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Functional Analysis and Control Theory
Mathematics and its Applications (East European Series)
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Editorial Board:
A. BIALYNICKI-BIRULA, Institute of Mathematics, Warsaw University, Poland H. KURKE, Humboldt University, Berlin, D.D.R. J. KURZWEIL, Mathematics Institute, Academy of Sciences, Prague, Czechoslovakia L. LEINDLER, Bolyai Institute, Szeged, Hungary L. LOY .Asz, Bolyai Institute, Szeged, Hungary D. S. MITRINOVIC, University of Belgrade, Yugoslavia S. ROLEWICZ, Institute of Mathematics, Polish Academy of Sciences, Warsaw,
Poland BL. H. SENDOV, Bulgarian Academy of Sciences, Sofia, Bulgaria I. T. TODOROV, Bulgarian Academy of Sciences, Sofia, Bulgaria H. TRIEBEL, University of Jena, D.D.R.
Stefan Rolewicz Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland
Functional Analysis and Control Theory Linear Systems
Springer-Science+Business Media, B.V.
Library of Congress Cataloging-in-Publicatlon Data
Rolewicz, Stefan, Functional analysis and control theory: linear systems. (Mathematics and its applications. East European series) Translation of: AnaliZa funkcjonalna i teoria sterowania. Bibliography: p. Includes index. 1. Functional analysis. 2. Control theory.
1. Title. IT. Series: Mathematics and its applications (D. Reidel Publishing Company). East European series. QA320.R6413 1986 515.7 85-23246 ISBN 978-90-481-8423-1 ISBN 978-94-015-7758-8 (eBook) DOI 10.1007/978-94-015-7758-8
Chapters I-VII translated by Ewa Bednarczuk
Ali Rights Reserved. Copyright© 1987 by Springer Science+Business Media Dordrecht
Originally published by D. Reidel Publishing Company in 1987 Softcover reprint ofthe hardcover 1st edition 1987
No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical including photocopying, recording or by any information storage and retrieval system, without writtea permission from the copyright owner.
Table of Contents
Editor's Preface IX
Introduction . . . . . . . xiii
CHAPTER 1. Metric spaces . . . . . . . . . 1 1.1. Definition and examples of metric spaces 1 1.2. Convergence and related notions 9 1.3. Continuous mappings 13 1.4. Semimetric spaces . . . . . 16 1.5. Completeness of metric spaces 17 1.6. Contractive mapping principle 23 1. 7. Sets of first and second category 27 1.8. Spaces of modulus integrable and square integrable func-
tions . . . . . . . . . . . . . . . . . . . . . . . 29 1.9. Basic facts of measure theory and the Lebesgue integral 32 1.10. Separable spaces . . . . . . . . . . . . . . 45 1.11. Sequentially compact spaces and compact spaces 49
CHAPTER 2. Linear spaces and normed spaces . 55 2.1. Basic facts about linear spaces . . . . . 55 2.2. Linear metric spaces and normed spaces . 62 2.3. Linear functionals 77 2.4. Finite-dimensional spaces 85 2.5. Extensions of functionals 90 2.6. General forms of continuous linear functionals in certain
spaces . . . . . . . . . . . . . . . . . . . . . 100
CHAPTER 3. Continuous linear operators in Banach spaces . 114 3.1. The Banach-Steinhaus theorem . . . . . . . . . 114
vi Table of Contents
3.2. The Banach theorem about the continuity of the inverse operator . . . . . . 118
3.3. Closed operators . . 122 3.4. Conjugate operators 133
CHAPTER 4. Weak topologies . . . . . . . . . . . . . . . 138 4.1. Why we introduce topological spaces? Basic topological
notions . . . . . . . . . . . . . . . . . . . 138 4.2. Sequentially compact spaces and compact spaces . 143 4.3. Linear topological spaces . . . . . . 146 4.4. Weak topologies . . . . . . . . . . 149 4.5. Reflexive spaces and weak compactness 156 4.6. Extreme points . . . . . . . . . . . 159
CHAPTER 5. Optimization and observability of linear systems . 165 5.1. Linear systems . . . . . . . . . . . . . . . . . . 165 5.2. Optimization of linear systems With a fixed input operator 179 5.3. Sufficient conditions for the existence of an optimal input 201 5.4. Minimum-time problem . . . . . . . . . . . . . . . 208 5.5. Reduction of the minimum-time problem to the minimum-
norm problem . . . . . . . . . . . . . . . 225 5.6. Observability of linear systems . . . . . . . . . . . . 231 5.7. Minimum-time problems of observability theory . . . . 249 5.8. Noncontinuous linear systems. Estimability, observability,
controllability . . . . . . . . . . . . . . . . . . . 253
CHAPTER 6. Systems governed by ordinary differential equations 262 6.1. Problems of minimization of convex functionals related
to systems governed by ordinary differential equations 262 6.2. Controllability of finite-dimensional systems 267 6.3. Minimum-norm problems for sup type norms 269 6.4. Criteria for the uniqueness of optimal control 274 6.5. Bang-bang principle . . . . . . . . . . . 277 6.6. Measurable multifunctions and their applications 285 6.7. Observability of systems described by ordinary differen-
tial equations . . . . . . . . . . . . . . . . . . . 300
Table of Contents vii
6.8. Optimal observation of stationary systems 316 6.9. Functionals observed by less than n measurements 318 6.10. Optimal integration and differentiation formulae 322
CHAPTER 7. Systems with distributed parameters 328 7.1. Bases in Banach spaces . . . . . . . . 329 7 .2. Eigenvalues and eigenvectors . . . . . . 340 7.3. Temperature distribution in a rod with zero boundary con-
ditions . . . . . . . . . . . . . . . . . . . . . . 345 7.4. Nonhomogeneous heat diffusion equation with zero bound-
ary conditions . . . . . . . . . . . . . . . . . . . 361 7.5. Homogeneous heat diffusion equation with nonhomogene-
ous boundary conditions . . . . . . . . . . . . 366 7.6. Rod heating control . . . . . . . . . . . . . . . . 376 7.7. Observability of temperature distribution in a rod 384 7.8. Other problems related to heat diffusion in one-dimensional
objects . . . . . . 394 7.9. Rod vibration control . . . . . . . . . . . . . . . . 403
CHAPTER 8. Differential and integral equations in Banach spaces 418 8.1. Derivatives and integrals of functions with values in
Banach spaces . . . . . . . . . . . . . . . . . . . 418 8.2. Volterra integral equations and differential equations with
continuous right-hand sides . . . . . . . . . . . . . 421 8.3. Autonomous differential equation with a discontinuous
right-hand side and semigroups of operators . . 428 8.4. Properties of the infinitesimal generator 440 8.5. Infinitesimal generator of bounded semigroups . 445 8.6. Nonautonomous differential equations in Banach spaces 449 8.7. Controllability and approximative controllability of linear
systems described by differential equations in Banach spaces . . . . . . . . . . . . . . . . . . . . . . 452
8.8. Long-term controllability and long-term approximative controllability . . . . . . . . . . . . 463
8.9. Linear systems with analytic semigroups 467
viii Table of Contents
8.10. Optimization problems for systems described by linear differential equations in Banach spaces . . . . . . . . 472
8.11. Duality between observability and controllability of sys-tems governed by autonomous linear differential equations in Banach spaces . . . . . . . . . . . . . . . . . 476
8.12. Stability of solutions of linear differential equations in Banach spaces . . . . . . . . . . . . . . . . . . . 481
APPENDIX. Necessary and sufficient condition for weak duality 495
Bibliography 505
Author index 513
Subject index 515
List of symbols 523
Editor's Preface
Approach your problems from the right end and begin with the answers. Then, one day, perhaps you will find the final question.
'The Hermit Clad in Crane Feathers' in R. Van Gulik's The Chinese Maze Murders.
It isn't that they can't see the solution. It is that they can't see the problem.
G.K. Chesterton, The Scandal of Father Brown 'The point of a Pin'.
Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related.
Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics. This programme, Mathematics and Its Applications, is devoted to new emerging (sub)disciplines and to such (new) interrelations as exempla gratia:
ix
X Editor's Preface
- a central concept which plays an important role in several different mathematical and/or scientific specialized areas;
- new applications of the results and ideas from one area of scientific endeavour into another;
- influences which the results, problems and concepts of one field of enquiry have and have had on the development of another.
The Mathematics and Its Applications programme tries to make available a careful selection of books which fit the philosophy outlined above. With such books, which are stimulating rather than definitive, intriguing rather than encyclopaedic, we hope to contribute something towards better communication among the practitioners in diversified fields.
Because of the wealth of scholarly research being undertaken in the Soviet Union, Eastern Europe, and Japan, it was decided to devote special attention to the work emanating from these particular regions. Thus it was decided to start three regional series under the umbrella of the main MIA programme.
System and central theory, a relative newcomer in the spectrum of mathematical specialisms, is also one of the fastest growing and most immediately applicable ones. Without a doubt, finite dimensional linear systems have at the moment the greatest applicability as models of all kinds of economic, biological, chemical, and engineering processes. Equally clear is that it is sorely needed to extend the class of potential models which can be handled "completely". One way to go is towards infinite dimensional linear systems, another is towards finite dimensional nonlinear ones (there is overlap of course). Both extensions tend to have something mysterious generating some unease in the minds of engineers and others who could use these more general models. The present book is about infinite dimensional linear systems (on Banach spaces rather than Hilbert spaces), certainly a topic that among practitioners of control and system theory is usually approached with some wariness. So much can go wrong in the infinite dimensional case! There are of course a number of infinite dimensional specialists in the control and system community and there do exist a number of books on the topic written by them. As far as I know, this is the first book on the topic written by a scientist whose background is in the first place in functional
Editor's Preface xi
analysis rather than control. This gives a unique flavour to the book and serves to make it even more evident just how powerful a tool functional analysis can be.
The German translation of this book was highly successful and I expect this improved and updated English language version to be even more so.
The unreasonable effectiveness of mathematics in science ...
Eugene Wigner
Well, if you know of a better 'ole, go to it.
Bruce Bairnsfather
What is now proved was once only imagined.
William Blake
Russum, July 1986
As long as algebra and geometry proceeded along separate paths, their advance was slow and their applications limited.
But when these sciences joined company, they drew from each other fresh vitality and thenceforward marched on at a rapid pace towards perfection.
Joseph Louis Lagrange
Michie) Hazewinkel
Introduction
It was in the early sixties that the language and methods of Functional Analysis began to penetrate more and more deeply into both control and optimization theory.
Several books have presented the various aspects of the applications of functional analysis to control and optimization theory (see, for instance, Kulikowski, 1965, 1970; Porter, 1966; Lions, 1968; Hermes-La Salle, 1969; Balakrishnan, 1976; Curtain-Pritchard, 1977, and others).
In this book the general theory of linear systems based on the theory of linear operators in Banach spaces is developed in a consistent way.
The book is addressed not only to mathematicians but also to engineers, economists and other specialists interested in optimization and control theory.
In view of the rather restricted knowledge of Functional Analysis among non-mathematicians, the book is written in such a way that no background in Functional Analysis is required to read it. The first four chapters of the book can be treated as a short course of Functional Analysis. However, it should be pointed out that this part of the book contains solely the material which is necessary for the understanding of the rest. Thus some important topics from Functional Analysis (for instance the theory of operators in Hilbert spaces) are not included.
The first chapter of the book contains also fundamental notions and theorems (some of them without proof) in Topology and Measure Theory.
I should also mention that, although the contents of the first four chapters can be found in any handbook on Functional Analysis, what distinguishes this book from any other are the Examples and the adopted method of presentation. The book gives engineering and economical
xiii
:xiv Introduction
interpretations of the fundamental notions of Functional Analysis. Also, I try to convince the reader that the notions and theorems included in the book have been necessary.
The most essential part of the book is Chapter 5, where the theory of general linear systems is developed. The theory seems rather abstract. However, it permits answering some fundamental questions, for instance, what is the meaning of the Pontryagin maximum principle, and to characterize systems satisfying this principle. The chapter contains important theorems concerning the existence of universal time for time-depending systems. Optimal observability is defined in a general way and the duality between optimal observability and optimal controllability is shown.
Chapter 6 contains applications of the theory developed in Chapter 5 to systems described by ordinary differential equations. It includes not only a new exposition of the classical results but also several new results concerning optimal observability and numerical differentiations and integrations.
Chapter 7 contains applications of the theory to systems described by partial differential equations.
In Chapter 8 the fundamental facts about differential equations in Banach spaces are given. Problems of stability, optimal controllability and optimal observation for systems described by such equations are considered.
An appendix containing necessary and sufficient conditions for weak duality obtained by the Dolecki-Kurcyusz method of multifunctions is also included in the book.
I wish to express my gratitude to Professor Kazimierz Malanowski for his valuable suggestions regarding the first Polish edition of this book. My gratitude extends also to Professor Diethard Pallaschke, the translator of the book into German. Discussions with him have permitted essential improvements in the German edition. I also thank Mrs. Ewa Bednarczuk, the translator of the present English edition, and to Doctor Phan Quoc Khanh for their valuable remarks concerning this edition.
I am also greatly indebted to the staff members of the Mathematical Department of the Monash University (Clayton, Victoria, Australia) who participated in my seminars on the subject of the book. Since at
Introduction XV
that time I was preparing this edition, their suggestions were extremely helpful and had a considerable influence on the final form of the book.
Warszawa, August 1985 Stefan Rolewicz
