Stabilization of Infinite Dimensional Systems (Studies in Systems, Decision and Control, 355) 3030685993, 9783030685997

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Studies in Systems, Decision and Control 355

El Hassan Zerrik Oscar Castillo

Stabilization of Infinite Dimensional Systems

Studies in Systems, Decision and Control Volume 355

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

The series “Studies in Systems, Decision and Control” (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control–quickly, up to date and with a high quality. The intent is to cover the theory, applications, and perspectives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the fields of engineering, computer science, physics, economics, social and life sciences, as well as the paradigms and methodologies behind them. The series contains monographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other. Of particular value to both the contributors and the readership are the short publication timeframe and the world-wide distribution and exposure which enable both a wide and rapid dissemination of research output. Indexed by SCOPUS, DBLP, WTI Frankfurt eG, zbMATH, SCImago. All books published in the series are submitted for consideration in Web of Science.

More information about this series at http://www.springer.com/series/13304

El Hassan Zerrik Oscar Castillo •

Stabilization of Infinite Dimensional Systems

With Contribution by Lahcen Ezzaki and Abderrahmane Ait Aadi

123

El Hassan Zerrik Faculty of Sciences Department of Mathematics Moulay Ismail University Meknes, Morocco

Oscar Castillo Division of Graduate Studies and Research Tijuana Institute of Technology Tijuana, Mexico

ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-3-030-68599-7 ISBN 978-3-030-68600-0 (eBook) https://doi.org/10.1007/978-3-030-68600-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Systems theory is a branch of applied mathematics, it is interdisciplinary and develops activities in fundamental research which are at the frontier of mathematics, automation and engineering sciences. It is everywhere, innumerable and daily, moreover is there something which is not system: it is present in medicine, in commerce, economy, in psychology, in biological sciences, in finance, in architecture (construction of towers, bridges, etc …), weather forecast, robotics, automobile, aeronautics, localization systems and so on. These are the few fields of application that are useful or even essential for our society. It is a question of studying the behavior of systems and acting on their evolution. Among the most important notions in system theory which attracted the most attention is stability. The notion of stability is crucial. Indeed, the evolution of a dynamic system is described by a system of differential equations, partial differential equations... These systems generally have an infinite number of solutions, unless initial conditions are fixed. Depending on the choice of these initial values, there can be an infinity of operating regimes. To illustrate this idea, we cite the example of a clock. This one works with a well-determined amplitude of the pendulum although at the starting time, the pendulum can deviate more or less from its vertical position. If at the starting time of the clock one does not deviate sufficiently the pendulum, it will stop after a few oscillations. On the other hand, if the deviation is large enough, the amplitude of the oscillations of the pendulum will become fully determined after a short time and the clock will operate with this amplitude, for an infinite time. So the equations describing the clock functioning have two stationary solutions: an equilibrium position corresponding to the pendulum at rest and a periodic solution corresponding to the normal operation of the clock. So to understand the evolution of any dynamic system, it is important to know all the stable solutions of its equations, these are equilibrium states. The existing literature on systems stability is quite important but disparate and the purpose of this book is to bring together in one document the essential results on the stability of infinite dimensional dynamical systems. In addition and as such systems evolve in time and space, explorations and research on their stability were mainly focused on the whole domain in which the system evolved. We have v

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strongly felt that, in this sense, important considerations were missing: those which are consist in considering that the system of interest may be unstable on the whole domain but stable in a certain region of the whole domain. This is the case in many applications ranging from engineering sciences to living science. For this reason, we have dedicated this work to extension of classical results on stability to the regional case. This book has its origins in the book I wrote with Professor El Jai [1]. An important consideration was that it should be accessible to mathematicians and to graduate engineering with a minimal background in functional analysis. Moreover, for the majority of the students this would be their only acquaintance with infinite dimensional system. It is organized by following increasing difficulty. The two first chapters deal with stability and stabilization of infinite dimensional linear systems described by partial differential equations. The following chapters concern original and innovative aspects of stability and stabilization of certain classes of systems motivated by real applications, that is to say bilinear and semilinear systems. The stability of these systems has been considered from a global and regional point of view. A particular aspect concerning the stability of the gradient has also been considered for various classes of systems. This book is for students of master’s degrees, engineering students and researchers interested in the stability of infinite dimensional dynamical systems, in various aspects. Our thanks to Prof. Abdelhaq El Jai the president of the “Systems Theory” network who initiated research on applied mathematic in Morocco in 1978. He has greatly contributed to the development of research in systems theory in most Moroccan universities. To all the researchers of MACS Laboratory particularly M. Ouzahra and Y. Benslimane with whom I initiated the concept of regional stabilization. This new concept has led to developments of important results and to openings on other lines of research that can enrich approaches in systems theory, as well as all the colleagues who were part of their constructive remarks. Finally we would like to thank the Hassan II Academy of Sciences and Techniques for its support, through the network Systems Theory, which made it possible to finalize this work. Meknes, Morocco August 2020

El Hassan Zerrik

Reference El Jai, A. and Zerrik, E. Stabilité des systèmes dynamiques, Presses Universitaires De Perpignan, 2014.

Contents

1 7

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

Stabilization of Infinite Dimensional Linear Systems . . . 2.1 Linear Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Stability of Infinite Dimensional Linear Systems . . . . 2.2.1 Definitions—Examples . . . . . . . . . . . . . . . . 2.2.2 Characterizations . . . . . . . . . . . . . . . . . . . . 2.3 Stabilization of Infinite Dimensional Linear Systems . 2.3.1 Characterizations . . . . . . . . . . . . . . . . . . . . 2.3.2 Optimal Stabilization Problem . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Stabilization of Infinite Dimensional Semilinear Systems . . . 3.1 Well Posed Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Stabilization of Infinite Dimensional Bilinear Systems . . . 3.3 Stabilization of Infinite Dimensional Semilinear Systems . 3.3.1 Existence and Uniqueness Results . . . . . . . . . . . 3.3.2 Stabilization Results . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Regional Stabilization of Infinite Dimensional Linear Systems 4.1 Internal Regional Stability . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Characterization of Regional Stability . . . . . . . . . 4.2 Internal Regional Stabilization . . . . . . . . . . . . . . . . . . . . . 4.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Characterizations . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Regional Stabilizing Control . . . . . . . . . . . . . . . . 4.2.4 Low Cost Regional Stabilization Problem . . . . . .

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73 73 73 74 76 84 85 86 89 92

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Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 One Dimensional Case . . . . . . . . . . . . . . . . 4.3.2 Two Dimensional Case . . . . . . . . . . . . . . . . 4.4 Regional Boundary Stabilization . . . . . . . . . . . . . . . 4.5 Optimal Control for Regional Boundary Stabilization 4.5.1 Direct Approach . . . . . . . . . . . . . . . . . . . . . 4.5.2 Internal Approach . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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95 95 96 98 103 104 107 109

Regional Stabilization of Infinite Dimensional Bilinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Regional Stabilization . . . . . . . . . . . . . . . . . . . . . 5.1.1 Regional Exponential Stabilization . . . . . 5.1.2 Regional Weak Stabilization . . . . . . . . . . 5.1.3 Regional Strong Stabilization . . . . . . . . . 5.2 Regional Stabilization Problem . . . . . . . . . . . . . . 5.2.1 Case of Isometries Semigroup . . . . . . . . . 5.2.2 Case of Contractions Semigroup . . . . . . . 5.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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111 111 112 115 117 120 120 124 132 136

Regional Stabilization of Infinite Dimensional Semilinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Regional Stabilization . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Regional Exponential Stabilization . . . . . . . . 6.1.2 Regional Strong Stabilization . . . . . . . . . . . . 6.1.3 Regional Weak Stabilization . . . . . . . . . . . . . 6.2 Regional Stabilization Problem . . . . . . . . . . . . . . . . . 6.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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137 137 138 142 145 147 154 157

Output Stabilization of Infinite Dimensional Semilinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Output Stabilization of Semilinear Systems . . . . . . 7.1.1 Output Exponential Stabilization . . . . . . . . 7.1.2 Output Strong Stabilization . . . . . . . . . . . . 7.1.3 Output Weak Stabilization . . . . . . . . . . . . 7.2 Output Stabilization of Bilinear Systems . . . . . . . . 7.2.1 Output Exponential Stabilization . . . . . . . . 7.2.2 Output Strong Stabilization . . . . . . . . . . . . 7.2.3 Output Weak Stabilization . . . . . . . . . . . . 7.2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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159 159 160 164 168 170 171 173 177 178 180 183

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Contents

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Stabilization of Infinite Dimensional Second Order Semilinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Stabilization of Bilinear Systems . . . . . . . . . . . . . . . . . . . . 8.1.1 Strong Stabilization . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Exponential Stabilization . . . . . . . . . . . . . . . . . . . 8.2 Stabilization of Semilinear Systems . . . . . . . . . . . . . . . . . . 8.2.1 Strong Stabilization . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Exponential Stabilization . . . . . . . . . . . . . . . . . . . 8.3 Regional Stabilization of Second Order Semilinear Systems 8.3.1 Regional Stabilization of Bilinear Systems . . . . . . . 8.3.2 Regional Stabilization of Semilinear Systems . . . . . 8.4 Simulations Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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185 185 186 189 192 192 195 199 199 205 210 212

Gradient Stabilization of Infinite Dimensional Linear Systems 9.1 Gradient Stability—Definitions—Examples . . . . . . . . . . . . 9.2 Characterizations of Gradient Stability . . . . . . . . . . . . . . . 9.3 Gradient Stabilization for Infinite Dimensional System . . . 9.3.1 State Space Decomposition . . . . . . . . . . . . . . . . . 9.3.2 Riccati Method . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Gradient Stabilization Control Problem . . . . . . . . . . . . . . 9.5 Numerical Algorithm and Simulations . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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213 213 215 219 220 223 223 224 228

10 Regional Gradient Stabilization of Infinite Dimensional Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Regional Gradient Stability . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.2 Characterizations . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Gradient Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Stabilizing Control . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 State Space Decomposition . . . . . . . . . . . . . . . . . . 10.2.3 Lower Cost Gradient Stabilization Problem . . . . . . 10.2.4 Numerical Approach and Simulations . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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229 229 229 232 237 237 240 242 244 249

11 Gradient Stabilization of Infinite Dimensional Bilinear Systems 11.1 Gradient Stabilization of Bilinear Systems . . . . . . . . . . . . . 11.1.1 Gradient Exponential Stabilization . . . . . . . . . . . . . 11.1.2 Gradient Strong Stabilization . . . . . . . . . . . . . . . . . 11.1.3 Gradient Weak Stabilization . . . . . . . . . . . . . . . . . 11.2 Regional Gradient Stabilization of Bilinear Systems . . . . . . 11.2.1 Regional Stabilizing Control . . . . . . . . . . . . . . . . . 11.2.2 State Space Decomposition . . . . . . . . . . . . . . . . . . 11.2.3 Problem of Regional Gradient Stabilization . . . . . .

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251 251 252 257 260 263 265 266 270

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11.3 Numerical Approach and Simulations . . . . . . 11.3.1 Regional Gradient Stabilization Case . 11.3.2 Gradient Stabilization Case . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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12 Regional Gradient Stabilization of Infinite Dimensional Semilinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Controls for Regional Gradient Stabilization . . . . . . 12.1.1 Regional Gradient Strong Stabilization . . . 12.1.2 Regional Gradient Weak Stabilization . . . . 12.2 Regional Gradient Stabilization Problem . . . . . . . . 12.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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13 Conclusion and Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

Notations

X @X f ð:; tÞ

open of Rn boundary of X function : x 2 X 7! f ðx; tÞ partial derivative with respect to xi

x vx c0 ry

subregion of X characteristic function of x  X trace application of order zero @y @y ; :::; @x Þ gradient of y : ry ¼ ð@x 1 n Pn @ 2 y Laplacien of y : Dy ¼ i¼1 @x2 i P 4 bilaplacien of y : D2 y ¼ ni¼1 @@xy4

@f @xi

Dy D2 y Dð AÞ NðAÞ Imð AÞ A1 A qð AÞ rðAÞ rðAÞ L2 ðXÞ H 1 ðXÞ H 10 ðXÞ

  1 H 2 ð@XÞ ¼ Im c0 H s ðXÞ jjj:jjj \; [ jj:jj jj:jjA Xw

i

domain of A linear operator kernel set of A images set of A Inverse of A adjoint operator of A resolvent set of A pointwise spectrum of A spectral radius of A space of square integrable functions on X   y 2 L2 ðXÞ j y' 2 L2 ðXÞ   y 2 H 1 ðXÞ j y ¼ 0 on @X Sobolev space of order 12 on @X Sobolev space of order s uniform norm on LðXÞ inner product defined on X norm defined on X graph norm X endowed with its weak topology

xi

xii

V ,! X yðtÞ uðtÞ; vðtÞ dyðtÞ dt @y @x @2 y @x2

X X0 LðX; YÞ LðXÞ Rðk; AÞ ! *

Notations

V is dense in X and the injection is continous solution of a system controls that act on the system derivative of the function y with respect to the variable t partial derivative of the function y with respect to the variable x second partial derivative of the function y with respect to the variable x the state space dual space of X bounded linear operators from X to Y bounded linear operators from X to X resolvent of the operator A strong convergence weak convergence

Chapter 1

Introduction

Systems theory has been extensively developed over the last few decades. It is present in various fields of research, and extends its applications in the development of many sectors of life such as medicine, economy or industry. Real systems, which have become more and more complex, are made up of a large number of components interconnected to each other following a determined structure. To modelize a system amounts to determine a virtual model which represents it by symbols and signs. We are looking for a model because we cannot repeat an experiment endlessly. In order to design and automatize these systems, it is obviously suitable to give a mathematical representation, i.e. a set of equations that can be joined to initial conditions (for evolution phenomena), and to boundary conditions (when speaking on a spatial domain where the phenomena is studied), whose the resolution makes possible to define the state of system. This state can be represented by a vector formed of a finite number of components, in this case we speak of localized systems, as it can be seen in a space of infinite dimension, we then speak of infinite dimensional systems. Modeling becomes an important area of scientific research in different disciplines. The goal is obviously to understand as best as possible the phenomenon we wants to study in view to predicting the behaviour of the state of the considered system. Many phenomena in different area, physical, biological, economic, ecological, . . . can be modeled by linear or non-linear partial differential equations, and the study of these equations partly goes through a better understanding of the properties of their solutions. Thus, most physical problems are represented by non-linear models. As an example, we quote Boltzmanns equation in statistical mechanics, Navier Stokes equations in fluid mechanics, the Von Karman equations of flat plates in large displacement, etc. . . . However, these models are generally difficult to study both theoretically or numerically. In this case we use other types of systems such as semi linear systems, which constitute a transition class between linear systems and non-linear ones. In © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. H. Zerrik and O. Castillo, Stabilization of Infinite Dimensional Systems, Studies in Systems, Decision and Control 355, https://doi.org/10.1007/978-3-030-68600-0_1

1

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1 Introduction

addition, a large number of industrial or natural processes inherently have a semi linear structure. For examples; the transfers of heat by conduction-convection, the kinetics of neutrons in a nuclear reactor, the dynamics of sense organs, the mass action law in chemistry, the diffusion of a semi-permeable membrane, cell multiplication by cell division, electric power production, direct current motors in mechanics, the economy of certain countries, etc. . . . The modeling of some physical problems can be directly obtained (without approximations) by Linear equations: this is the case of the transport equation of neutrons, the diffusion of heat in a metal bar, the propagation of an electromagnetic wave in a conductive material, certain chemical reactions, . . . Other linear models can be derived from non linear systems either by ignoring certain terms, which is valid in certain situations: small displacements, slow movements, . . . , either by linearizing around a special solution. In addition, the methods developed for solving linear systems are often useful for the study of non linear ones. It is classic, in any phase of modeling to describe dynamic systems following two approaches: – The first one, said external representation, and consists in modeling a system from an experimental study linking inputs and outputs magnitudes, which can be either controls or disturbances that the system undergoes, i.e. measurements, which correspond to the observations that can be made on the behavior of the system. – The other said, internal representation and allows to write, from the fundamental laws, the evolution equations and to analyze some intrinsic properties. Thus, a dynamical system can then be designed as a dynamic transformation of inputs into outputs. However, the modeling phase is still linked to an identification phase which consists of choosing from a class of specific models; the one that fits best with achieved experiences. It is a very broad notion that can be defined as the induction of mathematical models starting from experiences and a priori knowledge on systems. It is possible today, to modify the traditional scheme: modeling—identification— command, by the scheme Modeling −→ Analysis −→ Control where the analysis step consists in studying a set of notions and concepts that allows a better knowledge of the properties of the system. Among these notions, we note that of controllability, observability, stability, detectability, etalability . . . This analysis can be done through operators appearing in the model and also through the spatial distribution of actions or measures in the geometrical evolution domain representing the system. Any study concerning the analysis of a dynamical system is usually followed by a control step, which consists in determining a command which allows to drive the studied system to a certain objective. For example, the possibility of bringing a system from an initial state to a desired one or to reach its neighbors at a finite time,

1 Introduction

3

which respectively defines the notion of exact and weak controllability. In the case of infinite time, we obtain the notion of asymptotic controllability. By duality, the notion of exact or weak observability consists in reconstructing the state in a finite time, based on knowledge of dynamics and the output of the system. In the case of infinite time, we speak of the concept of detection. These problems have been the subject of numerous studies. Lions [1–3] considered many more or less standard situations. These results have been followed and completed by numerous developments concerning the different concepts of analysis and control of a system. In finite dimension, the controllability has been characterized by a rank condition which relates the dynamics to the controllability matrix, this algebraic condition has extensions to the case of infinite dimension, notably a temporal version involving the semigroup generated by the dynamic of the system (see for example [4] for the linear case, and [5] for the bilinear case). For some applications, we can be brought to design a control system that achieves an objective while minimizing a performance criterion. This kind of problem has dominated the literature: In the finite time case Pritchard [6, 7], used a dynamic programming to reduce the minimization problem from a quadratic cost to a resolution of a Riccati differential equation, and to the resolution of an algebraic Riccati equation in the case of infinite time. This study was completed by Zabczyk [8], then developed by other authors: Krasovski [9], Lions [2, 10], Curtain and Pritchard [11– 13], Lukes and Russell [14], Datko [15, 16], Quinn [17], Staffans [18], Weiss and Weiss [19]. In addition, control of a system is often sought by a feedback control requiring continuous knowledge of the state of the system, according to the used description. As an example, the problem of regulation which consists in finding a way to maintain the state of a system as close as possible to a desired state one. It is obviously possible to regulate in open loop, but such control is very sensitive to disturbances, so it is better to regulate by a closed loop, adjusting the control so that the state of the system be close to the desired one. We explore in this book the concept of stability which is one of the most important concepts in systems theory. And an unstable system is useless and potentially dangerous. Qualitatively a system is stable if whenever it is disturbed from its equilibrium state, it remains around this state thereafter. Analysis of the stability of a system induces two methods: the linearization method and the direct method of Lyapunov. The linearization method makes it possible to derive conclusions regarding the local stability of a dynamic system around an equilibrium state, from the properties of stability of its linear approximation. As for the direct method, introduced in the 19th century by Lyapunov, it is not restricted to a local character and makes it possible to determine the stability properties of a nonlinear system using an energy function. A dynamic system is always affected by disturbances, and for a good functioning of the system, it is therefore essential to remedy these disturbances, by considering the notion of stabilization, which consists in bringing the disturbed system back to its equilibrium state.

4

1 Introduction

By duality and by need to reconstruct the state of the system from measurements, we define a second system called observer which estimates, at each time, a part of the state, (or the whole state). It is about to determine an estimate of the state, in infinite time from the data of the equation defining the system, the control and the output. In finite dimension, the problem of the stabilization of the linear and nonlinear systems is today perfectly established, and has given rise to a rich literature (Whonham [20], Azzo-Houpis [21], Lasalle [22], Jurdjevic-Quin in [5], Quin [17], Slemrod [23], Gauthier-Bornard [24] and Outbib-Sallet [25], . . .). In the case of infinite dimensional systems, work on strong and or weak stabilization of such systems are mainly due to Slemrod [23, 26], Triggiani [27], Benchimol [28], Russell [29, 30], Balakrishnan [31], Ball [32], Ball-Slemrod [33], . . . The link between the asymptotic behavior of a system, the spectral properties of its dynamic and the existence of a Lyapunov function are explored in [34]. The exponential stabilization was studied in [27, 35] via an appropriate decomposition of the state space. Asymptotic and exponential stabilization were studied in [31] and [34] respectively, by means of the Riccati equation. Stabilization results for finite dimensional systems have always guided the study of the infinite dimensional case, and the transition from the first class to the second, leads to difficulties mainly due to the presence of spatial variable describing the geometrical domain of the system. The enhancement of this variable, conventionally hidden in spaces of infinite dimension, allows however the right understanding of the reality of the phenomenon. The stabilization of infinite dimensional semilinear systems has been the subject of many works. Stabilization of such systems was considered in [36], where one shows that a feedback control weakly stabilizes a bilinear system using the compactness of control operator and a weak controllability condition. In [37], the author proves exponential stabilization under the observability inequality. The same condition allows strong stabilization for a larger class of bilinear systems (see [38]). At the beginning of the 90s, and with motivations linked to real applications, El Jai and Zerrik [39–42] introduced the notion of regional analysis, it is a question of analyzing and controlling a system defined on a geometrical domain  in order to achieve a goal on a given region ω of . A new line of research was opened, and the analysis of the infinite dimensional systems can be treated in other way. The usual notions of controllability and of observability have been reconsidered with another point of view, indeed if we consider a region ω ⊂ , regional control over ω consists in driving the considered system from its initial state to any desired state on ω. It has been shown that a system can be controlled just on a region without being on its whole evolution domain. In addition, the cost of regional controllability is lower [42]. What has been done for an internal region of the spacial domain, has been extended thereafter in case where the region of interest is located on the boundary of  [43–45]. Then all these notions have been also studied through the input-output parametric structure, that is to say again, through the sensors and actuators structure [40, 42, 46, 47].

1 Introduction

5

u2

u1

Fig. 1.1 Stabilization on the boundary region ω

Thus, in the presence of a system evolving on a spatial domain, it became possible to analyze it or to control it only on a privileged region of the system evoling domain. As examples, we cite: – The example of certain oven tunnel in the ceramic industry, modeled by thermal equations, where the problem is to excite the system so to maintain, over a region of the oven, a prescribed temperature. – The problem of water purification of a river where the question is naturally to reach a minimum pollution degree in a given region. – The example of a thermal exchanger: It is about a heat exchanger (see Fig. 1.1) with parallel flow where two liquids are sent in the exchanger with speeds v1 (t) and v2 (t) with input temperatures T1 (t) and T2 (t) respectively. We want the output temperature z(t) = Ts (t) of the first fluid be as close as possible to a desired temperature Tc on  and for that it is possible to act on the flow rate of the second fluid by choosing v2 (t). The established results and the various examples prove that the regional approach in the study of infinite dimensional systems is a natural extension of the classical notions in systems theory. The question is naturally what becomes the notion of stability in this new design. Classically, any system is considered stable or else unstable. However, there are systems that are unstable on their geometrical domains , but do not be the same on all , indeed, they can be stable on certain regions ω of . In addition, you may need to stabilize a system or improve its degree of stability only on a part of its evolution domain, which will require a lower cost [48, 49]. The problem of regional stabilization consists in studying the asymptotic behavior of an infinite dimensional system only in a region ω interior to the evolution domain  or a part of the boundary ∂ of . Regional weak, strong and exponential stabilization for bilinear systems was studied in [49, 50] using various controls. In [51], authors proved regional strong and weak stabilization of bilinear systems with unbounded control operator. In [52], authors considered regional weak, strong and exponential stabilization of bilinear systems with control operator assumed to be bounded with respect to the graph norm of the dynamics operator. In [53, 54], authors considered exponential, strong and weak output stabilization of semilinear systems.

6

1 Introduction

Recently, the notion of gradient stabilization was introduced by Zerrik and coworkers for linear, bilinear and semilinear distributed systems (see [55–58]). There are many reasons to consider gradient stabilization, among others, there exist systems which are not stabilizable but gradient stabilizable ([58]), gradient stabilization may be realized by lower cost than the state stabilization and in many real problems the stabilization is not asked for the state but for gradient. As in the thermal insulation problem where the purpose is to keep a constant temperature in a local with regards to the outside environment, thus one has to regulate the local temperature in order to vanish the exchange thermal flux. On the other hand the strong and exponential stabilization for a class of bilinear and semilinear second order systems was proved in [59, 60] using the observability inequality, furthermore, the established results have been extended to regional case in [61]. We now give a general overview of the content of chapters. – In the first chapter, we recall the most important results on the theory of linear semigroups and their asymptotic behavior, it also concerns characterizations of the stability and stabilization of linear systems. – Chapter 2 concerns some results on the stabilization of semilinear systems, we then discuss the existence and the uniqueness of the solutions of bilinear and semilinear systems and then we give the main results of stabilization developed for such systems. – In chapter three, we introduce the notion of regional stabilization for linear systems. We give definition and characterizations of regional stability. And we present some motivating examples we give characterizations of the controls achieving regional stabilization, stabilizing controls that maintain the system state bounded over the whole domain and minimizes a performance criterion. The used techniques come from the classical case and are essentially based on spectral properties, Lyapunov function or Riccati equation [27, 31, 34, 62, 63]. Numerical examples illustrate the obtained results and extension to boundary region is also discussed. Chapter four focus on studying regional stabilization of infinite dimensional bilinear systems. Under sufficient conditions, we obtain exponential, strong and weak stabilization, also we discuss a regional optimal stabilization problem. The fifth chapter is devoted to regional stabilization of semilinear systems. So we give results about exponential, strong and weak stabilization. The strong stabilization is also obtained minimizing a functional cost. Chapter 6 deals with the output stabilization of bilinear and semilinear systems, then gives results on the exponential, strong and weak stabilisation. It is an extension of regional stabilization of such systems. Illustration by numerical example are also given. Chapter 7 discuss stabilization of infinite dimensional second order semilinear systems, so we establish results on strong and exponential stabilization for bilinear system, that we extend to semi linear ones. Moreover, we study regional stabilization of bilinear and semi linear second order systems.

1 Introduction

7

In Chap. 8, we introduce the notion of gradient stabilization of infinite dimensional linear systems, this requires a specific approach and some technical precautions. Thus we give definitions and some properties of gradient stability. Then we establish results on the gradient stabilization by three approaches: the state space decomposition, the Riccati method and minimizing a functional cost. We also give numerical algorithm that we illustrate by simulations. Chapter 9 discuss regional gradient stabilization of infinite dimensional linear systems. And stabilization results are established using stabilizing controls, state space decomposition method and by a control that minimizes a specific functional cost. In Chap. 10 we deal with the gradient stabilization of infinite dimensional bilinear systems, so we establish exponential, strong and weak gradient stabilization. Then we extend the obtained results to regional gradient stabilization. Chapter 11 is about regional gradient stabilization of infinite dimensional semilinear systems. Then we characterize gradient stabilizing controls, and the one that stabilizes the gradient and minimizes a given functional cost. The developed results are successfully illustrated by simulations. The last chapter presents possible perspectives to this work where we we give open questions useful for future developments.

References 1. Lions, J.L.: Contrôlabilité exacte. Perturbations et stabilisation des systèmes distribués, Tome 1, Contrôlabilité Exacte, Masson (1988) 2. Lions, J.L.: Contrôle optimal des systèmes gouvernés par des équations aux dérivées partielles. Dunod, Paris (1968) 3. Lions, J.L., et Magénes, E.: Problèmes aux limites non homogènes et application, vol. 1. Dunod, Paris (1968) 4. Dunford, N., Schwartz, J.T.: Linear Operators. Part II. Willy Classics Library, New York (1963) 5. Jurjevic, V., Quinn, J.P.: Controllability and stability. J. Differ. Equ. 28, 381–389 (1978) 6. Prichard, A.J.: Stability and control of distributed systems. Proc. IEEE., 1433–1438 (1969) 7. Prichard, A.J.: Stability and control of distributed systems governed by wave equations. In: Proceedings of IFAC. Conference on Distributed Parameter Systems, Banf, Canada (1971) 8. Zabczyk, J.: Remarks on the algebraic Riccati equation in Hilbert space. Appl. Math. Opt 3, 251–258 (1976) 9. Krasovskii, N.N.: On analytical design of optimum regulators in time-delay systems. Prikl. Mat. Mekh. 1, 39–52 (1962) 10. Lions, J.L.: Sur le contrôle optimal des systèmes decrits par des équations aux dérivées partielles linéaires. C.R. Acad. Sc. Paris 263, 661–663, 713–715, 776–779 (1966) 11. Curtain, R.F., Pritchard, A.J.: Infinite Dimensional Linear Systems Theory. Springer, Berlin (1978) 12. Curtain, R.F., Pritchard, A.J.: The infinite dimensional Riccati equation. J. Math. Anal. Appl. 47, 43–57 (1974) 13. Curtain, R.F., Pritchard, A.J.: The infinite dimensional Riccati equation for systems defined by evolution operators. SIAM J. Control and Optim. 14, 955–469 (1989) 14. Lukes, D.L., Russel, D.L.: The quadratic criterion for distributed systems. SIAM J. Control 7, 101–121 (1969)

8

1 Introduction

15. Datko, R.: A linear control problem in abstrat Hilbert space. J. Diff. Eq. 9, 346–359 (1971) 16. Datko, R.: unconstrained control problem with quadratic cost. SIAM J. Control 11, 32–52 (1973) 17. Quinn, J.P.: Stabilization of bilinear systems by quadratic feedback control. J. Math. Anal. Appl. 75, 66–80 (1980) 18. Staffans, O.: Quadratic optimal control of stable systems through spectral factorization. Math. Control Signals Syst. 8, 167–197 (1995) 19. Weiss, M., Weiss, G.: Optimal control of stable weakly regular linear systems. Math. Control Signals Syst. 10, 287–330 (1997) 20. Whonham, W.M.: Linear Multivariable Control : A Geometric Approach. Springer, Berlin (1979) 21. D’azzo, J.J., Houpis, C.H.: Linear Control System: Analysis and Design, 2nd edn. Mc GrawHill, New York (1981) 22. Lasalle, J.P.: Stability theory for ordinary differntial equations. J. Diff. Equ. 4, 307–324 (1968) 23. Slemrod, M.: A note on complete controllability and stability for linear control systems in Hilbert space. SIAM. J. Control 12, 500–508 (1974) 24. Gauthier, J.P., Bornard, G.: Stabilisation des systèmes nonlinéaires. Outils et modèles mathématiques pour l’automatique, l’analyse des systèmes et le traitement du signal. CNRS Edit (1981) 25. Outbib, R., Sallet, G.: Stabilizability of the angular velocity body revised. Syst. Control Lett. 18, 93–98 (1992) 26. Slemrod, M.: Feedback stabilisation of a linear control system in Hilbert space with a priori bounded control. Math. Control Sign. Syst. 2, 265–285 (1989) 27. Triggiani, R.: On the stabilizability problem in Banach space. J. Math. Anal. Appl. 52, 383–403 (1975) 28. Benchimol, C.: A note on weak stabilizability of contraction semi-groups. SIAM J. Control Optim. 16, 373–379 (1978) 29. Russell, D.L.: Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM J. Control Optim. 24, 797–816 (1986) 30. Russell, D.L. : Linear stabilization of the linear oscillator in Hilbert space. J. Math. Anal. Appl. 25, 663–675 (1969) 31. Balakrishnan, A.V.: Strong stability and the steady state Riccati equation. Appl. Math. Optim. 7, 335–345 (1981) 32. Ball, J.M.: On the asymptotic behavior of generalized processes with application to nonlinear evolutions equations. J. Diff. Eqs .27, 224–265 (1978) 33. Ball, J.M.: On the asymptotic behavior of generalized processes with application to nonlinear evolutions equations. J. Diff. Equ. 27, 224–265 (1978) 34. Ball, J.M., Slemrod, M.: Feedback stabilization of distributed semilinear control systems. Appl. Math. Optim. 5, 169–179 (1979) 35. Prichard, A.J., Zabczyk, J.: Stability and stabilizability of infinite dimensional systems. SIAM Rev. 23, 25–51 (1981) 36. Rabah, R., Ionescu, D.: Stabilization problem in Hilbert space. Int. J. Control 46, 2035–2042 (1987) 37. Ouzahra, M.: Global stabilization of semilinear systems using switching controls. Automatica 48(5), 837–843 (2012) 38. Ouzahra, M.: Strong stabilization with decay estimate of semilinear systems. Syst. Control Lett. 57(10), 813–815 (2008) 39. Amouroux, M., El Jai, A., Zerrik, E.: Regional observability of distributed systems. Int. J. Syst. Sci. 25, 301–313 (1994) 40. El Jai, A., Pritchard, A.J.: Sensors and actuators in distributed systems analysis. Ellis Horwood Series in Applied Mathematics, J. Wiley, Hoboken (1988) 41. El Jai, A., Simon, M.C., Zerrik, E., Pritchard, A.J.: Regional controllability of distributed parameter systems. Int. J. Control 62(6), 1351–1365 (1995)

References

9

42. Zerrik, E.: Analyse régionale des systèmes distribués. Université Mohammed V, Rabat, Thèse d’état (1993) 43. Zerrik, E., Boutoulout, A., El Jai, A.: Actuators and regional boundary controllability of parabolic system. Int. J. Syst. Sci. 31(1), 73–82 (2000) 44. Zerrik, E., Badraoui, L.El, Jai, A.: Sensors and regional boundary reconstruction for parabolic systems. Sens. Actuators J. 75, 102–117 (1999) 45. Zerrik, E., Boutoulout, A., Bourray, H.: Boundary strategic actuators. Sens. Actuators J. A 94, 197–203 (2001) 46. Berrahmoune, L.: Localisation d’actionneurs pour la contrôlabilité des systèmes paraboliques et hyperboliques. Université Med V, Application par dualité à la localisation de capteurs. Diplôme d’étude supérieures (1984) 47. Berrahmoune, L.: Actionneurs et capteurs dans la contrôlabilité et la stabilisation des systèmes distribués flexibles. Thèse d’état. Université Med V, Rabat (1990) 48. Zerrik, E., Ouzahra, M.: Regional stabilization for infinite-dimensional systems. Int. J. Control 76(1), 73–81 (2003) 49. Zerrik, E., Ouzahra, M., Ztot, K.: Regional stabilization for infinite bilinear systems. IEEE Control Theory. Appl. 151(1), 109–116 (2004) 50. Zerrik, E., Ezzaki, L.: Output stabilization of distributed bilinear systems. Control Theory Technol. 16(1), 58–71 (2018) 51. Zerrik, E., Ait Aadi, A., Larhrissi, R.: On the stabilization of infinite dimensional bilinear systems with unbounded control operator. J. Nonlinear Dyn. Syst. Theory 18, 418–425 (2018) 52. Zerrik, E., Ait Aadi, A., Larhrissi, R.: Regional stabilization for a class of bilinear systems. IFAC-PapersOnLine 50 4540–4545 (2017) 53. Zerrik, E., Ezzaki, L.: An output stabilization of infinite dimensional semilinear systems. IMA J. Math. Control. Inf. 36(1), 101–123 (2019) 54. Zerrik, E., Ait Aadi, A., Larhrissi, R.: On the output feedback stabilization for distributed semilinear systems. Asian J. Control (2019). https://doi.org/10.1002/asjc.2081 55. Zerrik, E., Ezzaki, L.: Stabilisation of the gradient of distributed bilinear systems. International Journal of Control (2019). https://doi.org/10.1080/00207179.2019.1652767 56. Zerrik, E., Ezzaki, L.: Regional gradient stabilization of semilinear distributed systems. J. Dyn. Control Syst. 23(2), 405–420 (2016) 57. Zerrik, E., Ouzahra, M.: Output stabilization for infinite bilinear systems. Int. J. Appl. Math. Comput. Sci. 15(2), 187–196 (2005) 58. Zerrik, E., Benslimane, Y., El Jai, A.: Regional gradient stabilization of distributed linear systems. Int. Rev. Autom. Control (IREACO) 4(5), 755–765 (2011) 59. Ezzaki, L., Zerrik, E.: Stabilization of second order bilinear and semilinear systems. Int. J. Control 2020, 1770335 (2020) 60. Zerrik, E., Ezzaki, L.: Strong and exponential stabilization for a class of second order semilinear systems. Recent Advances in Modeling, Analysis and Systems Control: Theoretical Aspects and Applications. Springer Nature Switzerland AG (2019). https://doi.org/10.1007/978-3-03026149-8 61. Ezzaki, L., Zerrik, E.: Regional gradient stabilization of semilinear distributed systems (2020) 62. Datko, R.: Extending a theorem of A. M. Liapunov to Hilbert Space. J. of Math. Anal. Appl. 32, 610–616 (1970) 63. Pazy, A.: On the applicability of Lyapunov’s theorem in Hilbert space. SIAM J. Math. Anal. 3(2), 291–294 (1972)

Chapter 2

Stabilization of Infinite Dimensional Linear Systems

In this chapter, we present the main results concerning the stabilization of infinite dimensional linear systems. The considered approach uses the semigroup theory.

2.1 Linear Semigroups In this section we recall the main results of semigroups that we will use in the sequel. Let us consider a linear system described by the equation: ⎧ dy(t) ⎪ ⎨ = Ay(t) dt ⎪ ⎩ y(., 0) = y0 ∈ X

(2.1)

where A is a linear operator defined on a complex Hilbert space X (state space) provided with a scalar product and the corresponding norm ||.||. To interpret (2.1), we assume the state y(t) depends only on the previous one. We can then define a linear operator of two parameters S(t, s) such that y(t) = S(t, s)y(s)

(2.2)

If we assume that the state y(t) depends only on the past and that the variation of the state from one moment to another depends only on the time difference, τ = t − s, then (2.2) is autonomous and becomes y(s + τ ) = S(τ )y(s) © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. H. Zerrik and O. Castillo, Stabilization of Infinite Dimensional Systems, Studies in Systems, Decision and Control 355, https://doi.org/10.1007/978-3-030-68600-0_2

(2.3)

11

12

2 Stabilization of Infinite Dimensional Linear Systems

It is then clear that S(0) = I d, the identity operator of X and y(s) = S(s)y0 , y0 ∈ X . It follows S(s + τ ) = S(τ )S(s), τ, s ≥ 0

Definition 2.1 A strongly continuous semigroup (C0 -semigroup) is an operator S(.) from R+ to L(X ), that satisfy the following properties: 1. S(t + s) = S(t)S(s) ∀ t, s ≥ 0; 2. S(0) = I d; 3. ||S(t)y − y|| −→ 0+ as t −→ 0, ∀y ∈ X. S(t) is a contraction semigroup if it is a C0 -semigroup that satisfy |||S(t)||| ≤ 1 for all t ≥ 0, where |||.||| is the uniform norm of L(X ). The above properties imply that S(t) is strongly continuous for every t > 0, and that there are constants M and σ such that |||S(t)||| ≤ Meσ t If M = 1 and σ ≤ 0, the semigroup S(t) is a contraction semigroup, in this case A is said a dissipative operator (see [1]): < Ay, y > + < y, Ay > ≤ 0, ∀y ∈ D(A) Definition 2.2 The infinitesimal generator A of a C0 -semigroup on a Hilbert space X is defined by S(t)y − y (2.4) Ay = lim t−→0 t whenever the limit exists, the domain of A, being the set D(A) of elements in X for which the limit (2.4) exists in X . We cannot, in general, derive S(t)y0 . However, if y0 ∈ D(A) then S(t)y0 ∈ D(A), and we can derive y(t) and we have d (S(t)y0 ) = AS(t)y0 = S(t)Ay0 , t ≥ 0 dt and therefore we deduce that 

t

S(t)y0 − y0 =

AS(s)y0 ds 0

and that y(t) = S(t)y0 is the solution to the linear system (2.1). In addition, S(t)y0 can be defined for every y0 ∈ X as a continuous function, and in this case it’s said weak solution to the system (2.1). Note that if an operator A generates a C0 -semigroup, then A is closed and has dense domain in X. There is a full characterization of the operators A that generate strongly

2.1 Linear Semigroups

13

continuous semigroups; this is given by the following theorem which remains even valid for a Banach space. Theorem 2.1 (Hille-Yosida [1]) A closed linear operator A with dense domain D(A) in X generates a strongly continuous semigroup S(t) if and only if there exist λ0 and M such that for every λ > λ0 , the operator λI − A is invertible, (λI − A)−1 ∈ L(X ) and M |||(λI − A)−r ||| ≤ , r = 1, 2, . . . (λ − λ0 )r let us recall the resolvent set of A, ρ(A) ⊂ C, the set of λ, such that (λI − A)−1 exists, R(λ, A) = (λI − A)−1 is called the resolvent operator of A. Proposition 2.1 ([1]) Let A be a closed linear operator with dense domain. If the operators A and A∗ are dissipative, then A is an infinitesimal generator of a contraction semigroup. For contraction semigroups, we have the characterization (see [1]). Proposition 2.2 A closed linear operator A with dense domain D(A) in X generates a contraction semigroup if and only if the resolvent set ρ(A) of A contains R+ and 1 for all λ > 0, we have |||R(λ, A)||| ≤ . λ A valid result in a Hilbert space [2] is as follows. Proposition 2.3 1. Let A be a closed operator with dense domain in a Hilbert space. If there exists β > 0 such that

and

Re(< Ay, y >) ≤ β||y||2 ,

∀ y ∈ D(A)

Re(< A∗ y, y >) ≤ β||y||2 ,

∀ y ∈ D(A∗ )

then A generates a C0 -semigroup S(t) such that |||S(t)||| ≤ eβt 2. If S(t) is a compact C0 -semigroup (i.e. S(t) is compact for every t > 0), then S(t) is continuous for the uniform norm |||.|||. 3. Let A be an infinitesimal generator of a C0 -semigroup S(t). S(t) is compact if and only if it is continuous for the uniform topology and the resolvent operator R(λ, A) is compact for every λ ∈ ρ(A). Example 2.1 Let (ϕn )n≥1 be an orthonormal basis of a Hilbert space X , and let (λn )n≥1 be a sequence complex numbers such that the sequence (eλn t )n≥1 is bounded (which is true if sup Re(λn ) < ∞). n≥1

Let us consider the diagonal operator defined by

14

2 Stabilization of Infinite Dimensional Linear Systems

Ay =

∞ 

λn < ϕn , y > ϕn

n=1

with domain D(A) = {y ∈ X /

∞ 

|λn < ϕn , y > |2 < ∞}. If moreover A is self-

n=1

adjoint and (λI − A)−1 is compact for some λ, then we have the following properties: 1. There exists a sequence (λn ) of distinct eigenvalues of A, such that: • |λn | −→ ∞; • the dimension of the eigenspace associated with λn is equal to its multiplicity order rn ; • the sequence (λn )n≥1 is strictly decreasing. 2. A has a complete set of eigenvectors (ϕ jk ) j,k≥1 . 3. The spectrum σ (A) is countable. 4. For every y ∈ D(A), we have Ay =

∞ 

λj

j=1

rj 

< ϕ jk , y > ϕ jk

k=1

(r j is the multiplicity order of λ j ). 5. The operator A generates the semigroup given by S(t)y =

∞  j=1

eλ j t

rj 

< ϕ jk , y > ϕ jk

k=1

The following result concerns the regularity of the solution of the system (2.1). Theorem 2.2 ([3]) For every y ∈ D(A), S(.)y ∈ C(R+ , D(A)) ∩ C 1 (R+ , X ) and d S(t)y = AS(t)y dt = S(t)Ay Proof Let y ∈ D(A), we set z(t) = S(t)y, and we have 1 lim (z(t + s) − z(t)) = S(t)Ay s→0 s As S(.)Ay ∈ C(R+ , X ), so z(t) ∈ C 1 (R+ , X ). In addition, we have

2.1 Linear Semigroups

15

z(t + s) − z(t)D(A) = z(t + s) − z(t) + Az(t + s) − Az(t) = S(t)(S(s)y − y) + S(t)(S(s)Ay − Ay) therefore lim z(t + s) − z(t)D(A) = 0 which completes the proof. s→0



The following result gives sufficient conditions so that a disturbed operator generate a semigroup [1, 4]. Theorem 2.3 ([3]) Let A be a generator of C0 -semigroup S(t) and B a bounded linear operator, then the operator K = A + B generates a C0 -semigroup S K (t) given by +∞  SkK (t) S K (t) = k=0

 with S0K (t) = S(t) and SkK (t) =

t 0

K S(t − s)B Sk−1 (s)ds for all k ≥ 1.

In addition if M and σ are such that |||S(t)||| ≤ Meσ t , then |||S K (t)||| ≤ Me(σ +M.|||B|||)t Some topological properties of S(t) remain true for S K (t). This is given by the following result. Theorem 2.4 ([3]) Let A be an infinitesimal generator of C0 -semigroup S(t) and B a bounded linear operator. If the semigroup S(t) is compact, then the operator K = A + B is an infinitesimal generator of a compact C0 -semigroup S K (t). We will examine the case of an operator A disturbed by a relatively bounded operator B. Let us first recall the following definition. Definition 2.3 Let two operators A and B of domains respective D(A) and D(B) with D(B) ⊂ D(A) in a Hilbert space X . We say that B is relatively bounded with respect to A if there exist a > 0 and b > 0 such that ∀y ∈ D(A)  By  ≤ a  Ay  + b  y  We then have the following result. Theorem 2.5 ([3]) Let A be an infinitesimal generator of a contraction semigroup. 1. If B is a dissipative operator and relatively bounded with respect to A with a < 1, then A + B generates a contraction semigroup. 2. If B is an operator relatively bounded with respect to A with a < 1 and A + B dissipative, then A + B generates a contraction semigroup. We give the following definitions.

16

2 Stabilization of Infinite Dimensional Linear Systems

Definition 2.4 Let K ∈ L(X ) be a bounded operator and Z a subspace of X. 1. We say that Z is a reduced subspace of K if Z is invariant by K and by its adjoint K ∗ , i.e. K Z ⊂ Z and K ∗ Z ⊂ Z . 2. K is said to be unitary if K K ∗ = K ∗ K = I d. The following result, classic and important, concerns the decomposition of the state space into a direct sum of two subspaces. Theorem 2.6 ([5] If A generates a contraction semigroup S(t) on X , then X can be decomposed as a direct subspaces sum X = X u ⊕ X s where X u and X s are two reduced subspaces by S(t). In addition, we have 1. The restriction of S(t) to X u , noted Su (t), is a unitary C0 -semigroup for all t > 0, 2. The restriction of S(t) to X s , noted Ss (t), is a completely non-unitary C0 -semigroup for all t > 0, 3. The decomposition is unique and X u can be written in the form X u = {y ∈ X ; S(t)y = S ∗ (t)y = y, t ≥ 0} and, if we consider the restriction K u of K , K u = X u ∩ D(A) and K u = X u . Proof Letting L(t) = I d − S ∗ (t)S(t) and L ∗ (t) = I d − S(t)S ∗ (t) we have, S(t)y = S ∗ (t)y = y ⇔ L(t)y, y = y, L ∗ (t) = 0

(2.5)

The operators L(t) and L ∗ (t) are self adjoint and positive. Indeed let y and z two elements of X , then we have L(t)y, z = y, z − S ∗ (t)S(t)y, z = y, z − S(t)y, S(t)z = y, z − y, S ∗ (t)S(t)z = y, L(t)z so L(t) is self adjoint. In addition, as S(t) is a contraction, then L(t) is positive, thus L(t)y, y = y2 − S(t)y2 ≥ 0 With the same techniques, we show that L ∗ (t) is self adjoint and positive, and there1 1 fore the operators L 2 (t) and L 2 ∗ (t) are well defined. Consequently (2.5) is equivalent to y ∈ K er L(t) ∩ K er L ∗ (t). The space 

K er L(t) ∩ K er L ∗ (t) Xu = t≥0

2.1 Linear Semigroups

17

is a closed subspace of X and the space X s = X u⊥ . The spaces X u and X s are invariant by S(t) and S ∗ (t), for all t ≥ 0. Indeed, y ∈ X u ⇔ S(t)S(s)y = S(t + s)y = y, ∀s, t ≥ 0.

(2.6)

thus X u is invariant by S(t) for all t ≥ 0. Since S(t) is a contraction we have S(t)y = y ⇒ S ∗ (t)S(t)y = y

(2.7)

since, S ∗ (t)S(t)y − y2 = S ∗ (t)S(t)y2 − 2ReS ∗ (t)S(t)y, y + y2 = S ∗ (t)S(t)y2 − 2S(t)y2 + y2 ≤ S ∗ (t)2 S(t)y2 − y2 ≤0 We conclude that X u is invariant by S ∗ (t) for all t ≥ 0. This gives S ∗ (s)S(t)y = S ∗ (s − t)S ∗ (t)S(t)y = S ∗ (s)S(s)S(t − s)y Using now (2.7), we obtain S ∗ (s)S(t)y =

⎧ ∗ ⎨ S (s − t)y = y if s > t ⎩

S(t − s)y = y if s ≤ t

and therefore X u is invariant by S ∗ (t). Let us show that X s is invariant by S(t) and S ∗ (t) for all t ≥ 0. Let y ∈ X s then for all z ∈ X u we have y, z = 0 and since X u is invariant by S ∗ (t), we also have S(t)y, z = y, S ∗ (t)z = 0 then X s is invariant by S(t). Similarly (X u is invariant by S(t)) S ∗ (t)y, z = y, S(t)z = 0 1. With the same techniques used to show (2.7), we have S ∗ (t)y = y ⇒ S(t)S ∗ (t)y = y

(2.8)

therefore (2.7) and (2.8) imply that Su (t)Su∗ (t) = Su∗ (t)Su (t) = I d X u 2. We suppose that there exists a subspace X 0 ⊂ X s such that the operator S X 0 (the restriction of S to X 0 ) is unitary. For y ∈ X 0 such that y = 0 we have

18

2 Stabilization of Infinite Dimensional Linear Systems

S(t)y2 = S ∗ (t)S(t)y, y = S(t)S ∗ (t)y, y = y2 thus S(t)y = S ∗ (t)y = y and therefore y ∈ X u , which is a contradiction. 3. If X admits another orthogonal decomposition X = X0 ⊕ X1 such that X 0 and X 1 verify the same assumptions as X u and X s respectively. Since S X 0 is unitary then for all y ∈ X 0 and for all t ≥ 0, we have S(t)y = S ∗ (t)y = y, hence X 0 ⊂ X u . Furthermore, since X 0 and X u are two reduced spaces of S(.), it will be the same for X u ∩ X 0⊥ . From X u ∩ X 0⊥ ⊂ X ∩ X 0⊥ = X 1 , the restriction of S(.) to X u ∩ X 0⊥ is unitary. But by hypothesis S X 1 is completely non-unitary, which implies that X u ∩ X 0⊥ = {0} and consequently X u = X 0 . Hence the uniqueness of the decomposition. It remains to show that X u = K u , with K u = X u ∩ D(A). Since X u is closed, then K u ⊂ X u . On the other hand for all y ∈ X u , we have  R(λ, A)y =

∞

e−λs S(s)yds ∈ X u

0

indeed,



∗

S(t)S (t)R(λ, A)y =

∞

S(t)S ∗ (t)S(s)yds, ∀t, s ≥ 0

0

 =

∞

S(s)yds, ∀s ≥ 0

0

= R(λ, A)y  =

∞

S ∗ (t)S(t)S(s)yds

0

= S ∗ (t)S(t)R(λ, A)y (because X u is invariant by S(t) and S X u is unitary). From where S(t)S ∗ (t)R(λ, A)y = S ∗ (t)S(t)R(λ, A)y = R(λ, A)y. Finally, we obtain S(t)R(λ, A)y2 = S ∗ (t)S(t)R(λ, A)y, R(λ, A)y = R(λ, A)y, R(λ, A)y Similarly, we have S ∗ (t)R(λ, A)y2 = S(t)S ∗ (t)R(λ, A)y, R(λ, A)y = R(λ, A)y, R(λ, A)y

2.1 Linear Semigroups

19

Thus R(λ, A)y ∈ X u ; and since R(λ, A)y ∈ D(A), then for all y ∈ X u , λR(λ, A)y ∈ Ku . Finally, we know that λR(λ, A)y → y, when Re(λ) → +∞ (see [6]), thus X u ⊂ K u . We conclude that X u = K u .  Lemma 2.1 Let S(t) be a contraction semigroup, then

and

S(t)[I d − S(s)S ∗ (s)]y  0 when t −→ +∞

(2.9)

S(t)[I d − S ∗ (s)S(s)]y  0 when t −→ +∞

(2.10)

Proof Let t and s be such that t ≥ s ≥ 0, and as S ∗ (t) is a contraction we have S ∗ (t)2 = S ∗ (t − s)S ∗ (s)2 ≤ S ∗ (s)2 . Hence the function t −→ |||S ∗ (t)|||2 is positive and decreasing, therefore for fixed s ≥ 0, f (t) = S ∗ (t)y2 − S ∗ (t + s)y2 −→ 0 when t −→ +∞. We have f (t) = S ∗ (t)y, S ∗ (t)y − S(s)S ∗ (s)S ∗ (t)y, S ∗ (t)y = (I d − S(s)S ∗ (s))S ∗ (t)y, S ∗ (t)y Since (I d − S(s)S ∗ (s)) is self adjoint and positive then f (t) = (I d − S(s)S ∗ (s))1/2 S ∗ (t)y2 Hence, for all y ∈ X and s ≥ 0, we have [I d − S(s)S ∗ (s)]S ∗ (t)y −→ 0 when t −→ +∞

(2.11)

and if we multiply the left term by S ∗ (s) we get S ∗ (s)[I d − S(s)S ∗ (s)]S ∗ (t)y = [I d − S ∗ (s)S(s)]S ∗ (t + s)y Therefore, for all y ∈ X and s ≥ 0, [I d − S ∗ (s)S(s)]S ∗ (t)y −→ 0 when t −→ +∞

(2.12)

We know that if Y (t) is a bounded linear operator, then Y (t)y → 0 when t → +∞, for all y ∈ X . If we apply this result to (2.11) and (2.12) we obtain (2.9) and (2.10).  Theorem 2.7 ([3]) Let S(t) be a contraction semigroup on X and the set W = {y ∈ X such that S(t)y  0 when t → +∞},

20

2 Stabilization of Infinite Dimensional Linear Systems

then 1. W is a closed and reduced subspace of S(t), for all t ≥ 0. 2. W ⊥ is a reduced subspace of S(t), and the restriction SW ⊥ (t) is a unitary C0 semigroup. 3. W = {y ∈ X such that S ∗ (t)y  0 when t → +∞}. Proof 1. W is a closed subspace of X. Let yn , y ∈ X be such that yn , z → 0 when n → +∞ for all z ∈ X ∗ and lim  n→+∞

yn − y = 0. Let z ∈ X ∗ , S(t)y, z = lim S(t)yn , z = 0. Which implies that y ∈ W, in n→+∞

other words W is closed in X. W is invariant by S(s) and by S ∗ (s), for all s ≥ 0. Indeed, let y ∈ W , for s ≥ 0, S(t)S(s)y = S(t + s)y  0 when t → +∞ and W is invariant by S(s), for all s ≥ 0. Now we will show that for all y ∈ W , then S ∗ (s)y ∈ W , for all s ≥ 0. For y ∈ W , S(t)y  0 when t −→ +∞ and with (2.9), we get −S(t)S(s)S ∗ (s)y = −S(t + s)S ∗ (s)y  0 when t −→ +∞ Which means that S(t)S ∗ (s)y  0 when t −→ +∞. Consequently W and W ⊥ are two reduced subspaces of S(s), for all s ≥ 0. 2. For this point, it suffices to show that W ⊥ ⊂ X u . By (2.9) and (2.10), for all s ≥ 0 we have I m L(s) ⊂ W and I m L ∗ (s) ⊂ W

(2.13)

and since L(s) and L ∗ (s) are bounded and self adjoint we have (I m L(s))⊥ = K er L(s) and (I m L ∗ (s))⊥ = K er L ∗ (s) ⊥ ⊥ ∗ ⊥ thus (2.13) implies that, for all s ≥ 0, W ⊂ (I m L(s)) ∩ (I m L (s)) . ∗ ⊥ Which is equivalent to W ⊂ (K er L(s) ∩ K er L (s)) = X u . s≥0

3. Let z ∈ X , z = z 1 + z 2 such as z 1 ∈ W and z 2 ∈ W ⊥ S ∗ (t)y, z = y, S(t)z 1  +  for all y ∈ W.y, S(t)z 2  As (z 1 , z 2 ) ∈ W × W ⊥ and W is invariant by S(t) then y, S(t)z 1  −→ 0 and y, S(t)z 2  −→ 0 when t −→ ∞. Hence S ∗ (t)y  0 when t → +∞, and by changing the roles of S(t) and S ∗ (t), and with the same techniques as above we show that S ∗ (t)y  0 when t −→ +∞, hence S(t)y  0 when t −→ +∞, which completes the proof.  Theorem 2.8 Let A be an operator of domain D(A) ⊂ X a Hilbert space. Let δ > 0 and the subsets of the spectrum points of A

2.1 Linear Semigroups

21

σu (A) = {λ ∈ σ (A) | Re(λ) ≥ −δ} (2.14) σs (A) = {λ ∈ σ (A) | Re(λ) < −δ} If the set σu (A) is bounded and separated from the set σs (A) by a simple closed curve (which is the case if σu (A) is finite), then the state space X can be decomposed into two subspaces in the form X = X u ⊕ X s with X u = X, X s = (I − )X

(2.15)

where ∈ L(X ) is a projection operator given by 1

= 2πi



(λI − A)−1 dλ,

(2.16)

C

and C is a curve surrounding σ (A).

2.2 Stability of Infinite Dimensional Linear Systems In this section we recall definitions and characterizations of stability of infinite dimensional system following by illustration with many examples.

2.2.1 Definitions—Examples Let us consider the linear system described by the equation ⎧ dy(t) ⎪ ⎨ = Ay(t) dt ⎪ ⎩ y(0) = y0 ∈ X

(2.17)

where A : D(A) ⊂ X −→ X is an infinitesimal generator of a C0 -semigroup S(t), t ≥ 0 in X. Definition 2.5 The system (2.17) is said 1. exponentially stable (or A generates a C0 -semigroup (S(t)) exponentially stable) if there exist ∃ α > 0, M ≥ 0 such that |||S(t)||| ≤ Me−αt . 2. uniformly stable if lim |||S(t)||| = 0. t→+∞

3. strongly stable if lim S(t)y = 0, for all y ∈ X. t→+∞

4. weakly stable if lim S(t)y, z = 0, for all (y, z) ∈ X × X ∗ . t→+∞

22

2 Stabilization of Infinite Dimensional Linear Systems

Remark 2.1 1. In the case of finite dimension, the three degrees of stability coincide. 2. Exponential stability =⇒ strong stability =⇒ weak stability. The converse is not true in the case where dim X = +∞. This is illustrated by the following examples. Example 2.2 On =]0, +∞[, we consider the system described by the equation ⎧ ∂ y(x, t) ⎨ ∂ y(x, t) = on ×]0, +∞[ ∂t ∂x ⎩ y(x, 0) = y (x), on ]0, +∞[ 0

(2.18)

∂ of domain D(A) = {y ∈ H 1 (]0, +∞[); y(0) = 0} genHere the operator A = ∂x erates C0 -semigroup given by (S(t)y)(x) = y(x + t),  For all y ∈ L (]0, +∞[), ||S(t)y|| = 2

2

+∞

t ≥0

|y(x)|2 d x → 0, when t → ∞, then

t

the system (2.18) is strongly stable. But |||S(t)||| = 1, for all t ≥ 0, so the system (2.18) is not uniformly neither exponentially stable. A system can be weakly stable without being strongly stable. Example 2.3 On =]0, +∞[ we consider the system ⎧ ∂ y(x, t) ⎨ ∂ y(x, t) =− on ×]0, +∞[ ∂t ∂x ⎩ y(x, 0) = y (x) ∈ L 2 (]0, +∞[)

(2.19)

0

∂ of domain D(A) = {y ∈ L 2 (]0, +∞[); y(0) = 0} generThe operator A = − ∂x ates a C0 -semigroup given by (S(t)y)(x) =

⎧ ⎨ y(x − t) if x ≥ t ⎩

0

if x < t

We have ||S(t)y|| = ||y||, for all t ≥ 0, therefore the system (2.19) is not strongly stable. However it is weakly stable, indeed, for all y, z ∈ L 2 (]0, +∞[), two functions with compact supports. For t large enough, the supports of S(t)y and z are disjoint, so  +∞ y(x − t)z(x)d x = 0 S(t)y, z = 0

Now let (y, z) ∈ (L 2 (]0, +∞[))2 then there exist two sequences of functions (yn ) and (z n ) with compact supports such that

2.2 Stability of Infinite Dimensional Linear Systems

y − yn  L 2 (]0,+∞[)
0; |||S(t)||| ≤ Meμt , ∀t ≥ 0} then we have the following result. Proposition 2.4 1. We have σ− ≤ σ+ , and if the system (2.17) is exponentially stable then σ− < 0. 2. If A generates a semigroup S(t) differentiable for t > 0, or if there exists t1 > 0, such that S(t1 ) be compact, then σ− = σ+ . Thus, the inequality σ− < 0 imply the exponential stability of the system (2.17). For the proof we refer to [7–9]. Theorem 2.9 ([3]) If A generates a C0 -semigroup S(t), then −∞ ≤ σ− ≤ σ0 . Definition 2.6 We say that the equality of the spectral decay is verified if σ0 = σ− . The following result gives a necessary and sufficient condition for the exponential stability [10].

24

2 Stabilization of Infinite Dimensional Linear Systems

Theorem 2.10 ([3]) The system (2.17) is exponentially stable if and only if 

∞

S(t)y2 dt < +∞ for all y ∈ X.

(2.21)

0

Proof If the system is exponentially stable, there exist M > 0 and α > 0 such that for t ≥ 0, |||S(t)||| ≤ Me−αt , thus for m ∈ N∗ , the operator Q m (y) = 1[0,m] S(t)y ∈ L(X, L 2 (R+ , X )) is well defined (see [11]) and then 

+∞

||Q m S(t)y||2 ≤ am ||y||2

0



m

where am =

|||S(t)|||2 dt.

0

According to Banach Steinhaus theorem, there exists γ > 0 such that ||Sm || ≤ γ , m ≥ 1. For t0 > 0, the family (|||S(t)|||)t≤t0 is bounded and, for t > t0 , we have 1 − e−2αt ||S(t)y||2 = 2α



t

e−2α(t−s) ||S(t)y||2 ds

0



t

≤

e−2α(t−s) ||S(s)y||2 |||S(t − s)|||2 ds

0

≤ M 2 γ 2 ||y||2 Hence there exists k > 0 such that |||S(t)||| ≤ k, for all t ≥ 0. In addition, we have 

t

t||S(t)y||2 ≤

|||S(s)|||2 ||S(t − s)y||2 ds

0

≤ k 2 γ 2 ||y||2 For τ large enough, |||S(τ )||| < 1, thus (ln(|||S(τ )|||) < 0 and by the Theorem 2.9, ˜ −αt . there exist M˜ > 0 and α > 0 such that |||S(t)||| ≤ Me The converse is immediate.  Corollary 2.1 If there exists t0 > 0 such that |||S(t0 )||| < 1 then the system (2.17) is exponentially stable. Proof The proof is immediate since |||S(t0 )||| < 1, then σ0 < 0.



2.2 Stability of Infinite Dimensional Linear Systems

25

Remark 2.2 1. In the case of finite dimension, the exponential stability is examined from the spectrum of the system dynamics so the system (2.17) is exponentially stable if and only if: sup{Re(λ), λ ∈ σ (A)} ≤ 0 (2.22) 2. In infinite dimension, we always have the inequality sup{Re(λ), λ ∈ σ (A)} ≤ σ0 . However, we do not need equality to obtain exponential stability as shown in the theorem above. Example 2.4 We consider the system described by the equation  ∂y

∂y (x, t) = (x, t) ∂t ∂x y(x, 0) = y0 ∈ X

on ]0, +∞[2

(2.23)

The state space X = {y ∈ L 2 (R+ )|y is absolutely continuous over [a, b], with 0 < a < b and  +∞ dy | (x)|2 e−2x < ∞} dt 0 endowed with the scalar product  y1 , y2  = 0

+∞

 y1 (x)y 2 (x)d x + 0

+∞

dy1 dy1 (x) (x)e−2x d x dx dx

is a Hilbert space. The operator Ay = ∂∂y with domain D(A) = {y ∈ X | ∂∂ xy ∈ X } and generates a C0 -semigroup given by S(t)y(x) = y(t + x), ∀t > 0 and y ∈ X. According to [12], any complex β ∈ C such that Re(β) ≥ 0, is an element of the resolvent of A, and we have σ0 = 1 (σ0 is defined in (2.20)). So the inequality (2.22) is strict, however the system (2.23) is exponentially stable. The following result characterizes the exponential stability of the system (2.17) [1]. Theorem 2.11 ([3]) Let A be the infinitesimal generator of a C0 -semigroup S(t) on X. The system (2.17) is exponentially stable if and only if there exists P ∈ L(X ) solution of Lyapunov’s equation Ay, P y + P y, Ay = −y, y, ∀ y ∈ D(A) Proof If the semigroup S(t) is exponentially stable, then the operator

(2.24)

26

2 Stabilization of Infinite Dimensional Linear Systems



+∞

Py =

S ∗ (s)S(s)y ds, y ∈ D(A)

0

is solution of the Eq. (2.24) and we have  ∞  S(t)y 2 dt ≥ 0, y ∈ D(A) y, P y = 0

Thus y, P y = 0 implies  S(t)y = 0, for all t ≥ 0 and since S(t) is strongly continuous semigroup, then y = 0; so P is a positive definite operator. Let y ∈ D(A) and let V (y) = P y, y. As P is solution of the Eq. (2.24), we have d V (S(t)y) = −S(t)y2 dt 

+∞

P is positive defined then

S(t)y2 ≤ P y, y.

0

The conclusion results from the density of D(A) in X and from the continuity of P.  Proposition 2.5 If the semigroup S(t) is differentiable for t > 0, then A verify the equality of the spectral decay σ0 = σ− and it follows that if σ− < 0, then the system (2.17) is exponentially stable. The proof is given in [3].

2.3 Stabilization of Infinite Dimensional Linear Systems We consider the controlled linear system ⎧ dy(t) ⎪ ⎨ = Ay(t) + Bv(t) dt ⎪ ⎩ y(., 0) = y0 ∈ X

(2.25)

where A is an infinitesimal generator of a C0 -semigroup S(t), t ≥ 0 in a Hilbert space X and B ∈ L(V, X ), V the controls space assumed to be a Hilbert space. We know that for all K ∈ L(X ), the operator A + B K generates a C0 -semigroup. The weak solution of the system (2.25), is given by  y(t) = S(t)y0 + 0

t

S(t − s)Bv(s)ds

2.3 Stabilization of Infinite Dimensional Linear Systems

27

We have the following definition. Definition 2.7 The system (2.25) is said to be weakly (respectively strongly, exponentially) stabilizable, if there exists a bounded operator K ∈ L(X, V ) such that the system ⎧ dy(t) ⎪ ⎨ = (A + B K )y(t) dt ⎪ ⎩ y(0) = y0 ∈ X be weakly (respectively strongly, exponentially) stable.

2.3.1 Characterizations There are mainly three approaches to tackle the problem of stabilization of the system (2.25). The first is based on the concept of controllability, the second by solving a Riccati equation and the third uses the spectral and structural properties of the operator A. Theorem 2.12 ([2]) The system (2.25) is exponentially stabilizable if and only if 

∞

T (t)y2 dt < +∞ for all y ∈ X,

(2.26)

0

where (T (t)) is a semigroup generated by the operator A + B K . 

Proof We use similar techniques to those for the proof of Proposition 2.10.

Corollary 2.2 ([2]) If there exists t0 > 0 such that |||S(t0 )||| < 1, then the system (2.25) is exponentially stabilizable. 

Proof The proof is similar to that of the Corollary 2.1.

A characterization of the stability of system (2.25) is given by the following result. Proposition 2.6 ([2]) The following properties are equivalent 1. The system (2.25) is exponentially stabilizable. 2. There exists a control v(t) such that the solution of (2.25) satisfy 

∞

 y(t) 2 +  v(t) 2 dt < ∞ for all y0 ∈ X

0

3. There exists an operator P positive defined, solution of the Riccati’s equation Ay, P y + P y, Ay − B ∗ P y, B ∗ P y + y, y , y ∈ D(A)

(2.27)

28

2 Stabilization of Infinite Dimensional Linear Systems

4. For all y0 ∈ X , there exists a control v(t) such that the associated solution y(t) tend to 0 when t → ∞. Remark 2.3 If the operator P is a solution of (2.27), the control v(t) = −B P y(t) minimizes the cost function  ∞ ||y(t)||2 + ||v(t)||2 dt < +∞ q(v) = 0

see [2]. In finite dimension, the stabilization of the system (2.25) is equivalent to the controllability of its unstable modes. In infinite dimension, the spectrum of an operator is not reduced to its punctual part and one cannot in general move the spectrum by a finite rank perturbation. In addition the position of the spectrum is not directly linked to the stability of the system. Proposition 2.7 ([13]) The system (2.25) is weakly stabilizable by the control v(t) = B ∗ y(t) if and only if the unstable states are controllable. One of the methods using the spectral properties of A is the decomposition method. Under the hypotheses of the Theorem 2.8, the system (2.25) can be decomposed into a stable system and an unstable one as follows: ⎧ dyu (t) ⎪ ⎨ = Au yu (t) + Bv(t) dt (2.28) ⎪ ⎩ y0u = y0 , yu = y and

⎧ dys (t) ⎪ ⎨ = As ys (t) + (I − )Bv(t) dt ⎪ ⎩ y0s = (I − )y0 , ys = (I − )y

(2.29)

where is defined in (2.16), As and Au are the restrictions of A to X s and X u respectively, and we have σ (As ) = σs (A), σ (Au ) = σu (A). In addition, Au is a bounded operator defined on X u . The solutions of (2.28) and (2.29) are given by  yu (t) = Su (t)y0u +

t

Su (t − τ ) Bv(τ )dτ

(2.30)

Ss (t − τ )(I − )Bv(τ )dτ

(2.31)

0



and

t

ys (t) = Ss (t)y0s + 0

where Su (t) and Ss (t) are the restrictions of S(t) on X u and X s , which are respectively the C0 - semigroups generated by Au and As .

2.3 Stabilization of Infinite Dimensional Linear Systems

29

We know ([7]) that if (2.25) is exactly controllable, then it is stabilizable. If the operator As verifies the following property of spectral decay lim

t→+∞

ln |||Ss (t)||| = sup Re(σ (As )) t

(2.32)

then stabilizing the system (2.25) turns up to stabilizing (2.28). Proposition 2.8 ([7]) If there exists K u ∈ L(X, V ) such that the control v = K u yu stabilizes the system (2.28), then the system (2.25) is exponentially stabilizable using the control operator K = (K u , 0). Proof According to the previous decomposition, we have sup Re(σ (As )) ≤ −δ. Since As verify (2.32), then there exists a > 0 and μ > 0 such that |||Ss (t)||| ≤ a e−μt , for t ≥ 0. The solution of system (2.28) is given by yu (t) = e Fu t y0u , with Fu = Au + B K u ∈ L(X u ) and there exist 0 < α < μ, b > 0 such that ||yu (t)|| ≤ b e−αt ||y0u ||. so for v = K u z u , we have ||v(t)||V ≤ b e−αt |||K u |||.||y0u || It follows  t −μt e−μ(t−τ ) e−ατ dτ ||ys (t)|| ≤ a e ||y0s || + c ||y0u || 0

≤ae

−δ  t

e−μt − e−αt ||y0s || + c ||y0u || α−μ

where c > 0. Therefore the state of the system (2.25) controlled by v(t) = K y(t) verifies e−μt − e−αt + b e−αt )||y0 || ||y(t)|| ≤ (a e−μt + c α−μ which leads to the exponential stabilization of the system (2.25).



The following result (see [7]) makes possible to bring back the problem of stabilization to that of the controllability of a finite dimensional system. Proposition 2.9 If the condition (2.32) is verified and if, moreover, 1. the space X u is of finite dimension, 2. the system (2.28) is controllable on X u , then the system (2.25) is exponentially stabilizable. Proof In finite dimension, there is equivalence between controllability and exponential stabilization. This implies that the system (2.28) is exponentially stabilizable. One of the results that characterize the weak stabilization of (2.25) is given by the following proposition (see [13]).

30

2 Stabilization of Infinite Dimensional Linear Systems

Proposition 2.10 The control v(t) = −B ∗ y(t) weakly stabilizes the system (2.25) if is only if the unstable states of (2.25) are controllable. We recall the definition of weak controllability Definition 2.8 If A generates a C0 -semigroup S(t) and yd ∈ X, 1. the system (2.25) is approximately controllable to yd , if for all ε > 0, there exist T > 0 and a control v ∈ L 2 ([0, T ], V ) such that: 

T

yd −

S(T − s)Bv(s)ds < ε

0

2. If C indicates the set of states approximately controllable, the system (2.25) is said to be approximately controllable if C = X .  t S(t − s)Bv(s)ds defined from V in X , we have the Let the operator L(t)v = following result.

0

Lemma 2.2 ([3]) 1. C is a closed subspace of X and C = 2. The set of non-controllable states is C ⊥ =



I m(L(t)).

 (K er [B S (t)]). t≥0 ∗ ∗

t≥0 ⊥

Proof Let T > 0 and y ∈ (I m(L(T )) , then for all v ∈ V , we have 

T

v(s), B ∗ S ∗ (s)yds = 0

0

which implies that B ∗ S ∗ (s)y = 0, 0 < s < T . Consequently y ∈ {I m(S(t)B), t ≤ T }⊥ , which gives L(t)⊥ ⊂ {I m(S(t)B), t ≤ T }⊥ Thus

V ect{I m(S(t)B), t ≤ T } ⊂ (L(T )⊥ )⊥ = L(T )

where V ect (E) indicates the subspace generated by E. And finally we show that



b

 S(t)Bydt, y ∈ X = X.

a

 Lemma 2.3 ([3]) Let K ∈ L(V, X ) be a bounded operator, and let T (t) be a C0 semigroup generated by A + B K . We have the following equivalence 

   ∀t ≥ 0, B ∗ S ∗ (t)y = 0 ⇔ ∀ t ≥ 0, B ∗ T ∗ (t)y = 0

2.3 Stabilization of Infinite Dimensional Linear Systems

31

Proof We have ∗

∗



T (t)y = S (t)y +

t

S ∗ (t − s)K ∗ B ∗ T ∗ (s)yds

0

and ∗

∗



S (t)y = T (t)y +

t

T ∗ (t − s)K ∗ B ∗ S ∗ (s)yds

0

Therefore B ∗ S ∗ (t)y = 0 ⇔ B ∗ T ∗ (t)y = 0.



Proposition 2.11 ([13]) If the semigroup S(t) is a contraction semigroup, then the two following conditions are equivalent: 1. The control v(t) = −B ∗ y(t) weakly stabilizes the system (2.25). 2. The unstable states of (2.25) are controllable. Proof Let W be the set of weakly stable states of the system (2.25), and C the set of weakly controllable states. Suppose that the system (2.25) is weakly stabilizable with the control v(t) = −B ∗ y(t). Property 2. is equivalent to W ⊥ ⊂ C, more again C ⊥ ⊂ W . Let y ∈ C ⊥ , according to the Lemma 2.2, for all t ≥ 0, we have B ∗ S ∗ y = 0. Using the Lemma 2.3, for all z ∈ X we have S ∗ (t)y, z = T ∗ (t)y, z = y, T (t)z −→ 0 when t −→ +∞ Therefore S ∗ (t)y  0 when t −→ +∞. As S ∗ (t) is a contraction semigroup, by Theorem 2.7 we have S(t)y  0 when t −→ +∞. Hence y ∈ W and therefore C ⊥ ⊂ W. Conversely, if we suppose that C ⊥ ⊂ W , B B ∗ is a dissipative bounded operator, then A − B B ∗ generates a contraction semigroup (T (t)) on X . Using Theorem 2.6, X = X u ⊕ X s with X u reduces T (t) to unitary C0 -semigroup, and X s reduces T (t) into a completely non-unitary C0 -semigroup. By Theorem 2.7 we have, for all y ∈ X s : T (t)y  0 when t → +∞ It remains to show that, for all y ∈ X u , T (t)y  0 when t → +∞ We note G u = X u ∩ D(A). For all y ∈ G u and t ≥ 0, we have d T ∗ (t)2 = (A∗ − B B ∗ )T ∗ (t)y, T ∗ (t)y) + T ∗ (t)y, (A∗ − B B ∗ )y) = 0 dt Since A∗ and B B ∗ are dissipative, then for all t ≥ 0, we have B ∗ T ∗ (t)y = 0 The Lemma 2.3 implies that for all t ≥ 0, B ∗ S ∗ (t)y = 0, but as C ⊥ ⊂ W , therefore Gu ⊂ W . With Lemma 2.3, we have, for all t ≥ 0 and y ∈ G u , T ∗ (t)y = S ∗ (t)y. Thus for y ∈ W, and by Theorem 2.7 we have S ∗ (t)y  0 when t → 0.

32

2 Stabilization of Infinite Dimensional Linear Systems

Consequently T ∗ (t)y  0 when t → 0, and with the Theorem 2.7 we have T (t)y  0 when t → 0. We deduce that, for all y ∈ G u , T (t)y  0 when t −→ +∞. Let y ∈ X u = G u (with Theorem 2.7), there exists a sequence (yn ) ∈ G u such that yn − y −→ 0 when n −→ +∞ and since T (t) is a contraction semigroup, we have |T (t)y, z| ≤ y − yn z + |T (t)yn , z| −→ 0 when n −→ +∞ 

which ends the proof.

The following result is very useful in the case where the controls space V = R p , and the control operator B[(v1 , v2 , . . . , v p )] =

p 

vj f j

j=1

with f j ∈ X , for j = 1, . . . , p. We also assume that the dynamics operator A of the system is self adjoint and has a compact resolvent.This implies that A admits a complete set of eigenvectors (m ), m ∈ N∗ , associated with eigenvalues (λm j ), j = 1, . . . , rm of multiplicity rm . If in addition A has a finite number of positive eigenvalues denoted λ1d , λ2d , . . . , λl j , in this case the hypothesis of the decomposition of the state space is verified and dim(X u ) < +∞. So we have the result. Proposition 2.12 The system (2.28) is exponentially stabilizable if ⎧ sup rm ⎪ ⎨ p ≥ 1≤m≤l ⎪ ⎩

rg(G m ) = rm ,

(2.33) m = 1, 2, . . . , l

where (Gm)i j =  f i , m j  Proof Au can be represented by a diagonal matrix such that: Au = diag(λ1 , . . . , λ1 , λ2 , . . . , λ2 , . . . , λl , . . . , λl )          r1 times

r2 times

rl times

We also have P B = t [t G 1 t G 2 · · ·t G l ]. The condition (2.33) implies that the system (2.28) is controllable. Consequently it is exponentially stabilizable (because X u is of finite dimension). Hence the existence of an operator K u ∈ L(X u , X ) which exponentially stabilizes the system (2.28). And Theorem 2.8 allows to conclude. 

2.3 Stabilization of Infinite Dimensional Linear Systems

33

2.3.2 Optimal Stabilization Problem The stabilization at lower cost remains a major problem which has been,among others, the subject of numerous works, see [12, 14]. In this subsection we propose to stabilize the system (2.25) while minimizing the functional 

∞

q(u) =

(||y(t)||2 + v(t), Rv(t))dt < +∞.

(2.34)

0

where v ∈ Vad = {v ∈ L 2 (R+ , X ) | q(v) < +∞} and R ∈ L(V ) a self adjoint and positive defined operator. Theorem 2.13 ([12]) The system (2.25) is exponentially stabilizable if and only if there exists P ∈ L(X ) solution of the Riccati equation Ay, P y + P y, Ay − B ∗ P y, B ∗ P y + Ry, y = 0

(2.35)

for all y ∈ D(A). In addition the control v∗ (t) = −P B ∗ y ∗ (t)

(2.36)

stabilizes the system (2.25) and is the unique control minimizing the functional (2.34). In reality the stabilizing control is of finite energy, below we assume that the control (2.36) is bounded. Now we consider a control v(t) = − f (B ∗ y(t)) where the function f : V → R to be determined, and the system ⎧ ∂y ⎪ ⎨ = Ay(t) − B f (B ∗ y(t)) ∂t ⎪ ⎩ y(0) = y0

(2.37)

The following lemma allows to state the main result. Lemma 2.4 ([15]) Let f 0 be a positive and increasing function such that f 0 (0) = 0. We note f 1 (t) = t − (I d + f 0 )−1 (t), and let (sk ) a sequence of positive numbers which verify f 0 (sk+1 ) + sk+1 ≤ sk , for all k ≥ 0. If U (t)ξ is solution of the system  dU (t)

= − f 1 (U (t)) dt U (0) = ξ ≥ 0

(2.38)

then sk ≤ U (k)s0 . Theorem 2.14 ([16]) If the function f satisfies the conditions: 1. there are two positive constants c, d such that for all v ∈ V with vV ≥ 1, we

34

2 Stabilization of Infinite Dimensional Linear Systems

have c v2V ≤  f (v), vV and || f (v)||2V ≤ d vV

(2.39)

2. there exists a strictly increasing concave function ξ : R+ −→ R such that ξ(0) = 0 and for all v such that v < 1, we have  f (v)2V + v2V ≤ ξ( f (v), v)

(2.40)

3. there exist T > 0, e > 0 such that 

T

0

B ∗ S(t)y0 2V dt ≥ ey0 

(2.41)

for all y0 ∈ X . Then the system (2.37) is strongly stable, moreover we have the estimate 1 y(t) = O( √ ) t More precisely, for all t > T , we have 1 1 y(t)2 ≤ U 2 2



 t −1 T

(2.42)

where U (t) → 0 when t → ∞, is a contraction semigroup solution of the differential equation ⎧ dU (t) ⎪ ⎨ + q(U (t)) = 0 dt (2.43) ⎪ ⎩ 1 2 U (0) = 2 y0  where q is given by

q(s) = s − (I + p)−1 (s)

(2.44)

Proof For all T > 0, the solution y(t) of the system (2.37) verify 1 y(T )2 + 2



T 0

 f (B ∗ y(t)), B ∗ y(t)V dt ≤

1 y0 2 2

(2.45)

Let φ and ψ be two functions such that y = ϕ + ψ, where ϕ is solution of the system ⎧ dϕ(t) ⎪ ⎨ = Aφ(t) dt (2.46) ⎪ ⎩ ϕ(0) = y0

2.3 Stabilization of Infinite Dimensional Linear Systems

35

and ψ solution of the system ⎧   dψ(t) ⎪ ⎨ = Aψ − B f B ∗ y(t) dt ⎪ ⎩ ψ(0) = 0

(2.47)

Using (2.45) and (2.41) we get 1 1 1 y(T )2 ≤ y(0)2 ≤ 2 2 2e



T 0

 B ∗ ϕ(t) 2V dt.

(2.48)

Since y = ϕ + ψ, we have    B ∗ ϕ(t) 2V ≤ 2  B ∗ ψ(t) 2V +  B ∗ y(t) 2V and 

T

0

 B ∗ ϕ(t) 2V dt ≤ 2



T 0

 B ∗ ψ(t) 2V dt +



T

0

(2.49)

 B ∗ y(t) 2V dt

 (2.50)

According to (2.47), for 0 ≤ t ≤ T0 , we have  ψ(t) =

t

e(t−s)A B f (B ∗ y(s))ds

(2.51)

0

As S(t) is a contraction semigroup, by the Cauchy–Schwarz inequality, we have



T

|||e(t−s)A |||  f (B ∗ y(s))  ds 0  T ≤ |||B ∗ |||2 |||B|||2 T  f (B ∗ y(s)) 2V ds

 B ∗ ψ(t) 2V ≤ |||B ∗ |||2 |||B|||2

2 (2.52)

0

Therefore there exists a constant C1 > 0 such that 

T 0

∗

 B ϕ(t)

 2V

dt ≤ C1 0

T

∗

 f (B y(t))

 2V

dt + 0

T

∗

 B y(t)

 2V

dt (2.53)

We consider the decomposition of [0, T0 ] = K 1 ∪ K 2 with K 1 = {t ∈ [0, T ]/  B ∗ y(t) V ≥ 1} K 2 = {t ∈ [0, T ]/  B ∗ y(t) V < 1} From the hypotheses (2.39) and (2.40), we obtain

(2.54)

36

2 Stabilization of Infinite Dimensional Linear Systems

 K1

 f (B ∗ y(t)) 2V dt +  ≤



and

K2

d2 1 + c c

 f (B ∗ y(t)) 2V dt +



 B ∗ y(t) 2V dt

K1



 K2

(2.55) T0

∗

∗

 f (B y(t)), B y(t)V dt

0

 B ∗ y(t) 2V dt (2.56)



T

≤

ξ( f (B ∗ y(t)), B ∗ y(t)V )dt

0

Applying Jensen’s inequality, we get 

T

ξ( f (B ∗ y(t)), B ∗ y(t)V )dt ≤ T ξ(

0

1 T



T

 f (B ∗ y(t)), B ∗ y(t)V )dt) (2.57)

0

Using inequalities (2.45) and (2.53), we have  1 y(T )2 2

≤ C0 {

T

 f (B ∗ y(t)), B ∗ y(t)V dt

0

(2.58)

 +ξ( T1

T

∗

∗

 f (B y(t)), B y(t)V dt)}.

0

If we note h(s) = s + ξ( Ts ), we have 1 1 1 y(T )2 ≤ C0 h( y0 2 − y(T )2 ) 2 2 2

(2.59)

t where  h(t) = t + h( ). The function p = (C0 h)−1 is increasing on [0, +∞) and T with (2.59), we get   1 1 1 y(T )2 + p y(T )2 ≤ y0 2 2 2 2

(2.60)

In addition, for k ∈ N, we have   1 1 1 2 2 y((k + 1)T ) + p y((k + 1)T ) ≤ y(kT )2 . 2 2 2

(2.61)

Since the estimate (2.60) is verified on intervals of the form [kT, (k + 1)T ]. 1 By applying Lemma 2.4 to the sequence sk = y(kT )2 , we get, for all k ∈ N, 2

2.3 Stabilization of Infinite Dimensional Linear Systems

1 y(kT )2 ≤ U (k) 2

37

(2.62)

For integer k = 0, for all t > T , and 0 ≤ τ < T , we have t = kT + τ , consequently we deduce, for all t > T ,     t 1 1 1 t −τ 2 2 ≤U y(t) ≤ y(kT ) ≤ U (k) ≤ U −1 (2.63) 2 2 2 T T  Remark 2.4 The stabilization problem can be seen as a special case of an output stabilization one. For this and for other results linking weak or strong stabilization of the output of the system (2.25) to that of (2.28), see, in particular, [17–19].

References 1. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983) 2. Prichard, A.J., Zabczyk, J.: Stability and stabilizability of infinite dimensional systems. Siam Rev. 23, 25–51 (1981) 3. Engel, K.J., Nagel, R.: One Parameter Semigroups for Linear Evolution Equations. Springer, New York (2000) 4. Kato, T.: Perturbation Theory for Linear Operators, 2nd edn. Springer, Berline (1980) 5. Nagy, S.Z.: Stability theory for ordinary differential equations. J. Differ. Equ. 4, 307–324 (1968) 6. Balakrishnan, A.V.: Strong stability and the steady state Riccati equation. Appl. Math. Optim. 7, 335–345 (1981) 7. Triggiani, R.: On the stabilizability problem in Banach space. J. Math. Anal. Appl 52, 383–403 (1975) 8. Hille, E., Phillips, R.: Functional Analysis and Semigroups. American Mathematical Society, Providence (1957) 9. Zabczyk, J.: Remarks on the algebraic Riccati equation in Hilbert space. Appl. Math. Optim. 3, 251–258 (1976) 10. Datko, R.: Extending a theorem of A. M. Liapunov to Hilbert Space. J. Math. Anal. Appl. 32, 610–616 (1970) 11. Datko, R.: A linear control problem in abstrat Hilbert space. J. Differ. Equ. 9, 346–359 (1971) 12. Curtain, R.F., Zwart, H.: An Introduction to Infinite Dimentional Linear System. Springer, New York (1995) 13. Benchimol, C.: A note on weak stabilizability of contraction semi-groups. SIAM J. Control Optim. 16, 373–379 (1978) 14. Banks, H., Kunisch, B.: The linear regulator problem for parabolic systems. SIAM J. Control Optim. 22, 684–696 (1984) 15. Lasiecka, I., Tataru, D.: Uniform boundary stabilisation of semilinear wave equation with nonlinear boundary damping. J. Differ. Integr. Equ. 6, 507–533 (1993) 16. Berrahmoune, L.: Asymptotic stabilization and decay estimate for distributed bilinear systems. Recerche di Matematica. fasc. 1, 89–103 (2001)

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17. Rabah, R., Ionescu, D.: Stabilization problem in Hilbert space. Int. J. Control 46, 2035–2042 (1987) 18. Staffans, O.: Quadratic optimal control of stable systems through spectral factorization. Math. Control Signals Syst. 8, 167–197 (1995) 19. Weiss, M., Weiss, G.: Optimal control of stable weakly regular linear systems. Math. Control Signals Syst. 10, 287–330 (1997)

Chapter 3

Stabilization of Infinite Dimensional Semilinear Systems

This chapter concerns the study of the stability of certain classes of infinite dimensional systems commonly encountered in applications: these are infinite dimensional semi linear systems.

3.1 Well Posed Systems In this section we present some results on the existence and the uniqueness of the solution of a class of non linear systems described by the equation ⎧ dy(t) ⎪ ⎨ = Ay(t) + g(y(t), t) dt ⎪ ⎩ y(0) = y0 ∈ X

(3.1)

where A : D(A) ⊂ X −→ X is a linear operator that generates a C0 -semigroup S(t), t ≥ 0, in X , and g : X × R+ −→ X a given function. First let recall some results on nonlinear semigroups. Definition 3.1 A nonlinear semigroup S(t) on X is a family of continuous operators S(t) : X −→ X, t ≥ 0, that verify: (i) S(0) = I d X (ii) S(t + s) = S(t)S(s), for all t, s ≥ 0 If in addition ||S(t)y − S(t)z|| ≤ ||y − z||, ∀y, z ∈ X, then S(t) is said to be a contraction semigroup.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. H. Zerrik and O. Castillo, Stabilization of Infinite Dimensional Systems, Studies in Systems, Decision and Control 355, https://doi.org/10.1007/978-3-030-68600-0_3

39

40

3 Stabilization of Infinite Dimensional Semilinear Systems

Also let recall that a set C is said to be positively invariant by S(t), if S(t)C ⊂ C, for all t ≥ 0, and it is said to be invariant if S(t)C = C, for all t ≥ 0. For φ ∈ X, we define the following sets:  S(t)φ. In particular, the pos– The positive orbit associated with φ by O+ (φ) = t≥0

itive orbit associated with an equilibrium state is reduced itself. – The set ω−limit of φ by ω(φ) = {ψ ∈ X such that there exists tn −→ +∞; S(tn )φ −→ ψ when n −→ +∞}. – The set weak ω−limit of φ by ωw (φ) = {ψ ∈ X such that ∃ tn −→ +∞; S(tn )φ  ψ when n −→ +∞}. The last two sets represent the states that can be approached to infinity by elements of O+ (φ). In general, we have ω(φ) ⊂ ωw (φ), with equality in the case of finite dimension. In the case of bounded orbits, we have ωw (φ) = ∅, which is not the case for the set ω(φ) which is positively invariant. The following result underlines this result Theorem 3.1 ([1, 2]) 1. If O+ (φ) is pre-compact then ω(φ) = ∅. 2. If for all t ≥ 0, S(t) is sequentially weakly continuous, then O+ (φ) bounded implies ωw = ∅ and invariant by S(t). The previous theorem can be used for the study of nonlinear semigroups of parabolic type and contraction semigroups of hyperbolic type with pre-compact orbits (see [3]). An important application concerns the Lasalle invariance principle which gives information on the structure of the sets ω− limit for dynamical systems having a Lyapunov function. Theorem 3.2 ([4, 5] (Lasalle principal)) Let V be a proper Lyapunov function. Then every Pre-compact trajectories of (3.1) tend strongly towards , the largest invariant set by the dynamic of the system (3.1) and contained in the set Mr =: {y ∈ X such that V (y(t)) = r, ∀t ≥ 0}, for r ≥ 0. To apply this principle in the study of the stability of a steady state y1 , it suffices to show that the set  is reduced to y1 . For the stabilization of such systems, let recall the definitions Definition 3.2 Let T > 0, a function y ∈ C([0, T ], X ) is said 1. weak solution to (3.1) on [0, T ] if the function g(y(.), .) ∈ L 1 (0, T ; X ) and for all ϕ ∈ D(A∗ ), the function t −→ y(t), ϕ is absolutely continuous on [0, T ] and verify d

y(t), ϕ = y(t), A∗ ϕ + g(y(t), t), ϕ a.e on [0, T ] dt 2. strong solution of system (3.1) on [0, T ] if for all y0 ∈ D(A), the solution y(.) is continuous on [0, T [, of class C 1 on ]0, T [ and verify (3.1). The system (3.1) is considered well posed if it has a unique global weak solution.

3.1 Well Posed Systems

41

The following result gives conditions for the existence of the weak solution to the system (3.1), see [6]. Theorem 3.3 ([6]) 1.If g(y(.), .) ∈ L 1 (0, T ; X ), then a solution of the integral equation  t S(t − s)g(y(s), s)ds a.e. on [0, T ] (3.2) y(t) = S(t)y0 + 0

is a weak solution of (3.1). 2. If g : X × [0, T ] → X is continuous on [0, T ], uniformly k-Lipshitz on X then the system (3.1) has a unique weak solution y ∈ C(0, T ; X ) and the application y0 → y is Lipshitz from X in C(0, T ; X ). Proof Let y0 ∈ X , we define the application F : C(0, T ; X ) → C(0, T ; X ) by  F(y)(t) = S(t)y0 +

t

S(t − s)g(y(s), s)ds

(3.3)

0

The function g is uniformly Lipshitz, there exists M > 0 such that  F(y)(t) − F(z)(t) ≤ Mk  y − z ∞ or again  F(y)m (t) − F(z)m (t) ≤

(MkT )m  y − z ∞ m!

(MkT )m < 1. Therefore F has a fixed point y ∈ m! C(0, T ; X ), solution of the system (3.1). Now let z be another weak solution of (3.1) associated to the initial condition z 0 , we have For m quite large we have

 y(t) − z(t)  ≤ S(t)y0 − S(t)z 0  

t

+

 S(t)(g(y(s), s) − g(z(s), s))  ds

0

 ≤ M  y0 − z 0  +Mk

t

 y(s) − z(s)  ds

0

By applying the Gronwall lemma we get  y(t) − z(t) ≤ Me MkT  y0 − z 0  from where  y − z ≤ Me MkT  y0 − z 0 . Therefore the solution is unique and continuous. 

42

3 Stabilization of Infinite Dimensional Semilinear Systems

Theorem 3.4 ([6]) Let g ∈ L 1 ([0, T ]; X ), and on [0, T ], we consider the function  t R(t) = S(t − s)g(s)ds 0

If one of the following two conditions is satisfied: d R(t) 1. R(t) is differentiable on [0, T ] and ∈ L 1 (0, T ; X ) dt 2. For all t ∈ [0, T ], R(t) ∈ D(A), then any solution y(t) of (3.2) is a strong solution of (3.1), for all y0 ∈ D(A). Theorem 3.5 ([6]) If g : X × [0, +∞[→ X is continuous for all t > 0 and uniformly continuous over any bounded interval of R+ and if, moreover, g is locally Lipschitz on X , then the system (3.1) has a unique weak solution y on [0, tmax [ with tmax ≤ +∞ and lim y(t) = +∞. t→tmax

Proof Let us first show that for t0 ≥ 0 and y0 ∈ X , the system (3.1) has a weak solution on [t0 , t1 ] such that t1 − t0 ≤ δ(t0 , y0 ) where  δ(t0 , y0 ) = min 1 , with

y0  K (t0 ) k(K (t0 ), t0 + 1) + N (t0 )

(3.4)

K (t0 ) = 2y0 M(t0 ) M(t0 ) = max{|||S(t)||| on [0, t0 + 1]} N (t0 ) = max{g(t, 0) : 0 ≤ t ≤ t0 + 1}

and k(c, t) is the Lipschitz constant of the function g. We consider the function (3.3) on C([t0 , t1 ]; X ) with t1 = t0 + δ(t0 , y0 ). We have F(B(0, K (t0 ))) = B(0, K (t0 )) is the ball centered in 0 and radius K (t0 ). Indeed 

t

(F y)(t) ≤ M(t0 )y0  +

|||S(t − s)|||(g(s, y(s)) − g(s, 0) + g(s, 0))ds

t0

≤ M(t0 )y0  + M(t0 )K (t0 )L(K (t0 ), t0 + 1))(t − t0 ) + M(t0 )N (t0 )(t − t0 ) ≤ M(t0 ){y0  + K (t0 )L(K (t0 ), t0 + 1)(t − t0 ) + N (t0 )(t − t0 )} ≤ 2M(t0 )y0  = K (t0 ) F is uniformly Lipschitz on B(0, k(t0 )) and of Lipschitz constant L = L(K (t0 ), t0 + 1) and similarly to the previous theorem, we show that F admits a fixed point y in this ball, solution of (3.1) on the interval [t0 , t1 ]. Therefore if y is a weak solution on the interval [0, τ ], it admits an extension on the interval [0, τ + δ] with δ > 0, defined on [τ, τ + δ], and solution of the following equation  y(t) = S(t − τ )y(τ ) +

τ

t

S(t − s)g(s, y(s))ds τ ≤ t ≤ τ + δ

3.1 Well Posed Systems

43

Moreover, δ depends only on y(τ ), k(τ ) and N (τ ). Let [0, tmax [ be the maximum interval of existence of the weak solution y of the system (3.1). If tmax < ∞, then lim y(t) = ∞, otherwise there exists an increasing sequence t→tmax

(tm ) which converges to tmax such that y(tm ) ≤ C, for all m. The solution y defined on [0, tm ] admits an extension on [0, tm + δ] with δ > 0 which does not depend on tm , thus y admits an extension for t > tmax which contradicts the definition of tmax . The uniqueness of the global solution of system (3.1), is a consequence of the Theorem 3.3.  Remark 3.1 If y is bounded, the weak solution y(t) is global (tmax = +∞). The result that follows gives conditions on the existence of a global solution [6]. Theorem 3.6 ([6]) 1. If A generates a compact C0 -semigroup S(t), t ≥ 0, on X and if g : [0, ∞[×X → X is such that the image of any bounded of [0, ∞[×X is a bounded of X, then for all y0 ∈ X , the system (3.1) has at most a weak solution over a maximum interval [0, tmax [. 2. If, moreover, tmax < ∞, then lim  y(t) = ∞. t→tmax

Proof First, we note that a weak solution y of (3.1) defined over a closed interval [0, t1 ] can be extended to a larger interval [0, t1 + δ], δ > 0. The function z(t) = y(t + t1 ) is a weak solution of the system ⎧ dz(t) ⎪ ⎨ − Az(t) = g(t + t1 , z(t)) dt ⎪ ⎩ z(0) = y(t1 )

(3.5)

defined on an interval of length δ [6]. Let [0, tmax [ be the maximum interval of existence of the weak solution y to (3.1), We will show that if tmax < ∞ then  y(t) → ∞ when t → tmax . For that, we will show that tmax < ∞ implies lim y(t) = ∞. Indeed t→tmax

if tmax < ∞ and lim y(t) < ∞, we can assume that |||S(t)||| ≤ M and  y(t) ≤ t→tmax

K for 0 ≤ t < tmax where M and K are constants. By assumption on g, there exists a constant N such that for 0 ≤ t < tmax ,  g(t, y(t)) ≤ N . Now if 0 < ρ < t < τ < tmax , then

44

3 Stabilization of Infinite Dimensional Semilinear Systems

 y(τ ) − y(t)  ≤ S(τ )y0 − S(t)y0   t− p  t +( + )(S(τ − s) − S(t − s))g(s, y(s))ds  t− p  0τ + S(τ − s)g(s, y(s))ds  t



t− p

≤ S(τ )y0 − S(t)y0  +N

 (S(τ − s) − S(t − s)  ds

0

+2M Nρ + (τ − t)M N . (3.6) Since 0 < ρ < t is arbitrary and S(t) is continuous for the operators uniform topology, the right term of (3.6) tends to zero when t and τ tend to tmax . So lim y(t) = y(tmax ) exists, and from the first part of the proof, y(t) can be extended

t→tmax

beyond tmax , which contradicts the fact that tmax is maximum. So the hypothesis tmax < ∞ implies lim y(t) = ∞. t→tmax

To finish the proof, we will show that lim y(t) = ∞. Otherwise, there exists a t→tmax

sequence (tn ) and a constant K such that  y(tn ) ≤ K for all n. Let M > 0 be such that  T (t) ≤ M for 0 ≤ t ≤ tmax and N = sup{ g(t, x) : 0 ≤ t ≤ tmax ,  x ≤ M(K + 1)} Since t → y(t)  is continuous and lim  y(t) = ∞, there exists a sequence t→tmax

(h n ) such that h n → 0 when n → ∞ and  y(t) ≤ M(K + 1) for tn ≤ t ≤ tn + h n and  y(tn + h n ) = M(K + 1). But we have: M(K + 1) = y(tn + h n ) ≤ S(h n )y(tn )   tn +h n |||S(tn + h n − s)|||ds +N tn

≤ M K + hn N M which is absurd when h n → 0 and so lim  y(t) = 0, which completes the proof. t→tmax



The corollary which follows gives another result of existence of a global solution of the system (3.1). Corollary 3.1 ([6]) Suppose that A generates a compact C0 -semigroup S(t) and that the function g is continuous in t and the image by g of any bounded of X × R+ is a bounded of X . If, in addition, one of the following conditions holds 1. There exists a function k : X × R+ −→ R+ , such that ||y(t)|| ≤ k(t), ∀t ∈ [0, t1 [. 2. There exist two locally integrable functions k1 and k2 : X × R+ −→ R+ , such that,

3.1 Well Posed Systems

45

for all y ∈ X , we have ||g(y, t)|| ≤ k1 (t)||y|| + k2 (t), ∀t ∈ [0, t1 [ then for all y0 ∈ X, the system (3.1) admits a global solution y(.). Proof 1. This point follows from the previous theorem. 2. The system (3.1) admits a weak solution y(t) defined on [0, tmax ]. We know that |||S(t)||| ≤ Meμt , where M ≥ 0 and μ ∈ R. The function  t Me−μs k2 (s)ds ψ(t) = M  y0  + 0

is continuous on R+ and we have  t  y(t)  e−μt ≤ e−μt  S(t)y0  +e−μt  S(t − s)g(y(s), s)  ds 0  t Mk1 (s)  y(s)  e−μs ds ≤ ψ(t) + 0

Applying the Gronwall lemma, we obtain  y(t)  e

−μt



t

≤ ψ(t) +



t

k1 (s)ψ(s) exp(M

0

k1 (r )dr )ds

s

Consequently  y(t)  is bounded by a function continuous, and the conclusion follows from 1.  We have another result about the existence of a global solution [6], given below. Theorem 3.7 If g is monotonous and does not depend on t, then the system (3.1) admits a unique and global weak solution. The following result gives sufficient conditions for the weak solution of the system (3.1) be a strong one. Theorem 3.8 ([6]) If g : X × [0, T ] → X is of class C 1 for every t ∈ [0, T ], then for all y0 ∈ D(A), the weak solution of system (3.1) is a strong one. Proof First if g is of class C 1 for all t ∈ [0, T ], then g is continuous with respect to t, uniformly continues on [0, T ], and Lipschitz with respect to the second variable. The Theorem 3.3 implies that the system (3.1) admits a unique weak solution. We show that the weak solution is of class C 1 on [0, T ]. Let the functions f (s) =

∂ g(s, y(s)) and ∂y 

h(t) = S(t)g(t, y(t)) − AS(t)y0 + 0

t

S(t − s)

∂ g(s, y(s))ds ∂s

46

3 Stabilization of Infinite Dimensional Semilinear Systems

By hypothesis and as s → f (s), is continuous on [0, T ], then h ∈ C([0, T ]; X ) and the function f (s)y is continuous on [0, T ], uniformly lipschitz in y, ends the proof.  In the previous result, we assume that g : X × [0, T ] → X is of class C 1 ; which is a too strong assumption. In the result that follows, a weaker condition on g ensures the existence of a strong solution. Theorem 3.9 ([6]) If X is a reflexive Banach space and g : X × [0, T ] → X is Lipschitz with respect to the two variables, g(x1 , t1 ) − g(x2 , t2 ) ≤ k(|t1 − t2 | + x1 − x2 ) for all t1 , t2 ∈ [0, T ] and for all x1 , x2 ∈ X . If for y0 ∈ D(A), y(.) is a weak solution of (3.1), then y(.) is a strong solution. The following result gives a characterization of the weak solution of the system (3.1) with respect to the weak topology of space X. Theorem 3.10 ([6]) If g is sequentially weakly continuous, (yn  y =⇒ g(yn )  g(y). ) and if the system (3.1) has a weak solution y(.) on [0, T ] and if there exist M0 > 0 and M1 > 0 such that ||y0 || ≤ M0 =⇒ ||y(t)|| ≤ M1 for all t ∈ [0, T ] then for all t ∈ [0, T ], the application y0 −→ y(t) is sequentially weakly continuous. Proof Let yn (t) = y(t; y0n ) and y(t) = y(t; y0 ). As (y0n ) is bounded then (yn (t)) is also bounded for each n and t ∈ [0, T ]. On the other hand, the image of any bounded by g remains bounded, thus there exists C > 0 such that g(yn (t)) ≤ C. Let tn → t in [0, T ], for z ∈ X , we set ar =

sup

φ≤1,0≤s≤t

| [S(t − s) − S(tr − s)]φ, z|

then ar → 0 when r → ∞. Otherwise there exist two sequences (φμ ) and (sμ ) such that φμ  φ, sμ → s in [0, T ], tμ → t, and a number ε > 0 with | [S(t − sμ ) − S(tμ − sμ )]φμ , z| ≥ ε But the application (t, φ) → S(t)φ is weakly sequentially continuous on R+ × X (see [4]) then S(t − sμ )φμ  S(t − s)φ and S(tμ − sμ )φμ  S(t − s)φ and therefore ar → 0 when r → ∞. By the formula of the variation of the constant, there exist two positive constants C1 and C2 such that

3.1 Well Posed Systems

47

| yn (tr ) − yn (t), z| ≤ | [S(t  t r ) − S(t)]y0n , z| | [S(tr − τ ) − S(t − τ )] f (yn (τ )), z|dτ + 0 tr | S(tr − τ )g(yn (τ )), z|dτ + t

≤ C1 + C2 |tr − t| Then yn (tr ) − yn (t), z → 0 uniformly when r → ∞. Hence yn (t) is equicontinuous in C([0, T ]; X σ (X,X  ) ). Since (yn (t)) is uniformly bounded with respect to n then for all t ∈ [0, T ], yn (t) belongs to a bounded set of X provided with the metric deduced by the weak topology of X . Applying the Ascoli-Arzela theorem, there exist y˜ ∈ C([0, T ]; X σ (X,X  ) ) and a sequence (yn (t)) such that yn (t)  y˜ (t) uniformly on [0, T ] when n → ∞. Since 

t

yn (t) = S(t)y0n +

S(t − s)g(yn (s))ds

0

then for all z ∈ X , we have 

t

yn (t), z = S(t)y0n , z +

S(t − s)g(yn (s)), zds

0

The function g is sequentially weakly continuous, by the Lebesgue dominated convergence theorem and when n → ∞, for all z ∈ X , we get 

t

y˜ (t), z = S(t)y0 , z +

S(t − s)g( y˜ (s)), zds

0



We deduce that

t

y˜ (t) = S(t)y0 +

S(t − s)g( y˜ (s))ds

0

By the uniqueness of the solution of (3.1) we have y(t) = y˜ (t) on [0, T ]. To conclude, we suppose that y0n  y0 and that (yn (t)) does not converge to y(t) in C([0, T ]; X σ (X,X  ) ), we can assume that there exists in C([0, T ]; X σ (X,X  ) ) / O. Consequently there exists a suba neighborhood O of y(t) such that (yn (t)) ∈  sequence of (yn (t)) which converges to y(t), which is a contradiction.

3.2 Stabilization of Infinite Dimensional Bilinear Systems Let X be a real and separable Hilbert space and we consider a system described by the equation:

48

3 Stabilization of Infinite Dimensional Semilinear Systems

⎧ dy(t) ⎪ ⎨ = Ay(t) + v(t)(By(t) + b) dt ⎪ ⎩ y(0) = y0 ∈ X

(3.7)

where A generates a contraction semigroup (S(t))t≥0 , B : X → X a linear operator, b ∈ X and v(.) is a real valued control. Definition 3.3 The system (3.7) is said to be globally strongly stabilizable if there exists a control v(.) : [0, ∞[→ R such that the system (3.7) admits a unique weak, global solution and lim y(t) = 0, for all y0 ∈ X. t→+∞

For all m ∈ N, let the operator 

∞

Rm (A) =

emt S(t)dt

0

In case b = 0, we have the following result ([7]). Proposition 3.1 If B is a self adjoint and dissipative operator (or positive defined) then the control vr (t) =

r By(t), y(t) with  = signe By(t), y(t) and r > 0, 1 +  By(t), y(t)

(3.8)

strongly stabilizes the system (3.7) if and only if {y ∈ X, B S(t)y + b, S(t)y = 0, t ≥ 0} ∩ G u = {0} where G u is defined in Theorem 2.6 Proof We have |vr (t)| ≤ r and according to the Theorem 3.10, the system (3.7) has a weak, unique and global solution. The operator

= −A + A

r B., . B 1 +  B., .

= D(A). Therefore is maximum monotonous with D( A)

=X D( A)

−1 is compact [8]. In addition, for all λ > 0 the operator λ(I d + A)

generates a nonlinear C0 -semi group

S(t), and the weak solution The operator − A of the system (3.7) is given by y(t) =

S(t)y0 = S(t)y0 +

 0

t

S(t − s)vr (s)By(s)ds

3.2 Stabilization of Infinite Dimensional Bilinear Systems

49

moreover lim y(t) ∈ ω(y0 ) ⊂ {y ∈ X | y = σ } and σ ≤ y0 . t→+∞

It remains to show that ω(y0 ) = {0}. For this, let y0 ∈ D(A), then ω(y0 ) ⊂ D(A), the weak solution y(t) coincides with the strong solution of the system (3.7) and y(t) ∈ D(A) [7]. S(t), let the function Let y ∈ ω(y0 ). Since ω(y0 ) is invariant by

f (t) =

1 1

 S(t)y2 = z2 2 2

Deriving the function f , we obtain r B

S(t)y,

S(t)y2 df =

S(t)y, A

S(t)y − =0 dt 1 +  B

S(t)y,

S(t)y

(3.9)

Therefore

S(t)y, A

S(t)y = 0 and r B

S(t)y,

S(t)y = 0 for all t ≥ 0.

(3.10)

From the equation (3.9), we get

S(t)y = S(t)z which implies that ω(y0 ) is invariant by S(t). The equation (3.10) implies that for all y ∈ D(A), we have d

S(t)y2 =

S(t)z, A

S(t)y = 0 and 

S(t)y2 = y2 dt

(3.11)

By the Theorem 2.6 there exist y1 ∈ X u and y2 ∈ X s such that y = y1 + y2 . As X u and X s are invariant by S(t) and X = X u ⊕ X s , we have y2 = y1 2 + y2 2 and S(t)y2 = S(t)y1 2 + S(t)y2 2

(3.12)

By (3.11) and (3.12) we obtain S(t)y2 2 = y2 2

(3.13)

For m integer large enough, Rm (A) is compact and y2 ∈ X s , the Theorem 2.6 imply lim Rm (A)S(t)y2 = 0.

m→+∞

In the other hand as S(t) is a contraction semigroup and Rm (A)S(t) = S(t)Rm (A), then m Rm (A)S(t)y2 converges uniformly to S(t)y2 when m → +∞, which gives lim

lim m Rm (A)S(t)y2 = lim

m→+∞ t→+∞

lim m Rm (A)S(t)y2

t→+∞ m→+∞

Therefore lim S(t)y2 = 0 and with (3.13) we obtain y2 = 0, which implies y = m→+∞

y2 ∈ {y ∈ X, B S(t)y + b, S(t)y = 0, t ≥ 0} ∩ G u = {0}. Thus ω(y0 ) = {0} for all y0 ∈ D(A).

50

3 Stabilization of Infinite Dimensional Semilinear Systems

Let y0 ∈ X , D(A) is dense in X , there exists a sequence y0m ∈ D(A) such that lim y0m − y = 0.

m→+∞

As S(t) is a contraction semigroup, by triangular inequality we deduce that ω(y0 ) = {0}. Conversely, let y0 ∈ {y ∈ X, B S(t)y + b, S(t)y = 0, t ≥ 0} ∩ G u and y(t) the solution of system (3.7) associated to y0 given by  y(t) = S(t)y0 +

t

S(t − s)vr (s)By(s)ds

(3.14)

0

As B S(t)y0 , S(t)y0  = 0, for all t ≥ 0, then S(t)y0 is also solution of the integral equation (3.14), and by the uniqueness of the solution we have y(t) = S(t)y0 . We have y0 ∈ G u and the system (3.7) is strongly stable then y0  = S(t)y0  = y(t). When t → ∞ we obtain y = 0.  In the case b = 0, we have the result (see [7]). Proposition 3.2 The control given by vr (t) =

r b, y(t) 1 + | b, y(t)|

(3.15)

strongly stabilizes the system (3.7) if and only if {y ∈ X, B S(t)y + b, S(t)y = 0, t ≥ 0} ∩ G u = {0} r b, . b is maximum 1 + | b, .| −1

is compact on X (see [9]). The operator − A

generates monotonous and (λI d − A) S(t), and the weak solution of the system (3.7) is given a nonlinear C0 -semigroup

by 

= −A + Proof (1) Sufficient condition: The operator A

t

y(t) =

S(t)y0 = S(t)y0 +

S(t − s)vr (s)By(s)ds

(3.16)

0

and we have ω(y0 ) ⊂ {y ∈ X | y = σ } and σ ≤ y0 , therefore lim y(t) ∈ ω(y0 ). t→∞

We will show that ω(y0 ) = {0}. Let y0 ∈ D(A), then ω(y0 ) ⊂ D(A) and the weak solution of system (3.7) is a strong solution and y(t) ∈ D(A) (see [7]). S(t). Deriving on ω(y0 ) the function Since ω(y0 ) is invariant by

f (t) = we obtain

1 1

 S(t)y 2 =  y 2 2 2

3.2 Stabilization of Infinite Dimensional Bilinear Systems

51

r b

S(t)y,

S(t)y

=0 f˙(t) =

S(t)y, A S(t)y − 1 +  b

S(t)y,

S(t)y 2

thus, for all t ≥ 0,



S(t)y, A S(t)y = 0 and r b

S(t)y,

S(t)y = 0

(3.17)

As ω(y0 ) is invariant by S(t), for all y ∈ D(A) we have ⎧ d  S(t)y 2 ⎪ ⎨ = 2 S(t)y, AS(t)y = 0 dt and ⎪ ⎩  S(t)y 2 =  y 2

(3.18)

There exist y1 ∈ X u and y2 ∈ X s such that y = y1 + y2 , thus  S(t)y 2 = y2 2 . For m integer large enough, Rm (A) is compact and since y2 ∈ X s , then Rm (A)S(t) y2 tends to 0 when m → ∞. But Rm (A)S(t) = S(t)Rm (A) and S(t) is a contraction semigroup, so m Rm (A)S(t) y2 converges uniformly to S(t)y2 when m → ∞ and therefore lim S(t)y2 = 0 and as  S(t)y2 = y2 , we deduce y2 = 0. This implies that

m→∞

y = y2 ∈ {y ∈ X, B S(t)y + b, S(t)y = 0, t ≥ 0} ∩ G u = {0} thus ω(y0 ) = {0} for all y0 ∈ D(A) and by density, it is also true for all y0 ∈ X . (2) Necessary Condition: Let y0 ∈ {y ∈ X, B S(t)y + b, S(t)y = 0, t ≥ 0} ∩ G u and y(t) the solution of system (3.7) associated to y0 , given by  y(t) = S(t)y0 +

t

S(t − s)vr (s)By(s)ds

0

As b, S(t)y0  = 0 for all t ≥ 0, therefore

S(t)y0 is also a solution of (3.7), and by the uniqueness of the solution we have y(t) = S(t)y0 , and since y0 ∈ G u , the system (3.7) is strongly stabilizable, then we have  y0 = S(t)y0 = y(t) → 0 when t → ∞ which completes the proof.



We consider the system (3.7) and assume that A is dissipative, (i.e Ay, y ≤ 0, for all y ∈ D(A)), that A is closed and D(A) = D(A∗ ). Furthermore we assume that B : X −→ X is anti adjoint ( By, y = − y, By). Let the control v(t) = − b, y(t) (3.19)

52

3 Stabilization of Infinite Dimensional Semilinear Systems

We show that the system (3.7) admits a unique and global weak solution. We need the following lemma. Lemma 3.1 1. If yn (t) the solution of the system (3.7) associated with the initial condition y0n and if y0n → y0 when n → +∞ then yn (t) → y(t) when n → +∞ on [0, T ], the solution of (3.7) associated with the initial condition y0 . 2. The system (3.7) admits a unique and global weak solution. If y0 ∈ D(A) then both weak and strong solutions of the system (3.7) coincide, and we have: d y(t)2 = 2( Ay(t), y(t) − b, y(t)2 ) ≤ 0 dt

(3.20)

We deduce that y(t) |2 ≤ y0 2 , so the solution is bounded. Proof 1. This point follows from the fact that y → − b, y(By + b) is locally Lipschitz and from the Gronwall lemma. 2. For y0 ∈ X , there exists y0n ∈ D(A) such that y0n → y0 when n → +∞, and in addition we have yn (t) ≤ y0n , for all t ∈ [0, T ] and n ∈ N, according to point 1, we have y(t) ≤ y0 , for all t ∈ [0, T ]. Therefore the solution of the system (3.7) is bounded for any initial condition y0 ∈ X, consequently the system (3.7) admits a weak and global solution.  The following result concerns sufficient conditions for the weak convergence of the solution of system (3.7). We need the following lemma. Lemma 3.2 We have the following two results: 1. lim b, y(t) = 0. t→+∞

2. For all t ≥ 0, S(t)ωw (y0 ) ⊆ ωw (y0 ). Proof 1. For y0 ∈ D(A), integrating (3.20) we have 

t

y(t) = y(0) + 2

2



Ay(s), y(s)ds −

0

 which gives  This gives

t

t

b, y(s)2 ds for all t ≥ 0

0

b, y(s)2 ds ≤ y0 2 , for all t ≥ 0.

0 +∞

b, y(s)2 ds < +∞, and moreover

0

Hence lim b, y(t) = 0.

d

b, y(t) is bounded. dt

t→+∞

Now for y0 ∈ X there exists a sequence (y0n ) ∈ D(A) such that y0n → y0 when n → +∞. From the above, for any n ≥ 0 we have 

+∞ 0

b, yn (s)2 ds ≤ y0n 2

3.2 Stabilization of Infinite Dimensional Bilinear Systems

 Fatou lemma gives

∞

53

b, y(s)2 ds ≤ y0 2 .

0

d

b, y(t) is bounded and therefore lim b, y(t) = 0. t→+∞ dt 2. We show that ωw (y0 ) = {0}. Let ψ ∈ ω(y0 ), and tn −→ +∞ such that y(tn )  ψ when n −→ +∞. Then we have  t S(t − s) b, y(s + tn )(By(s + tn ) + b)ds y(t + tn ) = S(t)y(tn ) − We also verify that

0

It follows | φ, y(t + tn ) − S(t)ψ, φ| ≤ | φ, y(t + tn ) − S(t)y(tn ), φ| We also have +| S(t)y(tn ), φ − S(t)ψ, φ| 

t

| φ, y(t + tn ) − S(t)y(tn ), φ| ≤ a|

b, y(s + tn )ds|

0

or when n → +∞, b, y(s + tn ) → 0 and | S(t)y(tn ), φ − S(t)ψ, φ| → 0. Therefore | φ, y(t + tn ) − S(t)y(tn ), φ| → 0 when n → +∞ which completes the proof.  The following theorem characterizes the weak stabilization of the system (3.7). Theorem 3.11 We assume that b ∈ D((A∗ )k ), for all k ∈ N. If, moreover, the subspace generated by {b, A∗ b, · · · , (A∗ )k b, · · · } is dense in X, then y(t)  0 when t → +∞. Proof We have ωw (y0 ) = {0}. Indeed, let ψ ∈ ωw (y0 ), by definition we have dk

S(t)ψ, b = S(t)ψ, (A∗ )k b = 0, for all t ≥ 0 and k ∈ N. dt k In particular, for t = 0, we have ψ, (A∗ )k b = 0 for all k ∈ N and therefore  ωw (y0 ) = {0}. Now we consider the autonomous bilinear system described by the equation ⎧ ⎨ dy(t) = Ay(t) + u(t)By(t) dt ⎩ y(0) = y ∈ X 0

(3.21)

where A is the generator of a contraction semigroup (S(t)), and we consider the control, u(t) = − K y(t), y(t) (3.22)

54

3 Stabilization of Infinite Dimensional Semilinear Systems

where K : X → X a continuous linear operator. The following result gives a sufficient condition ensuring the existence of a global solution of the system (3.21) excited by the control (3.22). Proposition 3.3 If the condition Re( K y, y y, By) ≥ 0

(3.23)

is satisfied for all y ∈ X , then the solution y(t) of the system (3.21) is global. In addition y(t) depends continuously on y0 . Proof As the application y −→ − < K y, y > By is monotonous, according to the Theorem 3.7, the system (3.21) admits a unique weak solution y ∈ C([0, tmax [; X ) defined on a maximum interval [0, tmax [. In addition, the mapping y0 −→ y(t) is continuous in C([0, tmax [; X ). To have tmax = +∞, it remains to show that y(t) is bounded on [0, tmax [. Let T ∈ [0, tmax ], on [0, T ] we consider the function F(t) = − < Dy(t), y(t) > By(t) Let a sequence of functions Fn ∈ C 1 ([0, T ]; X ) such that Fn −→ F in C([0, T ]; X ). For y0 ∈ X , there exists y0n ∈ D(A) such that y0n −→ y0 and we consider the system ⎧ dyn (t) ⎪ ⎨ = Ayn (t) + Fn (t) dt ⎪ ⎩ yn (0) = y0n

(3.24)

Since S(t) is a contraction semigroup, we have  ||yn (t) − y(t)|| ≤ ||y0n − y0 || +

T

||Fn (t) − F(t)||dt

0

≤ ||y0n − y0 || + T sup ||Fn (t) − F(t)||, ∀t ∈ [0, T ] t∈[0,T ]

It follows that yn −→ y in C([0, T ]; X ). As yn (t) ∈ D(A), for any t ∈ [0, T ] we have d ||yn (t)||2 = 2Re(< Ayn (t), yn (t) >) + 2Re(< Fn (t), yn (t) >) dt Since the operator A is dissipative, we get  ||yn (T )||2 − ||yn (0)||2 ≤ 2

T 0

Re(< Fn (t), yn (t) >)dt

3.2 Stabilization of Infinite Dimensional Bilinear Systems

55

Finally when n −→ +∞, we obtain  ||y(T )|| ≤ ||y0 || + 2 2

2

T

Re(< F(t), y(t) >)dt

0

and using (3.22), we deduce that ||y(T )|| ≤ ||y0 ||. So when T tends to tmax ; the solution y(t) remains bounded, and therefore it is global.  For weak stabilization, we have the following result [1]. Proposition 3.4 Suppose that the operator B is compact and that the condition { B S(t)y, S(t)y = 0} ⇒ y = 0

(3.25)

is verified, then the control (3.22) weakly stabilizes the system (3.21). Proof We set F(y) = − By, yBy, using the Proposition 3.3, the system (3.21) admits a weak and global solution. By Theorem 2.4 (see [1]), ωw (y0 ) = ∅, for all y0 ∈ X and ψ ∈ ωw (y0 ), we have

S(t)ψ, F(S(t)ψ) = 0 for all t ≥ 0 and by (3.25) we obtain ψ = 0.  For the strong stability of the system (3.21) we have the following result [8] . Theorem 3.12 If A is of compact resolvent and B self-adjoint and verify (3.25), if in addition one of the following two conditions is satisfied

By, y ≥ 0

(3.26)

By, y ≤ 0

(3.27)

or for all y ∈ X , then the control (3.22) strongly stabilizes the system (3.21). The following result (see [10]) gives conditions for the strong stabilization of the system (3.21). Theorem 3.13 If S(t) is a contraction semigroup and B is a self-adjoint operator verifying (3.26), if in addition, the condition  y ≤ 2

T

B S(t)y, S(t)ydt

(3.28)

0

is satisfied for some T > 0, then the system (3.21) is strongly stabilizable and we have the estimation 1 y(t) = O( √ ) when t → +∞ t

56

3 Stabilization of Infinite Dimensional Semilinear Systems

Proof We consider the system ⎧ dy(t) ⎪ ⎨ = Ay(t) − By(t), y(t)By(t) dt ⎪ ⎩ y(0) = 0

(3.29)

Since S(t) is a contraction semigroup we have 

1 2

T

y(t) ≤ B 

1

| By(s), y(s)|B 2 y(s)ds

0 1



T

≤ B 2 

1

B 2 y(s)3 ds

0

It gives 1 2

1 2



T

y(t) ≤ T B  ( 2

2

1

3

B 2 y(s)4 ds) 2

0

and with (3.28), there exists c > 0 such that y(0)2 ≤ 2(z(0)2 + y(0) − z(0)2 ) 

T

≤ c[(

1



3

T

B 2 y(s)4 ds) 2 + (

0

1

1

B 2 y(s)4 ds) 2 ]

0

With the same approach as for (3.20), we obtain  y(T ) + 2 2

T

By(s), y(s)2 ds ≤ y(0)2

0

and therefore we have 

T

y(T )2 ≤ c[(

1



3

T

B 2 y(s)4 ds) 2 +(

0

1

0



T

+

1

B 2 y(s)4 ds) 2

(3.30) 1 2

B y(s)4 ds]

0

Il follows that y(T )2 ≤ c f (y0 2 − y(T )2 ) 1

(3.31)

3

with f (t) = t 2 + t + t 2 . Thus 1 y(T )2 + f −1 (y(T )2 ) ≤ y(0)2 c

(3.32)

3.2 Stabilization of Infinite Dimensional Bilinear Systems

57

and moreover, we have f −1 (t) ∼ t 2 when t → 0. We set h(t) = y(t)2 and we use the same techniques on the intervals of type [kT, (k + 1)T ], k ∈ N∗ , we obtain 1 h(k(T + 1)) + f −1 (h(k(T + 1))) ≤ h(kT ) c

(3.33)

Now for k ∈ N we set sk = h(kT ). By Lemma 2.4 with (3.33) we obtain sk ≤ S(k)s0 , . For t > 0, we carry out the Euclidean division of t by T we obtain t = qT + r with q ∈ R and 0 ≤ r < T. Since the functions h(t) and S(t)ξ are decreasing, then h(t) ≤ h(qT ) ≤ S(

t t −r )h(0) ≤ S( − 1)S(0) T T

As f 1 (t + f 0 (t)) = f 0 (t), we have lim f 1 (t) = 0. t→+∞  ξ ds , t > 0 is decreasing with f 3 (ξ ) = 0 and f 3 (0+ ) = The function f 3 (t) = f 1 (s) t +∞. We deduce that f 3 ([0, +∞[) = [0, +∞[ and that the solution of system (2.38) is U (t) = f 3−1 (t), with lim U (t) = 0. t→+∞

Let 0 < ε < 1, there exists η > 0 such that if t < η we have: | f 1 (t) − ct 2 | < εct 2 . In addition there exists tε > 0 such that for all t > tε we have 0 < U (t) < η. For t ≥ tε we have − f 1 (U (t)) ≤ c(ε − 1)U (t)2 . thus dU (t) + c (ε − 1)U (t)2 ≤ 0 dt 1 1 Integrating over [tε , t] we obtain the estimation U (t) = O( ) and so h(t) = O( ). t t  In the following we present an extension of the space state decomposition method to the stabilization of bilinear system. We consider the system (3.21) and we assume that X can be decomposed as in (2.15), with dim(X u ) < +∞, Theorem 3.14 ([11]) If the semigroup Ss (t) verify (2.32) and B is a compact operator and verify the condition [ B S(t)y, S(t)y = 0] ⇒ y = 0

(3.34)

u(t) = r By(t), y(t)

(3.35)

then the control

strongly stabilizes the system (3.21).

58

3 Stabilization of Infinite Dimensional Semilinear Systems

Proof 1. Existence of a solution. The control (3.35) ensures the existence of a weak solution of the system (3.21) defined on a maximum interval [0, tmax [, y(t) given by  t y(t) = S(t)y0 + r By(s), y(s)S(t − s)By(s)ds (3.36) 0

Let the function F(y) = r By, yBy. Since S(t) is a contraction semigroup, for all y0 ∈ D(A) we have d y(t)2 ≤ −2 F(y(t)), y(t) ≤ y0 2 dt Furthermore considering (3.36) and by the fact that S(t) is a contraction semigroup and Gronwall’s lemma, the mapping y0 → y(t) is continuous. Therefore, for all y0 ∈ X , y(t) ≤ y0 , so the solution y(t) of the system (3.21) is global. 2. The system (3.21) is strongly stabilizable. Indeed, as X is reflexive and y(t) ≤ y0 , then there exists a sequence tn → +∞, such that y(tn )  y ∈ X when n → +∞. In addition for T > 0, there exists a constant c(T, y0 ) > 0 such that 

T



T

| B S(t)y0 , S(t)y0 |dt ≤ c(T, y0 )y0 

0

2 | F(y(t)), y(t)|dt

(3.37)

0

We take y0 = y(tn ) in (3.37), with dominated convergence Lebegue theorem we get

S(t)By, S(t)y rangle = 0, for all t ≥ 0, and with (3.34), it comes y = 0. Hence y(t)  0 when t → +∞. Finally since dim X u < +∞, yu (t) → 0 when t → +∞. It remains to study the behavior of ys (t). For that, with (2.32), there exists α > 0 such that, for all t ≥ 0, (3.38) Ss (t)ys  ≤ αe−δt Thus, for all 0 ≤

t ≤ t, we have ys (t) ≤ αe

−δ(t−

t)

ys (

t) + α



t

t

e−δ(t−τ ) F(y(τ ))dτ

(3.39)

Let  > 0 and

t > 0, since F is sequentially continuous then F(y(t)) <  and by (3.39) we obtain ys (t) ≤ αe

−δ(t−

t)

ys (

t) + α

≤ αe−δ(t− t) ys (

t) +

α δ



t

t

e−δ(t−τ ) dτ

3.2 Stabilization of Infinite Dimensional Bilinear Systems

59

Thus ys (t) → 0 when t → +∞. As y(t) = ys (t) + yu (t), the system (3.21) is strongly stabilizable.  Remark 3.2 We obtain the same result of the above theorem with the control ⎧ r ⎪

By(t), y(t) if y(t) = 0 ⎨ y(t)2 (3.40) u(t) = ⎪ ⎩ 0 otherwise For exponential stabilization we have the following result [11]. Theorem 3.15 If Su (t) is a C0 -semigroup of isometries and Ss (t) verify (2.32). If the condition (3.41) [ Bu Su (t)yu , Su (t)yu  = 0] ⇒ yu = 0 is verified, then the control

u(t) =

⎧ ⎪ ⎨ ⎪ ⎩

r

Byu (t), yu (t) if yu (t) = 0 yu (t)2 0

(3.42)

otherwise

exponentially stabilizes the system (3.21). Proof We set

F(yu (t)) =

then the system

⎧ ⎪ ⎨ ⎪ ⎩

r

Byu (t), yu (t) if yu (t) = 0 yu (t)2 0

(3.43)

otherwise

⎧ dyu (t) ⎪ ⎨ = Au yu (t) + F(yu (t))Bu yu (t) dt ⎪ ⎩ yu (0) = y0u

(3.44)

admits a unique and global weak solution yu (t) =

Su (t)y0u with

Su (t) is a non linear semigroup on X u . For m ∈ N, we integrate between m and m + 1 the relation d yu (t)2 ≤ −2|F(yu (t))|2 yu (t)2 dt we obtain  yu (m + 1) − yu (m) ≤ −2 2

m+1

2

m

|F(yu (τ ))|2 yu (t au)2 dτ

(3.45)

60

3 Stabilization of Infinite Dimensional Semilinear Systems

and since yu (t) is decreasing, we have  yu (m + 1) − yu (m) ≤ −2 2

m+1

2

|F(yu (τ ))|2  yu (τ ) 2 dτ

(3.46)

m

moreover  yu (m + 1) − yu (m) ≤ −2  yu (m + 1)  2

2

m+1

2

|F(yu (τ ))|2 dτ.

m

 We set η = 2 inf

y0u =1 0

1

|F(

Su (τ )y0u )|2 dτ ≥ 0.

Since Bu is linear, F(αyu ) = F(yu ), for all α ∈ C, yu ∈ X u . As the solution of the system (3.44) is unique, then for all t ≥ 0,

Su (t)yu Su (t)(αyu ) = α

Which implies that for all y0u ∈ X u such that y0u = 0 we have 

1

1 |F(

Su (τ )y0u )|2 dτ ≥ η 2

0

But 

Su (t)y0u  ≤ y0u  then Thus by (3.46) it follows



m+1

m

1 |F(

Su (τ )y0u )|2 dτ ≥ η, for all m ∈ N. 2

yu (m + 1)2 − yu (m)2 ≤ −ηyu (m + 1)2 and therefore we have yu (m) ≤ As yu (t) is decreasing we get

y0u  m. (1 + η) 2

ln(1 + η) ln(1 + η) t − 2 2 yu (t) ≤ yu (m) ≤ e e y0u  We  1 have η > 0, otherwise η = 0 then there exists yu1 ∈ X u with yu  = 1 and |F(

Su (τ )yu )|2 dτ = 0 since the mapping y0u → |F(

Su (τ )y0u )|2 dτ is con0

0

tinuous and dim X u < +∞. Su (t)yu = Su (t)yu ,for all 0 ≤ Thus F(

Su (t)yu ) = 0, for all 0 ≤ t ≤ 1. therefore

t ≤ 1. Thus < Bu Su (t)yu , Su (t)yu  = 0 for all 0 ≤ t ≤ 1. We deduce that (see [12]), for all yu ∈ X u ,

3.2 Stabilization of Infinite Dimensional Bilinear Systems



1

61

| < Bu Su (t)yu , Su (t)yu > |dt ≥ ηyu 2

0

Consequently yu = 0, contradiction. We conclude that yu (t) → 0 exponentially when t → +∞. Let us show that ys (t) → 0 exponentially when t → +∞. First we show that ys (t) is a weak, unique and global solution. The control (3.42) ensures the existence of a weak and unique solution of the system (3.21) over a maximum interval [0, tmax [, given by  ys (t) = Ss (t)y0s +

t

u(τ )Ss (t − τ )Bs ys (τ )dτ for all t ∈ [0, tmax [

(3.47)

0

By (3.38) and (3.47), for all t ∈ [0, tmax [, there exists β such that ys (t) ≤ β e

−δt



t

y0s  + αB

e−δ(t−τ ) |u(τ )|ys (τ )dτ

0



and thus z(t) ≤ βy0s  + rβB

2

t

z(τ )dτ

0

with z(t) = ys (t) eδt . 2 By Gronwall’s lemma we get z(t) ≤ βy0s er αB t . This means that for all t ≤ tmax , 2 (3.48) ys (t) ≤ βy0s e(rβB −δ)t For r such that rβB2 < δ, ys (t) is bounded therefore ys (t) is a global solution of the system (3.21). In addition, the estimate (3.48) is verified for all t ≥ 0, which allows to conclude. 

3.3 Stabilization of Infinite Dimensional Semilinear Systems This section concerns existence and uniqueness of the solution of semi linear systems and stabilization results.

3.3.1 Existence and Uniqueness Results Let us consider the system

62

3 Stabilization of Infinite Dimensional Semilinear Systems

⎧ ⎨ dy(t) = Ay(t) + f (y(t), t) dt ⎩ y(0) = y ∈ X 0

(3.49)

where A : D(A) ⊂ X −→ X is a linear operator that generates a C0 −semigroup S(t), t ≥ 0 X, and f : X × R+ −→ X a given function. Definition 3.4 1. A function y : [0, t1 ] −→ X is said a strong solution of (3.49) on [0, t1 ] if the function y(.) is of class C 1 on ]0, t1 [ and verify (3.49). 2. A function y ∈ C([0, t1 ]; X ) is said a weak solution to (3.49) on [0, t1 ] if f (y(.), .) ∈ L 1 (0, t1 ; X ), for all ϕ ∈ D(A∗ ), the function < y(t), ϕ > is absolutely continuous on [0, t1 ] and verify d < y(t), ϕ >=< y(t), A∗ ϕ > + < f (y(t), t), ϕ > p. p on [0, t1 ] dt The following result gives a characterization of the weak solution to the system (3.49). Theorem 3.16 ([6]) A function y : [0, t1 ] −→ X is a weak solution of system (3.49) on [0, t1 ] if and only if f (y(.), .) ∈ L 1 (0, t1 ; X ) and y verify the relationship (called formula of the constant variation )  y(t) = S(t)y0 +

t

S(t − s) f (y(s), s)ds p. p on [0, t1 ]

0

The following result gives sufficient conditions for the existence and uniqueness of a weak solution for system (3.49) as well as its dependence on the initial state. Theorem 3.17 ([6]) Suppose that f is continuous on [0, t1 ], and uniformly Lipschitz on X. For each y0 ∈ X, the system (3.49) has a weak solution y ∈ C([0, t1 ]; X ). In addition, the correspondence y0 −→ y(.) is continuous from X to C([0, t1 ]; X ). Previous assumptions may be relaxed and lead to local existence results. Theorem 3.18 ([1, 6]) Suppose that f is continuous with respect to t on IR + , uniformly continuous over all bounded intervals and locally Lipschitz with respect to y. Then for all y0 ∈ X, the system (3.49) has a weak solution y(.) on [0, tmax [, and if y(t) is bounded with respect to t then this solution is global (tmax = +∞). If in addition, f does not depend on t and monotonous, then for t1 ∈]0, tmax [ the mapping y0 −→ y(.) is continuous from X into C([0, t1 ]; X ). The above result gives information about the continuity of the application y0 −→ y(.) for the topology defined by the norm of X. The following theorem gives similar results for the weak topology of X.

3.3 Stabilization of Infinite Dimensional Semilinear Systems

63

Theorem 3.19 ([1]) Suppose that f is sequentially weakly continuous, (yn  y =⇒ f (yn )  f (y)) and that (3.49) has a weak solution y(.) on [0, T ]. If there exist M0 , M1 > 0; ||y0 || ≤ M0 =⇒ ||y(t)|| ≤ M1 , ∀t ∈ [0, T ], then for all t ∈ [0, T ]; the mapping y0 −→ y(t) is sequentially weakly continuous. The result that follows gives a sufficient conditions which allows regularity of the weak solution of the system (3.49). Theorem 3.20 ([6]) Suppose that f is continuously differentiable in t on IR + , uniformly over each bounded intervals and locally Lipschitz in x, then for each y0 ∈ D(A) the weak solution y(.) of the system (3.49) is a strong one. For stabilization problems, we need the global existence results that we give in the the following result Theorem 3.21 ([6]) Suppose that A generates a compact C0 −semigroup and that f is continuous and that the image of a bounded set of IR + × X be a bounded one of X . Then for each y0 ∈ X, the system (3.49) has a global solution y(.) in one of the following cases: 1. There exists a function k : IR + × X −→]0, +∞[, such that ||y(t)|| ≤ k(t), ∀t ∈ [0, t1 [. 2. There exist two functions locally integrable k1 , k2 : IR + × X −→]0, +∞[, such || f (t, y)|| ≤ k1 (t)||y|| + k2 (t), for all t ∈ [0, t1 [, y ∈ X. Let us consider the nonlinear autonomous evolution system

dy(t) dt

= Ay(t) + f (y(t)) y(0) = y0 ∈ D(A)

(3.50)

Definition 3.5 y1 ∈ D(A) is an equilibrium state if Ay1 + f (y1 ) = 0 To give an interpretation to the previous definition, we evoke the evolution of the temperature y(t) of a solid, which intuitively becomes stationary when t tends to dy infinity or again = 0. We are interested in the stability of a given equilibrium dt state y1 . Definition 3.6 1. An equilibrium state y1 is stable in the Lyapunov sense if ∀ε > 0, there exists η > 0 such that ||y0 − y1 || < η =⇒ ||y(t) − y1 || < ε, ∀t > 0. 2. An equilibrium state y1 is exponentially (resp. strongly, weakly) stable for the system (3.50) if it is stable in the Lyapunov sense and if there exists R > 0, such that ||y0 − y1 || < R implies that ||y(t) − y1 || tends to 0 exponentially (resp. strongly, weakly) when t −→ ∞. If this convergence holds for any initial state, the stability is said to be global.

64

3 Stabilization of Infinite Dimensional Semilinear Systems

Remark 3.3 1. An equilibrium state is stable in the Lyapunov sense if a weak disturbance in the initial conditions leads to a slight disturbance of the later trajectory. 2. The second definition consists in requiring in addition that the system returns asymptotically to the equilibrium state. The stability, as it was defined, is characterized in a different way according to the description given on the system. However, Lyapunov’s method remains the general approach allowing the study of the stability of any system without having to determine analytically (spectral approach) or numerically the trajectories. The original idea of this method comes from mechanic where we involve the concept of total energy to analyze the stability of a system. The generalization of this idea consists in constructing a so-called Lyapunov scalar function verifying conditions related to the decreasing of the system energy. Definition 3.7 ([4] (Lyapunov function)) A function V : X −→ IR continuously differentiable is called Lyapunov function if: 1. V (y) ≥ 0, ∀y ∈ X 2. < Ay + f (y), V  (y) >≤ 0, for all y ∈ D(A) where V  indicates the derivative of V. If in addition lim V (y) = ∞, then V is said proper. ||y||−→∞

Condition 2. imply that V (y) is decreasing along any trajectory of system (3.50). The problem of stabilizing infinite dimensional semi linear system was solved in the case of a contraction semigroup ([1]), and the used method is based on nonlinear semigroup tools.

3.3.2 Stabilization Results In this paragraph, we recall some results concerning the question of stabilization of the following homogeneous semi linear systems: ⎧ ⎨ dy(t) = Ay(t) + v(t)By(t) (3.51) dt ⎩ y(0) = y ∈ X 0

where A : D(A) ⊂ X −→ X is a Linear operator that generates a C0 − semigroup S(t), t ≥ 0 in X, B is an a nonlinear operator from X to X, and v : [0, +∞[ −→ C is a real valued control. Definition 3.8 The system (3.51) is said to be exponentially (resp. strongly, weakly) stabilizable if there exists a feedback control v(t) such that – For any initial state y0 , the system (3.51) has only one weak and global solution y(t). – {0} is a stable equilibrium state for (3.51).

3.3 Stabilization of Infinite Dimensional Semilinear Systems

65

– For all y0 ∈ X, y(t) −→ 0 exponentially (resp. strongly, weakly) when t −→ +∞. Note that the stabilization defined above is often called global stability and that we can consider any equilibrium state in place of 0. The natural approach for the stabilization uses the derivative (formally) of ||y(t)||2 along the trajectory of system (3.51), which provides d ||y(t)||2 = 2 < Ay(t), y(t) > +2v(t) < y(t), By(t) > dt It is therefore natural to consider the control v(t) = − < y(t), By(t) >, since this choice makes possible to obtain energy dissipation inequality d ||y(t)||2 ≤ −2 < y(t), By(t) >2 dt So, we are going to be interested in controls of type v(t) = − < y(t), K y(t) >

(3.52)

where K : X −→ X is a continuous operator. The following result gives a sufficient condition ensuring the existence of a global solution for system (3.51) excited by (3.52). This result was proved for K = B, a linear operator ([13]). Proposition 3.5 Suppose that S(t) is a contraction semigroup. If Re(< y, K y >< By, y >) ≥ 0, ∀y ∈ X

(3.53)

then the solution y(t) of the system (3.51) is defined for all t ≥ 0. Furthermore y(t) depends continuously on y0 . Proof We know from the previous existence and uniqueness results that the system (3.51) has only one weak solution y ∈ C([0, tmax [; X ) defined on a maximum interval [0, tmax [, and that the correspondence y0 −→ y(t) is continuous in C([0, tmax [; X ), since the application y −→ − < K y, y > By is monotone. We show that y(t) is bounded on [0, tmax [ to conclude that tmax = +∞. Let T ∈ [0, tmax ] and consider the function F(t) = − < K y(t), y(t) > By(t) for t ∈ [0, T ], and let Fn ∈ C 1 ([0, T ]; X ) such that Fn −→ F in C([0, T ]; X ). Let y0n −→ y0 with y0n ∈ D(A) and consider the following system ⎧ ⎨ dyn (t) = Ayn (t) + Fn (t) dt ⎩ y (0) = y n 0n The semigroup S(t) is a contraction, thus

(3.54)

66

3 Stabilization of Infinite Dimensional Semilinear Systems

 ||yn (t) − y(t)|| ≤ ||y0n − y0 || +

T

||Fn (t) − F(t)||dt

0

≤ ||y0n − y0 || + T sup ||Fn (t) − F(t)|| t∈[0,T ]

It follows that yn −→ y in C([0, T ]; X ). As yn ∈ D(A), for all t ∈ [0, T ] we have d ||yn (t)||2 = 2Re(< Ayn (t), yn (t) >) + 2Re(< Fn (t), yn (t) >) dt As the operator A is dissipative, gives 

T

||yn (T )|| − ||yn (0)|| ≤ 2 2

2

Re(< Fn (t), yn (t) >)dt

0

 When n −→ +∞, we obtain ||y(T )||2 ≤ ||y0 ||2 + 2

T

Re(< F(t), y(t) >)dt,

0

and using (3.53), we deduce that ||y(T )|| ≤ ||y0 ||, and when T tends to tmax ; the solution y(t) remains bounded, therefore it is global.  Ball and Slemrod [4] showed that it is possible to stabilize a large class of semi linear systems using the following feedback control v(t) = − < By(t), y(t) >

(3.55)

Theorem 3.22 Suppose that S(t) is a contraction semigroup and that B : X w −→ X is sequentially continuous (yn  y implies Byn −→ By), and that < B S(t)ψ, S(t)ψ > = 0, ∀t ≥ 0 =⇒ ψ = 0

(3.56)

then the control (3.55) weakly stabilizes system(3.51) Proof The existence and uniqueness of a global solution of the system (3.51) are assured by the previous proposition taking K = B. We consider the function F(t) = − < y(t), By(t) > By(t). where y ∈ X . With the same techniques used in the proof of previous proposition we have  t Re < F(t), y(t) > dt ≤ ||y0 ||2 (3.57) ||y(t)||2 ≤ ||y0 ||2 + 2 0

As the operator A is dissipative, we get ||y(t)|| ≤ ||y0 ||. The semigroup S(t) is sequentially continuous, and the second point of Theorem 3.1, imply that ωw (y0 ) is nonempty.

3.3 Stabilization of Infinite Dimensional Semilinear Systems

67

Let ψ ∈ ωw (y0 ), and a seauence (tn ) −→ ∞ such that S(tn )y0  ψ when n −→ ∞.  +∞ Using (3.57), we have Re(< F(t), y(t) >)dt < ∞, and by Cauchy crite0

ria, we deduce 

tn +t

lim

n−→∞ t n

< F(S(s)z 0 ), S(s)z 0 > ds = 0, t ∈ R+

By semigroup property we obtain  t lim < F(S(s)S(tn )y0 ), S(s)S(tn )y0 > ds = 0 n→∞ 0

Moreover, if yn  y, then (yn ) is bounded, and as B is sequentially continuous from X w to X, we have F(yn ) −→ F(y). Using (3.57), we deduce that the mapping y0 −→ y(t) is sequentially weakly continuous. consequently, for Therefore, for all s ∈ [0, t[, we have  lim

n→∞

=

t

0 t

< F(S(s)S(tn )y0 ), S(s)S(tn )y0 > ds < F(S(s)ψ), S(s)ψ > ds

0

The dominated convergence theorem allows 

t

< F(S(s)ψ), S(s)ψ > ds = 0

0

Since F is continuous, we have < F(S(t)ψ), S(t)ψ >= 0, for all t ∈ R+ , more again < B S(t)ψ, S(t)ψ >= 0, for all t ≥ 0 and (3.56) allows to conclude.  The following result gives a sufficient condition for strong stabilization of system (3.51) and allows an estimation of the stabilisation error (see [11]). Theorem 3.23 If B is locally lipschitz and (3.28) is verified, then the control (3.55) strongly stabilizes the system (3.51) and, for t large enough, we have the estimation 1 y(t)2 = O( ) t

(3.58)

Proof According to Proposition 3.3 system (3.51)we has a weak solution y(t) defined on a maximal interval [0, tmax [, and if this solution is bounded then it is global. With the same techniques using to prove (3.57) we have, for all t ∈ [0, tmax [,

68

3 Stabilization of Infinite Dimensional Semilinear Systems



t

y(t)2 ≤ y0 2 + 2

| By(s), y(s)|2 ds

(3.59)

0

therefore y(t) ≤ y0 , and we conclude that the solution y(t) is global. We set z(t) = y(t) − S(t)y0 and with the constant variation formula we obtain 

t

z(t) = −

By(s), y(s)S(t − s)By(s)ds

0

then there exists C1 > 0 that depends on B, T and y0  such that 

T

z(t) ≤ C1 2

| By(s), y(s)|2 ds, for all t ≤ T.

(3.60)

0

Using the triangular inequality gives: | B S(t)y0 , S(t)y0 | ≤ | B S(t)y0 − By(t), y(t) − z(t)| +| By(t), y(t)| + | By(t) − Bz(t), z(t)| +| Bz(t), z(t)| As B is locally Lipschitz and y(t) ≤ y0 , there exists β > 0 such that | B S(t)y0 , S(t)y0 | ≤ βS(t)y0 − y(t)y(t) − z(t) + | By(t), y(t)| + βy(t) − z(t)z(t) + βz(t)2 ≤ 2βz(t)y0  + | By(t), y(t)| + βz(t)2 and by (3.60), for all t ∈ [0, T ], we have the inequality | B S(t)y0 , S(t)y0 | ≤ 2βC1 C2 y0  + | By(t), y(t)| + βC12 α 2  with C2 =

T

(3.61)

| By(s), y(s)|2 ds depends on B, T and y0 .

0

Integrating (3.61) between 0 and T and using (3.28) and (3.60) we have  y(T )2 ≤ C2 ( C1 + C1 )

(3.62)

By (3.59) and (3.60) we obtain 2C2 ≤ y0 2 − y(T )2 , which gives the estimation y(T )2 ≤

1 C3 f 0 (y0 2 − y(T )2 ) 2

(3.63)

√ with f 0 (t) = t + t, for all t ≥ 0, and if f 1 (.) indicates the inverse of 21 C3 f 0 (.), then (3.63) becomes

3.3 Stabilization of Infinite Dimensional Semilinear Systems

69

f 1 (y(T )2 ) + y(T )2 ≤ y0 2 Instead of the interval [0, T ] we consider the intervals [kT, (k + 1)T ] with k ∈ N∗ to have the inequality f 1 (y((k + 1)T )2 ) + y((k + 1)T )2 ≤ y(kT )2 , for all k ∈ N∗

(3.64)

Following the same manner to prove Theorem 3.13, we deduce the desired estimate.  Remark 3.4 The above result is an improvement of the one of Theorem 3.13. Now we consider the system ⎧ dy(t) ⎪ ⎨ = Ay(t) + v(t) f (y(t)) dt ⎪ ⎩ y(0) = y0 ∈ X

(3.65)

where A : D(A) ⊂ X −→ X is a linear operator that generates a compact C0 semigroup S(t), t ≥ 0 in X and f : X → X . Let note B(0, a) the ball of center 0 and of radius a > 0, we have the result [8]. Theorem 3.24 If f is a function locally Lipschitz and r (.) : X → R+ is a function of class C p , p ≥ 1 verifying ∀a > 0 , ∃ra > 0 , y ∈ B(0, a) : r (y) ≥ ra If (S(t))t≥0 is compact and if {y ∈ H | f (S(t)y), S(t)y = 0} = {0} then the control v(t) = −r (y) < f (y), y > strongly stabilises the system (3.65). Proof If T (t) is a non linear semigroup of generator A + v(t) f (.), the system (3.65) has a unique weak and global solution y(t) = T (t)y0 . Let the function f (y) = 21  y 2 , as A est dissipative, then f (y) is a Lyapunov function f (T (t)y0 ) − f (T (s)y0 ) ≥ 0 for all (t, s), 0 ≤ t ≤ s Indeed, if y0 ∈ D(A), the solution y(t) = T (t)y0 of system (3.65) is absolutely continuous and y ∈ D(A). Deriving f (t) along the trajectories y(t) and since Ay, y ≤ 0 for all y ∈ D(A), we obtain

70

3 Stabilization of Infinite Dimensional Semilinear Systems

d f (T (t)y0 ) = AT (t)y0 + v(t) f (T (t)y0 ), T (t)y0  dt ≤ −r (T (t)y0 ) f (T (t)y0 ), T (t)y0 2 Therefore for all 0 ≤ t ≤ s, we have 

s

f (T (s)y0 ) − f (T (t)y0 ) ≤ −

r (T (τ )y0 ) T (τ )y0 , T (τ )y0 dτ

t

If y0 ∈ / D(A), the the above inequality is verified as the density of D(A) in X . It follows that (T (t)y0 )t≥0 is bounded and the system is stable in the Lyapunov sense, moreover we have  ∞ r (T (τ )y0 ) f (T (τ )y0 ), T (τ )y0 2 dτ < ∞ (3.66) 0

 For the proof we need the following Lemma [8]. Lemma 3.3 Under the hypothesis of Theorem 3.24, for all y0 ∈ X we have 1. The set ωw (y0 ) corresponding to the orbit (T (t)y0 )t≥0 is nonempty. 2. For all y ∗ ∈ ω(y0 ) and every sequence (tn ) of R+ , with lim tn = +∞, we have n→+∞

lim T (tn )y0 = y ∗ ⇒

n→+∞

lim T (tn + t)y0 = T (t)y ∗ , for all t ≥ 0.

n→+∞

In particular ω(y0 ) is positively invariant by T (t). Proof • We begin with the proof of the lemma. 1. We know that the system (3.65) has a bounded solution, then the set of positive orbit O + (y0 ) of system (3.65) is pre-compact [14] and the Theorem 3.1 imply that ω(y0 ) is a nonempty and invariant by T (t) in X . 2. Let y ∗ ∈ (y0 ) and a sequence (tn )n≥0 of R+ , such that lim tn = +∞, and n→+∞

lim T (tn )(y0 ) = y ∗ .

n→+∞

By the the constant variation formula we have  T (tn + t)y0 = T (t)T (tn )y0 +

t

T (t − s)vtn +s f (T (tn + s)y0 )ds

0

= T (t)T (tn )y0 

tn +t

− tn

T (t + tn − s)r (T (s)y0 ) f (T (s)y0 ), T (s)y0  f (T (s)y0 )ds

3.3 Stabilization of Infinite Dimensional Semilinear Systems

71

Applying Schwarz inequality gives 

tn +t

 T (tn + t)y0 − T (t)T (tn )y0  ≤ 2

r (T (s)y0 )  f (T (s)y0 )  ds 2

tn



tn +t

×

r (T (s)y0 ) f (T (s)y0 ), T (s)y0  ds 2

tn

therefore  T (tn + t)y0 − T (t)T (tn )y0 2 

tn +t

≤ ct

(3.67) r (T (s)(y0 )) f (T (s)(y0 )), T (s)(y0 )2 ds

tn

where c =

sup

y∈ f (0,y0 )

r (y)  f (y) 2 .

But as  S(t)y0 ≤ y0 , thus c < +∞. By formula (3.66) and (3.67) we obtain lim  T (tn + t)y0 − T (t)T (tn )y0 2 = 0

n→+∞

Finally lim T (tn )y0 = y ∗ , and (T (t))t≥0 is a C0 -semigroup uniformly bounded, n→+∞

and the triangular inequality gives: lim T (tn + t)y0 = T (t)y ∗ , for allt > 0

n→+∞

• Now we finish the proof of Theorem 3.24. Let y0 ∈ X , according to Theorem 3.1, we know that ω(y0 ) is nonempty. It remains to prove that ω(y0 ) = {0}. Let y0 ∈ X , y ∗ ∈ (y0 ), we consider y ∗ (t) = T (t)y ∗ the solution of system (3.65) that corresponds to the initial condition y ∗ , and the sequence (tn )n≥0 of R+ , such that lim (tn ) = +∞, we have lim S(tn )(y0 ) = y ∗ . n→+∞

n→+∞

By Theorem 3.1 we know that lim S(tn + s)y0 = S(t)y ∗ , for all s ≥ 0. n→+∞

By the continuity of y → r (y) f (y), y2 , we obtain lim r (S(tn + t)(y0 )) f (S(tn + t)(y0 )), S(tn + t)(y0 )2

n→+∞

= r (S(t)y ∗ ) f (S(t)y ∗ ), S(t)y ∗ 2 According to dominated convergence Lebesgue theorem, we deduce that

72

3 Stabilization of Infinite Dimensional Semilinear Systems



t

r (T (s)y ∗ ) f (T (s)y ∗ ), T (s)y ∗ 2 ds

0

 = lim

n→+∞ 0

 = lim

n→+∞ t n

t

r (T (tn + s)y0 ) f (S(tn + s)y0 ), T (tn + s)y0 2 ds

tn +t

r (T (s)(y0 )) f (T (s)y0 ), T (s)(y0 )2 ds = 0

As r (T (s)y ∗ ) < f (T (s)y ∗ ), T (s)y ∗  is a function continuous and positive then

f (T (s)y ∗ ), T (s)y ∗  = 0 for all s > 0. But {y ∈ X such that f (T (s)y), T (s)y = 0, s > 0} = {0} thus y ∗ = 0, and therefore ω(y0 ) = {0}.



References 1. Ball, J.M., Slemrod, M.: Feedback stabilization of distributed semilinear control systems. Appl. Math. Optim. 5, 169–179 (1979) 2. Dafermos, C.M.: Uniform processes and semicontinuous Liapunov functionals. J. Differ. Equ. 11, 401–415 (1972) 3. Konishi, C.M.: Sur la compacité des semi-groupes non linéaires dans les espaces de Hilbert. Proc. Japan Acad. 48, 278–280 (1972) 4. Ball, J.M.: On the asymptotic behavior of generalized processes with application to nonlinear evolutions equations. J. Differ. Equ. 27, 224–265 (1978) 5. Lasalle, J.P.: Stability theory for ordinary differential equations. J. Differ. Equ. 4, 307–324 (1968) 6. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer Verlag, New York (1983) 7. Bounit, H.: Comments in the feedback stabilization of bilinear control systems. Appl. Math. Lett. 16, 847 851 (2003) 8. Bounit, H., Hammouri, H.: Feedback stabilisation for class of distributed semilinear systems. Nonlinear Anal. Theory Methods Appl. 37, 953–969 (1999) 9. Bounit, H., Hammouri, H.: Bounded feedback stabilisation and globale separation principle of distributed systems. IEEE. Trans. Automat. Contr. 42 (1997) 10. Berrahmoune, L.: Asymptotic stabilization and decay estimate for distributed bilinear systems, Recerche di Matematica. fasc. 1, 89–103 (2001) 11. Ouzahra, M.: Exponential stabilization of distributed semilinear systems by optimal control. J. Math. Anal. Appl. 380, 117–123 (2011) 12. Quinn, J.P.: Stabilization of bilinear systems by quadratic feedback control. J. Math. Anal. Appl. 75, 66–80 (1980) 13. Berrahmoune, L., El boukfaoui, Y., Erraoui, M.: Remarks on the feedback stabilization of system affine in control. Eur. J. Control 7, 17–28 (2001) 14. Pazy, A.: A class of semilinear equations of evolution Isreal. J. Math. 20, 23–36 (1975)

Chapter 4

Regional Stabilization of Infinite Dimensional Linear Systems

This chapter, we concerns the concept of regional stability for infinite dimensional linear systems evolving on a spacial domain  ⊂ Rn : it is about studying the asymptotic behavior of the system not on its whole evolution domain but on an internal or boundary subregion ω ⊂ . In general, any system is considered to be stable or unstable. However, there exist systems that are unstable on their whole geometrical domains , but do not behave the same way on any subregion ω of . In addition, we can need to stabilize a system or improve its stability degree only on a subregion of its evolution domain, which will require a lower cost. Furthermore, when we have to stabilize the system on the whole domain, we will need to know its state on , while the regional stabilization requests the state only on the subregion of interest ω of  .

4.1 Internal Regional Stability In this section, we give the definitions and characterizations concerning the notion of regional stability. Next, we characterize a control of minimum cost that assures the stabilization on a target region, ω ,with a bounded state on the residual part  \ ω. The obtained results are illustrated by some examples.

4.1.1 Definitions Let  be a regular domain of Rn , n ≥ 1, and let Q = ×]0, ∞[.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. H. Zerrik and O. Castillo, Stabilization of Infinite Dimensional Systems, Studies in Systems, Decision and Control 355, https://doi.org/10.1007/978-3-030-68600-0_4

73

74

4 Regional Stabilization of Infinite Dimensional Linear Systems

We consider a system described by the following state equation  dy(t)

= Ay(t) dt y(., 0) = y0 ∈ X

Q 

(4.1)

where A an operator of domaine D(A) generates a C0 −semigroup S(t), t ≥ 0 on a functional state space X = L 2 (), endowed with the inner product , and the corresponding norm ||.||. Now lets consider an open internal region ω ⊂  of positive Lebesgue measure, and let χω be the operator restriction to ω. We note the operator i ω = χω∗ χω . Definition 4.1 The system (4.1) is said (regionally) 1. weakly stable (r.w.s.) on ω, if for any initial state y0 ∈ X < χω S(t)y0 , z > L 2 (ω) −→ 0 when t −→ ∞, ∀z ∈ X. 2. strongly stable (r.s.s) on ω, if for any initial state y0 ∈ X, we have ||χω S(t)y0 || L 2 (ω) −→ 0 when t −→ ∞. 3. exponentially stable (r.e.s.) on ω, if ∀ y0 ∈ X, ||χω S(t)y0 || L 2 (ω) exponentially tends to 0 when t −→ ∞. Remark 4.1 1. This concept is general because for ω =  we obtain the usual definitions of stability. 2. The above definitions show that we are interested by the asymptotic behavior of the system (4.1) only on ω. 3. Note that r.e.s. =⇒ r.s.s. =⇒ r.f.s. The converse is not true as shown by the following examples:

4.1.2 Examples Example 4.1 1. Let us consider the system defined on  =]0, 2[ by  dy(t)

= (x − 1)y on Q = ×]0, ∞[ dt y(., 0) = y0 (x) on 

(4.2)

It is clear that this system is unstable on , but exponentially stable on any subregion ω =]0, a[⊂]0, 1[. Indeed we have ||χω S(t)y0 || L 2 (ω) ≤ et (a−1) ||y0 ||, ∀ y0 ∈ L 2 (). 2. On  =]0, ∞[ Let us consider the system described by the equation

4.1 Internal Regional Stability

75

⎧ ∂ y(t) ∂ y(t) ⎪ ⎨ + = 0 on Q ∂t ∂x ⎪ ⎩ y(., 0) = y0 on 

(4.3)

∂ of domain D(A) = {y ∈ H 1 (); y(0) = 0} generates a ∂x C0 −semigroup S(t), t ≥ 0 given by The operator A = −

 (S(t)y0 )(x) =

y0 (x − t), if x ≥ t 0, if x < t

For a > 0 and ω =]a, +∞[, (a > 0) we have ||χω S(t)y0 || = ||χω y0 ||, ∀t ≥ 0, thus the system (4.3) is not strongly stable on ω. But it is weak stable on ω, indeed when t → ∞  +∞  +∞ y(x)2 d x z(x)2 d x → 0 | < χω S(t)y, z > |2 ≤ sup (t,a)

0

3. Let us consider the system defined on  =]0, 1[ by ⎧ dy(t) ⎪ ⎨ = Ay(t) on Q dt ⎪ ⎩ y(x, 0) = y0 ∈ L 2 ()

(4.4)

with Ay = λ1 < y, 1 > 1 +



λn,m < y, h (n,m) > h (n,m) , y ∈ L 2 (),

(4.5)

(n,m)∈

where λ1 = −1,  = {(n, m) ∈ IN 2 ; m < 2n } and (λn,m )(n,m)∈  are given by λ1,0 1 = 1, and λn,m = − n 2 +m 2 if (n, m)  = (1, 0), h (n,m) are the Haar functions defined for (n, m) ∈  and x ∈  by

h (n,m) (x) =

⎧ n ⎪ ⎪ 22 , ⎪ ⎪ ⎪ ⎨

if x ∈ [

m m + 21 , [ 2n 2n

m + 21 m + 1 ⎪ n ⎪ ⎪ 2 , if x ∈ [ −2 , [ ⎪ ⎪ 2n 2n ⎩ 0, otherwise

The family {1} ∪ {h (n,m) }(n,m)∈  forms an orthonormal basis of L 2 ().

(4.6)

76

4 Regional Stabilization of Infinite Dimensional Linear Systems

The system (4.4) is strongly stable on ω =] 21 , 1[. Indeed ||S(t)y0 ||2 = e−t y, 1111 + eλn,m t y, h (n,m) h (n,m) → 0, y ∈ L 2 (). (n,m)∈

Now for all y0 ∈ L 2 () we have ||S(t)z 0 || ≤ 1 and ||S(t)h (n,m) || → 1, when n, m → ∞ so ||S(t)|| = 1. This shows that the system (4.4) is not exponentially stable on ω. 4. Let us consider the system defined on  =]0, +∞[ by ⎧ ∂ y(t) ∂ y(t) ⎪ ⎨ − = 0 on Q ∂t ∂x ⎪ ⎩ y(., 0) = y0 ∈ H 1 ()

(4.7)

∂ of domain D(A) = {y ∈ H 1 (); y(0) = 0} generates a ∂x C0 −semigroup S(t), t ≥ 0 on H 1 () given by (S(t)y0 )(x) = y0 (x + t).  b+t For ω =]a, b[, and y ∈ L 2 (), ||χω S(t)y||2 = y(x)2 d x → 0, when t → The operator A =

a+t

∞. On the other hand we have ||χω S(t)|| = 1, ∀t ≥ 0, therefore the system (4.7) is not exponentially stable on ω. 5. The following example is about an unstable system on the whole domain  but stable on certain regions ω of the evolution domain . Let us consider a system of a spacial domain  =]0, 2[ and described by the equation ⎧ ∂ y(t) ⎪ ⎨ = (x − 1)y on Q ∂t (4.8) ⎪ ⎩ 2 y(., 0) = y0 ∈ L () It is clear that the system (4.8) is not stable on , while for a region ω =]0, a[⊂]0.1[, we have ||χω S(t)y0 || L 2 (ω) ≤ et (a−1) ||y0 ||, f or all y0 ∈ L 2 () So the system (4.8) is stable on ω.

4.1.3 Characterization of Regional Stability In this paragraph we give results that characterize the regional stability. We consider the sets σω1 (A) = {λ ∈ σ (A) | Re(λ) ≥ 0, K er (A − λI ) ⊂ K er (i ω )}

4.1 Internal Regional Stability

77

and σω2 (A) = {λ ∈ σ (A) | Re(λ) < 0, K er (A − λI ) ⊂ K er (i ω )} Proposition 4.1 1. If the system (4.1) is weakly stable on ω then σω1 (A) = ∅. 2. Let us suppose that the state space X has an orthonormal basis (ϕn )n of eigenfunctions of A. If σω1 (A) = ∅ and if there exists α > 0 such that, for all λ ∈ σω2 (A), Re(λ) ≤ −α, then the system (4.1) is exponentially stable on ω. Proof 1. Suppose there exist λ ∈ σ (A) and ϕ ∈ N (A − λI ) such that Re(λ) ≥ 0 and χω ϕ = 0. For y0 = ϕ, the solution of (4.1) is S(t)ϕ = etλ ϕ. So < χω S(t)ϕ, ϕ > L 2 (ω) ≥ ||χω ϕ|| L 2 (ω) = 0 and (4.1) is not strongly stable on ω. 2. Without loss of generality, we can assume that the eigenvalues of A are simple. In this case, for all y0 ∈ X, we have χω S(t)y0 =

+∞

eλn t < y0 , ϕn > χω ϕn

n=1

= χω



eλn t < y0 , ϕn > ϕn

λn ∈σω2 (A)

Now if there exists α > 0, such that Re(λ) ≤ −α, for all λ ∈ σω2 (A), then ||χω S(t)y0 || L 2 (ω) ≤ |||χω |||e−αt ||y0 || and therefore the system (4.1) is exponentially stable on ω.



Example 4.2 Lets consider the system (4.4) with ω =] 21 , 1[. We have σω1 (A) = ∅ and σω2 (A) = {−1} hence system (4.4) is exponentially stability on ω. Remark 4.2 For ω = , by the first point the stability is ensured if the dynamic operator of the system has no eigenvalues with positive real parts. In the global case (ω = ), the exponential stability is equivalent to the upper bound  ∞ ||S(t)y||2 dt < +∞ for all y ∈ X. (4.9) 0

Now, is the exponential stability over a region ω ⊂  equivalent to the regional upper bound  ∞ ||χω S(t)y||2 dt < for all y ∈ L 2 ()

0

We always have the result: exponential stability on ω implies

78

4 Regional Stabilization of Infinite Dimensional Linear Systems



∞

||χω S(t)y||2 dt < +∞ for all y ∈ L 2 ().

0

The converse is not always true as shown in the following example. Example 4.3 Let’s reconsider system (4.7), as we’ve seen the semigroup S(t), t ≥ 0 is given by (S(t)y0 )(x) = y0 (x + t). And for ω =]a, b[, a < b, we don’t have exponential stability on ω. However, for all y ∈ L 2 (), we have 



+∞

||χω S(t)y||2 dt =

0

b



a

+∞

y(t)2 dtd x < ∞.

x

Before characterizing regional stability, we introduce the following result. Proposition 4.2 The regional upper bound 

∞

||χω S(t)y||2 dt < +∞,

for all y ∈ L 2 (ω)

0

is true if and only if the Lyapunov equation < Ay, P y > + < P y, Ay > + < i ω y, y > = 0

(4.10)

admits a solution P, a self adjoint and positive operator.  ∞ ||χω S(t)y||2L 2 (ω) dt < ∞, for all y ∈ X , then the operator Proof Suppose that 0



∞

P(y) =

(χω S(t))∗ (χω S(t))y dt

(4.11)

0

is well defined and solution of (4.10). Conversely if (4.10) admits a solution, then 

∞

0

||χω S(t)y||2L 2 (ω) dt < ∞ f or ally ∈ X.

Proposition 4.3 Suppose that A generates a semigroup S(t), t ≥ 0 verifying ||χω S(t + s)y|| L 2 (ω) ≤ |||χω S(t)|||.||χω S(s)y|| L 2 (ω) for all t, s ≥ 0, y ∈ X. (4.12) then the system (4.1) is exponentially stable on ω, if and only if 

∞ 0

||χω S(t)y||2L 2 (ω) dt < ∞ for all y ∈ X.

4.1 Internal Regional Stability

79

Proof If the system (4.1) is exponentially stable on ω, then 

∞ 0

||χω S(t)y||2L 2 (ω) dt < ∞

for all y ∈ X . The operator 

∞

P(y) =

(χω S(t))∗ (χω S(t))y dt

0

is well defined and verify (4.10). Conversely, if (4.10) has a solution then  0

∞

||χω S(t)y||2L 2 (ω) dt < ∞ for all y ∈ X.

therefore for n ≥ 1, we can define the operator Sn : X −→ L 2 (0, ∞, L 2 (ω)) y −→ 1[0,n] (t)χω S(t)y 

n

It follows that ||Sn y||2 ≤ kn ||y||2 , where kn =

|||χω S(t)|||2 dt and according

0

to Banach Steinhaus’ theorem, there exists γ > 0 such that |||Sn ||| ≤ γ , n ≥ 1. Let α > 0, M1 > 0 be such that |||S(t)||| ≤ M1 eαt , t ≥ 0. For t0 > 0, the family (|||χω S(t)|||)t≤t0 is bounded and for t > t0 , we have 1 − e−2αt ||χω S(t)y||2L 2 (ω) = 2α 

t

≤ 0



t 0

e−2α(t−s) ||χω S(t)y||2L 2 (ω) ds

e−2α(t−s) ||χω S(s)y||2L 2 (ω) |||χω S(t − s)|||2 ds

≤ M12 γ 2 ||y||2 We deduce that there exists k > 0 such that |||χω S(t)||| ≤ k, t ≥ 0. On the other hand, by (4.12) we get  t||χω S(t)y||2L 2 (ω) ≤

t 0

||χω S(s)y||2L 2 (ω) |||χω S(t − s)|||2 ds.

then t||χω S(t)y||2L 2 (ω) ≤ k 2 γ 2 ||y||2 . Therefore there exists t0 such that ln |||χω S(t)||| < 0, for all t ≥ t0 . We deduce ln |||χω S(t)||| < 0. that σ0 =: inf t>0 t

80

4 Regional Stabilization of Infinite Dimensional Linear Systems

We show that σ0 = lim

t→+∞

ln |||χω S(t)||| t

(4.13)

Let t0 > 0, and set M2 = sup |||S(t)|||. For t ≥ t0 , there exists n ∈ N such that nt0 ≤ t < (n + 1)t0 . Thus

t∈[0,t0 ]

ln |||χω S(nt0 )||| ln |||S(t − nt0 )||| ln |||χω S(t)||| ≤ + t t t With the property (4.12) we have nt0 ln |||χω S(t0 )||| ln M2 ln |||χω S(t)||| ≤ + t t t0 t It follows that lim sup

t→∞

ln |||χω S(t0 )||| ln |||χω S(t)||| ≤ t t0

t0 being arbitrary, we get lim sup

t→∞

ln |||χω S(t)||| ln |||χω S(t)||| ln |||χω S(t)||| ≤ inf ≤ lim inf t→+∞ t>0 t t t

which shows (4.13) and therefore ∀σ ∈]0, −σ0 [, ∃ M3 / |||χω S(t)||| ≤ M3 e−σ t , t ≥ 0, Consequently the system is regionally exponentially stable on ω.



Remark 4.3 The condition (4.12) is not necessary for regional stability because any system exponentially stable on  is also on ω without condition (4.12). This condition somehow translates the continuity of χω S(t) with respect to χω y. Especially if ω = , then (4.12) is verified and the exponential stability on  is equivalent to the condition (4.9) (see [1]). Example 4.4 Let us consider the system defined on  =]0, +∞[ by ⎧ ∂ y(t) ∂ y(t) ⎪ ⎨ + =0 on ×]0, ∞[ ∂t ∂x ⎪ ⎩ y(., 0) = y0 = y0 ∈ H 1 ()

(4.14)

∂ of domain D(A) = {y ∈ H 1 (); y(0) = 0} generates a ∂x C0 -semigroup S(t), t ≥ 0 given by The operator A = −

4.1 Internal Regional Stability

81

(S(t)y0 )(x) =

⎧ ⎨ y0 (x − t) if x ≥ t ⎩

0

if x < t

On the region ω =]0, a[ (a > 0), the solution of (4.14) verify ||χω S(t)y||2L 2 (ω) ≤ ||χω y|| L 2 (ω) which gives (4.12) by applying it to y = S(t)y0 . On the other hand the operator P y = (a − x)i ω y is solution of (4.10). Indeed, for y ∈ D(A), we have + < (a − x)i ω y, − > ∂x ∂ x

 a  ∂ y(x) = −2Re (a − x)y(x)d x ∂x 0  a = −Re([(a − x)|y(x)|2 ]a0 + |y(x)|2 d x) = −||χω y||2L 2 (ω)

0

which shows that the system (4.14) is exponentially stable on ω. Remark  ∞ 4.4 1. Strong stability on ω does not cause convergence of the integral ||χω S(t)y||2 dt, as we can see by considering the system (4.7) and the region 0

ω =]a, ∞[, a ≥ 0. 2. If the system (4.7) is strongly stable on ω, this leads to the following question. Is 

∞ 0

||χω S(t)y||2L 2 (ω) dt < +∞ for all y ∈ L 2 ()

implies the strong stability on ω? This is the subject of next result. Proposition 4.4 If the system (4.1), with the output z(t) = χω y(t), is regionally observable on ω and if (4.10) admits a positive solution, then the system (4.1) is strongly stable on ω.  ∞ Proof If the operator P(y) = (χω S(t))∗ (χω S(t))y dt verify (4.10) then 0



∞ 0

||χω S(t)y||2L 2 (ω) dt < ∞ for all y ∈ X

and the operator P is well defined.

∞

We have < P S(t)y, S(t)y >= t

||χω S(τ )y||2L 2 (ω) dτ → 0 when t → ∞.

The system (4.1) is assumed to be observable on ω, we have

82

4 Regional Stabilization of Infinite Dimensional Linear Systems

< P y, y >≥ α||χω y||2L 2 (ω) 

which allows to conclude.

Proposition 4.5 If the Eq. (4.10) admits a positive solution and χω S(t) is uniformly bounded, then the system (4.1) is strongly stable on ω. Proof Let y ∈ D(A), according to the previous lemma we have < P S(t)y, S(t)y > → 0 when t → ∞. Which imply that P S(t)y → 0 when t → ∞, and with (4.10), we have ||χω S(t)y||2L 2 (ω) = − < AS(t)zy, P S(t)y > − < P S(t)y, AS(t)y > Thus lim ||χω S(t)y||L 2 (ω) = 0. t→∞

Now let y ∈ X, and (yn ) ⊂ D(A) such that yn → y when n → ∞. So there exists M > 0 such that ||χω S(t)y|| ≤ ||χω S(t)(y − yn )|| + ||χω S(t)yn || ≤ M||(y − yn )|| + ||χω S(t)yn ||. therefore lim sup ||χω S(t)y|| ≤ M||y − yn ||, for all n ≥ 1. This implies that ||χω t→+∞

S(t)y|| → 0 when t → ∞.



The following result characterizes the regional strong stability by a dissipativity condition. Let R ∈ L(X ), be a positive and self adjoint operator; there exists c > 0 such that

Ry, y ≥ c||χω y|| L 2 (ω) , y ∈ D(A) (4.15) and consider the Lyapunov equation

Ay, P y + P y, Ay + Ry, y = 0, y ∈ D(A)

(4.16)

So we have the following result. Proposition 4.6 Suppose there exists a positive and self-adjoint operator P ∈ L(X ) verifying (4.10). If, moreover, Re(< χω Ay, y > L 2 (ω) ) ≤ 0, y ∈ D(A)

(4.17)

then (4.1) is strongly stable on ω. Proof Let P be a solution of (4.16) and the function V (y) = < P y, y > . y ∈ L 2 (). For y0 ∈ D(A), we have

4.1 Internal Regional Stability

 0

t

83

||χω S(s)y0 ||2L 2 (ω) ds = V (y0 ) − V (y(t)).

and according to (4.15), we get  t V (y0 ) ||χω S(s)y0 ||2L 2 (ω) ds ≤ . c 0 d < i ω S(t)y0 , S(t)y0 > L 2 (ω) ≤ 0. Thus dt  t V (y0 ) t||χω S(t)y0 ||2L 2 (ω) ≤ ||χω S(s)y0 ||2L 2 (ω) ds ≤ c 0

The inequality (4.17) gives

Since V and i ω are continuous, then ||χω S(t)y0 ||2L 2 (ω) ≤ y0 ∈ X, hence (4.1) is strongly stable on ω.

V (y0 ) , for all t > 0 and c.t 

Remark 4.5 1. In the case where ω = , we have the above result without the condition (4.17). 2. The above results give necessary and sufficient conditions for regional stability of the system (4.1) on ω, without taking into account the behavior of the state of the system in the residual part \ω. The following proposition gives precisions on the behavior of the state on the residual part. Proposition 4.7 1. If the system (4.1) is exponentially stable on ω and if there exists d > 0 such that ||χω y||2L 2 (ω) ≥ d Re(< Ay, y >), y ∈ D(A)

(4.18)

then the state remains bounded on . 2. If the system (4.1) is weakly stable on ω and if there exists e > 0 such that Re(< χω Ay, y > L 2 (ω) ) ≥ e Re(< Ay, y >), y ∈ D(A) then the state remains bounded on . Proof 1. Let y0 ∈ D(A), we have Re (< AS(t)y0 , S(t)y0 >) =

1 d||S(t)y0 ||2 2 dt

According to (4.18), we have 

t 0

||χω S(s)y0 ||2L 2 (ω) ds ≥

d d ||S(t)y0 ||2 − ||y0 ||2 2 2

(4.19)

84

4 Regional Stabilization of Infinite Dimensional Linear Systems

and as the system (4.1) is exponentially stable on ω, then 

+∞ 0

thus ||S(t)y0 || ≤

||χω S(t)y0 ||2L 2 (ω) dt < ∞

2 < P y0 , y0 > +||y0 ||2 for all t ≥ 0, d

where P ∈ L(X ) is solution of (4.10). The density of D(A) in X. 2. Is deduced using the same approach as for the first point.



Remark 4.6 1. The condition (4.18) can be replaced by: ||χω y||2L 2 (ω) ≥ d Re (< χ\ω Ay, y >), y ∈ D(A) and the condition (4.19) can be replaced by: Re(< χω Ay, y > L 2 (ω) ) ≥ e Re(< χ\ω Ay, y >), y ∈ D(A) with e > 0. 2. In Lyapunov’s equation (4.10), the operator i ω can be replaced by a positive and self adjoint operator R ∈ L(X ) verifying < Ry, y > ≥ c ||χω y||2L 2 (ω) , y ∈ X with c > 0. 3. In the global case (ω = ), the stability exponential ensures the uniqueness of the solution of the Lyapunov equation < Ay, P y > + < P y, Ay > + < y, y > = 0, y ∈ D(A) In the regional case this result is not true, as we can see it in the case of the system (4.14) where (4.10) admits two distinct solutions P y = (a − x)i ω y and that given by (4.11). −1 χω y, ω is a region strictly We can also consider the example where Ay = 2 included in , in which case we have both solutions P = I d and P = i ω .

4.2 Internal Regional Stabilization In this section we will characterize a control which stabilizes an infinite dimensional linear system on a given region ω of the system evolution domain .

4.2 Internal Regional Stabilization

85

4.2.1 Definitions Lets consider the system ⎧ dy(t) ⎪ ⎨ = Ay(t) + Bv(t) dt ⎪ ⎩ y(., 0) = y0 ∈ X

on Q (4.20)

A generates a C0 -semigroup on X and B ∈ L(V, X ), V being the controls set assumed to be a Hilbert space. Definition 4.2 The system (4.20) is said regionally weakly (respectively strongly, exponentially) stabilizable on ω ⊂ , if there exists an operator K ∈ L(X, V ) such that the system ⎧ dy(t) ⎪ ⎨ = (A + B K )y(t) Q dt (4.21) ⎪ ⎩ y(., 0) = y0 ∈ X be regionally weakly (respectively strongly, exponentially) stable on ω. Remark 4.7 1. Regional stabilization can be seen as an output stabilization problem with a partial observation: z = χω y. 2. If the system (4.20) is stabilized on ω ⊂ , then it is stabilized on each ω ⊂ ω by applying the same control. 3. The cost of regional stabilization of a system is less than that of stabilization over the whole domain. Indeed, if we consider the functional cost  +∞ ||u(t)||2V dt (4.22) q(u) = 0

where u ∈ Vad (ω) = {u ∈ L 2 (0, +∞; V ) is a control that stabilizes (4.20) exponentially on ω and q(u) < ∞}, then we have Vad () ⊂ Vad (ω) hence min q(u) ≤ min q(u)

Vad (ω)

Vad ()

We may need to improve the degree of stability of a system on a privileged region of the system geometrical domain, as shown in the following example: Example 4.5 We consider the system defined on  =]0, +∞[ by ⎧ dy(t) ⎪ ⎨ = Ay(t) + Bv(t) on Q dt ⎪ ⎩ y(., 0) = y0 ∈ H 1 ()

(4.23)

86

4 Regional Stabilization of Infinite Dimensional Linear Systems

∂ , of domain D(A) = {y ∈ H 1 (); y(0) = 0} and the operator ∂x B ∈ L(V, H 1 ()). The control v(.) = 0 exponentially stabilizes the system (4.23) on ω =]0, a[, a > 0 with a bounded state on  (||y(t)|| = ||y0 ||), weakly stabilises the system but not exponentially on . On the other hand if we consider the cost function (4.22), then we have where A = −

min q(v) = 0 < min q(v)

Vad (ω)

Vad ()

4.2.2 Characterizations In this paragraph we characterize regional stabilization by an approach based on the decomposition of the system state space X. We consider the system (4.20) and assume that the space X = X u ⊕ X s and that (4.20) can be decomposed into two subsystems:

and

 dy (t) u = Au yu (t) + Bv(t) dt y0u = y0 , yu = y

(4.24)

 dy (t) s = As ys (t) + (I d − )Bv(t) dt y0s = (I d − )y0 , ys = (I d − )y

(4.25)

where the operators As and Au are restrictions of A to X s and X u respectively and

is the projection operator defined by (2.16) and we have σ (As ) = σs (A) , σ (Au ) = σu (A) and Au is a bounded operator. If Su (t) and Ss (t) are the restrictions of S(t) to X u and X s , which are respectively the C0 - semigroups generated by Au and As , the solutions of (4.24) and (4.25) are given by  t

yu (t) = Su (t)y0u +

Su (t − τ ) Bv(τ )dτ

(4.26)

Ss (t − τ )(I d − )Bv(τ )dτ

(4.27)

0



and ys (t) = Ss (t)y0s +

t

0

If the operator As verify the spectral growth assumption

4.2 Internal Regional Stabilization

87

ln |||Ss (t)||| = sup Re(σ (As )) t→+∞ t lim

(4.28)

then stabilizing the system (4.20) turns up to stabilizing (4.24) (see [2]). The following result extends this approach to the regional case. Proposition 4.8 Suppose that As verifies the inequality lim

t→+∞

ln |||χω Ss (t)||| ≤ sup Re(σ (As )) t

(4.29)

1. If there exists an operator K u ∈ L(X u , V ), such that the control v(t) = K u i ω yu (t), exponentially stabilizes the system (4.24) (respectively strongly) on ω, then the system (4.20) is exponentially (respectively strongly) stabilized on ω, by using the operator K = (K u i ω , 0). 2. If the system (4.24) is exponentially (respectively strongly) stabilized on  by a control v(t) = K u yu (t), with K u ∈ L(X, V ), then the system (4.20) is regionally exponentially (respectively strongly) stabilized on ω by using the operator K = (K u , 0). Proof 1. Exponential stabilization. According to the spectral decomposition, we have sup Re(σ (As )) ≤ −δ. As As verify (4.29) then there exist a > 0 and μ > 0 such that |||χω Ss (t)||| ≤ a e−μt , t ≥ 0

(4.30)

The solution of system (4.24) is therefore yu (t) = e Fu t y0u , with Fu = Au + B K u i ω ∈ L(X u ) and there exist 0 < α < μ, and b > 0 such that ||χω yu (t)|| L 2 (ω) ≤ b e−αt ||y0u || So for v(t) = K u i ω yu (t), we have ||v(t)||V ≤ b e−αt |||K u |||.||y0u || Using (4.27) and (4.30), there exists c > 0 such that ||χω ys (t)|| L 2 (ω) ≤ a e−μt ||y0s || + c ||y0u || ≤ a e−μt ||y0s || + c ||y0u ||

 e

t

e−μ(t−τ ) e−ατ dτ

0 −μt

− e−αt α−μ

Consequently the state of the system (4.20) excited by v(t) = K y(t) satisfy

88

4 Regional Stabilization of Infinite Dimensional Linear Systems



e−μt − e−αt + b e−αt ||y0 || ||χω y(t)|| L 2 (ω) ≤ a e−μt + c α−μ This shows that the system (4.20) is exponentially stabilizable on ω. 2. Strong stabilization. We have lim χω yu (t) L 2 (ω) = 0. Considering the control v(t) = K u i ω yu (t), t→+∞

we have ∀ε > 0, ∃ t0 > 0, t > t0 =⇒ v(t)V
0 such that χω ys (t)

L 2 (ω)

 We will show that lim

t→+∞ 0



t

e 0

−δ(t−τ )

t

≤ de

−μt



t

y0s  + d

e−μ(t−τ ) v(τ )V dτ

0

e−μ(t−τ ) v(τ )V dτ = 0. For t ≥ t0 , we have 

v(τ )V dτ =

t0

e

−μ(t−τ )

 v(τ )V dτ +

0

≤ γ e−μ(t−t0 ) +

ε 2

t

e−μ(t−τ ) v(τ )V dτ

t0

 t0 1 v(t)V dt, so lim ||χω ys (t)|| L 2 (ω) = 0. where γ = √ t→+∞ 2μ 0 Consequently the solution of the system (4.20) associated with the control v(t) = K y(t) satisfy ||χω y(t)|| L 2 (ω) ≤ ||χω yu (t)|| L 2 (ω) + ||χω ys (t)|| L 2 (ω) The system (4.20) is therefore strongly stabilizable on ω. 2. The second point is demonstrated by the same approach.



Remark 4.8 The inequality (4.29) may be strict, this is the case for example, if we consider the operator A defined by (4.5) with λ1,0 = −1 and λn,m = λ1 = −2, for (n, m) = (1, 0). The following result is a consequence of the previous proposition and the equivalence between the controllability and exponential stabilization of the system (4.24) (see [2]). Corollary 4.1 If the condition (4.29) is verified and that 1. the space X u is of finite dimension, 2. the system (4.24) is controllable on X u , then the system (4.20) is exponentially stable on ω.

4.2 Internal Regional Stabilization

89

The following example shows that it is possible that a system be regionally stabilized without being on the whole evolution domain. Example 4.6 Lets consider the system (4.4) excited by g(.)v(t) with g =< 1, h (1,0) > h (1,0) + < 1, h (2,0) > h (2,0) The system (4.4) can be decomposed into (4.24) and (4.25) with Au yu =< yu , h (1,0) > h (1,0) + < yu , h (2,0) > h (2,0) , y ∈ L 2 ()

and As ys = − < ys , 1 > 1 −

(4.31)

< ys , h (n,m) > h (n,m)

(n,m)∈−{(1,0),(2,0)}

By a rank argument, the unstable part (4.31) which is of finite dimension is not controllable, and therefore the system (4.4) cannot be stabilized on . But for ω = 1 ] , 1[, the operator As verify (4.29) and χω yu (t) = 0. So the system (4.4) may be 2 stabilized on ω by the null control.

4.2.3 Regional Stabilizing Control In this paragraph, we propose to look for a control ensuring the regional stabilization. The used approach is based on using the Riccati equation. Let us consider the system (4.20) with the same hypotheses on A and B. For an operator K ∈ L(X, V ), S K (t), t ≥ 0, indicates the semigroup generated by A K = A + BK. Let R ∈ L(X ) be a positive, self adjoint operator and verify ∃c > 0 | < Ry, y >≥ c ||χω y|| L 2 (ω) , y ∈ X

(4.32)

For y ∈ D(A), we consider the following Riccati equation < Ay, P y > + < P y, Ay > − < B ∗ P y, B ∗ P y > + < Ry, y >= 0

(4.33)

and suppose that there exists an operator P ∈ L(X ), positive and self adjoint, solution of the Eq. (4.33). Let note the operator K = −B ∗ P, The following results come from the Proposition 4.7. Proposition 4.9 (1) If the condition Re(< χω (A + B K )y, y > L 2 (ω) ) ≤ 0, y ∈ D(A) is verified, then (4.20) is strongly stabilizable on ω.

(4.34)

90

4 Regional Stabilization of Infinite Dimensional Linear Systems

If, in addition, we have Re(< χω (A + B K )y, y > L 2 (ω) ) ≥ e Re(< (A + B K )y, y >) then the state of the system (4.21) remains bounded on . (2) If S K (t) satisfies (4.12), then the system (4.20) is exponentially stabilizable on ω. If, moreover, there exists d > 0 such that < Ry, y > ≥ d Re(< (A + B K )y, y >)

(4.35)

then the state of the system (4.21) remains bounded on . In the case of conservative systems, consider the system (4.20) with < Ay, y > + < y, Ay >= 0, y ∈ D(A)

(4.36)

If the controls space is V = X and there exists c > 0 such that for all y ∈ X , ||B ∗ y|| ≥ c ||χω y|| L 2 (ω) Considering R = B B ∗ and P = I d, we have the following result. Corollary 4.2 1. If the condition (4.34) is verified, then the control v(t) = −B ∗ y(t) strongly stabilizes the system (4.20) on ω. 2. If S K (t) verify (4.12), then control v(t) exponentially stabilizes the system (4.20) on ω. The following result concerns the strong stabilization. Proposition 4.10 If (4.33) admits a positive solution P, then the control v(t) = −B ∗ P y(t) strongly stabilizes the system (4.20) on ω if and only if |||χω S K (t)| is bounded , with K = −B ∗ P. Proof Since (4.33) has a solution P ≥ 0, then

A K S K (t)y, P S K (t)y + P S K (t)y, A K S K (t)y + B ∗ P S K (t)y, B ∗ P S K (t)y + RS K (t)y, S K (t)y = 0 Thus

4.2 Internal Regional Stabilization

91

d

P S K (t)y, S K (t)y + ||B ∗ P S K (t)y||2 + RS K (t)y, S K (t)y = 0 dt Which gives 

t 0



∞

So



||B ∗ P S K (τ )y||2 dτ +

t

RS K (τ )y, S K (τ )ydτ

0

= P y, y − S K (t)y, S K (t)y ≤ P y, y

RS K (t)y, S K (t)ydt < ∞ and therefore the operator

0



∞

G(y) =



1

R 2 S K (t)

∗

1 R 2 S K (t) ydt

0



∞

is well defined and we have Gy, y =

RS K (t)y, S K (t)ydt. So

0



G S K (t)y, S K (t)y =

∞

RS K (τ )y, S K (τ )ydτ

0

 =

∞

RS K (τ )y, S K (τ )ydτ

t

We deduce that lim G S K (t)y, S K (t)y = 0, therefore lim G S K (t)y = 0. t→∞ t→∞ Since G is solution of Lyapunov’s equation

A K y, Gy + Gy, A K y + Ry, y = 0 it follows that for all y ∈ D(A), we have

A K S K (t)y, G S K (t)y + G S K (t)y, A K S K (t)y + RS K (t)y, S K (t)y = 0 When t → 0, G S K (t)y → 0, it follows RS K (t)y, S K (t)y → 0, which is equivalent to RS K (t)y → 0 (since R is positive and self-adjoint). Consequently ||χω S K (t)y||2 → 0, for all y ∈ D(A), and the conclusion follows with the same considerations as in the Proposition 4.5.  Example 4.7 We consider the system defined on  =]0, 1[ by ⎧ ∂ y(t) ∂ 2 y(t) ⎪ ⎨ = −i + v(t)i ω y(t) on Q ∂t ∂x2 ⎪ ⎩ y(., 0) = y0 on 

(4.37)

92

4 Regional Stabilization of Infinite Dimensional Linear Systems

∂2 The operator A = −i 2 with domain D(A) = H 2 () ∩ H01 (), B = i ω and i 2 = ∂x −1. The operator A is self adjoint and Re(< Ay, y >) = 0, for all y ∈ D(A) and so (4.36) is verified. On ω =]0, a[⊂ , for y ∈ D(A), we have  Re(< χω (A + B K )y, y > L 2 (ω) ) = −

ω

|y(x)|2 d x ≤ 0

thus (4.34) is verified. We therefore conclude that the control v(t) = −i ω y(t) strongly stabilizes (4.37) on ω.

4.2.4 Low Cost Regional Stabilization Problem We will look for the control ensuring stabilization of system (4.20) on a region ω and minimizing a functional cost. Let R ∈ L(X ) be a positive and self adjoint operator. The problem of regional stabilization at low cost consists in finding a control u ∈ L 2 (0, +∞; V ) which stabilizes (4.20) on ω minimizing the functional  q(v) =

+∞



+∞

< Ry(t), y(t) > dt +

0

0

||v(t)||2V dt

(4.38)

Suppose that for all initial state y0 , there exists a control v such that (4.38) is finite. It is known [3] that if the Riccati equation (4.33) has a positive definite solution P, then the control (4.39) v∗ (t) = −B ∗ P y ∗ (t) minimizes (4.38), which results exponential stabilization of the system (4.20) on . Proposition 4.11 If there exists e > 0 such that < Ry, y >≥ e ||χω y||2L 2 (ω) for all y ∈ X

(4.40)

and if the Eq. (4.33) has a positive solution P, then, noting K = −B ∗ P, we have 1. If the condition (4.34) is verified, the control (4.39) strongly stabilizes the system (4.48) on ω. 2. If S K (t) verify (4.12), then the control (4.39) exponentially stabilizes the system (4.20) on ω. The proof follows from the result of the previous proposition.

4.2 Internal Regional Stabilization

93

Remark 4.9 In the previous result if we suppose, moreover, that the operator R satisfies

Ry, y ≥ e Re( Ay, y) , y ∈ D(A) (4.41) and if B B ∗ P is positive, this is the case for example when P commutes with B B ∗ , then (4.41) implies (4.35); in this case the state of the system (4.20) remains bounded on the residual part \ω. Example 4.8 On  =]0, +∞[, we consider a system described by the equation ⎧ ∂ y(t) ∂ y(t) ⎪ ⎨ =− + v(x, t) on Q ∂t ∂x ⎪ ⎩ y(, 0) = y0 ∈ L 2 ()

(4.42)

∂ The operator A = − with domain D(A) = {y ∈ H 1 (); y(0) = 0}. ∂x On ω =]0, 1[ we consider the functional (4.38) with R = ((1 − x)2 + 1)i ω that verify (4.40). The operator P = (1 − x)i ω is solution of (4.33) and the operator K = −(1 − x)i ω satisfy (4.34). The control v∗ (x, t) = −(1 − x)i ω y(t) minimizes (4.38) and strongly stabilizes the system (4.42) on ω with a cost q(v∗ ) = < P y0 , y0 > =



1

(1 − x)|y0 (x)|2 d x

0

Example 4.9 On  =]0, ∞[, let a system described by the equation ⎧ ∂ y(x, t) ∂ y(x, t) ⎪ ⎨ =− + χω a(x)v(t) on Q ∂t ∂x ⎪ ⎩ y(., 0) = y0 ∈ L 2 ()

(4.43)

∂ with domain D(A) = {y ∈ H 1 () | y(x) = 0} and v ∈ V = ∂x L 2 (). Since y(∞) = 0, for all y ∈ D(A), we have

The operator A =



∞

Re( Ay, y) = − 0

dy(x) y(x)d x = 0 dx

94

4 Regional Stabilization of Infinite Dimensional Linear Systems

If ω = ]a, b[⊂  and a(.) ≥ α > 0, then the control v∗ (t) = −i ω y(t) strongly stabilizes the system (4.43) on ω and minimizes the functional  q(v) = 0

+∞

 ||χω y(t)||2L 2 (ω) dt

+ 0

+∞

||v(t)||2V dt

and q(u ∗ ) = ||y0 ||2 . Remark 4.10 We can also replace (4.36) by the condition

i ω Ay, y + y, i ω Ay = 0, ∀y ∈ D(A),

(4.44)

and if ||B ∗ i ω y|| ≥ α||χω y||, the control v∗ (t) = −B ∗ i ω y ∗ (t) stabilizes (4.20) on ω. However the relationship (4.36) can be verified without condition (4.44), as shown in the previous example since for any ω =]a, b[⊂ , there exists y ∈ D(A) such  b dy(x) y(x)d x = 0. that dx a Example 4.10 We consider the system defined on  =]0, 1[ by the Schrodinger equation ⎧ ∂ y(t) ∂ 2 y(t) ⎪ ⎨i = + u(t)i ω y(t) Q ∂t ∂x2 (4.45) ⎪ ⎩ 2 y(., 0) = y0 ∈ L () ∂2 The operator A = −i 2 with domain D(A) = H 2 () ∩ H01 (), is self adjoint ∂x and therefore verify (4.36). The eigenvalues and the eigenvectors of A are given by λ j = −iπ 2 j 2 and ϕ j (x) =

√ 2 sin( jπ x), j ≥ 1.

Let ω ⊂ , the control v∗ (t) = −i ω y(t) strongly stabilizes (4.45) on ω and minimizes the functional  +∞  +∞ 2 ||χω y(t)|| L 2 (ω) dt + ||v(t)||2V dt q(v) = 0

0

and q(u ∗ ) = ||χω y0 ||2L 2 (ω) . The stabilized system on ω becomes ⎧ ∂2 y ⎪ ⎪ −i ⎪ ⎨ ∂ x 2 + y(t) on ω

∂y ∂2 y = −i 2 + i ω y(t) = ⎪ ∂t ∂x 2 ⎪ ⎪ ⎩ −i ∂ y ∂x2

on  \ ω

4.2 Internal Regional Stabilization

95

Let y = y1 + i y2 , where y1 and y2 are the real and imaginary parts of the state y, we obtain ⎧ ∂ y1 (t) ⎪ ⎪ ⎪ ⎨ ∂t = y2 (t) − i ω y2 (t) on Q ⎪ ⎪ ∂ y (t) ⎪ ⎩ 2 = −y1 (t) + i ω y1 (t) on ∂t To summarize the previous approach, numerical resolution of an optimal stabilization problem is obtained through the following steps 1. 2. 3.

Resolution of the Eq. (4.33) Computation of the control given by (4.39) Resolution of the Eq. (4.20)

−→ −→ −→

P v∗ y ∗ (t)

4.3 Numerical Examples In this section we will present numerical examples illustrating the established results, in the case of one-dimensional and two-dimensional systems.

4.3.1 One Dimensional Case On  =]0, 1[, we consider the system described by the equation ⎧ ∂ y(t) ⎪ ⎨ = (1 − x)y(t) + v(x, t) on Q ∂t ⎪ ⎩ y(., 0) = y0 ∈ L 2 ()

(4.46)

where v(t) ∈ V = L 2 (), t ≥ 0 and y0 = x|0.5 − x|. The uncontrolled system (v(.) = 0) is unstable on  (see Fig. 4.1). We are looking for the control that stabilizes the system (4.46) on ω =]0, 0.5[ minimizing the functional (4.38)  in which we take R = i ω . The operator P = ((1 − x) + 1 + (1 − x)2 )i ω is solution of (4.33) and the control v∗ (t) = −P y(t) minimizes (4.38). The associated solution of (4.46) (see Fig. 4.2) is given by  √ −t 1+(1−x)2 y0 (x) if x ∈ ω y(x, t) = et (1−x) y0 (x) otherwise e The system (4.46) is exponentially stabilized on the region ω.

96

4 Regional Stabilization of Infinite Dimensional Linear Systems

Fig. 4.1 The evolution of the autonomous system on 

Fig. 4.2 The evolution of the system stabilized on ω

4.3.2 Two Dimensional Case On  =]0, 2[×]0, 2[, we consider the system  ∂ y(t)

= f (x, z)y(t) + v(t; x, z) on Q ∂t y(., 0) = y0 ∈ L 2 () 

with f (x, z) =

(x − 1)2 0

(4.47)

if (x, z) ∈ ]0, 1[×]0, 2[ otherwise

and we assume that v ∈ V = L 2 (). It is clear that the uncontrolled system (v(.) = 0) is instable on .

4.3 Numerical Examples

97

Fig. 4.3 The state of the system at t = 0

Fig. 4.4 Stabilized state on ω and bounded state on \ω at t = 1

We look for a control that minimizes (4.38) in which we take Ry = 3 f (.)2 y, verifying (4.40) that stabilizes the system (4.47) on the region ω =]0, 1[ × ]0, 2[ . The operator P y = 3 f (.)y is solution of the Eq. (4.33) and the solution of system (4.47) associated to the control v∗ (t) = −P y(t), is given by y(t; x, z) =

⎧ 2 ⎨ e−2t (x−1) y0 (x, z)

if (x, z) ∈ ]0, 1[×]0, 2[

⎩

otherwise

y0 (x, y)

Consequently the system (4.47) is exponentially stabilized on ω and the state remains bounded on \ω (see Figs. 4.3, 4.4, 4.5 and 4.6).

98

4 Regional Stabilization of Infinite Dimensional Linear Systems

Fig. 4.5 Stabilized state on ω and bounded state on \ω at t = 5

Fig. 4.6 Stabilized state on ω and bounded state on \ω at t = 10

4.4 Regional Boundary Stabilization In this section results of the previous section are expanded to the case where the target region  is located on the boundary ∂ of  ( ⊂ Rn−1 ). It is then a question of stabilizing an unbounded output. To make sense of this output, we’re going to use the notion of admissible region. Let  be an open bounded domain of Rn of boundary ∂. We still consider the system (4.1), with the same assumptions on the operator A and B. ⎧ dy(t) ⎪ ⎨ = Ay(t) on Q dt (4.48) ⎪ ⎩ y(., 0) = y0 ∈ X

4.4 Regional Boundary Stabilization

99

¯ a surface to stabilize and let the restriction operator Consider a region  ⊂  χ : X −→ L 2 () y −→ χ y = y| Let χ∗ be the adjoint operator of χ and i  = χ∗ χ .

 y¯ z dσ, We suppose that L 2 () is provided with its usual scalar product (y, z) = 

  21 the associated norm is given by ||z||L 2 () = z 2 dσ (dσ indicates the superficial 

measure defined on ∂ and induced by the Lebesgue measure). We are then interested in the asymptotic behavior of the output z(t) = χ y(t)

(4.49)

We will see that it is possible to make sense of z(t) for any trajectory y(t) of the system (4.1). Definition 4.3 The region  is said admissible for S(t) if 1. The operator χ : (D, ||.|| A ) −→ L 2 () is continuous, D(A) is provided with the graph norm ||.|| A ). 2. There exists α > 0 such that ||z(.)||L 2 (0,∞;L 2 ()) = ||χ S(.)y0 ||L 2 (0,∞;L 2 ()) ≤ α||y0 || for all y0 ∈ D(A).

(4.50)

Remark 4.11 The admissibility of  is in fact the admissibility of the operator χ in the sense of Weiss [4]. It allows to extend the application y0 −→ χ S(.)y0 in a continuous application from X to L 2 (0, ∞, X ), so (4.49) is well defined for any y0 ∈ X . Definition 4.4 If  is admissible, the system (4.48) with the output (4.49), is said weakly (respectively strongly, exponentially) stable on , if z(t) tends to 0 weakly (respectively strongly, exponentially) when t → ∞. Example 4.11 On  =]0, 2[ × ]0, 2[, we consider the system described par the following equation ⎧

 1 dy(t) ⎪ ⎪ = − z y(t) on ×]0, ∞[ ⎨ dt 2 ⎪ ⎪ ⎩ y(., 0) = y0 (x, z) ∈ H 1 () The operator Ay = ( 21 − z)y, with domain D(A) = {y ∈ H 1 () /y = 0 on 0 = {0} × [0, 1]}

(4.51)

100

4 Regional Stabilization of Infinite Dimensional Linear Systems

D(A) is a closed space in H 1 (), provided with its usual inner product , it is therefore a Hilbert space. 1 Let  = {0} × [0, 2] and ωa =]0, 2[×]a, 2[, with a > . 2 For y0 ∈ D(A), the solution of (4.51) verify  ||χ S(t)y0 ||L 2 () = 2

2

e

2( 21 −z)t

|y0 (0, z)| dz ≤ e 2

−t

1



2

|y0 (0, z)|2 dz

1

We deduce, by density and continuity of the trace operator on H 1 (), that the system (4.51) is strongly stable on . 1 Let ω be a region such that  ⊂ ∂ω, (there exists b > 0 with 0 < c < ) and 2 1 let ω0 =]0, b[×]c, [⊂ ω. We have ||χω y(t)|| ≥ ||χω0 y0 ||L 2 (ω ) , which shows that 0 2 system (4.51) is not strongly stable on ω. The following result establish the link between the regional stability of the system (4.1) on  and the properties of the spectrum point σ (A) of A. For that we suppose that A has a decreasing sequence of real eigenvalues(λn )n≥1 → −∞, this is the case for instance if A is self adjoint and has compact resolvent set. In this case A has a finite number of positive eigenvalues, each associated with a finite dimension proper subspace (see [2]). Proposition 4.12 1. If the system (4.48) is weakly stable on  then (∀λ ∈ σ (A)) ; Re (λ) ≥ 0 =⇒ ∀ϕ ∈ K er (A − λ I ), χ ϕ = 0

(4.52)

2. If the space X has an orthonormal basis (ϕn )n≥1 of eigenfunctions of A associated to eigenvalues (λn )n≥1 such that ∃ δ > 0 ∀λ ∈ σ (A), (∃ϕ ∈ K er (A − λ I ), | χ ϕ = 0) =⇒ Re (λ) ≤ −δ (4.53) then the system (4.1) is exponentially stable on . Proof 1. Suppose there exist λ ∈ σ (A) and ϕ ∈ K er (A − λI ) such that Re(λ) ≥ 0 and χ ϕ = 0, then

χ S(t)ϕ, ϕL 2 () ≥ ||χ ϕ||L 2 () = 0 , for all t ≥ 0 So the system (4.57) is not weakly stable on . 2. We can assume that the eigenvalues of A are simple. For y0 ∈ X and t ≥ 0, we have S(t)y0 =

λn ∈Sp(A)

eλn t y0 , ϕn ϕn

4.4 Regional Boundary Stabilization

101

By continuity of χ : (D(A), ||.|| A ) −→ L 2 (), there exists h > 0 such that

||χ S(t)y0 ||L 2 () ≤ h

exp{Re(λn )t}| y0 , ϕn |.||ϕn || A

λn ∈σ (A)

K er (A−λn I )⊂ K er (χ )

So

  Re(λn )t 2 exp{ } ||y0 ||.||ϕn || A 2



||χ S(t)y0 ||L 2 () ≤ h

λn ∈σ (A)

K er (A−λn I )⊂ K er (χ )

We deduce that for all t > 2, we have   δ ||χ S(t)y0 ||L 2 () ≤ h exp − t ||y0 || × 2



||ϕn || A e Re(λn )

λn ∈σ (A)

K er (A−λn I )⊂ K er (χ )

  δ ≤ h exp − t ||y0 || × 2



(1 + |λn |2 ) 2 e Re(λn ) 1

λn ∈σ (A)

ker (A−λn I )⊂ K er (χ )

But according to (4.53), we have

(1 + |λn |2 ) 2 e Re(λn ) < ∞ 1

λn ∈σ (A)

K er (A−λn I )⊂ K er (χ )

and so (4.48) is exponentially stable on .



Example 4.12 On  =]0, 1[×]0, 1[, we consider the system described by the equation ⎧ dy(t) ⎪ ⎨ = Ay(t) Q dt (4.54) ⎪ ⎩ y(0) = y0 ∈ H 2 () where Ay = y + 4π 2 y, cos(x) cos(π x)− < y, 11 > 11 with  is the Laplacian operator. The operator A of domain D(A) = {y ∈ H 2 () ∂y = 0}, generates a C0 -semigroup S(t). | ∂ν The eigenfunctions and the eigenvalues of A are given by ϕk, (x, z) =

⎧ ⎨ 2 cos(kπ x) cos(π z) if (k, l) ∈ N2 and (k, ) = (0, 0) ⎩

1

otherwise

102

4 Regional Stabilization of Infinite Dimensional Linear Systems

and λ(k,)

⎧ ⎨ −1 = 0 ⎩ −(k 2 + 2 )π 2

if (k, ) = (0, 0) if (k, ) = (1, 0) if (k, ) ∈ / {(0, 0), (1, 0)}

1 We will show that the region  = { } × [0, 1] is admissible for S(t). 2 The operator χ : (D(A), ||.|| A ) −→ (L 2 (), ||.||L 2 () ) is continuous, since the operator χ : H 1 () −→ L 2 () is continuous, there exists γ > 0 such that ||χ y|| L 2 () ≤ γ ||y|| H 1 () , for y ∈ D(A) ⊂ H 1 ()

(4.55)

According to Green’s formula, we have  ||∇ y||2(L 2 ())2

=−



Ay, yd x − | y, 11|2 + 4π 2 | y, ϕ(0,1) |2 ∀y ∈ D(A)

and using Schwartz’s inequality, we get ||∇ y||2(L 2 ())2 ≤ ||Ay||.||y|| + ||y||2 + 4π 2 ||y||2 ≤ (2 + 4π 2 )||y||2A So for y ∈ D(A), we have

21 ||χ y|| L 2 () ≤ γ ||y|| H 1 () ≤ γ ||y||2L 2 () + ||∇ y||2(L 2 ())2

21 ≤ γ ||y||2L 2 () + (2 + 4π 2 )||y||2A ≤ γ (3 + 4π 2 )||y|| A To show (4.50), we consider + y0 , ϕ(1,0) ϕ(1,0) S(t)y0 = e−t y0 , 1111 + eλ(k,l) t y0 , ϕ(k,l) ϕ(k,l) (k,l)∈{(0,0),(1,0)} /

As the series (4.56) is uniformly convergent in H 1 (), we can write χ S(t)y0 = e−t y0 , 11χ 11 + y0 , ϕ(1,0) χ ϕ(1,0)

+

(k,l)∈{(0,0),(1,0)} /

and since χ ϕ(1,0) = 0, it follows

eλ(k,l) t y0 , ϕ(k,l) χ ϕ(k,l)

(4.56)

4.4 Regional Boundary Stabilization

103



χ S(t)y0 = e−t y0 , 11χ 11 +

eλ(k,l) t y0 , ϕ(k,l) χ ϕ(k,l)

(k,l)∈{(0,0),(1,0)} /

According to (4.55), we have ||χ S(t)y0 || ≤ e−t | y0 , 11|.||χ 11|| L 2 () + eλ(k,l) t | y0 , ϕ(k,l) |.||χ ϕ(k,l) || L 2 () (k,l)∈{(0,0),(1,0)} /

≤ γ e−t | y0 , 11|.||11|| H 1 () +γ eλ(k,l) t | z 0 , ϕ(k,l) |.||ϕ(k,l) || H 1 () (k,l)∈{(0,0),(1,0)} /

It follows ||χ S(t)y0 || ≤ γ e−t | y0 , 11|||11|| H 1 ()

+γ

e−(k

2

+l 2 )π 2 t

 1 | y0 , ϕ(k,l) | 1 + 4π 2 (k 2 + l 2 ) 2

(k,l)∈{(0,0),(1,0)} /

⎡ ≤ γ ⎣e−t +



e−(k

2

+l )π 2

2

⎤ 1   t 1 + 4π 2 (k 2 + l 2 ) 2 ⎦ ||y0 ||

(k,l)∈{(0,0),(1,0)} /

 Consequently 0

+∞

||χ S(t)y0 ||2 dt < ∞ therefore  is admissible for S(t).

On the other hand, the only positive or zero eigenvalue is λ(1,0) = 0, and is such that χ ϕ(1,0) = 0, and λ(k,l) ≤ −1 for (k, l) = (1, 0). The previous proposition applies and we conclude that the system (4.54) is exponentially stable on . Remark 4.12 The system (4.54) is not stable on any neighborhood region ω ⊂  of .

4.5 Optimal Control for Regional Boundary Stabilization This section extends the previous work on regional stabilization on a region internal to the evolution domain to a boundary region. Consider the system ⎧ dy(t) ⎪ ⎨ = Ay(t) + Bv(t) dt ⎪ ⎩ y(., 0) = y0 ∈ D(A)

Q (4.57)

104

4 Regional Stabilization of Infinite Dimensional Linear Systems

where A is a linear operator with domain D(A) and generates a C0 -semi group S(t), t ≥ 0 in the state space X , assumed to be a Hilbert space, provided with the inner product ., . and the associated norm || · ||. The control v(.) belongs to an Hilbert space V and B ∈ L(V, X ). The problem of optimal stabilization consists in finding a control which ensures the regional stabilization of (4.57) on an admissible region , and solution of minimization problem ⎧ ⎨ ⎩



+∞

min q(u) =



+∞

Ry(t), y(t)dt +

0

0

||v(t)||2V dt

(4.58)

v ∈ Vad = {v ∈ L 2 (0, +∞; V ); q(v) < ∞}

where R : X −→ X is a bounded linear operator verifying

Ry, y ≥ c ||χ y|| L 2 () for all y ∈ D(A) and c > 0.

(4.59)

Remark 4.13 The cost of stabilizing a system on a boundary region  is less than the one on an internal region ω, such that  ⊂ ∂ω. Indeed, if we consider the minimization of (4.22) on the sets of admissible controls Vad () and Vad (ω), we have / Vad (ω). 0 ∈ Vad (), but for any internal region ω neighborhood of  we have 0 ∈ This imply the inequality min q(v) < min q(v) Vad ()

Vad (ω)

which shows that the cost is lower on . This remark apply to the case of system (4.51) with a boundary region  = {0} × [0, 2]. The problem (4.58) will be solved by two approaches: the first one is a direct and the second one is based on internal stabilization techniques.

4.5.1 Direct Approach The considered approach in this paragraph is based on the Lyapunov techniques using the Riccati equation. Theorem 4.1 Let K = −B ∗ P, where P is a positive solution of the following Riccati equation (4.60)

Ay, P y + P y, Ay − B ∗ P y, B ∗ P y + Ry, y = 0 and suppose that for y ∈ D(A),

Re χ (A + B K )y, χ yL 2 () ≤ 0 then

(4.61)

4.5 Optimal Control for Regional Boundary Stabilization

1. The control

105

v∗ (t) = −B ∗ P y ∗ (t)

(4.62)

is solution of the problem (4.58) and strongly stabilizes the system (4.57) on  with the estimation

 1 when t → +∞ χ y(t) = O √ t 2. If there exists β > 0, such that ||χ y||2L 2 () ≥ β Re ( (A + B K )y, y)

(4.63)

then the state of the system (4.57) is stabilized on  and remains bounded on . Proof 1. For y0 ∈ D(A), we have  d 

P S K (t)y0 , S K (t)y0  = P AS K (t)y0 , S K (t)y0  dt + P S K (t)y0 , AS K (t)y0  = − RS K (t)y0 , S K (t)y0  Integrating by parts, for any t ≥ 0 we get, 

t

P y0 , y0  − P S K (t)y0 , S K (t)y0  =

RS K (s)y0 , S K (s)y0 ds

0

This last relation is verified for all y0 of X, since the operators P, R and S K (t) are bounded. As P ≥ is positive, we deduce that q(v) < ∞, in other words Vad = ∅ for all initial state y0 , therefore (4.62) is the unique control solution of (4.58), and according to (4.59) we deduced that 

t 0

||χ S K (s)y0 ||2L 2 () ds ≤ P y0 , y0 ,

On the other hand, (4.61) allows to write d ||χ S K (t)y0 ||2L 2 () ≤ 0 dt  

It follows that t||χ S K (t)y0 ||2L 2 () ≤

0

t

||χ S K (s)y0 ||2L 2 () ds

106

4 Regional Stabilization of Infinite Dimensional Linear Systems

therefore ||χ S K (t)y0 ||2L 2 () ≤

P y0 , y0  >0 t

(4.64)

Let y0 ∈ X and (y0n )n≥1 ⊂ D(A) such that y0n −→ y0 in X . The Equation (4.60) implies the condition (4.50), thus for all n ≥ 1 

+∞

0

||χ S K (t)(y0n − y0 )||2L 2 () dt ≤ α||y0n − y0 ||2

when n → ∞, gives 

+∞ 0

||χ S K (t)(y0n − y0 )||2L 2 () dt −→ 0 a.e. t ≥ 0

Therefore there exists a subsequence (y0(n) )n≥1 of (y0n )n≥1 such that χ S K (t)y0(n) −→ χ S K (t)y0 , a.e. t ≥ 0, (see [5]). From (4.64), we get ||χ S K (t)y0(n) ||2L 2 () ≤ it follows ||χ S K (t)y0 ||2L 2 () ≤

P y0(n) , y0(n)  t

P y0 , y0  , a.e. , t > 0 t

(4.65)

(4.66)

 1 Therefore || χ S K (t)y0 || = O √ , for all y0 ∈ X . t  1 d  ||S K (t)y0 ||2 , 2. From (4.63) and using Re (A + B K )S K (t)y0 , S K (t)y0  = 2 dt we deduce that  t ||χ S K (s)y0 ||2L 2 () ds ≥ β||S K (t)y0 ||2 − β||y0 ||2 0

then ||S K (t)y0 || ≤ M(y0 ) := β1 P y0 , y0  + ||y0 ||2 , t ≥ 0. By the density of D(A) and the continuity of the mapping y0 −→ M(y0 ), we  deduce that ||S K (t)y0 || is bounded on . Remark 4.14 We can show that, under the condition Re( χ (A + B K )y, χ yL 2 () ) ≥ β.Re( (A + B K )y, y) for all y ∈ D(A), β > 0, the state of the system (4.57) is stabilized on  and remains bounded on . Example 4.13 Let a system be defined on the disk  = {(x, y) | x 2 + y 2 < 1} by

4.5 Optimal Control for Regional Boundary Stabilization

107

⎧ ∂ y(t) ⎪ ⎨ = f (x, z)y(t) + v(t)χ D ∂t ⎪ ⎩ y(0) = y0 where D = {(x, z) ∈ R2 | f (x, z) =

Q (4.67)

0.9 ≤ x 2 + z 2 ≤ 1} and ⎧ if (x, z) ∈ D ⎪ ⎨0 ⎪ ⎩ 1 (x 2 + z 2 − 0.9)2 otherwise 10

The solution of system (4.67) with v = 0 is given by y(t) =

⎧ ⎨ y0 ⎩

et (x

D 2

+y 2 −0.9)2

y0 otherwise

The system is unstable on  = {(x, z) | z ≥ 0 and x 2 + z 2 = 1}. We will see that we can stabilize (4.67) on  by a control which minimizes the functional (4.58) with R = i  . The operator P = i  is solution of (4.60), and the condition (4.61) is verified. The control v∗ (t) = −P y ∗ (t) strongly stabilizes (4.67) on  and minimizes the functional (4.58).

4.5.2 Internal Approach In this paragraph we develop an approach that characterizes an optimal control which stabilizes the system (4.57) on a boundary surface , considered as limit to a sequence of internal regions (ω j ) j≥1 ⊂  , with  ⊂ ∂ω j for all j ≥ 1. The control stabilizing the system on the boundary region  is then a limit to a sequence of controls stabilizing (4.57) on each of (ω j ). We consider the system (4.67) and the problem (4.58) with Ry, y = χ y2L 2 () . In what follows we assume that the Eq. (4.60) has a unique positive solution P. Let (ω j ) j≥1 be a sequence of neighborhoods of  defined by: 

1 ω j = x ∈  | dist(x, ) < j

 (4.68)

where “dist” indicates the distance given by: dist(x, ) = inf |x − ξ |Rn . ξ ∈  The sequence (ω j ) j≥1 is decreasing and converges to ω j = . j≥1

Consider the minimization problem

108

4 Regional Stabilization of Infinite Dimensional Linear Systems

⎧  ⎪ ⎪ ⎨ min q j (v) = v

⎪ ⎪ ⎩

+∞



+∞

R j y(t), y(t)dt +

0

0

||v(t)||2V dt

(4.69)

v ∈ V j = {v ∈ L 2 (0, +∞; V ); q j (v) < ∞}

where R j y, y = jχω j y2 . Then we have Theorem 4.2 If the system (4.57) is exponentially stabilized on an internal region ω ⊂  containing  on its boundary ( ⊂ ∂ω) and if the condition (4.61) is satisfied, then the sequence of controls v∗j (t) solutions of (4.69) converges to the control v∗ (t) = −B ∗ P y ∗ (t) solution of (4.58), and strongly stabilizes the system (4.57) on the boundary region . Proof Since (4.57) is exponentially stabilizable on ω, it is stabilizable on ω j ⊂ ω. We deduce that q j (v) < +∞, for all j ≥ 1. On the other hand, the problem (4.69) has a unique solution given by v∗j (t) = −B ∗ P j y ∗ (t), where P j ∈ L(X ) is a positive and self-adjoint operator verifying the Riccati equation:

Ay, P j y + P j y, Ay − B ∗ P j y, B ∗ P j y + R j y, y = 0

(4.70)

and we have q j (v∗j ) = P j y0 , y0 , ( see [6]). For y ∈ D(A), we have jχω j y2 −→ χ y2L 2 () when j −→ +∞ (Chilov

1967), it follows lim R j y, y = χ y2L 2 () , for all y ∈ D(A). j−→+∞

Now, for j large enough, we have

R j y, y ≤

3 2

lim R j y, y =

j−→+∞

3 χ y2 2  L 2 ()

Then for all v ∈ Va d ⊂ V q j (v) ≤ q(v) +

1 2



+∞ 0

χ y(t)2L 2 () dt

For v = v∗j , we obtain q j (v∗j )

≤

q(v∗j )

1 + 2

 0

+∞

χ y ∗j (t)2L 2 () dt

where y ∗j (t) is the associated solution to v∗j (t). Therefore 1 q(v∗j ) ≤ q(v∗ ) + q j (v∗j ), 2

4.5 Optimal Control for Regional Boundary Stabilization

109

and consequently P j y0 , y0  ≤ 2q(u ∗ ). This shows that the sequence P j y0 , y0  is bounded, and according to the Banach–Steinhaus theorem, the sequence (P j )n≥1 is uniformly bounded. Let P( j) be a subsequence which weakly converges (therefore converge strongly) to a positive and self-adjoint operator P verifying the Riccati equation

Ay, P y + P y, Ay − B ∗ P y, B ∗ P y + lim R j y, y = 0 j→+∞

(4.71)

The operator P is then the unique solution of the Eq. (4.60), therefore Pj converges strongly to P, and consequently the sequence v∗j (t) converges to v∗ (t) =  −B ∗ P y ∗ (t), as j → ∞ which is the solution of the problem (4.58). Example 4.14 We take again the system (4.67) and we will give an approximation of the optimal control v∗ (t) = −i  y ∗ (t) stabilizing the system (4.67) on the boundary region  = {(x, z) | z ≥ 0, x 2 + z 2 = 1}. Lets consider a sequence of regions ω j j ≥ 0 neighborhoods of  defined by ω j = {(x, z) ∈  | z ≥ 0 and (1 −

1 2 ) < x 2 + z 2 < 1} j

We propose to stabilize the system (4.67) on every ω j , j ≥ 1 by minimizing the functional  +∞  +∞ ||χω j y(t)||2 dt + ||v(t)||2V dt (4.72) q j (u) = j 0

0

√ The control v∗j (t) = − ji ω j y ∗j (t) minimizes (4.72) and strongly stabilizes (4.67) on ω j and the corresponding solution to y0 = 1 is given by y ∗j (t) =

⎧ −t ⎨e ⎩

et (x

on ω j 2

+z 2 −0.9)2

otherwise

√ Now by substituting y ∗j (t) in v∗j (t) we obtain v∗j (t) = je−t Pω j which is a sequence of controls that converges to v∗ (t). Therefore the sequence of optimal controls v∗j (t) converges to v∗ (t) the optimal control on .

References 1. Datko, R.: Extending a theorem of A. M. Liapunov to Hilbert space. J. Math. Anal. Appl. 32, 610–616 (1970) 2. Triggiani, R.: On the stabilizability problem in Banach space. J. Math. Anal. Appl. 52, 383–403 (1975) 3. Curtain, R.F., Zwart, H.J.: An Introduction to Infinite Dimensional Linear Systems Theory. Springer, New York (1995)

110

4 Regional Stabilization of Infinite Dimensional Linear Systems

4. Weiss, G.: Admissibility of unbounded control operators. SIAM J. Control Optim. 27, 527–545 (1989) 5. Brézis, H.: Analyse fonctionnelle, théorie et application. Masson, Paris (1992) 6. Zerrik, E., Ouzahra, M.: Regional stabilization for infinite-dimensional systems. Int. J. Control 76(1), 73–81 (2003)

Chapter 5

Regional Stabilization of Infinite Dimensional Bilinear Systems

The notion of regional stabilization of infinite dimensional systems consists in studying the asymptotic behaviour of such a system only within a subregion ω interior or in the boundary of the system evolution domain . This notion includes the classical one and enables us to analyse the behaviour of a distributed system in any subregion of its spatial domain. Also, it makes sense for the usual concept of stabilization taking into account the spatial variable and then becomes closer to real applications, where one wishes to stabilize a system in a critical subregion. The aim of this chapter is to study regional exponential, strong and weak stabilization of the bilinear system defined on a bounded and regular spatial domain  ⊂ Rn is described by  dy(t) = Ay(t) + u(t)By(t) (5.1) dt y(0) = y0 where the state space is L 2 () endowed with its usual inner product, denoted by ... and . the associated norm, the operator A generates a semigroup of contractions S(t) on L 2 (), u(t) ∈ L 2 (0, +∞) is a scalar valued control and B is a linear, bounded and positive operator mapping L 2 () into itself.

5.1 Regional Stabilization Consider system (5.1) evolving on an open and regular set  of Rn and let ω be a subregion of , χω : L 2 () −→ L 2 (ω) be the restriction operator to ω, while χω∗ denotes the adjoint operator of χω and given by

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. H. Zerrik and O. Castillo, Stabilization of Infinite Dimensional Systems, Studies in Systems, Decision and Control 355, https://doi.org/10.1007/978-3-030-68600-0_5

111

112

5 Regional Stabilization of Infinite Dimensional Bilinear Systems

(χω∗ y)(x)

 =

y(x), x ∈ ω 0, other wise

(5.2)

Assume that i ω B is positive where i ω = χω∗ χω . Regional stabilization consists in stabilizing the output z(t) = χω y(t). Let us recall a definition of regional stabilization for system (5.1). Definition 5.1 System (5.1) is said to be regionally weakly (respectively strongly, exponentially) stabilizable on ω, if for any initial condition y0 ∈ L 2 () the corresponding solution y(t) of (5.1) is unique and χω y(t) weakly (respectively strongly, exponentially) converges to 0 as t −→ +∞. Remark 5.1 1. The concept is general because when ω = , we obtain the classical definition of stability. 2. Note that regional exponential stabilization ⇒ regional strong stabilization ⇒ regional weak stabilization. In what follows, we will discuss regional exponential, strong and weak stabilization of system (5.1) considering the controls u(t) =

⎧ ⎨

i ω By(t), y(t) , y(t) = 0 y(t)2 ⎩ 0, y(t) = 0 −

(5.3)

and u(t) = −i ω By(t), y(t) Let us denote g(y) =

(5.4)

i ω By, y for all y = 0, g(0) = 0 and assume that y2 i ω Ay, y ≤ 0, ∀y ∈ D(A).

(5.5)

5.1.1 Regional Exponential Stabilization The following result gives sufficient conditions for regional exponential stabilization of system (5.1). Theorem 5.1 Let A generate a semigroup of contractions (S(t))t≥0 and B satisfy  0

T

|i ω B S(s)y, S(s)y|ds ≥ δχω y L 2 (ω) , ( f or some T, δ > 0)

then control (5.3) regionally exponentially stabilizes system (5.1) on ω.

(5.6)

5.1 Regional Stabilization

113

Proof System (5.1) has a unique mild solution y(t) on L 2 () (see [1]) defined on a maximal interval [0, tmax [ by the variation of constants formula 

t

y(t) = S(t)y0 −

g(y(s))S(t − s)By(s)ds

(5.7)

0

Let us consider the nonlinear semigroup (t)y0 := y(t). Since S(t) is a semigroup of contractions ( A is dissipative), then d (t)y0 2 ≤ −2g((t)y0 )B(t)y0 , (t)y0  dt Integrating this inequality, we get  (t)y0 2 ≤ y0 2 − 2

t

g((s)y0 )B(s)y0 , (s)y0 ds, ∀t ∈ [0, tmax [

0

Since B and i ω B are positive, then (t)y0  ≤ y0 .

(5.8)

For all y0 ∈ L 2 () and t ≥ 0, we have i ω B S(t)y0 , S(t)y0  = i ω B(S(t)y0 − (t)y0 ), S(t)y0  − i ω B(t)y0 , (t)y0 − S(t)y0  + i ω B(t)y0 , (t)y0  Since χω is continuous, then there exists α > 0 such that |i ω B S(t)y0 , S(t)y0 | ≤ 2αB(t)y0 − S(t)y0 y0  + |i ω B(t)y0 , (t)y0 |

(5.9)

It follows from (5.7), that 

t

(t)y0 − S(t)y0  ≤ B

|g((s)y0 )|(s)y0 ds

0

For a fixed T ∈]0, tmax [, Schwartz’s inequality yields   (t)y0 − S(t)y0  ≤ B T

T

|g((s)y0 )|2 (s)y0 2 ds

21

, ∀t ∈ [0, T ]

0

(5.10)

Using (5.8), we get |i ω B(t)y0 , (t)y0 | ≤ |g((t)y0 )|(t)y0 y0 , ∀t ∈ [0, T ]

(5.11)

114

5 Regional Stabilization of Infinite Dimensional Bilinear Systems

Integrating (5.9) over the interval [0, T ] and taking into account (5.10) and (5.11), we obtain  T  T 21 3 |i ω B S(s)y0 , S(s)y0 |ds ≤ 2αT 2 B2 y0  |g((s)y0 )|2 (s)y0 2 ds 0 0  T 21 √ + T y0  |g((s)y0 )|2 (s)y0 2 ds 0

(5.12) Replacing y0 by (t)y0 in (5.12), using a variable change and the superposition property of the semigroup (t), we deduce that 

T 0

3

|i ω B S(s)(t)y0 , S(s)(t)y0 |ds ≤ 2αT 2 B2 (t)y0  21   t+T |g((s)y0 )|2 (s)y0 2 ds × √t + T (t)y0  21   t+T |g((s)y0 )|2 (s)y0 2 ds × t

Using (5.6) and (5.8), we obtain δχω (t)y0 

L 2 (ω)

≤β



t+T

|g((s)y0 )|2 (s)y0 2 ds

21

t

√

where β = (2αT B2 + 1) T y0 . Then  2 2 2 δ χω (t)y0  L 2 (ω) ≤ β

t+T

|g((s)y0 )|2 (s)y0 2 ds

(5.13)

t

Using (5.5), we get d χω (t)y0 2L 2 (ω) ≤ −2|g((t)y0 )|2 (t)y0 2 dt

(5.14)

integrating this inequality from kT to (k + 1)T , (k ∈ N), we obtain χω (kT )y0 2L 2 (ω) − χω ((k + 1)T )y0 2L 2 (ω) ≥ 2

 (k+1)T kT

|g((s)y0 )|2 (s)y0 2 ds

Using (5.13) and (5.14), we deduce

δ 2  1+2 χω ((k + 1)T )y0 2L 2 (ω) ≤ χω (kT )y0 2L 2 (ω) β Then

5.1 Regional Stabilization

115

χω ((k + 1)T )y0  L 2 (ω) ≤ ρχω (kT )y0  L 2 (ω) 1

where ρ =

1 . (1 + 2( βδ )2 ) 2 By recurrence, we show that

χω (kT )y0  L 2 (ω) ≤ ρ k χω y0  L 2 (ω) Taking k = E( Tt ) the integer part of

t T

, we conclude that

χω (t)y0  L 2 (ω) ≤ K e−σ t y0 , ∀t ≥ 0 ln(1 + 2( βδ )2 ) 1 where K = α(1 + 2( βδ )2 ) 2 and σ = , which shows the regional expo2T nential stability of system (5.1) on ω. 

5.1.2 Regional Weak Stabilization Here, we give sufficient conditions that allow regional weak stabilization of system (5.1) using control (5.3). Theorem 5.2 Let A generate a semigroup of contractions (S(t))t≥0 and B be compact such that (5.15) i ω B S(t)z, S(t)z = 0, ∀t ≥ 0 =⇒ χω z = 0 then control (5.3) regionally weakly stabilizes system (5.1) on ω. Proof Let us consider the nonlinear semigroup (t)y0 = y(t) and let (tn ) be a sequence of real numbers such that tn −→ +∞ as n −→ +∞. By (5.8), (tn )y0 is bounded in the Hilbert space L 2 (), then there exists a subsequence (tφ(n) ) of (tn ) and z ∈ L 2 () such that (tφ(n) )y0 z as n −→ +∞ For all n ≥ 0, we set  λ(tφ(n) ) =

tφ(n) +T

|g((s)y0 )|2 (s)y0 2 ds.

tφ(n)

Integrating (5.14), we get 

t

2 0

|g((s)y0 )|2 (s)y0 2 ds ≤ χω y0 2L 2 (ω) − χω (t)y0 2L 2 (ω) .

116

5 Regional Stabilization of Infinite Dimensional Bilinear Systems



Then

t

|g((s)y0 )|2 (s)y0 2 ds ≤

0



It follows that

+∞

1 χω y0 2L 2 (ω) . 2

|g((s)y0 )|2 (s)y0 2 ds < +∞.

0

We deduce that λ(tφ(n) ) −→ 0 as n −→ +∞. Replacing y0 by (tφ(n) )y0 in (5.12) and using the superposition property of the semigroup (tφ(n) ), we deduce that 

T

|i ω B S(s)(tφ(n) )y0 , S(s)(tφ(n) )y0 |ds ≤ (2αB2 T + 1)y0  T λ(tφ(n) )

0

(5.16)

Since λ(tφ(n) ) −→ 0 as n −→ +∞, then  lim

T

n−→+∞ 0

i ω B S(s)(tφ(n) )y0 , S(s)(tφ(n) )y0 ds = 0

Since B is compact and S(s) is continuous ∀s ≥ 0, we have lim i ω B S(s)(tφ(n) )y0 , S(s)(tφ(n) )y0  = i ω B S(s)z, S(s)z

n−→+∞

By dominated convergence theorem, we get 

T

|i ω B S(s)z, S(s)z|ds = 0

0

It follows that i ω B S(s)z, S(s)z = 0, ∀s ∈ [0, T ] Using (5.15), we deduce that χω (tφ(n) )y0 0 as n −→ +∞ Which implies that ∀φ ∈ L 2 (), χω (tn )y0 , φ −→ 0 as n −→ +∞, and hence χω (t)y0 0 as t −→ +∞. In other words χω y(t) converges weakly to 0 as t −→ +∞, and then system (5.1) is regionally weakly stabilizable on ω.  Remark 5.2 In the case ω = , we retrieve the result established in [2] about the weak stabilization of system (5.1) on the whole domain .

5.1 Regional Stabilization

117

Example 5.1 On  =]0, 1[, we consider the heat equation defined by ⎧ ∂ y(x, t) ∂ 2 y(x, t) ⎪ ⎨ + u(t)By(x, t), ×]0, +∞[ = ∂t ∂x2 0) = y0 ,  ⎪ ⎩ y(x, ∂×]0, +∞[ y  (0, t) = y  (1, t) = 0,

(5.17)

Let us denote Ay = y, ∀y ∈ D(A) = {y ∈ H 2 ()| y  (0, t) = y  (1, t) = 0}. The eigenvalues and corresponding eigenfunctions of A are given by λk = −(kπ )2 √ and ϕk (x) = 2 cos(kπ x) for all k ∈ N. +∞  1 The operator of control defined by By = i ω y, ϕk ϕk is linear and compact. k2 k=1 The operator A generates the semigroup S(t)y =

+∞ 

eλk t y, ϕk ϕk .

k=0

For ω =]0, 21 [, we have

i ω B S(t)y, S(t)y = 0, ∀t ≥ 0 =⇒ χω y = 0. So (5.15) holds. Consequently, system (5.17) is regionally weakly stabilizable on ω by the control

u(t) = −

+∞  1 |i ω y(t), ϕk |2 2 k k=1

y(t)2

.

5.1.3 Regional Strong Stabilization In this subsection we give sufficient conditions for regional strong stabilization. Theorem 5.3 Suppose that A generates a semigroup of contractions (S(t))t≥0 and B satisfies  0

T

|i ω B S(s)y, S(s)y|ds ≥ δχω y2L 2 (ω) , ( f or some T, δ > 0)

then control (5.4) regionally strongly stabilizes system (5.1) on ω. Proof Using (5.5), we obtain d χω y(t)2L 2 (ω) ≤ −2|i ω By(t), y(t)|2 , ∀t ≥ 0. dt

(5.18)

118

5 Regional Stabilization of Infinite Dimensional Bilinear Systems

Integrating this inequality, we get 

t

|i ω By(s), y(s)|2 ds ≤

0



Hence

+∞

1 χω y0 2L 2 (ω) , ∀t ≥ 0. 2

|i ω By(s), y(s)|2 ds < +∞.

(5.19)

0

 Letting (t) =

t

S(t − s)u(s)By(s)ds, we obtain the following relation

0

i ω B S(t)y0 , S(t)y0  = i ω B(S(t)y0 − y(t)), S(t)y0  − i ω By(t), (t) +i ω By(t), y(t) Since χω is continuous, then there exists α > 0 such that |i ω B S(t)y0 , S(t)y0 | ≤ αB(t)(S(t)y0  + y(t)) +|i ω By(t), y(t)|

(5.20)

Then |i ω B S(t)y0 , S(t)y0 | ≤ 2αB(t)y0  + |i ω By(t), y(t)|, ∀t ∈ [0, T ]. Moreover, we have √  (t) ≤ By0  T

T

|i ω By(s), y(s)|2 ds

 21

(5.21)

0

Integrating (5.20) over the interval [0, T ] and taking into account (5.21), we obtain 

T

0

 T  21 3 |i ω B S(s)y0 , S(s)y0 |ds ≤ 2αT 2 y0 2 B |i ω By(s), y(s)|2 ds 0

 T  21 √ + T y0  |i ω By(s), y(s)|2 ds . 0

Replacing y0 by y(t) and using the superposition property of the solution, we get  0

T

 t+T  21 3 |i ω B S(s)y(t), S(s)y(t)|ds ≤ 2αT 2 y0 B |i ω By(s), y(s)|2 ds

 t+T t  21 √ + T y0  |i ω By(s), y(s)|2 ds . t

From (5.19), we have

5.1 Regional Stabilization



t+T

119

|i ω By(s), y(s)|2 ds −→ 0, as t −→ +∞.

t

It follows that 

T

|i ω B S(s)y(t), S(s)y(t)|ds −→ 0, as t −→ +∞.

(5.22)

0

From (5.18) and (5.22), we deduce that χω y(t) L 2 (ω) −→ 0, as t −→ +∞, which completes the proof.  Remark 5.3 When ω = , we retrieve the result established in [3] about the strong stabilization of system (5.1). Example 5.2 On  =]0, 1[, we consider the following beam equation ⎧ 2 ∂ y ∂4 y ∂y ⎪ ⎨ (x, t) = − (x, t) + u(t) (x, t) on ×]0, +∞[ 4 ∂t 2 ∂ x ∂t ∂2 y ⎪ ⎩ y(ξ, t) = (ξ, t) = 0, ξ = 0, 1 on ]0, +∞[ ∂x2

(5.23)

∂4 y ∂4 y ∂2 y 2 2 with domain D(P) = {y ∈ L (0, 1)/ ∈ L (0, 1), y(ξ, t) = ∂x4 ∂x4 ∂x2 (ξ, t) = 0, ξ = 0, 1}. 1 The set D(P 2 ) forms a Hilbert space under the inner product y1 , y2  P = 1 1 P 2 y1 , P 2 y2  L 2 (0,1) . 4 √ The eigenvalues of∗ P are λ j = ( jπ ) , corresponding to eigenfunctions ϕ j (x) = 2 sin( jπ x) ∀ j ∈ N . To write system (5.23) in the form (5.1) we set H = (H 2 (0, 1) ∩ H01 (0, 1)) × 2 L (0, 1) endowed with the inner product (y1 , z 1 ), (y2 , z 2 ) = y1 , y2  P + z 1 , z 2  L 2 (0,1) and the operators Let P =



0 I A= −P 0





00 and B = 0I

 .

A is skew-adjoint,   and B satisfies condition (5.18), indeed, for y ∈ H , we have ∞  αj 1 ϕ j , where (α j , β j ) ∈ R2 ∀ j ≥ 1. Then y= 2 λ β j j j=1 S(s)y =

∞ 

1

1

α j cos(λ j2 s) + β j sin(λ j2 s) 1

j=1

It follows that



1

1

1

β j λ j2 cos(λ j2 s) − α j λ j2 sin(λ j2 s)

 ϕ j , ∀s ≥ 0.

120

5 Regional Stabilization of Infinite Dimensional Bilinear Systems

i ω B S(s)y, S(s)y =

∞ 

1 1   λ j α j sin(λ j2 s) − β j cos(λ j2 s)

j=1

=

∞ 

  λ j α 2j sin2 ( jπ s) − β 2j cos2 ( jπ s) − sin(2 jπ s)β j α j .

j=1

Integrating this relation over the time interval [0, 2], we obtain 

2

|i ω B S(s)y, S(s)y|ds =

0

∞ 

λ j (α 2j + β 2j ) = χω y2L 2 (ω)

j=1

then (5.18) holds. We conclude that the control u(t) = −χω ∂t y(., t)2L 2 (ω) regionally strongly stabilizes system (5.23).



5.2 Regional Stabilization Problem The aim of this section is to characterize a control that stabilizes regionally system (5.1) and minimizes a given performance cost. We study the case when i ω A generates a semigroup of isometries and the one when A generates a semigroup of contractions.

5.2.1 Case of Isometries Semigroup Consider the problem ⎧ ⎨ ⎩



 +∞ i ω By(t), y(t)2 dt + y(t)2 |u(t)|2 dt y(t)2 0 0 = {u ∈ L 2 (0, +∞)|y(t) is a global solution and J (u) < +∞}. (5.24)

min J (u) = u ∈ Uad

+∞

Theorem 5.4 Let i ω A generate a semigroup of isometries (S(t))t≥0 and assume that 

T 0

|i ω B S(t)y, S(t)y|dt ≥ δχω y L 2 (ω) , ( f or some T, δ > 0)

then the control

(5.25)

5.2 Regional Stabilization Problem

121

u ∗ (t) = −

i ω By(t), y(t) y(t)2

(5.26)

is the unique solution of (5.24) and regionally exponentially stabilizes system (5.1) on ω. Proof Since S(t) is a semigroup of isometries, then d |i ω By(t), y(t)|2 χω y(t)2L 2 (ω) = −2 , ∀t ≥ 0 dt y(t)2

(5.27)

Integrating (5.27), we obtain 

t 0

|i ω By(s), y(s)|2 1 ds ≤ χω y0 2L 2 (ω) , ∀t ≥ 0 2 y(s) 2

(5.28)

which is verified ∀y0 ∈ L 2 (), then J (u ∗ ) < +∞ and u ∗ ∈ Uad . Let y0 ∈ D(A), from (5.27), we deduce that d i ω By(t), y(t) i ω By(t), y(t)2 χω y(t)2L 2 (ω) = y(t)2 (| + u(t)|2 − − u(t)2 ), 2 dt y(t) y(t)4

Then for all t ≥ 0, we have 

t

y(s)2 (

0

i ω By(s), y(s)2 + u(s)2 )ds + χω y(t)2L 2 (ω) − χω y0 2L 2 (ω) 4 y(s)  t i ω By(s), y(s) = y(s)2 | + u(s)|2 ds. 2 y(s) 0 (5.29)

The solution y(.) is continuous with respect to initial condition (see [1]) and D(A) is dense in L 2 (), then (5.29) holds ∀y0 ∈ L 2 (). From Theorem 5.1, we deduce that control (5.26) regionally exponentially stabilizes system (5.1) on ω. Thus, as t −→ +∞ in (5.29), we obtain  J (u) = χω y0 2L 2 (ω) +

0

+∞

y(s)2 |

i ω By(s), y(s) + u(s)|2 ds y(s)2

(5.30)

Then J (u) ≥ χω y0 2L 2 (ω) = J (u ∗ ), which implies that (5.26) is the unique solution of (5.24).  For the next result, we consider this problem ⎧ ⎨ ⎩

 min J (u) = 0

+∞

 i ω By(t), y(t)2 dt + 0

+∞

|u(t)|2 dt

u ∈ Uad = {u ∈ L 2 (0, +∞)|y(t) is a global solution and J (u) < +∞}. (5.31)

122

5 Regional Stabilization of Infinite Dimensional Bilinear Systems

Proposition 5.1 Suppose that i ω A generates a semigroup of isometries (S(t))t≥0 and  T |i ω B S(s)y, S(s)y|ds ≥ δχω y2L 2 (ω) , ( f or some T, δ > 0) (5.32) 0

then the control

u ∗ (t) = −i ω By(t), y(t)

(5.33)

is the unique solution of (5.31) and regionally strongly stabilizes system (5.1) on ω. Proof Since S(t) is a semigroup of isometries, then d χω y(t)2L 2 (ω) = −2|i ω By(t), y(t)|2 , ∀t ≥ 0 dt

(5.34)

Integrating (5.34), we obtain 

t

|i ω By(s), y(s)|2 ds ≤

0

1 χω y0 2L 2 (ω) , ∀t ≥ 0 2

which is verified ∀y0 ∈ L 2 (), then J (u ∗ ) < +∞ and u ∗ ∈ Uad . Let y0 ∈ D(A), from (5.34), we deduce that d χω y(t)2L 2 (ω) = |i ω By(t), y(t) + u(t)|2 − i ω By(t), y(t)2 − u(t)2 , dt Then for all t ≥ 0, we have 

t 0

(i ω By(s), y(s)2 + u(s)2 )ds + χω y(t)2L 2 (ω) − χω y0 2L 2 (ω)  t = |i ω By(s), y(s) + u(s)|2 ds.

(5.35)

0

The solution y(.) is continuous with respect to initial condition (see [1]) and D(A) is dense in L 2 (), then (5.35) holds ∀y0 ∈ L 2 (). To prove that (5.33) is optimal, let u ∈ Uad , t ≥ 0 and define z(τ ) = y(τ ) − S(τ − t)y(t), ∀τ ∈ [t, t + T ]. Applying the variation of constants formula with y(t) as initial state, 

τ

y(τ ) = S(τ − t)y(t) +

u(s)S(τ − s)By(s)ds, ∀τ ∈ [t, t + T ].

t

Therefore, ∀τ ∈ [t, t + T ], we have

(5.36)

5.2 Regional Stabilization Problem

123



τ

z(τ ) =

u(s)S(τ − s)By(s)ds

t

and using Schwartz’s inequality, we obtain z(τ ) ≤ By(t) T μ(t) ∀τ ∈ [t, t + T ], 

t+T

where μ(t) :=

(5.37)

|u(τ )|2 dτ .

t

Moreover, using (5.36) and Gronwall’s inequality, there exists u max ≥ |u(τ )| such that (5.38) y(τ ) ≤ y(t)eBT u max ∀τ ∈ [t, t + T ], Schwartz’s inequality and (5.38) give 

t+T

|i ω By(τ ), y(τ )|dτ ≤ My(t) γ (t),

(5.39)

t



t+T

where γ (t) = We have

|i ω By(τ ), y(τ )|2 dτ and M =

√

T eBT u max .

t

i ω B S(τ − t)y(t), S(τ − t)y(t) = i ω B(S(τ − t)y(t) − y(τ )), S(τ − t)y(t) − i ω Bz(τ ), y(τ ) + i ω By(τ ), y(τ ) Then there exists α > 0 such that |i ω B S(τ − t)y(t), S(τ − t)y(t)| ≤ αBy(t)z(τ ) + αBy(τ )z(τ ) + |i ω By(τ ), y(τ )|. Using (5.37) and (5.38), we deduce √ |i ω B S(τ − t)y(t), S(τ − t)y(t)| ≤ α(1 + eBT u max )y(t)B2 T μ(t) + |i ω By(τ ), y(τ )|. Integrating this inequality over [0, T] and using (5.39), we get 

T

|i ω B S(τ )y(t), S(τ )y(t)|dτ ≤ N y(t) μ(t) + My(t) γ (t),

0

√ where N = α(1 + eBT u max )B2 T T . It follows from (5.32) and (5.40) that δχω y(t)2L 2 (ω) ≤ y0 (N μ(t) + M γ (t)),

(5.40)

124

5 Regional Stabilization of Infinite Dimensional Bilinear Systems

Hence χω y(t) L 2 (ω) −→ 0 as t −→ +∞. Then control (5.33) regionally strongly stabilizes system (5.1) on ω. Thus, as t −→ +∞ in (5.35), we obtain  J (u) =

χω y0 2L 2 (ω)

+

+∞

|i ω By(s), y(s) + u(s)|2 ds

0

Then J (u) ≥ χω y0 2L 2 (ω) = J (u ∗ ), which implies that control (5.33) is the unique solution of (5.31). 

5.2.2 Case of Contractions Semigroup Here, we consider the problem ⎧ ⎨



 +∞ Pω By(t), y(t)2 min J (u) = dt + y(t)2 |u(t)|2 dt 2 y(t) 0 0 ⎩ u ∈ Uad = {u ∈ L 2 (0, +∞)|y(t) is a global solution and J (u) < +∞}. (5.41) +∞

where Pω = i ω Pi ω with P is a self-adjoint, positive and bounded operator which satisfies the equation Pω Ay, y + y, Pω Ay = 0, y ∈ D(A)

(5.42)

Theorem 5.5 Let A generate a semigroup of contractions (S(t))t≥0 and assume that 

T 0

|Pω B S(t)y, S(t)y|dt ≥ δχω y L 2 (ω) , ( f or some T, δ > 0).

(5.43)

and there exists η > 0 such that Pω By, y ≤ ηi ω By, y then the control u ∗ (t) = −

Pω By(t), y(t) y(t)2

(5.44)

(5.45)

is the unique solution of (5.41) and regionally exponentially stabilizes system (5.1) on ω. Proof Let us define the function V (y) = Pω y, y, ∀y ∈ L 2 ().

5.2 Regional Stabilization Problem

125

For all y0 ∈ D(A) and t ≥ 0, using (5.42), we obtain d V (y(t)) Pω By(t), y(t)2 = −2 dt y(t)2 Integrating this relation, we get 

t 0

Pω By(s), y(s)2 1 ds ≤ V (y0 ), t ≥ 0 y(s)2 2

(5.46)

The solution y(.) is continuous with respect to the initial condition y0 (see [1]) and D(A) is dense in L 2 (), then (5.46) holds for all y0 ∈ L 2 () so J (u ∗ ) is finite for all y0 ∈ L 2 (). Now, let us consider the nonlinear semigroup (t)y0 := y(t). For all y0 ∈ L 2 () and t ≥ 0, we have Pω B S(t)y0 , S(t)y0  = Pω B(S(t)y0 − (t)y0 ), S(t)y0  − Pω B(t)y0 , (t)y0 − S(t)y0  + Pω B(t)y0 , (t)y0  Since χω is continuous, there exists α > 0 such that |Pω B S(t)y0 , S(t)y0 | ≤ 2αPB(t)y0 − S(t)y0 y0  + |Pω B(t)y0 , (t)y0 |

(5.47)

Moreover, we have 

T

(t)y0 − S(t)y0  ≤ B 0

|Pω B(s)y0 , (s)y0 | ds (s)y0 

Schwartz’s inequality yields   (t)y0 − S(t)y0  ≤ B T

T 0

|Pω B(s)y0 , (s)y0 |2 21 ds , ∀t ∈ [0, T ] (s)y0 2 (5.48)

Using (5.8), we get |Pω B(t)y0 , (t)y0 | ≤

|Pω B(t)y0 , (t)y0 | y0 , ∀t ∈ [0, T ] (t)y0 

Integrating (5.47) over the interval [0, T ] and taking into account (5.48), we obtain

126

5 Regional Stabilization of Infinite Dimensional Bilinear Systems



T

3

|Pω B S(s)y0 , S(s)y0 |ds ≤ 2αPB2 T 2 y0  0   T |P B(s)y , (s)y |2 21 ω 0 0 ds × 2 (s)y  0 0  T  √ |Pω B(s)y0 , (s)y0 |2 21 + T y0  ds (s)y0 2 0 (5.49) Replacing y0 by (t)y0 in (5.49) and using the superposition property of the semigroup (t), we deduce that 

T 0

3

|Pω B S(s)(t)y0 , S(s)(t)y0 |ds ≤ 2αPB2 T 2 (t)y0    t+T |P B(s)y , (s)y |2 1 2 ω 0 0 × ds 2 (s)y  0 t   t+T |P B(s)y , (s)y |2 1 √ 2 ω 0 0 + T (t)y0  ds (s)y0 2 t

Using (5.43) and (5.8), we obtain δχω (t)y0  L 2 (ω) ≤ β



t+T t

|Pω B(s)y0 , (s)y0 |2 21 ds (s)y0 2

√

with β = (2αPB2 T + 1) T y0 . Then  δ 2 χω (t)y0 2L 2 (ω) ≤ β 2

t+T

t

|Pω B(s)y0 , (s)y0 |2 ds (s)y0 2

(5.50)

Using (5.44) and (5.5), we get d 2 |Pω B(t)y0 , (t)y0 |2 χω (t)y0 2L 2 (ω) ≤ − 2 dt η (t)y0 2 Integrating this inequality from kT to (k + 1)T , (k ∈ N), we obtain χω (kT )y0 2L 2 (ω) − χω ((k + 1)T )y0 2L 2 (ω) ≥

2 η2



(k+1)T

kT

|Pω B(s)y0 , (s)y0 |2 ds (s)y0 2

Using (5.50) and since χω (t)y0  L 2 (ω) decreases, we deduce

δ 2  1+2 χω ((k + 1)T )y0 2L 2 (ω) ≤ χω (kT )y0 2L 2 (ω) ηβ Then χω ((k + 1)T )y0  L 2 (ω) ≤ ρχω (kT )y0  L 2 (ω)

5.2 Regional Stabilization Problem

where ρ =

127

1

. δ 2 21 (1 + 2( ηβ ) ) By recurrence, we show that χω (kT )y0  L 2 (ω) ≤ ρ k χω y0  L 2 (ω)

Taking k = E( Tt ) the integer part of

t T

, we conclude that

χω (t)y0  L 2 (ω) ≤ K e−σ t y0 , ∀t ≥ 0 δ 2 21 2( ηβ ) )

(5.51)

δ 2 ln(1 + 2( ηβ ) )

where K = α(1 + , then control (5.45) regionally and σ = 2T exponentially stabilizes system (5.1) on ω. Now, to prove that (5.45) is the unique solution of (5.41), we have V (y(t)) ≤ αPχω y(t)2L 2 (ω) , it follows from (5.51) that V (y(t)) −→ 0 as t → +∞. Let y0 ∈ D(A), integrating the relation Pω By(t), y(t) Pω By(t), y(t)2 d V (y(t)) 2 = y(t)2 ([ + u(t)] − − u 2 (t)) dt y(t)2 y(t)4 (5.52) we have  +∞ Pω By(s), y(s) y(s)2 [ + u(s)]2 ds J (u) = V (y0 ) + y(s)2 0 then J (u) ≥ V (y0 ). Setting u = u ∗ , we obtain J (u ∗ ) = V (y0 ). Let y0 ∈ L 2 (), and a sequence y0n ⊂ D(A) such that y0n −→ y0 as n −→ +∞, we have  +∞ Pω Byn (s), yn (s) yn (s)2 [ + u(s)]2 ds J (u) = V (y0n ) + 2 y (s) n 0 Thus J (u) ≥ V (y0n ). We deduce that J (u) ≥ V (y0 ) = J (u ∗ ). Hence control (5.45) is the unique solution of the problem (5.41).  Proposition 5.2 Suppose that A generates a semigroup of contractions (S(t))t≥0 , P and B are compact such that Pω B S(t)z, S(t)z = 0, ∀t ≥ 0 =⇒ χω z = 0

(5.53)

then control (5.45) is the unique solution of (5.41) and regionally weakly stabilizes system (5.1) on ω.

128

5 Regional Stabilization of Infinite Dimensional Bilinear Systems

Proof Let us consider the nonlinear semigroup (t)y0 = y(t) and let (tn ) be a sequence of real numbers such that tn −→ +∞ as n −→ +∞. From (5.8), (tn )y0 is bounded in the Hilbert space L 2 (), then there exists a subsequence tφ(n) of (tn ) and z ∈ L 2 () such that (tφ(n) )y0 z as n −→ +∞. For all n ≥ 0, we set  f (tφ(n) ) =

tφ(n) +T

tφ(n)

|Pω By(s), y(s)|2 ds. y(s)2

From (5.46), we have 

+∞

0

Pω By(s), y(s)2 ds < +∞ y(s)2

We deduce that f (tφ(n) ) −→ 0 as n −→ +∞. Replacing y0 by (tφ(n) )y0 in (5.49) and using the superposition property of the semigroup (t), we deduce that  T 0

 |Pω B S(s)(tφ(n) )y0 , S(s)(tφ(n) )y0 |ds ≤ (2αPB2 T + 1)y0  T f (tφ(n) )

(5.54)

Since f (tφ(n) ) −→ 0 as n −→ +∞, then  lim

n−→+∞ 0

T

Pω B S(s)(tφ(n) )y0 , S(s)(tφ(n) )y0 ds = 0

Since B and P are compact and S(s) is continuous ∀s ≥ 0, then lim Pω B S(s)(tφ(n) )y0 , S(s)(tφ(n) )y0  = Pω B S(s)z, S(s)z

n−→+∞

By dominated convergence theorem, we obtain 

T

|Pω B S(s)z, S(s)z|ds = 0 andthen Pω B S(s)z, S(s)z = 0, ∀s ∈ [0, T ]

0

Using (5.53), we deduce that χω (tφ(n) )y0 0 as n −→ +∞ Which implies that ∀φ ∈ L 2 (), χω (tn )y0 , φ −→ 0 as n −→ +∞, and hence

5.2 Regional Stabilization Problem

129

χω (t)y0 0 as t −→ +∞. In means that χω y(t) converges weakly to 0 as t −→ +∞, and then system (5.1) is regionally weakly stabilizable on ω. Now, let us prove that (5.45) is the unique solution of (5.41). Since P is compact, it follows that V (y(t)) −→ 0 as t → +∞. Let y0 ∈ D(A), integrating the relation (5.52), we obtain 

+∞

J (u) = V (y0 ) + 0

y(s)2 [

Pω By(s), y(s) + u(s)]2 ds y(s)2

then J (u) ≥ V (y0 ), ∀u ∈ Uad . For u = u ∗ , we obtain J (u ∗ ) = V (y0 ). Let y0 ∈ L 2 (), and a sequence y0n ⊂ D(A) such that y0n −→ y0 as n −→ +∞, we have  +∞ Pω Byn (s), yn (s) yn (s)2 [ + u(s)]2 ds J (u) = V (y0n ) + yn (s)2 0 Thus J (u) ≥ V (y0n ). We deduce that J (u) ≥ V (y0 ) = J (u ∗ ), which implies that (5.45) is the unique solution of problem (5.41).  For the next result, we consider the problem ⎧ ⎨ ⎩

 min J (u) = 0

+∞



+∞

Pω By(t), y(t)2 dt +

|u(t)|2 dt

0

u ∈ Uad = {u ∈ L 2 (0, +∞)|y(t) is a global solution and J (u) < +∞}. (5.55)

Proposition 5.3 Let A generate a semigroup of contractions (S(t))t≥0 and assume that  T |Pω B S(t)y, S(t)y|dt ≥ δχω y2L 2 (ω) , ( f or some T, δ > 0) (5.56) 0

then the control

u ∗ (t) = −Pω By(t), y(t)

(5.57)

is the unique solution of (5.55) and regionally strongly stabilizes system (5.1) on ω. Proof Let us define the function V (y) = Pω y, y, ∀y ∈ L 2 (). For all y0 ∈ D(A) and t ≥ 0, using (5.42), we obtain

130

5 Regional Stabilization of Infinite Dimensional Bilinear Systems

d V (y(t)) = −2Pω By(t), y(t)2 dt Integrating this relation, we get 

t

Pω By(s), y(s)2 ds ≤

0

1 V (y0 ), t ≥ 0 2

(5.58)

The solution y(.) is continuous with respect to the initial condition y0 (see [1]) and D(A) is dense in L 2 (), then (5.58) holds for all y0 ∈ L 2 () so J (u ∗ ) is finite for all y0 ∈ L 2 (). For all y0 ∈ L 2 () and t ≥ 0, we have Pω B S(t)y0 , S(t)y0  = Pω B(S(t)y0 − y(t)), S(t)y0  −Pω By(t), (t) + Pω By(t), y(t) Since χω is continuous, there exists α > 0 such that |Pω B S(t)y0 , S(t)y0 | ≤ αPB(t)(S(t)y0  + y(t)) +|Pω By(t), y(t)|. 

t

where (t) =

(5.59)

S(t − s)Pω By(s), y(s)By(s)ds.

0

Moreover, we have √  (t) ≤ By0  T

T

|Pω By(s), y(s)|2 ds

 21

(5.60)

0

Integrating (5.59) over the interval [0, T ] and taking into account (5.60), we obtain 

T 0

 T  21 3 |Pω B S(s)y0 , S(s)y0 |ds ≤ 2αT 2 y0 2 B |Pω By(s), y(s)|2 ds 0

 T  21 √ + T y0  |Pω By(s), y(s)|2 ds . 0

Replacing y0 by y(t) and using the superposition property of the solution, we get

 t+T 1 3 2 |Pω B S(s)y(t), S(s)y(t)|ds ≤ 2αT 2 y0 B |Pω By(s), y(s)|2 ds 0 t

 t+T 1 √ 2 + T y0  |Pω By(s), y(s)|2 ds .

 T

t

From (5.58), we have

5.2 Regional Stabilization Problem



t+T

131

|Pω By(s), y(s)|2 ds −→ 0, as t −→ +∞.

t

It follows that 

T

|Pω B S(s)y(t), S(s)y(t)|ds −→ 0, as t −→ +∞.

(5.61)

0

From (5.56) and (5.61), we deduce that χω y(t) L 2 (ω) −→ 0, as t −→ +∞. Then the control (5.57) regionally strongly stabilizes system (5.1) on ω. Now, let us show that control (5.57) is the unique solution of (5.55). We have V (y(t)) ≤ αPχω y(t)2L 2 (ω) , then V (y(t)) −→ 0 as t → +∞. Let y0 ∈ D(A), integrating the relation d V (y(t)) = [Pω By(t), y(t) + u(t)]2 − Pω By(t), y(t)2 − u 2 (t) dt 

we obtain

+∞

J (u) = V (y0 ) +

[Pω By(s), y(s) + u(s)]2 ds

0

Setting u = u ∗ , we get J (u ∗ ) = V (y0 ). Let y0 ∈ L 2 (), and a sequence y0n ⊂ D(A) such that y0n −→ y0 as n −→ +∞, we have  +∞

J (u) = V (y0n ) +

[Pω Byn (s), yn (s) + u(s)]2 ds

0

thus J (u) ≥ V (y0n ). We deduce that J (u) ≥ V (y0 ) = J (u ∗ ), which implies that the unique solution of (5.55) is given by the control (5.57).  In order to illustrate the previous results numerically, we perform the following algorithm: Step 1: Initial data: initial condition y0 and subregion ω; Step 2: Solve the equation (5.42) using Bartels-Stewart method [4]; Step 3: Apply the control given by (5.3) or (5.45); Step 4: Solve system (5.1) using Petrov-Galerkin method given in [5]; Step 5:Calculate the stabilization cost J .

132

5 Regional Stabilization of Infinite Dimensional Bilinear Systems

5.3 Simulation Results On  =]0, 1[, we consider the system ⎧ ∂y ⎪ ⎪ ⎪ π 2 (x, t) = 0.01y(x, t) + u(t)By(x, t) ×]0, +∞[ ⎨ ∂t ∂y (1, t) = exp(−t), y(0, t) = 0 ⎪ ⎪ ⎪ ⎩ ∂x y(x, 0) = x(2x 2 − 3x + 1) where By =

(5.62)

+∞   1 2 i y, ϕ ϕ with ϕ (x) = cos( jπ x). ω j j j 1+( jπ)2 2 j j=1

1. Regional stabilizing control. The operator B is linear, compact and verifies the assumption (5.15), then control (5.3) regionally weakly stabilizes (5.62) on ω. Let ω =]0, 14 [ and applying the above algorithm, we obtain the following figures Figure 5.1 shows that system (5.62) is stabilized on the subregion ω, with a stabilization error equals to 1.02 × 10−4 (Fig. 5.2). For ω =]0, 21 [, we obtain Figure 5.3, shows that system (5.62) is stabilized on the subregion ω, with stabilization error 4.23 × 10−4 (Fig. 5.4). For ω =]0, 1[, we obtain Figure 5.5 shows that system (5.62) is stabilized on , with stabilization error 1.98 × 10−3 (Fig. 5.6). 2. Regional optimal stabilizing control. Consider system (5.62) and problem (5.41) such that Pω is the unique solution of the equation (5.42). Then control (5.45) is the unique solution of (5.41) and regionally

Fig. 5.1 Evolution of the state

5.3 Simulation Results Fig. 5.2 Evolution of the control

Fig. 5.3 Evolution of the state

Fig. 5.4 Evolution of the control

133

134

5 Regional Stabilization of Infinite Dimensional Bilinear Systems

Fig. 5.5 Evolution of the state

Fig. 5.6 Evolution of the control

weakly stabilizes system (5.62). Let ω =]0, 41 [, applying the above algorithm, we obtain the following figures Figure 5.7 shows how system (5.62) is stabilized by the control (5.45) on ω with a stabilization error equals to 8.87 10−5 and the cost J (u ∗ ) = 3.36 × 10−3 (Fig. 5.8). For ω = , Figure 5.9 shows that system (5.62) is stabilized on , with stabilization error 7.96 × 10−4 and the cost J (u ∗ ) = 1.64 × 10−2 (Fig. 5.10).

5.3 Simulation Results Fig. 5.7 Evolution of the state

Fig. 5.8 Evolution of the control

Fig. 5.9 Evolution of the state

135

136

5 Regional Stabilization of Infinite Dimensional Bilinear Systems

Fig. 5.10 Evolution of the control

The following table shows that there exists a relation between the area of target subregion ω, the cost and the error of stabilization. ω Error Cost

]0, 0.3[ 9.72×10−5 4.528×10−3

]0, 0.5[ 1.565×10−4 5.07×10−3

]0, 0.6[ 3.589×10−4 7.05×10−3

]0, 0.8[ 6.426×10−4 9.33×10−3

]0, 1[ 7.98×10−4 1.66×10−2

More the area of the subregion increases, more both the stabilization error and the cost increase.

References 1. Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Springer Verlag, New York (1983) 2. Ouzahra, M.: Exponential and weak stabilization of constrained bilinear systems. SIAM Journal on Control and Optimization 48(6), 3962–3974 (2010) 3. Berrahmoune, L. Asymptotic stabilization and decay estimate for distributed bilinear systems, Recerche di Matematica. 2001, fasc. 1, pp. 89-103 4. Penzl, T.: Numerical solution of generalized Lyapunov equations. Advances in Comp. Math 8, 33–48 (1998) 5. Skeel, R.D., Berzins, M.: A Method for the spatial discretization of parabolic equations in one space variable. SIAM J. ScI. Stat. Comput. 11, 1–32 (1990)

Chapter 6

Regional Stabilization of Infinite Dimensional Semilinear Systems

Semilinear systems is an important subclass of nonlinear systems and numerous realworld problems have a semilinear structure. They include applications in nuclear, thermal, chemical, social processes, etc [1]. Let  ⊂ Rn be an open bounded domain with regular boundary ∂, we consider the system  dy(t) = Ay(t) + u(t)By(t) (6.1) dt y(0) = y0 where the operator A generates a semigroup of contractions (S(t))t≥0 on the Hilbert space L 2 () with inner product ., . and norm ., u(t) ∈ L 2 (0, +∞) is a scalar valued control and B is a nonlinear operator mapping L 2 () into itself such that B(0) = 0. In this chapter, we study regional exponential, strong and weak stabilization of system (6.1). Moreover, we give a control that stabilizes regionally system (6.1) on ω minimizing a functional cost.

6.1 Regional Stabilization Here we deal with regional exponential, strong and weak stabilization of system (6.1). Let ω be an open subregion of  and denote by χω : L 2 () −→ L 2 (ω) the restriction operator to ω. Let us denote i ω = χω∗ χω with χω∗ is the adjoint operator. Assume that (6.2) i ω By, yBy, y ≥ 0, ∀y ∈ L 2 ().

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. H. Zerrik and O. Castillo, Stabilization of Infinite Dimensional Systems, Studies in Systems, Decision and Control 355, https://doi.org/10.1007/978-3-030-68600-0_6

137

138

6 Regional Stabilization of Infinite Dimensional Semilinear Systems

and i ω Ay, y ≤ 0, ∀y ∈ D(A).

(6.3)

6.1.1 Regional Exponential Stabilization The following result gives sufficient conditions for regional exponential stabilization of system (6.1). Theorem 6.1 Suppose that B is locally Lipschitz such that  0

T

|i ω B S(s)y, S(s)y|ds ≥ αχω y L 2 (ω) , ( f or some T, α > 0)

then the control

⎧ ⎨

i ω By(t), y(t) , y(t) = 0 u(t) = y(t)2 ⎩ 0, y(t) = 0 −

(6.4)

(6.5)

regionally exponentially stabilizes system (6.1) on ω. i ω By, y By is locally Lipschitz. y2 Since B is locally Lipshitz, for each R > 0 there exists a constant K > 0 such that

Proof First, let us show that the map h : y →

By − Bz ≤ K y − z, ∀y, z ∈ L 2 () : 0 < y ≤ z ≤ R.

(6.6)

Thus h(z)Bz − h(y)By =

y2 (i ω Bz, zBz − i ω By, yBy) (z2 − y2 )i ω By, yBy − . z2 y2 z2 y2

Since χω is continuous, there exists δ > 0 such that i ω Bz, zBz − i ω By, yBy y + K 2 δ|z2 − y2 |. z2 z2 i ω Bz, z(Bz − By) + (i ω Bz, z − y ≤ z2 i ω By − i ω Bz, y)By − + 2K 2 δz − y z2 ≤ K 2 δz2 z − y + K 2 δyz − y(y + z) + 2K 2 δz − y

h(z)Bz − h(y)By ≤

≤ 3K 2 δz2 z − y + 2K 2 δz − y.

We conclude that for all y, z ∈ L 2 (), we have h(z) − h(y) ≤ Lz − y where L = 3K 2 δ R 2 + 2K 2 δ.

6.1 Regional Stabilization

139

It follows that system (6.1) has a unique global mild solution y(t) defined on a maximal interval [0, tmax [ (see [2]) and given by the variation of constants formula 

t

y(t) = S(t)y0 +

S(t − s)u(s)By(s)ds.

(6.7)

0

Let us consider the nonlinear semigroup N (t)y0 := y(t), we have d N (t)y0 2 dt

= 2AN (t)y0 , N (t)y0  i ω B N (t)y0 , N (t)y0  −2 B N (t)y0 , N (t)y0 . 1 + |i ω B N (t)y0 , N (t)y0 |

Since S(t) is a semigroup of contractions, then d i ω B N (t)y0 , N (t)y0  N (t)y0 2 ≤ −2 B N (t)y0 , N (t)y0 . dt 1 + |i ω B N (t)y0 , N (t)y0 | Integrating this inequality, we get  N (t)y0 2 ≤ y0 2 − 2

t

h(N (s)y0 ), N (s)y0 ds, ∀t ∈ [0, tmax [.

0

Using (6.2), we obtain N (t)y0  ≤ y0 .

(6.8)

For all y0 ∈ L 2 () and t ≥ 0, we have i ω B S(t)y0 , S(t)y0  = i ω B S(t)y0 − i ω B N (t)y0 , S(t)y0  − i ω B N (t)y0 , N (t)y0 − S(t)y0  + i ω B N (t)y0 , N (t)y0 . Using (6.6) and the continuity of χω , we obtain |i ω B S(t)y0 , S(t)y0 | ≤ 2δ K N (t)y0 − S(t)y0 y0  + |i ω B N (t)y0 , N (t)y0 |. Moreover, from (6.7) we have  N (t)y0 − S(t)y0  ≤ K 0

t

|i ω B N (s)y0 , N (s)y0 | ds. N (s)y0 

For a fixed T ∈]0, tmax [, Schwarz’s inequality yields

(6.9)

140

6 Regional Stabilization of Infinite Dimensional Semilinear Systems

  N (t)y0 − S(t)y0  ≤ K T

|i ω B N (s)y0 , N (s)y0 |2  21 ds , ∀t ∈ [0, T ]. N (s)y0 2 (6.10)

T

0

Using (6.8), we get |i ω B N (t)y0 , N (t)y0 | ≤

|i ω B N (t)y0 , N (t)y0 | y0 , ∀t ∈ [0, T ]. N (t)y0 

(6.11)

Integrating (6.9) over the interval [0, T ] and using (6.10) and (6.11), we obtain   T |i B N (s)y , N (s)y |2  21 ω 0 0 |i ω B S(s)y0 , S(s)y0 |ds ≤ 2δ K T y0  ds 2 N (s)y0  0  T 0 1   2 √ |i ω B N (s)y0 , N (s)y0 | 2 + T y0  ds . 2 N (s)y0  0 (6.12) Replacing y0 by N (t)y0 in (6.12), we get 

T

3 2

2

 0

T

3

|i ω B S(s)N (t)y0 , S(s)N (t)y0 |ds ≤ 2δ K 2 T 2 N (t)y0    t+T |i B N (s)y , N (s)y |2  21 ω 0 0 ds × N (s)y0 2 √t + T N (t)y0    t+T |i B N (s)y , N (s)y |2  21 ω 0 0 ds . × 2 N (s)y  0 t

By (6.4) and (6.8), we obtain αχω N (t)y0  L 2 (ω) ≤ β



√ where β = (2δ K 2 T + 1) T y0 . Then  α 2 χω N (t)y0 2L 2 (ω) ≤ β 2

t+T

t

t+T t

|i ω B N (s)y0 , N (s)y0 |2  21 ds N (s)y0 2

|i ω B N (s)y0 , N (s)y0 |2 ds. N (s)y0 2

(6.13)

Using (6.3), we get d |i ω B N (t)y0 , N (t)y0 |2 χω N (t)y0 2L 2 (ω) ≤ −2 . dt N (t)y0 2 Integrating (6.14) from mT to (m + 1)T , (m ∈ N), we obtain

(6.14)

6.1 Regional Stabilization

141

χω N (mT )y0 2L 2 (ω) − χω N ((m + 1)T )y0 2L 2 (ω) ≥  (m+1)T |i ω B N (s)y0 , N (s)y0 |2 2 ds. N (s)y0 2 mT By (6.13) and (6.14), we deduce α 2

1+2 χω N ((m + 1)T )y0 2L 2 (ω) ≤ χω N (mT )y0 2L 2 (ω) . β Then χω N ((m + 1)T )y0  L 2 (ω) ≤ ρχω N (mT )y0  L 2 (ω) where ρ =

1

(1 + 2( βα )2 ) 2 By recurrence, we show that χω N (mT )y0  L 2 (ω) ≤ ρ m χω y0  L 2 (ω) . Taking m = E( Tt ) the integer part of Tt and remarking that m ≥ Tt − 1, we get

1

.

χω N (t)y0  L 2 (ω) ≤ ρ T −1 χω y0  L 2 (ω) , ∀t ≥ 0 t

It follows that

χω N (t)y0  L 2 (ω) ≤ Me−σ t y0 , ∀t ≥ 0

where M = δ(1 + 2( βα )2 ) 2 and σ = nential stability of system (1) on ω. 1

ln(1 + 2( βα )2 ) 2T

, which shows the regional expo

Example 6.1 Let  =]0, 1[, we consider the wave equation defined by ⎧ 2 ∂ y ∂y ⎪ ⎪ ⎪ ⎨ ∂t 2 (x, t) = y(x, t) + u(t) ∂t (x, t) on ×]0, +∞[ ∂y (x, 0) = y1 on  ⎪ y(x, 0) = y0 , ⎪ ⎪ ∂t ⎩ y(0, t) = y(1, t) = 0 on [0, +∞[

(6.15)

This system has the form of (6.1) if we set

0 I A= 0



00 and B = 0I

.

˜ = H 2 () ∩ H01 (). Let A˜ = −, with domain D( A) 1 2 Consider H = H0 () × L () the state space endowed with the inner product 1 1 (y1 , z 1 ), (y2 , z 2 ) =  A˜ 2 y1 , A˜ 2 y2  L 2 () + z 1 , z 2  L 2 () . 2 ˜ √ The eigenvalues of ∗A are λk = (kπ ) , corresponding to eigenfunctions φk (x) = 2 sin(kπ x), ∀k ∈ N . ∞ ∞    λk bk φk , where ak φk and y2 = We set y = (y1 , y2 ) ∈ H with y1 = k=1

(ak , bk ) ∈ R2 ∀k ≥ 1, and the semigroup is given by

k=1

142

6 Regional Stabilization of Infinite Dimensional Semilinear Systems

S(s)y =

∞  k=1

ak cos(kπ s) + bk sin(kπ s) φ , ∀s ≥ 0. bk kπ cos(kπ s) − ak kπ sin(kπ s) k

For ω =]0, 21 [, we have i ω B S(s)y, S(s)y =

∞   (kπ )2  2 2 ak sin (kπ s) − bk ak sin(2kπ s) + bk2 cos2 (kπ s) . 2 k=1

Taking T = 2, we get 

2

|i ω B S(s)y, S(s)y|ds =

0

where α =

∞  (kπ )2

2

k=1

(ak2 + bk2 ) ≥ αχω y L 2 (ω)

π√ η, (η = min{(ak2 + bk2 )|k ∈ N∗ }), thus (6.4) holds and the control 2 u(t) = −

χω ∂t y2L 2 (ω) y(t)2H 1 () + ∂t y(t)2 0

regionally exponentially stabilizes system (6.15) on ω.



6.1.2 Regional Strong Stabilization In this subsection we give sufficient conditions for regional strong stabilization. Theorem 6.2 Assume that B is locally Lipschitz and satisfies 

T 0

|i ω B S(s)y, S(s)y|ds ≥ αχω y2L 2 (ω) , (T, α > 0)

then the control u(t) = −

i ω By(t), y(t) 1 + |i ω By(t), y(t)|

(6.16)

(6.17)

regionally strongly stabilizes system (6.1) on ω. Proof By the same arguments using in the proof of Theorem 6.1, we can show that i ω By, y the map g : y → By is locally Lipschitz and we deduce that system 1 + |i ω By, y| (6.1) has a unique global mild solution y(t) (see [2]). It remains to show that χω y(t) −→ 0 as t −→ +∞. For all y0 ∈ L 2 (), we have

6.1 Regional Stabilization

143

i ω B S(t)y0 , S(t)y0  = i ω B S(t)y0 − i ω By(t), S(t)y0  + i ω By(t), S(t)y0  Using formula (6.7), we obtain i ω B S(t)y0 , S(t)y0  = i ω B S(t)y0 − i ω By(t), S(t)y0  − i ω By(t), (t) + i ω By(t), y(t) 

t

where (t) =

S(t − s)u(s)By(s)ds.

0

By (6.6) and the continuity of χω , we deduce that |i ω B S(t)y0 , S(t)y0 | ≤ δ K (t)(S(t)y0  + y(t)) + |i ω By(t), y(t)|. (6.18) Then |i ω B S(t)y0 , S(t)y0 | ≤ 2δ K (t)y0  + |i ω By(t), y(t)|, ∀t ∈ [0, T ]. Moreover, Schwarz’s inequality gives √  (t) ≤ K y0  T

T

|i ω By(s), y(s)|2 ds

21

.

(6.19)

0

Integrating (6.18) over the interval [0, T ] and using (6.19), we obtain 

T 0

 T

21 3 |i ω B S(s)y0 , S(s)y0 |ds ≤ 2δT 2 K y0 2 |i ω By(s), y(s)|2 ds  T 0

21 √ + T y0  |i ω By(s), y(s)|2 ds . 0

Replacing y0 by y(t), we get  0

T

 t+T

21 3 |i ω B S(s)y(t), S(s)y(t)|ds ≤ 2δT 2 K y0  |i ω By(s), y(s)|2 ds  t+T t

21 √ + T y0  |i ω By(s), y(s)|2 ds . t

Furthermore, using (6.3), we obtain d |i ω By(t), y(t)|2 χω y(t)2L 2 (ω) ≤ −2 , ∀t ≥ 0. dt 1 + |i ω By(t), y(t)| Integrating this inequality, we get 

t 0

|i ω By(s), y(s)|2 ds ≤

L y0  χω y0 2L 2 (ω) , ∀t ≥ 0. 2

(6.20)

144

6 Regional Stabilization of Infinite Dimensional Semilinear Systems

with L y0  =

sup (1 + |i ω By, y|).

y≤y0 



Hence

+∞

|i ω By(s), y(s)|2 ds < +∞.

(6.21)

0

From (6.21) and (6.20), we deduce that 

T

|i ω B S(s)y(t), S(s)y(t)|ds −→ 0, as t −→ +∞.

(6.22)

0

It follows from (6.16) and (6.22) that χω y(t) L 2 (ω) −→ 0, as t −→ +∞, which completes the proof.  Example 6.2 On  =]0, 1[, we consider the following beam equation ⎧ 2 ∂ y ∂4 y ∂y ⎪ ⎪ ⎪ (x, t) = − (x, t) + u(t) (x, t) on ×]0, +∞[ ⎪ 2 4 ⎪ ∂t ∂ x ∂t ⎨ ∂y y(x, 0) = y0 , (x, 0) = y1 on  ⎪ ∂t ⎪ ⎪ 2 ⎪ ∂ y ⎪ ⎩ y(ξ, t) = (ξ, t) = 0, ξ = 0, 1 on ]0, +∞[ ∂x2

(6.23)

4 2 ∂4 y ˜ = {y ∈ L 2 ()/ ∂ y ∈ L 2 (), y(ξ, t) = ∂ y (ξ, t) = 0, Let A˜ = 4 , with D( A) 4 ∂x ∂x ∂x2 ξ = 0, 1}. Setting H = (H 2 () ∩ H01 ()) × L 2 () the state space endowed with the inner 1 1 product (y1 , z 1 ), (y2 , z 2 ) =  A˜ 2 y1 , A˜ 2 y2  L 2 () + z 1 , z 2  L 2 () . of A˜ are λ j = ( jπ )4 , corresponding to eigenfunctions ϕ j (x) = √ The eigenvalues 2 2 sin(( jπ ) x), ∀ j ∈ N∗ . System (6.23) has the form of (6.1) if we take



0 I − A˜ 0

A=

and B =

00 . 0I

B satisfies condition (6.16), indeed, for y ∈ H , we have y =

∞ 



αj



1

j=1

λ j2 β j

(α j , β j ) ∈ R2 ∀ j ≥ 1 and the semigroup is given by S(s)y =

∞ 



1

1

j=1

For ω =]0, 21 [, we have

1

α j cos(λ j2 s) + β j sin(λ j2 s) 1

1

1

β j λ j2 cos(λ j2 s) − α j λ j2 sin(λ j2 s)

 ϕ j , ∀s ≥ 0.

ϕ j , where

6.1 Regional Stabilization

i ω B S(s)y, S(s)y = =

145 ∞  λj 

2 j=1 ∞  λj  j=1

2

1

1

α j sin(λ j2 s) − β j cos(λ j2 s)

2

α 2j sin2 ( jπ s) + β 2j cos2 ( jπ s) − α j β j sin(2 jπ s)



Integrating this relation over the time interval [0, 2], we obtain 

2

|i ω B S(s)y, S(s)y|ds =

0

∞  λj j=1

2

(α 2j + β 2j ) = χω y2

then (6.16) holds. We conclude that the control u(t) = −

χω ∂t y(., t)2L 2 (ω) 1 + χω ∂t y(., t)2L 2 (ω)

regionally strongly stabilizes system (6.23) on ω.



6.1.3 Regional Weak Stabilization The following result gives sufficient conditions for regional weak stabilization. Theorem 6.3 Let B be locally Lipschitz and weakly sequentially continuous such that i ω B S(t)z, S(t)z = 0, ∀t ≥ 0 =⇒ χω z = 0 (6.24) then control (6.17) regionally weakly stabilizes system (6.1) on ω. Proof Consider the nonlinear semigroup N (t)y0 = y(t) and let (tk ) be a sequence of real numbers such that tk −→ +∞ as k −→ +∞. From (6.8), N (tk )y0 is bounded in L 2 (), then there exists a subsequence (tφ(k) ) of (tk ) and z ∈ L 2 () such that N (tφ(k) )y0  z as k −→ +∞. For all y0 ∈ L 2 () and t ≥ 0, we have i ω B S(t)y0 , S(t)y0  = i ω B N (t)y0 , N (t)y0  − i ω B N (t)y0 , N (t)y0 − S(t)y0  + i ω B S(t)y0 − i ω B N (t)y0 , S(t)y0 . Using (6.6) and the continuity of χω , we get |i ω B S(t)y0 , S(t)y0 | ≤ |i ω B N (t)y0 , N (t)y0 | (6.25) +2δ K N (t)y0 − S(t)y0 y0 , (for some δ > 0).

146

6 Regional Stabilization of Infinite Dimensional Semilinear Systems

From (6.7) and Schwarz’s inequality, we have  N (t)y0 − S(t)y0  ≤ K y0  T f (0), ∀t ∈ [0, T ]  where f (t) =

t+T

(6.26)

|i ω B N (s)y0 , N (s)y0 |2 ds.

t

Integrating (6.25) over the interval [0, T ] and using (6.26), we obtain 

T

 |i ω B S(s)y0 , S(s)y0 |ds ≤ βy0  f (0)

(6.27)

0

√ with β = (2δ K 2 T y0  + 1) T . Replacing y0 by N (tφ(k) )y0 in (6.27), we get 

T

 |i ω B S(s)N (tφ(k) )y0 , S(s)N (tφ(k) )y0 |ds ≤ βy0  f (tφ(k) ).

0

It follows that  T |i ω B S(s)N (tφ(k) )y0 , S(s)N (tφ(k) )y0 |ds −→ 0 as k −→ 0. 0

Since B is weakly sequentially continuous and using the dominated convergence theorem, we deduce that i ω B S(s)z, S(s)z = 0, ∀s ∈ [0, T ]. From (6.24), it follows that χω N (tφ(k) )y0  0 as k −→ +∞, and then ∀φ ∈ L 2 (), χω N (tk )y0 , φ −→ 0 as k −→ +∞, hence χω N (t)y0  0 as t −→ +∞. In other words χω y(t) converges weakly to 0 as t −→ +∞, and then system (6.1) is regionally weakly stabilizable on ω.  Example 6.3 On  =]0, 1[, we consider the following equation ⎧ 2 ∂ y ∂4 y ∂y ⎪ ⎪ ⎪ 2 (x, t) = − 4 (x, t) + u(t) (x, t) on ×]0, +∞[ ⎪ ⎪ ∂t ∂ x ∂t ⎨ ∂y y(x, 0) = y0 , (x, 0) = y1 on  ⎪ ∂t ⎪ ⎪ 2 ⎪ ∂ y ⎪ ⎩ y(ξ, t) = (ξ, t) = 0, ξ = 0, 1 on ]0, +∞[ ∂x2

(6.28)

The state space is H = (H 2 () ∩ H01 ()) × L 2 (), this system has the form of (6.1) if we set

00 0 I and B = . A= 0I −2 0 For ω =]0, 21 [, by the same arguments using in Example 6.2, we show that

6.1 Regional Stabilization

147



2

|i ω B S(s)y, S(s)y|ds = χω y2

0

It follows that i ω B S(t)y, S(t)y = 0 =⇒ χω y = 0 Then, condition (6.24) holds. We conclude that the control u(t) = −

χω ∂t y(., t)2L 2 (ω) 1 + χω ∂t y(., t)2L 2 (ω)

regionally weakly stabilizes system (6.28) on ω.



6.2 Regional Stabilization Problem This section deals with regional stabilization of system (6.1) by considering the following minimization problem ⎧ ⎨ ⎩

 +∞  +∞ Pω By(t), y(t)2 dt + i Ry(t), y(t)dt + y(t)2 |u(t)|2 dt ω y(t)2 0 0 0 = {u ∈ L 2 (0, +∞)|y(t) is a global solution and J (u) < +∞}. 

+∞

min J (u) = u ∈ Uad

(6.29) where B is bounded, A satisfies (6.3) and Pω = i ω Pi ω with P ∈ L(L 2 ()) is a positive and bounded operator satisfying the equation Pω Ay, y + y, Pω Ay + i ω Ry, y = 0, y ∈ D(A)

(6.30)

where R ∈ L(L 2 ()) is a positive operator. Theorem 6.4 Suppose that B is locally Lipschitz and P is compact such that Pω B S(t)z, S(t)z = 0, ∀t ≥ 0 =⇒ χω z = 0 then the control u ∗ (t) = −

Pω By(t), y(t) y(t)2

(6.31)

(6.32)

is the unique solution of (6.29) and regionally weakly stabilizes system (6.1) on ω. Proof Let us define the function V (y) = Pω y, y, y ∈ L 2 (). Using (6.30), for all y0 ∈ D(A) and t ≥ 0, we have Pω By(t), y(t)2 d V (y(t)) = −2 − i ω Ry(t), y(t). dt y(t)2

(6.33)

148

6 Regional Stabilization of Infinite Dimensional Semilinear Systems

Integrating this relation, we get  0

t

Pω By(s), y(s)2 1 ds ≤ V (y0 ), t ≥ 0. 2 y(s) 2

(6.34)

The solution y(.) is continuous with respect to the initial condition y0 (see [2]) and D(A) is dense in L 2 (), then (6.34) holds for all y0 ∈ L 2 () so J (u ∗ ) is finite for all y0 ∈ L 2 (). Let (tk ) be a sequence of real numbers such that tk −→ +∞ as k −→ +∞. By (6.8), y(tk ) is bounded in L 2 (), then there exists a subsequence (tφ(k) ) of (tk ) and z ∈ L 2 () such that y(tφ(k) )  z as k −→ +∞. For all y0 ∈ L 2 () and t ≥ 0, we have Pω B S(t)y0 , S(t)y0  = Pω B S(t)y0 − Pω By(t), S(t)y0  − Pω By(t), y(t) − S(t)y0  + Pω By(t), y(t). Thus |Pω B S(t)y0 , S(t)y0 | ≤ 2δ 2 PK y(t) − S(t)y0 y0  + |Pω By(t), y(t)|. (6.35) Moreover, we have 

t

y(t) − S(t)y0  ≤ K y0 

|Pω By(s), y(s)|ds.

0

Schwarz’s inequality yields  y(t) − S(t)y0  ≤ K T λ(0), ∀t ∈ [0, T ]  where λ(t) =

t+T

(6.36)

|Pω By(s), y(s)|2 ds.

t

Integrating (6.35) over the interval [0, T ] and with (6.36), we obtain 

T

 |Pω B S(s)y0 , S(s)y0 |ds ≤ (2δ 2 K 2 T P + 1)y0  T λ(0).

(6.37)

0

Replacing y0 by y(tφ(k) ) in (6.37), we get  0

T

 |Pω B S(s)y(tφ(k) ), S(s)y(tφ(k) )|ds ≤ (2δ 2 K 2 T P + 1)y0  T λ(tφ(k) ). (6.38)

6.2 Regional Stabilization Problem



T

By (6.38), we get lim

k−→+∞ 0

149

Pω B S(s)y(tφ(k) ), S(s)y(tφ(k) )ds = 0.

Since P is compact and S(s) is continuous ∀s ≥ 0, we have lim Pω B S(s)y(tφ(k) ), S(s)y(tφ(k) ) = Pω B S(s)z, S(s)z.

k−→+∞

By dominated convergence theorem, we obtain 

T

|Pω B S(s)z, S(s)z|ds = 0 and then Pω B S(s)z, S(s)z = 0, ∀s ∈ [0, T ]

0

. Using (6.31), we deduce that χω y(tφ(k) )  0 as k −→ +∞. It follows that χω y(t) converges weakly to 0 as t −→ +∞, and system (6.1) is regionally weakly stabilizable on ω. Now, let us prove that (6.32) is the unique solution of (6.29). Since P is compact, it follows that V (y(t)) −→ 0 as t → +∞. Let y0 ∈ D(A), formula (6.33) may be written as   2 Pω By(t), y(t) d V (y(t)) Pω By(t), y(t)2 2 2 = y(t) + u(t) − − u (t) dt y(t)2 y(t)2 −i ω Ry(t), y(t) integrating this relation, we obtain 

+∞

J (u) = V (y0 ) + 0

 y(s)2

Pω By(s), y(s) + u(s) y(s)2

2 ds.

Then J (u) ≥ V (y0 ), ∀u ∈ Uad . For u = u ∗ , we get J (u ∗ ) = V (y0 ). Let y0 ∈ L 2 (), and a sequence (y0k ) ⊂ D(A) such that y0k −→ y0 as k −→ +∞, we have  2  +∞ Pω Byk (s), yk (s) J (u) = V (y0k ) + yk (s)2 + u(s) ds yk (s)2 0  +∞ − i ω Ryk (s), yk (s)ds. 0

Thus J (u) ≥ V (y0k ). We deduce that J (u) ≥ V (y0 ) = J (u ∗ ), so (6.32) is the unique solution of problem (6.29).  Proposition 6.1 Suppose that B is locally Lipschitz and satisfies

150

6 Regional Stabilization of Infinite Dimensional Semilinear Systems



T 0

|Pω B S(s)y, S(s)y|ds ≥ αχω y2L 2 (ω) , (T, α > 0)

(6.39)

then control (6.32) is the unique solution of (6.29) and regionally strongly stabilizes system (6.1) on ω. Proof For all y0 ∈ L 2 (), we have Pω B S(t)y0 , S(t)y0  = Pω B S(t)y0 − Pω By(t), S(t)y0  + Pω By(t), S(t)y0  Using (6.7), we obtain the following relation Pω B S(t)y0 , S(t)y0  = Pω B S(t)y0 − Pω By(t), S(t)y0  − Pω By(t), (t) +Pω By(t), y(t)  where (t) =

t

S(t − s)u(s)By(s)ds.

0

Since χω is continuous, then there exists δ > 0 such that |Pω B S(t)y0 , S(t)y0 | ≤ δ 2 K P(t)(S(t)y0  + y(t)) + |Pω By(t), y(t)|.

(6.40)

Then |Pω B S(t)y0 , S(t)y0 | ≤ 2δ 2 K P(t)y0  + |Pω By(t), y(t)|, ∀t ∈ [0, T ]. Moreover, we have √  (t) ≤ K y0  T

T

|Pω By(s), y(s)|2 ds

21

.

(6.41)

0

Integrating (6.40) over the interval [0, T ] and using (6.41), we obtain 

T

0

3 2



T

|Pω B S(s)y0 , S(s)y0 |ds ≤ 2δ T K Py0  |Pω By(s), y(s)|2 ds 0  T

21 √ + T y0  |Pω By(s), y(s)|2 ds . 2

2

21

0

Replacing y0 by y(t), we get  t+T

1 3 2 |Pω B S(s)y(t), S(s)y(t)|ds ≤ 2δ 2 T 2 K Py0  |Pω By(s), y(s)|2 ds 0 t  t+T

1 √ 2 + T y0  |Pω By(s), y(s)|2 ds .

 T

t

From (6.34) and (6.42), we deduce that

(6.42)

6.2 Regional Stabilization Problem



T

151

|Pω B S(s)y(t), S(s)y(t)|ds −→ 0, as t −→ +∞.

(6.43)

0

From (6.39) and (6.43), we deduce that χω y(t) L 2 (ω) −→ 0, as t −→ +∞, which completes the proof.  Proposition 6.2 Assume that B is locally Lipschitz such that  0

T

|Pω B S(t)y, S(t)y|dt ≥ δχω y L 2 (ω) , ( f or some T, δ > 0)

(6.44)

and there exists μ > 0 such that Pω By, y ≤ μi ω By, y

(6.45)

then control (6.32) is the unique solution of (6.29) and regionally exponentially stabilizes system (6.1) on ω. Proof Let us define the function F(y) = Pω y, y, ∀y ∈ L 2 (). For all y0 ∈ D(A) and t ≥ 0, using (6.30), we obtain Pω By(t), y(t)2 d F(y(t)) = −2 − i ω Ry(t), y(t). dt y(t)2 Integrating this relation, we get  0

t

Pω By(s), y(s)2 1 ds ≤ F(y0 ), t ≥ 0. y(s)2 2

(6.46)

By density, we deduce that J (u ∗ ) is finite for all y0 ∈ L 2 (). For y0 ∈ L 2 () and t ≥ 0, we have Pω B S(t)y0 , S(t)y0  = Pω B S(t)y0 − Pω By(t), S(t)y0  − Pω By(t), y(t) − S(t)y0  + Pω By(t), y(t). Since χω is continuous, there exists α > 0 such that |Pω B S(t)y0 , S(t)y0 | ≤ 2α K Py(t) − S(t)y0 y0  + |Pω By(t), y(t)|. (6.47) Moreover, we have  y(t) − S(t)y0  ≤ K 0

Schwarz’s inequality yields

T

|Pω By(s), y(s)| ds. y(s)

152

6 Regional Stabilization of Infinite Dimensional Semilinear Systems

  y(t) − S(t)y0  ≤ K T

T 0

|Pω By(s), y(s)|2  21 ds , ∀t ∈ [0, T ]. y(s)2

(6.48)

Using (6.8), we get |Pω By(t), y(t)| ≤

|Pω By(t), y(t)| y0 , ∀t ∈ [0, T ]. y(t)

Integrating (6.47) over the interval [0, T ] and taking into account (6.48), we obtain 

T

 √ |Pω B S(s)y0 , S(s)y0 |ds ≤ (2α K 2 T P + 1) T y0  f (0)

(6.49)

0



|Pω By(s), y(s)|2 ds. y(s)2 t Replacing y0 by y(t) in (6.49), we get

where f (t) = 

T

t+T

 √ |Pω B S(s)y(t), S(s)y(t)|ds ≤ (2αPK 2 T + 1) T y(t) f (t).

0

√ 2 2 Using √(6.44) and (6.8), we obtain δχω y(t) L (ω) ≤ β f (t) with β = (2α K T P + 1) T y0 . Then  (6.50) δ 2 χω y(t)2L 2 (ω) ≤ β 2 f (t). Using (6.45) and (6.3), we get 2 |Pω By(t), y(t)|2 d χω y(t)2L 2 (ω) ≤ − 2 . dt μ y(t)2 Integrating this inequality from mT to (m + 1)T , (m ∈ N), we obtain χω y(mT )2L 2 (ω) − χω y((m + 1)T )2L 2 (ω) ≥

2 μ2



(m+1)T mT

|Pω By(s), y(s)|2 ds. y(s)2

Using (6.50) and since χω y(t) L 2 (ω) decreases, we deduce δ 2

χω y((m + 1)T )2L 2 (ω) ≤ χω y(mT )2L 2 (ω) . 1+2 μβ Then χω y((m + 1)T ) L 2 (ω) ≤ σ χω y(mT ) L 2 (ω) where σ =

1 δ 2 2 (1 + 2( μβ ) )

1

.

6.2 Regional Stabilization Problem

153

This implies that χω y(mT ) L 2 (ω) ≤ σ m χω y0  L 2 (ω) . Taking m = E( Tt ) the integer part of

t T

and remarking that m ≥

t T

− 1, we obtain

χω y(t) L 2 (ω) ≤ σ T −1 χω y0  L 2 (ω) . t

We deduce that

χω y(t) L 2 (ω) ≤ Fe−ρt y0 , ∀t ≥ 0

(6.51)

δ 2 ln(1 + 2( μβ ) ) δ 2 21 where F = α(1 + 2( μβ , then control (6.32) regionally ) ) and ρ = 2T exponentially stabilizes system (6.1) on ω. It remains to show that (6.32) is the unique solution of (6.29). Remarking that F(y(t)) ≤ αPχω y(t)2L 2 (ω) , it follows from (6.51) that F(y(t)) −→ 0 as t → +∞. Let y0 ∈ D(A), integrating the relation



d F(y(t)) dt

Pω By(t), y(t) = y(t) + u(t) y(t)2 −i ω Ry(t), y(t) 2

2

Pω By(t), y(t)2 − − u 2 (t) y(t)4



(6.52) we have  J (u) = F(y0 ) +

+∞

 y(s)2

0

Pω By(s), y(s) + u(s) y(s)2

2 ds

then J (u) ≥ F(y0 ). Setting u = u ∗ , we obtain J (u ∗ ) = F(y0 ). Let y0 ∈ L 2 (), and a sequence y0k ⊂ D(A) such that y0k −→ y0 as k −→ +∞, we have  J (u) = F(y0k ) + 0

+∞

 yk (s)

2

Pω Byk (s), yk (s) + u(s) yk (s)2

2 ds.

Thus J (u) ≥ F(y0 ) = J (u ∗ ). Hence control (6.32) is the unique solution of the problem (6.29). In order to illustrate the previous results, we perform the following algorithm: Step 1: Initial data : initial condition y0 and subregion ω; Step 2: Solve equation (6.30) using Bartels-Stewart method given in [3]; Step 3: Apply the control given by (6.17) or (6.32); Step 4: Solve system (6.1) using Petrov-Galerkin method;



154

6 Regional Stabilization of Infinite Dimensional Semilinear Systems

6.3 Simulation Results Let  =]0, 1[, we consider the system ⎧ y(x, t) ∂y ⎪ ⎪ , ×]0, +∞[ ⎨ (x, t) = 0.01y(x, t) + u(t) ∂t 1 + y(x, t) y(0, t) = y(1, t) = 0, ]0, +∞[ ⎪ ⎪ ⎩ y(x, 0) = 2π sin(2π x), 

(6.53)

where the state space is L 2 (). 1. Regional stabilizing control. Let ω =]0, 21 [, we perform the above algorithm applying the control (6.17) to system (6.53), we obtain the following figures. Figure 6.1, shows that system (6.53) is stabilized on the subregion ω, with stabilization error equals to 3.84 × 10−4 (Fig. 6.2). For ω =]0, 1[, we obtain. Figure 6.3 shows that system (6.53) is stabilized on , with stabilization error equals to 9.83 × 10−4 (Fig. 6.4). 2. Regional optimal stabilizing control. Consider system (6.53) and problem (6.29) with R = I such that Pω is the unique solution of the equation (6.30). Let ω =]0, 21 [, we perform the above algorithm applying the control (6.32) to system (6.53), we obtain the figures below. Figure 6.5 shows how system (6.53) is stabilized by the control (6.32) on ω with a stabilization error equals to 2.1 10−4 and the cost J (u ∗ ) = 4.9 × 10−2 (Fig. 6.6). For ω = . Figure 6.7 shows that system (6.53) is stabilized on , with stabilization error equals to 8.23 × 10−4 and the cost J (u ∗ ) = 9.75 × 10−2 (Fig. 6.8).

Fig. 6.1 Evolution of the state

6.3 Simulation Results Fig. 6.2 Evolution of the control

Fig. 6.3 Evolution of the state

Fig. 6.4 Evolution of the control

155

156 Fig. 6.5 Evolution of the state

Fig. 6.6 Evolution of the control

Fig. 6.7 Evolution of the state

6 Regional Stabilization of Infinite Dimensional Semilinear Systems

6.3 Simulation Results

157

Fig. 6.8 Evolution of the control

The table below shows that there exists a relation between the area of subregion ω, the cost and the error of stabilization. Moreover, we remark that more the area of ω increases more the stabilization cost increases. ω Error Cost

]0, 0.2[ 1.33 ×10−4 2.38 ×10−2

]0, 0.4[ 1.73 ×10−4 3.84 ×10−2

]0, 0.6[ 3.95×10−4 5.63 ×10−2

]0, 0.8[ 6.526 ×10−4 7.56 ×10−2

]0, 1[ 8.23×10−4 9.75 ×10−2

References 1. Mohler, R.R.: Bilinear Control Processes with Application to Engineering, Ecology and Medicine. Academic, New York (1973) 2. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983) 3. Penzl, T.: Numerical solution of generalized Lyapunov equations. Adv. Comput. Math 8, 33–48 (1998)

Chapter 7

Output Stabilization of Infinite Dimensional Semilinear Systems

This chapter considers the output stabilization for a class of distributed bilinear and semilinear systems with stabilizing controls that depend on the output operator. We give sufficient conditions for exponential, strong and weak output stabilization of such system. The obtained results are illustrated by many examples.

7.1 Output Stabilization of Semilinear Systems We consider the following semilinear system ⎧ ⎨ dy(t) = Ay(t) + u(t)By(t), dt ⎩ y(0) = y , 0

(7.1)

where A : D(A) ⊂ X → X generates a strongly continuous semigroup of contractions (S(t))t≥0 on a Hilbert space X endowed with a norm and an inner product denoted, respectively, by . and ., ., u(.) ∈ Uad is a scalar valued control, where Uad is the admissible controls set, and B is a nonlinear operator from X to X with B(0) = 0 so that the origin be an equilibrium state of system (7.1). System (7.1) is augmented with the output z(t) := C y(t).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. H. Zerrik and O. Castillo, Stabilization of Infinite Dimensional Systems, Studies in Systems, Decision and Control 355, https://doi.org/10.1007/978-3-030-68600-0_7

(7.2)

159

160

7 Output Stabilization of Infinite Dimensional Semilinear Systems

Definition 7.1 The output (7.2) is said to be 1. exponentially stabilizable, if there exists a control u(t) = f (y(t)) such that for any initial condition y0 ∈ X , the corresponding solution y(t) of system (7.1) is global and there exist α, β > 0 such that C y(t)Y ≤ αe−βt y0 , ∀t > 0, 2. strongly stabilizable, if there exists a control u(t) = f (y(t)) such that for any initial condition y0 ∈ X , the corresponding solution y(t) of system (7.1) is global and verifies C y(t)Y → 0, as t → ∞, 3. weakly stabilizable, if there exists a control u(t) = f (y(t)) such that for any initial condition y0 ∈ X , the corresponding solution y(t) of system (7.1) is global and satisfies C y(t), ψY → 0, ∀ψ ∈ Y, as t → ∞. Remark 7.1 It is clear that exponential stability of (7.2) ⇒ strong stability of (7.2) ⇒ weak stability of (7.2).

7.1.1 Output Exponential Stabilization In this subsection, we develop sufficient conditions for exponential stabilization of the output (7.2). The following result concerns the exponential stabilization of (7.2). Theorem 7.1 Let A generate a semigroup (S(t))t≥0 of contractions on X and B be locally Lipschitz. If the conditions 1. C ∗ C Aψ, ψ ≤ 0, ∀ψ ∈ D(A), where C ∗ is the adjoint operator of C, 2. C ∗ C Bψ, ψBψ, ψ ≥ 0, ∀ψ ∈ X , 3. there exist T, γ > 0, such that 

T

0

|C ∗ C B S(t)ψ, S(t)ψ|dt ≥ γ Cψ2Y , ∀ψ ∈ X,

(7.3)

hold, then the control ⎧ ∗ ⎨− C C By(t), y(t) , y(t)2 u(t) = ⎩ 0,

if y(t) = 0, if y(t) = 0,

is in Uad = L ∞ (0, ∞), and exponentially stabilizes the output (7.2).

(7.4)

7.1 Output Stabilization of Semilinear Systems

161

Proof System (7.1) has a unique weak solution y(t) (see [1]) defined on a maximal interval [0, tmax ] by  y(t) = S(t)y0 +

t

g(y(s))S(t − s)By(s)ds,

(7.5)

0

where

⎧ ∗ ⎨− C C By(t), y(t) , y(t)2 g(y(t)) = ⎩ 0,

if y(t) = 0, if y(t) = 0.

Since (S(t))t≥0 is a semigroup of contractions, we have d y(t)2 ≤ 2g(y(t))By(t), y(t). dt Integrating this inequality, and using hypothesis 2 of Theorem 7.1, it follows that y(t) ≤ y0 .

(7.6)

For all y0 ∈ X and t ≥ 0, we have C ∗ C B S(t)y0 , S(t)y0  = C ∗ C By(t), y(t) − C ∗ C By(t), y(t) − S(t)y0  + C ∗ C B S(t)y0 − C ∗ C By(t), S(t)y0 . Since B is locally Lipschitz, there exists a constant positive L that depends on y0  such that |C ∗ C B S(t)y0 , S(t)y0 | ≤ |C ∗ C By(t), y(t)| + 2αLy(t) − S(t)y0 y0 , (7.7) where α is a positive constant. Using (7.6), we deduce |C ∗ C By(t), y(t)| ≤ |g(y(t))|y(t)y0 , ∀t ∈ [0, T ].

(7.8)

While from (7.5) and using Schwartz’s inequality, we obtain   y(t) − S(t)y0  ≤ L T

T

|g(y(t))| y(t) dt 2

2

 21

.

(7.9)

0

Integrating (7.7) over the interval [0, T ] and taking into account (7.8) and (7.9), we get

162

7 Output Stabilization of Infinite Dimensional Semilinear Systems



T

|C ∗ C B S(t)y0 , S(t)y0 |dt ≤ 2αT 2 L 2 y0 



T

3

0

|g(y(t))|2 y(t)2 dt

0



T

1 2

+ T y0 

|g(y(t))| y(t) dt 2

2

 21

 21

. (7.10)

0

Now, let us consider the nonlinear semigroup U (t)y0 := y(t) (see [2]). Replacing y0 by U (t)y0 in (7.10), and using the superposition properties of the semigroup (U (t))t≥0 , we deduce that 

T

|C ∗ C B S(s)U (t)y0 , S(s)U (t)y0 |ds ≤ 2αT 2 L 2 U (t)y0  3

0



t+T

×

(7.11)

|g(U (s)y0 )| U (s)y0  ds 2

 21

2

t 1

+ T 2 U (t)y0   21  t+T 2 2 |g(U (s)y0 )| U (s)y0  ds × t

Thus, by using (7.3) and (7.11), it follows that 

t+T

γ CU (t)y0 Y ≤ M

|g(U (s)y0 )|2 U (s)y0 2 ds

 21

,

(7.12)

t 1

where M = (2αT L 2 + 1)T 2 is a non-negative constant depending on y0  and T . From hypothesis 1 of Theorem 7.1, we have d CU (t)y0 2Y ≤ −2|g(U (t)y0 )|2 U (t)y0 2 . dt

(7.13)

Integrating (7.13) from nT and (n + 1)T, (n ∈ N), we obtain  CU (nT )y0 2Y − CU ((n + 1)T )y0 2Y ≥ 2

(n+1)T

|g(U (s)y0 )|2 U (s)y0 2 ds.

nT

Using (7.12), (7.13) and the fact that CU (t)z 0 Y decreases, it follows  1+2

 γ 2  M

CU ((n + 1)T )y0 2Y ≤ CU (nT )y0 2Y .

Then CU ((n + 1)T )y0 Y ≤ βCU (nT )y0 Y ,

7.1 Output Stabilization of Semilinear Systems

where β =

1  1 γ 2 2 1+2( M )

163

.

By recurrence, we show that CU (nT )y0 Y ≤ β n C y0 Y . Taking n = E( Tt ) the integer part of Tt , we deduce that CU (t)y0 Y ≤ Re−σ t y0 , 

γ 2 21 where R = α 1 + 2 M , with α > 0 and σ = the proof.

 γ 2 ln 1+2( M ) 2T

> 0, which achieves 

Corollary 7.1 Let  ⊂ Rn be the system evolution domain assumed to be regular and ω be a subregion of , if we take C = χω , where χω : X = L 2 () → Y = L 2 (ω) is the restriction operator to ω, and χω∗ denotes the adjoint operator of χω given by (χω∗ z)(x)

=

z(x), if x ∈ ω, 0, if x ∈ \ω.

If the assumptions of Theorem 7.1 hold, then the control ⎧ ∗ ⎨− χω χω By(t), y(t) , if y(t) = 0, y(t)2 u(t) = ⎩ 0, if y(t) = 0, exponentially stabilizes system (7.1) on ω. Example 7.1 Let us consider the system defined on an open and bounded domain  ⊂ Rn with C ∞ boundary ∂ by the equation ⎧ ∂ y(x, t) ⎪ ⎪ = −iy(x, t) + u(t)By(x, t), x ∈ , t > 0, ⎨ ∂t y(x, t) = 0, x ∈ ∂, t > 0, ⎪ ⎪ ⎩ y(x, 0) = y0 (x), x ∈ ,

(7.14)

where X = L 2 () endowed with its natural complex inner product, A = −i, (i ∈ C) with domain D(A) = H 2 () ∩ H01 () and the operator of control given by By = y . y+ 1 + y Let ω be a subregion of  that verifies the geometric control condition (GCC) (see [3]). System (7.14) is augmented with the output z(t) = χω y(t). The operator A generates a semigroup of isometries on L 2 (), and

(7.15)

164

7 Output Stabilization of Infinite Dimensional Semilinear Systems

    Re χω∗ χω Az, z = 0, ∀z ∈ D(A). For all y0 ∈ L 2 (), we have    1 χω∗ χω B S(t)y0 , S(t)y0 = S(t)y0 , S(t)y0 L 2 (ω) + 1 + S(t)y0    ≥ S(t)y0 , S(t)y0 L 2 (ω) .



 ω

|S(t)y0 |2 d x

Integrating this inequality, we get  0

T



 χω∗ χω B S(t)y0 , S(t)y0 dt ≥



T



S(t)y0 , S(t)y0



0

L 2 (ω)

dt.

Since the subregion ω verifies GCC, the inequality 

T



S(t)y0 , S(t)y0

0

 L 2 (ω)

dt ≥ αy0 2 ≥ αχω y0 2L 2 (ω) , for some α, T > 0,

holds (see [3, 4]), we deduce that 

T 0



 χω∗ χω B S(t)y0 , S(t)y0 dt ≥ αχω y0 2L 2 (ω) .

When we apply Theorem 7.1, we conclude that the control u(t) = −

χω y(., t)2L 2 (ω) y(., t)2

 1+

 1 11{t≥0, 1 + y(., t)

y(.,t)=0} ,

exponentially stabilizes the output (7.15).

7.1.2 Output Strong Stabilization The following result provides sufficient conditions for strong stabilisation of the output (7.2). Theorem 7.2 Let A generate a semigroup (S(t))t≥0 of contractions on X , B be locally Lipschitz and let assumptions 1, 2 and 3 of Theorem 7.1 hold, then the control u(t) = −C ∗ C By(t), y(t), is in Uad = L 2 (0, ∞), and strongly stabilizes the output (7.2).

(7.16)

7.1 Output Stabilization of Semilinear Systems

165

Proof From hypothesis 1 of Theorem 7.1, we obtain d C y(t)2Y ≤ −2|C ∗ C By(t), y(t)|2 . dt

(7.17)

Integrating this inequality, we deduce that 

t

2 0

Hence, we get



|C ∗ C By(s), y(s)|2 ds ≤ C y(0)2Y .

+∞

|C ∗ C By(s), y(s)|2 ds < +∞,

(7.18)

0

so u ∈ L 2 (0, ∞). From the variation of constants formula and using Schwartz’s inequality, we deduce  T  21 1 ∗ 2 2 |C C By(s), y(s)| ds . (7.19) y(t) − S(t)y0  ≤ L T 0

Integrating (7.7) over the interval [0, T ] and taking into account (7.19), we obtain 

T

|C ∗ C B S(s)y0 , S(s)y0 |ds ≤ 2αL 2 T 2 y0 2



T

3

0

0

 +T

|C ∗ C By(s), y(s)|2 ds

T

1 2

∗

|C C By(s), y(s)| ds

 21

2

 21

.

0

Replacing y0 by y(t) and using the superposition property of the solution, we get 

T

3

|C ∗ C B S(s)y(t), S(s)y(t)|ds ≤ 2αL 2 T 2 y0 2

0 1



+T2



t+T

|C ∗ C By(s), y(s)|2 ds

t t+T

|C ∗ C By(s), y(s)|2 ds

 21

.

 21

(7.20)

t

By (7.18), we get 

t+T

|C ∗ C B S(s)y(t), S(s)y(t)|ds → 0, as t → +∞.

(7.21)

t

From (7.3) and (7.21), we deduce that C y(t)Y → 0, as t → +∞, which completes the proof.  Proposition 7.1 If B is linear bounded and the assumptions 1, 2 and 3 of Theorem 7.1 hold, then the control (7.16) strongly stabilizes the output (7.2) with decay

166

7 Output Stabilization of Infinite Dimensional Semilinear Systems

estimate

C y(t)Y = O(t − 2 ), as t → +∞. 1

(7.22)

Proof Since B is bounded and while using (7.20), we get 

T

 |C ∗ C B S(s)U (t)y0 , S(s)U (t)y0 |ds ≤ θ ξ(t),

(7.23)

0





where θ = (2αT B2 y0 2 + 1)T 2 and ξ(t)= tt+T |C ∗ C BU (s)y0 , U (s)y0 |2 ds . From (7.3) and (7.23), we deduce that   ξ(nT ) ≥ CU (nT )y0 2Y , ∀n ≥ 0, (7.24) 1

where  = γ1 θ. Integrating the inequality d CU (t)y0 2Y ≤ −2|C ∗ C BU (t)y0 , U (t)y0 |2 , dt from nT to (n + 1)T , (n ∈ N) and using (7.24), we obtain CU (nT )y0 2Y − CU (nT + T )y0 2Y ≥ 2ξ(nT ), ∀n ≥ 0. Hence, we get 2 CU (nT + T )y0 2Y − 2 CU (nT )y0 2Y ≤ −2CU (nT )y0 4Y , ∀n ≥ 0. (7.25) Let us introduce the sequence rn = CU (nT )y0 2Y , ∀n ≥ 0. Using (7.25), we deduce that rn − rn+1 2 ≥ 2 , ∀n ≥ 0. 2 rn  Since the sequence (rn ) decreases, we get rn − rn+1 2 ≥ 2 , ∀n ≥ 0, rn .rn+1  and also

1 1 2 − ≥ 2 , ∀n ≥ 0. rn+1 rn 

We deduce that rn ≤

r0 , ∀n ≥ 0. +1

2r0 n 2

7.1 Output Stabilization of Semilinear Systems

167

Finally, introducing the integer part n = E( Tt ) and from (7.17), we get CU (t)y0 Y decreases. We deduce the estimate C y(t)Y = O(t −1/2 ), as t → +∞.  Corollary 7.2 Let A generate a semigroup of contractions on X and suppose that C ∗ C Az, z = 0, for all z ∈ D(A). If B is locally Lipschitz and condition (7.3) holds, then the control (7.16) strongly stabilizes the output (7.2) and it is the unique solution of problem ⎧  +∞  +∞ ⎨min Q(u) = ∗ 2 |C C By(t), y(t)| dt + |u(t)|2 dt, 0 0 ⎩ u ∈ Uad = {u ∈ L 2 (0, ∞) | y(t) is a global solution and Q(u) < +∞}. (7.26) Proof For y0 ∈ D(A), we get d C y(t)2Y = −2|C ∗ C By(t), y(t)|2 . dt

(7.27)

Integrating (7.27), we deduce 

t

|C ∗ C By(s), y(s)|2 ds ≤

0

1 C y0 2Y , ∀t ≥ 0. 2

This inequality holds for all y0 ∈ X , since the correspondence y0 −→ y(.) is continuous in C([0, t]; X ), then Q(u) < +∞, for all y0 ∈ X . In other word u ∈ Uad , so Uad = ∅. From (7.27), we get d C y(t)2Y = |C ∗ C By(t), y(t) + u(t)|2 − C ∗ C By(t), y(t)2 − u(t)2 . dt (7.28) Integrating (7.28) over (0, t), we obtain  0

t



 C ∗ C By(s), y(s)2 + u(s)2 ds + C y(t)2Y − C y0 2Y =



t

|C ∗ C By(s), y(s) + u(s)|2 ds.

(7.29)

0

From Theorem 7.2, we get C y(t)Y → 0, as t → +∞. Let t → +∞ in (7.29), we obtain  Q(u) =

C y0 2Y

+∞

+ 0

|C ∗ C By(s), y(s) + u(s)|2 ds,

(7.30)

168

7 Output Stabilization of Infinite Dimensional Semilinear Systems

which implies that Q(u) ≥ C y0 2Y , so (7.16) is an optimal control. Now, let vu i (t), i = 1, 2, be two solutions of problem (7.26). From (7.30), we deduce that u i (t) = C ∗ C Byi (t), yi (t), where yi (t) verifies

y˙ (t) = Ay(t) − C ∗ C By(t), y(t)By(t), y(0) = y0 ,

thus yi (t) = y(t), i = 1, 2 and u 1 (t) = u 2 (t).



Example 7.2 Consider the system defined on  ⊂ Rn by ⎧ ∂ y(x, t) ⎪ ⎪ = −iy(x, t) + u(t)By(x, t), x ∈ , t > 0, ⎨ ∂t y(x, t) = 0, x ∈ ∂, t > 0, ⎪ ⎪ ⎩ y(x, 0) = y0 (x), x ∈ ,

(7.31)

where X = L 2 () is endowed with its natural complex inner product, A = −i, (i ∈ C) with domain D(A) = H 2 () ∩ H01 (), B = χω∗ χω and the control function u(.) ∈ L 2 (0, +∞). Let ω be a subregion of  that verify GCC, and system (7.31) is augmented with the output (7.32) z(t) = χω y(t). The operator A generates a semigroup of isometries on L 2 (), and     Re χω∗ χω Az, z = 0, ∀z ∈ D(A). Since the subregion ω verifies GCC, so that the inequality (7.3) holds (see [3, 4]). We deduce that the control u(t) = −χω y(., t)2L 2 (ω) , strongly stabilizes the output (7.32) and minimizes the cost  Q(u) = 0

+∞

 χω y(., t)2L 2 (ω) dt +

+∞

|u(t)|2 dt.

0

7.1.3 Output Weak Stabilization The following result discusses the weak stabilization of the output (7.2). Theorem 7.3 Let A generate a semigroup (S(t))t≥0 of contractions on X , B be locally Lipschitz and weakly sequentially continuous. If assumptions 1, 2 of Theorem 7.1 and

7.1 Output Stabilization of Semilinear Systems

C ∗ C B S(t)ψ, S(t)ψ = 0, ∀t ≥ 0 =⇒ Cψ = 0, hold, then the control

u(t) = −C ∗ C By(t), y(t),

169

(7.33)

(7.34)

is in Uad = L 2 (0, ∞), and weakly stabilizes the output (7.2). Proof Let us consider ψ ∈ Y and (tn ) ≥ 0 be a sequence of real numbers such that tn → +∞, as n → +∞. Using (7.17), we deduce that the sequence h n = C y(tn ), ψY is bounded. Let h γ (n) be an arbitrary convergent subsequence of h n . From (7.6), we get y(tγ (n) ) is bounded in H , so we can extract a subsequence still denoted by y(tγ (n) ) such that y(tγ (n) )  ϕ ∈ X , as n → +∞. Since C is continuous, B is weakly sequentially continuous and S(t) is continuous ∀t ≥ 0, we get lim C ∗ C B S(t)y(tγ (n) ), S(t)y(tγ (n) ) = C ∗ C B S(t)ϕ, S(t)ϕ.

n→+∞

From (7.21), we have 

T

C ∗ C B S(s)y(tγ (n) ), S(s)y(tγ (n) )ds → 0, as n → +∞.

0

Using the dominated convergence Theorem, we deduce that C ∗ C B S(t)ϕ, S(t)ϕ = 0, for all t ≥ 0, which implies, according to (7.33), that Cϕ = 0, and hence h n → 0, as t → +∞. We deduce that C y(t), ψY → 0, as t → +∞. In other words C y(t)  0, as t → +∞, which achieves the proof.  Remark 7.2 Condition (7.33) may be verified for some subregion ω, but not on the whole domain , as illustrated below. Let us consider the system defined in  =]0, +∞[ by ⎧ ∂ y(x, t) ⎨ ∂ y(x, t) =− + u(t)By(x, t), x ∈ , t > 0, ∂t ∂x ⎩ y(x, 0) = y (x), x ∈ , 0

(7.35)

∂y with domain D(A) = {y ∈ H 1 () | y(0) = 0, y(x) ∂  x1   y(x)d x  the control operator and u(.) ∈ L 2 (0, +∞). → 0, as x → +∞}, By=

where X = L 2 (), Ay = −

0

The operator A generates a semigroup of contractions

170

7 Output Stabilization of Infinite Dimensional Semilinear Systems

(S(t)y0 )(x) =

y0 (x − t), if x ≥ t, 0, if x < t.

Let ω =]0, 1[ be a subregion of  and system (7.35) is augmented with the output z(t) = χω y(t).

(7.36)

Thus stabilizing the output (7.36) means the stabilization of system (7.35) on ω. We have  1   ∗ y  (x)y(x)d x χω χω Ay, y = − 0

y 2 (1) ≤ 0, =− 2 so, the assumption 1 of Theorem 7.1 holds. The operator B is sequentially continuous and verifies 

  χω∗ χω B S(t)y0 , S(t)y0 = 



1−t

 y0 (x)d x 



0

1−t

y0 (x)d x, 0 ≤ t ≤ 1.

0

Thus χω∗ χω B S(t)y0 , S(t)y0  = 0, ∀t ≥ 0 =⇒ y0 (x) = 0 a.e x ∈]0, 1[, i.e χ]0,1[ y0 = 0.

Then, the control

 u(t) = −



1

 y(x, t)d x 

0



1

y(x, t)d x,

(7.37)

0

weakly stabilizes the output (7.36).

7.2 Output Stabilization of Bilinear Systems We consider the following bilinear system ⎧ ⎨ dy(t) = Ay(t) + u(t)By(t) dt ⎩ y(0) = y , 0

(7.38)

where B : X → X is a linear bounded operator. The system (7.38) is augmented with the output z(t) := C y(t). (7.39)

7.2 Output Stabilization of Bilinear Systems

171

7.2.1 Output Exponential Stabilization In this subsection, we develop sufficient conditions for exponential stabilization of the output (7.39). The following result concerns the exponential stabilization of (7.39). Theorem 7.4 Let A generate a semigroup (S(t))t≥0 of contractions on X and B is a bounded control operator. If the conditions: 1. C ∗ C Aψ, ψ ≤ 0, ∀ψ ∈ D(A), where C ∗ is the adjoint of C, 2. C ∗ C Bψ, ψBψ, ψ ≥ 0, ∀ψ ∈ X , 3. there exist T, γ > 0, such that 

T

0

|C ∗ C B S(t)ψ, S(t)ψ|dt ≥ γ Cψ2Y , ∀ ψ ∈ X,

(7.40)

hold, then the control ⎧ ∗ ⎨− C C By(t), y(t) y(t)2 u(t) = ⎩ 0

if y(t) = 0

(7.41)

if y(t) = 0,

exponentially stabilizes the output (7.39). Proof System (7.38) has a unique weak solution y(t) (see [1]) defined on a maximal interval [0, tmax ] by  y(t) = S(t)y0 +

t

g(y(s))S(t − s)By(s)ds,

(7.42)

0

where g(y(t)) =

∗ By(t),y(t) − C Cy(t) 2 0

if y(t) = 0 if y(t) = 0.

Since (S(t))t≥0 is a semigroup of contractions, we deduce d y(t)2 ≤ 2g(y(t))By(t), y(t). dt Integrating this inequality over the interval [0, t], we have 

t

y(t) − y(0) ≤ 2 2

2

g(y(s))By(s), y(s)ds. 0

Using hypothesis 2 of Theorem 7.4, it follows that y(t) ≤ y0 .

(7.43)

172

7 Output Stabilization of Infinite Dimensional Semilinear Systems

For all y0 ∈ X and t ≥ 0, we have C ∗ C B S(t)y0 , S(t)y0  = C ∗ C By(t), y(t) − C ∗ C By(t), y(t) − S(t)y0  + C ∗ C B S(t)y0 − C ∗ C By(t), S(t)y0 . Since B is bounded, then |C ∗ C B S(t)y0 , S(t)y0 | ≤ |C ∗ C By(t), y(t)| + 2αBy(t) − S(t)y0 y0 , (7.44) where α is a positive constant. Using (7.43), we deduce |C ∗ C By(t), y(t)| ≤ |g(y(t))|y(t)y0 , ∀t ∈ [0, T ].

(7.45)

While from (7.42) and using Schwartz’s inequality, we obtain   y(t) − S(t)y0  ≤ B T

T

|g(y(t))| y(t) dt 2

 21

2

.

(7.46)

0

Integrating (7.44) over the interval [0, T ] and taking into account (7.45) and (7.46), we have 

T



∗

3 2

|C C B S(t)y0 , S(t)y0 |dt ≤ 2αT B y0  2

0

 1 2

+ T y0 

T

|g(y(t))| y(t) dt 2

0 T

|g(y(t))| y(t) dt 2

 21

2

2

 21

.

(7.47)

0

Let us consider the nonlinear semigroup U (t)y0 := y(t). Replacing y0 by U (t)y0 in (7.47), and using the superposition properties of the semigroup (U (t))t≥0 , we deduce that  T 0

3

|C ∗ C B S(s)U (t)y0 , S(s)U (t)y0 |ds ≤ 2αT 2 B2 U (t)y0   ×

t+T t

(7.48) 1 2

|g(U (s)y0 )|2 U (s)y0 2 ds

1



+ T 2 U (t)y0 

t+T t

1 |g(U (s)y0 )|2 U (s)y0 2 ds

2

.

Thus, by using (7.40) and (7.48), it follows that  γ CU (t)y0 Y ≤ M

t+T

|g(U (s)y0 )| U (s)y0  ds 2

t

2

 21

,

(7.49)

7.2 Output Stabilization of Bilinear Systems

173

1

where M = (2αT B2 + 1)T 2 is a positive constant depending on y0  and T . From hypothesis 1 of Theorem 7.1, we have d CU (t)y0 2Y ≤ −2|g(U (t)y0 )|2 U (t)y0 2 . dt

(7.50)

Integrating (7.50) from nT and (n + 1)T, (n ∈ N), we obtain  CU (nT )y0 2Y

− CU ((n +

1)T )y0 2Y

≥2

(n+1)T

|g(U (s)y0 )|2 U (s)y0 2 ds.

nT

Using (7.49), (7.50) and the fact that CU (t)y0 Y decreases, it follows  1+2

 γ 2  M

CU ((n + 1)T )y0 2Y ≤ CU (nT )y0 2Y .

Then CU ((n + 1)T )y0 Y ≤ βCU (nT )y0 Y , where β = Taking n =

1 1 . By recurrence, we show that CU (nT )y0 Y γ 2 2 1+2( M ) E( Tt ) the integer part of Tt , we deduce that



≤ β n C y0 Y .

CU (t)y0 Y ≤ Re−σ t y0 , 

γ 2 21 where R = α 1 + 2 M , with α > 0 and σ = the proof.

 γ 2 ln 1+2( M ) 2T

> 0, which achieves 

7.2.2 Output Strong Stabilization The following result will be used to prove strong stabilization of the output (7.39). Theorem 7.5 Let A generate a semigroup (S(t))t≥0 of contractions on X and B : X → X is a bounded linear operator. If the conditions:  1. C ∗ C Aψ, ψ ≤ 0, ∀ψ  ∈ D(A), 2. C ∗ C Bψ, ψ Bψ, ψ ≥ 0, ∀ψ ∈ X , hold, then control   ∗ C C By(t), y(t)   , u(t) = − (7.51) 1 + | C ∗ C By(t), y(t) | allows the estimate

174

7 Output Stabilization of Infinite Dimensional Semilinear Systems



T

  | C ∗ C B S(s)y(t), S(s)y(t) |ds

2



t+T

=

0

t

   | C ∗ C By(s), y(s) |2   ds , 1 + | C ∗ C By(s), y(s) | as t → +∞. (7.52)

Proof We have      1 d C y(t), C y(t) Y = C Ay(t), C y(t) Y + u(t) C By(t), C y(t) Y . 2 dt Then      1 d 1 d C y(t), C y(t) Y = C y(t)2Y = C ∗ C Ay(t), y(t) + u(t) C ∗ C By(t), y(t) . 2 dt 2 dt

From hypothesis 1 of Theorem 7.5, we have 1 d C y(t)2Y ≤ u(t)C ∗ C By(t), y(t). 2 dt In order to make the function 21 C y(t)2Y nonincreasing, we consider the control   ∗ C C By(t), y(t)   , u(t) = − 1 + | C ∗ C By(t), y(t) | so that the resulting closed-loop system is y  (t) = Ay(t) + f (y(t)), y(0) = y0 , 

 where f (z) = −

(7.53)

C ∗ C Bz,z Bz



 , ∀ z ∈ X.

1+| C ∗ C Bz,z |

Since f is locally Lipschitz, then system (7.38) has a unique mild solution y(t) (see [1]) defined on a maximal interval [0, tmax ] by 

t

y(t) = S(t)y0 +

S(t − s) f (y(s))ds.

(7.54)

0

  Because of the contractions of the semigroup (i.e Aψ, ψ ≤ 0, ∀ ψ ∈ D(A)), we have   ∗ C C By(t), y(t) By(t), y(t) d 2   y(t) ≤ −2 . dt 1 + | C ∗ C By(t), y(t) | Integrating this inequality over the interval [0, t], we deduce 

t

y(t) − y(0) ≤ −2 2

2

0

 ∗  C C By(s), y(s) By(s), y(s)   ds. 1 + | C ∗ C By(s), y(s) |

7.2 Output Stabilization of Bilinear Systems

175

Using condition 2 of Theorem 7.5, it follows that y(t) ≤ y0 .

(7.55)

From hypothesis 1 of Theorem 7.5, we have   | C ∗ C By(t), y(t) |2 d 2   . C y(t)Y ≤ −2 dt 1 + | C ∗ C By(t), y(t) | Integrating this inequality, we deduce  C y(t)2Y

−

C y(0)2Y

t

≤ −2 0

  | C ∗ C By(s), y(s) |2   ds. 1 + | C ∗ C By(s), y(s) |

(7.56)

While from (7.54) and using Schwartz inequality, we obtain    21 | C ∗ C By(s), y(s) |2   ds , ∀t ∈ [0, T ]. ∗ 0 1 + | C C By(s), y(s) | (7.57) Since B is bounded and C continuous, we have   y(t) − S(t)y0  ≤ By0  T

t

|C ∗ C B S(s)y0 , S(s)y0 | ≤ 2K By(s) − S(s)y0 y0  + |C ∗ C By(s), y(s)|, (7.58) where K is a positive constant. Replacing y0 by y(t) in (7.57) and (7.58), we deduce   t+T

|C ∗ C B S(s)y(t), S(s)y(t)| ≤ 2K B2 y0 2 T

t

  1 2 | C ∗ C By(s), y(s) |2   ds ∗ 1 + | C C By(s), y(s) |

+ |C ∗ C By(t + s), y(t + s)|,

∀t ≥ s ≥ 0.

Integrating this relation over [0, T ] and using Cauchy-Schwartz, we obtain 

T

   2 23 2 |C C B S(s)y(t), S(s)y(t)|ds ≤ 2K B T + T 1 + K By0  ∗

0

 × t

which achieves the proof.

t+T

   21 | C ∗ C By(s), y(s) |2   ds , 1 + | C ∗ C By(s), y(s) | 

The following result gives sufficient conditions for strong stabilization of the output (7.39).

176

7 Output Stabilization of Infinite Dimensional Semilinear Systems

Theorem 7.6 Let A generate a semigroup (S(t))t≥0 of contractions on X and B is a bounded linear operator. If the assumptions 1, 2 of Theorem 7.5 and  0

T

|C ∗ C B S(t)ψ, S(t)ψ|dt ≥ γ Cψ2Y , ∀ ψ ∈ X, ( f or some T, γ > 0),

(7.59) hold, then control (7.51) strongly stabilizes the output (7.39) with decay estimate   1 C y(t)Y =  √ , as t −→ +∞. t

(7.60)

Proof Using (7.56), we deduce  C y(kT )2Y − C y((k + 1)T )2Y ≥ 2

k(T +1)

kT

  | C ∗ C By(t), y(t) |2   dt, k ≥ 0. 1 + | C ∗ C By(t), y(t) |

From (7.52) and (7.59), we have C y(kT )2Y − C y((k + 1)T )2Y ≥ βC y(kT )4Y , where β =

2

γ2 3 2K B2 T 2



+T 1+K By0 2

(7.61)

2 . Taking sk = C y(kT )2Y , the inequality

(7.61) can be written as βsk2 + sk+1 ≤ sk , ∀k ≥ 0. Since sk+1 ≤ sk , we obtain 2 + sk+1 ≤ sk , ∀k ≥ 0. βsk+1

Taking p(s) = βs 2 and q(s) = s − (I + p)−1 (s) in Lemma 3.3 in [5], we deduce sk ≤ x(k), k ≥ 0, where x(t) is the solution of equation x  (t) + q(x(t)) = 0, x(0) = s0 . Since x(k) ≥ sk and x(t) decreases give x(t) ≥ 0, ∀t ≥ 0. Furthermore, it is easy to see that q(s) is an increasing function such that 0 ≤ q(s) ≤ p(s), ∀s ≥ 0. We obtain −βx(t)2 ≤ x  (t) ≤ 0, which implies that x(t) = (t −1 ), as t → +∞.

7.2 Output Stabilization of Bilinear Systems

177

Finally the inequality sk ≤ x(k), together with the fact that C y(t)Y decreases, we deduce the estimate   1 C y(t)Y =  √ , as t −→ +∞. t 

7.2.3 Output Weak Stabilization The following result provides sufficient conditions for weak stabilization of the output (7.39). Theorem 7.7 Let A generate a semigroup (S(t))t≥0 of contractions on X and B is a compact operator. If the conditions:   1. C ∗ C Aψ, ψ ≤ 0, ∀ψ  ∈ D(A), 2. C ∗ C Bψ, ψ Bψ, ψ  ≥ 0, ∀ψ ∈ X , 3. C ∗ C B S(t)ψ, S(t)ψ = 0, ∀t ≥ 0 =⇒ Cψ = 0, hold, then control (7.51) weakly stabilizes the output (7.39). Proof Let us consider the nonlinear semigroup (t)y0 := y(t) and let (tn ) be a sequence of real numbers such that tn −→ +∞ as n −→ +∞. From (7.6), (tn )y0 is bounded in X , then there exists a subsequence (tφ(n) ) of (tn ) such that (tφ(n) )y0  ψ, as n → ∞. Since B is compact and C continuous, we have lim C ∗ C B S(t)(tφ(n) )y0 , S(t)(tφ(n) )y0  = C ∗ C B S(t)ψ, S(t)ψ.

n→+∞

For all n ≥, we set  n (t) :=

φ(n)+t

φ(n)

  | C ∗ C B(s)y0 , (s)y0 |2  ds.  1 + | C ∗ C B(s)y0 , (s)y0 |

It follows that ∀t ≥ 0, n (t) → 0 as n → +∞. Using (7.52), we deduce  lim

n→+∞ 0

t

|C ∗ C B S(s)(tφ(n) )y0 , S(s)(tφ(n) )y0 |ds = 0.

Hence, by the dominated convergence Theorem, we have 

t 0

|C ∗ C B S(s)ψ, S(s)ψ|ds = 0.

178

7 Output Stabilization of Infinite Dimensional Semilinear Systems

We conclude that C ∗ C B S(s)ψ, S(s)ψ = 0, ∀s ∈ [0, t]. Using condition 3 of Theorem 7.7, we deduce that C(tφ(n) )y0  0, as n −→ +∞.

(7.62)

On the other hand, it is clear that (7.62) holds for each subsequence (tφ(n) ) of (tn ) such that C(tφ(n) )y0 weakly converges in Y . This implies that ∀ϕ ∈ Y , we have C(tn )y0 , ϕ → 0 as n −→ +∞ and hence C(t)y0  0, as t −→ +∞.

7.2.4 Examples Example 7.3 Let  denote a bounded open subset of Rn , and consider the following wave equation ⎧ 2 ∂ y(x, t) ∂ y(x, t) ⎪ ⎪ ×]0, +∞[ − y(x, t) = u(t) ⎪ ⎨ ∂t 2 ∂t y(x, t) = 0 ∂×]0, +∞[ ⎪ ⎪ ⎪ ⎩ y(x, 0) = y0 (x), ∂ y(x, 0) = y1 (x) . ∂t

(7.63)

This system has the form of equation (7.38) if we set  X = H01 () × L 2 ()  with  0 I 00 (y1 , z 1 ), (y2 , z 2 ) = y1 , y2  H 1 () + z 1 , z 2  L 2 () , A = and B = . 0 0I We consider the output operator C = I , we have A is skew-adjoint on X and the assumption (7.59) holds. Then the control ∂ y(., t) 2  L 2 (0,1) ∂t , u(t) = − ∂ y(., t) 2  L 2 (0,1) 1+ ∂t 

strongly stabilises system (7.63) with the decay estimate (y(., t),

1 ∂ y(., t) ) X = ( √ ), as t −→ +∞. ∂t t

Example 7.4 Let us consider a system defined on  =]0, 1[ by

(7.64)

7.2 Output Stabilization of Bilinear Systems

179

⎧ ⎨ ∂ y(x, t) = Ay(x, t) + u(t)a(x)y(x, t) ×]0, +∞[ ∂t ⎩ y(x, 0) = y (x) , 0

(7.65)

where X = L 2 (), Ay = −y, and a ∈ L ∞ (0, 1) such that a(x) ≥ 0 a.e on ]0, 1[ and a(x) ≥ c > 0 on subregion ω of  and u(.) ∈ L ∞ (0, +∞) is a control function. System (7.65) is augmented with the output z(t) = χω y(t),

(7.66)

where χω : L 2 () −→ L 2 (ω), the restriction operator to ω and χω∗ is the adjoint operator of χω . The operator A generates a semigroup of contractions on L 2 () given by S(t)y0 = e−t y0 . For all y0 ∈ L 2 () and T = 2, we obtain 

2 0



χω∗ χω B S(t)y0 ,





S(t)y0 dt =

2

e

−2t

 dt

0

ω

a(x)|y0 |2 d x

≥ βχω y0 2L 2 (ω) ,  with β = c

2

e−2t dt > 0.

0



Then the control

a(x)|y(x, t)|2 d x u(t) = −

ω

1+

, a(x)|y(x, t)|2 d x

ω

strongly stabilizes the output (7.15) with decay estimate  1 =  √ , as t −→ +∞. t 

χω y(t) L 2 (ω)

Example 7.5 Consider a system defined in  =]0, +∞[, and described by ⎧ ∂ y(x, t) ⎨ ∂ y(x, t) =− + u(t)By(x, t) ×]0, +∞[ ∂t ∂x ⎩ y(x, 0) = y (x) , 0 where Ay = −

(7.67)

∂y with domain D(A) = {y ∈ H 1 () | y(0) = 0, y(x) → 0 as x → ∂ x 1

+∞} and By =

y(x)d x. The operator A generates a semigroup of contractions 0

(S(t)y0 )(x) =

y0 (x − t) if x ≥ t 0 if x < t.

180

7 Output Stabilization of Infinite Dimensional Semilinear Systems

Let ω =]0, 1[ be a subregion of  and system (7.67) is augmented with the output z(t) = χω y(t).

(7.68)

We have 

 χω∗ χω Ay, y = − =−



1

y  (x)y(x)d x

0 2

y (1) ≤ 0, 2

so, the Assumption 1 of Theorem 7.7 holds. The operator B is compact and verifies 

χω∗ χω B S(t)y0 ,





1−t

S(t)y0 =

2 y0 (x)d x

, 0 ≤ t ≤ 1.

0

Thus χω∗ χω B S(t)y0 , S(t)y0  = 0, ∀t ≥ 0 =⇒ y0 (x) = 0, a.e on ω. Then, the control



2

1

y(x, t)d x 0

u(t) = −



2 ,

1

1+

(7.69)

y(x, t)d x 0

weakly stabilizes the output (7.68).

7.3 Simulations Consider system (7.67) with y(x, 0) = sin(π x), and augmented with the output (7.68). • For ω =]0, 2[, Fig. 7.1 shows that the state is stabilized on ω with error equals 3.4 × 10−4 , and the evolution of control function is given by Fig. 7.2. • For ω =]0, 3[, Fig. 7.3 shows that the state is stabilized on ω with error equals 7.8 × 10−4 and the evolution of control is given by Fig. 7.4. Remark 7.3 It is clear that the control (7.69) stabilizes the state on ω, but do not take into account the residual part \ω.

7.3 Simulations

Fig. 7.1 The stabilization of the state on ω =]0, 2[.

Fig. 7.2 Evolution of control function

181

182

7 Output Stabilization of Infinite Dimensional Semilinear Systems

Fig. 7.3 The stabilization of the state on ω =]0, 3[.

Fig. 7.4 Evolution of control function

References

183

References 1. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983) 2. Ball, J.M., Slemrod, M.: Feedback stabilization of distributed semilinear control systems. Appl. Math. Optim. 5, 169–179 (1979) 3. Bardos, C., Lebeau, G., Rauch, J.: Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim. 30, 1024–1065 (1992) 4. Lebeau, G.: Contrôle de l’équation de Schrödinger. J. Math. Pures Appl. 71, 267–291 (1992) 5. Lasiecka, I., Tataru, D.: Uniform boundary stabilisation of semilinear wave equation with nonlinear boundary damping. J. Differ. Integral Equ. 6, 507–533 (1993)

Chapter 8

Stabilization of Infinite Dimensional Second Order Semilinear Systems

Semilinear are special kinds of nonlinear systems capable of representing a variety of important physical processes. In this chapter we study strong and exponential stabilization of the following system  ytt (t) = Ay(t) + u(t)Byt (t), (8.1) y(0) = y0 , yt (0) = y1 , where A is an dissipative operator on the Hilbert space H endowed with the inner product ., . and the corresponding norm   H , B : H → H is a possibly nonlinear and locally Lipschitz operator, u denotes the control function, and X = V × H is 1 the state space with V = D((−A) 2 ) endowed with the norm vV .

8.1 Stabilization of Bilinear Systems In this section we give controls that ensure strong and exponential stabilization of system (8.1) when B is a linear operator. The system (8.1) can be written in the form  d y˜ (t) = A˜ y˜ (t) + B˜ y˜ (t) dt y˜ (0) = y˜0

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. H. Zerrik and O. Castillo, Stabilization of Infinite Dimensional Systems, Studies in Systems, Decision and Control 355, https://doi.org/10.1007/978-3-030-68600-0_8

(8.2)

185

186

8 Stabilization of Infinite Dimensional Second Order Semilinear Systems

where y˜ (t) = (y, yt ) and A˜ given by A˜ y˜ (t) =



 0 I y˜ (t) A0

and the operator B˜ is given by B˜ y˜ (t) =



 0 . u(t)Byt (t)

the system (8.2) has a unique weak solution in C([0, T ], X ) (see [1]) and given by the variation of constants formula  t S(t − s) B˜ y˜ (s)ds. (8.3) y˜ (t) = S(t) y˜0 + 0

˜ where S(t) is a semigroup generated by the operator A. We consider the following uncontrolled system 

z tt (t) = Az(t), z(0) = z 0 , z t (0) = z 1 ,

(8.4)

8.1.1 Strong Stabilization In this section we give sufficient conditions for strong stabilization of system (8.1). Theorem 8.1 Suppose that the solution z of system (8.4) satisfies  z 0 2V + z 1 2H ≤ α

T

|Bz t (t), z t (t)|dt, ( f or some T, α > 0)

(8.5)

0

then the control u(t) = −Byt (t), yt (t)

(8.6)

strongly stabilizes system (8.1). Proof From the relation Bz t , z t  = B(z t − yt ), z t  − Byt , yt − z t  + Byt , yt , and as B is bounded, we have |Bz t , z t | ≤ B H z t − yt  H z t  H + B H yt − z t  H yt  H + |Byt , yt | (8.7)

8.1 Stabilization of Bilinear Systems

187

1 Differentiating the energy E y (t) = {yt 2H + y2V } and since A is dissipative, we 2 get d E y (t) ≤ −Byt , yt 2 dt 

Then

t

E y (t) ≤ −

Bys (s), ys (s)2 ds + E y (0)

0

Which implies that E y (t) ≤ E y (0)

(8.8)

Moreover, we have yt 2H ≤ (y, yt )2X ≤ 2E y (t) and z t − yt 2H ≤ (ψ − φ)2X

(8.9)

where ψ = (z, z t ) and φ = (y, yt ). Using (8.7), (8.8) and (8.9), we obtain |Bz t , z t | ≤ 2B H ψ − φ X



E y (0) + |Byt , yt |

(8.10)

From (8.3), we have ψ − φ X ≤





t

E y (0)

 |Bys (s), ys (s)| ds 2

0

It follows from Schwartz’s inequality, that ψ − φ H ≤



 T E y (0)

T

|Bys (s), ys (s)| ds

 21

2

(8.11)

0

Integrating (8.10) over the interval [0, T ] and with (8.11), we obtain 

T

√



T

|Bz t , z t |dt ≤ (2E y (0) + 1)B H T

0

|Bys (s), ys (s)| ds

 21

2

0

Condition (8.5) gives z 0 2V

+

z 1 2H

√



≤ α(2E y (0) + 1)B H T

Taking (z 0 , z 1 ) = (y0 , y1 ), we have

T

|Bys (s), ys (s)| ds 2

0

 21

188

8 Stabilization of Infinite Dimensional Second Order Semilinear Systems

√ y0 2V + y1 2H ≤ α(2E y (0) + 1)B H T



T

|Bys (s), ys (s)|2 ds

 21

0

Now replacing (y0 , y1 ) by (y, yt ), we get  t+T

y(t)2V + yt (t)2H ≤ β

t

1 |Bys (s), ys (s)|2 ds

2

Then (y, yt ) X −→ 0, as t −→ +∞, which completes the proof.



Example 8.1 We consider on  =]0, 1[, the following system ⎧ ⎨ ytt (x, t) = yx x (x, t) + u(t)yt (x, t), ×]0, +∞[ y(x, 0) = y0 (x), yt (x, 0) = y1 (x),  ⎩ y(0, t) = y(1, t) = 0, ]0, +∞[

(8.12)

where u(t) ∈ L ∞ (0, +∞) is a real valued control. System (8.12) has a unique weak solution on the state space X = H01 () × L 2 () ([6]). The solution of the uncontrolled system ⎧ ×]0, +∞[ ⎨ z tt (x, t) = z x x (x, t), z(x, 0) = z 0 (x), z t (x, 0) = z 1 (x),  ⎩ z(0, t) = z(1, t) = 0, ]0, +∞[ is given by z(x, t) =

∞ (ak cos(kπ t) + bk sin(kπ t)) sin(kπ x). k=1

The assumption (8.5) is verified, indeed 

2

 Bz t (t), z t (t)dt =

0

=

2



1

|z t (x, t)|2 d xdt

0 0 ∞

(kπ )2 (ak2 + bk2 )

k=1

= z t (0)2L 2 () + z x (0)2L 2 () It follows that the control 

1

u(t) = − 0

strongly stabilizes system (8.12).

|yt (x, t)|2 d x

(8.13)

8.1 Stabilization of Bilinear Systems

189

8.1.2 Exponential Stabilization The following result provides sufficient conditions for exponential stabilization of system (8.1). Theorem 8.2 Suppose that the solution z of system (8.4) satisfies the condition (8.5), then the control ⎧ ⎨ − Byt (t), yt (t) , (y, y ) = (0, 0) t u(t) = (8.14) (y, yt )2X ⎩ 0, (y, yt ) = (0, 0) stabilizes exponentially system (8.1), in other word there exist M > 0 and λ > 0 such that for every (y0 , y1 ) ∈ X y(t)2V + yt (t)2H ≤ Me−λt (y0 2V + y1 2H ). Proof Let us consider the solution z of system (8.4) with z(0) = y0 , z t (0) = y1 and the natural energy associated with (8.1) E y (t) =

1 y(t)2V + yt (t)2H } 2

Since A is dissipative, we have d Byt , yt 2 E y (t) ≤ − dt y(t)2V + yt (t)2H

(8.15)

Integrating this inequality, we get 

t

E y (t) ≤ − 0

Bys (s), ys (s)2 ds + E y (0) y(s)2V + ys (s)2H

Then E y (t) ≤ E y (0)

(8.16)

Furthermore, we have yt 2H ≤ (y, yt )2X ≤ 2E y (t) and z t − yt 2H ≤ (ψ − φ)2X

(8.17)

where ψ = (z, z t ) and φ = (y, yt ). Using (8.7) and combining (8.16) and (8.17), we obtain  |Bz t , z t | ≤ 2B H ψ − φ X 2E y (0) + |Byt , yt |

(8.18)

190

8 Stabilization of Infinite Dimensional Second Order Semilinear Systems

Furthermore, from (8.3), we have 

ψ − φ X ≤ B H

t 0

 |Bys (s), ys (s)|  2E s (s)ds 2E y (s)

Schwartz’s inequality, gives √  ψ − φ X ≤ B H T

T 0

|Bys (s), ys (s)|2  21 ds 2E y (s)

(8.19)

Using (8.16), we have  |Byt , yt |  2E y (t) 2E y (0) 2E y (t)

|Byt , yt | ≤

(8.20)

Integrating (8.18) over the interval [0, T ] and taking into account (8.19) and (8.20), we obtain  T   T |By (s), y (s)|2  21  s s ds |Bz t , z t |dt ≤ (2B H + 1)B H T 2E y (0) 2E (s) y 0 0 It follows from (8.5) that z 0 2V

+

z 1 2H



≤ α(2B H + 1)B H 2T E y (0)



T 0

|Bys (s), ys (s)|2  21 ds 2E y (s)

Taking (z 0 , z 1 ) = (y0 , y1 ), we get y0 2V

+

y1 2H



≤ (2B H + 1)B H 2T E y (0)

 0

T

|Bys (s), ys (s)|2  21 ds 2E y (s)

Now replacing (y0 , y1 ) by (y, yt ), we get    t+T |By (s), y (s)|2  1 s s 2 ds y(t)2V + y(t)2H ≤ (2B H + 1)B H 2T E y (t) 2E (s) y t

Then



t+T

E y (t) ≤ γ t

|Bys (s), ys (s)|2 ds 2E y (s)

(8.21)

√ where γ = (2B H + 1)B H 2T . Integrating inequality (8.15) from nT to (n + 1)T , and since E y (t) decreases, we obtain  (n+1)T |Bys (s), ys (s)|2 ds E y (nT ) − E y ((n + 1)T ) ≥ 2E y (s) nT

8.1 Stabilization of Bilinear Systems

191

From (8.21), we get E y ((n + 1)T ) ≤ r E y (nT ) where r = γ1 . By recurrence, we show that E y ((n + 1)T ) ≤ r n E y (0) Since E y (t) decreases and taking n the integer part of

t T

, we deduce that

E y (t) ≤ Me−λt E y (0) √ √ ln((2B H + 1)B H T ) . where M = (2B H + 1)B H T and λ = T In other word system (8.1) is exponentially stabilizable by the control (8.14).  Example 8.2 On  =]0, 1[, we consider the following system ⎧ ×]0, +∞[ ⎨ ytt (x, t) + yx x x x (x, t) = u(t)yt (x, t),  y(x, 0) = y0 (x), yt (x, 0) = y1 (x), ⎩ y(0, t) = y(1, t) = yx x (0, t) = yx x (1, t) = 0, ]0, +∞[

(8.22)

where u(t) ∈ L ∞ (0, +∞) is a real valued control. System (8.22) has a unique weak solution on the state space X = (H 2 () ∩ 1 H0 ()) × L 2 () ([6]). Moreover, the solution of the following system ⎧ ×]0, +∞[ ⎨ z tt (x, t) + z x x x x (x, t) = 0,  z(x, 0) = z 0 (x), z t (x, 0) = z 1 (x), ⎩ z(0, t) = z(1, t) = z x x (0, t) = z x x (1, t) = 0, ]0, +∞[ takes the form z(x, t) =

∞

(am cos((mπ )2 t) + bm sin((mπ )2 t)) sin(mπ x).

m=1

The condition (8.5) is verified, indeed 

2

 Bz t (t), z t (t)dt =

0

=

2



1

|z t (x, t)|2 d xdt

0 0 ∞

(mπ )4 (am2 + bm2 )

m=1

= z t (0)2L 2 () + z x x (0)2L 2 ()

192

8 Stabilization of Infinite Dimensional Second Order Semilinear Systems

We conclude that the control  u(t) = −

1

|yt (x, t)|2 d x

0

yt 2L 2 () + yx x 2L 2 ()

(8.23)

ensures the exponential stabilization of system (8.22).

8.2 Stabilization of Semilinear Systems In this section we study the strong and exponential stabilization of the semilinear system (8.1) when B is nonlinear. In other hand, we consider the system 

z tt (x, t) = Az(t), z(0) = z 0 , z t (0) = z 1 ,

(8.24)

8.2.1 Strong Stabilization The following result gives sufficient conditions for strong stabilization of system (8.1). Theorem 8.3 If B is locally Lipschitz and there exist T > 0 and δ > 0 such that the solution (z, z t ) of system (8.24) verifies the inequality 

T 0

|Bz t (t), z t (t)|dt ≥ δ{z t (0)2H + z(0)2V },

(8.25)

then the control u(t) = −Byt (t), yt (t)

(8.26)

strongly stabilizes system (8.1). Proof We show first that h : yt → Byt , yt Byt is locally Lipschitz. Since B is locally Lipschitz, then for all R > 0, there exists L > 0 such that Byt − Bz t  H ≤ Lyt − z t  H , ∀yt , z t ∈ H : 0 < yt  H ≤ z t  H ≤ R. (8.27) Let us remark that

8.2 Stabilization of Semilinear Systems

193

Bz t , z t Bz t − Byt , yt Byt = Bz t , z t (Bz t − Byt ) + (Bz t , z t  − Byt , yt )Byt = (Bz t , z t − yt  − Bz t − Byt , yt )Byt + Bz t , z t (Bz t − Byt ) Using (8.27), we have h(z t ) − h(yt ) H ≤ L 2 z t 2H z t − yt  H + L 2 z t − yt  H (z t  H + yt  H )yt  H ≤ Lz t − yt  H where L = 3L R 2 . It follows that system (8.1) has a unique weak solution (y, yt ) (see [1]). Using (8.7) and (8.27), we obtain |Bz t , z t | ≤ Lz t − yt  H z t  H + Lyt − z t  H yt  H + |Byt , yt |

(8.28)

Now, we differentiate E y (t) d E y (t) ≤ −Byt , yt 2 dt 

Then

t

E y (t) ≤ −

Bys (s), ys (s)2 ds + E y (0)

0

It follows that E y (t) ≤ E y (0)

(8.29)

Moreover, we have yt 2H ≤ (y, yt )2X ≤ 2E y (t) and z t − yt 2H ≤ (ψ − φ)2X

(8.30)

where ψ = (z, z t ) and φ = (y, yt ). From (8.28) and combining (8.29) with (8.30), we get  |Bz t , z t | ≤ 2Lψ − φ X 2E y (0) + |Byt , yt |

(8.31)

From (8.3), we have ψ − φ X ≤

 2E y (0)



t

|Byt , yt |2 ds

0

For a fixed T > 0, Schwarz inequality, yields ψ − φ X ≤

 2T E y (0)g(0)

(8.32)

194

8 Stabilization of Infinite Dimensional Second Order Semilinear Systems



t+T

where g(t) =

|Bys (s), ys (s)|2 ds.

t

Integrating (8.31) over the interval [0, T ] and by (8.32), we obtain 

T

 |Bz t , z t |dt ≤ (2E y (0) + 1)L T g(0)

0

Condition (8.25) allows z 1 2H + z 0 2V ≤

√ (2E y (0) + 1)L T  g(0) δ

Replacing (z 0 , z 1 ) by (y0 , y1 ), we get y1 2H

+

y0 2V

√ (2E y (0) + 1)L T  ≤ g(0) δ

Now, replacing (y0 , y1 ) by (y, yt ), we obtain  yt 2H + y2V ≤ β g(t) √ (2E y (0) + 1)L T . where β = δ It follows that (y, yt ) X −→ 0 as t −→ +∞, we deduce that the control (8.26) strongly stabilizes system (8.1). Example 8.3 On  =]0, 1[, we consider the following system ⎧ ⎨ ytt (x, t) = yx x (x, t) + u(t)Byt (x, t), ×]0, +∞[ y(x, 0) = y0 (x), yt (x, 0) = y1 (x),  ⎩ y(0, t) = y(1, t) = 0, ]0, +∞[

(8.33)

where the operator Byt = (max{yt  L 2 () , 1})yt and u(t) ∈ L ∞ (0, +∞) is a real valued control. System (8.33) has a unique solution on state space X = H01 () × L 2 () ([6]). The solution of the uncontrolled system ⎧ ×]0, +∞[ ⎨ z tt (x, t) = z x x (x, t), z(x, 0) = z 0 (x), z t (x, 0) = z 1 (x),  ⎩ z(0, t) = z(1, t) = 0, ]0, +∞[ is given by z(t, x) =

∞ (ak cos(kπ t) + bk sin(kπ t)) sin(kπ x). k=1

8.2 Stabilization of Semilinear Systems

195

The assumption (8.25) is verified, indeed  0

2

 Bz t (t), z t (t)dt = (max{z t  L 2 () , 1}) ≥

2 0

∞ (kπ )2 (ak2 + bk2 )



1

|z t (x, t)|2 d xdt

0

k=1

≥ z t (0)2L 2 () + z x (0)2L 2 () We conclude that  u(t) = −(max{yt  L 2 () , 1})

1

|yt (x, t)|2 d x

(8.34)

0

strongly stabilizes system (8.33).

8.2.2 Exponential Stabilization Theorem 8.4 Assume that B is locally Lipschitz and there exist T > 0 and C > 0 such that the solution (z, z t ) of system (8.24) satisfy the following condition 

T 0

|Bz t (t), z t (t)|dt ≥ C{z t (0)2H + z(0)2V },

(8.35)

then the control ⎧ ⎨ − Byt (t), yt (t) , (y, y ) = (0, 0) t u(t) = (y, yt )2X ⎩ 0, (y, yt ) = (0, 0)

(8.36)

exponentially stabilizes system (8.1). Proof Let φ = (y, yt ) et ψ = (z, z t ). We show first that the operator f : φ → Byt , yt  Byt is locally Lipschitz. (y, yt )2X Since B is locally Lipschitz, then for all R > 0, there exists K > 0 such that Byt − Bz t  H ≤ K yt − z t  H , ∀yt , z t ∈ H : 0 < yt  H ≤ z t  H ≤ R. (8.37) Then f (ψ) − f (φ) =

φ2X (Bz t , z t Bz t − Byt , yt Byt ) ψ2X φ2X

−

(ψ2X − φ2X )Byt , yt Byt ψ2X φ2X

.

196

8 Stabilization of Infinite Dimensional Second Order Semilinear Systems

It follows that Bz t , z t Bz t − Byt , yt Byt  H

 f (ψ) − f (φ) X ≤

ψ2X

φ X ψ2X

Bz t , z t (Bz t − Byt ) + (Bz t , z t − yt 

≤

−

+ K 2 |ψ2X − φ2X |.

ψ2X Byt − Bz t , yt )Byt  H ψ2X

+ 2K 2 ψ − φ X

Furthermore, we have z t − yt 2H ≤ (ψ − φ)2X

(8.38)

Using (8.38), we get  f (ψ) − f (φ) X ≤ Lψ − φ X where L = 5K 2 . We conclude that system (8.1) has a unique weak solution (y, yt ) (see [1]). From (8.37) and (8.7), we get |Bz t , z t | ≤ K z t − yt  H z t  H + K yt − z t  H yt  H + |Byt , yt |

(8.39)

Moreover, we have yt 2H ≤ (y, yt )2X ≤ 2E y (t) Since A is dissipative, then Byt , yt 2 d E y (t) ≤ − dt yt 2H + y2V Thus



t

E y (t) = − 0

Bys (s), ys (s)2 ds + E y (0) ys (s)2H + y(s)2V

It follows that E y (t) ≤ E y (0)

(8.40)

Using (8.38), (8.39) and (8.40), we obtain  |Bz t , z t | ≤ 2K ψ − φ X 2E y (0) + |Byt , yt | From (8.3), we deduce 

t

ψ − φ X ≤ K 0

 | f (s)| 2E y (s)ds

(8.41)

8.2 Stabilization of Semilinear Systems

197

Bys (s), ys (s) . 2E y (s) Schwarz inequality, gives

where f (s) =

√ ψ − φ X ≤ K 2T



T

 1 | f (s)|2 E y (s)ds ) 2

(8.42)

0

Using (8.40), we get   |Byt , yt | ≤ | f (t)| 2E y (t) 2E y (0)

(8.43)

Integrating (8.41) on the interval [0, T ] and from (8.42) and (8.43), we obtain  T 0

|Bz s (s), z s (s)|ds ≤ 2(2K 2 + 1)



T E y (0)

 T 0

1 2

| f (s)|2 E y (s)ds

It follows from (8.35), that z t (0)2H

+

z(0)2V

2(2K + 1)  ≤ T E y (0) C



T

| f (s)| E y (s)ds

 21

2

0

Taking (z 0 , z 1 ) = (y0 , y1 ), we have y1 2H

+

y0 2V

2(2K + 1)  ≤ T E y (0) C



T

| f (s)| E y (s)ds

 21

2

0

Replacing (y0 , y1 ) by (y, yt ), we get yt 2H

+

y2V

2(2K + 1)  ≤ T E y (t) C

It follows that



t+T

E y (t) ≤ γ



t+T

t

| f (s)|2 E y (s)ds

t

2(2K + 1) √ T. C Integrating from kT to (k + 1)T the inequality

where γ =

d Byt , yt 2 E y (t) ≤ − dt 2E y (t) and since E y (t) decreases, we obtain

| f (s)| E y (s)ds

 21

2

(8.44)

198

8 Stabilization of Infinite Dimensional Second Order Semilinear Systems

 E y (kT ) − E y ((k + 1)T ) ≥

(k+1)T

| f (s)|2 E y (s)ds

kT

From (8.44), we get E y ((k + 1)T ) ≤ r E y (kT ) 1 . γ This implies that

where r =

E y ((k + 1)T ) ≤ r k E y (0)

(8.45)

t t Applying (8.45) to the integer part k = E( ) of , we deduce that T T E y (t) ≤ Me−λt E y (0) √

ln( 2 C T (2K + 1)) . where M = (2K + 1) and λ = T Then the system (8.1) is exponentially stabilizable by the control (8.36). √ 2 T C

Example 8.4 On  =]0, 1[, we consider the following system ⎧ ×]0, +∞[ ⎨ ytt (x, t) + yx x x x (x, t) = u(t)Byt (x, t),  y(x, 0) = y0 (x), yt (x, 0) = y1 (x), ⎩ y(0, t) = y(1, t) = yx x (0, t) = yx x (1, t) = 0, ]0, +∞[

(8.46)

where the operator Byt = (max{yt  L 2 () , 1})yt and u(t) ∈ L ∞ (0, +∞) is a real valued control. System (8.46) has a unique solution on the state space X = (H 2 () ∩ H01 ()) × 2 L () ([6]). The condition (8.35) is verified, indeed 

2

 Bz t (t), z t (t)dt = (max{z t 

0

≥

∞

L 2 ()

2

, 1}) 0



1

|z t (x, t)|2 d xdt

0

(mπ )4 (am2 + bm2 )

m=1

≥ z t (0)2L 2 () + z x x (0)2L 2 () where z(x, t) =

∞

(am cos((mπ )2 t) + bm sin((mπ )2 t)) sin(mπ x) is the solution of

m=1

the system ⎧ ×]0, +∞[ ⎨ z tt (x, t) + z x x x x (x, t) = 0,  z(x, 0) = z 0 (x), z t (x, 0) = z 1 (x), ⎩ z(0, t) = z(1, t) = z x x (0, t) = z x x (1, t) = 0, ]0, +∞[

8.2 Stabilization of Semilinear Systems

199

We deduce from Theorem 8.4 that the control  u(t) = −

(max{yt  L 2 () , 1})

1

|yt (x, t)|2 d x

0

yt 2L 2 () + yx x 2L 2 ()

(8.47)

exponentially stabilizes the system (8.46).

8.3 Regional Stabilization of Second Order Semilinear Systems In this section we study regional stabilization of the system defined on an open, regular and bounded domain  ⊂ IRn , n ≥ 0 by 

ytt (t) = Ay(t) + u(t)Byt (t), t ≥ 0 y(0) = y0 , yt (0) = y1 ,

(8.48)

where A is an unbounded dissipative operator on a functional space H = H () endowed with the inner product ., . and the corresponding norm   H , B : H → H is a bounded operator and possibly nonlinear, u denotes the control function. Let 1 1 V = D(A 2 ) = V () endowed with the norm vV = A 2 v H and X = V × H be the state space.

8.3.1 Regional Stabilization of Bilinear Systems In this section we study regional stabilization of system (8.48) that can be written in the form  d y˜ (t) = A˜ y˜ (t) + B˜ y˜ (t) (8.49) dt y˜ (0) = y˜0 where y˜ (t) = (y, yt ) and A˜ given by A˜ y˜ (t) =



 0 I y˜ (t) A0

and the operator B˜ is given by B˜ y˜ (t) =



 0 u(t)Byt (t)

200

8 Stabilization of Infinite Dimensional Second Order Semilinear Systems

System (8.49) has a unique weak solution in C([0, T ], X ) (see [1]), given by the variation of constants formula  t S(t − s) B˜ y˜ (s)ds. (8.50) y˜ (t) = S(t) y˜0 + 0

˜ where S(t) is the semigroup generated by the operator A. Let ω be an open subregion of ; V (ω) and H (ω) are respectively functional spaces of the restriction of the functions of V and H . We define by χω1 : V () −→ V (ω) z 1 −→ z 1 |ω and

χω2 : H () −→ H (ω) z 2 −→ z 2 |ω

the restrictions operators to ω while χω1∗ and χω2∗ denote respectively the adjoint operators of χω1 and χω2 . Let us denote i ω = χω1∗ χω1 and jω = χω2∗ χω2 . In what follows, we assume that i ω Byt , yt Byt , yt  ≥ 0, ∀yt ∈ H.

(8.51)

 jω Ay, y ≤ 0, ∀y ∈ D(A).

(8.52)

and

On the other hand, we consider the following uncontrolled system 

z tt (t) = Az(t), z(0) = y0 , z t (0) = y1 ,

(8.53)

Here we give sufficient conditions for regional strong stabilization of system (8.48) when B is linear. Theorem 8.5 Suppose that there exist some positive constants T and α such that the solution z of system (8.53) satisfies  χω1 y0 2V (ω) + χω2 y1 2H (ω) ≤ α

T

|i ω Bz t (t), z t (t)|dt

(8.54)

0

then the control u(t) = −i ω Byt (t), yt (t) stabilizes regionally strongly system (8.48).

(8.55)

8.3 Regional Stabilization of Second Order Semilinear Systems

201

Proof From the relation i ω Bz t , z t  = i ω B(z t − yt ), z t  − i ω Byt , yt − z t  + i ω Byt , yt , and since B and χω1 are bounded, then there exists δ > 0 such that |i ω Bz t , z t | ≤ δ 2 B H z t − yt  H z t  H +δ 2 B H yt − z t  H yt  H + |i ω Byt , yt |

(8.56)

1 Differentiating the energy E y (t) = {yt 2H + y2V } and since A is dissipative, we 2 get d E y (t) ≤ − f (yt (t)) dt where f (yt (t)) = i ω Byt , yt Byt , yt . Then  t E y (t) ≤ −

f (ys (s))ds + E y (0)

0

It follows from (8.51), that E y (t) ≤ E y (0)

(8.57)

Moreover, we have yt 2H ≤ (y, yt )2X ≤ E y (t)

and z t − yt 2H ≤ ψ − φ2X

(8.58)

where ψ = (z, z t ) and φ = (y, yt ). Using (8.56), (8.57) and (8.58), we obtain |i ω Bz t , z t | ≤ 2δ 2 B H ψ − φ X



E y (0) + |i ω Byt , yt |

(8.59)

From (8.50) and since A is dissipative, we have ψ − φ X ≤



 E y (0)

t

 i ω Bys (s), ys (s)ds

0

It follows from Schwartz’s inequality, that ψ − φ X ≤



 T E y (0)

T

|i ω Bys (s), ys (s)| ds

 21

2

0

Integrating (8.59) over the interval [0, T ], and with (8.60) we obtain

(8.60)

202

8 Stabilization of Infinite Dimensional Second Order Semilinear Systems



T 0

√ |i ω Bz t , z t |dt ≤ (2E y (0)B H + 1) T  21  T 2 |i ω Bys (s), ys (s)| ds × 0

Condition (8.54) gives  χω1 y0 2V (ω)

+

χω2 y1 2H (ω)

≤β

T

|i ω Bys (s), ys (s)| ds

 21

2

0

√ where β = α(2E y (0)B H + 1) T . Now replacing (y0 , y1 ) by (y, yt ), we get  χω1 y(t)2V (ω)

+

χω2 yt (t)2H (ω)

≤β

t+T

|i ω Bys (s), ys (s)| ds

 21

2

t

Then χω1 y(t)2V (ω) + χω2 yt (t)2H (ω) −→ 0, as t −→ +∞, which completes the proof. The following result provides sufficient conditions for regional exponential stabilization of system (8.48) when B is linear. Theorem 8.6 Assume that the solution z of system (8.53) satisfies the condition χω1 y0 V (ω) + χω2 y1  H (ω) ≤ α

T 0

|i ω Bz t (t), z t (t)|dt, ( f or some T, α > 0) (8.61)

then the control ⎧ ⎨ − i ω Byt (t), yt (t) , (y, y ) = (0, 0) t u(t) = (y, yt )2X ⎩ 0, (y, yt ) = (0, 0)

(8.62)

stabilizes regionally exponentially system (8.48), in other word there exist M > 0 and λ > 0 such that for every (y0 , y1 ) ∈ X χω1 y(t)2V (ω) + χω2 yt (t)2H (ω) ≤ Me−λt (y0 , y1 )2X . Proof Let us consider the solution z of system (8.53) with the natural energy associated with (8.48) 1 E y (t) = y(t)2V + yt (t)2H } 2 Since A is dissipative, we have d i ω Byt , yt  E y (t) ≤ − Byt , yt  dt 2E y (t)

(8.63)

8.3 Regional Stabilization of Second Order Semilinear Systems

203

Integrating this inequality, gives 

t

E y (t) ≤ − 0

i ω Bys (s), ys (s) Bys (s), ys (s)ds + E y (0) 2E y (s)

Then E y (t) ≤ E y (0)

(8.64)

Furthermore, we have yt 2H ≤ (y, yt )2X ≤ 2E y (t) and z t − yt 2H ≤ ψ − φ2X

(8.65)

where ψ = (z, z t ) and φ = (y, yt ). Using (8.56) and combining (8.64) and (8.65), we obtain  |i ω Bz t , z t | ≤ 2δ 2 B H ψ − φ H 2E y (0) + |i ω Byt , yt |

(8.66)

Furthermore, from (8.50), we have ψ − φ X ≤ B H



t 0

 |i ω Bys (s), ys (s)|  2E s (s)ds 2E y (s)

Schwartz’s inequality, gives   ψ − φ X ≤ B H 2T E y (0)



|i ω Bys (s), ys (s)|2  21 ds 2E y (s)

T

0

(8.67)

Using (8.64), we have |i ω Byt , yt | ≤

 |i ω Byt , yt |  2E y (t) 2E y (0) 2E y (t)

(8.68)

Integrating (8.66) over the interval [0, T ] and taking into account (8.67) and (8.68), we obtain    T |i By (s), y (s)|2  1 ω s s 2 ds |i ω Bz t , z t |dt ≤ (2B H + 1)B H T 2E y (0) 2E y (s) 0 0

 T

It follows from (8.61) that χω1 y0 V (ω)

+

χω2 y1  H (ω)

≤β



T 0

 where β = α(2B H + 1)B H 2T E y (0). Now replacing (y0 , y1 ) by (y, yt ), we get

|i ω Bys (s), ys (s)|2  21 ds 2E y (s)

204

8 Stabilization of Infinite Dimensional Second Order Semilinear Systems

χω1 yt (t)V (ω)

+

χω2 y(t) H (ω)

≤β



t+T

t

|i ω Bys (s), ys (s)|2  21 ds . 2E y (s)

Then  χω1 y(t)2V (ω) + χω2 yt (t)2H (ω) ≤ β 2

t+T t

|i ω Bys (s), ys (s)|2 ds 2E y (s)

(8.69)

Moreover, we have d |i ω Byt , yt |2 (χω1 y(t)2V (ω) + χω2 yt (t)2H (ω) ) ≤ − dt 2E y (t)

(8.70)

Integrating inequality (8.70) from nT to (n + 1)T , and since E y (t) decreases, we obtain χω1 y(nT )2V (ω) + χω2 yt (nT )2H (ω) − χω1 y((n + 1)T )2V (ω) − χω2 yt ((n + 1)T )2H (ω) ≥

 (n+1)T |i ω Bys (s), ys (s)|2 ds 2E y (s) nT

From (8.69), we get χω1 y((n + 1)T )2V (ω) + χω2 yt ((n + 1)T )2H (ω) ≤ r (χω1 y(nT )2V (ω) + χω2 yt (nT )2H (ω) ) where r = β12 . By recurrence, we show that χω1 y(nT )2V (ω) + χω2 yt (nT )2H (ω) ≤ r n (χω1 y0 2V (ω) + χω2 y1 2H (ω) ) Since E y (t) decreases and taking n the integer part of

t T

, it follows that

χω1 y(t)2V (ω) + χω2 yt (t)2H (ω) ≤ Me−λt (y0 , y1 )2X √ where M = (2B H + 1)δB √ H T and ln((2B H + 1)B H T ) . λ= T In other word system (8.48) is regionally exponentially stabilizable by the control (8.62). 

8.3 Regional Stabilization of Second Order Semilinear Systems

205

8.3.2 Regional Stabilization of Semilinear Systems In this section we study regional strong and exponential stabilization of system (8.48) when B is nonlinear. Let us consider the system 

z tt (t) = Az(t), z(0) = y0 , z t (0) = y1 ,

(8.71)

The following result gives sufficient conditions for strong stabilization of system (8.48). Theorem 8.7 Assume that B is locally Lipschitz and there exist T > 0 and δ > 0 such that the solution (z, z t ) of system (8.71) verifies the inequality  0

T

|i ω Bz t (t), z t (t)|dt ≥ δ{χω2 yt (0)2H (ω) + χω1 y(0)2V (ω) },

(8.72)

then the control u(t) = −i ω Byt (t), yt (t)

(8.73)

stabilizes regionally strongly system (8.48). Proof We show first that h : yt → i ω Byt , yt Byt is locally Lipschitz. Since B is locally Lipschitz, then for all R > 0, there exists L > 0 such that Byt − Bz t  H ≤ Lyt − z t  H , ∀yt , z t ∈ H : (8.74) 0 < yt  H ≤ z t  H ≤ R. Let remark that i ω Bz t , z t Bz t − i ω Byt , yt Byt = i ω Bz t , z t (Bz t − Byt ) +(i ω Bz t , z t  − i ω Byt , yt )Byt = (i ω Bz t , z t − yt  − i ω Bz t − i ω Byt , yt )Byt +i ω Bz t , z t (Bz t − Byt ) Using (8.74), we have h(z t ) − h(yt ) H ≤ L 2 z t 2H z t − yt  H + L 2 z t − yt  H (z t  H + yt  H )yt  H ≤ Lz t − yt  H where L = 3L R 2 . It follows that system (8.48) has a unique weak solution (y, yt ) (see [1]). Using (8.56) and (8.74), we obtain

206

8 Stabilization of Infinite Dimensional Second Order Semilinear Systems

|i ω Bz t , z t | ≤ Lz t − yt  H z t  H +Lyt − z t  H yt  H + |i ω Byt , yt |

(8.75)

Now, we differentiate E y (t) d E y (t) ≤ −i ω Byt , yt Byt , yt  dt 

Then

t

E y (t) ≤ −

i ω Bys (s), ys (s)Bys (s), ys (s)ds + E y (0)

0

It follows from (8.51), that E y (t) ≤ E y (0)

(8.76)

Moreover, we have yt 2H ≤ (y, yt )2X ≤ 2E y (t) and z t − yt 2H ≤ ψ − φ2X

(8.77)

where ψ = (z, z t ) and φ = (y, yt ). From (8.75) and combining (8.76) with (8.77), we get  |i ω Bz t , z t | ≤ 2Lψ − φ X 2E y (0) + |i ω Byt , yt |

(8.78)

From (8.50), we have ψ − φ X ≤





t

2E y (0)

|i ω Byt , yt |2 ds

0

For a fixed T > 0, Schwarz inequality, yields ψ − φ X ≤ 

t+T

where g(t) =

 2T E y (0)g(0)

|i ω Bys (s), ys (s)|2 ds.

t

Integrating (8.78) on the interval [0, T ] and using (8.79), we obtain 

T

 |i ω Bz t , z t |dt ≤ (2E y (0) + 1)L T g(0)

0

Condition (8.72) allows χω2 y1 2H (ω)

+

χω1 y0 2V (ω)

√ (2E y (0) + 1)L T  ≤ g(0) δ

Now, replacing (y0 , y1 ) by (y, yt ), we obtain

(8.79)

8.3 Regional Stabilization of Second Order Semilinear Systems

207

 χω2 yt 2H (ω) + χω1 y2V (ω) ≤ β g(t) √ (2E y (0) + 1)L T . δ It follows that χω1 y(t)2V (ω) + χω2 yt (t)2H (ω) −→ 0 as t −→ +∞, we deduce that the control (8.73) stabilizes regionally strongly system (8.48). 

where β =

The following result gives sufficient conditions for regional exponential stabilization of system (8.48). Theorem 8.8 Assume that B is locally Lipschitz and there exist T > 0 and C > 0 such that the solution (z, z t ) of system (8.71) satisfy the following condition 

T 0

|i ω Bz t (t), z t (t)|dt ≥ C{χω2 y1  H (ω) + χω1 y0 V (ω) },

(8.80)

then the control ⎧ ⎨ − i ω Byt (t), yt (t) , (y, y ) = (0, 0) t u(t) = (y, yt )2X ⎩ 0, (y, yt ) = (0, 0)

(8.81)

stabilizes exponentially system (8.48) on ω. Proof Let φ = (y, yt ) et ψ = (z, z t ). We show first that the operator f : φ → i ω Byt , yt  Byt is locally Lipschitz. (y, yt )2X Since B is locally Lipschitz, then for all R > 0, there exists K > 0 such that Byt − Bz t  H ≤ K yt − z t  H , ∀yt , z t ∈ H : 0 < yt  H ≤ z t  H ≤ R. (8.82) Then

φ2X (i ω Bz t , z t Bz t − i ω Byt , yt Byt ) ψ2X φ2X 2 2 (ψ X − φ X )i ω Byt , yt Byt − . ψ2X φ2X

f (ψ) − f (φ) =

It follows that i ω Bz t , z t Bz t − i ω Byt , yt Byt  H ψ2X φ X 2 2 +K |ψ X − φ2X |. ψ2X

 f (ψ) − f (φ) X ≤

(i ω Bz t , z t − yt  + ψ2X ψ2X i ω Byt − i ω Bz t , yt )Byt  H − + 2K 2 ψ − φ X ψ2X ≤

i ω Bz t , z t (Bz t − Byt )

208

8 Stabilization of Infinite Dimensional Second Order Semilinear Systems

Furthermore, we have z t − yt 2H ≤ (ψ − φ)2X

(8.83)

Using (8.83), we get  f (ψ) − f (φ) X ≤ Lψ − φ X where L = 5K 2 . We conclude that system (8.48) has a unique weak solution (y, yt ) (see [1]). Using (8.82) and (8.56), then there exists α > 0 such that |i ω Bz t , z t | ≤ K α 2 z t − yt  H z t  H +K α 2 yt − z t  H yt  H + |i ω Byt , yt |

(8.84)

Moreover, we have yt 2H ≤ (y, yt )2X ≤ 2E y (t) Since A is dissipative, then i ω Byt , yt Byt , yt  d E y (t) ≤ − dt yt 2H + y2 

Thus

t

E y (t) = − 0

i ω Bys (s), ys (s)Bys (s), ys (s) ds + E y (0) ys (s)2H + y(s)2

It follows from (8.51), that E y (t) ≤ E y (0)

(8.85)

Using (8.83), (8.84) and (8.85), we obtain  |i ω Bz t , z t | ≤ 2K α 2 ψ − φ X 2E y (0) + |i ω Byt , yt |

(8.86)

From (8.50), we deduce that 

t

ψ − φ X ≤ K

 | f (s)| 2E y (s)ds

0

i ω Bys (s), ys (s) . 2E y (s) Schwarz inequality, gives

where f (s) =

√ ψ − φ X ≤ K 2T



T

| f (s)| E y (s)ds 2

 21 (8.87)

0

Using (8.85), we get   |i ω Byt , yt | ≤ | f (t)| 2E y (t) 2E y (0)

(8.88)

8.3 Regional Stabilization of Second Order Semilinear Systems

209

Integrating (8.86) on the interval [0, T ] and using (8.87) and (8.88), we obtain 

T





T

|i ω Bz s (s), z s (s)|ds ≤ 2(2K α + 1) T E y (0) 2 2

| f (s)| E y (s)ds

 21

2

0

0

It follows from (8.80), that χω2 yt (0) H (ω)

+

χω1 y(0)V (ω)

≤

2(2K α 2 +1) C



 T E y (0)

T

| f (s)| E y (s)ds

 21

2

0

Replacing (y0 , y1 ) by (y, yt ), we get χω2 yt (t) H (ω)

+

χω1 y(t)V (ω)

≤

2(2K α 2 +1) C



 T E y (t)

t+T

| f (s)| E y (s)ds

 21

2

t

It follows that  χω1 y(t)2V (ω)

+

χω2 yt (t)2H (ω)

≤γ

t+T

| f (s)|2 E y (s)ds

t

2(2K α 2 + 1)  T E y (0). C Integrating from kT to (k + 1)T the inequality

where γ =

d |i ω Byt , yt |2 (χω1 y(t)2V (ω) + χω2 yt (t)2H (ω) ) ≤ − dt 2E y (t) and since (χω1 y(t)2V (ω) + χω2 yt (t)2H (ω) ) decreases, we obtain χω1 y(kT )2V (ω) + χω2 yt (kT )2H (ω) −χω1 y((k + 1)T )2V (ω) − χω2 yt ((k + 1)T )2H (ω)  ≥

(k+1)T

| f (s)|2 E y (s)ds

kT

From (8.89), we get χω1 y((k + 1)T )2V (ω) + χω2 yt ((k + 1)T )2H (ω) ≤ r (χω1 y(kT )2V (ω) + χω2 yt (kT )2H (ω) ) where r =

1 . γ

(8.89)

210

8 Stabilization of Infinite Dimensional Second Order Semilinear Systems

This implies that χω1 y((k + 1)T )2V (ω) + χω2 yt ((k + 1)T )2H (ω) ≤ r k (χω1 y0 2V (ω) + χω2 y1 2H (ω) ) (8.90) t t Applying (8.90) to the integer part k = E( ) of , we deduce that T T χω1 y(t)2V (ω) + χω2 yt (t)2H (ω) ≤ Me−λt (y0 , y1 )2X √

ln( 2 C T (2K + 1)) . where M = (2K + 1) and λ = T Then control (8.81) stabilizes exponentially system (8.48) on ω. √ 2α T C



8.4 Simulations Results In this section we illustrate the previous results by simulations. The developed results show that the stabilizing control is given by (8.14) for exponential stabilization and by (8.6) for strong stabilization. The computation of such control may be achieved using the following algorithm: Step 1: Initial data : threshold > 0 and initial conditiony0 ; Step 2: Apply the control u(ti ) = Byt (ti ), yt (ti ); Step 3: Solving the system (8.1) using Lax-Wendroff method given in [7], gives y(ti+1 ); Step 4: If y(ti+1 ) < stop, otherwise; Step 5: i = i + 1 and go to step 2. Example 8.5 On  =]0, 1[, consider the following system

Fig. 8.1 Initial position

8.4 Simulations Results

211

⎧ ⎪ ⎨ ytt (x, t) − yx x (x, t) = u(t)yt (x, t), ×]0, ∞[ y(x, 0) = sin(2π x), yt (x, 0) = 0,  ⎪ ⎩ y(0, t) = y(1, t) = 0, ]0, ∞[ where the control



1

u(t) = −

(8.91)

|yt (x, t)|2 d x.

0

System (8.91) has a unique solution on the state space H01 () × L 2 () ([6]). Figure 8.1 shows that the state is strongly stabilized on  =]0, 1[ with stabilization error equals to 1.03 × 10−4 . Figure 8.2 shows that the energy of the system decreases to zero (Fig. 8.3). Fig. 8.2 The energy of the system

Fig. 8.3 Evolution of the control

212

8 Stabilization of Infinite Dimensional Second Order Semilinear Systems

References 1. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983) 2. Ball, J.M., Slemrod, M.: Feedback stabilization of distributed semilinear control systems. Appl. Math. Optim. 5, 169–179 (1979) 3. Bardos, C., Lebeau, G., Rauch, J.: Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim. 30, 1024–1065 (1992) 4. Lebeau, G.: Contrôle de l’équation de Schrödinger. J. Math. Pures Appl. 71, 267–291 (1992) 5. Lasiecka, I., Tataru, D.: Uniform boundary stabilisation of semilinear wave equation with nonlinear boundary damping. J. Differ. Integr. Equ. 6, 507–533 (1993) 6. Komornik, V., Loreti, P.: Fourier Series in Control Theory. Springer Monographs in Mathematics (2005) 7. Lax, P., Wendroff, B.: Systems of conservation laws. Commun. Pure Appl. Math. 13, 217–237 (1960)

Chapter 9

Gradient Stabilization of Infinite Dimensional Linear Systems

The notion of gradient stabilization of infinite dimensional system is very important since it modelizes many real applications. For instance the problem of thermal insulation where the purpose is to keep a constant temperature of the system with regards to the outside environment assumed to be with fluctuating temperature. Thus one has to regulate the system temperature in order to vanish the exchange thermal flux. This is the case inside a car where one has to change the level of the internal air conditioning with respect to the external temperature. As we can not always have external measurements, we use a sensor to measure the flux, which is a transducer producing a signal that is proportional to the local heat flux. The goal of this chapter is to study the gradient stabilization of infinite dimensional linear systems. First we give definitions and properties of the gradient stability. Then we characterize controls that stabilize the gradient of such a system and the one that stabilizes the gradient and minimizes a functional cost. The obtained results lead to numerical approach successfully illustrated by simulations.

9.1 Gradient Stability—Definitions—Examples This section is devoted to some preliminaries concerning definition and characterization of gradient stability for infinite dimensional linear systems. Let  be an open regular subset of Rm and let us consider the state-space system ⎧ ⎨∂y = Ay (9.1) ∂t ⎩ y(0) = y ∈ X 0

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. H. Zerrik and O. Castillo, Stabilization of Infinite Dimensional Systems, Studies in Systems, Decision and Control 355, https://doi.org/10.1007/978-3-030-68600-0_9

213

214

9 Gradient Stabilization of Infinite Dimensional Linear Systems

where A : D(A) ⊂ X → X is a linear operator generating a strongly continuous semigroup S(t), t ≥ 0, on the state space X which is continuously embedded in H 1 (). The space X is endowed with its a complex inner product and the corresponding norm .. We define the gradient operator by: ∇ :X → (L 2 ())m  ∂y ∂y ∂y y → , ,..., ∂ x1 ∂ x2 ∂ xm

(9.2)

(L 2 ())m is endowed with its usual complex inner product m and the corresponding norm .m where: ., . (L 2 ())m × (L 2 ())m → C m   f i (x)gi (x)d x ( f, g) −→ i=1



with f = ( f 1 , f 2 , . . . , f m ) and g = (g1 , g2 , . . . , gm ) where f i , gi ∈ L 2 () i = 0, 1, . . . , m. The mild solution of (9.1) is given by y(t) = S(t)y0 . Let ∇ ∗ the adjoint operator of ∇, and we define the operator G = ∇ ∗ ∇, which is a bounded operator applying X into itself. Let L(X, (L 2 ())m ) be the space of bounded linear operators mapping X into 2 (L ())m endowed with the uniform norm of operators |||.|||. Definition 9.1 The system (9.1) is said to be 1. Gradient weakly stable (g.w.s) if ∀y0 ∈ X , the corresponding solution y(t) of (9.1) satisfies < ∇ y(t), z >m −→ 0 as t −→ ∞, ∀z ∈ (L 2 ())m

(9.3)

2. Gradient strongly stable (g.s.s) if for any initial condition y0 ∈ X the corresponding solution y(t) of (9.1) satisfies ||∇ y(t)||m −→ 0 as t −→ ∞

(9.4)

3. Gradient exponentially stable (g.e.s) if there exist M, α > 0 such that ∇ y(t)m ≤ Me−αt y0 , t ≥ 0, y0 ∈ X

(9.5)

Remark 9.1 From the above definitions we have: 1. g.e.s ⇒ g.s.s ⇒ g.w.s. 2. If the system (9.1) is stable then it is also gradient stable. 3. We can find systems gradient stable but not stable. This is illustrated in the following example.

9.1 Gradient Stability—Definitions—Examples

215

Example 9.1 Let  =]0, 1[, on H 1 () we consider the system ⎧ ∂y ⎪ ⎪ ×]0, +∞[ ⎪ ⎨ ∂t (t) = Ay(t) ∂y ∂y (0, t) = (1, t) = 0 ]0, +∞[ ⎪ ⎪ ∂t ∂t ⎪ ⎩ y(., 0) = y0 ∈ H 1 ()

(9.6)

d2 is the Laplace operator. dx2 The eigenpairs (λi , φi ), i ∈ N of A are given by:

where Ay = y + y and  =

λi

2 i ≥0 = 1 − (iπ ) 2 φi (x) = 1+(iπ)2 cos(iπ x) i ≥ 0

A generates a strongly continuous semigroup S(t) given by S(t)y0 =



eλi t y0 , φi φi

i≥0

Since λ0 > 0 then (9.6) isn’t stable but ∇ S(t)y0 2m ≤ e2λ1 t

 y0 , φi 2 ∇φi 2m i>0

≤ e2λ1 t y0 2 Therefore the system (9.6) is g.e.s.

9.2 Characterizations of Gradient Stability The following result links gradient stability of the system (9.1) to the spectrum properties of the operator A. Let us consider the sets σ 1 (A) = {λ ∈ σ (A), Re(λ) ≥ 0, N (A − λI ) ⊂ N (G)} and σ 2 (A) = {λ ∈ σ (A), Re(λ) < 0, N (A − λI ) ⊂ N (G)} where σ (A) and N (A) are the points spectrum and the kernel of the operator A. Proposition 9.1 (1) If the system (9.1) is g.w.s then σ 1 (A) = ∅. (2) Assume that the state space X has an orthonormal basis (φn )n of eigenfunctions of A, if σ 1 (A) = ∅ and, for some α > 0, Re(λ) ≤ −α for all λ ∈ σ 2 (A), then the system (9.1) is g.e.s.

216

9 Gradient Stabilization of Infinite Dimensional Linear Systems

Proof (1) Assume that there exists λ ∈ σ (A) such that Re(λ) ≥ 0 and there exists φ ∈ X such that Aφ = λφ for y0 = φ, the solution of (9.1) is S(t)φ = eλt φ, so | ∇ S(t)φ, ∇φ m | ≥ e Re(λ)t ∇φ2m ≥ ∇φ2m > 0 hence the system (9.1) is not g.w.s. (2) For y0 ∈ X we have ∇ S(t)y0 =



e λn t

rn  y0 , φn,k ∇φn,k

n≥0

k=1

where rn is the multiplicity of the eigenvalue λn . σ 1 (A) = ∅, gives ∇ S(t)y0 m ≤ Me−αt y0 , for some M > 0. 

So we have the g.e.s of the system (9.1).

As example we consider (9.6). We have σ 1 (A) = ∅ and ∀λ ∈ σ 2 (A), Re(λ) ≤ 1 − π 2 , then the system (9.6) is g.e.s. For the gradient exponential stability, we need the following lemma. Lemma 9.1 Assume that there exists a function M(t) ∈ L 2 (0, +∞; R+ ) such that |||∇ S(t + s)||| ≤ M(t)|||∇ S(s)||| ∀ t, s ≥ 0.

(9.7)

then the operators (∇ S(t))t≥0 are uniformly bounded. Proof Let us show that sup |||∇ S(t)||| < +∞. Otherwise there exists a sequence t≥0

(t1 + τk ), t1 > 0 and  τk → ∞ such that |||∇ S(t1 + τk )||| is increasing without bound. ∞

Now we have 0

∇ S(s + τk )y2m ds =

∞

τk

∇ S(s)y2m ds and the right-hand

side goes to zero when k → ∞. By Fatou’s Lemma lim inf ∇ S(s + τk )ym = 0 when k → ∞, almost everywhere 0 ≤ s < ∞. Hence for some s0 < t1 we can find a subsequence {τkn } such that lim ∇ S(s0 + n

τkn )ym = 0. But with (9.7) we have ∇ S(t1 + τkn )ym ≤ M(t1 − s0 )∇ S(s0 + τkn )ym → 0 when n → +∞, which is a contradiction. The conclusion follows from the uniform boundedness principle.  Proposition 9.2 If (9.7) is satisfied and |||∇ S(nt)||| ≤ |||∇ S(t)|||n ∀ t ≥ 0, ∀n ∈ N∗ then the system (9.1) is g.e.s if and only if

(9.8)

9.2 Characterizations of Gradient Stability



∞

0

217

||∇ S(t)y||2m dt < ∞, ∀y ∈ X

(9.9)

Proof We have  =

t∇ S(t)y2m

≤ ≤

t

0 t 0 t 0

∇ S(t)y2m ds ∇ S(s + t − s)y2m ds M 2 (s)∇ S(t − s)y2m ds

from (9.7) from lemma (9.1)

≤ N y

2

where N > 0, then ln |||∇ S(t)||| < 0, ∀t ≥ t0 for some t0 > 0, hence w0 = inf

t>0

ln |||∇ S(t)||| 0 and N = sup S(t), there exists n ∈ IN such that nt1 ≤ t < (n + 1)t1 Now we show that w0 = lim

t→+∞

for each t ≥ t1 , then

t∈[0,t1 ]

ln |||∇ S(nt1 )||| ln S(t − nt1 ) ln |||∇ S(t)||| ≤ + t t t with (9.8) we have nt1 ln |||∇ S(t1 )||| ln N  ln |||∇ S(t)||| ≤ + t t t1 t therefore lim sup t→∞

ln |||∇ S(t)||| ln |||∇ S(t)||| ln |||∇ S(t)||| ≤ inf ≤ lim inf t→∞ t>0 t t t

ln |||∇ S(t)||| . t→+∞ t Hence for all ω ∈]0, −ω0 [, there exists M  such that ||∇ S(t)y||m ≤ M  e−ωt y, ∀y ∈ X, t ≥ 0. So the system is g.e.s. The converse is immediate. 

then w0 = lim

Example 9.2 The system (9.6) satisfies the conditions (9.7) and (9.8). Indeed, let t > 0, and y ∈ H 1 (). We have ∇ S(t)y =

 i>0

eλi t y, φi ∇φi

218

9 Gradient Stabilization of Infinite Dimensional Linear Systems

which implies ∇ S(t)y2m ≤ e2λ1 t

 y, φi 2 ∇φi 2m i>0

≤ e2λ1 t y2  we can show that |∇ S(t)| = e

2λ1 t

+∞

. We have

∇ S(t)y2 dt < +∞. Therefore

0

the system (9.6) is g.e.s.

Corollary 9.1 Under conditions (9.7) and (9.8) and assume, in addition, that there exists a self-adjoint positive operator P ∈ L(X ) such that < Ay, P y > + < P y, Ay > + < Ry, y >= 0, y ∈ D(A)

(9.10)

where R ∈ L(X ) is a self-adjoint operator satisfying < Ry, y > ≥ c ||∇ y||2m , for some c > 0

(9.11)

then system (9.1) is g.e.s. Proof We define the function V (y) = < P y, y >, ∀y ∈ X. For y0 ∈ D(A), we have y(t) = S(t)y0 and d V (y(t)) = < P AS(t)y0 , S(t)y0 > + < P S(t)y0 , AS(t)y0 > dt = − < RS(t)y0 , S(t)y0 > 

Thus

+∞

< RS(s)y0 , S(s)y0 > ds ≤ V (y0 )

0



+∞

By (9.11), we obtain 0

||∇ S(s)y0 ||2m ds < ∞. Since D(A) is dense in X we can

extend this inequality to all y0 ∈ X, and the Proposition 9.2 gives the conclusion. For the gradient strong stability we have the following result. Proposition 9.3 Assume that the equation < Ay, P y > + < P y, Ay > + < Ry, y >= 0, y ∈ D(A)

(9.12)

has a self-adjoint positive solution P ∈ L(X ), where R ∈ L(X ) is a self-adjoint operator satisfying (9.11). Moreover if the following condition holds Re < G Ay, y > ≤ 0, y ∈ D(A) then (9.1) is g.s.s.

(9.13)

9.2 Characterizations of Gradient Stability

219

Proof Let us consider the function V (y) = < P y, y >, ∀y ∈ X, t ≥ 0 , For y0 ∈ D(A), we have y(t) = S(t)y0 and d V (y(t)) = < P AS(t)y0 , S(t)y0 > + < P S(t)y0 , AS(t)y0 > dt = − < RS(t)y0 , S(t)y0 > 

+∞

we obtain



+∞

< RS(s)y0 , S(s)y0 > ds ≤ V (y0 ). By (9.11),

0

0

||∇ S(s)y0 ||2m

∂ ds < ∞ and from (9.13), we have ||∇ S(t)y0 ||2m ≤ 0. ∂t  t  t 2 2 Then t||∇ S(t)y0 ||m = ||∇ S(t)y0 ||m ds ≤ ||∇ S(s)y0 ||2m ds. We deduce 0

||∇ S(t)y0 ||2m ≤

0

α(y0 ) , t > 0, y0 ∈ D(A) for some α(y). t

(9.14)

From the density of D(A) in X, and the continuity of α(.), then (9.14) is satisfied for all y0 ∈ X. This means that the gradient of system (9.1) is strongly stable. 

9.3 Gradient Stabilization for Infinite Dimensional System Let us consider the system

∂ y(t) = Ay(t) + Bu(t) ∂t y(., 0) = y0 ∈ X

(9.15)

with the same assumptions on A, and B is a bounded linear operator mapping U , the space of controls (assumed to be Hilbert space), into X. Definition 9.2 The system (9.15) is said to be gradient weakly (respectively strongly, exponentially) stabilizable if there exists a bounded operator K ∈ L(X, U ) such that the system

∂ y(t) = (A + B K )y(t) (9.16) ∂t y(., 0) = y0 ∈ X is g.w.s (respectively g.s.s, g.e.s). Remark 9.2 1. If a system is stabilizable, then it is also gradient stabilizable. 2. Gradient stabilization is cheaper than state stabilization. Indeed if we consider the cost functional

220

9 Gradient Stabilization of Infinite Dimensional Linear Systems

 q(u) =

+∞

u(t)2 dt

0

and the spaces Uad = {u ∈ L 2 (0, +∞; U ); u stabilizes the gradient} and 1 = {u ∈ L 2 (0, +∞; U ); u stabilizes the state} Uad

then we have 1 Uad ⊃ Uad

and therefore min q(u) ≤ min q(u)

u∈Uad

1 u∈Uad

3. The gradient stabilization may be seen as a special case of output stabilization with output operator ∇. In the following we give the feedback control which stabilizes the gradient of the system (9.15), by two approaches. The first is an extension of state space decomposition and the second one is based on algebraic Riccati equation.

9.3.1 State Space Decomposition Let δ < 0 be a fixed real and consider the subsets σu (A) and σs (A) of the spectrum σ (A) of A defined by σu (A) = {λ : λ ∈ σ (A), Re(λ) ≥ δ} and σs (A) = {λ : λ ∈ σ (A), Re(λ) < δ} Assume that σu (A) is bounded and is separated from the set σs (A) in such a way that a rectifiable simple closed curve can be drawn so as to enclose an open set containing σs (A) in its interior and σu (A) in its exterior. This is the case, for example, where A is self-adjoint with compact resolvent, there are at most finitely many nonnegative eigenvalues of A, and each with finite dimensional eigenspace. Then the state space X can be decomposed according to: X = Xu + Xs

(9.17)

with X u = P X, X s = (I − P)X, and P ∈ L(X ) is the projection operator given by

9.3 Gradient Stabilization for Infinite Dimensional System

P=



1 2πi

(λI − A)−1 dλ c

where C is a closed curve surrounding σs (A). The system (9.15) may be decomposed into the two subsystems ⎧ ∂ yu (t) ⎪ ⎨ = Au yu (t) + P Bv(t) ∂t y = P y0 ⎪ ⎩ 0u yu = P y and

221

⎧ ∂ ys (t) ⎪ ⎨ = As ys (t) + (I − P)Bv(t) ∂t y = (I − P)y0 ⎪ ⎩ 0s ys = (I − P)y

(9.18)

(9.19)

where As and Au are the restrictions of A to X s and X u , and are such that σs (A) = σ (As ), σu (A) = σ (Au ) and Au is a bounded operator on X u . The solutions of (9.18) and (9.19) are given by 

t

yu (t) = Su (t)y0u +

Su (t − τ )P Bv(τ )dτ

(9.20)

Ss (t − τ )(I − P)Bv(τ )dτ

(9.21)

0

and 

t

ys (t) = Ss (t)y0s + 0

where Su (t) and Ss (t) denote the restrictions of S(t) to X u and X s , which are strongly continuous semigroups generated by Au and As . For the system state, it is known (see [1]) that if the operator As satisfies the spectrum growth assumption lim

t→+∞

ln Ss (t) = sup Re(σ (As )) t

(9.22)

then stabilizing (9.15) comes back to stabilize (9.18). The following proposition gives an extension of this result to the gradient case. Proposition 9.4 Let the state space satisfy the decomposition (9.17) and As satisfy the following inequality lim

t→+∞

ln |||∇ Ss (t)||| ≤ sup Re(σ (As )). t

(9.23)

(1) If the system (9.18) is gradient exponentially (respectively strongly) stabilizable by a feedback control u¯ = K u Gyu , with K u ∈ L(X, U ), then the system (9.15)

222

9 Gradient Stabilization of Infinite Dimensional Linear Systems

is gradient exponentially (respectively strongly) stabilizable using the control v = (u, ¯ 0). (2) If the system (9.18) is gradient exponentially (resp strongly) stabilizable by the feedback control: v = K u yu , with K u ∈ L(X, U ) then the system (9.15) is gradient exponentially (respectively strongly) stabilizable using the feedback operator K = (K u , 0). Proof We give the proof for the exponential case. In view of the above decomposition, we have sup Re(σ (As )) ≤ δ. Hence if As satisfies (9.23) then for some M1 and β ∈]0, −δ[, we have: (9.24) |||∇ Ss (t)||| ≤ M1 e−βt , t ≥ 0. It follows that the system (9.19) is gradient exponentially stabilizable taking v(t) = 0. Let K u be such that yu (t) = e Fu t y0u , with Fu = Au + P B K u G ∈ L(yu ) and there exists α > 0, M2 > 0 such that ∇ yu (t)m < M2 e−αt y0u . Then with the feedback control v = K u Gyu we have v(t) ≤ M3 K u e−αt y0u , with M3 > 0. From (9.21) and (9.24) we have ∇ ys (t)m ≤ M1 e

−βt

 y0s  + M3 y0u 

≤ M1 e−βt y0s  + M4 y0u 

e

t

e−β(t−s) e−αs ds

0 −βt

− e−αt α−β

with M4 > 0. Thus the system (9.15) excited by v(t) = K y(t) satisfies ∇ y(t)m ≤ (M1 e−βt + M4

e−βt − e−αt + M2 e−αt )y0  α−β

which shows that the system (9.15) is gradient exponentially stabilizable. (2) The case of strong stabilization follows from similar above techniques.



Corollary 9.2 Let A satisfy the spectrum decomposition assumption (9.17) and suppose that (9.23) is satisfied. If in addition 1. X u is a finite dimensional space 2. The system (9.18) is controllable on X u then the system (9.15) is gradient exponentially stabilizable. Proof The system (9.18) is of finite dimension and is controllable on the space X u then it is stabilizable on the same space, hence it is gradient stabilizable, the conclusion is obtained with the Proposition 9.4. 

9.3 Gradient Stabilization for Infinite Dimensional System

223

9.3.2 Riccati Method Let us consider the system (9.15) with the same assumptions on A and B. We denote by SK (t), t ≥ 0 the strongly continuous semigroup generated by A + B K , where K is the feedback operator K ∈ L(X, U ). Let R ∈ L(X ) be a self-adjoint operator such that (9.11) is satisfied and suppose that the steady state Riccati equation < Ay, P y > + < P y, Ay > − < B ∗ P y, B ∗ P y > + < Ry, y >= 0, y ∈ D(A) (9.25) has a self-adjoint positive solution P ∈ L(X ), and let K = −B ∗ P. Proposition 9.5 1. If SK (t) satisfies the conditions (9.7) and (9.8), then the system (9.15) is gradient exponentially stabilizable by the control u ∗ (t) = K y(t). 2. If < G(A + B K )y, y > ≤ 0, y ∈ D(A) then the system (9.15) is gradient strongly stabilizable. 3. Suppose that the system (9.15) is gradient exponentially stabilizable. If in addition the feedback operator K satisfies < Gy, y >≥ c Re < (A + B K )y, y >, y ∈ D(A), for some c > 0

(9.26)

then the state of the system (9.16) remains bounded. Proof The first and second points are deduced from the second section. For the thirst point, let y0 ∈ D(A), we have Re( (A + B K )y(t), y(t) ) = 

t

and from (9.26) we obtain 0

∇ y(s)2m ds ≥

1 ∂ y(t)2 2 ∂t

c (y(t)2 − y0 2 ). 2 

+∞

Since the system is gradient exponentially stabilizable then 0

∇ y(t)2m dt
0 such that y(t) ≤ M, for all t ≥ 0 and by the density of D(A) in X we have the conclusion. 

9.4 Gradient Stabilization Control Problem Here we explore the control that stabilizes the gradient of the system (9.15) as a solution of the minimization problem

min q(u) u ∈ Uad

(9.27)

224

9 Gradient Stabilization of Infinite Dimensional Linear Systems



where

+∞

q(u) =

 Ry(t), y(t) dt +

0

+∞

u(t)2 dt

(9.28)

0

with Uad = {u ∈ L 2 (0, +∞; U ); q(u) < +∞} and R is a linear bounded operator mapping X into itself and satisfying (9.11). We recall the classical result known for state stabilization, if Uad = ∅ for each initial state y0 , then there exists a unique control u ∗ that minimizes (9.28) and given by u ∗ (t) = −B ∗ P y(t) where P is a positive operator, solution of the steady state Riccati equation (9.25). If in addition the operator R is coercive then the state of system (9.15) is exponentially stabilizable (see [2]). In the following we give an extension of the above result to the gradient case. We suppose that Uad = ∅ for each initial state y0 ∈ X , and R satisfies (9.11). Proposition 9.6 The control given by u ∗ (t) = −B ∗ P y ∗ (t) minimizes q(u) where P assumed to be a self-adjoint, positive operator, and satisfies the steady state Riccati equation (9.25), if in addition the semigroup SK (t) satisfies the conditions (9.7) and (9.8) then the same control stabilizes the gradient of system (9.15) Proof The proof follows from [2], and the Proposition 9.5.



9.5 Numerical Algorithm and Simulations In this section we present an algorithm which allows the calculation of the solution of problem (9.27) stabilizing the gradient of the system (9.15).By the previous result this control may be obtained by solving the algebraic Riccati equation (9.25). Let X n = span{ei , i = 1, 2, . . . , n} where {ei , i ≥ 1} is a hilbertian basis of X . X n is a subspace of X endowed with the restriction of the inner product of X . The projection operator n : X → X n is defined by n (y) =

n 

ei , y ei

∀y ∈ X

(9.29)

i=1

The projection of (9.25) on space X n , is given formally by: Pn An + An Pn − Pn Bn Bn∗ Pn + Rn = 0

(9.30)

where An , Pn and Rn are respectively the projections of A, P and R on X n , and Bn the projection of B which is mapping U the space of controls into X n .

9.5 Numerical Algorithm and Simulations

225

We have lim Pn n y − P y = 0, that is Pn n converges to P strongly in X, n→∞ (see [3]). We can write the projection of (9.15) like ∂ yn (t) = An yn (t) − Bn Bn∗ Pn yn (t) ; yn (0). ∂t

(9.31)

The solution of this system is given explicitly by: ∗

yn (t) = e(An −Bn Bn Pn )t yn (0)

(9.32)

To calculate the matrix exponential we use the Padé approximation with scaling and squaring ( see [4]). If we denote y˜ni (t) = yn (t), ei , we have ∇ yn (t) =

n 

y˜ni (t)∇ei

(9.33)

i=1

Let consider a time sequence ti = iδ, i ∈N where δ > 0 small enough. With these notations, the gradient stabilization control may be obtained following the algorithm steps. Remark 9.3 The crucial steps of the algorithm are the resolution of Riccati equation and system (9.31) using formula (9.32) where one have to be careful with respect to problem data. dy (0) = Example 9.3 Let  =]0, 1[, on the space X = {y ∈ H 1 () such that dx dy (1) = 0} we consider the following system dx ⎧ ∂ y(t) ⎪ ⎪ = Ay(t) + χ D u(t) ×]0, +∞[ ⎪ ⎨ ∂t ∂ y(1, t) ∂ y(0, t) (9.34) = =0 ∀t ≥ 0 ⎪ ⎪ ∂ x ∂ x ⎪ ⎩ y(x, 0) = 16 x 2 (3 − 2x)  where Ay = 0.02y + 0.5y, u(t) ∈ X ∀t ≥ 0, χ D is the restriction operator on D = ]0.2, 0.9[, and we consider the problem (9.27) with R = ∇ ∗ ∇. A generates a strongly continuous semigroup S(t) given by: S(t)y =

+∞ 

eλi t y, φi φi .

i≥0

where λi = −0.01(iπ )2 + 0.5 and φi (x) = αi cos(iπ x) with αi =



2 . 1+(iπ)2

226

9 Gradient Stabilization of Infinite Dimensional Linear Systems 0.15 t=2 t=3 t=7

0.1

0.05

flux

0

−0.05

−0.1

−0.15

−0.2

0

0.2

0.4

0.6

0.8

1

x

Fig. 9.1 The gradient evolution for the Neumann boundary condition case

The state and the gradient of system (9.34) are unstable since λ0 > 0 and λ1 > 0. Let consider the subspace X n = Span{αi−1 cos((i − 1)π x), 1 ≤ i ≤ n, x ∈ } Applying the algorithm taking the truncation at n = 5, we obtain Fig. 9.1 which illustrates the evolution of the system gradient and shows how the gradient evolves close to zero when the time t increases. The gradient is stabilized with error equals 9.9836 × 10−7 and cost equals 2.6982 × 10−4 . This shows the efficiency of the developed algorithm. In Table 9.1 we give the cost of gradient stabilization of system (9.34) for different supports control D. The Table 9.1 shows that there is relation between area of control support and the cost of gradient stabilization, more precisely more this area decreases more cost increases. Algorithm 1. Let ε > 0 the threshold, a time sequence ti = iδ, i ∈ N where δ > 0 small enough, n the dimension of the projection space. 2. Solve (9.30) using Schur-type methods (see [5])

Table 9.1 Support control-cost stabilization D ]0, 0.1[ ]0, 0.3[ ]0, 0.5[ Cost 9.0097 1.921 0.9408

]0, 0.7[ 0.817

]0, 0.9[ 0.2868

]0, 1[ 0.1636

9.5 Numerical Algorithm and Simulations Fig. 9.2 The gradient evolution for Dirichlet boundary condition

227 −4

6

x 10

t=3 t=5 t=13

4 2 0

flux

−2 −4 −6 −8 −10 −12 −14

0

0.2

0.4

0.6

0.8

1

x

3. Solve system (9.31) by formula (9.32) using the Padé approximation with scaling and squaring (see [4]). 4. Calculate ∇ yn (ti ) by the formula (9.33). 5. If  ∇ yn (ti ) <  stop, else 6. i=i+1 and go to 3 Example 9.4 Let  =]0, 1[, on X = H01 () we consider the system (9.34) with Dirichlet boundary conditions: ⎧ ∂ y(t) ⎪ ⎨ = Ay(t) + χ D u(t) ×]0, +∞[ ∂t y(0, t) = y(1, t) = 0 ∀t ≥ 0 ⎪ ⎩ y(x, 0) = (1 − x)2 x 2 x ∈

(9.35)

where Ay = 0.01y + 0.5y, u(t) ∈ X ∀ t ≥ 0, D =]0, 0.3[, and we consider the problem (9.27) with R = ∇ ∗ ∇. 2 The eigenpairs  of A are given by (λi , αi sin(iπ x)), i ≥ 1, with λi = −0.01(iπ ) 2 + 0.5 and αi = 1+(iπ) 2. The state and the gradient of system (9.35) are unstable since λ1 > 0.

We consider the subspace X n = {αi sin(iπ x), i = 1, . . . , n} with αi = 2 . 1 + (iπ )2 Applying the previous algorithm with truncation (n = 5), the Fig. 9.2 shows the gradient evolution at times t = 3, 5, and t = 13. The gradient is stabilized with error equals 9.7961 × 10−7 and cost equals 3.2689 × 10−4 .

228

9 Gradient Stabilization of Infinite Dimensional Linear Systems

Table 9.2 Support control-cost stabilization D ]0, 0.1[ ]0, 0.3[ ]0, 0.5[ ]0, 0.7[ Cost 0.0247 0.0039 0.0016 4.7255 × 10−4

]0, 0.9[ 2.6982 × 10−4

]0, 1[ 2.6081 × 10−4

In Table 9.2 we present the cost of gradient stabilization of system (9.34) for different zone control support D Also in this example, we remark that more the area of control support increases more the cost of gradient stabilization decreases.

References 1. Triggiani, R.: On the stabilizability problem in Banach space. J. Math. Anal. Appl. 52, 383–403 (1975) 2. Curtain, R.F., Zwart, H.J.: An Introduction to Infinite Dimensional Linear Systems Theory. Springer, New York (1995) 3. Banks, H., Kunisch, B.: The linear regulator problem for parabolic systems. SIAM J. Control Optim. 22, 684–696 (1984) 4. Higham, N.J.: The scaling and squaring method for the matrix exponential. SIAM Rev. 51(4), 747–767 (2009) 5. Arnold, W.F., Laub, A.J.: Generalized eigenproblem: algorithms and software for algebraic Riccati equations. Proc. IEEE 72, 1746–1754 (1984)

Chapter 10

Regional Gradient Stabilization of Infinite Dimensional Linear Systems

This chapter deals with regional gradient stabilization of infinite dimensional linear system evolving on a spacial domain . Then we study different degrees of the gradient stabilization on ω ⊂  and moreover we characterize the stabilizing control which minimizes a quadratic functional cost. The developed approaches are illustrated by examples and numerical simulations.

10.1 Regional Gradient Stability 10.1.1 Definitions Let  be an open regular subset of Rn and Q = ×]0, ∞[, we consider a system governed by the following state equation ⎧ ∂y ⎪ ⎨ = Ay on Q ∂t ⎪ ⎩ y(0) = y0 ∈ X

(10.1)

where A : D(A) ⊂ X −→ X a linear operator that generates a C0 -semigroup S(t), t ≥ 0, on a Hilbert state space X that is continuously injected in H 1 (). X is endowed with the induced inner product of H 1 () noted ., . and the corresponding norm . .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. H. Zerrik and O. Castillo, Stabilization of Infinite Dimensional Systems, Studies in Systems, Decision and Control 355, https://doi.org/10.1007/978-3-030-68600-0_10

229

230

10 Regional Gradient Stabilization of Infinite Dimensional Linear Systems

We define the gradient operator: ∇ : X → (L 2 ())n ∂y ∂y ∂y y → ( , ,··· , ) ∂ x1 ∂ x2 ∂ xn

(10.2)

(L 2 ())n provided with the usual inner product =

n   i=1



f i (x)gi (x)d x

and the associated norm . ; with f = ( f 1 , f 2 , · · · , f n ) and g = (g1 , g2 , · · · , gn ); f i , gi ∈ L 2 (). Let ω be a subregion of , we define the operator ∇ω by ∇ω = (χω

∂. ∂. ∂. , χω , · · · , χω ) ∂ x1 ∂ x2 ∂ xn

Let ∇ω∗ be it’s adjoint operator, and consider the operator G ω = ∇ω∗ ∇ω . Definition 10.1 The system (10.1) is said regionally 1. weakly gradient stable (R.G-stable) on ω if, for all y0 ∈ X , we have < ∇ω S(t)y0 , z > −→ 0 when t −→ ∞,

z ∈ (L 2 (ω))n

(10.3)

2. strongly gradient stable on ω if, for all y0 ∈ X , we have ||∇ω S(t)y0 || −→ 0 when t −→ ∞

(10.4)

3. exponentially gradient stable on ω if, for all y0 ∈ X , there exist M and α > 0 such that (10.5) ∇ω S(t)y0 ≤ Me−αt y0 , t ≥ 0, ∀y0 ∈ X Remark 10.1 1. In the case where ω = , the system (10.1) is said weakly (respectively strongly, exponentially) G-stable. 2. If the system (10.1) is stable then it is G-stable. 3. If the system (10.1) is G-stable on ω, then it is G-stable on any subregion ω ⊂ ω. 4. The notion of regional stability of the gradient is more general that the regional state stability. Indeed there exist unstable systems that are gradient stable on some subregion ω. Example 10.1 On  =]0, 1[×]0, 1[, we consider the system

10.1 Regional Gradient Stability

231

⎧ ∂y ⎪ ⎪ (x1 , x2 , t) = Ay(x1 , x2 , t) ×]0, +∞[ ⎪ ⎨ ∂t ∂y (ζ, ξ, t) = 0 ∂×]0, +∞[ ⎪ ⎪ ∂ν ⎪ ⎩ 1 y(., 0) = y0 ∈ H ()

(10.6)

where Ay = y + 6π 2 y, ϕ20 ϕ20 . The operator A has an orthogonal basis (ϕi j ) of eigenfunctions and (λi j ) the associated eigenvalues. We have ϕi j (x1 , x2 ) = 2ai j cos(iπ x1 ) cos( jπ x2 ) where ai j = (1 − λi j )− 2 . The eigenvalues are given by 1

 λi j =

−(i 2 + j 2 )π 2 (i, j) = (2, 0) (i, j) = (2, 0) 2π 2

A generates a C0 -semigroup (S(t))t≥0 on H 1 () given by S(t)y =

∞ 

eλi j t y, ϕi j ϕi j

(i, j)

The state and the gradient of system (10.6) are unstable on ω = { 21 } × [0, 1]. But ∇ω ϕ20 = 0 then ∇ω y(t) → 0 exponentially when t → ∞ and consequently the system (10.6) is exponentially G-stable on ω. Example 10.2 On  =]0, 1[ we consider the system described by the equation ⎧ ∂y ⎪ ⎪ ×]0, +∞[ ⎪ ⎨ ∂t (t) = Ay(t) ∂y ∂y (0, t) = (1, t) = 0 ]0, +∞[ ⎪ ⎪ ∂t ⎪ ⎩ ∂t y(., 0) = y0 ∈ H 1 ()

(10.7)

The operator Ay = y + y generates a C0 -semigroup given by S(t)y0 =



eλi t y0 , φi φi

i≥0



where φi (x) =

2 cos(iπ x) , λi = 1 − (iπ )2 1 + (iπ )2

Since λ0 > 0, the system (10.7) is unstable. However it is exponentially G -stable as

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10 Regional Gradient Stabilization of Infinite Dimensional Linear Systems

∇ S(t)y0 2 ≤ e2λ1 t

 y0 , φi 2 ∇φi 2 i>0

≤ e2λ1 t y0 2 Consequently (10.7) is exponentially G-stable on .

10.1.2 Characterizations We have seen that the exponential stability is obtained under the condition (2.21), the following result gives an extension to the gradient stability. Proposition 10.1 Suppose that there exists a function Mω ∈ L 2 (0, +∞; R+ ) such that, for all y ∈ X , we have

and

∇ω S(t + s)y ≤ Mω (t) ∇ω S(s)y , t, s ≥ 0

(10.8)

∇ω S(mt)y ≤ ∇ω S(t) m ∀ t ≥ 0 , m ∈ N∗

(10.9)

Then the system (10.1) is exponentially G-stable on ω if and only if 

∞

||∇ω S(t)y||2 dt < ∞ , y ∈ X

(10.10)

0

Proof 1. We show that the sequence of operators (∇ω S(t))t≥0 is uniformly bounded. Indeed, we have sup ∇ω S(t) < +∞, otherwise there exists a sequence (t1 + τk )k , t≥0

t1 > 0 and τk → ∞ when k → +∞, such that ∇ω S(t1 + τk ) → +∞ when k → ∞. So  ∞  ∞ ∇ω S(s + τk )y 2 ds = ∇ S(s)y 2 ds 0

τk

As the right term tends to 0 when k → ∞. Fatou lemma imply that lim inf ∇ω S(s + τk )y = 0 when k → ∞, almost for all s > 0. Consequently for s0 < t1 there exists a subsequence {τkm } such that lim ∇ω S(s0 + τkm )y = 0. But with (10.8) and when m → +∞, we have

m→∞

∇ω S(t1 + τkm )y ≤ M(t1 − s0 ) ∇ S(s0 + τkm )y → 0 which is impossible. Using Banach-Steinhaus theorem we deduce that (∇ω S(t))t≥0 is uniformly bounded. 2. We show that σ0 = inf

t>0

1 1 ln |||∇ω S(t)|||L(X,(L 2 ())n ) = lim ln |||∇ω S(t)|||L(X,(L 2 ())n ) t→+∞ t t

10.1 Regional Gradient Stability

233

Indeed, we have  t ∇ω S(t)y 2 = ≤

t

0 t

∇ω S(t)y 2 ds ∇ω S(s + t − s)y 2 ds

0

By the condition (10.8) and as the operator ∇ω S(t) is uniformly bounded, there exists a constant N > 0 such that  t 2 t ∇ω S(t)y ≤ M 2 (s) ∇ω S(t − s)y 2 ds 0

≤ N y 2 So there exists t0 > 0 such that, for all t ≥ t0 , ln |||∇ω S(t)|||L(X,(L 2 ())n ) < 0 We deduce that σ0 < 0.

1 ln |||∇ω S(t)|||L(X,(L 2 ())n ) . t  Let t1 > 0 and N = sup |||S(t)|||L(X ) , there exists p ∈ N such that pt1 ≤ t < It remains to show that σ0 = lim

t→+∞

t∈[0,t1 ]

( p + 1)t1 , for all t ≥ t1 . Then

1 1 1 ln |||∇ω S(t)||| ≤ ln |||∇ω S( pt1 )||| + ln |||S(t − pt1 )||| t t t and with (10.9) we obtain 1 pt1 1 ln N  ln |||∇ω S(t)||| ≤ ( ) ln |||∇ω S(t1 )||| + t t t1 t which gives lim sup t→∞

1 1 ln |||∇ω S(t)||| ≤ inf ln |||∇ω S(t)||| t>0 t t ≤ lim inf t→∞

1 ln |||∇ω S(t)||| t

1 ln |||∇ω S(t)|||L(X,(L 2 ())n ) . So for all σ ∈]0, −σ0 [, there t→+∞ t

Finally, this gives σ0 = lim exists M  such that

||∇ω S(t)y||n ≤ M  e−σ t y , y ∈ X, t ≥ 0

234

10 Regional Gradient Stabilization of Infinite Dimensional Linear Systems

Consequently the system is exponentially G-stable on ω. The converse is immediate.  Corollary 10.1 Let us assume that the conditions (10.8) and (10.9) hold and there exists t0 > 0, such that |||∇ω S(t0 )||| < 1, then the system (10.1) is exponentially G-stable on ω. Proof According to the second step of the Proposition 10.1, we have σ0 = lim

t→+∞

1 ln |||∇ω S(t) t

this imply that σ0 < 0. So (10.1) is exponentially G-stable on ω.



The following result gives a sufficient condition for strong stability of the gradient on a subregion ω. Proposition 10.2 Let us assume that there exists an operator P ∈ L(X ) positive and self adjoint auto-adjoint solution of the equation < Ay, P y > + < P y, Ay > + < Ry, y >= 0,

y ∈ D(A)

(10.11)

where R ∈ L(X ) is a self adjoint operator such that Ry, y ≥ α ∇ω y 2 ∀y ∈ X, for some real α > 0

(10.12)

If, moreover, the condition Re(< G ω Ay, y >) ≤ 0,

y ∈ D(A)

(10.13)

holds, then (10.1) is strongly G-stable on ω. Proof We consider the function V (y) = < P y, y >, for all y ∈ X. For y0 ∈ D(A), the relation d V (S(t)y0 ) = < P AS(t)y0 , S(t)y0 > + < P S(t)y0 , AS(t)y0 > dt = − < RS(t)y0 , S(t)y0 > gives



+∞

< RS(s)y0 , S(s)y0 > ds ≤ V (y0 )

0

 By (10.12), we have S(t)y0 ||2 ≤ 0.

0

+∞

||∇ω S(s)y0 ||2 ds < ∞. Using (10.13), we obtain

∂ ||∇ ∂t ω

10.1 Regional Gradient Stability

235

Thus  t||∇ω S(t)y0 ||2 =

0

t

 ||∇ω S(t)y0 ||2 ds ≤

t 0

||∇ω S(s)y0 ||2 ds

We deduce that for all y0 ∈ D(A), we have ||∇ω S(t)y0 ||2 ≤  with f (y0 , t) =

t

0

f (y0 , t) , t >0 t

(10.14)

||∇ω S(s)y0 ||2 ds which is continuous with respect to t.

It follows lim ∇ω S(t)y0 = 0. t→+∞

Since D(A) is dense in X, and the function y0 −→ f (y0 , t) is continuous, the  inequality (10.14) holds for all y0 ∈ X. So (10.1) is strongly G-stable on ω. Corollary 10.2 Let us assume there exists an operator P ∈ L(X ) self adjoint solution of the equation < Ay, P y > + < P y, Ay > + < G ω y, y >= 0, ∀ y ∈ D(A) If moreover Re(< G ω Ay, y >) ≤ 0 ∀ y ∈ D(A) then the system (10.1) is strongly G-stable on ω. The result deduced since G ω verify the conditions of the above proposition with (α = 1). The weak stability of system (10.1) is linked to the spectral properties of the dynamic of the system. Indeed, in finite dimension the system (10.1) is weakly stable if and only if all the poles of the system dynamic are located in the open left half plan. In infinite dimension we have the following result. Proposition 10.3 1. If the system (10.1) is weakly G-stable on ω then ∀λ ∈ σ (A), Re(λ) ≥ 0 ⇒ ∇ω φ = 0 , ∀φ ∈ K er (A − λI ) 2. Let us suppose that X is an Hilbert space which has an orthogonal basis (φm )m , of eigenfunctions of the operator A. If in addition ∃α > 0 such that, ∀λ ∈ σ (A), there exists φ ∈ K er (A − λI ) such that ∇ω φ = 0 ⇒ Re(λ) ≤ −α then the system (10.1) is exponentially G-stable on ω.

236

10 Regional Gradient Stabilization of Infinite Dimensional Linear Systems

Proof 1. We assume there exists λ ∈ σ (A) such that Re(λ) ≥ 0 and φ ∈ X satisfying Aφ = λφ and ∇ω φ = 0 The corresponding solution of the system (10.1) to the initial state y0 = φ is given by y(t) = eλt φ, so: ∇ω S(t)φ, ∇ω φ = e Re(λ)t ∇ω φ 2 ≥ ∇ω φ 2 > 0 consequently the system (10.1) is not weakly G-stable on ω. rm   e λn t y0 , φm,k ∇ω φm,k where rm is the 2. Let y0 ∈ X, we have, ∇ω S(t)y0 = m≥0

multiplicity of the eigenvalue λm , then

k=1

∇ω S(t)y0 ≤ e−αt y0 We conclude that the system (10.1) is exponentially G-stable on ω.



As illustration of the second point of the above result, we consider the system (10.6) and ω = { 21 } × [0, 1]. We have, for all i, j ∈ N, ∇ω ϕi j = 0 ⇒ Re(λi j ) ≤ −π 2 and so the system (10.6) is exponentially G-stable on ω. If the semigroup is compact, then the weak stability imply the exponential one. This is expressed by the following result. Proposition 10.4 We suppose that the semigroup S(t) verify the conditions (10.8) and (10.9) and there exists t0 such that S(t0 ) be compact. If the system (10.1) is weakly G-stable on ω then it is exponentially G-stable on ω. Proof We show that lim ∇ω S(t) = 0. t→∞

Since S(t0 ) is compact, then S(t) is compact for all t > t0 , and using ∇ω is continuous then ∇ω S(t) is compact for all t > t0 . If the system (10.1) is weakly G-stable then ∇ω S(t) → 0 when t → ∞. By Banach Steinhaus theorem there exists M > 0 such that ∇ω S(t) < M. Let ε > 0, as S(t0 ) is compact then there exist z 1 , z 2 , ..., z N in X such that {S(t0 )y, y ≤ 1} ⊂

i=1,N

B(z i ,

ε ) M

ε where B(z i , Mε ) is the ball of center 0 and radius . M Let tε such that ∇ω S(t)z i ≤ ε for all t > tε and i = 1, ..., N . For t > tε + t0 and y ≤ 1, we have

10.1 Regional Gradient Stability

237

∇ω S(t)y = ∇ω S(t − t0 )S(t0 )y − ∇ω S(t − t0 )z i + ∇ω S(t − t0 )z i ≤ ∇ω S(t − t0 )(S(t0 )y − z i ) +ε ≤M

ε + ε = 2ε M

So there exist t1 > 0 such that ∇ω S(t1 ) < 1 and by the corollary (10.1) the system (10.1) is exponentially G-stable on ω. 

10.2 Gradient Stabilization In the previous section we establish sufficient conditions and necessary and sufficient other ones for an infinite dimensional system be exponentially (resp. strongly, weakly) gradient stable on ω. The goal of this section is to characterize a control that stabilizes the gradient of the system (10.1) and minimizes a stabilisation cost.

10.2.1 Stabilizing Control We consider the system described by the equation  ∂ y(t) = Ay(t) + Bu(t) on Q ∂t y(., 0) = y0 ∈ X

(10.15)

where A generates a C0 -semigroup S(t), and B ∈ L(V, X ) a linear continuous operator; V is the space of controls supposed to be Hilbert. The considered controls are of feedback type u(t) = K y(t) with K ∈ L(X, V ). The system (10.15) is written in the form  ∂ y(t) = (A + B K )y(t) on Q ∂t y(., 0) = y0 ∈ X

(10.16)

In this case the operator A + B K generate a C0 -semigroup T (t). We consider the following definitions. Definition 10.2 1. The system (10.15) is said exponentially G-stabilizable on ω if there exists an operator K ∈ L(X, V ), such that the system (10.16) be exponentially G-stable on ω. 2. If there exists an operator K ∈ L(X, V ), such that the system (10.16) be strongly (resp. weakly) G-stable on ω, the system (10.15) is said strongly (resp. weakly) G-stabilizable on ω.

238

10 Regional Gradient Stabilization of Infinite Dimensional Linear Systems

Remark 10.2 1. The regional stabilisation of the gradient is a general notion that extends the stabilisation of gradient. 2. The cost of the regional stabilisation of the gradient lower than the one of the stabilisation of the gradient. 3. The cost of the gradient stabilisation is lower than the one of the state stabilisation. Indeed, if we consider the cost function  q(u) =

+∞

u(t) 2 dt

0

and the sets Vad = {u ∈ L 2 (0, +∞; V ); u stabilise the gradient on ω} 1 = {u ∈ L 2 (0, +∞; V ); u stabilise the gradient on } Vad 2 = {u ∈ L 2 (0, +∞; V ); u stabilise the state on ω} Vad 3 = {u ∈ L 2 (0, +∞; V ); u stabilise the state on } Vad 3 2 1 Then we have Vad ⊂ Vad ⊂ Vad ⊂ Vad . Consequently

min q(u) ≤ min q(u) and min q(u) ≤ min q(u) ≤ min q(u)

u∈Vad

1 u∈Vad

1 u∈Vad

2 u∈Vad

3 u∈Vad

In general, a characterization of a stabilizing control of the gradient on ω or on  may be obtained by two approaches, the first one leads to resolution of Riccati equation and the second one is based on the decomposition of the state space.

10.2.1.1

Riccati Approach

Let us recall that algebraic Riccati equation allows a control that stabilises exponentially the state of infinite dimensional system on ω. Here we give extension to the gradient stabilization . We suppose that algebraic Riccati equation (10.11) has a solution P ∈ L(X ) positive and self adjoint, if R verify the condition (10.12) then we have the following result. Proposition 10.5 If the semigroup T (t) verify the conditions (10.8) and (10.9) then 1. The system (10.15) is exponentially G-stabilizable on ω. 2. If there exists d > 0 such that < G ω y, y > ≥ d Re(< (A + B K )y, y >) , y ∈ D(A) then the state of the system (10.16) remains bounded on .

(10.17)

10.2 Gradient Stabilization

239

Proof 1. Since P is solution of (10.11) and R verify (10.12) then 

+∞

∇ω T (t)y 2 dt < +∞,

y∈X

0

and we apply the Theorem 10.1 to conclude. 2. The condition (10.17) gives 

t

d( y(t) 2 − y0 2 ) ≤

∇ω T (s)y0 2 ds

0

As system (10.15) is exponentially G-stabilizable on ω, then the application 

t

y0 −→

T (s)y0 2 ds

0

is uniformly bounded, this proves the state remains bounded on the whole evolution domain.  In the regional case the next result concerns the behaviour of the gradient on the residual part  \ ω of . Proposition 10.6 If the semigroup T (t) verify the conditions (10.8) and (10.9) then if there exists d > 0 such that: < G ω y, y > ≥ dRe < G  (A + B K )y, y > , y ∈ D(A)

(10.18)

then the gradient of the system (10.16) remains bounded on . Proof The semigroup (T (t)) verify (10.8) and (10.9), the previous proposition shows that the system (10.15) is exponentially G-stabilizable on the subregion ω. The condition (10.18) leads to, for all y0 ∈ D(A), we have 1 ∇ y(t) ≤ d



2

t

∇ω T (t)y0 2 dt

0

Since ∇ω and ∇ are continuous then the last inequality holds for all y0 ∈ X . As,  +∞ ∇ω T (t)y0 2 dt < +∞, we deduce the result.  moreover, 0

Remark 10.3 1. The condition (10.17) imply that the gradient of (10.16) remains bounded on  (since the state of (10.16) is bounded). 2. The condition (10.18) gives the same result as the condition (10.17), but without taking into account the behaviour of the state. 3. The condition (10.18) may be substituted by the following condition

240

10 Regional Gradient Stabilization of Infinite Dimensional Linear Systems

< G ω y, y > ≥ d 

d < (G \ω y, y > , y ∈ D(A) with d  > 0 dt

(10.19)

in view to limit the gradient of system (10.16) on the whole domain .

10.2.2 State Space Decomposition The aim of this paragraph is to extend the decomposition method used for the state stabilization to the gradient. Let δ > 0 and consider the subsets: σu (A) = {λ : Re(λ) ≥ −δ} and σs (A) = {λ : Re(λ) < −δ} We suppose that σu (A) is bounded and may be separated from σs (A) by a simple closed curve, then the state space X may be decomposed in sum of two orthogonal subspaces X = X u ⊕ X s with X u = X, X s = (I d − )X, and  ∈ L(X ) is the projection operator given by 1 = 2πi



(λI d − A)−1 dλ

(10.20)

C

where C is a curve surrounding σ (A) [1]. Thus the system (10.15) may be decomposed into the following subsystems:

and

⎧ dyu (t) ⎪ ⎨ = Au yu (t) + Bv(t) dt y = y0 ⎪ ⎩ 0u yu = y

(10.21)

⎧ dys (t) ⎪ ⎨ = As ys (t) + (I d − )Bv(t) dt y = (I d − )y0 ⎪ ⎩ 0s ys = (I d − )y

(10.22)

where As and Au are respectively the restrictions of A on X s and X u . Su (t) and Ss (t) are the restrictions of S(t) on X u and X s , which are respectively the C0 - semigroups generated by Au and As . We have the following result. Proposition 10.7 If the semigroup Ss (t) verify the inequality lim

t→+∞

1 ln ∇ω Ss (t) ≤ sup Re(σ (As )) t

(10.23)

10.2 Gradient Stabilization

241

and that the system (10.21) is exponentially (resp. strongly) G-stabilizable on ω by v = (v, 0), the control v = K u ∇ω yu , with K u ∈ L((L 2 (ω))n , V ), then the control exponentially (resp. strongly) G-stabilizes the system (10.15) on ω. Proof We show strong G-stabilization on ω. Indeed the previous decomposition gives sup Re(σ (As )) ≤ −δ, there exist M1 > 0 and β ∈]0, δ[, such that ∇ω Ss (t) ≤ M1 e−βt , t ≥ 0

(10.24)

Without control (v = 0), the system (10.22) is exponentially G-stabilisable on ω. If the system (10.21) is strongly G-stabilizable on ω, this means that lim ∇ω yu (t) = 0 on ω

(10.25)

t→+∞

β So for all  > 0 there exists t0 > 0 such that if t > t0 , we have v(t) < . The 2 solution of (10.22) is given by: 

t

ys (t) = Ss (t)y0s +

Ss (t − τ )(I d − )Bv(τ )dτ

(10.26)

0

Using (10.24), there exists M1 > 0 such that ∇ω ys (t) n ≤ M1 e−βt y0s + M1 |||(I d − )B|||



t

e−β(t−τ ) v(τ ) dτ

0

It remains to show that lim ∇ω ys (t) = 0. t→+∞

We have  t  e−β(t−τ ) v(τ ) dτ =

t0

e−β(t−τ ) v(τ ) dτ +

0

0

≤ γ e−β(t−t0 ) + 1 where γ = √ 2β As



t0

v(t) dt 2

0

t

e−β(t−τ ) v(τ ) dτ

t0

 21

 lim

 2



t−→+∞ 0

. t

e−β(t−τ ) v(τ ) dτ = 0

then lim ∇ω ys (t) = 0. Finally if we consider the system (10.15) controlled by t→+∞

v (t) we obtain

∇ω y(t) ≤ ∇ω yu (t) + ∇ω ys (t)

and so the system (10.15) is strongly G-stabilisable on ω.



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10 Regional Gradient Stabilization of Infinite Dimensional Linear Systems

10.2.3 Lower Cost Gradient Stabilization Problem In this paragraph we characterize a control that stabilises the gradient of the system (10.15), solution of the problem 



where

+∞

q(u) =

min q(u) u ∈ Vad

(10.27) 

Ry(t), y(t)dt +

0

+∞

u(t) 2 dt

(10.28)

0

with Vad = {u ∈ L 2 (0, +∞; V ); q(u) < +∞} and R : X → X is a bounded linear operator which verify (10.12). We suppose that (10.11) has a solution P, a linear positive and self adjoint operator, and ∀y0 ∈ X, Vad = ∅, then we have the following result. Proposition 10.8 If the semigroup T (t) satisfy the conditions (10.8) and (10.9), then the control u ∗ (t) = −B ∗ P y ∗ (t) (10.29) is solution of the problem (10.27) and exponentially stabilises the gradient of system (10.15) on ω. Proof 1. The control (10.29) stabilizes the gradient of system (10.15) on ω. Indeed, since T (t) verify (10.8) and (10.9), the system (10.15) is exponentially G-stabilisable on ω. 2. The control (10.29) is solution of problem (10.27). Let y0 ∈ D(A), since q(u) < ∞ then [2], the Riccati differential equation d y, P(t)y = P(t)Ay, y + y, P(t)Ay dt ∗

(10.30)

∗

−B P(t)y, B P(t)y + Ry, y admits a solution P(t) with P(0) = 0. The sequence P(m), m > 0, is uniformly bounded and for all y0 ∈ X , lim P(m)y0 = Pω y0 , where Pω is a solution of m−→+∞

equation (10.30). Moreover, 0 ≤ P(m) ≤ P(m + 1) and we conclude that [2] there exists a linear positive and self adjoint operator Pω ∈ L(X ) such that for all y0 ∈ X , lim P(m)y0 = Pω y0 and P(m) ≤ Pω for m ≥ 0.

m−→+∞

10.2 Gradient Stabilization

243

For all t ≥ 0, let m ∈ N such that m ≤ t ≤ m + 1, and according to [2], we have P(m) ≤ P(t) ≤ P(m + 1). Hence lim P(t)y0 = Pω y0

t−→+∞

Since Pω (t) satisfy (10.30) for all y ∈ D(A), and passing to limit in equation (10.30) we obtain Pω Ay, y + y, Pω Ay − B ∗ Pω y, B ∗ Pω y + Ry, y = 0 It remains to show that the control (10.29) is solution of problem (10.27). Let u ∈ L 2 (0, ∞, V ), with the Theorem 6.1.13 in [2], we have  q(y0 , u, m) =

m



m

Ry(t), y(t)dt +

0

u(t) 2 dt ≤ q(y0 , u)

0

Thus inf

u∈L 2 ([0,∞[,V )

q(y0 , v) ≥ =

inf

q(y0 , u, m)

inf

q(y0 , u, m)

u∈L 2 ([0,∞[,V )

u∈L 2 ([0,m],V )

= y0 , P(m)y0  When m → ∞, we have inf

u∈L 2 ([0,∞[,V )

q(y0 , u) ≥ y0 , P(m)y0 

(10.31)

Since Pω is positive, we have q(y0 , u, m) ≤ y0 , Pω y0  m + [u(s) + B ∗ Pω y(s)], [u(s) + B ∗ Pω y(s)]ds 0

Let y˜ (s) = T (s)y0 the solution of (10.15) controlled by u(s) ˜ = B ∗ Pω T (s)y0 and ˜ ≤ y0 , Pω y0 . with (10.31) we have q(y0 , m, v) We passe to limit when m → ∞ we have ˜ = q(y0 , u)

lim

m−→+∞

q(y0 , m, u) ˜ ≤ y0 , Pω y0 

By (10.31) and (10.32), we have y0 , Pω y0  ≤

inf

u∈L 2 ([0,∞[,V )

q(y0 , u) ≤ q(y0 , u) ˜ ≤ y0 , P(m)y0 

(10.32)

244

10 Regional Gradient Stabilization of Infinite Dimensional Linear Systems

We deduce that inf

u∈L 2 ([0,∞[,V )

q(y0 , u) = y0 , P(m)y0 

Consequently the control given by (10.29) is solution of problem (10.27).



10.2.4 Numerical Approach and Simulations Here we present a numerical approach that allows the computation of the control u ∗ solution solution of the problem (10.27) and stabilizes the gradient of system (10.15) on ω. We have seen that this control may be calculated by the resolution of algebraic Riccati equation (10.11). Let X N = V ect{ei , i = 1, 2, · · · , N } where {ei , i ≥ 1} a basis of the Hilbert space X . X N is a subspace of X endowed with the induced inner product of X . We consider the projection operator  N : X → X N defined by:  N (y) =

N  ei , yei , y ∈ X

(10.33)

i=1

The projection of (10.11) on X N , is formally given by PN A N + A N PN − PN B N B N∗ PN + R N = 0

(10.34)

where A N , PN and R N are respectively the projections of the operators A, P and R on HN , and B N is the projection of B : V → X N . So we have lim PN  N y − P y = 0, and hence PN  N strongly converges P N →∞

when N → +∞ (see [3]). The projection of system (10.15) is given by: ∂ y N (t) = A N y N (t) − B N B N∗ PN y N (t) ∂t with

∗

y N (t) = e(A N −BN BN PN )t y N (0)

(10.35)

(10.36)

To calculate the exponential of a matrix, we use the Padé approximations (see [4]). If we note y˜ Ni (t) = y N (t), ei , we have ∇ y N (t) =

N  i=1

y˜ Ni (t)∇ei

(10.37)

10.2 Gradient Stabilization

245

We consider a time regular subdivision ti = i h, i ∈ N, and h > 0 ( small enough) the subdivision step. The control that stabilizes the system (10.15) and solution of problem (10.27), may be obtained by the following algorithm; Algorithm Step 1 : Initial data: tolerance ε > 0 , target subregion ω, support of control D and N the dimension of the projection space. Step 2 : Solving equation (10.34) with the Schur method (see [5]) Step 3 : Solving equation (10.35) using the formula (10.36), gives y N (ti ) Step 4 : Computation of ∇ω y N (ti ) by formula (10.37). Step 5 : If ∇ω y N (ti ) < ε stop, otherwise i := i + 1 and go to step 3. Example 10.3 On  =]0, 1[, we consider the system ⎧ ∂ y(t) ⎪ ⎪ = Ay(t) + χ D u(t) ×]0, +∞[ ⎪ ⎪ ⎨ ∂t ∂ y(0, t) ∂ y(1, t) = =0 t ≥0 ⎪ ∂x ∂x ⎪ ⎪ 1 1 ⎪ ⎩ y(x, 0) = x 2 ( − x)  2 3

(10.38)

where Ay = 0.02 y + 0.5y. The state space is X = {y ∈ H 1 () such that

dy dy (0) = (1) = 0} dx dx

is of Hilbert and u(t) ∈ X , for all t ≥ 0. We take D =]0.2, 0.9[ and we consider the problem (10.27) with R = ∇ ∗ ∇. The operator A generates a C0 -semigroup given by S(t)y =

+∞ 

eλi t y, φi φi

i≥0

 2 where λi = −0.01(iπ )2 + 0.5 and φi (x) = αi cos(iπ x) with αi = 1+(iπ) 2. Since λ0 , λ1 > 0 the state and the gradient of system (10.38) are unstable. We consider the subspace X N = Span{αi−1 cos((i − 1)π x), 1 ≤ i ≤ N , x ∈ } We apply the above algorithm with truncation order N = 5. we obtain the following results.

246

10 Regional Gradient Stabilization of Infinite Dimensional Linear Systems

Fig. 10.1 Evolution of the gradient

0.15 t=2 t=3 t=7

0.1

0.05

flux

0

−0.05

−0.1

−0.15

−0.2

0

0.2

0.4

0.6

0.8

1

x

Global case. The next figure represents the gradient evolution of system (10.38) (Fig. 10.1). We remark that the goes close to 0 when time t increases. The stabilisation error is about 9.9836 × 10−7 and the cost equals to 2.6982 × 10−4 . Studying the variation of cost stabilization of the gradient with respect to the control zone area we obtain the following array. D Cost

]0,0.1[ 9.0097

]0,0.3[ 1.921

]0,0.5[ 0.9408

]0,0.7[ 0.817

]0,0.9[ 0.2868

]0,1[ 0.1636

This array shows that there exists a link between the zone area of the control and the cost of the gradient stabilization; more precisely more the area increases more the cost decreases. Regional case. We apply the above algorithm with ω =]0.3, 0.8[ and D =]0, 0.3[. We obtain the following figure (Fig. 10.2). The gradient is stabilised on ω =]0.3, 0.8[ with error equals 3.7933 × 10−9 and a cost equals 14.99 × 10−2 . Also in this case, for ω =]0, 0.3[ fixed, we have the following array. D Cost

]0.3,0.4[ 1.1802

]0.3,0.5[ 1.1801

]0.3,0.6[ 1.18

]0.3,0.7[ 1.1798

]0.3,0.8[ 1.1794

]0.3,1[ 1.1792

Once more we remark that the cost of the gradient stabilization and the zone area of the control action D are inversely proportional.

10.2 Gradient Stabilization

247 −3

x 10

1.4

0.025

t=5

t=3

1.2 0.02

The flux

The flux

1 0.015

0.01

0.8 0.6 0.4

0.005 0.2 0

0.2

0

0.6

0.4

0.8

1

0

0

0.2

0.6

0.4

0.8

1

x

x

Fig. 10.2 Evolution of the gradient

Now if we fix the zone of the control action and we compute the error and cost for the stabilization of the gradient for different subregions ω. The array bellow gives E = Error ×109 and J = cost ×102 . ω E J

]0,0.3[ 6.6039 117.83

]0,0.4[ 6.7428 135.17

]0,0.6[ 7.6575 155.13

]0,0.7[ 8.5953 159.52

]0,0.9[ 9.1983 162.03

]0,1[ 9.3011 165.18

We remark that if the area of the subregion ω increases the cost and the error of the stabilisation of the gradient increases. Example 10.4 On  =]0, 1[, we consider the system ⎧ ∂ y(t) ⎪ ⎨ = Ay(t) + χ D u(t) ×]0, +∞[ ∂t ⎪ ⎩ y(x, 0) = (1 − x)2 x 2 x ∈

(10.39)

where Ay = 0.01 y + 0.5y. The state space is X = H01 () and u(t) ∈ X , for all t ≥ 0. We take D =]0, 0.3[ and we consider the problem (10.27) with R = ∇ω∗ ∇ω . The eigenvalues and the eigenfunctions of A are given, for i ≥ 1, by ϕi = αi sin(iπ x) and λi = −0.01(iπ )2 + 0.5 As λ1 > 0, then the state and the gradient of system (10.39) are unstable. Weconsider the projection subspace X N = {αi sin(iπ x), i = 1, · · · , N } with 2 αi = 1+(iπ) 2.

248

10 Regional Gradient Stabilization of Infinite Dimensional Linear Systems

Fig. 10.3 Evolution of the gradient

−4

6

x 10

t=3 t=5 t=13

4 2 0

flux

−2 −4 −6 −8 −10 −12 −14

0

0.2

0.4

0.6

0.8

1

x

Global case. If we apply the previous algorithm with a truncation order N = 5, then we obtain the figure (Fig. 10.3). which shows the gradient evolves close to 0 when t increases. The stabilisation error is equal to 9.7961 × 10−7 and cost equals 3.2689 × 10−4 . The following array gives the cost of the gradient stabilisation of system (10.39) depending on different localisations of the control zone D. D ]0, 0.1[ ]0, 0.3[ ]0, 0.5[ ]0, 0.7[ ]0, 0.9[ ]0, 1[ Cost 0.0247 0.0039 0.0016 4.7255 10−4 2.6982 10−4 2.6081 10−4

We remark that if the area of control zone increases then the stabilization cost decreases. Regional case. Now we consider ω =]0, 0.45[, and D =]0.0.8[, we obtain the figure (Fig. 10.4) The cost and error of gradient stabilization are respectively 4.53 × 10−2 and 9.9659 × 10−6 . The next array gives errors and costs of gradient stabilization for different subregions ω. ω Error ×106 Cost

]0,0.17[ 9.9530 3.68 10−2

]0,0.33[ 9.9534 4.26 10−2

]0,0.36[ 9.9609 4.3 10−2

]0,0.45[ 9.9659 4.53 10−2

]0,0.78[ 9.9729 6.4 10−2

It is clear that the error and the cost increase when the area of the target subregion increases.

10.2 Gradient Stabilization

249

Fig. 10.4 Evolution du gradient

We fix ω =]0, 0.45[ and we study the link between cost of the gradient stabilisation and the area of the control support, we obtain the following array. D Cost ×102

]0,0.79[ 4.57

]0,0.75[ 4.74

]0,0.7[ 5

]0,0.65[ 5.37

]0,0.6[ 5.87

]0,55[ 6.38

]0,0.5[ 6.76

Which shows that are inversely proportional as in previous example.

References 1. Kato, T.: Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin (1980) 2. Curtain, R.F., Zwart, H.J.: An Introduction to Infinite Dimensional Linear Systems Theory. Springer, New York (1995) 3. Banks, H., Kunisch, B.: The linear regulator problem for parabolic systems. SIAM J. Control Optim. 22, 684–696 (1984) 4. Higham, N.J.: The scaling and squaring method for the matrix exponential. SIAM Rev. 51(4), 747–767 (2009) 5. Arnold, W.F., Laub, A.J.: Generalized eigenproblem: algorithms and software for algebraic Riccati equations. Proc. IEEE 72, 1746–1754 (1984)

Chapter 11

Gradient Stabilization of Infinite Dimensional Bilinear Systems

In this chapter we study gradient stabilization of infinite dimensional bilinear systems. Then under sufficient conditions, we establish exponential, strong and weak gradient stabilization of bilinear systems on the whole geometrical domain  ⊂ Rn and also on a subregion ω of .

11.1 Gradient Stabilization of Bilinear Systems In this section, we deal with the gradient stabilization of a bilinear system defined on a regular spatial domain  ⊂ Rn by  dy(t)

= Ay(t) + u(t)By(t), [0, +∞[ dt y(0) = y0

(11.1)

where the operator A generates a semigroup of contractions S(t) on H01 () endowed with the inner product, denoted by ., . and . the associated norm, u ∈ L 2 (0, +∞) is a scalar valued control and B is a linear and bounded operator mapping H01 () into itself. We define the gradient operator by ∇ : H01 () −→ y

(L 2 ())n ∂ y(x) ∂ y(x) ∂ y(x)

−→ ( , , ..., ) ∂ x1 ∂ x2 ∂ xn

(L 2 ())n is endowed with its usual inner product ., .n , and .n is the corresponding norm. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. H. Zerrik and O. Castillo, Stabilization of Infinite Dimensional Systems, Studies in Systems, Decision and Control 355, https://doi.org/10.1007/978-3-030-68600-0_11

251

252

11 Gradient Stabilization of Infinite Dimensional Bilinear Systems

Let us recall that the gradient of system (11.1) is exponentially (respectively strongly, weakly) stabilizable, if for any initial condition y0 ∈ H01 () the corresponding solution y(t) of (11.1) is unique and ∇ y(t) converges to zero exponentially (respectively strongly, weakly) as t −→ +∞. The purpose of this Chapter is to study the gradient stabilization of system (11.1) where operators A and B verify the conditions: ∇ By, ∇ yn By, y ≥ 0, ∀y ∈ H01 ().

(11.2)

∇ Ay, ∇ yn ≤ 0, ∀y ∈ D(A).

(11.3)

and

So using a nonlinear control, we establish exponential stabilization under a condition that we call the gradient’s observability inequality. Furthermore, under the same assumption, we obtain a strong stabilization using a feedback control, and the weak stabilization is obtained under a weaker condition.

11.1.1 Gradient Exponential Stabilization In this section, we study the gradient exponential stabilization of system (11.1) using the following control u(t) =

⎧ ⎨ ⎩

−

∇ By(t), ∇ y(t)n , y(t) = 0 y(t)2 0, y(t) = 0

(11.4)

∇ By, ∇ yn for all y = 0, h(0) = 0. y2 If we formally differentiate ∇ y(t)2n along the trajectories of (11.1), we obtain

Let us denote h(y) =

d ∇ y(t)2n ≤ 2∇ Ay(t), ∇ y(t)n + u(t)∇ By(t), ∇ y(t)n , ∀t ≥ 0. dt Using the assumption (3), we get d ∇ y(t)2n ≤ u(t)∇ By(t), ∇ y(t)n , ∀t ≥ 0. dt Then the control (11.4) yields the dissipating energy inequality d |∇ By(t), ∇ y(t)n |2 ∇ y(t)2n ≤ − , ∀t ≥ 0. dt y(t)2

11.1 Gradient Stabilization of Bilinear Systems

253

The following result gives sufficient conditions for the gradient exponential stabilization of the system (11.1). Theorem 11.1 Assume that A generates a semigroup of contractions (S(t))t≥0 and there exist T, μ > 0 such that 

T 0

|∇ B S(s)z, ∇ S(s)zn |ds ≥ μ∇z2n , ∀z ∈ H01 ()

(11.5)

then control (11.4) exponentially stabilizes the gradient of system (11.1). Proof We know from [1] that for each y0 ∈ H01 (), system (11.1) has a unique mild solution y ∈ C([0, tmax [, H01 ()) defined on a maximal interval [0, tmax [ and given by the variation of constants formula  y(t) = S(t)y0 +

t

u(s)S(t − s)By(s)ds

(11.6)

0

Since S(t) is a semigroup of contractions ( A is dissipative), then 

t

y(t)2 ≤ y0 2 − 2

h(y(s))By(s), y(s)ds 0

From (11.2), it follows that y(t) ≤ y0 .

(11.7)

Differentiating ∇ y(t)2n and using (11.3), we have d ∇ y(t)2n ≤ −2|h(y(t))|2 y(t)2 . dt Integrating this inequality from mT to (m + 1)T , (m ∈ IN), we obtain  ∇ y(mT )2n − ∇ y((m + 1)T )2n ≥ 2

(m+1)T

|h(y(s))|2 y(s)2 ds

mT

Moreover, for all y0 ∈ H01 () and t ≥ 0, we have ∇ B S(t)y0 , ∇ S(t)y0 n = ∇ By(t), ∇ y(t)n − ∇ By(t), ∇(y(t) − S(t)y0 )n + ∇ B(S(t)y0 − y(t)), ∇ S(t)y0 n Using (11.7) and since ∇ is bounded, there exists δ > 0 such that

(11.8)

254

11 Gradient Stabilization of Infinite Dimensional Bilinear Systems

|∇ B S(t)y0 , ∇ S(t)y0 n | ≤ 2δ 2 By(t) − S(t)y0 y0  (11.9) + |∇ By(t), ∇ y(t)n | It follows from (11.6), that 

t

y(t) − S(t)y0  ≤ B

|h(y(s))|y(s)ds

0

Schwarz’s inequality yields   y(t) − S(t)y0  ≤ B T

T

|h(y(s))|2 y(s)2 ds

 21

(11.10)

0

Using (11.7), we get |∇ By(t), ∇ y(t)n | ≤ |h(y(t))|y(t)y0 

(11.11)

Integrating (11.9) over the interval [0, T ] and taking into account (11.10) and (11.11), we obtain  T 3 |∇ B S(s)y0 , ∇ S(s)y0 n |ds ≤ 2δ 2 B2 T 2 y0  0

×



T

|h(y(s))|2 y(s)2 ds

 21

(11.12)

0

 √ + T y0 

T

|h(y(s))|2 y(s)2 ds

 21

0

Replacing y0 by y(t) in (11.12), we deduce that 

T

0

√ 3 |∇ B S(s)y(t), ∇ S(s)y(t)n |ds ≤ 2δ 2 B2 T 2 y(t) g(t) √ +y(t) T g(t) 

t+T

where g(t) =

|h(y(s))|2 y(s)2 ds.

t

From Poincarés inequality there exists a constant C > 0 such that  0

T

|∇ B S(s)y(t), ∇ S(s)y(t)n |ds ≤ (2δ 2 B2 T + 1) √ √ × T C∇ y(t)n g(t)

Condition (11.5), gives

11.1 Gradient Stabilization of Bilinear Systems

255

√ μ∇ y(t)n ≤ (2δ 2 B2 T + 1) T C g(t) Then, ρ∇ y(t)2n ≤ g(t)

(11.13)

μ2 . (2δ 2 B2 T + 1)2 T C 2 Combining (11.13) and (11.8), we deduce that

where ρ =

(1 + 2ρ)∇ y((m + 1)T )2n ≤ ∇ y(mT )2n Thus, ∇ y((m + 1)T )n ≤ β∇ y(mT )n 1 . where β = √ 1 + 2ρ By recurrence, we show that ∇ y(mT )n ≤ β m ∇ y0 n Taking m = E( Tt ) the integer part of

t T

, we conclude that

∇ y(t)n ≤ Me−αt y0 , ∀t ≥ 0 where M =



δ 2 (1 + 2( μρ )2 and α =

ln(1 + 2( μρ )2 ) 2T

, which completes the proof. 

Remark 11.1 1. Under the gradient’s observability inequality (11.5), exponential stabilization of the gradient may also be obtained using the following switching control u(t) = −r sign(∇ By(t), ∇ y(t)n ), for suitable r > 0. 2. If the system is stabilizable then the gradient is stabilizable. The converse is true for strong and exponential stabilization on H01 () (Poincaré’s inequality). 3. The gradient stabilization can be obtained under a lower cost than the state stabilization: let us consider the cost q(u) and 1 = {u ∈ L ∞ (0, +∞)|q(u) < +∞ and ∇ y(t) −→ 0 as t −→ +∞ on L 2 ()} Uad

and 2 = {u ∈ L ∞ (0, +∞)|q(u) < +∞ and y(t) −→ 0 as t −→ +∞ on L 2 ()} Uad 2 1 we have Uad ⊂ Uad , so

256

11 Gradient Stabilization of Infinite Dimensional Bilinear Systems

min q(u) ≤ min q(u).

1 u∈Uad

2 u∈Uad

Example 11.1 . On  =]0, 1[, the system, defined by ⎧ 2 ∂ y ∂y ⎪ ⎪ ⎪ 2 (x, t) = y(x, t) + u(t) (x, t) on ×]0, +∞[ ⎨ ∂t ∂t ∂y (x, 0) = y1 on  y(x, 0) = y0 , ⎪ ⎪ ⎪ ∂t ⎩ y(0, t) = y(1, t) = 0 on [0, +∞[

(11.14)

has a unique solution (y, ∂t y) in the state space X = H 2 () ∩ H01 () × H01 () endowed with the inner product (y1 , y2 ), (z 1 , z 2 ) = ∇ y1 , ∇z 1  H01 () + y2 , z 2  H01 () . ˜ = {y ∈ H01 (0, 1)/ − y ∈ H01 (0, 1)}. Let A˜ = − , with domain D( A) 2 ˜ The eigenvalues

of the operator A are λm = (mπ ) , corresponding to eigenfunc2 ∗ tions φm (x) = 1+(mπ)2 sin(mπ x), ∀m ∈ N . System (11.14) has the form of (11.1) if we set A=

0 I − A˜ 0

and B =

00 0I

y ˜ × D( A˜ 21 ), condition (11.3) is verified, indeed Let y = 1 ∈ D( A) y2 ∇ Ay, ∇ y =  y2 , y1  L 2 () −  y1 , y2  L 2 () = 0.   ∞  am z1 1 ∈ H , we have z = φm , where (am , bm ) ∈ R2 , ∀m ∈ N∗ . Let z = 2 z2 λ b m m m=1 The assumption (11.2) is satisfied, indeed ∇ Bz, ∇z H01 ()×L 2 () Bz, z = ∇z 2 2L 2 (0,1) z 2 2H 1 (0,1) ≥ 0. 0

The operator A generates a semigroup given by S(s)z =

+∞  m=1

Then

⎛ ⎝

am cos(mπ s) + bm sin(mπ s)

⎞

⎠ φm , ∀s ≥ 0. bm mπ cos(mπ s) − am mπ sin(mπ s)

11.1 Gradient Stabilization of Bilinear Systems

257 +∞ 

λ2m 1 + λm m=1 2 × am sin(mπ s) − bm cos(mπ s) +∞  λ2m  2 am sin2 (mπ s) = 1 + λ m m=1  −bm am sin(2mπ s) + bm2 cos2 (mπ s) .

∇ B S(s)z, ∇ S(s)z H01 ()×L 2 () =

Inequality (11.5) is verified for T = 2, indeed 

2 0

+∞ 

λ2m (am2 + bm2 ) 1 + λ m m=1 ≥ ∇z2L 2 ()

|∇ B S(s)z, ∇ S(s)z H01 ()×L 2 () |ds =

We deduce that, the gradient of system (11.14) is exponentially stabilizable by the control ⎧ ∇∂t y(t)2L 2 () ⎪ ⎪ ⎪ , i f (y(t), ∂t y(t)) = (0, 0) ⎨− ∂t y(t)2H 1 () + ∇ y(t)2H 1 () u(t) = 0 0 ⎪ ⎪ ⎪ ⎩ 0, i f (y(t), ∂t y(t)) = (0, 0)

11.1.2 Gradient Strong Stabilization Here, we study the gradient strong stabilization of system (11.1), and we give an illustrating example. Theorem 11.2 Let A generate a semigroup of contractions (S(t))t≥0 and assume that conditions (11.2), (11.3) and (11.5) hold, then the control u(t) = −∇ By(t), ∇ y(t)n strongly stabilizes the gradient of system (11.1) . Proof Differentiating ∇ y(t)2n and using (11.3), we obtain d ∇ y(t)2n ≤ −2|∇ By(t), ∇ y(t)n |2 , ∀t ≥ 0. dt Integrating this inequality over [0, t], we get 

t 0

|∇ By(s), ∇ y(s)n |2 ds ≤

1 ∇ y0 2n , ∀t ≥ 0. 2

(11.15)

258

11 Gradient Stabilization of Infinite Dimensional Bilinear Systems



Hence

+∞

|∇ By(s), ∇ y(s)n |2 ds < +∞.

(11.16)

0



t

Let (t) =

S(t − s)u(s)By(s)ds and using (11.6) we have

0

∇ B S(t)y0 , ∇ S(t)y0 n = ∇ By(t), ∇ y(t)n − ∇ B (t), ∇ S(t)y0 n −∇ By(t), ∇ (t)n In the previous result, we found out that (11.2) leads to (11.7), which gives |∇ B S(t)y0 , ∇ S(t)y0 n | ≤ 2δ 2 B (t)y0  (11.17) +|∇ By(t), ∇ y(t)n |, ∀t ∈ [0, T ]. Moreover, we have √   (t) ≤ By0  T

T

|∇ By(s), ∇ y(s)n |2 ds

 21

(11.18)

0

Integrating (11.17) over the interval [0, T ] and taking into account (11.18), we obtain 

T

|∇ B S(s)y0 , ∇ S(s)y0 n |ds ≤ (2δ 2 T B2 y0 2 + 1)

0

 √  × T

T

|∇ By(s), ∇ y(s)n |2 ds

 21

0

Replacing y0 by y(t), we get 

T

√ |∇ B S(s)y(t), ∇ S(s)y(t)n |ds ≤ (2δ 2 T B2 y0 2 + 1) T

0

×



t+T

|∇ By(s), ∇ y(s)n |2 ds

t

From (11.16), we have 

t+T t

It follows that

|∇ By(s), ∇ y(s)n |2 ds −→ 0, as t −→ +∞.

 21

11.1 Gradient Stabilization of Bilinear Systems



T

|∇ B S(s)y(t), ∇ S(s)y(t)n |ds −→ 0, as t −→ +∞.

259

(11.19)

0

From (11.5) and (11.19), we deduce that ∇ y(t)n −→ 0, as t −→ +∞, which completes the proof.  Remark 11.2 In fact, the gradient strong stabilization may be ensured by the controls of the form ∇ By(t), ∇ y(t)n u(t) = − g(y(t)) where g : H01 () −→ (0, +∞) is a positive function satisfying the dissipating energy inequality. Example 11.2 . On  =]0, 1[, the system, defined by ⎧ 2 ∂ y ∂4 y ∂y ⎪ ⎪ ⎪ (x, t) = − (x, t) + u(t) (x, t) on ×]0, +∞[ ⎪ 2 4 ⎪ ∂t ∂x ∂t ⎪ ⎪ ⎨ ∂y ⎪ y(x, 0) = y0 , (x, 0) = y1 on  ⎪ ⎪ ∂t ⎪ ⎪ 2 ⎪ ∂ y ⎪ ⎩ y(ξ, t) = (ξ, t) = 0, ξ = 0, 1 on ]0, +∞[ ∂x2

(11.20)

has only one solution (y, ∂t y) in the state space X = (H 3 (0, 1) ∩ H01 (0, 1)) × H01 (0, 1) endowed with the inner product (y1 , y2 ), (z 1 , z 2 ) =  y1 , z 1  H01 (0,1) + ∂4 y ∂4 y y2 , z 2  H01 (0,1) . Let F = 4 with domain D(F) = {y ∈ H01 (0, 1)/ 4 ∈ H01 (0, 1), ∂x ∂x ∂2 y y(ξ, t) = 2 (ξ, t) = 0, ξ = 0, 1}. ∂x 2 4 The eigenvalues  of the operator are λ j = ( jπ ) , corresponding to eigenfunc2 sin( jπ x), ∀ j ∈ N∗ . tions ϕ j (x) = 1 + ( jπ )2 We set

0 I 00 A= and B = . −F 0 0I 1 y Let y = 1 ∈ D(F) × D(F 2 ), condition (11.3) is satisfied, indeed y2 ∇ Ay, ∇ y = ∂x3 y2 , ∂x3 y1  L 2 () − ∂x3 y1 , ∂x3 y2  L 2 () = 0.   ∞  cj z1 1 ∈ H , we have z = ϕ j , where (c j , d j ) ∈ R2 ∀ j ≥ 1. Now, let z = z2 λ j2 d j j=1

The assumption (11.2) is verified, indeed

260

11 Gradient Stabilization of Infinite Dimensional Bilinear Systems

∇ Bz, ∇z(H 2 (0,1)∩H01 ())×L 2 () Bz, z = ∇z 2 2L 2 (0,1) z 2 2H 1 (0,1) 0 ≥ 0. The operator A generates a semigroup given by ⎛ S(s)z =

∞ 

⎜ ⎝

j=1

1

⎞

1

c j cos(λ j2 s) + d j sin(λ j2 s) 1 2

1 2

1 2

1 2

⎟ ⎠ ϕ j , ∀s ≥ 0.

d j λ j cos(λ j s) − c j λ j sin(λ j s)

Then ∇ B S(s)z, ∇ S(s)z(H 2 (0,1)∩H01 (0,1))×L 2 (0,1) =

+∞  j=1

( jπ )6 1 + ( jπ )4

 × c2j sin2 ( jπ s) −d j c j sin(2 jπ s)  +d 2j cos2 ( jπ s) . For T = 2, condition (11.5) holds, indeed 

2 0

=

|∇ B S(s)z, ∇ S(s)z(H 2 (0,1)∩H01 (0,1))×L 2 (0,1) |ds

+∞  j=1

( jπ )6 (c2 + d 2j ) 1 + ( jπ )4 j

≥ ∇z2L 2 () So the control u(t) = −

∂ 2 y(x, t) 2  L 2 () ∂ x∂t

strongly stabilizes the gradient of system (11.20).



11.1.3 Gradient Weak Stabilization This section is devoted to the gradient weak stabilization of system (11.1) . Theorem 11.3 Suppose that A generates a semigroup of contractions (S(t))t≥0 , conditions (11.2), (11.3) hold and B is compact such that

11.1 Gradient Stabilization of Bilinear Systems

∇ B S(t)z, ∇ S(t)zn = 0, t ≥ 0 =⇒ ∇z = 0

261

(11.21)

holds, then the control u(t) = −∇ By(t), ∇ y(t)n

(11.22)

weakly stabilizes the gradient of system (11.1). Proof System (11.1) has a unique mild solution given by  y(t) = S(t)y0 +

t

S(t − s)u(s)By(s)ds

0

Let (tn ) be a sequence of real numbers such that tn −→ +∞ as n −→ +∞. From (11.7), y(tn ) is bounded in the Hilbert space H01 (), then there exists a subsequence (tγ (n) ) of (tn ) such that y(tγ (n) )  z ∈ H01 () as n −→ +∞

(11.23)

The same arguments as in the proof of Theorem 11.2, gives 

T

|∇ B S(s)y(tγ (n) ), ∇ S(s)y(tγ (n) )n |ds   tγ (n)+T  21 ≤C |∇ By(s), ∇ y(s)n |2 ds 0

(11.24)

tγ (n)

√ where C = (2δ 2 T B2 y0 2 + 1) T . Let n −→ +∞ in (11.24), we obtain  lim

n−→+∞ 0

T

∇ B S(s)y(tγ (n) ), ∇ S(s)y(tγ (n) )n ds = 0

Since B is compact and S(s) is continuous ∀s ≥ 0, we have lim ∇ B S(s)y(tγ (n) ), ∇ S(s)y(tγ (n) )n = ∇ B S(s)z, ∇ S(s)zn

n→+∞

(11.25)

Moreover, by (11.7) and since the operators ∇ and B are bounded, there exists α > 0 such that |∇ B S(s)y(tγ (n) ), ∇ S(s)y(tγ (n) )n | ≤ αy0 2 (11.26) Using (11.25) and (11.26), we deduce from dominated convergence theorem (Bartle, 2014), that  T

0

which leads to

|∇ B S(s)z, ∇ S(s)zn |ds = 0

262

11 Gradient Stabilization of Infinite Dimensional Bilinear Systems

∇ B S(s)z, ∇ S(s)zn = 0, ∀s ∈ [0, T ] Using (11.21), it follows that ∇z = 0. Since ∇ is continuous, (11.23) leads to ∇ y(tγ (n) )  0 as n −→ +∞. Which implies that ∀ϕ ∈ (L 2 ())n , ∇ y(tn ), ϕn −→ 0 as n −→ +∞, and hence ∇ y(t)  0 as t −→ +∞. We conclude that control (11.22) weakly stabilizes the gradient of system (11.1).  Example 11.3 On  =]0, 1[, we consider the heat equation defined by ⎧ ∂ y(x, t) ∂ 2 y(x, t) ⎪ ⎨ = + y(x, t) + u(t)By(x, t), ×]0, +∞[ ∂t ∂x2  ⎪ ⎩ y(x, 0) = y0 , y(., t) = 0, ∂×]0, +∞[

(11.27)

+∞  1 y, ϕ j ϕ j , ∀ j ∈ N∗ . where By = 2 j j=1

System (11.27) has only one solution in the state space H01 (). 2 The eigenvalues of operator  Ay = y + y are λ j = −( jπ ) + 1 corresponding 2 sin( jπ x), for j ∈ N. 1 + ( jπ )2 Conditions (11.2) and (11.3) are verified, indeed

to eigenfunctions ϕ j (x) =

∇ Ay, ∇ y =

+∞  ( jπ )2 − ( jπ )4 j=1

and ∇ By, ∇ y L 2 () By, y =

1 + ( jπ )2 +∞  j=1

|y, ϕ j |2 ≤ 0

( jπ )2 |y, ϕ j |2 ≥ 0 + ( jπ )2 )

j 4 (1

The operator A generates the semigroup S(t)y =

+∞ 

eλ j t y, ϕ j ϕ j , ∀ j ∈ N∗ .

j=1

We have ∇ B S(t)y, ∇ S(t)y L 2 () = =⇒ ∇ y = 0.

+∞  j=1

π2 e2λ j t |y, ϕ j |2 = 0, ∀t ≥ 0 1 + ( jπ )2

then (11.21) holds, so we deduce that the control (11.22) weakly stabilizes the gradient of system (11.27).

11.2 Regional Gradient Stabilization of Bilinear Systems

263

11.2 Regional Gradient Stabilization of Bilinear Systems We study in this section the regional gradient stabilization of infinite dimensional bilinear systems and we give different characterizations of control that stabilizes the gradient on a subregion. Let  be a regular open domain of Rn , Q = ×]0, ∞[ and ω a non null measure subregion of . We consider a bilinear system described by the equation ⎧ ⎨ y˙ (t) = Ay(t) + u(t)By(t) on Q ⎩

(11.28) y(0) = y0 ∈ X = H 1 ()

where A is a linear operator of domain D(A) ⊂ H 1 () generates a C0 -semigroup (S(t))t≥0 on X = H 1 () endowed with its usual inner product < . > and the associated norm  . , B ∈ L(X ) is a bounded linear operator, and u(t) is a scalar control. We define the operator ∇ω : X −→ (L 2 (ω))n y −→ (χω

∂ y(x) ∂ y(x) ∂ y(x) , χω , · · · , χω ) ∂ x1 ∂ x2 ∂ xn

(11.29)

The space (L 2 ())n is endowed with its usual inner product < ., . > and the associated norm  . , and (L 2 (ω))n is endowed with the restriction < ., . >n . We note G ω = ∇ω∗ ∇ω , where ∇ω∗ is the adjoint operator of ∇ω . Definition 11.1 The gradient of system (11.28) is said to be 1. weakly stabilizable on ω, if for each initial condition y0 ∈ X, the corresponding solution y(t) of (11.28) is global and verify < ∇ω y(t), z >n −→ 0 when t −→ ∞, z ∈ (L 2 ())n . 2. strongly stabilizable on ω, if for each initial condition y0 ∈ X , the corresponding solution y(t) of system (11.28) is global, and ||∇ω y(t)||n −→ 0 when t −→ ∞. 3. exponentially stabilizable on ω, if for each initial condition y0 ∈ X, the corresponding solution y(t) of system (11.28) is global and there exist two reals M, α > 0 such that: ∇ω y(t)n ≤ Me−αt y0  , t ≥ 0 In the sequel a system of stabilizable gradient on ω is will be said G-stabilizable sur ω. If ω =  the system (11.28) will be said G-stabilizable.

264

11 Gradient Stabilization of Infinite Dimensional Bilinear Systems

Remark 11.3 If the system (11.28) is stabilizable, then it is G-stabilizable on all ω ⊂ . The converse is not true in general as illustrated by the following counter example. Example 11.4 On  =]0, 1[2 , we consider the system ⎧ ∂y ⎪ ⎪ (x, t) = y(x, t) + 10π 2 y(x, t) + u(t)By(x, t) on Q ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎨ ∂y (ξ, t) = 0 ∂×]0, +∞[ ⎪ ⎪ ⎪ ∂ν ⎪ ⎪ ⎪ ⎪ ⎩ y(0) = y0 ∈ H 1 ()

(11.30)

where is the Laplacien operator. The operator A = + 10π 2 I d has an orthogonal basis of eigenfunctions given by ϕi j (x, z) = 2ai j cos(iπ x) cos( jπ z) where ai j = (1 − λi j )− 2 . The associated eigenvalues are given by 1

λi j =

⎧ ⎨ −(i 2 + j 2 )π 2 + 10π 2 if (i, j) = (3, 0) ⎩

π2

otherwise

A generates a C0 -semigroup given by 

S(t)y = eπt y, ϕ30 ϕ30 +

eλi j t y, ϕi j ϕi j

(i, j)=(3,0)

The operator By =



−4 λi−4 j y, ϕi j ϕi j . Moreover, (λi j , ϕi j ), for (i, j)  = (3, 0),

(i, j)=(3,0)

are the eigenfunctions and the eigenvalues of the operator B that satisfy 

Bϕi j 2 =



Bϕi j 2

(i, j)=(3,0)

(i, j)∈N 2

=



|λi, j |−2 < +∞

(i, j)=(3,0)

which shows that B is a compact operator. We have B S(t)y, S(t)y =

 (i, j)=(3,0)

Hence

2λi j t λi−3 y, ϕi j 2 j e

11.2 Regional Gradient Stabilization of Bilinear Systems

BT (t)y, T (t)y = 0 , t ≥ 0 ⇒ y, ϕi j  = 0 , (i, j) = (3, 0)

265

(11.31)

As ∇ω ϕ30 = 0 according to (11.31) we have ∇ω y = 0. Consequently the system (11.30) is weakly G-stabilizable on ω. In the other hand, we remark that, for y0 = ϕ30 , the solution of (11.30) is written 2 y(t) = eπ t ϕ30 and so y(t) = eπ t ϕ30   0 when t → +∞ 2

Finally the system (11.30) is not weakly stabilizable on ω.

11.2.1 Regional Stabilizing Control In this paragraph we characterize the control that assure the weak gradient stabilization of the bilinear system (11.28) on a subregion of the system evolution domain The solution of system (11.28) is written  y(t) = S(t)y0 +

t

u(s)S(t − s)By(s)ds

(11.32)

0

Deriving formally ∇ω y(t)2 along the trajectories of system (11.28), we obtain d ∇ω y(t)2 = Re(G ω Ay(t), y(t)) + Re(u(t)G ω By(t), y(t)) dt If in addition the operator G ω A is dissipative (ReG ω Ay, y ≤ 0, for each y ∈ D(A)) then d ∇ω y(t)2 ≤ Re(u(t)G ω By(t), y(t)) dt The choice of control u(t) = −y(t), G ω By(t) allows the energy dissipation inequality. More general we are interested to the study of quadratic controls of type u(t) = y(t), K y(t) where K ∈ L(X ) is a linear bounded operator. The following result gives sufficient conditions assuring gradient weak stabilization of system (11.28) by u(t) = −K By(t), y(t) (11.33) Proposition 11.1 If the conditions 1. (S(t)) is a contraction semigroup, 2. B is compact, 3. Re(D By, yy, By) ≥ 0, for all y ∈ H 1 (),

266

11 Gradient Stabilization of Infinite Dimensional Bilinear Systems

4. D B S(t)y, S(t)yS(t)y, B S(t)y = 0, for all t ≥ 0 ⇒ ∇ω y = 0, are satisfied, then the control (11.33) weakly stabilize the gradient of system (11.28) on ω. Proof The existence of the solution is assured by the Proposition 3.3 and the continuity of the application y0 → y(t). Let the function F(y) = −y, K ByBy. The range of each ball of X by B is a ball of X , this imply that F is localy Lipshitz, and as B is compact, there exists y0 ∈ X such that y(t)  y0 when t → ∞ and F(y(t)), y(t) = 0, for all t > 0 (see [2]). Hence K BT (t)y0 , T (t)y0 T (t)y0 , BT (t)y0  = 0, for all t > 0 and with the hypothesis 4, we obtain ∇ω y0 = 0. Let z ∈ (L 2 ())n , as ∇ω is a continuous operator then ∇ω y(t), z → ∇ω y0 , z quand t → ∞ We deduce that system (11.28) is weakly G-stabilizable on ω.

 

Corollary 11.1 If the hypothesis 1 and 2 of the previous proposition and the following condition (11.34) B S(t)y, S(t)y = 0 , t ≥ 0 ⇒ ∇ω y = 0 hold, then the control u(t) = −y(t), By(t) weakly stabilizes the gradient of system (11.28) on ω. Proof For K = I d, the conditions 3. and 4. of the previous proposition hold.

 

Remark 11.4 1. If the condition (11.34) is replaced by the condition B S(t)y, S(t)y = 0 , t ≥ 0 ⇒ y = 0 then the control u(t) = −By(t), y(t) weakly stabilizes the system (11.28) (see [2]). But we also have, y = 0 ⇒ ∇ y = ∇ω y = 0. Hence the system (11.28) is weakly G-stabilizable.

11.2.2 State Space Decomposition In this paragraph we look for controls that stabilize the gradient of system (11.28), using the decomposition method. We consider the operator (10.20) and we suppose there exist two operators B1 and B2 in L(X ), such that 

B = B1  (I d − )B = B2 (I d − )

then the state space X = X u ⊕ X s and the system (11.28) is decomposed into two subsystems given by

11.2 Regional Gradient Stabilization of Bilinear Systems

and

267

⎧ ⎨ y˙u (t) = Au yu (t) + v(t)Bu yu (t) yu = y ⎩ y0u = y0 ∈ Hu

(11.35)

⎧ ⎨ y˙s (t) = As ys (t) + v(t)Bs ys (t) ys = (I d − )y ⎩ y0s = (I d − )y0 ∈ Hs

(11.36)

where Au = A, As = (I d − )A, Bu = B1  et Bs = B2 (I d − ). The solutions of (11.35) and (11.36) are given by 

t

yu (t) = Su (t)y0u +

v(τ )Su (t − τ )Bu yu (τ )dτ

(11.37)

v(τ )Ss (t − τ )Bs ys (τ )dτ

(11.38)

0



and

t

ys (t) = Ss (t)y0s + 0

where Su (t) and Ss (t) are respectively the restrictions of S(t) to Hu and to Hs , of generators Au and As . Let us recall that As verify the hypothesis of spectral decreasing ln Ss (t) = sup Re(σ (As )) t

(11.39)

Ss (t) ≤ a e−bt , t ≥ 0, 0 < b < δ

(11.40)

lim

t→+∞

which gives

We consider the control u(t) = −G ω K u G ω yu (t), yu (t)

(11.41)

where K u ∈ L(Hu ) and G ω = ∇ω∗ ∇ω , then we have the following result. Proposition 11.2 If the system (11.28) may be decomposed into subsystems (11.35) and (11.36) and if moreover, As verify (11.39), then we have the following two points: 1. If the control (11.41) exponentially stabilizes the gradient of system (11.35) ω, then the control (11.41) exponentially stabilizes the gradient of system (11.28) on ω. 2. If the state of system (11.35) is bounded, the the state of system (11.28) is bounded on the whole domain . Proof 1. As the system (11.35) is exponentially G-stabilizable on ω, then the solution yu is global and we have ∇ω yu (t) ≤ Me−ct y0u , for M, c > 0

(11.42)

268

11 Gradient Stabilization of Infinite Dimensional Bilinear Systems

The existence of a weak solution ys of system (11.36) is assured on an interval [0, tmax [. To show that the solution ys (t) is global, it’s sufficient to prove that ys (t) is bounded. By (11.38) and (11.40), we have ys (t) ≤ a e

−bt



t

y0s  + aBs 

|v(τ )|e−b(t−τ ) ys (τ )dτ

0

which gives 

t

ebt ys (t) ≤ a y0s  + aBs 

|v(τ )|ebτ ys (τ )dτ.

0

By Gronwall lemma, we have  e ys (t) ≤ ay0s  exp( bt

t

aBs |v(τ )|dτ )

(11.43)

0

From (11.41) and (11.42), it follows |v(t)| ≤ M 2 ∇ω D∇ω∗ e−2ct y0 2

(11.44)

using (11.43) and (11.44), there exists L > 0 such that ys (t) ≤ a Ly0s e−bt

(11.45)

Consequently ys (t) is bounded, so y(t) is a weak unique and global solution of system (11.28) (y = ys + yu ). According to the inequality (11.45), we have ys (t) → 0 exponentially when t → +∞, it follows ∇ω ys (t) → 0 exponentially when t → +∞. and using (11.35), (11.42) and the inequality ∇ω y(t) ≤ ∇ω yu (t) + ∇ω ys (t) we have the conclusion. 2. This point is immediate considering (11.45).

 

The spectral decreasing inequality (11.39) may be replaced by the less restrictive condition ln |||∇ω Ss (t)||| ≤ sup Re(σ (As )) (11.46) lim t→+∞ t We obtain the following result. Proposition 11.3 Let us suppose that the system (11.28) may be decomposed into two subsystems (11.35) and (11.36) and that As verify (11.46).

11.2 Regional Gradient Stabilization of Bilinear Systems

269

Assume also there exists a real N > 0, such that ∇ω Bs y ≤ N ∇ω y , y ∈ X

(11.47)

If in addition ys (t) is a weak, unique and global solution of system (11.36), then with the control (11.41), the results of the Proposition 11.2 hold. Proof If the system(11.35) is exponentially G-stabilizable on ω, then its weak solution yu (t) is unique and global. Moreover the weak solution of ys (t) of system (11.36) is unique and global, and y = yu + ys , then y(t) is a weak solution, unique and global of system (11.28). According to the spectrum decomposition of A, we have sup Re(σ (As )) ≤ −δ.  Using (11.46), it follows that ∇ω Ts (t)n ≤ a  e−b t , with t ≥ 0, 0 < b < δ and  a > 0. By (11.38) and (11.47), we obtain b t







e ∇ω ys (t)n ≤ a ∇ω y0s n + a N

t



|v(τ )|eb τ ∇ω ys (τ )n dτ

0

With the same techniques that used to prove the Proposition 11.2, we complete the proof.   Remark 11.5 1. If (Ss (t)) is contraction semigroup and Re(ys , Bs ys K u yu , yu ) ≥ 0 then ys (t) is a weak solution unique and global of system (11.36). 2. If Bs = I d and (11.47) is verified, if moreover (Ss (t)) is a contraction semigroup, and K u is a positive operator then ys (t) is a weak solution, unique and global of system (11.36). Proposition 11.4 If the system (11.28) may be decomposed in subsystems (11.35) and (11.36), if moreover the conditions (11.46), and (11.47) are verified and that ys (t) is a weak, unique and global solution of system (11.36) and (Su (t)) is a contraction semigroup. If the state yu is bounded, there exists M  > 0 such that yu  ≤ M  , if moreover Bu is compact and satisfy the condition: Bu Su (t)yu , Su (t)yu  = 0 , t ≥ 0 ⇒ ∇ω yu = 0 If Bs verify la condition K u M 2
0

(11.51)

11.2 Regional Gradient Stabilization of Bilinear Systems

271

We suppose there exists β > 0 such that Pω y, z ≤ β∇ω y∇ω z , (y, z) ∈ X × X

(11.52)

and reals a, b > 0, such that, for all t ≥ 0 and y ∈ X , we have ∇ω T (t)y ≤ a∇ω y

(11.53)

∇ω By ≤ b∇ω y

(11.54)

and

We consider the algebraic Riccati equation Pω Ay, y + y, Pω Ay + Ry, y = 0 , y ∈ D(A)

(11.55)

The main result of this section is based on the following lemma. Lemma 11.1 We note  1  μ= Pω By(t), y(t)dt and ν = 0

1

|u(t)|2 dt

0

Assume that (11.51), (11.52), (11.53) and (11.54) are verified, then there exist η > 0, and λ > 0 such that √ ν < η ⇒ ∇ω y0 2n < λ μ Proof Without loss of generality we assume that ν < 1. Let ψ(t) = y(t) − S(t)y0 . 1. We show the estimation √ there exists C0 > 0 such that ∇ω ψ(t) ≤ C0 ∇ω y0  ν , t ∈]0, 1[ 

t

ψ(t) =

 u(s)S(t − s)B S(s)y0 ds +

0

with (11.53) and (11.54) we obtain

t

u(s)S(t − s)B(y(s) − S(s)y0 )ds,

0

 t |u(s)|ds ∇ω ψ(t) ≤ a 2 b∇ω y0  0  t |u(s)|∇ω ψ(s)ds + ab 0  t √ |u(s)|∇ω ψ(s)ds ≤ a 2 b∇ω y0  ν + ab 0

Gronwall lemma allows to get

(11.56) and

272

11 Gradient Stabilization of Infinite Dimensional Bilinear Systems

√ ∇ω ψ(t) ≤ a 2 b∇ω y0  ν √ + a 3 b2 ν∇ω y0 



t



t

|u(s)| exp(

0

ab|u(τ )|dτ )ds

s

√ ≤ a 2 b∇ω y0  ν √ + a 3 b2 ν∇ω y0  exp(ab) foe all t ∈]0, 1[, and ν < 1. Thus (11.56) is verified with C0 = a 2 b∇ω y0  + a 3 b2 ∇ω y0  exp(ab). The triangular inequality gives 

1 0

 1 √ |Pω B S(t)y0 , S(t)y0 |dt ≤ μ + |Pω B S(t)ψ(t), S(t)y0 |dt 0  1  1 |Pω B S(t)y0 , ψ(t)|dt + |Pω Bψ(t), ψ(t)|dt + 0

0

By (11.52), (11.53), (11.54) and (11.56) there exist a, ˜ b˜ > 0 such that  1 √ √ 2 ˜ |Pω B S(t)y0 , S(t)y0 |dt ≤ μ + a˜ ν∇ω y0 2 + bν∇ ω y0  0

As ν < 1, with (11.51) we obtain √ ˜ ν)∇ω y0 2 ≤ √μ (α − (a˜ + b) It suffices to take η=

α2 1 inf(1, ) ˜ 2 2 (a˜ + b)  

to finish the proof.

Proposition 11.5 Assume that there exists Pω a positive and self adjoint operator, solution the algebraic Riccati equation (11.55), such that (11.51), (11.52), (11.53) and (11.54) are verified. If moreover the solution y ∗ (t) corresponding to the control u ∗ (t) = −B ∗ Pω y ∗ (t), y ∗ (t)

(11.57)

is global, then u ∗ (t) is the unique solution of problem (11.50) that strongly stabilises the gradient of system (11.28) on ω. In addition if there exists d > 0 such that 1 |Pω By, y|2 + Ry, y ≤ dRe(y, ByPω y, y − Ay, y) , y ∈ D(A) 2 (11.58)

11.2 Regional Gradient Stabilization of Bilinear Systems

273

then the state of system (11.28) remains bounded on . Proof We consider the function F(y) = Pω y, y, for each y ∈ X, we show that u ∗ ∈ Vad . For y0 ∈ D(A), with (11.55) we have ∂ F(y ∗ (t)) = −2|B ∗ Pω y ∗ (t), y ∗ (t)|2 − Ry ∗ (t), y ∗ (t) ∂t

(11.59)

Integrating (11.59) over [0, t] and since Pω is positive we obtain: 

t

∗



∗

t

|Pω By (s), y (s)| ds + 2

0

Ry ∗ (s), y ∗ (s)ds ≤ F(y0 )ds, t ≥ 0 (11.60)

0

As the application y0 −→ y ∗ (.) is continuous from X to C([0, t]; X ), then y −→ F(y) and y −→ Ry, y are continuous from H to C and hence (11.60) is verified for all y0 ∈ X. Which imply that for all initial condition y0 ∈ X, J (u ∗ ) < +∞ and consequently ∗ u ∈ Vad . 2. We show that each control u ∈ Vad stabilizes the gradient of system (11.28) on ω. Let  such that 0 <  < η. As J is finite, Cauchy criteria for integral convergence gives 

t+1

∃T > 0, ∀t > T :

 Pω By(s), y(s)ds <  and

t

t+1

|u(s)|2 ds < 

t

√ Applying Lemma 11.1 with y0 = y(t), we have ∇ω y(t) < λ , t > T. Then ∇ω y(t) → 0, when t → +∞. 3. The control u ∗ (t) is the unique solution of (11.50). Indeed, we have F(y(t)) ≤ M∇ω y(t)2 , for M > 0 Thus F(y(t)) → 0 when t → +∞. For y0 ∈ D(A), we integrate the relation ∂ F(y(t)) = |B ∗ Pω y(t), y(t) + u(t)|2 − |B ∗ Pω y(t), y(t)|2 ∂t −|u(t)|2 − Ry(t), y(t) and we obtain 

+∞

J (v) = F(y0 ) + 0

|B ∗ Pω y(t), y(t) + v(t)|2 dt

274

11 Gradient Stabilization of Infinite Dimensional Bilinear Systems

This leads to J (u ∗ ) ≤ J (u), for all u ∈ Vad . For y0 ∈ X, there exist a sequence (y0n ) ⊂ D(A) such that y0n → y0 when n → +∞. Let u ∈ Vad we have  J (u) = F(y0n ) +

+∞

|B ∗ Pω yn (t), yn (t) + v(t)|2 dt

(11.61)

0

So J (u) ≥ F(y0n ). As the application F is continuous we deduce that J (v) ≥ F(y0 ) = J (u ∗ ) which show that v∗ is a solution of problem (11.50). 4. The uniqueness of u ∗ (t) comes from the uniqueness of y(t) the solution of system (11.28). Indeed, according to (11.61), every optimal control is of form u(t) = −B ∗ Pω y(t), y(t) Let

u 1 (t) = −B ∗ Pω y1 (t), y1 (t) and u 2 = −B ∗ Pω y2 (t), y2 (t)

be two solutions of problem (11.50) where y1 (t) and y2 (t) are the corresponding solutions of system (11.28). As (11.28) has only one solution, then y1 (t) = y2 (t) and hence u 1 (t) = u 2 (t). 5. We show the state of system (11.28) remains bounded on . d ∂ By the inequality (11.58), we have F(y(t))2 ≥ d y(t)2 . And ∂t dt F(y(t)) − F(y0 ) ≥ d(y(t)2 − y0 2 ) , y0 ∈ D(A)

(11.62)

As y0 −→ y(.) is continuous from X to C([0, t], X ) and the application F is also continuous, the inequality (11.62) is satisfied for all y0 ∈ X. Moreover F(y(t)) → 0 when t → +∞.   Remark 11.6 We can take in condition (11.52), Pω = G ω P G ω where P ∈ L(X ) is a continuous and positive operator, solution of the equation (11.55). Proposition 11.6 If the conditions (11.51), (11.52), (11.53), and (11.54) are verified with Pω solution of the algebraic Riccati equation (11.55) and if there exists d > 0 such that Pω y, y ≥ d∇ω y2

(11.63)

there exists e > 0 such that Ry, y ≥ e∇ω y2

(11.64)

Re(R Ay, y) ≤ Re(Pω By, yR By, y) , y ∈ D(A)

(11.65)

and

are satisfied and that the solution y(t) is global, then for all initial condition y0 such that ∇ω y0 = 0, we have the estimation

11.2 Regional Gradient Stabilization of Bilinear Systems

∇ω y(t)2 = O(

275

1 ), when t → +∞ t3

Proof We consider the function 

t

W (y(t)) = Pω y(t), y(t) +

Ry(s), y(s)ds , t ≥ 0

0

and Wm =

1 1 Pω y ∗ (m), y ∗ (m) + 2 2



m

Ry ∗ (s), y ∗ (s)ds , t ≥ 0

0

We remark that Wm is a positive sequence. We integrate between m and m + 1 the relation ∂ W (y ∗ (t)) = −2|Pω By ∗ (t), y ∗ (t)|2 ∂t we obtain  Wm+1 − Wm = −

m+1

|Pω By ∗ (t), y ∗ (t)|2 dt = −μ

m

 =−

m+1

|u ∗ (t)|2 =: −ν

m

Applying the Lemma 11.1 with y0 = y ∗ (t) we obtain Wm+1 − Wm ≤ −η where Wm+1 − Wm
0 such that W0 =

1 Pω y0 , y0  ≤ M∇ω y0 2 . 2

The condition (11.63) leads to W0 > 0 and by (11.66) it follows Wm+1 − Wm ≤ − f (y0 )Wm2 with f (y0 ) = min( We note Vm =

1 λ2 M 2

,

η ). Thus Wm is a decreasing sequence. W02

1 , then Wm

Vm+1 − Vm =

Wm − Wm+1 Wm − Wm+1 ≥ Wm+1 Wm Wm2

(11.67)

276

11 Gradient Stabilization of Infinite Dimensional Bilinear Systems

By (11.67) we obtain Vm+1 − Vm ≥ f (y0 ) and so Vm+1 ≥ V0 + m f (y0 ). which gives Wm ≤

W0 , m≥0 1 + W0 f (y0 )m

and we have the estimation W (y ∗ (t)) ≤

2W (y0 ) , t ≥0 1 + W (y0 ) f (y0 )t

(11.68)

For y0 ∈ X such that ∇ω y0 = 0 then there exists a sequence y0n ∈ D(A) and y0n → y0 when n → +∞, ∇ω y0n = 0 (∇ω is a continuous operator). With the same above techniques, we show that(11.68) is verified for y0n . But as y0 −→ y ∗ (.) is continuous from X to C([0, t], X ), the function y0 −→ f (y0 ) is continuous. And the estimation (11.68) holds for all y0 ∈ X. ∂ The condition (11.65) leads to Ry(t), y(t) ≤ 0. It follows that ∂t W (y ∗ (t)) ≥ d∇ω y ∗ (t)2n + tRy ∗ (t), y ∗ (t) and by (11.64), we obtain W (y ∗ (t)) ≥ d∇ω y ∗ (t)2n + e∇ω y ∗ (t)2n t ≥ (d + et)∇ω y ∗ (t)2n We deduce that ∇ω y ∗ (t)2n ≤

2W (y0 ) (d + et)(1 + W (y0 ) f (y0 )t)

(11.69)

As f (y0 ), W (y0 ) and e are non zero, we obtain the desired estimation. If we consider the problem (11.50) with R = 0, we have the following result. Corollary 11.2 If the conditions (11.51), (11.52), (11.53), and (11.54) are verified with Pω , solution of the algebric Riccati equation (11.55) and satisfy the condition (11.63), then for all initial state y0 , such that ∇ω y0 = 0, we have the estimation ∇ω y ∗ (t)n = O(t − 2 ) when t −→ +∞ 1

Proof We note W (y(t)) = Pω y(t), y(t) et Wm =

1 W (y ∗ (m)) 2

With the same techniques as above we show that the estimation (11.68) and with (11.63), we deduce the desired estimation.  

11.3 Numerical Approach and Simulations

277

11.3 Numerical Approach and Simulations 11.3.1 Regional Gradient Stabilization Case We have seen that the control solution of problem (11.50) is given by (11.57) where Pω is the unique solution of the equation (11.55). We consider the system ⎧ ⎨ y˙ (t) = Ay(t) + u(t)By(t) t > 0 ⎩

(11.70) y(0) = y0

and the control u(t) = −Pω By(t), y(t)

(11.71)

Let us denote tm = mh, with h > 0 and m ∈ N. On [tm−1 , tm ], we take u(t) = u(tm−1 ) and the system (11.70) may be approached by the system ⎧ ⎨ y˙m (t) = Aym (t) + u(tm−1 )Bym (t) ⎩ and the control by

(11.72) y(0) = y0

⎧ ym (tm−1 ) = ym−1 (tm−1 ) ⎪ ⎪ ⎪ ⎪ ⎨ u(tm ) = −Pω By(tm ), y(tm ) ⎪ ⎪ ⎪ ⎪ ⎩ u(0) = −Pω By0 , y0 

(11.73)

Solving these equation gives y(tm ) and the control over the next interval is u(tm ) = −Pω By(tm ), y(tm ). The following result establish a relation between the gradient of system (11.70) and the one of system (11.72). Proposition 11.7 We assume that the weak solution of system (11.70) is global and bounded and there exist λ > 0 and M ≤ 1 such that ∇ω T (t)n ≤ Me−λt , if λ , then there exists c > 0 such that moreover B verify (11.54) and α2 ≤ M|||B||| ∇ω ym (t) − ∇ω y(t) < c h , tm−1 ≤ t ≤ tm

(11.74)

Proof 1. The solution of system (11.70) is bounded and there exists a real α1 > 0 such that y(t) ≤ α1 , this leads to |v(t)| ≤ α2 , where α2 = α12 |||Pω B|||. 2. We show that

278

11 Gradient Stabilization of Infinite Dimensional Bilinear Systems

∇ω y(t) ≤ M∇ω y0 e−(λ−Mb|||B|||α2 )t Indeed, let j (t) = ∇ω T (t), we have  ∇ω y(t) ≤  j (t)y0  +

t

|u(s)| j (t − s)By(s)ds  t −λt e−α(t−s) ∇ω y(s)n ds ≤ Me ∇ω y0 n + Mb|||B|||α2 0

0

More again eλt ∇ω y(t) ≤ M∇ω y0  + Mb|||B|||α2



t

eαs ∇ω y(s)ds

0

and Gronwall lemma gives eλt ∇ω y(t) ≤ M∇ω y0 e Mb|||B|||α2 t It follows ∇ω y(t) ≤ M∇ω y0 e−(λ−Mb|||B|||α2 )t . 3. We show that for m > 1, there exist two positive constants M1 and λ˜ such that ˜

m−1 ∇ω z mt  ≤ M∇ω z m−1  + M1 heλ(m−1)h

t

(11.75)

where z mt = ym (t) − y(t). Indeed, m−1  ∇ω z mt  ≤ Me−λ(t−tm−1 )∇ω z m−1

t

+bM|||B||| +bM|||B|||

t

tm−1 t tm−1

e−λ(t−s) |u(tm−1 ) − u(s)|∇ω y(s)ds e−λ(t−s) |u(tm−1 )|∇ω z ms ds

m−1   ≤ Me−λ(t−tm−1 ) ∇ω z m−1 t e−λ(t−s) e−(λ−Mb|||B|||α2 )s ds +2α2 M 2 b∇ω y0 |||B||| t m−1  t +α2 Mb|||B||| e−λ(t−s) ∇ω z ms ds

t

tm−1

Which gives t

m−1 eλt ∇ω z mt  ≤ Meλtm−1 ∇ω z m−1  + 2α2 M 2 b∇ω y0 |||B|||he Mb|||B|||α2 t

 +α2 Mb|||B|||

t

tm−1

Once again Gronwall lemma gives

eλs ∇ω z ms ds

11.3 Numerical Approach and Simulations

279

m−1 eλt ∇ω z mt  ≤ Meλtm−1 ∇ω z m−1  + 2α2 M 2 b∇ω y0 |||B|||he Mb|||B|||α2 t

t

+α2 M b|||B|||e 2

λtm−1

 tm−1 ∇ω z m−1 

t

e Mb|||B|||α2 (t−s) ds

tm−1

+2α22 M 3 b2 ∇ω y0 B2 h 2 e Mb|||B|||α2 t which leads to m−1 eλt ∇ω z mt  ≤ Meλtm−1 ∇ω z m−1 e Mb|||B|||α2 (t−tm−1 )

t

+2λM∇ω y0 he Mb|||B|||α2 t +2M∇ω y0 λ2 h 2 e Mb|||B|||α2 t We note α˜ = −λ + bMBα2 , it follows m−1 ˜ ˜ m−1 ) ∇ω z mt  ≤ M∇ω z m−1 n eα(t−t + 2λM∇ω y0 heαt

t

˜ +2M∇ω y0 λ2 h 2 eαt m−1 ˜ ≤ M∇ω z m−1  + 2λM∇ω y0 heα(m−1)h

t

˜ +2M∇ω y0 λ2 h 2 eα(m−1)h

We obtain

˜ m−1  + M1 heα(m−1)h ∇ω z mt  ≤ M∇ω z m−1 t

(11.76)

with M1 = 2λM∇ω y0 (1 + hλ). 4. We show the estimation (11.74). Following the same above techniques, we get ∇ω z 1t eλt

 ≤ 2λM∇ω y0 he

Mb|||B|||α2 t

t

+ Mb|||B|||α2 0

∇ω z 1t eλs ds

and with Gronwall lemma we have ∇ω z 1t eλt ≤ 2λM∇ω y0 n he Mb|||B|||α2 t + 2Mh 2 λ2 ∇ω y0 e Mb|||B|||α2 t hence ∇ω z 1t  ≤ M1 h. As −λ + α2 Mb|||B||| < 0, then lim (m + 1)2 e(−λ+α2 Mb|||B|||)mh = 0

m−→+∞

so there exists N1 > 0 such that for all m > N1 ,

280

11 Gradient Stabilization of Infinite Dimensional Bilinear Systems

e(−λ+α2 Mb|||B|||)mh )
N1 , we have t

m−1 + ∇ω z mt  ≤ M∇ω z m−1

thus t

m−2 ∇ω z mt  ≤ M 2 ∇ω z m−2 +M

so

M1 h m2

M1 h M1 h + 2 2 (m − 1) m

M1 h (N1 + 1)2 M h M1 h 1 +M m−N1 −2 + ··· + 2 (N1 + 2)2 m tN

∇ω z mt  ≤ M m−N1 ∇ω z N11  + M m−N1 −1

(11.77)

and by (11.76) for m  ≤ N1 , we have t



−1 ∇ω z mt   ≤ M∇ω z mm −1  + M1 h

t



−2  + M M1 h + M1 h ≤ M 2 ∇ω z mm −2 

≤

m 

M i M1 h

i=0 



then for all m ≤

t  N1 , ∇ω z mm 

m 

≤ h M1

M i . Replacing in (11.77), we obtain

i=0

∇ω z mt  ≤ M m−N1

N1 

≤ (M m−N1

M i M1 h +

i=0 N1  i=0

≤ (M m−N1

N1  i=0

M i M1 +

m−N 1 −1 i=0 m−N 1 −1 i=0

M i M1 h (m − i)2 M i M1 )h (m − i)2

m  M m−i Mi + )M1 h, i2 i=N +1 1

As the last serial is convergent, then the estimation (11.74) is verified for c > 0.   Now we solve the algebraic Riccati equation (11.55). We set H N = span{φi i = 1, · · · , N } is a subspace of H 1 () where {φi |i ∈ N ∗ } is an orthogonal basis of H 1 (). H N is endowed with the restriction of the inner product of H 1 ().

11.3 Numerical Approach and Simulations

281

We define the projection operator by:  N : H 1 () −→ H N N  y → y, φi φi (x)

(11.78)

i=1

Solving (11.55), turns to solving the algebraic Riccati equation: Pω N A N y N , y N  + y N , PωN A N y N  + R N y N , y N  = 0

(11.79)

where A N , PωN , and R N are respectively the projections of operators A, Pω and R. As for all y ∈ H 1 (), lim  N (y) − y = 0 then lim PωN  N y − P y = 0 N →∞

N →∞

or again PωN  N converges to Pω , (see [3]). For m ≥ 1 and tm−1 ≤ t ≤ tm , we consider the system

⎧ ∂ ym (t) ⎪ ⎪ = Aym (t) + u(tm−1 )Bym (t) ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ y (t ) = y (t ) m m−1 m−1 m−1 ⎪ ⎪ ⎪ ⎪ u(tm ) = −P N B N ymN (tm ), ymN (tm ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u(0) = −P N B N y0N , y0N 

(11.80)

where the operator (A + u N (tm−1 )B) generates a C0 -semigroup T m,N =



Tim,N

i≥0

with

⎧ ⎨ ⎩

 Tim,N (t)y = T m,N y + u(tm−1 ) T0m,N = T

t 0

m,N T m,N (t − s)BTi−1 yds

Then lim T m,N − T m  = 0, where T m is the C0 -semigroup of generator (A + N →+∞

u(tm−1 )B). So the weak solution of system (11.80) strongly converges to the weak solution of system (11.72) when N → +∞. This leads the the following algorithm. Step 1 Initialisation of data: A tolerance ε > 0, a target subregion ω, N the dimension of the projection space and y0N , the projection of the initial state. Step 2: Choice of a sequence (tm )m∈N , tm+1 = tm + h, with h > 0 small enough for numerical considerations.

282

11 Gradient Stabilization of Infinite Dimensional Bilinear Systems

Step 3 Solving (11.79) with the algorithm given in [4] gives PωN Step 4: 1. u(0) = −PωN B N y0N , y0N  Repeat. 2. u(t) = −PωN B N ymN (tm−1 ), ymN (tm−1 ) on [tm−1 , tm ]. 3. Solving (11.80) gives ymN (tm ) 4. If  ∇ω ym (tm ) ≤ , otherwise m = m + 1 and go to 2. We give illustrations through some examples. Example 11.5 On  =]0, 1[, we consider the system ⎧ ∂y ⎪ ⎪ (x, t) = 0.01 y(x, t) + 0.68y(x, t) + u(t)y(x, t) on Q ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎨ ∂ y(1, t) ∂ y(0, t) = = 0 on  ⎪ ⎪ ∂x ⎪ ∂x ⎪ ⎪ ⎪ ⎪ ⎩ y(x, 0) = x 2 (1 − x 3 )2

(11.81)

and a target subregion ω =]0, 0.4[. We consider the problem (11.50) with R = 5∇ω∗ ∇ω . The eigenfunctions of the operator A = 0.01 y(x, t) + 0.68y(x, t) are φi (x) = ai cos(iπ x), with ai =

2 for all i ≥ 0, and constitute an orthogonal basis 1 + (iπ )2

in H 1 (). We consider the space H N generated φi (x), i = 1, N , and for simulations we take N = 6. The projection of the operator A = 0.01 y(x, t) + 0.68y(x, t) is a matrix given by A6 = ((0.01 + 0.68)φi−1 , φ j−1 )1≤i, j≤6 with diag(A6 ) = (0.5813, 0.2852, −0.2083, −0.8991, −1.7874, −2.8731) Applying the previous algorithm, we obtain Pω6 = ( pi j )1≤i, j≤6 , a symmetric matrix with: ⎛

1.6725 ⎜ 0.6623 ⎜ ⎜ 0.2405 Pω6 = ⎜ ⎜ −0.1397 ⎝ −0.1499 −0.0089

0.6623 1.5564 0.5736 0.0236 −0.1218 −0.0444

0.2405 0.5736 0.8415 0.5082 0.1496 −0.0458

−0.1397 0.0236 0.5082 0.6683 0.4124 0.0814

−0.1499 −0.1218 0.1496 0.4124 0.4252 0.2482

⎞ −0.0089 −0.0444 ⎟ ⎟ −0.0458 ⎟ ⎟ 0.0814 ⎟ 0.2482 ⎠ 0.3048

and this gives the results represented by the following figures (Fig. 11.1).

11.3 Numerical Approach and Simulations

283

1

0

0.015

0.01 0

0.1

0.2

0.3

0.4

0.5 x

0.6

0.7

0.8

0.9

1 0.005

0

Flux

flux

−1

−2

0

0.1

0.2

0.3

0.4

0.5 x

0.6

0.7

0.8

0.9

1

−0.005 −3 −0.01 −4

t=1 t=2.5 t=4

−0.015 t=5

−5

−0.02

Fig. 11.1 Evolution du gradient Fig. 11.2 Evolution of control

0

−0.1

−0.2

v(t)

−0.3

−0.4

−0.5

−0.6

−0.7

−0.8

0

0.5

1

1.5

2

2.5

3

3.5

4

t

The above figures show that the system (11.81) is G-stabilizable on ω =]0, 0.4[ with error of 5.9678 10−7 and cost equals to 3.1 10−4 . The next figure presents the evolution of the stabilizing control (Fig. 11.2). We study the evolution of cost and error G-stabilization with respect to the area of the subregion target ω. We obtain the following array. ω Error ×106 Cost ×102

]0,0.3[ 0.0059 5.43

]0,0.4[ 3.8522 5.83

]0,0.65[ 3.5058 6.58

]0,0.7[ 4.7438 6.65

]0,0.9[ 26.048 6.79

]0,1[ 27.163 6.8

If we fix ω =]0.6, 1[, the error and cost of the G-stabilization on ω are respectively 1.6037 10−4 and 2.83 10−2 . The following figures show that at t = 4.5, the system (11.81) is G-stabilizable on ω (Fig. 11.3). The following figure shows that the evolution of the G-stabilizing control (Fig. 11.4).

284

11 Gradient Stabilization of Infinite Dimensional Bilinear Systems

Fig. 11.3 Evolution of the gradient

0.35 t=4.5 t=1.5 t=0.5

0.3 0.25

Flux

0.2 0.15 0.1 0.05 0

0

0.1

0.2

0.5

1

0.3

0.4

0.5 x

0.6

0.7

0.8

0.9

1

−0.05

Fig. 11.4 Evolution of control

0 −0.5 −1

v(t)

−1.5 −2 −2.5 −3 −3.5 −4

0

1.5

2

2.5

3

3.5

4

4.5

t

Example 11.6 On  =]0, 1[, we consider the system ⎧ ∂y ⎪ ⎪ (x, t) = 0.01 y(x, t) + 0.5y(x, t) + 0.7u(t)y(x, t) on Q ⎪ ⎪ ⎪ ⎨ ∂t y(0, t) = y(1, t) = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ y(x, 0) = 0.5x(1 − x) on 

(11.82)

We take ω =]0, 0.4[ and we consider the problem (11.50) with R = 3∇ω∗ ∇ω . For N = 5, A5 is the diagonal matrix projection of A = 0.01 + 0.5I d, given by diag(A5 ) = (0.4013, 0.10521, −0.3882, −1.07913, −1.9674) and

11.3 Numerical Approach and Simulations

285 0.005

0.15 t=3 t=4 t=5

0

0.1

−0.005

0.05

−0.015

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

t

v(t)

Flux

−0.01

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

−0.02 −0.025

1

x

−0.03

−0.05

−0.035

−0.1

−0.045

t=3 t=4 t=5 14

−0.04

Fig. 11.5 Evolution of the gradient Fig. 11.6 Evolution of the gradient on ω =]0, 0.4[

0.005 0

0

0.05

0.1

0.15

0.2 t

−0.005

0.25

0.3

0.35

0.4

−0.01

v(t)

−0.015 −0.02 −0.025 −0.03 t=3 t=4 t=5 14

−0.035 −0.04 −0.045

⎛

1.8654 ⎜ 0.4911 ⎜ Pω5 = ⎜ −0.0599 ⎝ −0.1141 −0.0101

0.4911 1.0613 0.6303 0.1827 −0.0805

−0.0599 0.63035 0.9209 0.5089 0.0623

−0.1141 0.1827 0.5089 0.5401 0.3287

⎞ −0.01013 −0.0805 ⎟ ⎟ 0.0623 ⎟ 0.3287 ⎠ 0.4292

The simulations results are given by the following figures (Fig. 11.5). The next figure indicates the gradient evolution of system (11.82) only on the target subregion ω =]0, 0.4[ (Fig. 11.6). We remark that at t = 14, the gradient evolves close to 0, on ω =]0, 0.4[, hence the system (11.82) is G-stabilizable on ω, with error equals 9.9235 10−4 and cost equals to 12.39 10−2 (Fig. 11.7). The following shows the relation between the area of the target subregion, the cost and the error of the G-stabilization. ω Error ×104 Cost ×102

]0,0.1[ 8.2691 11.9

]0,0.3[ 9.5172 12.22

]0,0.5[ 9.9658 13.01

]0,0.8[ 9.9878 13.5

]0,1[ 9.9954 14.29

286

11 Gradient Stabilization of Infinite Dimensional Bilinear Systems

Fig. 11.7 Evolution of control

0

−0.02

−0.04

−0.06

−0.08

−0.1

−0.12

0

4

2

6

8

10

12

14

11.3.2 Gradient Stabilization Case We have seen that control (11.4) stabilizes exponentially the gradient of system (11.1) and control (11.22) ensures strong and weak stabilization. The gradient stabilization of system (11.1) may follow the steps: Step 1: Initial data: threshold accuracy  > 0 and initial condition y0 ; Step 2: Apply u(ti ) = −∇ By(ti ), ∇ y(ti )n ; Step 3: Solving system (11.1) gives y(ti+1 ); Step 4: Calculate ∇ y(ti+1 ); Step 5: If ∇ y(ti+1 )n <  stop, else ; Step 6: i = i + 1 and go to 2. Example 11.7 On  =]0, 1[, we consider the system ⎧ ∂ y(x, t) ∂ 2 y(x, t) ⎪ ⎨ = + y(x, t) + u(t)By(x, t), ×]0, +∞[ ∂t ∂x2  ⎪ ⎩ y(x, 0) = x(1 − x), y(0, t) = y(1, t) = 0, ]0, +∞[ +∞  1 where By = y, ϕ j ϕ j , with ϕ j (x) = 2 j j=1

∀ j ∈ N∗ .

 2 sin( jπ x), 1 + ( jπ )2

(11.83)

11.3 Numerical Approach and Simulations

287

Fig. 11.8 Gradient evolution

Fig. 11.9 State evolution

Applying the above algorithm and solving (11.83) using Petrov-Galerkin method given in (Skeel and Berzins 1990), Fig. 11.8 shows that the gradient of system (11.83) is weakly stabilized at time t = 8 by control (11.22), with a stabilization error equals to 4.73 × 10−5 and Fig. 11.9 shows that the state of the system is stabilized at time t = 10 with stabilization error equals to 8.69 × 10−5 (Fig. 11.10). We remark that the state of the system is stabilized in time greater than the one needed for the stablization of the gradient of the system.

288

11 Gradient Stabilization of Infinite Dimensional Bilinear Systems

Fig. 11.10 Control evolution

References 1. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983) 2. Ball, J.M.: On the asymptotic behavior of generalized processes with application to nonlinear evolutions equations. J. Differ. Equ. 27, 224–265 (1978) 3. Banks, H., Kunisch, B.: The linear regulator problem for parabolic systems. SIAM J. Control Optim. 22, 684–696 (1984) 4. Arnold, W.F., Laub, A.J.: Generalized eigenproblem: algorithms and software for algebraic Riccati equations. Proc. IEEE 72, 1746–1754 (1984)

Chapter 12

Regional Gradient Stabilization of Infinite Dimensional Semilinear Systems

This chapter deals with regional gradient stabilization of a semilinear system defined on a bounded and regular spatial domain  ⊂ Rn by dy(t) = Ay(t) + u(t)By(t), dt

y(0) = y0

(12.1)

where the state space is considered to be complex-valued Hilbert space H 1 () endowed with its usual inner product, denoted by ., . and . the associated norm, A is a linear operator with domain D(A) ⊂ H 1 () and generates a semigroup of contractions S(t), u(t) is a scalar valued control and B is a nonlinear and bounded operator mapping H 1 () into itself and verifies B(0) = 0 (0 is an equilibrium state of system (12.1)).

12.1 Controls for Regional Gradient Stabilization In this section we study regional strong and weak gradient stabilization for distributed semilinear system (12.1) by considering the class of feedback controls of the form u(t) = −

y(t), By(t) , g(y(t))

(12.2)

where g : H 1 () −→ (0, +∞) is an appropriate function. Let ω be a subregion of  of positive Lebesgue measure and we define the operator ∇ω : H 1 () −→ y

(L 2 (ω))n ∂ y(x) ∂ y(x) ∂ y(x) −→ (χω , χω , ..., χω ) ∂ x1 ∂ x2 ∂ xn

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. H. Zerrik and O. Castillo, Stabilization of Infinite Dimensional Systems, Studies in Systems, Decision and Control 355, https://doi.org/10.1007/978-3-030-68600-0_12

289

290

12 Regional Gradient Stabilization …

where χω is the restriction operator to ω, defined by χω : L 2 () −→ L 2 (ω) y −→ y/ω

12.1.1 Regional Gradient Strong Stabilization In this subsection, we develop sufficient conditions for regional gradient strong stabilization of system (12.1). Theorem 12.1 Assume that g is locally Lipschitz such that g(.) ≥ m > 0 on H 1 (), B is locally Lipschitz and  0

T

|S(s)y, B S(s)y|ds ≥ δ∇ω y2(L 2 (ω))n , ∀y ∈ H 1 (), (T, δ > 0).

(12.3)

Then the control (12.2) regionally strongly stabilizes the gradient of system (12.1) on ω. y, By By is locally Lipshitz. g(y) 1 Let x, y, z ∈ H () and R > 0 such that x − y, x − z ≤ R. Without loss of generality, we can take x = 0 and we have

Proof Let us first show that the operator N (y) =

g(y) − g(z) ≤ L R (g)y − z and By − Bz ≤ L R (B)y − z Thus N z − N y = ≤

g(y)z, BzBz − g(z)y, ByBy g(z)g(y) 1 (g(y)(z, BzBz − y, ByBy) m2

+ (g(y) − g(z))y, ByBy) ≤

1 (g(y)z, Bz(Bz − By) + g(y)(z, Bz − y, By)By m2

+ (g(y) − g(z))y, ByBy) ≤

K R (g)R 2 L 2R (B) (3 + R)z − y. m2

12.1 Controls for Regional Gradient Stabilization

291

where K R (g) = sup(M R (g), L R (g)), M R (g) = sup g(x) and L R (g) = x≤R

|g(y) − g(z)| which implies that N is locally Lipschitz and y − z L R (N ) :=

≤

sup (y,z)∈B R2 ,y=z

|N (y) − N (z)| y − z (y,z)∈B R2 ,y=z sup

K R (g)R 2 L 2R (B) (3 + R). m2

Since N is locally Lipschitz and N y, y ≥ 0, ∀y ∈ H 1 (), system (12.1) has a unique global mild solution y(t). Since S(t) is a semigroup of contractions (so that A is dissipative), then d y(t)2 ≤ −2N y(t), y(t), ∀t ≥ 0. dt Integrating this last inequality, we get  y(t) − y(τ ) ≤ −2 2

2

τ

t

N y(s), y(s)ds, ∀t ≥ τ ≥ 0.

(12.4)

It follows that y(t) ≤ y0 , ∀t ≥ 0.

(12.5)

It remains to show that ∇ω y(t) −→ 0, as t −→ +∞ . From the relation N S(t)y0 , S(t)y0  = N S(t)y0 − N y(t), S(t)y0  − N y(t), (t) +N y(t), y(t) we obtain the following inequality |N S(t)y0 , S(t)y0 | ≤ L y0  (N )(t)(S(t)y0  + y(t)) + |N y(t), y(t)|, ∀t ∈ [0, T ]. 

t

where (t) =

S(t − s)N y(s)ds.

0

Moreover, since S(t) is a semigroup of contractions, then, for a fixed T > 0, Schwartz’s inequality yields √

 T  21 T By0  |y(s), By(s)|2 ds . (t) ≤ m 0

292

12 Regional Gradient Stabilization …

Furthermore, we have |N S(t)y0 , S(t)y0 | ≤ 2y0 L y0  (N )(t) + |N y(t), y(t)|, ∀t ∈ [0, T ]. Then, we deduce the following estimate 

T

3

|N S(s)y0 , S(s)y0 |ds ≤

0

×

2T 2 By0 2 L y0  (N ) m 

T

|y(s), By(s)|2 ds

 21

0

+

1 m



T

|y(s), By(s)|2 ds.

0

By remarking that |N S(s)y0 , S(s)y0 | ≥

|S(s)y0 , B S(s)y0 | , My0  (g)

we obtain 

T

3

|S(s)y0 , B S(s)y0 |ds ≤ My0  (g)

0

×



T

2T 2 By0 2 L y0  (N ) m

|y(s), By(s)|2 ds

 21

0

My0  (g) + m



T

|y(s), By(s)|2 ds.

0

Then, replacing y0 by y(t) and using (12.5), we get 

T

|S(s)y(t), B S(s)y(t)|ds ≤ K y0 

0



t+T t

My0  (g) + m

|y(s), By(s)|2 ds

t+T

(12.6) |y(s), By(s)| ds 2

t

3

2T 2 By0 2 L y0  (N ). where K y0  = m Setting τ = 0 in (12.4), we obtain 

t

2 0

 21

N y(s), y(s)ds ≤ y0 2 − y(t)2 , ∀t ≥ 0.

12.1 Controls for Regional Gradient Stabilization



then

t

2

293

N y(s), y(s)ds ≤ y0 2 , ∀t ≥ 0.

0

Since

|y(s), By(s)|2 |y(s), By(s)|2 ≤ , we get My0  (g) g(y(s)) 

t

|y(s), By(s)|2 ds ≤

0

which implies that



+∞

My0  (g) y0 2 , ∀t ≥ 0. 2

|y(s), By(s)|2 ds < +∞.

0

Hence



t+T

|y(s), By(s)|2 ds −→ 0, as t −→ +∞.

t

It follows from (12.6) that 

T

|S(s)y(t), B S(s)y(t)|ds −→ 0, as t −→ +∞.

(12.7)

0

From (12.3) and (12.7), we deduce that ∇ω y(t) −→ 0, as t −→ +∞. The following Corollary is a direct consequence of Theorem 12.1.



Corollary 12.1 Assume that B is locally Lipschitz and (12.3) holds. Then both controls y(t), By(t) u 1 (t) = − (12.8) 1 + |y(t), By(t)| and u 2 (t) = −

y(t), By(t) , sup(1, |y(t), By(t)|)

(12.9)

regionally strongly stabilize the gradient of system (12.1) on ω. Example 12.1 Let  =]0, 1[, we consider the following beam equation ⎧ 2 ∂ y ∂y ∂4 y ⎪ ⎪ (x, t) = − (x, t) + u(t)i ω (x, t) in ×]0, +∞[ ⎪ ⎨ ∂t 2 ∂x4 ∂t ⎪ ⎪ ⎪ ⎩

∂2 y y(ξ, t) = (ξ, t) = 0, ξ = 0, 1 on ]0, +∞[ ∂x2

where i ω = χω∗ χω , with χω is the restriction operator to ω.

(12.10)

294

12 Regional Gradient Stabilization …

Let P =

∂4 y ∂4 y 2 with domain D(P) = {y ∈ L (0, 1)/ ∈ L 2 (0, 1), y(ξ, t) = ∂x4 ∂x4

∂2 y (ξ, t) = 0, ξ = 0, 1}. ∂x2 1 The set D(P 2 ) forms a Hilbert space under the inner product y1 , y2  P = 1 1 P 2 y1 , P 2 y2  L 2 (0,1)√. The eigenvalues of P are λ j = ( jπ )4 , corresponding to eigenfunctions ϕ j (x) = 2 sin( jπ x) ∀ j ∈ N∗ . To write system (12.10) in the form (12.1) we set X = (H 2 (0, 1) ∩ H01 (0, 1)) × 2 L (0, 1) endowed with the inner product (y1 , z 1 ), (y2 , z 2 ) = y1 , y2  P + z 1 , z 2  L 2 (0,1)



0 I 0 0 . A= and B = −P 0 0 iω The operator A is skew-adjoint, and the operator B satisfies the condition (12.3), indeed, for y ∈ X , we have

∞  αj 1 y= ϕj λ j2 β j j=1 where (α j , β j ) ∈ R2 ∀ j ≥ 1. Then ⎛ ∞  ⎜ S(s)y = ⎝ j=1

1

1

α j cos(λ j2 s) + β j sin(λ j2 s) 1 2

1 2

1 2

1 2

⎞ ⎟ ⎠ ϕ j , ∀s ≥ 0.

β j λ j cos(λ j s) − α j λ j sin(λ j s)

It follows that S(s)y, B S(s)y =

∞ 

1 1   λ j α j sin(λ j2 s) − β j cos(λ j2 s)

j=1

=

∞ 

  λ j α 2j sin2 ( jπ s) − β 2j cos2 ( jπ s) − sin(2 jπ s)β j α j .

j=1

Integrating this relation over the time interval [0, 2], we obtain 

2 0

|S(s)y, B S(s)y|ds =

∞ 

λ j (α 2j + β 2j )

j=1

≥ δ∇ω y2(L 2 (ω))n −1

with δ = min{λ j 2 | j ∈ N∗ }, so that (12.3) holds.

12.1 Controls for Regional Gradient Stabilization

295

We conclude that the feedbacks u 1 (t) = −

χω ∂t y(., t)2 sup(1, χω ∂t y(., t)2 )

and u 2 (t) = −

χω ∂t y(., t)2 1 + χω ∂t y(., t)2

regionally strongly stabilize the gradient of system (12.10).



Remark 12.1 Note that this example has also been considered in [1] where a weak stabilization result using quadratic feedback has been established. To prove our next result, we will need the following lemma. Lemma 12.1 ([2]) Let p be a positive increasing function such that p(0) = 0 and set p(t) ˜ = t − (I + p)−1 (t), where I denotes the identity function. Let (z k )k≥0 be a sequence of positive numbers such that p(z k+1 ) + z k+1 ≤ z k , k ≥ 0 then z k ≤ S(k)z 0 where S(t)z 0 is the solution of  d x(t)

= − p(x(t)) ˜ dt x(0) = x0 ≥ 0.

(12.11)

Let G ω = ∇ω∗ ∇ω be a bounded operator applying H 1 () into itself, where ∇ω∗ denotes the adjoint operator of ∇ω . The following theorem characterizes the asymptotic behavior of the gradient and gives sufficient conditions for regional strong stabilization of the gradient on ω with explicit decay rate. Theorem 12.2 Assume that B is locally Lipschitz and satisfies the following inequality (12.12) y, By2 ≤ δy, Byy, G ω By, ( f or some δ > 0) moreover, if the following assumption holds  ∇ω y2(L 2 (ω))n

T

≤

|S(s)y, B S(s)y|ds, ( f or some T > 0).

(12.13)

0

Then the control u(t) = −y(t), By(t) regionally strongly stabilizes the gradient of system (12.1) on ω, and for all y0 we have the following decay estimate 1 ∇ω y(t)2(L 2 (ω))n = O( ) as t −→ +∞. t

296

12 Regional Gradient Stabilization …

Proof For all y0 ∈ H 1 (), the system (12.1) has a unique mild solution y(.) defined on a maximal interval [0, tmax [ (see [1]), given by the variation of constants formula 

t

y(t) = S(t)y0 +

S(t − s)v(s)By(s)ds.

0

Since S(t) is a semigroup of contractions (so that A is dissipative), then a standard approximation argument shows that 

t

y(t)2 ≤ y0 2 − 2

y(s), By(s)2 ds, ∀t ∈ [0, tmax [

(12.14)

y(t) ≤ y0 .

(12.15)

0

so for all y0 ∈ H 1 (), we have

Hence for all y0 ∈ H 1 (), the solution y(t) is global. For y0 ∈ H 1 (), we define the function z(t) = y(t) − S(t)y0 ∀t ≥ 0. Once more using the variation of constants formula, we have 

t

y(t) = −

y(s), By(s)S(t − s)By(s)ds, ∀t ≥ 0.

0

Using Schwartz’s inequality and the fact that S(t) is a semigroup of contractions (so that S(s) ≤ 1, ∀s ≥ 0), we get √  y(t) ≤ By0  T

T

|y(s), By(s)|2 ds

 21

, ∀t ∈ [0, T ]

0

Then

 y(t) ≤ C1 2

T

|y(s), By(s)|2 ds, ∀t ∈ [0, T ]

(12.16)

0

where C1 = B2 y0 2 T . From the relation |S(t)y0 , B S(t)y0 | ≤ |y(t) − z(t), B S(t)y0 − By(t)| + |y(t), By(t)| + |z(t), By(t) − Bz(t)| + |z(t), Bz(t)| and using (12.15) and the fact that B is locally Lipschitz, we deduce that

12.1 Controls for Regional Gradient Stabilization

297

|S(t)y0 , B S(t)y0 | ≤ k1 S(t)y0 − y(t)y(t) − z(t) + |y(t), By(t)| + k2 y(t) − z(t)z(t) + αz(t)2 ≤ 2kz(t) + |y(t), By(t)| + αz(t)2 with k = max{k1 , k2 }. From (12.16), for all t ∈ [0, T ] we have √ |S(t)y0 , B S(t)y0 | ≤ 2k C1 C2 + |y(t), By(t)| + αC1 C2  with C2 =

T

|y(s), By(s)|2 ds.

0

Integrating this last inequality over the interval [0, T ] and using (12.13) we obtain  ∇ω y(T )2(L 2 (ω))n ≤ C3 ( C2 + C2 ) √ where C3 = max{(2k C1 T + 1), αT C1 }. It follows from (12.12) that  ∇ω y(t)2(L 2 (ω))n ≤ ∇ω y0 2(L 2 (ω))n − 2δ

t

y(s), By(s)2 ds, ∀t ∈ [0, T ]

0

(12.17)

thus 2δC2 ≤ ∇ω y0 2(L 2 (ω))n − ∇ω y(T )2(L 2 (ω))n , hence 1 ∇ω y(T )2(L 2 (ω))n ≤ √ C3 h(∇ω y0 2(L 2 (ω))n − ∇ω y(T )2(L 2 (ω))n ) 2δ where h(t) = t +

(12.18)

√ t, ∀t ≥ 0.

1 Let p0 (.) denotes the inverse of √ C3 h(.), then (12.18) can be written 2δ p0 (∇ω y(T )2(L 2 (ω))n ) + ∇ω y(T )2(L 2 (ω))n ≤ ∇ω y0 2(L 2 (ω))n .

On the other hand, From (12.17), we obtain ∇ω y(t)2(L 2 (ω))n ≤ ∇ω y0 2(L 2 (ω))n ∀t ∈ [0, T ], instead of the interval [0, T ] we could just as well work on the interval [kT, (k + 1)T ], k ∈ N∗ , thus ∇ω y(t)2(L 2 (ω))n ≤ ∇ω y0 2(L 2 (ω))n ∀t ∈ [kT, (k + 1)T ], it follows that ∇ω y(kT )2(L 2 (ω))n ≤ ∇ω y0 2(L 2 (ω))n , then p0 (W ((k + 1)T )) + W ((k + 1)T ) ≤ W (kT ), ∀k ∈ N∗ with W (t) = ∇ω y(t)2(L 2 (ω))n . Let z k = W (kT ) and applying Lemma 12.1, we obtain

298

12 Regional Gradient Stabilization …

W (kT ) ≤ S(k)W (0), k ∈ N For any t > 0 we have t = kT + τ for some integer k ≥ 0 and τ ∈ [0, T [. Since W (t) and S(t)z 0 are nonincreasing, we have W (t) ≤ W (kT ) ≤ S( t−τ )W (0) T ≤ S( Tt − 1)W (0) On the other hand, the function p(t) ˜ = t − (I + p)−1 (t) satisfies p˜ 0 (t + p0 (t)) = p0 (t), so p˜ 0 (t) has the same asymptotic behavior as p0 (t) as t −→ 0. The function  ξ ds , t > 0, g(t) = p ˜ 0 (s) t is decreasing with g(ξ ) = 0 and g(0+ ) = +∞. Thus [0, +∞[ is in the range of g and the solution of (12.11) is given by z(t) = g −1 (t). Since g(0+ ) = +∞, lim z(t) = lim g −1 (t) = 0. t→+∞

η .

t→+∞

For 0 <  < 1, there exists η > 0 such that | p˜ 0 (s) − C3 s 2 | ≤ C3 s 2 for 0 < s < Moreover, there exists t > 0 such that 0 < z(t) < η for t ≥ t . Therefore, if t ≥ t , we have − p˜ 0 (z(t)) ≤ C3 ( − 1)z(t))2 . Then, dz(t) + C3 (1 − )(z(t))2 ≤ 0, t ≥ t . dt

1 1 It follows that z(t) = O( ) as t −→ ∞, hence ∇ω y(t)2(L 2 (ω))n = O( ) as t −→ t t +∞, which completes the proof. 

12.1.2 Regional Gradient Weak Stabilization In this subsection we give sufficient conditions for weak regional gradient stabilization of system (12.1). Theorem 12.3 Assume that B is Locally Lipschitz and weakly sequentially continuous, g is Locally Lipschitz such that g(.) ≥ m > 0 on H 1 () and S(t), B S(t) = 0, ∀t ≥ 0 =⇒ ∇ω  = 0.

(12.19)

Then the control (12.2) regionally weakly stabilizes the gradient of (12.1) on ω.

12.1 Controls for Regional Gradient Stabilization

299

Proof Let ψ ∈ H 1 () be fixed, we shall show that ∇ω y(t), ψ −→ 0, as t −→ +∞. Let {tn } be a sequence of positive numbers such that tn −→ +∞, as n −→ +∞. The sequence xn = ∇ω y(tn ), ψ is, by virtue of (12.5) bounded. Now, let xγ (n) be an arbitrary convergent subsequence of xn . Since y(tγ (n) ) is bounded in H 1 (), we can extract a subsequence also denoted y(tγ (n) ) such that y(tγ (n) )   ∈ H 1 (), as n −→ +∞. By similar arguments using in the proof of Theorem 12.1, we obtain the following inequality  T |S(s)y(tγ (n) ), B S(s)y(tγ (n) )|ds 0   tγ (n) +T  21 |y(s), By(s)|2 ds ≤ K y0  tγ (n)  My0  (g) tγ (n) +T + |y(s), By(s)|2 ds m tγ (n) It follows that  T

|S(s)y(tγ (n) ), B S(s)y(tγ (n) )|ds −→ 0, as n −→ +∞.

0

Using the dominated convergence theorem and the fact that B is weakly sequentially continuous, we deduce that S(s), B S(s) = 0 for all t ≥ 0, which implies, by virtue of (12.19), that ∇ω  = 0. Then ∇ω y(tγ (n) )  0. It follows that xn −→ 0, as t −→ +∞. We deduce that ∇ω y(t), ψ −→ 0, as t −→ +∞. In other words ∇ω y(t) converges weakly to 0 as t −→ +∞, which proves the theorem.  The next result is a direct consequence of the above theorem. Corollary 12.2 Assume that B is locally Lipschitz and (12.19) holds. Then both the control (12.8) and the control (12.9) regionally weakly stabilize the gradient of system (12.1) on ω. The following example illustrates the established results. Example 12.2 . Let  be an open regular subset of Rn such that ω ∈  and consider the following Schrodinger ¨ equation  ∂ y(x, t) = y(x, t) + u(t)By(x, t), ×]0, ∞[ i ∂t y(., t) = 0 ∂×]0, ∞[

(12.20)

300

12 Regional Gradient Stabilization … +∞

where the operator B is given by By = H 1 (), α j > 0, ∀ j ≥ 1,

+∞ 

 1 α j y, ϕ j |ω ϕ j |ω , where φ ∈ 1 + |y, φ| j=1

α 2j < ∞ and (ϕ j ) is an orthonormal basis of the state

j=1

space H 1 () endowed with its natural complex inner product. It is well known that the operator A = −i with domain D(A) = H 2 () ∩ 1 H0 () generates a semigroup of isometries (S(t)y = y). Also, it is easy to see that B is weakly sequentially continuous and locally Lipschitz and B(0) = 0, ( 0 is an equilibrium state for system(12.20)). For ω ⊂ , the condition (12.19) holds. Then the gradient of system (12.20) is regionally weakly stabilizable on the region ω by the controls +∞ 

u 1 (t) =

−1 × 2 j=1 |α j y(., t), ϕ j |ω | ) 1 + |y, φ|

+∞ sup(1,

and

j=1

+∞ 

u 2 (t) =

−1 × 2 j=1 |α j y(., t), ϕ j |ω |

+∞ 1+

|α j y(., t), ϕ j |ω |2 1 + |y, φ|

|α j y(., t), ϕ j |ω |2

j=1

1 + |y, φ|

.

1 + |y, φ| 

Now if we consider the control law v(t) = −y(t), K y(t)

(12.21)

where K : H 1 () −→ H 1 () is a continuous operator such that the following assumption holds (12.22) y, K yy, By ≥ 0, ∀y ∈ H 1 () then we have the following result. Proposition 12.1 1. Suppose that there exists η > 0 such that ∀y ∈ D(A) G ω Ay, y − y, Byy, G ω By ≥ η(Ay, y − y, By2 ), wher e G ω = ∇ω∗ ∇ω

If the control (12.21) regionally weakly stabilizes the gradient of system (12.1) on ω, then the corresponding state remains bounded on the whole domain .

12.1 Controls for Regional Gradient Stabilization

301

2. Suppose that B is locally Lipschitz and sequentially continuous. If (12.22) holds and S(t)ψ, K S(t)ψS(t)ψ, B S(t)ψ = 0, t ≥ 0 =⇒ ∇ω ψ = 0.

(12.23)

Then the control (12.21) regionally weakly stabilizes the gradient of (12.1) on ω, and the state remains bounded on . Proof 1. For all y0 ∈ D(A) we have d d ∇ω y(t)2(L 2 (ω))n ≥ η y(t)2 dt dt which implies that ∇ω y(t)2(L 2 (ω))n − ∇ω y0 2(L 2 (ω))n ≥ ηy(t)2 − ηy0 2 , t ≥ 0. This inequality holds for all initial states, as its terms are continuous in y0 ∈ H 1 (). The regional weak gradient stabilization of (12.1) on ω implies that ∇ω y(t)(L 2 (ω))n is bounded in t, then the above inequality ensures the boundedness of y(t). 2. First, we note that from the work of Ball and Slemrod [1], system (12.1) has a unique global solution y(t), which is bounded in t and there exists ψ ∈ H 1 () such that y(t)  ψ as t −→ +∞ and that S(t)ψ, B S(t)ψ = 0, t ≥ 0 From (12.23), it follows that ∇ω ψ = 0, and as the operator ∇ω is continuous, then ∀ϕ ∈ H 1 (), ∇ω y(t), ϕ −→ ∇ω ψ, ϕ, as t −→ +∞ which achieves the proof.



12.2 Regional Gradient Stabilization Problem The purpose of this section is to give the minimum energy control that achieves regional gradient stabilization of system (12.1) on a subregion ω ⊂  which may be formulated as ⎧  +∞  +∞ ⎪ ⎪ ⎪ min J (u) = y(t), Ry(t)dt + |y(t), Pω By(t)|2 dt ⎪ ⎨ 0 0 +∞ (12.24) + |u(t)|2 dt ⎪ ⎪ ⎪ 0 ⎪ ⎩ u ∈ Uad = {u|y(t) is a global solution and J (u) < +∞}.

302

12 Regional Gradient Stabilization …

where R ∈ L(H 1 ()) is a self-adjoint and positive operator and P is a self-adjoint and positive operator verifying the Lyapunov equation Pω Ay, y + y, Pω Ay + y, Ry = 0, y ∈ D(A)

(12.25)

with Pω = G ω P G ω and G ω = ∇ω∗ ∇ω . Proposition 12.2 Suppose that system (12.1) has a unique mild global solution y ∗ (t) which satisfies Pω y ∗ (t), y ∗ (t) −→ 0 as t −→ ∞. Then the control

u ∗ (t) = −y ∗ (t), Pω By ∗ (t)

(12.26)

is the unique control solution of problem (12.24). Proof Let us define the function F(y) = Pω y, y, y ∈ H 1 (). For all y0 ∈ D(A) and t ≥ 0, we have d F(y ∗ (t)) = −y ∗ (t), Ry ∗ (t) − 2y ∗ (t), Pω By ∗ (t)2 dt Integrating this relation, we obtain 

t

∗

∗



y (s), Ry (s)ds + 2

0

t

y ∗ (s), Pω By ∗ (s)2 ds ≤ F(y0 ), t ≥ 0

(12.27)

0

The solution y(.) is continuous with respect to the initial condition (see [3]), and since F(y) and y, Ry are continuous then (12.27) holds for all y0 ∈ H 1 (). It follows that J (u ∗ ) is finite for all initial condition y0 ∈ H 1 () and Uad = ∅. Let u ∈ Uad , for y0 ∈ D(A), integrating the relation d F(y(t)) = [y(t), Pω By(t) + u(t)]2 − y(t), Ry(t) − y(t), Pω By(t)2 − u 2 (t) dt



we have

+∞

J (u) = F(y0 ) +

[y(t), Pω By(t) + u(t)]2 dt

0

then J (u) ≥ F(y0 ), ∀u ∈ Uad . Setting u = u ∗ , we obtain J (u ∗ ) = F(y0 ). Let y0 ∈ H 1 (), and a sequence (y0n )n ⊂ D(A) such that y0n −→ y0 as n −→ +∞. we have

12.2 Regional Gradient Stabilization Problem

 J (u) = F(y0n ) +

+∞

303

[yn (t), Pω Byn (t) + v(t)]2 dt

0

thus J (u) ≥ F(y0n ). By the continuity of F we deduce that J (u) ≥ F(y0 ) = J (u ∗ ). In other words (12.26) is the optimal control solution of the problem (12.24). The uniqueness of u ∗ follows from the uniqueness of y(t) solution of system (12.1), which completes the proof.  The following result gives sufficient conditions for the optimal control (12.26) to be a stabilizing one. Proposition 12.3 Suppose that S(t) is a semigroup of contractions and P is compact such that K = Pω B verifies K S(t)ψ, S(t)ψS(t)ψ, B S(t)ψ = 0, t ≥ 0 =⇒ ∇ω ψ = 0.

(12.28)

Then (12.26) is the unique feedback control solution of (12.24) and stabilizes regionally weakly the gradient of system (12.1) on ω. Proof From Proposition 12.1 we deduce that the control (12.26) stabilizes regionally weakly the gradient of (12.1) on ω. Now as G ω is continuous and P is supposed compact, then Pω is also compact, so for any sequence of positive real number (tn ) verifying tn −→ +∞, the sequence Pω y(tn ) tends to 0 strongly and the real sequence F(y(tn )) converges to 0. Then from the above proposition, we deduce that (12.26) is the unique solution of problem (12.24). 

12.3 Simulation Results We have shown that the gradient of system (12.1) may be stabilized on a subregion by the controls (12.8) and (12.9) or by the control (12.26) solution of problem (12.24). In order to illustrate the previous results using numerical computations, we perform the following steps: Step 1: Solve the Lyapunov equation (12.25) using Bartels-Stewart method [4]; Step 2: Apply a feedback control given by (12.8), (12.9) or (12.26); Step 3: Solve system using Petrov-Galerkin method given in [5]; Step 4: Calculate ∇ω y(ti ); Step 5: Calculate the stabilization cost J . Example 12.3 . On  =]0, 1[, we consider the question of regional gradient stabilization of the following system

304

12 Regional Gradient Stabilization …

Fig. 12.1 The gradient evolution

⎧ ∂y 1 ⎪ ⎪ ⎪ (x, t) = 0.01y(x, t) + y(x, t) + u(t)|y(x, t)| ×]0, +∞[ ⎪ ∂t 2 ⎪ ⎪ ⎪ ⎨ ∂y ∂y (0, t) = (1, t) = 0 ∂×]0, +∞[ ⎪ ⎪ ∂ x ∂ x ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ y(x, 0) = x 2 (1 − x)3

(12.29)

We consider the problem (12.24) with R = 4∇ω∗ ∇ω and the subregion ω =]0, 0.4[. 1. Optimal stabilizing control. The solution of problem (12.24) is given by 1 u(t) = − Pω |y|, y, 2

(12.30)

the Fig. 12.1 shows how the gradient of system (12.29) is stabilized by the control (12.30) on ω with a stabilization error equals to 2.758 × 10−4 and minimizes the performance cost J that equals to 3.133 × 102 (Fig. 12.2). The following table shows that there exists a relation between the area of target subregion ω, the cost and the error of gradient stabilization. ω Error Cost

]0, 0.3[ 2.163×10−5 2.626×102

]0, 0.4[ 2.758×10−5 3.133×102

]0, 0.6[ 4.567×10−5 4.14×102

]0, 0.8[ 7.326×10−5 6.22×102

]0, 1[ 9.648×10−5 7.95×102

More the area of the target region increases, more both the gradient stabilization error and the cost increase.

12.3 Simulation Results

305

Fig. 12.2 Control evolution

Fig. 12.3 The gradient evolution

2. Stabilizing control. Following steps: 2, 3 and 4 we get the Fig. 12.3 which shows that the following control  21 |y|, y , (12.31) u(t) = − sup(1, | 21 |y|, y|) stabilizes the gradient of system (12.29) on ω with a stabilization error equals to 4.201 × 10−4 and cost J equals to 1.007 × 103 (Fig. 12.4). Remark 12.2 With respect to the error and cost, the optimal control (12.30) is better than the stabilizing one (12.31).

306

12 Regional Gradient Stabilization …

Fig. 12.4 Control evolution

References 1. Ball, J.M.: On the asymptotic behavior of generalized processes with application to nonlinear evolutions equations. J. Differ. Equ. 27, 224–265 (1978) 2. Lasiecka, I., Tataru, D.: Uniform boundary stabilisation of semilinear wave equation with nonlinear boundary damping. J. Differ. Integral Equ. 6, 507–533 (1993) 3. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983) 4. Penzl, T.: Numerical solution of generalized Lyapunov equations. Adv. Comput. Math 8, 33–48 (1998) 5. Skeel, R.D., Berzins, M.: A Method for the spatial discretization of parabolic equations in one space variable. SIAM J. ScI. Stat. Comput. 11, 1–32 (1990)

Chapter 13

Conclusion and Perspectives

In this work, we have considered the problem of the stability of the stabilization of infinite dimensional systems. We first recall the main results on the global stability and stabilization. Then we explore the concept of regional stability and stabilization for various classes of infinite dimensional systems. The gradient stabilization of such systems is also discussed and various approaches are developed. stabilization of such systems and various approaches are developed. Then we study the stabilization of a class of second order systems. This work shows that many technical difficulties have to be overcome. But the richness of this subject leads to many questions still open and may be the subjects of future explorations. This is the special case of the few problems posed below. 1. Exponential stabilization of a class of second order systems Let us consider the following system ⎧ ⎨ ytt (t) = Ay(t) + v(t)By(t), ⎩

(13.1) y(0) = y0 , yt (0) = y1 ,

where A is a dissipative operator defined on a Hilbert space X endowed with inner product ., . and the corresponding norm |.|, B is a linear and bounded operator on X and v ∈ L ∞ (0, +∞). The question is to study the exponential stabilization of (13.1), This problem could be resolved using a control verifying the dissipativity energy inequality and the observability assumption. 2. Boundary stabilization of infinite dimensional semilinear systems Let  ⊂ Rn be an open and bounded domain, we consider the following semilinear system © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. H. Zerrik and O. Castillo, Stabilization of Infinite Dimensional Systems, Studies in Systems, Decision and Control 355, https://doi.org/10.1007/978-3-030-68600-0_13

307

308

13 Conclusion and Perspectives

⎧ dy(t) ⎪ ⎪ = Ay(t) + g(y(t)) ⎪ ⎪ ⎪ ⎨ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

y = f (v(t), y(t))

(13.2)

y(0) = y0 ,

where A generates a semigroup of contraction S(t) on a Hilbert space X , v(t) is a real valued control, g is a nonlinear function on X and f is a linear function. The aim is to characterize the controls v(t) which stabilize exponentially, strongly or weakly the system (13.2). 3. On the stabilization of the gradient of linear systems and strategic regions We can study the notion of stabilization of the gradient through the admissibility notion. On  an open regular subset of Rn , let a system described by the equation ⎧ ∂y ⎪ ⎨ (t) = Ay(t) ∂t ⎪ ⎩ y(0, x) = y0 (x) ∈ X

(13.3)

where A generates a semi group S(t) on X = L 2 (). The system is augmented with the output z(t) = ∇ω y(t). Let ω ⊂  be a target subregion. and we recall the definition of the operator gradient ∇ω : X −→ (L 2 (ω))n y −→ (χω

∂ y1 ∂ y2 ∂ yn , χω , · · · , χω ) ∂t ∂t ∂t

The space (L 2 ())n is endowed with its usual inner product ., .n , and (L 2 (ω))n is endowed with the restriction of ., .n . To make sense to the output z(t), we suppose that D(A) is endowed with the graph norm ||.|| A . We can consider the following definitions. A region ω is said admissible for S(t) if (1) ∇ω : (D(A), ||.|| A ) −→ (L 2 (ω))n is continue. (2) there exists α > 0 such that ||z(.)||L 2 (0,∞;(L 2 (ω))n ) = ||∇ω S(.)y0 ||L 2 (0,∞;L 2 (ω)) ≤ α||y0 || , y0 ∈ D(A)

(13.4)

The question that remains is to characterize the stability of the output z(t) on admissible regions. Note that ω is said to be admissible, if the operator ∇ω is admissible in the meaning of Weiss (1989). This allows to extend the application y0 −→ ∇ω S(.)y0 in a

13 Conclusion and Perspectives

309

continuous map from X to (L 2 (0, ∞))n , thus the output z(t) is well defined for all y0 ∈ X . The notion of admissibility allows to work in a more general functional framework, (X = L 2 ()), consequently the existence of ∇ y will be discussed only on ω. 4. Exponential stabilization and stabilization with bounded controls of the gradient of bilinear systems. On  we consider a bilinear system described by the equation ⎧ dy ⎪ ⎨ = Ay(t) + v(t)By(t) dt ⎪ ⎩ y(0) = y0

(13.5)

Can we characterize the control v(t)? which exponentially stabilizes the gradient of the system (13.5)? We can also consider the stabilization of the gradient of the system (13.5) and minimizing a functional cost with bounded controls, (|v(t)| ≤ M for M > 0.) That is to consider the questions 1. Under what condition the system (13.5) with control u(t) admits a global and unique weak solution? 2. Characterize the stabilizing control at different degrees, the gradient of the system (13.5). 5. Regional detectability of the gradient. We consider the system (13.5) augmented with the output equation z(t) = C y(t)

(13.6)

where C ∈ L(X, O), with O the observation space supposed to be Hilbert space. We recall that the system (13.5), is said to be detectable, if there exists an operator F ∈ L(X, O) such that the operator A + FC generates a C0 -semigroup exponentially stable. It is said G-detectable on ω if there exists an operator F ∈ L(X, O) such that the system ⎧ ∂y ⎪ ⎨ (t) = (A + FC)y(t) ∂t (13.7) ⎪ ⎩ y(0, x) = y0 (x) be exponentially G-stable on ω. Can we then characterize the detectability of the gradient? Relation of Stability with Inputs and Outputs Structures Interesting work consists of taking up various notions of analysis of infinite dimensional systems and connecting them to actuators and sensors structure.

310

13 Conclusion and Perspectives

Several results would be interesting to explore, among others, the characterization of stability through the choice of the number and the spatial distribution of the actuators. It would also be interesting to take up the various criteria defining the cost of stabilization and to study their dependence with the number of actuators; whether for linear or non-linear systems or in internal and boundary cases. 6. Stabilization of a bilinear wave equation using distributed control We consider the wave equation in a smooth bounded domain  of Rn described by the equation ⎧ ⎪ ⎨ ytt (x, t) − y(x, t) + u(x, t)yt (x, t) = 0 Q = ×]0, +∞[ y(x, t) = 0 ∂×]0, +∞[ ⎪ ⎩ , y(x, 0) = y0 , yt (x, t) = y1 (x)

(13.8)

where u ∈ L ∞ (Q) are distributed controls. The goal is to: 1. characterize a control u and give sufficient conditions for weak, strong and exponential stabilisation of system (13.8), 2. study the strong stabilization of system (13.8) using a minimization problem  +∞  ⎧ ⎨min J (u) = (y(t), yt (t)2 dt + u 2 (x, t)d xdt 0 Q ⎩ u ∈ Uad := {u ∈ L ∞ (Q) | (y(t), yt (t)) is a global solution and J (u) < +∞}, 3. develop sufficient conditions for regional and gradient stabilization of system (13.8).

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Index

B Bilineare bilineare system, 263

R Riccati Riccati equation, 89

D Decomposition decomposition method, 240, 266 Dissipative, 51 dissipative operator , 12

I Infinitesimal generator of semigroup, 12

S Second order systems, 185 Semigroup, 11 compact Semigroup, 15 contraction semigroup, 16 C0 -semigroup, 12 Stabilization, 26 asymptotic stabilization , 27 exponential stabilization, 27 gradient stabilization of bilinear systems, 270 gradient stabilization of infinite dimensional bilinear systems, 263 internal regional stabilization, 84, 85 optimal stabilization , 104, 105 regional stabilization , 92 stabilizing control, 89 surface stabilization, 98, 99 weak stabilization, 27 System bilinear system, 47, 263 well posed system, 39

O Orthonormal basis , 13

U Unbounded, 185

G Gradient gradient stabilization of bilinear systems, 270 gradient stabilization of infinite dimensional bilinear systems, 263 regional gradient stabilization, 237

H Hille-Yosida, 13

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. H. Zerrik and O. Castillo, Stabilization of Infinite Dimensional Systems, Studies in Systems, Decision and Control 355, https://doi.org/10.1007/978-3-030-68600-0

315