Stabilization, Optimal and Robust Control: Theory and Applications in Biological and Physical Sciences (Communications and Control Engineering)

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Author: Aziz Belmiloudi

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Communications and Control Engineering

Series Editors

E.D. Sontag • M. Thoma • A. Isidori • J.H. van Schuppen

Published titles include: Stability and Stabilization of Infinite Dimensional Systems with Applications Zheng-Hua Luo, Bao-Zhu Guo and Omer Morgul Nonsmooth Mechanics (Second edition) Bernard Brogliato Nonlinear Control Systems II Alberto Isidori L2-Gain and Passivity Techniques in Nonlinear Control Arjan van der Schaft Control of Linear Systems with Regulation and Input Constraints Ali Saberi, Anton A. Stoorvogel and Peddapullaiah Sannuti Robust and H∞ Control Ben M. Chen Computer Controlled Systems Efim N. Rosenwasser and Bernhard P. Lampe Control of Complex and Uncertain Systems Stanislav V. Emelyanov and Sergey K. Korovin Robust Control Design Using H∞ Methods Ian R. Petersen, Valery A. Ugrinovski and Andrey V. Savkin Model Reduction for Control System Design Goro Obinata and Brian D.O. Anderson Control Theory for Linear Systems Harry L. Trentelman, Anton Stoorvogel and Malo Hautus Functional Adaptive Control Simon G. Fabri and Visakan Kadirkamanathan Positive 1D and 2D Systems Tadeusz Kaczorek Identification and Control Using Volterra Models Francis J. Doyle III, Ronald K. Pearson and Babatunde A. Ogunnaike Non-linear Control for Underactuated Mechanical Systems Isabelle Fantoni and Rogelio Lozano Robust Control (Second edition) Jürgen Ackermann Flow Control by Feedback Ole Morten Aamo and Miroslav Krstić

Learning and Generalization (Second edition) Mathukumalli Vidyasagar Constrained Control and Estimation Graham C. Goodwin, María M. Seron and José A. De Doná Randomized Algorithms for Analysis and Control of Uncertain Systems Roberto Tempo, Giuseppe Calafiore and Fabrizio Dabbene Switched Linear Systems Zhendong Sun and Shuzhi S. Ge Subspace Methods for System Identification Tohru Katayama Digital Control Systems Ioan D. Landau and Gianluca Zito Multivariable Computer-controlled Systems Efim N. Rosenwasser and Bernhard P. Lampe Dissipative Systems Analysis and Control (2nd Edition) Bernard Brogliato, Rogelio Lozano, Bernhard Maschke and Olav Egeland Algebraic Methods for Nonlinear Control Systems Giuseppe Conte, Claude H. Moog and Anna M. Perdon Polynomial and Rational Matrices Tadeusz Kaczorek Simulation-based Algorithms for Markov Decision Processes Hyeong Soo Chang, Michael C. Fu, Jiaqiao Hu and Steven I. Marcus Iterative Learning Control Hyo-Sung Ahn, Kevin L. Moore and YangQuan Chen Distributed Consensus in Multi-vehicle Cooperative Control Wei Ren and Randal W. Beard Control of Singular Systems with Random Abrupt Changes El-Kébir Boukas Nonlinear and Adaptive Control with Applications Alessandro Astolfi, Dimitrios Karagiannis and Romeo Ortega

Aziz Belmiloudi

Stabilization, Optimal and Robust Control Theory and Applications in Biological and Physical Sciences

123

Aziz Belmiloudi, PhD Institut de Recherche Mathématique de Rennes (IRMAR) Centre de Mathématiques Institut National des Sciences Appliquées (INSA) de Rennes 20 Avenue des buttes de Coesmes 35043 Rennes Cedex France

ISBN 978-1-84800-343-9

e-ISBN 978-1-84800-344-6

DOI 10.1007/978-1-84800-344-6 Communications and Control Engineering ISSN 0178-5354 British Library Cataloguing in Publication Data Belmiloudi, Aziz Stabilization, optimal and robust control : theory and applications in biological and physical sciences. (Communications and control engineering) 1. Robust control 2. Automatic control - Mathematical models 3. Differential equations, Partial 4. Differential equations, Nonlinear 5. Game theory I. Title 629.8'312 ISBN-13: 9781848003439 Library of Congress Control Number: 2008928175 © 2008 Springer-Verlag London Limited Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Cover design: le-tex publishing services oHG, Leipzig, Germany Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com

To my mother Fatima Elyoubi To my wife Géraldine and to our children Nora, Gwenda and Yann

To the memory of my grandmother

Preface

This book focuses on stabilization, control and fluctuation of systems governed by non-linear partial differential equations (PDEs), which arise in many applications. The developed approach is based on robust control theory in the dynamic non-cooperative optimization framework (i.e., game theory). Robust control problems are the subject of a huge and varied range of research worldwide. Research into the dynamical systems governed by PDEs is a relatively new area of study, but exciting and vitally important. This theory deals with optimization, identification, stability, robustness and regulation. It is impossible to mention all the applications of this theory, but they include: fluid flow in domains of variable configuration; advanced, composite and smart materials; aerospace and mechanical systems; electronic and optical devices; economic models; and biological, medical and chemical processes (for example, to maintain the various constituents at their appropriate levels). We can also mention the so-called delay systems, which are mathematical models used for many diffusion processes, in which time-delayed feedback signals are used to describe propagation phenomena, with applications in population dynamics, plasma physics, ecology, epidemiology, immunology, neural networks, etc. The objective of robust control theory, which generalizes optimal control theory, is to compensate for the undesirable effects of system uncertainties through control actions so that a cost function achieves its minimum for the worst uncertainties. In other words, the goal is to find the best control which stabilizes the fluctuations of the dynamic system with a limited control effort, by taking into account the worst-case disturbances which destabilize the dynamic behavior of the system. The existence, uniqueness, qualitative properties and good behavior under perturbation of solutions of the model are becoming an important prerequisite and are a research domain of their own, especially in view of the possibility of modeling by states and controls. The techniques developed in this book concern the robust control of infinite-dimensional dynamical systems. All these systems are derived from time-dependent coupled PDEs associated with boundary-valued problems that arise in the physical and biological sciences. It is clear that to make

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Preface

a rigorous analysis of these systems, it is necessary to take into account not only the non-linear dynamics of the problems but also the evolutionary and coupled behavior of the PDEs governing such systems. It is thus important to make a good choice of function spaces, both in terms of solvability (in general, a suitable treatment of such systems requires the use of several function spaces) and of the correct modeling of realistic problems. Therefore, our approach in this book is to combine the general theory of control, the optimization theory, the modeling process and the theory of time-dependent coupled PDEs into one complete unified theory. Mathematical foundations essential to the analysis of stabilization, robust control and fluctuation in the context of control systems described by dynamical coupled non-linear PDEs are provided, while remaining accessible to the non-specialist. Therefore, this book will be useful to researchers in mathematics, physics, biology and chemistry, and to professionals involved in complex problems in fluid mechanics, biological systems and material sciences. Most of the topics developed in this book are new or have been published recently. The book is divided roughly into three parts. In the first part, mathematical results necessary to control theory are presented. Proofs are only provided for those results that either cannot be easily found in the standard textbooks, or are useful in order to understand related problems or concern new results. Some essential results for convex functions are given in Chapter 2, and the basic features of Sobolev spaces with useful compactness results in Chapter 3. In Chapter 4, the convex conjugate duality theory (with the Legendre–Fenchel transformation) is developed. Chapter 5 discusses very important tools used in the study of non-linear systems: critical point, Lagrange duality theory and minimax principles. The minimax theorems have many useful applications and play a central role in the notion of stability and robust control theory. The non-convex parametric variational problem for a geometrically non-linear system, by introducing a new gap function, is also studied. This part is illustrated with different applications including the Navier–Stokes equations for fluid mechanics, Maxwell equations for electric and magnetic fields, Ginzburg–Landau equations for ferroelectric models, and the elasticity problem for deformation processes. In the second part of the book, classical optimal control theory and, the heart of this book, robust control theory (on PDEs) are developed. For both, several different realistic cases of observations and controls are analyzed. In the robust control approach, different cases of disturbances are also considered. Linear, bilinear and non-linear control problems for dynamical systems with or without time-varying delays are discussed. In Chapter 6, some elements of functional analysis are introduced: function spaces and linear evolution problems of first order in time; it is considered as a reference chapter. Chapter 7

Preface

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contains the general results and concepts for the optimal control problem of several time-dependent and differential systems, in different contexts. A very basic problem is first given to explain the theory as simply as possible. In each section, the existence, uniqueness and optimality conditions for the optimal solution (for different observation and control situations) are studied. Chapter 8 is devoted to stabilization and robust regulation problems, using some of the new mathematical objects that have been recently introduced in relation to the stabilization of dynamical systems. This chapter contains the essential and fundamental developments of the robust control theory of distributed parameter systems. This area concerns investigation into the control, stability and adjoint control optimization of infinite-dimensional dynamical systems. Moreover, we are interested in the robust regulation of the deviation of the systems from the desired target, by analyzing the full non-linear systems, which models large perturbations to the desired target. Several different mathematical applications are given in these two main chapters to illustrate optimal and robust control theory, such as a biochemical pollutant model (a non-linear problem) and a nuclear fission reactor model (a bilinear problem), the last model also describing cancer chemotherapy. A general case of time-varying delay in non-linear parabolic systems is also studied: the delays occur naturally in biological and chemical systems, in population dynamics, etc. Finally, the last chapter of this part briefly presents some numerical approaches. In the last part of the book, some applications to biological and physical sciences are given. Chapter 10 is devoted to vortex dynamics in superconducting films. In Chapter 11, the multiscale modeling solidification of binary alloys is studied. Chapter 12 concerns the large-scale ocean in the climate system. In Chapter 13, the impact of heat transfer laws on temperature distribution in biological systems with directional blood flow (with application in the cancer treatment) is analyzed. Chapter 14 concerns resource management problems and the stabilization of uncertain species resources (i.e., population dynamics). Chapter 15 presents two other interesting models, namely micropolar fluids (e.g., animal blood) and semiconductor melts. I would like to thank the editors of Springer, especially A. Doyle who proposed that I should write this book, and O. Jackson for his kind suggestions on the layout of the manuscript, and for support and patience. I thank the le-tex publishing services oHG and my PhD student, A. Rasheed, for their help in improving the English language. I am grateful to the commission of INSA of Rennes who allowed me a half-year of CRTC to write this book. I wish to thank the librarians of IRMAR (Institute of Mathematical Research of Rennes), V. Cohoner, A. Guillemer and D. Hervé, who have provided me with all the books and articles necessary for my research, and many friends and colleagues who encouraged me to complete this work. I thank also all the anonymous reviewers (who will recognize themselves, I hope) who, during their reading of my different articles, provided me with very helpful suggestions.

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Last, but not least, I would like to express my acknowledgment and thanks to my wife Géraldine for her patience, support and encouragement during this project and over the years.

Rennes (France), March 2008

Aziz Belmiloudi

Contents

Notation and Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix 1

General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Motivations and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 General Process of the Robust Control Theory . . . . . . . . . . . . . . 6 1.3 Applications to Biological and Physical Sciences . . . . . . . . . . . . . 7 1.3.1 Material Sciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.2 Fluid Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.3 Biological Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.4 Other Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Part I Convex Analysis and Duality Principles 2

Convexity and Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Topological Spaces and Properties . . . . . . . . . . . . . . . . . . . 2.1.3 Hahn–Banach and Separation Between Convex Sets . . . 2.2 Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Closure and Semi-continuous Functions . . . . . . . . . . . . . . 2.2.3 Weak Topologies and Dual Spaces . . . . . . . . . . . . . . . . . . . 2.2.4 Separable Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 Dual of Banach Spaces and Reflexivity . . . . . . . . . . . . . . . 2.2.6 Closure and Continuity of Convex Functions . . . . . . . . . . 2.3 Γ -Regularization and Continuous Affine Functions . . . . . . . . . . .

13 13 13 14 17 19 19 22 24 28 32 37 39

3

A Brief Overview of Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . 43 3.1 Tools and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.1.1 Definitions and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . 43

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3.1.2 Some Fundamental Inequalities and Convergence Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Definition of Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Some Properties of Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Density Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Embedding Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Compactness Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Trace Results and Green’s Formula . . . . . . . . . . . . . . . . . . 3.2.5 Truncation Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.6 Interpolation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

5

Legendre–Fenchel Transformation and Duality . . . . . . . . . . . . . 4.1 Fenchel Conjugate Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Definitions and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Subdifferentials and Superdifferentials of Extended-value Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Definition and Characterization . . . . . . . . . . . . . . . . . . . . . 4.2.2 General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Calculus Rules with Subdifferentials . . . . . . . . . . . . . . . . . 4.2.4 Connection with Directional Derivative . . . . . . . . . . . . . . . 4.3 Applications of the Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Fundamental Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Duality Mapping in Banach Spaces . . . . . . . . . . . . . . . . . . 4.3.3 Duality and Fundamental Equations . . . . . . . . . . . . . . . . . 4.3.4 Euler–Lagrange Equation and the Non-linear Operator . 4.3.5 Minimization of Convex Functions . . . . . . . . . . . . . . . . . . . 4.3.6 General Boundary Value Problems . . . . . . . . . . . . . . . . . .

45 47 49 49 49 50 50 53 54 57 57 57 61 62 62 66 68 70 77 78 79 82 86 93 95

Lagrange Duality Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.1 Frenchel–Rockafellar Duality in Optimization . . . . . . . . . . . . . . . 99 5.1.1 Primal and Dual Problems . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.1.2 Normal and Stability Problems . . . . . . . . . . . . . . . . . . . . . . 103 5.1.3 Optimality Conditions and Existence . . . . . . . . . . . . . . . . 106 5.1.4 Bidual Problem and Duality in Variational Inequalities . 107 5.2 Lagrange Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.2.1 Definitions and Critical Points of Lagrangians . . . . . . . . . 108 5.2.2 Lagrangian Duality and Saddle Points . . . . . . . . . . . . . . . 113 5.2.3 Application and Boundary-value Problems . . . . . . . . . . . . 116 5.3 Minimax Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.3.2 Saddle Point and Properties . . . . . . . . . . . . . . . . . . . . . . . . 127 5.3.3 Banach Spaces and Saddle Points . . . . . . . . . . . . . . . . . . . 131 5.3.4 Connection with Duality and Application . . . . . . . . . . . . . 140

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5.3.5 Ky Fan’s Minimax Inequality and Non-potential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.4 Duality and Parametric Variational Problems . . . . . . . . . . . . . . . 147 5.4.1 Abstract Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 5.4.2 Geometrically Non-linear Lagrangian Representation . . . 151

Part II General Results and Concepts on Robust and Optimal Control Theory for Evolutive Systems 6

Studied Systems and General Results . . . . . . . . . . . . . . . . . . . . . . 163 6.1 Hypotheses and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 6.2 Evolution Problems, Existence and Stability Results . . . . . . . . . 166 6.3 Regularity Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 6.4 Examples of Operators and Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 177 6.4.1 Dirichlet Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . 177 6.4.2 Neumann Boundary Condition . . . . . . . . . . . . . . . . . . . . . . 178 6.4.3 Robin Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . 179 6.4.4 Non-homogeneous Neumann and Dirichlet Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

7

Optimal Control Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 7.2 Basic Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 7.3 Linear Control Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 7.3.1 Position of the Problem, Existence and Uniqueness of the Optimal Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 7.3.2 Optimality Conditions and Identification of the Gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 7.4 Examples of Controls and Observations . . . . . . . . . . . . . . . . . . . . 193 7.4.1 Boundary Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 7.4.2 Pointwise Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 7.4.3 Pointwise Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 7.4.4 Boundary Controls and Boundary Observations . . . . . . . 199 7.4.5 Data Assimilation Problem and Initial Condition Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 7.5 Parameter Estimations and Bilinear Control Problems . . . . . . . 202 7.5.1 State Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 7.5.2 Existence of Optimal Solutions . . . . . . . . . . . . . . . . . . . . . . 203 7.5.3 First-order Optimality Conditions . . . . . . . . . . . . . . . . . . . 204 7.6 Non-linear Control for Non-linear Evolutive PDE Problems . . . 208 7.6.1 State Problem and Assumptions . . . . . . . . . . . . . . . . . . . . . 208 7.6.2 Existence and Uniqueness of the Solution . . . . . . . . . . . . . 210 7.6.3 The Control Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

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7.6.4 Initial Condition Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 7.6.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 8

Stabilization and Robust Control Problem . . . . . . . . . . . . . . . . . 227 8.1 Motivation and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 8.2 Basic Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 8.3 Linear Robust Control Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 232 8.3.1 Position of the Problem, and the Existence and Uniqueness of the Optimal Solution . . . . . . . . . . . . . . . . . . 232 8.3.2 Optimality Conditions and Identification of the Gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 8.4 Examples of Controls, Disturbances and Observations . . . . . . . . 240 8.4.1 Boundary Disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 8.4.2 Pointwise Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 8.4.3 Pointwise Controls and Pointwise Disturbances . . . . . . . . 246 8.4.4 Boundary Controls and Boundary Observations . . . . . . . 247 8.4.5 Data Assimilation Problem and Initial Condition Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 8.5 Bilinear-type Robust Control Problems . . . . . . . . . . . . . . . . . . . . . 253 8.5.1 State Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 8.5.2 Differentiability of the Mapping Solution . . . . . . . . . . . . . 257 8.5.3 Existence of an Optimal Solution . . . . . . . . . . . . . . . . . . . . 260 8.5.4 First-order Necessary Conditions . . . . . . . . . . . . . . . . . . . . 262 8.5.5 Other Situations and Applications . . . . . . . . . . . . . . . . . . . 263 8.6 Non-linear Robust Control for Non-linear Evolutive Problems . 266 8.6.1 State Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 8.6.2 The Perturbation Problem . . . . . . . . . . . . . . . . . . . . . . . . . 268 8.6.3 The Control Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 8.6.4 Initial Condition Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 8.6.5 A Remark on the Robust Boundary Control Problem . . 287 8.6.6 Contraction Mapping and Fixed-point Formulation . . . . 290 8.7 Non-linear Time-varying Delay Systems . . . . . . . . . . . . . . . . . . . . 296 8.7.1 Mathematical Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 8.7.2 Existence and Uniqueness of the Solution . . . . . . . . . . . . . 298 8.7.3 The Control Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 8.7.4 Remarks on Time-varying Delays and Control in the Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

9

Remarks on Numerical Techniques . . . . . . . . . . . . . . . . . . . . . . . . 319 9.1 Introduction and Studied Problem . . . . . . . . . . . . . . . . . . . . . . . . . 319 9.2 Continuous Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 9.2.1 Gradient Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 9.2.2 Conjugate Gradient Algorithm . . . . . . . . . . . . . . . . . . . . . . 322 9.2.3 Lagrange–Newton Method . . . . . . . . . . . . . . . . . . . . . . . . . . 324 9.3 Discrete Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

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xv

9.3.1 Approximation of Robust Control Problems . . . . . . . . . . 328 9.3.2 Discrete Gradient Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 329 9.3.3 Multi-grid Gradient Method . . . . . . . . . . . . . . . . . . . . . . . . 331

Part III Applications in the Biological and Physical Sciences: Modeling and Stabilization 10 Vortex Dynamics in Superconductors and Ginzburg– Landau-type Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 10.1.1 Assumptions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . 343 10.1.2 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 10.2 Existence and Uniqueness of the Solution of the MTDGL Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 10.3 The Perturbation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 10.3.1 Formulation of the Perturbation Problem . . . . . . . . . . . . . 346 10.3.2 Existence and Stability Results . . . . . . . . . . . . . . . . . . . . . . 347 10.4 Differentiability of the Operator Solution . . . . . . . . . . . . . . . . . . . 348 10.5 Robust Control Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 10.5.1 Control in the External Magnetic Field . . . . . . . . . . . . . . 350 10.5.2 Control in the Initial Condition of the Vector Potential . 360 11 Multi-scale Modeling of Alloy Solidification and Phase-field Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 11.1.1 Assumptions and Notations . . . . . . . . . . . . . . . . . . . . . . . . . 374 11.1.2 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 11.2 Existence, Uniqueness and a Maximum Principle . . . . . . . . . . . . 376 11.2.1 Existence and Uniqueness Results . . . . . . . . . . . . . . . . . . . 376 11.2.2 A Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 11.3 The Perturbation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 11.4 Differentiability of the Operator Solution . . . . . . . . . . . . . . . . . . . 380 11.5 Robust Control Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 11.5.1 Disturbance in the Forcing of the Phase-field Parameter 382 11.5.2 Distributed Disturbance in the Initial Condition of the Phase-field Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 12 Large-scale Ocean in the Climate System . . . . . . . . . . . . . . . . . . 395 12.1 Introduction and Formulation of the Problem . . . . . . . . . . . . . . . 395 12.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 12.1.2 Primitive Equations and Study Domain . . . . . . . . . . . . . . 397 12.2 The Perturbation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 12.2.1 Preliminary Results and Weak Formulations . . . . . . . . . . 400 12.2.2 Existence, Uniqueness and Regularity of the Solution . . 405 12.2.3 Comments on the Asymptotic Behavior . . . . . . . . . . . . . . 408

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Contents

12.3 Robust Control Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 12.3.1 Differentiability of the Operator Solution . . . . . . . . . . . . . 411 12.3.2 Existence of an Optimal Solution . . . . . . . . . . . . . . . . . . . . 413 12.3.3 Optimality Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 12.4 Primitive Ocean Equations with Vertical Viscosity . . . . . . . . . . . 418 13 Heat Transfer Laws on Temperature Distribution in Biological Tissues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 13.1.1 Motivation and Statement of the Problem . . . . . . . . . . . . 427 13.1.2 Thermal Damage Calculations . . . . . . . . . . . . . . . . . . . . . . 429 13.1.3 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . 430 13.1.4 Assumptions and Notations . . . . . . . . . . . . . . . . . . . . . . . . . 431 13.2 The State System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 13.2.1 Existence and Stability Results . . . . . . . . . . . . . . . . . . . . . . 432 13.2.2 A Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 13.3 The Perturbation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 13.3.1 Formulation of the Perturbation Problem . . . . . . . . . . . . . 437 13.3.2 Existence and Stability Results . . . . . . . . . . . . . . . . . . . . . . 438 13.4 Robust Control Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 13.4.1 Formulation of the Control Problem and Differentiability439 13.4.2 Existence of an Optimal Solution . . . . . . . . . . . . . . . . . . . . 442 13.4.3 Optimality Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 13.5 Other Situations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 13.5.1 Data Assimilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 13.5.2 Boundary Disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 13.5.3 Finite Number of Measurments . . . . . . . . . . . . . . . . . . . . . 447 13.5.4 Union of a Finite Number of Subdomains . . . . . . . . . . . . . 448 14 Lotka–Volterra-type Systems with Logistic Time-varying Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 14.1 Introduction and Mathematical Setting . . . . . . . . . . . . . . . . . . . . . 451 14.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 14.1.2 Studied Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 14.2 Existence and Uniqueness of the Solution . . . . . . . . . . . . . . . . . . . 454 14.3 The Perturbation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 14.4 Robust Control Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 14.4.1 Formulation of the Control Problem and Differentiability460 14.4.2 Existence of an Optimal Solution . . . . . . . . . . . . . . . . . . . . 462 14.4.3 Optimality Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 14.5 Other Situations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 14.5.1 Disturbance in the Parameter Function p . . . . . . . . . . . . . 468 14.5.2 Remarks on Boundary Control and Habitat Hostility . . 470

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15 Other Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 15.1 Micropolar Fluids and Blood Pressure . . . . . . . . . . . . . . . . . . . . . . 473 15.1.1 Introduction and Mathematical Setting . . . . . . . . . . . . . . 473 15.1.2 Fluctuation and Robust Regulation of the Blood Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 15.2 Semiconductor Melt Flow in Crystal Growth . . . . . . . . . . . . . . . . 478 15.2.1 Introduction and Mathematical Setting . . . . . . . . . . . . . . 478 15.2.2 Fluctuation and Robust Regulation of the Melt Flow Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499

Notation and Symbols

a.e.

Almost everywhere

♣ ♦

End End End End

IN = {0, 1, 2, . . .} IN∗ = {1, 2, . . .} IR IR+ C l IR = IR ∪ {−∞, +∞}

Set of integers Set of non-null integers Field of real numbers Field of non-negative real numbers Field of complex numbers −∞ and +∞ are allowed to the functions under consideration

∈ ∅ A⊂B A∩B A∪B

Belongs to Empty set A is a subset of B Intersection of subsets A and B Union of subsets A and B

|α| α I(α) R(α) max(a, b) min(a, b)

Absolute value of α ∈ C l Complex conjugate of α ∈ C l Imaginary part of α ∈ C l Real part of α ∈ C l The greater of a and b The lesser of a and b

clcoS coS int(C)

Closed convex hull of a set S Convex hull of a set S Interior of a set C

of of of of

definition example proof remark

xx

Notation and Symbols

XS

Indicator function of a set S

F : V −→ IR clF, F cvF dom(F ), D(F ) epiF hypoF lsF

Extended real-valued mapping on V Closure of a mapping F Convex envelope of a mapping F Effective domain of a mapping F Epigraph of a mapping F Hypograph of a mapping F Lower semi-continuous envelope of a mapping F

2V L(V ; W ) Γ (V )

Set of all subsets of V (power set of V ) Space of continuous linear functionals from V to W Set of extended real-valued mappings on V which are pointwise supremum of a family of continuous affine functions Adjoint of the linear operator Λ Dual of a space V Bidual of a space V Weak-star topology defined on V Weak topology defined on V Convex conjugate function of F : V −→ IR, defined as, F ∗ (f ) = sup ( f, uV ,V − F (u)), ∀f ∈ V

Λ∗ V V σ(V , V ) σ(V, V ) F ∗ : V −→ IR

u∈V

lim inf lim sup −→ ∗

Limit infimum Limit supremum Strong convergence in V Weak convergence in V Weak star convergence in V

∂Ω C(Ω) C k (Ω)

Boundary of a domain Ω Space of continuous functions on a domain Ω Space of k times continuously differentiable functions on Ω, k ∈ IN or k = +∞ Space of C ∞ functions on Ω with a compact support in Ω H¨ older space of order α ∈ (0, 1] on Ω i.e., the space of continuous functions u on Ω such that | u(x) − u(y) | sup λ}

are called open half-spaces bounded by H and {u ∈ V : L(u) ≤ λ},

{u ∈ V : L(u) ≥ λ}

are called closed half-spaces bounded by H. All these sets are convex.

Remark 2.23. (i) The hyperplane H is topologically closed if and only if the linear form L is continuous. (ii) The half-spaces bounded by H do not depend on the choice of the equation of the hyperplane H. (iii) If the function L is continuous then the open (respectively closed) halfspace bounded by H will be topologically open (respectively topologically closed). ♦

18

2 Convexity and Topology

Definition 2.24. (Internal point) Let C ⊂ V be a convex set and u be in C. The point u is called internal if every line passing through u meets C in a segment [u1 , u2 ] such that u ∈]u1 , u2 [. Remark 2.25. Every interior point is internal and if the interior of C is nonempty then every internal point is interior. ♦ Definition 2.26. (Closed convex hull) For S be any subspace of V , the intersection of all the closed convex subsets containing S is the smallest closed convex subset containing S. It is also the closure of the convex hull of S and it is called the closed convex hull of S denoted by clcoS. Remark 2.27. In general the closed convex hull of a subspace S of V is not the convex hull of the closure of S, i.e., clcoS = coS (where S is the closure of S). ♦ The proofs of all the following results of this section can be found, for example, in Trèves [283]. Next we recall the Hahn–Banach theorem in its analytical form (based on the well-known Zorn’s lemma argument). Theorem 2.28. (Hahn–Banach, analytical form) Let p : V −→ IR be a sublinear mapping such that p(αu) = αp(u), ∀u ∈ V and ∀α > 0, p(u + v) ≤ p(u) + p(v), ∀(u, v) ∈ V × V. Let also S be any vector subspace of V and let g : S −→ IR be a linear mapping such that g(u) ≤ p(u), ∀u ∈ S. Then there exists a linear form f on V which extends g such that g(u) = f (u), ∀u ∈ S and f (u) ≤ p(u), ∀u ∈ V.

Before giving the geometric form of the Hahn–Banach theorem, we introduce the following separate definition. Definition 2.29. (Separate sets by hyperplane) Let H be an affine hyperplane with equation L(u) = λ and let S1 , S2 be two subsets of V . (i) The hyperplane H is said to separate (strictly separate) S1 and S2 if S1 lies in one of the closed (open) half-spaces bounded by H and S2 in the other, i.e., (for example) L(u) ≤ λ, ∀u ∈ S1

and L(u) ≥ λ, ∀u ∈ S2 , (for separation),

L(u) < λ, ∀u ∈ S1

and L(u) > λ, ∀u ∈ S2 , (for strict separation).

(ii) The hyperplane H is said to separate properly S1 and S2 if S1 lies in one of the closed half-spaces bounded by H and S2 in the other and both are not contained in H.

2.2 Convex Functions

19

Definition 2.30. (Supporting hyperplane) Let S be a subset of V and H be a closed hyperplane which contains at least one point u ∈ S. The hyperplane H is said to be a supporting hyperplane of the subset S when S is completely contained in one of the two closed half-spaces bounded by H, and u is said to be a supporting point of S. We now recall the Hahn–Banach theorem in its geometrical form and its consequences for the separation of convex sets. Theorem 2.31. (Hahn–Banach, first geometric form) Let V be a topological vector space, C1 ⊂ V be an open non-empty convex subset and C2 ⊂ V be a non-empty convex subset which does not intersect C1 (i.e., C1 ∩ C2 = ∅). Then there exists a closed hyperplane H which separates C1 and C2 . Theorem 2.32. (Hahn–Banach, second geometric form) Let V be a locally convex topological vector space, C1 ⊂ V be a closed non-empty convex subset and C2 ⊂ V be a compact non-empty convex subset which does not intersect C1 . Then there exists a closed hyperplane H which strictly separates C1 and C2 . We can now give some useful corollaries (the first is an application of Theorem 2.31 and the second is an application of Theorem 2.32). Corollary 2.33. Let V be a topological vector space, C be a convex subset of V with non-empty interior and H be a supporting hyperplane of C. Then every boundary point of C is a supporting point of C. Corollary 2.34. Let V be a locally convex topological vector space, C be a closed convex subset of V and H be a supporting hyperplane of C. Then C is the intersection of the closed half-spaces which contain it. Let us give finally a very useful corollary that one seeks to prove the density of a topological subspace (an application of Theorem 2.32). Corollary 2.35. Let V be a Hausdorff locally convex space and S be a topological subspace of V such that S = V (S denotes the closure of S). Then there exists a non-zero continuous linear form f over V such that f (u) = 0,

∀u ∈ S.

2.2 Convex Functions 2.2.1 Definitions In the sequel we will work with the extended real-valued functions, that is IR-valued functions2 (i.e., −∞ and +∞ are allowed to the functions under consideration), except mentioned contrarily. 2

IR = IR ∪ {−∞, +∞}.

20


Definition 2.36. (Convex and concave functions) Let C be a non-empty convex subset of V and F be a mapping of C into IR. (i) The mapping F is said to be convex on C if, for all pair (u, v) in C × C and all α in ]0, 1[, the inequality (if the right-hand side is well defined) F (αu + (1 − α)v) ≤ αF (u) + (1 − α)F (v)

(Jensen s inequality)

(2.6)

holds. The function F is called strictly convex on C when (2.6) holds as a strict inequality if u = v with F (u) < ∞ and F (v) < ∞ . (ii) The function F is said to be (strictly) concave on C if the function −F is (strictly) convex on C. Remark 2.37. (i) If the mapping F : V −→ IR is convex, we obtain easily that, for every finite subset of elements (ui , αi )i=1,n of V × IR+ such that i=1,n αi = 1 (if the right-hand side is well defined) F(

αi ui ) ≤

i=1,n

αi F (ui ).

(2.7)

i=1,n

The inequality (2.7) is called the general inequality of Jensen. (ii) If the mapping F : V −→ IR is convex, the following level sets (for each λ ∈ IR) {u ∈ V : F (u) ≤ λ} and {u ∈ V : F (u) < λ} are convex (but the converse is false and such functions are called quasiconvex). ♦ Remark 2.38. (i) Let S be a non-empty subset of V and h be a mapping of S into IR. We can associate with it the extended mapping F on V by F (u) = h(u) if u ∈ S

and F (u) = +∞ if u ∈ / S.

(2.8)

The function F is convex if and only if the subset S is convex and the function h is convex. Therefore, we need only to consider functions defined on V everywhere. (ii) The study of a convex set is naturally reduced to the study of a convex function. Because for a subset S of V it is clear enough that the function XS : S −→ IR defined by XS (u) = 0 if u ∈ S

and XS (u) = +∞ if u ∈ /S

(called the indicator function of S) is a convex function if and only if S is a convex set. (iii) The notion of strict convexity is very important to derive the uniqueness of the minimum of a function (see below). ♦


21

Example 2.39. Let (V, . ) be a real normed vector space then the functions F : V −→ IR defined by F (u) := u p , 1 ≤ p < ∞, ∀u ∈ V ♣

are convex functions.

Let us now introduce the useful notions of effective domain, epigraph and hypograph of a function. Definition 2.40. (Effective domain of a function) Let F be a mapping of V into IR (extended real-valued functions). The domain (or also the effective domain) of F , domF (sometimes D(F )), is the non-empty set domF = {u ∈ V : F (u) < +∞}. It is clear that if the mapping F is convex then the domain domF is a convex set. Moreover, if F is given by (2.8), domF is a subset of S. Definition 2.41. (Epigraph and hypograph of a function) Given a mapping F : V −→ IR (extended real-valued function): (i) The epigraph of F is the non-empty set epiF = {(u, λ) ∈ V × IR : F (u) ≤ λ}. Its strict epigraph epis F is defined by epis F = {(u, λ) ∈ V × IR : F (u) < λ}. (ii) The hypograph of F is the non-empty set hypoF = {(u, λ) ∈ V × IR : F (u) ≥ λ}. Its strict hypograph hypos F is defined by hypos F = {(u, λ) ∈ V × IR : F (u) > λ}.

The following proposition shows that the epigraph of the pointwise supremum of a family of functions is the intersection of the epigraphs of these functions. Proposition 2.42. Let (Fi )i∈J be an arbitrary family of functions of V into IR and F their pointwise supremum F := supi∈J Fi . Then epiFi = epiF. i∈J

Remark 2.43. (i) The epigraph or hypograph of a function is a subset of V ×IR but not of V × IR. (ii) If (u, λ) ∈ V × IR is in the epigraph of a function then u ∈ V is in

22


the effective domain of the function under consideration. More precisely the projection of the epigraph of a function on V is its effective domain. ♦ Let us now give some characterizations and properties of convex functions. Proposition 2.44. Given a mapping F : V −→ IR. The following properties are equivalent: (i) F is a convex function on V (ii) its epigraph is a convex subset of V × IR (iii) its strict epigraph is a convex subset of V × IR.

Remark 2.45. (i) A similar characterization of concave function can be given in terms of its hypograph. (ii) The function which is identically the constant function +∞ is convex since epi(+∞) = ∅ is a convex set, and the constant function −∞ is also convex since its epigraph is V × IR. These functions are also concave. ♦ Proposition 2.46. (i) If F is a convex function from V into IR then λF is a convex function, ∀λ ∈ IR+ . (ii) If F and G are convex functions from V into IR then F + G is a convex function. (iii) Let (Fi )i∈J be an arbitrary family of convex functions of V into IR, then their pointwise supremum F := supi∈J Fi is a convex function. We can now introduce the following definition. Definition 2.47. (Convex envelope) Given a mapping F : V −→ IR. The convex envelope of F is the greatest convex function which is less than F , and is denoted by cvF . Otherwise cvF = sup{convex functions, G : V −→ IR : G ≤ F }.3

Let us now introduce an essential notion of regularity for an optimization problem, i.e., the semi-continuity. 2.2.2 Closure and Semi-continuous Functions Definition 2.48. (Semi-continuous function) Let us consider the mapping F : V −→ IR: (i) The function F is said to be lower semi-continuous (l.s.c), if for each u∈V lim inf F (v) ≥ F (u), (2.9) v−→u

where lim inf F (v) = v−→u

sup

neighborhoods of u. 3

inf F (v) and N (u) is the family of all open

O∈N (u) v∈O

That is G(u) ≤ F (u), ∀u ∈ V .


23

(ii) The function F is said to be upper semi-continuous (u.s.c), if −F is a lower semi-continuous function. In other words, if lim sup F (v) = v−→u

inf

sup F (v) ≤ F (u).

O∈N (u) v∈O

Remark 2.49. (i) Since the converse inequality of (2.9), i.e., lim inf F (v) ≤ v−→u

F (u) always holds, then the inequality (2.9) is equivalent to the equality F (u) = lim inf F (v). v−→u

(ii) A function F is continuous at point u if and only if F is both lower and upper semi-continuous at point u. ♦ Let us now give some geometrical characterizations and properties of semicontinuous functions. Proposition 2.50. For a function F : V −→ IR, the following three properties are equivalent: (i) F is lower semi-continuous on V (ii) the epigraph of the function F , epiF is a closed subset of V × IR (iii) the level sets {u ∈ V : F (u) ≤ λ} ⊂ V are closed, for each λ ∈ IR.

In particular we have Corollary 2.51. A subset C of V is closed if and only if its indicator function XC is lower semi-continuous. Proposition 2.52. Let (Fi )i∈J be an arbitrary family of lowersemi-continuous functions of V into IR, then their pointwise supremum F := supi∈J Fi is a lower semi-continuous function. We can now introduce the following definitions. Definition 2.53. (Lower semi-continuous envelope) Given a mapping F : V −→ IR. The lower semi-continuous envelope of F is the greatest lower semi-continuous function which is less than F , and is denoted by lsF . Otherwise lsF = sup{lower semi-continuous functions, G : V −→ IR : G ≤ F }.

Definition 2.54. (Closure of function) Given a mapping F : V −→ IR. The closure (or the lower semi-continuous regularization) of F is the function F : V −→ IR (sometimes clF ) defined by F (u) = lim inf F (v), v−→u

f or every u ∈ V,

or equivalently epiF = epiF (the closure of epiF ).

24


An important property of lower semi-continuous functions is given by the following well-known theorem of Weierstrass, which corresponds to the attainment of the minimum on a compact set (based on Propositions 2.14 and 2.50). Theorem 2.55. (Weierstrass) Let V be a Hausdorff topological space, K be a compact subset of V and F : V −→ IR be a given lower semi-continuous function. Then the function F takes a minimum value on K, i.e., there exists at least one point u ∈ K such that F (u) = inf F (v). v∈K

Moreover, if F takes only finite values then F is bounded from below.

As a consequence of the last result, we have (according to Lemma 2.13 and Proposition 2.50) the following corollary. Corollary 2.56. Let V be a Hausdorff topological space and F : V −→ IR be a given function such that all its level sets are compact (F is said to be inf-compact). Then the function F is lower semi-continuous and its infimum is attained. Let us now introduce the notion of weak topologies. 2.2.3 Weak Topologies and Dual Spaces Let V be a topological vector space. Definition 2.57. (Dual topological space) The vector space V (or V ∗ ) of continuous linear functionals over V is said to be the (topological) dual of V . For f in V and u in V we will denote, in general, the value at u of the linear functional f by f, uV ,V ( ., .V ,V is said to be the scalar product between V and V ). We can thus introduce over V (respectively over V ) the topology of weak convergence over V (respectively over V ). This will be termed weak topology of V (respectively of V ) associated with the duality between V and V and will be denoted by σ(V , V ) (respectively by σ(V, V )). Definition 2.58. (Weak topology) The topology σ(V, V ) (respectively σ(V , V )) is the weakest topology on V (respectively on V ) which makes the linear forms φf : u ∈ V −→ f, uV ,V ∈ IR, continuous for all f ∈ V (respectively the linear forms ψu : f ∈ V −→ f, uV ,V ∈ IR, continuous for all u ∈ V ). It is called the topology of V (respectively of V ) weakned by V (respectively by V ). Remark 2.59. (i) The functions pf : u ∈ V −→| f, uV ,V |∈ IR+ are continuous seminorms on V , for all f ∈ V . Moreover, if we assume that the family


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P of all seminorms defined by: P = {pf : f ∈ V } is separating, then the topology σ(V, V ), generated by the family P, makes V into a locally convex space: a fundamental system of neighborhoods of any u0 in V is given by ({u ∈ V : | fi , u − u0 V ,V |< , ∀i ∈ J}, where J is a finite set, fi ∈ V and > 0). (ii) If V is a locally convex space, then P is separating by the Hahn–Banach theorem (second geometric form) and consequently, (V, σ(V, V )) (V with the topology σ(V, V )) is a locally convex space. (iii) A sequence (un )n∈IN converges to u ∈ V in the topology σ(V, V ) if and ♦ only if for all f ∈ V , ( f, un − uV ,V )n∈IN converges to 0. In general the dual of V , denoted by V and called the bidual of V , and V are different. In the context of normed space, it is observed that, this last property characterizes a special class of normed spaces called reflexive. Moreover, in general the weak topologies σ(V, V ) and σ(V , V ) are different. For more clearness, we preserve the name of weak topology only for σ(V, V ) and the weak topology for σ(V , V ) will be called weak star topology (or weak* topology). In contrast to these topologies, the initial topologies on V (respectively on V ), will be called strong topology on V (respectively on V ). We denote by −→, and ∗ the strong convergence in V , the weak convergence in V and the weak star convergence in V . Similarly we will speak about strong neighborhood, strongly closed, strongly bounded, etc., and weak (or weak*) neighborhood, weakly (or weakly*) closed, weakly (or weakly*) bounded, etc. Weak Topology Proposition 2.60. If V is a (Hausdorff ) locally convex space, then: (i) The topology σ(V, V ) (respectively σ(V , V )) is a (Hausdorff ) locally convex topology on V (respectively on V ). (ii) The weak topology σ(V, V ) (respectively the weak star topology σ(V , V )) is always weaker than the strong topology on V (respectively on V ). The equality holds if and only if V is a finite dimensional space. Remark 2.61. (i) Every strongly convergent sequence is weakly convergent (the converse is, in general, false). (ii) The closed (respectively open) subsets of the weak topology σ(V, V ) are closed (respectively open) subsets of the strong topology (the converse is, in general, false). (iii) If V is a finite dimensional space, then every weakly convergent sequence is strongly convergent (and vice versa). ♦ Example 2.62. Suppose that (V, . ) is a Banach space, equipped with the norm . , then we can prove easily that:

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(i) The subset B = {u ∈ V : u = 1} is never a closed subset for the weak topology σ(V, V ) and the weak closure of B is the set D = {u ∈ V : u ≤ 1}. (ii) The subset D = {u ∈ V : u < 1} is never an open subset for the weak ♣ topology σ(V, V ). It is known that the closed subsets of the weak topology are closed subsets of the strong topology and the converse is, in general, false. The next result shows that these two concepts coincide, in the case of convex subsets (by applying Hahn–Banach theorem). Theorem 2.63. If V is a locally convex space and C ⊂ V is a convex subset, then C is weakly closed if and only if it is strongly closed. Proof. Let C be strongly closed and u0 be any point in the complement of C. We have to prove that there exists an open neighborhood of u0 , in the weak topology σ(V, V ), which doesn’t intersect C. According to the Hahn–Banach theorem (second geometric form), there exists a closed hyperplane which strictly separates {u0 } and C. So, there exists (f, λ) ∈ V × IR such that

f, u0 V ,V < λ < f, vV ,V

for all v ∈ C.

Consider the following open set (in the weak topology σ(V, V )): B = {v ∈ V : f, vV ,V < λ}, we have that u0 ∈ B, B ∩ C = ∅ and then the result of Theorem 2.63 follows. In particular, we have that: Corollary 2.64. (Mazur theorem) If V is a metrizable locally convex space, and the sequence (un )n∈IN converges weakly to u, then there exists a sequence of convex combinations (vk )k∈IN : vk = λk,n un , λk,n ≥ 0, λk,n = 1, with J a finite set n∈J

n∈J

which converges strongly to u. Proof. Let H be the convex hull of (un )n∈IN and C its weak closure, the weak limit of the sequence (un )n∈IN is in C. Because of Theorem 2.63, we can deduce that u is in the strong closure of H. Since V is metrizable then there is some sequence (vk )k∈IN converging strongly to u. Moreover, we have an interesting result pertaining to lower semi-continuous convex functions when the topology of V is weakened.


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Proposition 2.65. Let V be a locally convex space. Then any convex function F : V −→ IR is weakly lower semi-continuous (in the weak topology σ(V, V )) if and only if it is (strongly) lower semi-continuous. In particular, if V is metrizable and the sequence (un )n∈IN converges weakly to u then F (u) ≤ lim inf F (un ). n

Proof. (⇒) A simple consequence of the fact that σ(V, V ) is a subset of the original (strong) vector topology of V . (⇐) Suppose that F is lower semi-continuous and prove that F is weakly lower semi-continuous. For this it is sufficient to prove that the following subsets: Hλ = {v ∈ V : F (u) ≤ λ}, are closed in the weak topology σ(V, V ). Since F is lower semi-continuous and convex then Hλ is convex and strongly closed. According to Theorem 2.63, we have that Hλ is weakly closed. As an immediate corollary of the last proposition we have Corollary 2.66. Corollary 2.66. Let (V, . ) be a normed space, so that V is a Banach space with the dual norm . V defined by f V = sup | f, uV ,V | . u =1

If a sequence (un )n∈IN converges weakly to u, then (un )n∈IN is uniformly bounded in V . Moreover, u ≤ lim inf un . n

Proof. The first result is a consequence of the classical Banach Steinhaus’s theorem (or principle of uniform boundedness theorem). The second result is a simple consequence of the fact that the norm is a convex and continuous function. Weak* Topology On the dual space V , the family of seminorms P ∗ = {qu =| ψu |: u ∈ V } is separating, consequently the topology σ(V , V ), which is generated by P ∗ , makes V into locally convex space: a fundamental system of neighborhoods of any f0 in V is given by ({f ∈ V : | f − f0 , ui V ,V |< , ∀i ∈ J}, where J is a finite set, ui ∈ V and > 0). Moreover (as in the topology σ(V, V )), a sequence (fn )n∈IN converges to f in the topology σ(V , V ) if and only if ( fn − f, uV ,V )n∈IN converges to 0, for all u ∈ V . A priori, we can introduce the second dual Z of the locally convex space V with respect to (w.r.t in short) the topology σ(V , V ) (i.e., (V, σ(V , V ))) as follows:

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Z = {Λ : V −→ IR; Λ linear and continuous w.r.t the topology σ(V , V )}. The next proposition shows that the dual of V endowed with the topology σ(V , V ) can be identified with V . Proposition 2.67. Let Λ be an element of Z. Then there exists u ∈ V such that Λ(f ) = f, uV ,V , ∀f ∈ V . As a consequence, we have Corollary 2.68. Corollary 2.68. Let H be an affine hyperplane of V , closed with respect to the topology σ(V , V ). Then H = {f ∈ V : f, uV ,V = λ}, where u is a non-zero element of V and λ ∈ IR.

Remark 2.69. Weak topologies are interesting in regards to compactness. Indeed, weak topologies have less open sets and have on the other hand more compact sets than the finer (original) topology. Moreover, the compact sets play a fundamental role when we seek to analyze the existence results. ♦ Using Tychonoff’s theorem, which states that the product of an arbitrary number of compact spaces is still compact with respect to the corresponding product topology, we have the following fundamental result. Theorem 2.70. (Banach–Alaoglu) Let U be a neighborhood of 0 ∈ V . Then the set K = {f ∈ V : | f, uV ,V |≤ 1, ∀u ∈ U} is a compact subset of V with respect to the weak* topology σ(V , V ). As an immediate consequence, we have the following Banach–Alaoglu– Bourbaki’s theorem (in the case of normed spaces). Corollary 2.71. (Banach–Alaoglu–Bourbaki) If V is a normed space equipped with the norm . V , then the unit ball in V : BV (0, 1) = {f ∈ V : f V ≤ 1} is a compact subset of V with respect to the weak* topology σ(V , V ).

It is known that in general in a compact set there are sequences without converging subsequences. The next results show that, in a particular type of topological vector space, each sequence in a compact set has a converging subsequence. 2.2.4 Separable Spaces Definition 2.72. (Separability) Let V be a topological vector space. We say that V is separable if there exists a subset D of V countable and dense.


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Proposition 2.73. Let V be a separable metric space and U be a subspace of V . Then U is separable. Proof. Let (un )n∈IN be a countable and dense sequence in V and let (ηm )m∈IN be a real positive sequence converging to 0. If, arbitrarily, we choose the values dm,n ∈ B(un , ηm ) ∩ U , it is clear that the sequence (dm,n ) constitute a countable and dense subset in U . Proposition 2.74. Let V be a Banach space such that its dual V is separable. Then V is separable. The converse is false in general. Proof. Let (fn )n∈IN be a countable and dense sequence in V and consider the sequence (un )n∈IN in V such that un V = 1 and fn , un V ,V ≥

fn V . 2

It is clear that W0 the vector space on Q l generated by the sequence (un )n∈IN is countable and dense in W the vector space on IR generated by the sequence (un )n∈IN . Prove now that W is dense in V . For this, let f be in V such that

f, uV ,V = 0, for all u ∈ W and prove that f = 0 (see Corollary 2.35). Let > 0 be given then there exists n > 0 such that f − fn V < . Then (since 3 un ∈ W , un V = 1 and f, un V ,V = 0) fn V ≤ fn , un V ,V = fn − f, un V ,V + f, un V ,V ≤ . 2 3 Consequently, f V ≤ fn − f V + fn V ≤ and then f = 0. So, W is dense in V and then W0 is dense in V . We can conclude that the space V is separable. The following results show that the properties of separability are closely dependent on the metrizability of the weak topology. Definition 2.75. (Metrizability) A set D is said to be metrizable in the weak* topology σ(V , V ) if there exists a metric defined in D such that the associate topology coincides with σ(V , V ). Theorem 2.76. Let V be a Banach space and V be its dual. Then the closed unit ball of V , BV (0, 1), is metrizable in the weak* topology σ(V , V ) if and only if V is separable. Proof. Let (un )n∈IN be a subset countable and dense in BV (0, 1). For a couple (f, g) in BV (0, 1) we define the following function: d(f, g) =

∞ 1 | f − g, un V ,V | . n 2 n=1

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It is clear that the function d is a metric, since it satisfies all the metric axioms. (⇒) Suppose that BV (0, 1) is metrizable in the topology σ(V , V ). Let Un = {f ∈ BV (0, 1) : d(f, 0) < 1/n} and let Hn be a neighborhood of 0 for the topology σ(V , V ) in BV (0, 1) such that Hn ⊂ Un . We can assume that Hn = {f ∈ BV (0, 1) : | f, uV ,V |< n for all u ∈ Dn }, where Dn is a finite subset ∞ ∞of V and n > 0. We have that D := i=1 Dn is a countable set and i=1 Hn = {0}, and then If f, uV ,V = 0 for all u ∈ D then f = 0. Consequently, the vector space generated by D is dense in V and then the space V is separable. (⇐) Suppose now that the space V is separable. Prove now that the topology generated by the metric d coincides on BV (0, 1) with σ(V , V ). In order to deal with this issue, we will proceed in two stages: Stage 1: Let f0 ∈ BV (0, 1) and take a neighborhood H of f0 for the topology σ(V , V ) by H = {f ∈ BV (0, 1) : | f − f0 , vi V ,V |< for all i = 1, . . . , k}, where vi ∈ BV (0, 1) (i.e., vi ≤ 1), for all i = 1, . . . , k, and > 0. Prove that there exists η > 0 such that Uη = {f ∈ BV (0, 1) : d(f, f0 ) < η} ⊂ H. Since the sequence (un )n∈IN is dense in BV (0, 1), then, for each i, we can find an integer ni such that uni − vi < /4. Let η > 0 such that η < 2−(ni +1) , for all i = 1, . . . , k and prove that Uη ⊂ H. Let f be in Uη , we have f ∈ BV (0, 1) and d(f, f0 ) < η, and then | f − f0 , uni V ,V |< 2ni η
0, prove that there exists a neighborhood H of f0 for the topology σ(V , V ) in BV (0, 1) such that H ⊂ U = {f ∈ BV (0, 1) : d(f, f0 ) < η}. For this, we take H as H = {f ∈ BV (0, 1) : | f − f0 , ui V ,V |< for all i = 1, . . . , k},


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where ui ∈ BV (0, 1), and we prove the existence of > 0 and k ∈ IN such that H ⊂ U. If f is in H, we have then ∞ 1 | f − f0 , un V ,V | n 2 n=1 k ∞ 1 1 = |

f − f , u | + | f − f0 , un V ,V | . 0 n V ,V n n 2 2 n=1

d(f, f0 ) =

n=k+1

Since | f − f0 , un V ,V |< for all n = 1, . . . , k, un ∈ BV (0, 1), f ∈ BV (0, 1) and f0 ∈ BV (0, 1) we can deduce that d(f, f0 ) ≤

k ∞ 1 1 1 + 2 < + k−1 . n n 2 2 2 n=1 n=k+1

We can choose such that 0 < < η/2 and k sufficiently large such that 1/2k−1 < η/2, and we obtain the existence of H. We can obtain a generalization of the above result. Theorem 2.77. If the topological vector space V is separable, and K a subset of the dual space V is weakly* compact, then K is metrizable in the weak* topology σ(V , V ). As a consequence, each sequence in K has a convergent subsequence. Proof. The proof ∞ is leftn as an exercise (by replacing the distance d in V by d(f, g) = n=1 (1/2 ) min (| f − g, un V ,V |, 1), by proving the uniform continuity of any element f ∈ K and by using the same technique as to obtain the second result of the above theorem).

Theorem 2.78. Let V be a Banach space and V be its dual. Then the closed unit ball of V , BV (0, 1), is metrizable in the weak* topology σ(V, V ) if and only if V is separable. Proof. To prove the implication: V separable ⇒ BV (0, 1) is metrizable in σ(V , V ), we can use the proof of Theorem 2.76, by exchanging the roles of V and V . The converse of this theorem is more delicate and we can find the proof, e.g., in Dunford and Schwartz [108]. As a corollary, we have (according to Banach–Alaoglu–Bourbaki’s theorem) the following. Corollary 2.79. Let V be a Banach separable space and (fn ) be a bounded sequence in V . Then there exists a subsequence denoted also by (fn ) converging for the topology σ(V , V ).

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2.2.5 Dual of Banach Spaces and Reflexivity A particular case is when V is a normed topological space, equipped with the norm . . In this case the dual space V is a Banach space with the dual norm . V (that can be denoted by . ∗ ) defined by f V = sup | f, uV ,V | . u =1

Example 2.80. Let Ω be a domain in IRd , d ∈ IN∗ . Then: (i) The dual of the space V := C(Ω) (of real-valued and continuous functions on the domain Ω) is the space V := M(Ω) of Radon measures μ with

μ, uV ,V := udμ ∀u ∈ V. Ω

(ii) The dual of the space L (Ω), 1 ≤ p < ∞ (of real-valued and p-integrable, ∗ in the Lebesgue sense, functions on the domain Ω) is the space Lp (Ω), where p∗ is the conjugate to p, i.e., (1/p) + (1/p∗ ) = 1. p

(iii) The dual of the space L1 (Ω) is the space L∞ (Ω) (of real-valued and essentially bounded functions on the domain Ω). (iv) The dual of the space L∞ (Ω) is the space of Borel measures and contains ♣ the space L1 (Ω). We can introduce the bidual of V , i.e., the dual of V , denoted by V with the bidual norm ξ V = sup | ξ, f V ,V | . f V =1

We have a canonical immersion (or a natural embedding) J : V −→ V defined by: let u ∈ V be given, the application f ∈ V −→ f, uV ,V ∈ IR constitute a linear and continuous form, i.e., is an element of V , denoted by Ju. We then have

Ju, f V ,V = f, uV ,V , ∀u ∈ V, ∀f ∈ V . It is clear that the immersion J is linear and continuous, and J(V ) is a closed subset of V . Moreover, J is an isometric injection, i.e., Ju V = u , ∀u ∈ V . Remark 2.81. (i) The function J is not in general surjective, i.e., J(V ) = V . (ii) By using the function J, we can always identify the space V to a subspace ♦ of its bidual V . Definition 2.82. (Reflexive space) Let V be a Banach space and J a canonical immersion of V into V . The space V is said to be reflexive if J(V ) = V (J is surjective), i.e., it can be identified under J with its bidual V .


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The next result shows an important characterization of the reflexive spaces. Theorem 2.83. (Characterization of Kakutani) Let (V, . ) be a Banach space, with norm . . The space V is said to be reflexive if and only if its closed unit ball, BV (0, 1) = {u ∈ V ; u ≤ 1}, is compact in the weak topology σ(V, V ), where V is its dual. This implies that every bounded sequence admits a weakly converging subsequence. Proof. If J is reflexive then J is a linear, continuous and bijective function. Consequently, J −1 is linear and continuous with respect to the strong topologies V and V (then J −1 is also an isometric function). It is clear that (∀f ∈ V , and ∀η > 0) J({u ∈ V ; | f, uV ,V |≤ η}) = J({ξ ∈ V ; | ξ, f V ,V |≤ η}), so that the topology J −1 (σ(V , V )) coincides with the topology σ(V, V ). Since the closed unit ball in V is weak* compact, then it is the unit closed ball of V . Conversely, if the closed unit ball in V is compact (for the weak topology σ(V, V )), then as J(BV (0, 1)) is closed, and by using the result of Goldstine, based on the result of Helly (see below), it coincides with BV (0, 1) and then J(V ) = V . Now we give the results previously used in the proof of the theorem (that can be proved easily). Lemma 2.84. (Helley) Let (V, . ) be a Banach space, fi , i = 1, n, n elements in V and λi , i = 1, n, n real values. Then the following properties are equivalent: (i) for all η > 0, there exists uη in the closed unit ball of V ( uη ≤ 1) such that | fi , uη V ,V − λi |≤ η, ∀i = 1, n (ii) for all βi ∈ IR, i = 1, n, we have βi λi |≤ βi f i V . | i=1,n

i=1,n

Proof. The proof is left to the reader as an exercise.

Lemma 2.85. (Goldstine) Let V be a Banach space. Then J(BV (0, 1)) is dense in BV (0, 1) for the weak topology σ(V , V ).

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Proof. Let ξ ∈ BV (0, 1) and take a neighborhood of ξ (for the topology σ(V , V )) of the form Ω = {ψ ∈ V : | ψ − ξ, fi V ,V |< η, ∀i = 1, n}, where fi , i = 1, n are n elements in V . To prove the result of the lemma, it is sufficient to prove that J(BV (0, 1))∩ Ω = ∅. Otherwise, we need only to find u ∈ BV (0, 1) such that | fi , uV ,V − ξ, fi V ,V |< η, ∀i = 1, n. Let then λi be the values ξ, fi V ,V , for i = 1, n, then for βi ∈ IR, i = 1, n, we have (since ξ V ≤ 1) βi λi |=|

ξ, βi fi V ,V |≤ βi f i V . | i=1,n

i=1,n

i=1,n

By using Helley’s lemma, it follows that there exists an element uη of BV (0, 1) such that | fi , uη V ,V − λi |< η, ∀i = 1, n, and then J(uη ) ∈ J(BV (0, 1)) ∩ Ω. Now we give some elementary properties of the reflexive spaces (by using the characterization of Kakutani). Proposition 2.86. Let V be a reflexive Banach space and U be a closed subspace of V (with the norm induced by V ). Then U is reflexive. Proof. The proof follows by proving that the topology σ(U, U ) coincides with the trace on U of the topology σ(V, V ) (by using the restrictions and the prolongations of the linear forms), and by using Theorem 2.63 (closed for strong topology and convex), in order to obtain that BU (0, 1) is compact for the topology σ(V, V ) and then BU (0, 1) is compact for the topology σ(U, U ). As corollaries, we have the following. Corollary 2.87. Let V be a Banach space and let V be its dual. Then V is a reflexive space if and only if V is a reflexive space. Proof. (⇒) According to Banach–Alaoglu–Bourbaki’s theorem (Corollary 2.71), we have that BV (0, 1) is a compact subset of V with respect to the topology σ(V , V ). Moreover, as V is reflexive, σ(V , V ) = σ(V , V ”). Consequently, BV (0, 1) is a compact subset with respect to the topology σ(V , V ) and then V is reflexive (according to Kakutani’s theorem (Theorem 2.83)). (⇐) According to (⇒) we have that V is reflexive (since V is reflexive) and then, because of Proposition 2.86, J(V ) is reflexive (since J(V ) is a closed subset of V ). Hence, V is reflexive.


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Corollary 2.88. Let V be a reflexive Banach space and K be a closed, convex and bounded subset of V . Then K is compact for the topology σ(V, V ). Proof. Because of Theorem 2.63 and Theorem 2.83, we have first that K is closed for the topology σ(V, V ) and then K is compact for the topology σ(V, V ). Corollary 2.89. Let V be a reflexive Banach space and let V be its dual. Then V is separable if and only if V is separable. Proof. (⇒) If V is reflexive and separable then V = J(V ) is reflexive and separable. Consequently, V is separable (according to Corollary 2.87). (⇐) Because of Proposition 2.74 (the reflexivity assumption is not necessary). Now we give the interesting Eberlein–Smulian’s theorem, that shows the characterization of the reflexive space by sequences. Theorem 2.90. (Eberlein–Smulian) Let V be a Banach space. Then V is reflexive if and only if for any bounded sequence (un ) in V , there exists a subsequence, denoted also by (uk ), converging for the topology σ(V, V ). Proof. (⇒) Let U0 be the span of u1 , u2 , . . . , un , . . . , and U the closure of U0 . Then the subspace U of V is closed, separable and reflexive (according to Proposition 2.86), so U is separable (according to Corollary 2.89). Consequently, BU (0, 1) = BU (0, 1) is metrizable in the topology σ(U , U ) = σ(U, U ) (according to Theorem 2.78) and then BU (0, 1) is compact (characterization of Kakutani) and metrizable in the topology σ(U, U ). We can then extract a subsequence of (un ) (denoted also by (un )) which converges in the topology σ(U, U ) and then (un ) converges in σ(V, V ). (⇐) This result (the veritable result of Eberlein–Smulian) is more delicate to prove; see, e.g., Dunford and Schwartz [108] and Holmes [160] for details. Definition 2.91. (Uniformly convex Banach spaces) A Banach space (V, . ) is said to be uniformly convex, for the norm . , if for all η > 0, there exists δ > 0, such that if u and v are in BV (0, 1) : u − v > η then

u+v < 1 − δ. 2

The uniform convexity is a geometrical property of a Banach space. In the following theorem, we prove that this property involves a topological property. Theorem 2.92. (Milman–Pettis) If a Banach space (V, . ) is uniformly convex, then V is reflexive.

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Proof. Let ξ be in V such that ξ V = 1. We want to prove that ξ ∈ J(BV (0, 1)) or otherwise (since J(BV (0, 1)) is strongly closed in V ), to prove that ∀η > 0, ∃u ∈ BV (0, 1) : ξ − Ju V < η. For η > 0 fixed, consider the value δ > 0 which is chosen by the uniform convexity estimate (corresponding to η). Let f ∈ V be given such that f V = 1 and satisfies ξ, f V ,V > 1 − δ/2 (this inequality is valid, since ξ V = 1 and f V = 1). We consider the neighborhood of the value ξ by Ω = {ψ ∈ V : | ξ − ψ, f V ,V |< δ/2}. According to Goldstine’s lemma, we have that J(BV (0, 1)) ∩ Ω = ∅ and then there exists an element u ∈ BV (0, 1) such that Ju ∈ Ω, i.e., by the definition of J, | ξ, f V ,V − f, uV ,V |< δ/2. Prove now that ξ − Ju V ≤ η. Suppose that the result is false, then we can obtain a new neighborhood of ξ for the topology σ(V , V ) which does not contain u. By using again Goldstine’s lemma, we have the existence of v ∈ BV (0, 1), such that ξ − Jv V > η and | ξ, f V ,V − f, vV ,V |< δ/2. So, we have (by adding the two previous inequalities) 2 ξ, f V ,V ≤ f, u + vV ,V + δ ≤ u + v +δ (since f V = 1). Since ξ, f V ,V > 1 − δ/2, then (u + v)/2 ≥ 1 − δ and, consequently, according to the uniform convexity of V , u−v ≤ η, which is a contradiction. As a corollary, we have the following proposition. Proposition 2.93. If V is a Hilbert space, then V is a reflexive space. Proof. Clearly, a Hilbert space is a uniformly convex Banach space (by using the parallelogram law: 2( u 2 + v 2 ) = u + v 2 + u − v 2 , for all (u, v) ∈ V × V ). Consequently, because of Milman–Pettis’s theorem (Theorem 2.92), a Hilbert space is reflexive. Let us terminate this section with the following proposition. Proposition 2.94. Let V be an uniformly convex Banach space, and (un ) be a sequence of V converging weakly, for the topology σ(V, V ), into u such that lim sup un ≤ u . n−→∞

Then the sequence (un ) converges strongly to u. Proof. Suppose that the limit u is not null (else the proof is immediate). Let us introduce the following sequence (vn ) by vn =

un ∈ BV (0, 1), max( un , u )


37

then (vn ) converges weakly, for the topology σ(V, V ), into v = u/ u ( we have v = 1). Moreover, according to Corollary 2.66, we can deduce that 1 = v ≤ lim inf n−→∞

vn + v ≤ 1. 2

So, (vn + v)/2 converges into 1. Consequently, according to the uniform convexity of the space V , vn − v converges into 0. This completes the proof. Example 2.95. In view of Examples 2.80, the spaces Lp (Ω), 1 < p < ∞, are ♣ reflexive, but the spaces L1 (Ω) and L∞ (Ω) are not reflexive. 2.2.6 Closure and Continuity of Convex Functions The convex functions appreciate the remarkable properties of continuity: they are locally Lipschitzian on the relative interior of its effective domain. Before studying the continuity of convex functions, we introduce some definitions and give some results concerning the lower semi-continuous convex function. Definition 2.96. (Proper function) A function F : V −→ IR is said to be proper if it takes nowhere the value −∞ (i.e., F (u) > −∞, for every u ∈ V , and then F : V −→] − ∞, +∞]) and not identically equal to +∞ (i.e., F ≡ +∞ or also domF = ∅). Definition 2.97. (Closed function) A function F : V −→ IR is said to be closed if F = F (where F is the closure of F or the lower semi-continuous regularization). In particular we have (according to Proposition 2.50) the following corollary. Corollary 2.98. A proper function is closed if and only if it is lower semicontinuous, or if and only if its epigraph is closed, or if and only if its level sets are closed. The following proposition shows that if a convex function takes the value −∞, then the points where the function is finite are not numerous. More precisely: Proposition 2.99. If F : V −→ IR is a convex and lower semi-continuous function and takes the value −∞ (i.e., there exists a point u0 ∈ V such that F (u0 ) = −∞), then F cannot take any finite value. Proof. The proof is trivial and is left to the reader as an exercise.

Next we give an interesting result pertaining to lower semi-continuous, convex and proper functions for a minimization problem (according to Proposition 2.65 and Theorem 2.90).

38


Definition 2.100. (Minimizing sequence) A sequence (un )n∈IN in G (no structure is assumed on G) is said to be a minimizing sequence of some mapping F if lim F (un ) = inf F (v). n−→∞ v∈G Theorem 2.101. Let (V, . ) be a reflexive Banach space, F : V −→ IR be a proper, convex and lower semi-continuous function on V , and C a convex and closed subset of V . If we assume that C is bounded or that the functional F is coercive over C, i.e., F (v) −→ +∞ if v ∈ C, v −→ ∞,

(2.10)

then the function F assumes a minimum value on C. Otherwise there exists a point u ∈ C such that F (u) = inf F (v). (2.11) v∈C

Proof. Let (un ) be a minimizing sequence in C of F , then lim F (un ) = inf F (v).

n−→∞

v∈C

The sequence (un ) is uniformly bounded in V (since C is bounded or since F (un ) is bounded according to (2.10)), then we can extract from (un ) a subsequence, denoted also by (un ) which converges weakly in V (in the topology σ(V, V )) to a value u ∈ C (according to Eberlein–Smulian’s theorem (Theorem 2.90)). According to Proposition 2.65, we have that F is lower semicontinuous with respect to the weak topology σ(V, V ) and so, F (u) ≤ lim inf F (un ). n−→∞

Consequently, u is a solution of problem (2.11).

The next results of this section concern the continuity properties of convex functions. The proofs of these results can be found, for example, in Ekeland et al. [112]. The main result is contained in the following proposition. Proposition 2.102. Let V be a locally convex topological vector space and F : V −→ IR be a given convex function on V . If in the neighborhood of a point u ∈ V such that F (u) > −∞, the function F is bounded above by a finite constant, then F is continuous at point u. More generally we have the following proposition. Proposition 2.103. Let V be a locally convex topological vector space and F : V −→ IR be a given proper and convex function. The following statements are equivalent:

2.3 Γ -Regularization and Continuous Affine Functions

39

(i) there exists a non-empty open set V on which F is bounded from above by a finite constant (ii) F is continuous and locally Lipschitz over interior of its effective domain domF , which is non-empty. As a consequence we have with more precision, in many special situations, some results which clarify this principal result, as follows. Corollary 2.104. Let V be a locally convex topological vector space and F : V −→ IR be a proper and convex function. Then, if F is upper semi-continuous at a point which is interior to its effective domain domF , then F is continuous over interior of domF . Corollary 2.105. Let V be a Banach space and F : V −→ IR be a lower semi-continuous, proper and convex function. Then F is continuous over the interior of its effective domain domF . Corollary 2.106. Let V be a finite dimensional and separated topological space and F : V −→ IR be a proper and convex function. Then F is continuous over the interior of its effective domain domF .

2.3 Γ -Regularization and Continuous Affine Functions Definition 2.107. (Continuous affine function) A function f : V −→ IR on a vector space is a continuous affine function if it is of the form f (x) = l(x)+λ, for some linear continuous function l and some real λ. It is clear that every linear functional is affine, and every affine function is both convex and concave. Definition 2.108. (Pointwise supremum of continuous affine functions) The set of functions F : V −→ IR which are the pointwise supremum of a family of continuous affine functions is denoted by Γ (V ) and the subset of Γ (V ) other than the constant functions −∞ and +∞ is denoted by Γ0 (V ). From this definition we observe that all functions of Γ (V ) are convex (according to Proposition 2.46) and lower semi-continuous functions on V (according to Proposition 2.52). Moreover, if a function of Γ (V ) attains the value −∞ then this function is exactly the constant function −∞ (because the pointwise supremum of an empty family is −∞ and if the family under consideration is not empty, the pointwise supremum can not take the value −∞). More precisely we have the following characterization of Γ (V ). Proposition 2.109. (Characterization of Γ (V )) The function F : V −→ IR is in Γ (V ) if and only if F is a lower semi-continuous convex function, and if F takes the value −∞ then F is identically equal to −∞. Otherwise, Γ (V ) is the set of all closed convex functions on V .

40


Proof. Suppose that F is a lower semi-continuous convex function where F is different from the constant function −∞ and from the constant function +∞. We show that for every (u0 , λ0 ) ∈ V × IR such that F (u0 ) > λ0 , there exists a continuous affine function L such that λ0 ≤ L(u0 ) ≤ F (u0 ).

(2.12)

According to Propositions 2.44 and 2.50 we have that epiF is a closed convex / epiF . Then (because of Hahn–Banach’s second geometric set with (u0 , λ0 ) ∈ form) there exists a closed hyperplane H which strictly separates (u0 , λ0 ) and epiF , i.e., there exists a non-zero continuous linear function l and (α, β) ∈ IR2 such that H = {(u, λ) ∈ V × IR : l(u) + αλ = β} and l(u0 ) + αλ0 < β, l(u) + αλ > β ∀(u, λ) ∈ epiF.

(2.13)

To finish the proof we can distinguish the two following cases: • If F (u0 ) is finite, (u0 , F (u0 )) ∈ epiF and then α(λ0 − F (u0 )) < 0 (according to (2.13)). Since λ0 − F (u0 ) < 0, we can conclude that α > 0 and then λ0
0 and β − l(u) < 0 ∀u ∈ domF. Therefore, there exists a continuous linear function r and γ ∈ IR such that γ − r(u) < F (u) ∀u ∈ V and then, for all η > 0, the continuous affine functions Lη (.) := γ − r(.) + η(β − l(.)) satisfy Lη (u) < F (u) ∀u ∈ V. By taking η sufficiently large such λ0 < Lη (u0 ), we can deduce that the affine function Lη satisfies the relation (2.12). This completes the proof. A similar definition can be given for a concave function, as follows. Definition 2.110. (Closed concave function) The function F : V −→ IR is said to be a closed concave function if −F is a closed convex function and we denote the set of all closed concave functions on V by −Γ (V ) = {F : −F ∈ Γ (V )}.

2.3 Γ -Regularization and Continuous Affine Functions

41

Before giving a regularization of an extended real-valued function in Γ (V ), we introduce the following definition. Definition 2.111. We say that the extended real-valued function G dominates (or is greater than) the extended real-valued function F on V , written G ≥ F , if we have G(u) ≥ F (u), for every u ∈ V . In the same way we say that G is less than F , written G ≤ F , if we have G(u) ≤ F (u), for every u ∈ V . Definition 2.112. (Γ -regularization of a function) Let F and G be two functions of V into IR. The function G is said to be the Γ -regularization of F if G is the largest minorant of F in Γ (V ), or if G is the pointwise supremum of all the continuous affine functions less than F . In particular we have the following corollary. Corollary 2.113. If F : V −→ IR is in Γ (V ) then F coincides with its Γ -regularization. In general, we have the following properties of the Γ -regularization. Proposition 2.114. Let F : V −→ IR and G be its Γ -regularization. If there exists a continuous affine function less than F , we have that the epigraph of G is exactly the closed convex hull of the epigraph of F : epiG = clco(epiF ). Proof. The proof is left to the reader as an exercise.

Proposition 2.115. Let F : V −→ IR, F be its closure (or its lower semicontinuous regularization) and G be its Γ -regularization. Then the following statements are true: (i) G ≤ F ≤ F . (ii) If F is convex and there exists a continuous affine function less than F then F = G. Proof. This is a direct consequence of Proposition 2.114 and the definition of the closure of functions.

3 A Brief Overview of Sobolev Spaces

The aim of this chapter is to recall the basic features of Sobolev spaces and useful compactness results. For the proofs and more details, the reader is referred, for instance, to Adams [4], Lions and Magenes [204] and Maz’ja [221].

3.1 Tools and Definitions 3.1.1 Definitions and Notations We denote by Ω an open domain in IRn , n ≥ 1, with a smooth boundary Γ = ∂Ω. In general, some regularity of Ω will be assumed. We will suppose that either Ω is Lipschitz, (3.1) i.e., the boundary Γ is locally the graph of a Lipschitz function, or Ω is of class C r , r ≥ 1,

(3.2)

i.e., the boundary Γ is a manifold of dimension n − 1 of class C r (see below). In both cases we assume that Ω is totally on one side of Γ . These definitions mean that locally the domain Ω is below the graph of some function ψ, the boundary Γ is represented by the graph of ψ and its regularity is determined by that of the function ψ. Moreover, it is necessary to note that a domain with a continuous boundary is never on both sides of its boundary at any point of this boundary and that a Lipschitz boundary has almost everywhere a unit normal vector n. We will also use the following multi-index notation for partial differential derivatives of a function:

44


∂ku for all k ∈ IN and i = 1, . . . , n, ∂xki ∂ α1 +···+αn u α Dα u := ∂1α1 ∂2α2 · · · ∂j j · · · ∂nαn u = αn , 1 ∂xα 1 · · · ∂xn n α = (α1 , . . . αj , . . . , αn ) ∈ INn , [α] := αi . ∂ik u :=

(3.3)

i=1

We denote by C(D) (respectively C k (D), k ∈ IN or k = +∞) the space of real continuous functions on D (respectively the space of k times continuously differentiable functions on D), where D plays the role of Ω or its closure Ω. The space of real C ∞ functions on Ω with a compact support in Ω is denoted by C0∞ (Ω) or D(Ω) as in the theory of distributions of Schwartz [262] and the space of distributions on Ω is denoted by D (Ω), i.e., the space of continuous linear form over D(Ω) (for more details on the theory of distribution, see Schwartz [262, 263]). For p ∈ [1, +∞], we denote by Lp (Ω) the space of (class of) measurable functions v on Ω such that v Lp (Ω) = ( | v(x) |p dx)1/p < ∞ if p < ∞, v L∞ (Ω) =

Ω

sup ess | v(x) |< ∞ if p = +∞. x∈Ω

The space L (Ω) equipped with the norm . Lp (Ω) (for 1 ≤ p ≤ +∞) is a p Banach space: it is reflexive and separable for 1 < p < ∞ (its dual is L p−1 (Ω)), separable but not reflexive for p = 1 (its dual is L∞ (Ω)), and not separable, not reflexive for p = ∞ (its dual contains strictly L1 (Ω)). In particular the space L2 (Ω) is a Hilbert space for the scalar product (u, v) = (u, v)L2 (Ω) = u(x).v(x)dx. p

Ω

Lploc (Ω)

We denote by the space of functions which are Lp on any bounded subdomain of Ω. Of course, similar space can be defined on any open set other than Ω, in particular, on the cylinder set Ω×]a, b[ or on the set Γ ×]a, b[, where (a, b) ∈ IR and a < b. Let U be a Banach space, 1 ≤ p ≤ +∞ and −∞ ≤ a < b ≤ +∞, then Lp (a, b; U ) is the space of (class of) Lp functions v from (a,b) into U which is a Banach space for the norm

1/p b

v Lp (a,b;U) =

a

v(t) pU dt

if p < ∞

and for the norm v L∞ (a,b;U) = sup ess v(t) U t∈(a,b)

if p = +∞.

3.1 Tools and Definitions

45

Similarly, for a Banach space U , k ∈ IN and −∞ < a < b < +∞, we denote by C([a, b]; U ) (respectively C k ([a, b]; U )) the space of continuous functions (respectively the space of k times continuously differentiable functions) v from [a, b] into U , which are Banach spaces, respectively, for the norms ∂iv i C([a,b];U) . v C([a,b];U) = sup v(t) U and v C k ([a,b];U) = ∂t t∈[a,b] i=0,k

Finally, we define the H¨ older space C k,α (Ω), for k ∈ IN and α ∈ (0, 1], as the space of functions in C k (Ω) whose partial derivatives of order k, ∂j u, [j] = k, are H¨ older uniformly continuous with exponent α in Ω, i.e., if sup x,y∈Ω x=y

| ∂j u(x) − ∂j u(y) | < ∞, ∀j such that [j] = k. | x − y |α

It is a Banach space for the norm given by v C k,α (Ω) := v C k (Ω) + sup

x,y∈Ω x=y [j]=k

| ∂j u(x) − ∂j u(y) | . | x − y |α

3.1.2 Some Fundamental Inequalities and Convergence Criteria Some Fundamental Inequalities Our study involves the following fundamental inequalities, which are repeated here for review: (i)

H¨ older’s inequality : Πi=1,k fi dx ≤ Πi=1,k fi Lqi (D) , where D 1/qi 1 qi fi Lqi (D) = | fi | dx and = 1. qi D i=1,k

(ii) Young’s inequality (∀a, b > 0 and λ > 0): ab ≤

1 1 λ p λ−q/p q a + b , for p, q ∈]1, +∞[ and + = 1. p q p q

(iii) Gronwall’s lemma: dΦ ≤ g(t)Φ(t) + h(t), ∀t ≥ 0 then dt t t t g(s)ds + h(s) exp g(τ )dτ ds, ∀t ≥ 0. Φ(t) ≤ Φ(0) exp

If

0

0

s

46


Some Convergence Criteria The aim of this subsection is to recall some notions of the convergence of sequences on the Lp -spaces. For the proofs and more details, the reader is referred, for instance, to Hewitt and Stromberg [155]. Lemma 3.1. (Fatou’s lemma) Let (un )n≥1 be a sequence of measurable and positive functions. Then lim inf un (x)dx ≤ lim inf un (x)dx. n−→∞ Ω Ω n−→∞ Theorem 3.2. (Monotone convergence) Let (un )n≥1 be a decreasing sequence of measurable and positive functions. Then lim un (x)dx = lim un (x)dx. n−→∞ Ω Ω n−→∞ Theorem 3.3. (Lebesgue’s dominated convergence theorem) Let (un )n≥1 be a sequence of L1 -functions, which is almost everywhere (a.e.) convergent to a measurable function u. If there exists a positive function v ∈ L1 (Ω) such that | un |≤ v a.e. in Ω, then u ∈ L1 (Ω) and we have 1 un (x)dx = u(x)dx. lim un − u L (Ω) = 0, lim n−→∞ n−→∞ Ω Ω We give now some convergence results in Lp -spaces (see Brezis and Lieb [61]). Lemma 3.4. (Brezis–Lieb’s lemma) Let (un )n≥1 be a bounded sequence of Lp functions, 1 ≤ p < ∞, which is almost everywhere convergent to a measurable function u. Then u ∈ Lp (Ω) and lim ( un pLp (Ω) − un − u pLp(Ω) ) = u pLp(Ω) .

n−→∞

A direct corollary of Brezis–Lieb’s lemma is the following result. Corollary 3.5. Let u ∈ Lp (Ω), 1 ≤ p < ∞, and (un )n≥1 be a sequence of Lp -functions which is almost everywhere convergent to u. If

lim un Lp (Ω) = u Lp (Ω) then

n−→∞

lim un − u Lp(Ω) = 0.

n−→∞

Lemma 3.6. Let u ∈ Lp (Ω), 1 ≤ p < ∞, and (un )n≥1 be a sequence of Lp functions. If lim un − u Lp (Ω) = 0 then there exist v ∈ Lp (Ω) and a subn−→∞

sequence of (un )n≥1 , denoted also by (un )n≥1 , such that | un |≤ v and (un )n≥1 is almost everywhere convergent to u.

a.e. in Ω

3.1 Tools and Definitions

47

We finish this subsection by a proposition which gives a relation between the almost everywhere convergence and the weak-convergence results on Lp spaces, for 1 < p < ∞ (which are reflexive spaces). Proposition 3.7. Let (un )n≥1 be a bounded sequence of Lp -functions, 1 < p < ∞, which is almost everywhere convergent to u. Then un u weakly in Lp (Ω).

Remark 3.8. Since the space Lp (Ω), for 1 < p < ∞, is reflexive then, according to Eberlein–Smulian’s theorem (Theorem 2.90), for (un )n≥1 being a bounded sequence of Lp -functions, we can extract from (un ) a subsequence, denoted also by (un ) which converges weakly in Lp (Ω) to a function v ∈ Lp (Ω). If, ♦ moreover, (un ) is almost everywhere convergent to u then v = u. 3.1.3 Definition of Sobolev Spaces We introduce now the Sobolev spaces, which will be considered in more details in Adams [4]. Let Ω ⊂ IRn be an open domain (not necessarily bounded). For an integer m > 0 and 1 ≤ p ≤ ∞, the Sobolev space of order (m, p), denoted by W m,p (Ω), is defined as the space of functions in the space Lp (Ω) whose (distribution) derivatives of order ≤ m are also in Lp (Ω), i.e., W m,p (Ω) := {v ∈ Lp (Ω) : Dα v ∈ Lp (Ω) ∀α ∈ INn such that [α] ≤ m}. This is a Banach space for the Sobolev norm given by Dα v Lp(Ω) 1 ≤ p < ∞ v W m,p (Ω) := [α]≤m

(which is equivalent to the norm ( and, in the case of p = ∞,

Dα v pLp (Ω) )1/p )

[α]≤m

v W m,∞ (Ω) := max Dα v L∞ (Ω) . [α]≤m

Moreover, W m,p (Ω) is a reflexive space for 1 < p < ∞ and a separable space for 1 ≤ p < ∞. In particular, if p = 2, the Sobolev space W m,2 (Ω) denoted by H m (Ω) is a Hilbert and separable space for the following scalar product: (Dα u, Dα v)L2 (Ω) . (u, v)H m (Ω) := [α]≤m

The space H m (Ω) and W m,p (Ω) contain C ∞ (Ω) and C m (Ω). The closure of D(Ω) for the H m (Ω) norm (respectively W m,p (Ω) norm) is denoted by H0m (Ω) (respectively W0m,p (Ω)). In particular, the H 1 type spaces and the

48


corresponding dual spaces, will be frequently used. The space H 1 (Ω) denotes the space of L2 functions u on Ω such that ∂i u ∈ L2 (Ω), for i = 1, . . . , n, and H01 (Ω) is the closure of D(Ω) for the H 1 (Ω) norm. They are both Hilbert spaces for the following scalar product: (u, v)H 1 (Ω) := u(x)v(x)dx + ∇u(x).∇v(x)dx, Ω

Ω

where (∇u)i := ∂i u. The space H −1 (Ω) is defined as the dual space of H01 (Ω) for the duality between D (Ω) and D(Ω) and its norm is f H −1 (Ω) :=

sup 1 (Ω) v∈H0 v 1 ≤1 H (Ω)

| f, vD (Ω),D(Ω) | .

Let us recall the Poincaré inequality, where Ω is a bounded domain. Let v be in D(Ω) and write v(x) = v(x1 , x ), for x = (x1 , x ) and x = (x2 , . . . , xn ) as the form x1 v(x1 , x ) =

−∞

∂1 v(t, x )dt.

Then, by using the Cauchy–Schwartz inequality, we can deduce that | ∇v(t, x ) |2 dt | v(x1 , x ) |2 ≤ C(Ω) IR and then (by integrating) v 2L2 (Ω) ≤ C(Ω) ∇v 2L2 (Ω) . By density (by definition) of D(Ω) on H01 (Ω), we can deduce that for all v ∈ H01 (Ω) we have v 2L2 (Ω) ≤ C(Ω) ∇v 2L2 (Ω) and then we can define the norm in H01 (Ω) by v H01 (Ω) := ∇v L2 (Ω) , which is equivalent to the H 1 (Ω) norm (if Ω is a bounded domain). Consequently, H01 (Ω) is a Hilbert space for the following scalar product: (u, v)H01 (Ω) := ∇u(x).∇v(x)dx. Ω

If we identify L2 (Ω) to its dual, but if we do not identify H01 (Ω) and its dual H −1 (Ω), we have H01 (Ω) ⊂ L2 (Ω) ≡ (L2 (Ω)) ⊂ H −1 (Ω) = (H01 (Ω)) , where the injections are dense and continuous.

3.2 Some Properties of Sobolev Spaces

49

3.2 Some Properties of Sobolev Spaces Let us recall some important properties of Sobolev spaces, namely the density, embedding, compactness and trace theorems. Let p be a real value such that p ≥ 1 and m ≥ 1 be an integer value. Assume that the open domain Ω ⊂ IRn (not necessarily bounded) is sufficiently regular. It is clear that, if Ω is a bounded domain then Lp (Ω) ⊂ Lq (Ω) W

m,p

(Ω) ⊂ W

k,q

H m (Ω) ⊂ H k (Ω)

for all p ≥ q, (Ω)

for all m ≥ k, p ≥ q,

(3.4)

for all m ≥ k.

with continuous embedding. 3.2.1 Density Results If the open domain Ω is sufficiently regular, for example of class C m , then C m (Ω) is dense in W m,p (Ω) ∀1 ≤ p < ∞. This implies that (if Ω is sufficiently regular) W m,p (Ω) is dense in W m−1,p (Ω) ∀1 ≤ p < ∞, H m (Ω) is dense in H m−1 (Ω).

(3.5)

3.2.2 Embedding Results Assume that the open domain Ω is sufficiently regular, for example of class C m , we have if

if

if

m 1 m 1 1 − > 0 then W m,p (Ω) ⊂ Lq (Ω) where − = , and p n p q n u Lq (Ω) ≤ C u W m,p (Ω) (the embedding is continuous), 1 m − = 0 then W m,p (Ω) ⊂ Lqloc (Ω) where 1 ≤ q < ∞, and p n u Lq (ω) ≤ Cω u W m,p (Ω) for all bounded ω ⊂ Ω, n 1 m − < 0 then W m,p (Ω) ⊂ C k,α (ω) where m − = k + α, p n p n k is the integer part of m − and ∀j such that [j] = k p | ∂j u(x) − ∂j u(y) |≤ Cω | x − y |α u W m,p (Ω) for all bounded ω ⊂ Ω and x, y ∈ ω.

(3.6)

50


In particular, for m = 1, if the boundary Γ of Ω is bounded then 1 1 1 − = , p q n if p = n then W 1,p (Ω) ⊂ Lq (Ω) for all p ≤ q < ∞, if 1 ≤ p < n then W 1,p (Ω) ⊂ Lq (Ω) where

(3.7)

if p > n then W 1,p (Ω) ⊂ L∞ (Ω), with continuous embeddings. If the domain Ω is bounded, we have (because of (3.4)) n 1 m − < 0, then W m,p (Ω) ⊂ C k,α (Ω) where m − = k + α, p n p np 1 m m,p q > 0, then W ], (Ω) ⊂ L (Ω) where q ∈ [1, if − p n n − mp 1 m = 0, then W m,p (Ω) ⊂ Lq (Ω) where q ∈ [1, +∞[, if − p n

if

(3.8)

with continuous embeddings. 3.2.3 Compactness Results Assume now that m = 1 and the domain Ω is bounded and of class C 1 . Then we have the following results (due to Rellich–Kondrachov) if 1 ≤ p < n then W 1,p (Ω) ⊂ Lq˜(Ω) ∀1 ≤ q˜ < q where

1 1 1 − = , p q n

if p ≥ n then W 1,p (Ω) ⊂ Lq (Ω) for all 1 ≤ q < ∞,

(3.9)

n ˜ n then W 1,p (Ω) ⊂ C 0,α˜ (Ω) for all α p with compactness embeddings. 3.2.4 Trace Results and Green’s Formula Trace Results Suppose now that the domain Ω is of class C m+1 , then for a given function u ∈ W m,p (Ω), we can define its trace on Γ , u|Γ , which coincides with the value of u on Γ . More generally, we can define the linear and continuous operator trace γ : W m,p (Ω) −→ (Lp (Γ ))m by γ := {γ0 , γ1 , . . . , γm−1 } and γi u =

∂iu |Γ i = 1, . . . , m − 1, ∂ni

where n = (n1 , . . . , nn ) is the unit vector outward normal on Γ .

(3.10)


51

The space W0m,p (Ω) can be written as W0m,p (Ω) = kerγ (i.e., the kernel of γ).

(3.11)

So, ∂ iu |Γ = 0 for all i = 1, . . . , m − 1. ∂ni The dual spaces of W0m,p (Ω) are the following spaces: if u ∈ W0m,p (Ω) then

1 1 + = 1, p p∗ H −m (Ω) := (H0m (Ω)) (the case of p = 2), ∗

W −m,p (Ω) := (W0m,p (Ω)) where

(3.12)

where f W −m,p∗ (Ω) :=

sup

m,p v∈W0 (Ω) vW m,p (Ω) ≤1

| f, vW −m,p∗ (Ω),W m,p (Ω) | .

In particular: • For p = 2, γi (H m (Ω)), which is not the whole space L2 (Γ ), is denoted by H m−i−1/2 (Γ ) and is endowed with the following quotient norm w H m−i−1/2 (Γ ) :=

inf

v∈H m (Ω) γi v=w

v H m (Ω)

and its dual is denoted by H −m+i+1/2 (Γ ). For example, for m = 1, we have that H01 (Ω) = {v ∈ H 1 (Ω) : γ0 u = u|Γ = 0}, γ0 (H 1 (Ω)) = H 1/2 (Γ ) and (H 1/2 (Γ )) := H −1/2 (Γ ). • For m = 1, if Ω is bounded then ∗

W01,p (Ω) ⊂ L2 (Ω) ⊂ W −1,p (Ω) if

2n ≤ p < ∞, n+2

with continuous and dense embeddings. Moreover, because of Poincaré’s inequality, we have that for all u ∈ W01,p (Ω) (1 ≤ p < ∞), u Lp (Ω) ≤ C ∇u Lp (Ω) and then we can endowed the space W01,p (Ω) with the norm ∇u Lp (Ω) , which is equivalent to the norm u W m,p (Ω) . If Ω is not bounded then ∗

W01,p (Ω) ⊂ L2 (Ω) ⊂ W −1,p (Ω) if

2n ≤ p ≤ 2. n+2

Some Properties of Spaces Related to Divergence Operator We assume in this subsection that the domain Ω is a bounded subset of IRn with Lipschitz boundary Γ . For a vector function v := (vi )i=1,n , we define the divergence operator div by

52


div(v) =

∂vi . ∂xi i=1,n

Introducing now the following subspace of H 1 (Ω) by H(div; Ω) := {v ∈ L2 (Ω) : div(v) ∈ L2 (Ω)}, which is a Hilbert space for the norm 1/2

v H(div;Ω) = v 2L2 (Ω) + div(v) 2L2 (Ω) . The closure of D(Ω) for the H(div; Ω) norm is denoted by H0 (div; Ω). The following theorem concerns the normal component of boundary values of functions of the space H(div; Ω). Theorem 3.9. Let Ω be a bounded subset of IRn with Lipschitz boundary Γ , we have the following properties: D(Ω) is dense in H(div; Ω) (the boundedness of Ω is not necessary), (3.13) there exists a mapping γn : v −→ v.n|Γ defined on D(Ω) which can be extended by continuity to a linear and continuous mapping, denoted also by γn , from H(div; Ω) into H −1/2 (Γ ).

(3.14)

A consequence of the previous theorem is the following Green’s formula: for all vector functions v ∈ H(div; Ω) and scalar functions φ ∈ H 1 (Ω), v.∇φdx + φdiv(v)dx = v.n, φH −1/2 (Γ ),H 1/2 (Γ ) . (3.15) Ω

Ω

The next results give some properties of H0 (div; Ω). Theorem 3.10. Let Ω be a bounded subset of IRn with Lipschitz boundary Γ , we have the following properties: H0 (div; Ω) = kerγn = {v ∈ H(div; Ω) : γn v = v.n = 0 on Γ }, V is dense in H,

(3.16) (3.17)

where V = {v ∈ D(Ω) : div(v) = 0}, H = {v ∈ H0 (div; Ω) : div(v) = 0}. Green’s Formula Let us now give the interesting result of the integrating by parts for the Laplace operator, due to a trace theorem, namely the extended Green’s formula (which is a direct consequence of (3.15)). If u ∈ H 1 (Ω) and Δu ∈ L2 (Ω), then the traces on Γ , γ0 u = u|Γ is in 1/2 H (Γ ) and γ1 u = (∂u/∂n)|Γ is in H −1/2 (Γ ). Moreover, we have


(−Δu, v)L2 (Ω) = (∇u, ∇v)L2 (Ω) − γ1 u, γ0 uH −1/2 (Γ ),H 1/2 (Γ ) .

53

(3.18)

If we assume that γ1 u ∈ L2 (Ω) then the Green’s formula becomes (−Δu, v)L2 (Ω) = (∇u, ∇v)L2 (Ω) − (γ1 u, γ0 u)L2 (Γ ) .

(3.19)

More generaly, we can define the generalized form of Green’s theorem (see, for instance, Lions and Magenes [204]). For this, let aij : Ω −→ IR, for i, j = 1, . . . , n, be functions in L∞ (Ω) and satisfy the ellipticity conditions aij (x)ξi ξj ≥ α ξk2 ∀(ξ)i=1,n ∈ IRn , a.e. in Ω. (3.20) i,j=1,n

k=1,n

Consider the operator A by Au = −

∂ ∂u (aij (x) ). ∂x ∂x i j i,j=1,n

(3.21)

The normal derivative ∂u/∂ηA at Γ , directed towards the exterior of Ω, can be written in the following form: ∂u = aij (x) cos(η, xi )∂j u, ∂ηA i,j=1,n

(3.22)

where η is the unit normal vector at Γ exterior to Ω and cos(η, xi ) is an ith direction cosine of η. Finally, we introduce the following bilinear form a: a(u, v) = aij (x)∂i u∂j vdx, ∀u, v ∈ H 1 (Ω). (3.23) i,j=1,n

Ω

We can now give the generalized form of Green’s theorem: If u, v ∈ H 1 (Ω) and Au ∈ L2 (Ω), then the traces on Γ , γ0 v = v|Γ is in H 1/2 (Γ ) and γ1 u = (∂u/∂ηA )|Γ is in H −1/2 (Γ ). Moreover, we have (Au, v)L2 (Ω) = a(u, v) − γ1 u, γ0 vH −1/2 (Γ ),H 1/2 (Γ ) .

(3.24)

3.2.5 Truncation Operations Let us consider the following notations: • For r ∈ IR, we write classically r+ = max(r, 0), r− = (−r)+ and then r = r+ − r− and | r |= r+ + r− . • For u a real function on Ω, we define the functions u+ and u− by u+ (x) = (u(x))+ and u− (x) = (u(x))− , ∀x ∈ Ω, respectively.

54


According, for instance, to Gilbarg and Trudinger [132], we have that: (i) If u is in Lp (Ω), 1 ≤ p ≤ +∞, then u+ and u− are also in Lp (Ω) with u+ Lp (Ω) ≤ u Lp (Ω) and u− Lp (Ω) ≤ u Lp (Ω) . (ii) If u is in W 1,p (Ω), 1 ≤ p < +∞, then u+ and u− are also in W 1,p (Ω). Moreover, for a.e. x ∈ Ω, we have that (for i = 1, . . . , n) ∂i u+ (x) = ∂i u(x) if u(x) > 0 and ∂i u+ (x) = 0 if u(x) ≤ 0, ∂i u− (x) = ∂i u(x) if u(x) < 0 and ∂i u− (x) = 0 if u(x) ≥ 0,

(3.25)

and ∂i u+ Lp(Ω) ≤ ∂i u Lp (Ω) , u+ W 1,p (Ω) ≤ u W 1,p (Ω) , ∂i u− Lp (Ω) ≤ ∂i u Lp (Ω) , u− W 1,p (Ω) ≤ u W 1,p (Ω) .

(3.26)

3.2.6 Interpolation Theory Let X and Y be two Hilbert spaces, endowed respectively by the norm . X and . Y , such that X ⊂ Y , with the continuous and dense embedding. The interpolation theory gives a family of Hilbert spaces denoted by [X, Y ]θ , θ ∈ [0, 1], such that [X, Y ]0 := X, [X, Y ]1 := Y and X ⊂ [X, Y ]θ ⊂ Y,

(3.27)

with continuous and dense embeddings. The spaces [X, Y ]θ , θ ∈ [0, 1], are endowed with the norm . [X,Y ]θ such that θ ∀u ∈ X, ∀θ ∈ [0, 1]. (3.28) u [X,Y ]θ ≤ CΘ u 1−θ X u Y By using the interpolation between H m (Ω), 0 < m ∈ IN, and H 0 (Ω) = L (Ω) we can define, the following space by 2

H θm (Ω) := [H m (Ω), H 0 (Ω)]1−θ , for all θ ∈ (0, 1),

(3.29)

which gives the definition of H s (Ω), for a real s ≥ 0. The density, embedding, compactness and trace results given above, can be extended, without modifications to Sobolev spaces H s (Ω) = W s,2 (Ω), for a real s ≥ 0. Moreover, if Ω is bounded and sufficiently regular domain, we can complete the previous results by (for all s1 , s2 such that 0 ≤ s2 < s1 ) H s1 (Ω) ⊂ H s2 (Ω), with compactness embedding, H (1−θ)s1 +θs2 (Ω) := [H s1 (Ω), H s2 (Ω)]θ , for all θ ∈]0, 1[, with equivalent norms.

(3.30)


55

In particular, if the domain Ω is bounded, we have (because of (3.8)) the following useful estimates, namely the Gagliardo–Nirenberg inequalities: v Lp (Ω) ≤ C v θH q (Ω) v 1−θ L2 (Ω)

for all v ∈ H q (Ω),

(3.31)

where q is a non-negative integer, θ ∈ [0, 1[ and p = 2m/(m − 2θq) (with the exception that if q − m/2 is a non-negative integer, then θ is restricted to 0). Remark 3.11. (i) We can define H0s (Ω) as the closure of D(Ω) for the H s (Ω) norm and H −s (Ω) := (H0s (Ω)) . (ii) We can also define the family of Sobolev spaces W s,p (Ω), H s (Γ ) and W s,p (Ω), for s ∈ IR. (iii) In the whole space IRn and p = 2 (in (ii)), Sobolev spaces can be defined also in terms of integrability properties in frequency space by using the Fourier (denoted also by transformation. For all u ∈ D (IRn ), the Fourier transform u F u) is defined by u (y) := exp(−ix.y)u(x)dx for y ∈ IRn n IR and the inverse Fourier transform (which allows us to recover u from u ) is defined by 1 (x) := exp(ix.y) u(y)dy for y ∈ IRn . u(x) = F −1 u (2π)n IRn For all s ∈ IR, we can introduce the Sobolev spaces H s (IRn ) := {u ∈ S (IRn ) : u 2H s (IRn ) := (1+ | y |2 )s | u (y) |2 dy < ∞}, n IR and similarily the homogeneous Sobolev spaces H˙ s (IRn ) := {u ∈ S (IRn ) : u ∈ L1loc (IRn ), u 2H s (IRn ) :=

IRn

| y |2s | u (y) |2 dy < ∞},

where S (IRn ) is the space of tempered distributions, i.e., the dual of the Schwartz space S(IRn ) (the space of rapidly decreasing functions): S(IRn ) := {f ∈ C ∞ (IRn ) : xα Dβ f ∞ < ∞ for all multi-indices α, β}. Here, . ∞ is the supremum norm. We notice that H s (IRn ) is a Hilbert space for all s ∈ IR, but H˙ s (IRn ) is not a Hilbert space if s ≥ n/2. ♦ These last results, and others, will be recalled only when needed.

4 Legendre–Fenchel Transformation and Duality

For any physical or biological system, at least two types of dependent variables can be considered. The source variables, which represent the input variables of the system (for example the external force in mechanics, the distributed source energy or drug terms in medical treatment, . . ., etc.); the configuration variables, which represent the output variables of the system and describe the state of the system, are also said to be the state variables. These two types of variables, in most systems, usually appear in pairs. For each given configuration variable u ∈ V , where V is said to be the space of configuration or state variables, there exists a variable f ∈ V which is dual to u; the dual f is said to be the dual configuration variable and V is said to be the dual configuration space. Moreover, usually, the space of source variables is a subspace of V . In order to study the mathematical theory of duality in natural phenomena, we will study in this chapter the main part of convex conjugate duality theory, and we will present different applications.

4.1 Fenchel Conjugate Functions 4.1.1 Definitions and Properties Let V be a topological vector space and let V be its topological dual space (or conjugate space). We denote by ., .V ,V : V × V −→ IR the bilinear form with respect to the duality between V and V . We say that paired spaces V and V are placed in duality by the bilinear form ., .V ,V . Definition 4.1. (Conjugate function) Let F : V −→ IR be an extended realvalued function. The function F ∗ : V −→ IR defined by F ∗ (f ) = sup ( f, uV ,V − F (u)), ∀f ∈ V u∈V

(4.1)

58


is said to be the Fenchel (convex) conjugate, or simply conjugate (or also the polar) function of F . The mapping F −→ F ∗ is called the Legendre–Fenchel transformation. A direct consequence of the definition of F ∗ is the following result. Proposition 4.2. Let F : V −→ IR be a given extended real-valued function, the following statements are true: (i) (Fenchel’s inequality): F ∗ (f ) + F (u) ≥ f, uV ,V , ∀f ∈ V , ∀u ∈ V. (ii) Let f be in the dual V of V and λ ∈ IR, the continuous affine function u −→ f, uV ,V − λ is less than F if and only if F ∗ (f ) ≤ λ. (iii) If F is identically equal to +∞ then F ∗ is identically equal to −∞. Moreover, if the function F is proper, then the relation (4.1) may be restricted to the points u in the effective domain of F , domF . (iv) The function F ∗ is always in Γ (V ) (since F ∗ is the pointwise supremum of a family of affine continuous functions on V ). Therefore, F ∗ is always a lower semi-continuous convex function on V . Moreover, if F ∗ takes the value −∞ then F ∗ is identically equal to −∞. We can also deduce immediately the following properties. Proposition 4.3. (i) Let F and G be two given extended real-valued functions of V into IR, the following properties hold: (a) F ∗ (0) = − inf F (u). u∈V

(b) If F is less than G (F ≤ G) then G∗ is less than F ∗ (G∗ ≤ F ∗ ). (c) If G(u) = F (αu), ∀u ∈ V , with α = 0 then G∗ (f ) = F ∗ (f /α), ∀f ∈ V . (d) (αF )∗ (f ) = αF ∗ (f /α), ∀f ∈ V , ∀α > 0. (e) (F + β)∗ = F ∗ − β, ∀β ∈ IR. (ii) Given a family (Fi )i∈J of functions from V into IR, we have (inf Fi )∗ = sup Fi∗ , i∈J

i∈J

(sup Fi )∗ ≤ inf Fi∗ . i∈J

i∈J

(iii) For every a ∈ V , we denote by Fa the translated function (i.e., Fa (u) = F (u − a), ∀u ∈ V ). Then Fa∗ (f ) = F ∗ (f ) + f, aV ,V , ∀f ∈ V . We can also define the biconjugate or bipolar function of F .

4.1 Fenchel Conjugate Functions

Definition 4.4. (Biconjugate function) Let F : F ∗∗ : V −→ IR defined by

59

V −→ IR. The function

F ∗∗ (u) = sup ( f, uV ,V − F ∗ (f )), ∀u ∈ V

(4.2)

is said to be the biconjugate (or the bipolar) function of F .

f ∈V

Now, we recall some geometric characterizations of a biconjugate function. Theorem 4.5. (Fenchel–Moreau) Let F be a function of V into IR. Then its biconjugate F ∗∗ ∈ Γ (V ) is its Γ -regularization (i.e., the pointwise supremum of all affine functions which are less than F ). In particular the following results hold: (i) epiF ∗∗ = clco(epiF ) (the closed convex hull of epiF ). (ii) F ∗∗ ≤ F. (iii) If the function F is convex then F coincides with its closure, i.e., F ∗∗ = F . Proof. By the definition of F ∗∗ we have easily that F ∗∗ ∈ Γ (V ). Let now u be any element of V , then F ∗∗ = sup ( f, uV ,V − F ∗ (f )) = sup f ∈V

sup

( f, uV ,V − λ).

f ∈V {λ:F ∗ (f )≤λ}

Since for all λ ∈ IR, F ∗ (f ) ≤ λ if and only if the continuous affine functions u −→ f, uV ,V − λ are less than F (see Proposition 4.2), we can conclude that F ∗∗ is supremum of all the affine functions which are less than F and then F ∗∗ is the Γ -regularization of F . The last statements can be deduced directly from Proposition 2.115. A direct consequence is given by Corollary 4.6. Corollary 4.6. Let F be a function of V into IR. If the function F is in Γ (V ), then F ∗∗ = F . According to Theorem 4.5 and Proposition 4.3, we can deduce immediately that the repetition of the conjugation operation is limited. Corollary 4.7. Let F be a function of V into IR. Then F ∗∗∗ = F ∗ . In general manner, if we denote by F (n)∗ = (F (n−1)∗ )∗ the n-th repetition of the conjugation operation (n ≥ 2), we have F (n)∗ = F ∗∗ , if n is even, F (n)∗ = F ∗ , if n is odd.

(4.3)

60


Let us end this section with another property of the conjugate functions Theorem 4.8. (Fenchel–Rockafellar or Fenchel duality) Let V be a locally convex Hausdorff topological vector space over IR with its dual V . Let F and G be two proper convex functions of V into IR. Suppose that there exists u0 ∈ domF ∩ domG such that F is continuous in u0 . Then inf (F (u) + G(u)) = sup (−F ∗ (−f ) − G∗ (f )).

u∈V

f ∈V

Proof. It follows from Fenchels inequality that for any function H H ∗ (f ) + H(u) ≥ f, uV ,V ∀(f, u) ∈ V × V. Consequently, we have that sup (−F ∗ (−f ) − G∗ (f )) ≤ inf (F (u) + G(u)) u∈V

f ∈V

(this fact is usually referred to as weak duality). Denote p := inf u∈V (F (u) + G(u)), q := supf ∈V (−F ∗ (−f ) − G∗ (f )) and C := epiF . To complete the proof, we show that p ≤ q. If p = −∞ there is nothing to prove. Suppose now that p = −∞. It is clear that the interior of C: intC is not empty (because F is continuous in u0 ). We introduce now the following sets: A := intC, B := {(u, λ) ∈ V × IR : λ ≤ p − G(u)}. The set A and B are convex (since F and G are convex, Proposition 2.44) and disjoint (according to the definition of p), therefore, (because of Hahn– Banach’s first geometric form) there exists a closed hyperplane H which separates A and B and then separates C = A (see Proposition 2.3) and B, i.e., there exists a non-zero continuous linear function f ∈ V and (α, β) ∈ IR2 such that H = {(u, λ) ∈ V × IR : f, uV ,V + αλ = β} and

f, uV ,V + αλ ≥ β ∀(u, λ) ∈ C,

f, uV ,V + αλ ≤ β ∀(u, λ) ∈ B.

(4.4)

By taking u = u0 in the first part of (4.4) and by passing to the limit on λ (λ −→ +∞), we can deduce that α ≥ 0. Prove now that α = 0; for this we proceed by contradiction. Suppose that α = 0, then according to (4.4), we arrive at

f, uV ,V ≥ β, ∀u ∈ domF,

and f, uV ,V ≤ β, ∀u ∈ domG.

In particular f, u0 V ,V = β (since u0 ∈ domF ∩ domG) and then f, u − u0 V ,V ≥ 0 for all u in domF . Consequently, f = 0 since domF is a neighborhood of u0 . We thus have α > 0.

4.1 Fenchel Conjugate Functions

61

According to (4.4) and dividing by α > 0, we obtain easily that F ∗ (−fα ) ≤ −βα , G∗ (fα ) and then

≤ βα − p

−F ∗ (−fα ) − G∗ (fα ) ≥ p

where fα = f /α and βα = β/α. Therefore, p ≤ q. This completes the proof.

4.1.2 Examples 1. Let C be a non-empty subset of a topological vector space V and XC be its indicator function. Then the conjugate function XC∗ of XC is defined by XC∗ (f ) = sup f, uV ,V u∈C

and is called the support function of C. Moreover, if C is a closed and convex set, XC is closed and convex, and by the conjugacy theorem (Theorem 4.5) the conjugate of its support function is its indicator function. 2. Let (V, . ) be a Banach space, (V , . ∗ ) its dual, ψα : t ∈ IR −→ | t |α /α and Fα : V −→ IR such that Fα (u) = ψα ( u ), where 1 < α < ∞. Then Fα∗ (f ) = sup ( f, uV ,V − Fα (u)) u∈V u α = sup f, uV ,V − α u∈V u α = sup sup f, uV ,V − α λ≥0 u =λ λα = sup sup f, uV ,V − α λ≥0 u =λ

λα = sup sup ( f, uV ,V ) λ − α λ≥0 u =1 α λ = sup f ∗ λ − (by definition of . ∗ ). α λ≥0 Hence (by analyzing the function r(λ) := θλ − λα /α, where θ := f ∗ and ∗ α∗ λ ∈ [0, +∞[), Fα (f ) = ( f ∗ )/α∗ , where 1/α∗ + 1/α = 1. Consequently, Fα∗ (f ) = ψα∗ ( f ∗ ), with

1 1 + = 1. α∗ α

In the same way we have that F1∗ (f ) = XB∗ (f ) where B∗ is the unit closed ball BV (0, 1).

62


3. We finish with an interesting example for the boundary-valued problems in a lemma form. Lemma 4.9. Let (V, . ) be a Banach space, (V , . ∗ ) its dual and C be a non-empty closed and convex subset of V . Consider the convex and lower semi-continuous real-valued function F on V given by F (v) := f, vV ,V + XC (v − u) ∀v ∈ V, where u ∈ V and f ∈ V are given elements. Then the conjugate function of F is defined by F ∗ (g) = g − f, uV ,V + XC ∗ (g − f ) ∀g ∈ V , where C ∗ = {g ∈ V : g, vV ,V = 0 ∀v ∈ C} (which is said to be the polar set of C). Proof. Let g ∈ V , we have that F ∗ (g) = sup ( g, vV ,V − f, vV ,V − XC (v − u)) v∈V

= sup g − f, w + uV ,V w∈C

= g − f, uV ,V + sup g − f, wV ,V . w∈C

This completes the proof (since sup g − f, wV ,V = XC∗ (g − f ) = XC ∗ (g − f )). w∈C

4.2 Subdifferentials and Superdifferentials of Extended-value Functions Let V be a topological vector space and let V be its topological dual space. We denote by ., .V ,V the bilinear form with respect to the duality between V and V . 4.2.1 Definition and Characterization Definition 4.10. Let F be a function of V into IR. The continuous affine function L everywhere less than F (minorizing F on V ) is said to be exact at the point u ∈ V if L(u) = F (u). It is clear that we have the following proposition. Proposition 4.11. Let F be a function of V into IR and L a continuous affine function that is everywhere less than F . If L is exact at point u ∈ V then the value F (u) is finite and the affine function L will have the form:

4.2 Subdifferentials and Superdifferentials of Extended-value Functions

63

there exists f ∈ V such that L(v) = f, v − uV ,V + F (u), ∀v ∈ V.

(4.5)

Moreover, the function L is maximal and its constant term β = − f, uV ,V + F (u) is the greatest λ ∈ IR such that F (w) ≥ f, wV ,V + λ, i.e., β = − f, uV ,V + F (u) = −F ∗ (f ), where F ∗ is the conjugate of F .

(4.6)

Definition 4.12. (Subdifferential and subgradient) Let F be a function of V into IR (possibly non-convex): (i) The function F is said to be subdifferentiable at the point u ∈ V if it has a continuous affine minorant L which is exact at the point u (an affine function minorizing F and coinciding with F at u). (ii) A linear form f ∈ V defined by (4.5) is called a subgradient of F at u, and the set of all these subgradients of F at u is called the subdifferential of F at u and is denoted by ∂F (u). A direct corollary is the following characterization. Corollary 4.13. Let F be a function of V into IR. The function F is subdifferentiable at a point u ∈ V if ∂F (u) = ∅. Remark 4.14. (i) The subdifferential ∂F is a set-valued mapping. (ii) If F (u) = +∞ and F is not identically equal to +∞ then ∂F (u) is empty. On the other hand, if F is identically equal to +∞, the set ∂F (u) is nonempty, for all u ∈ V . (iii) Compared to the classical notion of the differentiability which is local and requires some regularity, the subdifferentiability concept is global and the function F may not be necessarily continuous at the point u so that ∂F (u) is non-empty. (iv) For u ∈ V , consider the function R by R(v) := F (u) + f, v − uV ,V , for all v ∈ V , then F is subdifferentiable at u and f ∈ ∂F (u) if and only if the graph of R (i.e., graph(R) = {(v, R(v)) : v ∈ V }) is a supporting hyperplane of the epigraph of F , epi(F ) in (u, F (u)) . ♦ We shall give now the characterization of the subdifferential of F at u, ∂F (u). Theorem 4.15. Let F : V −→ IR be a subdifferentiable function at point u ∈ V . Then

64


f ∈ ∂F (u) if and only if F (u) is finite and

f, v − uV ,V + F (u) ≤ F (v), ∀v ∈ V.

(4.7)

Otherwise, ∂F (u) = {f ∈ V : F (v) ≥ F (u) + f, v − uV ,V , for all v ∈ V }. Moreover,

if ∂F (u) is non-empty then F (u) = F ∗∗ (u), if F (u) = F

∗∗

(u) then ∂F (u) = ∂F

∗∗

(u).

(4.8)

(4.9) (4.10)

By using the definition of the conjugate function of F and the relation (4.6) we have also the following characterization of ∂F (u). Proposition 4.16. Let F be a function of V into IR and F ∗ its conjugate. Then f ∈ ∂F (u) if and only if F (u) + F ∗ (f ) = f, uV ,V . (4.11) Proof. Suppose that, for u ∈ V and f ∈ V we have that F (u) + F ∗ (f ) =

f, uV ,V . Then, by Fenchel’s inequality, i.e., F (v) + F ∗ (f ) ≥ f, vV ,V ∀v ∈ V , we can deduce that F (u) − F (v) ≤ f, u − vV ,V ∀v ∈ V . Consequently, according to Theorem 4.15, f ∈ ∂F (u). Conversely, consider u ∈ V and f ∈ ∂F (u) then, according to Theorem 4.15, F (u) − F (v) ≤ f, u − vV ,V ∀v ∈ V . So, F (u) + ( f, vV ,V − F (v)) ≤

f, uV ,V ∀v ∈ V and then F (u) + F ∗ (f ) = F (u) + sup ( f, vV ,V − F (v)) ≤ f, uV ,V . v∈V

Because of Fenchel’s inequality again we have that F (u) + F ∗ (f ) = f, uV ,V . Example 4.17. Let (V, . ) be a Banach space with its dual (V , . ∗ ) and F : V −→ IR such that F (u) := u for all u ∈ V . Then ∂F (0) = {f ∈ V : f ∗ ≤ 1}, ∂F (u) = {f ∈ V : f ∗ = 1, f, uV ,V = u }, if u = 0. The proof is left as an exercise.

(4.12) ♣

It is clear that the following statements hold. Corollary 4.18. Let F : V −→ IR be a subdifferentiable function at point u ∈ V . Then the set ∂F (u) is convex and closed in V with respect to the topology σ(V , V ).


65

Corollary 4.19. Let F : V −→ IR be any extended real-valued function on V . A point u ∈ V is a minimizer of F , i.e., F (u) = inf F (v) if and only if v∈V

0 ∈ ∂F (u).

Corollary 4.20. Let F be a proper function of V into IR and subdifferentiable at point u ∈ V . Then the following properties hold: (i) If f ∈ ∂F (u) then u ∈ ∂F ∗ (f ). (ii) If, furthermore, F ∈ Γ (V ) then: f ∈ ∂F (u) if and only if u ∈ ∂F ∗ (f ). Remark 4.21. The subgradient funtional ∂F is monotone, i.e.,

f1 − f2 , u1 − u2 V ,V ≥ 0 ∀ui ∈ dom(∂F ), fi ∈ ∂F (ui ), i = 1, 2. Indeed, by the characterization of ∂F , we have that F (u1 ) − F (u2 ) ≤ f1 , u1 − u2 V ,V and F (u2 ) − F (u1 ) ≤ f2 , u2 − u1 V ,V . Consequently, by adding the previous relations, we obtain that 0 ≤ f1 − f2 , u1 − u2 V ,V . This implies the result.

♦

In the case of convex functions we have the following criterion for subdifferentiability. Proposition 4.22. Let F be a proper and convex function of V into IR, finite valued and continuous at the point u ∈ V . Then ∂F (v) = ∅ for all v in the interior of the effective domain of F domF (in particular ∂F (u) = ∅). Proof. The main idea of the proof is based essentially on the usage of Hahn– Banach separation in order to separate the point (u, F (u)) (element of the boundary of epiF ) and the interior of epiF . The proof is left to the reader as an exercise (he can be inspired by the proof of Theorem 4.8). We can also introduce the notion of superdifferential and supergradient that occurs naturally in the case of concave functions and maximization problems. Definition 4.23. (Superdifferential and supergradient) Let G be a function of V into IR (possibly non concave). The function G is said to be superdifferentiable at the point u ∈ V if there exists a linear form f ∈ V such that G(v) ≤ G(u) + f, v − uV ,V , for all v ∈ V.

(4.13)

A linear form f ∈ V defined by (4.13) is called a supergradient of G at u, and the set of all these supergradients of G at u is called the superdifferential of G at u and is denoted by ∂G(u).

66


Definition 4.24. (Concave conjugate) Let G be a function of V into IR. The function G∗c : V −→ IR defined by G∗c (f ) = sup (G(u) − f, uV ,V ) for all f ∈ V ,

(4.14)

u∈V

is said to be the concave conjugate function of G.

Remark 4.25. A similar result, at the subdifferentiability case, can be given for the superdifferentiability case in terms of its concave conjugate because (−G)∗ (f ) = −G∗c (−f ), ∀f ∈ V and ∂(−G)(u) = −∂G(u), ∀u ∈ V. In particular, a point u ∈ V is a maximizer of an extended real-valued function ♦ G of V into IR, i.e., G(u) = sup G(v) if and only if 0 ∈ ∂G(u). v∈V

4.2.2 General Case In this section we give some generalization of the explicit results in Section 4.2.1 to the case of general functions c(., .). For the results we refer, for example, to Dietrich [103]. This notion plays, for example, an important role in the study of the Monge–Kantorovich mass transportation problem (see, for instance, Rachev and Ruschendorf [248]). Let V and X be two topological spaces and c : V × X −→ IR be an extended real-valued function. Example 4.26. A particular case is X = V the dual of V and c(u, f ) = f, uV ,V . Another particular interest is the case where (X, . ) = (V, . ) is a Banach space and 1 c(u, v) = u − v p , p ≥ 1. p ♣ Definition 4.27. (c-convex function) Let C be a non-empty convex subset of V and F be a mapping of C into IR. The mapping F is said to be c-convex on C if F can be written as F (u) := sup(c(u, fi ) + αi ), ∀u ∈ V

(4.15)

i∈J

for some index set J, fi ∈ X and αi ∈ IR for i ∈ J.

Definition 4.28. The set of functions F : V −→ IR which are c-convex and lower semi-continuous, such that if F takes the value −∞ then F is identically equal to −∞, is denoted by Γ c (V ) and the subset of Γ c (V ) other than the constant functions −∞ and +∞ is denoted by Γ0c (V ).


67

Definition 4.29. (c-conjugate and double c-conjugate function) Let us consider an extended real-valued function F : V −→ IR: (i) The function F c : X −→ IR defined by F c (f ) := sup (c(v, f ) − F (v)) for all f ∈ X,

(4.16)

v∈V

is said to be the c-conjugate or the c-polar of F . (ii) The function F cc : V −→ IR defined by F cc (u) := sup (c(u, g) − F c (g)) for all u ∈ V,

(4.17)

g∈X

is said to be the double c-conjugate of F .

We can now introduce the Fenchel-type inequality for the general case. Proposition 4.30. (Fenchel-type inequality) Let F : V −→ IR be a given extended real-valued function. Then F c (f ) + F (u) ≥ c(u, f ), ∀f ∈ X, ∀u ∈ V.

The following results are analogous to the results obtained in the case of the classical conjugate (which do not depend on the bilinearity of form ., .). Proposition 4.31. Let F and G be two given extended real-valued functions of V onto IR. Then the following properties hold: (i) If F is less than G then Gc is less than F c . (ii) (F + β)c = F c − β, ∀β ∈ IR. (iii) F c ∈ Γ c (X) and F cc ∈ Γ c (V ). cc (iv) F ⎧ is less than F . ⎨ F (n)c = F cc , if n is even, (v) ⎩ F (n)c = F c , if n is odd.

Proposition 4.32. Let F : V −→ IR be an extended real-valued function. Then the following statements are true: (i) F cc is the largest c-convex minorant of F . (ii) F ∈ Γ c (V ) if and only if F = F cc .

Definition 4.33. (c-subdifferentiability of function) Let F : V −→ IR be an extended real-valued function. The c-subdifferential of F at point u, the set-valued function ∂c F : V −→ X, is defined by ∂c F (u) = {f ∈ X : F (u) − F (v) ≤ c(u, f ) − c(v, f ), for all v ∈ V }. (4.18) Any element f ∈ ∂c F (u) is called a c-subgradient of F at u and if ∂c F (u) = ∅ then F is c-subdifferentiable at u.

68


It is obvious that the following statement holds. Proposition 4.34. Let F : V −→ IR be an extended real-valued function. Then the following properties hold: (i) f ∈ ∂c F (u) if and only if F (u) + F c (f ) = c(u, f ). (ii) If, furthermore, F ∈ Γ c (V ), then f ∈ ∂c F (u) if and only if u ∈ ∂c F c (f ). 4.2.3 Calculus Rules with Subdifferentials In this section, we give some results of calculus with subdifferentials which are the extension of the usual differential calculus. Proposition 4.35. (i) Let F : V −→ IR be a subdifferentiable function at point u ∈ V and α be positive. Then ∂(αF )(u) = α∂F (u). (ii) Let Fi : V −→ IR, i = 1, 2 be two subdifferentiable functions at point u ∈ V . Then (∂F1 (u) + ∂F2 (u)) ⊂ ∂(F1 + F2 )(u). Proposition 4.36. (Moreau–Rockafellar) Let Fi : V −→ IR, i = 1, 2 be two functions of Γ (V ) and αi , i = 1, 2 be positive. If there is a point u0 ∈ domF1 ∩ domF2 where F1 is continuous, then ∂(α1 F1 + α2 F2 )(u) = α1 ∂F1 (u) + α2 ∂F2 (u), ∀u ∈ V. Proof. We prove first that ∂(F1 + F2 )(u) = ∂F1 (u) + ∂F2 (u), ∀u ∈ V. According to the result (ii) of Proposition 4.35 we have that ∂F1 (u) + ∂F2 (u) ⊂ ∂(F1 + F2 )(u), ∀u ∈ V. We have now to show that the inverse inclusion holds, i.e., to prove that for each g ∈ ∂(F1 + F2 )(u), there exists (g1 , g2 ) ∈ ∂F1 (u) × ∂F2 (u) such that g = g1 + g2 . According to Theorem 4.15, we have that F1 (u) and F2 (u) are finite and that F1 (v) + F2 (v) ≥ F1 (u) + F2 (u) + g, v − uV ,V , ∀v ∈ V.

(4.19)

Consider the convex and continuous function at u0 G(.) := F1 (.) − g, .V ,V − F1 (u) + g, uV ,V and the convex set with a non-empty interior C := epiG. We now introduce the following sets A := intC, B := {(v, λ) ∈ V × IR : λ ≤ F2 (u) − F2 (v)}.


69

The set A and B are convex (since F and G are convex, see Proposition 2.44) and disjoints. To end the proof, we can use (again!) Hahn–Banach’s separation (which is left to the reader as an exercise). According to the result (i) of Proposition 4.35 and to the previous result, we can obtain easily that, for all positive constants αi , i = 1, 2 and for all u∈V, ∂(α1 F1 + α2 F2 )(u) = ∂(α1 F1 )(u) + ∂(α2 F2 )(u) = α1 ∂F1 (u) + α2 ∂F2 (u).

This completes the proof.

We shall now examine the subdifferential of a composition function with an affine mapping. For this, let us consider two pairs of topological vector spaces V , V and W , W in duality with respect to certain bilinear forms

., .V ,V and ., .W ,W , respectively. Let Λ : V −→ W be a continuous linear mapping (geometrical operator) with the transpose (adjoint) Λ∗ : W −→ V defined by

g, ΛuW ,W = Λ∗ g, uV ,V

∀(g, u) ∈ W × V.

(4.20)

Remark 4.37. (i) The two paired dual spaces V, V and W, W are linked respectively by a so-called geometrical equation p = Λu

(4.21)

and a so-called equilibrium equation g = Λ∗ f.

(4.22)

(ii) In calculus of variations, if Λ is a gradient-like operator “grad” then Λ∗ should be a divergence-like operator “div”. ♦ Proposition 4.38. Let F : W −→ IR be a function of Γ (W ) and suppose that there exists a point u0 ∈ V such that F is continuous and finite at point Λu0 . Then the function F ◦ Λ : V −→ IR (belong to Γ (V )) is a subdifferential function on V and satisfies ∂(F ◦ Λ)(u) = Λ∗ ∂F (Λu),

f or all u ∈ V.

(4.23)

Proof. The first inclusion, i.e., Λ∗ ∂F (Λu) ⊂ ∂(F ◦ Λ)(u) is based on the fact that for g ∈ Λ∗ ∂F (Λu), we have (by the definition of the subdifferential)

Λ∗ g, v − uV ,V + F ◦ Λ(u) = g, Λ(v − u)W ,W + F ◦ Λ(u) ≤ F ◦ Λ(v) ∀v ∈ V. The second inclusion ∂(F ◦Λ)(u) ⊂ Λ∗ ∂F (Λu) is based on the Hahn–Banach’s separation (which is left to the reader as an exercise).

70


Remark 4.39. The result of Proposition 4.38 holds in the case of affine functions. Precisely, let F : W −→ IR be a function of Γ (W ) and A be an affine mapping, i.e., A : V −→ W such that Au = Λu − μ, where μ ∈ W ; if there exists a point u0 ∈ V such that F is continuous and finite at point Au0 , then the function F ◦ A : V −→ IR (belong to Γ (V )) is a subdifferential function on V and satisfies ∂(F ◦ A)(u) = Λ∗ ∂F (Au),

for all u ∈ V.

♦

The subdifferentiability of a convex function F at a given point is connected with the directional derivative of F at this point. Precisely, it is closely connected with the other classical differentiability concepts such as Gˆ ateaux or Fréchet derivative. 4.2.4 Connection with Directional Derivative Differentiablility of Extended Real-valued Functions Definition 4.40. (Gˆ ateaux-differentiability or G-differentiability) Let F be a function of V into IR: (i) The limit as t −→ 0+ , if it exists, of the difference quotient of F F (u + tv) − F (u) for u, v ∈ V, t

(4.24)

is said to be the directional derivative (or Gˆ ateaux variation) of F at u in the direction v and is denoted by δF (u, v). (ii) If furthermore, there exists f ∈ V such that δF (u, v) = f, vV ,V for all v ∈ V, we say that F is Gˆ ateaux(or G)-differentiable at u, and the Gˆ ateaux(or G)-differential at u of F , is denoted by F (u) or by DF (u) (in finite dimensional space F (u) is denoted by ∇F or by grad F (u)) and is characterized by

F (u), vV ,V := δF (u, v) = lim

t−→0+

F (u + tv) − F (u) ∀v ∈ V. t

(4.25)

Remark 4.41. If the function F is convex, then the difference quotient (4.24) is a monotonically increasing function of the stepsize t ∈]0, +∞[ and its limit ♦ for t −→ 0+ exists, which, however, can be −∞ or +∞. Definition 4.42. (Fréchet-differentiability) Let F be a function of V into IR. The function F is said to be Fréchet(or F)-differentiable at a given point u ∈ V if the difference quotient (4.24) as a function of t converges uniformly on every bounded set.


71

We will note that the F-differentiability of a mapping F implies the G-differentiability of F , but the converse is not true. Moreover, the Fdifferentiability of a mapping F implies the continuity of F , but the Gdifferentiability of F does not imply necessarily the continuity of F . However, there are some cases where G-differentiability implies F-differentiability. Suppose now that (V, . ) is a Banach space and let F be a function of K (an open subset of V ) into IR. We say that F is a class C 1 functional on K if the F-derivative F exists at every point u ∈ K and the mapping u −→ F (u) is continuous from K into V , i.e., if limn−→∞ un = u ∈ K then lim F (un ) − F (u), vV ,V = 0 uniformly on BV (0, 1).

n−→∞

We have the following result: If F has a continuous G-derivative on K then F is F-differentiable on K and F is of class C 1 on K. This result follows by using the following simple manipulation for u, v ∈ K: 1

F (tu + (1 − t)v) − F (u), u − vV ,V dt. F (u) − F (v) − F (u), u − vV ,V := 0

Example 4.43. Let Ω be an open subset of IRd , d ≥ 1, with finite measure, i.e., | Ω |< ∞, and let a functional φ : V = Lp+1 (Ω) −→ IR, p ∈]1, ∞[, is defined by (∀u ∈ Lp+1 (Ω)) 1 | u(x) |p+1 dx. φ(u) = p+1 Ω Then the functional φ is of class C 1 on Lp+1 (Ω) and we have

φ (u), hV ,V = u(x) | u(x) |p−1 h(x)dx ∀h ∈ Lp+1 (Ω), Ω p+1

where V = L p (Ω). Indeed, let u and h be in Lp+1 (Ω) and t ∈ (0, 1), we have | u(x) + th(x) |p+1 − | u(x) |p+1 t(p + 1) 1 = | s(u(x) + th(x)) + (1 − s)u(x) |p 0

= 0

and then

sign(s(u(x) + th(x)) + (1 − s)u(x))h(x)ds 1

| u(x) + sth(x) |p sign(u(x) + sth(x))h(x)ds

72


|

| u(x) + th(x) |p+1 − | u(x) |p+1 |≤|| u(x) | + | h(x) ||p | h(x) | . t(p + 1)

Using Hölder’s inequality, we can deduce that | (| u(x) | + | h(x) |) |p | h(x) | dx Ω

≤

p 1 p+1 p+1 | (| u(x) | + | h(x) |) |p+1 dx | h(x) |p+1 dx

Ω

≤C u

p+1 Lp+1 (Ω)

+h

p+1 Lp+1 (Ω)

p p+1

Ω

h

p+1 Lp+1(Ω)
0 are real constants and a is a function in Lq (Ω) (1 ≤ q < ∞). Then Nf is a continuous function from Lp (Ω) into Lq (Ω), where p = rq, and maps bounded subsets into bounded subsets. Conversely, we have the following proposition. Proposition 4.50. Let Ω be an open and bounded domain in IRd , d ∈ IN∗ , and let f : Ω × IR −→ IR be a Carathéodory function. If the function Nf maps Lp (Ω) into Lq (Ω) (1 ≤ p, q < ∞), then Nf is a continuous function. The previous result is a particular case of the more general results of Krasnoselskii [182] Theorem 4.51. (Krasnoselskii’s theorem) Let V and W be two Banach spaces, D be a Borel subset of IRm , m ∈ IN∗ , and F be a Carathéodory mapping from D × V into W . For each function u ∈ M, let Gu : D −→ W be the measurable function, which is generated by the function F in the sense (for a.e. η ∈ D) Gu(η) := F (η, u(η)). (4.29) If G maps Lp (D; E) into Lq (D; E) (1 ≤ p, q < ∞), then Gu is a continuous function. We present now some results concerning differentiable functionals (defined on Lp -spaces), which are important in the calculus of variations. A direct corollary of Krasnoselskii’s theorem is the following result. Corollary 4.52. Let D be a bounded open subset of IRm , m ∈ IN∗ , and F be a Carathéodory mapping from D × IRl into IR, l ∈ IN∗ . If G, defined by (4.29),


75

maps (Lp (D))l into L1 (D) (1 ≤ p < ∞), then the function PG : (Lp (D))l −→ IR defined by PG (u) =

F (x, u(x))dx,

(4.30)

Ω

is continuous.

Concerning the potentiality of the Nemitskii operator, the following results hold. Proposition 4.53. Let Ω be an open and bounded domain in IRd , d ∈ IN∗ and let F : Ω × IR −→ IR be a Carathéodory function such that | F (x, s) |≤ C | s |p +a(x)

a.e. x ∈ Ω and f or all s ∈ IR,

(4.31)

where C ≥ 0, p ≥ 1 are real constants and a is a function in L1 (Ω). Then the functional PF : Lp (Ω) −→ IR defined by F (x, u(x))dx, (4.32) PF (u) = Ω

is continuous. In particular, for a Carathéodory function f : Ω × IR −→ IR such that | f (x, s) |≤ C | s |p−1 +b(x)

a.e. x ∈ Ω and f or all s ∈ IR,

(4.33)

where C ≥ 0, p > 1 are constants and b is a function in Lq (Ω), 1/p + 1/q = 1, s

we have that the function F : Ω×IR −→ IR defined by F (x, s) :=

f (x, τ )dτ 0

is a Carathéodory function and satisfies the condition (4.31). Moreover, the functional PF is continuously F-differentiable from Lp (Ω) into IR and PF (u) = f (., u(.)) = Nf u ∈ Lq (Ω).

According to Proposition 4.50 and Proposition 4.53, we can deduce the following proposition. Proposition 4.54. Let Ω be an open and bounded domain in IRd , d ∈ IN∗ , then the following results hold: (i) If F : Ω × IR −→ IR is a Carathéodory function such that NF maps Lp (Ω) into L1 (Ω), 1 ≤ p < ∞, then the functional PF defined by (4.32), is continuously on Lp (Ω). (ii) If f : Ω × IR −→ IR is a Carathéodory function such that the function Nf maps Lp (Ω) into Lq (Ω), 1 < p < ∞, 1/p + 1/q = 1, then the function s

F : Ω × IR −→ IR defined by F (x, s) :=

f (x, τ )dτ is a Carathéodory 0

function and NF maps Lp (Ω) into L1 (Ω). Moreover, the functional PF is continuously F-differentiable on Lp (Ω) and PF (u) = f (., u(.)) = Nf u ∈ Lq (Ω).

76


Differentiability of Operators and Second-order Derivative The G (or F)-differentiability also works for operators. Definition 4.55. (G-differentiability of an operator) Let V and W be locally convex topological vector spaces and K ⊂ V be an open subset. Let D be a function of V into W . (i) The limit as t −→ 0+ , if it exists, of the difference quotient of the operator D D(u + tψ) − D(u) (4.34) t is said to be the directional derivative (or Gˆ ateaux variation) of the operator D at point u ∈ K in the direction ψ and is denoted by δD(u, ψ). (ii) If the previous limit exists for all ψ ∈ V , we say that the operator D is Gˆ ateaux(or G)-differentiable operator at u. The Gˆ ateaux(or G)differential at u of the operator D is an operator from V to W denoted by D (u) (or by DD(u)) and defined by D (u) : V −→ W such that D (u)ψ := δD(u, ψ) = lim

t−→0+

D(u + tψ) − D(u) ∀ψ ∈ V. t

(4.35)

The operator D (u) is 1-homogeneous (i.e., D (u)(αψ) = αD (u)(ψ) for all (α, ψ) ∈ IR × V ), but is not additive in the general case and the linearity of G-derivative is a nontrivial requirement in the general case. Definition 4.56. (F-differentiability of operators) Let D be a function of V into W . The operator D is said to be Fréchet(or F)-differentiable at a given point u ∈ V if the difference quotient (D(u + tψ) − D(u))/t (as a function of t) converges uniformly on every bounded set. In this case the F-derivative of D at point u, denoted also by D (u), is a linear continuous operator on V . Remark 4.57. The F-differentiability of D implies the G-differentiability of D and the G-derivative D (u) of D at point u is a linear continuous operator on V . Conversely, if D is G-differentiable in an open subset K of V , if the G-derivative D (u) at point u is bounded and linear for all u ∈ K and if D : v ∈ V −→ D (v) ∈ L(V ; W ) 1 is continuous then D is F-differentiable (see, e.g., Schwartz [261]). ♦ Example 4.58. (i) Let Λ : V −→ W be a G-differentiable and linear operator, where V and W are two locally convex topological vector spaces. Then the G-derivative of Λ at point u is exactly the operator Λ, i.e., Λ (u) = Λ. 1

L(V ; W ) denotes the space of continuous linear functionals from V to W .

(4.36)

4.3 Applications of the Duality

77

In this case the operator Λ (u) is a linear operator. (ii) Let Λ : V −→ W be a G-differentiable and quadratic operator (i.e., there exists a bilinear operator B : V × V −→ W such that Λ(u) := B(u, u) for all u ∈ V ) then the G-derivative Λ at point u in the direction ψ is Λ (u)ψ := B(u, ψ) + B(ψ, u).

(4.37)

♣

Here also the operator Λ (u) is a linear operator.

We can now give the second-order G-derivative of a real valued on V . Let F : V −→ IR be a given G-differentiable functional. Then its G-derivative F (u) at a point u ∈ V is an operator from V into V . If now the operator F (u) is also G-derivative, then the second-order G-variation of F at point u ∈ V in the direction v ∈ V and ψ ∈ V is defined by F (u + tψ).v − F (u).v t−→0+ t = D2 F (u)ψ, vV ,V ,

δ 2 F (u, v, ψ) := lim

(4.38)

where D2 F (u) (or F (2) (u)) is said to be the second-order G-derivative of F at point u ∈ V . It is clear that δ 2 F (u, ., .) : V ×V −→ IR is a symmetric bilinear functional. As a direct consequence (because of Proposition 4.59), we have a characterization of the convexity of the second G-differentiable function F . Proposition 4.59. (Monotonicity of the second G-differential of a convex function) Let C be a non-empty convex subset of V and F be a G-differentiable function of C into IR. Then F is convex over C if and only if

D2 F (uα )(v − u), v − uV ,V ≥ 0 for all (u, v) ∈ C × C,

(4.39)

where uα = v + α(u − v), α ∈ (0, 1). Moreover, F is strictly convex over C if and only if (4.39) holds as a strict inequality if u = v. We finish this subsection with the following definitions: (i) The second F-derivative δ 2 D(u, ., .) : V ×V −→ W of the F-differentiable operator D : V −→ W (i.e., the F-derivative of the function D : v ∈ V −→ D (v) ∈ L(V ; W )) at point u ∈ V belongs to the space of symmetric bilinear functionals and is denoted by D2 D(u) (or D(2) (u)). (ii) Similarly as above, the d-th F -derivative D(d) (u, ., . . . , .) is a symmetric d-times

continuous d-linear function on V .

4.3 Applications of the Duality In this section we shall apply the duality results to different problems of calculus of variations arising from mechanics and physics.

78


Before presenting some applications of the duality, we need to make some classical definitions for general non-linear systems. 4.3.1 Fundamental Equations Assume that the paired topological vector space V and its dual V are placed in duality by the bilinear form ., .V ,V . The bilinear form ., .V ,V is defined on a space domain Ω ⊂ IRn , n ≥ 1 in static systems, and on a space-time domain Q := Ω × (0, T ) ⊂ IRn+1 (T is the final time), in dynamical systems. Consider an operator A : V −→ V , where its effective domain dom(A) is non-empty. The set {A(u) : u ∈ V } ⊂ V is said to be the range of the operator A and denoted by R(A). Then the operator A is a mapping from dom(A) into R(A) and is said to invertible, if there exists an inverse operator A−1 from R(A) into dom(A) such that A(u) = u∗ if and only if A−1 (u∗ ) = u.

(4.40)

Let S be a source space, assumed to be a subspace of V . For a given source f ∈ S the equation A(u) = f, (4.41) is called the fundamental equation and the operator A is said to be the fundamental operator. Generally, between the configuration and the source variables, there exist intermediate variables, which always appear in pairs placed in duality by other types of bilinear forms. These duality relations, between intermediate variables, describe different physical or biological interior properties or characteristics of the system (for example, the environmental, the physical or biological parameters). Consequently, the fundamental operator (respectively equation) is depending on certain intermediate equations (respectively operators). Assume now that the paired topological vector space W and its dual W are placed in duality by the bilinear form ., .W ,W . The paired spaces V, V and W, W are linked by the following operators: (i)

The geometrical operator Λ : U −→ W : this operator describes the geometrical transformation of the system (does not concern the physical or biological properties of the system). The associated geometrical equation (also called the kinematic equation, see Oden and Reddy [231]) is given by p = Λu. (4.42)

(ii) The constitutive operator C : W −→ W : this operator describes the physical or biological properties of the system. The associated constitutive equation (which gives the duality relation between the spaces W and W ) is given by (4.43) p∗ = Cp.


79

(iii) The balance operator B : W −→ V : this operator describes the local state of the system. The associated balance equation (also called the equilibrium equation) is given by f = Bp∗ .

(4.44)

Then, by taking account of (i)–(iii) we can deduce easily that the fundamental operator A = B ◦ C ◦ Λ : V −→ V and the fundamental equation Au = f can be written as p = Λu, (4.45) p∗ = Cp, ∗ f = Bp . 4.3.2 Duality Mapping in Banach Spaces Let (V, . ) be a real Banach space and its dual (V , . ∗ ). Before introducing the notion of duality mapping, we give a smoothness definition and a result concerning the G-differentiablity of a norm. Definition 4.60. (Smooth space) The real Banach space (V, . ) is said to be smooth, if for each non-null element u in V , there exists a unique element fu in V such that fu ∗ = 1 and fu , uV ,V = u . Definition 4.61. (Strictly convex space) The real Banach space (V, . ) is said to be strictly convex if for all u, v in V with u = v = 1 and u = v we have u + v < 2. It is obvious that a uniform convex Banach space is a strictly convex Banach space. Theorem 4.62. (G-differentiability of a norm) Let (V, . ) be a real Banach space. The norm . of V is G-differentiable if and only if (V, . ) is a smooth Banach space.

Proof. The proof can be found, e.g., in Diestel [102]. We can now study the duality mapping on a real Banach space.

Definition 4.63. (Gauge duality mapping) Let ϕ : IR −→ IR be a so-called gauge function, i.e., ϕ is continuous, strictly increasing , ϕ(0) = 0 and

lim ϕ(t) = ∞

t−→∞

and let 2V be the so-called power set of V , i.e., the set of all subsets of V . Then, the multivalued mapping Jϕ : V −→ 2V , defined as follows: Jϕ (0) := {0}, Jϕ (u) := {f ∈ V : f ∗ = ϕ( u ), f, uV ,V = ϕ( u ) u } if u = 0

(4.46)

is called the gauged (or weighted) duality mapping corresponding to the gauge function ϕ.

80


Remark 4.64. (i) Because of the Hahn–Banach theorem, it is easy to see that the domain of Jϕ is the whole space, i.e., dom(Jϕ ) = V . (ii) By the expression (4.46) we have that, for each u ∈ V , Jϕ (u) is bounded, closed and convex subset of V . (iii) According to (4.12) we have that Jϕ (u) := ϕ(ℵu)∂ℵ(u) ∀u ∈ V,

(4.47)

where ℵ : u ∈ V −→ ℵu := u and ∂ℵ is the subdifferential of ℵ. The mapping Jϕ can be also written as ℵu ϕ(t)dt ∀u ∈ V. Jϕ := ∂F, with F (u) := 0

♦

Proposition 4.65. Let (V, . ) be a real Banach space, then Jϕ is monotone, i.e.,

f1 − f2 , u1 − u2 V ,V ≥ 0, for all ui ∈ V, fi ∈ Jϕ (ui ), i = 1, 2. Moreover, if V is also strictly convex then Jϕ is strictly monotone and, in particular, Jϕ (u) ∩ Jϕ (v) = ∅ for u, v in V with u = v. Proof. Because of the expression (4.46) and the fact that ϕ is increasing, we obtain

f1 − f2 , u1 − u2 V ,V ≥ (ϕ( u1 ) − ϕ( u2 ))( u1 − u2 ) ≥ 0, for all ui ∈ V, fi ∈ Jϕ (ui ), i = 1, 2. Prove now the strict monotonicity of Jϕ . Suppose then, by contradiction, that there exist two elements u1 , u2 in V such that u1 = u2 and fi ∈ Jϕ (ui ), i = 1, 2 satisfying

f1 − f2 , u1 − u2 V ,V = 0. According to the first result we have that 0 = f1 − f2 , u1 − u2 V ,V ≥ (ϕ( u1 ) − ϕ( u2 ))( u1 − u2 ) ≥ 0 and then u1 = u2 = α = 0 (since u1 = u2 ). Now, putting vi = ui /α, i = 1, 2, then v1 = v2 = 1 and f1 − f2 , v1 − v2 V ,V = 0. Consequently (since, by definition, fi , vi V ,V = ϕ( ui ), i = 1, 2), 0 = [ϕ( u1 ) − f1 , v2 V ,V ] + [ϕ( u2 ) − f2 , v1 V ,V ]. So (since, by definition, ϕ( ui ) = fi ∗ ≥ fi , vj V ,V , i, j = 1, 2 with i = j) we have f1 ∗ = f2 ∗ = f2 , v1 V ,V = f1 , v2 V ,V Consequently, f1 , v1 + v2 V ,V = 2 f1 ∗ and then v1 + v2 ≥ 2, which is a contradiction with the fact that the space V is strictly convex.


81

It is clear that the following results hold (according to the definition of Jϕ , the expression (4.47), the convexity of ℵ, Proposition 4.44 and Theorem 4.62). Proposition 4.66. Let (V, . ) be a real Banach space. Then: (i) V is a smooth Banach space if and only if Jϕ is a singleton (i.e., Jϕ (u) is a singleton for all u in V ). (ii) If V is a smooth Banach space then Jϕ (u) = F (u). Otherwise Jϕ (0) = 0 and Jϕ (u) = ϕ(ℵu)ℵ (u) if u = 0, where ℵ (u) is the G-differential of ℵ at u.

Remark 4.67. If (V, . ) is a smooth Banach space then (since ℵ (u) satisfies ℵ (u) ∗ = 1, ℵ (u), uV ,V = u , for all u = 0 in V ) Jϕ (u) ∗ = ϕ(ℵu) and Jϕ (u), uV ,V = ϕ(ℵu) u ∀u ∈ V.

♦

Proposition 4.68. (Demi-continuous result) Let (V, . ) be a Banach space. If V is smooth and reflexive then Jϕ is demi-continuous, i.e., if un −→ u strongly on V then Jϕ (un ) Jϕ (u) weakly on V . Proof. Since (un ) is uniformly bounded then ϕ( un ) and Jϕ (un ) ∗ = ϕ(ℵun ) = ϕ( un ) is also bounded. Consequently, Jϕ (un ) is uniformly bounded on a reflexive Banach space V . Then, we can extract from Jϕ (un ) a subsequence, denoted also by Jϕ (un ), converging to f . Prove now that f = Jϕ (u). By the lower semi-continuity of the norm . we have that (according to the continuity of ϕ) f ∗ ≤ lim inf Jϕ (un ) ∗ = lim ϕ( un ) = ϕ( u ). n

n

(4.48)

Moreover, since

Jϕ (un ), un V ,V − f, uV ,V = Jϕ (un ) − f, uV ,V + Jϕ (un ), un − uV ,V , then | Jϕ (un ), un V ,V − f, uV ,V | ≤| Jϕ (un ), un − uV ,V | + Jϕ (un ) ∗ un − u . From un −→ u strongly in V , Jϕ (un ) f weakly in V and the boundedness of Jϕ un , it follows easily that Jϕ (un ), un V ,V −→ f, uV ,V . On the other hand, Jϕ (un ), un V ,V = ϕ( un ) un −→ ϕ( u ) u so, ϕ( u ) u = f, uV ,V and then ϕ( u ) ≤ f ∗ . Consequently, by (4.48) we can deduce that ϕ( u ) = f ∗ . Since ϕ( u ) = f ∗ and f, uV ,V = ϕ( u ) u , then, by definition, this implies that Jϕ (u) = f .

82


Remark 4.69. It is clear that Jϕ is coercive, because lim ϕ(t) = ∞ and so, t−→∞

Jϕ (u), uV ,V = ϕ( u ) −→ ∞ for u ∈ V, u −→ ∞. u

♦

The following theorem shows an important result namely the surjectivity of the gauge duality mapping Jϕ . Theorem 4.70. Let (V, . ) be a Banach space. If V is smooth and reflexive then Jϕ is surjective. Moreover, if V is also a strictly convex space, then Jϕ is a bijection function. Proof. The surjectivity result follows from the well-known theorem of Browder [64]: If V is a real Banach and reflexive space, then any monotone, coercive and demi-continuous operator T : V −→ V is surjective. Indeed, according to Proposition 4.65, Remark 4.69 and Proposition 4.68, Jϕ is monotone, coercive and demi-continuous. Consequently, by Browder’s theorem Jϕ is surjective. The injectivity of the functional Jϕ is a direct consequence of Proposition 4.65. Example 4.71. Let Ω be a domain in IRd (d ∈ IN∗ ) and V = Lp (Ω) (for 1 < p < ∞). The space V is a reflexive, smooth and uniformly convex Banach ∗ space and its dual is V = Lp (Ω), where 1/p + 1/p∗ = 1. It is clear that the function ϕ(t) = tp−1 is a gauge function. The gauge duality mapping Jϕ corresponding to ϕ is Jϕ (u)(x) =| u(x) |p−1 sign(u(x)), a.e. x ∈ Ω. ♣ 4.3.3 Duality and Fundamental Equations Let F and S be two real valued functions on two reflexive Banach spaces V and W respectively which are either convex or concave, and Gˆ ateaux-differentiable on the convex sets K ⊂ V and M ⊂ W , respectively. Then the two duality equations between the paired spaces V, V and W, W can be given by F (u) = g ∈ V , S (p) = f ∈ W ,

(4.49)

where F : K ⊂ V −→ V and S : M ⊂ W −→ W stand for the Gderivative of F and S, respectively. In mathematical modeling, the duality equation f = S (p) is known as the constitutive equation and the duality equation g = F (u) usually gives natural boundary conditions or external energy in variational boundary value problems.


83

Let now a so-called feasible set or admissible set (the set of possible solutions to a given problem (4.49)) Kad = {u ∈ V : u ∈ K and Λu ∈ M }, where Λ is a continuous linear mapping Λ : V −→ W with the adjoint Λ∗ : W −→ V defined by (4.20). According to (4.21) and (4.22), the system (4.49) can be written in a so-called fundamental equation F (u) = Λ∗ S (Λu).

(4.50)

If, for a given ξ ∈ V (external source) and a linear operator C : W −→ W , the functional F is written as F (u) := ξ, uV ,V for all u ∈ V and the functional S is written as S(p) := (1/2) Cp, pW ,W for all p ∈ W (i.e., S is a quadratic function), then Equation (4.50) becomes Λ∗ Cs Λu = ξ, where Cs := (C + C ∗ )/2 (the symmetric part of C) and C ∗ : W −→ W the adjoint operator of C is defined by

Cp, qW ,W = C ∗ q, pW ,W . It is clear that the operator A := Λ∗ Cs Λ is self-adjoint, i.e., A∗ = A. Example 4.72. Consider Maxwell’s system, which describes electric and magnetic fields in a homogeneous and isotropic medium, by (a.e. on [0, T ]) −div(νE) = f on Ω, 1 ∂E + curl( curlA) = g on Ω, ∂t μ ∂A E = ∇φ + on Ω, ∂t

(4.51)

where Ω is a bounded open subset of IRd (d ∈ IN∗ ), sufficiently regular with Γ = ∂Ω its boundary, T is the final time, φ is the scalar potential, B is the vector magnetic potential, E is the electric field intensity, f is the charge density, g is the current density, ν is the induction capacity of the medium and μ is the permeability of free space. We introduce now the following state variables: f D φ E ∗ ∗ ,p = , u= ,p= ,u = g H A B where D is the electric flux density, B is the magnetic flux density and H is the magnetic field intensity.

84


We can prove easily that the previous system (4.51) can be written in the form ∂ ∇ ∂t p = u on Ω, 0 curl ν 0 (4.52) p∗ = p on Ω, 0 μ1 −div 0 ∗ p∗ on Ω. u = ∂ − ∂t curl Let V, V and W, W be two pairs of real topological vector spaces, in duality with respect to the bilinear forms ., .V ,V and ., .W ,W , respectively defined by T (u, u∗ )2 dxdt,

u∗ , uV ,V := 0 Ω (4.53) T

p∗ , pW ,W :=

0

(p, p∗ )2 dxdt,

Ω

where (., .)2 is the euclidian scalar product in IR4 . Moreover, we suppose that W is a Hilbert space and can be identified to its dual W . By using Green’s formula and the identity H.curl(A) = div(A × H) + A.curl(H), we have that

H.curl(A)dx =

Ω

(A × H).ndx +

Γ

A.curl(H)dx,

where n is the unit outward normal on ∂Ω. If (A × H).n = 0 on Γ then H.curl(A)dx = A.curl(H)dx. Ω

Ω

Let now K = {u ∈ V : u = 0 on Γ, u(., t = 0) = u(., t = T )} and the admissible set Kad := {u ∈ K : Λu ∈ W }, where Λ : V −→ W is defined by Λu =

(4.54)

Ω

∂ ∇ ∂t 0 curl

u.

(4.55)


85

Then for a given u ∈ Kad and p∗ ∈ W such that p∗ (., t = 0) = p∗ (., t = T ) we have that (since u = 0 on Γ ) T ∂A ).D + H.curl(A))dxdt.

p∗ , ΛuW ,W = ((∇φ + ∂t 0 Ω By using Green’s formula and by integrating by parts in time, we can deduce that (according to (4.55)) T ∂D ∗ +curl(H)))dxdt = Λ∗ p∗ , uV ,V , (−div(D)φ +A.(−

p , ΛuW ,W = ∂t 0 Ω where Λ∗ : W −→ V is defined by −div 0 p∗ . Λ∗ p∗ := ∂ − ∂t curl Let now C be the following linear and symmetric matrix ν 0 C := . 0 1/μ For a given external source u∗ ∈ V , we introduce the functionals F : V −→ IR and S : W −→ IR defined respectively by F (u) := u∗ , uV ,V and S(p) :=

1

Cp, pW ,W . 2

The functionals F and S are G-differentiable and their G-differentials satisfy F (u) = u∗ , S (p) = Cp respectively. Consequently, according to (4.49) and (4.50), we can deduce that the fundamental system is Λ∗ CΛu = u∗ (which is an abstract form of System (4.51)), the constitutive system is Cp = p∗ , and finally the equilibrium system is Λ∗ p∗ = u∗ . ♣ Example 4.73. Let us consider the following mixed boundary problem in electrostatics −div(ν∇u) = f on Ω, (4.56) u = 0 on Γ0 , ν∇u.n = g on Γ1 , where Ω is a bounded open subset of IRd (d ∈ IN∗ ), sufficiently regular with ∂Ω = Γ0 ∪ Γ1 its boundary, n is the unit outward normal on ∂Ω, and ν > 0 is the dielectric constant. Let V, V and W, W be two pairs of real topological vector spaces, in duality with respect to the bilinear forms ., .V ,V and ., .W ,W , respectively. Let Λ := −∇, where p := Λu is the electric field intensity. The variational functionals are ν | p(x) |22 dx + XA (p), S(p) := Ω2 (4.57) f udx + gudx − XK (u), F (u) := Ω

Γ1

86


where A := H(div; Ω) ⊂ W , K := {u ∈ H 1 (Ω) : u = 0 on Γ0 } ⊂ V , | . |2 is the euclidian norm and XK (respectively XA ) is the indicator function of K (respectively of A). Then it is easy to prove that the functionals S and F are finite and Gdifferentiable on A and K respectively. Moreover, p∗ = S (p) = νp, u∗ = F (u) = f on Ω and u∗ = F (u) = g on Γ1 . By using Green’s formula we can deduce that

S (p), ΛuW ,W = p∗ , ΛuW ,W = −p∗ ∇udx Ω = udiv(p∗ )dx − p∗ .n, uH −1/2 (Γ ),H 1/2 (Γ ) Ω

= Λ∗ p∗ , uV ,V . Then we have the following abstract equilibrium equation: u∗ = Λ∗ p∗ = div(p∗ ) = f on Ω and u∗ = Λ∗ p∗ = −p∗ .n = g on Γ1 , and the fundamental operator in this problem is the operator −νΔ.

♣

Remark 4.74. More examples can be found, e.g., in Oden and Reddy [231] and Strang [275]. ♦ 4.3.4 Euler–Lagrange Equation and the Non-linear Operator Many non-linear boundary value problems can be written as Au = 0, where A : V −→ W is a non-linear operator between two topological vector spaces V and W . If the previous problem is variational then there exists a real-valued and G-differentiable function F on V , such that A := F , where the space W is a subset of V (the dual of V ). The operator A is called the potential operator and F is called the potential of A or the variational functional. Then the previous problem is equivalent to solving the so-called Euler–Lagrange equation ∀v ∈ V. (4.58) F (u) = 0 or F (u), vV ,V = 0 Definition 4.75. (Critical point of a function) A point uc ∈ V is said to be a critical point (or stationary point) of F if uc is a solution of (4.58), i.e., F (uc ) = 0 or F (uc ), vV ,V = 0 ∀v ∈ U . Its value α := F (uc ) is called a critical value of F and the set F −1 (α) is called a critical level of F .


87

Example 4.76. Consider the following semilinear elliptic problem −Δu + λu = f (., u) on Ω, αu + β∇u.n = 0 on ∂Ω.

(4.59)

where Ω is a bounded open subset of IRd (d ∈ IN∗ ), sufficiently regular with ∂Ω its boundary, n is the unit outward normal on ∂Ω, the constants α, β are such that α + β = 0, the constant λ is positive, f (., u) is a non-linear operator satisfying f (., 0) = 0 and other standard conditions. v

f (., w)dw. The variational functional is

Let F (., v) := F (u) :=

0

λ 1 (− Δu(x).u(x) + | u(x) |2 −F (x, u(x)))dx, 2 2 Ω

(4.60)

where V := H01 (Ω) if β = 0 and V := H 1 (Ω) if β = 0. Then it is easy to prove that the weak solutions of problem (4.59) coincide with critical points of F . ♣ There exist two classes of critical points, the first corresponds to the classical local extrema (infimum or supremum) of F and the second class corresponds to critical points which are not local extrema. Definition 4.77. (Saddle point of a function) A critical point uc of a realvalued G-differentiable function F that is not a local extremum is said to be a saddle point of F , i.e., for any neighborhood Kc of uc , there exist v, w of Kc such that F (v) < F (uc ) < F (w). (4.61) Remark 4.78. In theoretical chemistry and physics, saddle points appear as unstable equilibria or transient excited states (as they correspond to the transition states and lead to the minimum energy paths between reactant molecules and product molecules). A vast literature can be found in the theoretical and computational chemistry and physics. ♦ To study the critical point problem, which is generally motivated by the search of solutions to non-linear partial differential equations, to an infinite dimensional setting, requires certain compactness conditions in order to express the compacity of sequences (e.g., minimizing sequences) which converge to a point when we hope to be the critical point. Moreover, the compactness condition is a fundamental tool to carry out the deformation and then the well-known Mountain–Pass theorem of Ambrosetti-Rabinowitz, which is a minimax characterization of a critical point of a functional F in the sense of a minimax over a suitable non-empty class S of non-empty subsets of U c = inf sup F (u). A∈S u∈A

(4.62)

The one most frequently used is due to Palais and Small (see Palais [233]).

88


Definition 4.79. (Palais–Small condition) Let (U, . ) be a Banach space, (U , . ∗ ) be its dual space and F be a real-valued and G-differentiable function on a subset K of U . The function F is said to satisfy the so-called Palais– Small condition (denoted by (P S) for short), if every sequence (un ) in K, for which F (un ) is bounded and lim F (un ) ∗ = 0, has a convergent subsen−→∞ quence in K. A variant of Palais–Small condition is the following (see Brezis et al. [59]). Definition 4.80. (Palais–Small condition at level c) Let (U, . ) be a Banach space, (U , . ∗ ) be its dual space and F be a real-valued and Gdifferentiable function on a subset K of U . For a given c ∈ IR, the function F is said to satisfy the so-called Palais–Small condition at the level c (denoted by (P S)c for short), if every sequence (un ) in K, for which lim F (un ) = c n−→∞

and lim F (un ) ∗ = 0, has a convergent subsequence in K. n−→∞

Remark 4.81. (i) If a bounded function F satisfies the condition (P S), the set of critical points is compact. (ii) If a function F satisfies the condition (P S)c at a fixed level c, we have that the set of critical points at level c Z(c) := {u ∈ U : F (u) = c and F (u) = 0}, is compact. (iii) It is clear that the condition (P S) implies the condition (P S)c at a fixed level c. ♦ The following theorem shows the existence of the infimum of a function that satisfies the Palais–Small condition. Theorem 4.82. Let (U, . ) be a Banach space and F be a real valued, lower semi-continuous and G-differentiable function on a subset K of U . If the function F is bounded from below (i.e., the set of its values has a lower bound) and satisfies the Palais–Small condition (P S), then there exists at least one critical point uc ∈ K of F (i.e., F (uc ) = 0) such that F (uc ) = inf F (v). v∈K

This theorem is a consequence of Ekeland’s lemma in complete metric spaces (see Ekeland [111]), see below. Lemma 4.83. (Ekeland’s lemma) Let (X, d) be a complete metric space equipped with a metric d, φ : X −→ IR be a real-valued, lower semi-continuous function which is bounded from below on X and set by c := inf φ(u). Then for all > 0, there exists u ∈ X such that

u∈X


89

c ≤ φ(u ) ≤ c + , φ(u ) − φ(v) < d(v, u ) ∀v ∈ X, v = u .

(4.63)

We are now presenting a minimax characterization of a critical point of the type (4.62), namely the well-known Mountain–Pass theorem which has been proposed by Ambrosetti and Rabinowitz. This type of minimax problems, in which the basic ideas go back to Lusternik and Schnirelman [211], has been studied extensively in critical point theory since last few years (see Chang [71], Ekeland [113], Ghoussoub [131], Mawhin and Willem [220], Struwe [277]). Before giving this theorem, we introduce some definitions and a variant of the so-called deformation lemma. Definition 4.84. (Pseudo-gradient) Let (U, . ) be a Banach space and (U , . ∗ ) be its dual space. Let φ be a real-valued and continuously Fdifferentiable function on U and Kr (φ) := {u ∈ U : φ (u) = 0} be the set of non critical points of φ: (i) Let u ∈ U . An element v ∈ U is said to be a pseudo-gradient point of φ on u, if we have v ≤ 2 φ (u) ∗

and φ (u), v ≥ φ (u) 2∗ .

(4.64)

(ii) The function Vφ : Kr (φ) −→ U is said to be a pseudo-gradient field for φ on Kr (φ) if Vφ is a locally Lipschitz continuous mapping on Kr (φ). For all u ∈ Kr (φ), Vφ (u) is a pseudo-gradient point of φ on u. We have the following lemma. Lemma 4.85. Under the assumption of Definition 4.84, there exists a pseudogradient field for φ on Kr (φ). Proof. For the proof, we can refer, for example, to Rabinowitz [247] or Willem [301]. We can deduce easily the following corollary. Corollary 4.86. Let (U, . ) be a Banach space and (U , . ∗ ) be its dual space and φ be a real-valued, even and continuously F-differentiable function on U . Then there exists a pseudo-gradient field for φ on Kr (φ), which is an odd function. Proof. By using Lemma 4.85, we have the existence of a pseudo-gradient field Vφ for φ on Kr (φ). We verify easily that the mapping V˜φ defined by, for all u ∈ U, Vφ (u) − Vφ (−u) , V˜φ (u) := 2 is a pseudo-gradient field for φ on Kr (φ) and an odd function. This completes the proof.

90


Definition 4.87. (Homotopy) Let (U, . ) be a Banach space and η : [0, 1]× U −→ U be a continuous function: (i) The function η is called a homotopy function, if η(0, u) = u

for all u ∈ U.

(4.65)

(ii) The homotopy function η is said to be a homotopy of homeomorphisms if, for all t ∈ [0, 1], each map η(t, .) : [0, 1] × U −→ U is an homeomorphism function.

(4.66)

(iii) Let f be a real-valued and continuously F-differentiable function on U . The homotopy function η is called f -increasing (respectively f -decreasing) if for 0 ≤ s ≤ t ≤ 1 we have, for all u ∈ U , f (η(s, u)) ≤ f (η(t, u)) (respectively f (η(s, u)) ≥ f (η(t, u))).

(4.67)

We can now give the well-known deformation lemma (see Rabinowitz [247]). Lemma 4.88. (Deformation lemma) Let (U, . ) be a Banach space and (U , . ∗ ) be its dual space. Let φ be a real-valued and continuously Fdifferentiable function on U and c ∈ IR be a given value. If the function φ satisfies the Palais–Small (PS) condition at level c, then: (i) for a given 0 > 0, there exists a homotopy of homeomorphisms and φdecreasing function η 0 such that: (a) for all t ∈ [0, 1] and u ∈ U such that | φ(u) − c |> 0 , we have η 0 (t, u) = u (b) if, moreover, φ is even then, for all t ∈ [0, 1], each map η 0 (t, .) is odd (ii) if the set of critical points of φ at level c, Z(c) is empty then we can find 0 > 0 such that the corresponding homotopy of homeomorphisms and φ-decreasing function η 0 satisfies the following condition: (c) for all ∈]0, 0 [ and u ∈ U such that φ(u) ≤ c + , we have that φ(η 0 (1, u)) ≤ c − . Proof. The proof is based essentially on the existence of a pseudo gradient field Vφ , because of Lemma 4.85 (which can be chosen odd if the functional φ is even, by Corollary 4.86) and finally by considering the function η 0 (for a given 0 > 0), the unique solution of the following Cauchy problem:2 2

For the existence and uniqueness theorem see the general theory of ordinary differential equations in Banach spaces given, for example, in Cartan [69].


dη(t, u) = −Wφ (η(t, u)), dt with the initial condition η(0, u) = u, 1 Vφ and ξ : U −→ [0, 1] is given by where Wφ := ξ min 1, Vφ

P

d(u, P ) , d(u, P ) + d(u, Q) = {u ∈ U : φ(u) ≤ c − 0 } ∪ {u ∈ U : φ(u) ≥ c + 0 },

Q

= {u ∈ U : c − 0 ≤ φ(u) ≤ c + 0 },

91

(4.68)

ξ(u) =

(4.69)

and where d(u, v) := u − v is the distance in U . Since P and Q are closed disjoint non-empty subsets then the map ξ is locally Lipschistz continuous. Moreover, we have that ξ = 0 on P , ξ = 1 on Q and if the function φ is even, ξ is even also and then Wφ is odd (by the choice of Vφ , odd map if φ is even). We note also that the solution η 0 is locally Lipschistz continuous (since Wφ is locally Lipschistz continuous, see e.g., Cartan [69]) and satisfies η 0 (t, η 0 (s, u)) = η 0 (t + s, u), for all t, s ∈ IR, u ∈ U and then by the uniqueness of Cauchy problem (4.68) we can deduce that, for every t, each map η 0 (t, .) is a homeomorphism from U into U and (t, .) is η 0 (−t, .). Consequently, the function η 0 is a homotopy its inverse η −1 0 of homoemorphisms function. To obtain the property φ-decreasing of η 0 , we calculate the derivative by time of the function φ(η 0 (t, u)). Indeed (for simplicity we denote η 0 (., u) by η 0 (.)) d φ(η 0 (t)) dt dη 0 (t) U ,U = φ (η 0 (t)), dt 1 = −ξ(η 0 (t)) min 1,

φ (η 0 (t)), Vφ (η 0 (t))U ,U (4.70) Vφ (η 0 (t)) 1 ≤ −ξ(η 0 (t)) min 1, φ (η 0 (t)) 2∗ (by (4.64)) Vφ (η 0 (t)) ≤0 and then the φ-decreasing result. The point (b) is immediate and the point (a) is a direct consequence of the uniqueness of the Cauchy problem (4.68) and the fact that if u is such that | φ(u) − c |> 0 , Wφ (u) = 0 (η 0 (.) − u is a solution of (4.68) with null initial condition). For the point (c), since the set of critical points of φ at level c is empty, according to the Palais–Small condition, we have the existence of β > 0 and 0 < δ ≤ 1 such that ∀u ∈ {u ∈ U : | φ(u) − c |≤ β}, φ (u) ∗ ≥ δ

(4.71)

92


and we can take 0 := min(β, δ 2 /8). Let now be in ]0, 0 ] and u be in U such that φ(u) ≤ c + . We remark that if there exists t ∈ [0, 1[ such that φ(η 0 (t)) ≤ c − then φ(η 0 (1)) ≤ c − (since η 0 is a φ-decreasing function). Assume then, for all t ∈ [0, 1[, | φ(η 0 (t)) − c |≤ ≤ 0 ≤ β. So, according to relations (4.70), (4.71) and (4.64), and the fact that δ ≤ 1 and ξ ≤ 1, we can deduce that −δ 2 d φ(η 0 (t)) ≤ ≤ −20 (since 0 = min(β, δ 2 /8)) dt 4 and then φ(η 0 (1)) ≤ φ(η 0 (0)) − 20 ≤ φ(u) − 20 ≤ c + − 20 ≤ c − 0 . This completes the proof.

Let us now recall the well-known Mountain–Pass theorem in a useful and popular form. Theorem 4.89. (Mountain–Pass theorem) Let (U, . ) be a Banach space and (U , . ∗ ) be its dual space. Let φ be a real-valued and continuously Fdifferentiable function on U satisfying the Palais–Small (PS) condition. Assume that φ(0) = 0 and that: (i) there exist ρ > 0 and a > 0 such that if u ∈ U : u = ρ then φ(u) ≥ a (ii) there exists an element u0 ∈ U such that u0 > ρ and φ(u0 ) ≤ a. Then φ admits a critical value c ∈ IR, which is charaterized by the following minimax condition: a ≤ c := inf sup φ(u), A∈S u∈A

where S := {g([0, 1]) : g ∈ C([0, 1]; U ), g(0) = 0 and g(1) = u0 }. Proof. The proof is based on the deformation principle. It is clear that S = ∅ and, by convexity, for all A ∈ S, A ∩ {u ∈ U : u = ρ} = ∅. Consequently, max φ(w) ≥ a and c = inf sup φ(u) ≥ a. w∈A

A∈S u∈A

Prove now that c = inf sup φ(u) is a critical value of φ. For this suppose, A∈S u∈A

by contradiction, that the set of critical points of φ at level c, Z(c) is empty. Let 0 < ≤ a/2 then, from (i) and (ii), we can find A ∈ S such that A = g([0, 1]) and a ≤ c ≤ max φ(w) ≤ c + . w∈A

Consequently, according to (P S) condition, the Palais–Small condition at level c, (P S)c , holds. By using the deformation lemma, we can find 0 > 0 and a homotopy of homeomorphisms and φ-decreasing function η 0 such that the points (a)–(c) of Lemma 4.88 hold.


93

By taking f (τ ) := η 0 (1, g(τ )) we have that f (0) = η 0 (1, g(0)) = 0 and f (1) = η 0 (1, g(1)) = η 0 (1, u0 ) = u0 since | φ(0) − c |> 0 , | φ(u0 ) − c |> 0 . Then f ([0, 1]) ∈ S. By the deformation lemma (point (c)), we obtain that f ([0, 1]) ⊂ {u ∈ U : φ(u) ≤ c − }, which is a contradiction with the fact that

sup

φ(u) ≥ c (according to

u∈f ([0,1])

the definition of c). Therefore, Z(c) is non-empty. This completes the proof.

4.3.5 Minimization of Convex Functions In this section, we give some well-known results concerning the minimization of convex functionals in a reflexive Banach space – the existence of the minimum, the uniqueness and the characterization of the optimal solution – and some results concerning the variational inequalities. Existence, Uniqueness and Characterization of the Optimal Solution Let us consider a functional F of K ⊂ V into IR, where (V, . ) is a reflexive Banach space on IR with norm . ∗ , (V , . ∗ ) be the dual space of V with norm . ∗ , K is a non-empty closed convex subset of V , F is a convex, lower semi-continuous and proper function on K.

(4.72)

We consider the following minimization problem: find u ∈ K such that F (u) = inf F (v). v∈K

(4.73)

Remark 4.90. Problem (4.73) can be solved throughout the space V , by introducing the extended mapping F˜ : V −→ IR of F by F˜ (u) = F (u) if u ∈ K

and F˜ (u) = +∞ if u ∈ / K.

Otherwise, F˜ (u) = F (u) + XC (u).3 In this way we can relax the constraint u ∈ K on F to get a proper, convex and lower semi-continuous function F˜ on the whole space V . Therefore, the following problem and its set of solutions: F˜ (u) = inf F˜ (v), v∈V

are the same as problem (4.73) and its set of solutions. 3

(4.74) ♦

XC is a convex and lower semi-continuous function since C is a convex and closed set.

94


Proposition 4.91. The set of solutions of (4.73) with (4.72) is a closed convex set which is possibly empty. We give now simple conditions of the existence and the uniqueness of a solution of problem (4.73). Proposition 4.92. (Existence and uniqueness of an optimal solution) If we assume that the set K is bounded or that the functional J is coercive over K, i.e., satisfies (2.10), then there is at least one u ∈ K satisfying (4.73). Otherwise u ∈ K satisfies F (u) ≤ F (v)

for all v ∈ K.

(4.75)

Moreover, if we assume that the functional J is strictly convex over K then there is a unique u ∈ K satisfying (4.73). Proof. The existence result is given by Theorem 2.101. The uniqueness result is immediate by using the fact that if u1 and u2 are two solutions of (4.73) then (u1 + u2 )/2 is also a solution of (4.73) (because Proposition 4.91 holds). Let us now give a more analytic condition, for (4.75) to hold, which is said to be the characterization of a solution of (4.73). Proposition 4.93. (Characterization of an optimal solution I) If we assume that the functional F is G-differentiable with continuous derivative F , then if u ∈ K, the following conditions are equivalent: (i) u is a solution of (4.73) (ii) F (u), v − uV ,V ≥ 0 for all v ∈ K (iii) F (v), v − uV ,V ≥ 0 for all v ∈ K. Proof. (i) ⇔ (ii) is based on the fact that for u a solution of (4.73), F (u) ≤ F ((1 − t)u + tv) for all v ∈ K and t ∈]0, 1[, and the function F is G-differentiable. The proof of (ii) ⇔ (iii) is based on the monotonicity of the function F (because of Proposition 4.46) and on the continuity of F . The development of the proof is left to the reader as an exercise. Proposition 4.94. (Characterization of an optimal solution II) If we assume that the functional F = F1 + F2 , where F1 and F2 are convex and lower semicontinuous functions of K into IR, and F1 is G-differentiable with continuous derivative F1 (F2 is not necessarily differentiable), then if u ∈ K, the following conditions are equivalent: (i) u is a solution of (4.73) (ii) F1 (u), v − uV ,V + F2 (v) − F2 (u) ≥ 0 for all v ∈ K (iii) F1 (v), v − uV ,V + F2 (v) − F2 (u) ≥ 0 for all v ∈ K. Proof. The proof is obtained by using the same technique as used in the previous proposition.


95

Unilateral Problems or Variational Inequalities Let (V, . ) be a reflexive Banach space with norm . and let (V , . ∗ ) be its dual space with norm . ∗ . Let us consider a non-linear operator A from V into V , a convex, lower semi-continuous and proper function ϕ of V into IR and a given element f of V . We consider the following problem: find u ∈ V such that

Au − f, v − uV ,V + ϕ(v) − ϕ(u) ≥ 0 for all v ∈ V.

(4.76)

This is the so-called variational inequality. Theorem 4.95. We assume that the function ϕ is a convex, lower semicontinuous and proper function, and the operator A satisfies the following assumptions: A is weakly continuous over the subspaces of a finite dimension of V, (4.77) A is monotone, i.e., Au − Av, u − vV ,V ≥ 0 f or all u, v in V, and there exists an element u0 of the effective domain of ϕ domϕ such that

Av, v − u0 V ,V + ϕ(v) −→ +∞ f or v ∈ V, v −→ +∞. v

(4.78)

Then, for f given in V , there is at least one u ∈ V satisfying the variational inequality (4.76). Proof. The proof of this theorem and many other existence theorems can be find in Lions [203]. Let K be a closed and convex subset of V , we have the following special case as corollary of the previous theorem (by taking ϕ as the indicator function of K). Proposition 4.96. Let K be a closed and convex subset of V and suppose that the operator A satisfies the hypotheses (4.77) and (4.78). Then, for all f in V , there is an element u ∈ K such that

Au − f, v − uV ,V ≥ 0

f or all v ∈ K.

(4.79)

If, in addition, K = V , then u, the solution of (4.79), satisfies Au = f.

We are now giving some remarks on general boundary value problems. 4.3.6 General Boundary Value Problems For general boundary value, in addition to the paired function spaces U (Ω), U (Ω) and X(Ω), X (Ω) in duality with respect to certain bilinear forms

., .U ,U and ., .X ,X noted ., .U and ., .X respectively (to simplify), we introduce some appropriate boundary spaces UB (Γ ), UB (Γ ) and XB (Γ ), XB (Γ )

96


in duality with respect to certain bilinear forms (., .)Γ and ., .Γ respectively, where Γ is the boundary of the domain Ω. For simplicity, the trace of any element σ of the original spaces on its boundary spaces is also denoted by σ. We can now, for a given linear continuous operator Λ : U −→ X, introduce the associated boundary operator ΛB : UB −→ XB and the adjoint operator Λ∗ : X −→ U by

Λu, p∗ X := u, Λ∗ p∗ U + ΛB u, p∗ Γ ,

(4.80)

for all u in U and p∗ in X . We denote by Λ∗B : XB −→ UB the adjoint operator of ΛB defined by

ΛB u, p∗ Γ := (u, Λ∗B p∗ )Γ .

(4.81)

We introduce now the following operators on the closure of the domain Ω, denoted by Ω ΛT := −ΛB on Γ, ΛT := Λ on Ω, (4.82) ∗ ∗ ΛT := Λ on Ω, Λ∗T := Λ∗B on Γ. In mixed boundary condition value problems, for simplicity, we consider the situation when the boundary B is split into two parts Γu and Γs such that: (4.83) Γ = Γu ∪ Γs , Γu ∩ Γs = ∅ and mes(Γu ) = ∅ and

ΛB u, p∗ Γ := ΛB u, p∗ Γu + (u, Λ∗B p∗ )Γs .

(4.84)

We introduce the following bilinear forms:

ΛT u, p∗ X := Λu, p∗ X − ΛB u, p∗ Γu ,

u, Λ∗T p∗ U := u, Λ∗ p∗ U + (u, Λ∗B p∗ )Γs ,

(4.85)

where X = X ∩ XB and U = U ∩ UB . According to (4.81) and (4.84), we obtain the following equality (which is corresponding, for example, to the Green’s formula): (4.86)

ΛT u, p∗ X = u, Λ∗T p∗ U . Let JB be the so-called bilinear concomitant, evaluated on the boundary Γ , (the boundary term generated, for example, by integration by parts) defined by (4.87) JB (v, q ∗ ) := ΛB v, q ∗ Γ = ΛB v, q ∗ Γu + (v, Λ∗B q ∗ )Γs . for all v in UB and q ∗ in XB . Then, for all u, v in UB and p∗ , q ∗ ) in XB , we have ∂JB (u, p∗ )v = ΛB v, p∗ Γu + (v, Λ∗B p∗ )Γs = JB (v, p∗ ) = (v, Λ∗B p∗ )Γ , ∂u ∂JB (u, p∗ )q ∗ = ΛB u, q ∗ Γu + (u, Λ∗B q ∗ )Γs = JB (u, q ∗ ) = ΛB u, q ∗ Γ . ∂p∗


97

Consequently, ∂JB ∂JB (u, p∗ ) = Λ∗B p∗ and (u, p∗ ) = ΛB u. ∂u ∂p∗

(4.88)

In order to finish, we suppose that the boundary conditions Λ∗B p∗ = φ on Γs and ΛB u = ξ on Γu are given, then according to (4.88), we can deduce that ∂JB (u, p∗ ) = Λ∗B p∗ = φ on Γs , ∂u ∂JB (u, p∗ ) = ΛB u = ξ on Γu . ∂p∗ Consequently, we have the following abstract equilibrium equation: u∗ = Λ∗ p∗ = f on Ω, Λ∗B p∗ = φ on Γs , p = Λu on Ω, ΛB u = ξ on Γu .

(4.89)

We now introduce the following sets: Kad = {u ∈ U : ΛB u = ξ on Γu }, which is the admissibility set on u for the mixed boundary-value problem, Aad ⊂ X, which is the admissibility set on p, and Kkad = {u ∈ Kad : ΛT u ∈ Aad }, which is the kinematically admissible set. The external energy F : Kad ⊂ U −→ IR is defined by F (u) := f, uU + (u, φ)Γs , and the duality relation between u and u∗ is then given by u∗ = F (u) = f on Ω and u∗ = F (u) = φ on Γs . Suppose now there exists a G-differentiable and convex operator W : Aad ⊂ X −→ U such that (4.90) p∗ = W (p) on Ω. Thus the total potential energy J : Kkad ⊂ U −→ IR of the system is defined by J(u) := S(Λu) + F (u) = W (Λu), 1U + F (u). We can easily prove that, if Kkad is an open subset of Kad , uc is a solution of (4.89)-(4.90) if and only if uc is a minimizer of the total potential J, i.e., J(uc ) = inf J(u). Indeed, since the functional J is G-differentiable, then u∈Kkad

the critical point uc of J, i.e., J (uc ) = 0, satisfies (according to (4.80) and (4.84)) Λ∗ W (Λuc ) = f on Ω and Λ∗B W (ΛB uc ) = φ on Γs . This shows that the critical points of J are the solution of (4.89) and (4.90) and conversely (by the convexity of J). We end this chapter by the following remark.

98


Remark 4.97. The duality plays also a key role in optimal shape design problems. The idea is to find an optimal solution (Ω ∗ , u∗ ) ∈ Uad × Dad (Ω ∗ ) such that inf Φ(u(Ω), Λ(u(Ω)), Ω), Φ(u∗ , Λ(u∗ ), Ω ∗ ) = inf Ω∈Uad u∈Dad (Ω)

where Φ is a given performance (or cost) functional, Λ is some geometrical operator, Uad is the set of admissible configurations Ω ⊂ IRm and for any Ω ∈ Uad , and Dad (Ω) denotes the set of admissible deformations u. For more details, the reader can be referred, for example in the case of the linear geometrical operator, to Haftka and Gurdal [149], Pierre and Henrot [153] and Sokolowski and Zolesio [272]. ♦

5 Lagrange Duality Theory

In this chapter we are interested in the analysis of critical points of Lagrangians with duality. The critical point and duality theories have proven to be one of the most important tools in study of non-linear systems. Among the various critical point methods, minimax principles leading to the existence of saddle points have played an important role in mathematics and science, and together they play a central role for the notion of stability. This chapter describes, first, the Frenchel–Rockafellar duality which is based on a perturbation method and the extended Lagrange duality theory, second, a duality based on the classical minimax theorems, and, finally, the non-convex parametric variational problem for a geometrically non-linear system. For the latter, we have generalized the work of Strang and Gao [127], by introducing a new gap function.

5.1 Frenchel–Rockafellar Duality in Optimization The Frenchel–Rockafellar duality in convex optimization (see Rockafellar [251]), which is a special case of the generalized Lagrange duality, is based on perturbation techniques. In this section, we associate a so-called dual problem to a given optimization problem, by using conjugate convex function and a family of perturbed problems. Let U and X be locally convex topological vector spaces with the duals U and X respectively. We denote (to simplify the notation) by ., .U (respectively ., .X ) the canonical bilinear form ., .U ,U (respectively ., .X ,X ) with respect to the duality between U and U (respectively between X and X); the pairing between U × X and U × X is written (classically) as

(v ∗ , q ∗ ), (v, q)U×X = v ∗ , vU + q ∗ , qX .

100


5.1.1 Primal and Dual Problems Let us consider a functional J of U into IR and consider the following minimization problem (P). Find the infimum of the functional J, i.e., inf J(v).

v∈U

(5.1)

Definition 5.1. The problem (P) is called the primal problem, the infimum for problem (P) is denoted by inf(P) and every element u of U such that J(u) = inf(P) is called an optimal solution of (P). If there exists an element u0 of U such that J(u0 ) < +∞ (i.e., J is not the functional constant +∞), the problem (P) is said to be non-trivial. Before giving the dual problem associated with the primal problem (P), let us consider a functional φ of U × X into IR such that φ(u, 0) = J(u) ∀u ∈ U.

(5.2)

Its conjugate function, in the duality between U × X and U × X, φ∗ : U × X −→ IR in Γ (U × X ) is defined by (for (u∗ , p∗ ) ∈ U × X ) φ∗ (u∗ , p∗ ) =

( (u∗ , p∗ ), (v, q)U×X − φ(v, q)).

sup

(5.3)

(v,q)∈U×X

We also introduce, for all p ∈ X, the following sequence of minimization problems (Pp ): (5.4) inf φ(v, p). v∈U

Classically, the function φ is said to be the perturbation of the function J, the problem (Pp ) is called the perturbed problem and the value p is termed as the perturbation variable (or parameter). Remark 5.2. According to the hypothesis (5.2), it is clear that the problem ♦ (P0 ) is the primal problem (P). Example 5.3. Assume that there exists a continuous linear operator Λ : U −→ X, i.e., Λ ∈ L(U ; X), with its adjoint Λ∗ ∈ L(X ; U ) such that the functional J can be written in the form J(u) := Ψ (u, Λu), where Ψ is an extended real-valued function on U × X. Then the primal problem (P) becomes inf Ψ (u, Λu).

u∈U

5.1 Frenchel–Rockafellar Duality in Optimization

101

In this case the perturbation function φ can be written as φ(u, p) := Ψ (u, Λu − p). In a particular case, the functional Ψ can be decomposed into Ψ (u, Λu) := S(Λu) − F (u), and the perturbation function φ is then φ(u, p) := S(Λu − p) − F (u). For example, we can take the case of variational problems in the Sobolev spaces W 1,s (Ω) (1 ≤ s ≤ ∞), where Ω is an open bounded subset of IRd (d ∈ IN∗ ) with sufficiently regular boundary ∂Ω (for example of class C 1 ). Consider a functional F : Ω×IR×IRd −→ IR, which comes from the boundary value problems and satisfies some growth conditions, and a closed, convex and non-empty subset K of W 1,p (Ω), for example, K := {v ∈ W 1,p (Ω) : v = 0 on ∂Ω} = W01,s (Ω). F (x, u(x), Λu(x))dx. The perturbation funcLet Λ = grad and J(u) =

tional is then

Ω

F (x, u(x), Λu(x) − p(x))dx.

φ(u, p) := Ω

A simple example of J is 1 1 J(u) = ( | Λu(x) |2 + | u(x) |s+1 −f (x)u(x))dx s+1 Ω 2 where f is given in Lr (Ω), r ≥ max(1, 2d/(d + 2)). We can easily prove that the function J is strictly convex, continuously F-differentiable on U = H01 ∩ Ls+1 (Ω) and satisfies (by using Hölder’s and Sobolev’s inequalities) J(u) ≥

1 2

≥

1 4

1 1 r u 2H 1 (Ω) + s+1 u s+1 Ls+1 (Ω) −C0 f L (Ω) u H0 (Ω) 0

u

2H 1 (Ω) 0

1 2 + s+1 u s+1 Ls+1 (Ω) −C1 f Lr (Ω) .

Consequently, if u U −→ ∞ then J(u) −→ ∞ and then the infimum of J is attained on U at a unique critical point uc and we have 0 = J (uc ) = −Δuc + | uc |s−1 uc − f on Ω and uc = 0 on ∂Ω. In this case the perturbation functional is given by 1 | u(x) |s+1 −f (x)u(x))dx. φ(u, p) := (| Λu(x) − p(x) |2 + s + 1 Ω

♣

We are now be able to define a dual or adjoint problem. The dual problem associated with (P) with respect to φ is then the following problem (P ∗ ): sup (−φ∗ (0, q)).

q∈X

(5.5)

102


The supremum for problem (P ∗ ) is denoted by sup(P ∗ ) and every element p∗ of X such that −φ∗ (0, p∗ ) = sup(P ∗ ) is called an optimal solution of (P ∗ ). We shall now give some relationships between the primal problem (P) and its dual (P ∗ ). Proposition 5.4. (Weak duality) For problems (P) and (P ∗ ) we have: (i) −∞ ≤ sup(P ∗ ) ≤ inf(P) ≤ +∞. (ii) If (P) is non-trivial, then sup(P ∗ ) ≤ inf(P) < +∞. (iii) If (P ∗ ) is non-trivial, then −∞ < sup(P ∗ ) ≤ inf(P). (iv) If (P) and (P ∗ ) are non-trivial then −∞ < sup(P ∗ ) ≤ inf(P) < +∞. Proof. Let p∗ ∈ X , then we have, by definition, that φ∗ (0, p∗ ) =

sup

( p∗ , qX − φ(v, q))

(v,q)∈U×X

and, in particular, we have that for all v ∈ U φ∗ (0, p∗ ) ≥ p∗ , 0X − φ(v, 0) = −φ(v, 0). Consequently, sup(P ∗ ) ≤ inf(P) and then (i). If (P) is non-trivial, there exists u0 such that J(u0 ) < ∞ and then φ(u0 , 0) < +∞. Thus with (i) we can deduce the result (ii). In the same way we can prove the result (iii). The result (iv) follows from (ii) and (iii). Definition 5.5. (Dual gap) The difference between inf(P) and sup(P ∗ ): inf(P) − sup(P ∗ ) is said to be the dual gap of problem (P).

Since the technique used to introduce the dual problem corresponding to a minimization problem, is valid for a maximization problem, we can easily introduce the dual of the dual problem (P ∗ ) and then the bidual problem corresponding to the primal problem (P), (P ∗∗ ), is inf (φ∗∗ (u, 0)),

u∈U

(5.6)

where φ∗∗ is the conjugate function of φ∗ and then the biconjugate and the Γ -regularization function of φ. Remark 5.6. (i) Since the dual of φ∗∗ is none other than φ∗ (according to Corollary 4.7), then the repetition of the duality operation is limited and the dual problem corresponding to problem (P ∗∗ ) is the problem (P ∗ ). (ii) If φ∗∗ = φ (and then φ ∈ Γ (U × X)) then problem (P ∗∗ ) is identical to problem (P) and consequently (according to (i)) problem (P) is the dual of (P ∗ ) (and inversely). (iii) If φ ∈ Γ0 (U × X), i.e., the functional φ ∈ Γ (U × X) is not equal to the constant functions +∞ and −∞, then J ∈ Γ0 (U ). Consequently, problem (P) ♦ is non-trivial and (P) is the dual of (P ∗ ).


103

5.1.2 Normal and Stability Problems Let us consider a functional J of U into IR, and a functional φ of U × X into IR and its conjugate function φ∗ in the duality between U × X and U × X defined by (5.3). We make the following assumption for some of our results φ ∈ Γ0 (U × X).,

(5.7)

Let us consider now the infimal value function h of X into IR such that: for all element p ∈ X, the value h(p) is defined as the infimum for problem (5.4), i.e., h(p) := inf φ(v, p), v∈U

and h∗ be its conjugate function in the duality between X and X. It follows that h(0) = inf(P) = inf J(v). v∈U

Proposition 5.7. If the function φ is convex then the function h of X into IR is also convex. Proof. Let p, q be in X and suppose that there exist a, b in IR such that h(p) < a and h(q) < b. Let α be in (0, 1), then, since φ is convex on U × X, h(αp + (1 − α)q) = inf φ(δ, αp + (1 − α)q) δ∈X

≤ φ(αw + (1 − α)ζ, αp + (1 − α)q) ∀w, ζ ∈ X ≤ αφ(w, p) + (1 − α)φ(ζ, q) ∀w, ζ ∈ X. Consequently, h(αp + (1 − α)q) ≤ α inf φ(w, p) + (1 − α) inf φ(ζ, q) w∈X

ζ∈X

= αh(p) + (1 − α)h(q) and then the convexity result. If h(p) = +∞ or h(q) = +∞ the proof is trivial.

Remark 5.8. In particular, if φ satisfies the assumption (5.7), φ is convex and then h is convex, but in general (although φ ∈ Γ0 (U × X)) the function ♦ h∈ / Γ0 (X). Now we give the relation between the conjugate function h∗ of h and the conjugate function φ∗ of φ. Lemma 5.9. (i) For all q ∗ ∈ X , we have that h∗ (q ∗ ) = φ∗ (0, q ∗ ). (ii) Let h∗∗ be the biconjugate function of h, then we have that sup(P ∗ ) = sup (−h∗ (q ∗ )) = h∗∗ (0). q∗ ∈X

104



Lemma 5.10. If h is a subdifferentiable function at point 0, then the set of the solutions to problem (P ∗ ) is the set ∂h∗∗ (0). Proof. Let p∗ ∈ X be an optimal solution to problem (P ∗ ). It follows that φ∗ (0, p∗ ) ≤ φ∗ (0, q ∗ ) ∀q ∗ ∈ X . According to Lemma 5.9, we can deduce that h∗ (p∗ ) ≤ h∗ (q ∗ ) ∀q ∗ ∈ X and then

h∗ (p∗ ) ≤ − sup (−h∗ (q ∗ )) = −h∗∗ (0). q∗ ∈X

Therefore, h∗ (p∗ ) + h∗∗ (0) ≤ 0. Since h∗ (p∗ ) + h∗∗ (0) ≥ p∗ , 0X (Fenchel’s inequality), then h∗ (p∗ ) + h∗∗ (0) = 0 and consequently, p∗ ∈ ∂h∗∗ (0) (according to Proposition 4.16). Conversely, let p∗ be in ∂h∗∗ (0), then by using the same arguments as to obtain the previous inclusion (but in the reversed direction), it yields that p∗ is an optimal solution of (P ∗ ). Definition 5.11. (Normal problem) The problem (P) defined by (5.1) is said to be normal if h is a finite and lower semi-continuous function at point 0. The following proposition shows some characterization of a normal problem. Proposition 5.12. (Characterization of a normal problem) If φ satisfies the assumption (5.7), then the following conditions are equivalent: (i) problem (P) is normal (ii) problem (P ∗ ) is normal (iii) −∞ < inf(P) = sup(P ∗ ) < +∞. Proof. The equivalence (ii) ⇔ (iii) is immediate and results from the fact that (P) = (P ∗∗ ), i.e., (P) is the dual problem of (P ∗ ) (because of Remark 5.6). Prove now that (i) ⇔ (iii). Let h be the closure of the convex function h, then (according to Fenchel– Moreau’s theorem) we have h∗∗ = h ≤ h. Since h(0) = h(0) ∈ IR (by assumption) then (because of Lemma 5.9) inf(P) = h(0) = h∗∗ (0) = sup(P ∗ ) ∈ IR, and conversely.

(5.8)


105

Remark 5.13. In the proof of the equivalence (i) ⇔ (iii), we have only used the convexity of h and consequently, only the convexity of φ (according to Proposition 5.7). ♦ Let us now introduce the notion of stability. Definition 5.14. (Stable problem) The problem (P) defined by (5.1) is said to be stable if h is a finite and subdifferentiable function at point 0 (i.e., h(0) is finite and ∂h(0) = ∅). The following proposition shows the strong duality. Proposition 5.15. (Strong duality I) If φ satisfies the assumption (5.7), then the following conditions are equivalent: (i) problem (P) is stable (ii) problem (P) is normal and (P ∗ ) has at least one solution. Proof. The proof is a simple application of Theorem 4.15 by taking account of Lemma 5.10 and the relation (5.8). It is clear that we have the following result (by using Remark 5.6 and Proposition 5.15). Corollary 5.16. (Strong duality II) If φ satisfies the assumption (5.7), then the following conditions are equivalent: (i) problems (P) and (P ∗ ) are normal and have some solutions (ii) problems (P) and (P ∗ ) are stable (iii) problem (P) is stable and has some solutions.

To prove the stability by means of the definition is not so easy in general. We give now such a stability criterion. Proposition 5.17. (Stability criterion) Let φ be a convex function and let us assume that inf(P) is finite (i.e., −∞ < inf v∈U J(v) < ∞). Further, let us suppose that there exists an element u0 ∈ U such that the f unction p ∈ X −→ φ(u0 , p) ∈ IR is f inite and continuous at point 0,

(5.9)

then problem (P) is stable. Proof. According to assumptions we have that h(0) is finite and h is convex (because of Proposition 5.7). Since the function ψ0 : q ∈ X −→ ψ0 (q) = φ(u0 , q) ∈ IR is proper convex and continuous at point 0, then there exists a neighborhood of 0 in X on which ψ0 is bounded above by a finite constant. Consequently (according to Proposition 2.103), the function h that is bounded by the function ψ0 is then finite and continuous at point 0. Because of Proposition 4.22, we can conclude that h is finite and subdifferentiable at point 0 and then we have the stability of problem (P).

106


Now we give some extremality relations or optimality conditions and existence of solutions. 5.1.3 Optimality Conditions and Existence Proposition 5.18. (Extremal relations) If problems (P) and (P ∗ ) have both of the solutions and −∞ < inf(P) = sup(P ∗ ) < +∞,

(5.10)

then all solutions u ∈ U of (P) and all solutions p∗ ∈ X of (P ∗ ) satisfy the following optimality relations: φ(u, 0) + φ∗ (0, p∗ ) = 0,

(5.11)

(0, p∗ ) ∈ ∂φ(u, 0) or (u, 0) ∈ ∂φ∗ (0, p∗ ).

(5.12)

or (equivalently)

∗

Conversely, if u ∈ U and p ∈ X satisfy the relations (5.11) or (5.12), then u is a solution of problem (P), p∗ is a solution of (P ∗ ) and moreover, we have −∞ < inf(P) = sup(P ∗ ) < +∞. Proof. The proof is immediate by using the definition of the conjugate function φ∗ and the fact that inf(P) = φ(u, 0) = sup(P ∗ ) = −φ∗ (0, p∗ ). Remark 5.19. Because of Proposition 5.12 and Proposition 5.15, the relation (5.10) is true if problem (P) is normal or stable or if (P ∗ ) is normal. ♦ We give now a result of existence in the case of the reflexive Banach space. Proposition 5.20. Let (U, . ) be a reflexive Banach space. Under assumptions (5.7) and (5.9), and if φ(v, 0) −→ +∞ for v ∈ U, v −→ +∞,

(5.13)

then problems (P) and (P ∗ ) have at least one solution, −∞ < inf(P) = sup(P ∗ ) < +∞, and the optimality conditions (5.11) (or equivalently (5.12)) are satisfied. Proof. According to the assumption (5.13), the function J(.) = φ(., 0) satisfies the condition of the existence result of Proposition 4.92 (here K := U ) and then we have the existence of a solution of (P). Because of Proposition 5.17, we can deduce the existence of a solution of (P ∗ ) and then the optimality conditions (according to Proposition 5.18).


107

5.1.4 Bidual Problem and Duality in Variational Inequalities In this section we give some results which concern the bidual problem of a given problem and the duality in variational inequalities. The Bidual Problem In addition to the two pairs of topological vector spaces U, U and X, X in duality, we take two topological spaces U and X such that U and U (respectively X and X ) are in duality and U ⊂ U is dense in U (respectively X ⊂ X is dense in X ) where the injection is continuous.

(5.14)

Moreover, we assume that, the function φ satisfies (5.7), i.e., φ ∈ Γ0 (U × X). Let φ∗∗ be the biconjugate function of φ in the duality between U × X and U × X . Since φ ∈ Γ0 (U × X) then φ∗∗ ∈ Γ0 (U × X ). We can now give the dual of the dual problem (P ∗ ) between U × X and U × X and then the bidual problem corresponding to the primal problem (P), (P ∗∗ ), is inf (φ∗∗ (u, 0)). u∈U

According to Proposition 5.4, we have that (because of Frenchet–Moreau’s theorem) (5.15) −∞ ≤ sup(P ∗ ) ≤ inf(P ∗∗ ) ≤ inf P ≤ +∞. Remark 5.21. (i) If inf(P) = inf(P ∗∗ ) then any solution of (P) is a solution of (P ∗∗ ) and any solution in U ⊂ U of (P ∗∗ ) is a solution of (P). In this case the problem (P ∗∗ ) is called a weak formulation of problem (P) and each / U is called a weak solution of problem (P). solution u in U such that u ∈ (ii) If the assumption (5.9) holds then according to Propositions 5.17, 5.15 and 5.12, we have that inf(P) = sup(P ∗ ) and consequently (according to (5.15)), sup(P ∗ ) = inf(P ∗∗ ) = inf(P).

(5.16)

(iii) If we assume that U (respectively X) is a non-reflexive Banach space, U (respectively X ) its dual and U (respectively X ) its bidual, we can observe that the assumption (5.14) is satisfied. ♦ Proposition 5.22. Let us suppose that U (respectively X) is a non-reflexive Banach space, U (respectively X ) its dual and U (respectively X ) its bidual. If the function φ satisfies the conditions (5.7) and (5.9) and if domφ(., 0) is a non-empty subset of U such that φ(v, 0) −→ +∞ f or v ∈ domφ(., 0), v U −→ +∞,

(5.17)

108


then problem (P) admits weak solutions and all accumulation points1 of a minimizing sequence of (P) are weak solutions of (P). Proof. The proof is left to the reader as an exercise.

Duality in Variational Inequalities Let (U, . ) be a reflexive Banach space with norm . and let (U , . ∗ ) be its dual space with norm . ∗ . Let us consider an operator A from U into U satisfying the assumptions (4.77)–(4.78), a function ϕ that is a convex, lower semi-continuous and proper function of U into IR and a given element f of U . We consider the following variational inequality problem: find u ∈ U such that (5.18)

Au − f, v − uU + ϕ(v) − ϕ(u) ≥ 0 for all v ∈ U. According to Theorem 4.95, problem (5.18) admits at least one solution u in U . By setting g = Au − f we have that (according to (5.18))

g, vU + ϕ(v) ≥ g, uU + ϕ(u) for all v ∈ U.

(5.19)

Problem (5.19) is not an optimization problem because of the dependence of g on u, but once u is known then g (depending on the known u) can be considered as a known element of U and then problem (5.19) can be considered as an optimization problem (I) (for which u is one solution) inf G(v),

v∈U

(5.20)

where the functional G of U into IR is defined by G(v) = g, vU + ϕ(v). We can use the same technique as in previous section in order to obtain the dual problem corresponding to problem (I).

5.2 Lagrange Duality The Lagrange duality theory plays an important role in constrained optimization, in general variational problems and in control theory. In this section we examine the extremality relationship between primal and dual functionals, which are linked by an arbitrary extended Lagrangian function, by the comparaison of the infimum with the supremum. 5.2.1 Definitions and Critical Points of Lagrangians Let us now introduce the notions of saddle point, subcritical point and supercritical point for an extended real-valued function. 1

Also called cluster points.

5.2 Lagrange Duality

109

Definition 5.23. (Saddle point) Let U and Y be two topological vector spaces and L : U × Y −→ IR be an extended real-valued function. The pair (u, p∗ ) of U × Y is called a saddle point of the function L if L(u, q ∗ ) ≤ L(u, p∗ ) ≤ L(v, p∗ ), ∀(v, q ∗ ) ∈ U × Y.

(5.21)

Definition 5.24. (Supercritical point) Let U and Y be two topological vector spaces and L : U × Y −→ IR be an extended real-valued function. The pair (u, p∗ ) of U × Y is called a supercritical point of the function L if L(u, q ∗ ) ≤ L(u, p∗ ) ≥ L(v, p∗ ), ∀(v, q ∗ ) ∈ U × Y.

(5.22)

Definition 5.25. (Subcritical point) Let U and Y be two topological vector spaces and L : U × Y −→ IR be an extended real-valued function. The pair (u, p∗ ) of U × Y is called a subcritical point of the function L if L(u, q ∗ ) ≥ L(u, p∗ ) ≤ L(v, p∗ ), ∀(v, q ∗ ) ∈ U × Y, i.e., (u, p∗ ) is a supercritical point of the functional −L.

(5.23)

Definition 5.26. (Remarquable point) Let U and Y be two topological vector spaces and L : U × Y −→ IR be an extended real-valued function. A point (u, p∗ ) is said to be a remarquable point of the function L if (u, p∗ ) is either a saddle point of L or of −L, or the supercritical or the subcritical of L. We shall now give a characterization of remarquable points of an arbitrary given extended real-valued function L : U × X −→ IR by using its partial subdifferentials. For a given function L, their partial subdifferentials at point (u, p∗ ), ∂v L(u, p∗ ) and ∂q∗ L(u, p∗ ), are defined by v ∗ ∈ ∂v L(u, p∗ ) if v ∗ ∈ U and L(v, p∗ ) ≥ L(u, p∗ ) + v ∗ , v − uU , q ∗∗ ∈ ∂q∗ L(u, p∗ ) if q ∗∗ ∈ X and L(u, q ∗ ) ≥ L(u, p∗ ) + q ∗∗ , q ∗ − p∗ X ,

(5.24)

for all v in U and q ∗ in X (X is the bidual of X). According to the expressions (5.21), (5.23) and (5.22), we can deduce easily the following characterization. Lemma 5.27. Let L : U × X −→ IR be a given function and (u, p∗ ) be in U × X : (i) A point (u, p∗ ) is a saddle point of L if and only if 0 ∈ ∂v L(u, p∗ ) and 0 ∈ ∂q∗ (−L)(u, p∗ ).

(5.25)

110


(ii) A point (u, p∗ ) is a subcritical point of L if and only if 0 ∈ ∂v L(u, p∗ ) and 0 ∈ ∂q∗ L(u, p∗ ).

(5.26)

(iii) A point (u, p∗ ) is a supercritical point of L if and only if 0 ∈ ∂v (−L)(u, p∗ ) and 0 ∈ ∂q∗ (−L)(u, p∗ ).

(5.27)

Let us introduce now the Lagrangian forms of problem (P) (given by (5.1)) and give some results concerning the critical points. Definition 5.28. (Lagrangian types) Let L : function:

U × X −→ IR be a given

(i) L is said to be a Lagrangian function of type I of problem (P) (given by (5.1)), if for each v ∈ U J(v) = sup L(v, q ∗ ). q∗ ∈X

(5.28)

(ii) L is said to be a Lagrangian function of type II of problem (P) (given by (5.1)), if for each v ∈ U J(v) = ∗inf L(v, q ∗ ). q ∈X

(5.29)

If L is a Lagrangian function of type I, the dual problem associated with problem (P) is given by the following problem (PI∗ ): sup H(q ∗ )

q∗ ∈X

(5.30)

and if L is a Lagrangian function of type II, the dual problem associated with ∗ ): problem (P) is given by the following problem (PII inf H(q ∗ ),

q∗ ∈X

(5.31)

where H : X −→ IR is defined by H(q ∗ ) := inf L(v, q ∗ ). v∈U

(5.32)

Remark 5.29. (i) If L is a Lagrangian function of type I, the dual problem (PI∗ ) is a supremum problem and then the problem (PI∗ ) is corresponding to find saddle points of the Lagrangian L. On the other hand, if L is a Lagrangian ∗ ) is an infimum problem and then function of type II, the dual problem (PII ∗ the problem (PII ) is corresponding to another infimum problem (if u and p∗


111

∗ are solutions of (P) and (PII ) respectively, then (u, p∗ ) is an infimum of L on U × X ). (ii) If L is a Lagrangian function of type II, then ∗ inf(P) = inf(PII ), ∗ ∗ L(v, q ) = inf L(v, q ) . inf inf since inf ∗ ∗ v∈U

q ∈X

q ∈X

v∈U

♦

Proposition 5.30. Let L : U × X −→ IR be a Lagrangian function of type I of problem (P) and (PI∗ ) is the dual problem associated with (P). Then (us , p∗s ) is a saddle point of L on U × X if and only if us is a solution of (P), p∗s is a solution of (PI∗ ) and inf(P) = sup(PI∗ ). Proof. According to the definition of J and H, it is clear that for all (v, q ∗ ) given in U × X we have J(u) ≥ H(p∗ ) (5.33) (since J(u) − H(p∗ ) = sup sup (L(u, q ∗ ) − L(v, p∗ )) ≥ 0, because of (5.28) v∈U q∗ ∈X

and (5.32)). Let (us , p∗s ) be a saddle point of L. Then, according to the definition of J and H, and to the expression (5.21), we have that J(us ) ≤ L(us , p∗s ) ≤ H(p∗s ). Consequently, because of (5.33), we can deduce that J(us ) = L(us , p∗s ) = H(p∗s ) and then us is a solution of (P), p∗s is a solution of (PI∗ ) and inf(P) = sup(PI∗ ). Conversely, let us be a solution of (P), p∗s be a solution of (PI∗ ) and inf(P) = sup(PI∗ ). Then, from the definition of J and H sup L(us , q ∗ ) = J(us ) = H(p∗s ) = inf L(v, p∗s )

q∗ ∈X

v∈U

and we can deduce easily that the expression (5.21) holds, i.e., (us , p∗s ) is a saddle point of L. Similarly we have the following result concerning subcritical points of a Lagrangian function of type II of problem (P). Proposition 5.31. Let L : U × X −→ IR be a Lagrangian function of type ∗ ) is the dual problem associated with (P). Then the II of problem (P) and (PII following statements hold: (i) If (us , p∗s ) is a subcritical point of L on U × X then J(us ) = L(us , p∗s ) = H(p∗s ).

112


(ii) A point (us , p∗s ) minimizes L on U × X if and only if us is a solution of ∗ ). (P), p∗s is a solution of (PI∗ ) and inf(P) = inf(PII Proof. Let (us , p∗s ) be a subcritical point of L. According to (5.26) and (5.24), we have that L(v, p∗s ) ≥ L(us , p∗s ) and L(us , q ∗ ) ≥ L(us , p∗s ) for all v in U and q ∗ in X . Consequently, H(p∗s ) = inf L(v, p∗s ) = L(us , p∗s ) and J(us ) = ∗inf L(us , q ∗ ) = L(us , p∗s ). q ∈X

v∈U

This proves assertion (i). Assertion (ii) can be proved in the same way as the proof of Proposition 5.30. Therefore, the proof is left to the reader as an exercise. A direct consequence of Proposition 5.31 is the following result. Theorem 5.32. Let L : U × X −→ IR be a Lagrangian function of type II ∗ ) is the dual problem associated with (P) and (us , p∗s ) be of problem (P), (PII a subcritical point of L. Then the following conditions are equivalent: (i) us is a solution of (P) ∗ ) (ii) p∗s is a solution of (PII

(iii) L(us , p∗s ) =

inf

(v,q∗ )∈U×X

L(v, q ∗ ).


We finish this subsection by introducing the notion of critical points of a G-differentiable Lagrangian function. Definition 5.33. (Lagrangian-remarquable point) Let L : U × X −→ IR be a Lagrangian function. A point (u, p∗ ) is said to be a Lagrangian-remarquable point of L if (u, p∗ ) is a remarquable point of L. Definition 5.34. (Critical point of L) Let L : U ×X −→ IR be a Lagrangian function. A point (uc , p∗c ) ∈ U × X is said to be a critical point of L if L is partially G-differentiable at (uc , p∗c ) and ∂L ∂L (uc , p∗c ) = 0 in U and (uc , p∗c ) = 0 in X , ∂v ∂q ∗

(5.34)

where ∂L/∂v and ∂L/∂q ∗ denote partial derivatives on U and X , respectively. We can now give the critical point theorem.


113

Proposition 5.35. Let L : U × X −→ IR be a Lagrangian function of problem (P) (given by (5.1)) and (uc , p∗c ) ∈ U × X be a Lagrangian-remarquable point of L: (i) If L is partially G-differentiable at point (uc , p∗c ), then (uc , p∗c ) is a critical point of L. (ii) If J (respectively H) is G-differentiable at uc (respectively p∗c ), then J (uc ) = 0 (respectively H (p∗c ) = 0), where H is given by (5.32). Proof. The proof is left to the reader as an exercise2 .

In the sequel we will be interested essentially in saddle point results (and consequently, to the Lagrangian function of type I). For details on the supercritical or subcritical point results, the reader can be referred, for instance, to Auchmuty [15]. 5.2.2 Lagrangian Duality and Saddle Points In this section we introduce the so-called Lagrangian duality function and show the relation between the conjugate duality and the well-known Lagrange duality. Lagrangian Function Associated to a Perturbation Definition 5.36. (Lagrangian duality function) The functional L : U × X −→ IR such that L(u, q ∗ ) = − sup ( q ∗ , qX − φ(u, q)), ∀u ∈ U, ∀q ∗ ∈ X q∈X

= inf (φ(u, q) − q ∗ , qX ), ∀u ∈ U, ∀q ∗ ∈ X

(5.35)

q∈X

is said to be the Lagrangian (duality) function of problem (P) (given by (5.1)) with respect to the associated perturbation φ : U ×X −→ IR (defined by (5.2)). Moreover, L(u, q ∗ ) can be written as L(u, q ∗ ) = −ψu∗ (q ∗ ), ∀u ∈ U, ∀q ∗ ∈ X ,

(5.36)

where ψu is, for a fixed u ∈ U , the function: q −→ φ(u, q), and ψu∗ ∈ Γ (X ) is its conjugate function in the duality between X and X. Now, we give some properties of the Lagrangian function L. Lemma 5.37. (i) Let u be given in U . Then the function Ru : q ∗ ∈ X −→ L(u, q ∗ ) ∈ IR is a concave upper and semi-continuous function of X into IR. 2

By using the fact that if a real-valued function η is G-differentiable at point u0 and if ξ ∈ ∂η(u0 ) then ξ = η (u0 ).

114


(ii) If the function φ is convex then, for all q ∗ ∈ X , the function Sq∗ : u ∈ U −→ L(u, q ∗ ) ∈ IR is a convex function from U into IR. Proof. Since the function Ru = −ψu∗ ∈ Γ (X ), Ru is concave and upper semicontinuous on X and then the result (i). The result (ii) is a direct consequence of the convexity of φ and the expression of L given by (5.35). Characterization of Saddle Points In order to give a characterization of a saddle point of the Lagrangian function L, we first rewrite problem (P) and its dual problem (P ∗ ) by using the Lagrangian function L with respect to φ. We have easily that, for all (u∗ , q ∗ ) ∈ U × X (without supplementary condition on φ) φ∗ (u∗ , q ∗ ) = sup

( u∗ , uU + q ∗ , pX − φ(u, p))

(u,p)∈U×X ∗

= sup ( u , uU + sup ( q ∗ , pX − φ(u, p))) u∈U

p∈X

= sup ( u∗ , uU − L(u, q ∗ )) u∈U

and then φ∗ (0, q ∗ ) = sup (−L(u, q ∗ )) = − inf L(u, q ∗ ), ∀q ∗ ∈ X . u∈U

u∈U

(5.37)

According to the expression of problem (P ∗ ) in (5.5) and the previous relation (5.37), problem (P ∗ ) can be written as sup inf L(u, q ∗ ).

q∗ ∈X u∈U

(5.38)

In the same way, if we suppose that φ ∈ Γ0 (U × X) then ∀u ∈ U , the function ψu : q ∈ X −→ φ(u, q) ∈ IR belongs to Γ (X) and consequently, the bidual ψu∗∗ of ψu is exactly ψu . Thus (by Fenchel transformation) φ(u, p) = ψu (p) = ψu∗∗ (p) = sup ( q ∗ , pX − ψu∗ (q ∗ ))) q∗ ∈X

= sup ( q ∗ , pX + L(u, q ∗ )) (according to (5.36)) q∗ ∈X

and then

φ(u, 0) = sup L(u, q ∗ ), ∀u ∈ U. q∗ ∈X

(5.39)

According to expression of problem (P) in (5.1) with the conditions (5.2) and (5.39), problem (P) can be written as inf sup L(u, q ∗ ).

u∈U q∗ ∈X

(5.40)


115

Remark 5.38. By introducing the Lagrangian function, the weak duality relation sup(P ∗ ) ≤ inf(P) (corresponding to problems (P) and (P ∗ )) is exactly the inequality (5.41) sup inf L(u, q ∗ ) ≤ inf sup L(u, q ∗ ). q∗ ∈X u∈U

u∈U q∗ ∈X

Therefore, the inequality (5.41) is always true.

♦

The following theorem shows a characterization of a saddle point. Theorem 5.39. (Saddle point theorem) Let φ be a function satisfying φ ∈ Γ0 (U × X). Then the following results hold: (i) The pair (u, p∗ ) of U × X is a saddle point of the Lagrangian function L if and only if u is a solution of problem (P), p∗ is a solution of problem (P ∗ ) and inf(P) = sup(P ∗ ) (the strong duality). (ii) Moreover, if problem (P) is stable then an element u ∈ U is a solution of (P) if and only if there exists an element p∗ ∈ X such that (u, p∗ ) is a saddle point of L (according to (i), p∗ is then a solution of (P ∗ )). Proof. (i) (⇒) From the expressions (5.37), (5.39) and (5.40) we have that L(u, p∗ ) = inf L(v, p∗ ) = −φ∗ (0, p∗ ), v∈U

L(u, p∗ ) = sup L(u, q ∗ ) = φ(u, 0). q∗ ∈X

Therefore, we have the extremality condition φ(u, 0) + φ∗ (0, p∗ ) = 0 and then (because of Proposition 5.18) the result. (⇐) Conversely, because of Proposition 5.18 φ(u, 0) + φ∗ (0, p∗ ) = 0. Moreover, according to the relations (5.37) and (5.39) we have that −φ∗ (0, p∗ ) =

inf L(v, p∗ ) ≤ L(u, p∗ ),

v∈U

φ(u, 0) = sup L(u, q ∗ ) ≥ L(u, p∗ ). q∗ ∈X

Thus, inf L(v, p∗ ) = L(u, p∗ ) = sup L(u, q ∗ ) and then (u, p∗ ) is a saddle v∈U

q∗ ∈X

point of L. (ii) If u is a solution of (P) then (according to Corollary 5.16 and Proposition 5.12), we have that the dual problem (P ∗ ) has at least one solution p∗ and inf(P) = sup(P ∗ ). Therefore (according to (i)), (u, p∗ ) is a saddle point of L. The converse is immediate. Remark 5.40. According to (5.37) and (5.39), the Lagrangian duality function L is a Lagrangian function of type I of problem (P) and then the point (i) of Theorem 5.39 is a direct consequence of Proposition 5.30. ♦

116


5.2.3 Application and Boundary-value Problems Framework in Linear Geometric Operator Let U, U and X, X be two pairs of real topological vector spaces, in duality with respect to certain bilinear forms ., .U and ., .X , respectively. Let the geometric operator Λ : U −→ X be a linear and continuous operator of U into X, with the adjoint Λ∗ : X −→ U . Let F : U −→ IR and S : X −→ IR be two extended real-valued functions and finite on convex subsets K ⊂ U and A ⊂ X, respectively. By using the Fenchel transformation, the conjugate functions corresponding to S and F are defined respectively by S ∗ (q ∗ ) = sup ( q ∗ , qX − S(q)) ∀q ∗ ∈ X , q∈X

F ∗ (v ∗ ) = sup ( v ∗ , vU − F (v)) ∀v ∗ ∈ U .

(5.42)

v∈U

In this section, we analyze the infimum of the following functional Jν (v) = S(Λv − ν) + F (v),

(5.43)

where ν is a given distributed parameter. The parametric variational problem (P) is then inf Jν (v).

v∈K

(5.44)

Example 5.41. Let (U, . ) be a reflexive Banach space with norm . and let (U , . ∗ ) be its dual space with norm . ∗ . Let us consider an operator A from U into U satisfying the assumptions (4.77) and (4.78), a function ϕ that is a convex, lower semi-continuous and proper function of U into IR and a given element f of U . We have proved in Subsection 5.1.4 that the inequalities variational problem (5.19) can be considered as an optimization problem inf G(v),

v∈U

(5.45)

where the functional G of U into IR is defined by G(v) = g, vU + ϕ(v), with g = A˜ u − f , for some known solution u ˜ of (5.19). If we suppose that ϕ(v) = S(Λv − ν), where ν ∈ X is a given distributed parameter, Λ ∈ L(U ; X), X is a Banach space and S ∈ Γ0 (X) then the functional G is similar to Jν which is given by (5.43), where F is the finite linear function v ∈ U −→ F (v) = g, vU and problem (5.45) is similar to problem (5.44), with K = U . ♣


117

In order to study problem (5.44), we consider the perturbation function φ : U × X −→ IR, which is defined by φ(v, q) := S(Λv − ν − q) + F (v) and its conjugate φ∗ : U × X −→ IR, which is defined (using the Fenchel transformation) by φ∗ (v ∗ , q ∗ ) :=

sup

( (v ∗ , q ∗ ), (v, q)U×X − φ(v, q))

(v,q)∈U×X

and then φ∗ (0, q ∗ ) =

sup

( q ∗ , qX − φ(v, q))

(v,q)∈U×X

= sup sup ( q ∗ , qX − φ(v, q)) v∈U q∈X

=

sup

( q ∗ , Λv − ν − pX − S(p) − F (v))

(v,p)∈U×X ∗

= sup ( q , Λv − νX − F (v)) + sup (− q ∗ , pX − S(p)) v∈U

p∈X

= sup ( Λ∗ q ∗ , vX − F (v)) + sup ( −q ∗ , pX − S(p)) − q ∗ , νX v∈U

p∈X

= S ∗ (−q ∗ ) + F ∗ (Λ∗ q ∗ ) − q ∗ , νX . The dual problem (P ∗ ) can be written as sup (−φ∗ (0, q ∗ )) = sup (−S ∗ (−q ∗ ) − F ∗ (Λ∗ q ∗ ) + q ∗ , νX ).

q∗ ∈X

q∗ ∈X

(5.46)

Remark 5.42. Let (us , p∗s ) ∈ U × X such that φ(us , 0) + φ∗ (0, p∗s ) = 0 then F (us ) + F ∗ (Λ∗ p∗s ) − Λ∗ p∗s , us U = 0, i.e., Λ∗ p∗s ∈ ∂F (us ), S(Λus − ν) + S ∗ (−p∗s ) − −p∗s , Λus − νX = 0, i.e., −p∗s ∈ ∂G(Λus − ν). Indeed, 0 = φ(us , 0) + φ∗ (0, p∗s ) = S(Λus − ν) + F (us ) + S ∗ (−p∗s ) + F ∗ (Λ∗ p∗s ) − p∗s , νX = [F ∗ (Λ∗ p∗s ) + F (us ) − Λ∗ p∗s , us U ] +[S ∗ (−p∗s ) + S(Λus − ν) + p∗s , Λus − νX ]. According to 5.42, we can deduce that F ∗ (Λ∗ p∗s ) + F (us ) − Λ∗ p∗s , us U ≥ 0, S ∗ (−p∗s ) + S(Λus − ν) − −p∗s , Λus − νX ≥ 0 and then the results follow (because of Proposition 4.16).

♦

118


According to Propositions 5.17, 5.18 and 5.20, and the previous remark, respectively, we can obtain easily the following results. Theorem 5.43. Let F and S be convex functions and let us assume that inf(P) is finite (i.e., −∞ < inf Jν (v) < ∞). Further, let us suppose that v∈U

there exists an element u0 ∈ U such that F (u0 ) is finite and the function S is finite and continuous at point Λu0 − ν.

(5.47)

Then problem (P) is stable. Moreover, if F ∈ Γ0 (U ) and S ∈ Γ0 (X) then the dual gap of the problem (P) is null (i.e., inf(P) = sup(P ∗ )) and (P ∗ ) has at least one solution. Theorem 5.44. If problems (P) and (P ∗ ) both have solutions and the dual gap of the problem (P) is null then all solutions us ∈ K of (P) and all solutions p∗s ∈ X of (P ∗ ) satisfy the following optimality conditions F (us ) + F ∗ (Λ∗ p∗s ) − Λ∗ p∗s , us U = 0, S(Λus − ν) + S ∗ (−p∗s ) − −p∗s , Λus − νX = 0,

(5.48)

or (equivalently) Λ∗ p∗s ∈ ∂F (us ) and − p∗s ∈ ∂S(Λus − ν).

(5.49)

Conversely, if us ∈ U and p∗s ∈ X satisfy the relations (5.48) or (5.49), then us is a solution of problem (P), p∗s is a solution of (P ∗ ) and the dual gap of problem (P) is null. Theorem 5.45. Let (U, . ) be a reflexive Banach space. If F ∈ Γ0 (U ) and S ∈ Γ0 (X) and if S(Λv − ν) + F (v) −→ +∞ f or v ∈ U, v −→ +∞,

(5.50)

then, under assumption (5.47), problems (P) and (P ∗ ) have at least one solution, the dual gap of problem (P) is null and the optimality conditions (5.48) are satisfied. We shall now apply the previous study to an interesting problem of linear elasticity. Example: Elasticity Problem Consider a deformable elastic body, occupying in its undeformed state the bounded and open domain Ω ⊂ IRn with boundary Γ = ∂Ω = Γu ∪ Γs such that Γu ∩Γs = ∅ and mes(Γu ) = ∅ (see, e.g., Gurtin [147], Landau and Lifschitz [185] for more details). A deformation process of the body, under an external stress, is described by a smooth vector-valued mapping R from the reference


119

configuration Ω = Ω ∪ ∂Ω (the closure of Ω) to a configuration ω ⊂ IRm with boundary ∂ω at time t and is defined by R(x, t) = (Ri (xj , t))i=1,...,m;

j=1,...,n ,

with x = (xj )j=1,...,n .

The deformation is said to be admissible if R(., t) is invertible and then the inverse R−1 (., t) of R(., t) exists. The admissible configuration space, denoted by Dad , is defined by Dad = {R(., t) sufficiently regular on Ω : rank(∇R(., t)) = min(n, m) on Ω}. The deformation gradient tensor is defined on Dad by A(x, t) = ∇R(x, t) = (Ai (xj , t))i=1,...,m;

j=1,...,n ,

with Ai (xj , t) =

∂Ri (xj , t). ∂y

The metric tensor C is a symmetric Lagrangian tensor field, with rank(C) = min(n, m), defined by C = A A ∈ IR T

n×n

, with Ckl (x, t) =

m

Ai (xk , t)Ai (xl , t).

i=1

Suppose now that m = n, then the deformation of the elastic body is usually described in terms of a vector displacement field u(x, t) ∈ IRn , which is defined by u(x, t) = R(x, t) − x. In the linearized elasticity theory on the assumption of small strains, the changes in metric induced by the deformation are described by the linearized strain tensor (x, t) ∈ IRn×n defined by [u] =

∇u + ∇uT , 2

i.e., ij =

1 ∂uj ∂ui ( + ) 1 ≤ i, j ≤ n, 2 ∂xi ∂xj

and by the stress tensor σ(x, t) ∈ IRn×n , which depends linearly on (x, t) as σ = C : , i.e., σij = Cijkl kl 1 ≤ i, j ≤ n, 1≤k,l≤n

where C is the invertible elasticity tensor, which is supposed only spatially dependent. We assume that C satisfies the symmetric condition Cijkl = Cklij = Cjikl = Cijlk for 1 ≤ i, j, l, k ≤ n

(5.51)

(and then ij = ji , σij = σji for 1 ≤ i, j ≤ n) and the coercivity condition, i.e., there exists a constant α > 0 such that Cijkl ξij ξkl ≥ α ξij ξij ∀ξ ∈ E, (5.52) 1≤i,j,k,l≤n

1≤i,j≤n

120


where E is the symmetric second-order tensor space. In the case of isotropic homogeneous media, the elasticity tensor C can be defined in terms of the Lamé coefficients, the incompressibility λ and the rigidity μ by 2 Cijkl = λδij δkl + μ(δik δjl + δjk δil − δij δkl ) for all 1 ≤ i, j, k, l ≤ 3, 3 where δij is the kroneker delta, i.e., δij = 0 if i = j and δij = 1 if i = j. We introduce now the kinetic energy and the elastic strain energy, respectively by ∂u 2 | dx, ρ(x) | Ek (u) = ∂t Ω σ : dx, Es (u) = Ω

where ρ is the mass density distribution of the material and σij ji = tr(σ), σ:= 1≤i,j≤n

where tr(σ) is the matrix trace of the product of two tensors. The fundamental balance equation of the dynamics of deformable bodies is then ∂2u ρ 2 − div(σ) = f, ∂t where f is the distribution of body-force. We then have ρ

∂2u − div(C : ∇u) = f ∂t2

on Q = Ω × (0, T ),

where C : ∇u = σ. In order to complete the modeling, we give the following initial conditions: (u(., 0),

∂u (., 0)) = (u0 , v0 ) on Ω, ∂t

and the boundary conditions: u = ξ on Γu × (0, T ),

σ.n = φ on Γs × (0, T ),

(5.53)

where σ.n is a vector function with components (σ.n)i = σij nj i = 1, . . . , n. j=1,n

In the sequel, in order to simplify the presentation, we suppose that the system


121

is time independent and then the previous system becomes −div(σ) = f, u = ξ on Γu , σ.n = φ on Γs ,

(5.54)

with a given elasticity tensor C, a body field f in Ω, the traction φ on Γs and the displacement ξ on Γu , which satisfies ξ is the trace of a function uξ ∈ U0 on Γu , where U0 = {v ∈ H 1 (Ω) : v = 0 on Γs }.

(5.55)

Let U = H 1 (Ω) and X = L2 (Ω; E) = X . Similar to Subsection 3.2.4, we can consider the space Hσ (div; Ω) = {v ∈ L2 (Ω; E) : div(σ) ∈ L2 (Ω)}, which is a Hilbert space for the norm σ Hσ (div;Ω) = ( σ 2L2 (Ω) + div(σ) 2L2 (Ω) )1/2 , where div(σ) is a vector function with components (div(σ))i =

∂σij , i = 1, . . . , n. ∂xj j=1,n

The following theorem concerns the normal components of boundary values of functions of the space Hσ (div; Ω). Theorem 5.46. Let Ω be a bounded subset of IRn with Lipschitz boundary Γ , we have the following properties: D(Ω; E) is dense in Hσ (div; Ω) (the boundedness of Ω is not necessary), there exists a mapping γn : σ −→ σ.n|Γ defined on D(Ω; E) which can be extended by continuity to a linear and continuous mapping, denoted also by γn , from Hσ (div; Ω) into H −1/2 (Γ ), where (σ.n)i = j=1,n σij nj i = 1, . . . , n.

(5.56)

(5.57)

A consequence of the previous theorem is the following Green’s formula, for all tensor functions σ ∈ Hσ (div; Ω) and vector functions u ∈ H 1 (Ω): σ : ∇udx + udiv(σ)dx = σ.n, uH −1/2 (Γ ),H 1/2 (Γ ) . (5.58) Ω

Ω

Otherwise (for the strain tensor [u] ∈ L2 (Ω; E) if u ∈ H 1 (Ω)),

122


σ : [u]dx + Ω

Ω

udiv(σ)dx = σ.n, uH −1/2 (Γ ),H 1/2 (Γ ) .

(5.59)

We can also introduce the following space (corresponding to the strain tensor [u]) by H (Ω) = {u ∈ L2 (Ω) : [u] ∈ L2 (Ω; E)}, which is a Hilbert space for the norm u H (Ω) = ( u 2L2 (Ω) + [u] 2L2 (Ω) )1/2 . We have the following Korn’s inequality result. Theorem 5.47. Let Ω be a bounded subset of IRn with Lipschitz boundary Γ , we have the following property: D(Ω) is dense in H (Ω) (the boundedness of Ω is not necessary ),

(5.60)

and the classical Korn’s inequality: there exists a constant c0 > 0 (depending on Ω) such that u 2H 1 (Ω) ≤ c0 ( u 2L2 (Ω) + [u] 2L2 (Ω) ) = c0 u 2H (Ω) . Proof. For the proof see, e.g., Duvaut and Lions [109].

(5.61)

Let now Uad and Xad be the admissible displacement and stress sets, respectively, defined by Uad = {u ∈ U : u = ξ on Γu }, Xad = {σ ∈ X : −div(σ) = f,

σ.n = φ on Γs },

and let Λ : U −→ X be the geometrically linear operator, which is defined by ∇u + ∇uT Λu = − = −[u] on Ω. 2 The duality between X and X can be denoted by σ : dx.

σ, X = Ω

It is well known that the displacement u is a solution of the following problem (P): 1 Λv : C : Λvdx − f vdx − φvdΓ . (5.62) inf v∈Uad 2 Ω Ω Γs Otherwise, the displacement u minimizes the potential energy J(v) = S(Λv) + F (v) for v ∈ U,


where S() =

1 2

123

F (v) = Ω

: C : dx, −f vdx + −φvdΓ + XUad (v) (see Remark 4.90).

Ω

(5.63)

Γs

Then problem (5.62) is similar to problem (5.44), with the parameter ν = 0, and to obtain the dual problem it is sufficient to calculate the dual S ∗ and F ∗ of the functionals S and F at points σ and Λ∗ σ, respectively, for σ ∈ X. It is clear that F and S are G-differentiable functionals on X and Uad , respectively, and (since C satisfies 1 : C : 2 = 2 : C : 1 ) S () = C : and (F (u) = −f on Ω; F (u) = −φ on Γs ). From Fenchel transformation, the conjugate functions corresponding to S and F are defined respectively by S ∗ (σ) = sup ( σ, X − S()),

∈X

F ∗ (Λ∗ σ) = sup ( Λ∗ σ, vU − F (v)). v∈Uad

Calculate first S ∗ at point σ. Since the function 1 H : −→ H() = σ, X − S() = (σ : − : C : )dx, 2 Ω is strictly concave and upper semi-continuous on X, and H() −→ −∞ for, ∈ X, X −→ ∞ (by using the coercivity condition (5.52) and Korn’s inequality), then, according to Proposition 4.92, H admits a unique supremum s such that s is a critical point of H, i.e., H (s ) = 0. Consequently, s = C −1 : σ and then 1 σ : C −1 : σdx. (5.64) S ∗ (σ) = 2 Ω We shall now calculate F ∗ at point Λ∗ σ. Since the function v −→ −f vdx + −φvdΓ, Ω

Γs

is a linear and continuous mapping from U into IR then there exists f ∈ U such that −f vdx + −φvdΓ.

f , vU = Ω

Γs

Consequently (according to (5.55)), 0 (v − uξ ), F (v) = f , vU + XUad

0 where Uad = {v ∈ U : v = 0 on Γu }.

124


According to Lemma 4.9, we have that

0 (v − uξ ) F ∗ (Λ∗ σ) = sup Λ∗ σ, vU − f , vU − XUad v∈U

∗ ∗ (Λ σ − f ), = Λ∗ σ − f , uξ U + XUad ∗ 0 where Uad = {g ∈ U : g, vU = 0 ∀v ∈ Uad }. Otherwise,

∗

∗

F (Λ σ) =

∗

Λ∗ σ − f , uξ U if Λ∗ σ − f ∈ Uad

(5.65)

+∞ else. We remark, according to Theorem 5.46 and Green’s formula (5.59), that ∗ 0 and Λ σ − f ∈ Uad if (div(σ) + f )vdx + (−σ.n + φ)vdΓ = 0, ∀v ∈ Uad ∗

Ω

Γs

then, if σ ∈ Xad . Consequently,

F ∗ (Λ∗ σ) = Λ∗ σ − f , uξ U ∀σ ∈ Xad . Next calculate the term Λ∗ σ, uξ U . Since ∗ div(σ)uξ dx − σ.nuξ dΓ − σ.nuξ dΓ,

Λ σ, uξ U = σ, Λuξ X = Ω

Γu

Γs

then (because of uξ = 0 on Γs and uξ = ξ on Γu ), ∗ div(σ)uξ dx − σ.nξdΓ.

Λ σ, uξ U = Ω

Γu

Since σ ∈ Xad and uξ = 0 on Γs then Λ∗ σ − f , uξ U = −

σ.nξdΓ. Γu

This implies that F ∗ (Λ∗ σ) =

⎧ ⎪ ⎨ −σ.nξdΓ if σ ∈ Xad Γu

⎪ ⎩ +∞ else.

(5.66)

We can now give the dual problem (P ∗ ) corresponding to the primal problem (5.62). According to (5.46), (5.64) and (5.66), the dual problem (P ∗ ) can be written as 1 σ : C −1 : σdx − σ.nξdΓ . (5.67) sup − 2 Ω σ∈Xad Γu It is clear that F ∈ Γ0 (U ) and S ∈ Γ0 (X) and the conditions (5.50) hold (according to the coercivity condition (5.52) and Korn’s Inequality) then according to Theorem 5.45 there exists a solution us ∈ Uad of (5.62) (which


125

is unique since the functional S is strictly convex) and a solution σs ∈ Xad (which is also unique) of (5.67) satisfying −∞ < inf(P) = sup(P ∗ ) < ∞ and F (us ) + F ∗ (Λ∗ σs ) − Λ∗ σs , us U = 0, S(Λus ) + S ∗ (−σs ) + σs , Λus X = 0. Then (we denote by s := −Λus ) 1 1 (−σs : s + σs : C −1 : σs + s : C : s )dx = 0. 2 2 Ω

(5.68)

(5.69)

Let c := C −1 : σs then (since s : C : c = c : C : s ) 1 1 1 (−σs : s + σs : C −1 : σs + s : C : s ) = (c − s ) : C : (c − s ) ≥ 0. 2 2 2 Consequently, (c − s ) : C : (c − s ) = 0 a.e. in Ω and then c − s = 0 a.e. in Ω, i.e., s = C −1 : σs a.e. in Ω, which is called the Hooke’s law. Remark 5.48. In addition to the operator Λ, we introduce its boundary operator ΛB : UB −→ XB by ΛB u = −u on Γu , where (u)ij := ui nj , 1 ≤ i, j ≤ n, i.e., u = ut .n. We introduce now the adjoint operators of Λ and ΛB . By using Green’s formula we can deduce that (since σ : u = σ.nu) σ : Λudx

σ, ΛuX = Ω div(σ)udx − σ.nudΓ, = (5.70) Ω Γ = Λ∗ σ, uU + Λ∗B σ, uΓs − σ : udΓ, Γu

= Λ∗ σ, uU + Λ∗B σ, uΓs + σ, ΛB uΓu , where the adjoint operator Λ∗ : X = X −→ U and its boundary operator Λ∗B : XB −→ UB are defined by Λ∗ σ = div(σ) on Ω and Λ∗B σ = −σ.n on Γs . If we denote

σ, ΛT uX = σ, ΛuX − σ, ΛB uΓu ,

Λ∗T σ, uU = Λ∗ σ, uU + Λ∗B σ, uΓs , then

σ, ΛT uX = Λ∗T σ, uU .

126


The boundary conditions (5.53) can be written as Λ∗B σ = −σ.n = −φ on Γs and ΛB u = −ut .n = −ξ t .n on Γu and the total potential energy of the system J is defined by W (Λu)dx + F (u), J(u) = S(Λu) + F (u) = Ω

where W () := (1/2) : C : . According to the result of Section 4.3.6, we have that us is a critical point of J if and only if (see (4.89) and (4.90)) −s = Λus on Ω, uts .n = ξ t .n on Γu , −σs = W (−s ) on Ω, Λ∗ σs = −f on Ω, σs .n = φ on Γs , i.e., if and only if −s = Λus on Ω, us = ξ on Γu , σs = C : s on Ω, −div(σs ) = f on Ω, σs .n = φ on Γs .

(5.71) ♦

5.3 Minimax Duality The minimax duality theory plays an important role in constrained optimization, in general variational problems, in numerical analysis, in robust control theory and game theory. In this section, we analyze the connection between an optimization problem where the variable is some function u and some corresponding inf-sup problem of a functional type convex-concave, where the variables are the function u and some function p. The approach used is based on the classical minimax theorems of Ky-Fan and Sion, which are older than Rockafellar’s approach (for more details see Rockafellar [252]). 5.3.1 Motivation Let U and X be locally convex topological vector spaces and let us consider a function Ψ : U −→ IR. We consider the following minimization problem: inf Ψ (v),

(5.72)

v∈U

and we assume that we can write Ψ (v) as a supremum on the second variable of some function L : U × X −→ IR as Ψ (v) = sup L(v, q), q∈X

∀v ∈ U.

(5.73)

5.3 Minimax Duality

127

Then problem (5.72) becomes inf sup L(v, q),

v∈U q∈X

(5.74)

i.e., seek points us in U , which minimize Ψ on U , or points (us , ps ) ∈ U × X such that us minimizes Ψ on U and also Ψ (us ) = L(us , ps ) = sup L(us , q).

(5.75)

q∈X

Definition 5.49. (Minimax point) A point (us , ps ) is said to be a minimax point of L on U × X if us minimizes Ψ on U and (5.75) holds. Consequently, if (us , ps ) is a minimax point of L on U × X then L(us , q) ≤ L(us , ps ) = Ψ (us ) ≤ Ψ (v)

(5.76)

for all (v, q) ∈ U × X. Remark 5.50. For L given by (5.73), we have that if (us , ps ) is a saddle point of L then (us , ps ) is a minimax point of L (because of (5.76) and the fact that ♦ Ψ (v) ≤ L(v, ps ) for all v ∈ U ). But the converse is, in general, false. Since the convex lower semi-continuous functions are the pointwise supremum of affine continuous functions that they dominates, we can then, in general, rewrite Ψ (v) in the form (5.73). Therefore, a minimization problem becomes an inf-sup problem. Moreover, according to the study given in the previous section, it is clear that the study of the connection between sup inf L(v, q)

q∈X v∈U

(5.77)

and problem (5.74) is necessary and important. That is the goal of the study which will follow. First, we will be interested in saddle point results. Second, we will give a situation where a minimax point is not a saddle point. 5.3.2 Saddle Point and Properties Let K ⊂ U and A ⊂ X be two arbitrary sets and L : U × X −→ IR be a real-valued functional, finite on the set K × A ⊂ U × X. Game Theory Interpretation We recall that a point (us , ps ) ∈ K × A is a saddle point of L on K × A, if L attains at this point its maximum in p ∈ A and its minimum in u ∈ K, i.e., L(u, ps ) ≥ L(us , ps ) ≥ L(us , p).

(5.78)

128


Let us consider the so-called two-person zero-sum games, which play a central role in the development of the theory of games. We take the game in which players make payments only to each other. One player’s loss is exactly equal to the other player’s gain, so the total amount of “money” available remains constant. In order to analyze any game, we make the following assumptions about both players: (i) Each player makes the best possible strategy (or move). (ii) Each player knows that his (or her) opponent is also making the best possible strategy. This game can be described like this: player Pu chooses a strategy u and player Pp chooses a strategy p ; when the players have decided their strategy, player Pu makes a payment L(u, p) to player Pp . The goal for each player is of course to minimize his (or her) payment and to maximize his (or her) income. The question is to find the equilibrium of such a game, i.e., to know the strategy (u, p) of the two players such that every one of the players is not interested in varying his (or her) strategy independently whether the strategy of the other player is known. The response is that equilibria are exactly the saddle points of the cost functional L. Indeed, if (us , ps ) is such a point, then the first player shall not be interested in taking another strategy u, if the second player keeps his (or her) choice ps ; indeed, the first inequality in (5.78) shows that another strategy u cannot decrease the payment of the first player. Similarly, the second player shall not be interested in choosing something different from ps , if the first player keeps his (or her) choice us (because a different strategy from ps cannot increase the income of the second player). On the other hand, if a strategy (u, p) is not a saddle point, it is clear that the first player can decrease his payment passing from u to another choice (if the second player keeps his (or her) choice at p); we have similar analysis for the second player; thus, the equilibria are exactly the saddle points. Let us consider the following two scenarios: (I)

The first player choose first, and the second player makes his (or her) choice already knowing the choice of the first player. (II) Vice versa, the second player chooses first, and first player makes his (or her) choice already knowing the choice of the second player. In scenario (I) the reasoning of the first player is: if I choose some strategy u, then the second player will of course choose a strategy p which maximizes my payment L(u, p) which results in a payoff, for the second player, of A(u) = sup L(u, p). p∈A

Consequently, I should choose u which minimizes the function A, i.e., the one which solves the optimization problem

5.3 Minimax Duality

129

inf A(u).

u∈K

The resulting payoff, from the first player to the second player will then be inf A(u) = inf sup L(u, p).

u∈K

u∈K p∈A

(5.79)

In scenario (II), similar reasoning of the second player enforces him (or her) to choose p maximizing the profit function B(p) = inf L(u, p), u∈K

i.e., the one which solves the optimization problem sup B(p). p∈A

The resulting payoff, from the first player to the second player, will then be sup B(p) = sup inf L(u, p). p∈A

p∈A u∈K

(5.80)

The difference between the two payoffs can be interpreted as the advantage afforded to the player who makes the second move, with knowledge of the other player’s move. The loss of the first player in scenario (II) is less than or equal to the profit of the second player in scenario (I) (since the conditions of game (II) are favorable to the first player and those of game (I) to the second player). Thus, we may guess that, independently of the structure of the function L, there is the following inequality. Proposition 5.51. For all real-valued functional L, finite on the set K × A ⊂ U × X, we have that sup inf L(v, q) ≤ inf sup L(v, q).

q∈A v∈K

v∈K q∈A

Proof. The proof is trivial and is left to the reader as an exercise.

(5.81)

If a saddle point exists then games (I) and (II) are similar, i.e., there is no advantage anyway in making the second move (because L(us , ps ) is the same value of both payoffs (5.79) and (5.80)). Existence of Saddle Points Now we give an existence result of a saddle point of the function L. Theorem 5.52. The real-valued functional L, finite on the set K×A ⊂ U ×X, admits a saddle point (u, p) on K × A if and only if max inf L(v, q) = min sup L(v, q) q∈A v∈K

v∈K q∈A

(5.82)

and this quantity is equal to L(u, p). Here, min instead of inf (respectively max instead of sup) means that the extremum is taken.

130


Proof. Let us assume that there exists a saddle point (u, p) on K × A. It is clear that inf sup L(v, q) ≤ sup L(u, q), v∈K q∈A

q∈A

inf L(v, p) ≤ sup inf L(v, q).

v∈K

q∈A v∈K

Moreover, by the definition of saddle point, i.e., L(u, q) ≤ L(u, p) ≤ L(v, p),

∀(v, q) ∈ K × A,

we can deduce that sup L(u, q) = L(u, p) = inf L(v, p). v∈K

q∈A

(5.83)

Therefore, according to the inequality (5.81), we can deduce that sup inf L(v, q) = L(u, p) = inf sup L(v, q)

q∈A v∈K

v∈K q∈A

(5.84)

Precisely, according to the relation (5.83), we have L(u, p) = sup L(u, q) = min sup L(v, q) (i.e., the infimum is attained) v∈K q∈A

q∈A

= inf L(v, p) = max inf L(v, q) (i.e., the supremum is attained). v∈K

q∈A v∈K

Conversely, assume that the equality (5.82) holds and let u realize the minimum of sup L(., q) and p the maximum of inf L(v, .). Therefore, v∈K

q∈A

inf L(v, p) ≤ L(u, p) ≤ sup L(u, q)

v∈K

q∈A

and then (because of (5.82)) inf L(v, p) = L(u, p) = sup L(u, q),

v∈K

q∈A

meaning that (u, p) is a saddle point of L. This completes the proof.

As a direct consequence we have the following proposition. Proposition 5.53. Let L : U × X −→ IR be a real-valued functional finite on the set K × A ⊂ U × X. We then have if there exists a pair (u, p) ∈ K × A and a real value α such that L(u, q) ≤ α,

∀q ∈ A,

L(v, p) ≥ α,

∀v ∈ K,

(5.85)

then the pair (u, p) is a saddle point of L and α = sup inf L(v, q) = inf sup L(v, q) = L(u, p). q∈A v∈K

v∈K q∈A

(5.86)

The following proposition gives the form of the set of saddle points of a Lagrangian function.

5.3 Minimax Duality

131

Proposition 5.54. (Set of saddle points) Let L : U × X −→ IR be a realvalued functional, finite on the set K × A ⊂ U × X. The set of saddle points of L is of the form K0 × A0 ⊂ K × A . Proof. Let (ui , pi ), for i = 1, 2, be two saddle points of the function L on K × A and prove that, for example (u1 , p2 ) is also a saddle point of L. From the relation (5.82), we have that α = L(u1 , p1 ) = L(u2 , p2 ) and then (by the definition of saddle point) L(u1 , q) ≤ α, L(v, p2 ) ≥ α ∀(v, q) ∈ K × A. Therefore, because of Proposition 5.53, we can deduce that (u1 , p2 ) is a saddle point of L. 5.3.3 Banach Spaces and Saddle Points In this section we assume that the space U and X are two reflexive Banach spaces, and we make the following assumptions: (A1) The sets K and A verify K ⊂ U is convex, closed and non-empty, A ⊂ X is convex, closed and non-empty.

(5.87)

(A2) The function L : U × X −→ IR is finite on K × A and satisfies v ∈ K −→ L(v, q) is convex and lower semi-continuous ∀q ∈ A, q ∈ A −→ L(v, q) is concave and upper semi-continuous ∀v ∈ K.

(5.88)

Remark 5.55. According to Asplund [14] we have that for a reflexive Banach space, we can always find an equivalent norm which is strictly convex. ♦ We can now give some properties of the set of saddle points of the function L. Proposition 5.56. Under assumptions (A1) and (A2), the subset K0 × A0 of saddle points of L is convex. Moreover: (i) If ∀q ∈ A, the function v ∈ K −→ L(v, q) is strictly convex then K0 contains at most one point. (ii) If ∀v ∈ K, the function q ∈ A −→ L(v, q) is strictly concave then A0 contains at most one point.

132


Proof. If the subset K0 × A0 is empty the proof is immediate. Suppose now that the subset K0 × A0 is non-empty then L possesses at least one saddle point. Set α the value α := sup inf L(v, q) = inf sup L(v, q). q∈A v∈K

v∈K q∈A

For (ui , pi ) ∈ K0 × A0 , when i = 1, 2 and t ∈]0, 1[ we have that L(ui , q) ≤ α ≤ L(v, pi ) for all (v, q) ∈ K × A, for i = 1, 2 and (tu1 + (1 − t)u2 , tp1 + (1 − t)p2 ) ∈ K × A (because of the assumption (5.87)). Consequently (because of the assumption (5.88)), L(tu1 + (1 − t)u2 , q) ≤ α,

∀q ∈ A,

L(v, tp1 + (1 − t)p2 ) ≥ α,

∀v ∈ K.

With Proposition 5.53 this implies that (tu1 + (1 − t)u2 , tp1 + (1 − t)p2 ) is a saddle point of L and then (tu1 + (1 − t)u2 , tp1 + (1 − t)p2 ) ∈ K0 × A0 . If now L is strictly convex with respect to the first variable, then K0 is a singleton. Indeed, if K0 contains two distinct points u1 and u2 then for all t ∈]0, 1[ and a saddle point (u1 , p1 ), the points (u2 , p1 ) and (tu1 + (1 − t)u2, p1 ) are saddle points of L (since K0 × A0 is convex) and we have α = L(tu1 + (1 − t)u2 , p1 ) < tL(u1 , p1 ) + (1 − t)L(u2 , p1 ) = α, which is impossible, and we then obtain the result (i). Similarly, we can obtain the result (ii).

We shall now give some characterizations of a saddle point of the function L in the case where L is a differentiable function. Characterization of Saddle Points Theorem 5.57. Assume that the function L = S + R such that the function S satisfies v ∈ K −→ S(v, q) is convex and G-differentiable, ∀q ∈ A, q ∈ A −→ S(v, q) is concave and G-differentiable, ∀v ∈ K

(5.89)

and the function R satisfies v ∈ K −→ R(v, q) is convex, ∀q ∈ A, q ∈ A −→ R(v, q) is concave, ∀v ∈ K.

(5.90)

5.3 Minimax Duality

133

Then, the pair (u, p) ∈ K × A is the saddle point of L if and only if ∂S (u, p), v − uU + R(v, p) − R(u, p) ≥ 0, ∂u ∂S

(u, p), q − pX + R(u, p) − R(u, q) ≤ 0, ∂p

∀v ∈ K, (5.91) ∀q ∈ A,

where ∂S/∂u and ∂S/∂p denote the partial G-derivatives on U and X, respectively. Proof. (⇒) Let (u, p) be a saddle point of the function L, then for all t ∈]0, 1[ and for all v ∈ K (by (5.88) and (5.90), i.e., the convexity of R(., p)) S(u, p) + R(u, p) = ≤ = ≤

L(u, p) L(tv + (1 − t)u, p) S(tv + (1 − t)u, p) + R(tv + (1 − t)u, p) S(tv + (1 − t)u, p) + tR(v, p) + (1 − t)R(u, p).

Thus,

S(tv + (1 − t)u, p) − S(u, p) t and then by passing to the limit in t (because of the G-differentiability of S), we have the first relation of (5.91). In the same way, we can obtain the second relation of (5.91). (⇐) Let (u, p) be an element of K×A such that the inequalities (5.91) hold. Then, because of Proposition 4.45 (by (5.89), i.e., the convexity of S(., p)), we have that ∂S (u, p), v − uU + S(v, p) − S(u, p) ≥ 0, ∀v ∈ K. − ∂u Thus R(v, p) − R(u, p) + S(v, p) − S(u, p) ≥ 0, ∀v ∈ K, i.e., R(u, p) − R(v, p) ≤

L(u, p) ≤ L(v, p)

∀v ∈ K.

Similarly, we can prove that L(u, p) ≥ L(u, q)

∀q ∈ A.

This completes the proof. As a corollary we have the following theorem.

Theorem 5.58. In addition to Assumptions (A1) and (A2), we assume that v ∈ K −→ L(v, q) is G-differentiable, ∀q ∈ A, q ∈ A −→ L(v, q) is G-differentiable, ∀v ∈ K.

(5.92)

Then, the pair (u, p) ∈ K × A is the saddle point of L if and only if ∂L (u, p), v − uU ≥ 0, ∂u ∂L (u, p), q − pX ≤ 0,

∂p

∀v ∈ K, (5.93) ∀q ∈ A.

134


Existence of Saddle Points We shall now give some conditions of the existence of a saddle point in a minimax formulation (of Ky Fan–von Neumann). Theorem 5.59. In addition to Assumptions (A1) and (A2), we assume that the sets K and A are bounded. Then, the function L admits at least one saddle point (u, p) ∈ K × A and L(u, p) = max min L(v, q) = min max L(v, q). q∈A v∈K

v∈K q∈A

(5.94)

Proof. Suppose first that, ∀q ∈ A, the functions ψq : v ∈ K −→ L(v, q) are strictly convex.

(5.95)

Since U and X are Banach reflexive spaces then, since for all q ∈ A, the functions ψq are real-valued, convex and lower semi-continuous, and the set K is convex, closed, bounded and non-empty, the functional ψq is bounded and possesses at least one minimum uq in K (according to Proposition 4.92). Moreover, if the functional ψq is strictly convex then uq is unique (according again to Proposition 4.92) and we denote by r(q) the value r(q) := min ψq (v) = L(uq , q). v∈K

(5.96)

According to (5.88), the function r : q ∈ A −→ r(q) is real-valued, concave and upper semi-continuous. Since A is convex, closed, bounded and nonempty, we can deduce that r is bounded from above and attains its maximum at the point p such that r(p) = max r(q) = max min ψq (v) = max min L(v, q), q∈A

r(p) ≤ L(v, p)

q∈A v∈K

q∈A v∈K

∀v ∈ K.

(5.97)

According to the second part of the assumption (5.88) we have, for all (v, q) ∈ K × A and t ∈]0, 1[, that L(v, tq + (1 − t)p) ≥ tL(v, q) + (1 − t)L(v, p). In particular, for v = vt := utq+(1−t)p , we have (according to (5.97)) r(p) ≥ r(tq + (1 − t)p) = L(vt , tq + (1 − t)p) ≥ tL(vt , q) + (1 − t)L(vt , p) ≥ tL(vt , q) + (1 − t)r(p) and then r(p) ≥ L(vt , q) ∀q ∈ A.

(5.98)

According to Eberlein–Smulian’s theorem (Theorem 2.90), for (vt ) in K, we can extract from (vt ) a subsequence, denoted also by (vt ), converging weakly

5.3 Minimax Duality

135

in U (for the topology σ(V, V )) to a value u in K and then (because of Proposition 2.65 and the convexity of ψq ) L(u, q) = ψq (u) ≤ lim inf ψq (vt ) = lim inf L(vt , q) ∀q ∈ A. t−→0

t−→0

(5.99)

The limit u is exactly up . Indeed, by the definition of vt and from the second part of (5.88) (the concavity condition) we have that tL(vt , q) + (1 − t)L(vt , p) ≤ L(vt , tq + (1 − t)p) ≤ L(v, tq + (1 − t)p) ∀v ∈ K. Since L(vt , .) is uniformly bounded by r(p) then by passing to the limit (according to (5.99)), we can deduce that L(u, p) ≤ lim inf L(vt , p) ≤ lim sup L(u, tq + (1 − t)p) t−→0

t−→0

and then u = up (according to the strict convexity of ψp and Proposition 5.56). We can now pass to the limit in the inequality (5.98), and we obtain (according to (5.97)) L(u, q) ≤ r(p) = max min L(v, s) ∀q ∈ A, s∈A v∈K

max min L(w, q) = r(p) ≤ L(v, p) q∈A w∈K

∀v ∈ K.

Thus (according to Proposition 5.53) the point (u, p) is a saddle point L. Second, we suppose that the assumption (5.95) does not hold. To show the result, we suppose that U is equipped with a strictly convex norm . (see Remark 5.55) and we introduce the following perturbation Lμ , for μ > 0 by Lμ (v, q) = L(v, q) + μ v ∀(v, q) ∈ K × A. Since ψq is convex and . is strictly convex, then ψqμ = ψq +μ . is strictly convex and therefore, according to the previous study, we have the existence of a saddle point (uμ , pμ ) ∈ K × A such that L(uμ , q) + μ uμ ≤ L(uμ , pμ ) + μ uμ ∀q ∈ A, L(uμ , pμ ) + μ uμ ≤ L(v, pμ ) + μ v

∀v ∈ K.

(5.100)

According to Eberlein–Smulian’s theorem (Theorem 2.90), for (uμ , pμ ) in K × A, we can extract from (uμ , pμ ) a subsequence, denoted also by (uμ , pμ ), converging weakly in U × X to a value (u, p) in K × A and then (because of Proposition 2.65 and the assumption (5.88)) L(u, q) ≤ L(u, p) ≤ L(v, p) and then the point (u, p) is a saddle point L. This completes the proof.

∀(v, q) ∈ K × A

136


Remark 5.60. The previous theorem may easily be extended to the case where the sets K and A are compact subsets of separated (or Hausdorff) topological vector spaces (by using Weirstrass’s theorem), see e.g., Barbu and Precupanu [21]. ♦ Theorem 5.61. In addition to Assumptions (A1) and (A2), we assume that there exists (u0 , p0 ) ∈ K × A such that L(v, p0 ) −→ +∞ f or v ∈ K, v U −→ +∞, L(u0 , q) −→ −∞ f or q ∈ A, q X −→ +∞.

(5.101)

Then the function L admits at least one saddle point (u, p) ∈ K × A and L(u, p) = max inf L(v, q) = min sup L(v, q). q∈A v∈K

v∈K q∈A

(5.102)

Proof. Assume, for μ > 0 sufficiently large, the following subsets: Kμ := {v ∈ K : v U ≤ μ}, Aμ := {q ∈ A : q X ≤ μ}. Since the sets K and A are closed, convex and non-empty then the sets Kμ and Aμ are closed, convex, bounded and non-empty. Consequently, according to Theorem 5.59, the functional L admits a saddle point (uμ , pμ ) on Kμ × Aμ and L(uμ , q) ≤ L(uμ , pμ ) ≤ L(v, pμ )

∀(v, q) ∈ Kμ × Aμ .

(5.103)

Suppose now that μ is sufficiently large so that (u0 , p0 ) ∈ Kμ × Aμ then (according to (5.103)) L(uμ , p0 ) ≤ L(uμ , pμ ) ≤ L(u0 , pμ ).

(5.104)

According to assumptions (5.88) and (5.101), we have that the functions v ∈ K −→ L(v, p0 ) and q ∈ A −→ −L(u0 , q) are convex, lower semicontinuous and coercive and then (because of Proposition 4.92) there exists a pair of constants (a, b) ∈ IR2 such that −∞ < a ≤ L(v, p0 ) and L(u0 , q) ≤ b < +∞ ∀(v, q) ∈ K × A. In particular for all μ > 0 (v := uμ , q := pμ ) we have (according to (5.104)) −∞ < a ≤ L(uμ , p0 ) ≤ L(uμ , pμ ) ≤ L(u0 , pμ ) ≤ b < +∞ and then the sequence L(uμ , pμ ) is uniformly bounded. Moreover, according to (5.101), we can deduce that the sequence (uμ , pμ ) is also uniformly bounded, and then we can extract a subsequence, denoted also by (uμ , pμ ) converging weakly to a point (u, p) in K × A and L(uμ , pμ ) converging to a real point λ. By virtue of (5.104), we can deduce that L(u, q) ≤ λ ≤ L(v, p)

∀(v, q) ∈ K × A

and then (because of Proposition 5.53) (u, p) is a saddle point of L.

5.3 Minimax Duality

137

Lemma 5.62. Let assumptions (A1) and (A2) hold, then: (i) If K is bounded or else the first part of (5.101) is satisfied then sup inf L(v, q) = min sup L(v, q).

q∈A v∈K

v∈K q∈A

(ii) If A is bounded or else the second part of (5.101) is satisfied then max inf L(v, q) = inf sup L(v, q). q∈A v∈K

v∈K q∈A

Proof. The proof is left to the reader as an exercise (by considering, in order to prove for example the result (i), the penalized functional, for μ > 0 sufficiently small, Lμ (v, q) = L(v, q) − μ q X ). We can now give the following more general result than Theorems 5.59 and 5.61. Theorem 5.63. Let Assumptions (A1) and (A2) hold, then: (i) If the following condition is true: K is bounded or else the first part of (5.101) is satisfied, A is bounded or else inf L(v, q) −→ −∞, q ∈ A, q X −→ +∞,

(5.105)

v∈K

then the function L admits a saddle point on K × A. (ii) If the following condition is true: A is bounded or else the second part of (5.101) is satisfied, K is bounded or else sup L(v, q) −→ +∞, v ∈ K, v U −→ +∞,

(5.106)

q∈A

then the function L admits a saddle point on K × A. Proof. Since the functional ψ : q ∈ A −→ ψ(q) := inf v∈K L(v, q) is realvalued concave, lower semi-continuous and satisfies the same hypotheses as in Proposition 4.92 (according to (5.87) and (5.88), and the second part of (5.105)) we have that ψ attains its maximum on A and then (by the definition of ψ) sup inf L(v, q) = sup ψ(q) = max ψ(q) = max inf L(v, q). q∈A v∈K

q∈A

q∈A

q∈A v∈K

Moreover, from Lemma 5.62 and the first part of (5.105) we have that sup inf L(v, q) = min sup L(v, q).

q∈A v∈K

v∈K q∈A

Consequently, min sup L(v, q) = max inf L(v, q) v∈K q∈A

q∈A v∈K

and then the existence of saddle point of L (by Theorem 5.52). So, we have the result (i). In the same way we can prove the result (ii).

138


Example 5.64. Let us consider the following boundary problem: −Δu = au + f on Ω, u = 0 on Γ,

(5.107)

where Ω is an open and bounded subset of IRd , d ∈ IN∗ is sufficiently regular, ∂Ω = Γ is its boundary and n is the unit outward normal on ∂Ω. The domain of the operator −Δ is dom(−Δ) = H01 (Ω) ∩ H 2 (Ω) which is dense and compact in L2 (Ω) (see Yosida [306]). Moreover, the operator −Δ is auto-adjoint on dom(−Δ) (i.e.,

−vΔudx =

Ω (λi )i≥1 of

−uΔvdx, by Green’s Ω

eigenvalues such that 0 < λ1 ≤ formula) and admits a sequence λ2 ≤ · · · ≤ λk ≤ · · · , with the corresponding eigenfunctions (ϕi )i≥1 such that ϕi L2 (Ω) = 1, for all i. Our problem is to use the min-max results in order to prove the existence and the uniqueness of a solution of problem (5.107) in dom(−Δ). To show this result, we suppose that f ∈ L2 (Ω) and a ∈ L∞ (Ω) satisfying: there exist (a1 , a2 ) ∈ IR+ and la ∈ IN such that λla < a1 ≤ a ≤ a2 < λla +1 on Ω. We consider a Hilbert space Y equipped with the norm u H := ( −uΔudx)1/2 Ω

and we introduce the following functional: 1 1 −uΔudx − a | u |2 dx − f udx. J(u) := 2 Ω 2 Ω Ω

(5.108)

It is clear that the functional J is continuous F-differentiable on Y and, for all v ∈ Y , we have

J (u), vY = −vΔudx − auvdx − f vdx Ω Ω Ω (5.109) = (−Δu − au − f )vdx. Ω

If la ≥ 1, the operators J and −J are not monotone and then J is not convex and not concave (because of Proposition 4.46). We have then used the minimax duality by decomposing the space Y on the direct sum of two spaces U and X, i.e., Y := U ⊕⊥ X such that U := IRϕk = {u : u = αk ϕk , where αk ∈ IR}, k≤la

X :=

k≤la

IRϕk = {v : v =

k≥la +1

βk ϕk , where βk ∈ IR}.

(5.110)

k≥la +1

Let us consider the following function L : U × X −→ IR such that

5.3 Minimax Duality

139

L(v, q) := J(v + q) for all (v, q) ∈ U × X and prove the existence of a saddle point of L. The function L is strictly concave on U and L is strictly convex on X. Indeed, since λla < a1 ≤ a ≤ a2 < λla +1 and for all (w, r) ∈ U × X, −wΔwdx ≤ λla w 2L2 (Ω) and −rΔrdx ≥ λla +1 r 2L2 (Ω) , Ω

Ω

we can deduce, for all (u, v) ∈ U 2 and (p, q) ∈ X 2 , that ∂L ∂L (u, p) − (v, p), u − vU = (−Δ(u − v) − a(u − v))(u − v)dx

∂u ∂u Ω ≤ −(a1 − λla ) u − v 2L2 (Ω) , ∂L ∂L (u, p) − (u, q), p − qX = (−Δ(p − q) − a(p − q))(p − q)dx

∂p ∂p Ω ≥ (λla +1 − a2 ) p − q 2L2 (Ω) and then the strict concavity and convexity results on U and X, respectively hold (because of Proposition 4.46). Moreover, we have, for all (u0 , p0 ) ∈ U × X, that L(u, p0 ) −→ +∞ for v ∈ U, v U −→ +∞, L(u0 , q) −→ −∞ for q ∈ X, q X −→ +∞. Consequently, according to Theorem 5.61 and Proposition 5.56, the functional L admits a unique saddle point (u, p) on U × X and we prove easily that u + p is a solution of (5.107) (since J (u + p), vY = 0, because of saddle point characterization theorem 5.92). To prove the uniqueness of the solution, we can prove that the solution of (5.107) is a saddle point of L and by the uniqueness of the saddle point we can deduce the result. Here, we prove directly the uniqueness of the results. Let u ∈ Y and v ∈ Y be two solutions of (5.107) then w = u − v ∈ Y is a solution of (5.107) with the second member f = 0. Since Y := U ⊕⊥ X, then there exists a unique (w1 , w2 ) ∈ U × X such that w := w1 + w2 and −w1 Δw2 dx = −w2 Δw1 dx = 0. Ω

Ω

Consequently, −w1 Δw1 dx − a | w1 |2 dx − aw1 w2 dx 0 = (−Δw − aw)w1 dx = Ω Ω Ω Ω 2 ≤ −(a1 − λla ) w1 L2 (Ω) − aw1 w2 dx, Ω 0 = (−Δw − aw)w2 dx = −w2 Δw2 dx − a | w2 |2 dx − aw1 w2 dx Ω Ω Ω Ω 2 ≥ (λla +1 − a2 ) w2 L2 (Ω) − aw1 w2 dx Ω

140


and then 0 ≤ (λla +1 − a2 ) w2 2L2 (Ω) ≤ −(a1 − λla ) w1 2L2 (Ω) ≤ 0. Therefore, w1 = w2 = 0 and then w = 0. This completes the proof.

♣

5.3.4 Connection with Duality and Application We consider a minimization problem (P) inf Ψ (v),

(5.111)

inf Ψ (v),

(5.112)

v∈U

which we can write in the form v∈K

where K = domΨ , K ⊂ U . Let us assume that we can write Ψ (v) as a supremum on q of a function L(v, q) as (5.113) Ψ (v) = sup L(v, q), ∀v ∈ K. q∈A

The problem (5.111) then becomes inf sup L(v, q).

v∈K q∈A

(5.114)

The function Ψ can be written in the form (5.113) by using, for example, the theory of conjugate convex functions. By reducing the problem (P) to the form (5.114), the dual problem (P ∗ ) associated with problem (P) is then sup inf L(v, q).

q∈A v∈K

(5.115)

Connection with Duality In this subsection, we give the analogy between the results obtained in the section 5.3 and the results of Fenchel–Rockfellar given in the section 5.1. Precisely, we give the following interpretations and remarks: (i)

Proposition 5.51 means that −∞ ≤ sup(P ∗ ) ≤ inf(P) ≤ +∞ and it must be compared to Proposition 5.4.

(ii) Lemma 5.62 gives criteria which ensure that solutions of problems (P) and (P ∗ ) exist and the dual gap of (P) is null, i.e., sup(P ∗ ) = inf(P). It must be compared to the stability of problems (P ∗ ) and (P), respectively, given in Propositions 5.12 and 5.15. (iii) Theorems 5.59 and 5.61 give criteria which determine if problems (P) and (P ∗ ) are both stable, and it must be compared to Proposition 5.20.

5.3 Minimax Duality

141

(iv) In the case of the equality “inf sup = sup inf” of the Lagrangian L, the existence of a saddle point for L is equivalent to the existence of solutions us ∈ K, ps ∈ A, respectively, of problems (P ∗ ) and (P) with the stability of these problems. Moreover, in this situation, the set of saddle points of L is exactly the set of solutions (us , ps ) and we have the following extremality conditions L(us , ps ) = sup L(us , q), q∈A

L(us , ps ) = inf L(v, ps ).

(5.116)

v∈K

(v)

The minimax theorems play a key role in the control theory for problems governed by partial differential equations, in particular in the minimax control (see, e.g., Ahmed and Xiang [6], Arada and Raymond [10], Belmiloudi [44, 47], McMillan and Triggiani [222, 223], Mordukhovich and Zhang [226, 225], Papageorgiou and Yannakakis [234], and the references therein), and in the stability and robust control theory (which will be detailed in Chapter 8, Chapter 9 and Part III).

(vi) It is clear that the change from problem (5.111) (or (5.112)) to problem (5.114), can be envisaged, without difficulty, in non-convex optimizations (the duality theorems given in Section 5.3 are valid for non-convex systems) see the applications below. Applications In this subsection, we give two interesting applications. More precisely: 1. Suppose that the function Ψ is the sum in U of two functions Φ and ξ, i.e., Ψ = Φ + ξ, where Φ is convex, lower semi-continuous and proper on U . According to the Fenchel–Moreau theorem, we have that Φ∗∗ = Φ and then Φ(v) = sup ( g, vU − Φ∗ (g)) , ∀v ∈ U, g∈U

where Φ∗ ∈ Γ0 (U ) is the conjugate function of Φ and Φ∗∗ is the biconjugate function of Φ. So, for all u ∈ U , we have Ψ (v) = sup ( g, vU + ξ(v) − Φ∗ (g)) . g∈U

Put now

L(v, g) = g, vU + ξ(v) − Φ∗ (g),

where v ∈ K = domΦ and g ∈ A := U . Then Ψ can now be written in the form (5.113) and we have the formulation (5.114).

142


2. Let now X be another topological vector space, Λ be an operator of U into X (not necessarily linear) and suppose that Ψ = Φ + ξ, where Φ = F ◦ Λ and F ∈ Γ0 (X). In the same way as in the first example, we have that Ψ (v) = F (Λv) + ξ(v) = sup ( g, ΛvY,X + ξ(v) − F ∗ (g)), ∀v ∈ U, g∈Y

where X and Y are two locally convex spaces which are placed in duality by the bilinear form ., .Y,X , and F ∗ ∈ Γ0 (Y ) is the conjugate function of F . Consequently, Ψ can be written in the form (5.113), where A is a subspace of Y and L(v, g) := g, ΛvY,X + ξ(v) − F ∗ (g). A more precise and more detailed analysis of this type of non-convex problems will be considered in Section 5.4. 5.3.5 Ky Fan’s Minimax Inequality and Non-potential Operators Let U be a reflexive Banach space with its dual space U and φ : U −→ IR be a proper and weakly lower semi-continuous function on U . The goal of this subsection is to characterize the solutions of the differential inclusion f + F (u) ∈ ∂φ(u)

(5.117)

by variational problems. The functional F : U −→ U is given, f ∈ U and ∂φ is the subdifferential of φ on U . In order to simplify the presentation, we denote by G : U −→ U the function f + F (.), i.e., G(u) = f + F (u) for all u ∈ U . Remark 5.65. If the mapping G is a potential operator, there exists a realvalued and G-differentiable function R such that G = R and then problem (5.117) is equivalent to seek the minima of the functional φ − R on U . ♦ Assume now that the mapping G is a non-potential operator (i.e., not of “gradient-type”). The variational problem (P) associated with (5.117) is the following minimization problem: inf Ψ (u),

u∈U

(5.118)

where the functional Ψ : U −→ IR is defined as a supremum on v of a function L(u, v) as (5.119) Ψ (u) = sup L(u, v) ∀u ∈ U, v∈U

where L : U × U −→ IR is defined by L(u, v) = φ(u) − φ(v) + G(u), v − uU .

(5.120)

5.3 Minimax Duality

143

The problem (5.118) then becomes inf sup L(u, v).

(5.121)

u∈U v∈U

Moreover, according to the definition of the convex conjugate functional, the mapping Ψ can be written as Ψ (u) = φ(u) − G(u), uU + φ∗ (G(u)) ≥ 0

(5.122)

for all u in U .3 Consequently, because of Proposition 4.16, we can deduce that G(us ) ∈ ∂φ(us ) if and only if φ(us ) + φ∗ (G(us )) − G(us ), us U = 0, i.e., G(us ) ∈ ∂φ(us ) if and only if Ψ (us ) = 0 and us minimizes Ψ on U . Otherwise, us is a solution of (5.117) if and only if

(5.123)

Ψ (us ) = 0 and us is a solution of (5.118).

As a corollary of the existence result (5.123), we have that if (us , vs ) is a minimax point of L on U × U such that L(us , vs ) = 0 then us is a solution of (5.117). The following Ky Fan’s minimax inequality theorem [118], which is based on the intersection result due to Ky Fan [117], gives some sufficient conditions on −L and then on φ and G in order to obtain directly the existence of minimax points and then the existence of solutions of the differential inclusion problem (5.117). The intersection result of Ky Fan is known in the literature as Ky Fan’s lemma. Lemma 5.66. (Ky Fan’s lemma) Let Y be a Hausdorff topological vector space and K be a subset of Y . For each u ∈ K, consider a closed subset Ku of Y such that: (i) there exists u0 ∈ K such that the set Ku0 is compact (ii) for any finite set Sn = {u1 , . . . , un } of points in K, coSn ⊂

n

Kui , where

i=1

coSn is the convex hull of Sn . Then

u∈K

Ku = ∅.

Theorem 5.67. Let K be a non-empty closed, convex subset of a reflexive Banach space U and L : K × K −→ IR be such that: 3

Since, from the definition of φ∗ , we have that the inequality φ(v)+φ∗(g) ≥ g, v U holds, for all (v, g) ∈ U × U and in particular for all v ∈ U and g = G(v) ∈ U .

144


(i) −L(u, u) ≥ 0, for all u in K (ii) for each u ∈ K, the mapping v −→ −L(u, v) is quasi-convex on K (iii) for each v ∈ K, the mapping u −→ −L(u, v) is weakly upper semicontinuous on K (iv) there exists an element v0 such that K0 = {u ∈ K : −L(u, v0 ) ≥ 0} is a bounded subset of K. Then there exists an element us in K such that sup L(us , v) ≤ 0, i.e., −L(us , v) ≥ 0 f or all v ∈ K,

(5.124)

v∈K

i.e., there exists a solution us of the so-called equilibrium problem (5.124). Proof. For the details of the proof see, e.g., Auchmuty [16], Brezis et al. [58] and Ky Fan [118]. Remark 5.68. (i) Ky Fan’s minimax inequality has many applications in various other branches of mathematics: economy, game theory, fixed point theorems and variational inequalities. (ii) If L is defined by (5.120), L(u, u) = 0 for all u in K and then the assumption (i) of Theorem 5.67 is always true. ♦ As a direct consequence, we have the following result (because of (5.122) and (5.123)). Corollary 5.69. Let K be a non-empty closed, convex subset of a reflexive Banach space U and L be defined by (5.120) and satisfying assumptions (ii)– (iv) of Theorem 5.67. Then there exists a minimizer us of Ψ on K which is a solution of problem (5.117). Proof. The proof is left to the reader as an exercise.

Proposition 5.70. Let K be a non-empty closed, convex subset of a reflexive Banach space U and L be defined by (5.120). Assume that assumptions (iii)– (iv) of Theorem 5.67 hold and φ is a weakly lower semi-continuous, quasiconvex and coercive function on K. Then there exists a minimax point (us , vs ) of L on K × K and us is a solution of problem (5.117). Moreover, if we assume that the functional φ is strictly convex over K, then there exists a unique vs ∈ K depending on us (in a unique manner) such that Ψ (us ) = L(us , vs ). Proof. Since φ is a quasi-convex function, then L satisfies the assumption (ii) of Theorem 5.67 and from Corollary 5.69, there exists a minimizer us of Ψ on K and us is a solution of (5.117). Consider now the functional Gus : U −→ IR defined by Gus (v) := φ(v) − G(us ), vU .

5.3 Minimax Duality

145

Since φ is a weakly lower semi-continuous, quasi-convex and coercive function on K then Gus is a weakly lower semi-continuous and coercive function on K. So, the set Ks of minimizers of Gus is a non-empty bounded closed and convex subset of K. Moreover, Ks is also the set of maximizers of the mapping L(us , .). Then there exists an element vs ∈ K such that Ψ (us ) = L(us , vs ). This completes the proof of the existence of a minimax point. If φ is strictly convex, Gus is also strictly convex and Ks is a singleton. This completes the proof. Now we present an example of a minimax point problem which is not necessarily a saddle point problem (see Auchmuty [17]). Example 5.71. Let Ω be an open and bounded subset of IRd , d ≤ 3 sufficiently regular, ∂Ω = Γ be its boundary, n be the unit outward normal on ∂Ω. We seek a vector function u representing the velocity of the fluid and a scalar function p representing the pressure of the fluid, which are defined in Ω and satisfy the following stationary non-linear Navier–Stokes system, for the equilibrium of a viscous flow, under a force field f ∈ U and subject to no slip boundary conditions on Γ : −νΔu + ∇p = f − (u∇)u on Ω, div(u) = 0 on Ω, u = 0 on Γ,

(5.125)

where ν is the coefficient of kinematic viscosity (a positive constant). space. U is a reflexive Let U := {u ∈ H01 (Ω) : div(u) = 0} be a Hilbert | ∇u |2 dx)1/2 and satisfies

Banach space equipped with the norm u = ( Ω

U ⊂ L2 (Ω) ⊂ U . Let φ : U −→ IR be defined by ν φ(u) := 2

| ∇u |2 dx.

(5.126)

Ω

It is clear that φ is a continuous, coercive and strictly convex function on U .

(5.127)

Moreover, the stationary non-linear Navier–Stokes system (5.125) under the force f ∈ U can be written as f − (u∇)u ∈ ∂φ(u) for all u ∈ U. By using Green’s formula, we have that 1

(u∇)v, vU = u.∇(| v |2 )dx = 0 for all u, v ∈ U. 2 Ω

(5.128)

(5.129)

146


Consequently, when we replace v by v + w in (5.129), we have

(u∇)v, wU = − (u∇)w, vU for all u, v, w ∈ U.

(5.130)

Assume that the force f is in L2 (Ω) ⊂ U . The functionals G : U −→ U and L : U × U −→ IR corresponding to our problem are defined by (because of (5.129) and (5.130)) G(u) = f − (u∇)u,

f (v − u)dx −

L(u, v) := φ(u) − φ(v) + Ω

(u∇)v.udx,

(5.131)

Ω

for all u and v in V . Remark 5.72. For all v ∈ U the mapping u −→ L(u, v) is not, in general, convex. Moreover, from the classical result for Navier–Stokes system, we know (only) that (5.125), or equivalently (5.128), have a finite number of distinct solutions under various conditions on f and ν.4 Consequently, we can not expect to have a saddle point in U × U , on the other hand we will show the existence of a minimax point in U × U . ♦ Now we will study the existence of the minimax point of L in U × U . Since for all u ∈ U the mapping v −→ −L(u, v) − φ(v) is an affine functional then the mapping v −→ −L(u, v) is convex on V (since φ is convex). Moreover, we have that K0 = {u ∈ U : L(u, 0) = φ(u) − f udx ≤ 0} is bounded in U Ω

since, ∀u ∈ K0 , u 2 =

2 2 φ(u) ≤ f L2 (Ω) u L2 (Ω) ≤ C f L2 (Ω) u . ν ν

Consequently, L satisfies conditions (i) (by the definition of L), (ii) and (iv) of Ky Fan’s theorem 5.67. Prove now condition (iii) of Theorem 5.67, i.e., the weakly upper semi-continuity of the mapping u −→ −L(u, v) on U , for each fixed v ∈ U . Since the functions φ and u −→

f udx are lower Ω

semi-continuous, we need only prove the weak lower semi-continuity of the function u −→ − (u∇)v.udx. Ω

Let (un ) be a sequence of U which converges weakly to u in U . Then, from the compactness of Sobolev embeddings into Lp -spaces, (un ) converges strongly in Lp (Ω), for 1 ≤ p < 6 if d = 3 and p ∈ [1, ∞[ if d ≤ 2. Consequently, 4

For example, if ν is sufficiently large (or f is sufficiently small) so that ν 2 ≥ C(Ω, d) f U then there exists a unique solution for u ∈ U of (5.125), see, for instance, Temam [281].

5.4 Duality and Parametric Variational Problems

−

147

(un ∇)v.un dx −→ −

Ω

(u∇)v.udx Ω

and then condition (iii) of Theorem 5.67 follows. From (5.127) and the fact that L satisfies conditions (i)–(iv) of Ky Fan’s Theorem 5.67, we can apply Proposition 5.70 and we obtain the existence of us ∈ U and vs ∈ U such that vs depends in a unique manner on us , ♣ 0 = Ψ (us ) = L(us , vs ) and us minimizes Ψ on U .

5.4 Duality and Parametric Variational Problems In this work we have centered our analysis on the case of the non-convex parametric variational problem, namely the following primal problem (P). Find u ∈ Uad the infimum in Uad of the functional: Jν (v) = S(Λv − ν) + F (v),

(5.132)

where ν is a given distributed parameter and the geometric operator Λ is non-linear and of quadratic type. More precisely, Λ = Λ0 + ΛL , where ΛL is a linear and continuous operator and Λ0 is a non-linear operator as the form Λ0 v = B(v, v), with B a bilinear and continuous operator.

(5.133)

This type of problems appears in many physical and biological systems such as hysteresis and phase transitions (e.g., supraconductor, solidification, etc.), non-convex optimal design and control, non-linear bifurcation and stability analysis, and many others. The operator Jν is corresponding to the total potential, the operator S is corresponding to the internal energy and the operator F is corresponding to the external energy. Our approach is a generalization of the work of Strang and Gao [127], by introducing a new gap function. 5.4.1 Abstract Framework Let U, U and X, X be two pairs of reflexive Banach spaces, in duality with respect to certain bilinear forms ., .U and ., .X respectively. Let the geometric operator Λ : U −→ X be a continuous, G-differential non-linear and quadratic type operator from U into X. Then the so-called geometrical equation can be written as p = Λu. (5.134) Since Λ is G-differentiable then p is G-differentiable and his G-derivative at u is p (u) = Λt (u), where Λt (u) is the G-derivative of Λ at u that is corresponding to the tangent geometric operator. Moreover, since Λ is quadratic such that (5.133) then p (u)v = B(u, v) + B(v, u) + ΛL v because of (4.37), i.e., Λt (u)v = B(u, v) + B(v, u) + ΛL v.

(5.135)

148


Remark 5.73. Because of (5.135), we can deduce that the operator u −→ Λt (u)u = 2B(u, u) + ΛL u is non-linear and then the following relation holds. Λt (u)u = 2Λ0 u + ΛL u.

(5.136) ♦

Let the operator S : X −→ IR be an extended real-valued, finite and Gdifferentiable on a non-empty convex and closed subset A ⊂ X. In conservative systems, we can usually consider the operator Λ such that the function S is either convex or concave but the function S ◦ Λ may be non-convex and nonconcave (by the non-linearity of Λ). The dual relation between the pairing spaces X and X can be described by the so called constitutive equation p∗ = S (p − ν),

(5.137)

where S : A ⊂ X −→ X is the G-derivative of S. In the same way, we suppose that the operator F : U −→ IR is an extended real-valued, finite and G-differentiable functional on a non-empty convex and closed subset K ⊂ U . The dual relation between the pairing spaces U and U can be given by the following equation: u∗ = F (u),

(5.138)

where F : K ⊂ U −→ U is the G-derivative of F . Definition 5.74. The parametric variational problem (5.132) is said to be: (i) geometrically non-linear if the operator Λ is non-linear (ii) physically non-linear if at least one of the duality realations (5.137) and (5.138) is non-linear (iii) fully non-linear if it is both geometrically and physically non-linear.

For a given distributed parameter ν ∈ X, the total potential Jν : U −→ IR that is defined by Jν (u) = S(Λu − ν) + F (u) is finite and G-differentiable at u if and only if the element u is in the admissible space Kad , where Kad := {v ∈ K : Λv − ν ∈ A}. In particular, if u ∈ Kad is a critical point of Jν , i.e., Jν (u) = 0, then we have the so-called equilibrium equation (according to (5.137), (5.138) and (5.132)) Λ∗t (u)p∗ + u∗ = Λ∗t (u)S (Λu − ν) + F (u) = Jν (u) = 0,

(5.139)

where the operator Λ∗t (u) : X −→ U is the adjoint of Λt (u) defined by

p∗ , Λt (u)vX = Λ∗t (u)p∗ , vU ∀v ∈ Kad .


149

Remark 5.75. (i) The set of critical points of Jν is a subset of Kad and is denoted by Kc := {u ∈ Kad : Jν (u) = 0}. (ii) The set of the infimum of Jν is a subset of Kad and is denoted by Ks := {u ∈ Kad : Jν (u) = inf Jν (v)}. v∈U

♦

Remark 5.76. If Kad is a non-empty closed and bounded convex subset of a ♦ reflexive Banach space U then Ks is a non-empty convex subset of Kad . According to (5.134), (5.136) and (5.139), the relation between the pairing spaces U, U and X, X can be written as

p∗ , pX + u∗ , uU = p∗ , ΛuX − Λ∗t (u)p∗ , uU = p∗ , ΛuX − p∗ , Λt (u)uX = − p∗ , Λ0 uX

(5.140)

= −G(u, p∗ ), where

G(u, p∗ ) := p∗ , Λ0 uX .

(5.141)

G is said to be the complementary gap function, which is introduced by Strang and Gao in [127] (see also Gao [128] for different illustrations and physical applications). Theorem 5.77. Suppose that the function F is in Γ0 (U ), finite and Gdifferentiable on the subset K, and the function S is in Γ0 (X), finite and G-differentiable on the subset A. For uc be a critical point of Jν on Kad and p∗c = S (Λuc − ν) we have that if G(v, p∗c ) ≥ 0 for all v ∈ U , then uc is a minimizer of Jν on Kad . Proof. Let uc be a critical point of Jν then pc = Λuc , p∗c = S (Λuc − ν), u∗c

=

−Λ∗t (uc )p∗c

(5.142)

= F (uc ).

Since S and F are G-differentiable on A and K, respectively, we have that (because of the convexity of S and of F ) S(Λv − ν) − S(Λuc − ν) ≥ S (Λuc − ν), Λv − Λuc X ∀v ∈ Kad , F (v) − F (uc ) ≥ F (uc ), v − uc U ∀v ∈ K.

(5.143)

Since Λ is a quadratic operator then for any w and v in U , we can easily obtain the following equality (according to (5.133) and (5.135)):

150


Λ(v + w) = Λv + Λt (v)w + Λ0 w.

(5.144)

By the relations (5.142), (5.143) and (5.144) we can deduce that (for all v ∈ K) Jν (uc +w) − Jν (uc ) = (S(Λ(uc + w) − ν) − S(Λuc − ν)) + (F (uc + w) − F (uc )) ≥ S (Λuc − ν), Λ(uc + w) − Λuc X + F (uc ), wU = S (Λuc − ν), Λt (uc )w + Λ0 wX + F (uc ), wU = p∗c , Λt (uc )w + Λ0 wX + −Λ∗t (uc )p∗c , wU

(5.145)

= p∗c , Λ0 wX + p∗c − p∗c , Λt (uc )wX = p∗c , Λ0 wX = G(w, p∗c ), where w = v − uc ∈ U . Since G(z, p∗c ) ≥ 0 for all z ∈ U then (according to the previous result) Jν (v) − Jν (uc ) ≥ 0 for all v ∈ K. Consequently, uc is a minimizer of Jν and then the result of the theorem. We suppose now that F is a real-valued non-linear convex quadratic function on U (this mathematical formulation represents several realistic situations) in the sense F = F0 + FL , where FL is a linear continuous form and F0 is a non-linear and convex function in the form F0 (v) = H(v, v) ≥ 0, with H a bilinear positive and continuous form on U.

(5.146)

Therefore, we have a more precise result, by introducing a new gap function. Theorem 5.78. Assume that the assumptions of Theorem 5.77 hold and that the function F is a non-linear convex quadratic function such that (5.146). For uc be a critical point of Jν on Kad and p∗c = S (Λuc − ν) we have that if Gq (v, p∗c ) ≥ 0 for all v ∈ U , then uc is a minimizer of Jν on Kad , where Gq (v, p∗c ) := G(v, p∗c ) + F0 (v). This new gap function Gq will be called the quadratic complementary gap function. Proof. Let uc be a critical point of Jν . Since F is a quadratic form such that (5.146) then for any w and v on U , we can obtain easily the following equality: F (v + w) = F (v) + F (v), wU + F0 (w).

(5.147)

Applying the G-differentiability of S and F on A and K, respectively, we have that (because of the convexity of S and the expression (5.147))


S(Λv − ν) − S(Λuc − ν) ≥ S (Λuc − ν), Λv − Λuc X ∀v ∈ Kad , F (v) − F (uc ) = F0 (v − uc ) + F (uc ), v − uc U ∀v ∈ K.

151

(5.148)

By the relations (5.142), (5.148) and (5.144) we can deduce that (for all v ∈ K) Jν (uc +w) − Jν (uc ) = (S(Λ(uc + w) − ν) − S(Λuc − ν)) + (F (uc + w) − F (uc )) ≥ S (Λuc − ν), Λ(uc + w) − Λuc X + F (uc ), wU + F0 (w) = S (Λuc − ν), Λt (uc )w + Λ0 wX + F (uc ), wU + F0 (w) = p∗c , Λt (uc )w + Λ0 wX + −Λ∗t (uc )p∗c , wU + F0 (w)

(5.149)

= p∗c , Λ0 wX + p∗c − p∗c , Λt (uc )wX + F0 (w) = p∗c , Λ0 wX + F0 (w) = Gq (w, p∗c ), where w = v − uc . Since Gq (z, p∗c ) ≥ 0 for all z ∈ U then (according to the previous result) Jν (v) − Jν (uc ) ≥ 0 for all v ∈ K. Consequently, uc is a minimizer of Jν and the result of the theorem follows. 5.4.2 Geometrically Non-linear Lagrangian Representation In order to study the dual problem, we need to find the complementary energy of the non-linear system. From the Fenchel transformation, we known that the conjugate functionals, of given functionals S : X −→ IR and F : U −→ IR, are defined respectively by S ∗ (p∗ ) = sup ( p∗ , pX − S(p)) for all p∗ ∈ X , p∈X

F ∗ (u∗ ) = sup ( u∗ , uU − F (u)) for all u∗ ∈ U .

(5.150)

u∈U

We introduce now the functional Sν : p ∈ X −→ S(p − ν) ∈ IR. Then the conjugate functional of Sν is given, for all p∗ ∈ X , by Sν∗ (p∗ ) = S ∗ (p∗ ) + p∗ , νX .

(5.151)

Remark 5.79. (a) The function Sν is convex (respectively concave) if and only if the function S is convex (respectively concave). (b) From the theory of convex analysis, the functions Sν∗ and F ∗ are always convex and lower semi-continuous. (c) If the function S is strictly convex, G-differentiable on A, then the following conditions are equivalent:

152


(i) p∗ = S (p) (ii) p = (S ∗ ) (p∗ ) (iii) p∗ , pX = S ∗ (p∗ ) + S(p).

♦

Let K∗ ⊂ U and A∗ ⊂ X be non-empty, convex and closed subsets on which F ∗ and Sν∗ are finite and G-differentiable, respectively. Consider now the geometrical Lagrangian Lν : U × X −→ IR defined by Lν (v, q ∗ ) := q ∗ , ΛvX − Sν∗ (q ∗ ) + F (v),

(5.152)

which is finite and G-differentiable on K × A∗ . It is easy to obtain that, for all (u, p∗ ) ∈ K × A∗ , ∂Lν (u, p∗ ) = Λ∗t (u)p∗ + F (u), ∂v ∂Lν (u, p∗ ) = Λu − (Sν∗ ) (p∗ ), ∂q ∗

(5.153)

where ∂Lν /∂v and ∂Lν /∂q ∗ denote the partial G-derivatives on U and X , respectively. Remark 5.80. (i) Depending on the convexity of the functional Sν , there exists an interesting relation between the functional Jν and the Lagrangian Lν . More precisely, for any given function F we have that (by using the Fenchel transformation and Corollary 4.6), for all v ∈ U if Sν is in Γ (X) then Jν (v) = sup Lν (v, q ∗ ), q∗ ∈X

(5.154)

i.e., Lν is a Lagrangian of type I of problem (P) (given by (5.132)). (ii) According to the expressions (5.153) we can deduce that, if (uc , p∗c ) ∈ K × A∗ is a critical point of Lν then ∂Lν (uc , p∗c ) = Λ∗t (uc )p∗c + F (uc ) = 0, ∂v ∂Lν (uc , p∗c ) = Λ(uc ) − (Sν∗ ) (p∗c ) = 0. ∂q ∗

(5.155) ♦

Example 5.81. Let Ω be an open and bounded subset of IRd , d ≤ 3 sufficiently regular, ∂Ω = Γ be its boundary, n be the unit outward normal on ∂Ω. Let space. U is a reflexive Banach space equipped with U = H01 (Ω) be a Hilbert

| ∇u |2 dx)1/2 and U ⊂ L2 (Ω) ⊂ U . Moreover,

the norm u = ( Ω

according to Sobolev embeddings into Lp -spaces, we have the embedding U ⊂ Lp (Ω), for 1 ≤ p < 6 if d = 3 and p ∈ [1, ∞[ if d ≤ 2 with compactness. The Landau–Ginzburg energy Jν for a ferroelectric can be written as D(x) 1 | u |2 | ∇u |2 dx + ( − ν(x))2 dx − f udx, Jν (u) = 2 2 Ω Ω2 Ω


153

where u is a phase-field and scalar function whose values describe the phase of the system under consideration, D > 0 and ν > 0 are positive and bounded functions, and f is in L2 (Ω). We introduce now the following quadratic operator Λ : U −→ X by Λv =

1 | v |2 = Λ0 v for all v ∈ U, 2

where X = L2 (Ω) = X . The operator Λ is G-differentiable and its G-derivative at point u, Λt : U −→ X is given by Λt (u) = u = Λ∗t (u) for all u ∈ X.

(5.156)

Then the functional Jν : U −→ IR can be written as Jν (u) = S(Λu − ν) + F (u), where the functionals S : X −→ IR and F : U −→ IR corresponding to our problem are defined by 2 p dx for all p ∈ X, S(p) := Ω 2 (5.157) D(x) | ∇v |2 dx − f vdx for all v ∈ U. F (v) := 2 Ω Ω The functionals S and F are convex lower semi-continuous finite and Gdifferentiable, and their G-differentials at points p and u respectively are given by S (p) = p and F (u) = −div(D∇u) − f. (5.158) Moreover, the gap and the quadratic gap functions are given by (since F is a quadratic form) 1 G(v, p∗ ) = p∗ | v |2 dx, 2 Ω (5.159) D ∗ ∗ 2 | ∇v | dx. Gq (v, p ) = G(v, p ) + Ω 2 Let uc be a critical point of Jν . Then, according to (5.142), (5.156) and (5.158), we can deduce that uc is a solution of the well-known second-order Ginzburg–Landau equation 1 | uc |2 , 2 1 p∗c = | uc |2 −ν, 2

pc =

1 −div(D∇uc ) + uc ( | uc |2 −ν) = f on Ω 2 uc = 0 on Γ.

(5.160)

154


Moreover, according to Theorem 5.78, we have that if Gq (v, p∗c ) ≥ 0 for all v ∈ U then uc is a minimizer of Jν . We now give some sufficient conditions such that Gq (v, p∗c ) ≥ 0 for all v ∈ U . Let α > 0 be the smallest eigenvalue of the operator div(D∇.), i.e., D | ∇v |2 dx . α = inf Ω v∈U v 2L2 (Ω) We can deduce that if α + p∗c ≥ 0, i.e., if | uc |2 ≥ 2(ν − α) then Gq (v, p∗c ) ≥ 0. ♣ Consequently, uc is a minimizer of Jν . Next we give the properties of the critical points of Lν that depend on gap functions. Theorem 5.82. Assume that the function F is in Γ0 (U ), finite and Gdifferentiable on K, Λ : U −→ X is a quadratic operator and the function Sν is in Γ0 (X), finite and G-differentiable on A. Let (uc , p∗c ) be a critical point of Lν defined by (5.152). Then the following property holds: if G(v, p∗c ) ≥ 0 for all v in U then (uc , p∗c ) is a saddle point of Lν . Proof. Since F is convex then, for all v ∈ U , F (v) ≥ F (uc ) + F (uc ), v − uc U . According to (5.144) we have that Λ(uc + w) = Λuc + Λt (uc )w + Λ0 w, for all w ∈ U , and then, according to (5.155) (since (uc , p∗c ) is a critical point) Lν (v, p∗c ) − Lν (uc , p∗c ) ≥ Λ∗t (uc )p∗c + F (uc ), v − uc U + p∗c , Λ0 (v − uc )X ∂Lν (uc , p∗c ), v − uc U + G(v − uc , p∗c ) = ∂u = G(v − uc , p∗c ),

(5.161)

for all v in K. Consequently (since G(z, p∗c ) ≥ 0 for all z in U ), Lν (v, p∗c ) − Lν (uc , p∗c ) ≥ G(v − uc , p∗c ) ≥ 0 for all v ∈ K.

(5.162)

Since the function Sν∗ is convex then, for each given v ∈ U , the function ψv : q ∗ ∈ X −→ ψv (q ∗ ) = Lν (v, q ∗ ) is concave. Moreover, if (uc , p∗c ) ∈ K×A∗ is a critical point of Lν , then p∗c is a critical point of ψuc , i.e., ψu c (p∗c ) = 0. Thus, by Proposition 4.45, ψuc (p∗c ) ≥ ψuc (q ∗ ), for all q ∗ ∈ A∗ . Consequently, Lν (uc , p∗c ) ≥ Lν (uc , q ∗ ) for all q ∗ ∈ A∗ .

(5.163)

From (5.162) and (5.163) we obtain easily the result of the theorem.


155

Nota bene: In the sequel, we will suppose that F is a quadratic form in the sense (5.146). Theorem 5.83. Assume that the assumptions of Theorem 5.82 hold and that the function F is a non-linear convex quadratic function such that (5.146). Let (uc , p∗c ) be a critical point of Lν defined by (5.152). Then the following properties hold: (i) (uc , p∗c ) is a saddle point of Lν if and only if Gq (v, p∗c ) ≥ 0 for all v in U (ii) (uc , p∗c ) is a supercritical point of Lν if and only if Gq (v, p∗c ) ≤ 0 for all v in U . Proof. According to (5.144) and (5.147) we have, for all w ∈ U , that Λ(uc + w) = Λuc + Λt (uc )w + Λ0 w, F (uc + w) = F (uc ) + F (uc ), wU + F0 (w).

(5.164)

Then for all v ∈ K, according to (5.155) (since (uc , p∗c ) is a critical point) Lν (uc + w, p∗c ) − Lν (uc , p∗c ) = Λ∗t (uc )p∗c + F (uc ), wU + p∗c , Λ0 wX + F0 (w) = G(w, p∗c ) + F0 (w)

(5.165)

= Gq (w, p∗c ), where w = v − uc ∈ U . Consequently, Lν (v, p∗c ) − Lν (uc , p∗c ) = Gq (v − uc , p∗c ) for all v in K and then sign(Lν (v, p∗c ) − Lν (uc , p∗c )) = sign(Gq (v − uc , p∗c )) ∀v ∈ K,

(5.166)

where sign(x) = 1 if x > 0, sign(x) = −1 if x < 0 and sign(x) = 0 if x = 0. Since the function Sν∗ is convex then, for each given v ∈ U , the following function ψv : q ∗ ∈ X −→ ψv (q ∗ ) = Lν (v, q ∗ ) is concave. Moreover, if (uc , p∗c ) is a critical point of Lν , then p∗c is a critical point of ψuc , i.e., ψu c (p∗c ) = 0. Thus, by Proposition 4.45, ψuc (p∗c ) ≥ ψuc (q ∗ ), for all q ∗ ∈ A∗ . Consequently, Lν (uc , p∗c ) ≥ Lν (uc , q ∗ ) for all q ∗ ∈ A∗ .

(5.167)

From (5.166) and (5.167) we obtain easily results (i) and (ii) of the theorem. As a direct consequence we have the following corollary.

156


Corollary 5.84. Assume that the assumptions of Theorem 5.83 hold. If the quadratic gap function Gq is positive on Eq = U × A∗c , where

and

A∗c := {q ∗ ∈ A∗ | ∃uq ∈ K : (uq , q ∗ ) ∈ Ead }

(5.168)

Ead = {(v, q ∗ ) ∈ K × A∗ : Λ∗t (v)q ∗ + F (v) = 0},

(5.169)

then, all critical points of the Lagrangian Lν are saddle points of Lν . Similarly if Gq is negative on Eq then all critical points of the Lagrangian Lν are supercritical points of Lν . The following proposition gives some relations depending on the sign of Gq in the case where Ead is non-empty. Proposition 5.85. Assume that the assumptions of Theorem 5.83 hold. Let p∗ ∈ A∗c be given. Then there exists up ∈ K such that (up , p∗ ) is in Ead and we have the following properties if Gq (v, p∗ ) ≥ 0 ∀v ∈ U then Lν (up , p∗ ) = inf Lν (v, p∗ ), v∈K

if Gq (v, p∗ ) ≤ 0 ∀v ∈ U then Lν (up , p∗ ) = sup Lν (v, p∗ ).

(5.170)

v∈K

Proof. Let p∗ ∈ A∗c be given and let us introduce the following function: ψp∗ : v ∈ U −→ ψp∗ (v) = Lν (v, p∗ ). By hypothesis and the expression (5.153), there exists up ∈ K such that ψp ∗ (up ) =

∂Lν (up , p∗ ) = Λ∗t (up )p∗ + F (up ) = 0. ∂u

By the relations (5.164), we can deduce that (for all v ∈ K) ψp∗ (up +w) − ψp∗ (up ) = p∗ , Λ(up + w) − Λ(up )X + (F (up + w) − F (up )) = p∗ , Λt (up )w + Λ0 wX + F (up ), wU + F0 (w) = =

Λ∗t (up )p∗ Gq (w, p∗ )

(5.171)

∗

+ F (up ), wU + p , Λ0 wX + F0 (w) (since (up , p∗ ) ∈ Ead ),

where w = v − up . Thus, if Gq (z, p∗ ) ≥ 0 (respectively Gq (z, p∗ ) ≤ 0) ∀z ∈ U , then ψp∗ (v) ≥ ψp∗ (up ) (respectively ψp∗ (v) ≤ ψp∗ (up )) ∀v ∈ K. Consequently, the value up is a minimizer (respectively maximizer) of ψp∗ and inf Lν (v, p∗ ) = Lν (up , p∗ ) (respectively sup Lν (v, p∗ ) = Lν (up , p∗ )). v∈K

v∈K


157

Let A∗q ⊂ A∗ be the convex hull of the set A∗c . We introduce the so-called complementary energy Jν∗ : X −→ IR which is defined by, for p∗ ∈ A∗q , Jν∗ (p∗ ) := inf Lν (v, p∗ ) if Gq (z, p∗ ) ≥ 0 ∀z ∈ U, v∈K

Jν∗ (p∗ ) := sup Lν (v, p∗ ) if Gq (z, p∗ ) ≤ 0 ∀z ∈ U.

(5.172)

v∈K

Remark 5.86. If the function F is linear, we prove easily that A∗c is convex ♦ and then A∗q = A∗c . According to Proposition 5.85, it is interesting to study the following dual problems associated with problem (P) (given by (5.132)): find p∗ ∈ X such that Jν∗ (p∗ ) = sup Jν∗ (q ∗ ),

(5.173)

find p∗ ∈ X such that Jν∗ (p∗ ) = ∗inf Jν∗ (q ∗ ).

(5.174)

q∗ ∈X

and

q ∈X

We can now give the following lemma, which is a corollary of Propositions 5.30, 5.31 and 5.35. Lemma 5.87. Assume that the assumptions of Theorem 5.83 hold. Let (us , p∗s ) ∈ Ead be a saddle or a supercritical point of Lν and assume that Lν is partially G-differentiable at point (us , p∗s ). Then, if the functions Jν∗ and Jν are Gdifferentiable at us and p∗s respectively, then (Jν∗ ) (p∗s ) = 0, Jν (us ) = 0, Jν∗ (p∗s ) = Jν (us ) = Lν (us , p∗s ). Proof. Let (us , p∗s ) ∈ Ead . Then, for all v ∈ K we have Lν (v, p∗s ) − Lν (us , p∗s ) = Λ∗t (us )p∗s + F (us ), v − us U + Gq (v − us , p∗s ). Consequently, Lν (v, p∗s ) − Lν (us , p∗s ) = Gq (v − us , p∗s ) ∀v ∈ K.

(5.175)

Suppose that (us , p∗s ) is a saddle point of Lν . Then, because of the definition of saddle point and the relation (5.154), we have that Gq (w, p∗s ) ≥ 0 ∀w ∈ U (by (5.175) and (5.172)) and Jν (us ) = sup Lν (us , q ∗ ) q∗ ∈A∗

≤ Lν (us , p∗s ) ≤ inf Lν (v, p∗s ) = Jν∗ (p∗s ). v∈K

158


Moreover, according to (5.172), we have Jν (us ) − Jν∗ (p∗s ) = sup Lν (us , q ∗ ) − inf Lν (v, p∗s ) v∈K

q∗ ∈A∗

= sup sup (Lν (us , q ∗ ) − Lν (v, p∗s )) ≥ 0. q∗ ∈A∗ v∈K

Consequently,

Jν (us ) = Lν (us , p∗s ) = Jν∗ (p∗s ).

Because of Proposition 5.35, we can deduce that (Jν∗ ) (p∗s ) = 0 and Jν (us ) = 0. Suppose now that (us , p∗s ) is a supercritical point of Lν , then Lν (v, p∗s ) ≤ Lν (us , p∗s ) ≥ Lν (us , q ∗ ) ∀(v, q ∗ ) ∈ K × A∗ . Thus, because of the relation (5.154), we have that Gq (w, p∗s ) ≤ 0 ∀w ∈ U (by (5.175) and (5.172)) and Jν (us ) = sup Lν (us , q ∗ ) = Lν (us , p∗s ) = sup Lν (v, p∗s ) = Jν∗ (p∗s ). q∗ ∈A∗

Consequently,

v∈K

Jν (us ) = Lν (us , p∗s ) = Jν∗ (p∗s ).

Because of Proposition 5.35, we can deduce that (Jν∗ ) (p∗s ) = 0 and Jν (us ) = 0. This completes the proof. We end this section with the following result. Theorem 5.88. (Minimax duality theorem) Assume that the assumptions of Theorem 5.83 hold. Assume also that the set Ead is non-empty and the quadratic gap function Gq is positive on U × A∗q . Then a point (uc , p∗c ) ∈ Ead is a critical point of Lν if and only if Jν∗ (p∗c ) = sup Jν∗ (q ∗ ) = inf Jν (v) = Jν (uc ). q∗ ∈A∗ q

v∈K

(5.176)

Proof. Assume that the quadratic gap function Gq is positive on the space U ×A∗q then, because of Corollary 5.84, we have that all critical points (uc , p∗c ) of Lν are saddle points of Lν . Moreover, since the function ηv : q ∗ ∈ A∗q −→ ηv (q ∗ ) := Lν (v, q ∗ ) ∈ IR, for any v ∈ K, is concave, we can deduce that (since Sν is a convex function) Jν (v) = sup Lν (v, q ∗ ) = sup ηv (q ∗ ) = ηv (p∗c ) = Lν (v, p∗c ) ≥ Lν (uc , p∗c ). q∗ ∈A∗ q

q∗ ∈A∗ q

According to Lemma 5.87, we have that Lν (uc , p∗c ) = Jν (uc ) ≤ Jν (v) for all v ∈ K and then (5.177) Jν (uc ) = inf Jν (v). v∈K


159

Since Gq is a positive function on U × A∗q then, for any q ∗ ∈ A∗q (because of Proposition 5.85) Jν∗ (q ∗ ) = inf Lν (v, q ∗ ) = Lν (uc , q ∗ ) ≤ Lν (uc , p∗c ). v∈K

According to Lemma 5.87, we have that Lν (uc , p∗c ) = Jν∗ (p∗c ) ≤ Jν∗ (q ∗ ) for all q ∗ ∈ A∗q and then Jν∗ (p∗c ) = sup Jν∗ (q ∗ ). (5.178) q∗ ∈A∗ q

By the relations (5.177), (5.178) and according again to Lemma 5.87, we can deduce that inf Jν (v) = Jν (u) = Jν∗ (p∗c ) = sup Jν∗ (q ∗ ),

v∈K

q∗ ∈A∗ q

which gives the relations (5.176). Conversely, according to the expressions (5.154), (5.170) and since Ead is non-empty, we have that Jν (uc ) = sup Lν (uc , s∗ ) ≥ Lν (uc , q ∗ ) ∀q ∗ ∈ A∗q , s∗ ∈A∗ q

Jν∗ (p∗c )

= inf Lν (w, p∗c ) ≤ Lν (v, p∗c ) ∀v ∈ K. w∈K

Since the relations (5.176) hold then, for all (v, q ∗ ) in K × A∗q , α = Jν (uc ) = Jν∗ (p∗c ) ≤ Lν (uc , q ∗ ) and α ≥ Lν (v, p∗c ). Consequently, according to Proposition 5.53, we have that (uc , p∗c ) is a saddle point of Lν and then (because of Corollary 5.84) (uc , p∗c ) is a critical point. This completes the proof of the theorem.

Part II

General Results and Concepts on Robust and Optimal Control Theory for Evolutive Systems

6 Studied Systems and General Results

This chapter is devoted to general tools and basic results, on the existence, uniqueness and regularity of the solutions of linear time-dependent systems, which will be used frequently in later chapters. For the description of various function spaces including Sobolev spaces and different notations, the reader is referred to Chapter 3 for details. Our objective in this chapter is to recall the general framework and the basic results of existence of solutions for linear evolutive equations of the first order in time of the type ∂u + A(t)u = f (t). ∂t We consider some operator A (time dependent) and a right-hand side f which is allowed to depend on time. First we give some hypotheses and properties, and second we indicate in what sense such an equation is understood and give the existence, the uniqueness and the regularity of solutions.

6.1 Hypotheses and Properties Let V and H be two Hilbert spaces on IR. We denote by . (respectively | . |) the norm on V (respectively on H), and we denote by (( , )) (respectively ( , )) the inner (or scalar) product for V (respectively for H). We will add indices V or H (( , )V , . V , etc.) in the event of possible ambiguity. We suppose that V ⊂ H, with continuous embedding, (6.1) and V is dense in H. We identify the space H to its dual H , i.e., H ≡ H so that if V denotes the dual of V we have the following injections: V ⊂ H ⊂ V , with continuous and dense embedding.

(6.2)

164


We denote by t the time and suppose that t ∈ (0, T ), where T < +∞ is the final time. Let us now introduce the family of continuous bilinear forms on V by (u, v) ∈ V × V −→ a(t; u, v) ∈ IR, for any time t ∈ (0, T ).

(6.3)

We suppose that the forms a satisfy the following assumptions: for all u, v ∈ V, the function t ∈ (0, T ) −→ a(t; u, v) is measurable and there exists a constant M > 0 (independent of t, u, v) such that | a(t; u, v) |≤ M u v .

(6.4)

It results from these assumptions that the forms v ∈ V −→ a(t; u, v) (t ∈ (0, T ) and u ∈ V ) are linear and continuous on V and then there exists A(t) ∈ L(V, V ) (the space of linear and continuous mappings of V into V ) such that ∀(u, v) ∈ V × V, ∀t ∈ (0, T ),

a(t; u, v) = A(t)u, vV ,V sup A(t) L(V,V ) ≤ M,

(6.5)

t∈(0,T )

where the duality pairing , V ,V is the scalar product between V and V . We also suppose the following coercivity assumption: there exist constants λ ≥ 0, α > 0, such that a(t; u, u) + λ | u |2 ≥ α u 2 ∀u ∈ V, ∀t ∈ (0, T ).

(6.6)

Remark 6.1. If the operator A is independent of time t, we note its corresponding form a(t; u, v) by a(u, v). ♦ According to (6.5), the linear operator A defined by: if u ∈ L2 (0, T ; V ), A(.)u is the function t ∈ (0, T ) −→ A(t)u(t) ∈ V , is measurable and A ∈ L(L2 (0, T ; V ); L2 (0, T ; V )) and satisfies A(t)u(t) V ≤ M u(t) V . Moreover, for u ∈ L2 (0, T ; V ), we can define its time derivative ∂u/∂t. For this, we will need the following simple lemma. Lemma 6.2. Let X be a given Banach space with dual X and let u and g be two functions belonging to L1 (a, b; X). Then the following three conditions are equivalent: (i) u is almost everywhere (a.e.) equal to a primitive function of g, i.e., there exists ψ ∈ X such that t (6.7) u(t) = ψ + g(s)ds, for a.e. t ∈ [a, b]. 0

6.1 Hypotheses and Properties

165

(ii) For any test function φ ∈ D(]a, b[), we have

b

u(t)φ (t)dt = − a

b

g(t)φ(t)dt, (φ =

a

dφ ). dt

(6.8)

(iii) For each η ∈ X , we have d

u, η = g, η, in the scalar distribution sense on ]a, b[. dt

(6.9)

If one of the conditions (6.7)–(6.9) is satisfied, we say that the function g is the (X − valued) distribution derivative of u, and u is almost everywhere equal to a continuous function from [a, b] into X. Proof. For the proof see, e.g., Dautray and Lions [95] and Temam [282].

Lemma 6.2 gives a sense to Problem (6.11) if u is in L2 (0, T ; V ) (which will be the case below). Indeed, A being an isomorphism from V into V , Au ∈ L2 (0, T ; V ) by (6.5) and f ∈ L2 (0, T ; V ). Thus ∂u/∂t = f − Au in the distribution sense in V . In such a case, Lemma 6.2 also implies that u is almost everywhere equal to a continuous function from [0, T ] into V and u(0) = u0 makes sense too. As a direct consequence of Lemma 6.2, we have the following lemma. Lemma 6.3. Let V , H be two Hilbert spaces and V be the dual of V , such that Assumption (6.2) holds. If a function w belongs to L2 (0, T ; V ) and its derivative ∂w/∂t belongs to L2 (0, T ; V ), then w is almost everywhere equal to a continuous function from [0, T ] into H and we have the following equality which holds in the scalar distribution sense on (0, T ): ∂w d | w |2 = 2 , wV ,V . dt ∂t Corollary 6.4. Let V , H be two Hilbert spaces and V be the dual of V , such that Assumption (6.2) holds. If functions v and w belong to L2 (0, T ; V ) and their derivatives ∂v/∂t and ∂w/∂t belong to L2 (0, T ; V ), then the following equality holds in the scalar distribution sense on (0, T ):

∂w d ∂v , wV ,V + , vV ,V = (v, w). ∂t ∂t dt

Proof. According to assumptions, we can deduce that v − w belongs to L2 (0, T ; V ) and its derivative ∂(v − w)/∂t belongs to L2 (0, T ; V ). Then, because of Lemma 6.3, we have the following equalities for v, w and v − w:

166


∂v d | v |2 = 2 , vV ,V , dt ∂t 2 ∂w d|w| = 2 , wV ,V , dt ∂t ∂(v − w) d | v − w |2 = 2 , v − wV ,V . dt ∂t

(6.10)

By a simple manipulation, we can deduce easily the result of the corollary. We now give two other interesting lemmas in order to obtain the continuity result. Lemma 6.5. Let X, Y be two Banach spaces such that X ⊂ Y with a continuous embedding. If a function w belongs to L∞ (0, T ; X) and is weakly continuous with values in Y , then w is weakly continuous with values in X. Proof. For the proof see, e.g., Strauss [276].

The next lemma concerns the compact embedding results for time-dependent functions. Lemma 6.6. Let X0 , X and X1 be reflexive Banach spaces such that X0 ⊂ X ⊂ X1 , where the first embedding is compact and the second is continuous. Then, if T > 0 is finite, the following compact embeddings hold: ∂v ∈ Lp1 (0, T ; X1 )} ⊂ Lp0 (0, T ; X) ∂t ∂v ∈ Lp2 (0, T ; X1 )} ⊂ C([0, T ]; X) (ii) L∞ (0, T ; X0 ) ∩ {v : ∂t ∂v ∈ Lp0 (0, T ; X)} ⊂ C([0, T ]; X) (iii) Lp0 (0, T ; X) ∩ {v : ∂t for any 1 ≤ p0 , p1 ≤ ∞ and 1 < p2 ≤ ∞. (i) Lp0 (0, T ; X0) ∩ {v :

Proof. For the proof, we refer, e.g., to Lions [203] and Simon [271].

6.2 Evolution Problems, Existence and Stability Results We shall now study a linear evolutive problem. For this, let u0 ∈ H (for instance), f be a given function from [0, T ] in V satisfying, for example f ∈ L2 (0, T ; V ) and let a be a continuous bilinear form satisfying Assumptions

6.2 Evolution Problems, Existence and Stability Results

167

(6.4)-(6.6). We seek a function u from [0, T ] in V which is the solution to the initial-value problem ∂u(t) + A(t)u(t) = f (t) on (0, T ), ∂t u(0) = u0 .

(6.11)

Remark 6.7. If the final time T is finite, the hypothesis (6.6) can always be reduced to the case where λ = 0. Indeed, if we take w = exp(−λt)u, the problem (6.11) is equivalent to ∂w(t) ˜ + A(t)w(t) = g(t), w(0) = u0 , ∂t ˜ = A(t)+λI, for which its corresponding where g(t) = f (t)+exp(−λt) and A(t) bilinear form a, via (6.5), verifies Assumptions (6.3), (6.4) and (6.6) with λ = 0. ♦ In the sequel we suppose that the operator A(t) satisfies the hypothesis there exists a constant α > 0, such that

A(t)u, uV ,V = a(t; u, u) ≥ α u 2

∀u ∈ V, ∀t ∈ (0, T ).

(6.12)

We shall now give the existence and uniqueness results. Theorem 6.8. Suppose that the assumptions given in the previous section hold. Then, for u0 ∈ H and f ∈ L2 (0, T ; V ), there exists a unique solution u of problem (6.11) such that u ∈ L2 (0, T ; V ) ∩ C([0, T ]; H), ∂u ∈ L2 (0, T ; V ). ∂t

(6.13)

Moreover, the following estimates hold: u L2 (0,T ;V )∩L∞ (0,T ;H) ≤ C(| u0 | + f L2 (0,T ;V ) ),

∂u L2 (0,T ;V ) ≤ C(| u0 | + f L2 (0,T ;V ) ). ∂t

(6.14)

Proof. This result is proved, e.g., in Lions and Magenes [204]. We give only a sketch of the proof which emphasizes some points needed in the sequel. Nota bene: Throughout the present proof, we use C to denote a generic constant. The existence result is proved by using the Faedo–Galerkin method. For simplicity we suppose that the Hilbert space V is separable, then there exists (not in a unique manner) a linearly independent and total sequence w1 , w2 , . . . , wl , . . . , which is complete in V . For each l we denote by Vl the

168


space generated (or spanned) by the elements w1 , w2 , . . . , wl , i.e., the space Vl = span{w1 , w2 , . . . , wm }, and we define an approaching problem of (6.11) as follows: find ul (t) ∈ Vl such that (a.e. t ∈ (0, T )) d (ul (t), wj ) + a(t; ul (t), wj ) = f (t), wj V ,V , j = 1, . . . , l dt ul (0) = Pl u0 ,

(6.15)

where Pl is, for example, the H-projector onto Vl (the initial data u0 satisfies then Pl u0 H ≤ u0 H and Pl u0 −→ u0 strongly in H as l −→ ∞). Since the approximate solution ul is in Vl , we have that ul (t) = gil (t)wi , i=1,l

where gil , i = 1, . . . , l, are scalar functions on [0, T ]. The problem (6.15) is equivalent to an initial-value problem for a linear finite m-dimensional ordinary differential equation for the functions gil in the form: find Gl = (g1l , . . . , gll ) such that dGl + Al (t)Gl = Fl , dt Gl (0) = ηl , Ml

(6.16)

where Ml is the matrix [(wi , wj )]1≤i,j≤l , Al is the matrix [a(. ; wi , wj )]1≤i,j≤l , Fl is the vector [ f (.), wj V ,V ]1≤j≤l and ηl is the vector [ξil ]1≤i≤l when Pl u0 = i=1,l ξil wi . Clearly Ml is an invertible matrix (because (wj ) are linearly independent) then, by the theory of ordinary differential equations, Problem (6.16) has a unique solution Gl and consequently the problem (6.15) admits a unique solution ul ∈ C([0, T ]; Vl ) with dul /dt ∈ L2 (0, T ; Vl ). We will now show the convergence of the sequence ul as l −→ ∞, by using some a priori estimates and the weak compactness. Multiplying the system (6.15) by gjl and adding with respect to j, we obtain dul , ul V ,V + a(t; ul , ul ) = f (t), ul V ,V ,

dt and then (according to (6.12) and Lemma 6.3) d | ul |2 α + α ul 2 ≤ C f V ul ≤ C f 2V + ul 2 . 2dt 2 So,

d | ul |2 + α ul 2 ≤ C f 2V . dt

(6.17)

6.2 Evolution Problems, Existence and Stability Results

169

After integration with respect to time, we infer from (6.17) an a priori estimate (since Pl u0 H ≤ u0 H ): ul is uniformly bounded in L∞ (0, T ; H) ∩ L2 (0, T ; V ).

(6.18)

According to (6.18) and the relation (6.5), we can deduce that A(.)ul is uniformly bounded in L2 (0, T ; V ).

(6.19)

By weak compactness, we can extract from ul a subsequence, denoted also by ul such that ul u weakly in L2 (0, T ; V ), ul u weakly star in L∞ (0, T ; H), 2

(6.20)

A(.)ul A(.)u weakly in L (0, T ; V ). We are now going to prove that u is a solution of problem (6.11). In order to pass the limit in problem (6.15), let us consider a scalar function ϕ continuously differentiable on [0, T ], such that ϕ(T ) = 0. Multiplying the system (6.15) by ϕ and integrating with respect to time, we obtain (for i = 1, . . . , l) T T dϕ a(t; ul , ϕwi )dt )dt + − (ul , wi dt 0 0 (6.21) T =

f (t), ϕwi V ,V dt + (Pl u0 , ϕ(0)wi ). 0

According to (6.20) it is easy to pass the limit in (6.21) and we have that (since Pl u0 −→ u0 strongly in H as l −→ ∞) T T dϕ )dt + a(t; u, ϕwi )dt − (u, wi dt 0 0 (6.22) T

f (t), ϕwi V ,V dt + (u0 , ϕ(0)wi ). = 0

Taking now ϕ as a C ∞ function on (0, T ) with a compact support in Ω, then (6.22) gives (after adding) d (u, v) + a(t; u(t), v) = f (t), vV ,V , ∀l = 1, . . . , dt

∀v ∈ Vl

(6.23)

in the distribution sense in (0, T ). Since the terms of the system (6.23) depend continuously and linearly on the function v ∈ Vl , for all l = 1, . . ., then (6.23) remains valid for any function v ∈ V and then d (u, v) + a(t; u(t), v) = f (t), vV ,V , dt

(6.24)

170


in the distribution sense in (0, T ) and ∀v ∈ V . According to Lemma 6.2, we can deduce that ∂u + A(t)u = f ∂t

(6.25)

and that ∂u/∂t ∈ L2 (0, T ; V ). Moreover, u is almost everywhere equal to a continuous function from [0, T ] into V and the initial condition u(t = 0) = u0 follows by passing the limit in the relation ul = Pl u0 . We will now show that the solution u is unique and that u ∈ C([0, T ]; H). The continuity result is a direct consequence of Lemma 6.3. The proof of uniqueness follows easily also from Lemma 6.3. Indeed, let u and v be two solutions of Problem (6.11) satisfying the regularity (6.13). Then w = u − v satisfies the regularity (6.13) and is a solution of the following system: ∂w + A(t)w = 0, ∂t w(0) = 0.

(6.26)

Multiplying the first equality of (6.26) by w and using the result of Lemma 6.3, we obtain d | w |2 + 2a(t; w, w) = 0 dt and then (since a(t; w, w) ≥ 0) d | w |2 ≤ 0. dt By integrating with respect to time, we obtain | w(t) |2 ≤| w(0) |2 for all t ∈ (0, T ). According to the null initial condition of (6.26), we can deduce that w = 0 and then the uniqueness result. The estimates given in the theorem are a direct consequence of Proposition 6.9 (see below). We shall now give the Lipschitz continuity of the map solution. Proposition 6.9. Let (u0 , f ) and (v0 , g) be in H × L2 (0, T ; V ) and let u and v be the corresponding solutions of problem (6.11), respectively. Then u − v L2 (0,T ;V )∩L∞ (0,T ;H) ≤ C(| u0 − v0 | + f − g L2 (0,T ;V ) ),

∂(u − v) L2 (0,T ;V ) ≤ C(| u0 − v0 | + f − g L2 (0,T ;V ) ). ∂t

(6.27)

6.3 Regularity Results

171

Proof. Let w = u − v, w0 = u0 − v0 and h = f − g. Then w satisfies the regularity (6.13) and is a solution of the following system: ∂w + A(t)w = h, ∂t w(0) = w0 .

(6.28)

Multiplying the first equality of (6.28) by w and using the result of Lemma 6.3, we obtain d | w |2 + 2a(t; w, w) = 2 h, wV ,V . dt By using the same method to obtain (6.17), we have d | w |2 + α w 2 ≤ C h 2V . dt By integrating with respect to time, we can deduce that (according to the second equality of (6.28)) T T w 2 ds ≤ C h 2V ds+ | w0 |2 for t ∈ (0, T ) | w(t) |2 +α 0

0

and then the first result of (6.27). Taking now the scalar product of the first equality of (6.28) by z ∈ V , we have ∂w , zV ,V = −a(t; w, z) + h, zV ,V .

∂t According to Assumption (6.5), we can deduce that

∂w , zV ,V ≤ C( w + h V ) z ∂t

and then

∂w V ≤ C( w + h V ). ∂t Consequently, by integrating with respect to time and by using the first result of (6.27), we can deduce easily the second result of (6.27).

6.3 Regularity Results In this section, we shall give two regularity results for the solution u which will be used frequently in non-linear cases. The linear operator A(t) is an isomorphism from V onto V and from its domain D(A(t)) = dom(A(t)) ⊂ V onto H, where the domain D(A(t)) = {u ∈ V : the function v −→ a(t; u, v) is continuous on V for the topology of H}.

172


Theorem 6.10. Under the assumptions of Theorem 6.8, we suppose, furthermore, that the bilinear form a satisfies for every u, v in V , the function t ∈ [0, T ] −→ a(t; u, v) ∈ IR ∂ a(t; u, v), is absolutely continuous, its derivative ∂t for a.e. t ∈ [0, T ], is a bilinear continuous form on V , and there exists M1 > 0 such that for all u, v ∈ V and a.e. t ∈ [0, T ] |

(6.29)

∂ a(t; u, v) |≤ M1 u v , ∂t

and that the right-hand side f and the initial condition u0 satisfy ∂f ∈ L2 (0, T ; H), ∂t u0 ∈ D(A(0)). f,

(6.30)

Then the solution u of problem (6.11), satisfies the following regularity u(t) ∈ D(A(t)), ∀t ∈ [0, T ], t −→ A(t)u(t) is continuous from [0, T ] into H, ∂u ∈ L2 (0, T ; V ) ∩ C([0, T ]; H), ∂t ∂2u ∈ L2 (0, T ; V ). ∂t2

(6.31)

Moreover, if the operator A(t) is independent of time t, then the first result in (6.31) reduces to u ∈ C([0, T ]; D(A)). Proof. Introduce the following problem: ∂w , vV ,V + a(t; w, v) ∂t ∂ ∂f = ( , v) − a(t; u, v) ∀v ∈ V, a.e. t ∈ (0, T ), ∂t ∂t ∂u w(0) = (0) = f (0) − A(0)u0 . ∂t

(6.32)

Due to the first assumption of (6.30) and Lemma 6.2, the function f is in C([0, T ]; H) and f (0) ∈ H is well defined. Thus, with the second assumption of (6.30), we can deduce that f (0) − A(0)u0 ∈ H. Moreover, according to (6.29), we can write the right-hand side of (6.32) as g, vV ,V , with g ∈ L2 (0, T ; V ). According to Theorem 6.8, problem (6.32) admits a unique solution w such that ∂w ∈ L2 (0, T ; V ). w ∈ L2 (0, T ; V ) ∩ C([0, T ]; H), ∂t


Set

173

t

w(s)ds + u0 , ∀t ∈ [0, T ],

ω(t) = 0

then ω ∈ C 1 ([0, T ]; H) (since w ∈ C([0, T ]; H)). Integrating the first equality of (6.32) with respect to time gives (according to the second equality of (6.32)) ∂ω , vV ,V + a(t; ω, v) = (f, v) ∀v ∈ V, a.e. t ∈ (0, T ), ∂t ω(0) = u0 .

We have proved that ω is a solution of problem (6.11). Consequently, by the uniqueness result, we have u = ω. On the other hand, ∂u/∂t = ∂ω/∂t = w. Therefore, ∂u/∂t ∈ L2 (0, T ; V ) ∩ C([0, T ]; H) and ∂ 2 u/∂t2 ∈ L2 (0, T ; V ). Then, we have the second and the third results of (6.31). Thus, with the first equality of (6.11), we can deduce that A(t)u(t) = f −

∂u ∈ C([0, T ]; H) ∂t

and then the first result of (6.31). If now the operator A = A(t) is independent of t, and since A is an isomorphism from D(A) onto H, then u ∈ C([0, T ]; D(A)). The last regularity result is the following theorem. Theorem 6.11. Under the assumptions of Theorem 6.8, we suppose, furthermore, that the bilinear form a satisfies Assumptions (6.29) and a is symmetric, i.e., a(t; u, v) = a(t; v, u) the injection of V in H is compact

∀u, v ∈ V, ∀t ∈ [0, T ],

(6.33)

and that the right-hand side f and the initial condition u0 satisfy f ∈ L2 (0, T ; H), u0 ∈ V.

(6.34)

Then, the solution u of problem (6.11), satisfies the following regularity: A(.)u ∈ L2 (0, T ; H), u ∈ L∞ (0, T ; V ), ∂u ∈ L2 (0, T ; H). ∂t

(6.35)

Moreover, if the operator A(t) is independent of time t, then the first and second regularities given in (6.35) become u ∈ L2 (0, T ; D(A)) ∩ C([0, T ]; V ).

(6.36)

174


Remark 6.12. According to the first part of (6.33), the operator A is selfadjoint (from V onto V ) unbounded operator in H and its inverse A−1 is also self-adjoint in H. Moreover, because of the second part of (6.33), the operator A−1 can be considered as a self-adjoint compact operator in H and then according to Courant and Hilbert [89], there exist a sequence (λi (t))i≥1 of eigenvalues of the operator A(t) such that 0 < λ1 ≤ λ2 ≤ · · · ≤ λk ≤ · · · and the corresponding, smooth and complete orthonormal basis in H and orthogonal in V , eigenfunctions (wi )i≥1 , i.e., A(t)wi = λi wi in V and wi ∈ V

(for all i ≥ 1),

(wi , wj ) = δij = the Kronecker symbol (for all i, j ≥ 1), a(t; wi , wj ) = A(t)wi , wj = λi δij

(for all i, j ≥ 1).

(6.37) ♦

Proof of Theorem 6.11. The proof of the theorem is obtained by implementing the Faedo–Galerkin method, with a particular choice of elements wj . Taking advantage of the properties (6.33), we consider the sequence (λi , wi )i≥1 of eigenvalues–eigenfunctions of the operator A(t) such that (6.37), and we use these eigenfunctions to implement the Faedo–Galerkin method. We denote by Vl the space generated by (wi )1≤i≤l (∪l≥1 Vl is dense in H and also in V ), and we introduce the H-orthogonal (and also V -orthogonal) projector Pl on the space Vl . Remark 6.13. For all v ∈ H we have Pl v H ≤ C v H . Moreover, if v ∈ V ♦ we have Pl v ≤ C v (the positive constant C is independent of l). For all l ≥ 1, we consider ul (., t) = i=1,l gil (t)wi an approximation solution of problem (6.11) with the initial condition u0 as follows: find ul (t) ∈ Vl such that (a.e. t ∈ (0, T )) ∂ul , wj ) + a(t; ul (t), wj ) = (f (t), wj ), j = 1, . . . , l, ( ∂t ul (0) = Pl u0 .

(6.38)

Multiplying the first equality of (6.38) by λj gjl and adding these equalities for j = 1, . . . , l, we have that (according to (6.37)) (

∂ul , A(t)ul )+ | A(t)ul |2 = (f (t), A(t)ul (t)). ∂t

Since (d/dt)(a(t; ul , ul )) = (∂/∂t)a(t; ul , ul ) + 2a(t; ∂ul /∂t, ul ) (since a is symmetric) and (∂ul /∂t, A(t)ul ) = a(t; ∂ul /∂t, ul ), then ∂ d (a(t; ul , ul )) + 2 | A(t)ul |2 = 2(f (t), A(t)ul (t)) + a(t; ul , ul ). dt ∂t According to (6.29) and Young’s inequality, we have that d (a(t; ul , ul ))+ | A(t)ul |2 ≤ M1 ul 2 + | f |2 . dt

(6.39)


175

This implies (by integration with respect to time) t a(t; ul (t), ul (t)) + | A(s)ul (s) |2 ds ≤ C( ul 2L2 (0,T ;V ) + f 2L2 (0,T ;H) ) 0

+a(0; Pl u0 , Pl u0 ).

Since a(0; Pl u0 , Pl u0 ) ≤ M Pl u0 2 ≤ M u0 2 and according to (6.18), we can deduce that ul is uniformly bounded in L∞ (0, T ; V ), A(.)ul is uniformly bounded in L2 (0, T ; H).

(6.40)

By passing the limit, we can deduce that the limiting solution u belongs to L∞ (0, T ; V ) such that A(.)u belongs to L2 (0, T ; H). and adding these equalities Multiplying now the first equality of (6.38) by gjl for j = 1, . . . , l, we have that (according to (6.37)) |

∂ul ∂ul ∂ul 2 | = −a(t; ul , ) + (f (t), ) ∂t ∂t ∂t

and then (since a(t; ul , ∂ul /∂t) = (A(t)ul , ∂ul /∂t)) |

∂ul 2 | ≤ C(| A(t)ul |2 + | f |2 ). ∂t

By integrating with respect to time, according to (6.40) and passing the limit, we can deduce that ∂u/∂t ∈ L2 (0, T ; H). If now, the operator A is independent of time, then according to (6.40) and by passing the limit, we have that u ∈ L∞ (0, T ; V ) ∩ L2 (0, T ; D(A)). Prove now that u ∈ C([0, T ]; V ). From Lemma 6.5, we can deduce that t −→ (u(t), v) is continuous on [0, T ], for all v ∈ V , and t −→ a(u(t), v) is continuous on [0, T ], for all v ∈ V .

(6.41)

Moreover, we have for u a similar result to (6.39) (by using the same technique): d (a(u, u)) + 2 | Au |2 = 2(f, Au). dt Consequently, because of (6.41), we can deduce that t −→ a(u(t), u(t)) is a continuous function on [0, T ] and then u ∈ C([0, T ]; V ). This completes the proof. We shall now give the Lipschitz continuity of the map solution. Proposition 6.14. Under the assumptions of Theorem 6.11, we suppose, furthermore, that the bilinear form a is time-independent. Let (u0 , f ) and (v0 , g) be in V × L2 (0, T ; H) and let u and v in L2 (0, T ; D(A)) ∩ H 1 (0, T ; H) ∩ C([0, T ]; V ) be the corresponding solution of the problem (6.11), respectively.

176


Then w L2 (0,T ;D(A))∩L∞ (0,T ;V ) ≤ C( w0 + h L2 (0,T ;H) ), ∂w ≤ C( w0 + h L2 (0,T ;H) ), 2 ∂t L (0,T ;H) where w = u − v, w0 = u0 − v0 and h = f − g.

(6.42)

Proof. Let w = u − v, w0 = u0 − v0 and h = f − g. Then w satisfies the regularity w ∈ L2 (0, T ; D(A)) ∩ H 1 (0, T ; H) ∩ C([0, T ]; V ) and is a solution of the following system: ∂w + Aw = h, ∂t (6.43) w(0) = w0 . Multiplying the first equality of (6.43) by Aw and using the same method to obtain (6.39), we obtain d (a(w, w)) + 2 | Aw |2 = 2(h, Aw). dt This implies d (a(w, w))+ | Aw |2 ≤| h |2 . dt After integration with respect to time we obtain that, for all t ∈ (0, T ), t t 2 a(w(t), w(t)) + | Aw(s) | ds ≤ | h(s) |2 ds + a(w0 , w0 ). 0

0

By using the continuity and the coercivity of the bilinear form a, we can deduce that t t 2 2 | h(s) |2 ds + C w0 2 for all t ∈ (0, T ). α w(t) + | Aw(s) | ds ≤ 0

0

Consequently, w 2L2 (0,T ;D(A))∩L∞ (0,T ;V ) ≤ C( h 2L2 (0,T ;H) + w0 2 ).

(6.44)

Then the first result of (6.42) follows. Since w is a solution of (6.43) then ∂w = −Aw + h. ∂t Consequently (since Aw and h are in L2 (0, T ; H)), ∂w 2 | ≤ C(| Aw(t) |2 + | h(t) |2 ) ∂t and then (according to (6.44)) |

∂w 2 2 ≤ C( h 2L2 (0,T ;H) + w0 2 ). ∂t L (0,T ;H) Then, we have the second result of (6.42). This completes the proof.

(6.45)

6.4 Examples of Operators and Spaces

177

6.4 Examples of Operators and Spaces Let Ω be an open and bounded domain in IRn , n ≥ 1, with a smooth boundary Γ = ∂Ω with the regularity (3.1) or (3.2), and T > 0 be a fixed constant (a given final time). We take for V a closed subspace of H 1 (Ω) such that H01 (Ω) ⊂ V ⊂ H 1 (Ω) (with continuous injection) and H = L2 (Ω). We denote by Q the cylinder Ω × (0, T ) and by Σ the lateral (or side) boundary of Q, i.e., Γ × (0, T ). The generic vector point of IRn is denoted by x = (x1 , x2 , . . . , xn ) (or y = (y1 , y2 , . . . , yn ), etc.) and the Lebesgue measure on IRn is denoted by dx = dx1 dx2 · · · dxn . Let aij : Q −→ IR, for i, j = 1, . . . , n and a0 : Q −→ IR be functions in L∞ (Q) such that the functions aij satisfy the ellipticity conditions: there exists μ > 0 such that aij (x, t)ξi ξj ≥ μ ξk2 ∀ξ = (ξ)i=1,n ∈ IRn a.e. in Q (6.46) i,j=1,n

k=1,n

and a0 satisfies the pointwise constraint: there exist λi ∈ IR, for i = 1, 2 such that (6.47) λ1 ≤ a0 ≤ λ2 a.e. in Q. We introduce the bilinear form a1 by ∂u ∂v a1 (t; u, v) = aij (x, t) dx + a0 (x, t)uvdx, ∂xi ∂xj Ω i,j=1,n Ω

(6.48)

for all u and v in H 1 (Ω). According to the boundedness of aij and a0 , there exists M > 0 such that, a.e. t ∈ (0, T ), | a1 (t; u, v) |≤ M u v

for all u, v in V

and by the condition (6.46) we have, for λ ≥ 0 sufficiently large and a.e. t ∈ (0, T ), that a1 (t; u, u) + λ | u |2 ≥ ν u 2

for all u in H 1 (Ω), with ν > 0.

We will now present three simple examples, which will enable us to specify the functional spaces in which we work. 6.4.1 Dirichlet Boundary Condition We take V = H01 (Ω),1 H = L2 (Ω), V = H −1 (Ω), and we denote 1

The norm in H01 (Ω) is equivalent to the seminorm.

178


W = {w ∈ L2 (0, T ; V ) :

∂w ∈ L2 (0, T ; V )}. ∂t

Assume that the function a0 is positive,2 then the bilinear form a1 satisfies the assumptions of Theorem 6.8 (i.e., (6.3), (6.4) and (6.6)) and so there exists a unique solution u ∈ W ∩ C([0, T ]; H) of the problem ∂u + A1 (.)u = f on Q, ∂t subject to the homogeneous Dirichlet boundary condition u = 0 on Σ, with the initial condition u(0) = u0 on Ω, where A(t)u = −

(6.49)

∂ ∂u (aij (x, t) ), ∂x ∂x i j i,j=1,n

(6.50)

A1 (t)u = A(t)u + a0 (x, t)u

and f ∈ L (0, T ; V ) and u0 ∈ H. 2

Remark 6.15. (i) The homogeneous Dirichlet boundary condition given in (6.49) is included in the space of solution W, precisely in space L2 (0, T ; V ). (ii) for a given t ∈ (0, T ), v ∈ D(A(t)) is equivalent to v ∈ V = H01 (Ω) and ♦ A(t)v ∈ H = L2 (Ω). 6.4.2 Neumann Boundary Condition We take V = H 1 (Ω), H = L2 (Ω). So the space V = (H 1 (Ω)) is not a distribution space on Ω and then the space L2 (0, T ; V ) is not also a distribution space on Q. We denote W = {w ∈ L2 (0, T ; V ) :

∂w ∈ L2 (0, T ; V )}. ∂t

Then the bilinear form a1 satisfies the assumptions of Theorem 6.8 (i.e., (6.3), (6.4) and (6.6)) and so, “formally,” Theorem 6.8 gives the solution of the following problem ∂u + A1 (.)u = f on Q, ∂t subject to the homogeneous Neumann boundary condition ∂u = 0 on Σ, ∂ηA with the initial condition u(0) = u0 on Ω, 2

(6.51)

If a0 is negative, we put the term a0 u in the right-hand side of (6.49) and we can obtain similar results.


where A(t)u = −

179

∂ ∂u (aij (x, t) ), ∂xi ∂xj i,j=1,n

(6.52)

A1 (t)u = A(t)u + a0 (x, t)u, and the normal derivative ∂u/∂ηA at Γ , directed towards the exterior of Ω, is given by (3.22), f ∈ L2 (0, T ; V ) and u0 ∈ H. More precisely, Theorem 6.8 proves the existence and the uniqueness of the solution u ∈ W ∩ C([0, T ]; H) of the problem (the weak formulation of (6.51)) ∂u , vV ,V + a1 (.; u, v) = f, vV ,V ∂t with the initial condition u(0) = u0 on Ω.

a.e. in (0, T ), ∀v ∈ V, (6.53)

Remark 6.16. (i) The homogeneous Neumann boundary condition given in (6.51) is included (by using the Green’s formula) in the formulation (6.53). (ii) for a given t ∈ (0, T ), u ∈ D(A1 (t)) is equivalent to u ∈ V = H 1 (Ω), A1 (t)u ∈ H = L2 (Ω) and

A1 (t)uvdx = a1 (t; u, v) for all v ∈ V .

♦

Ω

6.4.3 Robin Boundary Condition We take V = H 1 (Ω), H = L2 (Ω) and V = (H 1 (Ω)) . We denote by W = {w ∈ L2 (0, T ; V ) :

∂w ∈ L2 (0, T ; V )}. ∂t

Then the bilinear form a satisfies the assumptions of Theorem 6.8 (i.e., (6.3), (6.4) and (6.6)) and so, “formally,” Theorem 6.8 gives the solution of the following problem ∂u + A1 (.)u = f on Q, ∂t subject to the homogeneous Robin boundary condition ∂u + αu = 0 on Σ, ∂ηA with the initial condition u(0) = u0 on Ω, where A(t)u = −

(6.54)

∂u ∂ (aij (x, t) ), ∂x ∂x i j i,j=1,n

A1 (t)u = A(t)u + a0 (x, t)u,

(6.55)

180


α is a positive constant, f ∈ L2 (0, T ; V ) and u0 ∈ H. More precisely, Theorem 6.8 proves the existence and the uniqueness of the solution u ∈ W ∩ C([0, T ]; H) of the problem (the weak formulation of (6.54)) ∂u , vV ,V + a ˜(.; u, v) = f, vV ,V ∂t with the initial condition u(0) = u0 on Ω, where a ˜(t; u, v) = a1 (t; u, v) + α uvdΓ.

a.e. in (0, T ), ∀v ∈ V, (6.56)

Γ

Remark 6.17. (i) The homogeneous Robin boundary condition given in (6.54) is included (by using the Green’s formula) in the formulation (6.56). (ii) for a given t ∈ (0, T ), u ∈ D(A(t)) is equivalent to u ∈ V = H 1 (Ω), A1 (t)u ∈ H = L2 (Ω) and

A1 (t)uvdx = a ˜(t; u, v) for all v ∈ V .

♦

Ω

6.4.4 Non-homogeneous Neumann and Dirichlet Boundary Conditions Suppose now that the domain Ω is bounded, C ∞ -manifold of dimension (n−1) and locally totally one side of the boundary Γ and that the coefficients aij for i, j = 1, n are sufficiently regular functions (for example are C ∞ functions on Q, the closure of Q) and a0 is in L∞ (Q). Let us consider the following problem ∂u + A(.)u + a0 u = f on Q, ∂t with the initial condition u(0) = u0 on Ω,

(6.57)

subject to one of the following boundary conditions: ∂u = g on Σ, ∂ηA

(6.58)

u = uB on Σ,

(6.59)

or where A(t)u = −

i,j=1,n

∂ ∂u (aij (x, t) ) ∂xi ∂xj

(6.60)

and the normal derivative ∂u/∂ηA at Γ , directed towards the exterior of Ω, is given by (3.22).


181

We shall formulate sufficient conditions for the existence of a unique solution of the mixed initial-boundary value problem (6.57) with boundary conditions (6.58) or (6.59). For this purpose, for any positive real numbers r and s, we introduce the Sobolev space H r,s (Q) by H r,s (Q) = L2 (0, T ; H r (Ω)) ∩ H s (0, T ; L2(Ω)), which is a Hilbert space equipped with the norm T u(t) 2H r (Ω) dt+ u 2H s (0,T ;L2 (Ω)) )1/2 u H r,s (Q) := (

(6.61)

(6.62)

0

where H s (0, T ; L2 (Ω)) = [H m (0, T ; L2(Ω)), L2 (0, T ; L2 (Ω))]θ (interpolation between H m (0, T ; L2 (Ω)) and L2 (0, T ; L2(Ω))), 0 < θ < 1, m is an integer such that s < m and (1 − θ)m = s, and H m (0, T ; L2(Ω)) := {u ∈ L2 (Q) :

∂ku ∈ L2 (Q), for all k = 1, . . . , m}. ∂tk

In the spaces H r,s (Q), we have the following trace theorem (for the proof see, for instance, Lions and Magenes [204]) Theorem 6.18. For u ∈ H r,s (Q) with r > 1/2 and s ≥ 0, we may define ∂j u j ∂ηA ∂j u j ∂ηA

on Σ

for j < r − 1/2 and j ∈ IN, (6.63)

∈ H rj ,sj (Σ),

where rj /r = sj /s = (r − j − 1/2)/r if s > 0 and sj = 0 if s = 0. Then j is a continuous linear mapping from H r,s (Q) into H rj ,sj (Σ). u −→ ∂ j u/∂ηA We can now give the results of existence and uniqueness of the solution to Problem (6.57). First, we give the following theorem concerning the existence of a unique solution in the case of non-homogeneous Neumann boundary conditions (6.58) (which can be found in Lions and Magenes [204]). Theorem 6.19. Let u0 , g and f be given: (i) If u0 ∈ H 1 (Ω), g ∈ H 1/2,1/4 (Σ) and f ∈ L2 (Q) then there exists a unique solution u ∈ H 2,1 (Q) for problems (6.57) and (6.58). (ii) If u0 ∈ H 1/2 (Ω), g ∈ L2 (Σ) and f ∈ H −1/2,−1/4 (Q) then there exists a unique solution u ∈ H 3/2,3/4 (Q) for problems (6.57) and (6.58). We may also formulate the theorem about the existence of a unique solution in the case of non-homogeneous Dirichlet boundary conditions (which can be found also in Lions and Magenes [204]).

182


Theorem 6.20. Let u0 , uB and f be given: (i) If u0 ∈ H 1 (Ω), uB ∈ H 3/2,3/4 (Σ) and f ∈ L2 (Q) and the compatibility condition u(t = 0) = uB (t = 0) on Γ holds, then there exists a unique solution u ∈ H 2,1 (Q) for problems (6.57) and (6.59). (ii) If u0 ∈ H −1/2 (Ω), uB ∈ L2 (Σ) and f ∈ H −3/2,−3/4 (Q) then there exists a unique solution u ∈ H 1/2,1/4 (Q) for problems (6.57) and (6.59). Remark 6.21. By similar technique as used in the proof of Propositions 6.9 and 6.14, we can obtain the uniform Lipschitz continuity of the map solution in the different situations enunciated in Theorems 6.19 and 6.20. ♦

7 Optimal Control Problems

The objective of this chapter is to describe optimal control theory in different situations. First we use a very basic problem in order to explain the theory as simply as possible, then we present the theory for general linear evolutive problems and for some classes of non-linear evolutive problems. In each section, we study the existence, the uniqueness and the optimality conditions for the optimal solution. We develop, our study for different realistic cases of observations and controls. The numerical aspect will be presented in a later chapter.

7.1 Introduction The theory of the optimal control of partial differential equations (PDEs) is based on the minimization (or maximization) of a calculus function, depending on the solution of the PDE, called the cost (or objective) functional. The control problem can be formulated, for example, like the minimization of the variation between the experimental observations and the corresponding quantities calculated by resolution of the system of equations. The control variables are the parameters or the functions to be identified (or estimate). They can intervene in the initial conditions, or in the boundary conditions or in the equation itself. The essential data used in optimal control problems are the following: 1. A “control” ϕ in a set Uad (known as set of “admissible controls”). 2. The state u(ϕ) of the system to be controlled, which is given, for a chosen control ϕ, by the resolution of the problem: F (t)(u(ϕ)) = “given function of ϕ”,

(7.1)

where F (.) is an (known) operator which represents the system to be controlled (F (.) is the model of the studied system). In this context problem (7.1) is called primal or direct problem.

184


3. An “observation” uobs which is supposed to be known exactly (for example given by measurements). 4. A “cost” functional J(ϕ) which is defined from a real-valued and positive function G(ϕ, ψ) by J(ϕ) = G(ϕ, u(ϕ)). We want to find the optimal control, i.e., the infimum of the functional J: inf J(ϕ).

ϕ∈Uad

This infimum is said to be the optimal control or optimal solution. We lay stress upon the fact that there is no general method to analyze the problems of control (it is necessary to adapt it to each situation). On the other hand, we can define a process to be followed for each problem: (i)

solve the direct problem (analysis of PDE, existence of solutions, stability according to the data, regularity, differentiability of the operators, etc.)

(ii) define the function or the parameter to be controlled (iii) define the cost functional (iv) obtain and analyse the necessary (and if possible the sufficient) conditions of optimality (v)

characterize the optimal solutions

(vi) define an algorithm allowing to solve numerically the control problem. We shall now present a basic framework for an optimal control problem.

7.2 Basic Framework In this section, we take an abstract boundary value problem. We give the space of controls, the admissibility set of controls, the control variable himself and the observation operator. The optimal control we consider is to maintain target state variables. We define the cost functional and the adjoint problem, corresponding to the primal problem. The main result of this study is the characterization of optimal solutions. Let F be a continuous linear partial differential operator from X into E, where X and E are, for example, Hilbert spaces. The space X contains in its definition some appropriate boundary conditions. We assume that the corresponding boundary value problem is well-posed (or correctly-set) in a Hadamard sense, i.e., F is an isomorphism from X onto E. Let U be the space of controls, which is assumed to be, for example, a Hilbert space. Let also Uad (the admissibility set of controls) be a closed convex non-empty subset of U and B be a linear and continuous operator (i.e., B ∈ L(U ; E)).

7.2 Basic Framework

185

For every ϕ ∈ U and f ∈ E, we consider the abstract boundary value problem by F (u) = f + Bϕ. (7.2) The problem (7.2) admits a unique solution u ∈ X such that u = u(ϕ) = F −1 (f + Bϕ).

(7.3)

Let now C ∈ L(X; M ) (where M is, for example, a Hilbert space), be the observation operator and we take ϕ as the control. The optimal control we consider, for example, is to maintain the target state variables while the desired power level and adjustment costs are taken into consideration. We will study the following optimal control problem. find (u, ϕ) ∈ X × U such that the following cost (or objective) functional, in the reduced form J(ϕ) =

1 1 Cu(ϕ) − uobs 2M + (N ϕ, ϕ)U . 2 2

(7.4)

is minimized subject to the problem (7.2), with uobs ∈ M is the target (the observation given, for example, by experiment measurements) and N ∈ L(U ; U ) is a symmetric positive definite operator, such that there exists a constant β > 0, (N ψ, ψ)U ≥ β ψ 2U , ∀ψ ∈ U. Remark 7.1. The cost functional J depends on the control ϕ and the state function u. In order to simplify the presentation we have used, in the expression (7.4), the reduced form of the functional, i.e., J(ϕ) in place of the classical form J(ϕ, u(ϕ)). ♦ Since v −→ Cv − uobs is a continuous affine function from X into M , the norm is a continuous function and (ϕ, ψ) −→ (N ϕ, ψ)U is a bilinear form and continuous on U , then the functional J is continuous. Moreover, according to the nature of J, the functional J is convex and even strictly convex, and we have J(ϕ) −→ +∞ if ϕ ∈ Uad , ϕ U −→ ∞. Consequently, there exists a unique optimal solution (u, ϕ) ∈ X × Uad such that ϕ is a solution of (see Proposition 4.92) (7.5) J(ϕ) = inf J(ψ) ψ∈Uad

and u = u(ϕ) is the solution of (7.2), corresponding to ϕ. It follows directly that the functional J is differentiable, and then the optimal control ϕ satisfies the following inequality (see Proposition 4.93) J (ϕ).(ψ − ϕ) ≥ 0

∀ψ ∈ Uad .

Since u(ϕ) satisfies (7.3) then, by differentiation, we have that u (ϕ).θ = F −1 (Bθ) ∀θ ∈ U,

(7.6)

186


and, in particular u (ϕ).(ψ − ϕ) = F −1 (B(ψ − ϕ)) = u(ψ) − u(ϕ) ∀ψ ∈ Uad .

(7.7)

According to (7.7), the condition (7.6) can be written as (Cu(ϕ) − uobs , C(u(ψ) − u(ϕ)))M + (N ϕ, ψ − ϕ)U ≥ 0 ∀ψ ∈ Uad .

(7.8)

In order to simplify the relation (7.8), we introduce the adjoint problem F ∗ (˜ u) = C ∗ Λ(Cu − uobs ),

(7.9)

where u ˜ is the adjoint state corresponding to the primal state u, C ∗ ∈ L(M , X ) is the adjoint of the operator C, F ∗ is the adjoint of the operator solution F and Λ is the canonical isomorphism from M onto M (such that p, q ∗ M,M = (Λp, q ∗ )M = (p, Λ−1 q ∗ )M , ∀(p, q ∗ ) ∈ M × M ), where M (respectively X ) is the dual of M (respectively of X). Then, (Cu(ϕ) − uobs , C(u(ψ) − u(ϕ)))M = C ∗ Λ(Cu(ϕ) − uobs ), u(ψ) − u(ϕ)X ,X = F ∗ (˜ u(ϕ)), u(ψ) − u(ϕ)X ,X = ˜ u(ϕ), F (u(ψ) − u(ϕ))E ,E = ˜ u(ϕ), B(ψ − ϕ)E ,E ∗ = (Λ−1 ˜(ϕ), ψ − ϕ)U , ∀ψ ∈ Uad , U B u

where B ∗ ∈ L(E , U ) is the adjoint of B, U (respectively E ) is the dual of U (respectively of E), ΛU is the canonical isomorphism from U onto U and u ˜(ϕ) is the solution of the adjoint problem (7.9) corresponding to the primal solution u(ϕ). Thus, the condition (7.8) can be written as ∗ ˜(ϕ) + N ϕ, ψ − ϕ)U ≥ 0, ∀ψ ∈ Uad . (Λ−1 U B u

Remark 7.2. Since the operator F ∗ is an isomorphism then Problem (7.9) admits a unique solution. ♦ We have proved that the optimal control ϕ ∈ Uad (the solution of the problem (7.5)) is characterized by the following optimality system which include the direct (or primal) problem and the adjoint problem, linked by an inequality F (u(ϕ)) = f + Bϕ, u(ϕ)) = C ∗ Λ(Cu(ϕ) − uobs ), F ∗ (˜ ∗ ˜(ϕ) (Λ−1 U B u

(7.10)

+ N ϕ, ψ − ϕ)U ≥ 0, ∀ψ ∈ Uad .

It is clear that the analysis of control problems depends naturally on the nature of the operator F . In the next study we consider problems described by partial differential operators. More precisely, we study systems whose state u(ϕ) is given by the resolution of linear evolutive partial differential equations to which we add the boundary conditions and the initial conditions.

7.3 Linear Control Problems

187

7.3 Linear Control Problems In this section, we consider linear control problems subject to parabolic type systems in a more general framework. The notations, the assumptions and the spaces are the same as in Chapter 6. 7.3.1 Position of the Problem, Existence and Uniqueness of the Optimal Solution Let U be the space of controls, which is assumed to be a Hilbert space, and let Uad (the admissibility set of controls) be a closed convex non-empty subset of U . Let also B be a linear continuous operator such that B ∈ L(U ; L2 (0, T ; V )). For f and u0 be given such that f ∈ L2 (0, T ; V ) and u0 ∈ H, and ϕ given in U , we consider the following linear evolutive equation (a.e. t ∈ (0, T )): ∂u(t) + A(t)u(t) = f + Bϕ (∈ L2 (0, T ; V )), ∂t u(0) = u0 (∈ H).

(7.11)

According to Theorem 6.8, the problem (7.11) admits a unique solution u ∈ L2 (0, T ; V ) ∩ C([0, T ]; H) such that ∂u/∂t ∈ L2 (0, T ; V ), for every f ∈ L2 (0, T ; V ), u0 ∈ H and ϕ ∈ U . Let us now introduce the space W = {u ∈ L2 (0, T ; V ) :

∂u ∈ L2 (0, T ; V )}, ∂t

and consider the observation operator C ∈ L(W; M ), where M is a Hilbert space. The optimal control that we consider is to find (u, ϕ) ∈ W × U such that the following cost (or objective) functional, in the reduced form J(ϕ) =

1 1 Cu(ϕ) − uobs 2M + (N ϕ, ϕ)U , 2 2

(7.12)

is minimized subject to problem (7.11), where uobs ∈ M is the target and N ∈ L(U ; U ) is a symmetric positive definite operator such that there exists a constant β > 0 such that (N ψ, ψ)U ≥ β ψ 2U , ∀ψ ∈ U . Since v −→ Cv − uobs is a continuous affine function from X into M , the norm is a continuous function and (ϕ, ψ) −→ (N ϕ, ψ)U is a bilinear form and continuous on U , then the functional J is continuous. Moreover, according to the nature of J, the functional J is strictly convex, and J(ϕ) −→ +∞ if ϕ ∈ Uad , ϕ U −→ ∞. Consequently, there exists a unique optimal solution (u, ϕ) ∈ W × Uad such that ϕ is a solution of J(ϕ) = inf J(ψ) ψ∈Uad

and u = u(ϕ) is the solution of (7.11), corresponding to ϕ.

(7.13)

188


7.3.2 Optimality Conditions and Identification of the Gradients As in the general framework (see Section 7.2), the functional J is differentiable and the control ϕ ∈ Uad is optimal if and only if J (ϕ).(ψ − ϕ) ≥ 0

∀ψ ∈ Uad ,

i.e., (Cu(ϕ) − uobs , C(u(ψ) − u(ϕ)))M + (N ϕ, ψ − ϕ)U ≥ 0 ∀ψ ∈ Uad . Otherwise for all ψ ∈ Uad , we have (C ∗ Λ(Cu(ϕ) − uobs ), u(ψ) − u(ϕ))W ,W + (N ϕ, ψ − ϕ)U ≥ 0,

(7.14)

where Λ is the canonical isomorphism from M onto M and C ∗ ∈ L(M , W ) is the adjoint of C. In order to simplify the relation (7.14), we introduce the adjoint problem associated with the primal problem (7.11). For this, we consider the following two situations: (i) the case of C ∈ L(L2 (0, T ; V ); M ) ⊂ L(W; M ) (ii) the case of the final observation: Cu = Du(t = T ), where D ∈ L(H; H) and M = H. Remark 7.3. It is clear that the operator C can be the linear combination of operators which correspond to (i) and (ii). ♦ The Case (i) and the Optimality System In this case, the adjoint of C, C ∗ is in L(M , L2 (0, T ; V )) and the inequality (7.14) becomes, for all ψ ∈ Uad

T

C ∗ Λ(Cu(ϕ) − uobs ), u(ψ) − u(ϕ)V ,V dt + (N ϕ, ψ − ϕ)U ≥ 0.

(7.15)

0

Let us introduce the following adjoint problem: ∂u ˜ u = C ∗ Λ(Cu − uobs ) ∈ L2 (0, T ; V ), + A∗ (t)˜ ∂t u ˜(t = T ) = 0 ∈ H, −

(7.16)

which admits a unique solution u ˜ ∈ W with A∗ the adjoint of A. To prove this result, we change the variables of System (7.16) by reversing the sense of time, i.e., t := T − t and we apply the same way to obtain the result of Theorem 6.8. Indeed, by setting w(., t) := u(., T −t), wobs (., t) := uobs (., T −t), w(., ˜ t) := u ˜(., T − t) and A˜∗ (t) := A∗ (T − t), problem (7.16) can be written as ∂w ˜ + A˜∗ (t)w˜ = C ∗ Λ(Cw − wobs ) ∈ L2 (0, T ; V ), ∂t w(0) ˜ = 0.

(7.17)


189

Since the operator A˜∗ satisfies the same assumptions as the operator A then problem (7.17) is similar to problem (7.11). Consequently, by Theorem 6.8, we have the existence and the uniqueness of the solution w ˜ in W and then the existence and the uniqueness of the solution u ˜ in W. Next we give the optimality conditions. Multiplying the first equation of (7.16) by u(ϕ) − u(ψ) and integrating with respect to time we have that T ∂u ˜(ϕ) , u(ϕ) − u(ψ)V ,V dt − ∂t 0 T (7.18) u(ϕ), u(ϕ) − u(ψ)V ,V dt + A∗ (t)˜ 0 T

C ∗ Λ(Cu(ϕ) − uobs ), u(ϕ) − u(ψ)V ,V dt, = 0

with u ˜(ϕ) solution of the adjoint problem (7.16), corresponding to the primal solution u(ϕ). By using the result of Corollary 6.4, we can deduce that T T ∂u ˜(ϕ) ∂u(ϕ) ∂u(ψ) , u(ϕ) − u(ψ)V ,V dt = − V,V dt

˜ u(ϕ), − ∂t ∂t ∂t 0 0 −(˜ u(ϕ)(T ), (u(ϕ) − u(ψ))(T )) + (˜ u(ϕ)(0), (u(ϕ) − u(ψ))(0)). Since u ˜(ϕ)(t = T ) = 0 and (u(ϕ) − u(ψ))(0) = u0 − u0 = 0 then T T ∂u ˜(ϕ) ∂u(ϕ) ∂u(ψ) − , u(ϕ) − u(ψ)V ,V dt = − ,u ˜(ϕ)V ,V dt.

∂t ∂t ∂t 0 0 Consequently, the relation (7.18) becomes T ∂(u(ϕ) − u(ψ)) ,u ˜(ϕ)V ,V dt

∂t 0 T + A(t)(u(ϕ) − u(ψ)), u˜(ϕ)V ,V dt 0 T

C ∗ Λ(Cu(ϕ) − uobs ), u(ϕ) − u(ψ)V ,V dt. =

(7.19)

0

Since u(ϕ) − u(ψ) is the solution of ∂(u(ϕ) − u(ψ)) + A(t)(u(ϕ) − u(ψ)) = B(ϕ − ψ), ∂t (u(ϕ) − u(ψ))(0) = 0, then, (7.19) gives T T

C ∗ Λ(Cu(ϕ) − uobs ), u(ϕ) − u(ψ)V ,V dt =

B(ϕ − ψ), u ˜(ϕ)V ,V dt 0

0

= B ∗ u˜(ϕ), ϕ − ψU ,U ∗ = (Λ−1 ˜(ϕ), ϕ − ψ)U , U B u

190


where B ∗ ∈ L(L2 (0, T ; V ); U ) is the adjoint of B and ΛU is the canonical isomorphism from U onto U . So, the inequality (7.15) can be written as ∗ ˜(ϕ) + N ϕ, ψ − ϕ)U ≥ 0 ∀ψ ∈ Uad (Λ−1 U B u

(7.20)

and then we have proved the following theorem. Theorem 7.4. Under the assumptions of Theorem 6.8, we suppose furthermore that the operator C ∈ L(L2 (0, T ; V ); M ) and the symmetric positive operator N ∈ L(U ; U ) satisfies (N ψ, ψ)U ≥ β ψ 2U , ∀ψ ∈ U , for a constant β > 0. Then there exists a unique optimal solution (ϕ∗ , u∗ ) ∈ U × W such that ∂u∗ + A(t)u∗ = f + Bϕ∗ , ∂t u∗ (0) = u0 and

∗ ∗ ˜ + N ϕ∗ , ψ − ϕ∗ )U ≥ 0 ∀ψ ∈ Uad , (Λ−1 U B u

˜∗ is the unique where ΛU is the canonical isomorphism from U onto U and u solution, which is in W, of the adjoint problem ∂u ˜∗ + A∗ (t)˜ u∗ = C ∗ Λ(Cu∗ − uobs ), ∂t u˜∗ (t = T ) = 0, −

where A∗ is the adjoint operator of A and Λ is the canonical isomorphism from M onto M . The Case (ii) and the Optimality System In this case (Cu = Du(t = T ) where D ∈ L(H; H) and M = H = M since H = H) the final observation uobs is in H, D∗ ∈ L(H; H), the cost functional is defined by J(ϕ) =

1 1 Du(ϕ)(T ) − uobs 2H + (N ϕ, ϕ)U 2 2

(7.21)

and the inequality (7.14) is equivalent, for all ψ ∈ Uad , to (D∗ (Du(ϕ)(T ) − uobs ), (u(ψ) − u(ϕ))(T ))H + (N ϕ, ψ − ϕ)U ≥ 0,

(7.22)

where u(ϕ) and u(ψ) are solutions of the primal problem (7.11), corresponding respectively with ϕ and ψ. Let us introduce the following adjoint problem: ∂u ˜ + A∗ (t)˜ u = 0, ∂t u˜(t = T ) = D∗ (Du(ϕ)(t = T ) − uobs ) (∈ H), −

(7.23)


191

which admits a unique solution u ˜ ∈ W. To prove this result, we change the variables of System (7.23) by reversing the sense of time, i.e., t := T − t and we apply the same way to obtain the result of Theorem 6.8. Multiplying the first equation of (7.23) by u(ϕ) − u(ψ) and integrating with respect to time we have that T T ∂u ˜(ϕ) , u(ϕ) − u(ψ)V ,V dt +

A∗ (t)˜ u(ϕ), u(ϕ) − u(ψ)V ,V dt = 0. − ∂t 0 0 By using the result of Corollary 6.4, we can deduce that T T ∂u ˜(ϕ) ∂u(ϕ) ∂u(ψ) , u(ϕ) − u(ψ)V ,V dt = − V,V dt −

˜ u(ϕ), ∂t ∂t ∂t 0 0 −(˜ u(ϕ)(T ), (u(ϕ) − u(ψ))(T ))H + (˜ u(ϕ)(0), (u(ϕ) − u(ψ))(0))H . Since u ˜(ϕ)(T ) = D∗ (Du(ϕ)(T ) − uobs ) and (u(ϕ) − u(ψ))(0) = u0 − u0 = 0 then T T ∂u ˜(ϕ) ∂u(ϕ) ∂u(ψ) , u(ϕ) − u(ψ)V ,V dt = − V,V dt

˜ u(ϕ), − ∂t ∂t ∂t 0 0 −(D∗ (Du(ϕ)(T ) − uobs ), (u(ϕ) − u(ψ))(T ))H and consequently T ∂u(ϕ) ∂u(ψ) − V,V dt

˜ u(ϕ), ∂t ∂t 0 T u(ϕ), A(t)(u(ϕ) − u(ψ))V,V dt + ˜

(7.24)

0

= (D∗ (Du(ϕ)(T ) − uobs ), (u(ϕ) − u(ψ))(T ))H . Since u(ϕ) − u(ψ) is the solution of ∂(u(ϕ) − u(ψ)) + A(t)(u(ϕ) − u(ψ)) = B(ϕ − ψ), ∂t (u(ϕ) − u(ψ))(0) = 0, then, (7.24) gives (D∗ (Du(ϕ)(T ) − uobs ), (u(ϕ) − u(ψ))(T ))H =

T

˜ u(ϕ), B(ϕ − ψ)V,V dt 0

= B ∗ u˜(ϕ), ϕ − ψU ,U ∗ = (Λ−1 ˜(ϕ), ϕ − ψ)U , U B u

where B ∗ ∈ L(L2 (0, T ; V ); U ) the adjoint of B and ΛU is the canonical isomorphism from U onto U . So, the inequality (7.22) can be written as ∗ (Λ−1 ˜(ϕ) + N ϕ, ψ − ϕ)U ≥ 0 ∀ψ ∈ Uad U B u

and then we have proved the following theorem.

(7.25)

192


Theorem 7.5. Under the assumptions of Theorem 6.8, we suppose furthermore that the cost J is given by (7.21) with the operator D ∈ L(H; H) and the symmetric positive operator N ∈ L(U ; U ) satisfies (N ψ, ψ)U ≥ β ψ 2U , ∀ψ ∈ U , for a constant β > 0. Then there exists a unique optimal solution (ϕ∗ , u∗ ) ∈ U × W such that ∂u∗ + A(t)u∗ = f + Bϕ∗ , ∂t u∗ (0) = u0 and

∗ ∗ ˜ + N ϕ∗ , ψ − ϕ∗ )U ≥ 0 ∀ψ ∈ Uad , (Λ−1 U B u

where ΛU is the canonical isomorphism from U onto U and u ˜∗ is the unique solution, which is in W, of the adjoint problem ∂u ˜∗ u∗ = 0, + A∗ (t)˜ − ∂t u ˜∗ (t = T ) = D∗ (Du∗ (t = T ) − uobs ). More generally, if we suppose that we have two observations: the final observation vobs , which is in H, and the state observation uobs , which is in M , the cost functional can be defined by θ2 1 θ1 J(ϕ) = Cu − uobs 2M + Du(T ) − vobs 2H + (N ϕ, ϕ)U , (7.26) 2 2 2 where θi ≥ 0, i = 1, 2, θ1 + θ2 > 0, C ∈ L(L2 (0, T ; V ); M ) and D ∈ L(H; H). Then we have the following theorem. Theorem 7.6. Under the assumptions of Theorem 6.8, we suppose furthermore that the cost functional is given by (7.26) with the operators D ∈ L(H; H) and C ∈ L(L2 (0, T ; V ); M ), and that the symmetric positive operator N ∈ L(U ; U ) satisfies (N ψ, ψ)U ≥ β ψ 2U , ∀ψ ∈ U , for a constant β > 0. Then there exists a unique optimal solution (ϕ∗ , u∗ ) ∈ U × W such that ∂u∗ + A(t)u∗ = f + Bϕ∗ , ∂t u∗ (0) = u0 and ∗ ∗ (Λ−1 ˜ + N ϕ∗ , ψ − ϕ∗ )U ≥ 0 ∀ψ ∈ Uad , U B u where ΛU is the canonical isomorphism from U onto U and u ˜∗ is the unique solution, which is in W, of the adjoint problem ∂u ˜∗ + A∗ (t)˜ u∗ = C ∗ Λ(Cu∗ − uobs ), − ∂t u˜∗ (t = T ) = D∗ (Du∗ (t = T ) − vobs ), where Λ is the canonical isomorphism from M onto M .

In the next section, we give some applications where we specify more precisely the control and the observation.

7.4 Examples of Controls and Observations

193

7.4 Examples of Controls and Observations In this section we present optimal control problems with different types of controls and observations, namely, problems with distributed control, boundary control, boundary observation, pointwise control, pointwise observation and/or data assimilation. For this, we consider the same operators and assumptions as in Section 6.4 and we take the following problem: ∂u + A(t)u = f + φ − a0 u on Q = Ω × (0, T ), ∂t subject to the linear Robin boundary condition ∂u + βu = αψ on Σ = Γ × (0, T ), α ∂ηA with the initial condition u(0) = u0 on Ω,

(7.27)

under the pointwise constraint r1 ≤ a0 ≤ r2

a.e. in Q,

(7.28)

where the normal derivative ∂u/∂ηA at Γ , directed towards the exterior of Ω, is given by (3.22), the functions f, φ, a0 are in L2 (Q), the function ψ is in L2 (Σ), the initial condition u0 is in L2 (Ω), the real constants α, β are such that αβ ≥ 0 and α + β = 0, the real constants r1 , r2 are such that r1 < r2 and ∂ ∂u (aij (x, t) ). (7.29) A(t)u = − ∂x ∂x i j i,j=1,n Next, let us denote by V the space H01 (Ω) in the case of homogeneous Dirichlet boundary condition (i.e., in the case of α = 0) and the space H 1 (Ω) in the case of linear Robin or Neumann boundary condition (i.e., if α = 0). We denote its dual by V and by ., .V ,V the duality product between V and V . Then the embedding V ⊂ L2 (Ω) ⊂ V are continuous. Finally, we denote by H = L2 (Ω) and by W = {w ∈ L2 (0, T ; V ) : ∂w/∂t ∈ L2 (0, T ; V )}. We introduce now the following form by: a0 u.vdx + θu.vdΓ, a ˜(t; u, v) = a(t; u, v) + Ω

Γ

where a is the bilinear form corresponding to the operator A (see section 6.4), θ = β/α ≥ 0 if α = 0 and θ = 0 else. Then the bilinear form a ˜ satisfies the assumptions of Theorem 6.8 (i.e., (6.3), (6.4) and (6.6), since v L2 (Γ ) ≤ v V , ∀v ∈ V , θ ≥ 0 and a0 satisfies (7.28)) and so, Theorem 6.8 gives the solution of problem (7.27). More precisely, Theorem 6.8 and Lemma 6.6 prove the existence and the

194


uniqueness of the solution u ∈ W ∩ C([0, T ]; H) of the problem, a.e. t ∈ (0, T ) (the weak formulation of (7.27)) ∂u ˜(t; u, v) = (f + φ, v) + α ψvdΓ ∀v ∈ V,

, vV ,V + a ∂t Γ (7.30) with the initial condition u(0) = u0 on Ω such that the following estimate holds. u 2W∩C([0,T ];H) ≤ C1 ( f 2L2 (Q) + φ 2L2 (Q) ) +C2 ( u0 2L2 (Ω) + ψ 2L2 (Σ) ).

(7.31)

7.4.1 Boundary Control In this subsection, we consider the optimal control problem where the control is both in the state equation and in the boundary condition. We suppose that the observation operator C is the injection of L2 (0, T ; V ) onto L2 (Q) (i.e., M = L2 (Q) = M , so the canonical isomorphism from M onto M is the identity operator) and consider the following two cases for the control: (i) The control is the function φ, i.e., the control is distributed in Q. (ii) The control is the function ψ, i.e., the control is on the boundary. Let K1 and K2 be given non-empty, closed and convex subsets of L2 (Q) and L2 (Σ) respectively, and the observation uobs ∈ L2 (Q). Our problem is then, to find Φ ∈ Uad such that the cost functional 1 δ1 δ2 Cu(Φ) − uobs 2L2 (Q) + φ 2L2 (Q) + ψ 2L2 (Σ) (7.32) 2 2 2 is minimized with respect to Φ subject to the problem (7.27), where δi ≥ 0, i = 1, 2 are fixed such that δ1 + δ2 > 0. The control Φ plays the role of: J(Φ) =

(i)

the function φ (i.e., Φ = φ) if δ2 = 0 and then Uad = K1

(ii) the function ψ (i.e., Φ = ψ) if δ1 = 0 and then Uad = K2 (iii) the function (φ, ψ) (i.e., Φ = (φ, ψ)) if δi > 0, i = 1, 2 and then Uad = K1 × K2 . The adjoint problem corresponding to the problem (7.27) is given by the following system: ∂u ˜(Φ) u(Φ) = −a0 u(Φ) + C ∗ (Cu(Φ) − uobs ) on Q, + A∗ (t)˜ ∂t subject to the linear Robin boundary condition ∂u ˜(Φ) + β u˜(Φ) = 0 on Σ, α ∂ηA∗ with the final condition u ˜(Φ)(t = T ) = 0 on Ω, −

(7.33)


195

where A∗ is the adjoint of the operator A. Moreover, the following optimality conditions apply: If δ2 = 0 (i.e., the control is distributed on Q), are given by (˜ u(φ∗ ) + δ1 φ∗ )(φ − φ∗ )dxdt ≥ 0 ∀φ ∈ K1 .

(7.34)

(ii) If δ1 = 0 (i.e., the control is in the boundary), are given by (˜ u(ψ ∗ ) + δ2 ψ ∗ )(ψ − ψ ∗ )dΓ dt ≥ 0 ∀ψ ∈ K2 .

(7.35)

(i)

Q

Σ

(iii) If δi > 0, i = 1, 2, are given by, ∀(φ, ψ) ∈ K1 × K2 (˜ u(Φ∗ ) + δ1 φ∗ )(φ − φ∗ )dxdt ≥ 0, Q (˜ u(Φ∗ ) + δ2 ψ ∗ )(ψ − ψ ∗ )dΓ dt ≥ 0.

(7.36)

Σ

Where (Φ∗ = (φ∗ , ψ ∗ ), u(Φ∗ )) is the optimal solution. Remark 7.7. In case (iii), for example, and without constraints, i.e., Uad = L2 (Q) × L2 (Σ), the optimality conditions (7.36) become u ˜(Φ∗ ) u˜(Φ∗ ) = −φ∗ , = −ψ ∗ δ1 δ2 and then, we can obtain the optimal control by the resolution of the following coupled system: u ˜∗ ∂u∗ + A(t)u∗ + a0 u∗ = f − on Q, ∂t δ1 ∂u ˜∗ u∗ + a0 u˜∗ = C ∗ (Cu∗ − uobs ) on Q, + A∗ (t)˜ − ∂t ∂u∗ α˜ u∗ α + βu∗ = − on Σ, ∂ηA δ2

(7.37)

∂u ˜∗ + β u˜∗ = 0 on Σ, ∂ηA∗ ˜∗ (T ) = 0 on Ω. u∗ (0) = u0 , u α

We can use the same method in order to study the case of the final observation. ♦ 7.4.2 Pointwise Observations In this subsection we consider pointwise observations (i.e., concentrated on internal points in the domain Ω) and a distributed control.

196


Let (xi )i=1,...,d be given d points of the domain Ω. We suppose that the observation is as the form (u(xi , t))i=1,...,d (if u(xi , t) has a sense). Suppose now that the operator A is independent of time, the space Ω ⊂ IRn with n ≤ 3, the boundary Γ of Ω and the coefficients aij of the operator A are sufficiently regular, and the function ψ = 0. Moreover, if u0 ∈ V , then the solution u of (7.27) is in H 2,1 (Q) = L2 (0, T ; H 2(Ω)) ∩ H 1 (0, T ; L2 (Ω)) and satisfies (7.38) u H 2,1 (Q) ≤ c( f L2 (Q) + φ L2 (Q) + u0 V ). Moreover, since Ω ⊂ IRn , n ≤ 3, we have that H 2 (Ω) ⊂ C(Ω) (the space of continuous functions on Ω) and then u(xi , t) ∈ L2 (0, T ), for i = 1, . . . , d, and T | u(xi , t) |2 dt ≤ c( f 2L2 (Q) + φ 2L2 (Q) + u0 2V ). (7.39) 0

Let K be a given non-empty, closed and convex subset of L2 (Q) and the observation uobs = (ui,obs )i=1,...,d ∈ (L2 (0, T ))d . Our problem is then, to find φ ∈ K such that the cost functional γ 1 u(xi , .) − ui,obs 2L2 (0,T ) + φ 2L2 (Q) 2 i=1 2 d

J(φ) =

(7.40)

is minimized with respect to φ subject to the problem (7.27) (with ψ = 0), where γ > 0 is a fixed constant. We have the existence and the uniqueness of the optimal control φ∗ which is characterized by the following optimality conditions, given by (˜ u(φ∗ ) + γφ∗ )(φ − φ∗ )dxdt ≥ 0 ∀φ ∈ K, (7.41) Q

∗

where u˜(φ ) is the solution of the following adjoint problem corresponding to problem (7.27) (with ψ = 0). d ∂u ˜ + A∗ u ˜ + a0 u ˜= (u(φ)(xi , .) − ui,obs )δxi on Q, ∂t i=1 subject to the linear Robin boundary condition

−

(7.42)

∂u ˜ + β u˜ = 0 on Σ, ∂ηA∗ with the final condition

α

u ˜(t = T ) = 0 on Ω, where A∗ is the adjoint of the operator A, δxi is the usual Dirac function at point xi and h(t)δxi is the distribution T S ∈ D(Q) −→ h(t)S(xi , t)dt. 0


197

The problem (7.42) admits a unique solution u ˜ on L2 (Q), given by transposition (see, e.g., Lions and Magenes [204]). For this let g ∈ L2 (Ω) be given and let w ∈ H 2,1 (Q) be the unique solution of ∂w + Aw + a0 w = g on Q, ∂t ∂w + βw = 0 on Σ, α ∂ηA w(0) = 0 on Ω.

(7.43)

w 2H 2,1 (Q) ≤ c g 2L2 (Q)

(7.44)

Moreover, we have that

and then (since H 2 (Ω) ⊂ C(Ω) and therefore w(xi , .) ∈ L2 (0, T ))

T

0

| w(xi , t) |2 dt ≤ c g 2L2 (Q) .

(7.45)

Multiplying now (7.42) by w, integrating over Q, integrating by parts in time, using Green’s formula and taking into account the boundary, initial and final conditions, we obtain T u˜gdxdt = 0

Ω

d i=1

T

(u(φ)(xi , t) − ui,obs (t))w(xi , t)dt.

(7.46)

0

Because of the relations (7.45), (7.39) and H¨ older’s inequality, we have that |

d i=1

T

(u(φ)(xi , t) − ui,obs (t))w(xi , t)dt |

0

≤ c1 g L2 (Q) ≤ c2 g L2 (Q)

d

( u(φ)(xi , .) L2 (0,T ) + ui,obs L2 (0,T ) )

(7.47)

i=1

and then the right-hand side of (7.46) is a linear and continuous form on H 2,1 (Q). Consequently, Problem (7.46) admits a unique solution u ˜ on L2 (Q). Remark 7.8. In the case without constraints, i.e., K = L2 (Q), the optimality conditions (7.41) become u˜(φ∗ ) = −φ∗ γ and then, we can obtain the optimal control by the resolution of the following coupled system:

198


∂u∗ u˜∗ + Au∗ + a0 u∗ = f − on Q, ∂t γ d ∂u ˜∗ + A∗ u˜∗ + a0 u − ˜∗ = (u∗ (xi , .) − ui,obs )δxi on Q, ∂t i=1 ∂u∗ + βu∗ = 0 on Σ, α ∂ηA ∂u ˜∗ + β u˜∗ = 0 on Σ, ∂ηA∗ u∗ (0) = u0 , u ˜∗ (T ) = 0 on Ω.

(7.48)

α

♦

7.4.3 Pointwise Controls In this subsection we consider pointwise controls (i.e., concentrated on internal points in the domain Ω) and a distributed observation. Let (xi )i=1,...,d be given d points of the domain Ω, and we suppose that the control function φ is as the form φ=

d

ϕi δxi

i=1

where ϕi ∈ L2 (0, T ), for i = 1, . . . , d and δxi is the usual Dirac function at point xi . Similarly as in the previous subsection, we suppose that the operator A is independent of time, the space Ω ⊂ IRn with n ≤ 3, the boundary Γ of Ω and the coefficients aij of the operator A are sufficiently regular, and the function ψ = 0. By using the same technique (the transposition technique) as to prove the unique solution of the adjoint problem (7.42), we obtain the existence and the uniqueness of the solution u in L2 (Q). Let K be a given non-empty, closed and convex subset of (L2 (0, T ))d and the observation uobs ∈ L2 (Q). Our problem is then, to find (ϕi )i=1,...,d ∈ K such that the cost functional 1 1 u(φ) − uobs 2L2 (Q) + γi ϕi 2L2 (0,T ) 2 2 i=1 d

J(φ) =

(7.49)

is minimized with respect to φ subject to the problem (7.27) (with ψ = 0), where γi > 0, for i = 1, . . . , d are fixed constants. ∗ dWe ∗have the existence and the uniqueness of the optimal control φ = i=1 ϕi δxi which is characterized by the following optimality conditions: d i=1

0

T

(˜ u(φ∗ )(xi , t) + γi ϕ∗i (t))(ϕi (t) − ϕ∗i (t))dt ≥ 0 ∀(ϕi )i=1,d ∈ K, (7.50)


199

where u ˜=u ˜(φ∗ ) is the solution of the following adjoint problem corresponding to problem (7.27) (with ψ = 0) ∂u ˜ + A∗ u˜ + a0 u˜ = u(φ∗ ) − uobs on Q, ∂t subject to the linear Robin boundary condition −

∂u ˜ + β u˜ = 0 on Σ, ∂ηA∗ with the final condition u ˜(t = T ) = 0 on Ω,

(7.51)

α

where A∗ is the adjoint of the operator A. Since u(φ) − uobs ∈ L2 (Q) and the final condition (i.e., the null function) is in V , then Problem (7.51) admits a unique solution u˜ in H 2,1 (Q). Moreover, ˜(xi , t) ∈ L2 (0, T ), for i = 1, . . . , d. since H 2 (Ω) ⊂ C(Ω), we have u Remark 7.9. In the case without constraints, i.e., K = (L2 (0, T ))d , the optimality conditions (7.50) become u˜(φ∗ )(xi , .) = −ϕ∗i γi and then we can obtain the optimal control by the resolution of the following coupled system: u ˜∗ (xi , .)δxi ∂u∗ + Au∗ + a0 u∗ = f − on Q, ∂t γi i=1 ∂u ˜∗ + A∗ u ˜∗ + a0 u˜∗ = u∗ − uobs on Q, − ∂t ∂u∗ α + βu∗ = 0 on Σ, ∂ηA d

∂u ˜∗ + β u˜∗ = 0 on Σ, ∂ηA∗ ˜∗ (T ) = 0 on Ω. u∗ (0) = u0 , u

(7.52)

α

♦

7.4.4 Boundary Controls and Boundary Observations In this subsection we consider a problem when controls and observations act both on the boundary during a time T . We denote, in the system (7.27), by g the sum of f and φ, i.e., g := f + φ, and we suppose that the parameter α = 0. The control is the function ψ defined on Σ and the observation is given by Cu = D(u|Σ ), (7.53) where u|Σ is the trace of u on Σ and D ∈ L(L2 (Σ); L2 (Σ)) (then Λ is the identity operator).

200


Example 7.10. If D is the identity, then we observe u on Σ. If D is the multiplication by the characteristic function of the subdomain Σ1 of Σ, then we observe u on Σ1 . ♣ Let K be a given non-empty, closed and convex subset of L2 (Σ) and the observation uobs ∈ L2 (Σ). Our problem is then, to find ψ ∈ K such that the cost functional γ 1 (7.54) J(ψ) = D(u|Σ ) − uobs 2L2 (Σ) + ψ 2L2 (Σ) 2 2 is minimized with respect to ψ subject to the problem (7.27), where γ > 0 is a fixed constant. We have the existence and the uniqueness of the optimal control ψ ∗ ∈ K, which is characterized by the optimality condition (˜ u(ψ ∗ ) + γψ ∗ )(ψ − ψ ∗ )dΓ dt ≥ 0 ∀ψ ∈ K, (7.55) Σ ∗

where u ˜(ψ ) is the solution of the following adjoint problem corresponding to the primal problem (7.27): ∂u ˜ + A∗ (t)˜ u + a0 u ˜ = 0 on Q, ∂t subject to the linear Robin boundary condition ∂u ˜ + β u˜ = αD∗ (D(u(ψ)|Σ ) − uobs ) on Σ, α ∂ηA∗ with the final condition u ˜(t = T ) = 0 on Ω, −

(7.56)

where A∗ is the adjoint of the operator A and D∗ is the adjoint of D. Since the function D∗ (D(u(ψ)|Σ ) − uobs ) is in L2 (Σ) then the problem (7.56) admits a unique solution in W ∩ C([0, T ]; H) (in the weak formulation sense). Remark 7.11. In the case without constraints, i.e., K = L2 (Σ), the optimality conditions (7.50) become u ˜(ψ ∗ )|Σ = −ψ ∗ γ and then, we can obtain the optimal control by the resolution of the coupled system: ∂u∗ + Au∗ + a0 u∗ = g on Q, ∂t ∂u ˜∗ + A∗ u˜∗ + a0 u − ˜∗ = 0 on Q, ∂t ∂u∗ α˜ u(ψ ∗ )|Σ (7.57) on Σ, α + βu∗ = − ∂ηA γ ∂u ˜∗ α + β u˜∗ = αD∗ (D(u(ψ ∗ )|Σ ) − uobs ) on Σ, ∂ηA∗ u∗ (0) = u0 , u ˜∗ (T ) = 0 on Ω.


201

♦ 7.4.5 Data Assimilation Problem and Initial Condition Control In this subsection we consider an application when the control is in the initial condition and the observation acts on the boundary. We denote, in the system (7.27), by g the sum of f and φ, i.e., g := f + φ, and we suppose that the parameter α = 0 and that the initial condition u0 is the sum of v0 (assumed to be known) and θ (assumed to be not well known), i.e., u0 = v0 + θ. The control is the function θ defined on Ω and the observation is given by Cu = D(u|Σ ),

(7.58)

where u|Σ is the trace of u on Σ and D ∈ L(L2 (Σ); L2 (Σ)) (then Λ is the identity operator). Remark 7.12. For the types of boundary observation operator D, see, e.g., Example 7.10. ♦ Let K be a given non-empty, closed and convex subset of L2 (Ω) and the observation uobs ∈ L2 (Σ). Our problem is then, to find ψ ∈ K such that the cost functional γ 1 (7.59) J(θ) = D(u|Σ ) − uobs 2L2 (Σ) + θ 2L2 (Ω) 2 2 is minimized with respect to θ subject to the problem (7.27), where γ > 0 is a fixed constant. We have the existence and the uniqueness of the optimal control θ∗ ∈ K, which is characterized by the optimality condition (˜ u(θ∗ ) + γθ∗ )(θ − θ∗ )dx ≥ 0 ∀θ ∈ K, (7.60) Ω ∗

where u ˜(θ ) is the solution of the following adjoint problem corresponding to the primal problem (7.27): ∂u ˜ + A∗ (t)˜ u + a0 u ˜ = 0 on Q, ∂t subject to the linear Robin boundary condition ∂u ˜ + β u˜ = αD∗ (D(u(θ)|Σ ) − uobs ) on Σ, α ∂ηA∗ with the final condition u ˜(t = T ) = 0 on Ω, −

(7.61)

where A∗ is the adjoint of the operator A and D∗ is the adjoint of the operator D. Since the function D∗ (D(u(ψ)|Σ ) − uobs ) is in L2 (Σ) then the problem (7.61) admits a unique solution in W ∩ C([0, T ]; H) (in the weak formulation sense).

202


Remark 7.13. In the case without constraints, i.e., K = L2 (Ω), the optimality conditions (7.60) become u ˜(θ∗ ) = −θ∗ γ and then, we can obtain the optimal control by the resolution of the following coupled system: ∂u∗ + Au∗ + a0 u∗ = g on Q, ∂t ∂u ˜∗ + A∗ u˜∗ + a0 u − ˜∗ = 0 on Q, ∂t ∂u∗ + βu∗ = αψ on Σ, α ∂ηA ∂u ˜∗ + β u˜∗ = αD∗ (D(u(ψ ∗ )|Σ ) − uobs ) on Σ, ∂ηA∗ u ˜(θ∗ ) u∗ (0) = v0 − , u ˜∗ (T ) = 0 on Ω. γ

(7.62)

α

♦

Remark 7.14. We can consider other control and observation cases, the technique remains the same (see, e.g., Lions [202] for other linear problems). ♦ In the next two sections we study two classes of non-linear control problems. More precisely, we analyze a class of bilinear problems (the primal problem is linear on the state variable when the control is fixed, and conversely) and a class of non-linear evolutive problems (which arise from the modeling, for example, of pollutants in liquid or atmospheric systems).

7.5 Parameter Estimations and Bilinear Control Problems In this section we present a first problem of non-linear control, namely an estimate parameter problem. The problem presented here is a simple but nontrivial application of the more general problems of estimate parameter models (see below for different biological and physical models). The problem is treated as an optimal control form with boundary observations. The control problems arising in this context are bilinear (this adds difficulties from a mathematical viewpoint).1 7.5.1 State Problem For this, we consider the linear primal problem used in Section 7.4, precisely the problem (7.27). In System (7.27) we use g to denote the sum of f and 1

For references on bilinear problems, see Section 8.5.

7.5 Parameter Estimations and Bilinear Control Problems

203

φ, i.e., g := f + φ, and we suppose that the parameter α = 1 and that the function θ := a0 is supposed to be not well known. The control is then the function θ defined on Q and the observation is given by Cu = D(u|Σ ),

(7.63)

where u|Σ is the trace of u on Σ and D ∈ L(L2 (Σ); L2 (Σ)). Then the problem (7.27) becomes ∂u + A(t)u = g − θ(x, t)u on Q, ∂t subject to the linear Robin boundary condition ∂u + βu = ψ on Σ, ∂ηA with the initial condition u(0) = u0 on Ω,

(7.64)

under the pointwise constraint r1 ≤ θ ≤ r2

a.e. in Q,

(7.65)

where the interval [r1 , r2 ] contains 0. The system (7.64) is linear on the state variable when the control is fixed, and linear on the control when the state variable is fixed, but the solution of (7.64) is depending non-linearily on the control θ. Consequently, the control problem is non-linear. Let Uad := {θ ∈ L2 (Q) : r1 ≤ θ ≤ r2 a.e. in Q} and F : θ ∈ Uad −→ W ∩ C([0, T ]; H) such that u = F (θ) is the unique solution of (7.64), corresponding to θ. Remark 7.15. (i) It is clear that Uad is a subset of L∞ (Q). (ii) According to Theorem 6.8 and Lemma 6.6 we have, for any given θ ∈ Uad , the following estimate: u 2W∩C([0,T ];H) ≤ C( g 2L2 (Q) + u0 2L2 (Ω) + ψ 2L2 (Σ) ).

(7.66)

(iii) For types of the boundary observation operator D, see, e.g., Example 7.10. ♦ Our problem is then, to find θ ∈ Uad such that the cost functional J(θ) =

γ 1 D(u|Σ ) − uobs 2L2 (Σ) + θ 2L2 (Q) 2 2

(7.67)

is minimized with respect to θ subject to the problem (7.64), where γ > 0 is a fixed constant and the function uobs ∈ L2 (Σ) is a given observation. 7.5.2 Existence of Optimal Solutions Now, let us study the following existence of an optimal solution.

204


Theorem 7.16. There exists θ∗ ∈ Uad and u∗ ∈ W such that θ∗ is defined by (7.67) and u∗ = F (θ∗ ) is a solution of (7.64). Proof. Let θk ∈ Uad be a minimizing sequence of J, i.e., lim inf J(θk ) = k−→∞

inf

θ∈L2 (Q)

J(θ).

Then, according to the nature of the cost function J, we can deduce that θk is uniformly bounded in Uad and we can extract from θk a subsequence also denoted by θk such that θk θ weakly in Uad . Therefore, uk = F(θk ) is uniformly bounded in W. Indeed, by using the weak form of (7.64), we can deduce easily that d uk 2L2 (Ω) 2 2 + a(t; uk , uk ) + θk | uk | dx + β | uk | dΓ = guk dx. 2dt Ω Γ Ω Then, by integrating with respect to time and because of the uniform boundedness of θk in L∞ (Q), we have that t t t 2 2 2 uk (t) L2 (Ω) + uk V ds+ uk L2 (Γ ) ds ≤ c1 uk 2L2 (Ω) ds+c2 . 0

0

0

By using Gronwall’s lemma, we can deduce first that uk is uniformly bounded in L∞ (0, T ; L2(Ω)) and finally that uk is uniformly bounded in L2 (0, T ; V ). Using the previous result and Equation (7.64) we obtain easily that ∂uk /∂t is uniformly bounded in L2 (0, T ; V ) and then uk is uniformly bounded in W. According to Lemma 6.6, the injection of W into L2 (Q) is compact. Therefore, this result makes it possible to extract from uk a subsequence also denoted by uk such that uk u˜ weakly in L2 (0, T ; V ), ˜ strongly in L2 (Q), uk −→ u

(7.68)

θk θ weakly in L (Q) and θ ∈ Uad . 2

Prove now that uk θk −→ uθ weakly in L2 (Q). Since uk θk − uθ = (uk − u)θk + u(θk − θ), then, according to the first and second parts of (7.68), we obtain the result. We can prove easily that u˜ is a solution of (7.64) with a parameter θ and according to the uniqueness of the solution of (7.64), we have then u ˜ = u = F (θ). Finally, since the norm is lower semi-continuous we have that the map J : θ −→ J(θ) is lower semi-continuous and then the function θ is an optimal solution, i.e., inf J(θ) = lim inf J(θk ) = J(θ∗ ). θ∈Uad k 7.5.3 First-order Optimality Conditions Before giving the optimality conditions for an optimal solution, we study the following G-differentiability results.


205

Proposition 7.17. The function F : θ −→ u = F (θ) solution of (7.64) is G-differentiable, with respect to θ from Uad to W ∩ C([0, T ]; H) where the Gderivative F (θ) : h −→ w = F (θ)h is the unique solution in W ∩C([0, T ]; H) of the following linear parabolic problem: ∂w + A(t)w = −θw − hu on Q ∂t ∂w + βw = 0 on Σ, ∂ηA w(0) = 0 on Ω.

(7.69)

Proof. Problem (7.69) is similar to Problem (7.64) (where the function −hu is in L2 (Q), since u ∈ L2 (Q) and h ∈ L∞ (Q), plays the role of g in (7.64)). Then Problem (7.69) admits a unique solution w ∈ W ∩ C([0, T ]; H) such that (because of the estimate (7.66)) w W∩C([0,T ];H) ≤ C u L2 (Q) . Let (θ, h) ∈ Uad × L∞ (Q) and > 0 such that h + θ ∈ Uad . Let u = F (θ) and u = F (θ + h). Step 1: Prove that u −→ u strongly in W ∩ C([0, T ]; H) as −→ 0. Let v := u − u, obviously, v satisfies ∂v + A(t)v = −(θ + h)v − hu on Q, ∂t ∂v + βv = 0 on Σ, ∂ηA v (0) = 0 on Ω.

(7.70)

Since h ∈ L∞ (Q) and u ∈ L2 (Q) then g := −hu is in L2 (Q). Consequently, since (θ + h) ∈ L∞ (Q), we have that, there exists a constant C > 0 (independent of ) such that v W∩C([0,T ];H) ≤ C g L2 (Q) and then v W∩C([0,T ];H) ≤ C1 u L2 (Q) . Consequently, v −→ 0 strongly in W ∩ C([0, T ]; H) as −→ 0. Step 2: Prove now that w :=

(u − u) v = −→ w strongly in W ∩ C([0, T ]; H) as −→ 0.

206


Obviously, w ˜ := w − w satisfies ∂w ˜ + A(t)w˜ = −θw ˜ − hv on Q, ∂t ∂w ˜ + βw ˜ = 0 on Σ, ∂ηA w ˜ (0) = 0 on Ω.

(7.71)

The problem (7.71) is similar to problem (7.70), so w ˜ W∩C([0,T ];H) ≤ C v L2 (Q) and then w ˜ W∩C([0,T ];H) −→ 0 as −→ 0 (since v L2 (Q) −→ 0 as −→ 0). This completes the proof. With the aid of this proposition, we can easily show the first-order necessary conditions (optimality conditions). Theorem 7.18. Let θ∗ ∈ Uad be an optimal control defined by (7.67) and u∗ ∈ W ∩ C([0, T ]; H) be the optimal state such that u∗ = F (θ∗ ) is the solution ˜∗ ∈ of (7.64), with the parameter θ∗ . Then there exists a unique solution u W ∩ C([0, T ]; H) for the following adjoint problem corresponding to the primal problem (7.64): ∂u ˜ + A∗ (t)˜ u + θ∗ u ˜ = 0 on Q, ∂t subject to the linear Robin boundary condition ∂u ˜ + β u˜ = D∗ (D(u∗ |Σ ) − uobs ) on Σ, ∂ηA∗ with the final condition u˜(t = T ) = 0 on Ω,

−

(7.72)

where A∗ is the adjoint of the operator A and D∗ is the adjoint of D. Moreover, ∗ ∗ u u ˜ , r2 θ∗ = max r1 , min γ or in the variational inequality formulation (7.73) T (−u∗ u ˜∗ + γθ∗ )(θ − θ∗ )dxdt ≥ 0 ∀θ ∈ Uad , 0

Ω

Proof. Since the function D∗ (D(u∗ |Σ ) − uobs ) is in L2 (Σ) then the problem (7.72) admits a unique solution in W ∩ C([0, T ]; H) (in the weak formulation sense). The cost functional J is a composition of G-differentiable mappings then J is G-differentiable and in particular the G-derivative of J at the optimal point θ∗ ∈ Uad is given by (for all h ∈ L∞ (Q) such that (θ∗ + h) ∈ Uad for small)


J(θ∗ + h) − J(θ∗ ) 0 ≤ J (θ∗ ) = lim

−→0 T T = (D∗ (D(u|Σ ) − uobs ).wdΓ ds + γ θ∗ hdxdt, 0

0

Γ

207

(7.74)

Ω

where w = F (θ∗ )θ. Multiplying (7.69) by u˜∗ and integrating over Q, this gives (using Green’s formula and integrating by parts in time) T T ∂u ˜∗ ∂u ˜∗ ∗ ∗ ∗ ∗ + A (t)˜ (− u +θ u ˜ )wdxdt + ( + β u˜∗ )wdΓ dt ∂t 0 Ω 0 Γ ∂ηA∗ T ∗ ∗ ∗ hu u ˜ dxdt + w(0)˜ u (0)dx − w(T )˜ u∗ (T )dx. =− 0

Ω

Ω

Ω

Since u ˜∗ is the solution of (7.72) and w(0) = 0, therefore we can deduce that T T ∗ ∗ D (D(u |Σ ) − uobs ).wdΓ ds = − hu∗ u˜∗ dxdt (7.75) 0

0

Γ

Ω

and then (according to (7.74)) T (γθ∗ − u∗ u ˜∗ )hdxdt. 0≤ 0

(7.76)

Ω

By using a standard control argument concerning the sign of the variation h (depending on the size of θ∗ ), we obtain that ∗ ∗ u u ˜ ∗ θ = max r1 , min , r2 . γ Indeed, by taking h = max (r1 , min (u∗ u ˜∗ /γ, r2 )) − θ∗ , we prove easily that ∗ ∗ ∗ h(γθ − u u˜ ) is negative, and then ∗ ∗ u u ˜ ˜∗ u∗ u , r2 max r1 , min − θ∗ θ∗ − = 0. γ γ So, (i)

˜∗ /γ, we have (r2 − θ∗ )(θ∗ − u∗ u ˜∗ /γ) = 0 and then (since if r2 ≤ u∗ u ∗ θ ≤ r2 ), θ∗ = r2

(ii) if r2 ≥ u∗ u ˜∗ /γ and r1 ≥ u∗ u ˜∗ /γ, we have (r1 − θ∗ )(θ∗ − u∗ u ˜∗ /γ) = 0 ∗ and then (since θ ≥ r1 ) θ∗ = r1 (iii) if r2 ≥ u∗ u ˜∗ /γ and r1 ≤ u∗ u ˜∗ /γ, we have (u∗ u˜∗ /γ−θ∗ )(θ∗ −u∗ u˜∗ /γ) = 0 and then u∗ u˜∗ θ∗ = . γ We can conclude that θ∗ = max(r1 , min(u∗ u˜∗ /γ, r2 )). This completes the proof.

208


7.6 Non-linear Control for Non-linear Evolutive PDE Problems In this section we analyze the full non-linear control problems. For this we consider, in this study, some non-linear evolutive problems where we can prove existence and uniqueness (under extra assumptions) theorems for the optimal control problems, and we give the optimality conditions characterizing the optimal solutions. In contrast to linear systems, the dynamic of evolutive non-linear systems obeys complicated laws that, in general, cannot be found by intuitive and direct calculations. The main result of the section includes the existence, the uniqueness and the first-order necessary conditions of optimality for the optimal controllers. The plan of this section is as follows. First, we give the existence and the uniqueness of the state equation and obtain some a priori estimates. Second, we formulate the optimal control problem, prove the existence and the uniqueness of the optimal solution, and give the appropriate optimality system. Next, we consider a data assimilation problem because in many situations the initial condition is not well known. We reformulate the optimal control problem. As in previous sections, the existence, the uniqueness and the optimality conditions are described. Finally, we present an example of convection–diffusion in the case of pollutants in liquid or atmospheric systems. 7.6.1 State Problem and Assumptions In this subsection we will be consider the non-linear parabolic partial differential equations of the form ∂u + Au + F (x, t, u) + K(, u) = f on Q = (0, T ) × Ω, ∂t u(0) = u0 on Ω,

(7.77)

where Ω is an open and bounded subset of IRm (m ≥ 1) sufficiently regular, T > 0 is a fixed constant (a given final time), A is an elliptic, selfadjoint operator, K(, .) is a linear operator with a given and sufficiently regular vector field, and F : Q × IR −→ IR is a Nemytsky operator on L2 (Q) for which we state the following hypothesis (see, e.g., Seidman and Zhou [265]): (i) F (., 0) = 0 (ii) F satisfies Carathéodory conditions and a one-sided Lipschitz condition: −2(F (., u) − F (., v))(u − v) ≤ γ0 | u − v |2 ∀(u, v) ∈ IR2 (iii) F is differentiable with G(., u) = F (., u) Lipschitz continuous: | G(., u) − G(., v) |≤ λ | u − v | ∀(u, v) ∈ IR2 .

(7.78)

7.6 Non-linear Control for Non-linear Evolutive PDE Problems

209

Remark 7.19. According to (7.78) we can deduce that the operator F is bounded and continuous, and that −2G(., u) ≤ γ0 ∀u ∈ IR.

♦

We assume that we can introduce a reflexive Banach space D of functions on Ω satisfying the boundary conditions such that D ⊂ L2 (Ω), with D dense in L2 (Ω) and the embedding of D in L2 (Ω) being compact. We can identify L2 (Ω) with its dual L2 (Ω) and then D ⊂ L2 (Ω) ⊂ D where the injections are continuous and each space is dense in the following one. We assume also that the linear and continuous operator A: D −→ D with Av, v ≥ ν v 2D ∀v ∈ D and Au, v ≤ M u D v D ∀u, v ∈ D ( ., . is the duality pairing on D and D, . D is the norm on D, . ∗ is the dual norm on D and ν, M > 0 are constants). Moreover, we impose the condition that D is embedded in L4 (Ω), i.e., D ⊂ L4 (Ω) and ∃ ce > 0 such that v L4 (Ω) ≤ ce v D ∀v ∈ D.

(7.79)

We introduce the following spaces: H = L∞ (0, T ; L2(Ω)), V = L2 (0, T ; D), H1 = H 1 (0, T ; D ) = {v : ∂v/∂t ∈ L2 (0, T ; D )}, W = H ∩ V and we denote . W = max( . H , ν . V ) and by cI the constant of the embedding map: D −→ L2 (Ω), i.e., (7.80) v L2 (Ω) ≤ cI v D ∀v ∈ D. Moreover, we assume that there exists a constant γ∞ ≥ 0 such that: √ K(, v) L2 (Ω) ≤ νγ∞ v D ∀v ∈ D.

(7.81)

Remark 7.20. (i) The parameter γ∞ , introduced in (7.81), is depending on the norm of the given field . (ii) If one has Av, v + θ v 2L2 (Ω) ≥ ν v 2D ∀v ∈ D, then we can shift the term θv from the expression of the operator F to the expression of the Av. Consequently, the value of γ0 in assumption (ii) of (7.78) is modified; however, the other assumptions of (7.78) remain unchanged. ♦ Remark 7.21. We denote by γ the value of the sum: γ0 + γ∞ , i.e., γ = γ0 + γ∞

(7.82)

and by K ∗ the adjoint of K, i.e.,

K ∗ (, u), v = K(, v), u ∀(u, v) ∈ D2 .

(7.83) ♦

In this work, the cost functional J describing the control problem depends on the control φ and the solution u(φ) in the domain Ω over the time interval under consideration [0, T ]. The optimal control corresponds to obtain the minimum point of a function J which measures the distance between the pronostic

210


variable u and the observation (uobs , vobs ). Precisely we will study the following optimal control problem: find (u, φ) such that the cost functional μ α 1 T C(u − uobs ) 2L2 (Ω) + u(T ) − vobs 2L2 (Ω) + φ 2Rs J(φ) = 2 0 2 2 is minimized with respect to φ subject to the solution of the problem (7.77), where the space Rs is a subset of some Banach space and C is an unbounded operator on L2 (Ω). 7.6.2 Existence and Uniqueness of the Solution Proposition 7.22. Assume that u0 ∈ L2 (Ω) and f ∈ L2 (Q). Then the problem (7.77) admits a unique solution u such that u ∈ W ∩ H1 and u 2W∩H1 ≤ C( u0 2L2 + f 2L2 (Q) ). Proof. We will just sketch the proof based on suitable a priori estimates. Multiplying (7.77) by u (u ∈ D) and integrating over Ω, this gives 1 d | u |2 + Au, u + F (., t, u), u + K(, u), u = f, u. 2 dt According to assumption (i) of (7.78), we obtain d | u |2 + 2 Au, u = −2 F (., t, u) − F (., t, 0), u − 0 − 2 K(, u), u + 2 f, u. dt By using the definition of the norm in D and according to assumption (ii) of (7.78) and assumption (7.81) we have d | u |2 + ν u 2D ≤ γ | u |2 +2 | f || u |, dt with γ given by the notation (7.82). Integrating with respect to time, over (0, t) for all t ∈ (0, T ), gives (by using H¨ older’s inequality) t t u 2D ds ≤ (1 + γ) | u |2 ds+ f 2L2 (Q) + | u0 |2 | u(t) |2 +ν 0

0

and then (according to Gronwall’s lemma) u 2W ≤ C( u0 2L2 + f 2L2 (Q) ).

(7.84)

Using the result (7.84) and (7.77), we prove easily that u satisfies the following estimate u 2H1 ≤ C( u0 2L2 + f 2L2 (Q) ). (7.85) The proof of the existence result can be completed by implementing the Galerkin method, by taking advantage of the above estimate and by using the continuity and the uniform boundedness of the operator F . Uniqueness of the solution of (7.77) is a corollary of Proposition 7.23 (see below).


211

Proposition 7.23. Let u01 , u02 be two functions in L2 (Ω) and let f1 , f2 be two functions in L2 (Q). If u1 (respectively u2 ) is a solution of (7.77) with data (f1 , u01 ) (respectively with data (f2 , u02 )) then u1 − u2 2W∩H1 ≤ C( u01 − u02 2L2 + f1 − f2 2L2 (Q) ).

(7.86)

Proof. Let (ui )i=1,2 be two solutions of (7.77), with the given data (fi , u0i )i=1,2 , respectively. We denote by u = u1 − u2 , u0 = u01 − u02 and f = f1 − f2 . Then u is the solution of the following problem ∂u + Au + F (x, t, u1 ) − F (x, t, u2 ) + K(, u) = f on Q, ∂t u(0) = u0 on Ω.

(7.87)

By using the same way to obtain the estimations (7.84) and (7.85) and the assumption (ii) of (7.78), we obtain the result of the proposition. 7.6.3 The Control Framework In the control framework, the value f is decomposed into a known function g ∈ L2 (Q) and the control φ ∈ L2 (Q), i.e., g = f + Bφ, where B is a given linear continuous operator on H = L2 (Ω): B ∈ L(H; H) such that ∀h ∈ L2 (Ω), Bh L2 ≤ b h L2 , for b > 0. (7.88) Then Problem (7.77) becomes ∂u + Au + F (x, t, u) + K(, u) = g + Bφ on Q, ∂t u(0) = u0 (given) on Ω.

(7.89)

We suppose the following hypothesis: u0 ∈ L2 (Ω), (g, φ) ∈ L2 (Q)2 . Let U : φ −→ u = U(φ) be the map: L2 (Q) −→ W defined by (7.89). We introduce the cost functional defined by J(φ) =

μ α 1 C(u − uobs ) 2L2 (Q) + u(T ) − vobs 2L2 + φ 2L2 (Q) , (7.90) 2 2 2

where μ ≥ 0 and α > 0 are fixed parameters, (uobs , vobs ) ∈ V × L2 (Ω) is the (given) observation, and C is an unbounded linear operator on L2 (Ω) satisfying the condition Cv 2L2 ≤ δ1 v 2L2 +δ2 v 2D

∀v ∈ D.

(7.91)

Then the optimal control problem is to minimize the functional J with respect to φ, i.e., to find φ∗ ∈ Uad such that J(φ∗ ) = inf J(φ), φ∈Uad

with Uad is a (given) non-empty, closed, convex subset of L2 (Q).

(7.92)

212


Proposition 7.24. Let F be an operator that satisfies assumptions (7.78). Then the function U : φ −→ u = U(φ) solution of (7.89) is continuously F-differentiable from L2 (Q) to W and its derivative w = U (φ)h at point φ ∈ L2 (Q) in the direction h ∈ L2 (Q) is given by the linear parabolic problem ∂w + Aw + G(x, t, U(φ))w + K(, w) = Bh on Q ∂t w(0) = 0 on Ω.

(7.93)

Moreover, for all φ ∈ L2 (Q) we have the following estimate: U (φ) L(L2 (Q);W) ≤ C. Proof. Let (φ, h) ∈ (L2 (Q))2 , let u = U(φ) and uh = U(φ + h) = u + wh . Let vh be the solution of ∂vh + Avh + G(x, t, u)vh + K(, vh ) = Bh on Q, ∂t vh (0) = 0 on Ω.

(7.94)

Multiplying (7.94) by vh and integrating over (0, t) × Ω, for all t ∈ (0, T ), we obtain (according to −2G(., u) ≤ γ0 ) t 2 vh (t) L2 +ν vh 2D ds 0 (7.95) t t vh 2L2 ds + 2 Bh L2 (Q) ( vh 2L2 ds)1/2 . ≤γ 0

0

By using Gronwall’s formula we then have (according to (7.88)) vh 2W ≤ c1 h 2L2 (Q) .

(7.96)

This shows that the map: h −→ vh solution of (7.94) is continuous. Moreover, by using Proposition 7.23 and (7.88) we deduce easily that: wh 2W ≤ c2 h 2L2 (Q) .

(7.97)

Let v = wh − vh and according to the equations satisfied by u, uh and vh we can deduce that v satisfies the following system: ∂v + Av + G(x, t, u)v + K(, v) = g on Q, ∂t v(0) = 0 on Ω,

(7.98)

where g = −(F (., u + wh ) − F (., u)) + G(., u)wh . Multiplying (7.98) by v and integrating over (0, t) × Ω, for all t ∈ (0, T ), we obtain (according to −2G(., u) ≤ γ0 ) t t t v(t) 2L2 +ν v 2D ds ≤ γ v 2L2 ds + 2 g, v(s)ds. (7.99) 0

0

0


213

t

g, v(s)ds for all t ∈ (0, T ). By using

We shall now estimate the term 2 0

a simple manipulation we obtain 1 (G(., u) − G(., u + swh ))wh ds. g= 0

Applying assumption (iii) of (7.78) we can deduce that 2 | g |≤ λ | wh |2 . So, t T T 2 g, v(s)ds ≤ 2 g, v(s)ds ≤ λ | wh |2 | v | dxds 0

and then

0

2 0

0

Ω

t

g, v(s)ds ≤ λ wh 2L2 (0,T ;L4 (Ω)) v L∞ (0,T ;L2 (Ω)) .

According to assumption (7.79) we obtain t 2 g, v(s)ds ≤ c3 wh 2W v W .

(7.100)

0

From (7.99) and (7.100), we deduce that t t v(t) 2L2 +ν v 2D ≤ γ v 2L2 +c3 wh 2W v W . 0

0

Using Gronwall’s formula we then have v 2W ≤ c4 wh 2W v W . According to (7.97) we can deduce that v W ≤ c5 h 2L2 (Q) and then v W = o( h L2 (Q) ). Therefore, U (φ) defined by (7.93) is the F-derivative of U at point φ and verifies U (φ) L(L2 (Q);W) ≤ C. Proposition 7.25. Let F be an operator that satisfies assumptions (7.78). Then for each t ∈ [0, T ], the function Vt : φ −→ u(t) = Vt (φ) solution of (7.89) is continuously F-differentiable from L2 (Q) to L2 (Ω) and its derivative w(t) = Vt (φ).h at point φ ∈ L2 (Q) in the direction h ∈ L2 (Q) is given by the linear parabolic problem ∂w + Aw + G(x, t, Vt (φ))w + K(, w) = Bh on Q, ∂t w(0) = 0 on Ω. Moreover, for all φ ∈ L2 (Q) the following estimate holds: Vt (φ) L(L2 (Q);L2 (Ω)) ≤ C.

(7.101)

214


Proof. The functional φ −→ u(t) is continuous from L2 (Q) to L2 (Ω) (the corollary of Proposition 7.23). The rest of the proposition is a corollary of Proposition 7.24. Before proving the existence of the optimal solution, we study the following result. Proposition 7.26. Let F be an operator that satisfies assumptions (7.78). Then the maps U : φ −→ u = U(φ) and Vt : φ −→ u(t) = Vt (φ), for t ∈ (0, T ) solutions of (7.89) are continuous from the weak topology of L2 (Q) to the strong topology of L2 (Q) and the weak topology of L2 (Ω), respectively. Proof. Let φ be given in L2 (Q) and let the sequence φk such that φk is weakly convergent in L2 (Q) to φ. Set u = U(φ), uk = U(φk ) and vk = u−uk . Since φk φ weakly in L2 (Q) then the sequence φk is uniformly bounded in L2 (Q) and therefore (according to Proposition 7.23) uk is uniformly bounded in W. By using assumption (ii) of (7.78) we deduce that F (., uk ) is uniformly bounded in L2 (Q). Using this result and Equation (7.89) we obtain easily that ∂uk /∂t is uniformly bounded in L1 (0, T ; D ). Let us introduce the space Y = {v ∈ L2 (0, T ; D),

∂v ∈ L1 (0, T ; D )}. ∂t

According to Lemma 6.6, the injection of Y into L2 (Q) is compact. Therefore uk is uniformly bounded in Y. This result makes it possible to extract from (uk , φk , F (., uk )) a subsequence also denoted by (uk , φk , F (., uk )) such that2 φk φ weakly in L2 (Q) uk u ˜ weakly in L2 (0, T ; D) uk −→ u˜ strongly in L2 (Q) F (., uk ) −→ F (., u ˜) strongly in L2 (Q).

(7.102)

It is easy for us to prove that u ˜ is a solution of (7.89) with a forcing φ and according to the uniqueness of the solution of (7.89), we have then u ˜=u= U(φ). In the same way we prove that for each t ∈ [0, T ], Vt (φk ) Vt (φ) weakly in L2 (Ω). Now, let us study the following existence of an optimal solution. Theorem 7.27. Let F be an operator that satisfies assumptions (7.78). Then there exists φ∗ ∈ L2 (Q) and u∗ ∈ W such that φ∗ is defined by (7.92) and u∗ = U(φ∗ ) is a solution of (7.89). 2

The operator F is continuous then F (., uk ) −→ F (., u ˜) strongly in L2 (Q) (according to assumption (iii) of (7.102)).


215

Proof. Let φk ∈ Uad be a minimizing sequence of J, i.e., lim inf J(φk ) = k−→∞

inf

φ∈L2 (Q)

J(φ).

Then φk is uniformly bounded in Uad and we can extract from φk a subsequence also denoted by φk such that φk φ weakly in Uad . By using Proposition 7.26 we then have U(φk ) u = U(φ) weakly in L2 (0, T ; D), U(φk ) −→ u strongly in L2 (Q), Vt (φk ) u(t) = Vt (φ) weakly in L2 (Ω), ∀t ∈ [0, T ] and u satisfies the system (7.89). Since the norm is lower semi-continuous therefore the map J : φ −→ J(φ) is lower semi-continuous and then the function φ is an optimal solution, i.e., inf J(ψ) = lim inf J(φk ) = J(φ).

ψ∈Uad

k

In order to characterize the solution of the optimal control problem, we introduce the “adjoint” problem corresponding to the primal problem (7.89) (we denote by u = U(φ) the solution of Problem (7.89) with the forcing φ): ∂u ˜ + A˜ u + (G(x, t, u))∗ u ˜ + K ∗ (, u ˜) = C ∗ C(u − uobs ) on Q, ∂t subject to the final condition u˜(t = T ) = μ(u(t = T ) − vobs ) on Ω,

−

(7.103)

where C ∗ (respectively (G(., u))∗ and K ∗ ) is the adjoint of the operator C (respectively G(., u) and K). Proposition 7.28. Let F be an operator that satisfies assumptions (7.78) and u ∈ W then the solution of (7.103) is in W and satisfies the following estimate: ˜ 2V ≤ Cd2 (μ2 u(T ) − vobs 2L2 + C(u − uobs ) 2L2 (Q) ). u ˜ 2H + u Proof. Multiplying (7.103) by u ˜ and integrating over (t, T ) × Ω, for all t ∈ (0, T ), we obtain for all r > 0 (according to −2G(., u) ≤ γ0 and to (7.81)) u ˜(t)

2L2

u ˜

+ν t

+

T

1 r2

2D

ds ≤ μ u(T ) − 2

vobs 2L2

T

C(u − uobs ) 2L2 ds + r2 t

T

u ˜ 2L2 ds

+γ t

T

Cu ˜ 2L2 ds. t

216


According to assumption (7.91) we obtain T u ˜ 2D ds u ˜(t) 2L2 +ν t

T u ˜ 2L2 ds ≤ μ2 u(T ) − vobs 2L2 +(δ1 r2 + γ) t T 1 2 2 u ˜ D ds + 2 C(u − uobs ) 2L2 (Q) . +δ2 r r t

(7.104)

Choosing r2 := ν/2δ2 we can deduce that ν T 2δ2 C(u − uobs ) 2L2 (Q) u ˜ 2L2 + u ˜ 2D ds ≤ μ2 u(T ) − vobs 2L2 + 2 t ν T νδ1 +( + γ) u ˜ 2L2 ds. 2δ2 t By using Gronwall’s formula we can conclude that ˜ 2V ≤ Cd2 (μ2 u(T ) − vobs 2L2 + C(u − uobs ) 2L2 (Q) ). u ˜ 2H + u This completes the proof.

˜ : φ −→ u˜ = U(φ) ˜ In the sequel, we will denote by U the map defined by the solution of the adjoint problem (7.103). We can now give the first-order optimality conditions for the optimal control problem (7.92). Theorem 7.29. Let F be an operator that satisfies assumptions (7.78) and (φ∗ , u∗ ) ∈ L2 (Q) × W such that φ∗ is defined by (7.92) and u∗ = U(φ∗ ) is a solution of (7.89). Then T (αφ∗ + B ∗ u ˜)(φ − φ∗ )dxdt ≥ 0, ∀φ ∈ Uad , 0

Ω

∗

˜ ) is a solution of the adjoint problem (7.103), corresponding where u˜ = U(φ to the primal solution u∗ . Moreover, the gradient of the cost functional J at φ∗ is given by J (φ∗ ) = αφ∗ + B ∗ u ˜. Proof. From Proposition 7.25 we know that U is differentiable. Therefore, the cost functional J is a composition of F-differentiable mappings and, then, J is F-differentiable and, for all h ∈ L2 (Q), we have T J (φ)h =

C(u − uobs ), CU (φ)hds + μ(u(T ) − vobs ), VT (φ)h 0 T + αφ, hds, 0


217

which implies

J (φ)h =

T

C ∗ C(u − uobs ), U (φ)hds + μ(u(T ) − vobs ), VT (φ)h 0 T + αφ, hds. 0

Multiplying (7.93) by u˜, integrating over Q, we obtain that, for all h ∈ L2 (Q), 0

T

B ∗ u ˜, hds = VT (φ∗ )h, u˜(T ) T ∂u ˜ + A˜ u + (G(., u∗ ))∗ u ˜ + K ∗ (, u˜), U (φ∗ )hds. + − ∂t 0

Since u ˜ is a solution of the adjoint problem (7.103) we can deduce that, for all h ∈ L2 (Q),

T

0

B ∗ u ˜, hds = VT (φ∗ )h, μ(u∗ (T ) − vobs ) T + C ∗ C(u∗ − uobs ), U (φ∗ )hds. 0

Using the previous equalities and the expression of J , we see that, for all h ∈ L2 (Q), T

αφ + B ∗ u ˜, hds. (7.105) J (φ).h = 0 ∗

Since φ is a solution of (7.92), we have J (φ∗ ).(φ − φ∗ ) ≥ 0 ∀φ ∈ Uad , and therefore, according to (7.105), we can deduce that T 0

(αφ∗ + B ∗ u ˜)(φ − φ∗ )dxdt ≥ 0, ∀φ ∈ Uad .

Ω

So the proof is complete.

In the sequel, we will assume that there exists a pair (φ∗ , u∗ ) such that φ∗ is defined by (7.92), u∗ = U(φ∗ ) is a solution of (7.89) and T 0

(αφ∗ + B ∗ u˜)(φ − φ∗ )dxdt ≥ 0

Ω

˜ ∗ ) is a solution of (7.103)). Now we give some ˜ = U(φ for all φ ∈ Uad , where u conditions to obtain the uniqueness of the optimal solution (φ∗ , u∗ ). Theorem 7.30. Suppose that F satisfies assumptions (7.78) and μ < 1 holds. If the following assumptions hold:

218


(i) θ = (ν − δ2 − c2I (γ + δ1 ) − 2b2 c2I /α) > 0

1/2 < θ, (ii) λc2e e(δ1 +γ+1)T u∗ (T ) − vobs 2L2 + C(u∗ − uobs ) 2L2 (Q) then the optimal solution (φ∗ , u∗ ), of Problem (7.92) subject to Problem (7.89), is unique. Proof. Assume that (φ∗1 , u∗1 ) is another solution. Then, φ∗1 satisfies (7.92), u∗1 = U(φ∗1 ) is a solution of (7.89) and T (αφ∗1 + B ∗ u ˜1 )(φ − φ∗1 )dxdt ≥ 0 0

Ω

˜ ∗1 ) is a solution of (7.103)). ˜1 = U(φ for all φ ∈ Uad , where u ∗ ∗ We set φ = φ − φ1 , v = u∗ − u∗1 and v˜ = u ˜−u ˜1 . Then we have ∂v + Av + (F (., u∗ ) − F (., u∗1 )) + K(, v) = Bφ on Q, ∂t v(0) = 0 on Ω, −

∂˜ v + A˜ v + (G(., u∗1 ))∗ v˜ + K ∗ (, v˜) ∂t ˜ on Q, = C ∗ Cv − (G(., u∗ ) − G(., u∗1 ))∗ u

(7.106)

(7.107)

v˜(T ) = μv(T ) on Ω and

T α φ 2L2 (Q) +

0

B ∗ v˜φdxdt ≤ 0.

(7.108)

Ω

According to assumptions (7.78) and (7.80) we have −2 F (., u∗ ) − F (., u∗1 ), v ≤ γ0 c2I v 2D , −2 (G(., u∗1 ))∗ v˜, v˜ ≤ γ0 c2I v˜ 2D , ∗ ∗ ∗ | (G(., u ) − G(., u1 )) u ˜, v˜ |≤ λ | v || v˜ || u˜ | dx.

(7.109)

Ω

Multiplying (7.106) by v, (7.107) by v˜ and integrating over Q, we obtain (according to (7.108), (7.109), and assumption (7.91)) T T T d v 2L2 ds + ν v 2D ds ≤ γc2I v 2D ds dt 0 0 0 2 ∗ + B v˜ L2 (Q) B ∗ v L2 (Q) , α T T T d 2 2 2 v˜ L2 ds + ν v˜ D ds ≤ γcI v˜ 2D ds − 0 dt 0 0 T T 2 2 2 |u ˜ | | v˜ | | v | dxds, +(δ1 cI + δ2 ) ( v D + v˜ D )ds + 2λ 0

v˜(T ) = μv(T ) and v(0) = 0 on Ω.

0

Ω


219

By using H¨ older’s inequality and the relationship (7.79) we obtain T d 2 2 v L2 ds + (ν − γcI ) v 2D ds 0 dt 0 2 2 2 T ( v 2D + v˜ 2D )ds, ≤ b cI α 0 T T d 2 2 v˜ L2 ds + (ν − γcI ) v˜ 2D ds − 0 dt 0 T 2 2 ≤ (δ1 cI + δ2 + λce u˜ H ) ( v 2D + v˜ 2D )ds,

T

(7.110)

0

v˜(T ) = μv(T ) and v(0) = 0 on Ω. Adding first and second inequalities of (7.110) we obtain 0

T

T d 2 2 ( v L2 − v˜ L2 )ds + θ ( v 2D + v˜ 2D )ds dt 0 T 2 ˜ H ( v 2D + v˜ 2D )ds, ≤ λce u

(7.111)

0

where θ = ν − δ2 − c2I (γ + (2b2 /α) + δ1 ) > 0 (according to assumption (i)). According to the third equalities of (7.110) we have (1 − μ2 ) v(T ) 2L2 + v˜(0) 2L2 +θ( v 2V + v˜ 2V ) ≤ λc2e u ˜ H ( v 2V + v˜ 2V ). Since 1 − μ2 > 0 then θ( v 2V + v˜ 2V ) ≤ λc2e u ˜ H ( v 2V + v˜ 2V ). By choosing r = 1 in (7.104) (since ν > δ2 , according to assumption (i)) and by using Gronwall’s formula, we can deduce that u˜ 2H ≤ e(δ1 +γ+1)T (μ2 u∗ (T ) − vobs 2L2 + C(u∗ − uobs ) 2L2 (Q) ), (7.112) and then θ∗ ( v 2V + v˜ 2V ) ≤ 0, where (since μ < 1)

1/2 . θ∗ = θ − λc2e e(δ1 +γ+1)T /2 u∗ (T ) − vobs 2L2 + C(u∗ − uobs ) 2L2 (Q) Since θ∗ > 0 (according to assumption (ii)), we have v 2V + v˜ 2V = 0 and then v = 0 and v˜ = 0. The proof of the uniqueness is complete. 7.6.4 Initial Condition Control In this subsection the objective of the optimal control problem is to find the best estimate of the initial state u0 .

220


We suppose now that the value u0 is decomposed into a known function g ∈ L2 (Ω) and the control φ ∈ L2 (Ω), i.e., u0 = g + Bφ, where B is a given bounded operator on L2 (Ω). So the function u is assumed to be related to the control φ through Problem (7.77): ∂u + Au + F (x, t, u) + K(, u) = f (given) on Q, ∂t u(0) = g + Bφ on Ω.

(7.113)

To obtain the regularity of Proposition 7.23, we suppose the following hypothesis: f ∈ L2 (Q), g ∈ L2 (Ω) and φ ∈ L2 (Ω). Let U : φ −→ u = U(φ) be the map: L2 (Ω) −→ W defined by (7.113) and let us introduce the cost function defined by J(φ) =

μ α 1 C(u − uobs ) 2L2 (Q) + u(T ) − vobs 2L2 + φ 2L2 , 2 2 2

(7.114)

where μ ≥ 0 and α > 0 are fixed parameters, (uobs , vobs ) ∈ V × L2 (Ω) is the observation and C is an unbounded linear operator on L2 (Ω) satisfying assumption (7.91). We want to minimize the functional J with respect to φ subject to Problem (7.113), i.e., to find φ∗ ∈ Uad such that J(φ∗ ) ≤ J(φ), ∀φ ∈ Uad

(7.115)

with Uad is a given non-empty, closed and convex subset of L2 (Ω). By using the same technique as used in Propositions 7.23 and 7.24 and Theorem 7.27, we have the following results (with no further estimates required). Proposition 7.31. Let F be an operator that satisfies assumptions (7.78). Then, the function U : φ −→ u = U(φ) solution of (7.113) is continuously F-differentiable from L2 (Ω) to W and its derivative w = U (φ)h at point φ ∈ L2 (Ω) in the direction h ∈ L2 (Ω) is given by the linear parabolic problem ∂w + Aw + G(x, t, U(φ))w + K(, w) = 0 on Q, ∂t w(0) = Bh on Ω.

(7.116)

Moreover, for all φ ∈ L2 (Ω) we have the following estimate: U (φ) L(L2 (Ω);W) ≤ C.

Proposition 7.32. Let F be an operator that satisfies assumptions (7.78). Then, for each t ∈ [0, T ], the function Vt : φ −→ u(t) = Vt (φ) solution of (7.113) is continuously F-differentiable from L2 (Ω) to L2 (Ω) and its derivative w(t) = Vt (φ).h at point φ ∈ L2 (Ω) in the direction h ∈ L2 (Ω) is given by


221

the linear parabolic problem ∂w + Aw + G(x, t, Vt (φ))w + K(, w) = 0 on Q, ∂t w(0) = Bh on Ω.

(7.117)

Moreover, for all φ ∈ L2 (Q) the following estimate holds. Vt (φ) L(L2 (Ω);L2 (Ω)) ≤ C.

Theorem 7.33. Let F be an operator that satisfies assumptions (7.78). Then, there exist φ∗ ∈ L2 (Ω) and u∗ ∈ W such that φ∗ is defined by (7.115) and u∗ = U(φ∗ ) is a solution of (7.113). In order to characterize the solution of the optimal control problem, we use the “adjoint” problem corresponding to the primal problem (7.113) (we denote by u = U(φ) the solution of problem (7.113) where the initial condition is φ): ∂u ˜ + A˜ u + (G(x, t, u))∗ u ˜ + K ∗ (, u ˜) = C ∗ C(u − uobs ) on Q, ∂t subject to the final condition −

(7.118)

u˜(T ) = μ(u(T ) − vobs ) on Ω, where C ∗ (resp. (G(., u))∗ ) is the adjoint of the operator C (resp. G(., u)). Remark 7.34. The adjoint problem (7.118) is the same as the problem (7.103), then the result of Proposition 7.28 remains valid. ♦ We can now give the optimality system for the optimal control problem (7.115). Theorem 7.35. Let F be an operator that satisfies assumptions (7.78), ν > δ2 and (φ∗ , u∗ ) ∈ Uad × W such that φ∗ is defined by (7.115) and u∗ = U(φ∗ ) is a solution of (7.113). Then (αφ∗ + B ∗ u ˜(0))(φ − φ∗ )dx ≥ 0 ∀φ ∈ Uad , Ω

˜ ∗ ) is a solution of the adjoint problem (7.118), corresponding where u˜ = U(φ to the primal solution u∗ . Moreover, the gradient of the cost functional J at φ∗ is given by J (φ∗ ) = αφ∗ + B ∗ u˜(0). Proof. The cost functional J is a composition of F-differentiable mappings then J is F-differentiable and for all h ∈ L2 (Ω) we have

222


J (φ).h =

T

C(u − uobs ), CU (φ)hds + μ(u(T ) − vobs ), VT (φ)h + αφ, h

0

and then J (φ).h =

T

C ∗ C(u − uobs ), U (φ)hds + μ(u(T ) − vobs ), VT (φ)h + αφ, h.

0

Multiplying (7.116) by u ˜ and integrating over Q , we obtain that for all h ∈ L2 (Q)

g + Bh, u˜(0) = VT (φ∗ )h, u˜(T ) T ∂u ˜ + A˜ u + (G(., u∗ ))∗ u ˜ + K ∗ (, u˜), U (φ∗ )hds. + − ∂t 0 Since u ˜ is a solution of the adjoint problem we can deduce that for all h ∈ L2 (Ω)

g + Bh, u ˜(0) = VT (φ∗ )h, μ(u∗ (T ) − vobs ) T + C ∗ C(u∗ − uobs ), U (φ∗ )hds. 0

Then we infer from the expression of J that for all h ∈ L2 (Ω) J (φ).h = αφ + B ∗ u˜(0), h.

(7.119)

Since φ∗ is a solution of (7.115) then J (φ∗ ).(φ − φ∗ ) ≥ 0, ∀φ ∈ Uad and therefore, from (7.119), we can deduce the result of this theorem. This completes the proof. In the sequel, we will assume that there exists a pair (φ∗ , u∗ ) such that φ∗ is defined by (7.115), u∗ = U(φ∗ ) is a solution of (7.113) and (αφ∗ + B ∗ u ˜(0))(φ − φ∗ )dx ≥ 0 Ω

˜ ∗ ) is a solution of (7.118). ˜ = U(φ for all φ ∈ Uad , where u We can now give some conditions to obtain the uniqueness of the optimal solution (φ∗ , u∗ ). Theorem 7.36. Suppose that F satisfies assumptions (7.78), α ≥ b2 and μ < 1 holds. If the following assumptions hold: (i) θ = (ν − δ2 − c2I (γ + δ1 )) > 0

1/2 (ii) λc2e e(δ1 +γ+1)T /2 u∗ (T ) − vobs 2L2 + C(u∗ − uobs ) 2L2 (Q) < θ, then the optimal solution (φ∗ , u∗ ), of the problem (7.115) subject to the problem (7.113), is unique.


223

Proof. Assume that (φ∗1 , u∗1 ) is another optimal solution, then φ∗1 satisfies (7.115), u∗1 = U(φ∗1 ) is a solution of (7.113) and (αφ∗1 + B ∗ u ˜1 (0))(φ − φ∗1 )dx ≥ 0 Ω

˜ ∗1 ) is a solution of (7.118). for all φ ∈ Uad , where u ˜1 = U(φ ∗ ∗ ˜−u ˜1 , and can deduce that v We set φ = φ − φ1 , v = u∗ − u∗1 and v˜ = u satisfies ∂v + Av + (F (x, t, u∗ ) − F (x, t, u∗1 )) + K(, v) = 0 on Q, ∂t (7.120) v(0) = Bφ on Ω, ∂˜ v + A˜ v + (G(., u∗1 ))∗ v˜ + K ∗ (, v˜) ∂t = C ∗ Cv − (G(., u∗ ) − G(., u∗1 ))∗ u˜ on Q, v˜(t = T ) = μv(t = T ) on Ω

−

and α φ 2L2 (Ω) +

B ∗ v˜(0)φdx ≤ 0,

(7.121)

(7.122)

Ω

Multiplying (7.120) by v, (7.121) by v˜ and integrating over Ω we obtain that (according to (7.122), (7.91) and (7.109)) d v 2L2 +ν v 2D ≤ γc2I v 2D , dt d − v˜ 2L2 +ν v˜ 2D ≤ γc2I v˜ 2D dt +(δ1 c2I + δ2 )( v 2D + v˜ 2D ) + 2λ

|u ˜ | | v˜ | | v | dx, Ω

v˜(T ) = μv(T ) and v(0) = Bφ on Ω. Using Hölder’s inequality and the relationship (7.79) we have that d v 2L2 +(ν − γc2I ) v 2D ≤ 0, dt d − v˜ 2L2 +(ν − γc2I ) v˜ 2D dt ≤ (δ1 c2I + δ2 + λc2e u ˜ H )( v 2D + v˜ 2D ),

(7.123)

v˜(T ) = μv(T ) and v(0) = Bφ on Ω. Summing the first and the second inequality of (7.123) we obtain that (according to ν > δ2 ) d ( v 2L2 − v˜ 2L2 ) + θ( v 2D + v˜ 2D ) dt ≤ λc2e u˜ H ( v 2D + v˜ 2D ), v˜(T ) = μv(T ) and v(0) = Bφ,

(7.124)

224


where θ = ν − δ2 − c2I (2γ + δ1 ) > 0 (according to the assumption (i)). Integrating over [0, T ] and according to the third part of (7.123) we have that (1 − μ2 ) v(T ) 2L2 + v˜(0) 2L2 +θ( v 2V + v˜ 2V ) ≤ λc2e u ˜ H ( v 2V + v˜ 2V ) +

b4 v˜(0) 2L2 . α2

Since 1 − μ2 > 0 and 1 − b4 /α2 ≥ 0 then ˜ H ( v 2V + v˜ 2V ). θ( v 2V + v˜ 2V ) ≤ λc2e u According to the relation (7.112) (see Remark 7.34) and that μ < 1 we can deduce that θ∗ ( v 2V + v˜ 2V ) ≤ 0, where

1/2 . θ∗ = θ − λc2e e(δ1 +γ+1)T /2 u∗ (T ) − vobs 2L2 + C(u∗ − uobs ) 2L2 (Q) Since θ∗ > 0 (according to assumption (ii)), we have v 2V + v˜ 2V = 0 and then v = 0 and v˜ = 0. The uniqueness result is proved. 7.6.5 Example We present here a more practical example dealing with a reaction–diffusion– transport system. The system is governed by: ∂u − div(β∇u) + .∇u + F (x, t, u) = f (x, t), ∂t u(., t) = 0 on Σ = (0, T ) × ∂Ω

(x, t) ∈ Q (7.125)

u(0) = u0 on Ω. Here, Ω is an open bounded set in IRm , m ≥ 1, with the boundary ∂Ω sufficiently regular, for example the oceanic (or atmospheric) domain: U represents the concentration, for example, of some biochemical pollutants in the studied domain Ω, β denotes the coefficient of eddy diffusivity, and is the three-dimensional velocity field of the fluid or the air. The second term in the equation accounts for pollutant movement by diffusion, the third term represents the transport of the pollutant by the flow field and the fourth term F (non-linear function) represents the reaction term which models the interaction between the pollutant and others biological elements (e.g., phytoplankton, zooplankton, nutrients, etc.). The function F is depended, in addition to the concentration U , on the concentrations of the other biological elements, represented by a vector function C, and on some parameters which describe


225

the interaction between the pollutant and these other biological elements, represented by a vector function P, i.e., F (x, t, U ) := F (C(x, t), P(x, t), U ). The right-hand side of the equation, f , may consist of agents (for example, biological agents capable of producing biodegradation of the pollutant), or chemical or physical extraction. Remark 7.37. The oceanic and atmospheric circulations are strongly sensitive to the parametrization of the vertical turbulent diffusion. The turbulent flux are usually modeled by the dissipative term (which corresponds to Reynolds stresses with a coefficient of eddy viscosity β) and linked to large-scale oceans (or atmospheres) by using the mixing coefficients βh and βv (which have very different behaviors): βh (resp. βv ) denotes the coefficient of horizontal (resp. vertical) eddy viscosity. ♦ For the mathematical setting, we take D = H01 (Ω) and D = H −1 (Ω). According to Sobolev embedding theorem, we have that D is embedded in L4 (Ω) provided m ≤ 4. Physically this condition on the dimension is not too restrictive, since in this case m = 3. The operator A is −div(β∇.) with Dirichlet boundary condition and K is the operator .∇. For the non-linear operators we suppose that they satisfy the hypothesis (7.78). If we assume that the coefficient β is positive and bounded function, then the operator A is continuous and coercive and we denote the coercivity constant by ν = min(β). Ω

Assume now that the velocity field is known and satisfies ∈ L∞ (Q) and div() = 0, so we have easily the estimate (7.81), where γ∞ = 2L∞ (Q) /ν and that K ∗ = −K. Remark 7.38. If we suppose that the initial condition is in H01 (Ω) (compatibility condition) we can obtain for the non-linear problem (7.125), by removing the terms F (., u) + K(, u) on the right-hand side of (7.125) and by using ♦ Theorem 6.20, the regularity result u ∈ H 2,1 (Q). In the cost functional the operator C represents the regional and temporal variation in the cost of pollutant extraction. The observation denotes the acceptable pollution level in the studied region. Since all the assumptions of our abstract results are satisfied by the example in this particular case, so our study applies. Remark 7.39. The vector function P, in the real application, is often badly known. So, we can use the parameter estimation technique in order to identify this vector by using, for example, satellite data or the ocean colour satellites, which measure the colour of the ocean’s surface (this instrument is often used to measure the concentration of chlorophyll in surface waters, from which the concentration of the desired biological or biochemical element can be inferred). ♦

8 Stabilization and Robust Control Problem

This chapter contains the essential and fundamental developments of the robust control theory of distributed parameter systems. This area concerns the investigation of the control, stability and adjoint control optimization of infinite-dimensional dissipative dynamical systems. The considered systems are derived from spatially and time-dependent partial differential equations associated with boundary-value problems. We recall that in our approach it is not assumed that the system is stabilizable or detectable. Moreover, we are interested in the robust regulation of the deviation of the systems from the desired target, by analyzing full non-linear systems, which models large perturbations to the desired target. The objective of this chapter is to describe robust control theory in different situations. First, we use a very basic problem to explain the theory as simply as possible. Second, we present the theory for general linear evolutive problems. Then, we present an estimate parameter problem in the case of bilinear systems. Finally, we analyze full non-linear robust control problems, with particular attention to time-delay problems. In each section, we study the existence, uniqueness and optimality conditions for the optimal solution (by using adjoint problems). We develop our study for different realistic cases of observations and controls. The numerical aspect (based on the adjoint problem) is presented in the next chapter.

8.1 Motivation and Objectives The goal of the robust control theory of partial differential equations (PDEs) is to take into account uncertainty (such as discrepancy or errors between reality and mathematical models used for controller design, and unmeasured noises and disturbances that act on the physical, biological or economical plants, fluctuations, etc.) and instability (because uncertainty modifies the system behavior).

228


In our approach, we transform robust stability and performance problems into constrained game-type minimax optimization ones, and in turn transform these into unconstrained game-type problems. The objective of robust control is then to compensate for the undesirable effects of system disturbances through control actions such that a cost function achieves its minimum for the worst disturbances, i.e., to find the best control which takes into account the worst-case disturbance. The problem is then to find a saddle point of a functional calculus (called cost or objective or performance functional) depending on the control and the disturbance (intervening either in the initial conditions, or in the boundary conditions or in the equation itself) and on the solution of the perturbed PDE. For more convenience, we recall here the data using in robust control problems (see Chapter 1): 1. A “control” variable ϕ in a set Uad (set of “admissible controls”) and a “disturbance” variable ψ in a set Vad (set of “admissible disturbances”). 2. The state u(ϕ, ψ) of the system to be controlled, which is given, for a chosen control-disturbance (ϕ, ψ), by the resolution of a perturbed equation F˜ u(ϕ, ψ) = “given function of (ϕ, ψ)” where F˜ is an operator (supposed to be known) which represents the system to be controlled (F˜ is the perturbation of the model F of the studied system). 3. An “observation” uobs which is supposed to be known exactly1 (for example the desired tolerance for the perturbation or the offset given by measurements). 4. A “cost” functional J(ϕ, ψ) which is defined from a numerical and positive function G(X, Y ) by J(ϕ, ψ) = G((ϕ, ψ), u(ϕ, ψ)). We want to find a saddle point of J, i.e., a solution (ϕ∗ , ψ ∗ ) ∈ Uad × Vad of J(ϕ∗ , ψ) ≤ J(ϕ∗ , ψ ∗ ) ≤ J(ϕ, ψ ∗ ) ∀ϕ ∈ Uad , ψ ∈ Vad . In a similar way to optimal control problems, it should be noted that there is no general method to analyze the problems of robust control but only a process to be followed for each situation: (i)

solve the initial problem (analysis of PDEs, existence of solutions, stability according to the data, regularity, differentiability of the operators, etc.)

(ii)

define the function or the parameter to be identified and the type of disturbance to be controlled

1

In order to simplify, the reader can assume that this function is null.

8.2 Basic Framework

229

(iii) introduce and solve the perturbed problem which plays the role of the primal problem (analysis of PDEs, existence of solutions, stability according to the data, regularity, differentiability of the operators, etc.) (iv) define the cost functional, which is dependent on control and disturbance functions (v)

obtain the existence of an optimal solution (as a saddle point of the cost functional) and analyze the necessary (and if possible the sufficient) conditions of optimality (which require to obtain before a very fine regularity on the state functions)

(vi) characterize the optimal solutions (vii) define an algorithm that solves numerically the robust control problem. We shall now present a basic framework for a robust control problem.

8.2 Basic Framework In this section, we take an abstract boundary value problem. We give the space of controls and disturbances, the admissibility set of controls and of disturbances, the control and the disturbance variables, and the observation operator. The robust control we consider is to maintain the target state variables by taking into account the influence of data noise, while the desired power level and adjustment costs are taken into consideration. We define the cost functional and the adjoint problem, corresponding to the primal problem. The main result of this study is the characterization of optimal solutions. Let F be a continuous linear partial differential operator from X into E, where X and E are, for example, Hilbert spaces. The space X contains in its definition some appropriate boundary conditions. We assume that the corresponding boundary value problem is well-posed (or correctly-set) in Hadamard sense, i.e., F is an isomorphism from X onto E. Let us consider the abstract boundary-value problem as the form F (w) = h ∈ E.

(8.1)

Problem (8.1) admits a unique solution w ∈ X. In the following, the solution w of problem (8.1) will be treated as the target function. We are then interested in the robust regulation of the deviation of the problem from the desired target w. We study the system, which models large perturbation u to the target w. Hence we consider the following perturbed system: F (w + u) = h + g, (8.2) where g is in E. Since the operator F is linear and w is a solution of (8.1) then the perturbation u satisfies F (u) = g, (8.3)

230


where g is in E. Let U1 and U2 be two spaces of controls and disturbances, respectively, which are assumed to be, for example, Hilbert spaces. Assume now that the value g is decomposed into the disturbance ψ ∈ U2 , the control ϕ ∈ U1 and a known function f ∈ E, i.e., g := f + B1 ϕ + B2 ψ, where Bi , for i = 1, 2, are bounded operators and such that Bi ∈ L(Ui ; E), for i = 1, 2. So, the function u is assumed to be related to the disturbance ψ and the control ϕ through the problem (8.3): F (u) = f + B1 ϕ + B2 ψ.

(8.4)

For every ϕ ∈ U1 , ψ ∈ U2 and f ∈ E, the problem (8.4) admits a unique solution u ∈ X such that u = F −1 (f + B1 ϕ + B2 ψ).

(8.5)

Let Uad be a closed convex non-empty subset of U1 (the admissibility set of controls), Vad be a closed convex non-empty subset of U2 (the admissibility set of disturbances) and C ∈ L(X; M ) (where M , for example, is a Hilbert space), be the observation operator. The robust control we consider, for example, is to maintain the target state variables while the desired power level and adjustment costs are taken into consideration. In particular, we will study the following robust control problem: find (u, ϕ, ψ) ∈ X × Uad × Vad such that the following cost functional, in the reduced form J(ϕ, ψ) =

β γ 1 Cu(ϕ, ψ) − uobs 2X + ϕ 2U1 − ψ 2U2 2 2 2

(8.6)

is minimized with respect to ϕ and maximized with respect to ψ subject to the problem (8.4), where uobs ∈ M is the target (the observation given, for example, by experiment measurements) and the constant β > 0 and γ > 0. Remark 8.1. (i) The coefficient β can be interpreted as the measure of the price of the control (that the engineer can afford) and the coefficient γ can be interpreted as the measure of the price of the disturbance (that the nature or environnement can afford). (ii) The cost functional J depends on the control ϕ, the disturbance ψ and the state function u(ϕ, ψ) (solution of problem (8.4) corresponding to the data (ϕ, ψ)). In order to simplify the presentation we have used, in the expression (8.6), the reduced form of the functional J, i.e., J(ϕ, ψ) in place of the form J(ϕ, ψ, u(ϕ, ψ)). ♦ Since v −→ Cv−uobs is a continuous affine function from X into M and the norm is a continuous function then the functional J is continuous. Moreover,

8.2 Basic Framework

231

according to the expression of J, the functional Jψ : ϕ ∈ U1 −→ J(ϕ, ψ) is strictly convex, the functional Jϕ : ψ ∈ U2 −→ J(ϕ, ψ) is strictly concave, and Jψ (ϕ) −→ +∞ if ϕ ∈ Uad , ϕ U1 −→ ∞, Jϕ (ψ) −→ −∞ if ψ ∈ Vad , ψ U2 −→ ∞. Consequently, there exists a unique optimal solution (see Section 5.3) (ϕ∗ , ψ ∗ , u∗ ) ∈ Uad × Vad × X such that (ϕ∗ , ψ ∗ ) is the saddle point of J, i.e., the solution of J(ϕ∗ , ψ) ≤ J(ϕ∗ , ψ ∗ ) ≤ J(ϕ, ψ ∗ )

∀ϕ ∈ Uad , ψ ∈ Vad

(8.7)

and u∗ = u(ϕ∗ , ψ ∗ ) is the solution of (8.4), corresponding to (ϕ∗ , ψ ∗ ). It follows directly that the functional J is G-differentiable, and then the saddle point solution (ϕ∗ , ψ ∗ ) satisfies the following inequalities (see also Section 5.3): ∂J ∗ ∗ (ϕ , ψ ).(ϕ − ϕ∗ ) ≥ 0 ∀ϕ ∈ Uad , ∂ϕ (8.8) ∂J ∗ ∗ (ϕ , ψ ).(ψ − ψ ∗ ) ≤ 0 ∀ψ ∈ Vad . ∂ψ Since u(ϕ, ψ) satisfies (8.5) then, by differentiation, we have that ∂u ∗ ∗ (ϕ , ψ ).θ = F −1 (B1 θ), ∀θ ∈ U1 , ∂ϕ ∂u ∗ ∗ (ϕ , ψ ).ζ = F −1 (B2 ζ), ∀ζ ∈ U2 ∂ψ and, in particular, for all ϕ ∈ Uad and ψ ∈ Vad , we have ∂u ∗ ∗ (ϕ , ψ ).(ϕ − ϕ∗ ) = F −1 (B1 (ϕ − ϕ∗ )) = u(ϕ, ψ ∗ ) − u(ϕ∗ , ψ ∗ ), ∂ϕ ∂u ∗ ∗ (ϕ , ψ ).(ψ − ψ ∗ ) = F −1 (B2 (ψ − ψ ∗ )) = u(ϕ∗ , ψ) − u(ϕ∗ , ψ ∗ ). ∂ψ

(8.9)

According to (8.9), the condition (8.8) can be written as (for all ϕ ∈ Uad and ψ ∈ Vad ) (Cu(ϕ∗ , ψ ∗ ) − uobs , C(u(ϕ, ψ ∗ ) − u(ϕ∗ , ψ ∗ )))X +β(ϕ∗ , ϕ − ϕ∗ )U1 ≥ 0, (Cu(ϕ∗ , ψ ∗ ) − uobs , C(u(ϕ∗ , ψ) − u(ϕ∗ , ψ ∗ )))X −γ(ψ ∗ , ψ − ψ ∗ )U2 ≤ 0.

(8.10)

In order to simplify the relation (8.10), we introduce the adjoint problem u(ϕ, ψ)) = C ∗ Λ(Cu(ϕ, ψ) − uobs ), F ∗ (˜

(8.11)

where u(ϕ, ψ) is the solution of the primal problem (8.4), corresponding to data (ϕ, ψ) ∈ U1 × U2 , C ∗ ∈ L(M , X ) is the adjoint of the operator C, F ∗ is the adjoint of the operator solution F and Λ is the canonical isomorphism from M onto M (such that p, q ∗ M,M = (Λp, q ∗ )M = (p, Λ−1 q ∗ )M , ∀(p, q ∗ ) ∈ M × M ), where M (respectively X ) is the dual of M (respectively of X).

232


Then (Cu(ϕ∗ , ψ ∗ ) − uobs , C(u(ϕ, ψ ∗ ) − u(ϕ∗ , ψ ∗ )))X = C ∗ Λ(Cu(ϕ∗ , ψ ∗ ) − uobs ), u(ϕ, ψ ∗ ) − u(ϕ∗ , ψ ∗ )X ,X = F ∗ (˜ u∗ ), u(ϕ, ψ ∗ ) − u(ϕ∗ , ψ ∗ )X ,X = (˜ u∗ , F (u(ϕ, ψ ∗ ) − u(ϕ∗ , ψ ∗ ))E ,E = (˜ u∗ , B1 (ϕ − ϕ∗ )E ,E ∗ ∗ = (Λ−1 ˜ , ϕ − ϕ∗ )U1 , ∀ϕ ∈ Uad U1 B1 u

and (in the same way) (Cu(ϕ∗ , ψ ∗ ) − uobs , C(u(ϕ∗ , ψ) − u(ϕ∗ , ψ ∗ )))X ∗ ∗ = (Λ−1 ˜ , ψ − ψ ∗ )U2 , ∀ψ ∈ Vad , U2 B2 u

where Bi∗ ∈ L(E , Ui ) is the adjoint of Bi , Ui (respectively E ) is the dual of Ui (respectively of E) and ΛUi is the canonical isomorphism from Ui onto Ui , ˜(ϕ∗ , ψ ∗ ) is the solution of the adjoint problem (8.11), for i = 1, 2, and u ˜∗ = u corresponding to the primal solution u(ϕ∗ , ψ ∗ ). Thus, the condition (8.10) can be written as ∗ ∗ ˜ + βϕ∗ , ϕ − ϕ∗ )U1 ≥ 0, ∀ϕ ∈ Uad , (Λ−1 U1 B1 u ∗ ∗ (Λ−1 ˜ − γψ ∗ , ψ − ψ ∗ )U2 ≤ 0, ∀ψ ∈ Vad . U2 B2 u

We have proved that the optimal solution (ϕ∗ , ψ ∗ , u∗ ) ∈ Uad × Vad × X, solution of the problem (8.7) subject to the problem (8.4), is characterized by the following optimality system which includes the direct (or primal) problem and the adjoint problem, linked by inequalities: F (u) = f + B1 ϕ + B2 ψ, u∗ ) = C ∗ Λ(Cu∗ − uobs ), F ∗ (˜ ∗ ∗ ˜ + βϕ∗ , ϕ − ϕ∗ )U1 ≥ 0, ∀ϕ ∈ Uad , (Λ−1 U1 B1 u

(8.12)

∗ ∗ ˜ − γψ ∗ , ψ − ψ ∗ )U2 ≤ 0, ∀ψ ∈ Vad . (Λ−1 U2 B2 u

8.3 Linear Robust Control Problems In this section, we consider problems of linear parabolic type in a more general framework (the same problems as in the previous chapter). The notations, the assumptions and the spaces are the same as in Chapter 6. 8.3.1 Position of the Problem, and the Existence and Uniqueness of the Optimal Solution For f and U0 given such that f ∈ L2 (0, T ; V ) and U0 ∈ H, and f0 given in

8.3 Linear Robust Control Problems

233

the space L2 (0, T ; V ), we consider the following linear evolutive equation: ∂U + A(t)U = f0 + f ∈ L2 (0, T ; V ), ∂t U (0) = U0 ∈ H.

(8.13)

According to Theorem 6.8 and Proposition 6.9, the problem (8.13) admits a unique solution U ∈ W ∩ C([0, T ]; H) (since f ∈ L2 (0, T ; V ), U0 ∈ H and g ∈ L2 (0, T ; V )), where W = {u ∈ L2 (0, T ; V ) : ∂u/∂t ∈ L2 (0, T ; V )}. Moreover, if v1 ∈ W (respectively v2 ∈ W) is the solution of (8.13), with data (f0 , f1 , v01 ) ∈ L2 (0, T ; V ) × L2 (0, T ; V ) × H (respectively with data (f0 , f2 , v02 ) ∈ L2 (0, T ; V ) × L2 (0, T ; V ) × H), then v1 − v2 2W ≤ C( f1 − f2 2L2 (0,T ;V ) + v01 − v02 2H ). The solution U with the corresponding forcing f will be referred as the target function for the control problem. We are interested in the robust regulation of the deviation of the problem from the desired target (U, f ). We analyze the full equation which models large perturbations (u, g) to the target (U, f ) with known initial conditions. Hence we consider the equation of the perturbation (since the problem (8.13) is linear) ∂u + A(t)u = g on Q, ∂t u(0) = u0 on Ω,

(8.14)

where g and u0 are perturbations of f and U0 respectively. The problem (8.14) is similar to (8.13) so we have the existence and the uniqueness of the solution u in W, and the following Lipschitz continuity result: u1 − u2 2W ≤ C( g1 − g2 2L2 (0,T ;V ) + u01 − u02 2H ),

(8.15)

where u1 ∈ W (respectively u2 ∈ W) is the solution of (8.14) with data (g1 , u01 ) ∈ L2 (0, T ; V ) × H (respectively with (g2 , u02 ) ∈ L2 (0, T ; V ) × H). In our control framework, the value g is decomposed into the disturbance ψ ∈ U2 and the control φ ∈ U1 . Thus, we write g = B1 φ + B2 ψ, where Bi are given in L(Ui , L2 (0, T ; V )) for i = 1, 2, U1 is the space of controls and U2 is the space of disturbances which are assumed to be Hilbert spaces. The objective in the robust control problem is to find the best control φ in the presence of the disturbance (or noise) ψ which maximally spoils the control objective. The function u is assumed to be related to the disturbance ψ and control φ through the problem (8.14)

234


∂u + A(t)u = B1 φ + B2 ψ on Q, ∂t u(0) = u0 on Ω.

(8.16)

For u0 given in H, we consider the mapping solution U : (φ, ψ) −→ u = U(φ, ψ) from U1 × U2 into W defined by (8.16). In this section, the cost functional is given, in the reduced form, by J(φ, ψ) =

1 β γ Cu(φ, ψ) − uobs 2M + φ 2U1 − ψ 2U2 , 2 2 2

(8.17)

where the scalar control parameters α, β > 0 are fixed and uobs ∈ M is a known observation. The observation operator C is such that C ∈ L(W; M ), with M a Hilbert space. The robust control problem corresponds to seek saddle points on Uad × Vad of the functional J, i.e., to find (φ∗ , ψ ∗ ) ∈ Uad × Vad such that J(φ∗ , ψ) ≤ J(φ∗ , ψ ∗ ) ≤ J(φ, ψ ∗ ) ∀(φ, ψ) ∈ Uad × Vad ,

(8.18)

with the admissibility set of controls Uad (respectively the admissibility set of disturbances Vad ) is (given) non-empty, closed, convex, bounded subsets of U1 (respectively of U2 ). Since v −→ Cv − uobs is a continuous affine function from L2 (0, T ; V ) into M , the norm is a lower semi-continuous function we have that the map Pψ : φ −→ J(φ, ψ) is lower semi-continuous for all ψ ∈ U2 and the map Qφ : ψ −→ J(φ, ψ) is upper semi-continuous for all φ ∈ U1 . Moreover, according to the nature of J, we have that the maps Pψ and Qφ are strictly convex and strictly concave respectively. Consequently, (by using the minimax theorems in infinite dimensions presented in Chapter 5) there exists a unique optimal solution (u∗ , φ∗ , ψ ∗ ) ∈ W × Uad × Vad such that (φ∗ , ψ ∗ ) is the solution of (8.18) and u∗ = U(φ∗ , ψ ∗ ) is the solution of (8.16), corresponding to (φ∗ , ψ ∗ ). Remark 8.2. In order to obtain the existence and the uniqueness of the optimal solution, it is not necessary to impose the boundedness condition on the spaces Uad and Vad . Indeed, Pψ (φ) −→ +∞ if φ ∈ Uad , φ U1 −→ ∞, and ♦ Qφ (ψ) −→ −∞ if ψ ∈ Vad , ψ U2 −→ ∞. 8.3.2 Optimality Conditions and Identification of the Gradients As in the basic framework (Section 8.2), the functional J is differentiable and (φ∗ , ψ ∗ , u∗ ) ∈ Uad × Vad × W is an optimal solution if and only if ∂J ∗ ∗ (φ , ψ ).(φ − φ∗ ) ≥ 0 ∂φ ∂J ∗ ∗ (φ , ψ ).(ψ − ψ ∗ ) ≤ 0 ∂ψ

∀φ ∈ Uad , ∀ψ ∈ Vad


235

and then ∂u ∗ ∗ (φ , ψ ).(φ − φ∗ ))M + β(φ∗ , φ − φ∗ )U1 ≥ 0 ∀φ ∈ Uad , ∂φ ∂u ∗ ∗ (φ , ψ ).(ψ − ψ ∗ ))M − γ(ψ ∗ , ψ − ψ ∗ )U2 ≤ 0 ∀ψ ∈ Vad , (Cu∗ − uobs , C( ∂ψ (Cu∗ − uobs , C(

where u∗ = U(φ∗ , ψ ∗ ) is the solution of (8.16), corresponding to the solution (φ∗ , ψ ∗ ). Since, for all (φ, ψ) ∈ U1 × U2 , the function U(φ, ψ) is a solution of (8.16) then the derivative of the operator solution U at point (φ∗ , ψ ∗ ) is defined by U (φ∗ , ψ ∗ ) : (h1 , h2 ) ∈ U1 × U2 −→ w ∈ W such that w=

∂u ∗ ∗ ∂u ∗ ∗ (φ , ψ ).h1 + (φ , ψ ).h2 ∂φ ∂ψ

is the unique solution of the following problem: ∂w + A(t)w = B1 h1 + B2 h2 on Q, ∂t w(0) = 0 on Ω.

(8.19)

The problem (8.19) is the same as the problem (8.13), with the null initial condition. Then we can deduce that ∂u ∗ ∗ (φ , ψ ).(φ − φ∗ ) = U(φ, ψ ∗ ) − U(φ∗ , ψ ∗ ) = U(φ, ψ ∗ ) − u∗ , ∂φ ∂u ∗ ∗ (φ , ψ ).(ψ − ψ ∗ ) = U(φ∗ , ψ) − U(φ∗ , ψ ∗ ) = U(φ∗ , ψ) − u∗ ∂ψ and then (Cu∗ − uobs , C(U(φ, ψ ∗ ) − u∗ ))M + β(φ∗ , φ − φ∗ )U1 ≥ 0 ∀φ ∈ Uad , (Cu∗ − uobs , C(U(φ∗ , ψ) − u∗ ))M − γ(ψ ∗ , ψ − ψ ∗ )U2 ≤ 0 ∀ψ ∈ Vad . Otherwise, for all φ ∈ Uad and ψ ∈ Vad ,

C ∗ Λ(Cu∗ − uobs ), U(φ, ψ ∗ ) − u∗ W ,W + β(φ∗ , φ − φ∗ )U1 ≥ 0,

C ∗ Λ(Cu∗ − uobs ), U(φ∗ , ψ) − u∗ W ,W − γ(ψ ∗ , ψ − ψ ∗ )U2 ≤ 0,

(8.20)

where Λ is the canonical isomorphism from M onto M and C ∗ ∈ L(M , W ) the adjoint of C. In order to simplify the relation (8.20), we introduce the adjoint problem associated with the primal problem (8.16). For this, we consider the two following situations (as in the previous chapter): (i) the case of C ∈ L(L2 (0, T ; V ); M ) ⊂ L(W; M ) (ii) the case of the final observation: Cu = Du(T ), where D ∈ L(H; H) and M = H = M , since H = H.

236


Case (i) and the Optimality System In the case of the operator C ∈ L(L2 (0, T ; V ); M ), the adjoint operator C ∗ of C is in L(M , L2 (0, T ; V )) and the inequality (8.20), for all φ ∈ Uad and ψ ∈ Vad ), becomes T

C ∗ Λ(Cu∗ − uobs ), U(φ, ψ ∗ ) − u∗ V ,V dt + β(φ∗ , φ − φ∗ )U1 ≥ 0, 0 (8.21) T ∗ ∗ ∗ ∗ ∗ ∗

C Λ(Cu − uobs ), U(φ , ψ) − u V ,V dt − γ(ψ , ψ − ψ )U2 ≤ 0. 0

Let us introduce the following adjoint problem: ∂u ˜ + A∗ (t)˜ u = C ∗ Λ(Cu − uobs ) ∈ L2 (0, T ; V ), ∂t u ˜(t = T ) = 0 ∈ H, −

(8.22)

which admits a unique solution u ˜ ∈ W, where u = U(φ, ψ) ∈ W and Λ is the canonical isomorphism from M onto M . To prove this result, we change the variables of System (8.22) by reversing the sense of time, i.e., t := T − t and we apply the same way to obtain the result of Theorem 6.8. Multiplying the first equation of (8.22) by u∗ − U∗ (ψ), where U∗ (ψ) = U(φ∗ , ψ), and integrating with respect to time we have that T ∗ T ∂u ˜ , u∗ − U∗ (ψ)V ,V dt + −

A∗ (t)˜ u∗ , u∗ − U∗ (ψ)V ,V dt ∂t 0 0 T

C ∗ Λ(Cu∗ − uobs ), u∗ − U∗ (ψ)V ,V dt, = 0 ∗

where u˜ is the solution of (8.22) which corresponds to the primal solution u∗ . By using the result of Corollary 6.4, we can deduce that T T ∗ ∂U∗ (ψ) ∂u ˜ ∂u∗ ∗ ∗ , u − U (ψ)V ,V dt = − V,V dt

˜ u∗ , − ∂t ∂t ∂t 0 0 u∗ (0), (u∗ − U∗ (ψ))(0))H . −(˜ u∗ (T ), (u∗ − U∗ (ψ))(T ))H + (˜ Since u ˜∗ (T ) = 0 and (u∗ − U∗ (ψ))(0) = u0 − u0 = 0 then T T ∂u ˜∗ ∗ ∂u∗ ∂U∗ (ψ) ∗ −

˜ u∗ , , u − U (ψ)V ,V dt = − V,V dt ∂t ∂t ∂t 0 0 and consequently T T ∂U∗ (ψ) ∂u∗ − V,V dt +

˜ u∗ ,

˜ u∗ , A(t)(u∗ − U∗ (ψ))V,V dt ∂t ∂t 0 0 T ∗ ∗ ∗ ∗

C Λ(Cu − uobs ), u − U (ψ)V ,V dt. = 0

(8.23)


237

Since u∗ − U∗ (ψ) is the solution of ∂(u∗ − U∗ (ψ)) + A(t)(u∗ − U∗ (ψ)) = B2 (ψ ∗ − ψ), ∂t (u∗ − U∗ (ψ))(0) = 0, then (8.23) gives T T ∗ ∗ ∗ ∗

C Λ(Cu − uobs ), u − U (ψ)V ,V dt =

˜ u∗ , B2 (ψ ∗ − ψ)V,V dt 0

0

= B2∗ u˜∗ , ψ ∗ − ψU2 ,U2 ∗ ∗ = (Λ−1 ˜ , ψ ∗ − ψ)U2 , U2 B2 u

where the operator Bi∗ ∈ L(L2 (0, T ; V ); Ui ) is the adjoint of Bi and ΛUi is the canonical isomorphism from Ui onto Ui , for i = 1, 2. Therefore, the second inequality of (8.21) can be written as ∗ ∗ ˜ − γψ ∗ , ψ − ψ ∗ )U2 ≤ 0 ∀ψ ∈ Vad . (Λ−1 U2 B2 u

(8.24)

In the same way we can prove also that the first inequality of (8.21) can be written as ∗ ∗ ˜ + βφ∗ , φ − φ∗ )U1 ≥ 0 ∀φ ∈ Uad (8.25) (Λ−1 U1 B1 u and then we have proved the following theorem, which gives the existence, the uniqueness and the first-order optimality conditions of the optimal solution. Theorem 8.3. Under the assumptions of Theorem 6.8, we suppose furthermore that the operator C is in L(L2 (0, T ; V ); M ) and that the operators Bi are in L(Ui ; L2 (0, T ; V )), for i = 1, 2. Then there exists a unique optimal solution (φ∗ , ψ ∗ , u∗ ) ∈ U1 × U2 × W such that ∂u∗ + A(t)u∗ = B1 φ∗ + B2 ψ ∗ , ∂t u∗ (0) = u0 and

∗ ∗ ˜ + βφ∗ , φ − φ∗ )U1 ≥ 0 ∀φ ∈ Uad , (Λ−1 U1 B1 u ∗ ∗ ˜ − γψ ∗ , ψ − ψ ∗ )U2 ≤ 0 ∀ψ ∈ Vad , (Λ−1 U2 B2 u

where ΛUi is the canonical isomorphism from Ui onto Ui and u ˜∗ is the unique solution, which is in W, of the adjoint problem ∂u ˜∗ + A∗ (t)˜ u∗ = C ∗ Λ(Cu∗ − uobs ), ∂t u˜∗ (t = T ) = 0, −

where Λ is the canonical isomorphism from M onto M .

238


Case (ii) and the Optimality System In the case of Cu = Du(T ), where D ∈ L(H; H) and M = H, the observation uobs is in H, D∗ ∈ L(H; H) and the cost functional is defined by J(φ, ψ) =

1 β γ Du(T ) − uobs 2H + φ 2U1 − ψ 2U2 2 2 2

(8.26)

and for all (φ, ψ) ∈ Uad × Vad the inequality (8.20) is equivalent to (D∗ (Du∗ (T ) − uobs ), (U(φ, ψ ∗ ) − u∗ )(T ))H + β(φ∗ , φ − φ∗ )U1 ≥ 0, (D∗ (Du∗ (T ) − uobs ), (U(φ∗ , ψ) − u∗ )(T ))H − γ(ψ ∗ , ψ − ψ ∗ )U2 ≤ 0

(8.27)

where u∗ = U(φ∗ , ψ ∗ ) is the primal solution which corresponds to the optimal solution (φ∗ , ψ ∗ ). Let us introduce the following adjoint problem: ∂u ˜ + A∗ (t)˜ u = 0, ∂t u˜(t = T ) = D∗ (Du(t = T ) − uobs ) ∈ H,

−

(8.28)

which admits a unique solution u˜ ∈ W, where u = U(φ, ψ) ∈ W. To prove this result, we change the variables of System (8.28) by reversing the sense of time, i.e., t := T − t and we apply the same way to obtain the result of Theorem 6.8. Multiplying the first equation of (8.28) by u∗ − U∗ (ψ), where U∗ (ψ) = U(φ∗ , ψ), and integrating with respect to time we have that

∂u ˜∗ ∗

, u − U∗ (ψ)V ,V dt + ∂t

T

− 0

T

A∗ (t)˜ u∗ , u∗ − U∗ (ψ)V ,V dt = 0,

0

where u˜∗ is the solution of (8.28) which corresponds to the primal solution u∗ = U(φ∗ , ψ ∗ ). By using the result of Corollary 6.4, we can deduce that − 0

∂u ˜∗ ∗ , u − U∗ (ψ)V ,V dt =

∂t

T

∗

∗

T

˜ u∗ ,

0

∂U∗ (ψ) ∂u∗ − V,V dt ∂t ∂t

∗

−(˜ u (T ), (u − U (ψ))(T ))H + (˜ u∗ (0), (u∗ − U∗ (ψ))(0))H , Since u ˜∗ (T ) = D∗ (Du∗ (T ) − uobs ) and (u∗ − U∗ (ψ))(0) = u0 − u0 = 0 then

T

− 0

∂u ˜∗ ∗ , u − U∗ (ψ)V ,V dt = ∂t

T

˜ u∗ ,

0

∂U∗ (ψ) ∂u∗ − V,V dt ∂t ∂t

−(D∗ (Du∗ (T ) − uobs ), (u∗ − U∗ (ψ))(T ))H .


Consequently, T T ∗ ∗ ∗ ∂(u − U (ψ)) V,V dt +

˜ u ,

˜ u∗ , A(t)(u∗ − U∗ (ψ))V,V dt ∂t 0 0 = (D∗ (Du∗ (T ) − uobs ), (u∗ − U∗ (ψ))(T ))H .

239

(8.29)

Since u∗ − U∗ (ψ) is a solution of ∂(u∗ − U∗ (ψ)) + A(t)(u∗ − U∗ (ψ)) = B2 (ψ ∗ − ψ), ∂t (u∗ − U∗ (ψ))(0) = 0, then (8.29) gives (D∗ (Du∗ (T ) − uobs , (u∗ − U∗ (ψ))(T ))H =

T

˜ u∗ , B2 (ψ ∗ − ψ)V,V dt

0

˜∗ , ψ ∗ − ψU2 ,U2 = B2∗ u ∗ ∗ = (Λ−1 ˜ , ψ ∗ − ψ)U2 , U2 B2 u

where the operator Bi∗ ∈ L(L2 (0, T ; V ); Ui ) is the adjoint of Bi and ΛUi is the canonical isomorphism from Ui onto Ui , for i = 1, 2. Therefore, the second inequality of (8.27) can be written as ∗ ∗ ˜ − γψ ∗ , ψ − ψ ∗ )U2 ≤ 0 ∀ψ ∈ Vad . (Λ−1 U2 B2 u

(8.30)

In the same way we can prove that the first inequality of (8.27) can also be written as ∗ ∗ ˜ + βφ∗ , φ − φ∗ )U1 ≥ 0 ∀φ ∈ Uad (8.31) (Λ−1 U1 B1 u and then we have proved the following characterization of the robust control theorem. Theorem 8.4. Under the assumptions of Theorem 6.8, we suppose furthermore that the operators Bi ∈ L(Ui ; L2 (0, T ; V )), for i = 1, 2. Then there exists a unique optimal solution (φ∗ , ψ ∗ , u∗ ) ∈ U1 × U2 × W such that ∂u∗ + A(t)u∗ = B1 φ∗ + B2 ψ ∗ , ∂t u∗ (0) = u0 and

∗ ∗ (Λ−1 ˜ + βφ∗ , φ − φ∗ )U1 ≥ 0 ∀φ ∈ Uad , U1 B1 u ∗ ∗ ˜ − γψ ∗ , ψ − ψ ∗ )U2 ≤ 0 ∀ψ ∈ Vad , (Λ−1 U2 B2 u

where ΛUi is the canonical isomorphism from Ui onto Ui and u ˜∗ is the unique solution, which is in W, of the adjoint problem ∂u ˜∗ + A∗ (t)˜ u∗ = 0, ∂t u ˜∗ (t = T ) = D∗ (Du(t = T ) − uobs ).

−

240


We conclude this section by the following results. Assume that we have two observations: the final observation vobs , which is in H, and the state observation uobs , which is in M , and therefore the cost functional can be defined by θ2 θ1 Cu − uobs 2M + Du(T ) − vobs 2H 2 2 (8.32) β γ 2 + φ U1 − ψ 2U2 , 2 2 where θi ≥ 0, i = 1, 2, θ1 + θ2 > 0, C ∈ L(L2 (0, T ; V ); M ) and D ∈ L(H; H). Then we have the following theorem. J(φ, ψ) =

Theorem 8.5. Under the assumptions of Theorem 6.8, we suppose furthermore that the cost J is given by (8.32) with the operators D ∈ L(H; H) and C ∈ L(L2 (0, T ; V ); M ), and that the operators Bi ∈ L(Ui ; L2 (0, T ; V )), for i = 1, 2. Then there exists a unique optimal solution (φ∗ , ψ ∗ , u∗ ) ∈ U1 ×U2 ×W such that ∂u∗ + A(t)u∗ = B1 φ∗ + B2 ψ ∗ , ∂t u∗ (0) = u0 and ∗ ∗ ˜ + βφ∗ , φ − φ∗ )U1 ≥ 0 ∀φ ∈ Uad , (Λ−1 U1 B1 u ∗ ∗ ˜ − γψ ∗ , ψ − ψ ∗ )U2 ≤ 0 ∀ψ ∈ Vad , (Λ−1 U2 B2 u

˜∗ is the unique where ΛUi is the canonical isomorphism from Ui onto Ui and u solution, which is in W, of the adjoint problem ∂u ˜∗ + A∗ (t)˜ − u∗ = C ∗ Λ(Cu∗ − uobs ), ∂t u˜∗ (t = T ) = D∗ (Du∗ (t = T ) − vobs ), where Λ is the canonical isomorphism from M onto M .

8.4 Examples of Controls, Disturbances and Observations In this section we present some robust optimal control problems with different types of controls, disturbances and observations. For this, we consider the same operators and assumptions as in Section 6.4 and take the following problem: ∂U + A(t)U = f + φ − a0 (x, t)U on Q, ∂t subject to the linear Robin boundary condition ∂U (8.33) + βU = αΨ on Σ, α ∂ηA with the initial condition U (0) = U0 on Ω,

8.4 Examples of Controls, Disturbances and Observations

241

under the pointwise constraint r1 ≤ a0 ≤ r2

a.e. in Q,

(8.34)

where f, φ, a0 ∈ L2 (Q), Ψ ∈ L2 (Σ), U0 ∈ L2 (Ω), the real constants α, β are such that αβ > 0, the real constants r1 , r2 are such that r1 < r2 , and A(t)U = −

∂ ∂U (aij (x, t) ). ∂x ∂x i j i,j=1,n

(8.35)

Next, let us denote by V the space H01 (Ω) in the case of homogeneous Dirichlet boundary condition (i.e., in the case of α = 0) and the space H 1 (Ω) in the case of linear Robin or Neumann boundary condition (i.e., if α = 0). We denote its dual by V and by ., .V ,V the duality product between V and V . Then the embedding V ⊂ L2 (Ω) ⊂ V are continuous. Finally, we denote by H = L2 (Ω) and by W = {w ∈ L2 (0, T ; V ) : ∂w/∂t ∈ L2 (0, T ; V )}. We introduce now the following form by a ˜(t; u, v) = a(t; u, v) + a0 u.vdx + θu.vdΓ, Ω

Γ

where a is the bilinear form corresponding to the operator A (see Section 6.4), θ = β/α ≥ 0 if α = 0 and θ = 0 otherwise. As in Section 7.4, problem (8.33) admits a unique solution U ∈ W ∩ C([0, T ]; H) of the problem, a.e. t ∈ (0, T ) (the weak formulation of (8.33)) ∂U , vV ,V + a ˜(t; U, v) = (f + φ, v) + α Ψ vdΓ ∀v ∈ V,

∂t Γ (8.36) with the initial condition U (0) = U0 on Ω and satisfies the following estimate: U 2W∩C([0,T ];H) ≤ C1 ( f 2L2 (Q) + φ 2L2 (Q) ) +C2 ( U0 2L2 (Ω) + Ψ 2L2 (Σ) ).

(8.37)

Moreover, for a0 and f be given, v1 ∈ W ∩ C([0, T ]; H) (respectively v2 ∈ W ∩ C([0, T ]; H)) is the solution of (8.33), with data (φ1 , Ψ1 , v01 ) ∈ L2 (Q)×L2 (Σ)× L2 (Ω) (respectively with data (φ2 , Ψ2 , v02 ) ∈ L2 (Q) × L2 (Σ) × L2 (Ω)), then v1 − v2 2W∩C([0,T ];H) ≤ C( φ 2L2 (Q) + Ψ 2L2 (Σ) + v0 2L2 (Ω) ), where φ = φ1 − φ2 , Ψ = Ψ1 − Ψ2 and v0 = v01 − v02 . In the following, the solution U of (8.33) will be treated as the target function. We are then interested in the robust regulation of the deviation of the problem from the desired target (U, φ, Ψ, U0 ). We analyze the full equation which models large perturbations (u, ϕ, ψ, u0 ) to the target (U, φ, Ψ, U0 ). We consider then the following equation (since the problem (8.33) is linear):

242


∂u + A(t)u = ϕ − a0 (x, t)u on Q, ∂t subject to the linear Robin boundary condition ∂u + βu = αψ on Σ, α ∂ηA with the initial condition u(0) = u0 on Ω,

(8.38)

under the pointwise constraint (8.34). The problem (8.38) is similar to (8.33), so we have the existence and uniqueness of the solution u ∈ W ∩ C([0, T ]; H), and the following Lipschitz continuity result holds: u1 − u2 2W∩C([0,T ];H) ≤ C( ϕ 2L2 (Q) + ψ 2L2 (Σ) + u0 2L2 (Ω) ), (8.39) where ui ∈ W ∩ C([0, T ]; H) is the solution of (8.38), with data (ϕi , ψi , u0i ) ∈ L2 (Q) × L2 (Σ) × L2 (Ω), for i = 1, 2 and ϕ = ϕ1 − ϕ2 , ψ = ψ1 − ψ2 and u0 = u01 − u02 . 8.4.1 Boundary Disturbance Suppose that the observation operator C is the injection of L2 (0, T ; V ) onto L2 (Q) (i.e., M = L2 (Q) = M and then the canonical isomorphism Λ from M onto M is the identity operator) and consider the case where the control is the function ϕ i.e., the control is distributed in Q, and the disturbance is the function ψ, i.e., the disturbance is on the boundary condition. Let K1 and K2 be given non-empty, closed and convex subsets of L2 (Q) and L2 (Σ), respectively, and the observation uobs ∈ L2 (Q). Our problem is then, to find (ϕ, ψ) ∈ K1 × K2 such that the following cost functional δ1 δ2 1 Cu − uobs 2L2 (Q) + ϕ 2L2 (Q) − ψ 2L2 (Σ) (8.40) 2 2 2 is minimized with respect to ϕ and maximized with respect to ψ subject to the problem (8.38), where δi > 0, i = 1, 2 are fixed parameters, i.e., to find a saddle point of J in K1 × K2 . The adjoint problem corresponding to problem (8.38) is given by the following system: J(ϕ, ψ) =

−

∂u ˜(ϕ, ψ) + A∗ (t)˜ u(ϕ, ψ) ∂t = −a0 u ˜(ϕ, ψ) + Cu(ϕ, ψ) − uobs on Q,

subject to the linear Robin boundary condition ∂u ˜(ϕ, ψ) + β u˜(ϕ, ψ) = 0 on Σ, α ∂ηA∗ with the final condition u˜(ϕ, ψ)(t = T ) = 0 on Ω,

(8.41)


243

where A∗ is the adjoint of the operator A. Moreover, the optimality conditions are given by, ∀(ϕ, ψ) ∈ K1 × K2 (˜ u∗ + δ1 ϕ∗ )(ϕ − ϕ∗ )dxdt ≥ 0, Q (8.42) (˜ u∗ − δ2 ψ ∗ )(ψ − ψ ∗ )dΓ dt ≤ 0, Σ

where (ϕ∗ , ψ ∗ ) is a saddle point of J, i.e., a solution of (8.40), u∗ = u(ϕ∗ , ψ ∗ ) ˜(ϕ∗ , ψ ∗ ) is the is the solution of (8.38) (corresponding to (ϕ∗ , ψ ∗ )) and u˜∗ = u solution of the adjoint problem (8.41) (corresponding to the optimal solution (ϕ∗ , ψ ∗ , u∗ )). Remark 8.6. If without constraints, i.e., K1 × K2 = L2 (Q) × L2 (Σ), the optimality conditions (8.42) become u ˜∗ u˜∗ = −ϕ∗ , = ψ∗ , δ1 δ2 and then we can obtain the optimal control by the resolution of the following coupled system u ˜∗ ∂u∗ + A(t)u∗ + a0 u∗ = − on Q, ∂t δ1 ∂u ˜∗ + A∗ (t)˜ u∗ + a0 u˜∗ = u∗ − uobs on Q, − ∂t ∂u∗ α˜ u∗ α + βu∗ = on Σ, ∂ηA δ2 ∂u ˜∗ + β u˜∗ = 0 on Σ, α ∂ηA∗ ˜∗ (T ) = 0 on Ω. u∗ (0) = u0 , u

(8.43)

We can use the same method in order to study the case of the final observation, or the case where the control and the disturbance are in the boundary condition, for example, for ϕ known, we suppose that ψ is decomposed into the disturbance ψd ∈ Vad ⊂ L2 (Σ) and the control ψc ∈ Uad ⊂ L2 (Σ) (i.e., ♦ ψ := ψc + ψd ). 8.4.2 Pointwise Observations In this subsection we consider the pointwise observation and distributed control and disturbance. Let (xi )i=1,...,d be given d points of the domain Ω, and we suppose that the observation is as the form (u(xi , t))i=1,...,d (if u(xi , t) has a sense). Suppose now that the operator A is independent of time, the space Ω ⊂ IRn with n ≤ 3, the boundary Γ of Ω and the coefficients aij of the operator A are sufficiently regular, and the function ψ = 0. Moreover, we suppose that

244


the function ϕ is decomposed into the control g ∈ L2 (Q) and the disturbance ξ ∈ L2 (Q), i.e., ϕ := g + ξ. The function u is assumed to be related to disturbance ξ and control g through the problem (8.38): ∂u + A(t)u = ξ + g − a0 (x, t)u on Q, ∂t subject to the linear Robin boundary condition ∂u α + βu = 0 on Σ, ∂ηA with the initial condition u(0) = u0 on Ω.

(8.44)

Moreover, if u0 ∈ V , then the unique solution of the problem (8.44) is in H 2,1 (Q) = L2 (0, T ; H 2 (Ω)) ∩ H 1 (0, T ; L2(Ω)) and we have the following Lipschitz continuity result: u 2H 2,1 (Q) ≤ c( g 2L2 (Q) + ξ 2L2 (Q) + u0 2V ).

(8.45)

where ui ∈ H 2,1 (Q) is the solution of (8.44), with data (gj , ξj , u0j ) ∈ L2 (Q) × L2 (Q) × L2 (Ω), for j=1,2 and u = u1 − u2 , g = g1 − g2 , ξ = ξ1 − ξ2 and u0 = u01 − u02 . Moreover, since Ω ⊂ IRn , n ≤ 3, we have that H 2 (Ω) ⊂ C(Ω) (the space of continuous functions on Ω) and then uj (xi , t) ∈ L2 (0, T ), for i = 1, . . . , d, j = 1, 2, and T | u(xi , t) |2 dt ≤ c( g 2L2 (Q) + ξ 2L2 (Q) + u0 2V ), (8.46) 0

where u = u1 − u2 . Let Kj , j = 1, 2 be given non-empty, closed and convex subsets of L2 (Q) and the observation uobs = (ui,obs )i=1,...,d ∈ (L2 (0, T ))d . Our problem, then, is to find (g, ξ) ∈ K1 × K2 such that the cost functional: 1 u(xi , .) − ui,obs 2L2 (0,T ) 2 i=1 (8.47) γ1 γ2 2 2 + g L2 (Q) − ξ L2 (Q) . 2 2 is minimized with respect to g and maximized with respect to ξ subject to the problem (8.44), where γj > 0 for j = 1, 2, are fixed constants, to find a saddle point of J in K1 × K2 . We have the existence and the uniqueness of the optimal solution (g ∗ , ξ ∗ ) which is characterized by the following optimality conditions, given by (˜ u∗ + γ1 g ∗ )(g − g ∗ )dxdt ≥ 0 ∀g ∈ K1 , Q (8.48) (˜ u∗ − γ2 ξ ∗ )(ξ − ξ ∗ )dxdt ≤ 0 ∀ξ ∈ K2 , d

J(g, ξ) =

Q


245

where u˜∗ = u˜(g ∗ , ξ ∗ ) is the unique solution of the following adjoint problem corresponding to problem (8.44): ∂u ˜ + A∗ u ˜ + a0 u ˜= (u∗ (xi , .) − ui,obs )δxi on Q, ∂t i=1 d

−

subject to the linear Robin boundary condition ∂u ˜ + β u˜ = 0 on Σ, α ∂ηA∗ with the final condition u ˜(t = T ) = 0 on Ω,

(8.49)

where u∗ = u(g ∗ , ξ ∗ ) is the solution of (8.44) (corresponding to (g ∗ , ξ ∗ )), A∗ is the adjoint of the operator A, δxi is the usual Dirac function at point xi and h(t)δxi is the distribution: S ∈ D(Q) −→

T

h(t)S(xi , t)dt. 0

As the problem (8.49) is similar to (7.42) then it admits a unique solution u ˜∗ on L2 (Q), given by the transposition technique. Remark 8.7. (i) In the case without constraints, i.e., K1 × K2 = (L2 (Q))2 , the optimality conditions (8.48) become u ˜∗ u ˜∗ = −g ∗ and = ξ∗ , γ1 γ2 so, we can obtain the optimal control by the resolution of the following coupled system: ∂u∗ u ˜∗ u ˜∗ + Au∗ + a0 u∗ = − + on Q, ∂t γ1 γ2 d ∂u ˜∗ + A∗ u˜∗ + a0 u ˜∗ = (u∗ (xi , .) − ui,obs )δxi on Q, − ∂t i=1 ∂u∗ ∗ α + βu = 0 on Σ, ∂ηA

(8.50)

∂u ˜∗ + β u˜∗ = 0 on Σ, ∂ηA∗ u∗ (0) = u0 , u ˜∗ (T ) = 0 on Ω. α

(ii) If we suppose that the boundary function ψ is sufficiently regular, such that the solutions of (8.33) are in H 2,1 (Q), we can use the same technique in order to study problems with boundary controls and/or boundary disturbances. ♦

246


8.4.3 Pointwise Controls and Pointwise Disturbances In this section we consider the pointwise controls and pointwise disturbances and a distributed observation, i.e., we consider the case where the value ϕ is decomposed into a pointwise disturbance ξ and a pointwise control g. More precisely, let (xi )i=1,...,dc (respectively (yi )i=1,...,dd ) be given dc points (respectively dd points) of the domain Ω, and suppose that the control function g and the disturbance function ξ are in the form g :=

dc

gi δxi , ξ :=

i=1

dd

ξj δyj and ϕ := g + ξ,

j=1

where gi , ξj ∈ L2 (0, T ), for i = 1, . . . , dc and j = 1, . . . , dd and δxi (respectively δyj ) is the usual Dirac function at point xi (respectively point yj ). Similarly, as in the previous subsection, we suppose that the operator A is independent of time, the space Ω ⊂ IRn with n ≤ 3, the boundary Γ of Ω and the coefficients aij of the operator A are sufficiently regular, and the function ψ = 0. Then the function u is related to disturbance ξ and control g through the problem (8.44). By using the same technique (the transposition technique) as used to prove the unique solution of the adjoint problem (7.42), we obtain the existence and the uniqueness of the solution u in L2 (Q). Let K1 (respectively K2 ) be a given non-empty, closed and convex subset of (L2 (0, T ))dc (respectively (L2 (0, T ))dd ) and the observation uobs ∈ L2 (Q). Our problem, then, is to find (gi )i=1,...,dc ∈ K1 and (ξj )j=1,...,dd ∈ K2 such that the cost functional: J(g, ξ) =

1 u − uobs 2L2 (Q) 2 dc dd 1 1 + αi gi 2L2 (0,T ) − βj ξj 2L2 (0,T ) , 2 i=1 2 j=1

(8.51)

is minimized with respect to g and maximized with respect to ξ subject to the problem (8.44), where αi , βj > 0, for i = 1, . . . , dc and j = 1, . . . , dd are fixed constants. We have the existence and the uniqueness of the optimal solution g ∗ = ∗ (gi )i=1,...,dc and ξ ∗ = (ξj∗ )j=1,...,dd which is characterized by the following optimality conditions: dc i=1

T 0

(˜ u∗ (xi , t) + αi gi∗ (t))(gi (t) − gi∗ (t))dt ≥ 0 ∀g = (gi )i=1,dc ∈ K1 ,

dd T

(8.52) ∗

(˜ u (yj , t) −

j=1

0

βj ξj∗ (t))(ξj (t)

−

ξj∗ (t))dt

≤ 0 ∀ξ = (ξj )j=1,dd ∈ K2 ,


247

where u ˜∗ = u˜(g ∗ , ξ ∗ ) is the solution of the following adjoint problem corresponding to problem (8.44): ∂u ˜ + A∗ u˜ + a0 u˜ = u∗ − uobs on Q, ∂t subject to the linear Robin boundary condition ∂u ˜ + β u˜ = 0 on Σ, α ∂ηA∗ with the final condition u ˜(t = T ) = 0 on Ω, −

(8.53)

where u∗ = u(g ∗ , ξ ∗ ) is the optimal state and A∗ is the adjoint of the operator A. Since u∗ −uobs ∈ L2 (Q) and the final condition (i.e., the null function) is in V , then the problem (8.53) admits a unique solution u ˜ in H 2,1 (Q). Moreover, 2 ˜(xi , t) (respectively u ˜(yj , t)) is in L2 (0, T ), for since H (Ω) ⊂ C(Ω), we have u i = 1, . . . , dc (respectively j = 1, . . . , dd ). Remark 8.8. In the case without constraints, i.e., K1 = (L2 (0, T ))dc and K2 = (L2 (0, T ))dd , the optimality conditions (8.52) become u ˜∗ (xi , .) u˜(yj , .) = −gi∗ and = ξj∗ for i = 1, . . . , dc ; j = 1, . . . , dd , αi βj so, we can obtain the optimal control by the resolution of the coupled system: dc dd u˜∗ (yj , .)δyj u ˜∗ (xi , .)δxi ∂u∗ + Au∗ + a0 u∗ = − + on Q, ∂t αi βj i=1 j=1 ∂u ˜∗ + A∗ u ˜ ∗ + a0 u ˜∗ = u∗ − uobs on Q, − ∂t ∂u∗ + βu∗ = 0 on Σ, α ∂ηA ∂u ˜∗ α + β u˜∗ = 0 on Σ, ∂ηA∗ u∗ (0) = u0 , u˜∗ (T ) = 0 on Ω.

(8.54)

♦

8.4.4 Boundary Controls and Boundary Observations In this subsection we consider a problem where controls and observations act on the boundary, and we suppose that the parameter α = 0. The disturbance can act on the boundary or on the state equation and the observation is given by Cu = D(u|Σ ), (8.55) where u|Σ is the trace of u on Σ and D ∈ L(L2 (Σ); L2 (Σ)). For the types of boundary observation operator D, see, e.g., Example 7.10.

248


Boundary Disturbances We assume that the function ψ is decomposed on the disturbance ξ and the control g defined on Σ, i.e., ψ := g + ξ. The function u is assumed to be related to disturbance ξ and control g through the problem (8.38): ∂u + A(t)u = ϕ − a0 (x, t)u on Q, ∂t subject to the linear Robin boundary condition ∂u + βu = α(g + ξ) on Σ, α ∂ηA with the initial condition u(0) = u0 on Ω.

(8.56)

Let Kj , for j = 1, 2, be given non-empty, closed and convex subsets of L2 (Σ) and the observation uobs ∈ L2 (Σ). Our problem, then, is to find (g, ξ) ∈ K1 × K2 such that the following cost functional: J(g, ξ) =

γ1 γ2 1 D(u|Σ ) − uobs 2L2 (Q) + g 2L2 (Σ) − ξ 2L2 (Σ) , (8.57) 2 2 2

is minimized with respect to g and maximized with respect to ξ subject to the problem (8.56), where γj > 0, for j = 1, 2 are fixed constants. We have the existence and the uniqueness of the optimal control (g ∗ , ξ ∗ ) ∈ K1 × K2 , which is characterized by the following optimality condition: (˜ u∗ + γ1 g ∗ )(g − g ∗ )dΓ dt ≥ 0 ∀g ∈ K1 , Σ (8.58) (˜ u∗ − γ2 ξ ∗ )(ξ − ξ ∗ )dΓ dt ≤ 0 ∀ξ ∈ K2 , Σ

where u ˜∗ = u˜(g ∗ , ξ ∗ ) is the solution of the following adjoint problem: ∂u ˜ + A∗ (t)˜ u + a0 u ˜ = 0 on Q, ∂t subject to the linear Robin boundary condition ∂u ˜ + β u˜ = D∗ (D(u∗ |Σ ) − uobs ) on Σ, α ∂ηA∗ with the final condition u ˜(t = T ) = 0 on Ω, −

(8.59)

where u∗ = u(g ∗ , ξ ∗ ) is the solution (8.38) (corresponding to (g ∗ , ξ ∗ )), A∗ is the adjoint of the operator A and D∗ is the adjoint of the operator D. Since the function D∗ (D(u∗ |Σ ) − uobs ) is in L2 (Σ) then the problem (8.59) admits a unique solution in W ∩ C([0, T ]; H) (in the weak formulation sense).


249

Remark 8.9. In the case without constraints, i.e., Kj = L2 (Σ), for j = 1, 2, the optimality conditions (8.58) become u ˜∗ |Σ u ˜∗ |Σ = −g ∗ , = ξ∗, γ1 γ2 so, we can obtain the optimal control by the resolution of the coupled system: ∂u∗ + A(t)u∗ + a0 u∗ = ϕ on Q, ∂t ∂u ˜∗ + A∗ (t)˜ − u ∗ + a0 u ˜∗ = 0 on Q, ∂t α˜ u∗ |Σ α˜ u∗ |Σ ∂u∗ + βu∗ = − + on Σ, α ∂ηA γ1 γ2 ∂u ˜∗ + β u˜∗ = D∗ (D(u∗ |Σ ) − uobs ) on Σ, ∂ηA∗ u∗ (0) = u0 , u ˜∗ (T ) = 0 on Ω.

(8.60)

α

♦

Distributed Disturbances We assume that the disturbance ξ, defined on Q, is in the function ϕ and the control g, defined on Σ, is in the boundary function ψ, i.e., ψ := g + ψ0 and ϕ := ξ + ϕ0 , where ψ0 and ϕ0 are supposed known. The function u is assumed to be related to disturbance ξ and control g through the problem (8.38): ∂u + A(t)u = ξ + ϕ0 − a0 (x, t)u on Q, ∂t subject to the linear Robin boundary condition ∂u α + βu = α(g + ψ0 ) on Σ, ∂ηA with the initial condition u(0) = u0 on Ω.

(8.61)

Let K1 (respectively K2 ) be a given non-empty, closed and convex subset of L2 (Σ) (respectively L2 (Q)) and the observation uobs ∈ L2 (Σ). Our problem, then, is to find (g, ξ) ∈ K1 × K2 such that the cost functional: J(g, ξ) =

γ1 γ2 1 D(u|Σ ) − uobs 2L2 (Q) + g 2L2 (Σ) − ξ 2L2 (Q) , (8.62) 2 2 2

is minimized with respect to g and maximized with respect to ξ subject to the problem (8.61), where γj > 0, for j = 1, 2 are fixed constants. We have the existence and the uniqueness of the optimal control (g ∗ , ξ ∗ ) ∈ K1 × K2 , which is characterized by the following optimality conditions:

250


Σ Q

∗

∗

(˜ u∗ + γ1 g ∗ )(g − g ∗ )dΓ dt ≥ 0 ∀g ∈ K1 , (˜ u∗ − γ2 ξ ∗ )(ξ − ξ ∗ )dxdt ≤ 0 ∀ξ ∈ K2 ,

(8.63)

∗

where u˜ = u ˜(g , ξ ) is the solution of the adjoint problem (8.59) corresponding to the primal problem (8.61). Moreover, in the case without constraints, i.e., K1 = L2 (Σ) and K2 = 2 L (Q) the optimality conditions (8.63) become u ˜∗ u ˜∗ |Σ = −g ∗ , = ξ∗, γ1 γ2 so, we can obtain the optimal control by the resolution of the coupled system: u ˜∗ ∂u∗ + Au∗ + a0 u∗ = ϕ0 + on Q, ∂t γ2 ∗ ∂u ˜ + A∗ u ˜∗ + a0 u˜∗ = 0 on Q, − ∂t u ˜∗ |Σ ∂u∗ + βu∗ = α(ψ0 − ) on Σ, α ∂ηA γ1

(8.64)

∂u ˜∗ + β u˜∗ = D∗ (D(u∗ |Σ ) − uobs ) on Σ, ∂ηA∗ ˜∗ (T ) = 0 on Ω. u∗ (0) = u0 , u α

Remark 8.10. We can replace in (8.56) and (8.61), for example, the control g by Bq, where B : U −→ L2 (Σ) denotes a bounded linear control operator which maps abstract controls of a Hilbert space U to feasible boundary controls. Typical control operators and control spaces are given by •

U = L2 (Σ) and B = Identity

•

U = (L2 (0, T ))d and Bq(t) =

d

qi (t)αi , where αi ∈ L2 (Γ ).

♦

i=1

8.4.5 Data Assimilation Problem and Initial Condition Control In this subsection, we suppose that the parameter α = 0 and we consider an application when the control is in the initial condition, the disturbance is in ϕ and the observation acts on the boundary. We assume that ϕ is the sum of ξ (assumed to be not well known) and ϕ0 (assumed to be known), i.e., ϕ := ξ + ϕ0 and that the initial condition u0 is the sum of v0 (assumed to be known) and θ (assumed to be not well known), i.e., u0 = v0 + θ. The control is the function θ defined on Ω, the disturbance is the function ξ defined on Q and the observation is given by Cu = D(u|Σ ),

(8.65)


251

where u|Σ is the trace of u on Σ and D ∈ L(L2 (Σ); L2 (Σ)). For the types of boundary observation operator D, see, e.g., Example 7.10. The function u is assumed to be related to disturbance ξ and control θ through the problem (8.38): ∂u + A(t)u = ξ + ϕ0 − a0 (x, t)u on Q, ∂t subject to the linear Robin boundary condition ∂u α + βu = αψ on Σ, ∂ηA with the initial condition u(0) = v0 + θ on Ω.

(8.66)

Let K1 (respectively K2 ) be a given non-empty, closed and convex subset of L2 (Ω) (respectively L2 (Q)) and the observation uobs ∈ L2 (Σ). Our problem, then, is to find (θ, ξ) ∈ K1 × K2 such that the following cost functional: J(θ, ξ) =

1 γ1 γ2 D(u|Σ ) − uobs 2L2 (Σ) + θ 2L2 (Ω) − ξ 2L2 (Q) , (8.67) 2 2 2

is minimized with respect to θ and maximized with respect to ξ subject to the problem (8.66), where γj > 0, for j = 1, 2 are fixed constants. We have the existence and the uniqueness of the optimal control (θ∗ , ξ ∗ ) ∈ K1 × K2 , which is characterized by the following optimality conditions: (˜ u∗ + γ1 θ∗ )(θ − θ∗ )dx ≥ 0 ∀θ ∈ K1 , Ω (8.68) (˜ u∗ − γ2 ξ ∗ )(ξ − ξ ∗ )dxdt ≤ 0 ∀ξ ∈ K2 , Q

where u ˜∗ is the solution of the following adjoint problem: ∂u ˜ + A∗ (t)˜ u + a0 u ˜ = 0 on Q, ∂t subject to the linear Robin boundary condition ∂u ˜ α + β u˜ = D∗ (D(u∗ |Σ ) − uobs ) on Σ, ∂ηA∗ with the final condition u ˜(t = T ) = 0 on Ω, −

(8.69)

where u∗ = u(θ∗ , ξ ∗ ), A∗ is the adjoint of the operator A and D∗ is the adjoint of the operator D. Since the function D∗ (D(u(ψ)|Σ ) − uobs ) is in L2 (Σ) then the problem (8.69) admits a unique solution in W ∩ C([0, T ]; H) (in the weak formulation sense).

252


Remark 8.11. (i) In the case without constraints, i.e., K = L2 (Ω), the optimality conditions (8.68) become u˜∗ u ˜∗ = −θ∗ and = ξ∗, γ1 γ2 so, we can obtain the optimal control by the resolution of the coupled system: ∂u∗ u˜∗ + A(t)u∗ + a0 u∗ = ϕ0 + on Q, ∂t γ2 ∂u ˜∗ + A∗ (t)˜ − u ∗ + a0 u ˜∗ = 0 on Q, ∂t ∂u∗ + βu∗ = αψ on Σ, α ∂ηA ∂u ˜∗ α + β u˜∗ = D∗ (D(u∗ |Σ ) − uobs ) on Σ, ∂ηA∗ u ˜∗ u∗ (0) = v0 − , u˜∗ (T ) = 0 on Ω. γ1

(8.70)

(ii) In the case where we consider the boundary disturbance, i.e., for example, we suppose that the function ϕ is completely known and that the boundary function ψ is the sum of the disturbance ξ and the known function ψ0 ∈ L2 (Σ): ψ := ξ + ψ0 , we obtain the same results. Indeed in this case we can give the cost functional J, on K1 × K2 , by J(θ, ξ) =

γ1 γ2 1 D(u|Σ ) − uobs 2L2 (Σ) + θ 2L2 (Ω) − ξ 2L2 (Σ) , 2 2 2

where γj > 0, for j = 1, 2 are fixed constants, K1 (respectively K2 ) is a given non-empty, closed and convex subset of L2 (Ω) (respectively L2 (Σ)), and the observation uobs ∈ L2 (Σ). We have the following optimality conditions: (˜ u∗ + γ1 θ∗ )(θ − θ∗ )dx ≥ 0 ∀θ ∈ K1 , Ω (8.71) (˜ u∗ − γ2 ξ ∗ )(ξ − ξ ∗ )dΓ dt ≤ 0 ∀ξ ∈ K2 , Σ ∗

where u˜ is the solution of the adjoint problem (8.69) corresponding to the primal problem: ∂u + A(t)u = ϕ − a0 u on Q, ∂t subject to the linear Robin boundary condition ∂u α + βu = α(ξ ∗ + ψ0 ) on Σ, ∂ηA with the initial condition u(0) = v0 + θ∗ on Ω.

8.5 Bilinear-type Robust Control Problems

253

Moreover, in the case without constraints, i.e., K1 = L2 (Ω) and K2 = L (Σ) the optimality conditions (8.71) become 2

u˜∗ |Σ u ˜∗ = −θ∗ , = ξ∗, γ1 γ2 so, we can obtain the optimal control by the resolution of the coupled system: ∂u∗ + A(t)u∗ + a0 u∗ = ϕ on Q, ∂t ∂u ˜∗ + A∗ (t)˜ − u ∗ + a0 u ˜∗ = 0 on Q, ∂t α˜ u∗ |Σ ∂u∗ + βu∗ = + αψ0 on Σ, α ∂ηA γ2 ∂u ˜∗ α + β u˜∗ = D∗ (D(u∗ |Σ ) − uobs ) on Σ, ∂ηA∗ u ˜∗ u∗ (0) = − + v0 , u ˜∗ (T ) = 0 on Ω. γ1

(8.72)

♦

In the two next sections we study two classes of non-linear robust control problems, more precisely a class of bilinear problems (the primal problem is linear on the state variable when the control is fixed, and conversely) and a class of non-linear evolutive problems (which come from the modeling, for example, of pollutants in liquid or atmospheric systems).

8.5 Bilinear-type Robust Control Problems In this section we present a first type of non-linear robust control problem. The robust control problems arising in this context are bilinear. This adds fundamental difficulties from a mathematical viewpoint and makes these problems extremely challenging. A solution of bilinear control problems was proposed at first for the investigation of the dynamic processes of nuclear reactors, the kinetics of neutrons and heat transfer. Further investigations show that many processes in engineering, biology, ecology, medicine, chemical reactions, human population growth and many other areas can be described by bilinear systems (the literature on such models and methods such as controllability, observability and stabilization is vast, see, e.g., Baciotti [18], Christensen et al. [84], Isidori [165], Khapalov [173] and the references therein). We can mention also Ball et al. [19], in which the authors study controllability for wave equations; Lenhart et al. [57, 195] in which the authors treat bilinear control for the Kirchhoff plate equation and for a wave equation with viscous damping; Sachkov [255] for the controllability of bilinear systems governed by ordinary differential equations; Zuazua [313] for the controllability of the Schr¨ odinger equation; Leung and Chen [198] and Sadek and Vedantham [256], in which

254


the authors consider the optimal control problem of nuclear fission reactors; and finally Belmiloudi [47], in which the author considers the minimax control problem of nuclear fission reactors. 8.5.1 State Problem For this, we consider the linear primal problem used in Section 8.4, precisely the problem (8.33). We suppose that the operator A is independent of time, the space Ω ⊂ IRn with n ≤ 3, the boundary Γ of Ω and the coefficients aij of the operator A are sufficiently regular. Moreover, we assume that the function Ψ = 0 and α = 1, and we denote by g the sum of f and φ, i.e., g := f + φ. More precisely, ∂U + AU + a0 U = g on Q, ∂t subject to the linear Robin boundary condition ∂U + βU = 0 on Σ, ∂ηA with the initial condition U (0) = U0 on Ω.

(8.73)

According to the regularity results of Theorem 6.19, we have that the solution U of problem (8.73) is in H 2,1 (Q) × L∞ (0, T ; V ) for any initial condition U0 ∈ V , the right-hand side g ∈ L2 (Q) and a0 ∈ L∞ (Q). In our study, the solution U of (8.73) will be treated as the target function. We are then interested in the robust regulation of the deviation of the problem from the desired target (U, g, U0 , a0 ). We analyze the full equation which models large perturbations (u, ϕ, u0 , θ) to the target (U, g, U0 , a0 ), i.e., we assume U + u satisfies problem (8.73) with the data (g + ϕ, U0 + u0 , a0 + θ). Here we consider the following system: ∂u + Au + a0 u = ϕ − θ(u + U ) on Q, ∂t subject to the linear Robin boundary condition ∂u + βu = 0 on Σ, ∂ηA with the initial condition u(0) = u0 on Ω,

(8.74)

In the following, we assume that the control is the parameter θ and the disturbance is in the initial condition u0 , i.e., u0 = Bξ. The functions ϕ ∈ L2 (Q) and a0 ∈ L∞ (Q) are given data, and B denotes the linear and bounded operator which maps L2 (Ω) into V . The control is then the parameter θ defined on Q, the disturbance is the function ξ defined on Ω.


255

Then the problem (8.74) becomes ∂u + Au + a0 u = ϕ − θ(u + U ) on Q, ∂t subject to the linear Robin boundary condition ∂u + βu = 0 on Σ, ∂ηA with the initial condition u(0) = Bξ on Ω,

(8.75)

under the pointwise constraint τ1 ≤ θ ≤ τ2

a.e. in Q,

(8.76)

where the interval [τ1 , τ2 ] contains 0. The systems (8.75) is linear on the state variable when the control θ is fixed, and linear on the control when the state variable u is fixed, but the solution of (8.75) is depending non-linearily on the control θ. Consequently, the robust control problem is non-linear. Let Uad = {θ ∈ L2 (Q) : τ1 ≤ θ ≤ τ2 a.e. in Q} ⊂ L∞ (Q), Vad := 2 L (Ω) and F : (θ, ξ) ∈ Uad × Vad −→ W such that u = F (θ, ξ) is the unique solution of (8.75), corresponding to (θ, ξ). More precisely, since Bξ ∈ V , ϕ ∈ L2 (Q), a0 ∈ L∞ (Q) and θ ∈ L∞ (Q), the solution u of (8.75) is in H 2,1 (Q) ∩ L∞ (0, T ; V ). Our problem is then: find (θ, ξ) ∈ Uad × Vad such that the cost functional 1 γ1 γ2 C(u − uobs ) 2L2 (Q) + θ 2L2 (Q) − ξ 2L2 (Ω) , 2 2 2 is minimized with respect to θ and maximized with respect to ξ subject to the problem (8.75),

J(θ, ξ) =

(8.77)

where γj > 0, for j = 1, 2 are fixed constants and the function uobs ∈ L2 (Q) is a given observation. Before studying the saddle point problem (8.77), we give the Lipschitz continuity result and study the G-differentiability of the mapping F . Proposition 8.12. Let ϕ ∈ L2 (Q) and a0 ∈ L∞ (Q) be fixed functions. Let ξi ∈ Vad , θi ∈ Uad and ui = F (θi , ξi ) be the solution of (8.75), corresponding to (θi , ξi ), for i = 1, 2. We have the following estimates: u 2W∩C([0,T ];H) ≤ C( θ 2L2 (Q) + ξ 2L2 (Ω) ).

(8.78)

where u := u1 − u2 , θ := θ1 − θ2 and ξ := ξ1 − ξ2 . Proof. Let ξi ∈ Vad , θi ∈ Uad and ui = F (θi , ξi ), for i = 1, 2 then u = u1 − u2 is a solution of the problem:

256


∂u + Au + a0 u = −θ2 u − θ(u1 + U ) on Q, ∂t subject to the linear Robin boundary condition ∂u + βu = 0 on Σ, ∂ηA with the initial condition u(0) = Bξ on Ω.

(8.79)

Multiplying now (8.79) by u and integrating over Ω (taking into account the boundary conditions), we obtain (by using H¨ older’s inequalities) d u 2L2 (Ω) +2 u 2V + u 2L2 (Σ) (8.80) dt ≤ c1 u 2L2 (Ω) +c2 θ L2 (Ω) u1 + U L4 (Ω) u L4 (Ω) . Since V ⊂ H 1 (Ω) ⊂ L4 (Ω) (because n ≤ 3) and according to the regularity of u1 + U (i.e., u1 + U ∈ L∞ (0, T ; V )), we can deduce that d u 2L2 (Ω) + u 2V + u 2L2 (Σ) dt ≤ c1 u 2L2 (Ω) +c3 θ 2L2 (Ω) . Integrating now with respect to time over (0, t), ∀t ∈ (0, T ), we can deduce that t t u 2L2 (Ω) + u 2V ds + u 2L2 (Σ) ds 0 0 t 2 u L2 (Ω) ds + c4 ( θ 2L2 (Q) + ξ 2L2 (Ω) ). ≤ c1 0

By using Gronwall’s formula we can deduce that u 2L∞ (0,T ;L2 (Ω))∩L2 (0,T ;V ) ≤ C( θ 2L2 (Q) + ξ 2L2 (Ω) ).

(8.81)

Multiplying now (8.79) by v ∈ V and integrating over Ω (taking into account the boundary conditions), we obtain (by using H¨ older’s inequalities) ∂u , vV ,V ≤ c1 u V v V +c2 θ L2 (Ω) u1 + U L4 (Ω) v L4 (Ω) . ∂t Since V ⊂ H 1 (Ω) ⊂ L4 (Ω) (because n ≤ 3) and according to the regularity of u1 + U (i.e., u1 + U ∈ L∞ (0, T ; V )), we can deduce that

∂u , vV ,V ≤ c1 u V v V +c3 θ L2 (Ω) v V . ∂t Integrating now with respect to time over (0, t), for t ∈ (0, T ), we can deduce that t ∂u 2 u 2V ds+ θ 2L2 (Q) ). 2 ≤ C( ∂t L (0,T ;V ) 0 According to (8.81), we can conclude that

u 2W∩C([0,T ];H) ≤ C( θ 2L2 (Q) + ξ 2L2 (Ω) ).


257

8.5.2 Differentiability of the Mapping Solution We will now state the G-differentiability of the mapping F which maps the source term (θ, ξ) of (8.75) to the corresponding solution u, and we will obtain some estimates for the derivative of the map F . Namely we will prove the following proposition. Proposition 8.13. The function F : (θ, ξ) −→ u = F (θ, ξ) solution of (8.75) is G-differentiable, with respect to (θ, ξ) from Uad × Vad to W ∩ C([0, T ]; H) where the G-derivative F (θ, ξ) : h = (h1 , h2 ) ∈ L∞ (Q) × Vad −→ w = F (θ, ξ)h is the unique solution in W ∩ C([0, T ]; H) of the following linear parabolic problem: ∂w + Aw + a0 w = −θw − h1 (u + U ) on Q ∂t ∂w + βw = 0 on Σ, ∂ηA w(0) = Bh2 on Ω

(8.82)

w 2W∩C([0,T ];H) ≤ C( h1 2L2 (Q) + h2 2L2 (Ω) ).

(8.83)

such that Moreover, for all (θi , ξi ) ∈ Uad × Vad , for i = 1, 2, we have the following estimate (for all h ∈ L∞ (Q) × Vad ): F (θ1 , ξ1 )h − F (θ2 , ξ2 )h 2W∩C([0,T ];H)

1/2 ≤ C1 θ 2L2 (Q) + ξ 2L2 (Ω) h1 2L2 (Q) + h2 2L2 (Ω) (8.84)

1/2 +C2 h1 2L2 (Q) + h2 2L2 (Ω) θ 2L2 (Q) + ξ 2L2 (Ω) . Proof. The problem (8.82) is similar to the problem ((8.75)) (with −h1 (u + U ) + θU ∈ L2 (Q), since (u, U ) ∈ L2 (Q)2 and (h1 , θ) ∈ (L∞ (Q))2 , plays the role of ϕ, and h2 ∈ L2 (Ω) plays the role of ξ in (8.75)). Then (8.82) admits a unique solution w ∈ W ∩ C([0, T ]; H) such that w W∩C([0,T ];H) ≤ C1 u L2 (Q) +C2 . Let (θ, h1 ) ∈ Uad × L∞ (Q), (ξ, h2 ) ∈ (Vad )2 and > 0 such that h1 + θ ∈ Uad . Let u = F (θ, ξ) and u = F (θ + h1 , ξ + h2 ). Step 1: Prove that u −→ u strongly in C([0, T ]; H) as −→ 0. Let v := u −u, obviously, v satisfies ∂v + Av + a0 v = −(θ + h1 )v − h1 (u + U ) on Q, ∂t ∂v + βv = 0 on Σ, ∂ηA v (0) = Bh2 on Ω.

(8.85)

258


Since h1 ∈ L∞ (Q), h2 ∈ L2 (Ω) and (u + U ) ∈ L2 (Q) then g := −h1 (u + U ) is in L2 (Q) and v0 := h2 is in L2 (Ω). Consequently, since (θ + h1 ) ∈ L∞ (Q) and the problem (8.85) is similar to problem (8.79), we have that, there exists a constant C > 0 (independent of ) such that v W∩C([0,T ];H) ≤ C( g L2 (Q) + v0 L2 (Ω) ) and then v W∩C([0,T ];H) ≤ C1 ( u + U L2 (Q) + h2 L2 (Ω) ). Consequently, v −→ 0 strongly in W ∩ C([0, T ]; H) as −→ 0. Step 2: Prove now that w := (u − u)/ −→ w strongly in W ∩ C([0, T ]; H) as −→ 0. Let w ˜ := w − w, obviously, w ˜ satisfies ∂w ˜ + Aw ˜ + a0 w ˜ = −θw ˜ − h1 v on Q, ∂t ∂w ˜ + βw ˜ = 0 on Σ, ∂ηA w ˜ (0) = 0 on Ω.

(8.86)

The problem (8.86) is similar to problem (8.85), so w ˜ W∩C([0,T ];H) ≤ C v L2 (Q) . Since v L2 (Q) −→ 0 as −→ 0, we can deduce the convergence result. The problem (8.82) is similar to problem (8.79), so we can apply the estimate (8.78) and we obtain the result w 2W∩C([0,T ];H) ≤ C( h1 2L2 (Q) + h2 2L2 (Ω) ). We shall now prove the estimate given in the second part of Proposition 8.13. For i = 1, 2, let (θi , ξi ) ∈ Uad × Vad , wi = F (θi , ξi )h (solution of (8.82)) and ui = F (θi , ξi )h (solution of (8.75)). Then w = w1 − w2 is the solution of ∂w + Aw + (a0 + θ2 )w = −θw1 − h1 u on Q ∂t ∂w + βw = 0 on Σ, ∂ηA w(0) = 0 on Ω.

(8.87)

where u = u1 − u2 , θ = θ1 − θ2 . Multiplying now (8.87) by w and integrating over Ω (taking into account the boundary conditions), we obtain (by using H¨ older’s inequalities) d w 2L2 (Ω) +2 w 2V + w 2L2 (Σ) dt ≤ c1 w 2L2 (Ω) +c2 θ L2 (Ω) w1 L3 (Ω) w L6 (Ω) +c3 h1 L2 (Ω) u L3 (Ω) w L6 (Ω) .

(8.88)


259

Since V ⊂ H 1 (Ω) ⊂ L6 (Ω) (because n ≤ 3), we can deduce that d w 2L2 (Ω) + w 2V + w 2L2 (Σ) dt ≤ c1 w 2L2 (Ω) +c4 θ 2L2 (Ω) w1 2L3 (Ω) +c5 h1 2L2 (Ω) u 2L3 (Ω) . Because of the Gagliardo–Nirenberg inequalities we have that, for all v ∈ H 1 (Ω), n

6−n

v 2L3 (Ω) ≤ C v V3 v L23(Ω) and then (since w1 and u are in L∞ (0, T ; H) = L∞ (0, T ; L2(Ω))) d w 2L2 (Ω) + w 2V + w 2L2 (Σ) dt 6−n n 3 ≤ c1 w 2L2 (Ω) +c4 θ 2L2 (Ω) w1 V3 w1 L∞ (0,T ;H) n

6−n

3 +c5 h1 2L2 (Ω) u V3 u L∞ (0,T ;H) .

Integrating now with respect to time over (0, t), ∀t ∈ (0, T ) we can deduce, in particular, that (by using H¨ older’s inequalities) t w 2V ds w(t) 2L2 (Ω) + 0 t w 2L2 (Ω) ds ≤ c1 0

6−n 6 T

0

n

12

θ L6−n 2 (Ω) ds

+c4 T

h1

+c5 0

12 6−n L2 (Ω)

6−n

3 w1 L6 2 (0,T ;V ) w1 L∞ (0,T ;H)

6−n 6

n

6−n

3 u L6 2 (0,T ;V ) u L∞ (0,T ;H) .

ds

Since (because L2 (0, T ; V ) and L∞ (0, T ; H) are subsets of W ∩ C([0, T ]; H)), n

6−n

2 3 w1 L6 2 (0,T ;V ) w1 L∞ (0,T ;H) ≤ C w1 W∩C([0,T ];H) , n

6−n

2 3 u L6 2 (0,T ;V ) u L∞ (0,T ;H) ≤ C u W∩C([0,T ];H) ,

6−n

12 6 T

0

T

h1 0

T

12

θ L6−n 2 (Ω) ds 12 6−n L2 (Ω)

≤C

6−n 6

ds

θ 4L2 (Ω) ds

0

≤C 0

,

12

T

h1 4L2 (Ω) ds

we can deduce that (since θ and h1 are in L∞ (Q))

,

260


w

2L2 (Ω)

t

w

+

2V

0

ds ≤ c1 0

t

w 2L2 (Ω) ds

+c6 w1 2W∩C([0,T ];H) θ L2 (Q) +c7 u 2W∩C([0,T ];H) h1 L2 (Q)

By using Gronwall’s formula and the estimates (8.78) and (8.83), we can deduce that w 2L∞ (0,T ;L2 (Ω))∩L2 (0,T ;V ) ≤ C1 ( θ 2L2 (Q) + ξ 2L2 (Ω) )( h1 2L2 (Q) + h2 2L2 (Ω) )1/2

(8.89)

+C2 ( h1 2L2 (Q) + h2 2L2 (Ω) )( θ 2L2 (Q) + ξ 2L2 (Ω) )1/2 . Because of Equation (8.88), and the previous estimate (8.89), we deduce easily the estimate w 2W∩C([0,T ];H) ≤ C1 ( θ 2L2 (Q) + ξ 2L2 (Ω) )( h1 2L2 (Q) + h2 2L2 (Ω) )1/2 +C2 ( h1 2L2 (Q) + h2 2L2 (Ω) )( θ 2L2 (Q) + ξ 2L2 (Ω) )1/2 .

This completes the proof. We can now study the following existence of an optimal solution. 8.5.3 Existence of an Optimal Solution

Let Kd be a convex, closed, non-empty and bounded subset of Vad . We have the following existence theorem. Theorem 8.14. For γi , i = 1, 2 sufficiently large, there exists (θ∗ , ξ ∗ ) ∈ Uad × Kd and u∗ ∈ W∩C([0, T ]; H) such that (θ∗ , ξ ∗ ) is the optimal solution of (8.77) and u∗ = F (θ∗ , ξ ∗ ) is the solution of (8.75). Proof. Let Pξ be the mapping: θ −→ J(θ, ξ) and Qθ be the mapping: ξ −→ J(θ, ξ). To obtain the existence of the robust control problem, we prove first that Pξ is convex and lower semi-continuous for all ξ ∈ Kd , second that Qθ is concave and upper semi-continuous for all θ ∈ Uad and finally we use the minimax theorems in infinite dimensions presented in Chapter 5. In order to prove the convexity, it is sufficient to show that for (θ1 , θ2 ) ∈ Uad × Uad , we have (Pξ (θ1 ) − Pξ (θ2 )).θ ≥ 0, where θ := θ1 − θ2 . From the expression of G-differentiable cost functional J (a composition of G-differentiable mappings), it follows that Pξ is G-differentiable and for i = 1, 2, J(θi + θ, ξ) − J(θi , ξ) Pξ (θi ).θ = lim

−→0 (ui − uobs )wi dxdt + γ1 θi θdxdt, = Q

Q


where ui = F (θi , ξ) and wi = F (θi , ξ).(θ, 0). Consequently, (u1 − u2 )w1 dxdt (Pξ (θ1 ) − Pξ (θ2 )).θ = γ1 θ 2L2 (Q) + Q + (u2 − uobs )(w1 − w2 )dxdt.

261

(8.90)

Q

The estimates (8.84), (8.83) and (8.78) imply that (u1 − u2 )w1 dxdt ≤ u1 − u2 L2 (Q) w1 L2 (Q) Q

Q

≤ C0 θ 2L2 (Q) ,

(8.91)

(u2 − uobs )(w1 − w2 )dxdt ≤ u2 − uobs L2 (Q) w1 − w2 L2 (Q) 3/2

≤ C1 θ L2 (Q) .

From (8.90) and the previous results (8.91), we can deduce that there exists a constant γ1l > 0 such that for γ1 ≥ γ1l we have (Pξ (θ1 ) − Pξ (θ2 )).θ ≥ (γ1 − C0 ) θ 2L2 (Q) −C1 θ L2 (Q) ≥ 0 3/2

and then the convexity of Pξ is established. In the same way, we can find γ2l > 0 such that for γ2 ≥ γ2l , Qθ is concave. We prove now that Pξ (respectively Qθ ) is lower (respectively upper) semicontinuous for all ξ ∈ Kd (respectively θ ∈ Uad ). Let θk ∈ Uad be a minimizing sequence of Pξ , i.e., lim inf J(θk , ξ) = k−→∞

inf

θ∈L2 (Q)

J(θ, ξ).

Then, according to the nature of the cost function J, we can deduce that θk is uniformly bounded in Uad and we can extract from θk a subsequence also denoted by θk such that θk θξ weakly in Uad . Therefore, by using the same technique as used in the proof of the estimate (8.78), uk = F (θk , ξ) is uniformly bounded in W ∩ C([0, T ]; H). Moreover, according now to Lemma 6.6, the injection of W into L2 (Q) is compact. Consequently, these results make it possible to extract from uk a subsequence also denoted by uk such that uk uξ weakly in L2 (0, T ; V ), uk −→ uξ strongly in L2 (Q), (8.92) 2 θk θξ weakly in L (Q) and θξ ∈ Uad . Prove now that uk θk −→ uξ θξ weakly in L2 (Q). Since uk θk − uξ θξ = (uk − uξ )θk + uξ (θk − θξ ), then according to the first and second parts of (8.92), we then obtain the result. Is is easy to prove that uξ is a solution of (8.75) with a parameter (θξ , ξ) and according to the uniqueness of the solution of (8.75), we have then uξ = F (θξ , ξ).

262


Since the norm is lower semi-continuous, therefore we have that the map Pξ is lower semi-continuous for all ξ ∈ Kd . By applying similar argument as in the proof of the previous result we obtain that Qθ is upper semi-continuous for all θ ∈ Uad . This completes the proof. We next wish to show the appropriate first-order necessary conditions (optimality conditions) of the saddle point problem (8.77). 8.5.4 First-order Necessary Conditions Theorem 8.15. Let γi , i = 1, 2 be sufficiently large, (θ∗ , ξ ∗ ) ∈ Uad × Kd be an optimal control defined by (8.77) and u∗ ∈ W ∩ C([0, T ]; H) be the optimal state such that u∗ = F (θ∗ , ξ ∗ ) is the solution of (8.75), with the data (θ∗ , ξ ∗ ). Then there exists a unique solution u ˜∗ ∈ W ∩ C([0, T ]; H) to the following adjoint problem corresponding to the primal problem (8.75): ∂u ˜ + A∗ u ˜ + a0 u ˜ = −θ∗ u ˜ + C ∗ C(u∗ − uobs ) on Q, ∂t subject to the linear Robin boundary condition ∂u ˜ + β u˜ = 0 on Σ, ∂ηA∗ with the final condition u ˜(t = T ) = 0 on Ω, −

(8.93)

where A∗ is the adjoint of A. Moreover, for all θ ∈ Uad and ξ ∈ Kd , u∗ /γ1 , τ2 )) θ∗ = max (τ1 , min ((u∗ + U )˜ ( or (γ1 θ∗ − (u∗ + U )˜ u∗ )(θ − θ∗ )dxdt ≥ 0), Q (−γ2 ξ ∗ + B ∗ u ˜∗ (0))(ξ − ξ ∗ )dx ≤ 0, Ω

where B ∗ is the adjoint of the operator B. Proof. Since the function C ∗ C(u∗ − uobs ) is in L2 (0, T ; V ) then the problem (8.101) admits a unique solution in W ∩ C([0, T ]; H) (in the weak formulation sense). The cost functional J is a composition of G-differentiable mappings then J is G-differentiable and in particular the G-derivative of J at the optimal point (θ∗ , ξ ∗ ) ∈ Uad × Kd is given by (for all h = (h1 , h2 ) ∈ L∞ (Q) × Kd such that (θ∗ + h1 ) ∈ Uad for small) J(θ∗ + h1 , ξ ∗ + h2 ) − J(θ∗ , ξ ∗ ) J (θ∗ , ξ ∗ )h = lim −→0 = C(u − uobs ).Cwdxdt Q +γ1 θ∗ h1 dxdt − γ2 ξ ∗ h2 dx, Q

Ω

(8.94)


263

where w = F (θ∗ , ξ ∗ )h is the solution of (8.82). Multiplying the first equation of (8.82) by u˜∗ and integrating over Q, this gives (using Green’s formula and integrating by parts in time)

T

− 0

T ∂u ˜∗ ∂u ˜∗ + A∗ u˜∗ + (a0 + θ∗ )˜ u∗ , wdt + ( + β u˜∗ )wdΓ dt ∗ ∂t ∂η A Γ 0 T ∗ ∗ h1 (u + U )˜ u dxdt + w(0)˜ u∗ (0)dx − w(T )˜ u∗ (T )dx. =− 0

Ω

Ω

Ω

∗

Since u ˜ is the solution of (8.101) and w(0) = Bh2 , therefore, we can deduce that C(u∗ − uobs ).Cwdxdt Q (8.95) =− h1 (u∗ + U )˜ u∗ dxdt + (Bh2 )˜ u∗ (0)dx Q

Ω

and then (according to (8.94)) ∗ ∗ J (θ , ξ )h = (γ1 θ∗ − (u∗ + U )˜ u∗ )h1 dxdt Q + (−γ2 ξ ∗ + B ∗ u˜∗ (0))h2 dx,

(8.96)

Ω

where B ∗ the adjoint operator of the linear and bounded operator B. Since (θ∗ , ξ ∗ ) is an optimal solution, then (γ1 θ∗ − (u∗ + U )˜ u∗ )(θ − θ∗ )dxdt ≥ 0 ∀θ ∈ Uad , Q (8.97) (−γ2 ξ ∗ + B ∗ u ˜∗ (0))(ξ − ξ ∗ )dx ≤ 0 ∀ξ ∈ Kd . Ω

Moreover, by using a standard control argument concerning the sign of the variation h (depending on the size of θ∗ ), we obtain that ∗ (u + U )˜ u∗ , τ2 . θ∗ = max τ1 , min γ1 This completes the proof.

8.5.5 Other Situations and Applications We assume now that the parameter θ is decomposed into the disturbance ξ and the control ϑ, i.e., θ = B1 ϑ + B2 ξ, where B1 : K1 −→ Uad (respectively B2 : K2 −→ Vad ) is the bounded and linear operator which maps abstract control (respectively disturbance) of the Hilbert space K1 (respectively K2 ) to feasible parameter control (respectively disturbance) space Uad (respectively Vad ), with

264


Uad = {ϑ ∈ K1 : τ11 ≤ B1 ϑ ≤ τ12 Vad = {ξ ∈ K2 : τ21 ≤ B2 ξ ≤ τ22

a.e. in Q}, a.e. in Q},

(8.98)

such that the intervals [τ11 , τ12 ] and [τ21 , τ22 ] contain 0. The function u is assumed to be related to the disturbance ξ and control ϑ through the problem (8.74): ∂u + Au + a0 u = ϕ − (B1 ϑ + B2 ξ)(u + U ) on Q, ∂t subject to the linear Robin boundary condition ∂u + βu = 0 on Σ, ∂ηA with the initial condition u(0) = u0 on Ω.

(8.99)

Our robust control problem is then: find (ϑ, ξ) ∈ K1ad × K2ad such that the cost functional 1 γ1 γ2 J(ϑ, ξ) = C(u − uobs ) 2L2 (Q) + ϑ 2K1 − ξ 2K2 , 2 2 2 is minimized with respect to ϑ and maximized with respect to ξ subject to the problem (8.99),

(8.100)

where γj > 0, for j = 1, 2 are fixed constants, K1ad (respectively K2ad ) is a non-empty, closed, convex, bounded subset of K1 (respectively of K2 ) and the function uobs ∈ L2 (Q) is a given observation. The arguments of the previous section extend directly to the present case without further estimates. We have then the following results Theorem 8.16. Let γi , i = 1, 2 be sufficiently large, (ϑ∗ , ξ ∗ ) ∈ K1ad × K2ad be an optimal control defined by (8.100) and u∗ ∈ W ∩ C([0, T ]; H) be the optimal state such that u∗ is the solution of (8.99), with the data (ϑ∗ , ξ ∗ ). Then there exists a unique solution u ˜∗ ∈ W ∩ C([0, T ]; H) to the following adjoint problem corresponding to the primal problem (8.99): ∂u ˜ + A∗ u˜ + a0 u˜ = −(B1 ϑ∗ + B2 ξ ∗ )˜ u + C ∗ C(u∗ − uobs ) on Q, ∂t subject to the linear Robin boundary condition ∂u ˜ + β u˜ = 0 on Σ, ∂ηA∗ with the final condition u ˜(t = T ) = 0 on Ω, −

(8.101)

where A∗ is the adjoint of the operator A. Moreover, u∗ ), ϑ − ϑ∗ )K1 ≥ 0 ∀ϑ ∈ K1ad , (γ1 ϑ∗ − B1∗ ((u∗ + U )˜ (−γ2 ξ ∗ − B2∗ ((u∗ + U )˜ u∗ ), ξ − ξ ∗ )K2 ≤ 0 ∀ξ ∈ K2ad , where Bi∗ is the adjoint of the operator Bi , for i = 1, 2.


265

Application: One-Neutron Diffusion Equation Here, we present an example illustrative of the abstract result of this section. We consider a one-neutron diffusion equation, describing the neutron reaction inside a nuclear fission reactor, when pointwise controllers are applied at locations x = (xi )i=1,l ∈ Ω, and pointwise disturbances are applied at locations y = (yj )j=1,m ∈ Ω. We suppose that the reactor core is a slab geometry with length L. The perturbed system that we consider is then l ∂2u ∂u − ν 2 = μu − fi (t)φ0 (x)δxi (u + U ) ∂t ∂x i=1

−

m

gj (t)φ1 (x)δyj (u + U ), in Q,

j=1

subject to the boundary conditions ∂u ∂u (L, t) + βu(L, t) = 0, t ∈ (0, T ) − (0, t) + βu(0, t) = 0, ∂x ∂x and the initial condition u(x, 0) = u0 (x ∈ (0, L)),

(8.102)

where δx is the usual Dirac function, Q = (0, L) × (0, T ), (φk , k = 0, 1) are given positive functions bounded above and below by two positive constants, ν = V D, μ = V (rσf − σa ), D is the neutron diffusion coefficient, and σf and σa denote the fission and absorption macroscopic cross-sections, respectively. The parameter V is the average neutron velocity and r denotes the number of neutrons generated per nuclear fission (for more details see, e.g., Christensen et al. [84]). The control (respectively disturbance) absorption cross-section at location xi (respectively yj ) is represented by fi (t) (respectively gj (t)) such that ai ≤ fi (t) ≤ bi , for i = 1, l, a.e. t ∈ (0, T ), cj ≤ gj (t) ≤ dj , for j = 1, m,

a.e. t ∈ (0, T ).

We propose the following objective functional: k0 J(ϑ, ξ) = 2 +

T

L

| u − uobs |2 dxdt 0

0

T γ1 T γ2 | fi |2 dt − | gj |2 dt, 2 2 0 j=1,m 0 i=1,l

where ϑ = (fi )i=1,l ∈ K1 and ξ = (gj )j=1,m ∈ K2 , with K1 = {ϑ = (fi )i=1,l ∈ [L2 (0, T )]l : ai ≤ fi (t) ≤ bi

a.e. in (0, T )},

K2 = {ξ = (gj )i=1,m ∈ [L2 (0, T )]m : cj ≤ gj (t) ≤ dj

a.e. in (0, T )}.

266


Since all the assumptions of the previous abstract results are satisfied by the model in this particular case, therefore we can apply the previous result of Theorem 8.16 and conclude the existence and the optimality conditions. Moreover, the gradients of J are ∂J (ϑ, ξ) ∂ϑ = [γ1 f1 (t) − φ0 (x1 )u(x1 , t)˜ u(x1 , t), . . . , γ1 fl (t) − φ0 (xl )u(xl , t)˜ u(xl , t)]T , ∂J (ϑ, ξ) ∂ξ T u(y1 , t), . . . , −γ2 gm (t) − φ1 (ym )u(ym , t)˜ u(ym , t)] , = [−γ2 g1 (t) − φ1 (y1 )u(y1 , t)˜ where u ˜ is the solution of following adjoint problem: −

∂2u ˜ ∂u ˜ − ν 2 = μ˜ u− fi (t)φ0 (x)δxi u ˜ ∂t ∂x i=1,l gj (t)φ1 (x)δyj u ˜ + k0 (u − uobs ), in Q , − j=1,m

subject to the boundary conditions ∂u ˜ ∂u ˜ (L, t) + β u˜(L, t) = 0, t ∈ (0, T ) − (0, t) + β u˜(0, t) = 0, ∂x ∂x and the final condition u˜(x, T ) = 0 (x ∈ (0, L)). Remark 8.17. (i) Another application is the cancer chemotherapy model (onecompartment) in which the function u describes the number of cancer cells and the function θ represents the drug dosage administered in the system (8.74). (ii) The same technique is valid in the case where the state variable u is a vector function (coupled system), see Belmiloudi [47]. We can then consider applications like the multi-neutron diffusion system (a problem in nuclear fission reactors) and the multi-compartment model for chemotherapy. ♦

8.6 Non-linear Robust Control for Non-linear Evolutive Problems In this section we analyze full non-linear robust control problems. For this we consider, the non-linear evolutive problems where we can prove the existence and stability (under extra assumptions) theorems. In contrast to linear systems, the dynamics of evolutive non-linear systems obey complicated laws that, in general, cannot be arrived at by intuitive and direct calculations. The main result of this study includes the existence, uniqueness and firstorder necessary conditions of optimality for the worst disturbance and optimal

8.6 Non-linear Robust Control for Non-linear Evolutive Problems

267

controllers. The plan of this study is as follows. First, we study the existence and the uniqueness of the perturbation of the problem (8.103) and obtain some a priori estimates. Second, we formulate the robust control problem in the case of the forcing f decomposed into a disturbance ψ and a control φ. We prove the existence and uniqueness and give the appropriate optimality system. Next, we consider an initial perturbation problem, i.e., the adjustment of initial conditions in order to obtain a system that agrees with a desired target system (because in many situations the initial condition is not well known). We reformulate the robust control problem for two cases: where the control is the initial condition and the disturbance is the forcing f , and where the initial condition is decomposed into a disturbance ψ and a control φ. As in previous sections we study the existence and uniqueness and give the optimality conditions. Finally, we present an example of convection–diffusion in the case of pollutants in liquid or atmospheric systems. The notations, assumptions and spaces are the same as in Section 7.6.1. 8.6.1 State Equations We will be consider the non-linear parabolic partial differential equations of the form (as in Section 7.6.1) ∂U + AU + F (x, t, U ) + K(, U ) = f on Q = (0, T ) × Ω ∂t U (0) = U0 on Ω,

(8.103)

where Ω is a boundary subset of IRm , m ≥ 1, with boundary Γ is sufficiently regular, A: D −→ D is an elliptic, selfadjoint operator with Av, v ≥ ν v 2D , Au, v ≤ M u D v D ∀u, v ∈ D ( ., . is the duality pairing on D and D, . D is the norm on D, . ∗ is the dual norm on D and ν, M > 0 are constants), K(, .): D −→ L2 (Ω) is a linear operator satisfying (7.81), is a given sufficiently regular function and F: Q×IR −→ IR is a Nemytsky operator on L2 (Q) satisfying Assumptions (7.78). The Banach space D of functions on Ω (satisfying the boundary conditions) such that D ⊂ L2 (Ω) ⊂ D satisfies Assumption (7.79). According to Theorems 7.22 and 7.23, the problem (8.103) admits a unique solution U ∈ W ∩ H1 for every U0 ∈ L2 (Ω) and f ∈ L2 (Q), where H1 = H 1 (0, T ; D ) = {v : ∂v/∂t ∈ L2 (0, T ; D )} and W = H ∩ V such that H = L∞ (0, T ; L2(Ω)) and V = L2 (0, T ; D). Moreover, if U1 (respectively U2 ) is a solution of (8.103) where the data in L2 (Q) × L2 (Ω) is (f1 , U0 ) (respectively (f2 , V0 )) then 1/2

. U1 − U2 W∩H1 ≤ C U0 − V0 2L2 + f1 − f2 2L2 (Q) In the following section, we formulate the perturbation problem and present the existence, uniqueness and regularity results of the perturbation solution.

268


8.6.2 The Perturbation Problem In the following, the solution U ∈ W of problem (8.103) will be treated as the target function. We are then interested in the robust regulation of the deviation of the problem from the desired target (U, f, U0 ). We analyze the full non-linear equation which models large perturbations (u, g, u0 ) to the target (U, f, U0 ). Hence we consider the equation: ∂u + Au + F (x, t, u + U ) − F (x, t, U ) + K(, u) = g on Q, ∂t u(0) = u0 on Ω.

(8.104)

If we set F˜ (., y) = F (., y + U ) − F (., U ) then (8.104) reduces to ∂u + Au + F˜ (x, t, u) + K(, u) = g on Q, ∂t u(0) = u0 on Ω.

(8.105)

Remark 8.18. (i) We easily verify that F˜ satisfies the same hypothesis that F , i.e., (7.78). (ii) For simplicity of future reference, we omit the “˜” on F˜ for (8.105). ♦ The problem (8.105) is of the same type as (8.103) so we have then the following proposition. Proposition 8.19. (i) Assume that g ∈ L2 (Q) and u0 ∈ L2 (Ω). Then the problem (8.105) admits a unique solution u such that u ∈ W and u 2W∩H1 ≤ C( u0 2L2 + g 2L2 (Q) ). (ii) Let u0 , v0 be two functions in L2 (Ω) and let g1 , g2 be two functions of L2 (Q). If u1 (resp. u2 ) is a solution of (8.105) with data (g1 , u0 ) (respectively with data (g2 , v0 )) then u1 − u2 2W∩H1 ≤ C( u0 − v0 2L2 + g1 − g2 2L2 (Q) ).

Remark 8.20. According to Lemma 6.6 we can deduce that W ∩ H1 ⊂ C([0, T ], H) and then (because of Proposition 8.19) for u (resp. v) the solution of (8.105) with data (g, u0 ) (resp. (g1 , v0 )) we have the following estimates u 2W∩H1 ∩C([0,T ];H) ≤ C( u0 2L2 + g 2L2 (Q) ). u − v 2W∩H1 ∩C([0,T ];H) ≤ C( u0 − v0 2L2 + g − g1 2L2 (Q) ).

(8.106) ♦

8.6.3 The Control Framework In the control framework, the value g is decomposed into the disturbance ψ ∈ L2 (Q) and the control φ ∈ L2 (Q), i.e., g = B1 φ + B2 ψ, where Bi , for i = 1, 2, are given (linear) bounded operators on L2 (Ω) such that (for i = 1, 2): there exists bi > 0 such that ∀hi ∈ L2 (Ω), Bi hi 2L2 ≤ b2i hi 2L2 . (8.107)


269

Remark 8.21. If we put the linear operator√B = (B1 , B2 ) defined by Bh = B1 h1 + B2 h2 where h = (h1 , h2 )t , and b = 2 max(b1 , b2 ) then Bh 2L2 ≤ (b1 h1 L2 +b2 h2 L2 )2 ≤ b2 h 2L2 , where h 2L2 = h1 2L2 + h2 2L2 .

(8.108) ♦

The objective in the robust control problem is to find the best control φ in the presence of the disturbance ψ which maximally spoils the control objective. The function u is assumed to be related to the disturbance ψ and control φ through the problem (8.105): ∂u + Au + F (x, t, u) + K(, u) = B1 φ + B2 ψ on Q, ∂t u(0) = u0 (given) on Ω.

(8.109)

To obtain the regularity of Proposition 8.19, we suppose the following hypothesis: u0 ∈ L2 (Ω), (φ, ψ) ∈ L2 (Q)2 . Let U : (φ, ψ) −→ u = U(φ, ψ) be the mapping: (L2 (Q))2 −→ W defined by (8.109). We introduce the cost functional defined by J(φ, ψ) =

μ 1 C(u − uobs ) 2L2 (Q) + u(T ) − vobs 2L2 2 2 α β 2 + φ L2 (Q) − ψ 2L2 (Q) , 2 2

(8.110)

where μ, α, β > 0 are fixed, (uobs , vobs ) ∈ V × L2(Ω) is the observation (given) and C is an unbounded, linear operator on L2 (Ω) satisfying the condition (7.91) Cv 2L2 ≤ δ1 v 2L2 +δ2 v 2D ∀v ∈ D. The robust control problem, then, is to minimize the functional J with respect to φ and maximize J with respect to ψ, i.e., to find (φ∗ , ψ ∗ ) ∈ Uad × Vad such that J(φ∗ , ψ) ≤ J(φ∗ , ψ ∗ ) ≤ J(φ, ψ ∗ ), ∀(φ, ψ) ∈ Uad × Vad ,

(8.111)

where Uad and Vad are non-empty, closed, convex, bounded subsets of L2 (Q). We are now going to show the differentiability result of the operator solution of (8.109). Proposition 8.22. Let F be an operator that satisfies Assumptions (7.78). Then the function U : (φ, ψ) −→ u = U(φ, ψ) solution of (8.109) is continuously F-differentiable from (L2 (Q))2 to W with the derivative U (φ, ψ) : (h1 , h2 ) −→ w given by the linear parabolic problem ∂w + Aw + G(., U(φ, ψ))w + K(, w) = B1 h1 + B2 h2 on Q, ∂t w(0) = 0 on Ω,

(8.112)

270


with G(., v) = F (., v). Moreover, for all Φi = (φi , ψi ) ∈ (L2 (Q))2 , i = 1, 2, the following estimates hold: (i) U (φ1 , ψ1 ) L((L2 (Q))2 ;W) ≤ C (ii) U (φ1 , ψ1 ) − U (φ2 , ψ2 ) L((L2 (Q))2 ;W) ≤ C Φ1 − Φ2 L2 (Q) . Proof. Let (φ, ψ, h1 , h2 ) ∈ (L2 (Q))4 , u = U(φ, ψ) and uh = u + wh = U(φ + h1 , ψ + h2 ). Let v = wh − vh , where vh is the solution of ∂vh + Avh + G(., u)vh + K(, vh ) = B1 h1 + B2 h2 on Q, ∂t vh (0) = 0 on Ω.

(8.113)

Then (according to the equations satisfied by u, uh and vh ) ∂v + Av + G(., u)v + K(, v) = g on Q ∂t v(0) = 0 on Ω,

(8.114)

where g = −(F (., u + wh ) − F (., u)) + G(., u)wh . By using similar argument as in the proof of Proposition 7.24, Chapter 7, and the estimate (8.108) we have the following estimates (where h = (h1 , h2 )): vh W ≤ C h L2 (Q)

(8.115)

v W ≤ C h 2L2 (Q) .

(8.116)

and Consequently, v W = o( h L2 (Q) ). Therefore, U (φ, ψ) defined by (8.112), is the F-derivative of U at point (φ, ψ) and verifies U (φ, ψ) L((L2 (Q))2 ,W) ≤ C. We shall now prove the second part of the proposition. Let Φi = (φi , ψi ) ∈ (L2 (Q))2 , for i = 1, 2, be given and wi = U (φi , ψi )h, for i = 1, 2, be the solution of the problem (8.112) (we denote by ui = U(φi , ψi ), for i = 1, 2, and by h = (h1 , h2 )). We set w = w1 − w2 . Then, according to the equations satisfied by w1 and w2 , we can deduce that w satisfies ∂w + Aw + G(., u1 )w + K(, w) = (G(., u2 ) − G(., u1 ))w2 on Q, ∂t (8.117) w(0) = 0 on Ω. Multiplying (8.117) by w and integrating over (0, t) × Ω, for all t ∈ (0, T ), we obtain (according to −2G(., u) ≤ γ0 and Assumptions (7.78) and (7.81))


t

w(t) 2L2 +ν 0t

w 2D ds

w

≤γ

271

2L2

t | u2 − u1 || w2 || w | dxds,

ds + 2λ

0

0

Ω

with γ given by (7.82). By using H¨ older’s inequality and the relationship (7.79), we have t w 2D ds w(t) 2L2 +ν 0t w 2L2 ds + 2λc2e u2 − u1 W w2 W w W , ≤γ 0

with γ given by (7.82). Using Gronwall’s formula we have then w W ≤ C u2 − u1 W w2 W . According to Proposition 8.19 and (8.115) we can deduce that w W ≤ C Φ1 − Φ2 (L2 (Q))2 h (L2 (Q))2 . Therefore, U (φ1 , ψ1 ) − U (φ2 , ψ2 ) L((L2 (Q))2 ,W) ≤ C Φ1 − Φ2 (L2 (Q))2

and then result (ii) of Proposition 8.22 follows.

Proposition 8.23. Let F be an operator that satisfies Assumptions (7.78). Then for each t ∈ [0, T ], the function Vt : (φ, ψ) −→ u(t) = Vt (φ, ψ) solution of (8.109) is continuously F-differentiable from (L2 (Q))2 to L2 (Ω) with the derivative Vt (φ, ψ) : (h1 , h2 ) −→ w(t) given by the linear parabolic problem ∂w + Aw + G(., Vt (φ, ψ))w + K(, w) = B1 h1 + B2 h2 on Q ∂t w(0) = 0 on Ω

(8.118)

which satisfies, for all Φi = (φi , ψi ) ∈ (L2 (Q))2 , i = 1, 2, the estimates: (i) Vt (φ1 , ψ1 ) L((L2 (Q))2 ;L2 (Ω)) ≤ C (ii) Vt (φ1 , ψ1 ) − Vt (φ2 , ψ2 ) L((L2 (Q))2 ;L2 (Ω)) ≤ C Φ1 − Φ2 L2 (Q) . Proof. The functional (φ, ψ) −→ u(t) is continuous from L2 (0, T ; L2 (Ω)) to L2 (Ω) (corollary of Proposition 8.19). The rest of this proposition is a corollary of Proposition 8.22. Proposition 8.24. Let F be an operator that satisfies Assumptions (7.78). Then the maps U and Vt defined by (8.109) are continuous from the weak topology of (L2 (Q))2 to the strong topology of L2 (Q) and the weak topology of L2 (Ω), respectively. Proof. Let Φ = (φ, ψ) be given in (L2 (Q))2 and let the sequence Φk = (φk , ψk ) such that Φk is weakly convergent in (L2 (Q))2 to Φ.

272


Set u = U(φ, ψ), uk = U(φk , ψk ) and vk = u − uk . Since Φk Φ weakly in (L2 (Q))2 then the sequence Φk is uniformly bounded in (L2 (Q))2 and therefore (according to Proposition 8.19) uk is uniformly bounded in W. By using Assumption 2 of (7.78) we deduce that F (., uk ) is uniformly bounded in L2 (Q). Using this result and Equation (8.109) we obtain easily that ∂uk /∂t is uniformly bounded in L1 (0, T ; D ). Let us introduce the space Y = {v ∈ L2 (0, T ; D), ∂v/∂t ∈ L1 (0, T ; D )}. According to Lemma 6.6, the injection of Y into L2 (0, T ; L2(Ω)) is compact. Therefore, uk is uniformly bounded in Y. This result makes it possible to extract from (uk , φk , ψk , F (., uk )) a subsequence also denoted by (uk , φk , ψk , F (., uk )) and such that:2 (φk , ψk ) (φ, ψ) weakly in (L2 (Q))2 , uk u ˜ weakly in L2 (0, T ; D), uk −→ u˜ strongly in L2 (Q), F (., uk ) −→ F (., u ˜) strongly in L2 (Q).

(8.119)

We easily prove that u ˜ is a solution of (8.109) with a forcing (φ, ψ) and according to the uniqueness of the solution of (8.109), we have then u˜ = u = U(φ, ψ). In the same way we prove that for each t ∈ [0, T ], Vt (φk , ψk ) Vt (φ, ψ) weakly in L2 (Ω). Theorem 8.25. Let F be an operator that satisfies Assumptions (7.78). Then, for α and β sufficiently large, there exists (φ∗ , ψ ∗ ) ∈ (L2 (Q))2 and u∗ ∈ W such that (φ∗ , ψ ∗ ) is defined by (8.111) and u∗ = U(φ∗ , ψ ∗ ) is a solution of (8.109). Proof. Let Pψ be the map: φ −→ J(φ, ψ) and Qφ be the map: ψ −→ J(φ, ψ). To obtain the existence of the robust control problem we prove that Pψ is convex and lower semi-continuous for all ψ ∈ Vad , and Pφ is concave and upper semi-continuous for all φ ∈ Uad and we use the minimax theorems in infinite dimensions presented in Chapter 5. First we prove that for α and β sufficiently large we have the convexity of the map Pψ and the concavity of the map Qφ . In order to prove the convexity, it is sufficient to show that for all (φ1 , φ2 ) ∈ Uad we have: (Pψ (φ1 ) − Pψ (φ2 )).φ ≥ 0, where φ = φ1 − φ2 . According to the definition of J, we have that T (Pψ (φ1 ) − Pψ (φ2 )).φ = α φ 2L2 (Q) + C(u1 − u2 ), Cw2 dt 0 T (8.120) + C(u1 − uobs ), C(w1 − w2 )dt 0

+μ u1 (T ) − u2 (T ), w2T +μ u1 (T ) − vobs , w1T − w2T , 2

The operator F is continuous then F (., uk ) −→ F (., u ˜) strongly in L2 (Q) (according to the thirth result of (8.119)).


273

where ui = U (φi , ψ), wi = U (φi , ψ).(φ, 0) and wiT = VT (φi , ψ).(φ, 0), for i = 1, 2. According to (7.80), (7.91) and the result of Propositions 8.19, 8.22 and 8.23, we have T

C(u1 − u2 ), Cw2 dt + μ u1 (T ) − u2 (T ), w2T 0

≤ C(u1 − u2 ) L2 (Q) Cw2 L2 (Q) +μ u1 (T ) − u2 (T ) L2 w2T L2

(8.121)

≤ c1 u1 − u2 V w2 V +μ u1 (T ) − u2 (T ) L2 w2T L2 ≤ C0 φ 2L2 (Q) and

T

C(u1 − uobs ), C(w1 − w2 )dt 0

+μ u1 (T ) − vobs , w1T − w2T ≤ C(u1 − uobs ) L2 (Q) C(w1 − w2 ) L2 (Q) +μ u1 (T ) − vobs L2 w1T − w2T L2

(8.122)

≤ c1 u1 − uobs V w1 − w2 V +μ u1 (T ) − vobs L2 w1T − w2T L2 ≤ c2 ( u1 − uobs V + u1 (T ) − vobs L2 ) φ 2L2 (Q) ≤ C1 C2 φ 2L2 (Q) . Remark 8.26. The generic constants C0 and C1 depend on the parameters μ, δ1 , cI , ce , λ, δ2 , γ, b and the final time T . The generic constant C2 depend on the observation (uobs , vobs ) and on the initial data u0 . ♦ From (8.120)–(8.122) we deduce that for α ≥ αl = C0 + C1 C2 we have (Pψ (φ1 ) − Pψ (φ2 )).φ ≥ 0 and then the convexity of Pψ . In the same way, we can find βl such that for β ≥ βl we have the concavity of Qφ . We prove now that Pψ is lower semi-continuous for all ψ ∈ Vad , and Pφ is upper semi-continuous for all φ ∈ Uad . Let φk be a minimizing sequence of J, i.e., J(φ, ψ) (∀ψ ∈ Vad ). lim inf J(φk , ψ) = min 2 k

φ∈L (Q)

Then φk is uniformly bounded in Uad and we can extract from φk a subsequence also denoted by φk such that φk φψ weakly in Uad . By using Proposition 8.24 we then have U(φk , ψ) uψ = U(φψ , ψ) weakly in L2 (0, T ; D), U(φk , ψ) −→ uψ strongly in L2 (Q), Vt (φk , ψ) uψ (t) = Vt (φψ , ψ) weakly in L2 (Ω), ∀t ∈ [0, T ].

274


Since the norm is lower semi-continuous, therefore the map Pψ : φ −→ J(φ, ψ) is lower semi-continuous for all ψ ∈ Vad . By using the same technique we then obtain that Pφ is upper semi-continuous for all φ ∈ Uad . In order to characterize the solution of the robust control problem, we introduce the adjoint problem corresponding to the primal problem (8.109) (we denote by u = U(φ, ψ) the solution of problem (8.109) where the forcing is (φ, ψ)): ∂u ˜ + A˜ u + (G(., u))∗ u˜ + K ∗ (, u˜) = C ∗ C(u − uobs ) on Q, ∂t u ˜(t = T ) = μ(u(t = T ) − vobs ) on Ω,

−

(8.123)

where C ∗ (resp. (G(., u))∗ ) is the adjoint of the operator C (resp. G(., u)) (the adjoint A∗ of A is itself, i.e., A∗ = A since A is a self-adjoint operator). The adjoint problem (8.123) is the same as the adjoint problem (7.103). Then, according to Proposition 7.28, the problem (8.123) admits a unique solution u ˜ ∈ W, if F satisfies Assumptions (7.78), u ∈ W, such that u ˜ 2H + u˜ 2V ≤ Cd2 (μ2 u(T ) − vobs 2L2 + C(u − uobs ) 2L2 (Q) ). (8.124) ˜ : (φ, ψ) −→ u ˜ In the sequel, we will denote by U ˜ = U(φ, ψ) the map defined by (8.123). We can now give the optimality system for the robust control problem (8.111). Theorem 8.27. Let F be an operator that satisfies Assumptions (7.78) and the optimal solution (φ∗ , ψ ∗ , u∗ ) ∈ (L2 (Q))2 × W such that (φ∗ , ψ ∗ ) is defined by (8.111) and u∗ = U(φ∗ , ψ ∗ ) is the solution of (8.109). Then, for α and β sufficiently large, we have for all (φ, ψ) ∈ Uad × Vad T 0

Ω

T 0

Ω

(αφ∗ + B1∗ u ˜)(φ − φ∗ )dxdt ≥ 0, (−βψ ∗ + B2∗ u˜)(ψ − ψ ∗ )dxdt ≤ 0,

˜ ∗ , ψ ∗ ) is a solution of adjoint problem (8.123). Moreover, where u ˜ = U(φ the gradient of the functional J at (φ∗ , ψ ∗ ) in any direction h = (h1 , h2 ) ∈ (L2 (Q))2 is given by J (φ∗ , ψ ∗ ).h =

T 0

Ω

(αφ∗ + B1∗ u ˜)h1 dxdt +

T 0

Ω

(−βψ ∗ + B2∗ u ˜)h2 dxdt.

Otherwise (in the weak sense), ∂J ∗ ∗ ∂J ∗ ∗ (φ , ψ ) = αφ∗ + B1∗ u (φ , ψ ) = −βψ ∗ + B2∗ u ˜ and ˜. ∂φ ∂ψ


275

Proof. Since the cost functional J is a composition of F-differentiable maps then J is F-differentiable and we have, for all h = (h1 , h2 ) ∈ (L2 (Q))2 ), T J (φ, ψ).h = C(u − uobs ), CU (φ, ψ)h + μ(u(T ) − vobs ), VT (φ, ψ)h 0

T

( αφ, h1 − βψ, h2 )

+ 0

and then

T

C ∗ C(u − uobs ), U (φ, ψ)h + μ(u(T ) − vobs ), VT (φ, ψ)h

J (φ, ψ).h = 0

T

( αφ, h1 − βψ, h2 ).

+ 0

Multiplying (8.112) by u˜, integrating over Q and (integrating) by parts in time t, we then obtain, for all h = (h1 , h2 ) ∈ (L2 (Q))2 , T ( B1∗ u ˜, h1 + B2∗ u ˜, h2 ) = VT (φ∗ , ψ ∗ )h, u˜(T ) 0 T ∂u ˜ + A˜ u + (G(., u))∗ u + − ˜ + K ∗ (, u ˜), U (φ∗ , ψ ∗ )h. ∂t 0 Since u ˜ is a solution of the adjoint problem (8.123) we then obtain, for all h = (h1 , h2 ) ∈ (L2 (Q))2 , T ( B1∗ u ˜, h1 + B2∗ u ˜, h2 )dt = VT (φ∗ , ψ ∗ )h, μ(u∗ (T ) − vobs ) 0 T (8.125) ∗ ∗ ∗ ∗ + C C(u − uobs ), U (φ , ψ )h. 0 ∗

∗

As (φ , ψ ) is a solution of (8.111), we have J (φ∗ , ψ ∗ )(φ − φ∗ , 0) ≥ 0, J (φ∗ , ψ ∗ )(0, ψ − ψ ∗ ) ≤ 0, for all (φ, ψ) ∈ Uad × Vad and, therefore (according to (8.125) and to the expression of J ), we deduce that T T ∗ ∗ ∗ (αφ + B1 u ˜)(φ − φ )dxdt ≥ 0, (−βψ ∗ + B2∗ u ˜)(ψ − ψ ∗ )dxdt ≤ 0 0

Ω

0

Ω

for all (φ, ψ) ∈ Uad × Vad . The proof is complete.

In the sequel, we will assume that there exists an optimal solution (φ∗ , ψ ∗ , u∗ ) such that (φ∗ , ψ ∗ ) is defined by (8.111), u∗ = U(φ∗ , ψ ∗ ) is a solution of (8.109) and T T (αφ∗ + B1∗ u ˜)(φ − φ∗ )dxdt ≥ 0, (−βψ ∗ + B2∗ u ˜)(ψ − ψ ∗ )dxdt ≤ 0 0

Ω

0

Ω

276


˜ ∗ , ψ ∗ ) is a solution of (8.123). for all (φ, ψ) ∈ Uad × Vad , where u ˜ = U(φ Now we give some conditions to obtain the uniqueness of the solution (φ∗ , ψ ∗ ). Theorem 8.28. Suppose that F satisfies the assumptions (7.78) and μ < 1 holds. Then, the solution (φ∗ , ψ ∗ , u∗ ) is unique if the following conditions hold: (i) θ = (ν − δ2 − c2I (γ + δ1 )) − 2b2 c2I (1/α + 1/β) > 0 1/2

(ii) σ u∗ (T ) − vobs 2L2 + C(u∗ − uobs ) 2L2 (Q) < θ, where σ = λc2e exp((δ1 + γ + 1)T /2). Proof. Assume that (φ∗1 , ψ1∗ , u∗1 ) is another solution. Then (φ∗1 , ψ1∗ ) satisfies (8.111), u∗1 = U(φ∗1 , ψ1∗ ) is a solution of (8.109), and T 0

Ω

0

Ω

T

(αφ∗1 + B1∗ u ˜1 )(φ − φ∗1 )dxdt ≥ 0, (−βψ1∗ + B2∗ u ˜1 )(ψ − ψ1∗ )dxdt ≤ 0,

˜ ∗1 , ψ1∗ ) is a solution of (8.123). for all (φ, ψ) ∈ Uad × Vad , where u ˜1 = U(φ ∗ ∗ ∗ ∗ ˜−u ˜1 . Then we We set φ = φ − φ1 , ψ = ψ − ψ1 , v = u∗ − u∗1 and v˜ = u have ∂v + Av + (F (., u∗ ) − F (., u∗1 )) + K(, v) = B1 φ + B2 ψ on Q, ∂t v(0) = 0 on Ω, −

∂˜ v + A˜ v + (G(., u∗ ))∗ v˜ + K ∗ (, v˜) ∂t = C ∗ Cv − (G(., u∗ ) − G(., u∗1 ))∗ u ˜ on Q,

(8.126)

(8.127)

v˜(t = T ) = μv(t = T ) on Ω and

T α φ 2L2 (Q) + βψ

2L2 (Q)

0

Ω

T

− 0

Ω

B1∗ v˜φdxdt ≤ 0, (8.128) B2∗ v˜ψdxdt ≤ 0.

According to Assumptions (7.78) and (7.80), we have −2 F (., u∗ ) − F (., u∗1 ), v ≤ γ0 c2I v 2D , −2 (G(., u∗ ))∗ v˜, v˜ ≤ γ0 c2I v˜ 2D , ˜, v˜ |≤ λ | v || v˜ || u ˜|. | (G(., u∗ ) − G(., u∗1 ))∗ u Ω

(8.129)


277

Multiplying (8.126) by v and (8.127) by v˜ and integrating over Q we obtain (according to (8.128), (8.129), (7.81) and (7.91))

T d 2 v L2 dt + ν v 2D dt 0 dt 0 T 2 2 v 2D dt + B1∗ v˜ L2 (Q) B1∗ v L2 (Q) ≤ γcI α 0 2 + B2∗ v˜ L2 (Q) B2∗ v L2 (Q) , β T T d v˜ 2L2 dt + ν v˜ 2D dt − 0 dt 0 T T 2 2 2 v˜ D dt + (δ1 cI + δ2 ) ( v 2D + v˜ 2D )dt ≤ γcI 0 0 T |u ˜ | | v˜ | | v | dxdt, +2λ T

0

Ω

v˜(T ) = μv(T ) and v(0) = 0. By using H¨ older’s inequality, the relationship (7.79), the assumption (8.107) and the estimate (8.108) we obtain

T d 2 2 v L2 dt + (ν − γcI ) v 2D dt 0 dt 0 1 2 2 T 1 ( v 2D + v˜ 2D )dt, ≤ 2( + )b cI α β 0 T T d 2 2 v˜ L2 dt + (ν − γcI ) v˜ 2D dt − 0 dt 0 T 2 2 ˜ H ) ( v 2D + v˜ 2D )dt, ≤ (δ1 cI + δ2 + λce u T

(8.130)

0

v˜(T ) = μv(T ) and v(0) = 0. Summing the first and the second part of (8.130) we obtain 0

T

T d 2 2 ( v L2 − v˜ L2 )dt + θ ( v 2D + v˜ 2D )dt dt 0 T ( v 2D + v˜ 2D )dt, ≤ λc2e u˜ H 0

where (according to assumption (i)) θ = ν − δ2 − c2I (γ +

2b2 2b2 + + δ1 ) > 0. α β

According to the third part of (8.130), we have

(8.131)

278


(1 − μ2 ) v(T ) 2L2 + v˜(0) 2L2 +θ( v 2V + v˜ 2V ) ˜ H ( v 2V + v˜ 2V ). ≤ λc2e u ˜ H ( v 2V + v˜ 2V ). Since 1 − μ2 > 0 then θ( v 2V + v˜ 2V ) ≤ λc2e u By applying the estimate (7.112) and μ < 1 we can deduce that θ∗ ( v 2V + v˜ 2V ) ≤ 0, where

1/2 θ∗ = θ − λc2e e(δ1 +γ+1)T /2 u∗ (T ) − vobs 2L2 + C(u∗ − uobs ) 2L2 (Q) . Since θ∗ > 0 (according to assumption (ii)), we have v 2V + v˜ 2V = 0 and then v = 0 and v˜ = 0. We obtain then the uniqueness result. 8.6.4 Initial Condition Control In this section the objective of the robust control problem is to find the best estimate of the initial state u0 in the presence of the disturbance which maximally spoils the control objective. For this, we study two problems: first, the case where the worst disturbance is in the initial condition, and, second, the case where the worst disturbance is in the forcing term. Distributed Disturbance in the Initial Condition We suppose now that the value u0 is decomposed into the disturbance ψ ∈ L2 (Ω) and the control φ ∈ L2 (Ω), i.e., u0 = B1 φ + B2 ψ, where Bi , i = 1, 2 are given bounded operators on L2 (Ω) satisfying the assumption (8.107) and the estimate (8.108). Therefore, the function u is assumed to be related to the disturbance ψ and control φ through the problem (8.105): ∂u + Au + F (x, t, u) + K(, u) = g (given) on Q, ∂t u(0) = B1 φ + B2 ψ on Ω.

(8.132)

To obtain the regularity of Proposition 8.19, we suppose the following hypothesis: g ∈ L2 (Q), (φ, ψ) ∈ L2 (Ω)2 . Let U : (φ, ψ) −→ u = U(φ, ψ) be the map: (L2 (Ω))2 −→ W defined by (8.132) and introducing the cost functional defined by J(φ, ψ) =

μ 1 C(u − uobs ) 2L2 (Q) + u(T ) − vobs 2L2 (Ω) 2 2 α β 2 + φ L2 (Ω) − ψ 2L2 (Ω) , 2 2

(8.133)


279

where μ, α, β > 0 are fixed, (uobs , vobs ) ∈ V × L2 (Ω) is the observation and C is an unbounded, linear operator on L2 (Ω) satisfying the hypothesis (7.91). We want to minimize the functional J with respect to φ and maximize J with respect to ψ, i.e., to find (φ∗ , ψ ∗ ) ∈ Uad × Vad such that J(φ∗ , ψ) ≤ J(φ∗ , ψ ∗ ) ≤ J(φ, ψ ∗ ), ∀(φ, ψ) ∈ Uad × Vad ,

(8.134)

where Uad and Vad are non-empty, closed, convex, bounded subsets of L2 (Ω). By using the same technique as used in the proof of Propositions 8.19 and 8.22 and Theorem 8.25, we have the following results (with no further estimates required). Proposition 8.29. Let F be an operator that satisfies the assumptions (7.78). Then the function U : (φ, ψ) −→ u = U(φ, ψ) solution of (8.132) is continuously F-differentiable from (L2 (Ω))2 to W with the derivative U (φ, ψ) : (h1 , h2 ) −→ w given by the linear parabolic problem ∂w + Aw + G(., U(φ, ψ))w + K(, w) = 0 on Q ∂t w(0) = B1 h1 + B2 h2 on Ω,

(8.135)

which satisfies, for all Φi = (φi , ψi ) ∈ (L2 (Ω)2 for i = 1, 2, the estimates: (i) U (φ1 , ψ1 ) L((L2 (Ω))2 ;W) ≤ C (ii) U (φ1 , ψ1 ) − U (φ2 , ψ2 ) L((L2 (Ω))2 ;W) ≤ C Φ1 − Φ2 L2 (Q) .

Proposition 8.30. Let F be an operator that satisfies the assumptions (7.78). Then, for each t ∈ [0, T ], the function Vt : (φ, ψ) −→ u(t) = Vt (φ, ψ) solution of (8.132) is continuously F-differentiable from (L2 (Ω))2 to L2 (Ω) with the derivative Vt (φ, ψ) : (h1 , h2 ) −→ w(t) given by the linear parabolic problem ∂w + Aw + G(., Vt (φ, ψ))w + K(, w) = 0 on Q ∂t w(0) = B1 h1 + B2 h2 on Ω,

(8.136)

which satisfies, for all Φi = (φi , ψi ) ∈ (L2 (Q))2 for i = 1, 2, the estimates: (i) Vt (φ1 , ψ1 ) L((L2 (Ω))2 ;L2 (Ω)) ≤ C (ii) Vt (φ1 , ψ1 ) − Vt (φ2 , ψ2 ) L((L2 (Ω))2 ;L2 (Ω)) ≤ C Φ1 − Φ2 L2 (Q) .

Theorem 8.31. Let F be an operator that satisfies the assumptions (7.78). Then, for α and β sufficiently large, there exists (φ∗ , ψ ∗ ) ∈ (L2 (Ω))2 and u∗ ∈ W such that (φ∗ , ψ ∗ ) is defined by (8.134) and u∗ = U(φ∗ , ψ ∗ ) is a solution of (8.132). Remark 8.32. The adjoint problem associated with the primal problem (8.132) is exactly the adjoint problem (8.123). ♦ Now we give the optimality system for the robust control problem (8.134).

280


Theorem 8.33. Let F be an operator that satisfies the assumptions (7.78), ν > δ2 , and the optimal solution (φ∗ , ψ ∗ , u∗ ) ∈ (L2 (Q))2 × W such that (φ∗ , ψ ∗ ) is defined by (8.134) and u∗ = U(φ∗ , ψ ∗ ) is the solution of (8.132). Then, for α and β sufficiently large, we have (∀(φ, ψ) ∈ Uad × Vad ) (αφ∗ + B1∗ u˜(0))(φ − φ∗ )dx ≥ 0, Ω (−βψ ∗ + B2∗ u ˜(0))(ψ − ψ ∗ )dx ≤ 0, Ω

˜ ∗ , ψ ∗ ) is a solution of the adjoint problem (8.123). Moreover, where u˜ = U(φ the gradient of the functional J at (φ∗ , ψ ∗ ) in any direction h = (h1 , h2 ) ∈ (L2 (Q))2 is given by ∗ ∗ ∗ ∗ (αφ + B1 u ˜(0))h1 dx + (−βψ ∗ + B2∗ u ˜(0))h2 dx. J (φ , ψ ).h = Ω

Ω

Otherwise (in the weak sense), ∂J ∗ ∗ ∂J ∗ ∗ (φ , ψ ) = αφ∗ + B1∗ u (φ , ψ ) = −βψ ∗ + B2∗ u ˜(0) and ˜(0). ∂φ ∂ψ Proof. The cost functional J is a composition of F-differentiable mappings then J is F-differentiable and we have (∀h = (h1 , h2 ) ∈ (L2 (Ω))2 ): T J (φ, ψ)h =

C(u − uobs ), CU (φ, ψ)h + μ(u(T ) − vobs ), VT (φ, ψ)h 0

+ αφ, h1 − βψ, h2

and then

J (φ, ψ)h =

T

C ∗ C(u − uobs ), U (φ, ψ)h + μ(u(T ) − vobs ), VT (φ, ψ)h

0

+ αφ, h1 − βψ, h2 .

Multiplying (8.135) by u˜, integrating over Q and (integrating) by parts in time t, we obtain (∀h = (h1 , h2 ) ∈ (L2 (Q))2 ):

B1 h1 + B2 h2 , u ˜(0) = VT (φ∗ , ψ ∗ )h, u˜(T ) T ∂u ˜ + A˜ u + (G(., u))∗ u ˜ + K ∗ (, u ˜), U (φ∗ , ψ ∗ )h. + − ∂t 0 Since u˜ is a solution of adjoint problem we then obtain (∀h = (h1 , h2 ) ∈ (L2 (Ω))2 ):

B1 h1 + B2 h2 , u ˜(0) = VT (φ∗ , ψ ∗ )h, μ(u∗ (T ) − vobs ) T + C ∗ C(u∗ − uobs ), U (φ∗ , ψ ∗ )h. 0

(8.137)


281

As (φ∗ , ψ ∗ ) is a solution of (8.134) then J (φ∗ , ψ ∗ )(φ − φ∗ , 0) ≥ 0, J (φ∗ , ψ ∗ )(0, ψ − ψ ∗ ) ≤ 0, for all (φ, ψ) ∈ Uad × Vad , and we can then deduce the result of this theorem (according to the expression of J ). The proof is complete. In the following, we will assume that there exists an optimal solution (φ∗ , ψ ∗ , u∗ ) such that (φ∗ , ψ ∗ ) is defined by (8.134), u∗ = U(φ∗ , ψ ∗ ) is a solution of (8.132) and (αφ∗ + B1∗ u˜(0))(φ − φ∗ )dx ≥ 0, (−βψ ∗ + B2∗ u ˜(0))(ψ − ψ ∗ )dx ≤ 0 Ω

Ω

˜ ∗ , ψ ∗ ) is a solution of (8.123). ˜ = U(φ for all (φ, ψ) ∈ Uad × Vad , where u Now we give some conditions to obtain the uniqueness of the optimal solution (φ∗ , ψ ∗ ). Theorem 8.34. Suppose that F satisfies the assumptions (7.78), ν > γ2 and μ < 1 holds. The optimal solution (φ∗ , ψ ∗ , u∗ ) is unique if the following conditions hold: (i) θ = (ν − δ2 − c2I (γ + δ1 )) > 0 and 1 − (b4 /α2 ) − (b4 /β 2 ) ≥ 0

1/2 < θ, (ii) λc2e cT u∗ (T ) − vobs 2L2 + C(u∗ − uobs ) 2L2 (Q) where cT = exp((δ1 + γ + 1)T /2). Proof. Suppose (φ∗1 , ψ1∗ , u∗1 ) is another solution, then (φ∗1 , ψ1∗ ) satisfies (8.134), u∗1 = U(φ∗1 , ψ1∗ ) is a solution of (8.132) and (αφ∗1 + B1∗ u ˜1 (0))(φ − φ∗1 )dx ≥ 0, (−βψ1∗ + B2∗ u ˜1 (0))(ψ − ψ1∗ )dx ≤ 0 Ω

Ω

˜ ∗1 , ψ1∗ ) is a solution of (8.123). ˜1 = U(φ for all (φ, ψ) ∈ Uad × Vad , where u ∗ ∗ ∗ ∗ ˜−u ˜1 . Then we We set φ = φ − φ1 , ψ = ψ − ψ1 , v = u∗ − u∗1 and v˜ = u have ∂v + Av + (F (., u∗ ) − F (., u∗1 )) + K(, v) = 0 on Q ∂t (8.138) v(0) = B1 φ + B2 ψ on Ω, −

∂˜ v + A˜ v + (G(., u∗ ))∗ v˜ + K ∗ (, v˜) ∂t ˜ on Q, = C ∗ Cv − (G(., u∗ ) − G(., u∗1 ))∗ u

(8.139)

v˜(t = T ) = μv(t = T ) on Ω

and αφ

2L2 (Ω)

+

β ψ 2L2 (Ω) −

Ω Ω

B1∗ v˜(0)φdx ≤ 0, (8.140) B2∗ v˜(0)ψdx ≤ 0.

282


Multiplying (8.138) by v, (8.139) by v˜ and integrating over Ω, this gives (according to (7.80), (7.81), (7.91) and (8.129)) d v 2L2 +ν v 2D ≤ γc2I v 2D , dt d − v˜ 2L2 +ν v˜ 2D ≤ γc2I v˜ 2D +(δ1 c2I + δ2 )( v 2D + v˜ 2D ) dt | u˜ | | v˜ | | v |, +2λ Ω

v˜(T ) = μv(T ) and v(0) = B1 φ + B2 ψ. By using H¨ older’s inequality and the relationship (7.79), we obtain d v 2L2 +(ν − γc2I ) v 2D ≤ 0, dt d − v˜ 2L2 +(ν − γc2I ) v˜ 2D dt ≤ (δ1 c2I + δ2 + λc2e u˜ H )( v 2D + v˜ 2D ),

(8.141)

v˜(T ) = μv(T ) and v(0) = B1 φ + B2 ψ. Summing the first and second parts of (8.141) we obtain d ( v 2L2 − v˜ 2L2 ) + θ( v 2D + v˜ 2D ) dt ≤ λc2e u ˜ H ( v 2D + v˜ 2D ),

(8.142)

v˜(T ) = μv(T ) and v(0) = B1 φ + B2 ψ where θ = ν − δ2 − c2I (γ + δ1 ) > 0 (according to assumption (i)). By integrating over [0, T ] the first part of (8.142) and according to the second and third parts of (8.142), to the assumption (8.107), and the relations (8.108) and (8.140) we can deduce that (1 − μ2 ) v(T ) 2L2 + v˜(0) 2L2 +θ( v 2V + v˜ 2V ) ≤ λc2e u˜ H ( v 2V + v˜ 2V ) + (

b4 b4 + 2 ) v˜(0) 2L2 . 2 α β

Since 1 − μ2 > 0 and 1 − (b4 /α2 ) − (b4 /β 2 ) ≥ 0 then ˜ H ( v 2V + v˜ 2V ). θ( v 2V + v˜ 2V ) ≤ λc2e u According to the estimate (7.112) and μ < 1 we have θ∗ ( v 2V + v˜ 2V ) ≤ 0, where θ∗ = θ−λc2e exp((δ1 +γ+1)T /2)( u∗ (T )−vobs 2L2 + C(u∗ −uobs ) 2L2 (Q) )1/2 . Since θ∗ > 0 (according to assumption (ii)), we have v 2V + v˜ 2V = 0 and then v = 0 and v˜ = 0. Therefore, the uniqueness result follows.


283

Distributed Disturbance in the Forcing In this section, the value g is the disturbance ψ ∈ L2 (Ω) and the initial condition u0 is the control φ ∈ L2 (Ω), i.e., g = B2 ψ, u0 = B1 φ, where Bi , i = 1, 2 are given bounded operators on L2 (Ω) such that Assumption (8.107) holds. The function u is assumed to be related to the disturbance ψ and control φ through the problem (8.105): ∂u + Au + F (., u) + K(, u) = B2 ψ on Q, ∂t u(0) = B1 φ on Ω.

(8.143)

To obtain the regularity of Proposition 8.19, we suppose the following hypothesis: (φ, ψ) ∈ L2 (Ω) × L2 (Q). Let U : (φ, ψ) −→ u = U(φ, ψ) be the map: L2 (Ω) × L2 (Q) −→ W defined by (8.143) and introducing the cost functional defined by J(φ, ψ) =

1 μ C(u − uobs ) 2L2 (Q) + u(T ) − vobs 2L2 2 2 α β 2 + φ L2 (Ω) − ψ 2L2 (Q) , 2 2

(8.144)

where μ, α, β > 0 are fixed, (uobs , vobs ) ∈ V × L2(Ω) is the observation (given) and C is unbounded, linear operator on L2 (Ω) satisfying the hypothesis (7.91). We want to minimize the functional J with respect to φ and maximize J with respect to ψ, i.e., to find (φ∗ , ψ ∗ ) ∈ Uad × Vad such that J(φ∗ , ψ) ≤ J(φ∗ , ψ ∗ ) ≤ J(φ, ψ ∗ ), ∀(φ, ψ) ∈ Uad × Vad ,

(8.145)

with Uad and Vad are (given) non-empty, closed, convex, bounded subsets of L2 (Ω) and L2 (Q) respectively. The arguments of the previous sections can be extended directly to the present case without further estimates. We have then the following results. Proposition 8.35. Let F be an operator that satisfies the assumptions (7.78). Then the function U : (φ, ψ) −→ u = U(φ, ψ) which is a solution of (8.143) is continuously F-differentiable from L2 (Ω) × L2 (Q) to W with the derivative U (φ, ψ) : (h1 , h2 ) −→ w given by the linear parabolic problem ∂w + Aw + G(., U(φ, ψ))w + K(, w) = B2 h2 on Q, ∂t w(0) = B1 h1 on Ω,

(8.146)

which satisfies, for all Φi = (φi , ψi ) ∈ L2 (Ω) × L2 (Q), for i = 1, 2, the estimates: (i) U (φ1 , ψ1 ) L(L2 (Ω)×L2 (Q);W) ≤ C (ii) U (φ1 , ψ1 ) − U (φ2 , ψ2 ) L(L2 (Ω)×L2 (Q);W) ≤ C Φ1 − Φ2 L2 (Ω)×L2 (Q) .

284


Proposition 8.36. Let F be an operator that satisfies Assumptions (7.78). Then for each t ∈ [0, T ], the function Vt : (φ, ψ) −→ u(t) = Vt (φ, ψ) which is a solution of (8.143) is continuously F-differentiable from (L2 (Ω))2 to L2 (Ω) with the derivative Vt (φ, ψ) : (h1 , h2 ) −→ w(t) given by the linear parabolic problem ∂w + Aw + G(., Vt (φ, ψ))w + K(, w) = B2 h2 on Q ∂t w(0) = B1 h1 on Ω

(8.147)

which satisfies, for all Φi = (φi , ψi ) ∈ L2 (Ω) × L2 (Q), for i = 1, 2, the estimates: (i) Vt (φ1 , ψ1 ) L(L2 (Ω)×L2 (Q);L2 (Ω)) ≤ C (ii) Vt (φ1 , ψ1 ) − Vt (φ2 , ψ2 ) L(L2 (Ω)×L2 (Q);L2 (Ω)) ≤ C Φ1 − Φ2 L2 (Ω)×L2 (Q) .

Theorem 8.37. Let F be an operator that satisfies the assumptions (7.78). Then, for α and β sufficiently large, there exists (φ∗ , ψ ∗ ) ∈ Uad × Vad and u∗ ∈ W such that (φ∗ , ψ ∗ ) is defined by (8.145) and u∗ = U(φ∗ , ψ ∗ ) is a solution of (8.143). Remark 8.38. The adjoint problem associated with the primal problem (8.143) is exactly the adjoint problem (8.123). ♦ Now we give the optimality system for the robust control problem (8.145). Theorem 8.39. Let F be an operator that satisfies Assumptions (7.78), ν > δ2 , and an optimal solution (φ∗ , ψ ∗ , u∗ ) ∈ Uad × Vad × W such that (φ∗ , ψ ∗ ) is defined by (8.134) and u∗ = U(φ∗ , ψ ∗ ) is the solution of (8.143). Then, for α and β sufficiently large, we have (∀(φ, ψ) ∈ Uad × Vad ) (αφ∗ + B1∗ u ˜(0))(φ − φ∗ )dx ≥ 0, Ω T (−βψ ∗ + B2∗ u˜)(ψ − ψ ∗ )dxdt ≤ 0, 0

Ω

˜ ∗ , ψ ∗ ) is a solution of the adjoint problem (8.123). Moreover, where u˜ = U(φ the gradient of the functional J at (φ∗ , ψ ∗ ) in any direction h = (h1 , h2 ) ∈ L2 (Ω) × L2 (Q) is given by T (αφ∗ + B1∗ u˜(0))h1 dx + (−βψ ∗ + B2∗ u˜)h2 dx. J (φ∗ , ψ ∗ ).h = Ω

0

Ω

Otherwise (in the weak sense), ∂J ∗ ∗ ∂J ∗ ∗ (φ , ψ ) = αφ∗ + B1∗ u (φ , ψ ) = −βψ ∗ + B2∗ u ˜(0) and ˜. ∂φ ∂ψ


285

Proof. We use the same technique as used in the proof of Theorem 8.33. So, we skip the details. In the following, we will assume that there exists an optimal solution (φ∗ , ψ ∗ , u∗ ) such that (φ∗ , ψ ∗ ) is defined by (8.145), u∗ = U(φ∗ , ψ ∗ ) is a solution of (8.143) and

∗

(αφ + Ω

B1∗ u˜(0))(φ

T

∗

− φ )dx ≥ 0, 0

Ω

(−βψ ∗ + B2∗ u ˜)(ψ − ψ ∗ )dxdt ≤ 0

˜ ∗ , ψ ∗ ) is a solution of (8.123). ˜ = U(φ for all (φ, ψ) ∈ Uad × Vad , where u Now we give some conditions to obtain the uniqueness of the solution (φ∗ , ψ ∗ ). Theorem 8.40. Suppose that F satisfies Assumptions (7.78), μ < 1 and α ≥ b2 holds. The optimal solution (φ∗ , ψ ∗ , u∗ ) is unique if the following conditions hold: (i) θ = (ν − δ2 − c2I (γ + δ1 ) − b2 c2I /β) > 0

1/2 (ii) λc2e cT u∗ (T ) − vobs 2L2 + C(u∗ − uobs ) 2L2 (Q) < θ, where cT = exp((δ1 + γ + 1)T /2). Proof. Suppose (φ∗1 , ψ1∗ , u∗1 ) is another solution, then (φ∗1 , ψ1∗ ) satisfies (8.134), u∗1 = U(φ∗1 , ψ1∗ ) is a solution of (8.143) and (αφ∗ + B1∗ u ˜1 (0))(φ − φ∗1 )dx ≥ 0, Ω T (−βψ1∗ + B2∗ u ˜1 )(ψ − ψ1∗ )dxdt ≤ 0, 0

Ω

˜ ∗1 , ψ1∗ ) is a solution of (8.123). ˜1 = U(φ for all (φ, ψ) ∈ Uad × Vad , where u ∗ ∗ ∗ ∗ ˜1 . We then We set φ = φ − φ1 , ψ = ψ − ψ1 , v = u∗ − u∗1 and v˜ = u˜ − u have ∂v + Av + (F (., u∗ ) − F (., u∗1 )) = B2 ψ on Q ∂t (8.148) v(0) = B1 φ on Ω, −

∂˜ v + A˜ v + (G(., u∗ ))∗ v˜ ∂t ˜ on Q, = C ∗ Cv − (G(., u∗ ) − G(., u∗1 ))∗ u

(8.149)

v˜(T ) = μv(T ) on Ω

and

B1∗ v˜(0)φdx ≤ 0, Ω T − B2∗ v˜ψdxdt ≤ 0.

α φ 2L2 (Ω) + β ψ 2L2 (Q)

0

Ω

(8.150)

286


Multiplying (8.148) by v, (8.149) by v˜ and integrating over Q, this gives (according to (8.150), (8.129), (7.80), (7.81) and (7.91)) T T d v 2L2 ds + ν v 2D ds 0 dt 0 T 2 v 2D ds + B2∗ v˜ L2 (Q) B2∗ v L2 (Q) , ≤ γc2I β 0 T T d v˜ 2L2 ds + ν v˜ 2D ds − 0 dt 0 T T v˜ 2D ds + (δ1 c2I + δ2 ) ( v 2D ds+ v˜ 2D ) ≤ γc2I 0 0 T |u ˜ | | v˜ | | v | ds, +2λ 0

Ω

v˜(T ) = μv(T ) and v(0) = B1 φ. By using H¨ older’s inequality, the relationship (7.79), the assumption (8.107) and the estimate (8.108) we obtain T T d v 2L2 ds + (ν − γc2I ) v 2D ds 0 dt 0 T 1 ( v 2D ds+ v˜ 2D ), ≤ b2 c2I β 0 T T (8.151) d v˜ 2L2 ds + (ν − γc2I ) v˜ 2D ds − 0 dt 0 T ≤ (δ1 c2I + δ2 + λc2e u˜ H ) ( v 2D ds+ v˜ 2D ), 0

v˜(T ) = μv(T ) and v(0) = B1 φ. Summing the first and second parts of (8.151) we obtain T T d 2 2 ( v L2 − v˜ L2 )ds + θ ( v 2D + v˜ 2D )ds 0 dt 0 T ˜ H ( v 2D + v˜ 2D )ds, ≤ λc2e u

(8.152)

0

where θ = ν − δ2 − + δ1 + b2 /β) > 0 (according to assumption (i)). According to third part of (8.151), the first part of (8.150) and the estimate (8.108), we have c2I (γ

(1 − μ2 ) v(T ) 2L2 + v˜(0) 2L2 +θ( v 2V + v˜ 2V ) ≤ λc2e u ˜ H ( v 2V + v˜ 2V ) + Since 1 − μ2 > 0 and 1 − b4 /α2 ≥ 0 then

b4 v˜(0) 2L2 . α2


287

θ( v 2V + v˜ 2V ) ≤ λc2e u ˜ H ( v 2V + v˜ 2V ). Applying the estimate (7.112) and μ < 1 we have θ∗ ( v 2V + v˜ 2V ) ≤ 0, where

1/2 . θ∗ = θ − λc2e e(δ1 +γ+1)T /2 u∗ (T ) − vobs 2L2 + C(u∗ − uobs ) 2L2 (Q) Since θ∗ > 0 (according to assumption (ii)), we have v 2V + v˜ 2V = 0 and then v = 0 and v˜ = 0. The uniqueness result is proved. 8.6.5 A Remark on the Robust Boundary Control Problem We conclude with an indication as to how the previous arguments can be made to apply to the robust boundary control problem. Let us consider the following problem: ∂u − div(η∇u) + F (x, t, u) + K(, u) = g on Q, ∂t ∂u + au = f on Σ = (0, T ) × ∂Ω, η ∂n u(0) = u0 on Ω,

(8.153)

where Ω is an open bounded in IRm , m ≥ 1, with the boundary ∂Ω sufficiently regular, g is in L2 (Q), u0 is in L2 (Ω) and F satisfies Assumptions (7.78) as before. We assume that f is in L2 (Σ), a is in L∞ (Σ) and non-negative, ∂u/∂n denotes the exterior normal derivative of u at the boundary ∂Ω, the function η is positive and bounded function above and below by two non-negative constants in Ω, and we denote ν = min(η). Ω

Here, we assume that m ≤ 4,3 D = H 1 (Ω) ⊂ L4 (Ω) and D = (H 1 (Ω)) . Multiplying now (8.153) by u, integrating over Ω, using Green’s formula and the first assumption of (7.78), we have that d | u |2 +2 η | ∇u |2 dx + 2 a | u |2 dΓ = −2 K(, u)udx dt Ω Γ Ω −2 (F (., t, u) − F (., t, 0))(u − 0)dx + 2 gudx + 2 f udΓ. Ω

Ω

Γ

According to the second assumption of (7.78), the assumption (7.81) and the trace theory we can deduce that d | u |2 + ν u 2D ≤ c1 | u |2 +c2 (| g |2 + f 2L2 (Γ ) ). dt 3

D ⊂ L4 (Ω) provided m ≤ 4 according to the Sobolev embedding theorem.

288


Integrating over (0, t), for all t ∈ (0, T ), gives t 2 u 2D ds | u(t) | +ν 0 t | u |2 ds + c3 ( g 2L2 (Q) + f 2L2 (Σ) + | u0 |2 ) ≤ c1 0

and then (according to Gronwall’s lemma) u 2W ≤ C( u0 2L2 + g 2L2 (Q) + f 2L2 (Σ) ).

(8.154)

Using the results (8.154) and (8.153), we prove easily that u satisfies also the following estimate: u 2H1 ≤ C( u0 2L2 + g 2L2 (Q) + f 2L2 (Σ) ).

(8.155)

Consequently, u ∈ W ∩ H1 . One similarly gets a uniform Lipschitz continuous solution for the mapping solution in W ∩ H1 . Remark 8.41. If we suppose that the initial condition is in H 1/2 (Ω) we can obtain for the solution u of the non-linear problem (8.153), by removing the terms F (., u)+K(, u) on the right-hand side of (8.153) and by using Theorem ♦ 6.19, the regularity result u ∈ H 3/2,3/4 (Q). We suppose now that the value g is the disturbance ψ ∈ L2 (Q) and the boundary condition f is the control φ ∈ L2 (Σ), i.e., f = B1 φ and g = B2 ψ, where B1 and B2 are given bounded operators on L2 (Σ) and on L2 (Q) respectively. So the function u is assumed to be related to the disturbance ψ and control φ through the problem (8.153): ∂u − div(η∇u) + F (x, t, u) + K(, u) = B2 ψ on Q, ∂t ∂u + au = B1 φ on Σ = (0, T ) × ∂Ω, η ∂n u(0) = u0 on Ω.

(8.156)

Let U : (φ, ψ) −→ u = U(φ, ψ) be the map: L2 (Σ)×L2 (Q) −→ W defined by (8.156) and introducing the cost functional defined by J(φ, ψ) =

μ 1 C(u − uobs ) 2L2 (Q) + u(T ) − vobs 2L2 (Ω) 2 2 α β 2 + φ L2 (Σ) − ψ 2L2 (Q) , 2 2

(8.157)

where μ, α, β > 0 are fixed, (uobs , vobs ) ∈ V × L2 (Ω) is the observation and C is an unbounded, linear operator on L2 (Ω) satisfying the hypothesis (7.91). We want to minimize the functional J with respect to φ and maximize J with respect to ψ, i.e., to find (φ∗ , ψ ∗ ) ∈ Uad × Vad such that J(φ∗ , ψ) ≤ J(φ∗ , ψ ∗ ) ≤ J(φ, ψ ∗ ), ∀(φ, ψ) ∈ Uad × Vad

(8.158)


289

with Uad and Vad are (given) non-empty, closed, convex, bounded subsets of L2 (Σ) and L2 (Q), respectively. By using the same technique as used in the proof of Propositions 8.19 and 8.22 and Theorems 8.25 and 8.27, we have the following results. Proposition 8.42. Let F be an operator that satisfies the assumptions (7.78). Then the function U : (φ, ψ) −→ u = U(φ, ψ) solution of (8.156) is continuously F-differentiable from L2 (Σ) × L2 (Q) to W with the derivative U (φ, ψ) : (h1 , h2 ) −→ w given by the linear parabolic problem ∂w − div(η∇w) + G(., U(φ, ψ))w + K(, w) = B2 h2 on Q ∂t ∂w + au = B1 h1 on Σ, η ∂n w(0) = 0 on Ω,

(8.159)

which satisfies, for all Φi = (φi , ψi ) ∈ L2 (Σ) × L2 (Q), for i = 1, 2, the following estimates: (i) U (φ1 , ψ1 ) L(L2 (Σ)×L2 (Q);W) ≤ C (ii) U (φ1 , ψ1 ) − U (φ2 , ψ2 ) L(L2 (Σ)×L2 (Q);W) ≤ C Φ1 − Φ2 L2 (Σ)×L2 (Q) . Proposition 8.43. Let F be an operator that satisfies the assumptions (7.78). Then for each t ∈ [0, T ], the function Vt : (φ, ψ) −→ u(t) = Vt (φ, ψ) solution of (8.156) is continuously F-differentiable from (L2 (Ω))2 to L2 (Ω) with the derivative Vt (φ, ψ) : (h1 , h2 ) −→ w(t) given by the linear parabolic problem ∂w − div(η∇w) + G(., U(φ, ψ))w + K(, w) = B2 h2 on Q ∂t ∂w + au = B1 h1 on Σ, η ∂n w(0) = 0 on Ω,

(8.160)

which satisfies, for all Φi = (φi , ψi ) ∈ L2 (Σ) × L2 (Q), for i = 1, 2, the following estimates: (i) Vt (φ1 , ψ1 ) L(L2 (Σ)×L2 (Q);L2 (Ω)) ≤ C (ii) Vt (φ1 , ψ1 ) − Vt (φ2 , ψ2 ) L(L2 (Σ)×L2 (Q);L2 (Ω)) ≤ C Φ1 − Φ2 L2 (Σ)×L2 (Q) .

Theorem 8.44. Let F be an operator that satisfies the assumptions (7.78). Then, for α and β sufficiently large, there exists (φ∗ , ψ ∗ ) ∈ L2 (Σ) × L2 (Q) and u∗ ∈ W such that (φ∗ , ψ ∗ ) is defined by (8.158) and u∗ = U(φ∗ , ψ ∗ ) is a solution of (8.156). Now we give the optimality system for the robust control problem (8.158).

290


Theorem 8.45. Let F be an operator that satisfies the assumptions (7.78) and an optimal solution (φ∗ , ψ ∗ , u∗ ) ∈ L2 (Σ) × L2 (Q) × W such that (φ∗ , ψ ∗ ) is defined by (8.158) and u∗ = U(φ∗ , ψ ∗ ) is the solution of (8.156). Then, for α and β sufficiently large, we have, for all (φ, ψ) ∈ Uad × Vad , T 0

Γ

T 0

Ω

(αφ∗ + B1∗ u ˜|Σ )(φ − φ∗ )dΓ dt ≥ 0, (8.161) (−βψ ∗ + B2∗ u ˜)(ψ − ψ ∗ )dxdt ≤ 0,

˜ ∗ , ψ ∗ ) is a solution of the following adjoint problem where u˜ = U(φ −

∂u ˜ − div(η∇˜ u) + (G(., u))∗ u ˜ + K ∗ (, u˜) ∂t = C ∗ C(u∗ − uobs ) on Q,

∂u ˜ + a˜ u = 0 on Σ, η ∂n u˜(T ) = μ(u∗ (T ) − vobs ) on Ω,

(8.162)

where C ∗ (resp. (G(., u))∗ ) is the adjoint of the operator C (resp. G(., u)). Moreover, the gradient of the functional J at (φ∗ , ψ ∗ ) in any direction h = (h1 , h2 ) ∈ L2 (Σ) × L2 (Q) is given by J (φ∗ , ψ ∗ ).h =

T 0

Γ

(αφ∗ + B1∗ u˜|Σ )h1 dΓ dt +

T 0

Ω

(−βψ ∗ + B2∗ u˜)h2 dxdt.

Otherwise (in the weak sense), ∂J ∗ ∗ ∂J ∗ ∗ (φ , ψ ) = αφ∗ + B1∗ u (φ , ψ ) = −βψ ∗ + B2∗ u˜. ˜|Σ and ∂φ ∂ψ

Remark 8.46. It is clear that we can consider other controls and disturbances (which can appear in the boundary condition or in the state system) and we can obtain the same results by using the same technique as in the previous work of this section. ♦ 8.6.6 Contraction Mapping and Fixed-point Formulation In this section we rewrite the robust control problem as a fixed-point problem (in the case of the equality-type constraints). Let Ui , i = 1, 2 be Hilbert spaces who will play the role of L2 (Q) or L2 (Ω) depending on different situations control-disturbance presented previously. For (φ, ψ) ∈ U = U1 × U2 , we set Gρ (φ, ψ) := (φ − ρ(αφ + B1∗ u˜), ψ − ρ(βψ − B2∗ u ˜)),

(8.163)


291

˜ ∗ , ψ ∗ ) is a solution of the where, ρ > 0 is a positive constant and u ˜ = U(φ adjoint problem (8.123). Our goal is to prove that Gρ is a contraction mapping on some subset Y of U. Let > 0 and Y be a ball of radius , i.e., Y = {(φ, ψ) ∈ U : (φ, ψ) U ≤ }. Proposition 8.47. Assume that α and β are sufficiently large and that is sufficiently small. Then there exists ρ0 > 0 (depending on , uobs , vobs , U and given fixed parameters) such that Gρ is a mapping from Y into Y , for all ρ ≤ ρ0 . Proof. Let (φ, ψ) ∈ Y . We have easily that φ − ρ(αφ + B1∗ u˜) 2U1 = ρ2 (α2 φ 2U1 + B1∗ u˜ 2U1 +2α(φ, B1∗ u ˜)U1 ) ˜)U1 )+ φ 2U1 −2ρ(α φ 2U1 +(φ, B1∗ u ψ − ρ(βψ − B2∗ u˜) 2U2 = ρ2 (β 2 ψ 2U2 + B2∗ u ˜ 2U2 −2β(ψ, B2∗ u ˜)U2 ) ˜)U2 )+ ψ 2U2 −2ρ(β ψ 2U2 −(ψ, B2∗ u and then ˜) 2U1 + ψ − ρ(βψ − B2∗ u˜) 2U2 Gρ (φ, ψ) 2U = φ − ρ(αφ + B1∗ u ≤ (1 + 2ρ2 α2 + (1 − 2α)ρ) φ 2U1 +ρ(2ρ + 1) B1∗ u ˜ 2U1 +(1 + 2ρ2 β 2 + (1 − 2β)ρ) ψ 2U2 +ρ(2ρ + 1) B2∗ u ˜ 2U2 . According to the continuity of the operators Bi∗ , i = 1, 2, we can deduce that (because of (8.124) which is valid for all the control-disturbance cases) Bi∗ u˜ 2Ui ≤ C0 u ˜ 2Ui ≤ C1 ( u(T ) − vobs 2L2 + C(u − uobs ) 2L2 (Q) ). According to (8.106) and (8.107) and Assumption (7.91), we can deduce that C1 ( u(T ) − vobs 2L2 + C(u − uobs ) 2L2 (Q) ) ≤ C2 ( φ 2U1 + ψ 2U2 ) + C3 , where the generic constant C3 depends on the observation (uobs , vobs ) and data (initial condition or forcing) and the generic constant C2 depends on the function U . Consequently, Gρ (φ, ψ) 2U ≤ 2ρ(2ρ + 1)C3 +(1 + 2ρ2 α2 + (1 − 2α)ρ + ρ(2ρ + 1)C2 ) φ 2U1 +(1 + 2ρ2 β 2 + (1 − 2β)ρ + ρ(2ρ + 1)C2 ) ψ 2U2 . Since (φ, ψ) ∈ Y , we can deduce that Gρ (φ, ψ) U ≤ if 2(2ρ + 1)C3 ≤ 0, 2 2(2ρ + 1)C3 2ρβ 2 + (1 − 2β) + (2ρ + 1)C2 + ≤ 0. 2 2ρα2 + (1 − 2α) + (2ρ + 1)C2 +

292


So, 2(2ρ + 1)C3 ≤ 0, 2 2(2ρ + 1)C3 ≤ 0. (2ρ + 1)β 2 + 2 − (1 + β)2 + C2 (2ρ + 1) + 2 Consequently, −2 + (1 + α)2 , (2ρ + 1) ≤ 2 α + C2 + 2C3 /2 (2ρ + 1)α2 + 2 − (1 + α)2 + C2 (2ρ + 1) +

(2ρ + 1) ≤

−2 + (1 + β)2 β 2 + C2 + 2C3 /2

and then 0 ≤ 2ρ ≤

−1 + 2α − C2 − 2C3 /2 −2 + (1 + α)2 − 1 = , α2 + C2 + 2C3 /2 α2 + C2 + 2C3 /2

−2 + (1 + β)2 −1 + 2β − C2 − 2C3 /2 0 ≤ 2ρ ≤ 2 −1= . 2 β + C2 + 2C3 / β 2 + C2 + 2C3 /2

(8.164)

Consequently, if min(α, β) is sufficiently large such that min(α, β) ≥ then for ρ0 =

1 min 2

C3 1 + C2 + 2, 2

−1 + 2α − C2 − 2C3 /2 −1 + 2β − C2 − 2C3 /2 , α2 + C2 + 2C3 /2 β 2 + C2 + 2C3 /2

we can deduce that Gρ maps Y into Y , for all ρ ≤ ρ0 . This completes the proof.

(8.165)

Remark 8.48. Since the result (8.165) is valid for min(α, β) sufficiently large. In order to avoid and then to compensate for this large value of min(α, β) (which makes expensive the optimization problem in practice), we can take the given data, the observations and sufficiently small (which is not very restrictive because our problem corresponds to a small perturbation). For this, we suppose that 2C3 ≤ 2+r and r ∈ (0, 1). Then (8.164) is valid if −1 + 2α − C2 − r , α2 + C2 + r −1 + 2β − C2 − r . 0 ≤ 2ρ ≤ β 2 + C2 + r 0 ≤ 2ρ ≤

Consequently, if


293

1 + C2 + r , min(α, β) ≥ 2 −1 + 2α − C2 − r −1 + 2β − C2 − r 1 ρ0 = min , 2 α2 + C2 + r β 2 + C2 + r then Gρ maps Y into Y , for all ρ ≤ ρ0 .

♦

Proposition 8.49. Assume that α and β are sufficiently large.Then there exists ρ2 > 0 such that Gρ is a K-contraction mapping from Y into Y , i.e., there exists 0 < K < 1 such that for all (φi , ψi ) ∈ Y , i = 1, 2, Gρ (φ1 , ψ1 ) − Gρ (φ2 , ψ2 ) U ≤ K (φ1 − φ2 , ψ2 − ψ2 ) U , for all 0 < ρ < ρ2 . Proof. Let (φi , ψi ) ∈ Y , ui = U(φi , ψi ) the solution of the primal problem ˜ i , ψi ) the adjoint solution corresponding to the primal solution and u ˜i = U(φ ui , for i = 1, 2. ˜1 − u˜2 . We can We set, (φ, ψ) = (φ1 , ψ1 ) − (φ2 , ψ2 ), u = u1 − u2 and u˜ = u prove easily that Gρ (φ1 , ψ1 ) − Gρ (φ2 , ψ2 ) = Gρ (φ, ψ) and that ˜)U1 ) φ − ρ(αφ + B1∗ u˜) 2U1 = ρ2 (α2 φ 2U1 + B1∗ u˜ 2U1 +2α(φ, B1∗ u −2ρ(α φ 2U1 +(φ, B1∗ u˜)U1 )+ φ 2U1 ψ − ρ(βψ − B2∗ u˜) 2U2 = ρ2 (β 2 ψ 2U2 + B2∗ u ˜ 2U2 −2β(ψ, B2∗ u ˜)U2 ) −2ρ(β ψ 2U2 −(ψ, B2∗ u˜)U2 )+ ψ 2U2 . Then ˜) 2U1 + ψ − ρ(βψ − B2∗ u˜) 2U2 Gρ (φ, ψ) 2U = φ − ρ(αφ + B1∗ u ˜ 2U1 ≤ (1 + 2ρ2 α2 + (1 − 2α)ρ) φ 2U1 +ρ(2ρ + 1) B1∗ u +(1 + 2ρ2 β 2 + (1 − 2β)ρ) ψ 2U2 +ρ(2ρ + 1) B2∗ u ˜ 2U2 . According to the continuity of the operators Bi∗ , i = 1, 2, we can deduce that (because of (8.124)) ˜ 2Ui ≤ C1 ( u(T ) 2L2 + Cu 2L2 (Q) ). Bi∗ u˜ 2Ui ≤ C0 u According to (8.106) and (8.107) and Assumption (7.91), we can deduce that C1 ( u(T ) 2L2 + Cu 2L2 (Q) ) ≤ C2 ( φ 2U1 + ψ 2U2 ), where the generic constant C2 depends on the U . Consequently, Gρ (φ, ψ) 2U ≤ (1 + 2ρ2 α2 + (1 − 2α)ρ + C2 ρ(2ρ + 1)) φ 2U1 +(1 + 2ρ2 β 2 + (1 − 2β)ρ + C2 ρ(2ρ + 1)) ψ 2U2

294


and then we have Gρ (φ, ψ) U ≤ K if ˜ ≤ 0, ρ(2ρα2 + (1 − 2α) + C2 (2ρ + 1)) + K ˜ ≤ 0, ρ(2ρβ 2 + (1 − 2β) + C2 (2ρ + 1)) + K ˜ = 1 − K 2 > 0. where K So, ˜ ≤ 0, 2ρ2 (α2 + C2 ) + (1 − 2α + C2 )ρ + K ˜ ≤ 0. 2ρ2 (β 2 + C2 ) + (1 − 2β + C2 )ρ + K Consequently, if min(α, β) is sufficiently large such that 1 + C2 , min(α, β) > 2 2 2 ˜ = δ 2 min (1 − 2α + C2 ) , (1 − 2β + C2 ) K < 1, where 0 < δ < 1, 8(α2 + C2 ) 8(β 2 + C2 ) then for

√ −(1 − 2α + C2 ) + Dα −(1 − 2β + C2 ) + Dβ , , α2 + C2 β 2 + C2

√ −(1 − 2α + C2 ) − Dα −(1 − 2β + C2 ) − Dβ 1 , , ρ1 = max 4 α2 + C2 β 2 + C2 1 ρ2 = min 4

where

˜ 2 + C2 ), Dα = (1 − 2α + C2 )2 − 8K(α ˜ 2 + C2 ), Dβ = (1 − 2β + C2 )2 − 8K(β

we have that Gρ is a contraction from Y into Y , for all ρ1 < ρ < ρ2 .

(8.166)

Since we can choose δ 2 as small as we want, then ρ1 can be taken as small as we want, this completes the proof. Remark 8.50. (i) Since the function Gρ is a K-contraction from Y into Y , for all 0 < ρ < ρ2 , then there exists a unique fixed-point (φ∗ , ψ ∗ ) of Gρ on Y , i.e., (φ∗ , ψ ∗ ) = Gρ (φ∗ , ψ ∗ ) = (φ∗ − ρ(αφ∗ + B1∗ u˜∗ ), ψ ∗ − ρ(βψ ∗ − B2∗ u˜∗ )), ˜ ∗ , ψ ∗ ) is a solution of the adjoint problem (8.123), correspondwhere, u ˜∗ = U(φ ing to the primal solution u∗ = U(φ∗ , ψ ∗ ). Consequently, (u∗ , u ˜∗ , φ∗ , ψ ∗ ) is a unique solution of the robust control problem. (ii) In order to approximate the solution (φ∗ , ψ ∗ ), we can use the well known fixed-point iterative algorithm: for k = 1, . . . , (iterative index), we denote


295

by (φk , ψk ) the numerical approximation of the optimal solution at the kth iteration of the algorithm. Initialization : (φ0 , ψ0 ) ∈ Y (given) (φk+1 , ψk+1 ) := Gρ (φk , ψk ).

(8.167)

Since the function Gρ is a K-contraction, we can obtain easily the classical error estimate:4 (φk − φ∗ , ψk − ψ ∗ ) U ≤

Kk (φ0 − φ∗ , ψ0 − ψ ∗ ) U 1−K

and then the convergence result (because K < 1). (iii) By taking into account the expression of the mapping Gρ , the fixed-point method is performed as follows: 1. Initialization: k = 0 and (φ0 , ψ0 ) ∈ Y (given). 2. Resolution of the direct problem with the source term (φk , ψk ), gives uk = U(φk , ψk ). 3. Resolution of the adjoint problem (based on (φk , ψk , uk )), gives ˜ k , ψk ). u ˜k = U(φ 4. Gradient of J at point (φk , ψk ): ⎧ def ∂J ∗ ⎪ ⎪ ˜k , ⎨ ck = ∂φ (φk , ψk ) = αφk + B1 u (GJ) ⎪ def ∂J ⎪ ⎩ dk = (φk , ψk ) = −βψk + B2∗ u˜k . ∂ψ 5. Determine (φk+1 , ψk+1 ):

φk+1 := φk − ρck , ψk+1 := ψk + ρdk .

6. Set k := k + 1 and goto 2. until convergence. The approximation of the optimal solution (φ∗ , ψ ∗ , u∗ ) is (φk , ψk , uk ). (iv) We remark that the fixed-point iterative method is a particular case of the general iterative methods. Consequently, a more precise and more detailed analysis of some iterative methods to solve robust control problem will be considered in Chapter 9. ♦ 4

The proof is left to the reader as an exercise.

296


8.7 Non-linear Time-varying Delay Systems In this section, we study a robust control problem for a class of systems governed by non-linear parabolic equations with multiple time-varying delays appearing in the state equation and in the boundary conditions. The introduction of retarded arguments is to reflect an after-effect. The delays occur naturally in biological and biochemical systems (e.g., the chemostat in which the delay is due to the cell cycle and to the fact that the organism stores the nutrient), in population modeling (e.g., the gestation period), in the area of plasma control (e.g., the confinement plasma in a bounded domain, where an electric potential barrier or magnetic mirror surrounds this domain, and where the particle reflection at this domain boundary does not act instantaneously), or in technical devices (e.g., a control circuit in which the delay is a measurable physical quantity). First, we introduce the studied model and analyze some well-posedness properties. Second, we introduce the perturbation problem and we formulate a distributed robust control problem when delays appear only in the state equation. We prove the existence and the uniqueness of the solution and we give necessary optimality conditions with the quadratic performance functionals. Finally, we consider a boundary control problem where delays appear in the state equation and in the boundary conditions. We reformulate a robust control problem, and derive the existence and the conditions of the uniqueness of the optimal solution, obtaining first-order necessary conditions of optimality. 8.7.1 Mathematical Setting Motivation The delay systems or systems with after-effects (delays in the state and/or in the input) represent a class of infinite-dimensional dynamical systems where the solution depends on an evolutive initial function (which includes informations on the past history) rather than an initial value. These equations constitute a universal mathematical model for many diffusion processes in which time-delayed feedback signals are introduced and are used to describe propagation phenomena, biological and chemical systems, population dynamics or mechanical engineering. The delay effects on the stability and control of dynamical systems are problems of recurring interest since the delay presence may induce complex behaviors, e.g., instability and poor performance, for the dynamical systems.5 Therefore, these behaviors and aspects motivate the study of time delay effects on properties of dynamical systems. 5

It turns out that time delay has been one of the major sources of instability in control.

8.7 Non-linear Time-varying Delay Systems

297

Our concern here is to present methods and techniques for the analysis and control of non-linear dynamical systems in the presence of multiple timevarying delays in the state systems and/or boundary conditions of systems. Various problems associated with theory of delay equations have been studied over the last few years see, e.g., the following references and the references therein: for the general theory see, e.g., Kolmanovskii and Myshkis [177], Hale and Verduyn-Lunel [150]; for problems of population dynamics and neutral equations see, e.g., Cushing [91, 92], Diekmann et al. [101], Kuang [183] and Banks [20]; for questions of stability, oscillations and control see, e.g., Gopalsamy [135], Gu et al. [143], Kolmanovskii and Shaikhet [178], MacDonald [212] and Niculescu [230]; for water quality management and pollution problems see, e.g., Lee and Leitmann [191], etc. Here we consider robust control problems for a class of non-linear parabolic systems with disturbances and controls in which multiple time-varying delays appear in the state systems and/or boundary conditions. For the optimal control problem of this type of system, for the linear case see Ichikawa [164], Kowalewski [179, 180] and Wang [292], and for the robust control problem in the non-linear case, see Belmiloudi [38, 39, 42, 43]. Studied Models with Delays In this section we consider non-linear parabolic partial differential delay equations of the form ∂U + AU + F (x, t, U ) + K(, U ) ∂t di (x, t)U (x, t − ei (t)) = f (x, t) (x, t) ∈ Q, +

i=1,n

U (x, t ) = R0 (x, t ) U (x, 0) = S0 (x)

(8.168)

(x, t ) ∈ Q0 = Ω × [−δ(0), 0),

x ∈ Ω,

where Q is the cylinder Ω × (0, T ) with Ω a boundary subset of IRm , m ≥ 1, Γ its sufficiently regular boundary, and Ω is totally on one side of Γ . The operator A: D −→ D is elliptic and selfadjoint with Av, v ≥ ν v 2D ,

Au, v ≤ M u D v D ∀u, v ∈ D ( ., . is the duality pairing on D and D, . D is the norm on D, . ∗ is the dual norm on D and ν, M > 0 are constants). The operator K(, .): D −→ L2 (Ω) is linear and satisfies (7.81), is a given sufficiently regular function. The operator F: Q × IR −→ IR is a Nemytsky operator on L2 (Q) satisfying Assumptions (7.78). The Banach space D of functions on Ω (satisfying the boundary conditions) such that D ⊂ L2 (Ω) ⊂ D satisfies Assumption (7.79). The functions (di , i = 1, n) : Q −→ IR are given C ∞ on Q, the functions (ei , i = 1, n) are sufficiently regular and represent multiple time-varying delays, and δ(0) = maxi=1,n (ei (0)). We introduce the following spaces: H = L∞ (0, T, L2 (Ω)), V = L2 (0, T, D), W = H ∩ V and H1 = H 1 (0, T ; D ) = {v : ∂v/∂t ∈ L2 (0, T ; D )}.

298


We state the following assumptions for the delay functions ei and for the functions ri defined by ri : t ∈ [0, T ) −→ ri (t) = t − ei (t), for i = 1, n: (ri , i = 1, n) are strictly increasing functions and (ei , i = 1, n) are C 1 non-negative functions on [0, T ).

(8.169)

Remark 8.51. According to Assumptions (8.169), we have the existence of the inverse functions (fi , i = 1, n) of the functions (ri , i = 1, n). ♦ Thus we define the following subdivision: s−1 = −δ(0), s0 = 0, sj = min (fi (sj−1 )) i=1,n

for all j ∈ IN − {0}.

(8.170)

Finally, we introduce the following notations: Ij =]s−1 , sj [ and Qj = Ω×Ij for j ∈ IN, and Tj = sj − sj−1 , for j ∈ IN − {0}. Remark 8.52. According to Assumptions (8.169), we can prove easily the following results: (i) the sequence (sj )j∈IN is strictly increasing and sj ≤ T, for all j ∈ IN (ii) for any integer j ≥ 2, if t ∈ (sj−1 , sj ) then for all i = 1, . . . , n, ri (t) ≤ sj−1 (iii) if t ∈ (s0 , s1 ) then for all i = 1, . . . , n, ri (t) ∈ (s−1 , s0 ).

♦

Since the functions (di , i = 1, n) are in C(Q) and (fi , i = 1, n) are in C([0, T ]), then we can introduce the following constants (Di , i = 1, n), C∞ and E∞ such that Di = di ∞ ( fi ∞ )1/2 for i = 1, . . . , n, n Di , C∞ =

(8.171)

i=1

E∞ = (1 + C∞ )2 . 8.7.2 Existence and Uniqueness of the Solution Theorem 8.53. Let F be an operator that satisfies the assumptions (7.78). Let S0 , R0 and f be given such that S0 ∈ L2 (Ω), R0 ∈ L2 (Q0 ) and f ∈ L2 (Q). Then the problem (8.168) admits a unique solution U such that U ∈ W ∩H1 ⊂ C([0, T ]; L2(Ω)) and there exists a constant C > 0 such that U 2W∩H1 ≤ C exp((γ + E∞ )T )( S0 2L2 + f 2L2 (Q) + R0 2L2 (Q0 ) ), where E∞ is given by (8.171).


299

Proof. The proof of this theorem can be found in Belmiloudi [38]. Let us recall the proof for the reader’s convenience. To prove the existence of a unique solution on Q, we first establish the existence of a unique solution on Qj , j ≥ 1 and obtain some estimates. We solve the problem on Q1 and obtain the existence of a unique solution on Q1 . Then, the existence of a unique solution on Q2 is proved by using the solution on Q1 to generate the initial data at s1 . This advancing process is repeated for Q3 , Q4 , . . . , Qj , Qj+1 , . . . until the final set is reached (in this way the solution in the step j determines the solution in the step j + 1). Hereafter, the solution on Qj will be denoted by Uj for j = 1, . . .. We take U0 (., s0 ) = S0 and U0 (., ri (t)) = R0 (., ri (t)), for all i = 1, n and t ∈ (s0 , s1 ).

(8.172)

Now we introduce the following problems (Pj ) for j ∈ IN − {0} ∂Vj + AVj + F (., Vj ) + K(, Vj ) = gj on Ω × (sj−1 , sj ), ∂t Vj (x, sj−1 ) = Uj−1 (x, sj−1 ) ∈ L2 (Ω), where gj (x, t) = f (x, t) − i=1,n di (x, t)Uj−1 (x, ri (t)) and Uj−1 ∈ L2 (Qj−1 ). Since f ∈ L2 (Q), Uj−1 ∈ L2 (Qj−1 ) and (di , i = 1, n) ∈ C ∞ (Q), and according to the remark (8.52) we have that gj ∈ L2 (sj−1 , sj , L2 (Ω)). According to Propositions 7.22 and 7.23 and Lemma 6.6, the problem (Pj ) (j) admits a unique solution Vj ∈ W (j) ∩ H1 ⊂ C([sj−1 , sj ]; L2 (Ω)), where H(j) = L∞ (sj−1 , sj , L2 ), V (j) = L2 (sj−1 , sj ; D), W (j) = H(j) ∩ V (j) , H1 = H 1 (sj−1 , sj ; D ) = {v : (j)

∂v ∈ L2 (sj−1 , sj ; D )}. ∂t

We can then extend the result to the cylinder set Qj+1 by taking Uj = Uj−1 on Qj−1 and Uj = Vj on Ω × (sj−1 , sj ). We shall now prove some estimates of Vj in Wj . Multiplying the first equation of (Pj ) by 2Vj and integrating over Ω × (sj−1 , t), for t ∈ (sj−1 , sj ), this gives, according to the relations (7.81) and (7.82), t t Vj 2D ds ≤ γ Vj 2L2 ds Vj (., t) 2L2 +ν sj−1 sj−1 t (8.173)

gj , Vj ds. + Uj−1 (., sj−1 ) 2L2 +2 sj−1

According to the regularity of di , i = 1, n and Remark 8.52 we obtain t

di (., s)Uj−1 (., ri (s)), Vj ds sj−1 t t 2 1/2 Uj−1 (., ri ) L2 ds) ( Vj 2L2 ds)1/2 . ≤ di ∞ ( sj−1

sj−1

300


Put a = ri (s), we have s = fi (a) and then ds = fi (a)da. So, according to Remarks 8.52 and the relations (8.171), t

di (., s)Uj−1 (., ri (s)), Vj ds sj−1 t (8.174) Vj 2L2 ds)1/2 . ≤ Di Uj−1 L2 (Qj−1 ) ( sj−1

Since gj (x, t) = f (x, t) −

di (x, t)Uj−1 (x, ri (t)),

i=1,n

we obtain then, according to (8.174) and (8.171) t t

gj , Vj ds ≤ f L2 (Qj ) ( Vj 2L2 ds)1/2 sj−1 sj−1 t +C∞ Uj−1 L2 (Qj−1 ) ( Vj 2L2 ds)1/2 . sj−1

Consequently, the relation (8.173) becomes t Vj 2D ds Vj 2L2 +ν sj−1 t Vj 2L2 ds+ Uj−1 (., sj−1 ) 2L2 ≤γ sj−1 t +2 f L2 (Qj ) ( Vj 2L2 ds)1/2 sj−1 t +2C∞ Uj−1 L2 (Qj−1 ) ( Vj 2L2 ds)1/2 .

(8.175)

sj−1

Applying Gronwall’s lemma now gives (j) Vj 2Wj ≤ D∞ ( Uj−1 (., sj−1 ) 2L2 + f 2L2 (Qj ) + Uj−1 2L2 (Qj−1 ) ), (j)

2 where D∞ = max(1, C∞ ) exp((γ + 2)Tj ).

We shall now prove the existence and uniqueness result of the problem (8.168). We observe that for j = 1, we have di (x, t)U0 (x, ri (t)) g1 (x, t) = f (x, t) − i=1,n

and according to (8.172) we can deduce that di (x, t)R0 (x, ri (t)). g1 (x, t) = f (x, t) − i=1,n


301

Consequently, g1 ∈ L2 (Q0 ). By using the previous result, the second and the third relations of (8.168) we can deduce that the problem (P1 ) admits a unique solution V1 such that V1 ∈ W1 and V1 ∈ C([s0 , s1 ], L2 (Ω)). We obtain then the solution U1 . We inject now U1 in the problem (P2 ) and by using the same approach, we obtain the existence and the uniqueness of V2 ∈ W2 and V2 ∈ C([s1 , s2 ], L2 (Ω)) (solution of (P2 )). We can now iterate the process for any domain Qj , for j ≥ 1 (until the procedure covers the whole cylinder Q) and we obtain the existence and the uniqueness of Vj ∈ Wj and Vj ∈ C([sj−1 , sj ], L2 (Ω)), solution of (Pj ). We deduce then the existence and the uniqueness of the solution U ∈ W of (8.168) such that U |Qj = Uj , j ≥ 1. We are now going to prove the estimate given in the proposition. Multiplying the first part of (8.168) by 2U and integrating over Ω × (0, t), for t ∈ (0, T ), this gives t U 2D ds U 2L2 +ν 0 (8.176) t t 2 2 U L2 ds + 2

g, U ds+ S0 L2 , ≤γ 0

where g(x, t) = f (x, t) −

0

di (x, t)U (x, ri (t)).

(8.177)

i=1,n

According to the expression (8.177) of the function g, the second relation of (8.168) and the estimate (8.180) given in Lemma 8.54 (see below), we can deduce that t t t 2 2

g, U ds ≤ C∞ U L2 ds+ f L (Q) ( U 2L2 ds)1/2 0 0 0 t 2 +C∞ R0 L2 (Q0 ) ( U L2 ds)1/2 . 0

Consequently, the relation (8.176) becomes t t 2 2 2 U D ds ≤ S0 L2 +(γ + 2C∞ ) U 2L2 ds U L2 +ν 0 0 t +2 f L2 (Q) ( U 2L2 ds)1/2 0 t +2C∞ R0 L2 (Q0 ) ( U 2L2 ds)1/2 . 0

By using Gronwall’s lemma we can deduce U 2W ≤ exp((γ + E∞ )T )( S0 2L2 + f 2L2 (Q) + R0 2L2 (Q0 ) ), (8.178) where E∞ is given by (8.171).

302


By using the system (8.168) and the relation (8.178), we can obtain easily that the solution U satisfies the following estimate: U 2H1 ≤ C exp((γ +E∞ )T )( S0 2L2 + f 2L2 (Q) + R0 2L2 (Q0 ) ). (8.179)

This completes the proof. We shall now prove the lemma previously used.

Lemma 8.54. Let v ∈ L2 (Q0 ∪ Q) such that v = k0 on Q0 , and w ∈ L2 (Q) we have for all t ∈ (0, T ) t

di (., s)v(., ri (s)), wds 0 (8.180) t t v 2L2 ds)1/2 ( w 2L2 ds)1/2 , ≤ Di k0 L2 (Q0 ) +( 0

0

for all i = 1, . . . , n. Proof. According to the regularity of di , i = 1, n and to Remark 8.52 we obtain t

di (., s)v(., ri (s)), wds 0 t t ≤ di ∞ ( v(., ri ) 2L2 ds)1/2 ( w 2L2 ds)1/2 . 0

0

Set a = ri (s), according to Remark 8.51 we can deduce that s = fi (a) and then ds = fi (a)da. So, according to (8.171) t−ei (t) t t 2 1/2

di (., s)v(., ri (s)), wds ≤ Di ( v L2 da) ( w 2L2 ds)1/2 . 0

−ei (0)

0

Since −δ(0) ≤ −ei (0), for all i = 1, . . . , n and according to the assumptions (8.169) we can deduce t

di (., s)v(., ri (s)), wds 0 t t ≤ Di k0 L2 (Q0 ) +( v 2L2 da)1/2 ( w 2L2 ds)1/2 . 0

0


The following proposition shows the Lipschiz continuous result. Proposition 8.55. Let F be an operator that satisfies the assumptions (7.78). Let S01 , S02 be two functions in L2 (Ω), R01 , R02 be two functions in L2 (Q0 ) and let f1 , f2 be two functions of L2 (Q). If U1 (respectively U2 ) is the solution of (8.168) with data (f1 , R01 , S01 ) (respectively with data (f2 , R02 , S02 )) then U 2W∩H1 ≤ C exp((γ + E∞ )T )( f 2L2 (Q) + S0 2L2 + R0 2L2 (Q0 ) ), where U = U1 − U2 , S0 = S01 − S02 , f = f1 − f2 and R0 = R01 − R02 .


303

Proof. Using the same way to obtain the estimates (8.178) and (8.179) and the second assumption of (7.78), we obtain the result of the proposition. So we omit the details. In the following, the solution U will be treated as the target function. We are then interested in the robust regulation of the deviation of the problem from the desired target U . We analyze the full non-linear equation which models large perturbations u to the target U . We assume that U satisfies the problem (8.168) with the data (f, R0 , S0 ) and U satisfies the problem (8.168) with the data (f + g, R0 + r0 , S0 + u0 ). Hence, we consider the system with multiple time-varying delays ∂u + Au + F (., u + U ) − F (., U ) + K(, u) ∂t di u(., ri ) = g on Q, + i=1,n

(8.181)

u(x, t ) = r0 (x, t ), (x, t ) ∈ Q0 , u(x, 0) = u0 (x), x ∈ Ω. If we set F˜ (., y) = F (., y + U ) − F (., U ) then (8.181) reduce to ∂u + Au + F˜ (., u) + K(, u) + di u(., ri ) = g on Q, ∂t i=1,n u(x, t ) = r0 (x, t ), (x, t ) ∈ Q0 ,

(8.182)

u(x, 0) = u0 (x), x ∈ Ω. In the following we assume that the solution U satisfies the regularity U ∈ W ∩ C([0, T ]; L2(Ω)). Similarly as in previous section, we verify easily that the operator F˜ satisfies the same hypotheses that the operator F , i.e., (7.78). For simplicity of future reference, we omit the “˜” on F˜ for (8.182). Problem (8.182) is similar to (8.168), and consequently the following proposition holds. Proposition 8.56. Let F be an operator that satisfies the assumptions (7.78). (i) Assume that g ∈ L2 (Q), r0 ∈ L2 (Q0 ) and u0 ∈ L2 (Ω). Then the problem (8.182) admits a unique solution u such that u ∈ W ∩ H1 with the following estimate: u 2W∩H1 ≤ C exp((γ + E∞ )T )( g 2L2 (Q) + u0 2L2 + r0 2L2 (Q0 ) ). (ii) Let u01 , u02 be two functions in L2 (Ω), r01 , r02 be two functions in L2 (Q0 ) and let g1 , g2 be two functions of L2 (Q). If u1 (respectively u2 ) is the solution of (8.182) with data (g1 , r01 , u01 ) (respectively with data (g2 , r02 , u02 )) then u 2W∩H1 ≤ C exp((γ + E∞ )T )( g 2L2 (Q) + u0 2L2 + r0 2L2 (Q0 ) ), where u = u1 − u2 , u0 = u01 − u02 , g = g1 − g2 and r0 = r01 − r02 .

304


8.7.3 The Control Framework In order to illustrate our robust control problem, we assume here that the disturbance ψ ∈ L2 (Ω) is in the initial condition u0 and the control φ ∈ L2 (Q) is in the forcing g, i.e., u0 = B2 ψ and g = B1 φ, where Bi , i = 1, 2 are given bounded operators on L2 (Q) and L2 (Ω), respectively, satisfying the assumption (8.107) and the estimate (8.108). Remark 8.57. For other types of controls and disturbances, the reader should refer to Belmiloudi [38]. ♦ The function u is then assumed to be related to the disturbance ψ and control φ through the problem (8.182) ∂u + Au + F (., u) + K(, u) ∂t di u(., ri ) = B1 φ on Q, + i=1,n

u(x, t ) = r0 (x, t ),

(8.183)

(x, t ) ∈ Q0 ,

u(., 0) = B2 ψ on Ω. To obtain the regularity of Proposition 8.56, we suppose that the given function r0 is in L2 (Q0 ). Let U : (φ, ψ) −→ u = U(φ, ψ) be the map: L2 (Q) × L2 (Ω) −→ W defined by (8.183) and introducing the cost functional J by J(φ, ψ) =

μ 1 C(u − uobs ) 2L2 (Q) + u(T ) − vobs 2L2 (Ω) 2 2 α β + φ 2L2 (Q) − ψ 2L2 (Ω) , 2 2

(8.184)

where μ, α, β > 0 are fixed, (uobs , vobs ) ∈ V × L2(Ω) is the observation (given) and C is an unbounded, linear operator on L2 (Ω) satisfying Assumption (7.91). We want to minimize the functional J with respect to φ and maximize J with respect to ψ, i.e., to find (φ∗ , ψ ∗ ) ∈ Uad × Vad such that J(φ∗ , ψ) ≤ J(φ∗ , ψ ∗ ) ≤ J(φ, ψ ∗ ), ∀(φ, ψ) ∈ Uad × Vad ,

(8.185)

where Uad and Vad are (given) non-empty, closed, convex, bounded subsets of L2 (Q) and L2 (Ω), respectively. Proposition 8.58. Let F be an operator that satisfies the assumptions (7.78). Then the function U : (φ, ψ) −→ u = U(φ, ψ) solution of (8.183) is continuously F-differentiable from L2 (Q) × L2 (Ω) to W with the derivative U (φ, ψ) : (h1 , h2 ) −→ w given by the linear parabolic problem with multiple time-varying delays:


∂w + Aw + G(., U(φ, ψ))w + K(, w) ∂t di (x, t)w(x, ri (t)) = B1 h1 on Q, + i=1,n

305

(8.186)

w(x, t ) = 0, (x, t ) ∈ Q0 , w(., 0) = B2 h2 on Ω, where the operator G is the differential of the operator F . Moreover, for all Φi = (φi , ψi ) ∈ L2 (Q) × L2 (Ω), for i = 1, 2, the following estimates hold: (i) U (φ1 , ψ1 ) L(L2 (Q)×L2 (Ω);W) ≤ C M∞ b (ii) U (φ1 , ψ1 ) − U (φ2 , ψ2 ) L(L2 (Q)×L2 (Ω);W) 2 ≤ CM∞ λc2e b2 Φ1 − Φ2 L2 (Q)×L2 (Ω) ,

where E∞ is given by (8.171), M∞ = exp((γ + E∞ )T ) and the constants λ, ce and b are given by the relations (7.78),(7.79) and (8.108). Proof. The proof is obtained by using similar technique as to obtain the results of Proposition 8.22 and by taking into account the estimate (8.180) given in Lemma 8.54. So we omit the details. Proposition 8.59. Let F be an operator that satisfies the assumptions (7.78). Then for each t ∈ [0, T ], the function Vt : (φ, ψ) −→ u(t) = Vt (φ, ψ) solution of (8.183) is uniformly Lipschitz continuous and continuously F-differentiable from L2 (Q)×L2 (Ω) to L2 (Ω) with the derivative Vt (φ, ψ) : (h1 , h2 ) −→ w(t), where the function w is given by the parabolic system with time-varying delays ∂w + Aw + G(., Vt (φ, ψ))w + K(, w) ∂t di w(., ri ) = B1 h1 on Q, + i=1,n

w(x, t ) = 0

(8.187)

(x, t ) ∈ Q0 ,

w(x, 0) = B2 h2

x ∈ Ω.

Moreover, for all Φi = (φi , ψi ) ∈ L2 (Q) × L2 (Ω), for i = 1, 2, the following estimates hold: (i) Vt (φ1 , ψ1 ) L(L2 (Q)×L2 (Ω),;L2 (Ω)) ≤ Cb M∞ (ii) Vt (φ1 , ψ1 ) − Vt (φ2 , ψ2 ) L(L2 (Q)×L2 (Ω);L2 (Ω)) 2 ≤ CM∞ λc2e b2 Φ1 − Φ2 L2 (Q)×L2 (Ω) ,

where E∞ is given by (8.171), M∞ = exp((γ + E∞ )T ) and the constants λ, ce and b are given by the relations (7.78),(7.79) and (8.108).

306


Proof. The proof of this proposition is a direct consequence of Propositions 8.56 and 8.58.

Remark 8.60. Problem (8.187) is exactly the problem (8.186).

Proposition 8.61. Let F be an operator that satisfies the assumptions (7.78). Then the maps U and Vt defined by (8.183) are continuous from the weak topology of L2 (Q) × L2 (Ω) to the strong topology of L2 (Q) and the weak topology of L2 (Ω), respectively. Proof. The proof of this proposition is similar to Proposition 8.24, so we omit the details. We can now obtain the existence of an optimal solution of the robust control problem. Theorem 8.62. Let F be an operator that satisfies the assumptions (7.78). Then, for α and β sufficiently large, there exists (φ∗ , ψ ∗ ) ∈ L2 (Q) × L2 (Ω) and u∗ ∈ W such that (φ∗ , ψ ∗ ) is defined by (8.185) and u∗ = U(φ∗ , ψ ∗ ) is a solution of (8.183). Proof. The proof of this theorem can be obtained by using similar existence result showed in Theorem 8.25, by taking advantage of the results given in Propositions 8.56–8.61. Now we give the characterization of the solution of the robust control problem. For simplicity we suppose that ei (T ) = δ(T ) for all i = 1, n, where δ(T ) is a given non-negative constant. In order to characterize the robust control,6 we introduce the following adjoint problem corresponding to the primal problem (8.183) (we denote by u = U(φ, ψ) the solution of problem (8.183) with the forcing (φ, ψ)): −

∂u ˜ + A˜ u + (G(., u))∗ u˜ + K ∗ (, u˜) ∂t ˜ di (., fi (t))˜ u(., fi (t))fi (t) = C ∗ C(u − uobs ) on Q, + i=1,n

(8.188)

∂u ˜ + A˜ u + (G(., u))∗ u˜ + K ∗ (, u˜) = C ∗ C(u − uobs ) on QT , − ∂t u ˜(T ) = μ(u(T ) − vobs ) on Ω, where C ∗ (respectively (G(., u))∗ ) is the adjoint of the operator C (respectively ˜ = Ω × (0, T − δ(T )) and QT = Ω × (T − δ(T ), T ). G(., u)), Q Proposition 8.63. Let F be an operator that satisfies Assumptions (7.78) and u ∈ W. Then the solution of (8.188) is in W and satisfies the estimate: ˜ 2V ≤ K∞ ( u(T ) − vobs 2L2 + C(u − uobs ) 2L2 (Q) ), u ˜ 2H + u where K∞ is a constant depending on the parameters ν, μ, δ1 , δ2 , γ and C∞ . 6

More precisely, in order to simplify the gradient of the cost functional J which depends on the derivative of the operator solution U.


307

Proof. To prove the existence of a unique solution u˜ of (8.188), we change the variables of problem (8.188) by reversing the sense of time, i.e., t := T − t, and we follow the same procedure used to obtain the result of Theorem 8.53. Setting U (., t) = u(., T − t), Uobs (., t) = uobs (., T − t), W (., t) = (., T − t), ˜ (., t) = u U ˜(., T − t), Ei (t) = fi (T − t) − (T − t), Ri (t) = t − Ei (t) and Fi (t) = T − ri (T − t), problem (8.188) can be written as ˜ ∂U ˜ + K ∗ (W, U ˜ + (G(., U ))∗ U ˜) + AU ∂t ˜ (., Ri (t)) di (., fi (T − t))fi (T − t)U + i=1,n ∗

= C C(U − Uobs ) on Ω × (δ(T ), T ),

(8.189)

˜ ∂U ˜ + (G(., U ))∗ U ˜) ˜ + K ∗ (W, U + AU ∂t = C ∗ C(U − Uobs ) on Ω × (0, δ(T )), ˜ (0) = μ(U (0) − vobs ) on Ω. U According to the properties of the functions (ri , i = 1, n) and (fi , i = 1, n) we can prove easily that (Ei , i = 1, n) are C 1 non-negative functions, (Ri , i = 1, n) are strictly increasing functions and (Fi , i = 1, n) are the inverse functions of (Ri , i = 1, n). Thus we define the following subdivision: S−1 = 0, S0 = δ(T ) and ∀j ≥ 1, Sj = min (Fi (Sj−1 )), and we denote by Ij =]S−1 , Sj [ i=1,n

and by Qj = Ω × Ij for j ≥ 0. We can now follow the same approach used to obtain the existence and uniqueness of the problem (8.168). Indeed, since C ∗ C(U − Uobs ) ∈ L2 (Q) and ˜ ∈W (U (0)−vobs ) ∈ L2 (Ω) then the existence and uniqueness of the solution U of the problem (8.189) hold and, therefore, the existence and uniqueness of u ˜ ∈ W solution of (8.188) hold also. Now we prove the estimate given in the proposition. Multiplying the first equation of (8.188) by u ˜ and integrating over (t, T ) × Ω, we obtain, according to −2G(., u) ≤ γ0 and to the second and third relations of (8.188), that (using Young’s inequality and H¨ older’s inequality) T T u˜ 2D ds ≤ μ2 u(T ) − vobs 2L2 +γ u ˜ 2L2 ds u˜ 2L2 +ν t t T 1 T C(u − uobs ) 2L2 ds + 2 Cu ˜ 2L2 ds + 2 (8.190) t t n T −δ(T ) +2

di (., fi (s))˜ u(., fi (s))fi (s), u ˜(., s)ds. i=1

min(t,T −δ(T ))

We can now estimate the term T −δ(T )

di (., fi (s))˜ u(., fi (s))fi (s), u˜(., s)ds. min(t,T −δ(T ))

308


According to the regularity of di and fi we then obtain

T −δ(T )

di (., fi (s))˜ u(., fi (s))fi (s), u˜(., s)ds

min(t,T −δ(T ))

≤ Di

T −δ(T )

min(t,T −δ(T ))

u˜(., fi (s)) L2

fi (s)ds

1/2 (

T

u˜ 2L2 ds)1/2 . t

Set a = fi (s), we have s = ri (a), da = fi (s)ds and then, since ri (T ) = T −δ(T ) and ri (t) ≤ t,

T −δ(T )

di (., fi (s))˜ u(., fi (s))fi (s), u˜(., s)ds

min(t,T −δ(T ))

≤ Di

(8.191)

T

u ˜

2L2

da.

t

According to (7.91) and (8.191) the relation (8.190) becomes

T

u ˜ 2L2 +ν

u ˜ 2D ds ≤ μ2 u(T ) − vobs 2L2 T +(γ + 2 δ1 + 2C∞ ) u ˜ 2L2 ds t T 1 T C(u − uobs ) 2L2 ds + 2 δ2 u ˜ 2D ds. + 2 t t t

(8.192)

By choosing 2 = ν/(2δ2 ), u ˜ 2L2 +

ν 2

T

u˜ 2D ds ≤ μ2 u(T ) − vobs 2L2 t T νδ1 2δ2 C(u − uobs ) 2L2 (Q) . + + γ + 2C∞ u ˜ 2L2 ds + 2δ2 ν t

By using Gronwall’s formula we then have ˜ 2V ≤ K∞ ( u(T ) − vobs 2L2 + C(u − uobs ) 2L2 (Q) ), u ˜ 2H + u where K∞ is a constant depending on the parameters ν, μ, δ1 , δ2 , γ and C∞ . This completes the proof. ˜ : (φ, ψ) −→ u ˜ We will denote in the sequel by U ˜ = U(φ, ψ) the map defined by (8.188). We can now give the optimality system for the robust control problem (8.185). Theorem 8.64. Let F be an operator that satisfies the assumptions (7.78), α and β sufficiently large, and an optimal solution (φ∗ , ψ ∗ , u∗ ) ∈ Uad × Vad × W


309

such that (φ∗ , ψ ∗ ) is defined by (8.185) and u∗ = U(φ∗ , ψ ∗ ) is a solution of (8.183). Then T (αφ∗ + B1∗ u ˜)(φ − φ∗ )dxdt ≥ 0, Ω 0 (8.193) ∗ ∗ ∗ (−βψ + B2 u ˜(0))(ψ − ψ )dx ≤ 0, Ω

˜ ∗ , ψ ∗ ) is the solution of the adjoint ˜ = U(φ for all (φ, ψ) ∈ Uad × Vad , where u problem (8.188). Proof. The cost functional J is a composition of differentiable maps then J is differentiable and we have, for all h = (h1 , h2 ) ∈ Uad × Vad J (φ, ψ)h =

T

C(u − uobs ), CU (φ, ψ)hdt + μ(u(T ) − vobs ), VT (φ, ψ)h 0 T + αφ, h1 dt − βψ, h2 , 0

where U (φ, ψ)h = w and VT (φ, ψ)h = w(T ), where w is the solution of (8.187). Then

J (φ, ψ)h =

T

C ∗ C(u − uobs ), U (φ, ψ)hdt + μ(u(T ) − vobs ), VT (φ, ψ)h T

αφ, h1 dt − βψ, h2 . + 0

0

˜, In order to simplify the expression of J (φ, ψ), multiplying (8.186) by u integrating over Q and (integrating) by parts in time t, we obtain, for all h = (h1 , h2 ) ∈ Uad × Vad

T 0

B1∗ u˜, h1 dt + B2∗ u˜(0), h2 ˜(T ) = VT (φ∗ , ψ ∗ )h, u T (8.194) ∂u ˜ + A˜ u + (G(., u∗ ))∗ u˜ + K ∗ (, u˜), U (φ∗ , ψ ∗ )hdt

− + ∂t 0 T

di (., t)˜ u(., t), U (φ∗ , ψ ∗ )h(., ri (t))dt. + i=1,n

0

We can now calculate the term T

di (., t)˜ u(., t), U (φ∗ , ψ ∗ )h(., ri (t))dt. 0

Let s = ri (t), then t = fi (s) and dt = fi (s)ds. So

310


0

T

di (., t)˜ u(., t), U (φ∗ , ψ ∗ )h(., ri (t))dt T −ei (T )

di (., fi (s))˜ u(., fi (s))fi (s), U (φ∗ , ψ ∗ )hds. = −ei (0)

Since ei (T ) = δ(T ), ∀i = 1, n and according to the second term of (8.186) we have T

di (., t)˜ u(., t), U (φ∗ , ψ ∗ )h(., ri (t))dt 0 (8.195) T −δ(T )

di (., fi (s))˜ u(., fi (s))fi (s), U (φ∗ , ψ ∗ )hds. = 0

Since u˜ is a solution of the adjoint problem (8.188) and according to (8.195), the equality (8.194) becomes T

B1∗ u ˜, h1 dt + B2∗ u ˜(0), h2 = VT (φ∗ , ψ ∗ )h, μ(u∗ (T ) − vobs ) 0 (8.196) T ∗ ∗ ∗ ∗

C C(u − uobs ), U (φ , ψ )hdt. + 0

Applying (8.196) and according to the expression of J one has, for all h = (h1 , h2 ) ∈ Uad × Vad , T (αφ∗ + B1∗ u˜)h1 dxdt + (−βψ ∗ + B2∗ u ˜(0))h2 dx. (8.197) J (φ∗ , ψ ∗ ).h = 0

∗

Ω

Ω

∗

As (φ , ψ ) is a solution of the saddle point problem (8.185) then J (φ∗ , ψ ∗ )(φ − φ∗ , 0) ≥ 0, J (φ∗ , ψ ∗ )(0, ψ − ψ ∗ ) ≤ 0, for all (φ, ψ) ∈ Uad × Vad and therefore, according to the expression (8.197), we can deduce that (αφ∗ + B1∗ u˜)(φ − φ∗ )dxdt ≥ 0, Q (−βψ ∗ + B2∗ u ˜(0))(ψ − ψ ∗ )dx ≤ 0, Ω

for all (φ, ψ) ∈ Uad × Vad . This completes the proof.

Now we give some conditions to obtain the uniqueness of the saddle point (φ∗ , ψ ∗ ). More precisely, we show that, for small data or large coefficient ν, α and β, one has uniqueness. Theorem 8.65. Assume that F satisfies the assumptions (7.78), ν > δ2 , μ < 1 and b2 /β ≤ 1 hold. Subject to the following conditions:


311

(i) θ = (ν − δ2 − c2I (γ + δ1 + 2C∞ )) − 2b2 c2I /α > 0

1/2 (ii) N∞ u∗ (T ) − vobs 2L2 + C(u∗ − uobs ) 2L2 (Q) < θ, where C∞ is given by (8.171), N∞ = λc2e exp((δ1 + γ + 2C∞ )T /2) and the constants λ, ce , cI , δ1 , δ2 and b are given by the relations (7.78), (7.79), (7.80), (7.82) and (8.108), the optimal solution (φ∗ , ψ ∗ , u∗ ) is unique. Proof. Suppose that there exist (φ∗ , ψ ∗ , u∗ ) and (φ∗1 , ψ1∗ , u∗1 ) two optimal solutions where (φ∗ , ψ ∗ ) and (φ∗1 , ψ1∗ ) satisfy (8.185), u∗1 = U(φ∗1 , ψ1∗ ) and u∗ = U(φ∗ , ψ ∗ ) are solutions of (8.183) such that the optimality conditions (8.193) hold, i.e., for all (φ, ψ) ∈ Uad × Vad , (αφ∗1 + B1∗ u ˜1 )(φ − φ∗1 )dxdt ≥ 0, Q (−βψ1∗ + B2∗ u ˜1 (0))(ψ − ψ1∗ )dx ≤ 0, Ω (8.198) (αφ∗ + B1∗ u ˜)(φ − φ∗ )dxdt ≥ 0, Q (−βψ ∗ + B2∗ u ˜(0))(ψ − ψ ∗ )dx ≤ 0, Ω

˜ ∗ 1 , ψ ∗ 1 ) (respectively u ˜ ∗ , ψ ∗ )) is a solution of the ˜ = U(φ where u ˜1 = U(φ adjoint problem (8.188) corresponding to the primal solution u∗1 (respectively u∗ ). Set φ = φ∗ − φ∗1 , ψ = ψ ∗ − ψ1∗ , v = u∗ − u∗1 and v˜ = u˜ − u˜1 . Then v satisfies the following system: ∂v + Av + (F (., u∗ ) − F (., u∗1 )) + K(, v) ∂t di v(., ri ) = B1 φ on Q, + v(., .) = 0 on Q0 , v(., 0) = B2 ψ on Ω,

(8.199)

i=1,n

v˜ satisfies the following system: −

∂˜ v + A˜ v + (G(., u∗ ))∗ v˜ + K ∗ (, v˜) + di (., fi )˜ v (., fi )fi ∂t i=1,n ˜ = C ∗ Cv − (G(., u∗ ) − G(., u∗1 ))∗ u ˜ on Q,

−

∂˜ v + A˜ v + (G(., u∗ ))∗ v˜ + K ∗ (, v˜) ∂t = C ∗ Cv − (G(., u∗ ) − G(., u∗1 ))∗ u ˜ on QT ,

v˜(., T ) = μv(., T ) on Ω

(8.200)

312


and (φ, ψ) satisfies the following inequalities: α φ 2L2 (Q) + B1∗ v˜φdxdt ≤ 0, Q β ψ 2L2 (Ω) − B2∗ v˜(0)ψdx ≤ 0.

(8.201)

Ω

By multiplying (8.199) by v, (8.200) by v˜ and integrating over Q, this gives, according to (8.201), the estimates (8.129) and (7.91) and the relations (7.81) and (7.82)

T ∂ 2 v L2 dt + ν v 2D dt ∂t 0 T 2 2 v 2D dt + B1∗ v˜ L2 (Q) B1∗ v L2 (Q) ≤ γcI α 0 T +2 |

di v(., ri ), vdt |,

T

0

0

T

− 0

i=1,n T ∂ 2 v˜ 2D dt v˜ L2 dt + ν (8.202) ∂t 0 T T ≤ γc2I v˜ 2D dt + (δ1 c2I + δ2 ) ( v 2D + v˜ 2D )dt 0 0 T |u ˜ | | v˜ | | v | dt +2λ 0 Ω T −δ(T )

di (., fi )˜ v (., fi )fi , v˜dt |, +2 | i=1,n

0

v˜(., T ) = μv(T ) and v(., 0) = B2 ψ. We can now estimate the terms T

di v(., ri ), vdt |, | | i=1,n

0

i=1,n

T −δ(T )

di (., fi )˜ v (., fi )fi , v˜dt | .

0

According to Lemma 8.54 and since v = 0 on Q0 , we can deduce that |

i=1,n

0

T

di v(., ri ), vdt |≤ C∞ v 2L2 (Q) ,

(8.203)

where C∞ is given in (8.171). Moreover, by using a similar technique to that given for Lemma 8.54, we can deduce that |

i=1,n

0

T −δ(T )

di (., fi )˜ v (., fi )fi , v˜dt |≤ C∞ v˜ 2L2 (Q) .

(8.204)


313

Applying H¨ older’s inequality and the relations (7.79), (8.108), (8.107), (8.203) and (8.204), the estimates (8.202) become T T ∂ v 2L2 dt + (ν − (γ + 2C∞ )c2I ) v 2D dt 0 ∂t 0 2b2 c2I T ( v 2D dt+ v˜ 2D )dt, ≤ α 0 T T (8.205) ∂ 2 2 v˜ L2 dt + (ν − (γ + 2C∞ )cI ) v˜ 2D dt − 0 ∂t 0 T ˜ H ) ( v 2D + v˜ 2D )dt, ≤ (δ1 c2I + δ2 + λc2e u 0

v˜(., T ) = μv(., T ) and v(., 0) = B2 ψ. Adding the first and the second estimate of (8.205) we obtain T T ∂ ( v 2L2 − v˜ 2L2 )dt + θ ( v 2D + v˜ 2D )dt ∂t 0 0 T 2 2 ˜ H ( v D + v˜ 2D )dt, ≤ λce u

(8.206)

0

where θ = ν − δ2 − (i).

c2I (γ

+ 2C∞ + (2b2 /α) + δ1 ) > 0, because of assumption

By using the third relation of (8.205) we can deduce that (1 − μ2 ) v(T ) 2L2 + v˜(0) 2L2 +θ( v 2V + v˜ 2V ) ≤ λc2e u˜ H ( v 2V + v˜ 2V )+ B2 ψ 2L2 . According to the relation (8.107) and the second estimate of (8.201), we can deduce that b4 B2 ψ 2L2 ≤ 2 v˜(0) 2L2 β and then b4 ) v˜(0) 2L2 +θ( v 2V + v˜ 2V ) β2 ( v 2V + v˜ 2V ).

(1 − μ2 ) v(T ) 2L2 +(1 − ˜ H ≤ λc2e u

Since 1 − μ2 > 0 and (1 − b4 /β 2 ) > 0 then ˜ H ( v 2V + v˜ 2V ). θ( v 2V + v˜ 2V ) ≤ λc2e u

(8.207)

By choosing 2 = 1 in the inequality (8.192) and by using Gronwall’s formula we can deduce, since μ < 1 and δ2 < ν, that

u˜ 2H ≤ e(δ1 + γ + 2C∞ )T u∗ (T ) − vobs 2L2 + C(u∗ − uobs ) 2L2 (Q) .

314


Therefore, according to (8.207), θ∗ ( v 2V + v˜ 2V ) ≤ 0, where

1/2 , θ∗ = θ − N∞ u∗ (T ) − vobs 2L2 + C(u∗ − uobs ) 2L2 (Q) (δ1 + γ + 2C∞ )T N∞ = λc2e exp . 2 Since θ∗ > 0 (according to assumption (ii)), we have v 2V + v˜ 2V = 0 and then v = 0 and v˜ = 0. We obtain then the uniqueness result. Remark 8.66. We can consider other time-varying delays, for example in the convolution (or integral) form or for some combination of the additional and the convolution forms. For more details in the case of the convolution form, for similar non-linear PDEs treated in this book, the reader is referred to Belmiloudi [39]. ♦ Example 8.67. We can consider the problem of biochemical pollutants presented in Section 7.6.5, by taking into account the time delays of pollution. In this case the concentration of pollutants can be governed by the following time delay reaction–diffusion–transport system ∂u − div(β∇u) + .∇u + F (x, t, u) ∂t + di (x, t)u(x, ri (t)) = f (x, t) (x, t) ∈ Q i=1,n

(8.208)

u = 0 on Σ = ∂Ω × (0, T ) u(x, t ) = r0 (x, t ) u(0) = u0 on Ω.

(x, t ) ∈ Q0 ,

In the cost functional J given by (8.184) the operator C represents the regional and temporal variation in the cost of pollutant extraction. Since all the assumptions of our abstract results are satisfied by the example in this particular case, our study applies. ♣ 8.7.4 Remarks on Time-varying Delays and Control in the Boundary Conditions We conclude with an indication as to how the previous arguments can be made to apply to robust boundary control problem with boundary multiple timevarying delays. Let us consider the following perturbed problem (by using a similar process as used to introduce the perturbation system in Section 8.7.2): ∂u − div(η∇u) + F (x, t, u) + K(, u) ∂t n + di (x, t)u(x, t − ei (t)) = g(x, t)

i=1

u(x, t ) = r0 (x, t ) u(x, 0) = u0 (x)

(x, t ) ∈ Q0 = (−δ(0), 0) × Ω,

x ∈ Ω,

(x, t) ∈ Q,

(8.209)


315

with the boundary conditions: ∂u = ai (x, t)u(x, t − ki (t)) + f (x, t) ∂n j=1 l

η

(x, t) ∈ Σ,

(8.210)

u(x, t ) = π0 (x, t ) (x, t ) ∈ Σ0 = (−δ(0), 0) × ∂Ω, where Σ = ∂Ω × (0, T ), Ω is open bounded in IRm , m ≥ 1, with the boundary ∂Ω sufficiently regular, g is in L2 (Q), u0 is in L2 (Ω) and F satisfies Assumptions (7.78) as before. Assume that f is in L2 (Σ), ∂u/∂n denotes the exterior normal derivative of u at the boundary ∂Ω, the function η is positive and bounded function above and below by two non-negative constants in Ω, and we denote by ν = min(η). The terms (di , i = 1, n) : Q −→ IR are given C ∞ funcΩ

tions on Q, (aj , j = 1, l) : Σ −→ IR are given C ∞ functions on Σ, (ei , i = 1, n) and (kj , j = 1, l) are sufficiently regular functions representing multiple timevarying delays and δ(0) = max(maxi=1,n (ei (0)), maxj=1,l (kj (0))). Remark 8.68. Such a problem arises for example in control of a collisiondominated plasma (for the linear case see, e.g., Wang [292]). The goal is to confine plasma in a bounded region Ω by introducing, for example, a finite electric potential barrier surrounding the domain Ω. In this case, the particle density is governed, in general, by parabolic equation. Because of the finiteness of electric potential the particle reflection is not instantaneous. Indeed the particle flux at the boundary of the considered domain at any time depends on particle flux which escaped earlies with various velocities and were reflected back into the domain Ω at later time. So in order to take into account this phenomenon, we introduce time delays in the boundary conditions. ♦ We state the following assumptions for the delay functions ei , ri which is defined by ri : t ∈ [0, T ) −→ ri (t) = t − ei (t), kj and qj which is defined by qj : t ∈ [0, T ) −→ qj (t) = t − kj (t), for i = 1, . . . , n and j = 1, . . . , l: (ri )i=1,n , (qj )j=1,l

are strictly increasing functions,

(ei )i=1,n , (kj )j=1,l

are C 1 non-negative functions on [0, T ).

(8.211)

Remark 8.69. According to Assumptions (8.211), we have the existence of the inverse functions (fi , i = 1, n) and (gj , j = 1, l) of the functions (ri , i = 1, n) ♦ and (qj , j = 1, l), respectively. Thus we define the following subdivision: s−1 = −δ(0), s0 = 0, sj = min( min (fi (sj−1 )), min (gk (sj−1 ))) i=1,n

k=1,n

for all j ∈ IN − {0}.

(8.212)

Finally, we introduce the following notations: Ij =]s−1 , sj [ and Qj = Ω×Ij for j ∈ IN, and Tj = sj − sj−1 , for j ∈ IN − {0}. We assume that m ≤ 4,7 D = H 1 (Ω) ⊂ L4 (Ω) and D = (H 1 (Ω)) . 7

D ⊂ L4 (Ω) provided m ≤ 4 according to the Sobolev embedding theorem.

316


By using similar technique as used to prove Theorem 8.53 and Remark 8.41, we obtain the following proposition. Proposition 8.70. Assume that F satisfies the assumptions (7.78). Let u0 , r0 , π0 , f and g be given such that u0 ∈ H 1/2 (Ω), r0 ∈ H 3/2,3/4 (Q0 ), π0 ∈ L2 (Σ0 ) ,f ∈ L2 (Σ) and g ∈ L2 (Q). Then, there exists a unique solution u ∈ H 3/2,3/4 (Q) of problem (8.209) with the boundary conditions (8.210). We suppose now that the disturbance ψ ∈ L2 (Q) is in the value g and the control φ ∈ L2 (Σ) is in the boundary condition f , i.e., f = B1 φ and g = B2 ψ, where B1 and B2 are given bounded operators on L2 (Σ) and on L2 (Q) respectively. So the function u is assumed to be related to the disturbance ψ and control φ through the problem (8.209) with the boundary conditions (8.210): ∂u − div(η∇u) + F (., u) + K(, u) ∂t di u(., ri ) = B2 ψ on Q, + i=1,n

u(x, t ) = r0 (x, t ) (x, t ) ∈ Q0 ,

(8.213)

u(., 0) = u0 on Ω, ∂u = ai u(., qi ) + B1 φ on Σ, η ∂n j=1,l

u(x, t ) = π0 (x, t )

(x, t ) ∈ Σ0 .

Let U : (φ, ψ) −→ u = U(φ, ψ) be the map: L2 (Σ)×L2 (Q) −→ W defined by (8.213) and introducing the cost functional defined by J(φ, ψ) =

μ 1 C(u − uobs ) 2L2 (Q) + u(T ) − vobs 2L2 (Ω) 2 2 α β 2 + φ L2 (Σ) − ψ 2L2 (Q) , 2 2

(8.214)

where μ, α, β > 0 are fixed, (uobs , vobs ) ∈ V × L2 (Ω) is the observation and C is an unbounded, linear operator on L2 (Ω) satisfying the hypothesis (7.91). We want to minimize the functional J with respect to φ and maximize J with respect to ψ, i.e., to find (φ∗ , ψ ∗ ) ∈ Uad × Vad such that J(φ∗ , ψ) ≤ J(φ∗ , ψ ∗ ) ≤ J(φ, ψ ∗ ), ∀(φ, ψ) ∈ Uad × Vad

(8.215)

with Uad and Vad are (given) non-empty, closed, convex, bounded subsets of L2 (Σ) and L2 (Q), respectively. By using the same technique as used in the proof of Theorems 8.62 and 8.64, we have the following results. Theorem 8.71. Let F be an operator that satisfies Assumptions (7.78). Then, for the coefficients α and β sufficiently large, there exists an optimal


317

solution (φ∗ , ψ ∗ , u∗ ) ∈ L2 (Σ) × L2 (Q) × W such that (φ∗ , ψ ∗ ) is defined by (8.215) and u∗ = U(φ∗ , ψ ∗ ) is the solution of (8.213), which satisfies the following optimality conditions, for all (φ, ψ) ∈ Uad × Vad , (αφ∗ + B1∗ u ˜|Σ )(φ − φ∗ )dΓ dt ≥ 0, Σ (8.216) (−βψ ∗ + B2∗ u˜)(ψ − ψ ∗ )dxdt ≤ 0, Q

˜ ∗ , ψ ∗ ) is the unique solution of the following adjoint problem: where u˜ = U(φ −

∂u ˜ − div(η∇˜ u) + (G(., u∗ ))∗ u˜ + K ∗ (, u˜) ∂t ˜ di (., fi )˜ u(., fi )fi = C ∗ C(u − uobs ) on Q, + i=1,n

∂u ˜ − div(η∇˜ u) + (G(., u∗ ))∗ u˜ + K ∗ (, u˜) − ∂t = C ∗ C(u − uobs ) on QT ,

(8.217)

u ˜(T ) = μ(u(T ) − vobs ) on Ω, η

∂u ˜ ˜ = ∂Ω × (0, T − δ(T )), = aj (., gj )˜ u(., gj )gj on Σ ∂n j=1,l

∂u ˜ = 0 on ΣT = ∂Ω × (T − δ(T ), T ). η ∂n Moreover, the gradient of the functional J at (φ∗ , ψ ∗ ) in any direction h = (h1 , h2 ) ∈ L2 (Σ) × L2 (Q) is given by J (φ∗ , ψ ∗ ).h = (αφ∗ + B1∗ u ˜|Σ )h1 dΓ dt + (−βψ ∗ + B2∗ u˜)h2 dxdt. Σ

Q

Otherwise (in the weak sense), ∂J ∗ ∗ ∂J ∗ ∗ ˜|Σ and (φ , ψ ) = αφ∗ + B1∗ u (φ , ψ ) = −βψ ∗ + B2∗ u˜. ∂φ ∂ψ

Remark 8.72. It is clear that we can consider other observations, controls and/or disturbances (which can appear in the boundary condition or in the state system) and we obtain similar results by using the same technique as used in the previous work of this section. ♦

9 Remarks on Numerical Techniques

We can now give some numerical strategies in order to solve robust control problems, by using the adjoint variables. In this chapter, we do not pretend to give a general theory for numerical resolution, but only assist the reader with possible strategies based on adjoint control optimization.

9.1 Introduction and Studied Problem We present algorithms where the descent direction is calculated using the adjoint variables, particularly by choosing an admissible step size. The primal and adjoint problems are discretized by a combination of Galerkin and finite element methods for space discretization and by using the classical first-order Euler method for time discretization. In order to illustrate the numerical methods, we consider the non-linear problem treated in Section 8.6, more precisely the non-linear robust control problem defined by the following systems: 1. The primal problem is given by ∂u + Au + F (., u) + K(, u) = B1 φ + B2 ψ on Q, ∂t u(0) = u0 (given) on Ω.

(9.1)

The operator solution U : (φ, ψ) ∈ (L2 (Q))2 −→ U(φ, ψ) is defined by U(φ, ψ) = u, which is a solution of (9.1). 2. The objective functional is given by J(φ, ψ) =

μ 1 C(u − uobs ) 2L2 (Q) + u(T ) − vobs 2L2 2 2 α β 2 + φ L2 (Q) − ψ 2L2 (Q) , 2 2

(9.2)

320


where μ, α, β > 0 are fixed, (uobs , vobs ) ∈ V × L2 (Ω) is the observation (given) and C is an unbounded, linear operator on L2 (Ω) satisfying (7.91). The robust control problem is to find a saddle point (φ∗ , ψ ∗ ) of J in a space Uad × Vad subject to the perturbation problem (9.1), where Uad and Vad are (given) non-empty, closed, convex, bounded subsets of L2 (Q), i.e., to find (φ∗ , ψ ∗ ) ∈ Uad × Vad such that J(φ∗ , ψ) ≤ J(φ∗ , ψ ∗ ) ≤ J(φ, ψ ∗ ), ∀(φ, ψ) ∈ Uad × Vad .

(9.3)

3. The derivative of the operator solution is given by ∂w + Aw + G(., U(φ, ψ))w + K(, w) = B1 h1 + B2 h2 on Q, ∂t w(0) = 0 on Ω, where w=

(9.4)

∂U ∂U (φ, ψ).h1 + (φ, ψ).h2 . ∂φ ∂ψ

˜ corresponding to the operator solution U is given 4. The adjoint operator U ˜ : (φ, ψ) −→ U(φ, ˜ by U ψ) = u ˜ such that u˜ satisfies ∂u ˜ + A˜ u + (G(., u))∗ u ˜ + K ∗ (V, u ˜) = C ∗ C(u − uobs ) on Q, ∂t u ˜(T ) = μ(u(T ) − vobs ) on Ω,

−

(9.5)

where u = U(φ, ψ) is the solution of problem (9.1) with the forcing (φ, ψ), C ∗ (resp. (G(., u))∗ ) is the adjoint of the operator C (resp. G(., u)) (the adjoint A∗ of A is itself, i.e., A∗ = A since A is a self-adjoint operator). 5. The optimality conditions are given by T (αφ∗ + B1∗ u ˜∗ )(φ − φ∗ )dxdt ≥ 0 Ω 0 T (−βψ ∗ + B2∗ u˜∗ )(ψ − ψ ∗ )dxdt ≤ 0, 0

(9.6)

Ω

˜ ∗ , ψ ∗ ) is the solution of the for all (φ, ψ) ∈ Uad × Vad , where u ˜∗ = U(φ ∗ ∗ adjoint problem (9.5) and (φ , ψ ) ∈ Uad × Vad is an optimal solution (a saddle point of the objective functional J). 6. The gradient of the objective functional J is given by ∂J ∂J (φ, ψ) = αφ + B1∗ u (φ, ψ) = −βψ + B2∗ u ˜ and ˜, ∂φ ∂ψ

(9.7)

˜ where u ˜ = U(φ, ψ) is the solution of the adjoint problem (9.5), corresponding to the solution u = U(φ, ψ) of the primal problem (9.1).

9.2 Continuous Case

321

9.2 Continuous Case In this section the descent method is formulated in terms of continuous variables that are independent of a specific discretization. We propose three classical algorithms: gradient algorithm, conjugate gradient algorithm and Lagrange–Newton algorithm. 9.2.1 Gradient Algorithm The gradient algorithm for the resolution of the optimization problem (9.3) is given by: for k = 1, . . . (the iteration index) we denote by (φk , ψk ) the numerical approximation of the control-disturbance at the kth iteration of the algorithm. 1. Initialization: k = 0 and (φ0 , ψ0 ) (given for example by (φ0 , ψ0 ) = (0, 0) on t ∈ [0, T ]). 2. Resolution of the direct problem (9.1) with the source term (φk , ψk ), gives uk = U(φk , ψk ). ˜k . 3. Resolution of the adjoint problem (9.5) (based on (φk , ψk , uk )), gives u 4. Local expression of the gradient of J at point (φk , ψk ): ⎧ def ∂J ⎪ ⎪ (φk , ψk ) = αφk + B1∗ u˜k , ⎨ ck = ∂φ (GJ ) ⎪ def ∂J ⎪ ⎩ dk = (φk , ψk ) = −βψk + B2∗ u ˜k . ∂ψ 5. Determine (φk+1 , ψk+1 ):

ψk+1 := ψk + δk dk , φk+1 := φk − λk ck ,

where 0 < m ≤ λk , δk ≤ M are the sequences of step lengths. 6. IF the gradient is sufficiently small (convergence) THEN end; ELSE set k := k + 1 and REPEAT from 2 UNTIL convergence. The approximation of the optimal solution (φ∗ , ψ ∗ , u∗ ) is then given by (φk , ψk , uk ). In order to obtain an algorithm that is numerically efficient, the best choice of λk and δk will be the result of a line minimization and maximization algorithm, respectively. Otherwise, at each iteration step k of the previous algorithm, we solve one-dimensional optimization problems of the parameters (λk , δk ): (9.8) min J(φk − λck , ψk ) λ>0

and max J(φk , ψk + δdk ). δ>0

(9.9)

322


From the numerical computational viewpoint, it is most efficient to compute (λk , δk ) only approximately, in order to reduce the computational cost. To derive an approximation for (λk , δk ) we can use, for example, the linearization of U(φk − λck , ψk ) and U(φk , ψk + δdk ) at (φk , ψk ) by ∂U (φk , ψk ).(λck ), ∂φ ∂U (φk , ψk ).(δdk ), U(φk , ψk + δdk ) ≈ U(φk , ψk ) + ∂ψ U(φk − λck , ψk ) ≈ U(φk , ψk ) −

otherwise

U(φk − λck , ψk ) ≈ uk − λwφk , k , U(φk , ψk − δdk ) ≈ uk + δwψ

(9.10)

(9.11)

k = (∂U/∂ψ)(φk , ψk ).dk are the soluwhere wφk = (∂U/∂φ)(φk , ψk ).ck and wψ tion of (9.4). According to the previous approximation (9.11), we can approximate the problems (9.8) and (9.9) by

min H(λ),

(9.12)

max R(δ),

(9.13)

λ>0

and δ>0

where 1 μ C(uk − uobs ) − λCwφk 2L2 (Q) + (uk (T ) − vobs ) − λwφk (T ) 2L2 2 2 α β 2 + φk − λck L2 (Q) − ψk 2L2 (Q) , 2 2 μ 1 k 2 k (T ) 2L2 R(δ) = C(uk − uobs ) + δCwψ L2 (Q) + (uk (T ) − vobs ) + δwψ 2 2 α β + φk 2L2 (Q) − ψk + δdk 2L2 (Q) . 2 2

H(λ) =

As the functions H and R are polynomial functions of degree 2, the problems (9.12) and (9.13) can be solved exactly. Consequently, we obtain explicitly the value of the parameters λk and δk . 9.2.2 Conjugate Gradient Algorithm Another strategy to numerically solve the optimization problem (8.111) is to use a conjugate gradient type algorithm (CG-algorithm) combined with the Wolfe–Powell line search procedure for computing admissible step sizes along the descent direction. The advantage of this method, compared to the gradient method, is that it performs a soft reset whenever the GC search direction yields no significant progress. In general, the method has the following form:

9.2 Continuous Case

Dk = Dzk =

323

−Gk for k = 0, −Gk + ξk−1 Dk−1 for k ≥ 1,

zk+1 = zk + γk Dk , where Gk denotes the gradient of the functional to be minimized at point zk , γk is a step length obtained by a line search, Dk is the search direction and ξk is a constant. Several varieties of this method differ in the way of selecting ξk . Some well-known formula for ξk are given by Fletcher–Reeves, Polak–Ribière, Hestenes–Stiefel and, recently, by Dai and Yuan [94]. The GC-algorithm for the resolution of the saddle point problem (8.111) is given below. For k = 1, . . . (the iteration index) we denote by Xk = (φk , ψk ) the numerical approximation of the control-disturbance at the kth iteration of the algorithm. 1. Initialization: (φ0 , ψ0 ) (given), ξ−1 = 0, η−1 = 0 and C0 = 0, D−1 = 0. (a) Resolution of the direct problem (9.1), where the source term is (φ0 , ψ0 ), gives u0 . (b) Resolution of the adjoint problem (9.5) (based on (φ0 , ψ0 , u0 )), gives u˜0 . (c) Gradient of J at (φ0 , ψ0 ), the vector (c0 , d0 ) is given by the system (GJ). (d) Determine the direction: C0 = −c0 , D0 = −d0 . (e) Determine (φ1 , ψ1 ): φ1 = φ0 + λ0 C0 and ψ1 = ψ0 − δ0 D0 . 2. Resolution of the problem (9.1) with the source term (φk , ψk ), gives uk . 3. Resolution of the problem (9.5) (based on (φk , ψk , uk )), gives u˜k . 4. Gradient of J at Xk , the vector (ck , dk ) is given by the system (GJ). 5. Determine ξk−1 , ηk−1 by one of the following expressions: ξk−1 =

ck 2Uad dk 2Vad , ηk−1 = (Fletcher–Reeves), 2 ck−1 Uad dk−1 2Vad

ξk−1 =

(ck − ck−1 , ck )Uad (dk − dk−1 , dk )Vad , ηk−1 = (Polak–Ribière), 2 ck−1 Uad dk−1 2Vad

ξk−1 =

(ck , ck − ck−1 )Uad , (Ck−1 , ck − ck−1 )Uad ηk−1=

ξk−1 =

(dk , dk − dk−1 )Vad (Hestenes–Stiefel), (Dk−1 , dk − dk−1 )Vad

ck 2Uad dk 2Vad , ηk−1 = (Dai–Yuan). (Ck−1 , ck − ck−1 )Uad (Dk−1 , dk − dk−1 )Vad

6. Determine the direction: Ck = −ck + ξk−1 Ck−1 , Dk = −dk + ξk−1 Dk−1 .

324


7. Determine (φk+1 , ψk ): φk+1 = φk + λk Ck , ψk+1 = ψk − δk Dk where 0 < m ≤ λk ≤ M are the sequences of step lengths. 8. IF the gradient is sufficiently small (convergence) THEN end; ELSE set k := k + 1 and GOTO 2. The approximation of the optimal solution (φ∗ , ψ ∗ , u∗ ) is (φk , ψk , uk ). These methods are convergent if the following line search conditions for λk are satisfied: J(φk + λk Ck , ψk ) ≤ J(φk , ψk ) + σ1 λk (ck , Ck )Uad + τ1k , J(φk , ψk − δk Dk ) ≤ J(φk , ψk ) − σ2 δk (dk , Dk )Vad + τ2k , ρ11 < ck , Ck >Uad ≤ (ck+1 , Ck )Uad ≤ −ρ12 (ck , Ck )Uad + τ3k , ρ21 < dk , Dk >Vad ≤ (dk+1 , Dk )Vad ≤ −ρ22 (dk , Dk )Vad + τ4k , where (τik )i=1,4 are the tolerances, 0 < σi < ρi1 < 1 and 0 < ρi2 < 1. In practice the tolerances (τik )i=1,4 are of the order of the roundoff error. For σi , i = 1, 2 and (ρij )i=1,2; j=1,2 , the choices σi = 10−p for p = 4, 5 or 6 and ρij = 10−q for q = 1 or 2 are effective when double precision arithmetic is used. Remark 9.1. We can adapt other strategies in order to solve the robust control problem by using descent algorithms combined with the line search procedure for computing admissible step sizes along the descent direction (in every iteration) (see, e.g., Gill et al. [133] and Hestenes [154]). 9.2.3 Lagrange–Newton Method The robust control problem (9.1), (9.5) and (9.7) with the unknown X = ˜ (φ, ψ, u, u ˜) with u = U(φ, ψ) and u ˜ = U(φ, ψ), can be reformulated as an inclusion, or the so-called generalized equation of the form 0 ∈ F(X) + T (X),

(9.14)

where F : M −→ E is F-differentiable and T : M −→ 2E (the power set of E) is a set-valued map with closed graph. More precisely, ∂u + Au + F (., u) + K(, u) − B1 φ − B2 ψ, ∂t F2 (X) := u(t = 0) − u0 ,

F1 (X) :=

∂u ˜ + A˜ u + (G(., u))∗ u ˜ + K ∗ (V, u˜) − C ∗ C(u − uobs ), ∂t F4 (X) := u˜(T ) − μ(u(T ) − vobs ), F3 (X) = −

F5 (X) := αφ + B1∗ u ˜, F6 (X) := −βψ + B2∗ u ˜,

(9.15)

9.2 Continuous Case

325

and T (X) := ({0}, {0}, {0}, {0}, N1(φ), N2 (ψ))

(9.16)

where the multi-valued maps N1 and N2 , called normal cone operators, are defined by T N1 (φ) = {z ∈ L2 (Q) : z(φ1 − φ)dxdt ≤ 0 ∀φ1 ∈ Uad }, 0

Ω

if φ ∈ Uad ,

N1 (φ) = ∅, if φ ∈ / Uad , T z(ψ1 − ψ)dxdt ≥ 0 ∀ψ1 ∈ Vad }, N2 (ψ) = {z ∈ L2 (Q) : 0

Ω

(9.17)

if ψ ∈ Vad ,

N2 (ψ) = ∅, if ψ ∈ / Vad , Remark 9.2. The cone-constrained problem (9.14) is an abstract model for various problems: • when T = 0, the problem (9.14) corresponds to finding a zero of a mapping F defined in a Banach space • when T is the positive orthant in IRp , p ≥ 1, the problem (9.14) is a system of inequalities • when T is the normal cone to a convex and closed set in a space M, the problem (9.14) may represent variational inequalities. For other situations the reader should refer, for instance, to Dontchev [104]. ♦ We assume that F is a twice-differentiable operator. The generalized Newton method for solving (9.14) is similar to the Newton method for non-linear equations in Banach spaces. More precisely, the Newton procedure for (9.14) is defined by constructing the sequence (Xk ) as follows: assume that we have already computed Xi , i = 1, k, the value Xk+1 is given by 0 ∈ F(Xk ) − F (Xk )(Xk+1 − Xk ) + T (Xk+1 ),

(9.18)

˜k ) is the approximation of X at the kth iteration of where Xk = (φk , ψk , uk , u the algoritm in M = (L2 (Q))2 × (W ∩ H1 )2 and X0 is given in M. In this work, we are not interested in the general cone constraints but we give the algorithm to resolve the problem (9.18) in the case of the equality-type constraints, i.e., when T = 0. In this case the generalized Newton method can be formulated as (we assume that F satisfies sufficient conditions such that the F-derivative of F is invertible) Xk+1 = Xk − (F (Xk ))−1 F (Xk ),

(9.19)

The Newton procedure (9.19) can be decomposed into the following two steps:

326


1. Find the solution Zk := (pk , qk , vk , v˜k ) of the problem F (Xk )Zk = F (Xk ).

(9.20)

Xk+1 = Xk − Zk .

(9.21)

2. Calculate Xk+1 by

By using (9.15) and (9.4), the mapping Zk satisfies the following system: ∂vk + Avk + G(., uk )vk + K(, vk ) − B1 pk − B2 qk ∂t ∂uk + Auk + F (., uk ) + K(, uk ) − B1 φk − B2 ψk , = ∂t vk (0) = uk (0) − u0 , −

∂˜ vk + A˜ vk + (G(., uk ))∗ v˜k + K ∗ (V, v˜k ) ∂t ∂u ˜k + A˜ uk ∂t +(G(., uk ))∗ u ˜k + K ∗ (V, u ˜k ) − C ∗ C(uk − uobs ),

+(G (., uk ))∗ (˜ uk )(vk ) − C ∗ Cvk = −

(9.22)

v˜k (T ) − μvk (T ) = u ˜k (T ) − μ(uk (T ) − vobs ), αpk + B1∗ v˜k = αφk + B1∗ u ˜k , −βqk + B2∗ v˜k = −βψk + B2∗ u ˜k . ˜k+1 ), According to (9.22) and (9.21) the mapping Xk+1 = (φk+1 , ψk+1 , uk+1 , u with φk+1 = φk − pk , ψk+1 = ψk − qk , uk+1 = uk − vk and u ˜k+1 = u˜k − v˜k , satisfies the following system: ∂uk+1 + Auk+1 + G(., uk )uk+1 + K(, uk+1 ) ∂t = (G(., uk )uk − F (., uk )) + B1 φk+1 + B2 ψk+1 , uk+1 (0) = u0 , −

∂u ˜k+1 + A˜ uk+1 + (G(., uk ))∗ u˜k+1 + (G (., uk ))∗ (˜ uk )(uk+1 ) ∂t −(G (., uk ))∗ (˜ uk )(uk ) + K ∗ (V, u˜k+1 ) = C ∗ C(uk+1 − uobs ),

(9.23)

u ˜k+1 (T ) = μ(uk+1 (T ) − vobs ), αφk+1 + B1∗ u ˜k+1 = 0, −βψk+1 + B2∗ u ˜k+1 = 0. Consequently, at each iteration of the Newton method, we have to solve the following system:

9.2 Continuous Case

327

∂w + Aw + G(., uk )w + K(, w) = fk + B1 Φ + B2 Ψ, ∂t w(0) = u0 , −

∂w ˜ ˜ + (G (., uk ))∗ (˜ uk )(w) + K ∗ (V, w) ˜ + Aw˜ + (G(., uk ))∗ w ∂t = gk + C ∗ C(w − uobs ),

(9.24)

w(T ˜ ) = μ(w(T ) − vobs ), ˜ = 0, αΦ + B1∗ w −βΨ + B2∗ w ˜ = 0. where fk = G(., uk )uk − F (., uk ) and gk = (G (., uk ))∗ (˜ uk )(uk ). Remark 9.3. (i) It was shown by Robinson [250], that Newton’s method applied to the general equation is locally quadratically convergent to the exact solution, provided that a propriety called strong regularity is satisfied. In our situation and in the case of the equality-type contraints we can prove the following result: Let X0 be given in M and M (X0 ) := {X ∈ M : X − X0 M < } be the open ball in M of radius centred on X0 . Under adequate assumptions, there exists h < 1, 1 < β ≤ 2 and c > 0 such the Newton sequence Xk is well defined and the following rate of convergence holds. X ∗ − Xk M ≤ chβ

k

−1

for k ≥ 2,

where X ∗ is the exact optimal solution. (ii) In general, the advantage of the Newton method is that the convergence is almost quadratic but its disadvantage resides in the fact that it is expensive (because it requires a lot of computing time) and if the structure is far from the optimal solution the method can become unstable. Therefore, Newton’s method is reserved mainly for cases of rapid convergence (i.e., the number of iterations is small) with a well-defined optimal solution. (iii) We can adapt other popular iterative techniques of Newton’s type to resolve the robust control problems, for example, SQP (sequential quadratic programming) (see, e.g., Alt [8, 9], Bergounioux and Kunisch [50], Ito and Kunisch [166, 167], Malanowski [214], Troltzsch [284], and the references therein), the Gauss–Newton method (see, e.g., Dennis and Schnabel [98], Le Dimet and Shutyaev [188]) and Landweber’s method (when the idea is to replace, in (9.19), the mapping (F (Xk ))−1 by the adjoint (F (Xk ))∗ of F (Xk ), which is calculated from the adjoint problem, and then for Landweber one need not to compute also the inverse of F (Xk )). For more discussion on this last method see, e.g., Engl and Scherezer [114] and Scherezer [260]. ♦

328


In the following, we will be only interested in the discretization of the robust control problem in the case of the gradient procedure (in order to numerically solve the optimization problem).

9.3 Discrete Problem 9.3.1 Approximation of Robust Control Problems Next we propose a finite element method combined with the first-order Euler scheme for solving the continuous optimization problem. Since the primal and dual systems are (linear or non-linear) PDEs, the approximation analysis of these problems is a non-negligible part of total numerical analysis. For space discretization, we give a space discretization parameter h (that is converging to 0). To describe the space discretization scheme, we introduce the finite dimensional subspace Vh of D associated with a partition Th of the domain Ω and piecewise polynomial functions of some fixed degree l. Example 9.4. In the case of D = H01 (Ω), we introduce the finite dimensional subspace Vh of D associated with a partition Th of the domain Ω and piecewise polynomial functions of some fixed degree l with all functions vanishing on the boundary ∂Ω. Before expressing the space–time discretization, we recall the following hypotheses (see, e.g., Ciarlet [85]): (H1E ) There exists C > 0, such that for all 1 ≤ r ≤ l and v ∈ H r+1 (Ω) inf v − vk H 1 ≤ Chr v H r+1 .

vh ∈Vh

(H2E ) There exists C > 0, such that for all (m, k) ∈ IN2 with 0 ≤ k ≤ m and vh ∈ Vh , vh H m ≤ Chk−m vh H k . (H3E ) The approximation u0h in Vh of u0 satisfies: if u0 ∈ H s+1 (Ω) then there exists C > 0, such that for all 1 ≤ r ≤ min(l, s), h u0 − u0h H 1 + u0 − u0h L2 ≤ Chr+1 .

♣

For the time discretization, we partition the interval (0, T ) by using the following points 0 = t0 ≤ t1 ≤ · · · ≤ tM = T with tn = nτ , for n = 0, . . . , M and τ = T /M . For a continuous mapping u : (0, T ) −→ L2 (Ω), we define un = u(., tn ) for n = 0, . . . , M . For a given sequence (un )n=0,...,M in L2 (Ω), we define its difference quotient as ∂τ un = (un − un−1 )/τ. The step is chosen sufficiently small to guarantee both the time accuracy and convergence of the solution.

9.3 Discrete Problem

329

With the above notation and the discretization of (9.2) and (9.1) by the composite trapezoidal rule in time, we formulate the finite approximation of the problems (9.2) and (9.1) as follows: find a saddle point in Uad,h × Vad,h of M τ θn | C(unh − unobs ) |2 dx Jh (φh , ψh ) = 2 n=0 Ω μ (9.25) + | uM − vobs |2 dx 2 Ω h M M τ τ n 2 θn α | φh | dx − θn β | ψhn |2 dx, + 2 n=0 2 n=0 Ω Ω with unh ∈ Vh satisfying (∀vh ∈ Vh ) n n n n ∂τ uh vh dx + a(uh , vh ) + K(Vh , uh )vh dx + Fhn (unh )vh dx Ω Ω Ω n n B1 φh vh dx + B2 ψh vh dx, = Ω

u0h = u0 on Ω,

(9.26)

Ω

for n = 1, 2, . . . , M . Here θ0 = θM = 1/2 and θn = 1 for n = 0 and n = M . In order to solve the discretized finite element minimization of Jh (φnh , ψhn ) over Uad,h × Vad,h , we need to calculate the partial derivatives of Jh (φnh , ψhn ) at direction pnh and qhn . According to (9.7) we have that the evaluation of the derivative of J(φnh , ψhn ) is M ∂J n n n (φh , ψh ).ph = τ θn (αφnh + B1∗ uñh )pnh dx, ∂φ Ω n=0 (9.27) M ∂J n n n n ∗ n n (φ , ψ ).q = τ θn (−βψh + B2 u ˜h )qh dx. ∂ψ h h h Ω n=0 where u ñh is the solution of following discrete adjoint problem (we use the discrete backward Euler approximation of time) n+1 n − ∂τ u˜h vh dx + a(˜ uh , vh ) + (Gnh (unh ))∗ u ñh vh dx Ω Ω (9.28) K ∗ (Vhn , u ñh )vh dx = C ∗ C(unh − unobs )vh dx, + Ω

M u ˜M h = μ(uh − vobs ) on Ω,

Ω

for n = 0, 1, . . . , M − 1. 9.3.2 Discrete Gradient Algorithm According to the previous discrete formula (9.27) we can now present the following gradient scheme in order to solve the discrete minimization problem (9.25)–(9.26).

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For k = 1, . . . (the iteration index) we denote by Xhk = (φh,k , ψh,k ) the numerical approximation of the control-disturbance at the kth iteration of the algorithm. 1. Initialization: (φh,0 , ψh,0 ) (given). 2. Compute the primal problem (9.26) with the source term (φh,k , ψh,k ), gives unh,k , for n = 1, 2, . . . , M . 3. Compute the adjoint problem (9.28) (based on (φh,k , ψh,k , uh,k )), gives u ñh,k , for n = M −1, M −2, . . . , 0. 4. Compute the components of the gradient of J at (φh,k , ψh,k ), corresponding to all the basic functions {φm } from Vh ck,h =αφk,h + B1∗ u˜k,h , (GJD) dk,h = − βψk,h + B2∗ u˜k,h . 5. Find λh,k > 0 and δh,k such that Jh (φh,k − λch,k , ψh,k ) is minimized over all λ > 0, Jh (φh,k , ψh,k + δdh,k ) is maximized over all δ > 0. 6. Compute (φh,k+1 , ψh,k+1 ): φh,k+1 := φh,k − λh,k ch,k , ψh,k+1 := ψh,k + δh,k dh,k . 7. IF the gradient is sufficiently small (convergence) THEN end; ELSE set k := k+1 and GOTO 2. The discrete approximation of the optimal solution (φ∗ , ψ ∗ , u∗ ) is (φh,k , ψh,k , uh,k ). For convenience, we denote the approximation of the optimal solution by the discrete gradient method, with the objective function Jh and initial guess (φh,0 , ψh,0 ), by ∗ , u∗h ) = Gradh (Jh , φh,0 , ψh,0 ). (φ∗h,0 , ψh,0

Remark 9.5. (i) By using the same discretization as in the gradient method we can obtain easily the conjugate gradient method in order to solve the discrete saddle point problem (9.25)–(9.26) by using the discrete formula (9.27). (ii) In order to accelerate the convergence of the non-linear gradient method, we propose in this section the non-linear multi-grid method (NMGM). Because in the iterative methods the frequency components of the residual error are obtained most rapidly on the grid corresponding to them, and the multi-grid method has the ability to use this characteristic in an effective way and thus to exploit many grids in order to converge rapidly. ♦


331

9.3.3 Multi-grid Gradient Method Next we propose a non-linear multi-grid gradient method (NMGM) as in Yamamoto and Zou [305] for solving the non-linear minimization problem (9.25)-(9.26) (for more details on the multi-grid methods see, e.g., Hackbusch [148], Wesseling [299] and the references therein). The basic element of NMGM is the well-known defect correction principle (see Bohmer and Steller [55]). The defect correction scheme starts with fine grid and iterates a few times using an iterative method (called pre-smoothing) and then goes to a coarser grid to solve the residual equation to achieve some coarse correction for the approximate solution obtained on the fine grid (called coarse-grid correction), again applying the same iterative method, a few iterations, for the residual equation (called post-smoothing). In order to develop the NMGM we introduce a set of regular partitions (Thj )j=0,N , with Thj+1 being the refinement of Thj and the spaces (Vhj )j=0,N that are the continuous piecewise linear finite element spaces defined on (Thj )j=0,N , such that Vh0 ⊂ Vh1 ⊂ . . . ⊂ Vhj ⊂ Vhj+1 ⊂ . . . ⊂ VhN −1 ⊂ VhN = Vh . Our goal is to solve the discrete saddle point problem (9.25)–(9.26), which is defined on the finest space Vh , by using the auxiliary spaces Vhj , for j = 0, . . . N − 1. For this, we introduce some more notations. For each partition Thj , we divide the time interval [0, T ] into Mj subintervals using the M −1 M following points: 0 = t0j < t1j . . . tkj < tk+1 . . . tj j < tj j = T , with tkj = kτj j and τj = T /Mj . Similar to Uad,h and Vad,h , we define constrained subsets j j Uad,h and Vad,h , respectively. (I) First step of NMGM: For the initialization step of the NMGM to be introduced below, we solve the saddle point problem on each coarse space Vhj j j as follows: find a saddle point in Uad,h × Vad,h of Jh0j (φhj , ψhj ) =

Mj τj θn | C(unhj − unobs ) |2 dx 2 n=0 Ω μ M + | u j − vobs |2 dx 2 Ω hj Mj τj + θn α | φnhj |2 dx 2 n=0 Ω M τj − θn β | ψhnj |2 dx, 2 n=0 Ω

(9.29)

where unhj ∈ Vhj solves (9.26) on Vhj , with the forcing (φhj , ψhj ), θ0 = θMj = 1/2 and θn = 1 if n = 0, Mj . (II) Second step of NMGM: We introduce the coarse grid correction, for which we define the following functional:

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Jh0j (φh + φhj , ψh + ψhj ) =

Mj τj θn | C(Uhnj − unobs ) |2 dx 2 n=0 Ω μ M + | U j − vobs |2 dx 2 Ω hj Mj τj θn α | φh + φnhj |2 dx + 2 n=0 Ω M τj − θn β | ψh + ψhnj |2 dx, 2 n=0 Ω

where Uhnj ∈ Vhj solves (∀vhj ∈ Vhj ) ∂τj Uhnj vhj dx + a(Uhnj , vhj ) + K(Vnhj , Uhnj )vhj dx Ω Ω n n + Fhj (Uhj )vhj dx = B1 (φnh + φnhj )vhj dx Ω Ω B2 (ψhn + ψhnj )vhj dx, +

(9.30)

(9.31)

Ω

Uh0j = u0 on Ω. (III) Third step of NMGM: Let us introduce the adjoint solution on each coarse space Vhj and the evaluation of the partial derivatives of the functional j j and qhj ∈ Vad,h , respectively: Jhj (φh + φhj , ψh + ψhj ) at direction phj ∈ Uad,h ∂Jh0j ∂φ

(φh + φhj , ψh + ψhj ).phj = τj

∂Jh0j ∂ψ

Mj

θn Ω

n=0

˜ n )pn dx, (α(φnh + φnhj ) + B1∗ U hj hj (9.32)

(φh + φhj , ψh + ψhj ).qhj = τj

Mj n=0

θn Ω

˜ nh )qhn dx, (−β(ψhn + ψhnj ) + B2∗ U j j

ñ U hj

is the solution of the following discrete adjoint problem (∀vhj ∈ Vhj ) ˜ n+1 vhj dx + a(U ˜ n vhj dx ˜ n , vhj ) + (Gn (U n ))∗ U − ∂τj U hj hj hj hj hj Ω Ω (9.33) ˜ n )vhj dx = + K ∗ (Vhnj , U C ∗ C(Uhnj − unobs )vhj dx, hj

where

Ω

˜ Mj = μ(U Mj − vobs ) on Ω, U hj hj for n = 0, 1, . . . , Mj − 1.

Ω


333

(IV) Fourth step of NMGM: The gradient method is used to solve the saddle point problem: find a saddle point of Jh0j (φnh +φnhj , ψhn +ψhnj ), over all (φnhj , ψhnj ) ∈

j j j j Uad,h × Vad,h such that (φnh + φnhj , ψhn + ψhnj ) ∈ Uad,h × Vad,h .

Gradient algorithm: For k = 1, . . . (the iteration index) we denote by the pair (φhj ,k , ψhj ,k ) the numerical approximation of the control-disturbance at the kth iteration of the algorithm. 1. Initialization: (φhj ,0 , ψhj ,0 ) (given). 2. Compute the primal problem (9.31) with the source term (φhj ,k +φh , ψhj ,k + ψh ), gives Uhnj ,k , for n = 1, 2, . . . , Mj . 3. Compute the adjoint problem (9.33) (based on (φhj ,k +φh , ψhj ,k +ψh , Uhj ,k )), ˜ n , for n = Mj −1, Mj −2, . . . , 0. gives U h,k

4. Compute the components of the gradient of J at (φhj ,k + φh , ψhj ,k + φh ), corresponding to all the basic functions {φm } from Vhj ⎧ ˜ hj ,k , ⎨ ck,h =α(φhj ,k + φh ) + B1∗ U (GJD) ⎩ d = − β(ψ ∗˜ k,h hj ,k + ψh ) + B2 Uhj ,k . 5. Find λhj ,k > 0 and δhj ,k such that Jh0j ((φhj ,k + φh ) − λchj ,k , ψhj ,k + ψh ) is minimized over all λ > 0, Jh0j (φhj ,k + φh , (ψhj ,k + ψh ) + δdhj ,k ) is maximized over all δ > 0. 6. Compute (φhj ,k+1 , ψhj ,k+1 ):

φhj ,k+1 := φhj ,k − λhj ,k chj ,k , ψhj ,k+1 := ψhj ,k + δhj ,k dhj ,k .

7. IF the gradient is sufficiently small (convergence) THEN end; ELSE set k := k+1 and GOTO 2. The discrete approximation of the optimal solution (φ∗hj , ψh∗j , Uh∗j ) is (φhj ,k , ψhj ,k , Uhj ,k ). Here, we denote the approximation of the optimal coarse correction of Xh = (φh , ψh ) by the gradient method, with the objective function Jhj and initial guess Xhj ,0 , by (φ∗hj , ψh∗j , Uh∗j ) = Gradhj (Jhj , φh , ψh , Xhj ,0 ). We can now formulate the algorithm of the non-linear multi-grid gradient method for solving the finite element optimization problem (9.25)-(9.26).

334

9 Remarks on Numerical Techniques (0)

(0)

0 0 The NMGM algorithm: Let (φh0 , ψh0 ) ∈ Uad,h × Vad,h be a given ini0 tial guess (on the coarse finite element Vh ).

(I) Initialization of the coarse grid: 1. For j = 0, . . . , N − 1, do: (0) (0) IF j = 0, calculate φhj = Πj−1,j φ∗hj−1 and ψhj = Πj−1,j ψh∗j−1 , Compute (φ∗hj , ψh∗j ) = Gradh (Jh0j , φhj , ψhj ) END (0) (0) 2. Compute φh = ΠN −1,N φ∗hN −1 and ψh = ΠN −1,N ψh∗N −1 , (0)

(0)

where the operator Πj−1,j can be any interpolation from Vhj−1 onto Vhj . (II) Smoothing and coarse grid correction: For k = 0, . . . (the iteration index): ˜ (0) = (φ˜(0) , ψ˜(0) ) := (φ(0) , ψ (0) ) 3. Set X h h h h h For j = N, . . . , 0, do: (0) (0) (0) IF j = N , calculate Xh = (φh , ψh ) = (φ∗h , ψh∗ ) = Xh∗ , Compute Xh∗ = (φ∗h , ψh∗ ) = Gradhj (Jhj , Xh , 0) END ˜ (0) is sufficiently small, stop 4. IF Xh∗ − X h (0)

ELSE, set Xh := Xh∗ , k := k + 1 and GOTO 3. (0)

Remark 9.6. We can consider, naturally, for the operator Πj−1,j the finite element injection of Vhj−1 into Vhj . For the starting point (φ0hj , ψh0 j ) we can take the value zero. ♦ Remark 9.7. In order to numerically solve the robust control problems (by taking into account the nature of the gradients), the reader can also adapt other numerical methods that exist in the literature for solving optimal control problems, in the context of the methods developed in this book. ♦

Part III

Applications in the Biological and Physical Sciences: Modeling and Stabilization

336

“As any human activity needs goals, mathematical research needs problems.” David Hilbert “Nature not only suggests to us problems, she suggests their solution.” Henri Poincaré “The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.” Eugene Wigner

We address three different topics in this part. The first is that of material sciences. We first study the phenomenon of vortex structure in the phase transitions taking place in superconductor films with variable thickness. The objective of this chapter is to control the motion of vortices by taking into account the influence of fluctuations and noises in data (for example, thermal fluctuations and material impurities). The next chapter is about the microstructures (dendrite) that appear during isothermal solidification of a binary alloy. The goal is to predict and stabilize the microstructure dynamics (in this case, as in the previous application, the thermal fluctuations and material impurities also affect considerably solidification microstructure dynamics). The second topic is that of oceanic currents, which have an influence on the climate system. Oceanic current movements play a key role in climate regulation. In the tropical zone, the oceanic circulation is characterized by steady zonal currents and by long waves propagating westward along the equator. These equatorial waves can be connected with strong vertical currents which are very sensitive to small changes in temperature (see, for example, the El N i˜ no phenomenon). Our purpose is motivated by the robust regulation of the deviation of circulation from the mean circulation by taking into account the worst-case noise caused by a small variation of the surface temperature. The third topic is that of biological systems. First, we analyze the transport of thermal energy in living tissues. A very important application of bioheat transfer models is in thermal therapy, especially in clinical cancer hyperthermia therapy (the goal is to destroy the pathological tissues, by applying

337

heat to these tissues, with minimal damage to the surrounding tissues). The thermal conductivity of living tissues is a very complex process which uses different phenomenogical mechanisms, including conduction, convection, radiation and metabolism evaporation. Moreover, the studied model takes into account blood flow and extra–cellular water which affect considerably both the heat transfer in the tissues and the tissues thermal properties. Our contribution is to control and stabilize the online temperature. The last chapter deals with some general equations called parabolic diffusive equations with multiple time-varying delays. These systems describe resource management problems in which the objective is the stabilization of uncertain biological species. The applications are very varied: we can cite, for example, fishery resource systems, wildlife damage management, harvesting species, and so on. Dynamical population models govern diffusive biological species with logistic growth terms and time delays (such as birth rate, finished period of gestation, etc.). Finally, we present other models which combine biology and fluid mechanics, namely micropolar systems (animal blood), and fluid mechanics and materials, namely semiconductors melts.

10 Vortex Dynamics in Superconductors and Ginzburg–Landau-type Models

In this chapter we formulate and study robust control problems for a twodimensional time-dependent Ginzburg–Landau (TDGL) type model with Robin boundary conditions on the phase-field parameter, which describes the phase transitions taking place in superconductor films with variable thickness. It is well known that thermal fluctuations and material impurities affect considerably the motion of vortices in superconductors. These effects are modeled by variants of the Ginzburg–Landau model containing either additive or multiplicative noise. The objective is then to control the motion of vortices by taking into account the influence of fluctuations and noises in data. First, a variant of TDGL model (MTDGL) is introduced and analyzed. Second, we introduce the perturbation problem of the non-linear governing coupled system of equations MTDGL (the deviation from the desired target). The existence and uniqueness of the solution of the perturbation are proved as well as stability under mild assumptions. Afterwards the robust control problems are formulated in the case when the control is in the external magnetic field and in the case when the control is in the initial condition of the vector potential. We show the existence of an optimal solution, and we also find the necessary conditions for a saddle point optimality. This work is a generalization of the recent research developed by Belmiloudi in [45].

10.1 Introduction The aim of this contribution is the study of robust control problems to describe the phenomenon of vortex structure in the superconducting phase transitions, using the time-dependent Ginzburg–Landau (TDGL) complex superconductivity model. This model was derived by Gor’kov and Eliashberg in [136] from the microscopic BCS (Bardeen–Cooper–Schieffer) theory (see Bardeen et al. [23]) for a superconductor with paramagnetic impurities. It involves the real

340


vector-valued potential U for the total magnetic field and a complex phasefield variable φ so that | φ |2 = φφ (φ is the complex conjugate of φ) gives the relative density of the superconducting charge carriers (Cooper pairs of electrons) which varies between 0 in the normal phase and 1 in the superconducting phase. The need for φ to be complex is associated with the macroscopic quantum nature of superconductivity. Here we are interested with the response of a superconducting material to an applied magnetic field under isothermal conditions below its critical temperature Tc (the transition from normally conducting to superconducting is usually associated with critical temperature).

Figure 10.1. The reaction of a superconducting material to the applied magnetic field.

Most applications of superconducting materials involve type-II superconductors in high magnetic fields. It is known that for the type-II superconductors, there is a critical magnetic field which splits into a lower critical field Hlc and an upper critical field Huc (cf. Figure 10.1). For the magnetic fields below Hlc the material is in the superconducting state and for magnetic field above Huc the material is in the normal state. For the magnetic fields between Hlc and Huc the material is in the mixed state. This mixed state is described by physicists as follows: around some isolated points (called vortices, which are most commonly arranged in a hexagonal arrangement, see Abrikosov [2]) inside the material, the superconducting property is destroyed and the magnetic field becomes stronger in the nearby regions surrounding these vortices. While, elsewhere, the superconducting property is still dominant and the magnetic field is excluded. Moreover, the motion of the vortices depends highly on

10.1 Introduction

341

the magnetization processes of the material (this is the result of the “Lorentz force”1 on the magnetic flux line carried by the vortex due to the transport current, for example). The motion of the vortices is undesirable, because this motion dissipate energy and leads to an electric field. Therefore, it is very interesting to study the applied magnetic in order to prevent their motion. In practice, attempts are made to pin down vortices to particular locations in the material. Pinning down vortices inside superconductors is achieved by the presence of any form of inhomogeneity (for example, point defects, impurities or a variation in the thickness of the sample of superconducting material, see Du and Gunzburger [106]). A popular way of modeling the effect of pinning in the Ginzburg–Landau framework is to allow the equilibrium density of superconducting electrons to be a function of position (see, e.g., Rubinstein [253]). In order to take into account the effect of inhomogeneities in superconducting thin films having variable thickness, we consider the following two-dimensional time-dependent and non-linear system defined on Ω:2 ∂φ − iηκdiv(ρU )φ + b(U ).(ρb(U ))(φ) + ρG(φ) = 0 on Q, ∂t ∂U ρ + curl(ρcurl(U )) − ∇(div(ρU )) ∂t +ρR(b(U )(φ)φ) = curl(ρH) on Q, ηρ

subject to the Robin-type boundary conditions

(10.1)

1 ∂φ = μφ, U.n = 0, curl(U ) = H on Σ, κ2 ∂n and the initial conditions φ(0) = φ0 , U (0) = U0 on Ω, where Q = Ω × (0, T ), Σ = ∂Ω × (0, T ) and the operator b (the covariant derivative) and the function G are defined by i i b(U ) = (− ∇ + U ), b(U ) = ( ∇ + U ), κ κ G(z) = (| z |2 −ϑ)z.

(10.2)

The equilibrium density of superconducting electron is denoted by the function 0 < ϑ(x) < 1 (spatially dependent). The function ϑ(x) can be thought of as measuring the quality of the superconductor. The domain Ω is an open bounded domain in IR2 with Lipschitz boundary ∂Ω, n is the unit normal to the surface of the superconductor Γ = ∂Ω and μ is an arbitrary real number (the boundary condition is appropriate for the superconductor interface with 1 2

This force is more important than the anchorage forces of vortices, and causes the displacement of the vortex. This problem is corresponding to three-dimensional time-dependent model defined on Ω × (− ρ(x), ρ(x)), when tends to zero.

342


vacuum (or insulator) if μ = 0 and for the superconductor interface with normal metal if μ = 0). R(.) (resp. I(.)) denotes the real part (resp. the imaginary part) of the quantity in (.) and curl denote the curl operators defined by (on the (x, y)-plane) curl(φ) = ( curl(U ) =

∂φ ∂φ T , − ) (φ is a scalar), ∂y ∂x

∂u1 ∂u2 − (U = (u1 , u2 ) is a vector). ∂x ∂y

(10.3)

curl(U ) is the induced magnetic field, J = curl(ρcurl(U )) is the current, H is the applied magnetic field, η is the non–dimensional diffusivity and ρ > 0 is a smooth function characterizing the vertical shape of the superconducting film and satisfying the following hypothesis: ρ ∈ C 1 (Ω) such that ρ0 ≤ ρ ≤ ρ1 and | ∇ρ |≤ ρd ,

(10.4)

where (ρ0 , ρ1 , ρd ) are non-negative constants. The positive constant κ is the Ginzburg–Landau parameter with κ = /", where " is a coherence length describing the size of thermodynamic fluctuations in the superconducting phase, and is the London penetration depth describing the depth to which an external magnetic field can penetrate the superconductor. The √ √ parameter κ determines the type of superconducting material: κ < 1/ 2 describes type-I superconductors, κ > 1/ 2 describes type-II superconductors. Various problems associated with the Ginzburg–Landau models in superconductivity have been studied recently (the literature on this model is vast, see, e.g., Bertuel et al. [51], Chapman et al. [72, 73], Chen et al. [79, 80], Coskun and Kwong [88], Deckelnick et al. [97], Du et al. [107, 106, 105], Gropp et al. [141], Sandier and Serfaty [258], Tang and Wang [279] and the references therein). For the optimal control problems associated with the TDGL models, we can mention Chen and Hoffmann [81] in which the authors studied the control of the vortices in superconducting films through the external magnetic field. These works are applicable only to highly idealized physical situations that do not take into account factors such as inhomogeneities and thermal fluctuations. For example, it is well known that thermal fluctuations and material defects play an important role in the pinning of vortex in typeII superconductors. Moreover, recently, it was experimentally observed that as the temperature approaches the transition temperature and the effects of thermal fluctuations increase, the vortex lattice melts and moves towards a vortex–liquid state (see, e.g., Ling et al. [200]). For the numerical analysis of vortexex in thermal equilibrium, by taking into account fluctuation, see, e.g., Deang et al. [96], Sasik et al. [259] and the references therein. It is then clear that it, in order to study the stability, dynamics and other properties of the vortex state, is very important to take into account the effects

10.1 Introduction

343

of thermal fluctuation and material defects on the vortex dynamics in typeII superconductors. In this study we consider, for thermal fluctuation, only the additive noise Ginzburg–Landau model, which is a simple modification of the Ginzburg–Landau model (10.1). More precisely, we introduce into the right-hand side of the first equation of (10.1) a complex-valued field in time and space. In this way, thermal fluctuations are modeled by deterministic Langevin-type dynamics. Thus, the additive deterministic noise Ginzburg– Landau model is given by ∂φ − iηκdiv(ρU )φ + b(U ).(ρb(U ))(φ) + ρG(φ) = ρβ on Q, ∂t ∂U ρ + curl(ρcurl(U )) − ∇(div(ρU )) ∂t +ρR(b(U )(φ)φ) = curl(ρH) on Q, ηρ

subject to the Robin-type boundary conditions

(10.5)

1 ∂φ = μφ, U.n = 0, curl(U ) = H on Σ, κ2 ∂n and the initial conditions φ(0) = φ0 , U (0) = U0 on Ω, where β is a deterministic complex-valued field in time and space which is temperature dependent (it is depending on ratio between the temperature and the critical transition temperature Tc ).3 The choice of a deterministic noise is motivated by the fact that, if we consider the individual motion of a molecule from a microscopic point of view, the Brownian motion is a deterministic motion (see Shimizu and Morioka [269] and Shimizu and Yaghi [270]). Here, we consider robust control problems, for the modified TDGL models (MTDGL) with Robin boundary conditions on the phase-field variable, which describes the phase transitions taking place in superconductor films, in order to take into account the influence of data noise. Indeed, such perturbations (noise) have the effect of impeding the ability of the material to become superconducting. 10.1.1 Assumptions and Notation We denote by Vn = {U ∈ H 1 (Ω); U.n = 0 on Ω} and Vn the dual of Vn . We denote by Vn ,Vn the duality product between Vn and Vn . For any pair of real number r, s ≥ 0, we introduce the Sobolev space H r,s (Q) defined by H r,s (Q) = L2 (0, T ; H r (Ω)) ∩ H s (0, T ; L2 (Ω)), which is a Hilbert space normed by

1/2 T 2 2 v H r (Ω) dt+ v H s (0,T ;L2 (Ω)) , 0

3

β is said to be the chaotic-type force.

344


where H s (0, T ; L2(Ω)) denotes the Sobolev space of order s of functions defined on (0, T ) and taking values in L2 (Ω). Remark 10.1. According to Girault and Raviart[134], we have the following embedding inequality on Vn : U 2H 1 ≤ C( U 2L2 + div(U ) 2L2 + curl(U ) 2L2 ), ∀U ∈ Vn .

(10.6) ♦

Since the unknown function φ is complex valued, for the mathematical setting, we are naturally led here to introduce complex valued spaces (this is the only occasion in this book in which such spaces appear). If X denotes some Banach space of real-valued functions, the corresponding space (its complexified space) of complex-valued functions will be denoted by X and the corresponding space of vector-valued functions, each of components belonging to X, will be denoted by X, and we use . X to denote the norms of spaces X, X or X . For example L2 (Ω) is the complexified space of L2 (Ω) and L2 (Ω) is the corresponding space of vector-valued functions, of L2 (Ω). Remark 10.2. The norm on L2 spaces is denoted here by . L2 (Ω) or . L2 , to avoid any confusion with the modulus | . |, which is frequently used. ♦ We can now introduce the following spaces Wn = L2 (0, T ; Vn ) ∩ H 1 (0, T ; Vn ), En = L2 (0, T ; Vn ) ∩ L∞ (0, T ; L2 (Ω)), W = L2 (0, T ; H1 (Ω)) ∩ H 1 (0, T ; (H1 ) (Ω)),

(10.7)

E = L2 (0, T ; H1 (Ω)) ∩ L∞ (0, T ; L2 (Ω)), L2∞ (D) = {φ ∈ L2 (D); | φ | is bounded a.e. in D}, where D = Ω or Q. Remark 10.3. (i) W and Wn are continuously embedded into C([0, T ]; L2(Ω)) and C([0, T ]; L2 (Ω)), respectively (see Lemma 6.6). (ii) Although L2∞ is a subset of L∞ , we have described this space using the ♦ standard norm of the space L2 . The weak formulation associated with problem (10.5) is then to find (φ, U ) ∈ W × Wn such that (a.e. t ∈ (0, T )) ∂φ η ρ qdx − iηκ div(ρU )φqdx + ρb(U )(φ)b(U )(q)dx Ω ∂t Ω Ω −iμ ρφqdΓ + ρG(φ)qdx = ρβqdx ∀q ∈ H1 (Ω), Γ Ω Ω ∂U (10.8) vdx + ρcurl(U )curl(v)dx + div(ρU )div(v)dx ρ Ω ∂t Ω Ω + ρR(b(U )(φ)φ)vdx = ρHcurl(v)dx ∀v ∈ Vn , Ω

φ(0) = φ0 , U (0) = U0 on Ω.

Ω

10.2 Existence and Uniqueness of the Solution of the MTDGL Model

345

10.1.2 Preliminary Results Lemma 10.4. (i) For all z ∈ C, l we have that G(z)z ∈ IR. (ii) For all (z1 , z2 ) ∈ Cl 2 , we have that G(z1 ) − G(z2 ) = (| z1 |2 + | z2 |2 −ϑ)(z1 − z2 ) + z1 z2 (z1 − z2 ). Lemma 10.5. For (ϕ, u) and (ψ, v) sufficiently regular, we have: (i) b(u)(ϕ) = b(v)(ϕ) + (u − v)ϕ (ii) b(u)(ϕ) − b(u)(ψ) = b(u)(ϕ − ψ) 1 2 (iii) b(u)(ϕ).b(u)(ϕ) = 2 | ∇ϕ |2 + | u |2 | ϕ |2 − I(ϕ∇ϕ).u. κ κ The proof of the previous lemmas are immediate.

Lemma 10.6. For (u, v, w, X) sufficiently regular, we have: (i) u H 1 v L4 X L4 ≤ C1 u 2H 1 v L2 +δ ∇X 2L2 +C2 ( v H 1 + v 2H 1 ) X 2L2 , with δ chosen suitably at each situation (ii) u L4 v L4 w L4 X L4 ≤ C1 u 2L4 v 2L4 +γ ∇X 2L2 +C2 ( w 4L4 + w 2L4 ) X 2L2 , with γ chosen suitably at each situation. Proof. For the prove of this lemma, see Belmiloudi [45].

10.2 Existence and Uniqueness of the Solution of the MTDGL Model The following results concern the existence and uniqueness of the solution of the modified Ginzburg–Landau model with Robin-type boundary conditions on the phase-field parameter (10.5). Theorem 10.7. For any function ϑ ∈ L2 (Ω) such that 0 < M1 ≤ ϑ ≤ M2 < 1 a.e. in Ω, H ∈ L2 (Q), and (φ0 , U0 , β) ∈ L2∞ (Ω)×L2 (Ω)×L2∞ (Q) satisfying | φ0 |≤ 1 a.e. in Ω and | β |< 1 − M2 a.e. in Q, there exists a unique solution (φ, U ) ∈ (W ∩ L∞ (Q)) × Wn of (10.8) satisfying | φ |≤ 1 a.e. in Q. Moreover the following estimate holds: φ 2W + U 2Wn ≤ C( H 2L2 (Q) + φ0 2L2 (Ω) + U0 2L2 (Ω) + β 2L2 (Q) ). Proof. The proof of this theorem is obtained by using the same technique as in Belmiloudi [45] (see also Hoffmann et al. [79, 81]). The existence of weak solutions and the estimate | φ |≤ 1 are obtained by constructing approximate

346


solutions of System (10.8) by the semidiscretized time approximation technique, by using the maximum principle (see Remark 10.8) and by employing standard convergence arguments. The uniqueness of the solution and the estimate given in the theorem can be obtained by the standard energy estimate arguments and Gronwall’s formula. So, we omit the details. Remark 10.8. (i) The estimate | φ |≤ 1 can be obtained by using the maximum principle as follows: by choosing q = φ(| φ |2 −1)+ = φr+ in the first part of (10.8) and by taking the real part we have (Ω + = {x ∈ Ω : | φ |2 −1 > 0}): ηρ0 d r+ 2L2 + ρ | φ |2 (| φ |2 −ϑ)r+ dx ≤ ρ | β || φ | r+ dx, 2 dt Ω+ Ω+ and then ηρ0 d r+ 2L2 + 2 dt

ρ | φ | (r ) dx ≤ 2

Ω

+ 2

ρ(| β | −(1 − ϑ) | φ |) | φ | r+ dx.

Ω+

Since | β |≤ 1− ϑ L∞ and ϑ L∞ < 1 then | β | −(1 − ϑ) | φ |≤ 0 on Ω + . Consequently, d r+ 2L2 /dt ≤ 0. By integrating with respect to time and by using the fact that (| φ0 |2 −1)+ = 0 (since | φ0 |≤ 1) we can deduce that | φ |≤ 1 a.e. in Q. (ii) Throughout the chapter, we suppose that the hypotheses of Theorem 10.7 are satisfied, to ensure that the solution of problem (10.5), is in (W ∩L∞ (Q))× ♦ Wn .

10.3 The Perturbation Problem 10.3.1 Formulation of the Perturbation Problem In the following, the solution (φ, U ) of problem (10.5) will be treated as the target function. We are then interested in the robust regulation of the deviation of the problem from the desired target (φ, U ). We analyze the full non-linear equation which models large perturbations (ϕ, u) to the target (φ, U ), i.e., we assume that (φ, U ) satisfies the problem (10.5) with the data (φ0 , U0 , H, ξ) and (φ + ϕ, U + u) satisfies the problem (10.5) with the data (φ0 + ϕ0 , U0 + u0 , h + H, ξ + β). Hence, we consider the following system (for (φ, U ) given satisfying the regularity of Theorem 10.7): ηρ

∂ϕ − iηκdiv(ρ(u + U ))ϕ − iηκdiv(ρu)φ ∂t −b(U ).(ρb(U ))(φ) + b(u + U ).(ρb(u + U ))(ϕ + φ) +ρ(G(φ + ϕ) − G(φ)) = ρξ on Q,

ρ

∂u + curl(ρcurl(u)) − ∇(div(ρu)) ∂t +ρR(b(u + U )(ϕ + φ)(ϕ + φ) − b(U )(φ)φ) = curl(ρh) on Q,

(10.9)

10.3 The Perturbation Problem

347

subject to the Robin-type boundary conditions 1 ∂ϕ = μϕ, u.n = 0, curl(u) = h on Σ, κ2 ∂n and the initial conditions

(10.10)

ϕ(0) = ϕ0 , u(0) = u0 on Ω. If we set F (ϕ) = G(ϕ + φ) − G(φ), B(u) = b(U + u) then (10.9) and (10.10) are reduced to ∂ϕ − iηκdiv(ρ(u + U ))ϕ − iηκdiv(ρu)φ + ρF (ϕ) ηρ ∂t +B(u).(ρB(u))(ϕ + φ) = B(0).(ρB(0))(φ) + ρξ on Q, ρ

∂u + curl(ρcurl(u)) − ∇(div(ρu)) + ρR(B(u)(ϕ + φ)(ϕ + φ)) ∂t = ρR(B(0)(φ)φ) + curl(ρh) on Q,

(10.11)

subject to the Robin-type boundary conditions 1 ∂ϕ = μϕ, u.n = 0, curl(u) = h on Σ, κ2 ∂n and the initial conditions ϕ(0) = ϕ0 , u(0) = u0 on Ω. Now we give the weak formulation associated with problem (10.11). Multiplying the first part of (10.11) by q ∈ H1 (Ω) and the second part by v ∈ Vn and integrating over Ω this gives (according to the third part of (10.11)) the weak formulation (a.e. t ∈ (0, T )) ∂ϕ η ρ qdx − iηκ div(ρ(u + U ))ϕqdx − iηκ div(ρu)φqdx Ω ∂t Ω Ω −iμ ρϕqdΓ + (ρB(u)(ϕ + φ)B(u)(q)dx Ω Γ + ρF (ϕ)qdx = ρB(0)(φ)B(0)(q)dx + ρξqdx, Ω Ω Ω ∂u (10.12) ρ vdx + ρcurl(u)curl(v)dx + div(ρu)div(v)dx ∂t Ω Ω Ω + ρR((ϕ + φ)B(u)(ϕ + φ))vdx Ω = ρR(φB(0)(φ))vdx + ρhcurl(v)dx, Ω

Ω

(ϕ(0), u(0)) = (ϕ0 , u0 ). 10.3.2 Existence and Stability Results Now we show the existence and uniqueness of the solution to problem (10.12),

348


and give some Lipschitz continuity results. Theorem 10.9. (i) For any (ϕ0 , u0 ) ∈ L2∞ (Ω) × L2 (Ω), ξ ∈ L2∞ (Q) and h ∈ L2 (Q), there exists a unique (ϕ, u) ∈ (W ∩ L∞ (Q)) × Wn solution of (10.12). Moreover, the following estimate holds: ϕ 2W + u 2Wn ≤ C( h 2L2 (Q) + ϕ0 2L2 (Ω) + u0 2L2 (Ω) + ξ 2L2 (Q) ). (ii) Let (u01 , ϕ01 , h1 , ξ1 ) and (u02 , ϕ02 , h2 , ξ2 ) be functions of the space L2 (Ω)× L2∞ (Ω) × L2 (Q) × L2∞ (Q). If (u1 , ϕ1 ) ∈ Wn × (W ∩ L∞ (Q)) (respectively (u2 , ϕ2 ) ∈ Wn × (W ∩ L∞ (Q))) is the solution of (10.12) with the given data (ϕ01 , u01 , h1 , ξ1 ) (respectively (ϕ02 , u02 , h1 , ξ2 )), then the following Lipshitz continuity result holds: ϕ 2W + u 2Wn ≤ C( ϕ0 2L2 + u0 2L2 + h 2L2 (Q) + ξ 2L2 (Q) ), where ϕ = ϕ1 − ϕ2 , u = u1 − u2 , ϕ0 = ϕ01 − ϕ02 , u0 = u01 − u02 , h = h1 − h2 and ξ = ξ1 − ξ2 . Proof. The proof of this result can be obtained by using a technique that is similar to the one used in Belmiloudi [45]. So, we omit the details. We are now going to study the differentiability of the operator solution of problem (10.12).

10.4 Differentiability of the Operator Solution Before proceeding to the investigation of the F-differentiability of the function F : (ϕ0 , u0 , h, ξ) −→ (ϕ, u), which maps the source term (ϕ0 , u0 , h, ξ) ∈ L2∞ (Ω) × L2 (Ω) × L2 (Q) × L2∞ (Q) of problem (10.12) into the corresponding solution (ϕ, u) ∈ E × En , we study, for (ψ0 , w0 , λ, k) be given data, the following problem (PI ): find (ψ, w) ∈ E × En such that (∀(q, v) ∈ H1 (Ω) × Vn and a.e. t ∈ (0, T )): ∂ψ qdx − iηκ div(ρU1 )ψqdx − iηκ div(ρw)φ1 qdx Ω ∂t Ω Ω −iμ ρψqdΓ + ρ(B(u)(ψ) + φ1 w)B(u)(q)dx Ω Γ + ρB(u)(φ1 )wqdx + ρ((2 | φ1 |2 −ϑ)ψ + φ21 ψ)qdx Ω Ω = ρλqdx,

η

ρ

Ω

10.4 Differentiability of the Operator Solution

349

∂w vdx + ρcurl(w)curl(v)dx + div(ρw)div(v)dx Ω ∂t Ω Ω + ρR(ψB(u)(φ1 ))vdx + ρ(R(φ1 B(u)(ψ)) + w | φ1 |2 )vdx Ω Ω = ρkcurl(v)dx,

ρ

Ω

(ψ(0), w(0)) = (ψ0 , w0 ), where U1 = U + u and φ1 = φ + ϕ. Remark 10.10. The problem (PI ) is the weak formulation of the problem: ηρ

∂ψ − iηκ(div(ρU1 )ψ + div(ρw)φ1 ) ∂t +B(u).(ρ(B(u)(ψ) + φ1 w)) + ρB(u)(φ1 )w +ρ((2 | φ1 |2 −ϑ)ψ + φ21 ψ) = ρλ on Q,

ρ

∂w + curl(ρcurl(w)) + ∇(div(ρw)) + ρR(ψB(u)(φ1 )) ∂t +ρ(R(φ1 B(u)(ψ)) + w | φ1 |2 ) = curl(ρk) on Q,

(10.13)

subject to the Robin-type boundary conditions 1 ∂ψ = μψ, w.n = 0, curl(w) = k on Σ, κ2 ∂n and the initial conditions (ψ(0), w(0)) = (ψ0 , w0 ) on Ω. Theorem 10.11. If (u, ϕ) and (U, φ) are in Wn ×(W ∩L∞ (Q)), the following results hold: (i) For any (ψ0 , w0 , k, λ) ∈ L2∞ (Ω) × L2 (Ω) × L2 (Q) × L2∞ (Q), there exists a unique couple of functions (ψ, w) ∈ E × En , solution of problem (PI ), such that ψ 2E + w 2En (10.14) ≤ Ce ( ψ0 2L2 + w0 2L2 + k 2L2 (Q) + λ 2L2 (Q) ). (ii) Let (ψ0i , w0i , ki , λi ), i = 1, 2 be in L2∞ (Ω) × L2 (Ω) × L2 (Q) × L2∞ (Q). If (ψi , wi ) is the solution of (PI ), corresponding to data (ψ0i , w0i , ki , λi ) for i = 1, 2, respectively, then ψ 2E + w 2En ≤ Ce ( ψ0 2L2 + w0 2L2 + k 2L2 (Q) + λ 2L2 (Q) )

(10.15)

where w = w1 − w2 , ψ = ψ1 − ψ2 , w0 = w01 − w02 , ψ0 = ψ01 − ψ02 , k = k1 − k2 and λ = λ1 − λ2 .

350


Proof. The existence, uniqueness and Lipschitz continuity results of problem (PI ) are obtained in the same way as used to prove Theorem 10.9 and by using the regularity of (U1 , φ1 ). For more details see Belmiloudi [45]. We are now going to study the F-differentiability of the operator solution F . For simplicity, we denote by X the data X := (ϕ0 , u0 , h, ξ), by Z the space Z := L2 (Ω) × L2 (Ω) × L2 (Q) × L2 (Q) and by Z∞ the space Z∞ := L2∞ (Ω) × L2 (Ω) × L2 (Q) × L2∞ (Q). The space Z and Z∞ are equipped with the norm X Z = ( ϕ0 2L2 + u0 2L2 + h 2L2 (Q) + ξ 2L2 (Q) )1/2 for all X = (ϕ0 , u0 , h, ξ) in Z (or in Z∞ ). Theorem 10.12. Let X = (ϕ0 , u0 , h, ξ) and Y = (p0 , q0 , z, π) be in Z∞ with F (X) and F (X + Y ) being the corresponding solutions of (10.12). Then F (X) − F(X + Y ) − F (X).Y E×En ≤ C Y Z , 3/2

(10.16)

where F (X) : Z∞ −→ E ×En is a linear operator such that (ψ, w) = Fϕ (X).Y is the solution of the problem (PI ) with the initial condition (ψ, w)(t = 0) = (p0 , q0 ) and the forcing (k, λ) = (z, π). Moreover, for all Xi = (ϕ0i , u0i , hi , ξi ) ∈ Z∞ , for i = 1, 2, we have the following estimate: F (X1 ).Y − F (X2 ).Y 2E×En ≤ Ce ( Y Z X 2Z + Y 2Z X Z ),

(10.17)

where ϕ0 = ϕ01 − ϕ02 , u0 = u01 − u02 , h = h1 − h2 , ξ = ξ1 − ξ2 and X = X1 − X2 = (ϕ0 , u0 , h, ξ). Proof. The proof of this theorem can be obtained by using the same technique as used to prove the results of Theorem 4.3 of Belmiloudi [45]. So, we omit the details.

10.5 Robust Control Problems The objective of the robust control problem is to find the best admissible control in the presence of the worst disturbance which maximally spoils the control objective. We formulate the problem for two situations: first, where the control is in the external magnetic field and, second, where the control is in the initial condition u0 (data assimilation). 10.5.1 Control in the External Magnetic Field In this section, we consider two situations: first, where the worst disturbance is in the chaotic-type force ξ and, second, where the disturbance is in the external magnetic field h.

10.5 Robust Control Problems

351

Distributed Disturbance in the Chaotic-Type Force We suppose now that the control is in the external magnetic field h and the disturbance is in the force ξ, that is, h = g (g ∈ L2 (Q)) and ξ = f (f ∈ L2∞ (Q)). Therefore, the function (ϕ, u) is assumed to be related to the disturbance f and control g through the problem (10.12) (∀(q, v) ∈ H1 (Ω)×Vn and a.e. t ∈ (0, T )): ∂ϕ η ρ qdx − iηκ div(ρ(u + U ))ϕqdx − iηκ div(ρu)φqdx Ω ∂t Ω Ω ρϕqdΓ + (ρB(u)(ϕ + φ)B(u)(q)dx −iμ Γ Ω + ρF (ϕ)qdx = ρB(0)(φ)B(0)(q)dx + ρf qdx, Ω Ω Ω ∂u (10.18) ρ vdx + ρcurl(u)curl(v)dx + div(ρu)div(v)dx Ω ∂t Ω Ω + ρR((ϕ + φ)B(u)(ϕ + φ))vdx Ω = ρR(φB(0)(φ))vdx + ρgcurl(v)dx, Ω

Ω

(ϕ(0), u(0)) = (ϕ0 , u0 ). To obtain the regularity of Theorem 10.9, we assume that (ϕ0 , u0 ) is in L2∞ (Ω) × L2 (Ω). Let P : (g, f ) −→ (ϕ, u) = P(g, f ) be the map: L2 (Q) × L2∞ (Q) −→ E × En defined by (10.18) and introducing the cost function defined by J(g, f ) =

a b | ϕ |2 −Λ 2L2 (Q) + u − uobs 2L2 (Q) 4 2 α γ 2 + g L2 (Q) − f 2L2 (Q) , 2 2

(10.19)

where a, b, α, γ are fixed such that α, γ > 0,4 a, b ≥ 0 and a + b > 0. The functions uobs ∈ L2 (Q) and Λ ∈ L∞ (Q) are given and represent the observation. Let K = K1 ×K2 such that K1 and K2 are given non-empty, closed, convex, bounded subsets of L2 (Q) and L2∞ (Q), respectively. We want to minimize the functional J with respect to g and maximize J with respect to f , i.e., to study the following problem (MP 1 ): find an admissible control g ∗ ∈ K1 and a disturbance f ∗ ∈ K2 such that: (g ∗ , f ∗ ) is a saddle point of the functional J on K, subject to system (10.18). 4

The parameters α can be interpreted as the price of the control to the engineer and the parameter γ as the price of the disturbance to the nature.

352


Proposition 10.13. The function P is continuously F-differentiable from L2 (Q)×L2∞ (Q) to E ×En with the derivative P (g, f ) : Y = (β1 , β2 ) −→ (ψ, w) given by the linear problem (PF 1 ) (∀(q, v) ∈ H1 (Ω) × Vn and a.e. t ∈ (0, T )): ∂ψ qdx − iηκ div(ρU1 )ψqdx − iηκ div(ρw)φ1 qdx η ρ Ω ∂t Ω Ω ρψqdΓ + ρ(B(u)(ψ) + φ1 w)B(u)(q)dx −iμ Ω Γ + ρB(u)(φ1 )wqdx + ρ((2 | φ1 |2 −ϑ)ψ + φ21 ψ)qdx Ω Ω = ρβ2 qdx, Ω ∂w ρ vdx + ρcurl(w)curl(v)dx + div(ρw)div(v)dx Ω ∂t Ω Ω + ρR(ψB(u)(φ1 ))vdx + ρ(R(φ1 B(u)(ψ)) + w | φ1 |2 )vdx Ω Ω = ρβ1 curl(v)dx, Ω

(ψ(0), w(0)) = (0, 0), where (U1 , φ1 ) = (u + U, ϕ + φ). Moreover, for all (fi , gi ) ∈ L2 (Q) × L2∞ (Q), for i = 1, 2, we have the following estimates: (i) P (g1 , f1 ) L(L2 (Q)×L2 (Q),E×En ) ≤ Ce (ii) P (g1 , f1 )Y − P (g2 , f2 )Y 2E×En ≤ Ce ( X L2 (Q)×L2 (Q) Y 2L2 (Q)×L2 (Q) + X 2L2 (Q)×L2 (Q) Y L2 (Q)×L2 (Q) ), where f = f1 − f2 , g = g1 − g2 and X = (g, f ). Proof. The proof of this proposition is the consequence of the result of Theorem 10.12. Here we omit the details. Proposition 10.14. The mapping P defined by (10.18) is continuous from the weak topology of L2 (Q) × L2∞ (Q) to the strong topology of L2 (Q) × L2 (Q). Proof. Let f = (g, f ) be given in L2 (Q) × L2∞ (Q) and let be a sequence fk = (gk , fk ) such that fk is weakly convergent in L2 (Q) × L2 (Q) to f . Set (ϕ, u) = P(g, f ) and (ϕk , uk ) = P(gk , fk ). Since fk f weakly in L2 (Q) × L2 (Q) then fk is uniformly bounded in L2 (Q) × L2 (Q). In view of Theorem 10.9, we can deduce that the sequence (ϕk , uk ) is uniformly bounded in E × En . Therefore, we can extract from (fk , ϕk , uk ) a subsequence also denoted by (fk , ϕk , uk ) and such that


(gk , fk ) (g, f ) weakly in L2 (Q) × L2 (Q), (ϕk , uk ) (ϕ, ˜ u˜) weakly in E × En , (ϕk , uk ) −→ (ϕ, ˜ u ˜) strongly in L2 (Q) × L2 (Q), ϕk ϕ˜ weakly in L2 (Σ).

353

(10.20)

We can easily prove that (ϕ, ˜ u ˜) = P(g, f ) and according to the uniqueness of the solution of (10.18), we then haveϕ˜ = ϕ and u ˜ = u. Theorem 10.15. For α and γ sufficiently large (i.e., there exists (αl , γl ) such that α ≥ αl and γ ≥ γl ) there exists (g ∗ , f ∗ ) ∈ K and (ϕ∗ , u∗ ) ∈ E × En such that (g ∗ , f ∗ ) is defined by (MP 1 ) and (ϕ∗ , u∗ ) = P(g ∗ , f ∗ ) is a solution of (10.18). Proof. Let Pf be the map: g −→ J(g, f ) and Qg be the map: f −→ J(g, f ). To obtain the existence of the robust control problem we prove that Pf is convex and lower semi-continuous for all f ∈ K2 , and Qg is concave and upper semicontinuous for all g ∈ K1 and we use minimax duality theorems for infinite dimensions presented in Chapter 5. First, we prove, for α and γ sufficiently large, the convexity of the map Pf and the concavity of the map Qg . In order to prove the convexity, it is sufficient to show that for all (g1 , g2 ) ∈ K1 we have (Pf (g1 ) − Pf (g2 )).g ≥ 0, where g = g1 − g2 (because Pf is G-differentiable). According to the definition of J, we have that (Pf (g1 ) − Pf (g2 )).g

2L2 (Q)

=αg +a R((| ϕ1 |2 − | ϕ2 |2 )ψ2 ϕ2 )dxdt Q +a R((| ϕ1 |2 −Λ)(ϕ1 − ϕ2 )ψ 1 )dxdt Q +b (u1 − u2 )w2 dxdt + b (u1 − uobs )(w1 − w2 )dxdt Q Q +a R((| ϕ1 |2 −Λ)(ψ1 − ψ2 )ϕ2 )dxdt,

(10.21)

Q

where (ϕi , ui ) = P(gi , f ) and (ψi , wi ) = P (gi , f ).(g, 0) (solution of problem (PF 1 )), for i = 1, 2. According to Theorem 10.9 and Proposition 10.13 we have (R((| ϕ1 |2 − | ϕ2 |2 )ψ2 ϕ2 )dxdt a Q +a R((| ϕ1 |2 −Λ)(ϕ1 − ϕ2 )ψ 1 ))dxdt Q (10.22) (u1 − u2 )w2 dxdt +b Q

≤ c1 ϕ L2 (Q) ( ψ1 L2 (Q) + ψ2 L2 (Q) ) +c2 u L2 (Q) w2 L2 (Q) ≤ C0 g 2L2 (Q)

354


and a

R((| ϕ1 |2 −Λ)((ψ1 − ψ2 )ϕ2 )dxdt +b (u1 − uobs )(w1 − w2 )dxdt

Q

Q

≤ c3 | ϕ1 |2 −Λ L2 (Q) ψ L2 (Q)

(10.23)

+c4 u1 − uobs L2 (Q) w L2 (Q) 3/2

≤ C1 g L2 (Q) , where u = u1 − u2 , ϕ = ϕ1 − ϕ2 , w = w1 − w2 and ψ = ψ1 − ψ2 . From (10.21), (10.22) and (10.23) we deduce that, for α ≥ αl such that 1/2 αl > C0 and (αl − C0 ) min g L2 (Q) = C1 , we have (Pf (g1 ) − Pf (g2 )).g ≥ 0 g∈K1

and then the convexity of Pf . In the same way, we can find γl such that for γ ≥ γl we have the concavity of Qg . We shall prove now that Pf is lower semi-continuous for all f ∈ K2 , and Qg is upper semi-continuous for all g ∈ K1 . Let gk be a minimizing sequence of J, i.e., lim inf J(gk , f ) = min J(g, f ) (∀f ∈ K2 ). Then gk is uniformly bounded k

g∈K1

in K1 and we can extract from gk a subsequence also denoted by gk such that gk gf weakly in K1 . By using Proposition 10.14 we have then P(gk , f ) −→ (ϕf , uf ) strongly in L2 (Q) × L2 (Q).

(10.24)

Therefore, since the norm is lower semi-continuous we have that the map Pf : g −→ J(g, f ) is lower semi-continuous for all f ∈ K2 . By using the same technique we obtain that Qg is upper semi-continuous for all g ∈ K1 . In order to obtain the necessary optimality conditions which have been satisfied by the solution of the robust control problem, we introduce the following adjoint problem corresponding to the primal problem (10.18) (we denote by (ϕ, u) = P(g, f ) and (φ1 , U1 ) = (ϕ + φ, u + U )): find (P, Q) ∈ E × En such that (∀(q, v) ∈ H1 (Ω) × Vn and a.e. t ∈ (0, T )): ∂P qdx − iηκ div(ρU1 )P qdx − iμ ρP qdΓ −η ρ Ω ∂t Ω Γ + ρB(u)(P )B(u)(q)dx + ρB(u)(φ1 )Qqdx Ω Ω −i + ( div(ρφ1 Q) + ρU1 φ1 Q)qdx (10.25) κ Ω 2 + ρ((2 | φ1 |2 −ϑ)P + φ1 P )qdx = a (| ϕ |2 −Λ)ϕqdx, Ω Ω ∂Q vdx + ρcurl(Q)curl(v)dx + div(Q)div(ρq)dx − ρ Ω ∂t Ω Ω


355

+

ρR(iηκ∇(φ1 P ) + P B(u)(φ1 ) + φ1 B(u)(P ))vdx Ω 2 + ρQ | φ1 | vdx = b (u − uobs )vdx, Ω

Ω

(P (T ), Q(T )) = (0, 0). Remark 10.16. (i) The adjoint problem (10.25) is a linear system. By reversing sense of time, i.e., t := T − t, and by applying the same way as to obtain the result of Theorem 10.11 we obtain the existence and uniqueness of (P, Q). (ii) The system (10.25) is the weak formulation of the following problem: −ηρ

−ρ

∂P − iηκdiv(ρU1 )P + B(u).(ρB(u)(P )) + ρB(u)(φ1 )Q ∂t i 2 − div(ρφ1 Q) + ρU1 φ1 Q + ρ((2 | φ1 |2 −ϑ)P + φ1 P ) κ = a(| ϕ |2 −Λ)ϕ on Q,

∂Q + curl(ρcurl(Q)) − ρ∇(div(Q)) + ρQ | φ1 |2 ∂t +ρR(iηκ∇(φ1 P ) + P B(u)(φ1 ) + φ1 B(u)(P ))

(10.26)

= b(u − uobs ) on Q, 1 ∂P = μP, Q.n = 0, curl(Q) = 0 on Σ, κ2 ∂n (P (T ), Q(T )) = (0, 0).

♦

We can now give the first-order optimality conditions for the robust control problem (MP 1 ). Theorem 10.17. Under the assumptions of Theorem 10.15, the optimal solution (g ∗ , f ∗ , u∗ , ϕ∗ ) ∈ K × En × E such that (g ∗ , f ∗ ) is defined by (MP 1 ) and (ϕ∗ , u∗ ) = P(g ∗ , f ∗ ) solution of (10.18), satisfies (ρcurl(Q∗ ) + αg ∗ )(g − g ∗ )dxdt ≥ 0, Q (10.27) R((ρP ∗ − γf ∗ )(f − f ∗ ))dxdt ≤ 0 ∀(g, f ) ∈ K, Q

∗

where (P , Q∗ ) is the solution of the adjoint problem (10.25) (corresponding to the primal solution (ϕ∗ , u∗ )). Proof. The cost function J is a composition of F-differentiable maps then J is differentiable and we have (∀Y = (β1 , β2 ) ∈ K ) 2 R((| ϕ | −Λ)ϕψ)dxdt + b (u − uobs )wdxdt J (g, f ).Y = a Q Q (10.28) +α gβ1 dxdt − γ R(f β2 )dxdt, Q

Q

356


where (ψ, w) = P (g, f ).Y is the solution of problem (PF 1 ). By taking (q, v) = (P, Q) in (PF 1 ), using Green’s formula and integrating with respect to time, we obtain (according to the initial conditions of (PF 1 )) ∂P ψdxdt + η ρP (T )ψ(T )dx − iηκ −η ρ div(ρU1 )P ψdxdt Q ∂t Ω Q +iηκ ρw∇(φ1 P )dxdt − iμ ρP ψdΓ dt Q Σ + ρB(u)(P )B(u)(ψ)dxdt + ρφ1 wB(u)(P )dxdt Q Q + ρ((2 | φ1 |2 −ϑ)P ψ + φ21 P ψ)dxdt Q ρB(u)(φ1 )wP dxdt = ρβ2 P dxdt, + (10.29) Q Q ∂Q wdxdt + ρQ(T )w(T )dx + ρ ρcurl(Q)curl(w)dxdt − Q ∂t Ω Q + div(Q)div(ρw)dxdt + ρR(ψB(u)(φ1 ))Qdxdt Q Q + ρ(R(φ1 B(u)(ψ))Q + w | φ1 |2 Q)dxdt Q ρβ1 curl(Q)dxdt. = Q

Since (P, Q) is a solution of (10.25), with null final conditions, we have that ρw∇(φ1 P )dxdt + ρφ1 wB(u)(P )dxdt iηκ Q Q 2 + ρB(u)(φ1 )wP dxdt + ρ(φ21 P ψ − φ1 P ψ)dxdt Q Q −i − ρB(u)(φ1 )Qψdxdt − ( div(ρφ1 Q) + ρU1 φ1 Q)ψdxdt κ Q Q 2 +a (| ϕ | −Λ)ϕψdxdt = ρβ2 P dxdt, (10.30) Q Q ρR(ψB(u)(φ1 ) + φ1 B(u)(ψ))Qdxdt + b (u − uobs )wdxdt Q Q − ρR(iηκ∇(φ1 P ) + P B(u)(φ1 ) + φ1 B(u)(P ))wdxdt Q = ρβ1 curl(Q)dxdt. Q

By adding the real part of the first part of (10.30) and the second part of (10.30), we obtain


R((| ϕ |2 −Λ)ϕψ)dxdt + b (u − uobs )wdxdt Q = ρβ1 curl(Q)dxdt + ρR(β2 P )dxdt,

a

Q

Q

since

357

(10.31)

Q

−i div(ρφ1 Q)+ρU1 φ1 Q)ψdx = Ω κ

(

ρφ1 B(u)(ψ)Qdx (because Q.n = 0), Ω

2

R(φ21 P ψ − φ1 P ψ) = 0 and R(−B(u)(φ1 )Qψ + B(u)(φ1 )Qψ) = 0. According to the expression of J (g, f ).Y we can deduce that (ρcurl(Q) + αg)β1 dxdt + R((ρP − γf )β2 )dxdt. (10.32) J (g, f ).Y = Q

∗

Q

∗

Since (f , g ) is an optimal solution we have ∂J ∗ ∗ ∂J ∗ ∗ (g , f ).(g − g ∗ ) ≥ 0, (g , f ).(f − f ∗ ) ≤ 0 ∀(g, f ) ∈ K ∂g ∂f and then

(10.33)

(ρcurl(Q∗ ) + αg ∗ )(g − g ∗ )dxdt ≥ 0,

Q

R((ρP ∗ − γf ∗ )(f − f ∗ ))dxdt ≤ 0

Q

(10.34) ∀(g, f ) ∈ K.

This completes the proof. Distributed Disturbance in the External Magnetic Field

In this section, the external magnetic field h is assumed to be decomposed into disturbance f ∈ L2 (Q) and the control g ∈ L2 (Q), i.e., h = f +g ∈ L2 (Q). So, the function (ϕ, u) is assumed to be related to the disturbance f and control g through the problem (10.12) (∀(q, v) ∈ H1 (Ω) × Vn and a.e. t ∈ (0, T )) ∂ϕ η ρ qdx − iηκ div(ρ(u + U ))ϕqdx − iηκ div(ρu)φqdx Ω ∂t Ω Ω −iμ ρϕqdΓ + (ρB(u)(ϕ + φ)B(u)(q)dx Ω Γ + ρF (ϕ)qdx = ρB(0)(φ)B(0)(q)dx + ρξqdx, Ω Ω Ω ∂u (10.35) ρ vdx + ρcurl(u)curl(v)dx + div(ρu)div(v)dx ∂t Ω Ω Ω + ρR((ϕ + φ)B(u)(ϕ + φ))vdx Ω = ρR(φB(0)(φ))vdx + ρ(f + g)curl(v)dx, Ω

(ϕ(0), u(0)) = (ϕ0 , u0 ).

Ω

358


To obtain the regularity of Theorem 10.9, we suppose that (ϕ0 , u0 ) ∈ L2∞ (Ω) × L2 (Ω) and ξ ∈ L2∞ (Q). Let P : (g, f ) −→ (ϕ, u) = P(g, f ) be the map: (L2 (Q))2 −→ E × En defined by (10.35) and introducing the cost function defined by J(g, f ) =

b a | ϕ |2 −Λ 2L2 (Q) + u − uobs 2L2 (Q) 4 2 α γ + g 2L2 (Q) − f 2L2 (Q) , 2 2

(10.36)

where α, β, a, b are fixed parameters such that α, β > 0, a, b ≥ 0 and a + b > 0. The pair of functions (uobs , Λ) is in L2 (Q) × L∞ (Q) and represents the observation. In this section we study the following robust control problem (MP 2 ): find a saddle point (g ∗ , f ∗ ) of the functional J on K, subject to (10.35), where K = K1 × K2 such that K1 and K2 are non-empty, closed, convex, bounded subsets of L2 (Q). The proof of the following propositions and existence theorem is similar to that of Propositions 10.13 and 10.14 and Theorem 10.15. Therefore, we will omit the details. Proposition 10.18. The function P is continuously F-differentiable from (L2 (Q))2 to E × En with the derivative P (g, f ) : Y = (β1 , β2 ) −→ (ψ, w) given by the linear problem (PF 2 ) (∀(q, v) ∈ H1 (Ω) × Vn and a.e. t ∈ (0, T )) ∂ψ qdx − iηκ div(ρU1 )ψqdx − iηκ div(ρw)φ1 qdx η ρ Ω ∂t Ω Ω −iμ ρψqdΓ + ρ(B(u)(ψ) + φ1 w)B(u)(q)dx Ω Γ + ρB(u)(φ1 )wqdx + ρ((2 | φ1 |2 −ϑ)ψ + φ21 ψ)qdx = 0, Ω Ω ∂w (10.37) udx + ρcurl(w)curl(v)dx + div(ρw)div(v)dx ρ Ω ∂t Ω Ω + ρR(ψB(u)(φ1 ))vdx + ρ(R(φ1 B(u)(ψ)) + w | φ1 |2 )vdx Ω Ω = ρ(β1 + β2 )curl(v)dx, Ω

(ψ(0), w(0)) = (0, 0), where (U1 , φ1 ) = (u + U, ϕ + φ). Moreover, we have the estimates (∀(gi , fi ) ∈ (L2 (Q))2 , i = 1, 2): (i) P (g1 , f1 ) L((L2 (Q))2 ,E×En ) ≤ Ce (ii) P (g1 , f1 )Y − P (g2 , f2 )Y 2E×En ≤ Ce ( X (L2 (Q))2 Y 2(L2 (Q))2 + X 2(L2 (Q))2 Y (L2 (Q))2 ), where f = f1 − f2 , g = g1 − g2 and X = (g, f ).


359

Proposition 10.19. The map P defined by (10.35) is continuous from the weak topology of (L2 (Q))2 to the strong topology of L2 (Q) × L2 (Q). Theorem 10.20. For α and γ sufficiently large, there exists (g ∗ , f ∗ ) ∈ K and (u∗ , ϕ∗ ) ∈ En × E such that (g ∗ , f ∗ ) is defined by (MP 2 ) and (ϕ∗ , u∗ ) = P(g ∗ , f ∗ ) is a solution of (10.35). Now we establish necessary optimality conditions for the robust control problem (MP 2 ). Theorem 10.21. Under the assumptions of Theorem 10.20, the optimal solution (g ∗ , f ∗ , u∗ , ϕ∗ ) ∈ K × En × E such that (g ∗ , f ∗ ) is defined by (MP 2 ) and (ϕ∗ , u∗ ) = P(g ∗ , f ∗ ) is a solution of (10.35), satisfies (ρcurl(Q∗ ) + αg ∗ )(g − g ∗ )dxdt ≥ 0, Q (10.38) (ρcurl(Q∗ ) − γf ∗ )(f − f ∗ )dxdt ≤ 0 ∀(g, f ) ∈ K, Q

∗

where (P , Q∗ ) is the solution of problem (10.25) (corresponding to (ϕ∗ , u∗ )). Proof. The cost function J is a composition of F-differentiable maps then J is differentiable and we have (∀Y = (β1 , β2 ) ∈ K ) 2 R((| ϕ | −Λ)ϕψ)dxdt + b (u − uobs )wdxdt J (g, f ).Y = a Q Q (10.39) +α gβ1 dxdt − γ f β2 dxdt, Q

Q

where (ψ, w) = P (g, f ).Y is the solution of problem (PF 2 ). By taking (q, v) = (P, Q) in (PF 2 ) and integrating with respect to time we obtain (according to the initial condition) ∂P ψdxdt + η ρP (T )ψ(T )dx − iηκ ρ div(ρU1 )P ψdxdt −η Q ∂t Ω Q +iηκ ρw∇(φ1 P )dxdt − iμ ρψP dΓ dt Σ Q + ρB(u)(P )B(u)(ψ)dxdt Q ρφ1 wB(u)(P )dxdt + ρB(u)(φ1 )wP dxdt + Q Q + ρ((2 | ϕ |2 −ϑ)P ψ + ϕ2 P ψ)dxdt = 0, Q ∂Q wdxdt + ρQ(T )w(T )dx + ρ ρcurl(Q)curl(w)dxdt − Q ∂t Ω Q div(Q)div(ρw)dxdt + ρR(ψB(u)(φ1 ))Qdxdt + Q Q + ρ(R(φ1 B(u)(ψ))Q + w | φ1 |2 Q)dxdt Q

360


=

ρ(β1 + β2 )curl(Q)dxdt.

Q

Since (P, Q) is a solution of (10.25) we have that ρw∇(φ1 P )dxdt + ρφ1 wB(u)(P )dxdt iηκ Q Q 2 + ρB(u)(φ1 )wP dxdt + ρ(φ21 P ψ − φ1 P ψ)dxdt Q Q −i − ( div(ρφ1 Q) + ρU1 φ1 Q)ψdxdt κ Q − ρB(u)(φ1 )Qψdxdt + a (| ϕ |2 −Λ)ϕψdxdt = 0, Q Q ρR(ψB(u)(φ1 ))Qdxdt + ρR(φ1 B(u)(ψ))Qdxdt Q Q − ρR(iηκ∇(φ1 P ) + P B(u)(φ1 ) + φ1 B(u)(P ))wdxdt Q +b (u − uobs )wdxdt = ρ(β1 + β2 )curl(Q)dxdt. Q

(10.40)

Q

By adding the real part of the first part of (10.40) and the second part of (10.40) we obtain (by using Green’s formula) R((| ϕ |2 −Λ)ϕψ)dxdt + b (u − uobs )wdxdt a Q Q = ρ(β1 + β2 )curl(Q)dxdt. Q

According to the expression of J (g, f ).Y we can deduce that J (g, f ).Y = (ρcurl(Q) + αg)β1 dxdt + (ρcurl(Q) − γf )β2 dxdt. Q

Q

Since (g ∗ , f ∗ ) is an optimal solution we then have (ρcurl(Q∗ ) + αg ∗ )(g − g ∗ )dxdt ≥ 0, Q (ρcurl(Q∗ ) − γf ∗ )(f − f ∗ )dxdt ≤ 0 ∀(g, f ) ∈ K.

(10.41)

Q


10.5.2 Control in the Initial Condition of the Vector Potential In this section, we formulate the problem in two situations: first, where the worst disturbance is in the chaotic-type force ξ and, second, where the disturbance is in the external magnetic field h.


361

Distributed Disturbance in the Chaotic-Type Term We suppose that the control is in the initial condition u0 , i.e., u0 = g (g ∈ L2 (Ω)) and the disturbance is in the force ξ, i.e., ξ = f (f ∈ L2∞ (Q)). So the function (ϕ, u) is assumed to be related to the disturbance f and control g through the problem (10.12) (∀(q, v) ∈ H1 (Ω) × Vn and a.e. t ∈ (0, T )): ∂ϕ η ρ qdx − iηκ div(ρ(u + U ))ϕqdx − iηκ div(ρu)φqdx Ω ∂t Ω Ω −iμ ρϕqdΓ + (ρB(u)(ϕ + φ)B(u)(q)dx + ρF (ϕ)qdx Ω Ω Γ = ρB(0)(φ)B(0)(q)dx + ρf qdx, Ω Ω ∂u (10.42) ρ vdx + ρcurl(u)curl(v)dx + div(ρu)div(v)dx Ω ∂t Ω Ω + ρR((ϕ + φ)B(u)(ϕ + φ))vdx Ω = ρR(φB(0)(φ))vdx + ρhcurl(v)dx, Ω

Ω

(ϕ(0), u(0)) = (ϕ0 , g). To obtain the regularity of Theorem 10.9, we suppose that h ∈ L2 (Q) and ϕ0 ∈ L2∞ (Ω). Let P : (g, f ) −→ (ϕ, u) = P(g, f ) be the map: L2 (Ω) × L2∞ (Q) −→ E × En defined by (10.42) and the cost function is defined by J(g, f ) =

a b | ϕ |2 −Λ 2L2 (Q) + u − uobs 2L2 (Q) 4 2 α γ + g 2L2 − f 2L2 (Q) , 2 2

(10.43)

where α, γ > 0, a, b ≥ 0 and a + b > 0. The functions (uobs , Λ) ∈ L2 (Q) × L∞ (Q) are given. In this section we study the following robust control problem (MP 3 ): find a saddle point (g ∗ , f ∗ ) of the functional J on K, subject to (10.42), where K = K1 × K2 such that K1 and K2 are non-empty, closed, convex, bounded subsets of L2 (Ω) and L2∞ (Q), respectively. The arguments of Section 10.5.1 extend directly to the present case without requiring further estimates. We have then the following results. Proposition 10.22. The function P is continuously F-differentiable from L2 (Ω)×L2∞ (Q) to E ×En with the derivative P (g, f ) : Y = (β1 , β2 ) −→ (ψ, w) given by the linear problem (PF 3 ) (∀(q, v) ∈ H1 (Ω) × Vn and a.e. t ∈ (0, T )): ∂ψ qdx − iηκ div(ρU1 )ψqdx − iηκ div(ρw)φ1 qdx η ρ Ω ∂t Ω Ω −iμ ρψqdΓ + ρ(B(u)(ψ) + φ1 w)B(u)(q)dx Γ

Ω

362


+ ρB(u)(φ1 )wqdx + ρ((2 | ϕ |2 −ϑ)ψ + ϕ2 ψ)qdx Ω Ω = ρβ2 qdx, Ω ∂w vdx + ρcurl(w)curl(v)dx + div(ρw)div(v)dx ρ Ω ∂t Ω Ω + ρR(ψB(u)(φ1 ))vdx + ρ(R(φ1 B(u)(ψ)) + w | φ1 |2 )vdx = 0, Ω

Ω

(ψ(0), w(0)) = (0, β1 ), where (U1 , φ1 ) = (u + U, ϕ + φ). Moreover, we have the following estimates, for all (gi , fi ) ∈ L2 (Ω) × 2 L∞ (Q), for i = 1, 2: (i) P (g1 , f1 ) L(L2 (Ω)×L2 (Q),E×En ) ≤ Ce (ii) P (g1 , f1 )Y − P (g2 , f2 )Y 2E×En ≤ Ce ( X L2 ×L2 (Q) Y 2L2 ×L2 (Q) + X 2L2 ×L2 (Q) Y L2 ×L2 (Q) ), where f = f1 − f2 , g = g1 − g2 and X = (g, f ).

Proposition 10.23. The mapping P defined by (10.42) is continuous from the weak topology of L2 (Ω) × L2∞ (Q) to the strong topology of L2 (Q) × L2 (Q). Theorem 10.24. For α and γ sufficiently large (i.e., there exists (αl , γl ) such that α ≥ αl and γ ≥ γl ) there exists (g ∗ , f ∗ ) ∈ K and (ϕ∗ , u∗ ) ∈ E × En such that (g ∗ , f ∗ ) is defined by (MP 3 ) and (ϕ∗ , u∗ ) = P(g ∗ , f ∗ ) is a solution of (10.42). Next we establish necessary optimality conditions for the robust control problem (MP 3 ). Theorem 10.25. Under the assumptions of Theorem 10.24, the optimal solution (g ∗ , f ∗ , u∗ , ϕ∗ ) ∈ K × En × E such that (g ∗ , f ∗ ) is defined by (MP 3 ) and (ϕ∗ , u∗ ) = P(g ∗ , f ∗ ) is a solution of (10.42), satisfies (ρQ∗ (0) + αg ∗ )(g − g ∗ )dx ≥ 0, Ω (10.44) ∗ ∗ ∗ R((ρP − γf )(f − f ))dxdt ≤ 0 ∀(g, f ) ∈ K, Q

where (P ∗ , Q∗ ) is the solution of the adjoint problem (10.25) (corresponding to the primal solution (ϕ∗ , u∗ )). Proof. Since the cost function J is a composition of F-differentiable maps, then J is differentiable and we have (∀Y = (β1 , β2 ) ∈ K)


J (g, f ).Y = a

R((| ϕ |2 −Λ)ϕψ)dxdt + b gβ1 dx − γ R(f β2 )dxdt,

Q +α

363

Ω

(u − uobs )wdxdt

Q

(10.45)

Q

where (ψ, w) = P (g, f ).Y is the solution of problem (PF 3 ). By taking (q, v) = (P, Q) in (PF 3 ) and integrating with respect to time we obtain (according to the homogeneous boundary conditions and initial condition) ∂P ψdxdt + ηρP (T )ψ(T )dx ρ −η Q ∂t Ω div(ρU1 )P ψdxdt + iηκ ρw∇(φ1 P )dxdt −iηκ Q Q ρψP dΓ dt + ρB(u)(P )B(u)(ψ)dxdt −iμ Σ Q + ρφ1 wB(u)(P )dxdt + ρB(u)(φ1 )wP dxdt Q Q + ρ((2 | φ1 |2 −ϑ)P ψ + φ21 P ψ)dxdt = ρβ2 P dxdt, Q Q ∂Q wdxdt + ρQ(T )w(T )dx − ρQ(0)β1 dx − ρ Q ∂t Ω Ω + ρcurl(Q)curl(w)dxdt Q + div(Q)div(ρw)dxdt + ρR(ψB(u)(φ1 ))Qdxdt Q Q + ρ(R(φ1 B(u)(ψ))Q + w | φ1 |2 Q)dxdt = 0. Q

Since (P, Q) is a solution of (10.25) we have that ρw∇(φ1 P )dxdt + ρφ1 wB(u)(P )dxdt iηκ Q Q 2 + ρB(u)(φ1 )wP dxdt + ρ(φ21 P ψ − φ1 P ψ)dxdt Q Q −i − ρB(u)(φ1 )Qψdxdt − ( div(ρφ1 Q) + ρU1 φ1 Q)ψdxdt κ Q Q (| ϕ |2 −Λ)ϕψdxdt = ρP β2 dxdt, +a (10.46) Q Q − ρQ(0)β1 dx + ρR(ψB(u)(φ1 ) + φ1 B(u)(ψ))Qdxdt Ω Q − ρR(iηκ∇(φ1 P ) + P B(u)(φ1 ) + φ1 B(u)(P ))wdxdt Q +b (u − uobs )wdxdt = 0. Q

364


By adding the real part of the first part of (10.46) and the second part of (10.46) we obtain 2 R((| ϕ | −Λ)ϕψ)dxdt + b (u − uobs )wdxdt a Q Q (10.47) = ρQ(0)β1 dx + ρR(P β2 )dxdt. Q

Ω

According to the expression of J (g, f ).Y we can deduce that R((ρP − γf )β2 )dxdt. J (g, f ).Y = (ρQ(0) + αg)β1 dx + Q

Ω ∗

∗

Since (g , f ) is an optimal solution we then have (ρQ∗ (0) + αg ∗ )(g − g ∗ )dx ≥ 0, Ω R((ρP ∗ − γf ∗ )(f − f ∗ ))dxdt ≤ 0 ∀(g, f ) ∈ K.

(10.48)

Q

This completes the proof. Distributed Disturbance in the External Magnetic Field

In this section, the disturbance is in the magnetic field h and the control is in the initial condition u0 , i.e., u0 = g (g ∈ L2 (Ω)), h = f (f ∈ L2 (Q)). So the function (ϕ, u) is assumed to be related to the disturbance f and control g through the problem (10.12)( ∀(q, v) ∈ H1 (Ω) × Vn and a.e. t ∈ (0, T )): ∂ϕ η ρ qdx − iηκ div(ρ(u + U ))ϕqdx − iηκ div(ρu)φqdx Ω ∂t Ω Ω −iμ ρϕqdΓ + (ρB(u)(ϕ + φ)B(u)(q)dx Γ Ω + ρF (ϕ)qdx = ρB(0)(φ)B(0)(q)dx + ρξqdx, Ω Ω Ω ∂u (10.49) ρ vdx + ρcurl(u)curl(v)dx + div(ρu)div(v)dx Ω ∂t Ω Ω + ρR((ϕ + φ)B(u)(ϕ + φ))vdx Ω = ρR(φB(0)(φ))vdx + ρf curl(v)dx, Ω

Ω

(ϕ(0), u(0)) = (ϕ0 , g). To obtain the regularity of Theorem 10.9, we suppose that ϕ0 ∈ L2∞ (Ω) and ξ ∈ L2∞ (Q). Let P : (g, f ) −→ (ϕ, u) = P(g, f ) be the map: L2 (Ω) × L2 (Q) −→ E × En defined by (10.49) and the cost function is defined by


J(g, f ) =

a b | ϕ |2 −Λ 2L2 (Q) + u − uobs 2L2 (Q) 4 2 α γ + g 2L2 − f 2L2 (Q) , 2 2

365

(10.50)

where α, γ > 0, a, b ≥ 0 and a + b > 0. The functions (uobs , Λ) ∈ L2 (Q) × L∞ (Q) are given. In this section we study the following robust control problem (MP 4 ): find a saddle point (g ∗ , f ∗ ) of the functional J on K, subject to (10.49), where, K = K1 × K2 such that K1 and K2 are non-empty, closed, convex, bounded subsets of L2 (Ω) and L2 (Q), respectively. The proof of the following propositions and existence theorem is obtained by using similar arguments as used in Section 10.5.1. Therefore, we will omit the details. Proposition 10.26. The function P is continuously F-differentiable from L2 (Ω)×L2 (Q) to E ×En with the derivative P (g, f ) : Y = (β1 , β2 ) −→ (ψ, w) given by the linear problem (PF 4 ) (∀(q, v) ∈ H1 (Ω) × Vn and a.e. t ∈ (0, T )): ∂ψ qdx − iηκ div(ρU1 )ψqdx − iηκ div(ρw)φ1 qdx η ρ Ω ∂t Ω Ω −iμ ρψqdΓ + ρ(B(u)(ψ) + φ1 w)B(u)(q)dx Ω Γ + ρB(u)(φ1 )wqdx + ρ((2 | φ1 |2 −ϑ)ψ + φ21 ψ)qdx = 0, Ω

Ω

∂w udx + ρcurl(w)curl(v)dx + div(ρw)div(v)dx ρ Ω ∂t Ω Ω + ρR(ψB(u)(φ1 ))vdx + ρ(R(φ1 B(u)(ψ)) + w | φ1 |2 )vdx Ω Ω = ρβ2 curl(v)dx, Ω

(ψ(0), w(0)) = (0, β1 ), where (U1 , φ1 ) = (u + U, ϕ + φ). Moreover, we have the following estimates, for all (gi , fi ) ∈ L2 (Ω)×L2 (Q), for i = 1, 2): (i) P (g1 , f1 ) L(L2 ×L2 (Q),E×En ) ≤ Ce (ii) P (g1 , f1 )Y − P (f2 , g2 )Y 2E×En ≤ Ce ( X L2 ×L2 (Q) Y 2L2 ×L2 (Q) + X 2L2 ×L2 (Q) Y L2 ×L2 (Q) ), where f = f1 − f2 , g = g1 − g2 and X = (g, f ). Proposition 10.27. The map P defined by (10.49) is continuous from the weak topology of L2 (Ω) × L2 (Q) to the strong topology of L2 (Q) × L2 (Q).

366


Theorem 10.28. For α and γ sufficiently large, there exists (g ∗ , f ∗ ) ∈ K and (u∗ , ϕ∗ ) ∈ En × E such that (g ∗ , f ∗ ) is defined by (MP 4 ) and (ϕ∗ , u∗ ) = P(g ∗ , f ∗ ) is a solution of (10.49). Next we give necessary optimality conditions for the robust control problem (MP 4 ). Theorem 10.29. Under the assumptions of Theorem 10.28, the optimal solution (g ∗ , f ∗ , u∗ , ϕ∗ ) ∈ K × En × E such that (g ∗ , f ∗ ) is defined by (MP 4 ) and (ϕ∗ , u∗ ) = P(f ∗ , g ∗ ) is a solution of (10.49), satisfies (ρQ∗ (0) + αg ∗ )(g − g ∗ )dx ≥ 0, Ω (10.51) (ρcurl(Q∗ ) − γf ∗ )(f − f ∗ ))dxdt ≤ 0 ∀(g, f ) ∈ K, Q

where (P ∗ , Q∗ ) is the solution of problem (10.25) (corresponding to (ϕ∗ , u∗ )). Proof. The proof is similar to that of Theorem 10.25, so we omit the details. We finish this section with the following remark. Remark 10.30. (i) We can consider other types of controls and disturbances, as the technique developed previously will be still valid. For example, in the case where the control is in the initial condition of the order parameter, i.e., g = ϕ0 and the distributed disturbance is in the chaotic-type force, i.e., ξ = f , we can prove, for α and γ sufficiently large, the existence theorem of the robust control problem and obtain necessary optimality conditions for its solution using the same method as previously. In this case the cost functional is given by J(g, f ) =

b α γ a | ϕ |2 −Λ 2L2 (Q) + u−uobs 2L2 (Q) + g 2L2 − f 2L2 (Q) , 4 2 2 2

where α, γ > 0, a, b ≥ 0 and a + b > 0. Let K = K1 × K2 such that K1 and K2 are non-empty, closed, convex, bounded subsets of L2∞ (Ω) and L2∞ (Q), respectively. For α and γ sufficiently large, there exists (g ∗ , f ∗ , ϕ, u) satisfying ∂ϕ η ρ qdx − iηκ div(ρ(u + U ))ϕqdx − iηκ div(ρu)φqdx Ω ∂t Ω Ω ρϕP dΓ + (ρB(u)(ϕ + φ)B(u)(q)dx −iμ Γ Ω + ρF (ϕ)qdx = ρB(0)(φ)B(0)(q)dx + ρf ∗ qdx, Ω

Ω

Ω


ρ Ω

367

∂u vdx + ρcurl(u)curl(v)dx + div(ρu)div(v)dx ∂t Ω Ω + ρR((ϕ + φ)B(u)(ϕ + φ))vdx Ω = ρR(φB(0)(φ))vdxdt + ρhcurl(v)dx, Ω

Ω

(ϕ(0), u(0)) = (g ∗ , u0 )

and the inequality R((ηρP ∗ (0) + αg ∗ )(g − g ∗ ))dx ≥ 0, Ω R((ρP ∗ − γf ∗ )(f − f ∗ ))dx ≤ 0 ∀(g, f ) ∈ K, Q

where (P ∗ , Q∗ ) is the solution of problem (10.25) associated with the solution (ϕ, u) corresponding to data (g ∗ , f ∗ ). (ii) We can consider also other types of observation, the technique developed previously will be still valid. For example, consider the following cost functional: J(g, f ) =

b a | ϕ + φ |2 −Λ˜ 2L2 (Q) + u − uobs 2L2 (Q) 4 2 α γ + g 2L2 − f 2L2 (Q) , 2 2

where Λ˜ = Λ+ | φ |2 , α, γ > 0, a, b ≥ 0 and a + b > 0. Let K = K1 × K2 such that K1 and K2 are non-empty, closed, convex, bounded subsets of L2∞ (Ω) and L2∞ (Q), respectively. For α and γ sufficiently large, there exists (g ∗ , f ∗ , ϕ, u) satisfying ∂ϕ η ρ qdx − iηκ div(ρ(u + U ))ϕqdx − iηκ div(ρu)φqdx Ω ∂t Ω Ω ρϕP dΓ + (ρB(u)(ϕ + φ)B(u)(q)dx −iμ Γ Ω + ρF (ϕ)qdx = ρB(0)(φ)B(0)(q)dx + ρf ∗ qdx, Ω Ω Ω ∂u (10.52) ρ vdx + ρcurl(u)curl(v)dx + div(ρu)div(v)dx Ω ∂t Ω Ω + ρR((ϕ + φ)B(u)(ϕ + φ))vdx Ω = ρR(φB(0)(φ))vdxdt + ρhcurl(v)dx, Ω

(ϕ(0), u(0)) = (g ∗ , u0 ) and the inequality

Ω

368


R((ηρP ∗ (0) + αg ∗ )(g − g ∗ ))dx ≥ 0,

Ω

R((ρP ∗ − γf ∗ )(f − f ∗ ))dx ≤ 0

Q

∀(g, f ) ∈ K,

where (P ∗ , Q∗ ) is the solution of the following adjoint problem (associated with the solution (ϕ, u) of (10.52) corresponding to data (g ∗ , f ∗ )): ∂P qdx − iηκ div(ρU1 )P qdx − iμ ρP qdΓ −η ρ Ω ∂t Ω Γ + ρB(u)(P )B(u)(q)dx + ρB(u)(φ1 )Qqdx Ω Ω −i + ( div(ρφ1 Q) + ρU1 φ1 Q)qdx κ Ω 2 2 ˜ 1 qdx, + ρ((2 | φ1 | −ϑ)P + φ1 P )qdx = a (| φ1 |2 −Λ)φ Ω Ω ∂Q − ρ vdx + ρcurl(Q)curl(v)dx + div(Q)div(ρq)dx Ω ∂t Ω Ω + ρR(iηκ∇(φ1 P ) + P B(u)(φ1 ) + φ1 B(u)(P ))vdx Ω + ρQ | φ1 |2 vdx = b (u − uobs )vdx, Ω

Ω

(P (T ), Q(T )) = (0, 0) where (φ1 , U1 ) = (ϕ + φ, u + U )).

♦

Remark 10.31. We recall that, for the numerical resolution, we can combine the optimal necessary conditions (which also give the gradients of the cost function) and, for example, the gradient-iterative algorithm (see Chapter 9). ♦

11 Multi-scale Modeling of Alloy Solidification and Phase-field Model

In this chapter we formulate and study robust control problems for a twodimensional, non-linear, time-dependent and solutal phase-field model of the Warren–Boettinger-type (TDWB), which describes the isothermal solidification of a binary alloy. This model contains two unknowns, the relative concentration and a non-conserved structural order parameter (which is said to be the phase-field variable) coming from thermodynamics. The phase-field theory is a direct consequence of the Cahn–Hilliard and Ginzburg–Landau type classical field theoretic approaches to phase boundaries. The order parameter describes the phase of the underlying substance: the order parameter is close to 1 if the system is in a liquid phase and is close to 0 if it is in a solid phase. It is well known that thermal fluctuations and material impurities affect considerably the solidification microstructure dynamics. These effects are modeled by variants of Warren–Boettinger model containing additive noise due to thermal fluctuations and by modification of some operators to take into account impurities. The objective is the prediction and stabilization of microstructure dynamics by taking into account the influence of fluctuations and data noises. First, a variant of TDWB model (MTDWB) is introduced and analyzed. Second, we introduce the perturbation problem of the non-linear governing coupled system of the MTDWB equations (the deviation from the desired target). The existence and uniqueness of the solution of the perturbation are proved as well as their stability under mild assumptions. Afterwards, some robust control problems are formulated for the cases when the control is in the initial condition of the concentration field and when the worst disturbance is the noise due to thermal fluctuations or is in the initial condition of the phasefield parameter. We show the existence of an optimal solution, and we also find the necessary conditions for a saddle point optimality. This work is a generalization of recent research developed by Belmiloudi in [40].

370


11.1 Introduction The aim of this chapter is the study of robust control problems related to the isotropic models for the isothermal solidification1 of a binary alloy (i.e., a mixture of two elements A and B), using a solutal phase-field model. This model was derived by Warren and Boettinger [293] to model dendrite growth in highly supersaturated binary melts. It involves the relative concentration U of solute B in solvent A and a phase-field variable φ with thermodynamically consistent evolution equations. The concentration U and the parameter φ vary sharply but smoothly between 0 in the solid phase and 1 in the liquid phase, over a thin layer which separates the phases. Our physical system is supposed to be closed, and no phase exchange takes place across its boundary. Therefore, we can close our system by considering Neumann boundary conditions and some initial physical conditions which are given between 0 and 1. The time evolution system of U = (U, φ) is governed by partial differential equations of the form ∂φ 2 − Δφ = F1 (t, φ) + U F2 (t, φ) on Q = Ω × (0, T ), ∂t τ ∂U − div(D1 (t, φ)∇U + D2 (t, U)∇φ) = 0 on Q, ∂t subject to the boundary conditions

(11.1)

∂φ ∂U = 0, = 0 on Σ = ∂Ω × (0, T ), ∂n ∂n and the initial conditions φ(0) = φ0 , U (0) = U0 on Ω, where Ω is an open bounded domain in IRm , m ≤ 2 with a smooth boundary ∂Ω of class C ∞ and n is the unit normal to ∂Ω. The positive constants τ and 2 are proportional to the relaxation time and interface thickness, respectively. In the sequel, we denote by ν the positive constant 2 /τ . The functions (Fi )i=1,2 and (Di )i=1,2 appearing in problem (11.1) have the following properties (see the sketch of the modeling below): 1. The functions Fi , for i = 1, 2, are regular, depending on the temperature function, and satisfy Fi (., φ = 0) = Fi (., φ = 1) = 0, for i = 1, 2. 2. The function D1 is positive regular and bounded above and below by two positive constants. 3. The function D2 is regular and satisfies D2 (., U = 0, .) = D2 (., U = 1, .) = 0. Next we give a brief review of the modeling leading to problem (11.1). We start with the choice of the free energy2 for the Warren–Boettinger model 1 2

So that the temperature is assumed to be externally imposed on the system. Generally, the choice of a model is conditioned by the choose form of the free energy functional.

11.1 Introduction

371

[293]. A free energy for a binary alloy, which must decrease during any process, can be written as F (t) = F (T (., t), φ(., t), U (., t)) 1 ( | φ ∇φ(., t) |2 +fAB (T (., t), φ(., t), U (., t)))dx, = 2 Ω

(11.2)

where | φ ∇φ(., t) |2 is a potential of Ginzburg–Landau type, the parameter φ is an energy scale for the order parameter interface and T is the temperature. √ The corresponding width of this interface is given by Wφ = φ / d, where d denotes the energy barrier between liquid and solid phases and is proportional to the liquid–solid surface energy, which is assumed to be the same for both A and B materials. The parameter φ is made isotropic (respectively anisotropic) by making it independent (respectively dependent) on the interface normal vector, which is given by ∇φ/ | ∇φ |. The function fAB denotes the bulk free energy density of the A–B mixture. It is designed to reproduce the thermodynamic phase diagram of the alloy and is built from the free energy densities fA and fB of the pure element (U = 0 and U = 1, respectively) and a mixing energy term. The simple form of fAB can be written as follows (see Warren et al. [293, 294]): fAB (T , φ, U ) = (1 − U )fA (T , φ) + U fB (T , φ) + H(T , U ), with H(T , U ) =

RT (U ln(U ) + (1 − U ) ln(1 − U )), vm

(11.3)

where R is the Bolzman constant, vm denotes the molar volume of solid A and B phases and the function H describes the energy of ideal mixing between A and B. By using some basic thermodynamical principles, we can obtain a general form for fA and fB (which are depending on the latent heats of fusion and melting temperature LA , TA and LB , TB of A and B, respectively) and then for fAB at any given temperature T . If we assume that at the melting temperature the functions fA and fB are of double-well potential type between the solid and liquid phases and if at any given temperature T , the functions fA and fB have each only two minima for the variable φ in [0,1], a particular explicit form for fAB can be given by fM (T , φ) = CM (T )g(φ) −

(T − TM )DM (T ) p(φ) + EM (T ), for M = A or B, TM

where the functions CM , DM and EM are positive such that EM (TM ) = 0 and CM is depending on Wφ . The function g(φ) = φ2 (1 − φ)2 provides a double-well potential between the solid and liquid phases, while p(φ) is, for example, p(φ) = φ2 (3 − 2φ) or p(φ) = φ3 (6φ2 − 15φ + 10) (which interpolates between the solid and liquid entropy states). Once the free energy function is defined, the equations of motion for the concentration and the phase-field parameter are obtained as follows:

372


∂F ∂φ =− , ∂t ∂φ ∂U ∂F = div(Mu (T , φ, U )∇( )) ∂t ∂U RT 1 ∂fAB = div(Mu (T , φ, U )(∇( )+ ∇U )), ∂U vm U (1 − U )

τφ

(11.4)

where τφ represents the interface kinetic attachment time scale (which can be dependent on the interface normal vector) and Mu is the solute mobility which is assumed to be in the form Mu (T , φ, U ) = D1 (T , φ)U (1 − U ), where D1 (T , φ) = D11 (T )D12 (φ) such that D11 is a non-negative function, D12 is an increasing smooth function for the variable φ such that D12 (0) > 0,3 D12 (1) > 0 and D1 is bounded above and below by two non-negative constants. According to the expression (11.3) of the free energy, we can easily prove that there exist functions F1 and F2 such that −

∂fAB = F1 (T , φ) + U F2 (T , φ). ∂φ

(11.5)

If we suppose, moreover, that the given temperature T is independent of variable space, we can deduce that ∇(

∂fAB ) = −F2 (T , φ)∇φ. ∂U

(11.6)

If we suppose now that τφ = τ and φ = are independent on φ then, according to (11.5) and (11.6) and the expression of the free energy function, System (11.4) becomes ∂φ = div(2 ∇φ) + F1 (T , φ) + U F2 (T , φ), ∂t ∂U = div(D1 (T , φ)∇U + D2 (T , U, φ)∇φ), ∂t

τ

(11.7)

where D2 (T , U, φ) = D1 (T , φ)U (1 − U )F2 (T , φ). If we suppose that τ and 2 are constants we can easily obtain System (11.1). Remark 11.1. In the general case the parameter φ is dependent on ∇φ. This variable is an anisotropic tensor and is introduced for the simulation of dentritic growth, and accounts for the existence of privileged directions for solidification due to microscopic crystal growth. ♦ 3

D12 is a function that interpolates between solid and liquid diffusivities.

11.1 Introduction

373

The greatest difficulty in this type of problem is the multiscale nature (for length and time scales) of solidification microstructures with thermal conditions and the material parameters used in experiments, which influence considerably the physical material properties (see, e.g., Askeland [12]). The interface between liquid and solid is a few nanometers thick and microstructural features are on the scale of tens to hundreds of microns. Moreover, the atomic attachment kinetics take a few picoseconds, whereas diffusion of heat and solute is of the order of seconds. Consequently, in order to take into account this multiscale nature, it is necessary to develop adapted mathematical and computational models, which help the engineer to complete the experiment as well as to define the pertinent material parameters to be analyzed. In recent years the so-called phase-field formulation has emerged as a powerful computational approach to modeling and predicting the range of phase transitions and complex “dendritic” growth structures occurring during solidification. This approach has proved to be an emerging technology that complements experimental research.4 Various problems associated with phase-field models have been studied to treat both pure materials and binary alloys, either from the theoretical or numerical point of view (see, e.g., Brochet et al. [62, 63], Caginalp [66], Grujicic et al. [142], Laurencot [186], Rappaz and Scheid [249], Warren et al. [293, 294] and the references therein). For control problems associated with phase-field models, refer to Hoffman and Jiang [159] (for optimal control problems) and Belmiloudi and Yvon [41] (for robust control problems). In these the authors have studied the problem of describing the phase transitions of pure materials due to thermal effects, and they have developed non-linear parabolic systems for the phase field and temperature. It is then clear that, in order to study the stability, dynamics and other properties of the microstructure state, it is necessary to take into account the effects of thermal fluctuation and material defects on the solidification dynamics, because for longer solidification times, the solid/liquid interface becomes unstable with respect to small perturbations caused by the introduction of noise and fluctuation terms. In this chapter, which generalizes the previous results of Belmiloudi [40], we consider, in order to take into account thermal fluctuations, an additive noise solidification model5 which is a modification of the model (11.1). More precisely, we introduce into the right-hand side of the first equation of (11.1) a real-valued field in time and space and assume that the operators appearing in (11.1) are modified.6 Thus, the modified system is 4

5 6

Because the experimental solidification research in metals that is required to link microstructural characteristics with processing regimes, is often limited in its ability to observe real-time development of microstructure and the associated segregation patterns. The thermal fluctuations can be modeled, for example, by deterministic Langevintype dynamics. The polynom functions and coefficients appearing in the modeling are dependent on the variable space x. For example, the function g defined previously can be replaced by g˜(x, φ) = φ2 (ϑ(x) − φ)2 with 0 < ϑ < 1 in Ω.

374


given by ∂φ − νΔφ = F˜1 (x, t, φ) + U F˜2 (x, t, φ) + λ on Q, ∂t ∂U ˜ 1 (x, t, φ)∇U + D ˜ 2 (x, t, U)∇φ) = 0 on Q, − div(D ∂t subject to the boundary conditions

(11.8)

∂φ ∂U = 0, = 0 on Σ, ∂n ∂n and the initial conditions φ(0) = φ0 , U (0) = U0 on Ω. In the following, we omit the “ ˜” on the operators appearing in problem (11.8). 11.1.1 Assumptions and Notations We denote by V = H 1 (Ω) and V the dual of V . We denote by , V ,V the duality product between V and V . For any pair of real number r, s ≥ 0, we introduce the Sobolev space H r,s (Q) defined by H r,s (Q) = L2 (0, T ; H r (Ω)) ∩ H s (0, T ; L2 (Ω)), which is normed by (see Chapter 6)

1/2

T

v 0

2H r (Ω)

dt+ v

2H s (0,T ;L2 (Ω))

.

We can now introduce the following spaces: Hi = L∞ (0, T ; H i−1 (Ω)), Vi = L2 (0, T ; H i (Ω)), Wi = Hi ∩ Vi , (for i = 1, 3), Hi1 = H 1 (0, T ; H i−2 (Ω)), (for i = 2, 3), H11 = H 1 (0, T ; V ), Wi1 = Hi1 ∩ Vi , (for i = 1, 3) and the space H02 = {v ∈ H 2 (Ω) : ∂v/∂n = 0}. Remark 11.2. (i) Wi1 is compactly embedded into Vi−1 , for i = 1, 3 (see Chapter 6). ♦ (ii) Wi1 ⊂ C([0, T ]; H i−1 (Ω)), for i = 1, 3 (see Chapter 6). We state the following hypotheses for the operators (Fi )i=1,2 and (Di )i=1,2 : (H1) F1 and F2 are Carathéodory functions from Q×IR into IR such that, for almost all (x, t) ∈ Q, F1 (x, t, .) and F2 (x, t, .) are Lipschitz and bounded functions with (i) | Fi (x, t, r) |≤ M1 ∀ i = 1, 2, ∀ r ∈ IR and a.e. (x, t) ∈ Q .n = 0, on Σ. (ii) Fix (H2) D1 is a Carathéodory function from Q×IR into IR such that, for almost all (x, t) ∈ Q, D1 (x, t, .) is Lipschitz positive and bounded function with 0 < D0 ≤ D1 (x, t, r) ≤ D1 ∀ r ∈ IR and a.e. (x, t) ∈ Q.

11.1 Introduction

375

(H3) D2 is a Carathéodory function from Q×IR2 into IR such that, for almost all (x, t) ∈ Q, D2 (x, t, .) is Lipschitz positive and bounded function with 0 < D0 ≤ D2 (x, t, r) ≤ M2 , ∀ r = (r, r ) ∈ IR2 and a.e. (x, t) ∈ Q. (H4) Fi , for i = 1, 2, and D1 are differentiable with Fix , Gi = Fir , D1x and H1 = D1r Lipschitz continuous a.e. in Q. , H2 = D2r Lipschitz continuous a.e. in Q. (H5) D2 is differentiable with D2x

Remark 11.3. The functions (Fi , i = 1, 2) and (Di , i = 1, 2) are depending on temperature. ♦ Remark 11.4. If u and φ are sufficiently regular and satisfy ∂u/∂n = 0 and ∂φ/∂n = 0, then according to the second assumption of (H1), we have ∇(F1 (x, t, φ)).n = ∇(uF2 (x, t, φ)).n = 0.

♦

In the following we will use C to denote some positive constant which may be different at each occurrence. 11.1.2 Preliminary Results Lemma 11.5. (Elliptic estimate) Let k ∈ IN and v ∈ H 2 (Ω) satisfy Δv ∈ H k and ∂v/∂n = 0 on ∂Ω. Then v ∈ H k+2 (Ω) and we have the following estimate: there exists C > 0 (independent of v) such that v H k+2 ≤ C( Δv H k + v H k ). Proof. For the proof of this lemma see, for example, Lions and Magenes [204]. Lemma 11.6. Let ul = (ul , ϕl ) be a sequence converging toward u = (u, ϕ) in W11 weakly and in L2 (Q) strongly. Then we have the following convergence results: (i) Fi (., ϕl ) −→ Fi (., ϕ),(i=1,2) in Lp (Q) strongly, ∀p ∈ [1, +∞) (ii) D1 (., ϕl ) −→ D1 (., ϕ) in Lp (Q) strongly, ∀p ∈ [1, +∞) (iii) ul F2 (., ϕl ) −→ uF2 (., ϕ) in Lq (Q) strongly, ∀q ∈ [1, 2) (iv) D1 (., ϕl )∇ul D1 (., ϕ)∇u in Lq (Q) weakly, ∀q ∈ [1, 2) (v) D2 (., ul )∇ϕl D2 (., u)∇ϕ in Lq (Q) weakly, ∀q ∈ [1, 2). Proof. For the proof of this lemma we can use a classical technique based on taking the difference between the sequence and its limit in the form of the sum of two terms such that the first uses the weak convergence result and the other uses the strong convergence result (for a similar result see, e.g., Rappaz et al. [249]). The results (i)–(iii) are a simple consequence of the strong convergence

376


in L2 (Q) and the assumptions (H1) and (H2). The proof of the results (iii) is similar to the proof of (iv). The proof of (iv) is based on the fact that Il =

∗

(D2 (x, t, ul )∇ϕl − D2 (x, t, u)∇ϕ)vdxdt, for all v ∈ Lq (Q), can be

Q

written as follows: Il =

D2 (x, t, u)(∇ϕl − ∇ϕ)vdxdt Q (D2 (x, t, ul ) − D2 (x, t, u))∇ϕl vdxdt +

(11.9)

Q

where the convergence to 0 of the first term is a consequence of the weak convergence in W11 and the convergence to 0 of the second term is a consequence of the strong convergence in L2 (Q) and the assumption (H3).

11.2 Existence, Uniqueness and a Maximum Principle 11.2.1 Existence and Uniqueness Results The following results concern the existence, uniqueness and regularity of a solution for problem (11.8). Proposition 11.7. Let Assumptions (H1)–(H3) be fulfilled. For any element (φ0 , U0 , λ) ∈ H02 (Ω) × H 1 (Ω) × L2 (0, T ; H 1 (Ω)), then there exists a unique global solution (φ, U ) of (11.8) satisfying φ ∈ W31 ⊂ C([0, T ]; H 2) and U ∈ W21 ⊂ C([0, T ]; H 1). Proof. The proof of this proposition can be obtained in the same way as used to prove the results of Section 11.3. So, we omit the details. The following results concern the maximum principle. 11.2.2 A Maximum Principle Assume that function λ is in the space Λ = {λ ∈ L2 (Q) : 0 ≤ λ(x, t) ≤ M

a.e. (x, t) ∈ Q}.

We establish now a maximum principle under extra assumptions on the nonlinear terms. So, in addition to Hypothesis (H1)–(H3), we assume that the non-linear terms F1 , F2 and D2 satisfy the following assumptions (a.e. (x, t) ∈ Q): F2 (x, t, .) = 0 in ] − ∞, 0] ∪ [1, +∞[, F1 (x, t, .) = 0 in ] − ∞, 0] and F1 (x, t, .) = −M in [1, +∞[, D2 (x, t, ., r ) = 0 in ] − ∞, 0] ∪ [1, +∞[ for all r ∈ IR. Then we have the following result.

(11.10)

11.2 Existence, Uniqueness and a Maximum Principle

377

Theorem 11.8. Let assumptions (H1)–(H3) and (11.10) be fulfilled. Assume that the initial data (φ0 , U0 ) ∈ (L2 (Ω))2 and the forcing λ ∈ Λ. Then every weak solution (φ, U ) ∈ (L2 (0, T ; H 1 (Ω)) ∩ H 1 (0, T ; V ))2 , satisfies for all t ∈ (0, T ) 0 ≤ φ(x, t) ≤ 1 and 0 ≤ U (x, t) ≤ 1 a.e. x ∈ Ω. Proof. Let us consider the following notations: r+ = max(r, 0), r− = (−r)+ and then r = r+ − r− . First, we prove that if φ0 ≥ 0 and U0 ≥ 0, a.e. in Ω then φ(., t) ≥ 0 and U (., t) ≥ 0, for all t ∈ (0, T ) and a.e. in Ω. According to Chapter 3, we have that φ− ∈ L2 (0, T ; H 1 (Ω)) with ∇φ− = −∇φ if φ > 0 and ∇φ− = 0 otherwise, a.e. in Q and the same properties hold for U − . Then, multiplying the first equation of System (11.8) by −φ− and the second equation of System (11.8) by −U − we have (a.e. in (0, T )) d φ− 2L2 + ν ∇φ− 2L2 = − (F1 (., t, φ) + U F2 (., t, φ))φ− dx − λφ− dx 2dt Ω Ω d U − 2L2 − 2 + D1 (., t, φ) | ∇U | dx = D2 (., t, U, φ)∇φ∇U dx, 2dt Ω Ω− where Ω − = {x ∈ Ω : U (t, x) < 0}. Using assumptions (11.10) on the non-linear terms F1 , F2 and D2 , respectively, we have (F1 (., t, φ) + U F2 (., t, φ))φ− = 0 and D2 (., t, U, φ) = 0 if U < 0, a.e. in Q. Consequently, for a.e. in (0, T ) (since 0 ≤ λ) d φ− 2L2 + ν ∇φ− 2L2 ≤ 0, 2dt d U − 2L2 + D1 (., t, φ) | ∇U − |2 dx = 0 2dt Ω

(11.11)

By integration over (0, t), for any t ∈ (0, T ), we can deduce that 2 − 2 φ− 2L2 ≤ φ− L2 ≤ U0− 2L2 . 0 L2 , U

Therefore, for all t ∈ (0, T ), φ− (., t) = U − (., t) = 0 a.e. in Ω (since φ− 0 = U0− = 0 a.e. in Ω). Next, we prove that if φ0 ≤ 1 and U0 ≤ 1, a.e. in Ω then φ(., t) ≤ 1 and U (., t) ≤ 1, for all t ∈ (0, T ) and a.e. in Ω. By multiplying the first equation of System (11.8) by (φ − 1)+ and the second equation of System (11.8) by (U − 1)+ we have (a.e. in (0, T )) d (φ − 1)+ 2L2 + ν ∇(φ − 1)+ 2L2 2dt = ((F1 (., t, φ) + λ) + U F2 (., t, φ))(φ − 1)+ dx Ω

378


and

d (U − 1)+ 2L2 + D1 (., t, φ) | ∇(U − 1)+ |2 dx 2dt Ω = D2 (., t, U, φ)∇φ∇U dx, Ω+

where Ω + = {x ∈ Ω : U (t, x) > 1}. Using assumptions (11.10) on the non-linear terms F1 , F2 and D2 , respectively, we have (since M ≥ λ), a.e. in Q (F1 (., t, φ) + λ + U F2 (., t, φ))(φ − 1)+ ≤ 0 and D2 (., t, U, φ) = 0 if U > 1. Consequently, for a.e. in (0, T ) d (φ − 1)+ 2L2 + ν ∇(φ − 1)+ 2L2 ≤ 0, 2dt d (U − 1)+ 2L2 + D1 (., t, φ) | ∇(U − 1)+ |2 dx = 0. 2dt Ω

(11.12)

By integration over (0, t), for any t ∈ (0, T ), we can deduce that (φ − 1)+ 2L2 ≤ (φ0 − 1)+ 2L2 , (U − 1)+ 2L2 ≤ (U0 − 1)+ 2L2 . Therefore, for all t ∈ (0, T ), (φ − 1)+ (., t) = (U − 1)+ (., t) = 0 a.e. in Ω (since (φ0 − 1)+ = (U0 − 1)+ = 0 a.e. in Ω). This completes the proof.

11.3 The Perturbation Problem In the following, the solution U = (U, φ) of problem (11.8) will be treated as the target function. We are then interested in the robust regulation of the deviation of the problem from the desired target U. We analyze the full nonlinear equation which models large perturbations u = (u, ϕ) to the target U, i.e., we assume that U satisfies the problem (11.8) with the data (U0 , φ0 , λ) and U + u satisfies the problem (11.8) with the data (U0 + u0 , φ0 + ϕ0 , λ + ξ). Hence, we consider the following system (for U = (U, φ) given satisfying the regularity of Proposition 11.7): ∂ϕ − νΔϕ = (F1 (x, t, ϕ + φ) − F1 (x, t, φ)) + uF2 (x, t, ϕ + φ) ∂t +U (F2 (x, t, ϕ + φ) − F2 (x, t, φ)) + ξ on Q, ∂u − div(D1 (x, t, ϕ + φ)∇u + D2 (x, t, u + U)∇ϕ) ∂t = div((D1 (x, t, ϕ + φ) − D1 (x, t, φ))∇U ) +div((D2 (x, t, u + U) − D2 (x, t, U))∇φ) on Q, ∂u ∂ϕ = 0, = 0 on Σ, ∂n ∂n (ϕ(0), u(0)) = (ϕ0 , u0 ) on Ω.

(11.13)


379

If we set F˜1 (x, t, ϕ) = F1 (x, t, ϕ + φ) − F1 (x, t, φ), ˜ 1 (x, t, ϕ) = D1 (x, t, ϕ + φ), F˜2 (x, t, ϕ) = F2 (x, t, ϕ + φ), D

(11.14)

˜ 2 (x, t, u) = D2 (x, t, u + U), D then System (11.13) reduces to ∂ϕ − νΔϕ = F˜1 (x, t, ϕ) + uF˜2 (x, t, ϕ) ∂t +U (F˜2 (x, t, ϕ) − F˜2 (x, t, 0)) + ξ on Q, ∂u ˜ 1 (x, t, ϕ)∇u + D ˜ 2 (x, t, u)∇ϕ) − div(D ∂t ˜ 1 (x, t, 0))∇U ) ˜ 1 (x, t, ϕ) − D = div((D

(11.15)

˜ 2 (x, t, u) − D ˜ 2 (x, t, 0))∇φ) on Q, +div((D ∂u ∂ϕ = 0, = 0 on Σ, ∂n ∂n (ϕ(0), u(0)) = (ϕ0 , u0 ) on Ω. ˜ i , i = 1, 2) Remark 11.9. (i) We can easily verify that (F˜i , i = 1, 2) and (D satisfy the same hypotheses that (Fi , i = 1, 2) and (Di , i = 1, 2), i.e., (H1)– (H5). ˜ for the (ii) For simplicity of future reference, we omit the “ ˜ ” on F˜ and D system (11.15). ♦ Now we give the weak formulation associated with the problem (11.15). Multiplying the first part of (11.15) by v ∈ V and the second part by q ∈ V and integrating over Ω this gives (according to the third part of (11.15)) the weak formulation (a.e. t ∈ (0, T )) ∂ϕ vdx + ν ∇ϕ.∇vdx = F1 (., t, ϕ)vdx + uF2 (., t, ϕ)vdx Ω ∂t Ω Ω Ω + U (F2 (., t, ϕ) − F2 (., t, 0))vdx + ξvdx, Ω Ω ∂u qdx + D1 (., t, ϕ)∇u.∇qdx = − D2 (., t, u)∇ϕ.∇qdx (11.16) Ω ∂t Ω Ω − (D1 (., t, ϕ) − D1 (., t, 0))∇U ∇qdx Ω − (D2 (., t, u) − D2 (., t, 0))∇φ∇qdx, (ϕ(0), u(0)) = (ϕ0 , u0 ).

Ω

Before giving the existence theorem, we study the following lemma. Lemma 11.10. Let assumptions (H1)-(H3) be fulfilled. For u = (u, ϕ) be sufficiently regular we have (for all i = 1, 2):

380


(i) | ∇(Fi (x, t, ϕ)) |≤ C(1+ | ∇ϕ |) (ii) | ∇(D1 (x, t, ϕ)) |≤ C(1+ | ∇ϕ |) (iii) | ∇(D2 (x, t, u)) |≤ C(1+ | ∇ϕ | + | ∇u |). (x, t, ϕ) + Fiϕ (x, t, ϕ)∇ϕ, for i = 1, 2, by Proof. Since ∇(Fi (x, t, ϕ)) = Fix 1,∞ we have the result (i). By using the same technique, we using Fi ∈ W obtain the results (ii) and (iii).

Now we show the existence, the regularity and the uniqueness of the solution for the problem (11.16). Theorem 11.11. Let assumptions (H1)–(H3) be fulfilled. The following results hold: (i) For any (ϕ0 , u0 ) in (L2 (Ω))2 and ξ in L2 (Q), there exists a couple of functions (ϕ, u) in W11 × W11 which is a solution of problem (11.16). (ii) For any (ϕ0 , u0 ) in H 1 (Ω) × L2 (Ω) and ξ in L2 (Q), the couple of functions (ϕ, u) which is a solution of problem (11.16) is in W21 × W11 . (iii) For any (ϕ0 , u0 ) in H02 (Ω) × H 1 (Ω) and ξ in L2 (0, T ; H 1(Ω)), there exists a unique couple of functions (ϕ, u) in W31 × W21 which is the unique solution of problem (11.16). Moreover, the Lipschitz continuity relation is satisfied, i.e., for any elements (ϕ01 , u01 , ξ1 ) and (ϕ02 , u02 , ξ2 ) in H02 × H 1 × L2 (0, T ; H 1(Ω)), we have ϕ1 − ϕ2 2W 1 + u1 − u2 2W 1 2 1 ≤ C( ϕ01 − ϕ02 2H 2 + u01 − u02 2H 1 + ξ1 − ξ2 2L2 (0,T ;H 1 ) ),

(11.17)

where (ϕ1 , u1 ) (respectively (ϕ2 , u2 )) is the solution of (11.15), which corresponds to the data (ϕ01 , u01 , ξ1 ) (respectively (ϕ02 , u02 , ξ2 )). Proof. The proof of this theorem can be obtained using the same technique as used in Belmiloudi [40]. For example, the existence and regularity results follow from the Faedo–Galerkin method and Lemmas 11.5, 11.6 and 11.10; according to the properties of the different forms appearing in the weak formulation, we obtain the a priori estimates necessary to prove the convergence of an approximate solution (um , ϕm ) for different necessary topologies. So, we omit the details. Remark 11.12. The results of (i) and (ii) are valid for any bounded and regular domain Ω ⊂ IRm with m ≥ 1 and the results of (iii) are valid for any bounded and regular domain Ω ⊂ IRm with m ≤ 3. The restriction m ≤ 2 does not occur in the proof. ♦

11.4 Differentiability of the Operator Solution Before proceeding to the investigation of the F-differentiability of the function F : (ϕ0 , u0 , ξ) −→ u = (u, ϕ), which maps the source term (ϕ0 , u0 , ξ) in H02 × H 1 × L2 (0, T ; H 1(Ω)) of problem (11.15) into the corresponding solution

11.4 Differentiability of the Operator Solution

381

(u, ϕ) in W21 × W31 , we study the following problem (PI ): find w = (w, ψ) such that, ∂ψ − νΔψ = G1 (x, t, ϕ)ψ + wF2 (x, t, ϕ) ∂t +U1 G2 (x, t, ϕ)ψ + h on Q, ∂w − div(D1 (x, t, ϕ)∇w + D2 (x, t, u)∇ψ) ∂t = div(H1 (x, t, ϕ)ψ∇U1 + (H2 (x, t, u).w)∇φ1 ) on Q,

(11.18)

∂w ∂ψ = 0, = 0 on Σ, ∂n ∂n ψ(0) = ψ0 , w(0) = w0 on Ω, where U1 = U + u, φ1 = φ + ϕ, and (Gi )i=1,2 , (Hi )i=1,2 are given in (H4)– (H5).7 Theorem 11.13. Let assumptions (H1)–(H5) be fulfilled. If u = (u, ϕ) and (U, φ) are in W21 × W31 , then the following results hold: (i) For any element (ψ0 , w0 , h) in H 1 (Ω) × L2 (Ω) × L2 (Q), there exists a unique couple of functions (ψ, w) in W21 × W11 solution of problem (PI ), such that ψ 2W 1 + w 2W 1 ≤ Ce ( ψ0 2H 1 + w0 2L2 + h 2L2 (Q) ). 2

1

(ii) Let (ψ0i , w0i , hi ), for i = 1, 2, be two elements from H 1 (Ω) × L2 (Ω) × L2 (Q). If (ψi , wi ) is the solution of (PI ), where the data is (ψ0i , w0i , hi ), for i = 1, 2, then ψ1 − ψ2 2W 1 + w1 − w2 2W 1 2 1 (11.19) ≤ Ce ( ψ01 − ψ02 2H 1 + w01 − w02 2L2 + h1 − h2 2L2 (Q) ). (iii) For any element (ψ0 , w0 , h) in H02 (Ω) × H 1 (Ω) × L2 (0, T ; H 1 (Ω)), the couple of functions (ψ, w) which is the unique solution of problem (PI ) is in W31 × W21 . Proof. The existence, uniqueness and Lipschitz continuity results of the problem (PI ) can be obtained in the same way as used to prove Theorem 11.11 and by using the regularity of (U1 , φ1 ). For more details see Belmiloudi [40]. We are now going to study the F-differentiability of the operator solution F . For simplicity, we denote by X the data X = (ϕ0 , u0 , ξ), by Z the space 7

Where H2 (x, t, u).w =

∂D2 ∂D2 (x, t, u).w + (x, t, u).ψ. ∂u ∂ϕ

382


Z = H02 (Ω) × H 1 (Ω) × L2 (0, T ; H 1 (Ω)). The space Z is equipped with the following norm X Z = ( ϕ0 2H 2 + u0 2H 1 + ξ 2L2 (0,T ;H 1 ) )1/2 , for all X = (ϕ0 , u0 , ξ) in Z. Theorem 11.14. Let X = (ϕ0 , u0 , ξ) and Y = (p0 , q0 , π) be in Z with F (X) and F (X + Y ) being the corresponding solutions of (11.16). Then F (X) − F(X + Y ) − F (X).Y W1 ×W2 ≤ C Y Z , 3/2

(11.20)

where F (X) : Z −→ W1 ×W2 is a linear operator such that (w, ψ) = F (X).Y is the solution of the problem (PI ) with the initial condition (ψ, w)(t = 0) = (p0 , q0 ) and the forcing h = π. Moreover, for all Xi = (ϕ0i , u0i , ξi ) ∈ Z, for i = 1, 2, we have the following estimate: F (X1 ).Y − F (X2 ).Y 2W1 ×W2 ≤ Ce ( Y Z X 2Z + Y 2Z X Z ),

(11.21)

where X = X1 − X2 = (ϕ01 − ϕ02 , u01 − u02 , ξ1 − ξ2 ). Proof. The proof of this theorem can be obtained by using the same technique as used to prove the results of Theorem 4.3 of Belmiloudi [40]. So, we omit the details.

11.5 Robust Control Problems The objective of the robust control problem is to find the best estimate of the initial state u0 in the presence of the disturbance which maximally spoils the control objective. We formulate the problem in two situations: where the worst disturbance is in the force ξ (i.e., if we take into account the temperature fluctuation) and where the disturbance is in the initial condition of the phasefield parameter ϕ0 . 11.5.1 Disturbance in the Forcing of the Phase-field Parameter We suppose now that the control is in the initial condition of the relative concentration u0 and the disturbance is in the force ξ, i.e., u0 = B1 g (g ∈ L2 (Ω)) and ξ = B2 f (f ∈ L2 (Q)), where B1 is a continuous and bounded operator from L2 (Ω) into H 1 (Ω) and B2 is a continuous and bounded operator from L2 (Q) into L2 (0, T ; H 1 (Ω)). Therefore, the function (ϕ, u) is assumed to be related to the disturbance f and the control g through the problem (11.15):


383

∂ϕ − νΔϕ = F1 (x, t, ϕ) + uF2 (x, t, ϕ) ∂t +U (F2 (x, t, ϕ) − F2 (x, t, 0)) + B2 f on Q, ∂u − div(D1 (x, t, ϕ)∇u + D2 (x, t, u)∇ϕ) ∂t = div((D1 (x, t, ϕ) − D1 (x, t, 0))∇U )

(11.22)

+div((D2 (x, t, u) − D2 (x, t, 0))∇φ) on Q, ∂u ∂ϕ = 0, = 0 on Σ, ∂n ∂n (ϕ(0), u0 ) = (ϕ0 , B1 g) on Ω. To obtain the regularity of Theorem 11.11, we suppose that (g, f ) ∈ L2 (Ω) × L2 (Q) and (U, φ) ∈ W21 × W31 . Let P : (g, f ) −→ (u, ϕ) = P(g, f ) be the map: L2 (Ω) × L2 (Q) −→ W21 × W31 defined by (11.22) and introducing the cost function defined by J(g, f ) =

b a ϕ − ϕobs 2L2 (Q) + u − uobs 2L2 (Q) 2 2 α β + g 2L2 − f 2L2 (Q) , 2 2

(11.23)

where a, b, α, β are fixed such that α, β > 0, a, b ≥ 0 and a + b > 0. The functions uobs ∈ L2 (Q) and ϕobs ∈ L2 (Q) are given and represent the observation. Let K = K1 × K2 such that K1 and K2 are (given) non-empty, closed, convex, bounded subsets of L2 (Ω) and L2 (Q), respectively. We want to minimize the functional J with respect to g and maximize J with respect to f , i.e., to find (g ∗ , f ∗ ) ∈ K such that J(g ∗ , f ) ≤ J(g ∗ , f ∗ ) ≤ J(g, f ∗ ), ∀(g, f ) ∈ K.

(11.24)

Proposition 11.15. The function P is continuously F-differentiable from L2 (Ω) × L2 (Q) to W1 × W2 with the derivative P (g, f ) : h = (h1 , h2 ) −→ w = (w, ψ) given by the following linear problem (PLP ) ∂ψ − νΔψ = G1 (x, t, ϕ)ψ + U1 G2 (x, t, ϕ)ψ ∂t +wF2 (x, t, ϕ) + B2 h2 on Q, ∂w − div(D1 (x, t, ϕ)∇w + D2 (x, t, u)∇ψ) ∂t = div(H1 (x, t, ϕ)ψ∇U1 + (H2 (x, t, u).w)∇φ1 ) on Q, ∂w ∂ψ = 0, = 0 on Σ, ∂n ∂n (ψ(0), w(0)) = (0, B1 h1 ) on Ω,

(11.25)

384


where (U1 , φ1 ) = (u + U, ϕ + φ). Moreover, ∀(gi , fi ) ∈ L2 (Ω) × L2 (Q), for i = 1, 2, we have the following estimates: (i) P (g1 , f1 ) L(L2 (Ω)×L2 (Q),W1 ×W2 ) ≤ Ce (ii) P (g1 , f1 )h − P (g2 , f2 )h 2W1 ×W2 ≤ Ce {( g 2L2 + f 2L2 (Q) )1/2 ( h1 2L2 + h2 2L2 (Q) ) +( g 2L2 + f 2L2 (Q) )( h1 2L2 + h2 2L2 (Q) )1/2 }, where f = f1 − f2 and g = g1 − g2 . Proof. The proof of this proposition is a consequence of the nature of the operators Bi , i = 1, 2 and the result of Theorem 11.14. So, we omit the details. Proposition 11.16. Let assumptions (H1)–(H3) be satisfied. Then the mapping P defined by (11.22) is continuous from the weak topology of L2 (Ω) × L2 (Q) to the strong topology of (L2 (Q))2 . Proof. Let f = (g, f ) be given in L2 (Ω) × L2 (Q) and let a sequence fk = (gk , fk ) be such that fk is weakly convergent in L2 (Ω) × L2 (Q) to f . Set Bf = (B1 g, B2 f ), (u, ϕ) = P(g, f ), Bfk = (B1 gk , B2 fk ) and (uk , ϕk ) = P(gk , fk ). Since fk f weakly in L2 (Ω) × L2 (Q) then fk is uniformly bounded in L2 (Ω) × L2 (Q) and then Bfk is uniformly bounded in H 1 (Ω) × L2 (0, T ; H 1(Ω)). In view of Theorem 11.11, we can deduce that the sequence (uk , ϕk ) is uniformly bounded in W21 × W31 . Therefore, we can extract from (fk , uk , ϕk ) a subsequence also denoted by (fk , uk , ϕk ) and such that (gk , fk ) (g, f ) weakly in L2 (Ω) × L2 (Q), ϕk ϕ˜ weakly in W3 , uk u ˜ weakly in W2 , ϕk −→ ϕ˜ strongly in L2 (Q), uk −→ u ˜ strongly in L2 (Q). We can easily prove that (˜ u, ϕ) ˜ = P(g, f ) and, according to the uniqueness of the solution of (11.22), we then haveϕ˜ = ϕ and u ˜ = u. Theorem 11.17. Let assumptions (H1)–(H5) be satisfied. Then for sufficiently large α and β (i.e., there exists (αl , βl ) such that α ≥ αl and β ≥ βl ) there exists (g ∗ , f ∗ ) ∈ K and (u∗ , ϕ∗ ) ∈ W21 × W31 such that (g ∗ , f ∗ ) is defined by (11.24) and (u∗ , ϕ∗ ) = P(g ∗ , f ∗ ) is a solution of (11.22). Proof. Let Pf be the mapping: g −→ J(g, f ) and Qg be the mapping: f −→ J(g, f ). To obtain the existence of the robust control problem we prove that Pf is convex and lower semi-continuous for all f ∈ K2 , and Qg is concave and


385

upper semi-continuous for all g ∈ K1 and we use the minimax theorems in infinite dimensions presented in Chapter 5. First, we prove for sufficiently large α and β, the convexity of the mapping Pf and the concavity of the mapping Qg . In order to prove the convexity, it is sufficient to show that for all (g1 , g2 ) ∈ K1 × K1 the estimate (Pf (g1 ) − Pf (g2 )).g ≥ 0 holds, where g = g1 − g2 (because Pf is G-differentiable). According to the definition of J, we have (ϕ1 − ϕ2 )ψ2 dxdt (Pf (g1 ) − Pf (g2 )).g = a Q +b (u1 − u2 )w2 dxdt + a (ϕ1 − ϕobs )(ψ1 − ψ2 )dxdt (11.26) Q Q +b (u1 − uobs )(w1 − w2 )dxdt + α g 2L2 , Q

where (ui , ϕi ) = P(gi , f ), (wi , ψi ) = P (gi , f ).(g, 0) (solution of problem (PLP )), for i = 1, 2). According to Theorem 11.11 and Proposition 11.15 we have (ϕ1 − ϕ2 )ψ2 dxdt + b (u1 − u2 )w2 dxdt | |a Q

Q

≤ a ϕ1 − ϕ2 L2 (Q) ψ2 L2 (Q) +b u1 − u2 L2 (Q) w2 L2 (Q) ≤ C0 g 2L2 ,

|a

(ϕ1 − ϕobs )(ψ1 − ψ2 )dxdt + b

Q

(11.27) (u1 − uobs )(w1 − w2 )dxdt |

Q

≤ a ϕ1 − ϕobs L2 (Q) ψ1 − ψ2 L2 (Q) +b u1 − uobs L2 (Q) w1 − w2 L2 (Q) 3/2

≤ C1 C2 (uobs , vobs ) g L2 . From (11.26) and (11.27) we can deduce that for α ≥ αl such that αl > C0 and 1/2 (αl − C0 ) min g L2 = C1 C2 g∈K1

(Pf (g1 )

Pf (g2 )).g

the estimate − ≥ 0 holds and, therefore, Pf is convex. In the same way, we can find βl such that Qg is concave for β ≥ βl . We shall prove now that Pf is lower semi-continuous for all f ∈ K2 , and Qg is upper semi-continuous for all g ∈ K1 . Let gk be a minimizing sequence of J, i.e.,

386


lim inf J(gk , f ) = min J(g, f ) ∀f ∈ K2 . k

g∈K1

Then gk is uniformly bounded in K1 and we can extract from gk a subsequence also denoted by gk such that gk gf weakly in K1 . By using Proposition 11.16, we then have P(gk , f ) −→ (uf , ϕf ) strongly in L2 (Q). Therefore, since the norm is lower semi-continuous, we have that the mapping Pf : g −→ J(g, f ) is lower semi-continuous for all f ∈ K2 . By using the same technique as above, we obtain that Qg is upper semi-continuous for all g ∈ K1 . In order to obtain the necessary optimality conditions which are satisfied by the solution of the robust control problem, we introduce the following adjoint problem corresponding to the primal problem (11.22) (we denote by u = (u, ϕ) = P(g, f ) and (U1 , φ1 ) = (u + U, ϕ + φ)): −

∂p − νΔp − G1 (x, t, ϕ)p − U1 G2 (x, t, ϕ)p ∂t +H1 (x, t, ϕ)∇q.∇U1 − div(D2 (x, t, u)∇q)

∂D2 (x, t, u)∇q∇φ1 = a(ϕ − ϕobs ) on Q, ∂ϕ ∂q ∂D2 − div(D1 (x, t, ϕ)∇q) + (x, t, u)∇q∇φ1 − ∂t ∂u −F2 (x, t, ϕ)p = b(u − uobs ) on Q, +

(11.28)

∂q ∂p = 0, = 0 on Σ, ∂n ∂n p(T ) = 0, q(T ) = 0 on Ω. Remark 11.18. The adjoint problem (11.28) is a linear system. By reversing sense of time, i.e., t := T − t, and by following the same procedure as used to derive the result of Theorem 11.13, we obtain the existence and uniqueness of (p, q). ♦ Now we can give the first-order optimality conditions for the robust control problem (11.24). Theorem 11.19. Under the assumptions of Theorem 11.17, the optimal solution (g ∗ , f ∗ , u∗ , ϕ∗ ) ∈ K × W21 × W31 , such that (g ∗ , f ∗ ) is defined by (11.24) and (u∗ , ϕ∗ ) = P(g ∗ , f ∗ ) is a solution of (11.22), satisfies (B1∗ q ∗ (0) + αg ∗ )(g − g ∗ )dx ≥ 0, Ω (11.29) (B2∗ p∗ − βf ∗ )(f − f ∗ )dxdt ≤ 0 for all (g, f ) ∈ K, Q

where (p∗ , q ∗ ) is the solution of the adjoint problem (11.28), corresponding to the primal solution (u∗ , ϕ∗ ).


387

Proof. The cost function J is a composition of F-differentiable mappings then J is F-differentiable and we have (∀h = (h1 , h2 ) ∈ K) J (g, f ).h = a (ϕ − ϕobs )ψdxdt + b (u − uobs )wdxdt Q Q (11.30) +α gh1 dx − β f h2 dxdt, Q

Ω

where (w, ψ) = P (g, f ).h is the solution of problem (PLP ). Multiplying the first part of (PLP ) by p and the second part by q, using Green’s formula, and integrating with respect to time, we obtain (according to the homogeneous Neumann boundary conditions and the fact that H2 (x, t, u).w = (∂D2 /∂u)(x, t, u)w + (∂D2 /∂ϕ)(x, t, u)ψ) ∂p − νΔp − G1 (., t, ϕ)p − U1 G2 (., t, ϕ)p)ψdxdt (− ∂t Q = F2 (., t, ϕ)pwdxdt + B2∗ p.h2 dxdt − p(T )ψ(T )dx, Q

Q

Ω

∂q ∂D2 + (., t, u)∇q.∇φ1 − div(D1 (., t, ϕ)∇q))wdxdt (− ∂t ∂u Q = B1∗ q(0).h1 dx − q(T )w(T )dx Ω Ω div(D2 (., t, u)∇q)ψdxdt +

(11.31)

Q

−

(H1 (., t, ϕ)∇q.∇U1 +

Q

∂D2 (., t, u)∇q.∇φ1 )ψdxdt. ∂ϕ

Since (p, q) satisfies System (11.28), with null final conditions, we have ∂D2 (., t, u)∇q∇φ1 )ψdxdt (−H1 (., t, ϕ)∇q.∇U1 + div(D2 (., t, u)∇q) − ∂ϕ Q + a(ϕ − ϕobs )ψdxdt = (F2 (., t, ϕ)p)wdxdt + B2∗ p.h2 dxdt, Q Q Q (F2 (., t, ϕ)p)wdxdt + b(u − uobs )wdxdt = B1∗ q(0).h1 dx Q Q Ω ∂D2 (., t, u)∇q.∇φ1 )ψdxdt + (−H1 (., t, ϕ)∇q.∇U1 + div(D2 (., t, u)∇q) − ∂ϕ Q and, therefore,

b(u − uobs )wdxdt + a(ϕ − ϕobs )ψdxdt Q Q = B1∗ q(0).h1 dx + B2∗ p.h2 dxdt. Q

Ω

According to the expression of J (g, f ).h we can deduce that

(11.32)

388


J (g, f ).h =

(B1∗ q(0) + αg).h1 dx +

Ω

(B2∗ p − βf ).h2 dxdt.

Q

Since (g ∗ , f ∗ ) is a saddle point of J we have ∂J ∗ ∗ ∂J ∗ ∗ (g , f ).(g − g ∗ ) ≥ 0 and (g , f ).(f − f ∗ ) ≤ 0, ∀(g, f ) ∈ K ∂g ∂f

and then

(B1∗ q ∗ (0) + αg ∗ )(g − g ∗ )dx ≥ 0,

Ω

(B2∗ p∗ − βf ∗ )(f − f ∗ )dxdt ≤ 0 ∀(g, f ) ∈ K,

(11.33)

Q ∗

∗

where (p , q ) is the solution of the problem (11.28), corresponding to the primal solution (u∗ , ϕ∗ ) = P(g ∗ , f ∗ ). This completes the proof. Remark 11.20. In the case where we take into account the final observation, we obtain the same results. In this case the cost functional can be given as the form a2 a1 ϕ − ϕobs 2L2 (Q) + u − uobs 2L2 (Q) J(g, f ) = 2 2 a3 a4 2 + ϕ(T ) − ηobs L2 + u(T ) − vobs 2L2 (11.34) 2 2 α β + g 2L2 − f 2L2 (Q) , 2 2 i = 1, 4, α and β are fixed and such that α, β > 0, ai ≥ 0, for where ai , for i = 1, 4, and i=1,4 ai > 0. The functions (uobs , ϕobs ) ∈ L2 (Q) × L2 (Q) and (ηobs , vobs ) ∈ H 1 (Ω) × 2 L (Ω) are given and represent the observation. We can prove also an existence theorem for the robust control problem and obtain the necessary optimality conditions for its solution using the same method. Let K = K1 × K2 such that K1 and K2 are non-empty, closed, convex, bounded subsets of L2 (Ω) and L2 (Q), respectively. Let assumptions (H1)–(H5) be satisfied. Then for sufficiently large α and β, there exists (u, ϕ, p, q, g ∗ , f ∗ ) satisfying ∂ϕ − νΔϕ = F1 (x, t, ϕ) + uF2 (x, t, ϕ) ∂t +U (F2 (x, t, ϕ) − F2 (x, t, 0)) + B2 f ∗ on Q, ∂u − div(D1 (x, t, ϕ)∇u + D2 (x, t, u)∇ϕ) ∂t = div((D1 (x, t, ϕ) − D1 (x, t, 0))∇U ) +div((D2 (x, t, u) − D2 (x, t, 0))∇φ) on Q,


−

389

∂p − νΔp − G1 (x, t, ϕ)p − U1 G2 (x, t, ϕ)p ∂t +H1 (x, t, ϕ)∇q.∇U1 − div(D2 (x, t, u)∇q)

∂D2 (x, t, u)∇q∇φ1 = a1 (ϕ − ϕobs ) on Q, ∂ϕ ∂D2 ∂q − div(D1 (x, t, ϕ)∇q) + (x, t, u)∇q∇φ1 − ∂t ∂u −F2 (x, t, ϕ)p = a2 (u − uobs ) on Q, +

∂u ∂p ∂q ∂ϕ = 0, = 0, = 0, = 0 on Σ, ∂n ∂n ∂n ∂n (ϕ(0), u0 ) = (ϕ0 , B1 g ∗ ) on Ω, p(T ) = a3 (ϕ(T ) − ηobs ), q(T ) = a4 (u(T ) − vobs ) on Ω and inequalities

(B1∗ q(0) + αg ∗ )(g − g ∗ )dx ≥ 0, Ω (B2∗ p − βf ∗ )(f − f ∗ )dxdt ≤ 0 Q

∀(g, f ) ∈ K. ♦

11.5.2 Distributed Disturbance in the Initial Condition of the Phase-field Variable In this section, the disturbance is in the initial condition of the phase-field variable ϕ0 and the control is in the initial condition of the concentration u0 , i.e., ϕ0 = B2 f (f ∈ L2 (Ω)), u0 = B1 g (g ∈ L2 (Ω)), where B1 is a bounded operator from L2 (Ω) into H 1 (Ω) and B2 is a bounded operator from L2 (Ω) into H02 (Ω). So the function (ϕ, u) is assumed to be related to the disturbance f and control g through the problem (11.15): ∂ϕ − νΔϕ = F1 (x, t, ϕ) + uF2 (x, t, ϕ) ∂t +U (F2 (x, t, ϕ) − F2 (x, t, 0)) on Q, ∂u − div(D1 (x, t, ϕ)∇u + D2 (x, t, u)∇ϕ) ∂t = div((D1 (x, t, ϕ) − D1 (x, t, 0))∇U )

(11.35)

+div((D2 (x, t, u) − D2 (x, t, 0))∇φ) on Q, ∂u ∂ϕ = 0, = 0 on Σ, ∂n ∂n ϕ(0) = B2 f, u(0) = B1 g on Ω. To obtain the regularity of Theorem 11.11, we suppose that (g, f ) ∈ L2 (Ω)2 and (U, φ) ∈ W21 ×W31 . Let P : (g, f ) −→ (u, ϕ) = P(g, f ) be the map:

390


(L2 (Ω))2 −→ W21 × W31 defined by (11.35). The cost function and the saddle point problem that we want to study are the same as in the previous section, i.e., the functional defined by (11.23) and the problem defined by (11.24). The arguments of the previous sections extend directly to the present case without further estimates. We have then the following results. Proposition 11.21. The function P is continuously F-differentiable from (L2 (Ω))2 to W1 × W2 where the derivative P (g, f ) : h = (h1 , h2 ) −→ (w, ψ) is a linear mapping from (L2 (Ω))2 into W1 × W2 such that P (g, f ).h satisfies the linear problem (PLPB ) ∂ψ − νΔψ = G1 (x, t, ϕ)ψ + uG2 (x, t, ϕ)ψ ∂t +wF2 (x, t, ϕ) + U G2 (x, t, ϕ)ψ on Q, ∂w − div(D1 (x, t, ϕ)∇w + D2 (x, t, u)∇ψ) ∂t = div(H1 (x, t, ϕ)ψ∇U1 + (H2 (x, t, u).w)∇φ1 ) on Q, ∂w ∂ψ = 0, = 0 on Σ, ∂n ∂n ψ(0) = B2 h2 w(0) = B1 h1 on Ω, where (U1 , φ1 ) = (u + U, ϕ + φ). Moreover, we have the estimates (for all (fi , gi ) ∈ (L2 (Ω))2 , for i = 1, 2): (i) P (f1 , g1 ) L((L2 (Ω))2 ,W1 ×W2 ) ≤ Ce (ii) P (f1 , g1 ) − P (f2 , g2 ) 2W1 ×W2 ≤ Ce {( f 2L2 + g 2L2 )1/2 ( h1 2L2 + h2 2L2 ) + ( f 2L2 + g 2L2 )( h1 2L2 + h2 2L2 )1/2 }, where f = f1 − f2 and g = g1 − g2 .

Proposition 11.22. Let assumptions (H1)–(H5) be satisfied. Then the map P defined by (11.35) is continuous from the weak topology of (L2 (Ω))2 to the strong topology of (L2 (Q))2 . Theorem 11.23. Let assumptions (H1)–(H5) be satisfied. Then, for α and β sufficiently large, there exist (g ∗ , f ∗ ) ∈ K and (u∗ , ϕ∗ ) ∈ W21 × W31 such that (g ∗ , f ∗ ) is defined by (11.24) and (u∗ , ϕ∗ ) = P(g ∗ , f ∗ ) is a solution of (11.35). Theorem 11.24. Let assumptions (H1)–(H5) be satisfied, α and β sufficiently large, (g ∗ , f ∗ ) ∈ K and (u∗ , ϕ∗ ) ∈ W21 × W31 such that (g ∗ , f ∗ ) is defined by (11.24) and (u∗ , ϕ∗ ) = P(g ∗ , f ∗ ) is a solution of (11.35). Then


391

(B1∗ q ∗ (0) + αg ∗ )(g − g ∗ ) ≥ 0,

Ω

(11.36)

(B2∗ p∗ (0) Ω

∗

∗

− βf )(f − f ) ≤ 0, ∀(g, f ) ∈ K,

where (p∗ , q ∗ ) is the solution of the adjoint problem (11.28), corresponding to the primal solution (u∗ , ϕ∗ ). Remark 11.25. We can consider other types of controls and disturbances, the technique developed previously is still valid. For example in the case where the control is in the initial condition of the order parameter, i.e., g = ϕ0 and the distributed disturbance is in the forcing ξ, i.e., ξ = f , we can prove, under the assumptions (H1)–(H5) and for α and γ sufficiently large, the existence theorem of the robust control problem and obtain necessary optimality conditions for its solution using the same method as previously. In this case the cost functional can be given as the form (if we take also into account the final observation) a2 a1 ϕ − ϕobs 2L2 (Q) + u − uobs 2L2 (Q) J(g, f ) = 2 2 a3 a4 2 + ϕ(T ) − ηobs L2 + u(T ) − vobs 2L2 (11.37) 2 2 α β + g 2L2 − f 2L2 (Q) , 2 2 where ai , for i = 1, 4, α and β are fixed and such that α, β > 0, ai ≥ 0, for i = 1, 4, and i=1,4 ai > 0. The functions (uobs , ϕobs ) ∈ L2 (Q) × L2 (Q) and (ηobs , vobs ) ∈ H 1 (Ω) × 2 L (Ω) are given and represent the observation. Let K = K1 × K2 such that K1 and K2 are non-empty, closed, convex, bounded subsets of L2 (Ω) and L2 (Q), respectively. For sufficiently large α and β, there exists (u, ϕ, p, q, g ∗ , f ∗ ) satisfying ∂ϕ − νΔϕ = F1 (x, t, ϕ) + uF2 (x, t, ϕ) ∂t +U (F2 (x, t, ϕ) − F2 (x, t, 0)) + B2 f ∗ on Q, ∂u − div(D1 (x, t, ϕ)∇u + D2 (x, t, u)∇ϕ) ∂t = div((D1 (x, t, ϕ) − D1 (x, t, 0))∇U ) +div((D2 (x, t, u) − D2 (x, t, 0))∇φ) on Q, ∂p − νΔp − G1 (x, t, ϕ)p − U1 G2 (x, t, ϕ)p ∂t +H1 (x, t, ϕ)∇q.∇U1 − div(D2 (x, t, u)∇q) ∂D2 (x, t, u)∇q∇φ1 = a1 (ϕ − ϕobs ) on Q, + ∂ϕ ∂D2 ∂q − div(D1 (x, t, ϕ)∇q) + (x, t, u)∇q∇φ1 − ∂t ∂u −F2 (x, t, ϕ)p = a2 (u − uobs ) on Q,

−

392


∂u ∂p ∂q ∂ϕ = 0, = 0, = 0, = 0 on Σ, ∂n ∂n ∂n ∂n (ϕ(0), u0 ) = (B1 g ∗ , u0 ) on Ω, p(T ) = a3 (ϕ(T ) − ηobs ),

q(T ) = a4 (u(T ) − vobs ) on Ω

and inequalities

(B1∗ p(0) + αg ∗ )(g − g ∗ )dx ≥ 0, Ω (B2∗ p − βf ∗ )(f − f ∗ )dxdt ≤ 0 Q

∀(g, f ) ∈ K. ♦

Remark 11.26. (i) To obtain more realistic simulations of dendrites using the phase-field system, it is interesting to include the anisotropy in our studied system. The most widely used method to include this anisotropy for twodimensional case is to assume that Wφ = φ / (given in (11.2)) depends on an angle θ. The angle θ is corresponding to the orientation of the normal to the interface with respect to the x-axis given by tan θ = (∂φ/∂y)/(∂φ/∂x). According to Kobayashi [175] (see also Taylor and Cahn [280] for a discussion about the choice of the anisotropy formulation), the coefficient can be written as φ = (1 + 0 cos(Ns θ)), where Ns ∈ IN is corresponding to the number of branching directions and 0 ∈ [0, 1[ is corresponding to the anisotropy amptitude and is assumed to be sufficiently small. Consequently, the problem (11.8) can be reformulated, by taking into account the anisotropy, as follows: ∂φ − div(A(∇φ)∇φ) = F1 (x, t, φ) + U F2 (x, t, φ) + λ on Q, ∂t ∂U − div(D1 (x, t, φ)∇U + D2 (x, t, U)∇φ) = 0 on Q, ∂t subject to the boundary conditions

(11.38)

A(∇φ)∇φ.n = 0, (D1 (x, t, φ)∇U + D2 (x, t, U)∇φ).n = 0 on Σ, and the initial conditions φ(0) = φ0 , U (0) = U0 on Ω, where A is an anisotropy tensor. If we assume some strong regularity for the non-linear operator A, we can extend directly the arguments of the previous sections to the present case. For example, in order to have a well-posedeness problem (11.38), we can make the following assumptions:


393

(A1) There exists ν > 0 such that ν v1 − v2 2L2 ≤ (A(v1 )v1 − A(v2 )v2 )(v1 − v2 )dx, Ω

for all (v1 , v2 ) ∈ L (Ω) × L (Ω). (A2) There exists M > 0 such that 2

2

A(v1 )v1 − A(v2 )v2 L2 ≤ M v1 − v2 L2 , for all (v1 , v2 ) ∈ L2 (Ω) × L2 (Ω). (ii) It is clear that we can treat, by using the same technique developed in this chapter and more generally in this book, (at least from the “formal” viewpoint), different general physical models concerning the solidification process, for example the problem presented by Granasy et al. [137, 138, 139] and Warren et al. [293, 294, 295]. ♦ Remark 11.27. As indicated in Remark 10.31, for numerical resolution of the robust control problems, the reader is referred to Chapter 9. ♦

12 Large-scale Ocean in the Climate System

In this chapter, we study robust control problems arising from oceanic currents. The equations are non-linear, time-dependent and coupled, and are of Navier–Stokes type for the velocity and pressure, and of transport-diffusion type for the temperature and the salinity, with Robin-type boundary conditions. The objective is the prediction and robust regulation of the circulation from the mean circulation by taking into account the worst disturbance caused by small variations of the surface temperature. First, the mathematical models are introduced and the existence, uniqueness and regularity results of the solution of the state systems are studied. The asymptotic behavior is also considered. Afterwards, robust control problems are formulated. The controls and disturbances are of Robin type and act on a part of the boundary during a time T . The existence of a robust control in the admissible sets is proved and, finally, first-order necessary conditions of optimality are obtained. The problem is considered first for a system with Boussinesq approximation and second for a system with hydrostatic approximation with vertical viscosity.

12.1 Introduction and Formulation of the Problem 12.1.1 Motivation There are two types of currents whose main causes are solar radiation (the warming of the surface by downwelling radiation coming from the atmosphere), wind and gravity. The Earth unevenly receives solar energy, which is not the same in the polar region as in the equatorial region. Consequently, the intertropical zone receives so much more energy than the rest of the planet, which creates a thermal imbalance. This imbalance is the origin of the two types of oceanic currents (surface currents and deep currents), and, indeed, puts in motion the atmosphere and oceanic currents which trying to thermally readjust the whole system. These movements are influenced by a force

396


caused by the rotation of the Earth, called the Coriolis force. Moreover, it generates winds, which are the main causes of surface currents (which affect approximately 10% of the water of the oceans and are generally limited to the top 400 meters of the oceans). The location of these surface currents changes significantly with the seasons; this phenomenon is particularly sensitive for the equatorial currents. The thermal imbalance leads also to differences in temperature depending on latitude1 (between the various layers of the ocean). This temperature difference leads to a difference in salinity and thus in density, creating deep oceanic currents (the difference in density is a function of the difference in temperature and salinity). Consequently, the surface currents and deep currents are interconnected. Thanks to the heat capacity of water (its thermal inertia is much larger than that of the air), the ocean tempers the seasonal temperature changes of the air masses, which otherwise would be much more important. Thus currents of the warm surface layers can warm a region’s climate. In contrast, the rise of cold water from the sea depths to the surface moderates water temperatures in the equatorial region.2 Therefore, the surface temperature variation plays an important role in oceanic current movements, which play a key role in the regulation of the climate, and ensure that a heat transport from the equator to the poles is as important as the atmosphere. We can not neglect the fluctuation caused by this variation (even if the variation is small) and consequently the analysis of its influence is very important. The phenomenon that we want to model occurs in the tropical Atlantic and Pacific oceans: the circulation there is characterized by steady zonal currents and by long waves propagating westward along the equator and superposed on the mean currents. The equatorial waves are connected with strong vertical currents, called convective currents, which are very sensitive to small changes in temperature and induce “upwellings” or “downwellings.” These phenomena modify the properties of the sea water near the surface –decreasing or increasing the temperature, supply of plankton etc.– and are therefore of great importance for climate, fishing etc. This climate variability can be illustrated, for example, by the El N i˜ no phenomenon, which occurs every two to seven years. El N i˜ no warming begins with a modest temperature elevation (2 or 3◦ C) of surface waters in the Pacific ocean along the equatorial coast of South America. The El N i˜ no current then cuts the cold and deep upwelling (it results in a reduction of food, and, consequently, causes fish populations to decline). This phenomenon is accompanied with a sharp increase in rainfall and flooding. In contrast, around Australia and Asia, the high pressures and cold water temperature decrease the precipitation, and then cause severe drought, which often lead to a series of fires. 1 2

The deep waters sink in the ocean basins located at high latitudes, where temperatures are low enough so that the density increases. But this movement is still poorly understood because it is difficult to measure directly.

12.1 Introduction and Formulation of the Problem

397

These equatorial waves have been evidenced from “in situ” observations, and more recently from altimetric measurements3 and data assimilation in connection with control theory, see Belmiloudi et al. [27, 28, 30, 31, 33, 34, 35, 36, 37], in which the authors analyze the process of these tropical instability waves. The equations of the large-scale systems of the ocean are derived from Navier–Stokes equations but take into account oceanographical assumptions such as Boussinesq approximation, i.e., the density variations ρ are neglected in the system except in the buoyancy term (on the other hand, density is assumed to be constant and equal to a mean value ρav in the equations describing the horizontal motion) and in the equation of state. Of course, there is a vast literature concerning the control theory and inverse problems in connection with fluid mechanics problems. The reader is referred, for example, to Blayo et al. [53], Gejadze et al. [129, 130], Gunzburger et al. [144, 145], Marchuk [217, 218], Parmuzin et al. [235], Sritharan et al. [119, 120, 273], Wunsch [303] and the references therein. For robust control analysis of Navier–Stokes equations in the Riccati operator, see Barbu and Srithran [22], in which the authors assume appropriate detectability and stability constraints on the system; while for a study using a similar approach as developed in this book, see Bewley et al. [52]. 12.1.2 Primitive Equations and Study Domain We consider a time interval (0, T ) and an oceanic domain Ω extending on both sides of the equator (10◦ S–10◦ N), and of constant depth H (for example H = 3000 m). The curvature of the Earth is neglected. The vertical extension of Ω (−H ≤ z ≤ 0) corresponds to a part of the physical domain where the variability of the mean current is large. The studied model includes the following unknown functions: the velocity u, the pressure p, the temperature T and the salinity S. Taking the explicit equation of state given by Washington and Parkinson [296] into consideration: ρ = δ0 − δT T + δS S, the total Boussinesq equations of the ocean are as follows: ∂u 1 ρ + (u.∇)u + F ∧ u − div(ν1 ∇u) + ∇p = G on Q, ∂t ρav ρav div(u) = 0 on Q, ∂T + (u.∇)T − div(ν2 ∇T ) = 0 on Q, ∂t ∂S + (u.∇)S − div(ν3 ∇S) = 0 on Q, ∂t ρ = δ0 − δT T + δS S, 3

(12.1)

Altimetric measurements give the distance between the satellite and the sea surface.

398


where Q = Ω × (0, T ), G = (0, 0, −g) is the gravity force, (x, y, z) are the cartesian coordinates: x, y are measured in the horizontal plane of the undisturbed sea-surface (x towards the east, y towards the north) and z is vertically ascendant. The constants δT and δS are expansion coefficients. The function F ∧ u = (0, 0, 2ω sin φ) ∧ u is the Coriolis acceleration, ω the rotation rate of Earth, φ the latitude (this acceleration becomes dominant for large-scale flows (see, e.g., Pedlosky [236]), because the relative acceleration becomes small when the scale of the motion is large). Afterwards, we will suppose that ρav is equal to 1 (ρav =1 g cm−3 ). The sensitivity of ocean general circulation models to the parametrization of the vertical turbulent diffusion has been known for a long time and has been proved by Bryan [65]. The turbulent flux is usually modeled by the dissipative terms and linked to large-scale oceans by using the mixing coefficients: ν1h , ν2h and ν3h (resp. ν1v , ν2v and ν3v ) denote the coefficients of horizontal (resp. vertical) eddy viscosity and diffusivity. Moreover, the turbulent mixing is high above the thermocline and low in stratified regions. According to the parametrization described in Philander et al. [239, 240], the coefficients ν1v , ν2v and ν3v are variable, positive and bounded functions. They are deduced from the given mean circulation. The coefficients ν1h , ν2h and ν3h are constant and positive. The total circulation (u, p, T , S) is the sum of the mean circulation ˜ u, p˜, T˜ , S), (u0 , p0 , T0 , S0 ), which is given, and a variability (a perturbation) (˜ which is corresponding to small deviations from the target flow-temperaturesalinity (u0 , p0 , T0 , S0 ). This expansion is justified in tropical region: the steady mean circulation (u0 , p0 , T0 , S0 ) is known for each tropical season; ˜ is made of westward propagating waves. We asthe variability (˜ u, p˜, T˜ , S) sume that, at initial time (t = 0) the flow in the domain Ω is the mean circulation (u0 , p0 , T0 , S0 ) which satisfies steady-state Equations (12.1), i.e., (u0 , p0 , T0 , S0 ) satisfies the system (since ρav = 1) −ν1h Δ2 u0 −

∂u0 ∂ (ν1v ) + (u0 .∇)u0 ∂z ∂z +F ∧ u0 + ∇p0 = ρ0 G on Ω,

div(u0 ) = 0 on Ω, ∂T0 ∂ (ν2v ) + (u0 .∇)T0 = 0 on Ω, ∂z ∂z ∂S0 ∂ (ν3v ) + (u0 .∇)S0 = 0 on Ω, −ν3h Δ2 S0 − ∂z ∂z ρ0 = δ0 − δT T0 + δS S0 .

(12.2)

−ν2h Δ2 T0 −

˜ to The full non-linear system which models large perturbation (˜ u, p˜, T˜ , S) the target (u0 , p0 , T0 , S0 ) can be deduced from (12.1) and (12.2):

12.1 Introduction and Formulation of the Problem

399

∂u ˜ + (˜ u.∇)u0 + (u0 .∇)˜ u + (˜ u.∇)˜ u + F ∧ u˜ ∂t ∂ ∂u ˜ ˜ (ν1v ) + ∇˜ p = (−δT T˜ + δS S)G on Q, ˜− −ν1h Δ2 u ∂z ∂z div(˜ u) = 0 on Q, ∂ T˜ + (˜ u.∇)T0 + (u0 .∇)T˜ + (˜ u.∇)T˜ ∂t ∂ ∂ T˜ (ν2v ) = 0 on Q, −ν2h Δ2 T˜ − ∂z ∂z ∂ S˜ + (˜ u.∇)S0 + (u0 .∇)S˜ + (˜ u.∇)S˜ ∂t ∂ ∂ S˜ (ν3v ) = 0 on Q, −ν3h Δ2 S˜ − ∂z ∂z

(12.3)

where Δ2 . = ∂ 2 ./∂x2 + ∂ 2 ./∂y 2 . Nota bene: The notation “˜” used for the perturbation of the mean circulation will now be omitted. The flow domain Ω can be defined as Ω = ]0, Lx[ × ]−Ly, Ly[× ]−H, 0[ . Γ denotes its boundary: Γ = Γ0 ∪ Γ5 ∪ Γ1 ∪ Γ2 ∪ Γ3 ∪ Γ4 where Γ0 denotes the surface (z = 0), Γ5 the bottom (z = −H), Γ1 and Γ2 the eastern and western boundaries, Γ3 and Γ4 the northern and southern boundaries. We note that the vertical extension −H ≤ z ≤ 0 of the domain Ω doesn’t correspond to the physical sea water but only to the layer where the variability is computed. We assume that for depths greater than H the perturbation is negligible. The perturbation of the current is made of zonal propagating waves. Therefore, we can choose the zonal extension of Ω as the greatest wavelength, and impose periodic conditions on the eastern and western boundaries. To take into account the phenomena we want to describe, we set mixed boundary conditions: • the flow is periodic in the x-direction: u |Γ1 = u |Γ2 , T |Γ1 = T |Γ2 and S |Γ1 = S |Γ2 , • on the northern and southern boundaries we impose a sliding condition for the flow and an homogeneous Dirichlet condition (12.4) for the temperature-salinity: ∂w ∂u = 0 and = 0 on Γ3 and Γ4 , v = T = S = 0, ∂y ∂y • the perturbation vanishes at z = −H: u = 0 and T = S = 0 on Γ5 , where u = (u, v, w). In the surface Γ0 the systems (12.3) with (12.4) are supplemented with the following Robin-type boundary conditions

400


∂u ∂v = f1 , ν1v = f2 , w = 0, ∂z ∂z ∂T (12.5) + δT = , ν2v ∂z ∂S ν3v = 0, ∂z where u = (u, v, w), δ is a positive constant related with the turbulent heating on the surface of the ocean, f = (f1 , f2 ) is the variability of the wind stress on the surface of the ocean and is the disturbance of the heat stress on the surface of the ocean. We assume that, at initial time t = 0, the mean circulation is not disturbed. Therefore, the initial condition is: ν1v

(u, T , S)(t = 0) = 0.

(12.6)

Remark 12.1. The perturbation of the mean circulation is driven by the perturbation of the surface stress, which depends, for example, on the perturbation of the velocity of the atmosphere. If we suppose that Γ0 is the upper surface of the ocean, the perturbation of the vertical velocity w (with w = u.n) vanishes on Γ0 . This condition is classically named the impermeability condition on the sea surface. Moreover, the perturbation is not computed in the thin surface layer 0 ≤ z ≤ ξ, where z = ξ is the free sea surface observed by the satellite. ♦ Our purpose is motivated by the robust regulation of the deviation of circulation from the mean circulation (u0 , T0 , S0 , , p0 ), by taking into account the worst disturbance caused by the small variation of the surface temperature. The variability of the surface stress f acts as the forcing of the perturbation and is unknown, and the variability of the surface temperature acts as the worst disturbance (generated by the warming of surface water). We take f as the control and as the disturbance in our robust control problem.

12.2 The Perturbation Problem In this section we develop some results of the existence, uniqueness and regularity of the solutions for the non-linear and three-dimensional cases. These results will still be valid for the non-linear and two-dimensional, and for the linear and three-dimensional case. After introducing some notations and preliminary results useful in the linear as well as in the non-linear situation (in two or three-dimensional cases), we give the weak (or variational) formulations of the problem and the main existence, uniqueness and regularity results. 12.2.1 Preliminary Results and Weak Formulations In order to study the problem (12.3) with the boundary conditions (12.4) and (12.5), and the initial condition (12.6), we introduce the following functional


401

spaces: V0 = {v ∈ (H 1 (Ω))3 : div(v) = 0, v.n = 0 on Γ }, H1 = {v ∈ (L2 (Ω))3 : div(v) = 0, v.n = 0 on Γ0 ∪ Γ3 ∪ Γ4 ∪ Γ5 , v.n|Γ1 = −v.n|Γ2 }, W1 = {v ∈ (H 1 (Ω))3 : v = 0 on Γ5 , v.n = 0 on Γ0 ∪ Γ3 ∪ Γ4 , v|Γ1 = v|Γ2 }, V1 = {v ∈ W1 : div(v) = 0}, H2 = L2 (Ω), V2 = {φ ∈ H 1 (Ω) : φ = 0 on Γ3 ∪ Γ4 ∪ Γ5 , φ|Γ1 = φ|Γ2 }, H = H1 × H2 × H2 ,

V = V 1 × V2 × V2 ,

where n is the unit outward vector normal to Γ . Nota bene: In the following, if X denotes some Banach space of real-valued functions, the corresponding space of vector-valued functions, each of components belonging to X, will be denoted, for simplicity, by the same notation X, and we use . X to denote the norms of spaces of real-valued functions or of vector-valued functions X. For example L2 (Ω) (respectively L2 (Γ0 )) is the corresponding space of vector-valued functions of L2 (Ω) (respectively L2 (Γ0 )), etc. We use the following notations: | . |1,Ω and . 1,Ω denote, respectively the semi-norm and the norm on H 1 (Ω). They are equivalent on W1 , V1 and V2 , and we set u =| u |1,Ω = u W1 = u V1 , φ =| φ |1,Ω = φ V2 . We denote by | u | and | φ | the norm in L2 (Ω), and by | . |Γ0 and (., .)Γ0 the norm and the scalar product in L2 (Γ0 ). Remark 12.2. (i) If v ∈ H(div; Ω), then v.n makes sense on the boundary Γ (see Chapter 3). (ii) If v ∈ H1 (respectively v ∈ V1 ), then v can be extended as a free divergent, 1 ). x-periodic function, which belongs to the space L2loc (resp. Hloc (iii) If φ ∈ V2 , then φ can be extended as a x-periodic function, which belongs 1 . ♦ to the space Hloc We now define the following forms: ∂u ∂v ∂u ∂u , ), with ∇2 u = ( , ), ∂z ∂z ∂x ∂y ∂T ∂φ , ) + δ(T , φ)Γ0 , a2 (T , φ) = ν2h (∇2 T , ∇2 φ) + (ν2v ∂z ∂z ∂S ∂ψ , ), d(u, v) = (F ∧ u, v), a3 (S, ψ) = ν3h (∇2 S, ∇2 ψ) + (ν3v ∂z ∂z b1 (u, v, w) = ((u.∇)v, w), b2 (u, φ, ψ) = ((u.∇)φ, ψ), a1 (u, v) = ν1h (∇2 u, ∇2 v) + (ν1v

l0 (u, v) = ((u0 .∇)u, v) + ((u.∇)u0 , v) + d(u, v),

402


l1 (T , S, v) = −((−δT T + δS S)G, v), l2 (u, T , φ) = ((u0 .∇)T , φ) + ((u.∇)T0 , φ), l3 (u, S, ψ) = ((u0 .∇)S, ψ) + ((u.∇)S0 , ψ), c(v, p) = −(div(v), p). We now recall some properties of the previous operators, which can be found in Belmiloudi [35]. Lemma 12.3. For the operators d, (ai )i=1,3 and (bi )i=1,2 the following properties hold: (i) a1 (respectively ai ,for i = 2, 3) is a bilinear continuous and coercive form on W1 × W1 (respectively on V2 × V2 ), where the coercivity constants are denoted by αi , for i = 1, 3 (ii) d is a bilinear continuous form on W1 × W1 (iii) b1 (respectively b2 ) is a trilinear continuous form on W1 × W1 × W1 (respectively W1 × V2 × V2 ). By using the interpolation theory and Gagliardo–Nirenberg inequalities (see Chapter 3), we can easily prove the following classical estimates (see, for example, Belmiloudi [32] and Temam [281]). Lemma 12.4. There exists a positive constant C (depending on Ω) such that the following estimates hold: 1/4

1/4

| b1 (u, v, w) |≤ C u L2 (Ω) u 3/4 v w L2 (Ω) w 3/4 for all u, v, w ∈ W1 , 1/2

| b1 (u, v, w) |≤ C u v 1/2 v H 2 (Ω) w L2 (Ω) for all u ∈ W1 , v ∈ H 2 (Ω) ∩ W1 , w ∈ H1 , 1/2

| b1 (u, v, w) |≤ C u 1/2 u H 2 (Ω) u w L2 (Ω)) for all u ∈ H 2 (Ω) ∩ W1 , v ∈ W1 , w ∈ H1 , 1/4

1/4

| b2 (u, ψ, φ) |≤ C u L2 (Ω) u 3/4 ψ φ L2 (Ω) φ 3/4 for all u ∈ W1 , ψ, φ ∈ V2 ,

(12.7)

1/2

| b2 (u, ψ, φ) |≤ C u ψ 1/2 ψ H 2 (Ω) φ L2 (Ω) for all u ∈ W1 , ψ ∈ H 2 (Ω) ∩ V2 , φ ∈ H2 , 1/2

| b2 (u, ψ, φ) |≤ C u 1/2 u H 2 (Ω) ψ φ L2 (Ω)) for all u ∈ H 2 (Ω) ∩ W1 , ψ ∈ V2 , φ ∈ H2 .

In order to introduce the weak formulations associated with the perturbation problems (12.3)–(12.6), we give the following lemma. Lemma 12.5. Suppose that the function f = (f1 , f2 ) is in L2 (0, T ; L2(Γ0 )), the function is in L2 (0, T ; L2(Γ0 )) and that (u, p, T , S) is sufficiently regular and satisfies the boundary conditions (12.4) and (12.5), then the following relations hold:


(i)

403

For all v ∈ W1 , −ν1h (Δ2 u, v)−(

∂u ∂ (ν1v ), v)+(∇p, v) = a1 (u, v)+c(v, p)−(fT , v)Γ0 , ∂z ∂z

where fT = (f , 0). Moreover, if v ∈ V1 , then −ν1h (Δ2 u, v) − (

∂ ∂u (ν1v ), v) + (∇p, v) = a1 (u, v) − (fT , v)Γ0 . ∂z ∂z

(ii) For all φ ∈ V2 , −ν2h (Δ2 T , φ) − (

∂T ∂ (ν2v ), φ) = a2 (T , φ) − (, φ)Γ0 . ∂z ∂z

(iii) For all ψ ∈ V2 , −ν3h (Δ2 S, ψ) − ( (iv) For all u, v in W1 ,

∂S ∂ (ν3v ), ψ) = a3 (S, ψ). ∂z ∂z

d(u, v) = −d(v, u).

(v) For all u in V1 and v, w in W1 , b1 (u, v, v) = 0 and b1 (u, v, w) = −b1 (u, w, v). (vi) For all u in V1 and φ, ψ in V2 , b2 (u, φ, φ) = 0 and b2 (u, φ, ψ) = −b2 (u, ψ, φ). (vii) Since u0 ∈ V0 we have, for all v, w in W1 and φ, ψ in V2 , the following identities: b1 (u0 , v, v) = 0, b2 (u0 , φ, φ) = 0, b1 (u0 , v, w) = −b1 (u0 , w, v), b2 (u0 , φ, ψ) = −b2 (u0 , ψ, φ). Proof. Results (i)–(iv) are deduced from the definition of the spaces W1 , V1 and V2 , and from the boundary conditions satisfied by (u, T , S) on Γ . The orthogonality identities (v) and (vi) are consequences of div(u) = 0 and Green’s formula. Finally, result (vii) is a direct consequence of (v)–(vi). This completes the proof. According to Lemma 12.5, the problems (12.3)–(12.6) satisfied by the perturbation (u, p, T , S) of the mean flow admits the two following equivalent weak formulations: find (u, T , S) ∈ L2 (0, T ; V) such that, for all (v, φ, ψ) in V, a.e. in (0, T )

404


(

∂u , v) + a1 (u, v) + l0 (u, v) + l1 (T , S, v) + b1 (u, u, v) ∂t = (fT , v)Γ0 ,

∂T , φ) + a2 (T , φ) + l2 (u, T , φ) + b2 (u, T , φ) = (, φ)Γ0 , ∂t ∂S , ψ) + a3 (S, ψ) + l3 (u, S, ψ) + b2 (u, S, ψ) = 0, ( ∂t (u, T , S)(t = 0) = 0 (

(12.8)

and find (u, T , S, p) ∈ L2 (0, T ; W1 × V2 × V2 ) × D (Q) such that, for all (v, φ, ψ, q) in W1 × V2 × V2 × L2 (Ω), a.e. in (0, T ) (

∂u , v) + a1 (u, v) + l0 (u, v) + l1 (T , S, v) + b1 (u, u, v) ∂t +c(v, p) = (fT , v)Γ0 ,

∂T , φ) + a2 (T , φ) + l2 (u, T , φ) + b2 (u, T , φ) = (, φ)Γ0 , ∂t ∂S , ψ) + a3 (S, ψ) + l3 (u, S, ψ) + b2 (u, S, ψ) = 0, ( ∂t c(u, q) = 0, (

(12.9)

(u, T , S)(t = 0) = 0, where fT is the vector (f , 0). We end this section by the following lemma which shows the convergence results by applying the limit in the non-linear terms b1 and b2 (necessary to prove the existence and some regularity results). Lemma 12.6. If the sequence (uk , ψk )k converges towards (u, ψ) in the space L2 (0, T ; V1 × V2 ) weakly and in L2 (0, T ; H1 × H2 ) strongly, then for any vector function (λ(t), v, φ) such that λ ∈ C 1 (0, T ) and (v, φ) ∈ V1 × V2 we have: T T (i) b1 (uk (., t), uk (., t), λ(t)v)dt converges to b1 (u(., t), u(., t), λ(t)v)dt 0

T

0 T

b2 (uk (., t), ψk (., t), λ(t)φ)dt converges to

(ii) 0

b2 (u(., t), ψ(., t), λ(t)φ)dt. 0

Proof. (i) By using Lemma 12.5 we can deduce that T T b1 (uk (., t), uk (., t), λ(t)v)dt = − b1 (uk (., t), λ(t)v, uk (., t))dt. 0

0

According to the assumptions of the lemma, the following convergence result holds:


T

T

b1 (uk (., t), λ(t)v, uk (., t))dt −→ 0

405

b1 (u(., t), λ(t)v, u(., t))dt. 0

Consequently (by using again Lemma 12.5),

T

T

b1 (uk (., t), uk (., t), λ(t)v)dt −→ 0

b1 (u(., t), u(., t), λ(t)v)dt 0

and then result (i) follows. By using the same arguments as in (i), we obtain result (ii).

12.2.2 Existence, Uniqueness and Regularity of the Solution Theorem 12.7. Let u0 , T0 , S0 and (f , ) be given such that (T0 , S0 ) ∈ H 1 (Ω), u0 ∈ V0 and (f , ) ∈ L2 (0, T ; L2(Γ0 )). Then there exists a solution (u, T , S) of the problem (12.8) satisfying X = (u, T , S) ∈ L2 (0, T ; V) ∩ C([0, T ]; H) and

∂X ∈ L2 (0, T ; V ). ∂t

Proof. The proof of this theorem is similar to the one used to obtain Theorem 2.1 in Belmiloudi [35]. The existence results are obtained by constructing approximate solutions of system (12.8) by the classical Faedo–Galerkin method, by using Lemma 12.3 and the estimates given in Lemma 12.4, and applying the limit. For applying the limit we use the convergence results of Lemma 12.6 and the compactness arguments. So, we omit the details. From now on we assume the following regularity for the surface-stress ∂f ∈ L2 (0, T ; L2 (Γ0 ))}, ∂t ∂ ∈ L2 (0, T ; L2(Γ0 ))}. = { ∈ L2 (0, T ; H 1 (Γ0 )) : ∂t

f ∈ Uf = {f ∈ L2 (0, T ; H 1 (Γ0 )) : ∈ U

(12.10)

The regularity (12.10) implies that (f , ) ∈ C([0, T ]; L2(Γ0 )), a.e. on [0, T ] (see Lemma 6.6). Moreover, since the data (f , ) must be consistent with the initial and boundary conditions imposed to the velocity u and temperature T on the open set Γ0 =]0, Lx[×] − Ly , Ly [×{0}, we have to impose the following compatibility conditions (f , )(t = 0) = 0 on Γ0 , ∂f1 = 0, = 0 on γ3 ∪ γ4 , ∂y (f , )|γ1 = (f , )|γ2 , f2 = 0,

(12.11)

where γ = ∪i=1,4 γi denotes the boundary of Γ0 , with γ1 and γ2 the western and eastern boundaries, and γ3 and γ4 the southern and northern boundaries.

406


Theorem 12.8. Assume that the data (f , ) satisfies the conditions (12.10) and (f , )(0) = 0. Then there exists Rα > 0 such that if X0 ≤ Rα ,4 where X0 = (u0 , T0 , S0 ), and if (f , ) or the final time T is small, there exists a unique solution X = (u, T , S) of the problem (12.8) such that ∂X ∈ L2 (0, T ; V) ∩ L∞ (0, T ; H) and X ∈ L∞ (0, T ; V). ∂t Proof. The proof of this theorem is similar to the one used to obtain Theorem 2.2 in Belmiloudi [35]. So, we omit the details (the proof is based on the classical Faedo–Galerkin method, Lemmas 12.3 and 12.6, the estimates given in Lemma 12.4 and the result of Theorem 12.7). Remark 12.9. (i) The existence, uniqueness and regularity given, for the nonlinear and three-dimensional case, in Theorem 12.8 are still valid for the nonlinear and two-dimensional case, and for the linear and three-dimensional case (see Belmiloudi [31]) with no smallness assumptions on the data or on the final time. (ii) In order to obtain the uniqueness and the regularity of the solution (u, T , S) of the non-linear and three-dimensional problem (12.8), the variability of the wind stress and the disturbance of heat stress on the surface of the ocean must be small enough. Physically, these conditions are not too restrictive, because they do not concern the total wind stress and heat stress on the surface, but concern only the variability of the wind stress and the small disturbance of the heat stress. ♦ The surface pressure is of great interest for oceanographic interpretation. To define it, it is necessary to have some regularity H (Ω), for > 0. The following theorem shows that the pressure is in H 1 (Ω). Theorem 12.10. Under the assumptions of Theorem 12.8, assume moreover that X0 is in (H 2 (Ω)∩V0 )×H 2 (Ω)×H 2 (Ω), and that the data (f , ) satisfies assumptions (12.10) and (12.11), then the solution (u, T , S, p) = (X, p) of problem (12.9) is such that X ∈ L2 (0, T ; H 2 (Ω)), p ∈ L2 (0, T ; H 1 (Ω)). Proof. The shape of the oceanic domain Ω, and especially the presence of corners, prevents us from applying a standard theorem of regularity. We use an extension method, with even-odd reflection, to prove this result. The idea is the following: by using the classical arguments (see, e.g., Agmon et al. [5]) we can obtain the regularity on the open set Ω except near the corners. For the regularity in the corners we proceed in two steps. First, since the solution is periodic in the x-direction, the open set Ω can be extended from x = −Lx to x = 2Lx and the regularity near the western and eastern ˜ u, p˜, T˜ , S), boundaries Γ1 and Γ2 is automatically obtained; second we define (˜ 4

Rα depends on the coefficients νih , νiv , for i = 1, 3.


407

˜ =]0, Lx[×] − 2Ly , 2Ly [×] − H, 0[ such that extension of (u, p, T , S) in Ω ˜ ˜ ˜ verifies the equations and (˜ u, p˜, T , S) = (u, p, T , S) on Ω and (˜ u, p˜, T˜ , S) boundary conditions similar to (12.3),(12.4) and (12.5). Then we can apply ˜ on Ω, ˜ except near the corners of Ω. ˜ This the regularity result for (˜ u, p˜, T˜ , S) implies the regularity of (u, p, T , S) near the corners of Ω. For more details the reader can be referred to Belmiloudi et al. [31, 28]. So, we omit the details. Remark 12.11. (i) Since the pressure is in H 1 (Ω), we can now define the trace of p on the boundary Γ . The pressure p is defined regardless of any time-dependent function. This function can be fixed for example by setting the condition 1 pdΓ = pd where pd is given. (12.12) mesΓ0 Γ0 (ii) According to the first equation of System (12.8) and Green’s formula, we can deduce that p|Γ1 = p|Γ2 . (iii) According to Theorems 12.8 and 12.10, we have that X ∈ L2 (0, T ; H 2 (Ω)) ∩ L∞ (0, T ; V), ∂X ∈ L2 (0, T ; V) ∩ L∞ (0, T ; H), ∂t p ∈ L2 (0, T ; H 1(Ω))

(12.13)

and then, according to Lemma 6.6, that X ∈ C([0, T ]; V) ∩ H 2,1 (Q) and

∂X ∈ C([0, T ]; H). ∂t

♦

Proposition 12.12. Under the assumptions of Theorem 12.8, assume moreover that X0 is in (H 2 (Ω)∩V0 )× H 2 (Ω)× H 2 (Ω). Let (f1 , 1 ) and (f2 , 2 ) be two functions in Uf × U such that (12.11) holds. If (ui , Ti , Si , pi ) = (Xi , pi ) is the solution of problem (12.9), corresponding to the forcing (fi , i ), for i = 1, 2, then the following estimate holds: X 2H 2,1 (Q) + p 2L2 (0,T ;H 1 (Ω)) ≤ C( f 2Uf + 2U ),

(12.14)

where X = X1 − X2 , p = p1 − p2 , f = f1 − f2 and = 1 − 2 . Proof. By using a similar technique as used in Belmiloudi [31] (see also, for the more general case, Agmon et al. [5]) we can obtain the desired estimate. So, we omit the details. Remark 12.13. (i) In a more general case for the desired target (X0 , p0 ) = (u0 , T0 , S0 , p0 ), i.e., if we assume that the target (X0 , p0 ) satisfies the complete evolution Equations (12.1) and such that

408


X0 ∈ C([0, T ]; V) ∩ H 2,1 (Q),

∂X0 ∈ C([0, T ]; H) and p0 ∈ L2 (0, T ; H 1 (Ω)), ∂t

then the previous arguments and results (in particular the results of Theorem 12.10 and Proposition 12.14) extend directly to the present situation without further estimates. (ii) We can consider other domains Ω and other boundary conditions, the previous arguments can be extended to these new situations. ♦ The following theorem shows more regularity for the pressure, which can be useful in the case of the use of the pressure, for example, as observation in robust control problems. Theorem 12.14. Under the assumptions of Theorem 12.10, we have that ∂p/∂t ∈ L2 (Q). Proof. Since X = (u, T , S) satisfies X ∈ L2 (0, T ; H 2 (Ω)) ∩ L∞ (0, T ; V), we can deduce that ((∂u/∂t)∇)(u+u0 ), ((u+u0 )∇)(∂u/∂t) and (−δT (∂T /∂t)+ δS (∂S/∂t))G are in L2 (0, T ; (H 1 (Ω)) ). Consequently, by differentiating the first equation of System (12.8) and by using the regularity of ∂f /∂t (i.e., ∂f /∂t ∈ L2 (0, T ; L2 (Γ0 )),5 we conclude that ∂ 2 u/∂t2 ∈ L2 (0, T ; (H 1 (Ω)) ). Finally, the first equation of (12.3) written in L2 (0, T ; (H 1 (Ω)) ) can be differentiated in D (0, T ; (H 1 (Ω)) ) and we obtain that ∂2u ∂u ∂u ∂u ∂p − ( .∇)(u + u0 ) − F ∧ ∇( ) = − 2 − ((u + u0 ).∇) ∂t ∂t ∂t ∂t ∂t ∂ ∂ ∂u ∂u ˜ (ν1v ( )))(−δT T˜ + δS S)G, + ν1h Δ2 ( ) + ∂t ∂z ∂z ∂t in D (0, T ; (H 1(Ω)) ). The regularity of the right-hand side obtained before, yields ∇(∂p/∂t) ∈ L2 (0, T ; (H 1 (Ω)) ) and then the result of the theorem. This completes the proof. We finish this section by the following comments, which gives some asymptotic behavior of the perturbation under extra assumptions. 12.2.3 Comments on the Asymptotic Behavior In order to give the stability result, we assume that there exist sufficiently regular functions (g, #) on Q such that6 (f , v)Γ0 = (g, v), (, v)Γ0 = (#, φ) for all (v, φ) ∈ V1 × V2 , 5 6

The function v −→ (∂f /∂t, v)Γ0 is a linear continuous application from V1 into IR, which implies that there exists g ∈ V1 such that (∂f /∂t, v)Γ0 = g, v V1 ,V1 . g and can represent the wind stress and the heat stress, respectively, which can be considered as body forces.


409

the current (u0 , T0 , S0 ) is given and real-valued, the functional spaces Vi,i=1,2 , Hi,i=1,2 , V, H and all Sobolev spaces, will be considered as complex spaces and finally we introduce the following forms: h1T (φ, v) = (−δT φG, v), h1S (ψ, v) = (δS ψG, v), h2T (v, φ) = b2 (v, T0 , ψ), h2S (v, ψ) = b2 (v, S0 , ψ), a ˜1 (u, v) = a1 (u, v) + l0 (u, v), a ˜2 (T , φ) = a2 (T , φ) + ((u0 ∇)T , φ), a ˜3 (S, ψ) = a3 (S, ψ) + ((u0 ∇)S, ψ), a ˜(X, Y ) = a ˜1 (u, v) + a ˜2 (T , φ) + a ˜3 (S, ψ), with X = (u, T , S), Y = (v, φ, ψ). Now we formulate the problem (12.8) in the abstract form useful for the mathematical analysis. For this let us consider the operators A˜i associated with the forms a ˜i , for i = 1, 3, respectively and Ai the operators associated with the forms ai for i = 1, 3, respectively. By noting P the well-known Leray–Hopf projector which is the orthogonal projector of L2 (Ω) onto the divergence-free space H1 , we have then A˜1 u = A1 u + L1 u, A˜2 T = A2 T + L2 T and A˜3 S = A3 S + L3 S, where L1 u = P ((u0 ∇)u + (u∇)u0 + F ∧ u), L2 T = (u0 ∇)T and L3 S = (u0 ∇)S. Let HiT and HiS be the operators associated with the forms hiT and hiS , for i = 1, 2, respectively. ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ A˜1 H1T H1S L1 H1T H1S A1 0 0 Let A = ⎝ 0 A2 0 ⎠, C = ⎝ H2T L2 0 ⎠, A˜ = ⎝ H2T A˜2 0 ⎠. 0 0 A3 H2S 0 L3 H2S 0 A˜3 The domain of the operator A˜1 is defined by: D1 = H 2 (Ω) ∩ V1 and the domain of the operator A˜2 (respectively A˜3 ) is defined by: D2 = H 2 (Ω) ∩ V2 . Consequently, the domain of the operator A˜ is defined by: D = D1 × D2 × D2 . We can easily verify that System (12.8), can be written as the following Cauchy differential form: dX ˜ =g ˜, + AX (12.15) dt X(t = 0) = 0, ˜ = (g1 , g2 , g3 ), g1 = g − (u∇)u, g2 = # − (u∇)T and where X = (u, T , S), g g3 = −(u∇)S. By using similar techniques as used in Belmiloudi [32], which are based on a method introduced by Prodi [245], the following asymptotic result holds: If the wind stress f and the heat stress act only during a finite time and ˜ which are are small enough, and if the eigenvalues λA˜ of the operator A, situated inside a parabola of the complex plane, have positive real parts, then, the perturbation X = (u, T , S) tends to 0 as time tends to +∞ and X is a strong solution on [0, +∞[ of the differential problem (12.15).

410


The assumptions of the previous result depend on the initial situation (u0 , T0 , S0 ) and on the eddy viscosity and diffusivity (which can be deduced from the characteristics of the initial situation (u0 , T0 , S0 ), see Philander and Pacanowski [238]).

12.3 Robust Control Problems In the control framework, our objective is to find the best estimate of the variability of the wind stress f in the presence of the worst disturbance in the heat stress which maximally spoils the control objective. We assume then the control is in the variability of the wind stress and the disturbance is in heat stress in the context of non-cooperative game discussed in Chapter 8. Thus, we write f and as f = B1 ξ,

fT = B1T ξ = (B1 ξ, 0),

= B2 η,

(12.16)

where B1 (respectively B2 ) is taken here as a given linear continuous and bounded operator from L2 (Σ0 ) (respectively L2 (Σ0 )) into Uf (respectively U ), with Σ0 = Γ0 × (0, T ). The function (u, T , S) is assumed to be related to the disturbance η and control ξ through the problem (12.8): (

∂u , v) + a1 (u, v) + l0 (u, v) + l1 (T , S, v) + b1 (u, u, v) ∂t = (B1T ξ, v)Γ0 ,

∂T , φ) + a2 (T , φ) + l2 (u, T , φ) + b2 (u, T , φ) = (B2 η, φ)Γ0 , ∂t ∂S , ψ) + a3 (S, ψ) + l3 (u, S, ψ) + b2 (u, S, ψ) = 0, ( ∂t (u, T , S)(t = 0) = 0,

(

(12.17)

for all (v, φ, ψ) in V and a.e. in (0, T ). The cost functional considered in the present work is of the form 1 J(ξ, η) = | C(u − uobs ) |2 dxdt 2 Q (12.18) β α 2 | ξ | dxdt − | η |2 dxdt + 2 2 Σ0 Σ0 where α, β > 0 are fixed, uobs ∈ L2 (0, T ; V1) is the observation and C is unbounded, linear operator on L2 (Ω) satisfying the condition (7.91) (here V1 plays the role of D). In particular, we can consider the following two cases: (i) C(.) = a ∗ Id(.) which corresponds to the regulation of turbulent kinetic energy


411

(ii) C(.) = a ∗ ∇ × . = a ∗ curl(.) which corresponds to the regulation of the vorticity. The robust control problem is then to minimize the functional J with respect to ξ and maximize J with respect to η, i.e., find (ξ ∗ , η ∗ ) ∈ Uad × Vad such that J(ξ ∗ , η) ≤ J(ξ ∗ , η ∗ ) ≤ J(ξ, η ∗ ), ∀(ξ, η) ∈ Uad × Vad ,

(12.19)

where Uad (respectively Vad ) is a given non-empty, closed, convex and bounded subsets of vector space L2 (Σ0 ) (respectively scalar space L2 (Σ0 )). We are now going to show the differentiability result of the operator solution of problem (12.17) F : (ξ, η) −→ X = (u, T , S), which maps the source term (ξ, η) ∈ L2 (Σ0 ) × L2 (Σ0 ) into the corresponding solution X ∈ Z, where Z = {Y ∈ L2 (0, T ; V) ∩ C([0, T ]; H) :

∂Y ∈ L2 (0, T ; V )}. ∂t

12.3.1 Differentiability of the Operator Solution Proposition 12.15. Under the assumptions of Proposition 12.12, the operator solution F is continuously F-differentiable from L2 (Σ0 ) × L2 (Σ0 ) to Z where the derivative F (ξ, η) : (h, κ) −→ (w, θ, ϑ) is a linear mapping from L2 (Σ0 ) × L2 (Σ0 ) to Z such that F (ξ, η).(h, κ) = (w, θ, ϑ) satisfies, a.e. in (0, T ) and for all (v, φ, ψ) ∈ V (

∂w , v) + a1 (w, v) + d(w, v) + l1 (θ, ϑ, v) + b1 (u1 , w, v) ∂t +b1 (w, u1 , v) = (B1T h, v)Γ0 ,

∂θ , φ) + a2 (θ, φ) + b2 (w, T1 , φ) + b2 (u1 , θ, φ) = (B2 κ, φ)Γ0 , ∂t ∂ϑ ( , ψ) + a3 (ϑ, ψ) + b2 (w, S1 , ψ) + b2 (u1 , ϑ, ψ) = 0, ∂t (w, θ, ϑ)(t = 0) = 0,

(

(12.20)

where (u, T , S) = F (ξ, η) is the solution of (12.17), u1 = u + u0 , T1 = T + T0 and S1 = S + S0 . Moreover, for any (ξi , ηi ) ∈ L2 (Σ0 ) × L2 (Σ0 ), for i = 1, 2, we have the following estimates: (i) F (ξ1 , η1 ) L(L2 (Σ)×L2 (Σ);Z) ≤ C (ii) F (ξ1 , η1 ) − F (ξ2 , η2 ) L(L2 (Σ)×L2 (Σ);Z) ≤ C( ξ 2L2 (Σ) + η 2L2 (Σ) ), where ξ = ξ1 − ξ2 and η = η1 − η2 .

412


Proof. The existence of the F-derivative and its characterization by (12.20) as well as estimates (i) and (ii) can be obtained by using a technique similar to that employed in Section 8.6 and by using the estimates given in Lemma 12.4, the results of Lemma 12.3 and the regularity of the solution X. So, we omit the details. Remark 12.16. (i) A similar result concerning the existence of the F-derivatives for Navier–Stokes system can be found, e.g., in Abergel and Temam [1]. (ii) System (12.20) is the weak formulation of the following system (on Q): find (w, π, θ, ϑ) such that ∂w + (w.∇)u1 + (u1 .∇)w + F ∧ w ∂t ∂w ∂ (ν1v ) + ∇π = (−δT θ + δS ϑ)G, −ν1h Δ2 w − ∂z ∂z div(w) = 0, ∂θ ∂θ ∂ + (w.∇)T1 + (u1 .∇)θ − ν2h Δ2 θ − (ν2v ) = 0, ∂t ∂z ∂z ∂ϑ ∂ ∂ϑ + (w.∇)S1 + (u1 .∇)ϑ − ν3h Δ2 ϑ − (ν3v ) = 0, ∂t ∂z ∂z with the initial condition

(12.21)

(w, θ, ϑ)(t = 0) = 0, subject to the boundary conditions (12.4) and (12.5) where B1T h (respectively B2 κ) plays the role of f (respectively ), with u1 = u + u0 , T1 = T + T0 and S1 = S + S0 .

♦

Concerning the system of Equations (12.21) (and then (12.20)) we have the following result. Theorem 12.17. Let (ξ, η), (h, κ) ∈ L2 (Σ0 )×L2 (Σ0 ) and (u, T , S) = F (ξ, η) be the solution of the non-linear problem (12.17). Then there exists a unique Y = (w, θ, ϑ) ∈ Z and π ∈ D (Q) (up to the addition of a distribution in (0, T )) solution of the linear system (12.20). Moreover, if (h1 , κ1 ) and (h2 , κ2 ) are two functions in L2 (Σ0 ) × L2 (Σ0 ) and (wi , θi , ϑi , πi ) = (Yi , πi ), for i=1,2, are weak solutions of problem (12.20), corresponding to the forcing (hi , κi ), respectively, the following estimate holds. Y 2Z ≤ C( h 2L2 (Σ) + κ 2L2 (Σ) )

(12.22)

where Y = Y1 − Y2 , h = h1 − h2 and κ = κ1 − κ2 . Proof. The existence of solutions Y = (w, θ, ϑ) ∈ Z of the linear system (12.20) can be proved by using the classical Faedo–Galerkin method as in the


413

existence of the problem (12.8). The uniqueness of solutions can be obtained by the standard energy estimate arguments. For the proof of the estimate (12.22) we take (v, φ, ψ) = (w, θ, ϑ) as the test function in (12.20), integrate over (0, t), for all t in (0, T ) and use the classical technique to obtain some energy inequalities. The existence of the pressure can be obtained by using the standard technique to those for the Navier–Stokes equations (see, e.g., Temam [281]). Therefore we omit the details. We are now going to study the existence of a solution for the robust control problem. 12.3.2 Existence of an Optimal Solution Theorem 12.18. For α and β sufficiently large (i.e., there exists (αl , βl ) such that α ≥ αl and β ≥ βl ) there exist (ξ ∗ , η ∗ ) ∈ U = Uad × Vad and X ∗ = (u∗ , T ∗ , S ∗ ) ∈ H 2,1 (Q) ∩ L∞ (0, T ; V) such that (ξ ∗ , η ∗ ) satisfies (12.19) and (u∗ , T ∗ , S ∗ ) = F (ξ ∗ , η ∗ ) is a solution of the primal problem (12.17). Proof. Let Pη be the map: ξ −→ J(ξ, η) and Qξ be the map: η −→ J(ξ, η). To obtain the existence of the robust control problem we prove that Pη is convex and lower semi-continuous for all η ∈ Vad , and Qξ is concave and upper semicontinuous for all ξ ∈ Uad and we use the minimax duality theorems in infinite dimensions presented in Chapter 5. First we prove, for α and β sufficiently large, the convexity of the map Pη and the concavity of the map Qξ . In order to prove the convexity, it is sufficient to show that for all (ξ1 , ξ2 ) ∈ Uad we have (Pη (ξ1 ) − Pη (ξ2 )).ξ ≥ 0, where ξ = ξ1 − ξ2 (because Pη is G-differentiable). According to the definition of J, we have that (Pη (ξ1 ) − Pη (ξ2 )).ξ 2L2 (Σ)

T

=αξ + (C(u1 − u2 ), Cw2 )L2 (Ω) dt 0 T (C(u1 − uobs ), C(w1 − w2 ))L2 (Ω) dt, +

(12.23)

0

where (ui , Ti , Si ) = F (ξi , η) and (ηi , wi ) = (∂F /∂ξ)(ξi , η).ξ = F (ξi , η).(ξ, 0), the solution of the problem (12.20), for i = 1, 2. According to Proposition 12.12, Proposition 12.15 and the relation (7.91), satisfied by the operator C, we can deduce that there exists a constant C0 > 0 such that7

7

The constant C0 depends on the given data: the steady mean circulation X0 , the final time T , the domain Ω, the observation uobs and the coefficients of horizontal and vertical eddy viscosity and diffusivity νih , νiv , for i = 1, 3.

414


T

0

(C(u1 − u2 ), Cw2 )L2 (Ω) dt T (C(u1 − uobs ), C(w1 − w2 ))L2 (Ω) dt ≤ C0 ξ 2L2 (Σ) . +

(12.24)

0

From (12.23) and (12.24) we deduce that for α ≥ αl such that αl ≥ C0 we have (Pη (ξ1 ) − Pη (ξ2 )).ξ ≥ (α − C0 ) ξ 2L2 (Σ) ≥ (αl − C0 ) ξ 2L2 (Σ) ≥ 0 and then the convexity of Pη . In the same way, we can find βl such that for β ≥ βl we have the concavity of Qξ . We shall now prove that Pη is lower semi-continuous for all η ∈ Vad , and Qξ is upper semi-continuous for all ξ ∈ Uad . Let ξk be a minimizing sequence of J, i.e., lim inf J(ξk , η) = min J(ξ, η) (∀η ∈ Vad ). Then ξk is uniformly k

ξ∈Uad

bounded in Uad and we can extract from ξk a subsequence also denoted by ξk such that ξk ξη weakly in Uad . Let us consider (uk , Tk , Sk ) = F (ξk , η) which is the solution of (12.17), i.e., (

∂uk , v) + a1 (uk , v) + l0 (uk , v) + l1 (Tk , Sk , v) ∂t +b1 (uk , uk , v) = (B1T ξk , v)Γ0 ,

∂Tk , φ) + a2 (Tk , φ) + l2 (uk , Tk , φ) + b2 (uk , Tk , φ) = (B2 η, φ)Γ0 , ∂t ∂Sk , ψ) + a3 (Sk , ψ) + l3 (uk , Sk , ψ) + b2 (uk , Sk , ψ) = 0, ( ∂t (uk , Tk , Sk )(t = 0) = 0. (

(12.25)

By taking (v, φ, ψ) = (uk , Tk , Sk ) as the test function in (12.25), integrating over (0, t), for all t in (0, T ) and using the classical technique we obtain that the sequence Xk = (uk , Tk , Sk ) is uniformly bounded in L∞ (0, T ; H) ∩ L2 (0, T ; V) and ∂Xk /∂t is uniformly bounded in L2 (0, T ; V ). This result, the compactness results and the fact that ξk is uniformly bounded in Uad makes it possible to extract from (Xk , ξk ) a subsequence also denoted by (Xk , ξk ) such that ξk ξη weakly in Uad , Xk Xη = (uη , Tη , Sη ) weakly in L2 (0, T ; V),

(12.26)

Xk −→ Xη strongly in L (0, T ; H). 2

Using the same techniques as to prove the existence of a solution to problem (12.8), we may apply the limit in (12.25) to conclude that (uη , Tη , Sη ) = F (ξη , η) satisfies the problem (12.17). Therefore, since the norm is lower semi-continuous we have that the mapping Pη : ξ −→ J(ξ, η) is lower semicontinuous for all η ∈ Vad . By using the same technique we obtain that Qξ is upper semi-continuous for all ξ ∈ Uad . This completes the proof.


415

We next wish to show the appropriate first-order necessary conditions for the robust control problem. 12.3.3 Optimality Conditions In order to obtain the necessary optimality conditions which have been satisfied by the solution of the robust control problem, we introduce the following adjoint problem corresponding to the primal problem (12.17): find (R, D, P ) ∈ Z such that, for all (v, φ, ψ) ∈ V and a.e. in (0, T ), −(

∂R , v) + a1 (R, v) + b1 (v, u1 , R) − b1 (u1 , R, v) − d(R, v) ∂t +b2 (v, T1 , D) + b2 (v, S1 , P ) = (C ∗ C(u − uobs ), v)L2

∂D , φ) + a2 (D, φ) − b2 (u1 , D, φ) + (−δT G.R, φ)L2 = 0, ∂t ∂P , ψ) + a3 (P, ψ) − b2 (u1 , P, ψ) + (δS G.R, ψ)L2 = 0, −( ∂t (R, D, P )(t = T ) = 0, −(

(12.27)

where (u, T , S) = F (ξ, η), (ξ, η) ∈ L2 (Σ0 ) × L2 (Σ0 ), with u1 = u + u0 , T1 = T + T0 and S1 = S + S0 . Remark 12.19. (i) The adjoint problem (12.27) is a linear system. By reversing sense of time, i.e., t := T − t, and by applying the same way as to obtain the result of Theorem 12.17 we obtain the existence and uniqueness of the solution (R, D, P ) in Z. (ii) The system (12.27) is the weak formulation of the following system: find (R, Π, D, P ) such that ∂R − (u1 .∇)R + (∇R)t u1 − F ∧ R + (∇T1 )t D + (∇S1 )t P ∂t ∂R ∂ (ν1v ) + ∇Π = C ∗ C(u − uobs ), −ν1h Δ2 R − ∂z ∂z div(R) = 0,

−

∂D ∂ ∂D − (u1 ∇)D − ν2h Δ2 D − (ν2v ) = δT G.R, ∂t ∂z ∂z ∂P ∂ ∂P − (u1 ∇)P − ν3h Δ2 P − (ν3v ) = −δS G.R, − ∂t ∂z ∂z with the initial condition (R, D, P )(t = T ) = 0, subject to the boundary conditions (12.4) and (12.5) where the vector 0 (respectively the scalar 0) plays the role of f (respectively ), −

(12.28)

416


with u1 = u + u0 , T1 = T + T0 and S1 = S + S0 .

♦

We can now give the first-order optimality conditions for the robust control problem (12.19). Theorem 12.20. Under the assumptions of Theorem 12.18, the optimal solutions (ξ ∗ , η ∗ ) ∈ U and X ∗ = (u∗ , T ∗ , S ∗ ) ∈ H 2,1 (Q) ∩ L∞ (0, T ; V) such that (ξ ∗ , η ∗ ) satisfies (12.19) and (u∗ , T ∗ , S ∗ ) = F (ξ ∗ , η ∗ ) is a solution of the primal (12.17), satisfy ∗ (αξ ∗ + B1T R∗ ).(ξ − ξ ∗ )dΓ dt ≥ 0, Σ0 (12.29) (−βη ∗ + B2∗ D∗ )(η − η ∗ )dΓ dt ≤ 0 ∀(ξ, η) ∈ U, Σ0

where (R∗ , D∗ , P ∗ ) is the adjoint solution of the problem (12.27) corresponding to the primal solution F (ξ ∗ , η ∗ ). Proof. The cost function J is a composition of F-differentiable mappings then J is differentiable and we have (∀(h, κ) ∈ U ) C ∗ C(u − uobs )wdxdt J (ξ, η).(h, κ) = Q (12.30) ξ.hdΓ dt − β κηdΓ dt, +α Σ0

Σ0

where (w, θ, ϑ) = F (ξ, η).(h, κ) is the solution of problem (12.20). By taking (v, φ, ψ) = (R, D, P ) in (12.20), and integrating with respect to time, we obtain (according to the initial conditions of (12.20))

T

! ∂R , w) + a1 (R, w) − d(R, w) − b1 (u1 , R, w) + b1 (w, u1 , R) dt ∂t T T ∗ l1 (θ, ϑ, R)dt + (B1T R, h)Γ0 dt, = −(R(T ), w(T ))L2 − 0 0 ! ∂D , θ) + a2 (D, θ) − b2 (u1 , D, θ) dt (− ∂t T T b2 (w, T1 , D)dt + (B2∗ D, κ)Γ0 dt, = −(D(T ), θ(T ))L2 − 0 0 ! ∂P , ϑ) + a3 (P, ϑ) − b2 (u1 , P, ϑ) dt (− ∂t T b2 (w, S1 , P )dt. = −(P (T ), ϑ(T ))L2 −

(− 0

T

0

0

T

0

Since (R, D, P ) is a solution of (12.27), with null final conditions, we can deduce that (since l1 (θ, ϑ, R) = ((−δT θ + δS ϑ)G, R)L2 )


T

T

417

T

b2 (w, T1 , D)dt + b2 (w, S1 , P )dt − (C ∗ C(u − uobs ), w)L2 dt 0 0 0 T T ∗ l1 (θ, ϑ, R)dt − (B1T R, h)Γ0 dt, = 0 0 (12.31) T T T ∗ (δT θG, R)L2 dt = b2 (w, T1 , D)dt − (B2 D, κ)Γ0 dt, − 0 0 0 T T b2 (w, S1 , P )dt. (δS ϑG, R)L2 dt = 0

0

By adding the first, the second and the third part of (12.31), we obtain that

T

(C ∗ C(u − uobs ), w)L2 dt =

0

0

T

∗ (B1T R, h)Γ0 dt +

0

T

(B2∗ D, κ)Γ0 dt. (12.32)

According to the expression of J (ξ, η).(h, κ) we can deduce that ∗ J (ξ, η).(h, κ) = (αξ + B1T R).hdΓ dt − (βη − B2∗ D)κdΓ dt, (12.33) Σ0

Σ0

Since (ξ ∗ , η ∗ ) is an optimal solution we have ∂J ∗ ∗ ∂J ∗ ∗ (ξ , η ).(ξ − ξ ∗ ) ≥ 0, (ξ , η ).(η − η ∗ ) ≤ 0 ∂ξ ∂η and then

∀(ξ, η) ∈ U

(12.34)

∗ (αξ ∗ + B1T R∗ ).(ξ − ξ ∗ )dΓ dt ≥ 0,

Σ0 (−βη ∗ + B2∗ D∗ )(η − η ∗ )dΓ dt ≤ 0

(12.35) ∀(ξ, η) ∈ U,

Σ0

where (R∗ , D∗ , P ∗ ) is the adjoint solution corresponding to the primal solution F (ξ ∗ , η ∗ ). This completes the proof. We end this section by the following results, which correspond to the case where we take into account the final observation. In this case the cost functional can be given by 1 J(ξ, η) = | C(u − uobs ) |2 dxdt 2 Q 1 | C1 (u(., T ) − vobs ) |2 dx + (12.36) 2 Ω β α | ξ |2 dΓ dt − | η |2 dΓ dt + 2 2 Σ0 Σ0

418


where α, β > 0 are fixed, (uobs , vobs ) ∈ L2 (0, T ; V1 ) × V1 is the observation and the operators C and C1 are unbounded and linear operators on L2 (Ω) satisfying the condition (7.91) (here V1 plays the role of D). The arguments of the previous results extend directly to the present case without further estimates. Therefore, we have the following results. Theorem 12.21. Under the assumptions of Theorem 12.18, the robust control problem (12.19) admits a solution (ξ ∗ , η ∗ ) ∈ U, which is characterized by the following necessary optimality conditions: ∗ (αξ ∗ + B1T R∗ ).(ξ − ξ ∗ )dΓ dt ≥ 0, Σ0 (12.37) (−βη ∗ + B2∗ D∗ )(η − η ∗ )dΓ dt ≤ 0 ∀(ξ, η) ∈ U, Σ0

where (R∗ , D∗ , P ∗ ) is the unique solution in Z of the following adjoint problem (corresponding to the unique solution (u∗ , T ∗ , S ∗ ) = F (ξ ∗ , η ∗ ) ∈ H 2,1 (Q) ∩ L∞ (0, T ; V) of the primal problem (12.17)): −(

∂R∗ , v) + a1 (R∗ , v) + b1 (v, u1 , R∗ ) − b1 (u1 , R∗ , v) ∂t +b2 (v, T1 , D∗ ) + b2 (v, S1 , P ∗ ) −d(R∗ , v) = (C ∗ C(u∗ − uobs ), v)L2

∂D∗ , φ) + a2 (D∗ , φ) − b2 (u1 , D∗ , φ) + (−δT G.R∗ , φ)L2 = 0, ∂t ∂P ∗ , ψ) + a3 (P ∗ , ψ) − b2 (u1 , P ∗ , ψ) + (δS G.R∗ , ψ)L2 = 0, −( ∂t (R∗ , D∗ , P ∗ )(t = T ) = C1∗ C1 (u∗ (., T ) − vobs ),

(12.38)

−(

for all (v, φ, ψ) ∈ V and a.e. in (0, T ), with u1 = u + u0 , T1 = T + T0 and S1 = S + S0 .

12.4 Primitive Ocean Equations with Vertical Viscosity In this section we focus our attention on different physical assumptions. In the equatorial ocean the horizontal scale is much bigger than the vertical one.8 Therefore, we can simplify the total Boussinesq equations of the ocean (12.1) by neglecting some terms. However, we must take into account the viscosity, which plays a very important role in the dynamics of the ocean in such region. The velocity solution is now denoted by u = (u , w) = (u, v, w). Therefore, we can replace the equation describing the vertical motion by the following 8

The value of the ratio between the vertical and horizontal scales of the oceanic domain is approximately 10−3 for the physical problems considered here.

12.4 Primitive Ocean Equations with Vertical Viscosity

419

equation called the hydrostatic approximation with vertical viscosity9 (see Lions et al. [205]): −ν1h Δw −

1 ∂p ρ ∂w ∂ (ν1v )+ + g = 0. ∂z ∂z ρav ∂z ρav

(12.39)

Applying these physical assumptions to (12.1), we obtain the following equations (since ρav = 1): ∂u ∂ ∂u ∂u + (u .∇2 )u + w − ν1h Δu − (ν1v ) ∂t ∂z ∂z ∂z +F ∧2 u + ∇2 p = 0, ∂p ∂w ∂ (ν1v )+ + ρg = 0, ∂z ∂z ∂z ∂w div2 (u ) + = 0, ∂z ∂T ∂T ∂ ∂T + (u .∇2 )T + w − ν2h Δ2 T − (ν2v ) = 0, ∂t ∂z ∂z ∂z ∂S ∂S ∂S ∂ + (u .∇2 )S + w − ν2h Δ2 S − (ν2v ) = 0, ∂t ∂z ∂z ∂z ρ = δ0 − δT T + δS S,

−ν1h Δ2 w −

(12.40)

with F ∧2 u = F ∧ (u , 0), ∇2 . = (∂./∂x, ∂./∂y) and div2 (u ) = ∇2 .u . Nota bene: The notation “ ” used for the separation between the vertical and the horizontal velocity will now be omitted. We now split up the circulation (u, w, T , S, p) into a given mean value ˜ p˜) that will be computed by u, w, ˜ T˜ , S, (u0 , w0 , T0 , S0 , p0 ) and a variability (˜ the model. The mean circulation (u0 , w0 , T0 , S0 , p0 ) has to satisfy the steady state Equations (12.40). ˜ p˜) The full non-linear system which models large perturbation (˜ u, w, ˜ T˜ , S, to the target (u0 , w0 , T0 , S0 , p0 ) can be deduced from (12.40). It can be written as ∂˜ u ∂˜ u ∂˜ u + (˜ u.∇2 )˜ + (u0 .∇2 )˜ + (˜ u.∇2 )u0 u+w ˜ u + w0 ∂t ∂z ∂z ∂˜ u ∂u0 ∂ + F ∧2 u (ν1v ) + ∇˜ p = 0, +w ˜ ˜ − ν1h Δ2 u ˜− ∂z ∂z ∂z ∂w ˜ ∂ p˜ ∂w ˜ ∂ ˜ = 0, −ν1h Δ2 w (ν1v )+ + + g(−δT T˜ + δS S) ˜− ∂z ∂z ∂z ∂z ∂ T˜ ∂ T˜ ∂T0 + (˜ u.∇2 )T˜ + w + (˜ u.∇2 )T0 + w + (u0 .∇2 )T˜ ˜ ˜ ∂t ∂z ∂z 9

In contrast with the classical hydrostatic approximation, where the vertical motion is replaced by the hydrostatic pressure equation: ∂p/∂z = −ρg.

420


∂ ∂ T˜ ∂ T˜ − ν2h Δ2 T˜ − (ν2v ) = 0, ∂z ∂z ∂z ∂ S˜ ∂ S˜ ∂S0 + (˜ u.∇2 )S˜ + w + (˜ u.∇2 )S0 + w + (u0 .∇2 )S˜ ˜ ˜ ∂t ∂z ∂z (12.41) ˜ ∂ ∂ S˜ ∂ S − ν2h Δ2 S˜ − (ν2v ) = 0, +w0 ∂z ∂z ∂z ∂w ˜ div2 (˜ = 0. u) + ∂z Nota bene: The notation “˜” used for the perturbation of the mean flow will now be omitted. +w0

To take into account the phenomena we want to describe, we assume that the circulation (u, w, T , S, p) satisfies the boundary conditions (12.4) and (12.5), and the initial conditions (12.6). Equations (12.41) are not of a Cauchy–Kovalevsky type with respect to all variables because of the continuity equation and of the hydrostatic equation with vertical viscosity (12.39). The method proposed by Lions et al. in [205, 206] is the integration of these two diagnostic equations with respect to the vertical variable, and then we obtain another formulation of the primitive equations, which is a three dimensional evolution system. Therefore we can use Faedo–Galerkin method to solve the weak formulations. By integrating the continuity equation with respect to z and taking the boundary conditions for w into account, we obtain z u(x, y, s; t)ds , w(x, y, z; t) = W (u)(x, y, z; t) = −div −H 0 (12.42) u(., ., z; .)dz = 0 −div −H

By integrating the hydrostatic equation with vertical viscosity in (12.41), we obtain p=P +g z

0

(−δT T (s) + δS S(s))ds +

0

L(u)(s)ds

(12.43)

z

where L(u) = −ν1h ΔW (u) − (∂/∂z) (ν1v (∂W (u)/∂z)) and P is the pressure of the sea water on the surface of the ocean. Then by using (12.42) and (12.43) we obtain the following reformulation of Equations (12.41) ∂u ∂u ∂u + (u.∇2 )u + W (u) + (u0 .∇2 )u + w0 + (u.∇2 )u0 ∂t ∂z ∂z ∂u0 ∂ ∂u + F ∧2 u − ν1h Δ2 u − (ν1v ) +W (u) ∂z ∂z ∂z


421

+∇P + g∇(−δT R(T ) + δS R(S)) + ∇L(u) = 0, ∂T ∂T0 ∂T + (u.∇2 )T + W (u) + (u.∇2 )T0 + W (u) + (u0 .∇2 )T ∂t ∂z ∂z ∂T ∂T ∂ − ν2h Δ2 T − (ν2v ) = 0, +w0 ∂z ∂z ∂z ∂S ∂S ∂S0 + (u.∇2 )S + W (u) + (u.∇2 )S0 + W (u) + (u0 .∇2 )S ∂t ∂z ∂z ∂S ∂S ∂ − ν2h Δ2 S − (ν2v ) = 0, +w0 ∂z ∂z ∂z 0 div u(z)dz = 0,

(12.44)

−H

where

R(φ) =

0

φ(s)ds, L(u) =

z

0

L(u)(s)ds. z

Remark 12.22. The above system is three-dimensional, but the unknown function P is only a function of x, y and time t. ♦ For obtaining the variational formulation of System (12.44) with Conditions (12.4), (12.5) and (12.6), we use the following spaces: H1 = {v ∈ (L2 (Ω))2 : (v, 0).n = 0 on Γ0 ∪ Γ3 ∪ Γ4 ∪ Γ5 , (v, 0).n|Γ1 = −(v, 0).n|Γ2 }, z div(v)ds ∈ H 1 (Ω), v = 0 on Γ5 , W1 = {v ∈ (H 1 (Ω))2 : W (v) = − −H

(v, 0).n = 0 on Γ0 ∪ Γ3 ∪ Γ4 , v|Γ1 = v|Γ2 }, 0 V1 = {v ∈ W1 : div(v)dz = 0}, −H

H2 = L2 (Ω), V2 = {φ ∈ H 1 (Ω) : φ = 0 on Γ3 ∪ Γ4 ∪ Γ5 , φ|Γ1 = φ|Γ2 }, H = H1 × H2 × H2 ,

V = V 1 × V2 × V2 ,

where n is the unit outward vector normal to Γ . Remark 12.23. (i) The spaces W1 and V1 are equipped with the norm z div(u)ds 2 for all u ∈ W1 (respectively V1 ) u 2W = u 2 + −H

and V2 is equipped with the norm φ , for all φ ∈ V2 . (ii) The space V is equipped with the norm X 2W = u 2W + T 2 + S 2

for all X = (u, T , S) ∈ V.

♦

422


We now define the following forms: ∂u ∂v , ), ∂z ∂z a12 (u, v) = ν1h (∇2 W (u), ∇2 W (v)) + (ν1v div2 (u), div2 (v)), a11 (u, v) = ν1h (∇2 u, ∇2 v) + (ν1v

a1 (u, v) = a11 (u, v) + a12 (u, v), ∂T ∂φ , ), ∂z ∂z ∂S ∂ψ , ), a3 (S, ψ) = ν3h (∇2 S, ∇2 ψ) + (ν3v ∂z ∂z d(u, v) = (F ∧2 u, v),

a2 (T , φ) = ν2h (∇2 T , ∇2 φ) + (ν2v

∂v , w), ∂z ∂φ b2 (u, φ, ψ) = ((u.∇2 )φ, ψ) + (W (u) , ψ), ∂z

b1 (u, v, w) = ((u.∇2 )v, w) + (W (u)

∂u , v) + d(u, v), l0 (u, v) = b1 (u, u0 , v) + ((u0 .∇2 )u, v) + (w0 ∂z z l1 (T , S, v) = g(−δT T + δS S, div(v)ds), −H

∂T , φ), ∂z ∂S , ψ). l3 (u, T , ψ) = b2 (u, S0 , ψ) + ((u0 .∇2 )S, ψ) + (w0 ∂z l2 (u, T , φ) = b2 (u, T0 , φ) + ((u0 .∇2 )T , φ) + (w0

In order to introduce the weak formulations associated with the perturbation problem (12.44), (12.4), (12.5), (12.6), we give the following lemmas, which can be found in Belmiloudi [32, 36]. Lemma 12.24. For all (u, v) in (W1 ∩ H 2 (Ω))2 , the following relation holds: z z (∇L(u), v) = ν1h (∇( div(u)ds), ∇( div(v)ds))+(ν1v div(u), div(v)) −H −H 0 − (ν1v div(u), div(v)dz)Γ0 . −H

Moreover, if v ∈ V1 , the previous relation becomes z z (∇L(u), v) = ν1h (∇( div(u)dz ), ∇( div(v)dz )) −H

−H

+ (ν1v div(u), div(v)) = a12 (u, v).

Lemma 12.25. Suppose that f = (f1 , f2 ) ∈ L2 (Γ0 ), ∈ L2 (Γ0 ) and (u, T , S, P ) is sufficiently regular and satisfies the boundary condition (12.4), (12.5). Then the following relations hold:


423

∂u ∂ (ν1v ), v) = a11 (u, v) − (f , v)Γ0 , for all v ∈ W1 . ∂z ∂z ∂T ∂ ), φ) = a2 (T , φ) − (, φ)Γ0 , for all φ ∈ V2 . (ii) −ν2h (Δ2 T , φ) − ( (ν2v ∂z ∂z ∂ ∂S ), ψ) = a3 (S, ψ), for all ψ ∈ V2 . (iii) −ν2h (Δ2 S, ψ) − ( (ν2v ∂z ∂z 0 div(v)dz)Γ0 , for all v ∈ W1 . (iv) (∇P, v) = −(P, (i) −ν1h (Δ2 u, v) − (

−H

(v) (∇P, v) = 0, for all v ∈ V1 . (vi) (∇(

0

φ(s)ds), v) = −(

z

0

φ(s)ds, div(v)). z = (φ, div(v)ds), for all v ∈ W1 , φ ∈ V2 . z

−H

We now recall some properties of the previous operators. Proposition 12.26. The following properties hold: (i) a1 (respectively ai , i = 2, 3) is a bilinear continuous and coercive form on W1 × W1 (respectively on V2 × V2 ). (ii) d is a bilinear continuous form on W1 × W1 . (iii) The operator W is linear and satisfies: W (u) L2 (Ω) ≤ C u . z (iv) | (φ, div(v)dz ) |≤ c φ L2 (Ω) v , −H

≤ c v L2 (Ω) φ , for all (v, φ) ∈ W1 × V2 .

Proof. The proof of this proposition may be found in Belmiloudi [32, 36]. Remark 12.27. The trilinear continuous forms b1 and b2 satisfy the same estimates and relations as in Lemmas 12.4 and 12.5. ♦ According to Lemma 12.25, the problem (12.44) with the boundary conditions (12.4), (12.5) and the initial conditions (12.6) satisfied by the perturbation (u, p, T , S) of the mean flow admits the following weak formulation: find (u, T , S) ∈ L2 (0, T ; V) such that, for all (v, φ, ψ) in V, a.e. in (0, T ) (

∂u , v) + a1 (u, v) ∂t

+ l1 (T , S, v) + b1 (u, u, v) + l0 (u, v) = (f , v)Γ0 ,

∂T , φ) + a2 (T , φ) + l2 (u, T , φ) + b2 (u, T , φ) = (, φ)Γ0 , ∂t ∂S , ψ) + a3 (S, ψ) + l3 (u, S, ψ) + b2 (u, S, ψ) = 0, ( ∂t (u, T , S)(t = 0) = 0. (

(12.45)

424


From the properties of the forms (ai )i=1,3 , (bi )i=1,2 and (li )i=0,3 (Proposition 12.26, Lemmas 12.24 and 12.25 and Lemma 12.4), we derive the following existence and uniqueness results (see Belmiloudi [32, 36] for more details). Proposition 12.28. Under the assumptions of Theorem 12.8, assume moreover, that X0 = (u0 , w0 , T0 , S0 ) is in (H 2 (Ω) ∩ V0 ) × H 2 (Ω) × H 2 (Ω). The problem (12.45) admits a unique solution (u, T , S) ∈ H 2,1 (Q) ∩ C([0, T ]; V). Moreover, if (f1 , 1 ) and (f2 , 2 ) are two functions in Uf × U such that (12.11) holds, and Xi = (ui , Ti , Si ) is the solution of problem (12.45), corresponding to the forcing (fi , i ), for i = 1, 2, then the following estimate holds: X 2H 2,1 (Q) ≤ C( f 2Uf + 2U ), (12.46) where X = X1 − X2 , f = f1 − f2 and = 1 − 2 .

For the robust control problem, we state similar problems as in Section 12.3. More precisely, we assume that the control is in the variability of the wind stress and the disturbance is in heat stress in the context of the noncooperative game discussed in Chapter 8. Thus, we write f and as in (12.16). The function (u, T , S) is then assumed to be related to the disturbance η and control ξ through the problem (12.45): (

∂u , v) + a1 (u, v) ∂t

+ l1 (T , S, v) + b1 (u, u, v) + l0 (u, v) = (B1 ξ, v)Γ0 ,

∂T , φ) + a2 (T , φ) + l2 (u, T , φ) + b2 (u, T , φ) = (B2 η, φ)Γ0 , ∂t ∂S , ψ) + a3 (S, ψ) + l3 (u, S, ψ) + b2 (u, S, ψ) = 0, ( ∂t (u, T , S)(t = 0) = 0. (

(12.47)

for all (v, φ, ψ) in V and a.e. in (0, T ). The cost functional considered here is the same as in (12.18), i.e., of the form 1 J(ξ, η) = | C(u − uobs ) |2 dxdt 2 Q β α | ξ |2 dΓ dt − | η |2 dΓ dt, + 2 2 Σ0 Σ0 where α, β > 0 are fixed, uobs ∈ L2 (0, T ; V1 ) is the observation and C is an unbounded, linear operator on L2 (Ω) satisfying the condition (7.91). The arguments of Section 12.3 extend directly to the present case. So, we omit the details. Therefore, we have the following existence and first-order optimality condition results.


425

Theorem 12.29. Under the assumptions of Theorem 12.18, the robust control problem (12.19) admits a solution (ξ ∗ , η ∗ ) ∈ U, which is charaterized by the following necessary optimality conditions: (αξ ∗ + B1∗ R∗ ).(ξ − ξ ∗ )dΓ dt ≥ 0, Σ 0 (12.48) (−βη ∗ + B2∗ D∗ )(η − η ∗ )dΓ dt ≤ 0 ∀(ξ, η) ∈ U, Σ0

where (R∗ , D∗ , P ∗ ) is the unique solution in Z of the following adjoint problem (corresponding to the unique solution (u∗ , T ∗ , S ∗ ) = F (ξ ∗ , η ∗ ) ∈ H 2,1 (Q) ∩ L∞ (0, T ; V) of the primal problem (12.47)): 10 −(

∂R∗ , v) + a1 (R∗ , v) + b1 (v, u1 , R∗ ) − b1 (u1 , R∗ , v) ∂t −d(R∗ , v) + b2 (v, T1 , D∗ ) +b2 (v, S1 , P ∗ ) = (C ∗ C(u∗ − uobs ), v)L2 ,

∂D∗ , φ) + a2 (D∗ , φ) − b2 (u1 , D∗ , φ) − (gδT W (R∗ ), φ)L2 = 0, ∂t ∂P ∗ , ψ) + a3 (P ∗ , ψ) − b2 (u1 , P ∗ , ψ) + (gδS W (R∗ ), ψ)L2 = 0, −( ∂t (R∗ , D∗ , P ∗ )(t = T ) = 0,

(12.49)

−(

for all (v, φ, ψ) ∈ V and a.e. in (0, T ), with u1 = u + u0 , T1 = T + T0 and S1 = S + S0 . Remark 12.30. As indicated in Remark 10.31, for numerical resolution of the robust control problems, the reader is referred to Chapter 9. ♦ Remark 12.31. It is clear that we can treat, by using the techniques developed in this chapter and more generally in this book, other physical models concerning the ocean, atmosphere or coupled ocean–atmosphere, for example the problems presented by Lions et al. [205, 206, 207] and Belmiloudi [37]. ♦

10

Where Z = {Y ∈ L2 (0, T ; V) ∩ C([0, T ]; H) : ∂Y /∂t ∈ L2 (0, T ; V )}.

13 Heat Transfer Laws on Temperature Distribution in Biological Tissues

This chapter considers non-linear robust control problems governed by some generalized transient bioheat transfer type models in biological systems with directional blood flow and Robin boundary conditions. The model equation depends on the blood perfusion rate, the heat transfer parameter, the distributed energy source terms and the heat flux due to the evaporation, which affect the effects of thermal and physical properties on the transient temperature of biological tissues. The result can be very beneficial for thermal diagnostics and treatments in medical practices, for example for laser surgery, and photoand thermotherapy for regional hyperthermia, oftenly used in the treatment of cancer. First, the mathematical models are introduced and the existence, uniqueness and regularity of the solution of the state equation are proved as well as the stability and maximum principle under extra assumptions. Second, the control problems are formulated for different situations in order to control and stabilize the desired online temperature. An optimal solution is proven to exist and finally the necessary optimality conditions are given. Our problem incorporates the effect of blood flow in the heat transfer equation in a way that captures the directionality of the blood flow and incorporates the convection features of the heat transfer between blood and solid tissue.

13.1 Introduction 13.1.1 Motivation and Statement of the Problem The mathematical problem studied in this chapter is derived from the modeling of the transport of thermal energy in living tissues. The evaluation of thermal conductivities in living tissues is a very complex process which uses different phenomenological mechanisms including conduction, convection, radiation, metabolism, evaporation and others. Moreover, blood flow and extracellular water affect considerably both the heat transfer in the tissues and the

428


tissues thermal properties. The bioheat transfer process in tissues is dependent on the behavior of blood perfusion along the vascular system. The first model, taking account of the blood perfusion, was introduced by Pennes [237] in 1948. The model is based on the classical thermal diffusion system, by incorporating the effects of metabolism and blood perfusion. The Pennes model has been adapted by many biologists for the analysis of various heat transfer phenomena in a living body. Others, after evaluating the Pennes model in specific situations, have concluded that many of the hypotheses (which are foundational to the model) are not valid. They then modified and generalized the model for these systems, see, e.g., Arkin et al. [11], Chato [75], Chen and Holmes [77], Hirst [158], Valvano et al. [287] and Weinbaum and Jiji [298] (see also, e.g., Charney [74] for a review on the mathematical modeling of the influence of blood perfusion). The goal of this chapter is to use the robust control technique in order to analyze the effects of the fluctuations and the uncertainties on the transient temperature of biological tissues. To treat the system of motion in living body, we consider the following transient bioheat transfer type model introduced by Belmiloudi [48]:1 ∂U = div(κ(x)∇U ) − p(U − Ua ) − K(, U ) + F (x, t, U ) + f on Q, ∂t subject to the heat flux boundary condition (κ∇U ).n = −q(U − Ub ) + g on Σ = ∂Ω × (0, T ),

(13.1)

and the initial condition U (0) = U0 on Ω, under the pointwise constraints a1 ≤ p ≤ a2 a.e. in Q,

(13.2)

where the cylinder Q = Ω × (0, T ), the state function U is the temperature distribution, T > 0 is a fixed constant (a given final time), the body Ω is an open bounded domain in IRm , m ≤ 3 with a smooth boundary Γ = ∂Ω which is sufficiently regular, and Ω is totally on one side of Γ , n is the unit outward normal to Γ and ai , for i = 1, 2, are given positive constants. The quantity p is the blood perfusion rate and q describes the heat transfer coefficient. The heat capacity is assumed to be constant and thermal conductivity of tissue κ is assumed to be variable and satisfies ν ≥ κ = σ 2 ≥ μ > 0 (where ν and μ are two positive constants). The second term on the right-hand side of the state Equation (13.1) describes the heat transport between the tissue and microcirculatory blood perfusion, the third term K corresponds, for example, to the directional convective mechanism of heat transfer due to blood flow, the fourth term F is the body heating function which describes the physical 1

To simplify the presentation we have neglected the radiative terms.

13.1 Introduction

429

properties of material (depending for example on the thermal absorptivity).2 The source term f describes a distributed energy source which can be generated through a variety of sources, such as focused ultrasound, radio-frequency, microwave, resistive heating, laser beams and others (depending on the difference between the energy generated by the metabolic processes and the heat exchanged between, for example, the electrode and the tissue). The first term on the right-hand side of the boundary condition in (13.1) describes the convective component and the second term g is the heat flux due to evaporation. The function Ua is the blood temperature, and the function Ub is the bolus temperature, and they are assumed to be in L∞ (Q) and L∞ (Σ), respectively. The initial value U0 is assumed to be in L2 (Ω). 13.1.2 Thermal Damage Calculations After obtaining the temperature distribution, we can calculate the accumulation of thermal damage, which is associated with injury to tissue. For this, we can use the well known Arrhennuis damage integral formulation (see, e.g., Tropea and Lee [285]): τexp C(0) −E )=A )dt, (13.3) exp( D(x, τexp ) = ln( C(τexp ) RU (x, t) 0 where D is the non-dimensional degree of tissue injury, C is the concentration of living cells, τexp is the duration of the exposure, A is the molecular collision frequency (s−1 ), E is the denaturation activation energy (J mol−1 ) and R is the universal gas constant, equal to 8.314 J mol−1 K−1 . The parameters A and E are dependent on the type of tissue, and the cumulative damage can be interpreted as the fraction of hypothetical indicator molecules that are denatured. Example 13.1. (Examples of models) 1. For the Pennes model, the operator K is null and the operator F is constant. 2. In the Chen–Holmes model [77] (the model has been formulated after the analyzing of blood vessel thermal equilibration length), the model incorporates the effect of blood flow in the heat transfer equation in a way that captures the directionality of the blood flow and incorporates the convection features of the heat transfer between blood and solid tissue,3 see Figure 13.1. The operator K is then the transport operator .∇ and it is 2

3

The introduction of the non-linear term F in the bioheat system is very important, because the physical properties of material have power law dependence on temperature. The blood-perfused tumor tissue volume, including blood flow in microvascular bed with the blood flow direction, contains many vessels and can be regarded as a porous medium consisting of a tumor tissue (a solid domain) fully filled with blood (a liquid domain).

430


corresponding to a directional convective term due to the net flux of the equilibrated blood. The operator F = F (x, t) is independent of the temperature. The conductivity term is the sum of two terms: κ = κt + κp , where div(κp ∇T ) is corresponding to the enhancement of thermal conductivity in tissue due to the flow of blood within thermally significant blood vessels and the term div(κt ∇T ) is similar to the Pennes model. Moreover, if we assume that the velocity field is known and satisfies ∈ L∞ (Q) and div() = 0, so we have easily the estimate given in hypothesis (H2) (given below) where γv = L∞ (Q) , and that K ∗ = −K. 3. In biological modeling, the non-linear operator F can be chosen as a polynomial functions of the temperature. ♣

Figure 13.1. Relationship between tumor vascular and blood flow direction.

13.1.3 Background and Motivation The mathematical modeling of cancer treatments (chemotherapy, thermotherapy, etc.) is an highly challenging area of applied mathematics. Recently, a large number of studies and research related to the cancer treatments, in particular by chemotherapy or thermotherapy, have been the object of numerous developments. Various problems associated with cancer chemotherapy (drug treatment) have been intensively studied (see, e.g., De Pillis et al. [99], Liu and Freedman [209], Mishra et al. [224], and the references therein). In order to control the drug dosage administered, many control problems have been studied (see, e.g., Alamir and Chareyron [7], Belmiloudi [47], Fister and Panetta [124], Jackson and Byrne [168], Kimmel and Swierniak [174], Ledzewicz and Sachttler [189, 190], and the references therein). As an alternative to the traditional surgical treatment or to enhance the effect of conventional chemotherapy, various problems associated with localized thermal therapy have been intensively studied, see e.g., Deuflhard and Seebass [100], Hill and Pincombe [156], Liu et al. [208], Marchant and Lui

13.1 Introduction

431

[216], Martin et al. [219], Pincombe and Smyth [242], Seip and Ebbini [266], Shih et al. [268], Sturesson and Engels [278], Tropea and Lee [285], and the references therein. In order to improve the treatments, several approaches have been proposed recently to control the temperature during thermal therapy. We can mention, e.g., Belmiloudi [46, 48], Bohm et al. [54], Hutchinson et al. [163], Kohler et al. [176], Kowalski and Jin [181], Malinen et al. [215], Ganzler et al. [126], Vanne and Hynynen [289], and the references therein. The essential aspect of these contributions has been the numerical analysis, MRI-based optimization techniques and mathematical analysis. An important application of all bioheat transfer models in interdisciplinary research areas, joining the mathematical, biological and medical fields, is the analysis of the temperature field which develops in living tissue when heat is applied to the tissue, especially in clinical cancer therapy hyperthermia and in accidental heating injury, such as burns (in hyperthermia, tissue is heated to enhance the effect of an accompanying radio or chemotherapy). Indeed, thermal therapy (performed with a laser, focused ultrasound or microwaves) gives the possibility of destroying the pathological tissues with minimal damage to the surrounding tissues. Moreover, due to the self-regulating capability of the biological tissue, the blood perfusion depends on the evolution of the temperature and vary significantly between different patients, and between different therapy sessions (for the same patient). Consequently, in order to have optimal thermal diagnostics and maximize the benefit of the therapy to the patient, it is necessary to study the value of the perfusion parameter. The new feature introduced in this chapter concerns the study of robust control problems for a generalized non-linear evolutive bioheat transfer system, where the goal is to stabilize and control the desired online temperature control m provided by magnetic resonance imaging (M RI) measurements (MRI is a new efficient tool in medicine in order to control surgery and treatments). The online temperature can be modeled by the relation m ≈ γU + δP , where the non-negative functions δ and γ satisfy γ ≈ δ in muscle and γ ≤ δ in fat. 13.1.4 Assumptions and Notations First, we state the main hypotheses on the body heating coefficient F and the operator K: (H1) F is a Carathéodory function from Q × IR into IR such that (i) for almost all (x, t) ∈ Q, F (x, t, .) is Lipschitz and bounded function with | F (x, t, r) |≤ M1 , ∀ r ∈ IR and a.e. in Q, (ii) F is differentiable and the partial derivatives Fx (., r) = (∂F /∂x)(., r) and G = Fr (., r) = (∂F /∂r)(., r) are Lipschitz continuous in Q for all r ∈ IR, and are globally bounded in Q × IR.

432


(H2) For all sufficiently regular functions , K(, .) is a linear operator and satisfies the following estimate: there exists a constant γv ≥ 0 such that K(, v) L2 (Ω) ≤ γv v H 1 (Ω) , ∀v ∈ H 1 (Ω).4 We denote by K ∗ the adjoint of K, i.e.,

K ∗ (, u), v = K(, v), u, ∀(u, v) ∈ (H 1 (Ω))2 . We can now introduce the following spaces: H = L∞ (0, T ; L2(Ω)), V = H 1 (Ω), V = L2 (0, T ; V ), ∂w ∈ L2 (0, T ; V )} ∂t and the set of the admissible functions describing the constraints

W = {w ∈ L2 (0, T ; V ) :

Cad = {p ∈ L2 (Q) : a1 ≤ p ≤ a2 a.e. in Q}, where ai , for i = 1, 2, are two constants. In the rest of this application, we may omit the variables x and t in their functions without ambiguity. Lemma 13.2. Let ul be a sequence converging toward u in W weakly and in L2 (Q) strongly we have F (., ul ) −→ F (., u) in Lp (Q) strongly ∀p ∈ [1, +∞). Proof. We prove easily the result by using the convergence result of ul and assumption (i) of (H1).

13.2 The State System In this section we present the existence, uniqueness and regularity of the solution of the problem (13.1) and obtain stability and maximum principle results. 13.2.1 Existence and Stability Results A function U ∈ W is a weak solution of system (13.1) provided, for all v ∈ V and a.e. t ∈ (0, T ), ∂U , v + κ∇U.∇vdx + K(, U )vdx

∂t Ω Ω q(U − Ub )vdΓ + p(U − Ua )vdx + (13.4) Ω Γ = F (x, t, U )vdx + f vdx + gvdΓ, Ω

Ω

Γ

U (0) = U0 on Ω, where ., . denotes the duality between V and V . 4

The constant γv depends on the norm of the given function .

13.2 The State System

433

Theorem 13.3. Let assumptions (H1)–(H2) be fulfilled: (i) Let the initial condition U0 be given in L2 (Ω) and the source term (p, q, f, g) be in Cad × L∞ (Σ)× L2 (Q)× L2 (Σ). Then there exists a unique solution U in W ∩ H of problem (13.1). (ii) Let q ∈ L∞ (Σ) and let (pi , fi , gi , U0i ), for i = 1, 2, be two functions of Cad × L2 (Q) × L2 (Σ) × L2 (Ω). If Ui ∈ W ∩ H is the solution of (13.1) corresponding to data (pi , fi , gi , U0i ), for i = 1, 2, then, a.e. t ∈ (0, T ) U 2W∩H ≤ c( p 2L2 (Q) + f 2L2 (Q) + g 2L2 (Σ) + U0 2L2 (Ω) ), (13.5) where U = U1 − U2 , p = p1 − p2 , f = f1 − f2 , g = g1 − g2 and U0 = U01 − U02 . Proof. To prove the existence of a solution, we use the Faedo–Galerkin method. Set (v1 , . . . , vl , . . .) a total and free sequence in V . Set ul = li=1 gil (t)vi (the functions gil are scalar functions defined on [0, T ]) as an approximation of the solution of problem (13.4), verifying the following problem ∂ul , vk + κ∇ul .∇vk dx + K(, ul )vk dx

∂t Ω Ω + q(ul − Ub )vk dΓ + p(ul − Ua )vk dx (13.6) Ω Γ = F (x, t, ul )vk dx + f vk dx + gvk dΓ, Ω

Ω

Γ

ul (0) = Pl U0 on Ω, where Pl is the L2 -projector onto the space spanned by v1 , . . . , vl (the initial data U0 satisfies then Pl U0 L2 (Ω) ≤ U0 L2 (Ω) and Pl U0 −→ U0 in L2 (Ω) as l −→ ∞). We can deduce from (13.6) that 1 d 2 2 ul L2 + κ | ∇ul | dx + K(, ul )ul dx + q(ul − Ub )ul dΓ 2 dt Ω Ω Γ + p(ul − Ua )ul dx = F (x, t, ul )ul dx + f ul dx + gul dΓ Ω

Ω

Ω

Γ

and then 1 d ul 2L2 + κ | ∇ul |2 dx + q | ul |2 dΓ + p | ul |2 dx 2 dt Ω Γ Ω K(, ul )ul dx + qUb ul dΓ =− Γ Ω pUa ul dx + F (x, t, ul )ul dx + f ul dx + gul dΓ. + Ω

Ω

Ω

Γ

434


According to the positivity of the function q, the boundedness of the functions p, Ua , Ub and assumptions (H1)–(H2) we can deduce by using Young’s inequality that d ul 2L2 +μ ∇ul 2L2 dt ≤ c1 ul 2L2 +c2 ( p 2L2 + f 2L2 + g 2L2 (Γ ) ), which implies that ul is bounded in L2 (0, T ; V ) ∩ L∞ (0, T ; L2(Ω)). Prove now that ∂ul /∂t ∈ L2 (0, T ; V ). According to (13.6) we have that, for all v in V , ∂ul , v = − κ∇ul .∇vdx − K(, ul )vdx

∂t Ω Ω − q(ul − Ub )vdΓ − p(ul − Ua )vdx Γ Ω + F (x, t, ul )vdx + f vdx + gvdΓ. Ω

Ω

Ω

Thanks to the boundedness of p, q, Ua and Ub we then have |

∂ul , v |≤ c1 ul H 1 v H 1 +c2 ( f L2 + g L2 (Γ ) ) v H 1 . ∂t

Applying this result (according to Young’s formula) we can deduce that

∂ul V ≤ c1 ul H 1 +c2 ( f L2 + g L2 (Γ ) ). ∂t

(13.7)

Since ul is bounded in L∞ (0, T ; L2(Ω))∩L2 (0, T ; H 1 (Ω)), we can deduce that ∂ul /∂t is bounded in L2 (0, T ; V ). According to Lemma 6.6 the injection of W into L2 (0, T ; L2(Ω)) is compact. So this result makes it possible to extract from ul a subsequence also denoted by ul and such that ul −→ U strongly in L2 (0, T ; L2(Ω)), ul U weakly in L2 (0, T ; H 1(Ω)).

(13.8)

By using the previous results we can prove easily that U is the solution of problem (13.1) and verifies the regularity U ∈ L2 (0, T ; V ) ∩ L∞ (0, T ; L2(Ω)) and ∂U /∂t ∈ L2 (0, T ; V ). Now we prove the uniqueness and the stability results given in (ii). To obtain the uniqueness result, we suppose that there exist two solutions U1 , U2 of (13.1). Then U = U1 − U2 is a solution of the following problem: ∂U − div(κ∇U ) + K(, U ) + pU = F (., U1 ) − F (., U2 ) on Q, ∂t (κ∇U ).n = −qU on Σ, U (0) = 0 on Ω.

13.2 The State System

435

Multiplying the previous system by U and integrating over Ω × (0, t), for t ∈ (0, T ) (by using Green’s formula), gives t t t 1 U (t) 2L2 + σ∇U 2L2 ds + p | U |2 dxds + q | U |2 dΓ ds 2 0 0 Ω 0 Γ t t (F (x, s, U1 ) − F (x, s, U2 ))U dxds − K(, U )U dxds. = 0 Ω

0

Ω

By using the hypotheses (H1)–(H2), the regularity of p and q, and the positivity of q we have that t t 2 2 ∇U L2 ds ≤ c1 U 2L2 ds. U L2 (t) + μ 0

0

According to Gronwall’s formula (since U (0) = 0) we prove easily that U1 = U2 and then the uniqueness result follows. To prove the estimate given in (ii), we set p = p1 − p2 , f = f1 − f2 , g = g1 − g2 , U0 = U01 − U02 and U = U1 − U2 . Then U is the solution of ∂U − div(κ∇U ) + p(U2 − Ua ) + p1 U + K(, U ) ∂t = F (., U1 ) − F (., U2 ) + f on Q, (κ∇U ).n = −qU + g on Σ, U (0) = U0 on Ω. Multiplying the previous system by U and integrating over Ω × (0, t), for t ∈ (0, T ) (by using Green’s formula), gives t t 1 2 2 U (t) L2 + σ∇U L2 ds + p1 | U |2 dxds 2 0 0 Ω t t 2 q | U | dΓ ds = − K(, U )U dxds + 0 Ω 0 tΓ p(u2 − Ua )U dxds − 0 Ω t (F (x, s, U1 ) − F (x, s, U2 ))U dxds + 0 Ω t t 1 f U dxds + gU dΓ ds. + U0 2L2 + 2 0 Ω 0 Γ According to the regularity of Ui , pi , fi , gi (for i = 1, 2), q, Ua and Ub , and to the hypotheses (H1)–(H2) we have, by using Gronwall’s formula, U 2V∩H ≤ c( p 2L2 (Q) + f 2L2 (Q) + g 2L2 (Σ) + U0 2L2 (Ω) ).

(13.9)

By using similar arguments as used to obtain the estimates (13.7) and (13.9)

436


we can deduce that ∂U 2 2 ∂t L (0,T ;V ) (13.10) ≤ c( p 2L2 (Q) + f 2L2 (Q) + g 2L2 (Σ) + U0 2L2 (Ω) ). This completes the proof.

13.2.2 A Maximum Principle In this section we establish a maximum principle under extra assumptions on the data. In addition to (H1)–(H2), we suppose, for a couple of constants (uf , us ) such that 0 ≤ uf < us , the following assumptions (H3) F (x, t, .) = 0 in ] − ∞, uf ] ∪ [us , +∞[, for almost all (x, t) ∈ Q. (H4) K(, s) = 0, for all constant s ∈ IR and K(, v)vdx ≥ 0 for all v ∈ H 1 (Ω). Ω

(H5) uf ≤ Ua ≤ us and uf ≤ Ub ≤ us for all in Q and in Σ, respectively. Then we have the following theorem. Theorem 13.4. Let (H1)–(H5) be fulfilled. Suppose that the initial data U0 is such that uf ≤ U0 ≤ us , a.e. in Ω and f, g are positive functions. Then, the weak solution U ∈ W ∩H of (13.1) satisfies, for all t ∈ (0, T ), uf ≤ U (., t) ≤ us a.e. in Ω. Proof. Let us consider the following notations: r+ = max(r, 0), r− = (−r)+ and then r = r+ − r− . We prove now that if U0 ≥ uf , a.e. in Ω then U (., t) ≥ uf , for all t ∈ [0, T ] and a.e. in Ω. According to Chapter 3, we have that (U − uf )− ∈ L2 (0, T ; H 1(Ω)) with ∇(U − uf )− = −∇(U − uf ) if (U − uf ) > 0 and ∇(U − uf )− = 0 otherwise, a.e. in Q. Then, taking v = −(U − uf )− in Equation (13.4), we have (a.e. in (0, T )) d κ | ∇(U − uf )− |2 dx (U − uf )− 2L2 + 2dt Ω K(, (U − uf )− )(U − uf )− dx + Ω + q(Ub − uf )(U − uf )− dΓ + q((U − uf )− )2 dΓ Γ Γ + p(Ua − uf )(U − uf )− dx + p((U − uf )− )2 dx Ω Ω − =− F (x, t, U )(U − uf ) dx − f (U − uf )− dx Ω Ω g(U − uf )− dΓ. − Γ


437

According to hypothesis (H3), and the assumptions Ua ≥ uf and Ub ≥ uf , we can deduce that d (U − uf )− 2L2 ≤ 0 2dt and then, for all time t ∈ (0, T ) (by integrating with respect to time), (U − uf )− 2L2 (t) ≤ (U0 − uf )− 2L2 . Using the assumption U0 ≥ uf we can deduce that U (t, .) ≥ uf for all t ∈ (0, T ) and a.e. in Ω. To prove that, for all t ∈ (0, T ), U (., t) ≤ us a.e. in Ω, we take v = (U −us )+ in Equation (13.4) and use the same technique as before. This completes the proof.

13.3 The Perturbation Problem In this section we formulate the perturbation problem and present the existence, uniqueness and regularity results of the perturbation solution. 13.3.1 Formulation of the Perturbation Problem In the following, the solution U of problem (13.1) will be treated as the target function. We are then interested in the robust regulation of the deviation of the problem from the desired target U . We analyze the full non-linear equation which models large perturbations u to the target U , i.e., we assume that U satisfies the problem (13.1) with the data (U0 , P, f, g, Ua , Ub ) and U + u satisfies problem (13.1) with the data (U0 + u0 , P + p, f + ξ, g + η, Ua + ua , Ub + ub ). Hence we consider the following system (for a given U satisfying the regularity of Theorem 13.3): ∂u = div(κ∇u) − P (u − ua ) − p(u − ua + va ) ∂t −K(, u) + (F (., u + U ) − F (., U )) + ξ on Q, subject to the heat flux boundary condition

(13.11)

(κ∇u).n = −q(u − ub ) + η on Σ, and the initial condition u(0) = u0 on Ω, where va = U − Ua . If we set F˜ (., u) = F (., u + U ) − F (., U ) then (13.11) is reduced to the following system:

438


∂u = div(κ∇u) − P (u − ua ) − p(u − ua + va ) ∂t −K(, u) + F˜ (x, t, u) + ξ on Q, subject to the heat flux boundary condition

(13.12)

(κ∇u).n = −q(u − ub ) + η on Σ, and the initial condition u(0) = u0 on Ω. Remark 13.5. (i) We can easily verify that F˜ satisfies the same hypotheses that F , i.e., (H1)–(H2). (ii) For simplicity of future reference, we omit the “˜” on F˜ . ♦ Next we give the weak formulation associated with the problem (13.12). Multiplying the first part of (13.12) by v ∈ V and integrating over Ω this gives (according to the second part of (13.12)) the weak formulation (a.e. t ∈ (0, T )) ∂u κ∇u.∇vdx + K(, u)vdx + q(u − ub )vdΓ

, v + ∂t Ω Ω Γ P (u − ua )vdx + p(u − ua + va )vdx + (13.13) Ω Ω = F (., u)vdx + ξvdx + ηvdΓ, Ω

Ω

Γ

u(0) = u0 on Ω. 13.3.2 Existence and Stability Results Now we show the existence and uniqueness of the solution to the problem (13.13), and give some Lipschitz continuity results. Theorem 13.6. Let assumptions (H1)–(H2) be fulfilled: (i) Let the initial condition u0 be given in L2 (Ω) and the source term (p, ξ, η) be in L∞ (Q) × L2 (Q) × L2 (Σ). Then there exists a unique solution u in W ∩ H of (13.12). (ii) Let (pi , ξi , ηi , u0i ), for i = 1, 2, be two functions of L∞ (Q) × L2 (Q) × L2 (Σ). If ui ∈ W ∩ H is the solution of (13.12) corresponding to data (pi , ξi , ηi , u0i ), i = 1, 2, then the following Lipshitz continuity result holds: u 2W∩H ≤ c( p 2L2 (Q) + ξ 2L2 (Q) + η 2L2 (Σ) + u0 2L2 (Ω) ),

(13.14)

where u = u1 −u2 , p = p1 −p2 , ξ = ξ1 −ξ2 , η = η1 −η2 and u0 = u01 −u02 . Proof. The proof of this result can be obtained by using a similar technique as in the proof of Theorem 13.3. So, we omit the details.


439

13.4 Robust Control Problems In this section we formulate the robust control problem and study the existence and necessary optimality conditions for an optimal solution. 13.4.1 Formulation of the Control Problem and Differentiability Our problem, in this section, is to find the best admissible perfusion function in the presence of the worst disturbance in the distributed energy sources. We then suppose that the control is in the perfusion function p and the disturbance is in the force ξ, that is, p = ϕ (ϕ ∈ L∞ (Q)) and ξ = ψ (ψ ∈ L2 (Q)). Therefore, the function u is assumed to be related to the disturbance ψ and control φ through the problem ∂u = div(κ∇u) − P (u − ua ) − ϕ(u − ua + va ) ∂t −K(, u) + F (., u) + ψ on Q, subject to the heat flux boundary condition

(13.15)

(κ∇u).n = −q(u − ub ) + η on Σ, and the initial condition u(0) = u0 on Ω, under the pointwise constraint τ1 ≤ ϕ ≤ τ2

a.e. in Q,

(13.16)

where [τ1 , τ2 ] contains 0 and, u0 ∈ L2 (Ω) and η ∈ L2 (Σ) are assumed to be given. Let Uad = {ϕ ∈ L2 (Q) : τ1 ≤ ϕ ≤ τ2 a.e. in Q} ⊂ L∞ (Q). Our control problem is then, find (ϕ, ψ) ∈ Uad × Vad such that the following cost functional : 1 γu + δϕ − mobs 2L2 (Q) 2 (13.17) α β + ϕ 2L2 (Q) − ψ 2L2 (Q) , 2 2 is minimized with respect to ϕ and maximized with respect to ψ subject to the problem (13.15),

J(ϕ, ψ) =

where Vad is a convex, closed, non-empty and bounded subset of L2 (Q), the function mobs is in L2 (Q), the parameters α and β are fixed constants and the functions γ, δ are positive with space-time dependent entries such that 0 < γ1 ≤ γ ≤ γ2 , 0 < δ1 ≤ δ ≤ δ2 on Q.

440


We are now going to study the differentiability of the operator solution of problem (13.15). Before proceeding to investigation of the differentiability of the function F : (ϕ, ψ) −→ u, which maps the source term (ϕ, ψ) ∈ L∞ (Q) × L2 (Q) of problem (13.15) into the corresponding solution u ∈ W ∩ H, we study, for (h, ϑ, κ, w0 ) given data, the following problem: find w ∈ W ∩ H such that, for a.e. t ∈ (0, T )) ∂w = div(κ∇w) − (P + ϕ)w − h(u − ua + va ) ∂t −K(, w) + G(., u)w + ϑ on Q, subject to the heat flux boundary condition

(13.18)

(κ∇w).n = −qw + θ on Σ, and the initial condition w(0) = w0 on Ω, where u = F (ϕ, ψ) and G is the partial derivative function of F . Theorem 13.7. If u is in W ∩ H, the following results hold: (i) For any (h, ϑ, θ, w0 ) ∈ L∞ (Q) × L2 (Q) × L2 (Σ) × L2 (Ω), there exists a unique function w ∈ W ∩ H, solution of problem (13.18), such that w 2W∩H ≤ Ce ( h 2L2 (Q) + ϑ 2L2 (Q) + θ 2L2 (Σ) + w0 2L2 (Ω) ).

(13.19)

(ii) Let (hi , ϑi , θi , w0i ), for i = 1, 2 be in L∞ (Q) × L2 (Q) × L2 (Σ) × L2 (Ω). If wi is the solution of (13.18), corresponding to data (hi , ϑi , θi , w0i ) for i = 1, 2, respectively, then w 2W∩H ≤ Ce ( h 2L2 (Q) + ϑ 2L2 (Q) + θ 2L2 (Σ) + w0 2L2 (Ω) ),

(13.20)

where w = w1 − w2 , h = h1 − h2 , w0 = w01 − w02 , ϑ = ϑ1 − ϑ2 and θ = θ1 − θ2 . Proof. The existence, uniqueness and Lipschitz continuity results of problem (13.18) are obtained in the same way as used to prove Theorem 13.6 and by using the regularity of u. So, we omit the details. We are now going to study the differentiability of the operator solution F . Theorem 13.8. Let assumptions (H1)–(H2) be fulfilled, the initial condition u0 be in L2 (Ω) and the perturbation of the heat flux g be in L2 (Σ). Then the function F is differentiable with respect to X = (ϕ, ψ) ∈ Uad × Vad where


441

its derivative F (X) : L∞ (Q) × L2 (Q) −→ H ∩ V is a linear operator such that w = F (X).Y is the unique solution of the problem (13.18), with data Y = (h, ϑ), the given initial condition w(t = 0) = 0 and the boundary forcing θ = 0, i.e., ∂w = div(κ∇w) − (P + ϕ)w − h(u − ua + va ) ∂t −K(, w) + G(., u)w + ϑ on Q, subject to the heat flux boundary condition (κ∇w).n = −qw on Σ,

(13.21)

and the initial condition w(0) = 0 on Ω. Moreover, we have the estimate (13.20), i.e., for all wi is the solution of (13.18), with data (hi , ϑi ) ∈ L∞ (Q) × L2 (Q), the given initial condition w(t = 0) = 0 and the boundary forcing θ = 0, for i = 1, 2, respectively, the following estimate holds: w 2H∩V ≤ Ce ( h 2L2 (Q) + ϑ 2L2 (Q) ),

(13.22)

where w = w1 − w2 , h = h1 − h2 and ϑ = ϑ1 − ϑ2 . Moreover, for all Xi = (ϕi , ψi ) ∈ Uad × Vad , for i = 1, 2, we have the following estimate, for all Y = (h, ϑ) ∈ L∞ (Q) × Vad : F (X1 ).Y − F (X2 ).Y 2H∩V ≤ Ce ( Y L2 (Q)×L2 (Q) X 2L2 (Q)×L2 (Q) +Y

2L2 (Q)×L2 (Q)

(13.23)

X L2 (Q)×L2 (Q) ),

where ϕ = ϕ1 − ϕ2 , ψ = ψ1 − ψ2 and X = X1 − X2 = (ϕ, ψ). Proof. The proof of this theorem can be obtained by using the same technique as used to prove the results of Proposition 8.13, by taking into account the properties of the non-linear operator F and by using the following expression (obtained by a simple manipulation):5

1

F (., v) − F (., u) =

G(., u + s(v − u))(v − u)ds

0

(for the differentiability of F see also Belmiloudi [48], in which the studied model take into account the non-linear radiative terms in boundary). So, we omit the details. In the next subsection we study the existence of an optimal solution. 5

The operator F is similar to F1 in Section 11.3.

442


13.4.2 Existence of an Optimal Solution Theorem 13.9. For α and β sufficiently large, there exists (ϕ∗ , ψ ∗ ) ∈ Uad × Vad and u∗ ∈ W ∩ H such that (ϕ∗ , ψ ∗ ) is the optimal solution of (13.17) and u∗ = F (ϕ∗ , ψ ∗ ) is the solution of (13.15). Proof. Let Pψ be the mapping: ϕ −→ J(ϕ, ψ) and Qϕ be the mapping: ψ −→ J(ϕ, ψ). To obtain the existence of the robust control problem, we prove first that Pψ is convex and lower semi-continuous for all ψ ∈ Vad , second that Qϕ is concave and upper semi-continuous for all ϕ ∈ Uad and, finally, we use the minimax theorems in infinite dimensions presented in Chapter 5. In order to prove the convexity, it is sufficient to show that for (ϕ1 , ϕ2 ) ∈ Uad × Uad , we have (Pψ (ϕ1 ) − Pψ (ϕ2 )).ϕ ≥ 0, where ϕ = ϕ1 − ϕ2 . From the expression of G-differentiable cost functional J (a composition of G-differentiable mappings), it follows that Pψ is G-differentiable and, for i = 1, 2, (γui + δϕi − mobs )(γwi + δϕ)dxdt Pψ (ϕi ).ϕ = Q ϕi ϕdxdt, +α Q

where ui = F (ϕi , ψ) and wi = F (ϕi , ψ).(ϕ, 0). Consequently, (Pψ (ϕ1 ) − Pψ (ϕ2 )).ϕ

= α ϕ 2L2 (Q) + δ 2 | ϕ |2 dxdt + γδuϕdxdt (13.24) Q Q + (γu + δϕ)γw1 dxdt + (γu2 + δϕ2 − mobs )γwdxdt, Q

Q

where u = u1 − u2 and w = w1 − w2 . The estimates (13.23), (13.22) and (13.14) imply that γδuϕdxdt + (γu + δϕ)γw1 dxdt ≤ C0 ϕ 2L2 (Q) , Q Q (13.25) 3/2 (γu2 + δϕ2 − mobs )γwdxdt ≤ C1 ϕ L2 (Q) . Q

From (13.24) and the previous results (13.25), we can deduce that there exists a constant αl > 0 such that, for α ≥ αl , we have (Pψ (ϕ1 ) − Pψ (ϕ2 )).ϕ ≥ (δ 2 + αl − C0 ) ϕ 2L2 (Q) −C1 ϕ L2 (Q) ≥ 0 3/2

and then the convexity of Pξ is established. In the same way, we can find βl > 0 such that, for β ≥ βl , Qϕ is concave. We prove now that Pψ (respectively Qϕ ) is lower (respectively upper) semi-continuous for all ψ ∈ Vad (respectively ϕ ∈ Uad ).


443

Let ϕk ∈ Uad be a minimizing sequence of Pψ , i.e., lim inf J(ϕk , ψ) = k−→∞

inf

ϕ∈L2 (Q)

J(ϕ, ψ).

Then, according to the nature of the cost function J, we can deduce that ϕk is uniformly bounded in Uad and we can extract from ϕk a subsequence also denoted by ϕk such that ϕk ϕξ weakly in Uad . Therefore, by using the same technique as to obtain the estimate (13.14), the function uk = F (ϕk , ψ) is uniformly bounded in W ∩ H. Moreover, according now to Lemma 6.6, the injection of W into L2 (Q) is compact. Consequently, these results make it possible to extract from uk a subsequence also denoted by uk such that uk uψ weakly in L2 (0, T ; V ), uk −→ uψ strongly in L2 (Q), ϕk ϕψ weakly in L2 (Q) and ϕψ ∈ Uad .

(13.26)

Prove now that uk ϕk −→ uψ ϕψ weakly in L2 (Q). Since uk ϕk − uψ ϕψ = (uk − uψ )ϕk + uψ (ϕk − ϕψ ), then according to the first and second parts of (13.26), we then obtain the result. It is easy to prove that uψ is a solution of (13.15) with data (ϕψ , ψ) and according to the uniqueness of the solution of the problem (13.15), we then have uψ = F (ϕψ , ψ). Since the norm is lower semi-continuous, therefore we have that the map Pψ is lower semi-continuous for all ψ ∈ Vad . By applying similar argument as in the proof of the previous result we obtain that Qϕ is upper semi-continuous for all ϕ ∈ Uad . This completes the proof. We next wish to show the appropriate first-order necessary conditions (optimality conditions) of the saddle point problem (13.17). 13.4.3 Optimality Conditions Theorem 13.10. Let assumptions (H1)–(H2) be fulfilled, (ϕ∗ , ψ ∗ , u∗ ) ∈ Uad × Vad × (W ∩ H) be an optimal solution such that (ϕ∗ , ψ ∗ ) is defined by (13.17) and u∗ = F (ϕ∗ , ψ ∗ ) is a solution of (13.15). Then, for (h, ϑ) ∈ Uad × Vad , ((u∗ − ua + va )˜ u∗ + αϕ∗ )(h − ϕ∗ )dxdt Q + (δ(γu∗ + δϕ∗ − mobs ))(h − ϕ∗)dxdt ≥ 0, (13.27) Q (−˜ u∗ − βψ ∗ )(ϑ − ψ ∗ )dxdt ≤ 0. Q

Moreover, the gradient of J at (ϕ∗ , ψ ∗ ) in direction (h, ϑ) is given by

444


J (ϕ∗ , ψ ∗ ).(h, ϑ) =

((u∗ − ua + va )˜ u∗ + αϕ∗ + δ(γu∗ + δϕ∗ − mobs ))hdxdt

Q +

(−˜ u∗ − βψ ∗ )ϑdxdt.

Q

Otherwise (in the weak sense), ∂J ∗ ∗ (ϕ , ψ ) = (u∗ − ua + va )˜ u∗ + αϕ∗ + δ(γu∗ + δϕ∗ − mobs ), ∂ϕ ∂J ∗ ∗ (ϕ , ψ ) = −˜ u∗ − βψ ∗ , ∂ψ where u ˜∗ is the solution of the adjoint problem (13.29) (given below), corresponding to the primal solution u∗ . Proof. Let (ϕ∗ , ψ ∗ ) ∈ Uad × Vad be a saddle point of J, i.e., the corresponding solution of problem (13.17) and u∗ = F (ϕ∗ , ψ ∗ ) be the solution of problem (13.1). From Theorem 13.8 we know that F is differentiable. Therefore d J(ϕ + λh, ψ + λϑ)|λ=0 dλ = (γu + δϕ − mobs )(γw + δh)dxdt Q +α ϕhdxdt − β ψϑdxdt,

J (ϕ, ψ).(h, ϑ) =

Q

(13.28)

Q

where w = F (ϕ, ψ).(h, ϑ) is the solution of problem (13.21). In order to simplify (13.28), we introduce the following adjoint problem corresponding to the primal problem (13.15) (we denote by u = F (ϕ, ψ)): −

∂u ˜ − div(κ∇˜ u) + γ(γu + δϕ − mobs ) ∂t +K ∗ (, u ˜) + (P + ϕ)˜ u = (G(., u))∗ u ˜ on Q,

subject to the boundary condition

(13.29)

(κ∇˜ u).n = −q˜ u on Σ, and the final condition u ˜(T ) = 0 on Ω. To prove the existence of a unique solution u ˜ ∈ W ∩H, we change the variables of problem (13.29) by reversing the sense of time, i.e., t := T −t, and we apply a similar argument as used in the proof of Theorem 13.7. Due to (13.21) and (13.29) (multiplying (13.21) by the solution of (13.29) and integrating with respect to space and time) and using Green’s formula we obtain according to the second part of (13.21) and of (13.29) that

13.5 Other Situations

445

T

u˜(T )w(T )dx − Ω

u ˜(0)w(0)dx + h(u − ua + va )˜ udxdr Ω 0 Ω T T ϑ˜ udxdr = γ(γu + δϕ − mobs )wdxdr. − 0

0

Ω

Ω

Since u ˜(T ) = 0 and w(0) = 0, then h(u − ua + va )˜ udxdt − ϑ˜ udxdt = γ(γu + δϕ − mobs )wdxdt. Q

Q

Q

According to the expression of J (ϕ, ψ) we can deduce that ((u − ua + va )˜ u + αϕ)hdxdt J (ϕ, ψ).(h, ϑ) = Q + δ(γu + δϕ − mobs )hdxdt Q + (−˜ u − βψ)ϑdxdt. Q

Since (ϕ∗ , ψ ∗ ) is a saddle point we can deduce easily the optimality conditions (13.27) given in this theorem. This completes the proof. Remark 13.11. We can also prove by using the sign of the variation h (depending on the size of ϕ∗ ) that δ(γu∗ − mobs ) + (u∗ − ua + va )˜ u∗ ∗ , τ2 ϕ = max τ1 , min − . α + δ2

13.5 Other Situations 13.5.1 Data Assimilation If we want to take into account the initial condition disturbance ψ = u0 (data assimilation), we obtain the same results as in Section 13.4. In this case the cost functional can be given by J(ϕ, ψ) =

α β 1 γu + δϕ − mobs 2L2 (Q) + ϕ 2L2 (Q) − ψ 2L2 (Ω) . 2 2 2

We can also prove an existence theorem of the control problem and obtain necessary optimality conditions for its solution using the same method as in Section 13.4. Let now K = Uad × Vad such that Vad is a non-empty closed convex and bounded subset of L2 (Ω) and Uad is similar to previous section. Then, for α and β sufficiently large, there exist (ϕ∗ , ψ ∗ ) ∈ K and u∗ ∈ W ∩ H that satisfy

446


∂u∗ = div(κ∇u∗ ) − P (u∗ − ua ) ∂t −ϕ∗ (u∗ − ua + va ) − K(, u∗ ) + F (., u∗ ) + ξ on Q, (κ∇u∗ ).n = −q(u − ub ) + η on Σ,

(13.30)

u∗ (0) = ψ ∗ on Ω and, for (h, ϑ) ∈ Uad × Vad , we have the optimality conditions ((u∗ − ua + va )˜ u∗ + αϕ∗ + δ(γu∗ + δϕ∗ − mobs ))(h − ϕ∗ )dxdt ≥ 0, Q (−˜ u∗ (0) − βψ ∗ )(ϑ − ψ ∗ )dx ≤ 0, Ω ∗

where u ˜ is the solution of the adjoint problem (13.29) corresponding to the solution u∗ . 13.5.2 Boundary Disturbance If we want to take into account the heat flux disturbance (due to evaporation) ψ = η, we obtain the same results as in Section 13.4. In this case the cost functional can be given by J(ϕ, ψ) =

1 α β γu + δϕ − mobs 2L2 (Q) + ϕ 2L2 (Q) − ψ 2L2 (Σ) . 2 2 2

We can also prove an existence theorem of the control problem and obtain necessary optimality conditions for its solution by using the same method as in Section 13.4. Let now K = Uad × Vad such that Vad is a non-empty closed convex and bounded subset of L2 (Σ). Then, for α and β sufficiently large, there exist (ϕ∗ , ψ ∗ ) ∈ K and u∗ ∈ W ∩ H that satisfy ∂u∗ = div(κ∇u∗ ) − P (u∗ − ua ) ∂t −ϕ∗ (u∗ − ua + va ) − K(, u∗ ) + F (., u∗ ) + ξ on Q, (κ∇u∗ ).n = −q(u − ub ) + ψ ∗ on Σ,

(13.31)

u∗ (0) = u0 on Ω and, for (h, ϑ) ∈ Uad × Vad , we have the optimality conditions ((u∗ − ua + va )˜ u∗ + αϕ∗ + δ(γu∗ + δϕ∗ − mobs ))(h − ϕ∗ )dxdt ≥ 0, Q (−˜ u∗ − βψ ∗ )(ϑ − ψ ∗ )dΓ dt ≤ 0, Σ ∗

where u ˜ is the solution of the adjoint problem (13.29) corresponding to the solution u∗ .


447

13.5.3 Finite Number of Measurments In many biological situations, we can measure u in only some points in spacetime domain. Let now some points be in Ω × (0, T ) where we assume that we can measure u. Let xi ∈ Ω, i = 1, . . . l such that xi = xj if i = j, 0 < t1 < t2 < · · · < T , and assume that we measure quantities ij which are meant to be the tolerance uncertainty of the function M = γu + δp at point (xi , tj ), for i = 1, . . . , l and j = 1, . . . , N . Let (Ωi )i=1,l be a sequence of disjoint small balls in Ω such that xi ∈ Ωi , ∀i = 1, . . . , l. Let (Ij )j=1,N be also a sequence of disjoint intervals in (0, T ) such that tj ∈ Ij , ∀j = 1, . . . , N . We denote the average operator over the domain Qij = Ωi ×Ij (for i = 1, . . . , l, j = 1, . . . , N ) by 1 v(x, t)dxdt < v >ij = | Qij | Qij and propose the following cost function: 1 α β J(ϕ, ψ) = |< M >ij −ij |2 + ϕ 2L2 (Q) − ψ 2L2 (Q) . 2 2 2 i=1,l j=1,N

Let χD be the usual characteristic function of a domain D, i.e., χD = 1 on D, and 0 outside of D. Let S : L2 (Q) −→ L2 (Q) be defined by (∀v ∈ L2 (Q)) 1 χQ < v >ij S(v) = | Qij | ij i=1,l j=1,N

and let m be the following value: m= i=1,l j=1,N

1 χQ ij . | Qij | ij

By using the same technique as used in the proof of the previous results of Section 13.4, we can prove the existence theorem of the control problem and obtain necessary optimality conditions. Then, for α and β sufficiently large, there exist (ϕ∗ , ψ ∗ ) ∈ Uad × Vad and u∗ ∈ W such that u∗ = F (ϕ∗ , ψ ∗ ) is a solution of (13.11) and, for (h, ϑ) ∈ Uad × Vad , ((u∗ − ua )˜ u∗ + αϕ∗ + δ(S(M ∗ ) − m))(h − ϕ∗ )dxdt ≥ 0, Q (−˜ u∗ − βψ ∗ )(ϑ − ψ ∗ )dxdt ≤ 0, Q

˜∗ is the solution of the following adjoint problem where M ∗ = γu∗ + δϕ∗ and u (corresponding to the solution u∗ ): −

∂u ˜∗ − div(κ∇˜ u∗ ) + γ(S(M ∗ ) − m) ∂t u∗ = (G(., u∗ ))∗ u ˜∗ on Q, +K ∗ (, u˜∗ ) + (P + ϕ∗ )˜

u∗ on Σ, (κ∇˜ u∗ ).n = −q˜ u ˜∗ (T ) = 0 on Ω.

448


Remark 13.12. We obtain, by using the same techniques, similar results if we want to take into account the disturbance in the initial condition. ♦ 13.5.4 Union of a Finite Number of Subdomains Suppose now that the body is made of different types of tissue which occupy finitely many disjointed subdomains Ωi , i = 1, . . . , ND , of Ω, such that Ω = i=1,ND Ω i . Moreover, we assume that the perfusion acts continuously according to the temperature in each domain Ωi and discontinuously at the tissue boundaries. We propose the following cost function: J(ϕ, ψ) =

α β 1 γu + δϕ − mobs 2L2 (Q) + ϕ 2U˜ − ψ 2L2 (Ω) , ad 2 2 2

ãd = L2 (0, T ; R) ∩ Uad and R is the Hilbert space ãd × Vad , U where (ϕ, ψ) ∈ U R = {q ∈ L2 (Ω) : q|Ωi ∈ H 1 (Ωi ), for i = 1, . . . , ND } equipped with the following norm ⎛ ⎞1/2 (α1 q 2L2 (Ωi ) +α2 ∇q 2L2 (Ωi ) )⎠ , q R = ⎝ i=1,ND

with the fixed constants αi > 0 for i = 1, 2, and ϕ Uãd = ϕ L2 (0,T ;R) . Let Λ : L2 (Ω) −→ R be a linear operator such that, for all v ∈ L2 (Ω), the function π = Λ(v) is the solution of −α2 Δπi + α1 πi = v|Ωi on Ωi , (∇πi ).n = 0 on ∂Ωi ,

(13.32)

where πi = π, a.e. in Ωi and v|Ωi denotes the restriction of v on the subdomain Ωi , for i = 1, . . . , ND . By using the same technique as in the proof of the results of Section 13.4, we can prove the existence of the control problem and obtain the necessary optimality conditions. More precisely, for α and β sufficiently large, there ãd × Vad × W such that (ϕ∗ , ψ ∗ ) is exists an optimal solution (ϕ∗ , ψ ∗ , u∗ ) ∈ U defined by (13.17), u∗ = F (ϕ∗ , ψ ∗ ) is the solution of (13.1) and, for (h, ϑ) ∈ ãd × Vad U T

Λ((u∗ − ua + va )˜ u∗ + δ(M ∗ − mobs )) + αϕ∗ , h − ϕ∗ R dt ≥ 0, 0 (−˜ u∗ − βψ ∗ )(ϑ − ψ ∗ ) ≤ 0, Q

where M ∗ = γu∗ + δϕ∗ and u˜∗ is the solution of the adjoint problem (13.29) (corresponding to the solution u∗ ).


449

Remark 13.13. (i) We can also combine the different situations presented in this section and obtain the existence of the control problem and necessary optimality conditions without major difficulties. (ii) It is very interesting to study the case where the disturbance is in the bolus temperature, because during, for example, hyperthermia with prolongated laser induction heat treatment, the exchange mechanisms at the body–air interface play a very important role on the total tissue temperature distribution and, moreover, small fluctuations of the bolus temperature can affect considerably the resulting temperature distribution and thus the treatment. (iii) As indicated in Remark 10.31, for numerical resolution of robust control problems, the reader is referred to Chapter 9. ♦

14 Lotka–Volterra-type Systems with Logistic Time-varying Delays

In this chapter, we consider a resource management problem in which the management objectives are the stabilization of uncertain biological species. Such situations may arise in many different fields of resource economics and wildlife management. The stabilizing management consists of robust control strategies for a class of systems governed by parabolic equations governing diffusive biological species with logistic growth terms and multiple time-varying delays. First, we prove the existence, uniqueness and regularity results for these parabolic systems. Then, we introduce the perturbation problem and analyze its well-posedness. Second, we formulate the control problem in different situations. The existence and condition of uniqueness of the optimal solution are derived and first-order necessary conditions of optimality, characterizing the optimal solution, are obtained.

14.1 Introduction and Mathematical Setting 14.1.1 Motivation The aim of this chapter is the study of optimal control problem of a biological species, whose logistic growth is governed by a degenerate parabolic diffusive equation with multiple time-varying delays. Various problems associated with biology and economics behind diffusion population models with logistic growth terms have been studied extensively in the literature since few last years (see, e.g.,, Clark [87], Cushing [91], Edelstein [110], Murray [227], Waltman [291], and the references therein). For the optimal control and PDEs (partial diferential equations) see, e.g., Fife [122]. We can mention also Leung and Stojanovic [197] in which the authors present optimal results for uniformly elliptic PDEs with logistic growth terms; He et al. [151, 152] in which the authors study optimal harvesting control of a periodic parabolic system

452


with maximization of profit; Lenhart et al. [192, 193, 194] in which the authors treat a wildlife management problem in the case of constant diffusion and in the case of degenerate parabolic equation; Belmiloudi [42, 43] in which the author considers a wildlife management problem for time-varying delay non-linear and degenerate parabolic equations with logistic growth terms; and Belmiloudi [44] in which the author treats a minimax optimal control problem, for two parabolic competition systems with logistic growth terms, in order to minimize damage and trapping costs of the first species, and to maximize the difference between economic revenue and cost of the second species. In this chapter, we consider resource management problems of some timevarying delay non-linear parabolic equations governing diffusive biological species with logistic growth terms. The management objectives are the stabilization of uncertain biological species in order to take into account the influence of the population at earlier times on the regulatory effect. The incorporation of multiple time-varying delays is motivated by the fact that the birth rate does not act instantaneously, since there is a time delay to take into account the time to reach maturity, the finished period of gestation, etc. 14.1.2 Studied Equations In this contribution we consider non-linear parabolic partial differential delay equations of the form ∂U + F U = U (d0 − d U − di U (., t − ei (t)) − P ) on Q, ∂t i=1,n U = H0 on Q0 = Ω × [−δ(0), 0),

(14.1)

U (., 0) = U0 on Ω, subject to homogeneous Neumann boundary conditions (no-flux boundary conditions). The cylinder Q is Q = Ω ×(0, T ), the domain Ω (the habitat sits) is a bounded subset of IRm , m ≥ 1, where its boundary Γ is sufficiently regular and Ω is totally on one side of Γ . The given final time T > 0 (the planning period) is a fixed constant. The function U (x, t) is the species concentration and the function P (x, t) denotes the trapping rate of the species or the harvesting effort on the species. The function d0 is the growth parameter and d is the crowding effect. The functions (ei , i = 1, n) are sufficiently regular functions representing multiple time-varying delays and δ(0) = maxi=1,n (ei (0)). The function U0 gives the initial condition of a given population and the function H0 is the initial function on Q0 of this given population. The operator F has the form m ∂v ∂ aij (x) . (14.2) Fv = − ∂xi ∂xj i,j=1 We state the following hypotheses for the operator F :

14.1 Introduction and Mathematical Setting

453

(A1 ) The functions aij : Ω −→ IR are C 1 and the matrix A(.) = (aij (.))1≤i,j≤m is symmetric and satisfies the ellipticity conditions (for all x ∈ Ω and (yi )i=1,m ∈ IRm ): μ

m i=1

yi2 ≥

m

aij (x)yi yj ≥ ν

i,j=1

m

yi2 , with ν > 0 and μ > 0

(14.3)

i=1

and there exists a C 1 function σ : Ω −→ IRm×m , such that A(x) = σ ∗ (x)σ(x), x ∈ Ω,

(14.4)

where σ ∗ is the dual of the matrix σ. According to the assumption (A1 ) we can now express the boundary conditions satisfied by the problem (14.1). Problem (14.1) is then subject to homogeneous Neumann boundary conditions ∂U = (σ ∗ σ∇U ).n = 0 on Σ = Γ × (0, T ), ∂na

(14.5)

with n being the outward normal to Γ . We make the following assumptions (similar to Section 8.7): (A2 ) The functions (di )i=1,n , d0 and d : Q −→ IR are non-negative and C ∞ functions on Q. (A3 ) The functions ri : t ∈ [0, T ) −→ ri (t) = t − ei (t) are strictly increasing functions and the functions ei are C 1 non-negative on [0, T ). Remark 14.1. According to the assumptions (A3 ), the inverse functions fi of ri exist, for all i and the properties given in Remark 8.52 hold. ♦ We recall now the notations used in Section 8.7: we take the following subdivision s−1 = −δ(0), s0 = 0 and ∀j ∈ IN − {0}, sj = mini=1,n (fi (sj−1 )), the time intervals Ij =]s−1 , sj [ and the cylinders Qj = Ω × Ij for j ≥ 0. Next, let the Hilbert space Hσ (Ω) be the completion of C ∞ (Ω), under norm

v Hσ = v 2L2 (Ω) + σ∇v 2L2 (Ω)

1/2

,

(14.6)

which is equivalent to the standard H 1 (Ω)-norm v H 1 (Ω) . We denote its dual by Hσ∗ (Ω). Then the embeddings Hσ (Ω) ⊂ L2 (Ω) ⊂ Hσ∗ (Ω) are continuous. Now we introduce the following sets: U = {v ∈ L∞ (Ω) : 0 < v in Ω},

B = {f ∈ L∞ (Q) : 0 < f in Q},

∞ L∞ + (D) = {h ∈ L (D) : 0 ≤ h in D},

454


where D is a bounded subset of Ω × (−∞, +∞) and spaces: H(Q) = L∞ (0, T ; L2 (Ω)),

Vσ (Q) = L2 (0, T ; Hσ (Ω)),

Wσ (Q) = Vσ (Q) ∩ H 1 (0, T ; Hσ∗ (Ω)). According to Lemma 6.6, the space Wσ is continuously embedded in C([0, T ]; L2(Ω)).

14.2 Existence and Uniqueness of the Solution In this section, we prove the existence and uniqueness of the solution of the state equation. First, we state the following theorem. Theorem 14.2. Let assumptions (A1 )–(A3 ) hold. Then, for any Uint ∈ U, g ∈ L∞ + (Q1,2 ) (Q1,2 = Ω × (T1 , T2 )) and P ∈ B such that 0 0 be a constant such that

14.2 Existence and Uniqueness of the Solution

C1 d0 − dC12 ≥ 0 and C1 ≥ Uint L2 (Ω)

455

(14.11)

and let C2 such that sup (2dC1 ) + sup (P + g − d0 ) < C2 .

Q1,2

Q1,2

(14.12)

We define the initial iterate by V0 = C1

(14.13)

and we obtain inductively Vk , k = 1, 2, . . . , as the unique non-negative solution in Wσ (Q1,2 ) of the problem: ∂Vk + F Vk + C2 Vk ∂t = Vk−1 (d0 − g − d Vk−1 − P ) + C2 Vk−1 on Q1,2 ,

(14.14)

Vk (., T1 ) = Uint on Ω, subject to homogeneous Neumann boundary conditions. By the choice of C2 in (14.12), the right-hand side of the first part of (14.14) is an increasing function of Vk−1 for 0 ≤ Vk−1 ≤ V0 . By comparing the righthand side of (14.14) for Vk and Vk−1 , we obtain that Vk ≤ Vk−1 ≤ V0 , for k = 2, 3, . . . . Consequently, using the regularity of different data appearing in (14.14) and the uniform boundedness of the right-hand side of (14.14), we can easily obtain that the solution Vk is uniformly bounded in Wσ (Q1,2 ). By using the classical argument, we can deduce the weak convergence of Vk in Wσ (Q1,2 ) and its strong convergence in L2 (Q) to V where V is a solution of (14.8) and satisfies the estimate (14.10) given in this theorem. In order to obtain (14.9), we introduce the following problem (see, e.g., Chipot [82] for the existence of such solution) F W ≤ 0 on Ω, Uint ≤ W ≤ Uint on Ω, 2 subject to the same boundary conditions as V and by using the maximum principle we obtain that V ≥ e−δt W ≥

e−δt Uint in Q1,2 , 2

where δ = C0 d ∞ + g ∞ + P ∞ − inf Q (d0 ).

456


Remark 14.3. By taking into account the regularity of the right-hand side and the estimate (14.10), we can deduce that the solution U is in H 2,1 (Q1,2 ) and satisfies an estimate similar to (14.10) in H 2,1 (Q1,2 ) by mean of parabolic estimates. ♦ We can now prove the main result of this section. Theorem 14.4. Let assumptions (A1 )–(A3 ) hold: (i) Assume that U0 ∈ U, H0 ∈ L∞ + (Q0 ) and P ∈ B such that P L∞ (Q) ≤ inf (d0 ) − D∞ max(C0 , H0 L∞ (Q0 ) ), Q

where

C0 = max

d0 ∞ , U0 L∞ (Ω) inf Q (d)

and D∞ =

(14.15)

di ∞ .

i=1,n

Then there exists a unique solution U ∈ Wσ (Q) ∩ L∞ + (Q), of the state Equation (14.1) that satisfies 0 < U (x, t) ≤ max(C0 , H0 L∞ (Q0 ) ), a.e. (x, t) ∈ Q. Moreover, we have the following a priori estimate: U 2Wσ (Q) ≤ C( U0 2L2 (Ω) + P 2L2 (Q) + H0 2L2 (Q0 ) ).

(14.16)

(ii) Let (U0,i , H0,i , Pi ), for i = 1, 2 be two smooth data of U × L∞ + (Q0 ) × B. If U1 (respectively U2 ) is the solution of (14.1) corresponding to data (U0,1 , H0,1 , P1 ) (respectively (U0,2 , H0,2 , P2 )) then U 2Wσ (Q) ≤ C( U0 2L2 (Ω) + P 2L2 (Q) + H0 2L2 (Q0 ) ),

(14.17)

where U = U1 − U2 , P = P1 − P2 , U0 = U0,1 − U0,2 and H0 = H0,1 − H0,2 . Proof. In order to prove the existence of a unique solution on Q, we first establish the existence of a unique solution on Qj , j ≥ 1 and obtain some estimates, like in Section 8.7. We solve the problem on Q1 and obtain the existence of a unique solution on Q1 . Then, the existence of a unique solution on Q2 is proved by using the solution on Q1 to generate the initial data at s1 . This advancing process is repeated for Q3 , Q4 , . . . until the final set is reached. Hereafter, the solution ˜j for j = 1, . . .. on Qj will be denoted by U We shall now introduce the following problems (Pj ) for j ∈ IN − {0}: ∂Vj + F Vj = Vj (d0 − gj − d Vj − P ) on Ω × (sj−1 , sj ), ∂t ˜j−1 (., sj−1 ) on Ω, Vj (., sj−1 ) = U subject to homogeneous Neumann boundary conditions,

14.2 Existence and Uniqueness of the Solution

where gj =

457

˜j−1 (., ri ), U ˜j−1 ∈ Wσ (Qj−1 ) ∩ C([s−1 , sj−1 ]; L2 (Ω)) and di U

i=1,n

˜j−1 (., sj−1 ) ∈ U. We take U ˜0 (., s0 ) = U0 and for any i, U ˜0 (., ri ) = H0 (., ri ) U on Ω × (s0 , s1 ). Since P ∈ B, (di , i = 0, n) ∈ C ∞ (Q) and according to the remark 14.1, we can use the result of Theorem 14.2 and we obtain that the problem (Pj ) admits a unique solution Vj ∈ Wσ (Ω×]sj−1 , sj [) ∩ C([ss−1 , sj ]; L2 (Ω)), such that d0 ∞ ˜j−1 (., sj−1 ) L∞ (Ω) , 0 < Vj ≤ max , U inf Q (d) where gj L∞ (Ω×(sj−2 ,sj−1 )) + P L∞ (Q) − inf Q (d0 ) ≤ 0. Then we can ˜j = U ˜j−1 on Qj−1 and extend the result to the cylinder set Qj+1 by taking U ˜ Uj = Vj on Ω × (sj−1 , sj ). We can now prove the existence and uniqueness result of the problem (14.1). ˜0 (., ri ) = di U di H0 (., ri ) We observe that, for j = 1, we have g1 = i=1,n

i=1,n

L∞ + (Ω

˜0 (., s0 ) = U0 is in U. Then, by using The× (s−1 , s0 )) and U is in orem 14.2, the problem (P1 ) admits a unique solution V1 such that V1 ∈ ˜1 such that Wσ (Ω×]s0 , s1 [) ∩ C([s0 , s1 ]; L2 (Ω)), and we obtain the solution U ˜1 ≤ max(C0 , H0 L∞ (Q ) ) 0 0 and β > 0. We are now going to study the differentiability of the operator solution of the problem (14.27). Before proceeding with investigation of the differentiability of the function ∞ F : (ϕ, ψ) −→ u, which maps the source term (ϕ, ψ) ∈ L∞ + (Q) × L+ (Ω) of ∞ the problem (14.27) into the corresponding solution u ∈ Wσ (Q) ∩ L+ (Q), we study, for (ξ, η0 , ϑ) given data, the following problem: find w ∈ Wσ (Q) such that ∂w ˜ + F w = w(d0 − du − di u(., ri ) − ϕ − G) ∂t i=1,n −U1 (dw + di w(., ri ) + ξ) on Q, w = η0 on Q0 ,

(14.30)

i=1,n

w(., 0) = ϑ on Ω, subject to homogeneous Neumann boundary conditions, where u = F (ϕ, ψ) and U1 = u + U . Theorem 14.6. If u and U are in Wσ (Q)∩L∞ + (Q), the following results hold: (i) For any (ξ, η0 , ϑ) ∈ L∞ (Q) × L∞ (Q0 ) × L∞ (Ω), there exists a unique function w ∈ Wσ (Q), solution of the problem (14.30), such that w 2Wσ (Q) ≤ Ce ( ξ 2L2 (Q) + η0 2L2 (Q0 ) + ϑ 2L2 (Ω) ).

(14.31)

(ii) Let (ξi , η0i , ϑi ), for i = 1, 2 be in L∞ (Q) × L∞ (Q0 ) × L∞ (Ω). If wi is the solution of (14.30), corresponding to data (ξi , η0 , ϑi ), for i = 1, 2, respectively, then w 2Wσ (Q) ≤ Ce ( ξ 2L2 (Q) + η0 2L2 (Q0 ) + ϑ 2L2 (Ω) ),

(14.32)

where w = w1 − w2 , ξ = ξ1 − ξ2 , η0 = η01 − η02 , ϑ = ϑ1 − ϑ2 . Proof. The existence, uniqueness and Lipschitz continuity results of problem (14.30) are obtained in the same way as used to prove Theorem 14.4 and by using the regularity of u and U . So, we omit the details. We shall now study the differentiability of the operator solution F .

462


Theorem 14.7. Let assumptions (A1 )–(A3 ) be fulfilled and the perturbation of the initial function h0 be in L∞ + (Q0 ). Then the following results hold: (i) The function F is differentiable with respect to X = (ϕ, ψ) ∈ Uad ×Vad and its derivative F (X) : L∞ (Q) × L∞ (Ω) −→ Wσ (Q) is a linear operator such that w = F (X).Y is the unique solution of the problem (14.30), with data Y = (ξ, ϑ) and the given initial function η0 = 0, i.e., ∂w ˜ + F w = w(d0 − du − di u(., ri ) − ϕ − G) ∂t i=1,n −U1 (dw + di w(., ri ) + ξ) on Q, i=1,n

w = 0 on Q0 ,

(14.33)

w(., 0) = ϑ on Ω, subject to homogeneous Neumann boundary conditions. Moreover, the Lipschitz continuous result (14.32) holds, i.e., for all wi solution of (14.33), with data (ξi , ϑi ) ∈ L∞ (Q) × L∞ (Q), for i = 1, 2, respectively, we have the estimate w 2Wσ (Q) ≤ Ce ( ξ 2L2 (Q) + ϑ 2L2 (Ω) ),

(14.34)

where w = w1 − w2 , ξ = ξ1 − ξ2 and ϑ = ϑ1 − ϑ2 . (ii) For all Xi = (ϕi , ψi ) ∈ Uad × Vad , for i = 1, 2, we have the following estimate, for all Y = (h, ϑ) ∈ L∞ (Q) × L∞ (Ω): F (X1 ).Y − F (X2 ).Y 2Wσ (Q) ≤ Ce ( Y L2 (Q)×L2 (Ω) X 2L2 (Q)×L2 (Ω)

(14.35)

+ Y 2L2 (Q)×L2 (Ω) X L2 (Q)×L2 (Ω) ), where ϕ = ϕ1 − ϕ2 , ψ = ψ1 − ψ2 and X = X1 − X2 = (ϕ, ψ). Proof. The proof of this theorem is obtained by using a similar technique as used to prove the results of Proposition 8.131 (for the differentiability of F see also Belmiloudi [38], in which the author considers a bioeconomic model). So, we omit the details. Next, we will study the existence of an optimal solution. 14.4.2 Existence of an Optimal Solution Theorem 14.8. For α and β sufficiently large, there exists an optimal solution (ϕ∗ , ψ ∗ , u∗ ) ∈ Uad × Vad × Wσ (Q) such that (ϕ∗ , ψ ∗ ) is a solution of (14.29) and u∗ = F (ϕ∗ , ψ ∗ ) is the solution of (14.27). 1

See also Belmiloudi [44], in which minimax problems are considered for periodic competing systems.


463

Proof. Let Pψ be the mapping: ϕ −→ J(ϕ, ψ) and Qϕ be the mapping: ψ −→ J(ϕ, ψ). In order to obtain the existence of the robust control problem, we prove first that Pψ is convex and lower semi-continuous for all ψ ∈ Vad , second that Qϕ is concave and upper semi-continuous for all ϕ ∈ Uad and, finally, we use the minimax theorems in infinite dimensions presented in Chapter 5. In order to prove the convexity, it is sufficient to show that for (ϕ1 , ϕ2 ) ∈ Uad × Uad , we have (Pψ (ϕ1 ) − Pψ (ϕ2 )).ϕ ≥ 0, where ϕ = ϕ1 − ϕ2 . From the expression of G-differentiable cost functional J (a composition of G-differentiable mappings), it follows that Pψ is G-differentiable and, for i = 1, 2, (ui − uobs )wi dxdt + α ϕi ϕdxdt, Pψ (ϕi ).ϕ = Q

Q

where ui = F (ϕi , ψ) and wi = F (ϕi , ψ).(ϕ, 0). Consequently, (Pψ (ϕ1 )

−

Pψ (ϕ2 )).ϕ

2L2 (Q)

=αϕ + uw1 dxdt Q + (u2 − uobs )wdxdt,

(14.36)

Q

where u = u1 − u2 and w = w1 − w2 . The estimates (14.35), (14.34) and (14.26) imply that uw1 dxdt ≤ C0 ϕ 2L2 (Q) , Q (14.37) 3/2 (u2 − uobs )wdxdt ≤ C1 ϕ L2 (Q) . Q

From (14.36) and the previous results (14.37), we can deduce that there exists a constant αl > 0 such that for α ≥ αl we have (Pψ (ϕ1 ) − Pψ (ϕ2 )).ϕ ≥ (αl − C0 ) ϕ 2L2 (Q) −C1 ϕ L2 (Q) ≥ 0 3/2

and then the convexity of Pξ is established. In the same way, we can find βl > 0 such that for β ≥ βl , Qθ is concave. We shall now prove that Pψ (respectively Qϕ ) is lower (respectively upper) semi-continuous for all ψ ∈ Vad (respectively ϕ ∈ Uad ). Let ϕk ∈ Uad be a minimizing sequence of Pψ , i.e., lim inf J(ϕk , ψ) = inf J(ϕ, ψ). k−→∞

ϕ∈Uad

Then, according to the nature of the cost function J, we can deduce that ϕk is uniformly bounded in Uad and we can extract from ϕk a subsequence also denoted by ϕk such that ϕk ϕξ weakly in Uad . Therefore, by using the same technique as to obtain the estimate (14.16), uk = F (ϕk , ψ) is uniformly

464


bounded in Wσ (Q). Moreover, according now to Lemma 6.6, the injection of Wσ (Q) into L2 (Q) is compact. Consequently, these results make it possible to extract from uk a subsequence also denoted by uk such that uk uψ weakly in Wσ (Q), uk −→ uψ strongly in L2 (Q), ϕk ϕψ weakly in L2 (Q) and ϕψ ∈ Uad .

(14.38)

We shall now prove that uk ϕk −→ uψ ϕψ weakly in L2 (Q). Since uk ϕk − uψ ϕψ = (uk − uψ )ϕk + uψ (ϕk − ϕξ ), and according to the first and second parts of (14.38), we then obtain the result. Is is easy to prove that uψ is a solution of (14.27) with data (ϕψ , ψ) and according to the uniqueness of the solution of the problem (14.27), we then have uψ = F (ϕψ , ψ). Since the norm is lower semi-continuous, therefore we have that the map Pψ is lower semi-continuous for all ψ ∈ Vad . By applying similar arguments as in the proof of the previous result we obtain that Qϕ is upper semi-continuous for all ϕ ∈ Uad . This completes the proof. We next wish to show the appropriate first-order necessary conditions (optimality conditions) of the saddle point problem (14.29). 14.4.3 Optimality Conditions For simplicity we suppose that ei (T ) = δ(T ) for all i = 1, n, where δ(T ) is a given non-negative constant. In order to characterize the optimal control, we introduce the following adjoint problem corresponding to the primal problem (14.27) (we denote by u = F (ϕ, ψ) the solution of the problem (14.27) where the forcing is (ϕ, ψ)): ∂u ˜ ˜ u + Fu ˜ = (d0 − d(U1 + u) − di u(., ri ) − ϕ − G)˜ − ∂t i=1,n ˜ − di (., fi )˜ u(., fi )U1 (., fi )fi + (u − uobs ) on Q, i=1,n

∂u ˜ ˜ u + Fu ˜ = (d0 − d(U1 + u) − di u(., ri ) − ϕ − G)˜ − ∂t i=1,n

(14.39)

+(u − uobs ) on QT ,

u˜(., T ) = 0 on Ω, subject to homogeneous Neumann boundary conditions, ˜ = Ω × (0, T − δ(T )) and QT = Ω × (T − δ(T ), T ). where Q Proposition 14.9. Let assumptions (A1 )–(A3 ) hold and let u be in Wσ (Q)∩ L∞ (Q). Then the problem (14.39) has a unique solution in Wσ (Q) ∩ L∞ (Q) with u˜ satisfying u ˜ Wσ (Q) ≤ C u − uobs L2 (Q) .


465

Proof. In order to prove the existence of a unique solution u˜ , we change the variables of problem (14.39) by reversing sense of time, i.e., t := T − t and we apply the same approach (a constructive method) as to obtain the existence and uniqueness result of the problem (14.39), given in Theorem 14.4. So, we omit the details. Next, we prove the estimate given in the proposition. Multiplying (14.39) by u˜ and integrating over (t, T ) × Ω, we obtain (since u˜(T ) = 0) T σ∇˜ u 2L2 ds u ˜(., t) 2L2 +2 t T T u ˜ 2L2 ds + u − uobs 2L2 ds ≤ c1 (14.40) t t T −δ(T ) di (., fi )U1 (., fi )˜ u(., fi )fi u ˜dxds. +2 i=1,n

min(t,T −δ(T ))

We shall now estimate T −δ(T ) IT = 2

Ω

min(t,T −δ(T ))

di (., fi )U1 (., fi )˜ u(., fi )fi u ˜dxds.

Ω

According to the regularity of di , fi and U1 we then obtain IT ≤ Ci (

T −δ(T )

u ˜(., fi ) 2L2 fi ds)1/2 (

min(t,T −δ(T ))

T

u ˜ 2L2 ds)1/2 , t

where Ci = c2 di ∞ ( fi ∞ )1/2 . If we set a = fi (s), we have da = fi (s)ds and then, according to ri (T ) = T − δ(T ) and ri (t) ≤ t, T IT ≤ Ci u ˜ 2L2 ds. (14.41) t

According to (14.41) the estimate (14.40) becomes T T σ∇˜ u 2L2 ds ≤ c3 u ˜ 2L2 ds + u ˜(., t) 2L2 +2 t

t

T

u − uobs 2L2 ds.

t

By using Gronwall’s formula we then have u ˜ 2H(Q) + u˜ 2Vσ (Q) ≤ C u − uobs 2L2 (Q) .

(14.42)

Using this result and Equation (14.39) we prove easily that u ˜ satisfies u˜ 2H 1 (0,T ;Hσ∗ (Q)) ≤ C u − uobs 2L2 (Q) .

(14.43)

Applying (14.42) and (14.43), we obtain the second result of the proposition.

466


We can now give the optimality system for the saddle point problem (14.29). Theorem 14.10. Let assumptions (A1 )–(A3 ) hold, and (ϕ∗ , ψ ∗ , u∗ ) ∈ Uad × ∗ ∗ ∗ Vad × (Wσ (Q) ∩ L∞ + (Q)) such that (ϕ , ψ ) is defined by (14.29) and u = ∗ ∗ F (ϕ , ψ ) is a solution of (14.27). Then, for all (ϕ, ψ) ∈ Uad × Vad , (αϕ∗ − u ˜∗ U1∗ )(ϕ − ϕ∗ )dxdt ≥ 0, Q (14.44) ∗ ∗ ∗ (˜ u (., 0) − βψ )(ψ − ψ )dx ≤ 0, Ω

U1∗

∗

= u + U and u ˜∗ is the solution of the adjoint problem (14.39), where corresponding to the primal solution u∗ . Otherwise, ∗ ∗ u ˜ U1 ∗ , τ1 ϕ = min τ2 , max , α (14.45) ∗ u ˜ (., 0) ∗ , π1 . ψ = min π2 , max β Moreover, the gradient of the functional J, in the weak sense, at point (ϕ∗ , ψ ∗ ) is ∂J ∗ ∗ ∂J ∗ ∗ (ϕ , ψ ) = αϕ∗ − u (ϕ , ψ ) = u ˜∗ U1∗ and ˜∗ (., 0) − βψ ∗ . ∂ϕ ∂ϕ Proof. The cost functional J is a composition of differentiable maps then J is differentiable and, for all (ξ, ϑ) ∈ L∞ (Q) × L∞(Ω) such that (ϕ + ξ, ψ + ϑ) ∈ Uad × Vad for small, we have 1 J (ϕ, ψ).(ξ, ϑ) = lim (J(ϕ + ξ, ψ + ϑ) − J(ϕ, ψ))

−→0 = (u − uobs )wdxdt + α ϕξdxdt Q Q −β ψϑdxdt,

(14.46)

Q

where w is the solution of (14.33). Multiplying (14.30) by u˜, integrating over Q and integrating by parts with respect to time t, we obtain T ∂u ˜ ˜ + Fu ˜−u ˜(d0 − d(U1 + u) − (− di u(., ri ) − ϕ − G))wdxdt ∂t 0 Ω i=1,n T T U1 ξ u ˜dxdt − di w(., ri )U1 u ˜dxdt =− 0 Ω 0 Ω i=1,n u(., T )dx + ϑ˜ u(., 0)dx. − w(., T )˜ Ω

Ω


T

Next, we shall calculate the term A = then t = fi (s) and dt =

T −ei (T )

di w(., ri )U1 u ˜dxdt. Let s = ri (t), 0

Ω

Therefore,

di (., fi (s))˜ u(., fi (s))U1 (., fi (s))fi (s)w(., s)dxds.

A= −ei (0)

fi (s)ds.

467

Ω

Since ei (T ) = δ(T ), ∀i = 1, n and according to the second part of (14.30) we have T −δ(T ) (14.47) di (., fi (s))˜ u(., fi (s))U1 (., fi (s))fi (s)w(., s)dxds. A= 0

Ω

Since u˜ is a solution of the adjoint problem (14.39) and according to (14.47), we obtain that (u − uobs )wdxdt = − ξu ˜U1 dxdt + ϑ˜ u(., 0)dx. (14.48) Q

Q

Ω

Applying (14.48) and according to the expression of J (ϕ, ψ) given by (14.46) we then have (αϕ − u ˜U1 )ξdxdt + (˜ u(., 0) − βψ)ϑdx. J (ϕ, ψ).(ξ, ϑ) = (14.49) Q

Ω

Since (ϕ∗ , ψ ∗ ) is an optimal solution we have ∂J ∗ ∗ ∂J ∗ ∗ (ϕ , ψ ).ξ ≥ 0 and (ϕ , ψ ).ϑ ≤ 0 for all (ξ, ϑ) ∈ L∞ (Q) × L∞ (Ω) ∂ϕ ∂ψ and so we obtain, for all (ξ, ϑ) ∈ L∞ (Q) × L∞ (Ω), (αϕ∗ − u˜∗ U1∗ )ξdxdt ≥ 0, Q (˜ u∗ (., 0) − βψ ∗ )ϑdx ≤ 0,

(14.50)

Ω

where U1∗ = u∗ + U . By using a classical control argument concerning the sign of the variations ξ and ϑ (depending on the size of ϕ∗ and ψ ∗ respectively), we obtain that ∗ ∗ u ˜ U1 , τ1 ϕ∗ = min τ2 , max , α ∗ u ˜ (., 0) ∗ , π1 ψ = min π2 , max . β This completes the proof.

468


Remark 14.11. (i) By similar arguments as used in Belmiloudi [42, 43], the uniqueness of the optimal solution can be proved under, for example, the condition that the final time is small enough or the parameters α and β are large enough. (ii) We can treat, by using the same technique developed in this chapter, other dynamical population models. For example, coupled cooperative diffusion systems of various populations (for a well-posedness problem, without delay, see e.g., Jia and Feng [169]). (iii) We can also consider the situation of heterogeneous environment with homogeneous Dirichlet2 or Neumann boundary conditions. In this case, the diffusion can be degenerated in some areas and we can assume that the diffusion operator F given by (14.2) may be degenerate and satisfies the following (see, e.g., Belmiloudi [42, 43] and Lenhart and Yong [193]): The functions aij : Ω −→ IR are C 1 and the matrix A(.) = (aij (.))1≤i,j≤m is symmetric positive semidefinite in Ω. Moreover, meas{x ∈ Ω such that det(A(x)) = 0} = 0,

(14.51)

and there exists a C 1 function σ : Ω −→ IRm×m , such that A(x) = σ ∗ (x)σ(x), x ∈ Ω,

(14.52)

where σ ∗ is the dual of the matrix σ. For the existence result, we can use the vanishing viscosity method (see, e.g., Crandall et al. [90]) in the adequate spaces and for the robust control problem we can use a similar technique to the one developed in this chapter. ♦

14.5 Other Situations 14.5.1 Disturbance in the Parameter Function p If we assume that the control and the disturbance are in the parameter p, i.e., p = ϕ + ψ, we obtain the same results as in Section 14.4. In this case, the cost functional can be given by J(ϕ, ψ) =

1 α β u − uobs 2L2 (Q) + ϕ 2L2 (Q) − π2 − ψ 2L2 (Q) . 2 2 2

We can also prove an existence theorem of the saddle point problem and obtain necessary optimality conditions for its solution by using the same method as in Section 14.4. Let now 2

The given species does not live on the edge of the region.


469

Uad = {ϕ ∈ L2 (Q) : τ1 ≤ ϕ ≤ τ2 a.e. in Q}, Vad = {ψ ∈ L2 (Q) : 0 ≤ ψ ≤ π2 a.e. in Q}. Then, for α and β sufficiently large, there exist (ϕ∗ , ψ ∗ ) ∈ Uad × Vad and u∗ ∈ Wσ (Q) ∩ L∞ (Q) satisfying ∂u∗ ˜ + F u∗ = u∗ (d0 − du∗ − di u∗ (., ri ) − ϕ∗ − ψ ∗ − G) ∂t i=1,n −U (du∗ + di u∗ (., ri ) + ϕ∗ + ψ ∗ ) on Q, i=1,n (14.53) u∗ = h0 on Q0 , u∗ (., 0) = u0 on Ω, subject to homogeneous Neumann boundary conditions, and for all (ϕ, ψ) ∈ Uad × Vad , we have the optimality conditions (αϕ∗ − u ˜∗ U1∗ )(ϕ − ϕ∗ )dxdt ≥ 0, Q (−˜ u∗ U1∗ − β(−π2 + ψ ∗ ))(ψ − ψ ∗ )dxdt ≤ 0, Q

otherwise,

∗ ∗ u ˜ U1 , τ1 , ϕ = min τ2 , max α u˜∗ U1∗ ∗ ,0 , ψ = min π2 , max π2 − β where u∗ is the solution of the problem (14.53) with the forcing (ϕ∗ , ψ ∗ ), U1∗ = ˜∗ is the solution of the following adjoint problem corresponding u∗ + U and u to the primal solution u∗ ∂u ˜∗ ˜ u∗ + Fu ˜∗ = (d0 − dU1∗ − − di u∗ (., ri ) − ϕ∗ − ψ ∗ − G)˜ ∂t i=1,n ˜ − di (., fi )˜ u∗ (., fi )U1 (., fi )fi + (u∗ − uobs ) on Q, ∗

i=1,n

∂u ˜∗ ˜ u∗ + Fu ˜∗ = (d0 − dU1∗ − di u∗ (., ri ) − ϕ∗ − ψ ∗ − G)˜ − ∂t i=1,n

(14.54)

+(u∗ − uobs ) on QT ,

u˜∗ (., T ) = 0 on Ω, subject to homogeneous Neumann boundary conditions, ˜ = Ω × (0, T − δ(T )) and QT = Ω × (T − δ(T ), T ). with Q Moreover, the gradient of the functional J, in the weak sense, at point (ϕ∗ , ψ ∗ ) is ∂J ∗ ∗ ∂J ∗ ∗ (ϕ , ψ ) = αϕ∗ − u˜∗ U1∗ and (ϕ , ψ ) = −˜ u∗ U1∗ − β(−π2 + ψ ∗ ). ∂ϕ ∂ϕ

470


14.5.2 Remarks on Boundary Control and Habitat Hostility If we want to take into account the influence of the hostility of the boundary environment, we can use a similar consideration as in Lenhart at al. [194]. More precisely, the hostility of the boundary environment can be represented by some parameter η, multiplying the state variable in the Robin boundary condition. For a given η, the corresponding concentration of species U satisfies the following system delay system: ∂U + F U = U (d0 − dU − di U (., ri ) − P ) on Q, ∂t i=1,n U = H0 on Q0 = Ω × [−δ(0), 0),

(14.55)

U (., 0) = U0 on Ω, (σ ∗ σ∇U ).n = −ηU on Σ = Γ × (0, T ). The solution U will be treated as the target function and we analyze the non-linear equation which models large perturbations u to the target U , i.e., we assume that U satisfies the problem (14.55) with the data (P, H0 , U0 , η) and U +u satisfies the problem (14.55) with the data (P +p, H0 +h0 , U0 +u0 , η+ξ). Hence, we consider the system with multiple time-varying delays ∂u ˜ + F u = u(d0 − du − di u(., ri ) − p − G) ∂t i=1,n −U (du + di u(., ri ) + p) on Q, i=1,n (14.56) u = h0 on Q0 , u(., 0) = u0 on Ω, (σ ∗ σ∇u).n = −ξ(U + u) − ηu on Σ, ˜ = dU + di U (., ri ) + P ∈ L∞ where G + (Q). i=1,n

Our problem, in this section, is to find the best admissible parameter function p in the presence of the worst disturbance in the hostility boundary parameter. We then suppose that the control is in the parameter function p and the disturbance is in the boundary parameter η, i.e., p = ϕ (ϕ ∈ L∞ (Q)) and ξ = ψ (ψ ∈ L∞ (Σ)). So the function u is assumed to be related to the disturbance ψ and control ϕ through the problem ∂u ˜ + F u = u(d0 − du − di u(., ri ) − ϕ − G) ∂t i=1,n −U (du + di u(., ri ) + ϕ) on Q, i=1,n (14.57) u = h0 on Q0 , u(., 0) = u0 on Ω, (σ ∗ σ∇u).n = −ψ(U + u) − ηu on Σ,


471

under the pointwise constraints 0 ≤ τ1 ≤ ϕ ≤ τ2 0 ≤ ψ ≤ π2

a.e. in Q,

a.e. in Σ.

(14.58)

We obtain the same results as in Section 14.4. In this case, the cost functional can be given by J(ϕ, ψ) =

1 α β u − uobs 2L2 (Q) + ϕ 2L2 (Q) − π2 − ψ 2L2 (Σ) . 2 2 2

We can also prove an existence theorem of the control problem and obtain necessary optimality conditions for its solution using the same method as in Section 14.4. Let now Uad = {ϕ ∈ L2 (Q) : 0 ≤ τ1 ≤ ϕ ≤ τ2 a.e. in Q}, Vad = {ψ ∈ L2 (Σ) : 0 ≤ ψ ≤ π2 a.e. in Σ}. Then, for α and β sufficiently large, there exists an optimal solution (ϕ∗ , ψ ∗ , u∗ ) in Uad × Vad × (Wσ ∩L∞ (Q)) such that u∗ is a solution of the problem (14.57), with the forcing (ϕ∗ , ψ ∗ ) and, for all (ϕ, ψ) ∈ Uad × Vad , the following optimality conditions hold: (αϕ∗ − u˜∗ U1∗ )(ϕ − ϕ∗ )dxdt ≥ 0, Q (−˜ u∗ U1∗ − β(−π2 + ψ ∗ ))(ψ − ψ ∗ )dΓ dt ≤ 0, Σ

otherwise,

∗ ∗ u ˜ U1 ϕ∗ = min τ2 , max , , τ1 α u˜∗ U1∗ ∗ |Σ , 0 , ψ = min π2 , max π2 − β

where u∗ = F (ϕ∗ , ψ ∗ ) is the solution of the problem (14.57) with the forcing ˜∗ is the solution of the following adjoint problem (ϕ∗ , ψ ∗ ), U1∗ = u∗ + U and u corresponding to the primal solution u∗ ∂u ˜∗ ˜ u∗ + Fu ˜∗ = (d0 − dU1∗ − − di u∗ (., ri ) − ϕ∗ − G)˜ ∂t i=1,n ˜ − di (., fi )˜ u∗ (., fi )U1 (., fi )fi + (u∗ − uobs ) on Q, i=1,n

∂u ˜∗ ˜ u∗ + Fu ˜∗ = (d0 − dU1∗ − di u∗ (., ri ) − ϕ∗ − G)˜ − ∂t i=1,n +(u∗ − uobs ) on QT ,

u˜∗ (., T ) = 0 on Ω, u∗ on Σ, (σ ∗ σ∇u).n = −(η + ψ ∗ )˜

(14.59)

472


˜ = Ω × (0, T − δ(T )) and QT = Ω × (T − δ(T ), T ). with Q Moreover, the gradient of the functional J, in the weak sense, at point (ϕ∗ , ψ ∗ ) is ∂J ∗ ∗ ∂J ∗ ∗ (ϕ , ψ ) = αϕ∗ − u˜∗ U1∗ and (ϕ , ψ ) = −˜ u∗ U1∗ |Σ − β(−π2 + ψ ∗ ). ∂ϕ ∂ϕ

15 Other Systems

The methods developed in this book can be applied to various physical, biological and chemical systems. We will not detail all the possible applications and will only quote two very interesting systems arising in micropolar fluids and semiconductor melts, namely the motion of animal blood, which is described by using the model of micropolar fluids, and Czochralski growth configurations and semiconductor melts in zone-melting. The main theorems of this book can be extended to these situations.

15.1 Micropolar Fluids and Blood Pressure 15.1.1 Introduction and Mathematical Setting A micropolar fluid is a viscous, non-Newtonian fluid with local microstructure, which contains suspensions of rigid particles. A mathematical description of such fluids, which cannot be described, rheologically, by classical Navier– Stokes systems (especially when the diameter of the domain of flow becomes small), was first given by Eringen [115]. Animal blood, liquid crystals (with dumbbell type molecules), polymeric fluids, and certain colloidal fluids, whose fluid elements exhibit microrotations and complex biological structures, are examples of fluids modeled by micropolar fluid theory (see Eringen [116] and Popel et al. [243]). Various problems associated with the micropolar fluid model have been studied recently (see, e.g., Calmelet and Rosenhaus [67], Lukaszewicz [210], Yamaguchi [304]). For the optimal control problems, see Stavre [274], in which the author controls the blood pressure. In the motion of blood, a problem of physical and biological interest is to control and stabilize the blood pressure. The goal of our study is to stabilize a desired blood pressure, by taking into account disturbances in the external forces. For this we consider the following constitutive system for twodimensional incompressible time-dependent micropolar fluids given by Eringen [115]:

474

15 Other Systems

∂U + (U.∇)U − (μ + χ)ΔU + ∇P − χcurl() = f on Q, ∂t div(U) = 0 on Q, ∂ + ξ(U.∇) − νΔ + 2χ − χcurl(U) = g on Q, ∂t subject to homegenous Dirichlet boundary conditions U = 0, = 0 on Σ, and the null initial conditions U(., 0) = 0, (., 0) = 0 on Ω,

ξ

(15.1)

where Ω is the domain occupied by the blood with the boundary Γ , T is the final time, Q = Ω×(0, T ), Σ = Γ ×(0, T ) denotes the space-time cylinders and the parameters χ, μ, ξ and ν are positive given constants associated with the properties of the material. Furthermore, the functions f and g are the external fields, and the functions U, and P are the velocity, the microrotation (which describes the skew-symmetric gyration tensor in the two-dimensional case) and the blood pressure, respectively. We now introduce the following spaces: V = {v ∈ H01 (Ω) : div(v) = 0}, H = {v ∈ L2 (Ω) : div(v) = 0, v.n = 0 on Γ }, L20 (Ω) = {p ∈ L2 (Ω) : pdx = 0}. Ω

By using a similar technique as used in Chapter 12, we can prove the following: For given g ∈ L2 (Q) and f ∈ L2 (0, T ; H), there exists a unique solution (U, , P ) of the problem (15.1) such that U ∈ H 2,1 (Q) ∩ C([0, T ]; V ), ∈ H 2,1 (Q) ∩ C([0, T ]; H01(Ω)), P ∈ L2 (0, T ; H 1 (Ω) ∩ L20 (Ω)). Moreover, if f is in R(Q), where R(Q) = {f ∈ L2 (0, T ; H) :

∂f ∈ L2 (0, T ; H −1(Ω)), f (., 0) ∈ H}, ∂t

then the pressure satisfies ∂P ∈ L2 (Q). ∂t

15.1 Micropolar Fluids and Blood Pressure

475

15.1.2 Fluctuation and Robust Regulation of the Blood Pressure In order to study the robust regulation of deviation of the blood pressure from the desired target P , we introduce the perturbation problem, which models small fluctuations (u, ω, p, ˜f , g˜) to the target micropolar flow (U, , P, f , g) with Dirichlet boundary condition and null initial condition (we assume that (u + U, ω + , p + P, ˜f + f , g˜ + g) is also a solution of (15.1)). Then the perturbation (u, ω, p, ˜f , g˜) satisfies the following micropolar type system: ∂u − (μ + χ)Δu + (u.∇)u + (u.∇)U + (U.∇)u ∂t +∇p − χcurl(ω) = ˜f on Q, div(u) = 0 on Q, ∂ω − νΔω + ξ(u.∇)ω + ξ(U.∇)ω + ξ(u.∇) ∂t +2χω − χcurl(u) = g˜ on Q, subject to homegenous Dirichlet boundary conditions u = 0, ω = 0 on Σ, and the null initial conditions u(., 0) = 0, ω(., 0) = 0 on Ω, ξ

(15.2)

where the regularity required on ˜f , g˜ and (U, ) are (U, ) ∈ H 2,1 (Q) ∩ C([0, T ]; H01(Ω)), g˜ ∈ L2 (Q), ˜f ∈ R(Q).

(15.3)

By again using a similar technique as used in Chapter 12 and according to the regularity (15.3), we can prove the following: There exists a unique solution (u, ω, p) of the problem (15.2) such that U ∈ H 2,1 (Q) ∩ C([0, T ]; V ), ∈ H 2,1 (Q) ∩ C([0, T ]; H01(Ω)),

(15.4)

p ∈ P(Q) ⊂ C([0, T ]; L2 (Ω)) (see Lemma 6.6), where P(Q) = {p ∈ L2 (0, T ; H 1 (Ω) ∩ L20 (Ω)) :

∂p ∈ L2 (Q)}. ∂t

Our purpose is to stabilize a desired blood pressure: for this we assume that the external field g˜ is decomposed into a disturbance ψ ∈ L2 (Q) and a control ϕ ∈ L2 (Q). Thus, we write g˜ as g˜ = B1 ϕ + B2 ψ, where B1 and B2 are taken as given bounded operators on L2 (Ω).

476

15 Other Systems

The function (u, ω, p) is assumed to be related to the disturbance ψ and control ϕ through the problem (15.2): ∂u − (μ + χ)Δu + (u.∇)u + (u.∇)U + (U.∇)u ∂t +∇p − χcurl(ω) = ˜f on Q, div(u) = 0 on Q, ξ

∂ω − νΔω + ξ(u.∇)ω + ξ(U.∇)ω + ξ(u.∇) ∂t +2χω − χcurl(u) = B1 ϕ + B2 ψ on Q,

(15.5)

subject to homegenous Dirichlet boundary conditions u = 0, ω = 0 on Σ, and the null initial conditions u(., 0) = 0, ω(., 0) = 0 on Ω. The cost functional considered in the present work is of the form: 1 J(ϕ, ψ) = | p − pobs |2 dxdt 2 Q (15.6) β α | ϕ |2 dxdt − | ψ |2 dxdt, + 2 2 Q Q where α, β > 0 are fixed, and pobs ∈ P(Q) is the observation. We shall consider the following robust control problem: find (ϕ∗ , ψ ∗ ) ∈ Uad × Vad such that J(ϕ∗ , ψ) ≤ J(ϕ∗ , ψ ∗ ) ≤ J(ϕ, ψ ∗ ), ∀(ϕ, ψ) ∈ Uad × Vad ,

(15.7)

where Uad and Vad are given non-empty, closed convex and bounded subsets of L2 (Q). The arguments of Chapter 12 extend directly to the present chapter. So, we omit the details. Therefore, we have the following existence and firstoptimality conditions results. Theorem 15.1. For α and β be sufficiently large, there exist (ϕ∗ , ψ ∗ ) ∈ Uad × Vad and X ∗ = (u∗ , ω ∗ , p∗ ) ∈ H 2,1 (Q)∩C([0, T ]; H01 (Ω)) such that (ϕ∗ , ψ ∗ ) satisfies (15.7) and (u∗ , ω ∗ , p∗ ) is the solution of the primal problem (15.5) with data (ϕ∗ , ψ ∗ ). Moreover, the optimal solution (ϕ∗ , ψ ∗ , X ∗ ) is characterized by the following necessary optimality conditions, for all (ϕ, ψ) ∈ Uad × Vad : (αϕ∗ + B1∗ ϑ∗ )(ϕ − ϕ∗ )dxdt ≥ 0, Q (15.8) (−βψ ∗ + B2∗ ϑ∗ )(ψ − ψ ∗ )dxdt ≤ 0, Q

15.1 Micropolar Fluids and Blood Pressure

477

where (w∗ , ϑ∗ , π ∗ ) is the unique solution in Wu (Q) × Wr (Q) × D (Q) (π is unique up to the addition of a distribution in (0, T )) of the following adjoint problem: −

∂w − (μ + χ)Δw − (u1 .∇)w + (∇u1 )t w − ξω1 ∇ϑ ∂t +∇π − χcurl(ϑ) = 0 on Q,

div(w) = p − pobs on Q, ∂ϑ − νΔϑ − ξ(u1 .∇)ϑ + 2χϑ − χcurl(w) = 0 on Q, ∂t subject to homegenous Dirichlet boundary conditions w = 0, ϑ = 0 on Σ, and the final conditions w(., T ) = w0 , ϑ(., T ) = 0 on Ω, −ξ

(15.9)

where u1 = u + U, ω1 = ω + , Wu (Q) = {w ∈ L2 (0, T ; H01 (Ω)) ∩ C([0, T ]; L2(Ω)) :

∂w ∈ L2 (0, T ; V )}, ∂t

∂ϑ ∈ L2 (0, T ; H −1 (Ω))} ∂t and w0 ∈ L2 (Ω) is the unique solution of the following problem: Wr (Q) = {ϑ ∈ L2 (0, T ; H01 (Ω)) :

−div(w0 ) = pobs (T ) − p∗ (T ), w0 .n = 0 on Γ.

(15.10)

Remark 15.2. The proof of the existence and uniqueness of the solution of the problem (15.9) can be obtained by introducing a divergence-free non-linear system, which is equivalent to (15.9). For this we can consider, for a.e. in (0, T ), functions T v(s)ds ∈ L2 (0, T ; H01(Ω)), wL = − t

with wL ∈ L2 (0, T ; H01 (Ω)) and wT ∈ H01 (Ω) such that v(t) and wT are solutions, respectively, of the problems −div(v) = and

∂ (pobs − p∗ ), ∂t

−div(wT ) = pobs (T ) − p∗ (T ).

(15.11) (15.12)

˜ = w − wL − wT , then div(w) ˜ = 0, w(T ˜ ) = w0 − wT and w(T ˜ ).n = Set w ˜ ϑ), is similar to (15.1) where the 0. The obtained system, satisfied by (w, corresponding right-hand sides, f and g, depend on wL and are in L2 (0, T ; V ) and L2 (Q), respectively. Therefore, by using similar technique as used to obtain the existence and the uniqueness of (15.1), we can prove the existence ˜ ϑ) and then those of (w, ϑ). and uniqueness of (w, ♦

478

15 Other Systems

15.2 Semiconductor Melt Flow in Crystal Growth 15.2.1 Introduction and Mathematical Setting Liquid encapsulate Czochralski crystal growth is a major technique used to produce crystals, for example, GaAs and InP single crystals used in electronics and optoelectronics (see, e.g., Neubert and Rudolph [229]). During the whole growth process, the behavior of the crystal depends considerably on the thermal regime and formation of the crystallization front geometry. Moreover, during experimental study of these Czochralski growth processes in axisymetric zone–melting devices, a transition from the two-dimensional incompressible turbulent flow regime (melt) to an unsteady three-dimensional behavior is observed. The distribution temperature in the melt and at the crystal growth interface causes fluctuations in the microscopic growth rate of the crystal. These fluctuations (in the impurity segregation) decrease the material quality of the grown crystal. They are known to cause the so-called striations which transform into clusters of point defects. To make crystals striation-free, it is necessary to reduce the influence of the thermal convection in the melt. Therefore, in order to suppress or to reduce the convection and, in particular, thermal convection in the melt, during practical crystal production processes, magnetic fields are often applied by the physicists and engineers. Various problems associated with the semiconductor melts model have been studied recently (see, e.g., Choe [83], Fedoseyev and Alexander [121], Prasad et al. [244], Watanabe et al. [297]). For the optimal control problems, see Barwolff and Hinze [25] and Gunzburger et al. [146] in which the authors develop computational techniques and optimal strategy for the suppression of turbulent motions in the melt. The goal of our study is to avoid such crystal defects by stabilizing the melt flow motion during the growth process. For this, we use a robust regulation method in order to maintain the desired state (the growth of crystal with desired properties) which represents a physically favourable situation, by taking into account the worst disturbances in the temperature flux on the wall of the crucible. The control function is in body forces due to a magnetic field. The unsteady model of the melt flow motion as well as heat transfer in the crystal and crucibles is governed by the following incompressible viscous fluid with a Boussinesq approximation: 1 T ∂U + (U.∇)U − div(ν1 ∇U) + ∇P + ξ G = F on Q, ∂t ρav ρav div(U) = 0 on Q, ∂T + (U.∇)T − div(ν2 ∇T ) = 0 on Q, ∂t with the initial conditions U(., 0) = u0 , T (., 0) = T0 on Ω,

(15.13)

15.2 Semiconductor Melt Flow in Crystal Growth

479

where Ω ⊂ IRm , for m = 2 or 3, is the flow domain with the boundary Γ = Γc ∪ Γd , where Γc corresponds to the crucible walls and Γd corresponds to the solid–liquid interface. Q = Ω × (0, T ) denotes the space-time cylinder, with T the final time, G = (0, 0, −g) is the gravity force, the density ρav is a constant mean value which is supposed to be equal to 1 and ξ is a given coefficient depending on the Prandtl–Rayleigh and thermal expansion coefficient. The function U is the velocity, P is the pressure, T is the temperature and F is a body force due to the magnetic field (the Lorentz force). The turbulent flux is usually modeled by the dissipative terms which correspond to Raynolds stress with the variable coefficient of eddy viscosity ν1 and the variable coefficient of eddy diffusivity ν2 . The functions ν1 and ν2 are assumed to be positive and bounded functions above and below by non-negative constants in Ω. On account of the phenomena we want to describe, the system (15.13) is supplied with the following mixed boundary conditions: U = uB on Σ, ∂T + γ(T − Te ) = τ on Σc , ∂n T = Tsl on Σd . ν2

(15.14)

where Σ = Γ × (0, T ), Σc = Γc × (0, T ) and Σd = Γd × (0, T ). The function Te is a given environmental temperature, γ denotes some physical constant, and uB and Tsl are given boundary functions such that uB .n = 0. 15.2.2 Fluctuation and Robust Regulation of the Melt Flow Motion In order to study the robust regulation of deviation of the flow from the desired state, we introduce the perturbation problem, which models small fluctuations (u, θ, p, f , η) to the target flow (U, T , P, F, τ ) with the same Dirichlet boundary conditions and initial conditions (we assume that (u + U, T + θ, p + P, F + f , τ + η) is also a solution of (15.13) and (15.14)). Then the perturbation (u, θ, p, f , η) satisfies the following non-linear system: ∂u + (u.∇)u + (U.∇)u + (u.∇)U ∂t −div(ν1 ∇u) + ∇p + ξθG = f on Q, div(u) = 0 on Q, ∂θ + (u.∇)θ + (u.∇)T + (U.∇)θ − div(ν2 ∇θ) = 0 on Q, ∂t with the initial conditions u(., 0) = 0, θ(., 0) = 0 on Ω, and the boundary conditions u = 0 on Σ, ∂θ + γθ = η on Σc , θ = 0 on Σd . ν2 ∂n

(15.15)

480

15 Other Systems

Assume that the functions (U, T ) ∈ H 2,1 (Q) ∩ C([0, T ]; H 1(Ω)), f ∈ L (0, T ; L2(Ω)) and η ∈ Uc = {η ∈ L2 (0, T ; H 1(Γc )) : ∂η/∂t ∈ L2 (0, T ; L2 (Γc ))} then, by using a similar technique as used in Chapter 12, we can prove (under the constraint of a small data for the non-linear 3D case) the existence and uniqueness of the solution (u, θ) ∈ H 2,1 (Q) ∩ C([0, T ]; H 1(Ω)) of the problem (15.15). Our purpose is to stabilize the desired flow, and for this we assume that the control ϕ ∈ L2 (Q) is in the external magnetic field and the disturbance ψ ∈ L2 (Σc ) is in the temperature flux on the wall of the crucible, i.e., 2

f = B1 ϕ and η = B2 ψ, where B1 (respectively B2 ) is taken here as given linear continuous and bounded operator from L2 (Q) (respectively L2 (Σc )) into L2 (Q) (respectively Uc ). The function (u, θ, p) is assumed to be related to the disturbance ψ and control ϕ through the problem (15.15): ∂u + (u.∇)u + (U.∇)u + (u.∇)U ∂t −div(ν1 ∇u) + ∇p + ξθG = B1 ϕ on Q, div(u) = 0 on Q, ∂θ + (u.∇)θ + (u.∇)T + (U.∇)θ − div(ν2 ∇θ) = 0 on Q, ∂t with the initial conditions u(., 0) = 0, θ(., 0) = 0 on Ω, and the boundary conditions u = 0 on Σ, ∂θ ν2 + γθ = B2 ψ on Σc , θ = 0 on Σd . ∂n

(15.16)

The cost functional considered in the present work is of the form 1 | Cu |2 dxdt J(ϕ, ψ) = 2 Q (15.17) β α | ϕ |2 dxdt − | ψ |2 dΓ dt + 2 2 Q Σc where α, β > 0 are fixed parameters and C is similar as in (12.18). We shall consider the following robust control problem: find (ϕ∗ , ψ ∗ ) ∈ Uad × Vad such that J(ϕ∗ , ψ) ≤ J(ϕ∗ , ψ ∗ ) ≤ J(ϕ, ψ ∗ ), ∀(ϕ, ψ) ∈ Uad × Vad ,

(15.18)

where Uad (respectively Vad ) is a given non-empty, closed convex and bounded subset of L2 (Q) (respectively L2 (Σc )).

15.2 Semiconductor Melt Flow in Crystal Growth

481

The arguments of Chapter 12 extend directly to the present work. So, we omit the details. Therefore, we have the following existence and firstoptimality conditions results. More precisely, for α and β be sufficiently large, there exist (ϕ∗ , ψ ∗ ) ∈ Uad × Vad and X ∗ = (u∗ , T ∗ ) ∈ H 2,1 (Q) ∩ C([0, T ]; H 1(Ω)) such that (ϕ∗ , ψ ∗ ) satisfies (15.18) and (u∗ , T ∗ ) is the solution of the primal problem (15.16) with data (ϕ∗ , ψ ∗ ). Moreover, the optimal solution (ϕ∗ , ψ ∗ , X ∗ ) is characterized by the following necessary optimality conditions, for all (ϕ, ψ) ∈ Uad × Vad : (αϕ∗ + B1∗ w∗ )(ϕ − ϕ∗ )dxdt ≥ 0, Q (15.19) (−βψ ∗ + B2∗ ϑ∗ )(ψ − ψ ∗ )dΓ dt ≤ 0, Σc

where (w∗ , ϑ∗ , π ∗ ) is the unique solution of the following adjoint problem: −

∂w − div(ν1 ∇w) − (u∗1 .∇)w + (∇u∗1 )t w ∂t +θ1∗ ∇ϑ + ∇π = C ∗ Cu∗ on Q,

div(w) = 0 on Q, ∂ϑ − div(ν2 ∇ϑ) − (u∗1 .∇)ϑ = ξG.w on Q, − ∂t with the final conditions w(., T ) = 0, ϑ(., T ) = 0 on Ω, and the boundary conditions w = 0 on Σ, ∂ϑ ν2 + γϑ = 0 on Σc , ϑ = 0 on Σd . ∂n

(15.20)

where u∗1 = u∗ + U and θ1∗ = θ∗ + T . Remark 15.3. It is clear that we can consider other control and disturbance functions and obtain the same results by using the same techniques. We can also consider other observations, for example, temperature gradients in the crystal in Czochralski crystal growth processes. ♦

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Index

adjoint operator, 69 adjoint problem, 101 admissible set, 83, 183 anisotropic, 371 anisotropic tensor, 372 asymptotic behavior, 408 balance equation, 79 balance operator, 79 Banach vector space, 16 Banach–Alaoglu theorem, 28 Banach–Alaoglu–Bourbaki theorem, 28 Bardeen–Cooper–Schieffer theory, 339 biconjugate function, 59 bidual problem, 102, 107 binary alloy, 370 blood perfusion rate, 427 Boussinesq approximation, 397 Brezis–Lieb lemma, 46 Browder theorem, 82 c-conjugate function, 67 c-convex function, 66 c-subdifferentiability of function, 67 c-subgradient, 67 Cahn–Hilliard approach, 369 canonical immersion, 32 canonical isomorphism, 186 Carathéodory function, 73, 374, 431 Cauchy–Kovalevsky system, 420 characterization of Kakutani, 33 characterization of optimal solution, 94 characterization of the subdifferential, 63

characterizations of a saddle point, 132 closed concave function, 40 closed convex function, 39 closed convex hull subset, 18 closed function, 37 closure of function, 23 coercive function, 38 compact set, 15 complementary energy, 157 complementary gap function, 149 complete metric space, 16 concave conjugate function, 66 concave function, 20 cone-constaints, 325 conjugate function, 57 constitutive equation, 78, 82 constitutive operator, 78 continuous affine function, 39 convex conjugate function, 58 convex envelope, 22 convex function, 20 convex hull, 14 convex set, 13 Cooper pairs of electrons, 340 cost functional, 184 critical level of function, 86 critical point of function, 86 critical point of Lagrangian, 112 critical value of function, 86 Czochralski crystal growth, 478 demi-continuous, 81 dentritic, 373

500

Index

difference quotient, 70 directional derivative, 70 dominated function, 41 double c-conjugate function, 67 dual gap, 102 dual problem, 101 dual topological space, 24 Eberlein–Smulian theorem, 35 effective domain, 21 Ekeland lemma, 88 El Ni˜ no phenomenon, 396 elasticity problem, 118 electrostatics, 85 ellipticity conditions, 53 epigraph of function, 21 equality-type constaints, 290, 325 equilibrium, 128 equilibrium admissible space, 156 equilibrium equation, 69, 79, 148 Euler–Lagrange equation, 86 exact affine function, 62 external energy, 82 extremality relations, 106 f -decreasing homotopy function, 90 f -increasing homotopy function, 90 F-differentiable, 70 operator, 76 Faedo–Galerkin, 167 Fatou’s lemma, 46 feasible set, 83 Fenchel’s inequality, 58 Fenchel–Moreau theorem, 59 Fenchel–Rockafellar theorem, 60 Fenchel-type inequality, 67 fixed-point, 294 Fréchet-differentiable, 70 operator, 76 fully non-linear, 148 fundamental equation, 78, 83 fundamental operator, 78 G-differentiable, 70 operator, 76 Gˆ ateaux variation, 70 Gˆ ateaux-differentiable, 70 operator, 76 Gˆ ateaux-differential, 70, 76

Gagliardo–Nirenberg inequalities, 55 game theory interpretation, 128 Γ -regularization of function, 41 gauge function, 79 gauged (or weighted) duality mapping, 79 generalized equation, 324 geometric characterizations of biconjugate function, 59 geometrical equation, 69, 78 geometrical Hahn–Banach theorem, 19 geometrical operator, 69, 78 geometrically non-linear, 148 Ginzburg–Landau equation, 153 Goldstine lemma, 33 graph of function, 63 Green’s formula, 52, 121 Gronwall’s lemma, 45 H¨ older’s inequality, 45 Hahn–Banach theorem, 18 half-spaces bounded by hyperplane, 17 Hausdorff space, 15 Helley lemma, 33 Hilbert vector space, 17 homotopy function, 90 Hooke’s law, 125 hydrostatic approximation with vertical viscosity, 419 hypograph of function, 21 indicator function , 20 inf-compact function, 24 input variable, 57 internal point, 18 interpolation, 181 invertible operator, 78 isotropic, 371 Jensen’s inequality, 20 general inequality, 20 kinematic equation, 78 Korn’s inequality, 122 Krasnoselskii theorem, 74 kroneker delta, 120 Ky Fan’s lemma, 143 Ky Fan–von Neumann formulation, 134 Lagrangian duality function, 113

Index Lagrangian-remarquable point, 112 Landau–Ginzburg energy, 152 Lebesgue dominated convergence theorem, 46 Legendre–Fenchel transformation, 58 Leray–Hopf projector, 409 locally convex space, 15 London penetration depth, 342 Lorentz force, 341, 479 lower semi-continuous envelope, 23 lower semi-continuous function, 22 magnetic field, 340, 478 Magnetic Resonance Imaging, 431 Maxwell system, 83 Mazur theorem, 26 micropolar fluids, 473 Milman–Pettis theorem, 35 minimax point, 127 minimizing sequence, 38 Monge–Kantorovich, 66 monotone convergence theorem, 46 monotone mapping, 73 Moreau–Rockafellar theorem, 68 Mountain–Pass theorem, 92 natural boundary conditions, 82 natural imbedding, 32 Navier–Stokes, 145 Nemytskii operator, 72, 74 non-potential operator, 142 non-trivial problem, 100 normal cone operator, 325 normal problem, 104 normed vector space, 16 objective functional, 184 observation function, 184 observation operator, 185 online temperature, 431 open cover, 14 optimal solution, 100, 102 optimality conditions, 106 optoelectronics, 478 output variable, 57 Palais–Small (or (P S)) condition, 88 Palais–Small condition at level c (or (P S)c ), 88

501

parallelogram law, 36 phase-field method, 373 phase-field variable, 369 physically non-linear, 148 Poincaré’s inequality, 48 pointwise supremum, 21–23, 39 polar set, 62 potential of Ginzburg–Landau, 371 potential operator, 86 power set, 79 primal problem, 100 proper function, 37 pseudo-gradient field, 89 pseudo-gradient point, 89 quadratic complementary gap function, 150 quasi-convex function, 20 range of operator, 78 reflexive, 107 reflexive space, 25, 32 remarquable point, 109 saddle point, 87, 109, 115 saddle point theorem, 115 second-order G-derivative, 77 self-adjoint operator, 83 semiconductor melts, 478 separable space, 28 separate sets, 18 separated points, 15 separated space, 15 skew-symmetric gyration tensor, 474 smooth Banach space, 79 Sobolev spaces, 47 stability criterion, 105 stable problem , 105 stationary point of a function, 86 strict epigraph of function, 21 strict hypograph of function, 21 strictly convex Banach space, 79 strictly monotone mapping, 73 strictly separate sets, 18 strong convergence, 25 strong duality, 105 strong topology, 25 subcritical point, 109 subdifferentiable function, 63

502

Index

subdifferential set, 63 subgradient form, 63 superconducting materials, 340 supercritical point, 109 superdifferentiable function, 65 superdifferential set, 65 supergradient form, 65 support function, 61 supporting hyperplane of subset, 19

upper semi-continuous function, 23 variational functional, 86 variational inequality, 95, 108 vortex pinning, 341 vortices, 340

tangent geometric operator, 147 time-varying delays, 296 topological vector space (t.v.s), 14 translated function, 58 Tychonoff’s theorem, 28

Warren–Boettinger model, 369 weak convergence, 25 weak duality, 60 weak formulation of problem, 107 weak solution of problem, 107 weak star convergence, 25 weak topology, 24 Weierstrass theorem, 16, 24

uniformly convex space, 35

Young’s inequality, 45

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