Countable Systems of Differential Equations [Reprint 2011 ed.] 9783110942033

Table of contents :
PREFACE
1. GENERAL CONCEPTS OF THE THEORY OF INFINITE SYSTEMS OF DIFFERENTIAL EQUATIONS
1.1. Theorems on Existence and Uniqueness of Solutions
1.2. Truncation Method
1.3. Solutions of the Linear System
1.4. Matrizant of a Linear System
1.5. Normal Autonomous Systems
1.6. Periodic Solutions
2. INVARIANT TORI
2.7. Green Function
2.8. Existence of a Smooth Invariant Torus
2.9. Cl-Differentiability of the Invariant Torus
2.10. The Case of Infinitely Many Angular Variables
2.11. Theorem on Convergence of the Sequence of Invariant Tori
2.12. Invariant Tori of Nonlinear Systems
2.13. Exponential Attraction of Motions in a Neighborhood of the Invariant Torus of a System of Equations to Its Motions on the Torus
3. REDUCIBILITY OF LINEAR SYSTEMS
3.14. Erugin and Floquet–Lyapunov Theorems
3.15. Periodic Systems
3.16. Systems with Almost Periodic Coefficients
3.17. Quasiperiodic Systems with Unbounded Right-Hand Side
3.18. Decomposition of Countable Systems
4. IMPULSIVE SYSTEMS
4.19. Some Results of the Theory of Linear Systems
4.20. Integral Sets and Invariant Tori
4.21. Periodic Solutions for Impulsive Systems with Small Parameter
4.22. Approximate Solution of the Periodic Problem of Control
REFERENCES

Citation preview

Countable Systems of Differential Equations

COUNTABLE SYSTEMS OF DIFFERENTIAL EQUATIONS

A.M. Samoilenko and Yu.V. Teplinskii

mySPm

Utrecht · Boston, 2003

VSP

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an imprint o f Brill A c a d e m i c P u b l i s h e r s

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© VSP 2003 First p u b l i s h e d in 2 0 0 3 ISBN 90-6764-393-9

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

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Contents

PREFACE

vii

1. GENERAL CONCEPTS OF THE THEORY OF INFINITE SYSTEMS OF DIFFERENTIAL EQUATIONS 1.1. Theorems on Existence and Uniqueness of Solutions 1.2. Truncation Method 1.3. Solutions of the Linear System 1.4. Matrizant of a Linear System 1.5. Normal Autonomous Systems 1.6. Periodic Solutions

1 1 11 15 20 27 32

2. INVARIANT TORI 2.7. Green Function 2.8. Existence of a Smooth Invariant Torus 2.9. C^-Differentiability of the Invariant Torus 2.10. The Case of Infinitely Many Angular Variables 2.11. Theorem on Convergence of the Sequence of Invariant Tori . . . . 2.12. Invariant Tori of Nonlinear Systems 2.13. Exponential Attraction of Motions in a Neighborhood of the Invariant Torus of a System of Equations to Its Motions on the Torus

49 49 68 75 99 115 119

3. REDUCIBILITY OF LINEAR SYSTEMS 3.14. Erugin and Floquet-Lyapunov Theorems 3.15. Periodic Systems 3.16. Systems with Almost Periodic Coefficients 3.17. Quasiperiodic Systems with Unbounded Right-Hand Side 3.18. Decomposition of Countable Systems

151 151 155 163 175 189

ν

133

vi

4. IMPULSIVE SYSTEMS 4.19. Some Results of the Theory of Linear Systems 4.20. Integral Sets and Invariant Tori 4.21. Periodic Solutions for Impulsive Systems with Small Parameter 4.22. Approximate Solution of the Periodic Problem of Control

REFERENCES

197 197 207 231 252

271

PREFACE

The present book is devoted to the solution of various problems in the theory of differential equations in a space T t of bounded numerical sequences (called countable systems). In particular, we deal with the general theory of countable systems, the theory of oscillating solutions, and the theory of countable systems with pulse action. Our main attention is given to the generalization of the results of numerous authors obtained in recent years for finite-dimensional systems of differential equations to the case of systems from the analyzed class. It is clear that countable systems form a class of systems of differential equations in Banach spaces and, thus, many issues of the theory of equations of this sort are well known. At the same time, differential equations in the space 9JI have many specific features, which caused the necessity of development of the theory of these equations originated by A. N. Tikhonov and K. P. Persidskii [Tik, Perl, Per2]. As a result of the systematic investigation of countable systems, K. P. Persidskii [Per4] proved theorems on the existence and uniqueness of a solution of the Cauchy problem, developed methods for their approximate solution, and created the theory of stability of these systems. Extensive subsequent investigations of these systems were carried out by O. A. Zhautykov and K. G. Valeev [Zhal-Zha3, VaZ]. A great number of problems connected with the development of asymptotic methods and the investigation of oscillatory processes was solved by the followers of the scientific school of Ν. N. Bogolyubov and Yu. A. Mitropolsky. The proposed monograph is based on the original results obtained by the authors in [Satl-Sat8, STA, STL, STT1, STT2, Tepl-TeplO, TeAl-TeA3, TeT, TeL] and consists of four chapters. The first chapter is devoted to the presentation of the general concepts of the theory of infinite systems of differential equations and is partially based on the results obtained by K. P. Persidskii. vii

viii

Preface

The second chapter deals with the theory of covariant toroidal manifolds of linear and nonlinear countable systems similar to the theory of extensions of dynamical systems on a torus developed recently for finite-dimensional systems by Yu. A. Mitropolsky, A. M. Samoilenko, and V. L. Kulik [MSK2, SamlO, Sam8, SaK2, SaK3]. The third chapter contains results on the reducibility of linear systems with periodic and quasiperiodic coefficients. In the fourth chapter, the theory of pulse systems [SaPl-SaP3, Per] is generalized to the case of countable systems of differential equations. Throughout the book, we use double enumeration of formulas and statements (definitions, lemmas, theorems, etc.). The first number is the number of a given section and the second number is the ordinal number of a formula or statement (definition, lemma, theorem, etc.) in this section.

The Authors

1. GENERAL CONCEPTS OF THE THEORY OF INFINITE SYSTEMS OF DIFFERENTIAL EQUATIONS

In Chapter 1, we present various facts from the theory of countable systems of differential equations used in the subsequent chapters. The major part of these facts can be found in the works [Perl-Per4, Tik], which are well known to the experts in various fields of mathematics, physics, and engineering dealing with systems with infinitely many degrees of freedom. At present, there are several directions of investigation of these systems, including their general theory, systems of partial differential equations, characteristic numbers and the stability of solutions, averaging, differential equations in normed spaces, "multiperiodic" solutions, etc. [Bro, VaZ, Gor, DaK, DaKr, Erm, Zhal-Zha3, Res, Khal, Kha2, Kha]. In our book, we present only some results from the theory of systems of ordinary differential equations: theorems on the existence and uniqueness of solutions, properties of solutions of linear and nonlinear systems (including conditions for the continuous dependence of solutions on the initial data), and matrizants of linear systems and their properties. Moreover, we consider analogs of some properties of normal autonomous systems [Pon] and present the fundamentals of the numerical-analytic method aimed at finding periodic solutions of nonlinear countable systems [SaR].

1.1. Theorems on Existence and Uniqueness of Solutions Consider a system of differential equations of the form = fi(x,yi,y2,---,yn,···), ι

»= 1,2,....

(l.i)

2

General Concepts of the Theory of Infinite Systems

Chapter 1

Definition 1.1. A function / ( y i , y2, · • ·) is called strongly continuous i f , for any ε > 0, there exist No and δ > 0 such that the inequality \y[ — y"\ < δ, i — 1 , 2 , . . . , No, implies the estimate \f(y'i,y'2,···)-fivlvl

•••)]< e.

Theorem 1.1. Assume that the right-hand sides of the system of equations (1.1)

(a) are defined for any Vi e R\ i — 1 , 2 , . . . , and all χ G Tq = [xq, xq + a] C Ä1; (b) are strongly continuous in yi, y2, • • • for fixed χ and measurable in χ for fixed yi, i = 1 , 2 , . . . ; (c) satisfy the inequalities \fi(x,yuV2,

···)!
(s)

.··

y^\x)

y^\x)

.·.

y^\x)

···

y^\x)

y^\x)

·..

y^\x)

···

and rewrite the set of sequences row by row: y[ai\x)

y) ^ H· It is easy to see that the series ys(x) =y°s + (x + x0)y°s+l

+ ^

y°s+2 + ... ,

s = l,2,... ,

define the required solution. Thus, in particular, the solution of this sort that passes through the point (0; 0 , 0 , 0 , . . . ) e Η has the form ys{x) = 0, s = 1 , 2 , . . . . Consider a function

(θ,

χ = 0.

/ χ dsf{x) The vector yis(z) = — s , s = 1 , 2 , . . . , is also a solution of the system of dx equations (1.5) that passes through the point (0; 0 , 0 , 0 , . . . ) € H. The presented solutions have the same initial conditions but do not coincide, i.e., the theorem does not guarantee the uniqueness of a solution for certain initial conditions. The following theorem is presented here without proof: Theorem 1.2. Let the system of equations (1.1) be such that oo 1- | / η Ο , 2 / ι , · · · , 2 / ή > · · · ) - / η ( ζ , ! / ΐ , · · · , ί / η , · · · ) Ι < Σ Kni\y'i ~ y"\, Tl = 1,2,... ;

i=l

Section 1

Theorems on Existence and Uniqueness of Solutions

1

00

2. Σ Κ™ — Ai < A < oo, Kni = const > 0, A — const > 0 , n=1 const > 0 , ζ = 1,2,

Ai

=

77ien exists at most one solution (y\(x), 1/2(2), • • •) of the system of equations (1.1) satisfying given initial conditions and such that 00

< Β — const < 00 .

y^ \yn(x)\ n=l

We now consider a domain h defined by the inequalities 0 < χ < r, \ys\ < R°, s = 1 , 2 , . . . , and the domain Η defined in the example considered above. Assume that the right-hand sides of the system of equations (1.1) in the domain Η satisfy the following conditions: 1*. The functions fo, i — 1 , 2 , . . . , are continuous in χ at any point, i.e., fi(x + &x,yi,y2,...)

- fi(x,yi,y2,...)

-»· 0

as

Ax - » 0.

2*. The functions fi, i = 1 , 2 , . . . , satisfy the Cauchy-Lipschitz condition in variables y \ , y 2 , . . . , i . e . ,

\fi(x,y[,y'2,···)

- fi(x,y",V2,···)]

< 0 as g —• oo, for any fixed ν > 0 there exists a number g' — g(u) such that es(g') < v. We set g — g'. One can also find a number N(s, v) such that J'sn < α sup{|ui(x) - vltn(x)\,...,

\ug(x) - fg,n(z)|}

< ai/

for η > N(s, u). Then, for η > N(s, u), we have Jsn < (2αη + a)ν for any χ e σ. This means that, for any χ e σ, X

X

lim η—»oo Consequently, χ Us(x)=y°s+

J /e(r,ui(r),u2(r),...)dr,

s = 1,2,..

This yields us(x) = ys(x), s = 1 , 2 , . . . , i.e., for any fixed natural g, max(sup{|yi(:r) - f i , n ( x ) | , · · . , | y g { x ) IfciT

0

as η —» oo, which contradicts inequality (2.9). Theorem 2.1 is proved. Corollary 2.1. Assume that σ C [0, r] and the quantities XQ and σ are chosen so that the quantity 7 satisfies the condition 7 < R. Then Theorem 2.1 is true in the domain h.

Section 3

Solutions of the Linear

System

15

1.3. Solutions of the Linear System Consider a linear system of equations of the form dxs

(3.1)

dt

i=1

satisfying the following conditions [called conditions ( P ) ] : In the domain H, the functions pSi(t), p8{t) = \Psj(t)\ are continuous for all s, i = 1 , 2 , . . . , and satisfy the inequalities ps(t) < a(t), where a(t) is a continuous function of t. Assume that a sequence of vector functions (xn

(t)\

X21 ((r)||dr,

m

to

where we have introduced the following compatible norms of vectors and matrices: ||χ(-)(ί)||=8υρ{|^(ί)|,...}

and

||P(i)|| = sup £ i

\Pij(t)\.

3=1

By using the Gronwall-Bellman inequality, we arrive at the formula t

||x (m) (i)ll < c ^ e x p j J a ( r ) d r | } ,

tea,

m = 1,2,....

to

Consequently, (m)

X

(t)| < cß exp|| J o ( r ) dr |}

(3.4)

ίο for all t e σ. In expression (3.4), we fix the values of s and t. Without changing the absolute values of c\,..., Cm, we choose them to guarantee the validity of the inequalities cgxsg(t) > 0, g = 1 , 2 , . . . , m. Then ^m

Σ

Ν

t.

\ ^ c/3exp| / α(τ to

i = 1

whence

™

ι

t

< 0exp{ / a(r)dr|}. i = 1

(3.5)

to

Since the quantities s and t were chosen arbitrarily, inequality (3.5) is true for all s = 1 , 2 , . . . and t 6 σ.

Section 3

17

Solutions of the Linear System

By virtue of (3.5), we obtain

} M t ) | < Σ

Ν

x

\ si(t)\

i=1

< c/?exp{ / a ( r ) d r | } , xo

5 = 1,2,...,

and, consequently, t ||x(i)|| < cßexp|

Jα(τ)άτ

|},

t e σ.

(3.6)

This means that series (3.3) are absolutely convergent and, in view of the well-known Lebesgue theorem, can be integrated term by term. This yields m oo Y,cg(j2psk(t)xkg{t)) 9=1 fc=l

=

J2cgx'sg(t)

1= 1,2,...,«,....

(6.4)

Assume that the right-hand side of system (6.4) is defined for any real t and all χ = (χι, X2, • • ·) from a domain D. Moreover, for any fixed t and χ from their domains of variation, the function f(t,x) = (fi(t, χ), /2(i, x), •..) is a point of the space Wl. We represent the system of equations (6.4) in the form of a differential equation in the space SOt:

dx

— = /(*,*).

(6.5)

The solution of (6.5) is understood as a function x(t) = (xi(t),x2(t),...) defined for t from an interval (a, 6), continuously differentiable with respect to t in this interval, satisfying equation (6.5), and belonging to the space DJl for all

t

g ( a , 6).

Assume that the function f(t,x) is a continuous periodic function of t with period Τ that satisfies the inequalities

\f(t,x)\

< M, \f(t,x')-f(t,x")\ · · • is defined by the recurrence relations t

awt-i(f)

=

(l

-

~ ψ ~ )

τ+Τ

J

J

τ

t

(6.9)

Oim(s)ds,

m = 0,1,2,... . Then there exist positive

where

πι

2.

and ( ^

constants

) are, respectively, the integral and fractional parts of the

number —, such that

am+i(t)