Introduction to the theory of stability

223 Pages·1970·3.545 MB·English

by  E. A Barbashin

INTRODUCTION TO THE THEORY
OF STABILITY

E. A. BARBASHIN
Sverdlov Department of the Institute for Mathematics of the Academy of Sciences of the USSR
INTRODUCTION
TO THE THEORY OF STABILITY
translated from Russian by
TRANSCRIPTA SER VICE, LONDON
edited by T. LUKES
WOLTERS-NOORDHOFF PUBLISHING GRONINGEN
THE NETHERLANDS
© WOLTERS-NOORDHOFF PUBLISHING.1970
No part of this book may be reproduced in any form by print, photoprint,
microfilm or any other means without written permission from the publisher.
This translation first published in 1970
Library of Congress Catalog Card No. 70-90843
ISBN 90 01 05345 9
PRINTED IN THE NETHERLANDS
BY NEDERLA.NDSE BOEKDRUK INDUSTRIE N.V. - DEN BOSCH
CONTENTS
Editor's foreword 7
Preface ... 9
Chapter 1
The Method of Lyapunov Functions . 13
1. Estimate of the variation of the solutions . 14
2. Definition of stability. Derivation of equations for disturbed mo-
tion. . . . . . . . . . . . . . . 19
3. Lyapunov functions . . . . . . . . 20
4. The stability theorems of Lyapunov . 23
5. The asymptotic stability theorem . 26
6. Instability theorems . 29
7. Examples . . . . . . . . . . . 32
8. Linear systems . . . . . . . . . 34
9. Construction of Lyapunov functions in the form of quadratic
forms for linear systems of differential equations . . . . 38
10. Estimates of the solutions of linear systems. . . . . . 41
11. Stability theorems according to the first approximation . 43
12. Stability on the whole . 48
13. Aizerman's problem. 50
14. Examples . . . . . . 53
Chapter 2
Stability of Control Systems with Variable Structure 62
1. Preliminary remarks. Statement of the problem . 62
2. Stabilization of a second order system . . . . . 68
6 Contents
3. Stabilization of a third order system. Conditions for the existence
of the sliding mode . . . . . . . . . . . . . . . . . . . 74
4. Stabilization of a third order system. The stability of the system 77
5. Stabilization of an n-th order system . . . . . . . . . . . . 85
6. Stability of a system incorporating a limiter in the critical case
of a single zero root . . . . . . . . . . . . . . . . . . . 89
7. Non-linear systems with variable structure. Control of the co
ordinate x . · . . . . . . . . . . . . . . . . . . . . . . 94
8. Non-linear systems with variable structure. Control with respect
to the x-coordinate and its derivatives . . . . . . . . . . . 103
9. Investigation of a third order system with a discontinuous
switching surface . . . . . . . . . . . . . . . . . . · 108
I O. A system operating under forced sliding conditions . . . 122
11. An example of third order focussing under forced sliding . 131
Chapter 3
Stability of the Solutions of Differential Equations in Banach Space 139
l. Banach space . . . . . . . . . . . . . . . . . 139
2. Differential equations in Banach space. . . . . . . 145
3. Examples of differential equations in Banach spaces . 154
4. Problem of perturbation build-up over a finite interval of time . 159
5. Problem of perturbation build-up over an infinite interval of time.
Stability theorems for the zero solution of a homogeneous linear
equation . . . . . . . . . . . . . . . . . . . . . . . . 162
6. Theorems about the stability of solutions of non-linear equations 179
7. Stability with respect to impulse perturbations. . . . . . 191
8. Problem of realising a motion along a specified trajectory . . . 197
References ..... . 211
Additional Bibliography . 221
EDITOR'S FOREWORD
Considerable contributions to the theory of stability have been made in
the USSR starting with the work of the mathematicians Lurie and Lya
punov about eighteen years ago. In this book readers will find a clear
and concise account of this work, to which Dr. Barbashin has himself
made original contributions.
T. Lukes

PREFACE
This book is based on a course of lectures given by the author to ad
vanced mathematics students at the "A. M. Gor'kii" Urals State Uni
versity. The lectures were directed also to scientific workers and engineers
interested in applying the methods of stability theory, a circumstance that
explains certain features of the contents of the course. On the one hand
the author aimed at giving the mathematicians a picture of the present
day level of development of stability theory while at the same time point
ing out the relation of this theory to other fields of mathematics. The
author wished also to acquaint this section of his audience with the most
recent methods of investigation, including an account of his own work
and of the work of his students. On the other hand, he did ·not want his
audience to go away from the lecture hall with their minds filled only with
bare mathematical constructions. Each mathematical fact is therefore
considered from the point of view of its applicability and importance to
practical problems. Unfortunately, it was not possible to include all such
discussion in the book, but the choice of materjal reflects the above situa
tion well enough.
The first chapter gives an account of the method of Lyapunov functions
originally expounded in a book by A. M. Lyapunov with the title The
general problem of stability of motion which went out of print in 1892.
Since then a number of monographs devoted to the further development
of the method of Lyapunov functions has been published: in the USSR,
those by A. I. Lurie (22], N. G. Chetaev (26], I. G. Malkin [8], A. M.
Letov [23], N. N. Krasovskii [7], V. I. Zubov [138]; and abroad, J. La
Salle and S. Lefshets [11], W. Hahn [137].
Our book certainly does not pretend to give an exhaustive account of these
methods; it does not even cover all the theorems given in the monograph
by Lyapunov. Only autonomous systems are discussed and, in the linear
case, we confine ourselves to a survey of Lyapunov functions in the form
of quadratic forms only. In the non-linear case we do not consider the
question of the invertibility of the stability and instability theorems.
10 Preface
On the other hand, Chapter 1 gives a detailed account of problems per
taining to stability in the presence of any initial perturbation, the theory
of which was first propounded during the period 1950-1955. The first
important work in this field was that of N. P. Erugin [133-135, 16] and
the credit for applying Lyapunov functions to these problems belongs to
L'!lrie and Malkin. Theorems of the type 5.2, 6.3, 12.2 presented in Chap
ter 1 played a significant role in the development of the theory of stability
on the whole. In these theorems the property of stability is explained by the
presence of a Lyapunov function of constant signs and not one of fixed
sign differentiated with respect to time as is required in certain of Lya
punov's theorems. The fundamental role played by these theorems is
explained by the fact that almost any attempt to construct simple
Lyapunov functions for non-linear systems leads to functions with the
above property.
In presenting the material of Chapter 1, the method of constructing the
Lyapunov functions is indicated where possible. Examples are given at
the end of the Chapter, each of which brings out a particular point of
interest.
Chapter 2 is devoted to problems pertaining to systems with variable
structure. From a mathematical point of view such systems represent a
very narrow class of systems of differential equations with discontinuous
right-hand sides, a fact that has enabled the author and his collaborators
to construct a more or less complete and rigorous theory for this class of
systems. Special note should be taken of the importance of studying the
stability of systems with variable structure since such systems are capable
of stabilising objects whose parameters are varying over wide limits.
Some of the results of Chapter 2 were obtained jointly with the engineers
who not only elaborated the theory along independent lines but also con
structed analogues of the systems being studied.
The method of Lyapunov function finds an application here also but the
reader interested in Chapter 2 can acquaint himself with the contents
independently of the material of the preceding Chapter.
In Chapter 3 the stability of the solutions of differential equations in
Banach space is discussed. The reasons for including this chapter are the
following. First, at the time work commenced on this chapter, no mono
graph or even basic work existed on this subject apart from the articles
by L. Massera and Schaffer [94, 95, 139, 140]. The author also wished
to demonstrate the part played by the methods of functional analysis in