 1], where M (t) = * 2 + 2 H 20 P P ∗ Dxu P ∗ Q˙ + QDx f0 + Dx f0∗ Q + Dxx H0 + QDu f0 (t). 2 2 H Duu Dux H0 + Du f0∗ Q P 0 (18) Let π(S) denote the number of positive eigenvalues of a symmetric matrix S. Condition (17) takes the form π(M (t) − δI) = n + m − (t), where I is the unit matrix of the appropriate dimension. We assume: Copyright © 2005 Marcel Dekker, Inc.

(19)

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Second Order Suﬃcient Conditions

(A5) The following matrix Riccati equation 2 H 20 Q˙ + QDx f0 + Dx f0∗ Q + Dxx 2 −1 2 2 Dux H0 + Du f0∗ Q = S − Dxu H0 + QDu f0 Duu H0 (20) has a solution uniformly bounded on (0, 1). Here S ∈ L∞ (0, 1; Rn×n ) is any symmetric matrix function such that P ∗ (t)S(t)P(t) = 0 for a.a. t ∈ (0, 1). (21)

Remark If S(t) ≡ 0, then (20) reduces to the classical Riccati equation, used in second order suﬃcient optimality conditions for optimal control problems (see e.g., [5]). Clearly, (A5) is weaker than the latter condition, since S can be any arbitrary matrix function satisfying (21). Thus, in (A5) strongly active state constraints are taken into account.

Lemma 7 If (A1) - (A3) and (A5) hold, then (C) is satisﬁed. Proof Let π(S) denote the number of positive eigenvalues of a symmetric matrix S. We will need the following result from linear algebra (see Theorem 1 in [2]):

Lemma 8 Let A, B, C be n×n, m×n, and m×m dimensional matrices, respectively, where A and C are symmetric and C is invertible. Deﬁne A B∗ D= . B C Then π(D) = π(C) + π(A − B ∗ C −1 B), where A − B ∗ C −1 B is the Schur complement of C. In view of (A3) we have 2 H0 − δI) = m π(Duu

for δ ∈ (0, γ).

(22)

Hence, by (A5) and by well known stability results for ODEs, the following perturbed Riccati equation 2 H 20 Q˙ + QDx f0 + Dx f0∗ Q + Dxx 2 −1 2 2 − Dxu H0 + QDu f0 Duu H0 − δI Dux H0 + Du f0∗ Q = S + 2δI

(23)

has a bounded solution for any δ ∈ (0, γ) suﬃciently small. Multiplying (23) from the left and right by P(t)∗ and P(t), respectively, and taking Copyright © 2005 Marcel Dekker, Inc.

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into account (21), we get 2 H 20 R := P ∗ Q˙ + QDx f0 + Dx f0∗ Q + Dxx 2 2 −1 2 ( ∗ Dux H0 + Du f0 Q P − Dxu H0 + QDu f0 Duu H0 − δI −δP ∗ P = δP ∗ P. (24) Simple calculations show that R(t) is the Schur complement of the sub2 H (t) − δI of the matrix M (t) − δI, where M (t) is deﬁned matrix Duu 0 in (18). Clearly, (24) yields that π(R(t)) = n − (t). Hence, by Lemma 8 and by (22) we get π(M (t) − δI) = n + m − (t). It implies that (C) holds. Using Theorem 4 and Lemma 7 we arrive at the principal result of this paper:

Theorem 9 (Suﬃcient Optimality Condition) Suppose that conditions (A1) – (A3) are satisﬁed, and there exists a symmetric matrix function Q ∈ W 1,∞ (0, 1; Rn×n ) such that (A4) and (A5) hold. Then there exist c > 0 and > 0, such that F (x, u) − F (x0 , u0 ) ≥ c(x, u) − (x0 , u0 )X 2 ∞ for all admissible (x, u) ∈ B X ((x0 , u0 )), i.e., (x0 , u0 ) is a locally isolated strong local minimizer of (O).

4.

Example

Consider the following variational problem in a time interval [0, T ] with ﬁxed endtime T > 0 but free endpoint x(T ): T (u(t)2 − x(t)2 ) dt (25) Minimize F (x, u) = 0.5 0

subject to

x(t) ˙ = u(t), x(0) = x0 , t ∈ [0, T ], −a ≤ x(t) ≤ b for t ∈ [0, T ], 0 < a < b .

(26) (27)

The initial value x0 is supposed to satisfy −a < x0 < b. It is the same problem as Example 7.1 in Kawasaki, Zeidan [3] except that we consider a free endpoint. Consider the Hamiltonian (5) 2 u, p) = 0.5(u2 − x2 ) + pu + νa (−x − a) + νb (x − b) H(x,

(28)

with two multipliers νa , νb associated with the state constraints −x−a ≤ 0 and x − b ≤ 0. The adjoint equation (6) is p˙ = x + νa − νb . Copyright © 2005 Marcel Dekker, Inc.

(29)

Second Order Suﬃcient Conditions

303

2 u = u + p = 0, the optimal control is given by In view of H u = −p .

(30)

It is obvious that the strengthened Legendre-Clebsch condition (A3) holds and that the state constraint is of order one. Depending on the initial value x0 ∈ (−a, b), the optimal solution is composed by an interior arc with −a < x(t) < b in [0, t1 ) and a boundary arc either on the lower boundary, x(t) ≡ −a, or on the upper boundary, x(t) ≡ b, for t1 ≤ t ≤ T . The solutions on the boundary arcs are as follows: x(t) ≡ −a : u = 0, p = 0, νa = a > 0, x(t) ≡ b : u = 0, p = 0, νb = b > 0.

(31)

To determine the unknown entry time t1 , we use the continuity of the control and obtain the entry conditions x(t1 ) ∈ (−a, b), p(t1 ) = 0. Thus, the following boundary value problem has to be solved on the interior arc: (32) x˙ = −p, p˙ = x, x(0) = x0 , x(t1 ) ∈ (−a, b), p(t1 ) = 0. The solution is x(t) = α sin(t) + β cos(t) with β = x0 , α = x0 tan(t1 ). The entry time is determined by the condition cos(t1 ) = x0 /b on the upper boundary, resp., cos(t1 ) = −x0 /a on the lower boundary. Thus, for initial values with −a < x0 < a the last equation is solvable with 0 < t1 < π both for the upper and the lower boundary. Hence, we have found two extremal solutions. For x0 = a, we also get two extremals with x(t1 ) = −a for t1 = π and x(t1 ) = b with t1 < π. Finally, for all initial values a < x0 < b there exists only one extremal joining the upper boundary with x(t1 ) = b for t1 < π. We are going to use the SSC derived in Section 3 to check optimality of all these extremals. On the interior arc [0, t1 ), the function S in the Riccati equation (20) becomes zero and the equation takes the form (33) Q˙ − Q2 − 1 = 0. This equation has a bounded solution Q(t) = tan(t − t1 /2) for t1 < π. However, there is no bounded solution of (33) in the whole interval [0, T ] for an endtime T ≥ π. So, in particular, the classical SSC condition with S ≡ 0 fails. Here, we can take advantage of the freedom in choosing any S satisfying (21). In our case, S ∈ W 1,∞ (t1 , T ) can be arbitrary. Setting S(t) ≡ (−Q(t1 ) − 1) we ﬁnd that (20) reduces to Q˙ = 0. The boundary condition for Q(0) and Q(T ) are both vacuous since the only admissible variations in (A4) are y 0 = y 1 = 0. Hence, the solution of (20) needed in Theorem 9 takes the form: for t ∈ [0, t1 ], tan(t − t1 /2) Q(t) = for t ∈ [t1 , T ], tan(t1 /2) and by that theorem the extremal is the solution of the optimal control problem. Copyright © 2005 Marcel Dekker, Inc.

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References [1] W.W. Hager. Lipschitz continuity for constrained processes. SIAM J. Control and Optimization, 17:321–338, 1979. [2] E.V. Haynsworth. Determination of the inertia of a partitioned hermitian matrix. Linear Algebra Appl., 1:73–81, 1968. [3] H. Kawasaki and V. Zeidan. Conjugate points for variational problems with equality and inequality state constraints. SIAM J. Control and Optimization, 39:433–456, 2000. [4] K. Malanowski. On normality of Lagrange multipliers for state constrained optimal control problems. Optimization, 52:75–91, 2003. [5] H. Maurer. First and second order suﬃcient optimality conditions in mathematical programming and optimal control. Mathematical Programming Study, 14:163–177, 1981. [6] H. Maurer and S. Pickenhain. Second-order suﬃcient conditions for control problems with mixed control-state constraints. J. Optim. Theory Appl., 86:649– 667, 1995. [7] L.W. Neustadt. Optimization: a theory of necessary conditions. Princeton University Press, Princeton, NJ, 1976. [8] S. Pickenhain and K. Tammer. Suﬃcient conditions for local optimality in multidimensional control problems with state restrictions. Zeitschrift f¨ ur Analysis und ihre Anwendungen, 11:397–405, 1994. [9] S.P. Sethi, R.F. Hartl, and R.G. Vickson. A survey of the maximum principle for optimal control problems with state constraints. SIAM Review, 37:181–218, 1995. [10] V. Zeidan. The Riccati equation for optimal control problems with mixed state control problems: necessity and suﬃciency. SIAM J. Control and Optimization, 32:1297–1321, 1994.

Copyright © 2005 Marcel Dekker, Inc.

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