Stability Theory by Liapunov's Direct Method

Transcription

1 N. Rouche P. Habets M. Laloy Stability Theory by Liapunov's Direct Method Applied Mathematical Sciences 22 Springer-Verlag New York Heidelberg Berlin

2 N. Rouche P. Habets M. Laloy U.C.L. Institut de Mathematique Pure et Appliquee Chemin du Cyclotron 2 B-1348 Louvain-la-Neuve Belgium AMS Subject Classifications: 34D20, 93D05 (Primary), 34Dxx, 34H05 Library of Congress Cataloging in Publication Data Rouche, Nicolas. Stability theory by Liapunov's direct method. (Applied mathematical sciences ; v. 22) Bibliography: p. Includes indexes. 1. Differential equations. 2. Stability. 3. Liapunov functions. I. Habets, P., joint author. II. L al oy, M., j oint author. III. Title. IV. Series. QAl.A647 vol. 22 [QA372] 5l0'.8s [515'.352] All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag. ~ 1 9by 7Springer-Verlag, 7 New York Inc ISBN DOl / ISBN (ebook)

3 PREFACE This monograph is a collective work. The names appearing on the front cover are those of the people who worked on every chapter. very important: But the contributions of others were also C. Risito for Chapters I, II and IV, K. Peiffer for III, IV, VI, IX R. J. Ballieu for I and IX, Dang Chau Phien for VI and IX, J. L. Corne for VII and VIII. The idea of writing this book originated in a seminar held at the University of Louvain during the academic year Two years later, a first draft was completed. However, it was unsatisfactory mainly because it was exc e ~ s iabstract v e l y and lacked examples. It was then decided to write it again, taking advantage of -some remarks of the students to whom it had been partly addressed. text is this second version. The actual The subject matter is stability theory in the general setting of ordinary differential equations using what is known as Liapunov's direct or second method. We concentrate our efforts on this method, not because we underrate those which appear more powerful in some circumstances, but because it is important enough, along with its modern developments, to justify the writing of an up-to-date monograph. Also excellent books exist concerning the other methods, as for example R. Bellman [1953] and W. A. Coppel [1965]. Liapunov's second method has the undeserved reputation of being mainly of theoretical interest, because auxiliary

4 iv PREFACE functions appear to be so difficult to construct. We feel this is the opinion of those people who have not really tried. Indeed, many mathematicians have tackled only theoretical problems. On the other hand, too many of those involved in applications are unaware of the useful theorems or are victims of the myth of the elusive Liapunov function. Our aim, in writing this book, has been twofold: to describe the present state of the most useful Parts of the theory, and to appeal to the practical man with a wealth of applications taken from many varied fields. Chapters I and II constitute an elementary self-contained treatment of stability theory. They should normally be read first. Almost every other chapter can be studied without further prerequisite, except that some definitions or propositions of Chapter VI are needed in Chapters VII, VIII, and IX. The whole of Chapter VI is used in Section IX.6. We are also grateful to H. Everard, S. Spinacci and Kate MacDougall for their particularly expert typing of successive versions of the manuscript. Finally, it is a pleasure to acknowledge the financial support of this work by "Fonds National de 1a Recherche Scientifique". Louvain-1a-Neuve, October, 1975

5 SOME NOTATIONS AND DEFINITIONS This books requires a familiarity with some basic concepts from the theory of ordinary differential equations. As a general rule we have used symbols which are common place in mathematics. notations:!jr, the set of real numbers, Let us however point out the following!jr, the extended real number system, a ::: 0, a is a positive real number, a > 0, a is a strictly positive real number, [a,b], closed interval, ]a,b[, open interval, (alb) or atb, according to context, scalar product in!jrn, Ilxll, norm of point x in ~, d (x, M) = inf I I x-y II, distance from x E!Jfn to M C!Jfn, yem Be: = {x E!Jfn, II x II < d, open ball with center at the origin and radius e: > 0, B(a,e:) {x E ~, II x-a II < d, open ball with center a E!JRn a.nd radius e: > 0, B (M, e:) = {x E!JRn, d (x,m) < d, e: - neighborhood of the set M C!JR n, M e: B(M,e:) n n, e: - neighborhood of M E!JRn with respect to n C!JRn, E, unit n x n matrix, x = dt dx ' time derivative of the function ~!' jacobian matrix of the function f:!jfn ~!Jfm, x ~ f(x), J+, see p. 7, ~, see definition p. 12.

6 vi SOME NOTATIONS AND DEFINITIONS 'ttx, universal quantifier; read "for all x" or "given x", 3x, existential quantifier; read "for some x" or "there exist x". For g.eneral concepts on differential equations which are not defined in this text we refer to Ph. Hartman [1964], E. Coddington and N. Levinson [1965] or N. Rouche and J. Mawhin [1973]. The following definitions might be useful. Let A C 9f and f: A + 9f, x + f (x) be a real valued function. The function f is said ~ o be: increasing if 'ttx E A, 'tty E A, x < y implies f (x) ~ f (y) ; i. e for. ~ all x and y in A, x < y implies f(x) ~ f(y). strictly inc;j;'easing if 'ttx E A, 'tty E A, x < y f (x) < f (y), implies decreasing if 'ttx E A, 'tty EA, x < y implies f(x) ~ fey), strictly decreasing if 'ttx E A, 'tty E A, x < y implies f (x) > f (y), monotonic if it is increasing on A or decreasing on A. Let a E A, the extended closure of A. superior (upper limit) of f at a is Then the limit lim sup f(x) x+a inf{sup{f(x): x EB(a,o), x t- a}}e 9f. 0>0 Similarly the limit inferior (lower limit) of f at a is lim inf f(x) x+a sup{inf{f(x): x E B(a,o), x ;, an E!:if. 0>0 If a EA, the function f is said to be lower semicontinuous at a if lim inf f(x) ~ f(a). Y+rv If

7 SOME NOTATIONS AND DEFINITIONS vii lim sup f(x) < f(a), the function f is said to be upper x->-a - semi-continuous at a. It is easy to verify that a function f is continuous at a if and only if it is lower and upper semi-continuous at a. A function V: ~ l ->- + n ~,(t,x) ->- V(t,x) is said to be positive definite (with respect to x) if there exists a function a E Je such that (i) V(t,O) 0 (ii) V(t,x) > a(lixll). If -V is positive definite, the function V is said to be negative definite (with respect to x). If V(t,O) = 0 and V(t,x) ~ 0 the function V is said to be positive semidefinite (with respect to x). A function V: ~ l + ->- n + ~ m, (t,x,y) ->- V(t,x,y) is said to be positive definite with respect to x (1) V(t,O,O) = 0 if for some function a E Je (ii) V (t, x, y) > a (I I x I I ) An important class of positive definite functions are the positive quadratic forms where A is a symmetric positive definite matrix (T denotes transpose)

8 TABLE OF CONTENTS CHAPTER I. ELEMENTS OF STABILITY THEORY 1 1. A First Glance at Stability Concepts 1 2. Various Definitions of Stability and Attracti vi ty Auxiliary Functions Stability and Partial Stability Instability Asymptotic Stability 7. Converse Theorems 8. Bibliographical Note CHAPTER II. SIMPLE TOPICS IN STABILITY THEORY Theorems of E.A. Barbashin and N.N. Krasovski for Autonomous and Periodic Systems A Theorem of V.M. Matrosov on Asymptotic Stability Introduction to the Comparison Method Total Stability The Frequency Method for Stability of Control Systems Non-Differentiable Liapunov Functions 7. Bibliographical Note CHAPTER III. STABILITY OF A MECHANICAL EQUILIBRIUM Introduction The Lagrange-Dirichlet Theorem and Its Variants Inversion of the Lagrange-Dirichlet Theorem Using Auxiliary Functions Inversion of the Lagrange-Dirichlet Theorem Using the First Approximation 108

9 x 5. Mechanical Equilibrium in the Presence of Dissipative Forces Mechanical Equilibrium in the Presence of Gyroscopic Forces Bibliographical Note CHAPTER IV. STABILITY IN THE PRESENCE OF FIRST INTEGRALS Introduction General Hypotheses How to Construct Liapunov Functions Eliminating Part of the Variables Stability of Stationary Motions Stability of a Betatron Construction of Positive Definite Functions Bibliographical Note 165 CHAPTER V. INSTABILITY Introduction 2. Definitions and General Hypotheses 3. Fundamental Proposition Sectors Expellers Example of an Equation of Nth Order Instability of the Betatron Example of an Equation of Third Order Exercises....; Bibliographical Notes CHAPTER VI. A SURVEY OF QUALITATIVE CONCEPTS Introduction A View of Stability and Attractivity Concepts

10 xi 3. Qualitative Concepts in General Equivalence Theorems for Qualitative Concepts A Tentative Classification of Concepts Weak Attractivity, Boundedness, Ultimate Boundedness Asymptotic Stability Bibliographical Note 238 CHAPTER VII. ATTRACTIVITY FOR AUTONOMOUS EQUATIONS Introduction 2. General Hypotheses 3. The Invariance Principle An Attractivity and a Weak Attractivity Theorem Attraction of a Particle by a Fixed Center A Class of Nonlinear Electrical Networks The Ecological Problem of Interacting Populations Bibliographical Note 269 CHAPTER VIII. ATTRACTIVITY FOR NON AUTONOMOUS EQUATIONS Introduction, General Hypotheses The Families of Auxiliary Functions Another Asymptotic Stability Theorem Extensions of the Invariance Principle and Related Questions The Invariance Principle for Asymptotically Autonomous and Related Equations Dissipative Periodic S ystems Bibliographical Note 310

11 xii CHAPTER IX. THE COMPARISON METHOD Introduction Differential Inequalities A Vectorial Comparison Equation in Stability Theory Stability of Composite Systems An Example from Economics A General Comparison Principle Bibliographical Note 342 APPENDIX I. DINI DERIVATIVES AND MONOTONIC FUNCTIONS The Dini Derivatives Continuous Monotonic Functions The Derivative of a Monotonic Function Dini Derivative of a Function along the Solutions of a Differential Equation APPENDIX II. THE EQUATIONS OF MECHANICAL SYSTEMS APPENDIX III. LIMIT SETS 363 LIST OF EXAMPLES 368 BIBLIOGRAPHY 370 AUTHOR INDEX 386 SUBJECT INDEX 393