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

375 Pages·1973·3.75 MB·English

by  L.E. El'sgol'ts and S.B. Norkin (Eds.)

lntroduction to the
Theory and Application of
Differential Equations with
Deviating Arguments
ACADEMIC PRESS RAPID MANUSCRIPT REPRODUCTION
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Introduction to the
Theory andApplication of
Differential Equations with
Deviating Arguments
L. E. El’sgol’ts
and
S. B. Norkin
Translated by John L. Casti
University of Arizona
Tucson, Arizona
1973
ACADEMIC PRESS New York London
A Subsidiary of Harcourt Brace Jovanovich, Publishers
COPYRIGHQT 1 973, BY ACADEMPICR ESSI,N C.
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LILIRARYO F CONGRESCSA TALOQ CARDN UMBER:1 3-8 11
Introduction to the Theory and Application of Differential
Equations with Deviating Arguments. Translated from the
original Russian edition entitled Vvdenie V Teoriyu Differen-
cial’Nyh Uravneni! S Otklonyayu~imsyaA rgumentom, pub-
lished by “Nauka”Press, Moscow, 1971.
PRINTED IN THE UNITED STATES OF AMERICA
CONTENTS
PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . vii
TRANSLATORS NOTE . . . . . . . . . . . . . . . . . . . . . xi
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . xiii
Chapter I. Basic Concepts and Existence Theorems
1. Statement of the Basic Initial Value Problem.
Classifications . . . . . . . . . . . . . . . . . . 1
2. The Method of Steps . . . . . . . . . . . . . . . 6
3. Integrable Types of Equations with a Deviating
Argument . . . . . . . . . . . . . . . . . . . 15
4. Existence and Uniqueness Theorems for the Solution
of the Basic Initial Value Problem . . . . . . . . . . 19
5. Some Specific Singularities of the Solutions of Equations
with a Deviating Argument . . . . . . . . . . . . . 29
Chapter 11. Linear Equations
1. Some Properties of Linear Equations . . . . . . . . . 57
2. Linear Equations with Constant Coefficients and
Constant Deviating Arguments . . . . . . . . . . . 62
3. The Characteristic Quasipolynomial . . . . . . . . . 69
4. The Expansion of the Solution into a Series of Basic
Solutions . . . . . . . . . . . . . . . . . . . . 80
5. Two-sided Solutions . . . . . . . . . . . . . . . 83
6. The Homogeneous Initial Value Problem. . . . . . . . 91
7. Some Types of Linear Equations with Variable
Coefficients and Variable Deviating Arguments . . . . . 98
Chapter 111. Stability Theory
1. Basic Concepts. . . . . . . . . . . . . . . . . . 119
2. The Stability of Solutions to Stationary Linear
Equations . . . . . . . . . . . . . . . . . . . 121
3. Conditions for Negativity of the Real Parts of All Roots
of the Quasipolynomial . . . . . . . . . . . . . . 126
4. The Case of Small Deviating Arguments . . . . . . . . 139
5. The Case of Large Deviating Arguments . . . . . . . . 143
6. Lyapunov's Second Method. . . . . . . . . . . . . 144
7. Stability in the First Approximation . . . . . . . . . 159
8. Stability under Constantly Acting Disturbances . . . . . 162
V
CONTENTS
9 . Lyapunov's Second Method for Equations of
Neutral Type . . . . . . . . . . . . . . . . . . 165
10. Absolute Stability . . . . . . . . . . . . . . . . 175
Chapter IV . Periodic Solutions
1 . Some Properties of Periodic Solutions and Existence
Theorems . . . . . . . . . . . . . . . . . . . 183
2 . Periodic Solutions of Stationary. Linear. Homogeneous
Equations . . . . . . . . . . . . . . . . . . . 187
3 . Periodic Solutions of Linear Inhomogeneous Equations
with Stationary Homogeneous Parts . . . . . . . . . 191
4 . Periodic Solutions of Linear Equations with Variable
Coefficients and Deviating Arguments . . . . . . . . 196
5 . Periodic Solutionsof Quasilinear Equations . . . . . . 200
6 . Functionally Equivalent Systems of Differential
Equations with a Deviating Argument . . . . . . . . 210
Chapter V . Stochastic Differential Equations with a Retarded
Argument
1 . Basic Concepts . . . . . . . . . . . . . . . . . 219
2 . Stability . . . . . . . . . . . . . . . . . . . . 223
3 . Stationary Solutions of Equations with a Delay . . . . . 229
Chapter VI . Approximate Methods for the Integration of Differential
Equations with a Deviating Argument
1 . General Remarks about the Application of Approximate
Integration Methods . . . . . . . . . . . . . . . 235
2 . Euler's Method and Parabolic Methods . . . . . . . . 237
3 . Expansion in Powers of the Retardation . . . . . . . 243
4 . Asymptotic Methods for Equations with Small
Deviating Argument . . . . . . . . . . . . . . . 244
5. Iterative Methods . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . 251
2 . Periodic Solutions . . . . . . . . . . . . . . . . 256
3 . Boundary-Value Problems . . . . . . . . . . . . . 265
4 . Optimal Processes with a Retardation . . . . . . . . 275
5 . Stationary Points . . . . . . . . . . . . . . . . 285
BIBLIOGRAPHY
I . Monographs . . . . . . . . . . . . . . . . . . 293
I1 . Survey Articles . . . . . . . . . . . . . . . . . 294
111. Journal Articles . . . . . . . . . . . . . . . . . 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. Exten-
sions of the theory of differential equations with deviating argument as well as the
stimuli of developments within various fields of science and technology contribute
to the need for a new edition. This theory in recent years has attracted the atten-
tion of vast numbers of researchers, interested both in the theory and its applica-
tions.
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 do 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 to 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 Me-
chanics-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 of control systems, and
because of the intrinsic richness and beauty of such equations. However, this do-
main 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 to 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 corre-
sponding results carrying over to the theory of differential equations with a de-
viating argument, and the new characteristics that appear because of this trans-
ference; 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 acces-
sible 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 in-
formation 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. How-
ever, 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 im-
mense 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