Introduction to the Theory and Application of Differential Equations with Deviating Arguments (Mathematics in Science & Engineering, 105)

lntroduction to the Theory and Application of Differential Equations with Deviating Arguments ACADEMIC PRESS RAPID MA...

Author: L.E. El'sgol'ts | S.B. Norkin

303 downloads 719 Views 4MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

lntroduction to the Theory and Application of Differential Equations with Deviating Arguments

ACADEMIC PRESS RAPID MANUSCRIPT REPRODUCTION

This is Volumc 105 in MATHEMATICS IN SCIENCE AND ENGINEERING A series of monographs and textbooks Edited by RICHARD BELLMAN, Univcr.\ity of Soirthcrn Cdifornici The complete listing of books i n this series is available from the Publisher upon request.

Introduction to the Theory andApplication of Differential Equations with Deviating Arguments L. E. El’sgol’ts and S. B. Norkin

Translated b y John L. Casti University of Arizona Tucson, Arizona

1973

ACADEMIC PRESS

New York

A Subsidiary of Harcourt Brace Jovanovich, Publishers

London

COPYRIGHT Q 1973, BY ACADEMIC PRESS,INC.

ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMIITED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITlNG FROM THE PUBLISHER.

ACADEMIC PRESS, INC.

111 Fifth Avenue, New York, New York 10003

United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road. London NW1

LILIRARYOF CONGRESS CATALOQ CARDNUMBER:13-8 11

Introduction t o t h e Theory and Application of Differential Equations with Deviating Arguments. Translated from the original Russian edition entitled Vvdenie V Teoriyu Differencial’Nyh Uravneni! S O t k l o n y a y u ~ i m s y aArgumentom, published by “Nauka”Press, Moscow, 1971.

PRINTED IN THE UNITED STATES OF AMERICA

CONTENTS PREFACE . . . . TRANSLATORS NOTE INTRODUCTION

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii xi xiii

. . . . . . . . . . . . . . . . . . . . . . .

Chapter

I. Basic Concepts and Existence Theorems 1. Statement of the Basic Initial Value Problem. Classifications . . . . . . . . . . . . . . . . . . 2. The Method of Steps . . . . . . . . . . . . . . . 3. Integrable Types of Equations with a Deviating Argument . . . . . . . . . . . . . . . . . . . 4. Existence and Uniqueness Theorems for the Solution of the Basic Initial Value Problem . . . . . . . . . . 5. Some Specific Singularities of the Solutions of Equations with a Deviating Argument . . . . . . . . . . . . .

Chapter 11. Linear Equations 1. Some Properties of Linear Equations . . . . . . 2. Linear Equations with Constant Coefficients and Constant Deviating Arguments . . . . . . . . 3. The Characteristic Quasipolynomial . . . . . . 4. The Expansion of the Solution into a Series of Basic Solutions . . . . . . . . . . . . . . . . . . 5. Two-sided Solutions . . . . . . . . . . . . 6. The Homogeneous Initial Value Problem. . . . . 7. Some Types of Linear Equations with Variable Coefficients and Variable Deviating Arguments . .

15 19 29

. . .

57

. . . . . .

62 69

. . . . . . . .

80 83 91

. . .

98

Chapter 111. Stability Theory 1. Basic Concepts. . . . . . . . . . . . . . . . . 2. The Stability of Solutions t o Stationary Linear Equations . . . . . . . . . . . . . . . . . . 3. Conditions for Negativity of the Real Parts of All Roots of the Quasipolynomial . . . . . . . . . . . . . 4. The Case of Small Deviating Arguments . . . . . . . 5. The Case of Large Deviating Arguments . . . . . . . 6. Lyapunov's Second Method. . . . . . . . . . . . 7. Stability in the First Approximation . . . . . . . . 8. Stability under Constantly Acting Disturbances . . . .

V

1 6

.

119

.

121

. . . . .

126 139 143 144 159 162

.

CONTENTS

9 . Lyapunov's Second Method for Equations of Neutral Type . . . . . . . . . . . . . 10. Absolute Stability . . . . . . . . . . .

. . . . . . . . . .

Chapter IV . Periodic Solutions 1 . Some Properties of Periodic Solutions and Existence Theorems . . . . . . . . . . . . . . . . . . . 2. Periodic Solutions of Stationary. Linear. Homogeneous Equations . . . . . . . . . . . . . . . . . . . 3 . Periodic Solutions of Linear Inhomogeneous Equations with Stationary Homogeneous Parts . . . . . . . . . 4. Periodic Solutions of Linear Equations with Variable Coefficients and Deviating Arguments . . . . . . . . 5. Periodic Solutionsof Quasilinear Equations . . . . . . 6. Functionally Equivalent Systems of Differential Equations with a Deviating Argument . . . . . . . . Chapter

V . Stochastic Differential Equations with a Retarded Argument 1 . Basic Concepts . . . . . . . . . . . . . . . . . 2 . Stability . . . . . . . . . . . . . . . . . . . . 3. Stationary Solutions of Equations with a Delay . . . . .

Chapter VI . Approximate Methods for the Integration o f Differential Equations with a Deviating Argument 1. General Remarks about the Application of Approximate Integration Methods . . . . . . . . . . . . . . . 2 . Euler's Method and Parabolic Methods . . . . . . . . 3 . Expansion in Powers of the Retardation . . . . . . . 4. Asymptotic Methods for Equations with Small Deviating Argument . . . . . . . . . . . . . . . 5. Iterative Methods . . . . . . . . . . . . . . . .

165 175

183 187 191 196 200 210

219 223 229

235 237 243 244 248

Chapter VII . Some Generalizations and a Brief Survey of Work in Other Areas of the Theory of Differential Equations with a Deviating Argument 1 . Some Generalizations . . . . . . . . . . . . . . . 2. Periodic Solutions . . . . . . . . . . . . . . . . 3. Boundary-Value Problems . . . . . . . . . . . . . 4. Optimal Processes with a Retardation . . . . . . . . 5 . Stationary Points . . . . . . . . . . . . . . . .

251 256 265 275 285

BIBLIOGRAPHY I . Monographs . . . . . . . . . . . . . . . . . . I1. Survey Articles . . . . . . . . . . . . . . . . . 111. Journal Articles . . . . . . . . . . . . . . . . .

293 294 296

vi

This book is a revised and substantially expanded edition of the well-known book of L. E. El’sgol’ts published under this same title by Nauka in 1964. Extensions of the theory of differential equations with deviating argument as well as the stimuli of developments within various fields of science and technology contribute t o the need for a new edition. This theory in recent years has attracted the attention of vast numbers of researchers, interested both i n the theory and its applications. The first edition of this book acquainted a wide circle of readers with the theory of differential equations with a deviating argument. On the other hand, intensive development of the theory required inclusion of new material in an introductory book reflecting this development. Therefore, the question was naturally raised about the preparation of a new, expanded edition. But El’sgol’ts was not able to d o this. Tragically, the life of Lev Ernestovich El’sgol’ts was cut offin the full bloom of its creative strength: on October 24, 1967. in his 59th year Lev Ernestovich El’sgol’ts perished in an automobile accident. In his initial activity, El’sgol’ts directed his attention t o problems of the calculus of variations in the large. However, his principal contributions were related to the theory of differential equations with a deviating argument, in which he became interested in 1949. In 1950 in the correspondence division of the Mechanics-Mathematical faculty of MGU, he organized a seminar on the theory of differential equations with a deviating argument. In this period, there was considerably heightened interest on the part of mathematicians in the investigation of differential equations with a deviating argument, both in connection with problems in the theory o f control systems, and because of the intrinsic richness and beauty of such equations. However, this domain was intensively developed only in a small number of directions by the efforts of a very few people. The fundamental contribution of El’sgol’ts is that he, one of the first t o realize the value and scope of this field, restricted his comprehensive studies. In later years this resulted in the appearance of the theory of differential equations with a deviating argument as an independent domain of mathematical analysis. Through the initiative of El’sgol’ts, his students and collaborators, and many other mathematicians associated with him, various parts of the theory of ordinary differential equations were analyzed, clarifying in each case the corresponding results carrying over t o the theory of differential equations with a deviating argument, and the new characteristics that appear because of this transference; for the discovery of such characteristics, El’sgol’ts initiated his detailed vii

PREFACE

investigations. This promoted regular departures by him from analytic problems in the given domain. As a result of this work, El’sgol’ts’ seminar on the theory of differential equations with a deviating argument became a universally recognized center of research in this field. In the U.S.S.R. the scale of research on the theory of differential equations with a deviating argument has increased significantly in recent years and the investigations of our scholars in this domain have indisputably assumed leadership in the world. Certainly, an outstanding role in this success was played by the tireless activity of El’sgol’ts.* The works of L. E. El’sgol’ts on the theory of differential equations with a deviating argument include, for example, LI.11, [1.131, [II.6]-[11.8], [II.13], III.141, [II.16]-[11.18], [132.11-132.281. We offer the book, as in its first edition, in a self-contained, brief, and accessible form in order to acquaint the reader with the basic theory of differential equations with a deviating argument. The book does not encompass an extensive scope of material nor present far-reaching generalizations; it is restricted to the simplest cases. Not infrequently, in order to avoid cumbersome details of a proof, we only point out the idea of the proof or give a brief sketch. Much detailed information on various aspects of the theory may be obtained from the primary sources in the bibliographic literature given: lI.91, lI.31 -[I.51, [I.14], [I.81, [I.101 -[I.121. It is assumed that the reader is familiar with ordinary differential equations without a deviating argument, the simplest properties of analytic functions, and (for Chapter V) the basic theory of probability. The development of the foundations of the theory of differential equations with a deviating argument is still far from complete. This situation, of course, leaves its mark on our suggestions to the reader of the book and prevents as orderly and systematic a presentation as is usual for mathematical literature. However, it is hoped that in spite of these deficiencies the book will prove useful as a first acquaintanceship with the theory of differential equations with a deviating argument. It was my fortunate experience to be a pupil of L. E. El’sgol’ts, and to work with him for many years. In addition, I was the editor of the first edition. The second edition of the book makes use of comments from a very broad circulation since the book was translated and published in many countries. In preparing the second edition without the aid of the author, I felt an immense responsibility to lighten the memory of L. E. El’sgol’ts’ very costly death for my people. Thus, in order to answer to the reader for all possible deficiencies connected with the selection of new material and its presentation (written by me, to the extent the new text exceeds the old), I added my name to the title page.

*Lev Ernestovich El’sgol’ts, Necrolog, UspekhiiMufh. Nuuk. 23,2 (140)( 19681, 193-200. A complete list of his work is cited.

viii

PREFACE

I consider it a pleasurable debt to convey thanks to A. D. Myshkis, a very attentive reader of the manuscript of the second edition and the contributor of much valuable counsel. I take this opportunity to thank V. B. Kolmanovskii, V. R. Nosov, and the editor of the book, L. A. Zhivotovskii, for a series of useful comments. S. B. Norkin

1970

ix

This page intentionally left blank

TRANSLATOR’S NOTE In the preparation of this translation, I wish t o acknowledge the invaluable assistance of S. Chebotarev of the Institute for Control Problems, Moscow, who supplied the initial impetus for the work and constant encouragement and assistance; of Professor Richard Bellman, who immediately recognized the need for a book of this type in the English-speaking mathematical community and readily welcomed the book into his Applied Mathematics and Engineering Series; and, most importantly, of Gayle Wood, who painstakingly typed version after version of the manuscript t o put it into its final polished form.

xi


INTRODUCTION Differential equations with a deviating argument are differential equations in which the unknown function and its derivatives enter, generally speaking, under different values of the argument. F o r example, x ( t ) = f ( t , x ( t ) , x (t-7)),

7>0

The first isolated equation of such type appeared in the literature in the second half of the eighteenth century (Kondorse, 1771), but a systematic study of equations with a deviating argument began only in the twentieth century (especially in the last forty years-A. D. Myshkis in the Soviet Union, E. M. Wright and R. Bellman in other countries) in connection with the requirements of applied science. Differential equations with a deviating argument have many applications in the theory of automatic control, the theory of self-oscillating systems, the study of problems connected with combustion in rocket motion, the problem of long-range planning in economics, a series of biological problems, and in many other areas of science and technology, the number of which is steadily expanding. The abundance of applications is stimulating a rapid development of the theory of differential equations with a deviating argument and, a t present, this theory is one of the most rapidly developing branches of mathematical analysis. Equations with a deviating argument describe many processes with an aftereffect; such equations appear, for example, any time when in physics o r technology we consider a problem of a force, acting o n a material point, that depends on the velocity and position of the point not only at the given moment but at some moment preceding the given moment. The presence of a deviation-delay in the systems studied often turns out t o be the cause of phenomena substantially affecting the motion of the process. For example, in automatic regulators, the delay is the interval of time, always present, which the system needs t o react t o the input impulse. The presence of a delay in an automatic regulator system may cause the appearance of a self-exciting oscillation, o f increase of overregulation, and even of the instability of the system. The cause of unstable combustion in liquid rocket engines is, as is customary t o assume, the existence of a delay time, the time necessary for conversion of the fuel mixture into the products of combustion, etc. Technological and engineering improvements require accounting for the phenomenon of aftereffect in the traditional domains of technology. For example, for modern high-speed diesels, approximately a fifty centimeter intake pipe, with respect t o the time of suction turns out t o be a rather lengthy line, and to describe the process of fuel injection it is necessary t o use differential equations with a deviating argument (of neutral type). xiii

INTRODUCTION

For study of a real system with aftereffect, as an initial approximation it is assumed that the delay 7 is constant. Such an assumption represents a step forward compared with the model of an “ideal” process which is obtained if it is presupposed that the aftereffects do not exist, that the “operation” takes place instantaneously. In a series of cases, the assumption that T = constant is good, reflecting the true situation. For example, when the delay is connected with the transmission of sonic signals, with hydraulic thrust, or other wave processes. In other cases, such an assumption describes the process only approximately. A more complete analysis shows that in a series of important cases of real systems the delay T depends not only on the time but also on the unknown function, and even its derivatives. In such cases it is natural to suppose that the dependence relations, in general, are not determinate and have a random character. The book addresses readers encountering differential equations with a deviating argument for the first time. Thus, it is assumed that its readers will be not only mathematicians, but also physicists and engineers concerned with the study of systems with aftereffects. The bibliography at the end of the book consists of three sections: I. Monographs; 11. Survey articles; 111. Journal articles. It covers all areas of the theory of ordinary differential equations with a deviating argument and presents interest for specialists in this area. The bibliography is sufficiently large, but in no way does it pretend t o be complete. In particular, additional information about older literature may be obtained in the reviews 111.61, 111.71, and 111.121. Much work of an applied character is indicated in the bibliography of the monograph [I. 1 11 .

xiv

lntroduction to the Theory and Application of Differential Equations with Deviating Arguments


Chapter I B a s i c C o n c e p t s and E x i s t e n c e Theorems

1.

S t a t e m e n t o f t h e B a s i c I n i t i a l Value.-P Classifications.

For t h e s i m p l e s t d i f f e r e n t i a l e q u a t i o n w i t h a d e v i a t i n g argument

k(t)

= f(t,X(t),X(t-T)),

(1)

where t h e d e l a y T , f o r t h e t i m e b e i n g , w i l l be assumed t o be a p o s i t i v e c o n s t a n t , t h e b a s i c i n i t i a l v a l u e problem c o n s i s t s i n t h e d e t e r m i n a t i o n o f a c o n t i n u o u s s o l u t i o n x ( t ) o f e q u a t i o n (1) f o r t > t Ou,n d e r t h e c o n d i t i o n t h a t x ( t ) = + ( t ) f o r t - - r < t < t where O, + ( t )i s a g i v e n c o n t i n u o u s

--

0

function c a l l e d the i n i t i a l function (Fig. 1). The c l o s e d i n t e r v a l t -TLtLto, on which t h e i n i t i a l f u n c t i o n i s g i v e n , i s 0 c a l l e d t h e i n i t i a l s e t and denoted E ; t h e p o i n t t i s c a l l e d t h e i n i t i a l 0 point. U s u a l l y , i t i s assumed t h a t x ( t + 0 ) = + ( t o ) . 0

Under some r e s t r i c t i o n s , t h e e x i s t e n c e o f a s o l u t i o n t o t h i s i n i t i a l v a l u e problem w i l l be established. T h i s s o l u t i o n w i l l o f t e n be d e n o t e d x+( t )

.

I f i n e q u a t i o n (1) and i n t h e i n i t i a l c o n d i t i o n x ( t ) , f , and 4 are r e g a r d e d a s v e c t o r f u n c t i o n s , t h e n w e o b t a i n t h e b a s i c i n i t i a l v a l u e problem f o r a s y s t e m of e q u a t i o n s . I n t h e c a s e of a v a r i a b l e d e l a y T = T ( ~ ) > O i n Eq. ( l ) ,it i s a l s o r e q u i r e d t o f i n d a s o l u t i o n o f t h i s e q u a t i o n f o r t > t where on t h e i n i t i a l s e t E t , c o n s i s t i n g o f tdle p o i n t t o , and of t h o s e 0

1

D I F F E R E N T I A L EQUATIONS WITH D E V I A T I N G ARGUMENTS

va.lues t - T ( t ) , which a r e less t h a n t f o r tit , x ( t ) i s t o c o i n c i d e with t h e given i a i t i a l f u g c t i o n $(t)*. For example, i n t h e e q u a t i o n x ( t ) = f ( t , x ( t ) , x ( t - c o s 2 t ) ) , f o r t =O t h e i n i t i a l f u n c t i o n 0 ( t ) 0 m u s t be g i v e n on t h e i n i t i a l s e t E which i s t h e 0’

i n t e r v a l - l O , s i n c e i n t h i s case it i s p o s s i b l e t o a p p l y t h e method of s t e p s w i t h t h e s t e p d . I f t h e lengths of t h e i n t e r v a l s 6

n-,

i n o t h e r words,

+O a s

Y,..(t,) if t h e r e e x i s t s Wig y n ( t )

=

t,

t h e n , by v i r t u e o f t h e c o n t i n u i t y of t h e r e t a r d a t i o n w e o b t a i n T (&)=O and t h e s i n g u l a r case may T (t), occur a t t h e point t. W e o b s e r v e t h a t f o r T ( ~ ) = O t h e s i n g u l a r case d o e s not necessarily occur s i n c e , i f i n t h e right-half neighborhood of t h i s p o i n t , r ( t ) i n c r e a s e s more r a p i d l y t h a n t, t h e n w e a p p l y t h e method o f s t e p s a t t h e p o i n t t . The s i n g u l a r case f o r T ( t ) = O o c c u r s o n l y u n d e r t h e c o n d i t i o n t h a t y ( t ) = t , i . e . when i n a s u f f i c i e n t l y s m a l l r i g h t - h a l f neighborhood o f t h e p o i n t t, T ( t ) i n c r e a s e s m o r e s l o w l y t h a n t.

I f t h e s i n g u l a r case is e x c l u d e d t h e n , a p p l y i n g t h e method o f s t e p s , w e e s t a b l i s h t h e e x i s t e n c e and u n i q u e n e s s o f t h e s o l u t i o n o f Eq. ( 1 0 ) w i t h t h e i n i t i a l f u n c t i o n g i v e n on E under t h e c o n d i t i a n

I1

DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

t h a t a l l e q u a t i o n s 2 ( t ) = f ( t , x ( t ) , $Ik

(t-T

(where t l i e s i n t h e i n t e r v a l y k ( t ) i t ' y k + l

( t )) ) ( t ),

and $Ik(t) i s t h e c o n t i n u a t i o n of t h e i n i t i a l f u n c t i o n by t h e s o l u t i o n o f Eq. ( 1 0 ) on t h e i n t e r v a l s a t i s f y t h e c o n d i t i o n s of t h e 'k-1 ( t ) i t ' y k ( t ) )

t h e o r e m o f e x i s t e n c e and u n i q u e n e s s and a d m i t c o n t i n u a t i o n of t h e s o l u t i o n t o t h e e n t i r e i n t e r v a l yk ( t ) c t c-y k + l ( t ) '

Under t h e same a s s u m p t i o n s , w e e v i d e n t l y f i n d t h a t t h e s o l u t i o n o f Eq. ( 1 0 ) w i l l have c o n t i n u o u s d e r i v a t i v e s o f o r d e r up t o p a t t h e p o i n t s , k - 1 ) , i f t h e f u n c t i o n f and t > y p - l ( t o )( p = l , 2 , t h e r e t a r d a t i o n T (t)are s u f f i c i e n t l y d i f f e r e n t i a b l e , b u t t h e d e r i v a t i v e of o r d e r k a t t h e p o i n t yk-l(tO)

...

w i l l , i n g e n e r a l , have a d i s c o n t i n u i t y of t h e f i r s t kind. If the equation contains d i f f e r e n t delays, f o r example, it h a s t h e form

then t h e i n t e r v a l 5

= [ t O , y ( t O ) ] is the largest

i n t e r v a l w i t h l e f t e n d p o i n t t o , o n which a l l o f the differences t - r i ( t ) < t (i=1 , 2 , . . . , m ) . - 0

5-

,

I n t h e same words,we d e f i n e t h e i n t e r v a l s on which it i s p o s s i b l e t o c o n t i n u e , v i a

t h e method o f s u c c e s s i v e i n t e g r a t i o n s a t e a c h s t e p , t h e s o l u t i o n of a s i n g l e e q u a t i o n of t h e n t h o r d e r o r a s y s t e m of s u c h e q u a t i o n s w i t h a r e t a r d e d argument. I n t h e same way, i f 6- r e d u c e s t o a t s i n g l e p o i n t F, t h e n f u r t h e r a p p l i c a t i o n of t h e method of s u c c e s s i v e i n t e g r a t i o n s becomes i m p o s s i b l e . W e c a l l t h i s case s i n g u l a r . I f w e a p p l y t h e method o f s u c c e s s i v e i n t e g r a t i o n s t o t h e i n t e g r a t i o n of a n e q u a t i o n w i t h o u t r e t a r d a t i o q w h i c h reduces a t each s t e p t o t h e s o l u t i o n o f Eq. ( 1 0 ) ( o r a s y s t e m of e q u a t i o n s w i t h r e t a r d a t i o n ) s a t i s f y i n g t h e conditions of t h e 12

I. BASIC CONCEPTS

t h e o r e m o f e x i s t e n c e and u n i q u e n e s s o f s o l u t i o n and t h e c o n t i n u o u s dependence o f t h e s o l u t i o n on p a r a meters, t h e n t h e s o l u t i o n o f t h e o r i g i n a l e q u a t i o n s with delay a l s o possesses those properties. In t h i s case, from t h e method o f o b t a i n i n g t h e s o l u t i o n , it i s immediately a p p a r e n t t h a t t h e s o l u t i o n continuo u s l y d e p e n d s on t h e c h o i c e o f t h e i n i t i a l f u n c t i o n and on t h e d e l a y . r ( t ) i n a s p a c e C i f t h e r i g h t 0

hand s i d e o f t h e e q u a t i o n % ( t ) = f( t , x ( t ) , x ( t - T ( t ) ) ) ( o r an e q u a t i o n of more g e n e r a l form) i s c o n t i n u o u s and s a t i s f i e s a L i p s c h i t z c o n d i t i o n i n i t s second argument. W e may, of c o u r s e , a p p l y t h e method of s t e p s

t o e q u a t i o n s of n e u t r a l t y p e i f t h e set 5- d o e s t n o t reduce t o a s i n g l e p o i n t . For s i m p l i c i t y , we c o n s i d e r a n e q u a t i o n of n e u t r a l t y p e of t h e f i r s t o r d e r w i t h a s i n g l e c o n s t a n t d e v i a t i n g argument T : k(t) = f

( t , X ( t ) ,X(t-T)

,%(t-T)).

(11)

I n c o n t r a s t t o e q u a t i o n s w i t h a r e t a r d e d argument o f t h e f i r s t o r d e r , t h e i n i t i a l f u n c t i o n $ ( t ) f o r t h e s o l u t i o n o f Eq. (11) must be n o t o n l y cont i n u o u s b u t h a v e c o n t i n u o u s (or p i e c e w i s e c o n t i n u o u s ) d e r i v a t i v e s . On t h e i n i t i a l s t e p w e o b t a i n t h e e q u a t i o n w i t h o u t a d e v i a t i n g argument k ( t ) = f ( t t x ( t )t $ O ( t - T ) r 6 0 ( t - T ) ) f o r t < t < +tT . 0- - 0

(12)

On t h e n e x t s t e p for

tO+Tzt(t0+2T,

and so on.

The c o n t r a s t t o an e q u a t i o n w i t h a d e l a y cons i s t s of t h e f a c t t h a t t h e s o l u t i o n i s n o t smoothed. In f a c t , not only a t t h e point t i s t h e l e f t d e r i v a t i v e 4 ( t O - O ) , g e n e r a l l y spgaking , n o t equal t o x ( t O + O ) b u t a l s o a t t h e p o i n t t O + T , a s i s obv i o u s from Eq.

(13).

% (t)w i l l be, i n general,

4

d i s c o n t i n u o u s b e c a u s e of t h e d i s c o n t i n u i t y of t h e l a s t argument 2 ( t - T ) a t t = to+'. These same 4

13


a r g u m e n t s show t h a t t h e s o l u t i o n x ( t ) h a s , i n g e n e r a l , c o r n e r p o i n t s a t t < t O + k . c 4 (k=O ,1 , 2 ,

... .

Smoothing does n o t o c c u r a t t h e i n t e r i o r p o i n t s o f t h e i n t e r v a l s as i s o b v i o u s from Eqs. (12), (13) and t h o s e s i m i l a r t o them o b t a i n e d on t h e k t h step. The method o f s t e p s a l s o a p p l i e s t o e q u a t i o n s of advanced t y p e if St d o e s n o t c o n s i s t o f a s i n g l e point. For t h i s i n general, lose i n i t i a l function t h e s o l u t i o n may

0

it i s found t h a t s u c h e q u a t i o n s , t h e i n h e r i t e d smoothness o f t h e a n d , a f t e r some number o f s t e p s , n o t even e x i s t .

F o r example, w e c o n s i d e r t h e e q u a t i o n X ( t )

= f

(t,X(t-T)

,?(t-T))

I

(141

where t h e c o n s t a n t T > O . The i n i t i a l v a l u e p r o b l e m i s g i v e n as €or a n e q u a t i o n w i t h r e t a r d a t i o n , i.e. it i s r e q u i r e d t o f i n d t h e s o l u t i o n of t h e equation f o r t > t Oi,f on t h e i n i t i a l s e t t the

o-~zt(to

s o l u t i o n i s equal t o t h e given i n i t i a l function I $ ( t ) . On t h e f i r s t s t e p , Eq. ( 1 4 ) t u r n s i n t o t h e f i n i t e equation x(t) =

f(t,I$(t-T)

(t-T))

(15)

€or t < t < t+ T . Consequently, a t t h e p o i n t t=tO, 0 - 0 i n g e n e r a l , $ ( t o #) x ( t + 0 ) a n d , i n t h i s s e n s e , 4 0 t h e solution is discontinuous. I f the function f i s d i f f e r e n t i a b l e a s u f f i c i e n t number of t i m e s ( n o t less t h a n k-1 t i m e s ) and $ i s k t i m e s d i f f e r e n t i a b l e , t h e n t h e s o l u t i o n o f Eq. ( 1 3 ) i s d i f f e r e n t i a b l e , i n g e n e r a l , o n l y k-1 t i m e s ( s i n c e x(IC-1) ( t ) contains the t e r m 7 ak-lf (t-T) and f o r (k-1) (t-T t h e e x i s t e n c e o f t h e n e x t d e r i v a t i v e , i would be n e c e s s a r y t o r e q u i r e t h e e x i s t e n c e of + l k + l ) ( t - T ) , which w e d o n o t a s s u m e ) .

14

I. BASIC CONCEPTS

On t h e n e x t s t e p , f o r t h e above r e a s o n , a t t h e boundary p o i n t t = t + T t h e s o l u t i o n w i l l b e 0

discontinuous, while i n t h e i n t e r i o r of t h e interval t + - r < t < t 0 + 2 ~ ,it i s p o s s i b l e t o g u a r a n t e e o n l y t h e 0 - e x i s t e n c e o f d e r i v a t i v e s up t o o r d e r k-2 i n c l u s i v e . On t h e kth s t e p , t h e s o l u t i o n , i n g e n e r a l , w i l l n o t b e d i f f e r e n t i a b l e and f o r l a r g e v a l u e s o f t may n o t e x i s t . The method o f s t e p s may o f t e n be a p p l i e d f o r t h e s o l u t i o n o f e q u a t i o n s w i t h a d e v i a t i o n dependi n g upon t h e unknown f u n c t i o n . For example,

ic(t)

=

f ( t , x ( t ) ,x(t-T ( t , x ( t )1 ) )

.

(16)

The method of s t e p s t r i v i a l l y a p p l i e s i f i n f . r ( t , x ( t ) ) = d>O, e x i s t s f o r t > t and f o r a n y v a l u e s o f t h e second 15 g h i s c a s e , it i s p o s s i b l e t o advance argument. forward by " s t e p s " e q u a l t o d , o b t a i n i n g from Eq. ( 1 6 ) f o r e a c h s t e p d i f f e r e n t i a l e q u a t i o n s w i t h o u t d e v i a t i n g a r g u m e n t s o f t h e form k ( t ) = f(t,x(t),ok(t-r(t,x(t)))). 3.

I n t e g r a b l e Types o f E q u a t i o n s w i t h a Deviatincr Aruument

The method o f s t e p s i s one o f t h e b a s i c methods of i n t e g r a t i n g d i f f e r e n t i a l e q u a t i o n s w i t h a deviat i n g argument. A p p l i c a t i o n o f t h i s method t o a n e q u a t i o n o f t h e form k ( t ) = f ( t , x ( t ) , x ( t - . r f t )) ) r e d u c e s t o t h e i n t e g r a t i o n of e q u a t i o n s w i t h o u t retardation

(17)


I t i s n a t u r a l t o c a l l Eq. ( 1 7 ) i n t e g r a b l e by q u a d r a t u r e s i f w e a p p l y t h e method o f s t e p s t o it and Eq. ( 1 8 ) i s i n t e g r a b l e i n q u a d r a t u r e s f o r any cont i n u o u s f u n c t i o n $ n ( t ) .An a n a l o g o u s d e f i n i t i o n i s appropriately introduced f o r equations of f i r s t order o f t h e more g e n e r a l form

G ( t ) = f ( t I x ( t )I x ( t - T l ( t ) ) I

- - - I X ( ~ - T ~ ( ~ )I ) )

and a l s o f o r e q u a t i o n s o f h i g h e r o r d e r w i t h a r e t a r d e d argument and € o r s y s t e m s of s u c h e q u a t i o n s . Examples o f e q u a t i o n s o f t h e f i r s t o r d e r w i t h a r e t a r d e d a r g u m e n t ? i n t e g r a b l e i n q u a d r a t u r e s , are:

1) G ( t ) = f ( t , x ( t - r l ( t )1 I . .

., x ( t - T m ( t ) ) ) .

T h i s t y p e o f e q u a t i o n is i n t e g r a b l e p a r t i c u l a r l y simply s i n c e , f o r each s t e p , t h e right-hand s i d e i s a known f u n c t i o n of t .

I . BASIC CONCEPTS

I n a l l t h e s e e x a m p l e s , i t i s assumed t h a t t h e method o f s t e p s i s a p p l i c a b l e . W e note t h a t t h e i n t e g r a b i l i t y of some e q u a t i o n s i n q u a d r a t u r e s does n o t u s u a l l y f a c i l i t a t e i n v e s t i g a t i o n of t h e a s y m p t o t i c b e h a v i o r of t h e i r s o l u t i o n , and n o t i n f r e q u e n t l y , even on a f i n i t e i n t e r v a l , c h a n g e s o f i n d e p e n d e n t v a r i a b l e a n d t h e u s e o f them e i t h e r by o t h e r a p p r o x i m a t i o n s o r q u a l i t a t i v e methods of i n v e s t i g a t i o n t u r n o u t t o b e more a p p r o p r i a t e t h a n a p p l i c a t i o n of t h e method of s t e p s . T o t h e number o f i n t e g r a b l e t y p e s o f e q u a t i o n s w i t h a d e v i a t i n g argument may b e added t o t a l d i f f e r e n t i a l equations.

I f i n some domain D t h e e q u a t i o n

with continuously d i f f e r e n t i a b l e functions M, N, P, where N f O , and t h e c o n s t a n t T > O s a t i s f y t h e c o n d i tion r o t where F = M ( t , x , y ) & + N ( t ! x , y ) j + p ( t , x , y ) k , t h e n it may be represented in t h e form

c=O*,

d L J ( t , x ( t ) , x ( t - T ) ) = 0.

(20)

I n t h i s case, it i s n a t u r a l t o c a l l E q . ( 1 9 ) a t o t a l d i f f e r e n t i a l e q u a t i o n w i t h a d e v i a t i n g argument. The b a s i c i n i t i a l v a l u e p r o b l e m f o r Eq. ( 1 9 ) o r ( 2 0 ) is posed i n t h e f o l l o w i n g form: on t h e i n i t i a l set tO-TLtct t h e r e i s given a continuous, 0

p i e c e w i s e smooth f u n c t i o n $ ( t )and it i s r e q u i r e d t o determine a continuous, piecewise d i f f e r e n t i a b l e f u n c t i o n x ( t ) f o r t O - T i t t O + H , H>O, s a t i s f y i n g Eq.

+

(18) a t a l l p o i n t s of t h e e x i s t e n c e of

0

(t)

f o r t O - T i t L t +H and a g r e e i n g w i t h t h e f u n c t i o n + ( t ) 0

*

W e recall t h a t

17


for t -r O be arbitrarily small and A(y)=F. Then for sufficiently large n

since P ( Y , Y ~ ) < E / ~ because , y=lim n- Y,,

P lynryn+l)< ~ / 3 ,

and, since the sequence (25) is fundamental, finally

Thug, the=distance between the two fixed points, y and y, may be made less than any positive numbgr=e, consequently, the distance equals zero and y=yI i.e. A(y)=y. 3) The fixed point is unique. In fact, under the action of the operator A, the distance between any two points y and z is strictly reduced meaning that both of these points cannot be fixed points.

22

I. BASIC CONCEPTS

Theorem of Existence and Uniqueness of the Solution to the Basic Initial Value Problem for Eq.

(29)

If in Eq. (29) all .ri(t) are continuous for

fies a Lipschitz condition in all arguments beginning with the second, the initial function +(t) is continuous on Et , then there exists a unique solution 0

--ox (t) of the basic initial value problem for Eq. (29)

a .

for t0 0

f o r t > t O .Consequently, f o r any E>O, f o r t - t O < T , it is p o s s i b l e t o choose ~ ( E ) > O so s m a l l t h a t

Proof. max [to I t l

It

m

N C i=O

max l x 4 1 ( t - r i ( t ) ) [toft]

*Cf. Math. S b o r n i k 22 (19481, 193-204

28

-

I . BASIC CONCEPTS

Solving the polynomial inequality relative to

we will have (in fact, setting u(t) = max J x (t)-x (t)I [tO#t1 $1 $2 we obtain u(t) O s u c h t h a t

2 ) t - T ( t , x , y ) 5 t o f o r a l l t,x,y€K1.

I f under t h e s e c o n d i t i o n s

l O , nLm, an>O, bm#O. For t h e v a l u e of z w i t h l a r g e s t modulu8, t h e p r i n c i p a l t e r m of t h e f i r s t sum w i l l be a n z , and , Consequently, f o r 1 z I >>1 t h e second sum bmzme-". t h e c h a r a c t e r i s t i c e q u a t i o n O ( z ) = 0 may be r e p l a c e d by t h e approximate e q u a t i o n n m -TZ a z + bmz e = 0. n To b e g i n w i t h w e c o n s i d e r t h e case n=m, c o r r e s ponding t o a d i f f e r e n t i a l e q u a t i o n of n e u t r a l type. Then, f r o m ( 1 9 ) w e o b t a i n

hence f o r b < O n

71


zk

a

%

-

-

1

a ~nl-l+ bn

2Tki T

...) ,

(k=OIfl,+2,

(20)

and for bn>O

Consequently, if n>m, (18) is an equation with a retarded argument, and from the equation anz n + bm zm e-Tz = 0 (22) we find eTzzP

=

a,

(23)

. Assuming z=z =x +iy where p = n-m>O, a = -bm a k k k n and, having taken the modulus of the left and right TX parts in (23), we obtain e kl zkpI=lal, or raising 2 power, we will have to the -

-

P

It is known

that €or p>O, lim xk= -m, con2XkT k+m = 0 and (24) may be written sequently, lim ex p k+m in the form

where lirn k+w

E~

= 0, or

12

1 1 . LINEAR EQUATIONS

2 TX Tk= 0 a n d a#O, t h e n 2 yk - m. lim 2 -

S i n c e l i m xk2 e k+m

k-tw

Xk

The r o o t s o f Eq. ( 2 3 ) a r e e n c o u n t e r e d i n c o n j u g a t e p a i r s ; t h e r e f o r e , it i s p o s s i b l e t o t a k e t h e r o o t w i t h yk > O and t h e n

= T/2 + a r g zk = a r c t a n yk X k where

E~

+

0 as k

-+

m.

T a k i n g t h e l o g a r i t h m of imaginary p a r t s , w e o b t a i n T

Z

+~

plnlzkl

+

E2,

(23) and comparing t h e

i p a r g zk = i a r g a + 2 k a i

+

p arg z

or

- 7 (2kn - p

yk -

k

= arg a

5) +

E~

+

+ lnlal,

2ka

f o r a>O

(26)

and yk -

7

( 2 k ~- p

where l i m t = 0 . 3 k+m

73

+

T)

+

E~

for a < O ,

(27)


Taking the logarithm of (25), we find

X

k

= 1 (-pin yk + Inla]) + c 4 r Xk T where E 4 - + O as k+-. In the right-side yk may be replaced by (26) or (27). From this we obtain 1 ai for a>O, 'L + (2k-P) + h/al k ' 'L 'I -pln(2k- $) 2 T

5

-

(28)

k '

1 T

'L

-pln(Zk+l-f)

ni

+ Inla( + (2k+l-$) Ifor a 4 (29)

The case m>O, corresponding to a differential equation of advanced type, may also be studied by this method. If the equation contains more than one deviating argument, then leaving only the dominant terms we also obtain, generally speaking, an equation of the form (22), but, in exceptional cases, it is also possible to obtain an equation containing more than two terms. For example, let the equation contain the three dominant terms -TIZ -T2Z a z n + bmzm e +czPe = 0, n P

(30)

where p =

nln\zl = mln)z)-.rlRe z = plnlzl

- T Rez. ~

(31)

Rewriting Eq. (30), we have by virtue of Eq. (31)

74

II. LINEAR EQUATIONS

lnlzl -zln I z I -2 nz In(zL Rez m Re z e [an(ze )” + bm(ze 1

+

-2

c (ze

P

.

and, finding the roots ui (i=1,2,. .I of the algebraic equation a un + bmum + c up = 0 8 n P we obtain an equation of the form zeqz = ui, (i-1,2,...) for defining z , where in correspondence with (31)

which was considered above. We treat a greater number of dominant terms in a similar fashion. The method given above allows the possibility of the approximate determination of the roots of largest modulus of the characteristic equation. These roots are often called the asymptotic roots of the equation. The non-asymptotic roots of the characteristic equation may be found from other approximation methods, for example, Newton’s method. As a first approximation to the non-asymptotic roots it is sometimes convenient to take values of the roots calculated by the asymptotic formulas

(20) (21) (28) (29)

In the formulas (201,(211,(28), (291, it is not difficult to estimate the order of error, it equalling respectively (cf. [I.171)

.


It is also possible to obtain the next terms of the asymptotic representations. We noce that, as is obvious from the asymptotic formulas ( 2 0 1 , (211,(281, (29)I all roots of the quasipolynomial are separated from one another by some positive distance d and therefore it is possible to circumscribe around each root, with the root as its center, a circle of radius rzd, non-overlapping with the circles of the other roots. It is easy to show that this property is preserved for all characteristic quasipolynomials corresponding to equations with a single deviating argument and also €or a wide class of characteristic quasipolynomials corresponding to equations with several deviations. However, there exist equations with a deviating argument for which the asymptotic roots of the characteristic quasipolynomials become arbitrarily close. In this case, it is possible to find a number d, such that all roots of the characteristic quasipolynomial will be inside non-overlapping discs Ci (i=1,2,. .) of radius rn from ,:L for study of the behavior of solutions of Eq. (54) with a deviating argument it is possible to use the qualitative results of the theory of differential equations without a deviating argument of the form (57) (the theorem of equivalence to an equation without a deviating argument p. 40). Results of such type are obtained for an equation of the second order with a retarded argument in f89.31 , [89.8] , [89.10], tI.101 and in one special case, for an equation of neutral type [66.1]. The possibilities of such an approach are still far from being exhausted. On the other hand, the separation of the finite-dimensional subspaces admits in a series of cases, the study of specific properties of the solutions, connected with the presence of a deviating argument (for example, the structure of the set of zeroes of the solutions, the comparison of two different solutions on the whole interval, specific abilities to oscillate, and so on). For equations with a retarded argument, results of such type are obtained in 189.141 , [89.16] , L89.171 , [I.lOl.

97


7.

Some Types of Linear Equations with Variable Coefficients and Variable Deviating Arguments

In this paragraph, we briefly pause to look at some more frequently studied types of linear equations with variable coefficients and variable deviating arguments. 1. Equations analogous to Euler's equation. The equation of the form

..

where all a and ps are constants, l=po>pl>. > rs pm>O, is natural to regard as an analog to the Euler equation since by the change of independent variable U U t = e (or t= -e ) , Eq. (63) is transformed into a linear equation with constant coefficients and constant retardations. It is possible, of course, to directly seek a particular solution of the homogeneous equation corresponding to Eq. (63), in the form x(t)=tk. Then for the determination of k we obtain the equation

or, letting ps = e-T s ' n m c c a k(k-l)--*(k-r+l)e (k-r)7s = 0. (65) r=O s=O rs

-

To the root ki of multiplicity ai of Eq. (64) or (65) corresponds the solutions us ekiu (s=O,l,..., (ai-l)) of the transformed homogeneous equation, or the solutions k (ai-1)) t ilnSJtl, ( s = o , ~ ,

...,

98

11. LINEAR EOUATIONS

-

of the homogeneous equation, corresponding to Eq. (63)

2. Reducible Systems of Equations. homogeneous equation %(t) = Ao(t)x(t)

The linear

+ Al(t)X(t-T)

(66)

is called reducible if there exists a non-singular differentiable linear transformation of the unknown function x (t) = B (t)Y (t)t

(67)

transforming Eq. (66) into a linear equation with constant coefficients and constant deviating arguments. In (66) and (671, it is possible to assume that x(t) and y(t) are n-vectors, while Ao(t) ,Al(t), and B(t) are nxn matrices. The transformation (67) transforms Eq. (66) into B (t)3 (t) = [Ao(t)B (t)-6 (t)I y (t) + A1 (t)B (t-T)y (t-T) P(t) = B-l(t) [Ao(t)B(t)-i(t)ly(t)

+

B-1 (t)Al(t)B(t-r)y(t-T). Consequently, for reducibility, it is necessary and sufficient that the matrices B-’(t) [Ao (t)B(t)(t)I and B-l (t)A1 (t)B (t-T) be constant. The first of these two conditions signifies the reducibility of an equation without a deviating argument A(t) = Ao(t)x(t).

(68)

However, for the reducibility of Eq. (661, it is necessary and sufficient that Eq. (68) be reducible and the matrix

99


3. Linear Equations with Periodic Coefficients Equation ( 6 8 ) , if the matrix Ao(t) is periodic, as is knowi., is reducible (Lynpunov's Theorem) , hut F,q. (66) with periodic matrices Ao(t) and Al(t) with a general period T is, generally speaking, not reducible since the matrix B(t), with the aid of which Eq. (68) is reduced, although defined with some degree of arbitrariness is, generally speaking, not sufficient for the satisfaction of the condition

B-1 (t)Al(t)F(t-T) = constant. In the scalar case, for Eq. (66) with periodic coefficients of period T, and even for the more general equation m (69) 3(t) = C ap(t)x(t-pT), p=o where a (t) is a periodic function of period T, there P exists, generally speaking, an infinite set of solutions of the form f (u)du x(t) = e 0 (70)

'1

where f(u) is a periodic function of period T. In fact, substituting (70) into (69) and intro-

rf

li'

-T

ducing the notation k = lation of

e 0

f (u)du, after cancel-

0 we obtain

( ' ) d u I

f(t) =

m C

p=o

a (t)e-PTk.

100

1 1 . LINEAR EQUATIONS

For the determination of k, we multiply (71) by dt and integrate between 0 and T: = 0.

where

2p =

lo T

$ i

...,m).

ap(t)dt, (p=O,l,

Simple roots k . of Eq. (72) correspond, by virtue of ( 7 0 ) , to $he solutions

x. (t) = e 1 where

It

0

f (u,ki)du I

m -pTk f (t,k) = C ap(t)e p=o

It is possible to prove that multiple roots ki of Eq. (72) of multiplicity a i correspond to the solutions t Xij(t) =

[A 8k’

j0 f(u,k)du

...,(ai-l)i

(j=O,l,

I

k=ki,

(73)

...) .

i=1,2,

The system of solutions (73) of Eq. (59) is not, in the general case, fundamental. For example, f o r an equation of the type (69) k(t) = a(t)x(t-T),

101


as is easily verified, the system (73) consists, in jta (u)du general, only of the single solution x(t)=e 0 Nevertheless, in many cases, the solutions (73) allow us to decide about the stability and other asymptotic properties of the solutions of Eq. (69) (cf. A.M. Zverkin 132.11, i32.71). From (70) it follows that x(t+T) = ekTx(t)

.

(74)

It is easy to show that each function satisfying the funct.iona1 equation ( 7 4 ) is representable in the form x(t) = ekt$ (t),

(75)

where $(t) is a periodic function of period T. (73) with j=O we obtain then, that in (75) $(t) = @(t,k) = e-kt+

1;

f(u,k)du.

From

(76)

by virtue of (7!3,(76), and (73), for a root ki of Eq. (72) of multiplicity ai, differentiating according to Leibniz' rule, we obtain a i linearly independent solutions of the form Now,

102

I I . LINEAR EQUATIONS

where

1s ( t + T )

JI.

=

qis ( t )

S o l u t i o n s of t h e form ( 7 7 ) are c a l l e d s o l u t i o n s of F l o q u e t t y p e . According t o F l o q u e t ' s t h e o r y , * f o r a l i n e a r homogeneous d i f f e r e n t i a l e q u a t i o n w i t h o u t a d e v i a t i n g argument w i t h p e r i o d i c c o e f f i c i e n t s , s o l u t i o n s of t h e form ( 7 7 ) form a b a s i s f o r t h e s p a c e of s o l u t i o n s . The m o s t complete r e s u l t , connected w i t h t h e t r a n s f e r e n c e o f F l o q u e t t h e o r y t o d i f f e r e n t i a l equat i o n s w i t h a r e t a r d e d argument w i t h p e r i o d i c c o e f f i c i e n t s and c o n s t a n t r e t a r d a t i o n s , belong t o A. S t o k e s [184.1] ( c f . a l s o t h e works [131.11], i131.131, [159.10], [159.18], [9.3]). Theorem. L e t x ( t ) be a s o l u t i o n of t h e b a s i c 4 i n i t i a l v a l u e problem f o r t h e l i n e a r homogeneous d i f f e r e n t i a l e q u a t i o n ( 6 6 ) w i t h p e r i o d i c matrices of p e r i o d T and c o n s t a n t r e t a r d a t i o n T > O , d e f i n e d by a c o n t i n u o u s v e c t o r f u n c t i o n $ ( t )on t h e i n i t i a l s e t Et Then f o r any a > O , it i s p o s s i b l e t o f i n d 0 an N and c o n t i n u o u s v e c t o r f u n c t i o n s @ , ( tand ) @O(t) on t h e i n i t i a l s e t E

.

x (t)= x

cb

ON

( t ) + x , ( t ), toftO, and all sufficiently large t,we have 2 then the solution x (t) shall be called rapidly damp4 ed. It turns out that, under condition (861, the totality of solutions of Eq. (81) of stable type decompose into those which are slowly damped and rapidly damped. There are no solutions with intermediate velocities of damping - see the remark on p. 35 about the separation of principal solutions. All the results presented for Eq. (81) are given in the monograph of A . D . Myshkis 11.91 in terms of an equation with a distributed delay.* Far more recent results connected with Eq. (8U, we mention the comparison theorems of V.R. NOSOV [91.1]. In these theorems, an arbitrary positive solution xl(t) (xl(t)>O on the interval tOct<m) of Eq. (81) is selected and it is compared with an arbitrary oscillatory solution x2(t) of this equation (the solution x2(t) is called oscillatory if it changes *Equations with a distributed delay are treated in Ch. VII, §l. 109


s i g n on any i n t e r v a l ( a , = ) , where a i s an a r b i t r a r y number). For comparison of t h e speed of growth of t h e c o n s i d e r e d s o l u t i o n s , t h e f o l l o w i n g method t u r n s o u t t o be e f f e c t i v e . W e c o n s i d e r t h e f u n c t i o n x2 ( t ) The f u n c t i o n V ( t ) i s a c o n t i n u o u s l y V(t) = x , ( t ) d i f f e r e n t i a b l e o s c i l l a t o r y s o l u t i o n of t h e e q u a t i o n

.

v

I

(t)=

M ( t ) x1

(t-T

( t )1

(t)

[V(t-T (t) )-V(t)

1

(87)

For Eq. ( 8 1 ) , l e t it be p o s s i b l e t o f i n d a t > t o ,such t h a t i n t h e i n t e r v a l t 0< t < =- c,( t ) > O , t - T ( t ) > t O , and M ( t ) # O . Then t h e f u n c t i o n V ( t ) i s bounded f o r t < t < = . 0

0-

I n f a c t , w e c o n s i d e r t h e sequence z o f s e m i k c y c l e s of t h e f u n c t i o n V ( t ) , on which V i s p o s i t i v e and l e t tk be t h e p o i n t where V ( t ) a t t a i n s i t s The sequence V ( t k ) i s bounded g r e a t e s t v a l u e on zk. 0 from above. Otherwise, t h e r e e x i s t s a p o i n t tk > t , 0 such t h a t V ( t k ) > V ( t k ) f o r a l l k < k o , a s V ( t k ) > V ( t ) , 0 1 0 ( t o i t < t1. k But, V ( t k ) = 0 and, by ( 8 7 1 , 0

0

W e o b t a i n a c o n t r a d i c t i o n , p r o v i n g t h e boundedn e s s of t h e f u n c t i o n V ( t ) from above. The boundedn e s s from below i s shown a n a l o g o u s l y .

Thus, under s u i t a b l e a s s u m p t i o n s , any o s c i l l a t o r y s o l u t i o n x 2 ( t ) of Eq. ( 8 1 ) i n c r e a s e s no more r a p i d l y t h a n any p o s i t i v e s o l u t i o n of t h i s e q u a t i o n . I f i n Eq. ( 8 1 ) t h e c o e f f i c i e n t M ( t ) o s c i l l a t e s , t h e n it i s n o t d i f f i c u l t t o c o n s t r u c t examples when an o s c i l l a t o r y s o l u t i o n of t h i s e q u a t i o n i n c r e a s e s more r a p i d l y t h a n a p o s i t i v e s o l u t i o n . However, i n t h i s case it i s p o s s i b l e t o compare t h e speed of its increase. 110

( I . LINEAR EQUATIONS

Let there exist positive constants C1 and C2 and t0>to such that for a positive solution of Eq. 0 (811, x,(t), on the interval t O , 4 there exists a 6 ( ~ ) > 0 ,such that from the inequality I $ (t)- $ (t)I < 6 ( € 1 on the initial set, there follows Ix4 (t)-x$ (t) I < E for all tLto, where JI (t) is any continuous initial €unction.

Solutions not possessing this property are called unstable. 119


11. Definition of Asymptotic Stability. A stable solution x (t) is called asymptotically 4 stable, if & i g Ix (t1-x (t)I = 0 for any continuous 4 JI initial function $(t), satisfying for sufficiently small 6 1> O the condition 1 (I (t)-$(t) 1 0 , such that for each E > O there exists a T(E) such that for for any continuous init>tl+T(E), Ix4 (t)-xJ,(t) tial function $(t) satisfying the inequality 14 (t)-J,(t)I < 6 on the initial set Et , where 6 does 1 >t0' not depend on the choice of t1IV. Definition of Exponential Asymptotic Stability. The solution x (t) of Eq. (1) is called 4 exponentially asymptotically stable if there exist constants 6>0, u > O , B > 1 such that from the inequality 14 (t)-J, (t)1 < 6 , there follows Ix (t)-x (t)I < 4 J, B max I + - ~ , l-a(t-tO) e for t>T. tEEt

I<E

0

V. Definition of Asymptotic Stability in the Large. The solution of Eq. (1) x (t) is called 4 asymptotically stable in the large, if it is stable and'

for all continuous initial functions $(t). All definitions remain unchanged if, in Eq. (11, x(t) is an n-vector or even an element of a Banach space. In this case, only the modulus sign I I need be changed to the norm sign 1 1 I I.

120

Ill. STABILITY THEORY

Sometimes in the definitions of stability and asymptotic stability it is expedient to use not the metric space C but some other (cf. 11.131 1. 0' For an equation of neutral type, 9 (t) = f (t,X(t),X (t'T1

.

(t)) I . . ,X (t-T (t)) m

all definitions given above remain unchanged, but in place of 'the requirement 1 4 (t)-J, (t)I < 6 it is usually necessary to require nearness in the space C1 : I+(t)-J,(t)1.6

and I@'(t)- ~l'(t)1.6.

For investigations into the stability of some solution x (t) of Eq. (1) or (21, it is possible by the change4 of variable y(t)=x(t)-x (t) to transform 4 the discussion to the stability of the solution x (t) into that of y(tIr0. Therefore, in the fol4 lowing discussion on stability, we study only the trivial solution.

2.

The Stability of Solutions of Stationary Linear Equations

All solutions of the linear equations with a deviating argument L(x(t)) = f(t)

(3)

with fixed initial point t , as for a linear equation without a deviating apgument, are stable or unstable simultaneously. In fact, any solution x (t) of Eq. (3) by the 4 change of variable y(t)=x(t)-x (t) turns into the 4 trivial solution of the corresponding homogeneous equation L(Y(t)) = 0.

121

(4 1


Consequently, all solutions of Eq. (31, in the sense of stability, will lead in the same way to the trivial solution of Eq. ( 4 ) . In particular, for f (t)-0 all solutions of the homogeneous equation, in the sense of stability, will lead in the same way to the trivial solution of this equation. Particularly simple is the study of the stability of the solution of linear equations with constant coefficients and constant deviating arguments:

where apj and

T

j

are constants, T m > T ~ - ~ > . . . > T ~ > T ~ = ~ .

..

In Chapter II,§3, it was shown that if in Eq. # 0 and a = 0, j = O,l,. ,m-1 (in this nj case Eq. (5) is of advanced type), then there always exists a solution of the form Cekt, where c is an arbitrary constant with Re k>O. Consequently, all solutions of Eq. ( 5 ) in this case are unstable, since,for arbitrarily small IC1,the solution of the form Cekt is either unboundedly increasing in modulus as t- or for complex k , the solutions CRe ekt and C Im ekt oscillate with unboundedly increasing amplitudes. (51, anm

If Eq. ( 5 ) is an equation with a retarded argument (a # 0 and ani= 0, i=l,2,...,m~, then any nO solution x (t) for t +T(tT may be decomposed into 0 0 an absolutely and uniformly convergent series of basic solutions.

where p.(t) is a polynomial of degree less than or 3 equal to c1 -1, c1 is the multiplicity of the root j j k . of the characteristic quasipolynomial 7

122


-? .z m c a zpe 3 p=o j=O pj Re kl > - Re k&. Re knl..

n

(7)

1

..)

.

If all roots of the characteristic quasipolynomial have negative real parts, then the remainder in the series may be represented in the form -knt e Rn(t) , where IRn(t) I < € for n>N(E). Thus, there follows the asymptotic stability of solutions of Eq. (5) t'tO+'

Moreover, it is possible to prove that for (Re kn+l+E)t c pj(t)ekjt/ < BRe I j=n+l

1

where B and E are constants, E > O and arbitrarily small (see [1.17]). Consequently, for Re kld; conse3 j quently, max Re k = Re kl exists and, if Re kl O , such that Re kl+EO.

In the case under consideration, the characteristic equation has the form

+

be-'IZ

wo(z) =

-b z+a -

z

+

a

=

0,

The limiting characteristic is the form of the imaginary axis under the fractional linear transformation (12). Under this mapping, the imaginary axis transforms into the circle of radius lb/2a] with center at the point z=-b/2a, the equation of which has the form

Let a>O, then the function wT(z) has no poles in the half-plane Re z>O and, if IblO, w e c h o o s e sup V < i;f W , where U6 i s t h e U6 , t > t

> O such t h a t

- 0

SE

&-neighborhood o f t h e o r i g i n , and S E i s t h e s p h e r e o f r a d i u s E w i t h c e n t e r a t t h e o r i g i n . By t h e e x i s t e n c e o f t h e i n f i n i t e l y s m a l l u p p e r l i m i t , 6 ( ~ )may be chosen independently of to. T r a j e c t o r i e s beginning i n U

6

d o n o t r e a c h t h e l i m i t S E , and must p a s s

i n t o a n a r b i t r a r i l y s m a l l n-neighborhood of t h e origin. I n f a c t , i n t h e o p p o s i t e case, -dV < - a dt -

< 0

(34)

a l o n g a t r a j e c t o r y , where a i s a c o n s t a n t . Multip l y i n g ( 3 4 ) by d t and i n t e g r a t i n g from t o t o t > t O , we obtain V - V ~ 5 -a ( t - t O ) .

(35)

From ( 3 5 ) , f o r s u f f i c i e n t l y l a r g e t , w e o b t a i n V < O , which c o n t r a d i c t s t h e f i r s t c o n d i t i o n o f t h e theorem. A s a r e s u l t of t h e a r b i t r a r y c h o i c e of n , asymptotic s t a b i l i t y i s proved. As a r e s u l t o f t h e i n d e p e n d e n c e o f 6 ( ~ from ) t o , uniform asymptotic s t a b i l i t y i s a l s o proved.

146

I l l . STABILITY T H E O R Y

The theorem can be inverted. Chetayev‘s Instability Theorem. The trivial solution of the system (31) is unstable if,in an arbitrarily small neighborhood of the origin, for t>t there exists a neighborhood U, not depending on -0 t, in which the function V(xl, x-,t) satisfies the conditions:

...,

1. v>o; 2.

3 dt

> 0

dV > ’ dt-

3.

in such a way that

B > 0 in the region Vza>Q;

in the neighborhood of the origin for txt the function V is bounded.

Idea of the proof. We choose an initial point such that V ( ~ ~ ~ , . . . , x ~ ~ =, tal>O. ~) Then by condition 2, for t’tO the trajectory remains in the region V>al>O, and therefore along the trajectory, dV dt 2 B1

’ 0.

Multiplying (36) by dt and integrating from tO,to t>tO, we obtain

v-vo 2

B1 (t-tO).

(37)

From (37) we find that V-m for t-, which contradicts condition 3. Therefore, for t- the trajectory leaves the neighborhood of the origin. Since on the basis of condition 1 the initial point may be chosen arbitrarily close to the origin, the instability is proven. Obviously, the formulation and proof of these three theorems is almost unchanged if the system (31) is changed to the system of differential equations with deviating arguments,

147


jCi

...,xn (t),x1 (t-.cl(t)), ...,XI(t-Tm(t)1 ...,xn (t'Tl (t),...,xn (t"Trn(t)1 ,t) ,

(t)=fi (xl(t),

(i=1,2,. where

T

T

i (t)LO.

..,n),

I

(38)

In this case,

dV dt (xl,...,xn' t)

=

n av c i=l axi fi

av at +

becomes a function of the n(m+l)+l arguments t,Xl(t) ,...,xp ,x1(t-Tl(tH

r...rX

n (t-Tm(W

I

and its non-positiveness in the first theorem (or negative definiteness in the second theorem, and positiveness in the instability theorem) may be understood as non-positiveness for independently changing arguments, or else it may be borne in mind that xi(t-Tk(t)) is one of the previous values of the function xi (t). tem

For example, the trivial solution of the sys-

?(t) = -x(t)-y(t)x

2

(t-T2(t)) ,

T . > O , ]=1,2,is asymptotically stable since for a 3Lyapunov function,it is possible to take V = x2+y 2 For this

.

However, for equations with a deviating argument such a direct transfer of Lyapunov's second method is impossible to consider as a general technique for stability investigations, since the theorems of this method do not admit conversion for equations with a deviating argument,

148


I n f a c t , if t h e s e theorems a d m i t t e d c o n v e r s i o n , then f o r the equations

2 ( t ) = f ( x ( t ) , x ( t - r ) ) and & ( t ) = kf ( x ( t ) , x ( t - T )

,

where k i s a p o s i t i v e c o n s t a n t , s t a b i l i t y must o c c u r f o r b o t h s i n c e , i f t h e t r i v i a l s o l u t i o n of t h e f i r s t e q u a t i o n i s s t a b l e , t h e n by t h e c o n v e r s e theorem, t h i s e q u a t i o n must have a Lyapunov f u n c t i o n V ( t , x ) , and t h e n i t i s p o s s i b l e t o choose V ( k t , x ) as a Lyapunov f u n c t i o n f o r t h e second e q u a t i o n . However, e l e m e n t a r y examples show t h a t t h i s may n o t be so. C o n s i d e r , f o r example, t h e e q u a t i o n s

and k(t)

+

k [ a x ( t ) + b x ( t - T ) l = 0.

I t i s e a s y t o show t h a t t h e s t a b i l i t y r e g i o n s f o r t h e s e e q u a t i o n s do n o t c o i n c i d e . On p. 1 6 t h e domain of s t a b i l i t y f o r Eq. ( 1 7 ) w a s c o n s t r u c t e d by t h e method o f D - p a r t i t i o n s ( t h e domain I, p=O - F i g . 11). For i t s upper boundary, upon e l i m i n a t i n g t h e parameter y, we o b t a i n t h e equation

=-T

arccos (-a/b)

.

Now it i s o b v i o u s t h a t t h e s u b s t i t u t i o n o f a and b

by ka and kb a l t e r s t h i s boundary.

N e v e r t h e l e s s , i n a series of cases, a p p l i c a t i o n o f Lyapunov f u n c t i o n s f o r s t u d y o f t h e s t a b i l i t y of s o l u t i o n s o f e q u a t i o n s w i t h a d e v i a t i n g argument have proved t o be e f f e c t i v e . Some m o d i f i c a t i o n s of t h i s method are g i v e n i n [ 1 0 1 . 2 1 , D01.41. An i d e a which h a s proven t o be much more f r u i t f u l i n t h e g e n e r a l case, i s t h a t o f N.N. K r a s o v s k i i ( c f . [ 5 2 . 1 ] , [ 1 . 3 ] ) , who has c o n s i d e r e d , i n p l a c e of Lyapunov f u n c t i o n s , f u n c t i o n a l s w i t h analogous properties.

149


...

We consider a vector function x(s) with compo,xn ( s ) I , defined on the internents {xl ( s ) ,x2 ( s ) , val - T < s < O (in what follows, s always varies within the inaicated limits). For each tLto, there is defined on the vector function x(s) a functional V[Xl(S) I . . . I Xn ( s ) ,tl = V[X(S) ,tl Definition I. The functional

is called positive-definite if there exists a continuous function +(r)>O such that for r # 0

A negative-definite functional is analogously defined.

The norm of the vector-function x(s) may be taken in various spaces. In what follows, we shall need norms in the spaces C0 and L2, and sometimes, particularly for equations of neutral type, in the space C1' We introduce the following notation:

11x1

l2

=I : i=l

150

Xi2]

i

I l l . STABILITY THEORY

U E w i l l mean t h e &-neighborhood o f t h e e q u i l i brium p o i n t x =x = * * - = x= O i n t h e m e t r i c o f C 1 2 n 0; S E i s t h e € - s p h e r e which i s t h e boundary o f UE.

D e f i n i t i o n 11. The f u n c t i o n a l V [ x ( s ) , t ] has an i n f i n i t e l y s m a l l upper l i m i t i f t h e r e e x i s t s a c o n t i n u o u s f u n c t i o n c $ ( r~) ) O w i t h q1 (O)=O s u c h t h a t V[x(s),tl 5 41(llx(S)IIr)' S t a b i l i t y Theorem. The t r i v i a l s o l u t i o n o f t h e s y s t e m (38) i s s t a b l e i f t h e r e e x i s t s a c o n t i n u o u s vositive-definite functional

whose d e r i v a t i v e a l o n g i n t e g r a l c u r v e s i s non-positive.

where x ( t + s ) i s t h e s o l u t i o n o f Eq.

(38) defined by 4 t h e i n i t i a l v e c t o r f u n c t i o n $ ( t ) ,( t O - T & t z t 0 ) 2

Proof. such t h a t

For a given E > O

151

(E O

DIFFERENTIAL EOUATIONS WITH DEVIATING ARGUMENTS

(on t h e s t r e n g t h o f t h e p o s i t i v e - d e f i n i t e n e s s of t h e f u n c t i o n a l V [ x ( s ) , t ], and t h e c o n t i n u i t y of t h e f u n c t i o n a l V [ x ( s ) , t ] i n t h e neighborhood of x ( s ) =O). 0

For such a c h o i c e of & ( E ) , any i n i t i a l v e c t o r f u n c t i o n 4 ( t ) s a t i s f y i n g t h e c o n d i t i o n I 14 ( t O + sI )I T < & ( E ) d e t e r m i n e s a s o l u t i o n x (t) f o r t t o s u c h t h a t QI

I

s i n c e t h e f u n c t i o n i n t o which t h e f u n c t i o n a l V is converted along an i n t e g r a l curve does n o t i n c r e a s e a l o n g a t r a j e c t o r y , and t h e r e f o r e by i n e q u a l i t y (39) I I x 4 ( t ) I I Tc a n n o t e q u a l E . K r a s o v s k i i ' s Asymptotic S t a b i l i t y Theorem. The t r i v i a l s o l u t i o n o f Eq. (38) is u n i f o r m l y a s y m p t o t i c a l l y s t a b l e if f o r t2t0 and I Ix ( s ) I I T+ where H > O , t h e r e e x i s t s a c o n t i n u o u s p o s i t i v e - d e f i n i t e f u n c t i o n a l V [x ( s ) , t ] w i t h an i n f i n i t e l y s m a l l upper l i m i t such t h a t t h e d e r i v a t i v e w i t h r e s p e c t t o t of V [ x ( s ) , t ] i s n e g a t i v e - d e f i n i t e . Here x@!t+s) i s t h e s o l u t i o n o f Eq. ( 3 8 ) d e f i n e d b t h e i n i t i a l vector-function & Q.> .Io.t( is sufficiently s m a l l . I.

N.N.

U

Proof. For g i v e n € 7 0 , w e choose 6 ( ~ ) 7 Osuch that=>$* ( 6 ) (see t h e d e f i n i t i o n s on pp.32). Then as a consequence of t h e i n e q u a l i t i e s

I 10 ( t O + sI)I r < & , t h e

f u n c t i o n V ( t ) = V[x@( t + s ), t ] Therefore, f o r t > t t h e t r a j e c t o r y x = x ( t ) remains i n t h e

For

i s a monotone d e c r e a s i n g f u n c t i o n o f t. - 0

Q

region 1 Ix,(t) 1 I T < € , s i n c e i n t h e o p p o s i t e case i n e q u a l i t y ( 4 0 ) would be v i o l a t e d . By t h i s , t h e s t a b i l i t y of t h e s o l u t i o n x ( t ) = O is proved.

152

I l l . STABILITY THEORY

We choose an arbitrarily small n>OI and for such that this we choose S,(n)>O

1 Ix(s)

SUP

I lT (l-a)[V[x@ (to),tO]+atOl). Therefore, there exists a point

such that

I Ix,(t*) 1 l T < A l ( r l ) , *

IlxQ(t)]l,t

and consequently,

+T.

Since TI is arbitrary, asymptotic stability is proved. Since 6 ( r l ) is independent of to and the estimate ( 4 2 ) , the asymptotic stability is uniform. Remark I. The proof is essentially unchanged if the derivative in condition 3 is replaced by the AV right-hand derivative lim sup (see lI.31). Such a t+O a change is also possible in the stability theorem. Remark 11. N.N. Krasovskii's asymptotic stability theorem admits inversion, i.e. if the system of equations ( 3 8 ) has a uniformly asymptotically stable trivial solution, then there exists a functional V[x(s),t] satisfying all the conditions of 153


Krasovskii's asymptotic stability theorem, and a Lipschitz condition in the first argument: IV[X,(S) rtl-V[x2(s) It1

lJ'I

Ix2(s)-x1(s)

I IT.

Scheme of the proof. From the uniform asymptotic stability follows the existence of a continuous monotone-decreasing function $(t) such that lim $ (t)=O, and satisfying the inequality t-+m for any initial funcI Ixo(t+s)I 1 ' 1 ~ $(t-to) for t>t - 0 tion I I@(to+S) I I t0' it is often expedient to separate out the linear part and represent the system in the form n m (t)X (t-TQ(t) +Ri (t,xl(t)I I A+) = c 7 j=1 R = O aijll Xn (t),X1 (t-Tl(t))

1 . .

.I Xn (t-Tl(t)

I

n (t-Tm (t) I (49) n ) , ~ ~ = 0T,11-> 0 1

...,

(i=1,2,

159

IX


where Ri is of greater than first order in the set of all arguments beginning with the second. In many cases, the investigation of stability of the null solution of the system ( 4 9 ) is equivalent to the investigation of stability of the null solution of the simpler linear system n m k. (t) = c C aij,(t)x.(t-r,(t)), (i=1,2 n),(50) 1 7 j=1 k = O

,...,

which is called the first approximation for the system ( 4 9 ) . The case of variable coefficients and variable retardations ~,(t) in the linear part of the system ( 4 9 ) is still insufficiently worked out. Only systems ( 4 9 ) for which the system (50) has constant coefficients and constant retardations have been studied in much detail. Such systems are called stationary in the first approximation. The following theorems analogous to the corresponding theorems of Lyapunov have been proved: Theorem I.

The null solution of the system n

m

is asymptotically stable if: 1. all roots of the characteristic equation for the first approximation system for ( 4 9 1 ,

where A k are the matrices)kj&=aA which, for fixed 8 , have roots w i t h negative real parts:

160

111. STABILITY THEORY

.

where a>O is a sufficiently small constant, all Iui are sufficiently small, lui I O . 1

It is not difficult to verify that €or sufficiently small c1 this functional will satisfy the conditions of the asymptotic stability theorem I ( p . 34) for the system (51) also. Example 1. Investigate the stability of the trivial solution of the equation ?(t)

+

+ 2x(t-.r) =

3 sinx(t)

0.

(56)

The first approximation equation has the form ?(t) + 3X(t)

+ 2X(t-T)

= 0.

Its trivial solution is asymptotically stable for any T>O. Therefore, the solution of Eq. (56) is asymptotically stable for any T , O < T < ~ . Example 2. Investigate the stability of the solution x-0 of the equation A(t) + 2x(t) -sinh x(t)-2x(t-~)+cos x(t-T)=l ( 5 7 ) for T > O . The first approximation equation A(t)

+ x(t)

-

Sx(t-r) = 0

has roots with a positive real part €or any r > Q ; consequently, the solution x-0 of Eq. (57) is unstable. 8.

Stability Under Constantly-Acting Disturbances

The solution xiEO (i=1,2,...,n) of the system of equations Ki(t)=fi(t,x

j

(t-Tk(t)),

(i,j=1,2

,...,n;k=1,2,...,m) (58)

162


is called stable under constantly acting disturbances if for every E > O I there exists C ~ ~ ( E ) > Oand 6 2 ( ~ ) > 0such that solutions of the perturbed system jri(t) = f. (tIX (t-~k(t)))+Ri(t,X.(t-T~(t)f)I I j 3

...,m)

(i,j=l,Z,...,n; k=1,2,

(59)

satisfy the inequalities Ixi(t)

(i=l121...rn)Itlto

for

Theorem. If the solution x.: O (i=l12,...,n) of 1 the system ( 5 8 ) is uniformly asymptotically stable and the tunctions fi satisfy Lipschitz conditions in all arguments beginning with the second, then this solution is stable under constantly acting disturbances. Scheme of the proof. For the system ( 5 8 ) there exists a functional V satisfying the conditions of the asymptotic stability theorem. In an arbitrarily small neighborhood of the trivial solution, this functional satisfies the condition < - a < O I where a is a constant for the lim+ sup At At+O system (58), on the basis of the negative-definiteness of lim+ sup AV At At+O

a

-.

Examining this functional along solutions of the perturbed system ( 5 9 ) for sufficiently small A,, we obtain lim sup - < -a+KsuplRil , where K is a& + At At+O positive constant. I63


Choosing 61

a 2K'

c

(60)

AV 2 - a / 2 < 0 . Therefore, by the we obtain lim+ sup At At+O L1V negative-definiteness of lim+ sup =,in an arbitrarAt+o ily small neighborhood of the trivial solution a trajectory may not pass outside the limits of a sufficiently small neighborhood of the trivial solution, which signifies stability under constantlyacting disturbances. 1 - 7

Remark I. If the function fi(t,x.) and ~,(t) 3 are periodic functions of t with some period T (or, in particular, independent of t), then the asymptotic stability is always uniform. Remark 11. Sometimes the concept of stability under constantly-acting disturbances is also taken to include stability with respect to disturbances of the deviations of the arguments; that is, small variations of the functions T k (t) are allowed in the perturbed system ( 5 9 ) . In this case, the system ( 5 9 ) assumes the form: Zi(t) = fi(t,xj(t-hk(t)))+Ri(t,xj (t-hk(t))) (i=1,2,...,n) where all the differences where 63>0.

,

all hk(t),O,

I hk ft)-

I ~6~

T ft) ~

(€1,

The formulation of the theorem and scheme of its proof remain unchanged in this case, the only difference being that in the estimate (601, new terms appear which are arbitrarily small for sufficiently small 1 5 ~ . For more details, see [I.3].

164


Is t h e t r i v i a l s o l u t i o n o f t h e

Example 1. equation

+

2 x ( t ) +x(t--r) = 0 , r > o ,

s t a b l e under c o n s t a n t l y a c t i n g d i s t u r b a n c e s ? The t r i v i a l s o l u t i o n of t h e c o n s i d e r e d e q u a t i o n i s u n i f o r m l y a s y m p t o t i c a l l y s t a b l e and t h e r e f o r e i s

s t a b l e under constantly a c t i n g dis t u r b a n c e s . 9.

Lyapunov's Second Method f o r E q u a t i o n s o f N e u t r a l Type

I n t h i s p a r a g r a p h i t w i l l b e shown how N . N . K r a s o v s k i i ' s form of t h e theorems o f Lyapunov's second method may b e g e n e r a l i z e d f o r t h e s y s t e m o f d i f f e r e n t i a l e q u a t i o n s w i t h a d e v i a t i n g argument o f neutral type

zi ( t ) = x

n

f i ( t r x l ( t ),

(t-T

( t ))

,Rl(t-T

...,xn ( t )'XI ( t )1,.

(i=1,2,.

.

.. , n )

( t - r ( t )1 I . .

,fin(t-T

(t)

I

0

I

(61)

I

f o r s i m p l i c i t y , w i t h o n e r e t a r d a t i o n Olr(t)'-r, ([132.10] I [ 7 7 . 2 ] ) .

As a l r e a d y shown, t h e meaning o f t h e d e f i n i t i o n s I - V of s t a b i l i t y , a s y m p t o t i c s t a b i l i t y , u n i form a s y m p t o t i c s t a b i l i t y , and so on remain unchanged. Only, i n c o n t r a s t t o e q u a t i o n s w i t h a r e t a r d e d argument, a l l of t h e s e d e f i n i t i o n s m u s t now b e f o r m u l a t e d i n terms o f t h e s p a c e C1. I n accordance w i t h t h i s , w e i n t r o d u c e t h e notation:

165


D e f i n i t i o n I.

The f u n c t i o n a l

( i n what f o l l o w s s a l w a y s v a r i e s w i t h i n t h e i n d i cated l i m i t s ) , is c a l l e d positive-definite i f there e x i s t s a c o n t i n u o u s f u n c t i o n cp ( r ) s u c h t h a t cp ( r ) > O f o r r # O and V[x(~),?(s),tI

$41I x ( s ) I

Ilr).

functional i s analogously

A negative-definite

defined

'

.

D e f i n i t -i.o n 11. The f u n c t i o r . a l V [ x ( s ) , 2 ( s ) , t ] h a s a n i n f i n i t e l y s m a l l u p p e r bound i f t h e r e e x i s t s > 0 , cpl(0) = 0 , s u c h a c o n t i n u o u s f u n c t i o n cp 1 ( r ) that V[X(S) ,jr(s)

,tl 2

cp10I x ( s ) I Ilr).

S t a b i l i t y Theorem. The t r i v i a l s o l u t i o n o f t h e system ( 6 1 ) i s s t a b l e i f t h e r e e x i s t s a continuous positive-definite functional

such t h a t along an i n t e g r a l curve x o ( t + s ) of t h e system ( 6 1 )

lim+

At+O

sup

AV a t5

0.

The p r o o f i s a n a l o g o u s t o t h e p r o o f of t h e s t a b i l i t y o f t h e t r i v i a l s o l u t i o n of t h e system of d i f f e r e n t i a l e q u a t i o n s w i t h a r e t a r d e d argument.

166


W e n o t e t h a t f o r systems of e q u a t i o n s of n e u t r a l t y p e , n a t u r a l l y a l l t h e o r e m s on Lyapunov's s e c o n d method a r e f o r m u l a t e d i n terms of d e r i v e d numbers and d e r i v a t i v e s , s i n c e t h e p e r t u r b e d s o l u t i o n s , g e n e r a l l y s p e a k i n g , d o n o t s a t i s f y t h e agreed condV d i t i o n s and, consequently, t h e d e r i v a t i v e - a l o n g dt an i n t e g r a l curve does n o t e x i s t a t each p o i n t .

Theorem on Asymptotic S t a b i l i t y I. The t r i v i a l s o l u t i o n of t h e s y s t e m ( 6 1 ) i s u n i f o r m l y a s y m p t o t i c a l l y s t a b l e i f t h e r e e x i s t s a continuous, positived e f i n i t e f u n c t i o n a l V [ x ( s ) , k ( s ) , t ] f o r t>t, and

I ] X ( S ) ] ] ~ ~ < HH>O, ,

Y

a d m i t t i n g an i n f i n i t e l y s m a l l

u p p e r bound and s u c h t h a t f o r t h e f u n c t i o n V ( t ) = AV

V[xo ( t + s ),Ao ( t + s ), t ], l i m At+O-

sup - i s negative-defiAt nite. Here x @ ( t + s ) is t h e s o l u t i o n o f t h e s y s t e m (611, d e f i n e d by t h e i n i t i a l v e c t o r - f u n c t i o n @(t) , where I 10 ( t o + s )I I 0 i s s u f f i c i e n t l y small. T h i s theorem i s proven analogously t o t h e c o r r e s p o n d i n g theorem f o r a s y s t e m o f e q u a t i o n s w i t h a r e t a r d e d argument (Theorem I o f N . N . K r a s o v s k i i ) . The t h e o r e m of a s y m p t o t i c s t a b i l i t y a d m i t s a c o n v e r s e - i n a n a l o g y t o t h e c o n v e r s e theorem ment i o n e d e a r l i e r (see [77.21)

.

F o r t h e s y s t e m o f e q u a t i o n s of n e u t r a l t y p e (61), it i s p o s s i b l e t o r e f o r m u l a t e a l l t h e theorems p r o v e n i n t h e p r e c e d i n g p a r a g r a p h s f o r a s y s t e m of e q u a t i o n s w i t h a r e t a r d e d argument ( 3 8 1 , [ 7 7 . 2 ] . For t h i s , t h e i d e a of t h e p r o o f o f t h e c o r r e s p o n d i n g t h e o r e m r e m a i n s unchanged. W e have

Theorem. I f t h e t r i v i a l s o l u t i o n o f t h e s y s t em ( 6 1 ) i s u n i f o r m l y a s y m p t o t i c a l l y s t a b l e and t h e functions fi s a t i s f y Lipschitz conditions i n a l l

167


arguments beginning w i t h t h e second, t h e n t h i s s o l u t i o n i s s t a b l e under c o n s t a n t l y a c t i n g d i s t u r b a n c e s . For a system o f e q u a t i o n s o f n e u t r a l t y p e , t h e r e h o l d t h e o r e m s a n a l o g o u s t o t h e o r e m s I and I1 a b o u t s t a b i l i t y i n t h e f i r s t approximation b u t c o n d i t i o n 1) o f t h e o r e m I i s r e p l a c e d by t h e r e q u i r e m e n t : a l l R e k . < - y < 0 , where y i s a c o n s t a n t . In the

7 -

work [77.11, t h e r e i s a l s o s t u d i e d some c r i t i c a l cases. I n [77.2] i s p r o v e n an a n a l o g o f t h e ChetayevShimanov t h e o r e m o f i n s t a b i l i t y . Thus, t h e problem o f g e n e r a l i z i n g t h e whole c i r c l e o f t h e o r e m s c o n n e c t e d w i t h Lyapunov's second method t o a s y s t e m o f d i f f e r e n t i a l e q u a t i o n s w i t h d e v i a t i n g a r g u m e n t s o f r e t a r d e d and n e u t r a l t y p e s i s , i n p r i n c i p l e , complete. Nevertheless, t h e p r a c t i c a l a p p l i c a t i o n of t h e s e theorems i s complic a t e d by t h e a b s e n c e o f any g e n e r a l methods f o r t h e c o n s t r u c t i o n of Lyapunov-Krasovskii f u n c t i o n a l s having t h e necessary properties. I n t h i s r e s p e c t , s y s t e m s o f e q u a t i o n s of neut r a l t y p e f u r n i s h many d i f f i c u l t i e s s i n c e t h e f u n c t i o n a l s depend n o t o n l y o n x ( s ) , b u t a l s o upon k ( s ) . Connected w i t h t h i s i s t h e s e a r c h f o r d i f f e r e n t m o d i f i c a t i o n s o f Lyapunov's s e c o n d method a d m i t t i n g t h e u s e , i n s o m e s e n s e , of m o r e c o n v e n i e n t f u n c t i o n als. For a c o n c r e t e system of d i f f e r e n t i a l e q u a t i o n s o f n e u t r a l t y p e , it sometimes t u r n s o u t t o b e conv e n i e n t t o u s e t h e metric

i=l

168


Under t h e a s s u m p t i o n t h a t t h e r i g h t - s i d e o f ( 6 1 ) i s c o n t i n u o u s and s a t i s f i e s a L i p s c h i t z c o n d i t i o n i n a l l arguments beginning w i t h t h e second, it i s p o s s i b l e t o prove t h e A s y m p t o t i c S t a b i l i t y Theorem 11. I f t h e r e e x i s t s a functional V[x(s) , G ( s ) , t l s a t i s f y i n g the conditions

AV l i m + sup < A t t+O

where W1 ( r ) and W2 ( r ) a r e m o n o t o n i c a l l y i n c r e a s i n g f u n c t i o n s f o r r > O , where W1 (0)=W, ( 0 ) =O

,

W ( r ) and

$ ( r ) a r e c o n t i n u o u s and p o s i t i v e f c r r > O , t h e n t h e s o l u t i o n of t h e s y s t e m ( 6 1 ) i s a s y m p t o t i c a l l y s t a b l e . The p r o o f i s a n a l o g o u s t o t h e proof o f N.N. K r a s o v s k i i ' s Theorem 11. However, i n c o n t r a s t t o a s y s t e m o f e q u a t i o n s w i t h a r e t a r d e d argument, f o r a s y s t e m o f e q u a t i o n s o f n e u t r a l t y p e such a n approach t u r n s o u t , i n p r a c t i c e , t o b e n o t so e f f e c t i v e . The d i f f i c u l t i e s c o n n e c t e d w i t h t h i s are t h a t i n t h e c o n d i t i o n ( 6 3 ) , (and n o t 11x1 1 , a s i n t h e r e e n t e r s t h e norm 11x1 (44) a b o v e ) .

Il

For surmounting t h e s e d i f f i c u l t i e s , w e i n t r o duce t h e following a u x i l i a r y n o t i o n s . W e w i l l say t h a t t h e t r i v i a l s o l u t i o n of t h e s y s t e m ( 6 1 ) i s s t a b l e i n t h e m e t r i c C i f , f o r any 0

w e may f i n d a 6 ( ~ ) > 0 , s u c h t h a t t h e i n e q u a l i t y 1 I @ ( t O +I sllTtO. A s y m p t o t i c s t a b i l i t y i n t h e metric C o i s c o r r e s p o n d E>O,

ingly defined.

-

Here

\

1x1

I

169

i s t h e norm i n t h e s p a c e


Usually, i f w e succeed i n e s t a b l i s h i n g e f f e c t i v e conditions €or the s t a b i l i t y of solutions of systems o f d i f f e r e n t i a l e q u a t i o n s of n e u t r a l type i n t h e metric C o , a n d t h e n e s t a b l i s h a c o n n e c t i o n between s t a b i l i t y i n t h e m e t r i c s C

0

and C1, t h e n t h i s

i t s e l f w i l l be a s o l u t i o n about s t a b i l i t y of t h e b a s i c i n i t i a l v a l u e problem i n t h e metric C 1' W e show t h a t u n d e r t h e c o n d i t i o n s o f t h e p r e v i o u s theorem, t h e r e o c c u r s t h e

5.

Theorem on A s y m p t o t i c S t a b i l i t y i n t h e Metric I f t h e r e e x i s t s a f u n c t i o n a l V [ x ( s ) ,?(,I , t ] ,

where W (r) and W2 ( r ) a r e m o n o t o n i c a l l y i n c r e a s i n g

1

f o r r > O and W (O)=W,(O)=O,

a n d +(r) are con-

W(r)

1 -

t i n u o u s and p o s i t i v e f o r r > O , t h e n t h e s o l u t i o n o f t h e s y s t e m ( 6 1 ) is a s y m p t o t i c a l l y s t a b l e i n t h e metric C o . Proof.

For given E > O , w e choose 6>0 such t h a t

W1(6)+W2(26&)

c

W(E).

Then by (62)

V[Q ( s ) I 6 ( s ) , t o ] < f o r 1 j e ( t O + s )1 t h e function

IlT

#

(62')

Since along a trajectory,

< 6.

V ( t ) = V[x ( t + s ),it (P

is non-increasing,

W(E)

(P

( t + s ), t l

from (62) it follows t h a t

V[X@(t+s)'9, (t+s), t l

170

tO, and keeping in mind that I 1x1 1x1 11, by (64) we obtain 1 Ix,(t) I I < € . This alone proves the stability of the solution in the metric C 0' Asymptotic stability is also proven as in Theorem I of N.N. Krasovskii. On the other hand, we have the Theorem. For the system (611, let T(t)=T>O, and let the €unctions €.(t,xl, xn-1 ,y ,...,y , 1 11 gl, z n satisfy Lipschitz conditions in all arguments beginning with the second with constants Lis(XI, (i,s=1,2,. ,n) in the region < H, t 0, such t h a t f o r 1 I @ ( t o + s I) IlT O are c o n s t a n t s , b ( t ) i s a c o n t i n u o u s f u n c t i o n f o r t 0-< t < w . The i n i t i a l f u n c t i o n is an a r b i t r a r y continuously d i f f e r e n t i a b l e function.

172

(69)

Ill. STABILITY T H E O R Y

lo

We consider the functional V[X(S),i(S) ,tl = x 2 (t) +

a

Z2(t+s) ds.

For a>O, this functional satisfies the conditions: 2 v[x(s),~(s),tl 5 r2 + CP I

V[x(s) , g ( s ) ,tl

2 r

.

. W e calculate the derivative along an integral curve. * dV dt =

-ax2 (t)

-

1-b2(t) ?2 (t-.c). a

For negative-definiteness of this quadratic form in the variables x(t) and Z(t--T), it suffices to satisfy the conditions a>O, Ib(t)l

0.

(70)

Consequently, by the asymptotic stability theorem in the metric Co and the theorem of connection between stability in the metrics Co and C1, under condition ( 7 0 ) , the trivial solution of Eq. (69) is asymptotically stable in the metric C1. By the same method, it is possible to obtain sufficient conditions for asymptotic stability for the system

*

At points of the form t=tO+k , the derivative k(t), generally speaking, does not exist. Therefore, we will assume that at these points, we consider not the derivative but the right upper derived numAVdt I berA t+8+ li sup At' 173


2 . (t) = 1

n

m n a x. (t) + c c bijk(t)x. (t-'ck) j=1 ij 3 k = l j=1 3 C

m

+

n

For this, the functional V may be chosen in the following form:

v

=

n

c

i,j=l

6 . .x.x 11 1

m

j

m

+ c

n

c

k = l j=l

n

1 E Y k = l j=1 kj

0

akj

j

0

x j

(t+s)ds

+

' k

k j 2 (t+s)ds.

-Tk

Example 2. &(t)

+

ax(t) = 4 (x(t-'c),ir(t-.c),t),

(711

where a and T > O are constants, the nonlinear function 4 satisfies the condition

uniformly in t. It is possible to show that the functional 1 x2 (t) + V[x(s),k(s),t] = x2(t+s)ds 2a 2

'1

+

-'c

satisfies the conditions of the theorem on asymptotic stability in the metric C o , if

174

1 1 1 . STABILITY T H E O R Y

On the other hand, by (72) and (731, the asymptotic stability of the trivial solution of Eq. (71) in the metric C0 implies by itself asymptotic stability in the metric C1. 10.

Absolute Stability We again consider the equation G(t)

+ ax(t) + bx(t-r)

= 0,

(74)

where a r b , and T Z O are constants. Earlier, Eq. (74) was studied for stability by the method of D-partitions and the domain of asymptotic stability of the solution of this equation was constructed. It is bounded by the lines

and the line a+b=O (Fig. 16; and Fig. 11). Of basic interest is the part of this domain represented by the inequalities

I

a>O, bl -O . Definition. The solution x @ (t) of the differential equation with one o r several deviating arguments ?(t) = f(t,x(t) ,x(t-Tl) ,...,x(t--rrnH

(76)

(or analogously an equation of neutral type) we will call absolutely stable (absolutely asymptotically stable), if it is stable (asymr,totically stable) for arbitrary, constant, non-negative values T

j’

175


Thus, the conditions (75) define the domain of absolute asymptotic stability in the plane (a,b) of parameters for Eq. (74). We consider the linear equation n-1 n m x ( ~ )(t) + c akx(kl (t) + c c b . x ( ~(t-T.)=O, ) (77) k=O k=O j=1 k3 3 with constant coefficients ak,bkj and retardations r . > O (if for some j, T . > O , b # 0, then Eq. (77) is 37 nj of neutral type; if there is no such j,then (77) is an equation with a retarded argument). To Eq. (77) corresponds the characteristic quasipolynomial -T .z m @ ( z ) = P(z) + C Q..(z)e 7 j=1 I where P(z) = zn

n-1

+

C

k=O

akzk, Q . ( z ) = 3

n Z

k=O

k bkjz

.

We have the following Theorem.

(L.A. Zhivotovskii [29.2]).

Let in

(77)I

Then, in order that the solution of Eq. (77) be absolutely asymptotically stable, it is necessary and sufficient that the following two conditions be satisfied. 1. The real parts of all roots of the polynomial P ( z ) be negative. 2.

For any y > O , m C lQj(iY)I j=1

N (N is easy to estimate), Eq. 17) has no real roots.

190

IV. PERIODIC SOLUTIONS

3.

Periodic Solutions of Linear Homogeneous Parts Periodic solutions of the equation n m c c aijx(J)(t-ri) = f(t), j=O i=O

where all aij and T~ are constants, a =O, O=T < T On 0 1 tO is a continuous periodic function of period 2n, expanded into the Fourier series W f(t) = E CL epit , p=-w P where

(It is possible, of course, to write the Fourier series in the real form W

f(t) = c ( B cos pt+h sin pt), P P p=o however, computation of the coefficients of the solution of the equation in this case is rather complicated). If the period of the function f(t) equals T, T then after the transformation t = 7tl, we obtain an equation, the right side of which has period 2 n with respect to the variable tl.

191


If the characteristic quasipolynomial cp(z) =

n

c

m

x

j = O i=O

a

.

z ' e ij

-T.Z J

does not have purely imaginary integer roots, i.e., the case of no resonance, then using the principle of superposition, we seek a solution in the form ~ApePit, for each term in (151, A = ap and, sumP $(Pi) ming these, we obtain the periodic solution

The series (16) converges and admits n term-by-term differentiations, since the coefficients of this a series A,by comparison with the coefficients 4 (Pl) a of the uniformly convergent series (IS), conP tain in the denominator 4 (pi)=O(pn) for real p+m. Even if one root of the quasipolynomial $ ( z ) is close to mi, where m is an integer, then for am#O or a-m#O, resonance appears: the coefficient am sharply increases in modulus in comparison cp (mi) with the case of the absence of roots of $(z) near mi. If one of the roots of the quasipolynomial 4 ( z ) equals mi and a #O or a #O, then there do not exist m -m periodic solutions. If one of the roots of the quasipolynomial c p ( z ) equals mi and a = a =0, and there are no other purely m m imaginary integral roots, then there exists a twoparameter family of periodic solutions of the form (16), but only the coefficients of eimt and e-imt in (16) remain arbitrary.

192

PERIODIC SOLUTIONS

IV

Example 1.

The equation

k(t) where

+

ax(t)

+

f (t) = a,U

+

bx(t-T) = f(t),

(17)

(ancos nt+Bnsin nt) ,

W

C

n=1 in the non-resonance case, has only the single periodic solution m

x(t)=

ao

[crn(a+bcos n7)-Bn(n-bsin nr)] cos nt

+ z

2(a+b) n=l

(a+bcos nrI2

+ [ B,

+ (n-bsin n7)2

(a+bcos n7)+an (n-bsin n ~ 1) sin nt (a+bcos n

2

~ +(n-bsin ) ~ n.r)

Resonance is possible only at one frequency. In the resonant case, i.e., if the characteristic equation z+a+be-Tz = 0 has the integral purely imaginary root +mi, a periodic solution exists only for m = j2'f

a

(t)cos mt dt = 0

0

and

2n

f(t)sin mt dt = 0,

6, =

0

and has the form (18), but the coefficients of cos mt and sin mt are arbitrary constants. Example 2. x(t)

+

The equation

a S(t)+a2x(t)+blk(t-r)+b2x(t-r) = f(t), (19) 1

where m

f(t) = '0 + z (ancos nt+Bnsin nt). n=l

*

193


i n t h e non-resonant case h a s o n l y t h e s i n g l e p e r i o d i c solution

co -+

x(t) =

m

C

n=l

(Cncos nt+Dnsin n t ) ,

(20)

where

-

P n 8 n +Qnan

DnPn = a +b

2

1

Qn = aln+b

nsin nT ncos n.r

2

'n

+

2 +Qn

b2 cos n r

-

b

2

-

n

2

,

s i n nT.

Resonance i s p o s s i b l e a t one o r two f r e q u e n c i e s . Remark. The p e r i o d i c s o l u t i o n s of E q s . ( 1 7 ) o r ( 1 9 ) a r e s i m p l e r t o w r i t e i n t h e complex form ( 1 6 1 , b u t i n p r a c t i c a l problems it i s o f t e n c o n v e n i e n t t o make u s e of t h e n o t a t i o n i n r e a l form ( 1 8 ) and ( 2 0 ) . Example 3 .

The e q u a t i o n

Z(t)-x(t)-k(t-2a)+x(t-27r) = f ( t )

i l l u s t r a t e s t h e e x c e p t i o n a l case, mentioned above. Resonance i s o b s e r v e d a t a l l i n t e g r a l f r e q u e n c i e s . For any c h o i c e o f p e r i o d i c f u n c t i o n f ( t ) w i t h p e r i o d 27~ ( w i t h t h e e x c e p t i o n o f f ( t ) : O ) r e s o n a n c e i s obs e r v e d and t h e r e e x i s t s no p e r i o d i c s o l u t i o n . I n problems c o n n e c t e d w i t h t h e e x i s t e n c e of p e r i o d i c s o l u t i o n s and w i t h t h e a p p e a r a n c e o f r e s o nance, i t i s o f t e n i m p o s s i b l e t o n e g l e c t even a r b i t r a r i l y s m a l l retardations.

194

I V . PERIODIC SOLUTIONS

Example 4 . k(t)

The e q u a t i o n

+ ax(t) +

bx(t-r) = 0,

where a,b#O, a n d T > O a r e c o n s t a n t s , may h a v e a p e r i o d i c s o l u t i o n f o r a r b i t r a r i l y s m a l l T , but for T = O t h i s e q u a t i o n h a s no p e r i o d i c s o l u t i o n . Example 5. k(t)

For t h e e q u a t i o n

+ a x ( t ) + bx(t-T)

= A sin wt,

where a , b # O , A # O , w > O , and T > O are c o n s t a n t s , resonancemay a p p e a r f o r a r b i t r a r i l y s m a l l T , b u t when T=O resonance i s impossible. Finally, we s h a l l find t h a t i n l i n e a r (or q u a s i l i n e a r ) systems w i t h a d e v i a t i n g argument, t h e r e is also observed t h e appearance of s p e c i f i c resonances: t h e resonance of t h e i n i t i a l functions and t h e r e s o n a n c e o f t h e d e v i a t i n g arguments [132.19]. Example 6 . x(t)

The e q u a t i o n

+ x(t)

= 2x(t-2mn),

where m i s a n a t u r a l number, w i t h t h e i n i t i a l cond i t i o n s x(O)=O, 3 r ( O ) = 1 , x ( t ) = s i n t f o r -2mn - T , y ( t ) a s o l u t i o n of t h e syst e m (231, d e f i n e d f o r t O . We seek a solution in the form x(t,p) = x,,(t)+pxl(t)+p

2x2(t)+**-+p nxn(t)+Rn(t,u).

The coefficients xo(t) , x1 (t),

the equations

x2 (t) are defined by

x1 (t)+a1k 1 (t)+a2x1 (t)+blkl (t-T)+b2xl(t-r) =

202


Example 2. To approximately find the periodic solution of the equation ll 2 k(t) + x(t- -) 4 = sin t+px (t), under the constraint that only two terms of the expansion in powers of u are to be used. It is easy to be convinced that all conditions mentioned on p.18 are satisfied. In particular, €or checking condition 2), it is sufficient to see that the point (0;l) belongs to the domain I (a=O,b=l, . r = . r r / l ) of stability n X(t,p) = xo(t)+~x1(t)+..*+~xn(t)+R(t,v) %o (t)+xo(t-

TI 7) =

sin t.

This case is one of non-resonance and the solution may be expressed in the form x0 (t) = A cos t + B sin

t

hence xo(t) =

-

1 cos t + 1+fi sin t 2 2

Further

203


or j,

1( t ) + x l ( t -

TTI )

2 + n = 4

'+LT (cos 2t+sin 2 t ) . 4

and, by t h e s a m e method, w e f i n d

+-

4

(cos 2 t - s i n 2 t ) .

Consequently,

%

x(t,p) =

-

1 cos t + 2

sin 2

t

Asymptotic methods u s e d f o r t h e d e t e r m i n a t i o n of p e r i o d i c s o l u t i o n s of q u a s i l i n e a r e q u a t i o n s w i t h a d e v i a t i n g argument a r e , i n t h e f i n a l a n a l y s i s , m o d i f i c a t i o n s o f t h e a s y m p t o t i c methods o f n o n l i n e a r mechanics o f N.M. Krylov-N.N. Bogoliubov, Yu.A. Mitropolskii. Restricting ourselves t o oscillatory s y s t e m s w i t h a s i n g l e d e g r e e o f freedom, w e c o n s i d e r t h e i d e a o f a p p l y i n g t h e s e methods € o r s e v e r a l b a s i c cases ( f o r d e t a i l s , see t h e r e v i e w [ 1 1 . 1 1 1 , see a l s o t h e monograph [I.111). I n t h e s i m p l e s t case o f a n autonomous o s c i l l a t o r y s y s t e m , it i s d e s c r i b e d by a d i f f e r e n t i a l equat i o n o f t h e form 2

X(t&+L! X ( t )

= pf ( X ( t )

,X(t-T)

,?(t) , ? ( t - T ) ) ,

(32)

is a s m a l l with a constant r e t a r d a t i o n . H e r e p o s i t i v e p a r a m e t e r , and t h e n o n l i n e a r f u n c t i o n f h a s a s u f f i c i e n t number o f c o n t i n u o u s . p a r t i a l d e r i v a t i v e s w i t h r e s p e c t t o a l l arguments i n a s u f f i c i e n t l y l a r g e region. Problem S t a t e m e n t : d e t e r m i n e a p e r i o d i c s o l u t i o n o f Eq. ( 3 2 ) , x = x ( t , p ) , c o r r e s p o n d i n g t o t h e s t e a d y - s t a t e behavior i n t h e o s c i l l a t o r y system, which f o r u+O r e d u c e s t o t h e p e r i o d i c s o l u t i o n x (t)= a cos(wt+$) of t h e generating equation 0

2 X(t)+w x ( t ) = 0 .

204


I n [120.21 t h e indicated s o l u t i o n i s sought i n t h e form x ( t ) = z (Gt+$) I where z ( $ 1 i s a p e r i o d i c f u n c t i o n o f J, o f p e r i o d IT. The s u b s t i t u t i o n o f x ( t ) i n t o Eq. ( 3 2 ) l e a d s t o t h e d i f f e r e n t i a l equation f o r t h e function z ( $ ) -2

-.

2

w Z($)+w z ( $ ) = Pf(z($),z(J,-vT),~~(J,),

32

(JI-WT)).

(33)

The s o l u t i o n o f Eq. ( 3 3 ) and t h e f r e q u e n c y 3 i s s o u g h t i n t h e form o f an e x p a n s i o n i n powers o f a

I n t h i s c o n n e c t i o n , i t i s r e q u i r e d t h a t t..e f u n c t i o n s (n= ~ ~ ( $ 1 l I 2 , . . . ) be p e r i o d i c f u n c t i o n s o f J, w i t h period 2n. S u b s t i t u t i n g (34) i n t o (33) , expanding t h e r i g h t - s i d e i n t o a series o f powers o f u, and equat i n g c o e f f i c i e n t s o f l i k e powers of p , w e o b t a i n a system of o r d i n a r y l i n e a r d i f f e r e n t i a l e q u a t i o n s w i t h o u t a d e v i a t i n g a r g u m e n t , from which i n t u r n a r e d e f i n e d t h e unknown f u n c t i o n s z ( $ ) and t h e n q u a n t i t i e s c1 ( n = lI 2 , . I .n

.. .

W e now c o n s i d e r t h e a n a l o g o u s p r o b l e m f o r t h e more g e n e r a l e q u a t i o n

w i t h a s t a t i o n a r y l i n e a r p a r t o r , more b r i e f l y ,

205


and l e t t h e c h a r a c t e r i s t i c e q u a t i o n , composed f o r the generating equation

have o n l y a s i n g l e p a i r o f p u r e l y i m a g i n a r y r o o t s These r o o t s c o r r e s p o n d t o a f a m i l y of z 1 , 2 = +-i w 0 '

p e r i o d i c s o l u t i o n s of t h e g e n e r a t i n g e q u a t i o n ( 3 6 ) xo ( t ) = a c o s ( c o o t + $ ) ,

(37)

where a and $ are a r b i t r a r y c o n s t a n t s . For c o n s t r u c t i o n of t h e p e r i o d i c s o l u t i o n x = x ( t , p ) o f Eq. (351, r e d u c i n g f o r p+O t o t h e gene r a t i n g Eq. (371, t h e f o l l o w i n g method may be a p p l i e d 11.131. The p e r i o d o f t h e unknown s o l u t i o n x ( t , p ) i s set equal t o T = - 2, TicL wO

where a = 1+pa1+p

t h e q u a n t i t i e s al,a2,...

2

a

2

+-.a

r

t o be d e t e r m i n e d .

The change o f i n d e p e n d e n t v a r i a b l e tl = wOt a r e d u c e s Eq. ( 3 5 ) t o t h e e q u a t i o n

2

E q u a t i o n ( 3 8 ) i s also o f where Ti = T~ (i=1,2). t h e t y p e (351, b u t i t s p e r i o d i c s o l u t i o n x ( t l l p ) has t h e constant period 2a. This s o l u t i o n i s sought i n t h e form o f t h e f o r m a l series

.

where x n ( t l ) ( n = 0 , 1 , 2 , . .) are p e r i o d i c f u n c t i o n s of p e r i o d 2 a , which are s o l u t i o n s o f t h e l i n e a r d i f f e r e n t i a l equations obtained a f t e r s u b s t i t u t i n g ( 3 9 ) i n t o (381, e x p a n d i n g t h e r i g h t - s i d e i n a series o f powers o f p , and e q u a t i n g c o e f f i c i e n t s o f l i k e powers o f p. I n t h i s connection, t h e generating 206


e q u a t i o n E ( x ( t l )1 =O h a s t h e f a m i l y o f p e r i o d i c s o l u t i o n s x O ( t l ) = a c o s (t,+$). Periodic s o l u t i o n s of t h e equation

d e f i n i n g x l ( t l ) , e x i s t o n l y i n t h e case of t h e abs e n c e o f r e s o n a n t terms i n t h e F o u r i e r series expans i o n of i t s r i g h t - s i d e , i.e. under t h e c o n d i t i o n s

J’ = 0

Qlcos t l d t l

s i n 4 d t l = 0.

= 0,

From t h e s e e q u a t i o n s t h e q u a n t i t i e s a and a l a r e d e f i n e d and so f o r t h .

W e a p p l y an a n a l o g o u s method f o r non-autonomous q u a s i l i n e a r e q u a t i o n s o f t h e form L ( x ( t ) ) = f ( t ) + p F ( t , X ( t )t X ( t - T 2 ) r k ( t ) r ? ( t - T l ) where f and F are p e r i o d i c f u n c t i o n s o f t. H e r e t h e r e i s p o s s i b i l i t y o f b o t h t h e r e s o n a n t and nonr e s o n a n t case. I n [ 106.11 , a s y m p t o t i c methods of n o n l i n e a r mechanics a r e u s e d f o r t h e d e t e r m i n a t i o n of p e r i o d i c s o l u t i o n s of Eq. (35) for t h e c o r r e s p o n d i n g nons t a t i o n a r y o s c i l l a t i o n s . I n t h e a b s e n c e of p e r t u r b a t i o n ( p = O ) , l e t t h e s y s t e m p e r f o r m t h e harmonic o s c i l l a t i o n s x=a c o s I), I)= wet+$ w i t h c o n s t a n t amp l i t u d e and u n i f o r m l y i n c r e a s i n g p h a s e a n g l e

da = dt

0,

dJ, dt

=

Under a s m a l l i n i t i a l p e r t u r b a t i o n , (l.if0) , t h e s y s t e m w i l l p e r f o r m o s c i l l a t i o n s n e a r t o harmonic, b u t t h e amplitude of o s c i l l a t i o n w i l l , g e n e r a l l y s p e a k i n g , be t i m e - d e p e n d e n t and t h e i n s t a n t a n e o u s In t h i s f r e q u e n c y w i l l depend o n t h e a m p l i t u d e .


case, a p e r i o d i c s o l u t i o n o f Eq. t h e form o f t h e f o r m a l series x = a cos $

+

(35) i s sought i n

2

uul(a,$)+p u 2 ( a , $ ) + * - - ,

i n which u l ( a , $ ) , u 2 ( a , $ ) ,

... are p e r i o d i c

(40)

functions

of $ of p e r i o d 2 ~ ,w h i l e t h e q u a n t i t i e s a and $, as f u n c t i o n s o f t i m e , a r e d e f i n e d by t h e s y s t e m o f d i f f e r e n t i a l equations

2

-

= pAl ( a ) + p A2 ( a ) + -

I n t h e work [ 1 2 0 . 6 ] a s y m p t o t i c methods a r e used f o r i n v e s t i g a t i o n s o f t h e n o n - s t a t i o n a r y o s c i l l a t i o n s i n q u a s i l i n e a r s y s t e m s , d e s c r i b e d by equat i o n s of t h e type (35) with slowly-varying c o e f f i c i e n t s and r e t a r d a t i o n s . In t h e papers [71.1], [ 1 0 6 . 2 ] methods a r e i n d i c a t e d f o r t h e c o n s t r u c t i o n o f a s y m p t o t i c a p p r o x i m a t i o n s f o r cases o f q u a s i l i n e a r o s c i l l a t o r y systems w i t h r e t a r d a t i o n s w i t h many d e g r e e s o f freedom. W e n o t e , however, t h a t f o r m a l l y o b t a i n e d series of t h e t y p e (341, (391, (40) r e p r e s e n t a n unknown p e r i o d i c s o l u t i o n o n l y u n d e r t h e a s s u m p t i o n of t h e e x i s t e n c e of a unique p e r i o d i c s o l u t i o n of t h e c o r r e s p o n d i n g e q u a t i o n , c o n v e r g i n g a s u+O t o some p e r i o d i c s o l u t i o n of t h e g e n e r a t i n g equation.

The f i r s t work s p e c i f i c a l l y d e a l i n g w i t h t h e b a s i c a s y m p t o t i c methods i s t h e work o f A. Halanay [159.3]. I n t h i s work i s c o n s i d e r e d t h e b a s i s o f t h e a v e r a g i n g method, t o which r e d u c e many asymptoti c methods. I n [159.31 it i s shown t h a t , along w i t h t h e given system o f e q u a t i o n s reduced t o t h e s o - c a l l e d s t a n d a r d form ( i n v e c t o r - m a t r i x n o t a t i o n ) k ( t ) = p X ( t , x ( t ), x ( t - . r ) )

208

I

(41)


it is useful to examine the average system 9(t) = "X0 (x(t) IY (t)1 I where

I

T Xo(x,y) = lim 1 X(t,x,y)dt T-- T O

If the functions X(t,x,y) and Xo(x,y) satisfy uniformly in t a Lipschitz condition in x and y r then for any T>O and p > O , there exists p o > O such that T we have for 0 < p < p and t€[O,-] 0 lJ Ix(t,v)-y(t,d

I

< n,

where x(t,p) and y(t,p) are the solutions of the systems (41) and (42) respectively, with the same initial function. Also, it is shown that the system 9 (t) = lJz (ttz(t)1 2 (t-lJT) ,!J)

,

where Z is periodic in t of period w , has a periodic solution of the same period converging as p+O to the solution c o of the finite equation z o ~ u , u , o ) = 0,

if the characteristic values of the matrix a Zo( O , the constants q.>O 3 (j=ll2,...,s), x(t) is an n-vector, and f is a continuous vector-function. Generally, the n-times differentiable periodic solutions of period w of Eq. ( 4 4 ) are also solutions of the equation

obtained from ( 4 4 ) with pi'O, and the n-times differentiable periodic solutions of period w of Eq. ( 4 5 ) satisfy Eq. ( 4 4 ) . In this (and only in this) sense, Eqs. ( 4 4 ) and ( 4 5 ) are equivalent. Such type of equivalence we shall call functional equivalence relative to the functional relation

-

We note that if Eq. ( 4 4 ) is of neutral type, then' its periodic solutions may be n-times piecewise differentiable functions. Then, as for Eq. ( 4 5 ) , there is a possibility of the case when its order turns out to be less than n; consequently, some of its solutions may remain differentiable a smaller number of times and may even be only piecewise continuous. There is also the possibility of the case (for example, if Eq. ( 4 5 ) has order n and s=O) for which all solutions of Eq. ( 4 5 ) are n-times differentiable. Consequently, generally speaking, it is impossible to assert that Eqs. ( 4 4 ) and ( 4 5 ) have in common all periodic solutions of periodic w. If Eq. ( 4 5 ) is an equation with a retarded argument (i.e. the right side of this equation does (t-q.)), not explicitly depend on x(n) (t-pi) , and x ( ~ ) 3 then all solutions of period w of Eqs. ( 4 4 ) and ( 4 5 ) are differentiable no more than n times and, consequently, Eqs. ( 4 4 ) and ( 4 5 ) are equivalent relative to any solution of period w.

21 1


From the trivial remarks about equivalence in the considered sense of Eqs. (44) and (451, follow numerous non-trivial consequences [132.28]. 1.

The scalar nth order equation

(or the system of equations of the first order

k (t) = f (trX(t)tX (t-Ti) here x(t) where the multiples parameter

I

is an n-vector, f is a vector-function), constant deviations T i are positive and of a number w>O, has not more than an nfamily of solutions of period w .

2. A scalar equation of neutral type of the form (44) for s=O, also has not more than an nHowever, parameter family of solutions of period w . there is a possibility of an exceptional case - Eq. (45), corresponding to the considered Eq. (441, is transformed to an identity and then any n-times piecewise differentiable periodic function of period w will be a solution of Eq. (44).

3.

The quasipolynomial n Q ( Z )

=

c

m

z

j=1 k = l

a

jk

.

-Tkz

z’e

I

(47)

where T~ is a multiple of w, has no more than n different roots of the form z = 2rrri, where r is an w integer.

-

The exceptions are composed of only those quasipolynomials of the form (47), which for - r k E O are identically zero. This case usually may not occur if the quasipolynomial (47) is the characteristic quasipolynomial of a linear stationary equation with a retarded argument.

212


4.

The l i n e a r system

m

I ( t ) + A o ( t ) x ( t ) +X s=l

As(t)X(t-Ts)

= f(t) I

are m u l t i p l e s of w, h a s a unique p e r i o d i c s o l u t i o n of m p e r i o d w i f t h e m a t r i x C A ( t ) i s p e r i o d i c of

where F ( t ) i s a v e c t o r - f u n c t i o n of p e r i o d w ,

p e r i o d w and t h e system

A(t)

+

m C

s=o

s=o

S

S

As(t)x(t)

= 0

h a s no p e r i o d i c s o l u t i o n s of p e r i o d w , t r i v i a l one.

5.

T

except t h e

The s c a l a r e q u a t i o n

m

x ( t ) + a o ( t ) x ( t )+ C

s=l

where T ~ > Oare m u l t i p l e s of .m .

as(t)x(t-Ts) = 0, W,

h a s no p e r i o d i c s o l u m

t as(t)fO. s=o s=o ~. -~ Analogous a s s e r t i o n s hold € o r a s i m i l a r e q u a t i o n o f n e u t r a l type, i f t h i s equation does n o t reduce t o an i d e n t i t y € o r ~ ~ - 0 . t i o n s o f p e r i o d w if

C

as(t)iO, but

6. I n t h o s e c a s e s when Eq. (45) e q u i v a l e n t , i n t h e c o n s i d e r e d s e n s e , t o Eq. (44) i s of i n t e g r a b l e t y p e , t h e p e r i o d i c s o l u t i o n s of Eq. (44) may be o b t a i n e d by q u a d r a t u r e s . W e c o n s i d e r t h e e q u a t i o n of n e u t r a l t y p e *

m I ( t ) +C

n ai(t)I(t-pi)+ 1 bj (t)x(t-q.)=O, 3 i=l j =O

* t h e c o r r e s p o n d i n g e q u a t i o n w i t h a r e t a r d e d argu-

ment i s s t u d i e d i n [117.4].

213

(48)


where t h e c o e f f i c i e n t s ai ( t ), b . ( t ) are d e f i n e d and 3

c o n t i n u o u s on - - m < t < m , t h e c o n s t a n t d e v i a t i o n s p i , q j a r e non-negative and m u l t i p l e s of a number w > O . Let

m 1+ c ai(t) # 0 i=l

(-m O , under c o n d i t i o n ( 4 9 ) r e d u c e s t o a l i n e a r inhomogeneous e q u a t i o n o f t h e f i r s t o r d e r w i t h o u t a d e v i a t i n g argument. Analog o u s l y t o t h e p r e v i o u s arguments, i t is n o t d i f f i c u l t t o show c o n d i t i o n s s u f f i c i e n t f o r t h e e x i s t e n c e of a s o l u t i o n of p e r i o d w f o r Eq. (53) and t o o b t a i n t h i s s o l u t i o n i n quadratures. I t i s also p o s s i b l e t o c o n s t r u c t e q u a t i o n s w i t h a d e v i a t i n g argument, e q u i v a l e n t r e l a t i v e t o t h e functional r e l a t i o n (46) t o a Bernoulli equation, t o a n e q u a t i o n w i t h s e p a r a b l e v a r i a b l e s and so f o r t h .

I n t h e c o n c l u s i o n of t h i s p a r a g r a p h , w e n o t e t h a t a n a l o g o u s l y it i s p o s s i b l e t o c o n s i d e r s y s t e m s of t h e form ( 4 4 ) , ( 4 5 ) e q u i v a l e n t r e l a t i v e t o a f u n c t i o n a l r e l a t i o n d i f f e r e n t from ( 4 6 ) . For example [ 1 1 7 . 4 ] ,

f o r the equation

where t h e numbers O l s i n c e

243


t h i s transition is equivalent t o the rejection i n a d i f f e r e n t i a l equation of unstable type with a s m a l l c o e f f i c i e n t f o r t h e highest d e r i v a t i v e , of t e r m s w i t h d e r i v a t i v e s o f maximal o r d e r . However, from t h e t h e o r y o f s u c h e q u a t i o n s ( s e e [ 1 . 1 3 ] ) it f o l l o w s t h a t i n t h e g e n e r a l case €or m > l , r e j e c t i n g t e r m s by no means s m a l l , o f c o u r s e , i s i m p o s s i b l e . T r a n s i t i o n t o Eq. ( 1 6 ) f o r s m a l l T i s a d m i t t e d o n l y f o r m = l s i n c e i n t h i s case, g e n e r a l l y s p e a k i n g , t h e r e arises an equation w i t h a s m a l l c o e f f i c i e n t f o r t h e h i g h e s t d e r i v a t i v e and t h e c o n s i d e r e d method g i v e s good r e s u l t s . 4.

A s y m p t o t i c Methods f o r E q u a t i o n s w i t h S m a l l D e v i a t i n g Arguments W e c o n s i d e r t h e s o l u t i o n of t h e e q u a t i o n

w i t h a s m a l l r e t a r d a t i o n T > O and w i t h t h e i n i t i a l co n d it i o n X ( t )

= 4 ( t )I

(Olt(T)

(18)

A l r e a d y i n t h e work of A.D. Myshkis 111.121 it was shown t h a t i f i n ( 1 7 ) w e p l a c e T = O , t h e n t h e s o l u t i o n of t h e e q u a t i o n without r e t a r d a t i o n

A ( t ) = f ( t , x ( t ), x ( t ) ), ( O l t ' " ) w i t h t h e i n i t i a l c o n d i t i o n x ( O ) = $( 0 ) , u n d e r s u f f i c i e n t l y g e n e r a l a s s u m p t i o n s , o n some f i n i t e t i n t e r v a l w i l l be n e a r t h e s o l u t i o n o f t h e i n i t i a l v a l u e problem ( 1 7 ) , ( 1 8 ) i f T i s s u f f i c i e n t l y s m a l l a n d , c o n s e q u e n t l y , may be c o n s i d e r e d a s t h e z e r o t h term of t h e asymptotic expansion of t h e l a t t e r s o l u t i o n . The method o f e x p a n s i o n i n powers of t h e ret a r d a t i o n , considered i n t h e previous paragraph, g i v e s an a s y m p t o t i c f o r m u l a o f t h e f i r s t o r d e r ( i n T ) and o n l y i n i s o l a t e d cases of t h e s e c o n d o r d e r .

244

V I . APPROXIMATE INTEGRATION METHODS

For o b t a i n i n g a s y m p t o t i c formulas o f h i g h e r ordel; it i s p o s s i b l e t o u s e p e r t u r b a t i o n methods, t h e i d e a of a p p l i c a t i o n of which t o e q u a t i o n s w i t h a r e t a r d e d argument b e l o n g s t o A.B. V a s i l e v and A.M. Rodinov [ 1 2 . 1 1 , [11.11. W e d e n o t e t h e r i g h t s i d e of ( 1 7 ) by f ( t , x , y ) and, assuming t h a t t h e s o l u t i o n of t h e i n i t i a l v a l u e problem ( 1 7 ) , (18) i s s u f f i c i e n t l y smooth ( u n d e r u s u a l assumptions on f and 4 , t h e d e r i v a t i v e s x ( ~()t ) w i l l be c o n t i n u o u s i n t and uniformly bounded i n ( T < T ) on t h e i n t e r v a l kTctLT) , w e expand t h e r i g h t - 0 s i d e of ( 1 7 ) i n powers o f T : 2 A ( t ) = f ( t , x ( t ), x ( f ) - T A ( t ) + X ( t ) ...)

5

y=x ( t ) and w e w i l l s e e k a formal s o l u t i o n of t h i s e q u a t i o n i n t h e form of an expansion i n powers of 2 X ( t ) = X o ( t ) + TX,(t) + TT X 2 ( t ) + . . * l (20) by s u b s t i t u t i n g ( 2 0 ) i n t o ( 1 9 ) and e q u a t i n g c o e f f i c i e n t s o f l i k e powers o f T . I t i s important t h a t f o r t h e d e t e r m i n a t i o n o f x i ( t ) ( i-> l l)i n e a r f i r s t order equations without r e t a r d a t i o n s a r e obtained. For computing x i ( t ) , it i s n e c e s s a r y t o s e t i n i -

t i a l conditions.

For t h e c o n s t r u c t i o n of an asympk+l t o t i c approximation of o r d e r k ( a c c u r a c y t o T )

w e s e t t h e s e c o n d i t i o n s a t t h e p o i n t t 0 = ( k + 2 ) ~ ,t o

t h e c o n t i n u i t y and uniform boundinsure f o r t>to'T e d n e s s i n T and t , r e s p e c t i v e l y , o f t h e d e r i v a t i v e s

245


..

k,. , x ( k + l ) o f t h e c o n s i d e r e d s o l u t i o n , which a r e n e c e s s a r y f o r e s t i m a t e s o f t h e a c c u r a c y o f t h e asymptotic formulas. For o b t a i n i n g t h e i n i t i a l v a l u e s a t t h e p o i n t t o = ( k + 2 ) ~ o, n e must compute t h e s o l u t i o n o f t h e problem ( 1 7 ) , ( 1 8 ) on t h e i n t e r v a l [ T , ( k + 2 ) ~ by ] some o t h e r method: f o r example, t h e method o f s t e p s . I t i s shown t h a t f o r t h e s o l u t i o n x ( t ) o f E q . ( 1 7 ) , s a t i s f y i n g t h e i n i t i a l c o n d i t i o n (181, t h e r e occurs t h e asymptotic expansion

u n i f o r m l y i n t and

T

f o r ~ ( k + 2) O ,

u(t,x) = $(t,x) for OztLT, O<x O t h e cond i t i o n (34) o f t h e m a x i m u m p r i n c i p l e d e t e r m i n e s t h e

-

i n i t i a l v a l u e of t h e s o l u t i o n o f t h e system (32) w i t h a c c u r a c y up t o a c o n s t a n t m u l t i p l i e r C>O. Hence, a c c o r d i n g t o ( 4 2 ) and (40)

281


'L

uo =

l< -t -< 2 ,

-1,

l< -t

Introduction to the Theory and Application of Differential Equations with Deviating Arguments (Mathematics in Science & Engineering, 105)

Introduction to the theory and application of differential equations with deviating arguments

Oscillation Theory of Differential Equations With Deviating Arguments (Pure and Applied Mathematics)

Riccati Differential Equations (Mathematics in Science & Engineering, volume 86)

Differential-difference Equations (Mathematics in Science & Engineering)

Stochastic Differential Equations in Science And Engineering

Stochastic Differential Equations In Science And Engineering

Random differential equations in science and engineering

Stochastic Differential Equations in Science And Engineering

Stochastic Differential Equations in Science And Engineering

Random differential equations in science and engineering

Riccati differential equations, Volume 86 (Mathematics in Science and Engineering)

Theory of Partial Differential Equations (Mathematics in Science & Engineering, 93)

Introduction to the theory and applications of functional differential equations

Comparison and Oscillation Theory of Linear Differential Equations. (=Mathematics in Science and Engineering; Vol. 48)

Introduction to partial differential equations with applications

Introduction to Partial Differential Equations with Applications

Introduction to Partial Differential Equations with MATLAB

Introduction to Partial Differential Equations with MATLAB

Nonlinear Partial Differential Equations in Engineering: v. 1 (Mathematics in Science & Engineering Volume 18)

Differential Equations & Control Theory

Ordinary differential equations, with introduction to Lie theory

Introduction to the Algebraic Theory of Invariants of Differential Equations (Nonlinear Science : Theory and Applications)

Engineering differential equations: Theory and applications

Delay Differential Equations: With Applications in Population Dynamics (Mathematics in Science and Engineering)

Introduction to the Theory of Linear Partial Differential Equations (Studies in mathematics and its applications)

Introduction to the Theory of Linear Partial Differential Equations (Studies in mathematics and its applications)

Random Integral Equations with Applications to Life Sciences and Engineering (Mathematics in Science and Engineering 108)

Numerical solution of partial differential equations in science and engineering

Numerical Solution of Partial Differential Equations in Science and Engineering

Quantum Detection and Estimation Theory (Mathematics in Science & Engineering)

Introduction to the Theory and Application of Differential Equations with Deviating Arguments (Mathematics in Science & Engineering, 105)