Table Of ContentIMPULSIVE DIFFERENTIIIL EQUATIONS
WORLD SCIENTIHC SERIES ON NONUNEAR SCIENCE — SERIES A
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Series A Vol. 14
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Series Editor: Leon O. Chua
IMPULSIVE DIFFERENTIAL EQUATIONS
A M Samoilenko & N A Perestyuk
Institute of Mathematics
National Academy of science of Ukraine
3, Tereshchenkivska str
252601 Kiev, Ukraine
Translated from the Russian by
Yury Chapovsky
Y(fe World Scientific
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IMPULSIVE DIFFERENTIAL EQUATIONS
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Preface
To describe mathematically an evolution of a real process with a short-
term perturbation, it is sometimes convenient to neglect the duration of the
perturbation and to consider these perturbations to be "instantaneous". For
such an idealization, it becomes necessary to study dynamical systems with
discontinuous trajectories or, as they might be called, differential equations
with impulses.
By itself, it is not a new idea to make an exhaustive study of ordi
nary differential equations with impulses. Such problems were considered at
the beginning of the development of nonlinear mechanics and attracted the
attention of physicists because they gave the possibility to adequately
describe processes in nonlinear oscillating systems. A well known example
of such a problem is the model of a clock. Using this elegant example,
N.M. Kruylov and N.N. Bogolyubov have shown in 1937 in their classical
monograph Introduction to Nonlinear Mechanics that for a study of systems
of differential equation with impulses, it is possible to apply approximation
methods used in nonlinear mechanics.
The interest in systems with discontinuous trajectories have grown in
recent years because of the needs of modern technology, where impulsive
automatic control systems and impulsive computing systems became very
important and are intensively developing broadening the scope of their ap
plications in technical problems, heterogeneous by their physical nature and
functional purpose. As a natural response to this, the number of mathe
matical works on impulsive differential equations has increased in different
mathematical schools both in our country and abroad. However, the most
systematic and in-depth studies were made in the Kiev school of Nonlinear
Mechanics. It is the mathematicians of this school, who could broadly ap
proach this problem, consider it in a general form, formulate and solve a
number of problems which are important for applications but have not been
studied before. With every reason one can say that, as a result of the efforts
v
vi PREFACE
of this group of Kiev mathematicians, there arose a mathematical theory
of impulsive ordinary diiferential equations, which has its methods, general
and deep results, and specific problems.
This monograph is written by the representatives of the Kiev school of
Nonlinear Mechanics, who fruitfully work in the area of impulsive differential
equations, and give a systematic and sufficiently complete treatment of the
subject.
It contains a sufficiently complete study of systems of impulsive linear dif
ferential equations. It was shown in the monograph that the classic theory of
the first Lyapunov's method can be naturally carried over to the considered
systems. This substantially enriches and develops the fundamental research
in differential equations as such. Deep results are obtained for the stability of
solutions of impulsive systems. Again (how many times!) one sees that the
idea of the direct Lyapunov method is universal and can be applied not only
to classic differential equations but to more general classes of mathematical
objects.
In addition to this, the problem of the existence of integral sets of im
pulsive differential equations is solved and properties of the integral sets are
investigated in the monograph. An important class of discontinuous almost
periodic systems is studied and the problem of optimum control is solved for
impulsive systems.
An indubitable merit of this book is that it contains a large number of
worked out nontrivial examples which could serve as guidelines for solving
other particular applied problems.
Undoubtedly, this monograph would be interesting not only to specialists
in the theory of differential equations but to many high technology specialists
who work in the areas of applied mathematics, computer technology, and
automatic control. The book will also be useful to instructors of courses in
differential equations. Already, it is impossible today not to offer elements of
the theory of impulsive differential equations in special courses for students
specializing in differential equations, theoretical and applied mechanics.
Academician Yu.A. Mitropol'skii
Contents
Preface v
1 General Description of Impulsive Differential Systems 1
1.1 Description of mathematical model 2
1.2 Systems with impulses at fixed times 7
1.3 Systems with impulses at variable times 19
1.4 Discontinuous dynamical systems 28
1.5 Motion of an impulsive oscillator under the effect of an im
pulsive force 34
2 Linear Systems 43
2.1 General properties of solutions of linear systems 44
2.2 Linear systems with constant coefficients 52
2.3 Stability of solutions of linear impulsive systems 56
2.4 Characteristic exponents of functions and matrices of functions 61
2.5 Adjoint systems. Perron theorem 69
2.6 Reducible systems 80
2.7 Linear periodic impulsive systems 85
2.8 Linear Hamiltonian systems of impulsive differential equations 92
2.9 Periodic solutions of a certain second order equation 101
3 Stability of solutions 107
3.1 Linear systems with constant and almost constant matrices . 108
3.2 Stability criterion based on first order approximation 120
3.3 Stability in systems with variable times of impulsive effect . . 130
3.4 Direct Lyapunov method for studying stability of solutions of
impulsive systems 139
vii
viii CONTENTS
4 Periodic and almost periodic impulsive systems 151
4.1 Nonhomogeneous linear periodic systems 152
4.2 Nonlinear periodic systems 158
4.3 Numerical-Analytical method for finding periodic solutions . 166
4.4 Almost periodic sequences 183
4.5 Almost periodic functions 201
4.6 Almost periodic differential systems 207
4.7 Homogeneous linear periodic systems 218
5 Integral sets of impulsive systems 220
5.1 Bounded solutions of nonhomogeneous linear systems 230
5.2 Existence of bounded solutions of nonlinear systems 238
5.3 Integral sets of systems with hyperbolic linear part 243
5.4 Integral sets of a certain class of discontinuous dynamical
systems 267
6 Optimum control in impulsive systems 275
6.1 Formulation of the problem. Auxiliary results 276
6.2 Necessary conditions for optimum 284
6.3 Impulsive control with fixed times 290
6.4 Necessary and sufficient conditions for optimum 295
7 Asymptotic study of oscillations in impulsive systems 207
7.1 Formulation of the problem 298
7.2 Formulas for an approximate solution of a non-resonance
system 302
7.3 Substantiation of the averaging method for a non-resonance
system 308
7.4 Averaging in a resonance system and its substantiation . . .. 314
7.5 Formulas for approximate solutions for an impulsive effect
occuring at fixed positions 326
7.6 Substantiation of the averaging method for systems with im
pulses occurring at fixed positions 337
7.7 General averaging scheme for impulsive systems 341
7.8 On correspondence between exact and approximate solutions
over an infinite time interval 351
A Periodic and almost periodic impulsive systems 361
(by S.I. Trofimchuk)
CONTENTS ix
A.l Impulsive systems with generally distributed impulses at fixed
times 362
A.2 Periodic impulsive systems with impulses located on a surface 370
A.3 Unbounded functions with almost periodic differences . . .. 374
A.4 Spaces of almost periodic functions on the line 381
A.5 Spaces of piecewise continuous almost periodic functions . . . 389
A.6 Almost periodic measures on the line 404
A.7 Almost periodic solutions of impulsive ordinary differential
equations 415
A.8 Linear abstract impulsive systems and their almost periodic
solutions 426
Bibliographical Notes 437
Bibliography 443
Subject Index 459
Chapter 1
General Description of
Impulsive Differential
Systems
