Differential and Integral Inequalities: Ordinary Differential Equations v. 1: Theory and Applications: Ordinary Differential Equations v. 1

DIFFERENTIAL AND INTEGRAL INEQUALITIES Theory and Applications Volume I ORDINARY DIFFERENTIAL EQUATIONS

This is Volume 55 in R’IATHEMA’I’ICS I N SCIENCE AND ENGINEERING A series of monographs and textbooks Edited by RICHARD BELLMAN, University of Southern California A complete list of the boolts in this series appears at the end of this volume.

DIFFERENTIAL AND

INTEGRAL INEQUALITIES Theory and Applications Volume I ORDINARY DIFFERENTIAL EQUATIONS

V. LAKSHILIIKANTHAM and S. LEELA {Jnioersity of Rho& Islmiii Kiiqstow, Rliotlc Islaiid

A C A D E RI I C P R E SS

New J’ork and London

1969

COPYRIGHT 0 1969, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS BOOK MAY BE REPRODUCED I N ANY FORM, BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

ACADEMIC PRESS, INC. 111 Fifth Avenue, New York, New York 10003

United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. Berkeley Square House, London W.l

LIBRARY OF CONGRESS CATALOG CARDNUMBER: 68-8425

PRINTED I N THE UNITED STATES OF AMERICA

Preface

This volume constitutes the first part of a monograph on theory and applications of differential and integral inequalities. 'The entire work, as a whole, is intended to be a research monograph, a guide to the literature, and a textbook for advanced courses. T h e unifying theme of this treatment is a systematic development of the theory and applications of differential inequalities as well as Volterra integral inequalities. T h e main tools for applications are the norm and the Lyapunov functions. Familiarity with real and complex analysis, elements of general topology and functional analysis, and differential and integral equations is assumed. T h e theory of differential inequalities depends on integration of differential inequalities or what may be called the general comparison principle. T h e treatment of this theory is not for its own sake. 'The essential unity is achieved by the wealth of its applications to various qualitative problems of a variety of differential systems. T h e material of the present volume is divided into two sections. T h e first section consisting of four chapters deals with ordinary differential equations while the second section is devoted to Volterra integral equations. T h e remaining portion of the monograph, which will appear as a second volume, is concerned with differential equations with time lag, partial differential equations of first order, parabolic and hyperbolic respectively, differential equations in abstract spaces including nonlinear evolution equations and complex differential equations types. T h e vector notation and vectorial inequalities are used freely throughout the book. Also, because of the several allied fields covered, it becomes convenient to use the same letter with different meanings in different situations. This, however, should not cause confusion, since it is spelled out wherever necessary. T h e notes at the end of each chapter indicate the sources which have V

vi

PREFACE

been consulted and those whose ideas are developed. Some sources which are closely related but not included in the book are also given for guidance. We wish to express our warmest thanks to our colleague Professor C. Corduneanu for reading the manuscript and suggesting improvements. Our thanks are also due to Professors J. Hale, N. Onuchic, and C. Olech for their helpful suggestions. We are immensely pleased that our monograph appears in a series inspired and edited by Professor R. Bellman and we wish to express our gratitude and warmest thanks for his interest in this book.

V. LAKSHMIKANTHAM S. LEELA Kingston, Rhode Island December, I968

Contents

PREFACE

V

ORDINARY DIFFERENTIAL EQUATIONS Chapter 1 . 1.0. Introduction 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.8. 1.9. 1.10. 1.11.

Chapter 2. 2.0.

2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8. 2.9. 2.10. 2.11. 2.12. 2.13. 2.14. 2.15 2.16.

3

Existence and Continuation of Solutions Scalar Differential Inequalities Maximal and Minimal Solutions Comparison Theorems Finite Systems of Differential Inequalities Minimax Solutions Further Comparison Theorems Infinite Systems of Differential Inequalities Integral Inequalities Reducible to Differential Inequalities Differential Inequalities in the Sense of Caratheodory Notes

3 7 11 15 21 25 27 31 37 41 44

Introduction Global Existence Uniqueness Convergence of Successive Approximations Chaplygin’s Method Dependence on Initial Conditions and Parameters Variation of Constants Upper and Lower Bounds Componentwise Bounds Asymptotic Equilibrium Asynlptotic Equivalence A Topological Principle Applications of Topological Principle Stability Criteria Asymptotic Behavior Periodic and Almost Periodic Systems Notes

45 45 48 60 64 69 76 79 84 88 91 96 100 102 108 120 129

vi i

...

CONTENTS

Vlll

Chapter 3. 3.0. Introduction 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.1. 3.8. 3.9. 3.10. 3.1 I . 3. I ? . 3.13. 3.14. 3.15. 3.16. 3.11. 3.18. 3.19. 3.20. 3.21.

Basic Comparison Theorems Definitions Stability Asymptotic Stability Stability of Perturbed Systems Convcrse Theorems Stability by the First Approximation Total Stability Integral Stability I,”-S ta hi I i ty Partial Stability Stabilit) of Differential Inequalities Boundcdness and Lagrange Stability Eventual Stability Asymptotic Behavior Relative Stability Stability with Respect to a Manifold .-\lmost Periodic Systems Uniqueness and Estimates Continuous Dependence and the Method of Averaging Notes

Chapter 4 . 4.0. Introduction 4.1. 4.2. 4.3. 4.4. 4.5. 4.6. 4.7. 4.8. 4.9. 4.10.

Main Comparison Theorem Asymptotic Stability Instability Conditional Stability and Boundedness Converse Theorcms Stability in Tube-like Domain Stability of Asymptotically Self-Inwriant Sets Stability of Conditionally Invariant Sets Existence and Stability of Stationary Points Notes

131 131 135 138 145 155 158 177 186 191 199 205 209 212 222 229 24 1 244 245 254 257 264 267 267 269 273 277 284 293 291 305 308 311

VOLTERRA INTEGRAL EQUATIONS Chapter 5. 5.0. Introduction

Integral Inequaiitics 5.2. Local and Global Existence 5.3. Comparison Theorems 5.4. Approximate Solutions, Bounds, and Uniqueness 5.5. Asymptotic Behavior 5.6. Perturbed Integral Equations 5.7. Admissibility and Asymptotic Behavior 5.8. Integrodifferential Inequalities 5.9. Notes 5.1.

31 5 315 319 322 324 327 333 340 350 354

CONTENTS

ix

Bibliography

355

AUTHORINDEX

385

SUBJECT INDEX

388

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DIFFERENTIAL AND INTEGRAL INEQUALITIES Theory and Applications Volume I

ORDINARY DIFFERENTIAL EQUATIONS

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ORDINARY DIFFERENTIAL EQUATIONS

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Chapter 1

1 .O. Introduction This chapter is an introduction to the theory of differential inequalities and therefore forms a basis of the remaining chapters. After sketching the preliminary existence and continuation of solutions of an initial value problem for ordinary differential equations, we develop fundamental results involving differential inequalities. Basic comparison theorems that form the core of the monograph are treated in detail. While considering the system of differential inequalities (finite or infinite), we find it convenient to utilize the notion minimax solutions, and consequently our treatment rests on this notion. Certain useful integral inequalities that can be reduced to the theory of differential inequalities are also presented. Some results on differential inequalities of Caratheodory's type are also dealt with.

1.1. Existence and continuation of solutions Let R" denote the real n-dimensional, euclidean space of elements u = ( u l ,u2 ,..., un). Sometimes, we shall denote also the (a + 1)-tuple (t, u l , u, ,..., un) as an element, and Rn+I shall denote the space of elements ( t , u l , uz ,..., un) or ( t , u). Let 11 u 11 be any convenient norm. As usual, we shall use R instead of R'. Let E be an open (t, u)-set in I?"+'. We shall mean by C [ E ,R"] the class of continuous mappings from E into R". Iff is a member of this class, one writes f E C[E, R"]. Let us consider a system of first-order differential equations with an initial condition u' = g(t, u ) , u(t0) = u 0 , (1.1.1) where u' = du/dt,u0 = ( u I 0 ,u,,, ,..., unO),and g E C [ E ,R"]. A solution of the initial value problem (1.1.1) is a differentiable function of t such 3

4

CHAPTER

1

that ~ ( t , , = ) ZL, , ( t , u(t))E E, and u'(t) = g(t, u(t)) for a t-interval J containing t,, . This means that u ( t ) has a continuous derivative. From these requirements on the continuous function u(t), it follows that it satisfies the integral equation

I n order to prove the classical Peano's existence theorem, we have to introduce the notion of an equicontinuous family of functions. 1)EFINITION 1.1.1. A family of functions F = { f ( u ) } defined on some ti-set E C R" is said to be equicontinuous if, for every E > 0, there exists a S = S ( E ) , independent of f E F and also u l , u2 E E, such that 11 f ( u J -f(u2)ll -E, whenever / / u1 - u2 I/ < 6. T h e following theorem shows the fundamental property of such a family of functions, the proof of which will be omitted.

THEOREM 1.1.1. (Ascoli-Arzela). Let F = ( f ] be a sequence of functions defined on a compact u-set E C R", which is equicontinuous and equibounded. Then, there exists a subsequence ( f n } , n = 1, 2, ..., which is uniformly convergent on E. THEOREM 1.1.2. (Peano's Existence Theorem). Let g E CIRo , Rn], whcrc R,, is the set [ ( t ,u ) : to t to a, 11 u - u, /I 61; 11 g(t, u)\l M on R,, . Then, the initial value problem (1.1.1) possesses at least one solution u ( t ) on t, t t, 01, where 01 = min(a, 6 / M ) .

< < +


0, there is an n 3 a(.) such that

1 k ( t , un-1(t)I
to

E

(1.4.3)

a).

Let to < T < to + a. By Lemma 1.3.1, the maximal solutions (1.3.2) exist on [to , T ] for all E > 0 sufficiently small, and O E'

m(t) < r(t, E ) ,

t

of

(1.4.4)

~ ( t= ) lim r ( t , c)

uniformly on [ t o ,TI. Using (1.3.2) and Theorem 1.2.1, we derive that

u ( t , c)

(1.4.3)

[ t o , 71.

and

applying (1.4.5)

T h e last inequality, together with (1.4.4), proves the assertion of the theorem.

16

1

CHAPTER

REMARK1.4.1. If the inequality (1.4.1) is reversed and m(t,) 2 u,, , then we have to replace the conclusion (1.4.2) by m(t) 3 p ( t ) , where p ( t ) is the minimal solution of (1.2. I).

Theorem 1.4.1 can also be proved under a weaker hypothesis.

THEORFM 1.4.2.

2

Let m(t),r ( t ) be as in Theorem 1.4.1, and =

[t t [t,,, t,

+ a) :

Y(t)

< m ( t ) < r ( t ) + €01,

for some c0 > 0. If (1.4.1) is satisfied for t E 2 at-most countable subset of 2,then (1.4.2) holds.

-

9,where 3

(1.4.6)

is an

Proof. I t is enough to prove (1.4.5). As before, Lemma 1.2.2 implies that (1.4.3) is satisfied for t E Z. Proceeding as in the proof of Theorem 1.2.1, we arrive at a t, such that m(td

In view of (1.4.4), there exists an r ( t , 6)

=Y(t, en

9

.).

> 0 such that

< y(t) i€0,

t

E

[to

, 71.

RIoreover, we have r ( t ) -'r(t, , E), and hence there results the inequality r ( t ) < r ( t , €)



[ 1.4.7)

D-w(t> 3 d t , w ( t ) )

(1.4.8)

f)-v(t)

1.4.

17

COMPARISON THEOREMS

+

for t E ( t o ,to u). Assume further that, for each t E [to , TI,g satisfies the condition g( t >~

2

1 - )g ( t > ~ 2 )

where G E C[[t,, to of

-G(T

+

-

T E

t , ~1 - uZ),

( t o ,t,

~1

3~

+ u ) and 2

,

(1.4.9)

+ a) x R, R], and r(t) = 0 is the maximal solution G(t,u),

U' =

~ ( t ,= ) 0.

T h e n (1.2.5) holds. Proof. Proceeding as in the proof of Theorem 1.2.1, there exists a t, E (t, , to a) such that 4 t l ) = W(tl>, (1.4.10)

+

and

~ ( t< ) ~ ( t ) , to

+

tll.

Setting m ( t ) = w l ( t ) vl(t), the definitions of n, , w1 and the assumptions (1.4.7) and (1.4.8) imply the inequality ~

D-m(t)

=

D-w,(t)

where gl(t, u ) = -g(tl (1.4.9) to arrive at

D-s(t)

~

+ to

D-m(t)

-

< g1(t, W l ( t ) )

- gl(4

s(t)),

t, u). Since (1.4.13) holds, we can use

-< G(t,m(t)),

t E [to

9

tll.

By Theorem 1.4.1, we have m(t)

< r(t),

t

E

[to , tll,

(1.4.14)

where r ( t )is the maximal solution of u' = G(t,u), such that r(t,) = m(t,). From the definition of m ( t ) and (1.4.12) and (1.4.13), we deduce that m ( t ) 3 0, t E [t, , t l ] , and nz(t,) = 0. Then, the inequality (1.4.14) and the assumption r ( t ) = 0 show that v ( t ) = ~ ( t ) , t E [to , ti],

which, however, is contrary to the assumption (1.4.1 1) and the definition of t , . Hence, the set 2 is empty, and the theorem is proved.

18

CIIAPTER

1

T o give another comparison theorem that, in certain situations, is more useful than Theorem 1.4.1, we require the following result:

+

THEOREM 1.4.4. Let E be the product space [to, to a ) x R2 and g E C [ E , R ] . Assume that g is nondecreasing in v for each t and u. Suppose that ~ ( tis) the maximal solution of the differential equation u' = g(t, u, u ) ,

existing on [to , to

u ( f o )= u,

+ a ) , and

t € [to, t,

r ( t ) 3 0,

(1.4.1 5)

2; 0

(1.4.16)

4-u).

Then, the maximal solution r l ( t ) of u(t,) = uo ;3 0,

u' = gl(t, u ) ,

where g l ( t , 21)

= g(t, u, ~ ( t ) )exists ,

r ( t ) = rdt),

on [to, t, t

E

[t" , to

(1.4.17)

+ u ) and

+ a).

(1.4.18)

Proof. By Theorems 1.3.1 and 1.3.2, the maximal solution r l ( t ) of (1 .4.17) exists on an interval [t,,, to $- (Y], (Y a, which can be extended to the boundary of E. This implies that either r l ( t ) is defined over [ t o ,t, a ) or there exists a t , < t, a such that

+

+

and this yields, from Theorem 1.4.1, that

-

as far as r l ( t ) exists. I t follows from (1.4.16), (1.4.19), and (1.4.20) that Yl(t,)

(1.4.21)

-km

as t,. -+ t l - . We shall show that (1.4.21) cannot be true. For this purpose, consider the maximal solution r ( t , E) of u' = g (t, u, u )

+

E,

u(to)= u g

+

6,

ug

2 0,

(1.4.22)

1.4.

19

COMPARISON THEOREMS

+

+

which, by Lemma 1.3.1, exists on [to, t, v ] , v > 0, and t , v < to for sufficiently small E > 0. Moreover, we have from (1.4.22) that

+ a,

and Hence, one gets, from Theorem 1.2.1, the inequality r(t) < r(t,E),

t

-t .I.

[to , t ,

E

(1.4.24)

Since g is nondecreasing in o, (1.4.23) and (1.4.24) lead to

> g d t , r ( t , .I),

r'(4 .)

t

E

[to , t,

+ .I.

) ri(to, €1. But ri(t) = g(t, ri(t), r(t)) gi(t, ri(t)), t E [to > ti), and ~ i ( t o < An application of Theorem 1.2.1 again shows that ri(t> < r ( t , E ) ,

t

E [to

7

( I -4.25)

ti)-

Since r ( t , €) exists on [to , t , + v], v > 0, (1.4.2t) leads us to a contradiction because of (1.4.25), and this proves the existence of r l ( t ) on [to to 9

+ a).

T o prove (1.4.1 S), we now see that (1.4.20) is true for t Furthermore,

E

[t, , t,

+ a).

r;(t> = A t 7 rdt>>= d t t rl(t>?+>)-

From the monotonic character of g in

ZI

and (1.4.20), one gets

G s(4 r d t ) , ri(t))* Theorem 1.4.1 now gives that ri(t)

G ~ ( t ) , t~

[ t o , to

+ a).

This inequality, together with (1.4.20), proves (1.4.18), as is desired.

THEOREM 1.4.5. Let the hypothesis of Theorem 1.4.4 hold; m E C [ [ t o, to a), R ] such that ( t , m ( t ) , o) E E, t E [to , to a ) , and m(t,) < uo . Assume that for a fixed Dini derivative the inequality

+

+

Dm(t>

is satisfied for t E [to , to we have

< g(t, m(t),4

(1.4.26)

+ a ) - 5'. Then, for all < r ( t ) , t E [to , to + a ) ,

m(t)

ZI




+

74,d t )

> g,(t, 4 t ) )

hold for t E [t,, , to a). If v(t) satisfies the reversed inequalities, it is said to be a k over ( n - k) under-function. These definitions clearly include the definitions of under- and overfunctions as special cases, viz., K = 0 or k = n. We require that the function g(t, u ) should satisfy certain monotonic properties, which are listed below. DEFINITION 1.5.2. T h e function g(t, u ) is said to possess a mixed quasimonotone property if the following conditions hold:

(i) gp(t,u ) is nondecreasing in u j , j = 1, 2,..., k , j f p , and nonincreasing in uQ. (ii) g,(t, u ) is nonincreasing in up and nondecreasing in u j , j = k + 1 , k + 2 ,..., n, j # q . Evidently, the particular cases k = n and k = 0 in the mixed quasimonotone property correspond to quasi-monotone nondecreasing and quasi-monotone nonincreasing properties of the function g(t, u), respectively. Furthermore, g(t, u ) is said to possess mixed monotone property if, in conditions (i) and (ii), j # p , j f q are not demanded.

22

CHAPTER

1

An extension of Theorem 1.2.1 which plays an equally important role is the following: T H E O R E M 1.5.1. Let (i) g E C[E, R"], where E is an open ( t , u)-set in R"-I'; (ii) zi, w E C[[t,, to a), R"], ( t , v(t)), ( t , w(t)) E E for t E [t,,, t,, a ) ;and (iii)g(t, u ) possess a mixed quasi-monotone property. Assume further that

+

+

(1 5 3 ) (1 5 4 ) (1 5 5 ) (1 S.6)

Proof. Define mlJ(t)= wIJ(t) because of (1.5.2), ~

n J t ) and m J t )

m,(t*) 1 0 ,

=

vq(t) - wq(t). Then,

i = 1, 2)...)n.

(1.5.8)

Suppose that the assertion (1.5.7) is not true. Then, the set

z

(J [ t E [to , to 4-a): m,(t) < 01 11

=

2

1

is nonernpty. Let t , = inf 2. By (1.5-8), it is obvious that t, > t o . Since the set Z is closed, t , E Z , and consequently there exists a j such that m,(tl) = 0.

(1 5 9 )

If (1.5.9) is not true, one would have m,(t,) &(tl

7

Ntl)),

using the relations (1.5.4), (1.5.6), (1.5.9), (1.5.10), (1.5.11), and the mixed quasi-monotone property of g(t, u ) in u. Hence the set 2 is empty, and (1.5.7) is proved.

COROLLARY I .5.1. Let conditions (i), (ii), and (iii) of Theorem 1.5.1 be satisfied. Assume that, for t E (to, to a ) , the inequalities

+

DWt)

< g(t, W ) ,

D-m(t)

> g(t, 4))

hold. Then, v(to) < w(to)implies v(t) < 4 t h

t

E

[to , t,

+ a).

REMARK1.5.1. Notice that the proof of Theorem 1.5.1 remains unchanged even when the inequalities (1.5.3)-( 1.5.6) are replaced by

< sdt, f4th D-.,(t) 2 g*(t, 4 t ) ) , D--Wp(t) 2 g p ( t ,w ( t ) ) , D-w*(t) < g,(C 4 t ) ) . D-.,(t)

REMARK1.5.2. One can, in Theorem 1.5.1 and the following corollary, use any fixed Dini derivative D in place of D-, the corresponding inequalities being satisfied only for t E [to , to + a) - S. This follows from Lemmas 1.2.1 and 1.2.2. T h e next theorem is an analog of Theorem 1.2.3.

24

CHAPTER

1

THEOREM 1.5.2. I,et v(t),w ( t ) be k under (n - k) over-, k over ( n - k) under-functions, respectively, for t E [t,,, t, + a ) , with respect to the initial value problem (1.5.1). Assume that g ( t , u ) has mixed quasimonotone property. Let u ( t ) be any solution of (1.5.1) existing on [ t o ,to a ) such that

+

v(t,)

Then

for t

E

[to , to

+ a).

(1.5.14)

= U" = W ( t " ) .

% ( t ) < %(t> < zuv(t),

( 1.5.15)

%(t) > 4

(1.5.16)

4 > %(t)

Pmof. If (1.5.15) and (1.5.16) hold for t, < t < fo , fo sufficiently close to t o , then one can deduce the assertion of the theorem by the application of Theorem I .5. I and the subsequent Remark 1.5.1. Indeed, such a f, exists. For, defining 7n,(t)

=

%(t)

~

m,(q

%(t),

% ( t )- % ( t )

and noting thdt m,(t,,) = 0 because of (1.5.14), it is easy to deduce that mi, +(to)> 0, which implies m,(t) is increasing in a sufficiently small neighborhood of t o , say t, t t , . Similar argument with

<
‘

-t-

€2))

, €2) > .,(to

, €1);

;

€2

& ( t , u(t, 4) - €2 ;

s,(t,

ZGf,€1) < g,(C

44 €1)) u(t, 4)

+

~

;

€2 €2.

An application of Theorem 1.5.1 yields

+

%At,

€2)

< %(t, 4

udt, €2)

> ua(t,€1).

for t E [t, , t , 771. Since the family of functions u ( t , E ) is equicontinuous and uniformly bounded, one can establish that lim u(t, E , )

=

en 4

+

~(t)

71 and that r ( t ) is a solution of (1.5.1) on uniformly on [to , t, [fn > 4, 171. T o show that r ( t ) is a k max ( n - k) mini-solution of (1.5.1) on [t,,, t, 171, we have to prove that (1.6.1) is satisfied. Let u(t) be any solution of (1.5.1) existing on [ t o ,t, -k 171. Then,

+

+

< U J t n , e), uo(tn) > ug(tn uC(t) < R B ( t , 4 t ) ) + E ;

Z4tn)

gq(t,u ( t ) )

uXt)

~

u x t , €1 2 g,(t, u ( t , 6 ) )

4 ( t , .) < g,(4

for

E

< b/2. By Theorem

u(t,

e;

+

E;

-E

1.5.1, it follows that

4 4 < %(t, E ) , % ( t ) > uo(t, €1

for t E [to, t,

+ 71. Consequently,


mc(to) 2 *o,c

?

(1.7.1)

and, for a fixed Dini derivative, the inequalities (1.7.2) hold for t E [to, to

+ u)

-

S. Then,

28

CHAPTER

1

Proof. By Lemma 1.2.2, it follows that (1.7.2) is equivalent to the inequalities (1.7.4)

+

+

for t E ( t o ,t, u). Lct T E ( t o ,to a). Then, the existence of k max - k ) mini-solution r ( t , c) of (1.6.3) on [ t o ,71,for all E > 0 sufficiently small, satisfying r ( t ) = lim r ( t , 6) (I -7.5)

mdt)

where p ( t ) is the k mini ( n requires obvious changes.

-

< PQ(t),

t E It” > to

+ 4,

k ) max-solution of (1.5.1). T h e proof

T h e following corollary of Theorem 1.7.1 is important in later applications,

COROLLARY 1.7.1. Let condition (i) of Theorem 1.5.1 be satisfied. Suppose that g is quasi-monotone nondecreasing in u. Let [ t o ,to a ) be the largest interval of existence of the maximal solution r ( t ) of (1.5.1). Let m E C [ [ t o to a), and, for a , a),R”],( t , m ( t ) )E E, t E [ t o ,to fixed Dini derivative, the inequality

+

-+

+

Dm(t)

< g(t, m(t))

(1.7.7)

holds for t E [to, to -{- a ) - 5’. Then, 4to)

implies m(t)

mj(tl)

if

if

K

1

<j < k

+ 1 < j < n.

This implies, from the definitions of Zi and di(t), that t , E Zj . Hence, the j t h inequality in (1.7.10) is satisfied for such a t , . T h e rest of the proof is identical with the proof of Theorem 1.5.1 in order to arrive at (1.7.6), and this is sufficient to draw the conclusion (1.7.3).

30

CHAPTER

1

THEOREM 1.7.3. Let the hypothesis of Theorem 1.5.1 hold, except that the inequalities D-.*(t) D-wD(t)

+

3 g,(t,

( I .5.4*)

.(t)),

> g,(t, w ( t ) )

(1.5.5*)

are satisfied for t E (to, to a ) , instead of (1.5.4) and (1.5.5). Assume further, for each T E ( t , , t, a), t E [ t o ,71, and for each i, that g satisfies the condition gi(t,U)

-

+

g,(t, U) 3 -G(T

+ to

ui 3 z& , u j = zZi (i # j ) , where G E C[[to, to is the maximal solution of U'

=

G(t,u),

-

t , ui - Ci),

(1.7.13)

+ a ) x R, R], and r(t) = 0

~(t,= ) 0.

Then (1.5.7) holds. Proof. Following the proof of Theorem 1.5.1, we arrive at a t, E ( t o, to u ) and a j ( 1 j TZ) satisfying

+

<
4 f l ( t )= ) YY {$j(f))

($l(t),

1

%z(t),..*,

% q ( t ) ) =--V *

I,et E l , E, stand for systems of sequences y = {qh,(t)},ZI = {$j(t)}.If $#p(t)], y, = {$:."(t)} are two sequences such that, if y 1 ,y z E E, , then yl y 2 implies +;"(t) $ y ) ( t ) on [to, to a] and for each i. Similarly for z', , u2 E E, , v 1 3 v, means that $ j l ) ( t )3 $iz)(t) on [to , f , -1- a ] and for each j . Let F , , P, stand for the systems of sequences of continuous functions on [t,,, t , + a] which take values in the real extended line, the order relations in F , , F2 being the same as those in El , E, , respectively. It is easy to verify that conditions (1.8.1)-( I .8.4) and (1.8.1 *)-( 1.8.4") are satisfied. Let us now define the operators P I ,P, and the functions Q1 , 0,as follows:

3'1

~


..*,$j-l(t)>

~j

1

&dt),..*,d ’ ~ ( ~ ) ) ;

d’i+l(t),...,d’n(t)>.

Let, for each i, ri(t) be the maximal solution of Y’ =

h(t,Y),

Yi(t0) == uio,

and, for eachj, pj(t) be the minimal solution of P’ = Ej(G P ) ,

Pj(t0)

=

vjo.

Since the functions f i and gj satisfy (ii), the existence of r i ( t ) and pj(t) on [to , to u ] is ensured. Let us now define the functions z, , z2 by

+

%({$At)))= {fj(t)>.

~ , ( ~ g l i ( l ) )= ) {Yi(t)}>

From Theorem 1.4.1, it follows that the functions z, , z2 satisfy (1.8.13). Moreover, the sets U , , U , are nonempty, since uio - Mi(t - to)E U , and zjjO + Ni(t - t o )E U , , because of (ii). Furthermore,

I Yk(t)l

< Mi

9

I P;(t)l

< Nj .

Therefore, the family of functions {ri(t)},{ p j ( t ) } are equicontinuous and uniformly bounded. This proves that sup z1( Ul)=: {sup Y i ( t ) S , inf z,( U,)

+

= {inf p j ( t ) }

are continuous on [t,, t, u ] . T h e assertion of Theorem 1.8.2 now follows from Theorem 1.8.1.

1.9, Integral inequalities reducible to differential inequalities We shall consider, in this section, only those integral inequalities that are reducible to differential inequalities. We begin with one of the simplest and most useful integral inequalities.

THEOREM 1.9.1.

+

Let m, v E C[[t,, t, a), I?+], where R , denotes the nonnegative real line. Suppose further that, for some nonnegative constant C , we have (1.9.1)

38

CHAPTER

1

Then m(t)

< C exp j t

v(s) ds,

t

[to, to

E

t,l

Proof.

If C

+ u).

(1-9.2)

> 0, it follows from (1.9.1) that

which, by integration, yields C

+ J:vv(s)m(s) dsI

~

log C


0, and therefore the previous argument gives (1.9.2) with C = C, . Letting C, + 0 implies m ( t ) 7 0. This proves the theorem.

COROLLARY 1.9.1. 1,et m, u E C[[t,, to and satisfy the inequality m(t)

< n(t) -1-

It

+ a ) , R,], n E C [ [ t ,, to + a), R],

v(s)m(s)ds,

to

t

E

[ t o , to

+ u).

Then we have

If, in addition, the derivative n'(t) exists for t

E

[to, to + a ) , then

A generalization of Theorem 1.9.1 is the following analog of Theorem 1.4.1 which, however, requires the monotony of g with respect to u.

THEOREM I .9.2. Let E be an open ( t , u)-set in R2 and g E C[E,R]. Suppose that g(t, u ) is monotonic nondecreasing in u for each t. Let m E C"t0 to a), RI, (4 4 4 ) E E, t f [to , t o a), m(t,) %J , and 9

+

+


U0.Q

3

4 s ) ) ds

to

m,(t) b .dt>

(1.9.6)

the K max (n - k) mini-solution of

1 .lo.

Proof.

41

DIFFERENTIAL INEQUALITIES IN THE SENSE OF CARATHEODORY

Let the vector function v(t) be equal to

so that mP(t)

and

< %(t), v'(4

%(t)

24

(1.9.7)

2 )

= g(t, v ( t ) ) .

T h e mixed monotonic character of g in u shows, in view of the inequalities (1.9.7), that % (t)

< g,(t, 4 t ) ) l

VXt)

>, gdt, n ( t ) ) ,

t E [ t o t" 9

+ 4.

Theorem 1.7.1 is now applicable, and we get %(t)


K is a constant. It is enough to show that n ( t ) = 0 on t, t to a. Suppose, on the contrary, that max n(t) > 0, t o < t 4t o f a

< < +

and that the maximum occurs at t = u. We have, at On the other hand, using (2.2.2), we obtain n'(o)eLu

+ Ln(a)eLu= m'(a) < llf(% x d 4 ) - f ( % xz(4)ll < Kn(u)eLu.

u, n'(o) =

0.

50

CHAPTER

< L , that n‘(n) < 0, contradicting

This implies, because of the choice K n’(o) 0. Thus, n(t) 0 on t, t ~

2

< < to + a.

~

T h e next result is known as liamke’s uniqueness theorem, which is, evidently, more general than that of Perron and is sufficient for many practical cases, since it includes as special cases many known criteria.

THFOKEM 2.2.2.

Assume that (i) the function g(t, u)is continuous and nonnegative for t, < t t, -1- a, 0 u 2b, and, for every t, , t, 2, to a, u ( t ) .~ 0 is the only function differentiable on t, t < t , and continuous on t,, t -.t , , for which e .

-_


0 and proceeding as in the proof of Theorem 2.2.3, making use of relations (2.2.18) and (2.2.19), we obtain PAt)

< r(t),

(2.2.20)

54

CHAPTER

2

as far as p 2 ( t ) exists to the left of u, where p 2 ( t ) is the minimal solution of (2.2.15) such that pL(~) = r(u). As before, we can continue p2(t) up to t, by defining pz(to) : 0. Since p2(t) 0, we have

+

t-tO+ lim

# 0,

pz(t)P(t)

which, in view of (2.2.20), implies that lim+v(t)/B(t) # 0.

t-tto

This, together with assumption (2.2.13), shows that there exists a t, such that r(t1)

>4tJ.

(2.2.21)

Let p l ( t ) be the minimal solution of (2.2.14) such that pl(tl) = r ( t l ) . Then it can be shown, arguing similarly, that p l ( t ) can be continued up to t, , pl(t,) = 0, and

0

0.

Pmof qf Theorem 2.2.4. Consider the function g(t, u ) gf(t, u), where gf(t, u ) is the function defined by (2.2.12). By combining the respective -~

arguments in the proofs of Theorems 2.2.2 and 2.2.5, it is easy to show that gf(t, u ) verifies Perron's uniqueness conditions of Theorem 2.2.1, which is sufficient to establish the uniqueness of solutions. REMARK2.2.1. Whenever f ( t , x) is assumed to be continuous on R, , it follows from the foregoing considerations that the uniqueness conditions of Theorems 2.2.2 and 2.2.4 can be reduced to that of Perron's condition. If the pair of functions g,(t, u), g 2 ( t ,u ) satisfies the hypotheses of Theorem 2.2.4, we can also show that there exists a function g(t, u )

2.2.

55

UNIQUENESS

that fulfills the uniqueness criteria of Kamke as given in Theorem 2.2.2. This is the content of the following:

THEOREM 2.2.6. Let the functions A(t), B(t), gl(t, u), and gz(t,u ) satisfy hypotheses (i), (ii), (iii), and (iv) of Theorem 2.2.4. Then, there exists a function g(t, u ) verifying assumption (i) of Theorem 2.2.2.

Proof. Define the function g(t, u ) by g(t,

4 = min[g,(t,

u), gdt,

41.

(2.2.23)

Then g satisfies (2.2.18). T o prove the stated result, it is enough to show that no nontrivial solution of (2.2.4) fulfills the limiting conditions (2.2.5). I n fact, the assumption that there exists a differentiable function u(t) satisfying the differential equation (2.2.4) and the conditions (2.2.5) for which u(a) > 0, to < CT < to a, leads, following the proof of Theorem 2.2.5, to the contradiction that u(t,) > 0.

+

COROLLARY 2.2.6. T h e functions gl(t, u ) = K1ua, gz(t,u ) = K,(u/t) are admissible in Theorem 2.2.4, if 0 < 01 < I , K2(1 - a ) < 1, with A(t) = K,(l - ~ ~ ) t l / ( l and - ~ ) , B(t) = t K 2 . We shall now show that, if certain conditions of Theorem 2.2.2 are violated, Eq. (2.2.3) has nonunique solution. We prove this for the case n = 1 and t, = 0.


0 for u > 0. 0 < t , < a, u(t) 0 is a differentiable function on 0 < t < t , , and continuous on 0 < t < t, for which uL(0) exists,

+

0 < t < t, ,

u’ = g ( t , a),

and u(0) = u i ( 0 ) = 0.

Let f E C I R o ,R], where R, : 0 ( t ,Y ) E R, t # 0,

< t < a, 1 x I < b,

and, for ( t , x),

7

I f ( t , x) -f(t,Y)I

>g(t,

I x -Y I ) .

(2.2.24)

Then, the scalar differential equation x’ = f ( t , x),

has at least two solutions on 0

x(0)

< t < a.

=0

(2.2.25)

56

CHAPTER

2

Pmf. Let us first suppose that f ( t , 0) = 0, so that, putting y we obtain the inequality

=

0,

lf(t,41 > d t l I x I), because of condition (2.2.24). Sincef(t, x) is continuous and g(t, u) > 0 for u > 0, it follows that eitherf(t, x) < 0 or f ( t , x) > 0, for x .f 0. This implies that either (2.2.26)

or (2.2.27)

By hypothesis, there exists a u, 0 < u < a, such that u(u) > 0. Let y ( t ) be the minimal solution of x’ = f ( t , x), y(a) = u(u). Then, using an argument similar to that in the proof of Theorem 2.2.3 and the inequality (2.2.26), it can be shown that y ( t ) u ( t ) to the left of (T, as far as y ( t ) exists. Moreover, y ( t ) can be continued u p to t = 0 and


- Y(t)Il.

Observe that m(0) = 0, and, because of the assumed continuity of x) at (0, 0), we have lim m ( t ) / t = 0.

f(t,

t-.o+

Setting B(t) = sups

tQ

m(s)/s, it is easily verified that lim m ( t ) / B ( t )= 0. -o+

I

+

This is possible if m(s) 0 in some neighborhood of the origin; otherwise, the existence of B(t) is trivial. We notice that the continuity requirement off(t, x) at (0,O) is stronger than the condition (2.2.30). T o see this, definef(t, x) as follows:

I

1, f ( t , x) = x / t ,

0,

x > t, 0 < x < t, x < 0.

2.2.

59

UNIQUENESS

<
%+l) =

W

n

>

741

of functions satisfying the following relations:

6) (ii) (iii)

uXt)

4 ( t ) > f ( t , v,(t)), u,(t)

(3 4 + * ( t ) (v)

< .f(t,un(t)),

%(to)

=

xo ;

vn(to) = xn ;

< ~ , + ~ (< t ) x ( t ) < ~ , + ~ (< t )v,(t), = f d t 7

t

E

( t o , to

+ a];

%+At); u,(t>, vn(t));

4 + 1 ( t ) = f d t , 7-5n+1(t);

% ( t ) >vn(t))-

It is clear from (iii) that the sequences {urn},{urn}are monotonic and uniformly bounded on [ t o ,to a ] . Furthermore, they are equicontin-

+

2.4.

67

CHAPLYGIN'S METHOD

uous, in view of the fact that, for each fixed n, u, , v, are solutions of linear equations. Hence, an application of Theorem 1.1.1 proves the uniform convergence of un(t),vn(t)to x ( t ) as n --f 00. Let (2.4.17)

and




- un+1(t)l

+ H22"+1?

68

2

CHAPTER

which, in view of Theorem 1.4.1, yields

Since

S eK(lPs)ds < meKa, we get 1

to

Thus, by induction, the relation (2.4.4) is true for all n, and consequently we have, by (iii),

I x ( t ) - U,(t)i

and

G 2/3/2'"

1 x ( t ) - v,(t)l ,< 2/3/22". This completes the proof. Let us now consider the differential system x' = f ( t , x),

x(t,) = x,

.

(2.4.22)

I n this case, we shall be able to demonstrate only the lower Chaplygin's sequence {un},under some additional restrictions.

THEOREM 2.4.2.

Let f E C[R,, Rn],where R, is the set,

< t G to + a ,

R, :


g(t, 4 t h

and, by Theorem 1.4.1, we obtain where r(t, to , 11 xo - y o 11) is the maximal solution of (2.5.4) such that u(to)= 11 xo - yo 11. Since the solutions u(t, to , uo) of (2.5.4) are assumed to be continuous with respect to the initial values, it follows that lim r ( t , to , I/ xo

J(o-yo

-

Yo I l l

= r ( t , to

7

01,

and, by hypothesis, r ( t , t o ,0) = 0. This, in view of the definition of m ( t ) , yields that lim x ( t , to , xo)

xo-Yo

= r(t,t o

,Yo),

which shows the continuity of x ( t , t o ,xo) with respect to x,, . We shall next prove the continuity with respect to initial time t o . If x(t, t o , xo), y ( t , t, ,xo), t , > t o , are the solutions of (2.5.1) through ( t o ,xo), ( t l , xo), respectively, then, as before, we obtain the inequality D+m(t)

where Also,

< gft, W ) ) ,

m(t> = /I x ( t , to > xo) - Y ( t , t , 4 t l ) = ll X ( t ,

?

>

to xo) - xo It. ?

Hence, by Lemma 2.5.1, m(t1)

and, consequently, m(t)

< r*(t,

< r"(t),

x0)Il.

7

to 9 01,

t

> t, ,

12

CHAPTER

2

where f(t) =

q t , tl , Y * ( t ,

7

to ,O))

is the maximal solution of (2.5.4) through (tl , r * ( t l , t o , 0)). Since = 0, we have

r * ( t , , t o , 0)

lim f ( t , t, , ~ * ( t,,t o ,0 ) ) = f ( t , t o ,0 ) ,

tl-fo

and, by hypothesis, T ( t , t o , 0) is identically zero, thus proving the continuity of x ( t , t o , xu) with respect to to .

COROLLARY 2.5.1. T h e function g ( t , u ) in Theorem 2.5.1.

= Lu,

L

> 0,

is admissible

THEOREM 2.5.2. Let f E C[E, Rn],where E is an open ( t , x, p)-set in R"+"L+l,and for p = po , let xo(t) = x(t, to , xo , p,,) be a solution of x' = f ( t , x,Po), existing for t

4 t O ) = xo

7

(2.5.5)

3 to . Assume further that lim f ( t ,x , Y ) = f ( t ,x, Po),

w-wo

(2.5.6)

uniformly in ( t , x), and, for ( t , x1 , p), ( t , x2,p ) E E, I l f ( t 7

x1

9

Y) -f(t,

x2

, P)Il

e At, /I

x1 - xzll),

(2.5.7)

whereg E C[/ x R, , R,]. Suppose that u(t) = 0 is the unique solution of (2.5.4) such that u(tn) = 0. Then, given E > 0, there exists a S ( E ) > 0 such that, for every p, 11 p p,, I/ < S ( E ) , the differential system ~

x' = f ( t , x,P),

x(tn) = xo

(2.5.8)

admits a unique solution x ( t ) = x(t, t o ,x,, , p ) satisfying

I1 ~

( t ) xo(t)ll ~


to

Proof. T h e uniqueness of solutions is obvious from Theorem 2.2.1. From the assumption that u(t) = 0 is the only solution of (2.5.4), it

+

follows, by Lemma 1.3.1, that, given any compact interval [to , to U ] contained in J and any E > 0, there exists a positive number 7 = V(E) such that the maximal solution r(t, to , 0, 7) of 24'

= g(t, 24)

+7

2.5.

exists on to

DEPENDENCE ON INITIAL CONDITIONS AND PARAMETERS

< t < to + a and satisfies r ( t , to , 079)

< 6,

t

E

[to , to

+ 4.

Furthermore, because of the condition (2.5.6), given 7 a 6 = 6 ( q ) > 0 such that

Ilf ( t , X, P ) - f ( t ,

3 9

PJI

13

> 0, there exists

< 17

provided

I1 P Now, let

E

> 0 be

- Po

II < 8.

given, and define

Nt)= I1

- XO(~)lL

where x(t), x,(t) are the solutions of (2.5.8) and (2.5.5), respectively. Then, using the assumption (2.5.7), we get D+m(t)

< g ( t , 4 t ) ) + llf ( t , xo(%

From this, it turns out that, whenever D+m(t)

11 p

P ) - f(t, xo(t), P0)Il.

- po 11

< 6,

< g(t, 4 t ) ) + ?-

By Theorem 1.4.1, we have

< r(t,t o , O,d,

t

3 to,

II x ( t ) - x,(t)Il < e ,

t

2 to,

m(t)

and hence provided that

/I P

- Po

II < 8.

Clearly, 6 depends on E since q does. T h e proof is complete.

LEMMA 2.5.2. Let f E C [ J x D,R"],where D is an open, convex set in R",and let af /ax exist and be continuous. Then,

74

CHAPTER

2

the convexity of D implies that F ( s ) is defined. Hence, (2.5.9)

SinceF(1) = f ( t , x2) andF(0) (2.5.9) from 0 to I .

=f

( t , xl),the result follows by integrating

T H E O R E M 2.5.3. Assunic that f E C [ J x Rn,Rn] and possesses continuous partial derivatives af /& on J x Rn. Let the solution xO(t)= x(t, t o , x,,)of (2.5.1) exist for t 3 t o , and let

Then

exists and is the solution of Y'

=

H ( t , t o , x0)y

(2.5.10)

such that O(to, to , xo) is the unit matrix; (ii)

ax(t, f, , XI)) at,

exists, is the solution of (2.5.10), and satisfies the relation

First wc shall prove conclusion (i). Let h be a scalar and e, = (eTcl, ..., ek?() be the vector such that ekj = 0 if j f k and ekk = 1. Then, for small h, let

Proof.

x(t, h )

=

x(t, t o ,

x0

+ e&),

which is defined on J , and lim x(t, h ) = xo(t) h-0

uniformly on J . Since

2.5.

DEPENDENCE ON INITIAL CONDITIONS AND PARAMETERS

75

applying Lemma 2.5.2 with x2 = x(t, h), x1 = x,(t), we have

If we write

the existence of ax(t, t o ,xo)/2xo is equivalent to the existence of the limit of x h ( t ) as h 3 0, since ~ ( t, ,h ) = xo ekh, xh(t,) = ek . Thus, xh(t) is the solution of the initial value problem

+

y‘ == H ( t , ‘ 0 ,

9

h)y,

Y(tO) = ek

(2.5.12)

7

where

As x(t, h) + x,(t) as h + 0, by the continuity of lim H(t, to , xo , h) h-0

=

af /ax,

it follows that

H(t, t o ,xo)

uniformly on J . Considering (2.5.12) as a family of initial value problems depending on a parameter h, where H(t, to , xo , h) is continuous for t E J , h being small and y arbitrary, and observing that the solutions of (2.5.12) are unique, it is clear that the general solution of (2.5.12) is a continuous function of h. In particular, limb+,, xl,(t) = x ( t ) exists and is the solution of (2.5.10) on J. This implies that ax(t, t o ,xo)/axo exists and is the solution of (2.5.10). T o prove (ii), define

Since (2.5.1) has unique solutions, we have x ( t , to

+ h, xo) = x(t, to , “(to , to + h,

and therefore hGhh(t)= x(t, tn >

-

.To)),

to , tn + h, xo)) - x ( t , to , xo).

Because ax(t, t o ,xo)/axoexists and is continuous and .(to

, to

+ h, xo)

.(to , to , xo) = xo

as

h

-

(2.5.13)

0,

76

CHAPTER

2

it follows from (2.5.13) that hqt)

xO],

=

0. By the mean value theorem, there exists a 0 ash such that --f

where 0

as h

--f

=

0, ,k

=

(2.5.14) 1 , 2,...,n

< B < 1. Notice that, for each k,

0. Thus, (2.5.14) shows that

0, which implies that ax(t, to , xo)/ato = limh+oi r l ( t )exists and satisfies (2.5.1 1). This completes the proof.

as h

--f

2.6. Variation of constants Let us prove some elementary facts about linear differential systems,

x'

=

(2.6.1)

A(t)x,

where A ( t ) is a continuous n x n matrix on J. Let U ( t ) be the n x n matrix whose columns are the n-vector solutions x ( t ) , x(t), being so chosen to satisfy the initial condition U(t,) = unit matrix. Since each column of U ( t )is a solution of (2.6. l ) , it is clear that U satisfies the matrix difierential equation U'

=

U(to)= unit matrix.

d(t)U,

THEOREM 2.6.1.

(2.6.2)

Let A ( t )be a continuous n x n matrix on J. T h e n the fundamental solution U ( t )of (2.6.2) is nonsingular on J , More precisely, det U ( t )

exp

1 :

/ t t0

where tr A ( t ) =

CF=laii(t).

tr A ( s ) ds,

t E J,

2.6. Proof.

VARIATION OF CONSTANTS

77

T h e proof depends on the following two facts:

(i) d(det U ( t ))/d t= sum of the determinants formed by replacing the elements of one row of det U ( t )by their derivatives. (ii)

T h e columns of U ( t )are the solutions of (2.6.1).

Simplifying the determinants obtained in (i) by the use of (ii), we get d dt

- det U ( t ) = tr A(t)det U ( t ) .

T h e result follows, since U(to)= unit matrix.

THEOREM 2.6.2.

Let y ( t ) be a solution of Y’

=

A(t)Y + F ( t , Y ) ,

(2.6.3)

where F E C[J x Rn,R”],such that y(to)= y o . If U ( t ) is the matrix solution of (2.6.2), then y ( t ) satisfies the integral equation

This, because of (2.6.2), yields z’(t) = U-’(t)F(t,Y ( t ) ) ,

whence z ( t ) = Yo

+

Jt

U-l(s)F(s,y(s))ds.

t0

Multiplying this equation by U ( t )gives (2.6.4). COROLLARY 2.6.1. Let A(t) be a continuous n x n matrix on J such that every solution x ( t ) of (2.6.1) is bounded for t 3 to . Let U ( t )be the fundamental matrix of (2.6.1). Then, U-l(t) is bounded if and only if

is bounded from below.

78

2

CHAPTER

We shall now consider the nonlinear differential system (2.5.1). T h e following theorem gives an analog of variation of parameters formula for the solutions y(t, to , x,,)of

Y' = . f ( t , y ) +F(t,Y).

(2.6.5)

THEOREM 2.6.3.

Let ~ , F CE[ J x RrL,R"], and let af /ax exist and be continuous on J x R". If x ( t , t o , x,,) is the solution of (2.5.1) and (2.5.2) existing for t >, t o , any solution y ( t , t o , xo) of (2.6.5), with ~ ( t , ,= ) xo , satisfies the integral equation Y ( t , to .o> 7

=

x ( t , to 7 %>

+ It @ ( t ,

s,

to

Y ( S , to , x,,))F(s,Y ( S , to , x,,)) ds

(2.6.6)

for t 2 t,, , where @(t,t o , x,,) = ax(t, to , xo)/axo. Proof.

Write y ( t ) = y(t, t o , x,,). Then, dx(t, s , y ( s ) ) ds

-

ax(t,s,y(s))

+

q t ,S?Y(S))

aY

as

= @ ( t ?s,Y N ) [ Y ' ( S )

.Y" (2.6.7)

YG))l,

using Theorem 2.5.3. Noting that x(t, t , y ( t , to , x,,)) = y(t, t,, , x,,) and y'(s) - f ( s , y(s)) = F(s, y(s)), by integrating (2.6.7) from to to t, the desired result (2.6.6) follows.

THEOREM 2.6.4. Let f E C [ J x Rn, R"], and aflax exist and be continuous on J x R". Assume that x ( t , t o , x,,) and x(t, t,, ,y o ) are the solutions of (2.5.1) through (to , x,,) and (to ,yo), respectively, existing for t 3 t o , such that x,,, y o belong to a convex subset of R". Then, for t >, t o , x ( t , t" , Y") - x ( t , to xo) = 7

[I;

@(G to , xo

+

S(Y0 - XI)))

4.

(Yo - xo).

(2.6.8)

Proof. Since xo,y o belong to a convex subset of R", x ( t , to,x,, is defined for 0 s I . Thus,

<
to and x, y E Q. If x ( t ) , y ( t ) be a solution and a &approximate solution of (2.7.1) and (2.7.3), respectively, on [ t o , co) such that

(2.7.6)

Proof. If (2.7.6) is not true, the set

80

CHAPTER

2

is nonempty. Arguing as in the proof of Theorem 1.2.1, we arrive at a t , > to such that either or In either case, it follows from the definition of 9 that, at t = t, , x(tl), y ( t l )E $2, and therefore, defining m ( t) = I/ x ( t ) - y ( t ) 11, we get

I m;(tl)l

< I/ x’(t1) -Y‘(tl)ll G

Ilf(t1

x(t1))

-fAh ?Y(tl))ll

+ I/ Y’(t1) -fdG

?Y(tl))ll.

This, together with (2.7.2) and (2.7.5), implies

A repetition of the rest of the proof of Theorem 1.2.1, with appropriate changes, proves (2.7.6).

THEOREM 2.7.2.

Let the assumptions of Theorem 2.7.1 hold except that (2.7.4) and (2.7.5) are replaced, respectively, by (2.7.7)

for t > to , x, y E Rtl. Suppose further that, for each t E [to , 71,g satisfies the condition

1 g(t7 ~

1 - )g ( f ,

uZ)I

< G(7 + t o

-

t , ~1 - uz),

7E

~1

[to , 00) and

2~

2 , (2.7.9)

where G E C [ J x R, , R,] and r ( t ) = 0 is the maximal solution of U’ =

G(t,u),

~ ( t , )= 0.

Then, the inequality (2.7.6) remains valid.

(2.7.10)

2.7.

81

UPPER AND LOWER BOUNDS

Proof. By a repeated application of Theorem 1.4.3, we can prove (2.7.6). For this purpose, it is enough to see that g 6 and -(g 6) satisfy the condition (1.4.9), in view of (2.7.9). Also, (2.7.8) implies that

+

+

for t

> to.

+ S(t)l < m;(t) < g(t7 4 t ) ) + s(t)>

-[g(t, 4 t ) )

THEOREM 2.7.3.

Let g E C [ J x R, , R,], 6 E C [ J ,R,], and r ( t ) , p ( t ) be the maximal and the minimal solutions of

respectively, existing on [to, 00). Let Q = [ x7 Y and

where

c1

E

R" :P ( t ) - €2 II x - Y

r(t)


II < ~ ( t ) el , t 2 to],

+

, c2 > 0. Assume that, for t > to , x, y

E 9,

IIf(t! x) - f d ~ > Y ) l l G g(t, II x - Y 11).

(2.7.12)

If x(t), y ( t ) be a solution and a &approximate solution of (2.7.1) and (2.7.3), respectively, on [ t o , 00) such that

< II -2^o- Y o II < uo ,

no

then p(t)

G II 4 t ) - Y(t)ll

< r(t),

t

(2.7.13)

2 to.

Proof. Let T E [to, 03). By Lemma 1.3.1, the maximal and the minimal solutions r ( t , E ) and p ( t , c) of u' = g(4 u )

v'

=

+ s(t) +

-[g(t, v)

exist for sufficiently small E

4tO)

€7

+ 8 ( t ) + €1,

= UO

v(to)= vo

+

~

E, E

> 0, and r ( t ) = lim r(t, E ) , 6-0

p(t)

=

lim p ( t , t-0

E)

uniformly on [ t o ,TI. In view of this, there exist r ( t , €1 < r ( t ) p(t9

c)

+

€1

> p ( t > - €2

,

, c2 > 0 such that

82

CHAPTER

for t for d

2

E

[t,, , TI. Furthermore, an application of Theorem 1.2.1 yields that,

E

[f,

, 71,

r ( t ) < r(4 € ) r

f(t)

At,

4.

It now follows that, f o r t E [ t o ,TI, (2.7.14)

T o prove (2.7.13), it is enough to show that

Assuming the contrary and arguing as in Theorem 2.7.1, we get either

or These relations show, because of the inequalities (2.7.14), that, in either case, x ( t l ) , y(tJ E 8. By following the rest of the standard argument, it is easy to prove (2.7.15). This completes the proof.

REMARK2.7.1. Evidently, Theorem 2.7.3 holds when the condition (2.7.12) is satisfied for all x,y E Rn instead of 8. Similar comment is valid for Theorem 2.7.1 also. T h e bounds obtained in the foregoing theorems are on a general setup. They include a number of special cases. For instance, if 6(t) I= 0, we get the estimates of the difference of solutions of (2.7.1) and (2.7.3), respectively; whereas if, in addition, f ( t , x) = fl(t, x), the same results yield the growth conditions between any two solutions of the system (2.7.1). On the other hand, if fl(t, x) f ( t , x), error estimates between a solution and a 6-approximate solution of the system (2.7.3) are obtained. Furthermore, if S ( t ) E 0 and fi(t, x) = 0, these results provide the upper and lower bounds of solutions of the system (2.7.1). ~

For future use, the following well-known result is stated as

COROLLARY 2.7.1.

Let f~ C [ J x IIf(f,

).

-

f ( f >Y)II

R",R"],and, for t 3 @,

< L(f>llx

~

Y

I/

1

x, y E R",

2.7. where L

E

83

UPPER AND LOWER BOUNDS

C [ J ,R,]. Then, for t >, to ,

where x ( t ) , y ( t ) are any two solutions of the system (2.7.1), through ( t o xo), ( t o Yo), respectively. I n the foregoing results, the upper bounds obtained are increasing functions of t, since the assumptions demand that g(t, u) 3 0 and 8 ( t ) > 0, and therefore give very little information about the growth of solutions for large time. We give below a different set of assumptions that yield sharper bounds because the function g(t, u ) need not be restricted to be positive. 9

7

THEOREM 2.7.4. Let g E C [ J x R, , R],6 E C [ J ,R,], and r ( t ) be the maximal solution of u' = g(t, u )

+qt),

u(t,) = u g ,

existing on [ t o , 00). Assume that, for t

II X - Y

+ h[f(t,

X) -fi(t,

Y>III

for all sufficiently small h

< /I x

-Y

II

E

J , x, y

+ hg(t,I/ x

E

Rn,

-

y 11)

+ O(h),

(2.7.17)

> 0. Then, II xo - Yo II < uo

implies

I/

-Y(t)ll

< r(t)l

t 3 to

1

(2.7.18)

x(t), y ( t ) being a solution and a &approximate solution of (2.7.1) and (2.7.3), respectively, existing on [ t o , 00). Proof.

Consider the function

44 We have, for small h

> 0,

=

II x(t>-Y(t>ll.

84

CHAPTER

where r ( h ) / h+ 0 as h (2.7.17), that

2

It therefore follows, using (2.7..2) and

+ 0.

D+m(t)

< g(t, m(t))+ q t ) ,

which, by Theorem 1.4. I , yields the estimate (2.7.1 8). 2.8. Componentwise bounds

Instead of the differential inequality (2.7.2), we shall be considering a system of differential inequalities given by

I Y’

- f A t , Y)l

< s(4,

(2.8.1)

where 6 E C [ ] ,R,”]. Here and in what follows, we mean by j x 1 a vector whose components are I x1 1, I x2 1, ..., I x, I for any x E R”. Note that 8(t) is a scalar function in (2.7.2), whereas it is a vector in (2.8.1). In this case, the &approximate solution of (2.7.3) must satisfy (2.8.1) in place of (2.7.2).

THEOREM 2.8.1. Let g E C [ ] x R,”, R,”] and possess the quasimonotone nondecreasing property. Let u,v, S E C [ J , R,”] such that, for t > t , , (2.8.2)

Suppose that f,fl E C [ ] x R”, RTL], and, for t

> t, , x,y E Qi, (2.8.3)

If x ( t ) , y ( t ) be a solution and a 8-approximate solution of (2.7.1) and (2.7.3), respectively, on [ t o ,a)such that

(2.8.4)

2.8.

85

COMPONENTWISE BOUNDS

Proof. T h e proof runs parallel to that of Theorem 2.7.1. However, in this situation, the assumption that the set n

2=

(J [t 3 to : q ( t ) < 1 Xi(t) -yi(t)i

< Ui(t)j

i=l

is nonempty leads to the existence of an index j , 1 t , > to such that either

< j < n, and

a

or which shows that x ( t l ) ,y ( t l )E Qj . Consequently, as in Theorem 2.7.1, it is easy to show, using (2.8.3), that - M t l9

+ %(tl)l G 4 A t d < &(tl

"1))

9

m(t1))

+ Utl).

Making use of the quasi-monotone property of g(t, u)and the arguments of Theorem I .5. I , we can prove (2.8.4). The next theorem is analogous to Theorem 2.7.2 for componentwise bounds, the proof of which can be deduced from Theorem 1.7.3, with an observation similar to that of Theorem 2.7.2.

THEOREM 2.8.2. Assume that, in place of (2.8.2) and (2.8.3), we have

for t > t o , x,y E R",other assumptions being the same as in Theorem 2.8.1. Moreover, let, for each T E [to, a),t E [to, T] and for each i - 1 , 2,..., n,

I gdt, u ) u,

>, ii, ,

4

< G(7 + to u, = U, ,

-

t , u,

-

4,

i #j,

where G E C[J x R, , R,], and r ( t ) = 0 is the maximal solution of (2.7.10). Then, the assertion of Theorem 2.8.1 remains true.

86

CHAPTER

2

THEOREM 2.8.3. Let g E C [ J x Rn+,R,"] and possess the quasimonotone nondecreasing property. Assume that r ( t ) ,p(t) are the maximal and the minimal solutions of

and, for each t

> tn , x,y I f i ( t ,x)

E Qi

-fi.i(t?Y)I

Then an

implies P(t)

,

< Ri(G I x -Y

I).

< I xn -yo I < un

< I x ( t ) -Y(t)l < r ( t ) ,

t

3 to

7

x ( t ) , y ( t ) being a solution and a &approximate solution of (2.7.1) and

(2.7.3), respectively, existing on [t, , a). T h e proof of this theorem can be constructed by following the respective arguments of Theorems 2.7.3, 2.7.1, and 1.5.1 with necessary modifications.

THEOREM 2.8.4. I,et S E C [ J ,Rtn], g E C [ / x R+n, R"], and g possess the mixed quasi-monotone property. Suppose that f,fiE C [J x Rn, Rn], a n d , f o r e a c h t > , t , , , p - 1 , 2,..., k , q = k + l , k + 2 ,..., n,

and x ( t ) , y ( t ) are any solution and a &approximate solution of (2.7.1) and (2.7.3), respectively, on [ t o , a). (i) If r ( t ) is the k max(n u'

-

= g(t, ).

k) mini-solution of

+qt),

u(t,)

=

u,

2.8.

then, for t

87

COMPONENTWISE BOUNDS

> t o ,we have (2.8.5)

whenever

for t > t o ,then (2.8.6) implies (2.8.5) provided that, for each T E [ t o ,GO), t E [to,TI, and for each i = 1, 2,...,n,

+

g i ( t , u ) -gg,(t, c)

3 -G(T to - t , ui - $), ui 3 zii , ui = z i i , i #j,

where G E C [ J x R, , R,], (2.7.10).

and r ( t ) = 0 is the maximal solution of

THEOREM 2.8.5. Let g E C [ J x R+",R,"] and possess the quasimonotone nondecreasing property. Let r ( t ) be the maximal solution of u' = g(t, 4

+qt),

existing on [to, co),where S E C [ J ,R,"].

I x -Y

+ h [ f ( t ,x) -fdt,r)ll

4td

=

uo

9

Assume that, for t

< I x -Y

I

E

J, x,y

E

+ k ( t , I x -Y I) + O(h)

R",

88

CHAPTER

for all sufficiently small h

2

> 0. Then,

~ ( t )y,( t ) being a solution and a 6 approximate solution of (2.7.1) and

(2.7.3), respectively, existing on [to, a).

2.9. Asymptotic equilibrium

We shall continue to consider the differential system (2.7.1). DEFINITION 2.9.1. We shall say that the differential system (2.7.1) has asymptotic equilibrium if every solution of the system (2.7.1) tends to a finite limit vector f as t 03 and to every constant vector .$ there is a solution x ( t ) of (2.7.1) on to t < co such that limt+mx(t) = 4. T h e following theorem gives sufficient conditions for the system (2.7.1) to have asymptotic equilibrium: ---f

THEOREM 2.9.1.


= X(t)&u),

2 0 is continuous for t~ J and +(u) 3 0 is continuous for 3 0, +(0) = 0, +(u) > 0, u > 0, and montonic nondecreasing in u,

where A ( t ) u

and if

(2.9.8)

the conclusion of Theorem 2.9.1 holds.

2.10.

91

ASYMPTOTIC EQUIVALENCE

Theorem 2.9.1 has a corollary for the case that (2.7.1) is replaced by x'

(2.9.9)

A(+ + F ( t , x),

=

where A(t) is a continuous n x n matrix and F Let X ( t ) be a fundamental matrix for

E

C [ J x R", R"].

X ( t o ) = unit matrix,

x' = A(t)x,

(2.9.10)

so that the transformation

x

reduces (2.9.9) to y'

=

X(t)y

(2.9.1 1)

= x-1(t )F(t, X ( t ) Y ) .

Thus, an application of Theorem 2.9.1 to (2.9.11) gives

COROLLARY 2.9.2. Let A(t) be a continuous matrix for t E J and X ( t ) be a fundamental matrix for (2.9.10). Let F E C[J x R", R"], and, for ( t ,Y ) E J x R", /I X-Yt)F(t, X(t)Y)lI < h(t)llY 11, (2.9.12) where h(t) >, 0 is continuous for t

E

J , and

s z X ( s ) ds

< a.

Furthermore, let x ( t ) be a solution of (2.9.9) on some t-interval to the right of to . T h e n x ( t ) exists for all t 2 to , lim X-l(t)x(t) = [,

(2.9.13)

t-m

and, conversely, given a constant vector f , there is a solution x ( t ) of (2.9.9) satisfying (2.9.13). An interesting special case in which the hypotheses of Corollary 2.9.1 are satisfied is that of the linear homogeneous system (2.9.10), where

1; I/

4 s ) l l ds

< a-

I t is enough to take h(t) = 11 A(t)11 and $(u) = u.

2.10, Asymptotic equivalence Suppose we are given the following two differential systems:

x'

=f1(t,

x),

Y'

=f i ( t , Y ) ,

x(h)

=

xo

Y(to) = Yo

1

(2.10.1 ) (2.10.2)

92

CHAPTER

where fi, f 2 valence.

E

2

C [ J x R”, R”]. We shall first define asymptotic equi-

DEFINITION 2.10.1. T h e differential systems (2.10.1) and (2.10.2) are said to be asymptotically equivalent if, for every solution y ( t ) of (2.10.2) [a(t) of (2.lO.l)], there is a solution x ( t ) of (2.10.1) [ y ( t ) of (2.10.2)] such that as

x(t) -y(t)+O

THEOREM 2.10.1

2)

co.

>g(t,u)

for t 3 to such that u ( t ) + 0 as t Suppose further that

+ h(fi(t,

+

Let zc(t) be a positive solution of u’

II X -Y

t

4

< I/ x -Y

-fZ(t,Y))Il

I1

GO,

where g E C [ J x R, , R].

+ hg(t, II x - y 11) + O(h)

(2.10.3)

for all sufficiently small h, t 3 to , and I] x - y 11 = u(t). Then, the systems (2.10.1) and (2.10.2) are asymptotically equivalent. If, in addition, one of the systems has asymptotic equilibrium, then the other system also has asymptotic equilibrium.

Proof. Let us first suppose that y ( t ) is a solution of (2.10.2) defined for t >, t,, . Let x ( t ) be a solution of (2.10.1), defined on some t-interval

to the right oft,, such that

/I 4 t o )

- Y(4J)ll

< .(to>.

Clearly, such a solution exists. Define Then as far as x ( t ) exists. If this assertion is false, let t , be the greatest lower bound of numbers t > t o , for which m ( t ) u(t) does not hold. Since m ( t ) and u(t) are continuous functions, we have, at t = t, ,


U(t,

+ h),

h

> 0.

This implies the inequality (2.10.4)

2.10.

93

ASYMPTOTIC EQUIVALENCE

I n view of the condition (2.10.3), one also gets, at t

=

t, ,

D+m(t,) G d t l 7 m(tl)),

which is a contradiction to (2.10.4). Hence,

/I x(t) -y(t)ll

< u(t)

is true as far as x(t) exists. Now, using Corollary 1 .I .2, it follows that x(t) exists for all t 3 to , since y ( t ) and u ( t ) are assumed to exist for t 3 to . Moreover, as limL+m u(t) = 0, lim I/ x(t) -y(t)ll

t--.m

=

(2.10.5)

0.

On the other hand, if x ( t ) is a solution of (2.10.1) existing on [ t o , a), arguing as before, we can conclude that there exists a solution y ( t ) of (2.10.2) on [ t o , KI) such that (2.10.5) is satisfied. It therefore follows that the systems (2.10.1) and (2.10.2) are asymptotically equivalent. If one of the systems has asymptotic equilibrium, the asymptotic equilibrium of the other system is a consequence of (2.10.5). T h e proof is complete. T h e next theorem gives sufficient conditions for the asymptotic equivalence of the systems (2.9.9) and (2.9.10).

THEOREM 2.10.2. Let A(t) be a continuous matrix for t E J and F E C [J x Rn,R"]. Suppose that

I1F(t>41 < Yt)ll x It. where A ( t )

3 0 is continuous for

t

E

J , such that

s I h ( s ) ds

< 03.

Assume that all the solutions of (2.9.10) are bounded as t lim inf t-m

J:o

(2.10.6)

tr A(s)ds

>

+ CQ

-03.

and (2.10.7)

Then, the systems (2.9.9) and (2.9.10) are asymptotically equivalent. Proof.

Let Y ( t )be a fundamental matrix of (2.9.10). Setting Y(t),(t) = r(t),

94

2

CHAPTER

it is easy to verify that x ( t ) is a solution of (2.9.9) if and only if v ( t ) satisfies a' = Y-'(t)F(t, Y(t),). (2.10.8) Using (2.10.6), (2.10.7), and the assumption that all the solutions of (2.9.10) are bounded, we get

II W f ) F ( t ,y(wll < II W ~ ) l l IY(t)llll l 21 Ilqq

< KII

?J

llA(9,

where K is some constant. Hence, Corollary 2.9.1 implies that (2.10.8) has asymptotic equilibrium. Now, any solution y(t) of (2.9.10) can be written as Y ( t ) = Y(t)S,

4 being a constant column vector. Therefore, 44

-y(t)

=

Y(t)[,(t>- [I,

and the desired result follows, since Y ( t )is bounded on ( t o , a). T h e asymptotic equivalence of the systems (2.10.1) and (2.10.2) can also be considered on the basis of the variation of parameters formula for nonlinear systems developed in Theorem 2.6.3.

THEOREM 2.10.3.

Assume that (i)fi

,.f2

E

C [ J x Rn,Rn],

exists and is continuous on J x R";(ii) dj1(t, t, , x,)(Q2(t, t o ,yo))is the fundamental matrix solution of the variational system

2.10.

ASYMPTOTIC EQUIVALENCE

95

Then, there exists a solution x ( t ) of (2.10.1) [ y ( t ) of (2.10.2)] on [to , a) satisfying the relation lim x ( t ) - y ( t ) = 0. (2.10.10) i-m Pyoof. Let y ( t ) = y ( t , to , y o ) be a given solution of (2.10.2) existing on [ t o , a).Define a function x ( t ) by the relation x(t) =Y(t)

+\

W

' t

@l(hs,y(s))[f,(s,y(s)) -fl(s>Y(S))lds.

(2.10-11)

Since the integral converges by assumption (iii), it follows that x ( t ) is well defined, and, consequently, (2.10.10) is satisfied. It therefore suffices to prove that x(t) is a solution of (2.10.1). For this purpose, we observe, as in Theorem 2.6.3, that

Here use is made of the relation (2.5.11) and the fact that

T h e relations (2.10.1 1) and (2.10.12) yield

Moreover, we have

96

CHAPTER

2

1,et us differentiate (2.10.1 l), recalling that y ( t ) ,Q l ( t , t, , x,) are the solutions of (2.10.2) and (2.10.9), respectively, and using (2.10.12) to obtain

This reduces to, in view of (2.10.14),

T h e relation (2.10.13) implies that x ( t ) is a solution of (2.10.1) with

On the other hand, if x ( t ) is a solution of (2.10.1) existing on [ t o , a), we can show exactly in a similar way that there exists a solution y ( t ) of (2.10.2) on [ t o , m) such that (2.10.10) holds. Thus the theorem is established.

2.1 1 . A topological principle

This topological principle is concerned with the differential system Y'

= f ( t , x),

.(to)

= X"

,

to 2 0,

(2.1 I. 1)

where f E C [ E ,RT1],E being an open ( t , x)-set in Rn+l. Let E, be an open subset of E , r?E, the boundary, and l?, the closure of E, .

DEFINITION 2.1 1.1. A point ( t o , x,) E E n aE, is said to be an egress point of E,) with respect to the system (2.11.1) if, for every solution x ( t )

2.1 1.

97

A TOPOLOGICAL PRINCIPLE

<
0 such that ( t , x ( t ) ) E Eo for to E t to . An egress point ( t o ,xo) of Eo is called a strict egress point of E, if to E , for a small E > 0. ( t , x ( t ) ) E, for to < t T h e set of all points of egress (strict egress) is denoted by S(S*). It is clear that S* C S. ~

< +

DEFINITION 2.1 I .2. If A C B are any two sets of a topological space and T : B 4 A is a continuous mapping from B onto A such that W( p ) = p for every p E A , then T is said to be a retraction of B onto A. When there exists a retraction of B onto A , A is called a retract of B. T h e following examples would sufflce to illustrate the concept of retraction.


0, it

f ( t , x) = J;.(t, 0)x

is possible to find a S ( E ) > 0 (2.13.12)

4-F ( t , x),

wherc

~ l F ( t , x ) l ~ cllx~l if 11 x / j < 6 (2.13.1 3) uniformly in t. Let E '> 0 and t, E be given. By the condition (2.13.1 l ) , it follows that we have, for large t > t,, , =
t, with the property that 11 'Y(t,)lJ

=

6,

11 x(t)ll

< 6,

t"

< t < t, .

(2.13.16)

Defining nr(t) = / / x ( t ) 11, we observe that

I! , x ( t ) I- h f ( t , s(t)lI < 11 I

+ 4fz(fv0)Iiil 4t)JI + hll F ( f ,x(t))ll,

for t E [t, , t,], because of (2.13.2), and hence ?at)

.< p[f&

O)l.z(t)

+ 11 F ( t , .2.(t))ll,

which, in view of thc relations (2.13.16) and (2.13.13), yields m; ( t )

,< [ P [ f n . ( f ,0)l

I - EIm(t),

t

E [to

t

tll.

Theorem 1.4.1 then implies, choosing u ~= , m(t,), that, for t m(t)

.< m(to) exp [ ~ (--t to)-1-

it '0

p [ f z ( s , O)] ds],

(2.1 3.17) E

[to , t J ,

(2.13.18)

2.13.

107

STABILITY CRITERIA

and we are led to an absurdity:

because of relations (2.13.15), (2.13.16), and the fact that /I xo 1) < 6,. This proves that, whenever 11 xo ( 1 0, and therefore m(t,) > 0. Thus, the relations (2.13.20), (2.13.21), and (2.13.22) lead to the contradiction g(t2 , u(t2 , .))

proving m(t)

+ eg(t2 , m(t2)), E

u(t, E ) , t E [to , t J , which implies (2.13.5) because of the

,
0,

I1 .t'

~

Assume that F


0. T h e conclusion is now immediate.

THEOREM 2.14.5. Let the assumptions of Theorem 2.14.4 hold except that the inequalities (2.14.13) are replaced by

11

u(t)il< M ,

11 u(t)U-l(s)II < N ,

to

< s < t.

(2.14.17)

Then, all solutions x ( t ) of (2.14.9) exist for t 3 to and verify the estimate

/I 4t)Il

< KIIx,

11,

t 3 to

7

(2.14.18)

for some K > 0. If, in addition, y ( t ) is the solution of the linear system (2.14.6) with y(to) = xo such that limt+wy ( t ) = 0, then 1imL.,%x ( t ) = 0.

I12

CHAPTER

2

Pmof. T h e integral equation (2.14.16) gives, using the conditions (2.14.17) and (2.14.14),

I/ ?c(t)Il /
0, which is independent of that lirnl ,I, x ( t ) = 0.

E

and T . This proves

THEOKIJM 2.14.6. Assume that (i) A is an n x n constant matrix and the characteristic roots of A have negative real parts; (ii) F E C[J x R", Rn], and, given any E > 0, there exist 6(t), T ( E )> 0 such that

I/ F ( t , .)I1

< €11

X

I/

provided 11 x 11 < 6 ( ~ )and t 3 T ( E ) (iii) ; G E C [ J x Rn,R"] and there OL and t E J , exists an .I >- 0 such that, if 11 x /I i

I1 G(t,xlll where y

E

< 74%

C [ J ,R,] and tfl

p(t) =

j t

y(s) cis -+

o

as

t

+

co.

2.14.

113

ASYMPTOTIC BEHAVIOR

Then, there exist To 3 0, 6 > 0 such that, for every to 2 To and I/ xo 11 < 6, any solution x ( t ) = x(t, to , xo) of the differential system X' =

A X + F ( t , X) + G(t,x),

satisfies

lim x ( t )

t t m

=

x(to) = xo

(2.14.19)

0.

If, in particular, (2.14.19) possesses trivial solution, then the trivial solution is asymptotically stable.

Proof. o

By assumption (i), it follows that there exist constants K

> 0 such that

11 eAt Ij

< KcUt,

t

0.

2 1 and (2.14.20)

Choose E so that 0 < E < min(a/K, a). Because of assumption (ii), we can choose T ( E )3 1 and 6 ( ~ ) E . Let To 3 T ( E )be so large that t To implies that


0,

whence e-Bt

J' eBsy(s) ds < e-Bt 1

Applying L'Hospital's rule on

for t

> to > 1.

I14

CHAPTER

2

it can be easily verified that

j t essy(s)ds

lim

t-m

for all /3

=0

1

> 0. T h e validity of (2.14.21) is now clear.

Let to 2 To and 11 x,,// x ( t ) = exp[A(t

-

-26, .

to)]x,

Then, as long as 11 x(t)/(< 6(~), we have

+

/:o

eA(t-s)[F(s,x(s)

+ G(s,x(s))]

ds,

from which, using assumptions (ii), (iii), and the estimate (2.14.20), it follows that

II x(t)lleuf

+

< Kal exp(ot,)

By Corollary 1.9.1, we obtain /I x(t)/jeot < kT6, exp(ut,,) exp[Kc(t

~

so that

I/ x(t)ll

< KS, e x p - ( o

-

Kc)(t - t,,)]

i:

to)]

[ell

x(s)ll

+

1'

+ y(s)]KeuSds.

Keusy(s)exp[Kc(t - s)] ds,

to

+ K It

exp[-(a

-

Kc)(t

~

(2.14.22)

which, using (2.14.21), yields

11 x(t)]l < KSl


=

0.

Consider the function m ( t ) = IIy(t)ll. Observe that

where e(h)/h-+ 0 as h -+ 0. Furthermore, from the definition of p ( A ) given by (2.13.6), the inequality (2.13.10), and the assumption (2.14.24), we obtain

~ [ f d 4t))l t, Hence, it follows that

< &it7

0)l + Kli 4t)ll.

(2.14.29)

116

CHAPTER

2

-4s thc hypotheses of Theorem 2.13.2 are satisfied, the trivial solution of (2.14.27) is asymptotically stable, and therefore, if 11 x, I/ is small t 2 t, . Consequently, choosing 11 x, I/ sufficiently enough, 11 x(t)ll small, we have, from (2.14.30),

v,

which, by Theorem 1.4. I , leads to the estimate (2.14.28). Moreover, by condition (2.14.25), it results that, if 7 is small,

This, together with (2.14.28), implies that liml+my ( t ) = 0.

REMARK2.14.1. T h e condition (2.14.25) implies that the solutions yo(t)of the variational system

have the property that liml+myo(t)= 0. For, setting m(t) = IIy,(t)ll,

we obtain

D+m(t)

< P [ f Z ( t , O)lm(t),

and hence, by Theorem 1.4.1,

T h e assumption (2.14.25) assures that liml->va yo(t) = 0. Thus, in essence, Theorem 2.14.7 guarantees the asymptotic behavior of the solutions of (2.14.26), whenever there exists a similar behavior for the solutions of (2.14.31). From these considerations, we infer the following lemma, which is interesting in itself. T,EMMA 2.14.1. Let A(t) be a continuous n x n matrix on J . If x(t) is the solution of s' = A ( t ) x ,

we have

x(t,) = .xo ,

2.14.

COROLLARY 2.14.2.

Under the assumptions of Theorem 2.14.7, if ,t

1

then u

117

ASYMPTOTIC BEHAVIOR

(2.14.32)

< 0.

Proof. Since the trivial solution of (2.14.27) is asymptotically stable by Theorem 2.13.2, choosing (1 xo 11 small, we can make 11 x(t)ll < 7, t 3 t o , and hence we have 0


0, there exists a 6 = S ( E ) > 0 such that

II G(t,Y)ll

0,

Thus, (2.14.36) is valid for t 3 t, , and the asymptotic stability of the trivial solution of (2.14.35) follows.

THEOREM 2.14.10. Assume that (i) f E C [ J x R",R"],f ( t , 0)= 0, and f z ( t , x) is continuous on J x Rn;(ii) p [ f z ( t , O)] -u, u > 0, t E J ; (iii) G E C [ J x Rn,R"],G(t,0) = 0, and there exists an a > 0 such that, if 11 x 11 < a, t E J , 11 G(t,x)II y(t), where y E C [ J ,R,] and


0 be given such that 0 < E < min(u, a ) . Choose To 3 1

so large that, for t

s:

T o ,we have

exp[-(cr

-

c)(t

- s)]y(s)

ds

< YE) - = 6, , 2

(2.14.37)


, to . For, otherwise, there would exist a t, > to 3 'Tosuch that

II 4tl)ll Define m ( t ) inequality

=

= 6(E),

ll 4t)ll

< 6(c),

t

E

[to >

4.

11 x ( t ) 11. Then, for t E [ t o ,t l ] , we obtain the differential

m;(t>

< P [ f Z ( t ? O)lm(t) + /I F(t7 4t))ll + It G(t,4t))ll < -(a 4 4 t ) + y(t). -

Here we have used assumptions (ii) and (iii) of the theorem, in addition to the relations (2.13.12) and (2.13.13) and the argument employed in Theorem 2.13.2.

120

CHAPTER

2

An application of Theorem 1.4. I gives

/I 4t)ll

< /I xo I/ exP[-(u

+ for t

E

[t, , t,]. At t S(c)

=

Jl0

exP[-(a

-

.)(t

-

441

- E)(t -

s)1 A s )

(2.14.38)

t , , there arises an absurdity

< 6, -k j:'

< 6,

+ 6,

exp[-(o =

-

c)(tl

-

s)] y(s) ds

S(E),

because of (2.14.37). This proves that, if 11 x,,11 < a,, Ij x(t)ll < a(€), t 2 t, , which, in its turn, implies the inequality (2.14.38) for all t 3 to . Since S(E) E , the stated result follows, as in Theorem 2.14.6.


0. Setting

because of the nondecreasing nature of g(t, u ) we get

According to Theorem 1.2.1, we infer that m(t) < p(t, E),

t

But limp(t, E ) 0,

II x - y I W t , 4

-f(t,y)lll

< II x - y

11[1

-

ah]

+ O(h),

t

2 0, (2.15.1 1)

2.15.

125

PERIODIC AND ALMOST PERIODIC SYSTEMS

where 01 > 0, and that the almost periodic system (2.15.4) admits a bounded solution x ( t , t o , x,) with a uniform bound B. Then, (2.15.4) admits an almost periodic solution that is uniformly asymptotically stable.

Proof. Let x(t) be the bounded solution of (2.15.4), defined on [ t o ,co) so that I/ x(t)ll B, t 3 t o , where B does not depend on t o . Let rk be a sequence of numbers such that rk -+ co as k -+ co and


0,

we have

where ~ ( h ) /+ h 0 as h + 0. Since V ( t ,x) is locally Lipschitzian in x, we get, using (3.1.3), the inequality L> ' m ( t ) < s(t,m ( t ) ) ,

t E J.

Applying Theorem 1.4.1, we obtain the desired result (3.1.6). REMARK 3.1.1.

Let S, = [x E Rn : 11 x

I/ < p ] ,

and assume that the condition (3.1.3) holds for ( t , x) E J x S, . If x ( t ) is any solution of (3.1.1) such that 11 xo 11 < p, then (3.1.5) implies (3.1.6) as far as x ( t ) remains in S, to the right of to . COROLLARY 3.1. I . zero, i.e.,

If the function g(t, u ) in Theorem 3.1.1 is identically W V ( t ,x)

< 0,

( t , X) E J x

Rn,

3.1.

BASIC COMPARISON THEOREMS

133

then the function V ( t ,x ( t ) ) is nonincreasing in t , and V ( t ,X(t))

< V(to

9

t

Xo),

3to.

COROLLARY 3.1.2. Assume that the hypotheses of Theorem 3.1.1 hold except that the condition (3.1.3) is to be satisfied only for ( t , x) E J x 52, where SZ = [X E R" : ~ ( t< ) V ( t ,X) < ~ ( t+ ) co , t 3 to], c0 being some positive number. Then, the conclusion of Theorem 3.1.1 is true.

Proof. We choose uo = V ( t , , xo) and proceed as in the proof of Theorem 3.1.1 to obtain Z being the set

D+m(t)

< g(t, m ( t ) ) ,

t

E

z,

z= [t E J : r ( t ) < m ( t ) < Y ( t ) + 61. Theorem 1.4.2 now assures the stated result. Sometimes, the following variants of Theorem 3.1.1 are more useful in applications.

THEOREM 3.1.2. Assume that the hypotheses of Theorem 3.1.1 hold except that the inequality (3.1.3) is replaced by A ( t P + V t X) ,

+ v, X)D+A(t)< g(t, V ( t ,x)A(t)),

(3.1.7)

for ( t , x) E J x R", where the function A(t) > 0 is continuous for t E J , and 1 D+A(t) = lim sup - [A(t h++ h

Then U t o xo)A(to)

imp1ies q t , X(t))A(t)

Proof.

+ h)-

A(t)].

< uo

< r(t),

t

(3.1.8)

2 to -

(3.1.9)

Defining VAt, X)

=

V ( t ,X)A(t),

it is easy to show that Vl(t,x) satisfies the assumptions of Theorem 3.1.1.

134

For, if h

CHAPTER

3

> 0 is sufficiently small,

and therefore, using the assumption (3.1.7), we get

T h e estimate (3.1.9) follows immediately from Theorem 3.1.1.

THEOREM 3.1.3. Let the hypotheses of Theorem 3.1.1 hold except that, instead of the inequality (3.1.3), we now assume

o+v(t, x) + 9(ll

.X

Ill < R(& v(t,x)),

( t , x) E I x I?”,

(3.1.10)

where +(u) 3 0 is continuous for u 3 0, $(O) = 0, and $(u) is strictly increasing in u. Suppose further that g(t, u ) is nondecreasing in u for each t E J . T he n (3.1.5) implies that

Proof.

Consider the function


0, V ( t ,x) + as 11 x 11 co uniformly for t E [0, TI. T h e mild unboundedness of V ( t ,x) guarantees that, whenever V ( t ,x ( t ) ) is finite, 11 x(t)ll is also finite. T h e assumption that the solutions x(t) of (3.1.1) exist for all t 3 t o , therefore, becomes superfluous, if V ( t ,x) is further assumed to be mildly unbounded in the foregoing theorem. From this observation stems the following global existence theorem. ---f

3.2.

135

DEFINITIONS

THEOREM 3.1.4. Let V EC [ J x Rn, R,], V(t,x) be mildly unbounded and locally Lipschitzian in x. Suppose that g E C [ J x R, , R] and r ( t ) is the maximal solution of (3.1.4) defined for t 3 t, . Assume that (3.1.3) holds. Then, every solution x(t) of (3.1.1) exists in the future, i.e., for all t 3 to , and (3.1.5) implies (3.1.6). Proof. Suppose that the assertion that every solution x(t) of (3.1.1) exists for all t >, t, is false. Then, by Corollary 1.1.2, there exists a t, > t, such that x(t) cannot be extended to the closed interval t, t t , , which implies that there cannot exist an increasing sequence {t,} + t,- such that 11 x(tn)1l is bounded. This, in its turn, yields that 11 x(tn)lI + 00 as t, + t,- . On the basis of Theorem 3.1.1, it follows that (3.1.5) implies (3.1.6) for to t t, . By the assumption that V ( t ,x) is mildly unbounded, the fact that r ( t ) exists for all t >, t, , and (3.1.6), there arises a contradiction as t, + t,- . Hence, the global existence of solutions x(t) of (3.1.1) is proved, which, in turn, assures the estimate (3.1.6) for t 3 t, whenever (3.1.5) holds. T h e proof is complete.

<
> g ( t , V ( t ,x))

2 0,

( t ,x) E G .

(iii) For to > T , the solutions u(t, t o , uo) of (3.2.3), for arbitrarily small uo > 0, are either unbounded or indeterminate, for t 2 t o . Then the trivial solution of (3.2.1) is unstable. There exists a point ( t o ,x,,)E G, xo f 0, in the vicinity of ( T , 0). Let x(t) = x(t, t o ,x,,) be any solution of (3.2.1). Then, the Lipschitzian nature of V(t,x) and condition (ii) yield Proof.

V ( t ,x ( t ) ) 2 V ( t , , xg)

=

uo

> 0,

(3.3.10)

for all t >, 0, for which ( t , x ( t ) ) E G. Since V ( t ,x) = 0 for all (t,x) E - G, it follows from (3.3.10) that (t, x(t)) E G for t 2 t o . Moreover, we also have

e

which, in view of Remark 1.4.1, implies that V t , 4 t ) ) 2 d t , to

7

U"),

t

2 to ,

(3.3.11 )

where p ( t , t o , uo) is the minimal solution of (3.2.3). Since V ( t ,x) is bounded by assumption, the estimate (3.3.1 1) leads to an absurdity,

3.3.

143

STABILITY

if we assume the trivial solution of (3.2.1) is stable. This proves the theorem.

THEOREM 3.3.7. L e t f E C[(-co, co) x S o ,R"] andf(t, x) be periodic in t with a period w . Then, under the hypotheses of Theorem 3.3.1, the trivial solution of (3.2.1) is equistable for to E (- 00, co). Proof. Let 0 < E < p and to E (- co, co) be given. I t is possible to choose an integer k such that to kw 2 0. Since the solution u = 0 of (3.2.3) is equistable, given b ( ~ > ) 0, t, kw >, 0, there exists a positive function 6 = 6 ( t o , c) such that uo 6 implies

+

u(t

u(t

+

+ kw, to + kw, uo) < We),


0 satisfying


0.

(iii) D+V(t,x) d g(t, V ( t ,x)), ( t , x) E J x S,, . (iv) For any function X E C [ J ,R,],

Then, the trivial solution of (3.2.1) is uniformly asymptotically stable. Proof. Let x ( t ) = x ( t , t o ,xo) be any solution of (3.2.1). Defining m(t) = I/ x(t)ll, we obtain, from (3.4.17), m;(t)

< I1 x'(t)ll < Ilf(4 x(t))ll < Y t ) m ( t ) .

By Theorem 1.4.1, it follows that

152

3

C'IIAPTER

as far as (1 x(t)(\ p. Consider thc interval [ t o ,T] for some p. Then, since which 11 .z(t)l/
, to for

-< j'h(s) ds, 0

I1 4f)ll< I1 X" II e N T ,

[to , TI,

t

(3.4.18)

where N N ( T )= supo A(t). Since V ( t ,,x) is positive definite and ctescresccnt, (3.4.15) holds. Let 0 < c c p , toE J be given. Choose 8, S , ( c ) such that 481) < 4 6 ) . -

-

It is clcar that s1 E. Let 8 : B[a(S,), A(€)], tZ = k[a(S,), b ( ~ ) ] and , 8 ( ~= ) 8,ecN0. Wc choosc I/ xo 11 6 so that (3.4.18) assures that 11 x(t)ll < 6 , , to t B, which implies that

<
0.

(3.4.20)

On the other hand, as t , 2 8 and (3.4.19) holds, we obtain, from conditions (ii) and (iii) and the fact that V ( t ,x) is locally 1,ipschitzian in x, the inequality I ) ' l'(f,

>

dtl))

< A t , , Vtl :< -k < 0 ,

7

x(t1)))

which contradicts (3.4.20), thus proving that V ( t ,x ( t ) ) < A(€) for t 3 8. It therefore follows that, if (1 xl, (1 -'8, V ( t ,x ( t ) ) , t o , and

3.4.

153

ASYMPTOTIC STABILITY

consequently, in view of (3.4.15), the uniform stability of the trivial solution of (3.2.1) is proved. Let us denote by So the number 6 ( p ) obtained by setting E = p, and let 0 < E < p . Let 6 = S ( E ) be the same function as before. Assume 11 x,,11 < 8,. Choose

where

T o prove uniform asymptotic stability, it is enough to show that there exists a t, E [to , to TI satisfying /I x(t,)ll < S ( E ) ; since it would then follow that 11 x(t)ll < c, t 3 to T in any case. Suppose there is no such t , ; then,

+

+

a(€) G I1 4t)ll G P ,

t

E

[t" t o 9

+ TI,

which, because of (3.4.15), shows that

Using assumptions (i) and (iii), we get

This absurdity proves that 11 x(t)iI < S ( E ) < E , t 3 t, and the theorem is completely proved.

11 x,,11 < 8,,

-t T , whenever

We have seen in Theorem 3.3.5 that the uniform stability of the trivial solution of (3.2.1) can be formulated by means of monotone functions. We may likewise state the following theorem with respect to uniform asymptotic stability.

154

3

CHAPTER

THEOREM 3.4.1 1. T h e trivial solution of (3.2.1) is uniformly asymptotically stable if and only if there exist functions a E X , (r E 9such that

I/ . ~ ( tto, , xo)ll for

1) xo I/

< 41xn I l ) ~ ( f

-

t

to),

2 to >

(3.4.21)

-, p.

Pyoof. If (3.4.21) is satisfied, it is easy to verify that (S,) holds, and hence sufficiency of the condition is evident. Assume now that the trivial solution of (3.2.1) is uniformly asymptotically stable, so that (S,) holds. Let { e n } be a positive, monotonic sequence, converging to zero as n + GO. Let T , ( E )= inf T ( C ? ~ + ~for ) en+,

Then,

I/ X" I/ < 6 0 ,

t

I1 .x(t, to , xojll < [E,,+~,

en],

TI(€,),

E

E,

.

> t" iTI(€)

assures that

Define T*(e) linear in T,(E,, , l ) . Note that

<


( t ,x) €

J x so,

(t,

J

'R

>

where ~ ( tu ,) is the same function given by (3.5.2). T h e conclusion of the theorem is then a direct conseqeunce of Theorems 3.3.3, 3.3.4, 3.4.3, and 3.4.4. COROLLARY 3.5.1. Let the assumptions of Theorem 3.5.1 hold with 2, m = 1, and w ( t , u ) = X(t)u, where h(t) 3 0 is continuous on J such that N =

Then, the solution x

=

0 of (3.5.1) is asymptotically stable.

THEOREM 3.5.2. Let there exist functions V(t,x), g l ( t , u), and w(t, u ) fulfilling the following conditions: (i) V t C [ J x SR, R ,], V ( t ,x) is Lipschitzian in x for a function k ( t ) 3 0 continuous on J , and

411x II)

< L'(t, x) < 411x II),

t , x) E J

x s, ,

(3.5.3)

where a, h E f . (ii) g,

E

C [ J x R, , R],g,(t,0)

= 0, and

n + l 7 ( t 4? ( 3 2.1) < gl(t? f7(k x)),

( t ,x) E J x

s,

1

(iii) w E C [ J x R, , R,], w(t, 0) = 0, w ( t , u ) is nondecreasing in u for each t, and I / R(t,.y)II

< w ( t , II x II)?

( t ,).

E

Then, the stability properties of the solution u

At, ).

= Rl(t>).

J x s,. =

0 of (3.2.3) with

4- k ( t ) w ( t ,W U ) ) ,

(3.5.4)

where b ' ( u ) IS the inverse function of b(u), imply the same kind of stability properties of the trivial solution of (3.5.1).

3.5.

157

STABILITY OF PERTURBED SYSTEMS

Proof. Let us define the function D+V(t,x) with respect to the perturbed differential system (3.5.1) as follows:

using assumptions (ii) and (iii). This, together with (3.5.3) and the monotonicity of w ( t , u ) in u, leads to the differential inequality D'

v(t,

X)(:5.5.1)

< g(t, V ( t ,x)),

( t ,).

E

J

x s, ,

where g(t, u ) is given by (3.5.4). Now, it only remains to apply Theorems 3.3.3, 3.3.4, 3.4.3, and 3.4.4 to get the desired result. COROLLARY 3.5.2. T h e functions b(u) = u,gl(t, u ) = -au, a > 0, and w(t, u ) = X(t)u, X ( t ) 3 0 being continuous on J and satisfying

are admissible in Theorem 3.5.2, to guarantee the uniform asymptotic stability of the solution x = 0 of (3.5.1) provided k is the Lipschitz constant for V ( t ,x). COROLLARY 3.5.3. T h e functions w ( t , u ) = h ( t ) $ ( u ) , gl(t, u ) = 0, where X E C [ J ,I?,], 4 E X , are admissible in Theorem 3.5.2 to yield the uniform stability of the trivial solution of (3.5.1)) provided that k is the 1,ipschitz constant for V(t,x) and :J- X(s) ds < GO. COROLLARY 3.5.4. T h e functions w(t, u ) = h(t)$(u), gl(t, u ) = -C(u), where h E C [ J ,I?+], 4,C E .Y, are admissible in Theorem 3.5.2 to assure that the trivial solution of (3.5.1) is uniformly asymptotically stable, provided that Fz is the Lipschitz constant for V ( t ,x), k 4 ( k 1 ( u ) )< ,C(u), for some 01 > 0, and limt+m[-t 01 h(s) ds] = -a for all to 3 0.

+ J-7'

158

CIIAPTER

3

3.6. Converse theorems This section will be devoted to a variety of results concerning the construction of Lyapunov functions. Let us first define the notion of generalizcd exponential asymptotic stability. DEFINITION 3.6.1. T h c trivial solution of (3.2.1) is said to be (,Sll) generalized exponentially asymptotically stable if

where K ( t ) > 0 is continuous for t E J , p E X' for t E J , and p ( t ) + GO as t + co. I n particular, if K ( t ) = K > 0 , p ( t ) = at, a: > 0. We have the exponential asymptotic stability of the trivial solution of (3.2.1)

THEOREM 3.6.1. Assume that the solution x = 0 of (3.2.1) is generalized exponentially asymptotically stable and thatf(t, x) is linear in x. Suppose further that p'(t) exists and is continuous on J . T h e n there exists a function V ( t ,x) satisfying the following properties: (i) V E C [ J x S o , R,], and V ( t ,x) is Lipschitzian in x for the function K(t).


0 such that 0, K ( t ) = K > 0, then T = log Klqct, and M = KL+(-q)a/qa, where n/l is the Lipschitz constant for V ( t ,x), are admissible in Theorem 3.6.2. It is possible to prove the previous theorems, under milder assumptions, in a different way.

162

3

CHAPTER

THEOREM 3.6.3.

Assume that

(i) f E C [ J x S, , R”],f(t,0) = 0, andf(t, x) satisfies

< L ( t ) I/ x

IIf(t, x ) -f(t,Y)II

-y

11,

( t , x ) , ( t ,y ) E

1X

L(t) 0 being continuous on J ; (ii) There exists a p E X’ for t E J , p ( t ) 4 co as t exists, and

11 x ( t , 0, xo)ll

< K II xo II exp[-P(t)l,

t

S,

, (3.6.5)

-+00,

2 t o , K > 0,

p’(t)

(306.6)

where x(t, 0, xo) is the solution of (3.2.1) through (0, xo). Then, there exists a function V ( t ,x) enjoying the following properties: (1) V E C [ J x S o ,R,], and V ( t ,x) is Lipschitzian in x for a continuous function K ( t ) > 0.


0. We have already seen that the concepts of stability and asymptotic stability can be defined by means of simple inequalities involving certain monotone functions. We give below some converse theorems in terms of differential inequalities. As will be seen, the approach depends upon the differentiable properties of solutions with respect to the initial values and yields, in a unified way, a method of constructing Lyapunov functions.


, 0, L,(t) 2 0 being continuous on J ; (iv) the solution u(t, 0, u,,) of (3.2.3) fulfill the estimate n(.o)

< 4 t >0, a,),

t

2 0,

Y1 €37.

3.6.

167

CONVERSE THEOREMS

Then, there exists a function V(t,x) with the following properties: (I) V E C [ j x S, , R,], and V ( t ,x) is positive definite and satisfies a Lipschitz condition for a continuous function K ( t ) 3 0.


u(t + k , 0, /I 40, t , u(t

1 D+V(t,x ( t ) ) = lim sup - [ V ( t &+Of k

X)II)>

+ k , x ( t + h, t , x))

-

V ( t ,x)]

= u’(t,0 , !I 40,t , .)!I) = g(t,

V t , XI).

According to Corollary 2.7.1, if we let x = x(t, 0, x,,), y condition (i) implies, as in Theorem 3.6.3, the inequality

Moreover, assumption (iii) also implies that

Thus, for ( t , x), ( t , y ) E J x S o ,we have

(3.6.14) =

x(t, 0, yo),

168

CHAPTER

3

+

if K ( t ) = exp[Ji (L,(s) L2(s))ds]. This, together with (3.6.14), enables us to deduce, as in Theorem 3.6.1, that

where P(T) = /322c22(0)T. Moreover, using the lower estimate of (3.6.20), it follows that

COROLLARY 3.6.4. Theorem 3.6.5.

The

function

g(t, u) = 0

is

admissible

in

T h e next theorem deals with the converse problem for asymptotic stability.

THEOREM 3.6.6.

Let assumptions (i) and (iv) of Theorem 3.6.4 hold. Suppose that the solution x ( t , 0, xo) of (3.2.1) satisfies

< P,(I

II x ( t , 0, .T")ll

t

X" Il)o(t),

where p2 E X , c E 9'. Assume that there exist functions y such that y'(u)

3K

b

0,

and r(u")qt)

t

E

.X, 6 E 9

K, > 0,

8 ( t ) 3 klU(t),

< u ( t , 0, uo),

(3.6.15)

2 0,

3 0.

(3.6.16)

Then, there exists a function V(t, x) satisfying V ( t ,0 ) 0, (1) V E C [ J x s, , R,], V ( t ,x) is positive definite and possesses continuous partial derivatives with respect to t and the components of x; :

(2)

V'(4

x) = g(t,

V t , x)),

(4 x) E

1 x so.

Pmof. Let x ( t , 0, x,,), u ( t , 0, uo) be the solutions of (3.2.1), (3.2.3) obeying the estimates (3.6.15), (3.6.16), respectively. Choose any

a

continuous function p(x) having continuous partial derivatives ap(x)/ax for x E S,, such that y(0) = 0 and

Bz(ll x II)

Defining

V(4 ).

0, y(0) = 0, and u(t, 0,O) = 0. From the assumption y’(u) there results Y(u1uz) ku1uz. (3.6.18)

z

Furthermore, by virtue of the fact that x the inequality (3.6.15) yields

=

x(t, 0, xo) and xo = x(0, t, x),

(3.6.19)

where P(T) = /322c22(0)T. Moreover, using the lower estimate of (3.6.20),

which implies, on account of the assumption 8 ( t ) is positive definite. T h e proof is complete.

2 Rla(t), that V(t,x)

COROLLARY 3.6.5. If u E 2 is a differentiable function for t E J , then the function g ( t , u ) = [a’(t)/u(t)]uis a candidate in Theorem 3.6.6. REMARK 3.6.1. Notice that, in Theorems 3.6.5 and 3.6.6, we have not assumed that the trivial solution of (3.2.3) is stable and asymptotically stable, respectively, since we do not need, in the proof, such

170

3

CHAPTER

a specific assumption. However, these hypotheses are required to prove

direct theorems. Nevertheless, the lower estimates of the solutions ~ ( t0,, u0) of (3.2.3) are compatible with the corresponding stability requirements. It can be seen from the proof of Theorem 3.6.4 that the lower estimate on x ( t , 0, xo) and the upper estimate on u(t, 0, uo) are useful only to prove the dccrescent nature of V ( t ,x). Observe also that we need only the stability information of solutions starting at to = 0, and this is a definite advantage. Undcr the rather general assumptions of Theorem3.6.6, it is not possible to show that. V ( t ,x) is decrescent. This can, however, be achieved in the following: 'rHE O R EM 3.6.7. Let assumptions (i) and (iv) of Theorem 3.6.4 hold. Suppose that, in place of (3.6.15), we have

Plll %J Y 4 t ) < I1 x ( t , 0, X")ll

< All

where R, p1 , p2 > 0 are constants, and of (3.2.3) allow the estimate 4u"qt)

where A, , A,

p > 0,

G

E

Xo

t >, 0,

Ilao(t),

2. Let the solution u(t, 0, uo)

< 4 t , 0, uo) < huo8(t),

t

z 0,

> 0 are constants, and 6 E 9 such that,

for some constant

syt) = U B ( t ) .

Then, there exists a function V ( t ,x) that is decrescent and that obeys (I), (2) of Theorem 3.6.6.

Proof.

By choosing a continuous function p ( x ) so as to satisfy

kill X /I0 < ("(x)

< k,lI

.^c

k , I R,

I T 3

I

B

0,

and following the proof of Theorem 3.6.6 with necessary modifications, we can easily construct the proof of this theorem.

THEOREM 3.6.8. Let assumption (i) of Theorem 3.6.4 hold, and let there exist functions u l , u2 E 9 such that

PlII xo / / ~ i (-t to)

< /I

4)

~ ( t l

~o)ll

< Bzll

Xo

IluAt

-

to),

t

> to

(3.6.20)

p,, p2 > 0 being constants and x ( t , t o ,XJ being the solution of (32.1). Then, there exists a function V ( t ,x) satisfying the following with

properties:

3.6.

171

CONVERSE THEOREMS

(1) V E C [ J x S, , R,], and V ( t ,x) is positive definite, decrescent, and possesses continuous partial derivatives with respect to t and the components of x. -eV(t, x), N > 0, ( t , X) E J x S o . (2) V'(t,X)


0 that we choose later,

Because of assumption (i), one can argue, as before, to show that V E C [ J x S, , R,] and V ( t ,x) is continuously differentiable. Furthermore, from the upper estimate of (3.6.20), we have

and hence

(3.6.21)

where P(T) = /322c22(0)T. Moreover, using the lower estimate of (3.6.20), it follows that

Thus, we have shown that (1) is verified. To prove (2), observe that

O n the other hand, using relation (3.6.1 l), we see that

Consequently,

172

CHAPTER

3

We now fix T by choosing it so large that

This is possible, since upE 2.Evidently, from this choice results the inequality V ' ( t ,x ) < - g 11 x 112, which, in view of (3.6.21), leads to

_-=

setting

&i=

~

d ( t ,x ) ,

1/[2P(T)]. T h e theorem is proved.

COROLLARY 3.6.6. condition

Instead of the lower estimate in (3.6.20), the llf(t,

1)I.

< -%)I1

ZL'

I/,

( t ,x ) E

I

x

s,

is admissible in Theorem 3.6.8.

TIIEOREM 3.6.9. Let the trivial solution of the system (3.2.1) be uniformly asymptotically stable. Suppose that

Ilf(t, XI)

-f(t,

%>ll

< L(t)/l

x1

for ( t ,q),( t , xg)E J x S, , where L(t)

~

xz

I1

0 is continuous on J , and

Then, there exists a function V ( t ,x) with the following properties:

, V ( t ,x) is positive definite, decrescent, (1) V E C [ J x S o ,R ! ] and and satisfies 1

v(t,.TI)

Y t , %)I

< MI1x1

-

% II

for ( t , xi), ( t , EJ x . D + V ( t , x ) C ( V ( t , x ) ) , ( t , x )J~ x S,,CE%. (2) C y J


0, G"(r)> 0, and lct cx > 1. Since G (T)=

J: du

G"(v)dv 0

=

0, G'(0)

=

0,

3.6.

173

CONVERSE THEOREMS

and

we have, setting u

w/a,

=

( 1) = - i: dw [w'm G"(v)dv

G-

0



Consequently, observing that ( 1

we have

t,4 11 < 41* 11))

1

t , X)ll)

< G(4l * 11))-

+ ma)/( 1 + a) < a, it follows that

v(t*x) < m G ( 4 x 11)).

Since u

> T ( E )implies 11 x ( t + u,t , *)I1 I1 x (t

if

u

2 T(lj x li/a). Thus, G(ll x(t

which, in turn, leads to G(I/4 t

+

D?

t >.)lo

+

D7

t,

< E,

we get

x>ll < II x llb

+ u>t?x)II) < G(ll x IIbh 1+ I +o< olG *I( 01u

< G(ll x 11)

< V(t,21,

because of relations (3.6.22) and (3.6.23). This shows that

1 74

3

CHAPTER

T h e continuity of the function V ( t ,x) implies that there exists a such that v(t,4 = G(llx(t 0 1 t , "411)1 a(J1.

+

If we let x = x(t, to , xJ, x* of (3.2.1) shows that V(t

Denote cr* 1

1

+h

+ au*

+

+ h, x * )

u*

=

-

-

(5.

(1 (1

1

x(t

=

+

7

+ h, t, x), the uniqueness of solutions

+ + u*, t + h, x*) I )

G(/I~ ( t h

Then

+ au*)(l +

+ U*)(l +

0) -

-

1

(a

(1

-

1 +a,* 1 +u*

+ u* + au + auu* - ah + h

0)

+ au

u1

+ o*)(l + ).

1-

1)h

1 +o

It therefore follows that

< V ( t ,x) [I

-

(a

(1

+ u*)(l -

I)h +am)

1'

using (3.6.23), it is easy to obtain D+V(t,x ( t ) )

=

1 lim sup - [ V ( t h-Q+ h

+ h, x ( t + h, t , x))

-

V ( t ,x)]

(3.6.24)

3.6.

because of the fact that limb+*+ I( x* 11 function. We have seen previously that

where

11 x2 /I < S(p),

175

CONVERSE THEOREMS

=

(1 x (1 and that T ( E is) a decreasing

be such that

so that the solutions x ( t , to , xl), x ( t , to , x2) remain in

Let

s,.

If r2 3 rl , we have G(r,) 3 G(r& and hence

On the other hand, if r2

(3.6.25) Since f(t, x) satisfies the Lipschitz condition, by Corollary 2.7.1, we obtain

Furthermore, choosing

the relation (3.6.25) leads to

using the monotonic decreasing character of T ( E ) It . follows from these relations that

176

CHAPTER

3

and thus that

,,

V ( t ,xl) - aAl/ XI

~

x2 11.

These considerations show that, in all cases, V ( t ,xZ)

-

V ( t ,xl)

2

-

o~i4/lx1 - x2 11.

By interchanging the roles of x l ,x2 , we obtain V ( t ,XI) - V ( t ,x2) 3 --olR/l XI - x2 I/,

and therefore there results (3.6.27)

provided xl,x2 f 0. If xg = 0, (3.6.26) yields 0

< V ( t ,21) < 4 1x1 11,

and hence (3.6.27) is true even when x2 = 0. If x1 = x2 = 0, the relation (3.6.27) is trivially satisfied. As previously, it is now easy to obtain (2) from relations (3.6.24) and (3.6.27) and the descrescent character of V ( t , x). Finally, it remains to prove that we can choose G(r) satisfying the required conditions. For this purpose, we may take G(r) == A

J: exp [-KT(!E1)]

dr.

One can easily verify that G(0)

=

0,

G’((Y)= A exp[--KT(S(r)/a)

> 0,

G’(0) = 0,

0, T(0) = a,G’(r) is monotone increasing, and thus since 6(0) G”(r)exists almost everywhere and is positive. T h e proof is complete. Although we have used Theorem 3.6.9 only t o consider stability properties of perturbed systems, we give below a result that makes such a treatment easier.

THEOREM 3.6.10.

Under the assumptions of Theorem 3.6.9, there exists a function w(t, x) satisfying (1) and D+w(t, x) -w(t, x).


0;


, T, such that

II 4 t J

= 8(€),

II 4t)ll

< S(E),

t

[to

6

ti].

In view of conditions (i) and (ii) and (3.7.5), there results the differential inequality D+m(t)

< -(a

-

+

K€)W(t) K y ( t ) ,

t


1, and the trivial solution of x' = f ( x ) (3.7.8) be asymptotically stable. Then, the trivial solution of the system

(3.7.9)

is exponentially asymptotically stable.

Proof.

Let y(s, s o , x,,) be a solution of (3.7.9) and

s ( t ) being the inverse function of t(s). Set x( t) = y(s(t),so , xo). Then,

3.7.

STABILITY BY THE FIRST APPROXIMATION

183

Furthermore, x(to) = x o , where to = t(so). Since the solution x = 0 of (3.7.8) is assumed to be asymptotically stable, the functionf(x) being autonomous, we have

/I 4t)ll

< 4 xo Il)o(t

-

t

to),

3 to >

where a E X , (T E 9. It therefore follows that

/I Y ( S , so

x0)ll

< 4 xo Il)u[t(s)

- t(s0)l

< 411xo Il)aCP(s - so)] =

4 xo I l ) 4

-

(3.7.10)

so),

where o1 E 9, using the fact that y(u, so , xo) is bounded, and so

From the evaluation (3.7.1 O), the uniform asymptotic stability of the solution y = 0 of (3.7.9) is evident. Clearly, F( y ) is homogeneous in y of first degree. Hence, because of uniqueness of solutions, it results that Y ( &10 , axe)

=

U Y ( % so xo). 9

Moreover, using (3.7.10), we derive that

which implies that a*(.) is linear in u. One can now conclude, on the basis of Corollary 3.6.6 and the facts that a*(.) is linear in u and F ( y ) is homogeneous in y of first degree, that the trivial solution of (3.7.9) is exponentially asymptotically stable.

THEOREM 3.7.7.

Let f E C [ S ,, R"],f ( 0 ) = 0, d)Lf(x) = f ( a x ) , rn > 1, and the trivial solution of (3.7.8) be asymptotically stable. Assume that R E C [ J x S, , R"] and

II Nt,.)I1

< CII x 1Irn,

(1, x)

E

1 x s, ,

(3.7.1 1)

184

CHAPTER

3

C being a sufficiently small constant. Then, the trivial solution of the system x'

=f(.)

+ R(t,).

(3.7.12)

is uniformly asymptotically stable.

Pyoof.

Let x ( t , t o , xo) be a solution of (3.7.12). Define

and let t ( s ) be the inverse function of s ( t ) . Setting y ( t ) = x(t(s),to , xo), it is easy to check that

verifying that y(s ) satisfies the system +ids

=F(Y)

+ R*(s,y),

T h e conditions of Theorem 3.7.1 being fulfilled, it follows that

/I Y(S)ll G k'll X" I/ exp[--or(s whence

-

dl

/I 4 t ( s ) , to , xo)ll < KII2, /I exp[--n(s

(S"

= s(f,)),

-

dl,

and therefore

Since the solution y(s) is defined for all s 2 so , limt+ms ( t ) = 00. This shows that the integral in (3.7.13) is divergent, proving the exponential asymptotic stability of the trivial solution of (3.7.12). T h e theorem is proved.

3.7.

STABILITY BY THE FIRST APPROXIMATION

I85

T h e next theorem is of less general character, which may prove effective in certain concrete cases. T h e importance of the theorem, however, is that a judicious selection of V ( t ,x), reflecting more closely particular properties of the given system, frequently leads to much more precise results rather than yielding to the temptation of choosing V(t,x) as simple as possible, such as V(t,x) = // x /I.

THEOREM 3.7.8. Let the following assymptions hold: (i) There exists a continuously differentiable matrix G(t), which is self-adjoint and positive, that is, the Hermitian form (Gx, x) is positive definite, and A,, A, > 0 are the smallest and the largest eigenvalues of G(t ). (ii) T h e function q E C [ j ,R] is the largest eigenvalue of the matrix G-I(t)Q(t), where

ec4

+ G(t)A(t)+ A*(t)G(t),

=dG(t)

dt

A(t) being a continuous matrix on J and A*(t) its transpose. (iii)

R E C [ J x S,, , R"],and

< P(t)ll x /la,

/I R ( t , .)I1

0, V(t,x ( t ) ) , which implies, in view of Theorem 1.4.1, that z(t)

< J-"J(V(to .")I T

~

(t

- to)],

t

E

[ t o ? tll.

I90

CHAPTER

3

Note that the maximal solution of u' = -C(u), u(t,) = V(t, , x,) is just the right-hand side of the foregoing inequality. Thus, it follows that




4 < B1,

t

2 to

7

< a1 and, for every T > 0, /l:T4(s)

ds

< al.

Since assumption (ii) holds, it is possible to choose a satisfying the relation b(B) b B1 7

p

=

&to, a) (3.9.4)

where p1 is the function occurring in (IF). Evidently, /3 is continuous in t,, for each a and p E X for each to . We claim that, with this p, definition (I,) holds. If this is not true, there would exist a t , > to such that (3.9.5)

194

3

CHAPTER

We extend $ ( t ) continuously for all t 3 to such that

("a(.)

ds

< a1 .

to

> t , , satisfying the inequality

T o do this, it is enough to take t,

t,

-

t,


, E. As in the proof of Theorem 3.9.1, condition (iii), in view of the fact that V ( t ,x) is Lipschitzian, gives

+


0.


0 be such that, if 11 xo I/ < So, then ( t ,x(t, to,x,,)) E J x S,, for t 3 t o . This is possible because (S,) is valid. Define m(t) =

v(t,x ( t , to , xu)) + c

1'I/ to

x ( s , to

3

X~)IIP ds.

3.10.

20 1

LP-STABILITY

Then, condition (ii), in view of Theorem 3.1.3, gives the inequality m(t)

< m(to),

2 to,

t

from which there further results ..W

J

to

/I x(s, t” > x0)Il” ds

It is clear that the null solution x

=


xo).

0 is equi-Lp stable.

THEOREM 3.10.3. Let the assumptions of Theorem 3.10.1 Suppose further that V ( t ,4

< 411xll),

( t ,4 E

f

x

s, ,

a

hold.

C3Y.

Then, the uniform-L1 stability of the solution u = 0 of (3.2.3) implies the uniform-LP stability of the solution x = 0 of (3.2.1). Proof. By the assumption, the null solution of (3.2.3) is uniformly independent of to such that (3.10.3) holds stable, and there exists a uniformly in to whenever u, 8,. Consequently, the uniform stability of the solution x = 0 of (3.2.1) follows from Theorem 3.3.4. T o prove that (L,) holds, we follow the proof of Theorem 3.10.1 and we choose uo = u(\\ xo I]), thereby deducing So = u - ~ @ , ) . I t is evident that 6, is independent of to and the integral (3.10.1) is uniformly convergent in to . This proves the theorem.

so


0.

(3.10.5)


0, ( t ,x) E J x S o . (ii) D+V(t,x) Then, the null solution of (3.2.1) is uniform-LP stable. Proof.

Assumption (ii), in virtue of (3.10.5), reduces to

D V ( t ,4 < g(t, v t , XI), where g ( t , u) = -Cu/B, and hence it is easy to check that the solution u = 0 of (3.2.1) is uniform-L1 stable. Now, the assertion of the corollary is a consequence of Theorem 3.10.3.

202

CHAPTER

3

Although L?-'stability and asymptotic stability are different concepts, under certain conditions Lr' stability implies asymptotic stability, as shown in the following:

THEOREM 3.10.4. Let the trivial solution of (3.2.1) be let there exist a constant M > 0 such that

Then, the trivial solution x

=

L p

stable, and

0 of (3.2.1) is asymptotically stable.

Proof. Assume that there is a solution x(t, t o , xo) of (3.2.1) such that I/ xo I/ &,(to) and Iimi+%x ( t , t o , xo) # 0. Then, there exists an E > 0 and a sequence {t,;},t,; + co as k co such that


~ o ) / l

Y O Il)u(t -

3 t o , p E .X, and u E 9. Let R(t,Y ) = H ( t , x(t)7 Y )

~

H ( t , 0, Y ) .

to)

(3.11.12)

3.12.

STABILITY OF DIFFERENTIAL INEQUALITIES

209

Then, because of (3.11.10) and (3.11.12), we deduce that

/I R(t,Y)ll ,< KP(ll xo /I + I1Yo Il)u(t - to).

(3.11.13)

Consider the perturbed system Y'

=

H(t, 0,Y)

+ R(t,Y),

(3.11.14)

and let V(t,y ) be the Lyapunov function constructed according to Theorem 3.6.9. If 11 y o 11 < S(8,) = po , it follows that Dfv(t,y)(3.11.14)

+

< -c[v(t,Y)l +

R(t,y)ll,

(3.11*15)

where C E X . Let 11 xo 11 11 y o 11 < So , where 8, is the number occuring in the definition of partial uniform asymptotic stability. It is easy to deduce, taking into account relations (3.11.13) and (3.1 1.15), that D+v(t, Y ) ( 3 . 1 1 . 1 4 ) < dt,v(t, Y))l

where

g(t, ).

= -C(U)

+ MKP(So)o(t).

Notice that R(t, y ) satisfies a Lipschitz condition in y because of conp < po be given, and let KI(ci,p) = $C(U). dition (i). Let 0 < 01 T h e fact that u E 2 shows that there exists a O(N, p) 3 0 such that


, t" i- 7';

3.13.

213

BOUNDEDNESS AND LAGRANGE STABILITY

(B4)quasi-uniform-ultimately bounded if the T in (BJ is independent of t o ;

(B5)equi-ultimately bounded if (B,) and (B3)hold at the same time; (B,) uniform-ultimately bounded if (B,) and (B4)hold simultaneously ; (B,) equi-Lagrange stable if (23,) and (S,) hold simultaneously; (B,) uniform-Lagrange stable if (B,) and (S,) hold simultaneously;

PROPOSITION 3.13.1. If f ( t , 0) = 0, t E J , and p occurring in (B,) and (B,) has the property that /I+ 0 as 01 + 0, then the definitions (B,), (B,) imply the definitions (S,), (S,), respectively. T h e proof of the statement is obvious.

PROPOSITION 3.13.2. boundedness if

Quasi-equi-ultimate boundedness implies equi-

llf(t, .)I1

G g(t, II x

(3.13.2)

II)7

where g E C [J x R, , R,].

Proof. Consider the function m ( t ) = /I x ( t , t o ,xo)ll, where x ( t, t o , xo) is any solution of (3.13.1). Then, D+m(t)

< It

X'(t,

to , xo)ll

= Ilf(t,

x ( t , to , %))It

< g(t, m(t)),

using assumption (3.13.2). By Theorem 1.4.1, we have

whenever

1) xo 11

< a , where r ( t , t o , a ) is the maximal solution of u' = g ( t , u ) ,

(3.13.4)

u(to)= a.

By the quasi-equi-ultimate boundedness, given 3 0 and to E J , there exist two positive numbers N and T = T(to, a ) such that the inequality 11 xo 11 < 01 implies (I:

Since g(t, u ) 2 0, the solution r ( t , t o , a ) of (3.13.4) is monotonic nondecreasing in t , and therefore we have, from (3.13.3), that

/I x ( t , to

1

xo)ll

G r(to

+ T , to

7

a),

t

6

[to , to

+ 77-

214

3

CHAPTER

It then follows that

/I x ( t , to , xJl

+

< max",

~ ( t , T , t o , a)],

t

3 to,

and this proves (&). Analogous to the group of definitions (Ell)-(&), we can define the concepts of boundedness and Lagrange stability with respect to the scalar differential equation (3.2.3) and designate them by (BT)-(B$).

THEOREM 3.13.1. Assume that there exist functions V ( t ,x) and g(t, u ) with the following properties: (i) g E C [ J x R, , RI. (ii) V EC [ J x R", R,], V ( t ,0) = 0, V ( t ,x) is locally Lipschitzian in x, and, for ( t , x) E J x Rn, V ( t ,4 3 4 1x I/),

where h E N on the interval 0 u + 03.

(iii)

D+V(t,x)

< g(t, V ( t ,x)),

(3.13.5)

< u < co

and b(u) + co as

( t , x) E J x R".

Then, the equiboundedness of Eq. (3.2.3) implies the equiboundedness of the system (3.13.1).


30)

< a1

together. Assume that Eq. (3.2.3) is equibounded. Then, given al 2 0 and t, E J , there exists a PI = Pl(to, a ) that is continuous in to for each rn such that r(t,t o ,

U")


70,

providcd 11 xo 11 6, ; (E,) eaentzially uniformly asymptotically stable if (E,) and ( E 2 ) hold simultancously; (E,) eventually exponentially asymptotically stable if there exist constants L 5 0, (Y > 0 such that

/I

r(f7

t o , .xo)II

-< LII xu /I c T b ( t

-

to)],

t 3 to,

(3.14.1)

d ( ~ ) where , d(r) is a monotonic Y provided 0 / ( x, I/ -, p and to decreasing function of Y for 0 . Y < p.

KFMARK 3.14.1. Notice that, if (E,) holds and if x = 0 is a trivial solution of (3.2. l ) , then the uniform Lyapunov stability (S,) results from the continuity of solutions with respect to the initial values, provided the unicity of solutions of (3.2.1) is assured. Similarly, ( E 3 )

3.14.

223

EVENTUAL STABILITY

implies, in such a case, uniform asymptotic stability of the trivial solution of (3.2.1). As usual, let us denote by (ET)-(E$) the corresponding notions of the set u = 0 with respect to the differential equation (3.2.3).

THEOREM 3.14.1. Assume that there exist functions V(t,x) and g(t, u ) verifying the following properties: (i)

I/ E

C [ J x S o ,R,], V ( t ,x) is locally Lipschitzian in x, and

4 x 11)

< v(t,4 < 4 x II),

for 0 < Y < 11 x 11 < p and t >, O(Y), where a, 6 E X and B(Y) is continuous and monotonic decreasing in Y for 0 < Y < p. (ii) g E C [ J x R,., R], and the set u stable with respect to (3.2.3).

=

0 is eventually uniformly

(iii) f E C[J x S, , R"],and D+V(t,4

for 0

< Y < 11 x I/ < p

Then, the set x system (3.2.1).

=

and t

< g(t, V ( t ,XI),

3 O(r).

0 is eventually uniformly stable with respect to the

Proof. Let 0 < E < p. Since the set u 0 is eventually uniformly stable, given b ( ~ > ) 0, there exist a 6, = 6 , ( ~ )> 0 and T , = T , ( E ) > 0 such that (3.14.2) u(t, t o , uo) < b ( E ) , t > t o > TI(€), 1


t, > to 3 T ,

II ~

( t, ,to , xo)ll = 6,

I/ x ( t 2 , to , x0)li = E,

and 8

< II x ( t , to > x0)Il < E ,

t

E

(tl

,tz).

Choose u,, = a(\\x1\I), where x1 = ~ ( t, ,to , x,,). Then, condition (iii) and Theorem 3.1 1.1 show that V ( t , y ( t ,tl

7

XI))

< r(t,t,

>

uo),

t

t

[tl

, f21,

(3.14.3)

where y ( t , t , , xl) is any solution of (3.2.1) through (t, , xl), r ( t , t , , u,,) being the maximal solution of (3.2.3) through ( t l , uo). Thus, (3.14.3)

224

CIIAPTER

3

is also true for x(t, t,, , x,,) on the interval t , obtain h(E)

< f’ftz,

< Y(t,

4 t , , t o , 4)

7

< t < t, . We

therefore

t , 3 uo) < @ E ) ,

taking into account the uniformity of the relation (3.14.2) and the fact t, t , > t, 3 T. This absurdity that we are led to prove ( E , ) is true, and the proof is complete. ‘ 2

COROLLAKY 3.14.1. If, instead of the eventual uniform stability of the set ti - 0, it is assumed that the trivial solution u = 0 is uniformly stable, the conclusion of Theorem 3.14.1 remains the same. I n particular, g ( t , ZL) -= 0 is admissible.

THFOREM 3.14.2. Suppose that there exists a function V ( t ,x) such that V E C [ J Y S,,, R-1, V ( t ,x) is Lipschitzian in x for a constant (1) L > 0, and h(l’

li)

+ 17(t, x) < .(I1

x II),

for 0 Y ,/ / s I/

0 and to E J be given. Because of condition (3.15.3), we have, if E is small enough,

232

CHAPTER

If we choose

11 xo 11

J

3

6, , where K,S,B

< ,

K , ~ ( Eand )

then wc: get 11 .r(t)il 0,

IIfAG x) - f d t , 0111 < LII x II.

(3.1 5.7)

If y ( t ) = y ( t , t o ,x,,)is the solution of the variational system -Y’

=f,(f,X ( ~ ) ) Y >

to)

(3.15.8)

= xg,

where x ( t ) = x ( t , t,, , xo) is the solution of (3.15.2), I/ xo 11 being sufficiently small, then linit+Ty ( t ) = 0. L’yoof.

Let us first observe that 1)’ l’(f>Y)(3.15.8)

< Di b’(f>Y) + KZllfdf,4 t ) ) - f A t ,

0)Il IIY II.

If we now set m(t) = V ( t , y ( t ) ) ,we readily obtain, in view of (3.15.7) and condition (ii), the inequality D.44

< a ( t ) m ( t )+ LKII 4t)lI I/ Y(t)ll.

(3.15.9)

3.15.

233

ASYMPTOTIC BEHAVIOR

Since the hypotheses of Theorem 3.15.3 hold, if 11 xo 11 is small enough, we have // x(t)/\ < E, t >, t o . Consequently, choosing I/ x, // sufficiently small and using the relation Kl 11 x 11 V ( t ,x), it follows that

< D+m(t) < [a@) + L 551 m(t), Kl

which leads to the estimate

for t >, t o . If E is small enough, the condition (3.15.3) assures that lim,,,y(t) = 0. We shall next consider a theorem on the dependence of solutions on the initial values, which is useful in what follows.

THEOREM 3.15.5. Suppose that f ( t , x) is continuous on an open set D in J x Rn and that every solution of (3.15.2), (to , xo) E D is continuable to t = t, > 0. Let E be the set of all the points consisting of the solution curves for [ t o ,t J starting from ( t o ,xo), and let E be contained in a compact set in D.Then, to each E > 0, there exists a S > 0 such that every solution x*(t, t$, x;), t$ E [ t o ,t l ] , of x'

=At,).

+ g(th

(3.15.10)

< 8,

(3.1 5.1 1)

where g E C [ J ,R] and satisfies f ; d s ) ds

passing through [tf, t,] and obeys

p*

=

(t?, x?) such that d(p*, E )

I1 X * ( t ,

to*, x,+) - x ( t , to 9 xo)ll

< 8,

exists on

< c,

x ( t , t o ,xo) being a solution of (3.15.2) contained in E, which may ,):x depend on x * ( t , t z .

Proof. Suppose that, for some E > 0, there is no S such that it satisfies the condition in Theorem 3.15.5. We may assume that U ( E , c) C D, -~ where U ( E , 6) = [x: d(x, E ) < €1. Since U(E, c) is a compact set, there is a functionf*(t, x) that is continuous and bounded on (~ 03, 03) x Rn -~ and is equal to f ( t , x) on U ( E ,c). A solution of (3.15.2) remaining in U(E,E) is a solution of x' = f * ( t , x),

(3.15.12)

234

CHAPTER

3

and the set of all the points consisting of the solution curves for t E [to, tl] coincides with E. We may therefore assume that, for E > 0 and the equation of (3.15.12) through ( t , , x,)

s'

(3.15.13)

= f * ( t , x) + g ( t ) ,

the conclusion of the theorem is not verified. Every solution of (3.15.13) exists for all t. By hypothesis, there are a sequence of points {pr = (t,, , xJ} and a sequence of functions {gx(t)}such that d(p, , E ) tends to zero,

J;)mdt

+

0

as

k

+

m,

< t < t , , of

and a solution $ h ( t ) ,t,&

x' = f * ( t , x)

+ g&)

through p,> such that there is no solution curve of (3.15.12) lying in E with the property that the distances of all the points on the arc of the former to the latter are smaller than E . Since $,,(t)is defined on t,, t t , ,

<
0, there is a T > 0 with the property that, for all t > T , the points x(t) are contained in S ( A , c).

THEOREM 3.15.6.

Assume that the functions f ( t , x), R(t, x), and V ( t ,x) satisfy the following conditions for ( t ,x) E J x E:

(i) f E C [ J x E, Rn], and f ( t , x) is bounded for all t E J when x belongs to an arbitrary compact set in E. (ii) R E C [ J x E , R"], and, if x ( t ) is continuous and bounded on to t < co, that is, x ( t ) CQ, Q being a compact set in E, then R(t, x) satisfies the inequality


f,

t

.

If we suppose that this solution does not approach Q as t --t co, then, for some E > 0, there exists a sequence {tr>, t, co as k + 00 such that --f

.\(t,, t,, , x,) E S(Q, E)" n Q.

T h e assumption that f ( t , x) is bounded when x E Q assures that llf(t, .v)II


t, , we have

/I 4 t l ) -Y(tdll where, we may assume, B -: p such that

E

=

< t , , there exists a constant

< p. For to < t

II 4t)ll

< B,

€7

IIY(t)ll

< B.

By Lemma 1.3.1, given $I(€) and a compact set a(€) > 0 such that p(t,

T o , 0 , 6)

< Me),

t

E [To,

, TI, there exists

[ T ~

a6 =

TI,

(3.18.2)

where r ( t , T~ , 0, 8) is the maximal solution of u' = g(t, u )

+ 6,

U(T0) = 0.

(3.18.3)

Let 8 be a 6(~)/2M-translationnumber for f ( t , x) such that to + 6 0, that is, Ilf(t 87-4 - A t , .)I1 < 6(€)/2M, (3.1 8.4)

+

provided x E a compact set S C S,. Consider the function m(t) = V ( t 8, x ( t ) , y ( t ) ) for t E [ t o , t,]. For small h > 0, we have, using Lipschitz condition on V ( t ,x,y ) ,

+

m(t -1- 12)

< Mh [lIJ(t'x(t)) - f ( t

+ Y [ t + 8 + h,

X(t)

+ 0, x(t))ll

+ hf(t + 0, x ( t ) ) , Y ( t ) + hf(t + 0, y(t))],

where t,(h)/h,~ , ( h ) / h+ 0 as h -+ 0. It then follows, because of assumption (iii) and relation (3.18.4), that D'm(t)

< g(t + 0, m ( t ) ) + 6,

t

E

[t" , tll.

3.18.

Defining ro = to inequality

+

ALMOST PERIODIC SYSTEMS

247

+ 8, we get, on the strength of

Theorem 1.4.1, the

4 t )

< r(t + 8,

70

0, a),

9

t

[to , tll,

E

where r(t 8, T ~ 0,, 6) is the maximal solution of (3.18.3). At t we obtain, using relations (3.18.2) and (3.18.5), the estimate 4tl)

d

r(t1

+ 8,

70

9

0,q

(3.18.5) =

t,

,

< ib(€),

which is a contradiction to the fact that m(t1)

3b(E).

Hence, it follows that the system (3.18.1) has at most one solution to the right of to .

COROLLARY 3.18.1. T h e function g(t, u) = 0 is admissible in Theorem 3.18.1. As observed earlier, in the stability results that follow, we allow to E (- GO, GO). Then, the corresponding notions will be designated as perfect stability concepts to distinguish them from the previous stability definitions. We need the following notions with respect to the scalar differential equation (3.2.3).

DEFINITION 3.18.1. The trivial solution of (3.2.3) is said to be strongly equistable if, given any E > 0, r,,E J , and any compact interval K = [T,,,tl], there exist an 7 = V ( E ) > 0 and a positive function 6 = 6(r0 , E ) that is continuous in T,,for each E such that, if u,, 6,


0, and, for ( t , x, y ) E J x Sox S, ,

< v t 9 x,Y ) < 4x D+V(t, x,y ) < g(t, V ( t ,x,y ) ) , b(ll x

-

Y

II)

~

Y

ll),

a, ZJ

E

-x;

where t E J , x, y E S, , (iii) g E C [ J x R, , R l ; (iv) there exists a solution x ( t ) = x(t, t o , xo) of (3.18.1) such that

I/ x(t)II < B,

t

3to,

t o E (-a, a), B

0, and T~ E J , there exist positive numbers (v) given b ( ~ > T ( E ) , T = T ( Ea , ) such that, if uo N and T T~ T,

7 =


t o ] ;

(iii) f E C [ J x RT1, R"],x ( t ) is a 6-approximate solution, and y ( t ) a solution of (3.19.3), defined for t 2 t o . Then, V ( t , , x,,

~

y o ) < u(to)implies V ( t ,~ ( t- )y ( t ) )

cc ~ ( t ) ,

t

to

.

3.20.

CONTINUOUS DEPENDENCE AND THE METHOD OF AVERAGING

257

Proof. Consider the function m(t) = V(t,x(t) - y(t)). If, for a t = t, , x ( t l ) , y(tl) E 8, then we obtain, using assumption (ii), the differential inequality D+m(t,)

0 such that

E

V(t,x)

0

?

<
0, a constant 6 = S ( E ) > 0 such that, whenever v ( t ) is a step function in [0, To] with v(0) = xo(0) and 11 v(t) - x,(t)ll < 6 in [0, To],there is a neighborhood F = F ( E )C Rm of y o for which y E F implies

I n view of (3.20.4), this means that, for each y

ll F ( 4 v(t),Y ) - xo(t)l/ < E ,

t

E

r,

E

10, To].

This fact will be used prominently in the following

THEOREM 3.20. I .

Assume that

(i) V E C [ J x R’l, R,], V ( t ,0) = 0, and V ( t ,x) is positive definite and satisfies the Lipschitz condition in x for a constant M > 0; (ii) g E C [ J x R+ , R ] ,g(t, 0) = 0, and r ( t ) = 0 is the maximal solution of u’ = g(t, u ) , (3.20.8) passing through (0, 0);

260 t

3

CHAPTER

(iii) for any step function v(t) on J , with values in S, , and for every J , x E S, ,y E RJJ1,

E

' U(t,x, r) < g(tt q t , v ( t ) - 4);

(3.20.9)

I)

(iv) the relation (3.20.7) holds. Then, given any compact interval [0, To]C J and any E > 0, there exists a neighborhood T ( E )of yo such that, for every y E (3.20.1) admits a unique solution x ( t ) with x(0) = xo(0), which is defined in [0, To]and satisfies

r,

II x ( t )

~


= T(y)l c J. From hypothesis (ii) and Lemma 1.3.1, we deduce that, given any compact interval [0, To]C J and any p > 0, there is an q = q(p) > 0 such that the maximal solution r ( t , 0, 0, q) of u' = s(t, u )

exists for t

E

+ $7

(3.20.1 1)

[0, T , ] and satisfies r(4 o,o, 7)


) 0 such that, whenever V ( t ,x) p, we have 11 x 11 < Let q(c) > 0 be the constant referred to previously. Choose a constant OL > ME. By the continuity of g on x R , , there exists a S ( E ) > 0 such that

4.

1

I

%)

- R(t3

%)I

< '27

for t E [0, T o ]u1 , , u2 E [0, n], and I u1 - u2 1 < S(E). For every y E Rm and every step function v in J with values in S, , we have, for every t E J and x t P i ,

I

L'(t,

U ( t ) - 2) -

V(t,r;.(t,.(t),y)

Mlli 4 t ) - %(t)ll

~

)I.

+ II q t , .(t), Y )

-

%(~)lll9

where F ( t , v(t), y ) is defined as in (3.20.4). Hence, as observed earlier, we can select a positive constant P ( c ) < E and a step function v in [O, To]

3.20.

CONTINUOUS DEPENDENCE AND THE METHOD OF AVERAGING

261

with v(0) = xo(0) and 11 v(t) - xo(t)lI < fi in [0, To] such that there is a neighborhood T ( E of ) y o for which y E T ( Eimplies ) and

II F(t, 4 t h Y ) - xo(t)ll < BE,

t

E

[O, To],

(3.20.12)

I q t , v ( t ) - 4 - V(t,F(t,W , Y ) - 4 < 8, for every t E [0, To] and x E Rn. Let us now take some y E T ( E )and , let us consider the unique solution x ( t ) of (3.20.1), with x(0) = xo(0), which exists on some interval J ( y ) = [0, T ( y ) ]contained in J . Defining for t E J ( y ) n [0, To], m(t)

=

V ( t ,F(t, +), Y ) - x(t)).

We deduce from (3.20.9) that DWt)

< g(t, q t , u ( t )

-

x(t))).

Hence, for all those t E J ( y )n [0, To] for which

This implies, by Theorem 1.4. I, that

where r ( t , 0, 0, 7) is the maximal solution of (3.20.1 I ) through (0, 0). Since r(t, 0, 0, 7) < p for every t E [0, To], we infer that m ( t ) < p as long as (3.20.13) holds, and therefore

Thus, using (3.20.12), we obtain, for sufficiently small t E J ( y ) n [O, To],

1144 - xo(t)ll
0 such that, for each t t [0, To],the open ball B , in R" of center x,(t) and radius y is contained in S o , we shall have C E S, provided E > 0 was chosen sufficiently small. Therefore, x ( t ) can be continued as a solution of (3.20.1) to the compact interval [0, T * ] ,which contradicts the definition of ]*. This completes the proof. If the assumption (3.20.7) is replaced by the stronger requirement lim f ( 4 x,Y ) = f ( t ,x,Y") T,,

Y

uniformly in J x S , , then we can prove the conclusion (3.20.10) without the use of approximating step functions. This we state in the form of a corollary, observing that it is a generalization of Theorem 2.5.2. COROLLARY 3.20.1. Let assumptions (i) and (ii) of Theorem 3.20.1 hold. Suppose that, for each t E 1,x1 , x2E S , , and y E Rm,

Then, the conclusion of Theorem 3.20.1 is true. \.lie next consider the problem of continuity of solutions with respect to initial values. We first prove the following

LEMMA 3.20.2.

Suppose that

(i) I;E C [ ] x R", R+],and V ( t ,x) satisfies a Lipschitz condition in x locally; (ii) f E C [ ] x R", R"], and G(t,m )

=

max

Y(t,x-r")

m

D+V(t,x

-

x,,),

3.20.

CONTINUOUS DEPENDENCE AND THE METHOD OF AVERAGING

263

where D+V(t,x

(iii)

-

1 h

x,,) = lim sup - [V(t + h, x h-Of

x,,

~

+ hf(t, x)))

-

V(t,x

- xO)];

r * ( t , t o , 0) is the maximal solution of U' =

G(t,u),

~ ( t ,= ) 0

existing for t >, t o . Then, if x ( t ) is any solution of x),

x' = f ( t ,

existing for t

x(to) = xg

(3.20.16)

2 t o , we have V(t,x(t)

- xo)

< Y*(t,

Proof. Define m ( t ) = V ( t ,x ( t )

-

t" , 01,

=

*

xo). Then, it is readily seen that

D+m(t) < D+V(t, x ( t )


).

< un . Hence, by Corollary 1.7.1, we have m(t)

< r ( t , to

7

uo)

as far as x ( t ) exists to the right of to , proving the desired relation (4.1.6). We can now state a global existence theorem analogous to Theorem 3. I .4.

4.1.2. Assume that b' E C [ J x Rn, R + N ] , V ( t ,x) is locally Lipschitzian in x, and Cr=l Vi(t, x) is mildly unbounded. Suppose that g E C [ J x R+N,R N ] ,g(t, u ) is quasi-monotone nondecreasing in u for each fixed t E J , and r ( t , t, , uo) is the maximal solution of (4.1.4) existing for t >, to . I f f € C [ J x R7L,Rn] and 'rlEOREhl

D+V(t,x)

< g(t, I'(t, x)),

( t , x) E J x R",

4.2.

ASYMPTOTIC STABILITY

269

then every solution x ( t ) == x ( t , to , x), of (4.1.1) exists in the future, and (4.1.5) implies (4.1.6) for all t 3 to . By repeating the arguments used in the proof of Theorem 3.1.4, with appropriate changes, this theorem can be established. On the basis of Corollary 1.7.1 and the remark that follows, we can prove the following:

THEOREM 4.1.3. Let V E C [ J x S, , R+N] and V ( t ,x) be locally Lipschitizan in x.Suppose that g, ,g, E C [J x R+N,RN], gl(t, u), g2(t,u ) possess quasi-monotone nondecreasing property in u for each t E J , and, for ( t , x) E J x S , ,

< D-'V ( t ,x) ,< gz(t, V ( t ,4).

gdt, V ( t ,4)

Let r ( t , to , uo),p(t, t o , vo) be the maximal, minimal solutions of u' = gz(t,

4,

v' = gl(C v),

4 t " ) = uo , v(t0) = U o ,

respectively, such that

Then, as far as x ( t ) = x(t, t o , xo) exists to the right of to , we have p(t, t o

,4

< V ( t ,4 t ) )
8 for

tt

/ / x 11

p and d(x, E ) < 7, t

,

R

=

2 0, where

[x E s, : W(.)

= 01

and d(s, E) is the distance between the point x and the set E. Then, the trivial solution of (4.1.1) is uniformly asymptotically stable.

0 and t,, E J be given. Since V l ( t ,x) is positive definite Proof. Let t and decrescent, there exist functions a, b E .f such that A ;

We choose 6

6 ( ~ so ) that

:

h ( € ) > a(6).

(4.2.2)

Then, arguing as in the first part of the proof of Theorem 3.4.9, we can conclude that the trivial solution of (4.1.1) is uniformly stable. 1,ct u s now fix t p and define 6, = S(p). Let 0 'c E p, to E J , and S -= S(t) be the same 6 obtained in (4.2.2) for uniform stability. Assume that 11 A,, 11 .S,, . T o prove uniform asymptotic stability of the solution .T = 0, it is enough to show that there exists a T = T ( E )such that, for some t* E [t,,, t,, 7'1, we have ~

+

I1 x(t*, t"

9

x0)ll

< 8.

This we achieve in a number of stages: (1)

If d[x(t,), x ( t 2 ) ] > Y

> 0, t , > t , , then Y

< M?P(t,

-

tl),

(4.2.3)

4.2. where IIf(t, x)II

ASYMPTOTIC STABILITY

< M , ( t , x) E J

x S, . For, consider

I "Atl) - xi(tz)/< fz I x:(s)l

ds ,
t, ,

D+m(t) b D Vz(t,x ( t ) ) > 6,

because of condition (iii) and the fact that V2(t,x) satisfies a Lipschitz condition in x locally. Thus, m(t) - m(tl) =

j t D-Im(s)ds, tl

and hence m(t)

+ m(Q

2

it

D'-m(s)ds

fl

>at

2 f lD+V,(s, x(s))ds

- tl)

as long as x ( t ) remains in U. This inequality can simultaneously be realized with m ( t ) L only if




d(x, E )

=

J

(4.3.2)

a, 7, t

E

J, (4.3.3)

where Then, thc trivial solution of (4.1.1) is unstable.

Proof. T h e proof of this theorem closely resembles that of Theorem 4.2.1, and hence we shall be brief. Suppose that, under the conditions of thc theorem, the trivial solution is stable. T h a t is, given 0 E p, t , E J , there exists a 6 > 0 such that I] x, I/ < 6 implies 6, t 3 to . II x( t , t,, , %,)I1 According to assumption (ii), a point ( t o ,x$) can be found such that I/ .x(,’ / / 6 and tT1(t,,, x$) > 0. We shall consider the motion x(t) = a ( t , t,, , x;) and its properties: I

‘

( I ) d ( A ( t ) , x ( T ) ) 3 7 , t > T ; then t from ( 1 ) in the proof of Theorem 4.2.1. (2)

For every t

- T

2 q/Mn1/2.This

is clear

3 t, , there will be a positive number 01 such that 01

< [I x(t)ll < E < p .

(4.3.4)

4.3. INSTABILITY

275

This is compatible with the assumption of stability, that is, 11 x(t)ll t 3 to . However, since D+V,(t, x) 3 0, it follows that V1(4

< E,

40) 2 Vl(t,, .,*I > 0.

Since Vl(t,x) is decrescent, for numbers V l ( t o x$) , > 0, a number a: > 0 can be found such that, for all t 3 t o , (1 x 11 a, we shall have


that (4.3.2) and (4.3.3) hold. ) ,) < 7, then a t* > 7 can be found such that (3) If ~ ( x ( T E d(x(t*),E )

=

(4.3.5)

7.

Suppose that d(x(t), E ) < 7 for all t 3 7. Letting m ( t ) = Vz(t,x(t)), we obtain, using the Lipschitzian character of V2(t,x) in x,the inequality D+m(t) 2 D'Vz(f,

44) >, U t ) ,

and hence

Since Vz(t,x) is assumed to be bounded, the relation (4.3.2) shows that d(x(t),E ) < 7 cannot hold for all t 3 7. Hence, there exists a t* > T such that (4.3.5) is satisfied. ) ,) < 7 / 2 , then, for t = t*, when d(x(t*), E ) = 7, (4) If ~ ( x ( T E we have Vl(t*,x ( t * ) ) 2

where 7

= inf[w(x),

~

01

< 11 x /I < p, d(x, E ) 2 $71 > 0.

In fact, under the given conditions,

T

d(x(f*),E )

and, for t ,

ds,

rl < t** === t* - 2Mn1I2

and E

4.))+ 6 J t** C&) t*

Vl(T,

< t < t*, we shall have 4,

< t , < t* can be found such that = $7,

< d(x(t),E ) < 7.

216

4

CHAPTER

Hence, by (iii), it follows that 4 t ) ) 3 4At) w ( x ( t ) )3 E # b ( t ) ,

D"'l(t,

using the fact that V,(t, x) is locally Lipschitzian in x, and, consequently,

Observing, however, that d(x(t*),x(t*)) >, $77, we get, in view of (I), that

( 5 ) There is no number t,

have

t,, such that, for all t

d(x(t),E ) 3

=

would

11.

> t, , we should have

Indeed, if such a t , exists, then, for all t 1 71(t,x(t))

> t, , we

+ J" n+Vl(s, x(s)) ds

V,(tl , ~ ( t , ) )

tl

Ry (4.3.1), this implies that V,(t, x ( t ) )+ co as t -+ co, which is absurd because of the relation (4.3.4) and the fact that V,(t, x) is decrescent. Thus it follows that, for any t:, a T ~ > , t: can be found such that

,

d(x(72+1),

< frl,

and, according to ( 3 ) , there corresponds a tz, > T ~ + ,satisfying d(x(tz*,,),E )

= rl-

1,et us consider the infinite sequence of numbers to

< T I < t: < '.' < Tz < t,* < "'.

I n view of assumption (iii) and (4), we have


0 and to E J , there exists a positive function 6 = 6 ( t o ,6) that is continuous in to for each E such that x ( t , to , xo)

provided

c

S(C),

t

3 to ,

x0 E S(8)n MG-k) .

Evidently, if k = 0 so that M(7L-!L) = R",definitions (Cl)-(C16)coincide with the stability and boundedness notions ( S1)-(S,) and (B1)-(B8). Analogous to the definitions (C,)-(C,,), we need some kind of of conditional stability and boundedness concepts with respect to the auxiliary differential system (4.1.4). Perhaps the simplest type of definition is the following. DEFINITION 4.4.2. T h e trivial solution of the system (4. I .4) is said to be (CF) conditionally equistable if, for each E > 0, to E J , there exists a positive function S = S ( t o , 6) that is continuous in t,, for each E such that the condition

C uio < 6, N

i=l

and

uio = O

(i = I , 2,..., K)

218

CHAPTER

implies

c N

% ( t , t o , %)

4

< 6,

t

2=1

> to

'

Definitions (Ca )-( C&) are to be understood in a similar way.

THEOREM 4.4.1.

I'\ssume that

(i) g E C[/ x R+N,K N ] ,g(t, 0) = 0, and g(t, u) is quasi-monotone nondecreasing in 11 for each t E 1; (ii) V E C [ J x S o ,R+N], V ( t ,x) is locally Lipschitzian in x, CL, Vi(t,x) is positive definite, and

C T'i(t, x) N

--f

0

as

/ / x / ---f (

0 for each t~ J ;

i-I

(iii) Vi(t, x) = 0 (z' = I , 2,..., k), k < n, if x E M(n--k), where M(,,_,, is an ( n -- k) dimensional manifold containing the origin; (iv) f E C [ J x S o ,R " ] , f ( t ,0) = 0, and I ) ; q t , x)

< g(t, V ( t ,x ) ) ,

( t , x) E

J x

s,.

Then, if the trivial solution of (4.1.4) is conditianally equistable, the trivial solution of the system (4.1.1) is conditionally equistable. Proof. Let 0 :' E < p and t , , J~ be given. Since positive definite, there exists a b E .X such that b(l/x 11)

N

0 and t,, E ], there exists a 6: 6 ( t , , E ) that is continuous in to for each E , so that (4.4.2)

provided

c u,o < 8, v

u,o

=0

(i

=

1 , 2 )...,k ) .

(4.4.3)

a=1

Let us choose ui0 = V , ( t o ,xo) (i = I, 2, ..., N ) and x,,E M(n-fc)so that 0 (z' = 1, 2, ..., k), by condition (iii). Furthermore, since Cf=l V L ( tx,) + 0 as 11 x 11 + 0 for each t E J , and V ( t ,x) is continuous,

uio =

4.4.

279

CONDITIONAL STABILITY AND BOUNDEDNESS

it is possible to find a 6, verifying the inequalities

=

S,(t, , e ) that is continuous in to for each

E,

(4.4.4) simultaneously. With this choice, it certainly follows that x0 E S(8,) n M(n-k)

>

implies x(t, to , xo)C S(E),t to . If this were not true, there would exist a t, > to and a solution x ( t , t o ,xo) of (4.1.1) such that, whenever xo E s(S,) n M(n-lc), we have x(t, t o , xo) C S ( E ) , t E [to , t,), and x ( t , , to , xo) lies on the boundary of S ( E ) This . means that

II 4 4

to

7

> x0)Il


.")I1

= €7

t

P7

E

[to

9

t119

and, consequently, N

(4.4.5)

Moreover, for t E [to , t J , we can apply Theorem 4.1 .I to obtain V( t,x ( t , t o , xo))

< r ( t , to ,uo),

[to

t

7

tll,

where r(t, to , uo) is the maximal solution of (4.1.4), which implies that N

N

1 V,(t,x ( t , t ,

7

xo))

< 1 rdt, to ,uo),

t

i-1

i-1

6

[4J,tll.

(4.4.6)

Notice that, from the choice uio = V i ( t o ,xo) and the relation (4.4.4), xo E s(6,) n M(n-k)assures that (4.4.3) is satisfied. Hence, (4.4.2) and (4.4.6) yield the inequality

c N

xo))

ri(t1

i=l

, t"

7

uo)

, t o , provided xo E s(6,) n M(n-k), and the theorem is proved.

THEOREM 4.4.2.

Let assumptions (i), (ii), (iii), and (iv) of Theorem 4.4.1 hold. Suppose further that

c Vi(t, N

i=l

x) + 0

as

11 x 11

--t

0 uniformly in t .

(4.4.7)

280

CHAPTER

4

Then the conditional uniform stability of the solution u = 0 of (4.1.4) guarantees the conditional uniform stability of the trivial solution of (4.1.1).

Proof. By definition ( C z ) , it is evident that 6 occurring in (4.4.3) is independent of t o . I n view of (4.4.7), this makes it possible to choose 6, also independent of t o , according to (4.4.4). Noting these changes, the theorem can be proved as in Theorem 4.4.1.

THEOREM 4.4.3.

Under assumptions (i), (ii), (iii), and (iv) of Theorem 4.4.1, the conditional equi-asymptotic stability of the trivial solution of (4.1.4) implies the conditional equi-asymptotic stability of the trivial solution of the system (4.1.1).

Proof. Assume that the trivial solution of the auxiliary system (4.1.4) is conditionally equi-asymptotically stable. Then, it is conditionally equistable and conditionally quasi-equi-asymptotically stable. Since, by Theorem 4.4.1, the conditional equistability of the trivial solution of (4.1.1) is guaranteed, we need only to prove the conditional quasi-equi0 of (4.1.1). For this purpose, asymptotic stability of the solution x suppose that we are given 0 < E < p and to E J . Then, given 6 ( ~ )> 0 and t,, E J , there exist two positive numbers 6, = S,(t,) and T = T ( t o ,6) such that, if the condition

c N

1

=I

U,"

.< 8" ,

U," = 0

(2-

=

1 , 2 ,..., k)

(4.4.8)

As previously, the choice uio = Vi(t,, x,,) and x,,E implies uio == 0 (i = 1, 2, ..., k). Also, there exists a So = S,(t,) satisfying

II xo /I , to -1 T whenever xo E s(8,) wise, suppose that there exists a sequence {tx-},t, > to + T , and t, 0 and to E J , there exists a positive function 8(t,) e ) that is continuous in to for each E such that the inequality

=

implies

As a typical example, we shall merely state a theorem that gives sufficient conditions, in terms of any Lyapunov function, for the equistability of the trivial solution of (4.1.1). THEOREM 4.4.6.

Suppose that

(i) g E C [ J x R+N,R N ] g(t, , 0) = 0, and g(t, u) is quasi-monotone nondecreasing in u for each t E J; (ii) V E C [ J x S o ,R + N ] ,V ( t ,x) is locally Lipschitzian in x, C;"=, Vi(t,x) is positive definite, and N

L'*(t, X)

+

0

as

(1 x 11 -+ 0

for each t

E

J;

2=1

(iii) f

E

C [ J x So,R n ] , f ( t0) , = 0, and

D+F7(f,).

< g(t, V ( t ,x)),

( t , x)

E

J x

s,.

Then the definition (Cf*) implies that the trivial solution of (4.1.1) is equistable. T o exhibit the fruitfulness of using vector Lyapunov function, even in the case of ordinary stability, we give the following example.

4.4.

283

CONDITIONAL STABILITY AND BOUNDEDNESS

Let us consider the two systems

Example.

+ y sin t y’ = x sin t + e+y x’ = e+x

+ xy2)sin2 t,

-

(x3

-

(x2y

+ y 3 )sin2 t .

( 4 . 4 . 1 1)

Suppose we choose a single Lyapunov function V given by V ( t ,x)

= x2

+

y2.

Then, it is evident that D+V(t,x)

< 2 ( r t + j sin t 1)

V ( t ,x),

< u2 + b2 and observing that [x2 + y2I2sin2 t 3 0.

using the inequality 21 ab I

Clearly, the trivial solution of the scalar differential equation u’ = 2(e+

+ I sin t I) u,

u(to) = uo 3 0

is not stable, and so we cannot deduce any information about the stability of the trivial solution of (4.4.1 1) from Theorem 3.3.1, although it is easy to check that it is stable. On the other hand, let us attempt to seek a Lyapunov function as a quadratic form with constant coefficients V(t,X)

=

+[x2

+ 2Bxy + Ay2].

(4.4.12)

Then, the function D+V(t,x) with respect to (4.4.1 1) is equal to the sum of two functions q ( t ,x), w,(t, x), where q ( t , x) = x2[e+

+ B sin t] + xy[2Be+ + (A + 1 ) sin t ]

+ y2[Aect+ B sin t ] , wz(t,x) = -sin2 t [ ( x 2+ y2)(x2+ 2Bxy + Ay2)].

For arbitrary A and B, the functions V ( t ,x) defined in (4.4.12) does not satisfy Lyapunov’s theorem (Corollary 3.3.2) on the stability of motion. Let us try to satisfy the conditions of Theorem 3.3.3 by assuming w,(t, x) = h(t) V(t,x). This equality can occur in two cases:

+

+

1, B , = 1, h,(t) = 2[c1 sin t] when V,(t, x) = &(x Y)~. 1, B, = -1, h,(t) = 2[ect - sin t] when V,(t, x) = +(x - y)2.

(i) A, (ii) A,

=

=

284

CHAPTER

4

T h e functions V, , V, are not positive definite and hence do not satisfy Theorem 3.3.3. However, they do fulfill the conditions of Theorem 4.4.6. I n fact,

x:=l

(a) the functions Vl(t,x) >, 0, V,(t, x) >, 0, and Vi(t,x) = x2 + y2, and therefore C Z , Vi(t,x) is positive definite as well as decrescent;

< +

(b) the vectorial inequality D+V(t,x) g(t, V ( t ,x)) is satisfied with the functions g l ( t , u1 , uz) = 2(e+ sin t ) u1 , g2(t,uI, u2) = 2(e+

-

sin t ) uz .

I t is clear that g(t, u ) is quasi-monotone nondecreasing in u, and the null solution of u' = g(t, u ) is stable. Consequently, the trivial solution of (4.4.11) is stable by Theorem 4.4.6.

4.5, Converse theorems We shall consider the converse problem of showing the existence of several Lyapunov functions, whenever the motion is conditionally stable or asymptotically stable. T h e techniques employed in the construction of a single Lyapunov function earlier in Sect. 3.6 do not right away extend to this situation. As will be seen, the results rest heavily on the choice of special solutions of a certain differential system and the chain of inequalities among them, a kind of diagonal selection of the components of these solutions, and the quasi-monotone property. With a view to avoid interruption in the proofs, let us first exhibit some properties of certain solutions of the system (4.1.4) and its related system u' = g*(t, u ) , (4.5.1)

Assume that g E C [ J x R+N,A"], g(t, 0 ) E 0, ag(t, u ) / & exists and is continuous for ( t ,u)E J x R+N,and g(t, u ) is quasi-monotone nondecreasing in u for each t E /. Evidently, g*(t, u ) also satisfies these

4.5.

285

CONVERSE THEOREMS

assumptions. Moreover, since ui > 0 (i = 1, 2,..., N ) , it follows, in view of the quasi-monotone property of g(t, u), that

< g(t, 4.

g*(t, u )

(4.5.2)

Observe that the hypothesis on g(t, u ) guarantees the existence and uniqueness of solutions of (4.1.4) as well as their continuous dependence on initial values. Also, the solutions u(t, to , uo) are continuously differentiable with respect to the initial values. Furthermore, u s 0 is the trivial solution of (4.1.4). Clearly, similar assertions can be made with respect to the related system (4.5.1). If U ( t) = U(t,0, uo) and U*(t) = U*(t, 0, uo) are the solutions of (4.1.4) and (4.5.1), through the same point (0, uo),respectively, it follows, from Corollary 1.7.1, that U*(t)

< U(t>,

t

(4.5.3)

2 0,

in view of (4.5.2). Consider next the N initial vectors, with uio > 0 (i = 1, 2, ..., N ) defined by

o,..., 01, u20 o,..., 01,

Pl

= (u10 9

P,

=h10

Pi

= (u10 u20

,*-., uio, o,..., O ) ,

= (%I3

,..-,U N O ) .

P,

9

, 9

9

...

...

a20




(4.5.12)

290

CHAPTER

4

(b) the solution U(t,0, p,) of (4.1.4) verifies the estimate (4.5.13) where y, E X , 6, E 27, and, whenever uio = 0, i = 1 , 2,..., I z ; (c) the solution U,$(t,0, p,) of (4.5.1) is such that (4.5.14) and whenever ui0 = 0, i = I, 2,..., k ; where y1 E X ,6,E 9, (d) y l ( y ) is differentiable, and y;(r) 3 m 3 0; (e) 8,(t) and a2(t)are such that 6,(t) 3 mla2(t), m, > 0. Then, there exists a function V ( t ,x) with the properties (I), (2), (3) of Theorem 4.5.1 and

41x 11) ,


z=1

4

< a ( t , II x ll),

( 4 x) E 1 x

s,,

where 0 t X and a(t, Y) belongs to class X for each fixed t E J and is continuous in t for each Y.

Proof. Let x ( t , 0, x"), U ( t ,0, p,), and U$(t, 0, p,) be the solutions of (4.1.1), (4.1.4), and (4.5.1) satisfying (4.5.12), (4.5.13), and (4.5.14), respectively. Choose any continuous function p ( x ) E R,N possessing continuous partial derivatives with respect to the components of x, such that (4.5.9) and

all x II)

< c P Z ( 4 < 41x Ill? N

1 1

B2

1

01

E

.x,

(4.5.15)

hold. Using the same definition (4.5.10) for V(t,x) and proceeding as in Theorem 4.5. I , it can be easily shown that (l), (2), and (3) are valid. Assumption (d) implies that YI(YlY2)

2 mYlYZ.

(4.5.16)

T h e inequality (4.5.12), in view of the fact that x = x(t, 0, xo) and x,,= x(0, t , x), yields that (4.5.17) where p ~ ' ,1,3;1 both belong to class X .

4.5.

29 1

CONVERSE THEOREMS

As in Theorem 4.5.1, using the definition (4.5.10) and the nonnegative character of U$(t), we get N

1vZ(t,

i=l

).

2 u N N [ t , O, P1(x(O,t ,

x))>***7

PN(x(o,

t , x))],

which, by virtue of (4.5.14), the lower estimates in (4.5.15) and (4.5.17), the relation (4.5.16), and the assumption ( e ) , gives successively

Again, as before, making use of the definition of V(t,x) and the relation (4.5.4) and (4.5.3), we obtain N

1

Z=l

N

vi(t,

< 1 uZ(t, i-1

Pl(x(o,t , x)),***, PN(x(o, t , x))],

which, in its turn, allows the following estimates successively,

because of (4.5.13) and the upper estimates in (4.5.15) and (4.5.17). The theorem is proved. Under the general assumptions of Theorem 4.5.2, it is not possible Vi(t,x) .(\I x 11). This can, to prove the stronger requirement that however, be done if the estimates (4.5.12). (4.5.13), and (4.5.14) are modified as in the following:

xr=l


0 being constants, N

1

t

3 0,

i=l

O, $ N )

(4.5.18)

> 0,

(4.5.19)

and u E 9; N

ui(t,

xo E M ( n - k ) ,

< y2 1

*iO

i=l

s(t),

where y 2 > 0 is a constant, 6 E 9, and, whenever uio = 0, i N

~ “ ( t 0, , pjv)

2 yi C uio s(t),

t

i=l

2 0,

=

1, 2 ,..., k; (4.5.20)

where y1 > 0 is a constant and uio = 0, i = 1, 2, ..., k; respectively. Furthermore, let the functions 6 ( t ) and a ( t ) be related by S=(t)

=

d(t),

for some constant p > 0. Then, there exists a function V ( t ,x) with the properties (I), (2), (3) of Theorem 4.5.1, and M I

/I x’ /I”

N

< 1 Vi(t,). < M , II x’ llP, 2=1

where MI = ylA1/3;”, M , = y2A,/3yp, p suitable positive constants.

=

Pla,

and A,,

A, are some

PToof. By choosing the continuous function p(x) E R,N that satisfies (4.5.9) and

4I1 x’ /ID
a ] , Z(a) = [x E s : [I x I/ 2 a], Z(a) = [x E

respectively, and let S ( a ) , S(ol), and M(n--k)have the same meaning as in Sect. 4.4. Let x(t, to , xo) be any solution of (4.1.1).

DEFINITION 4.6.1.

T h e trivial solution of (4.1.1) is said to be

(CS,) conditionally strictly equistable if, for any E , > 0, to E 1,it is possible t o find positive functions 6, = 6,(t0, el), 6, = ?&(to,E , ) , and E, = €,(to , el) that are continuous in to for each el , such that €2

provided

< 8,

< 6,

< €1 ,

x(4 t o , xo>c q . 1 ) n Z ( E 2 ) , xo E

W,) n Z(8,) n

t

2 to,

M(T2-k) ;

(CS,) conditionally strictly unijormly stable if 6, , 6, , and E, in (CS,) are independent of to ; (CS,) conditionally quasi-equi-asymptotically stable if, given E , > 0, 01, > 0, and to E 1, it is possible to find, for every 01, satisfying 0 < 01, < 01, ,

294

4

CHAPTER

positive numbers c2 , T , that Ti


0, to E J , it is possible to find, for every rx, satisfying 0 < a, a1 , positive functions PI = &(t,, , q),p2 = &(tn, q) that are continuous in to for each al , such that




t

3 to ,

S(al) n Z(a2)n M ( ~ -;~ )

(CS,) conditionally strictly uniform bounded if p1 , p, in (CS,) are indepcndent of to . We observe that the foregoing notions assure that the motion remains in tube-like domains. I n order to obtain the sufficient conditions for the stability of motion in tube-like domains, we have to estimate simultaneously both lower and upper bounds of the derivatives of Lyapunov functions and use the theory of differential inequalities. We are thus led to consider the two auxiliary systems uo

u'

=: g l ( t , u),

up,)

v'

= g2(t, ?I),

v(to)= uo

=

3 0, 2 0,

(4.6.1)

(4.6.2)


g , E C [ J x R+N,R N ) ,g2(t, 4 g d t , u), and g1(t, u), gz(4 u> possess the quasi-monotone nondecreasing property in u for each t E J. Then as a consequence of Corollary 1.7.1, we deduce that p(t9 to

7

< ~ ( tto,

~ 0 )

7

uo),

t

>, to

9

4.6.

295

STABILITY IN TUBE-LIKE DOMAIN

provided where r ( t , t o , uo), p ( t , to , vo) are the maximal, minimal solutions of (4.6.1), (4.6.2), respectively. Corresponding to definitions (CS,)-( CS,), we may formulate (CSf)-(CSZ) with respect to the system (4.6.1) and (4.6.2). For example, (CSF) would imply the following:

(CSf) Given el > 0, to E J , there exist positive functions 6, = a1(t0, el), 6, = S,(to , el), eg = e z ( t o , el) that are continuous in to for each el such that €2 < 6 , e 6, < €1 , €2


< €1

,

t b to

7

if uio = vi0 = 0 (i = 1, 2, ..., k) and

Let us restrict ourselves to proving conditional strict equistability only. Similar arguments with necessary modifications yield any desired result.

THEOREM 4.6.1. Assume that (9 g,

gz(t,0)

, g2 E C [ J x R+N,RN1,

gz(4

4 < gdt, 4,

gdt, 0) = 0,

= 0, and g l ( t , u), g2(t,u ) possess the quasi-monotone nondecreasing property in u for each t E J ; (ii) V E C [ J x S,, , R+N],V(t,x) is locally Lipschitzian in x, and, for ( t , x) E J x S , ,

(iii) Vi(t,x) = 0 (i = 1, 2,..., k), k < n, if x E M(n-k), where M(n-k) is an ( n - k) dimensional manifold containing the origin; (iv) f E C [ J x S,, , R n ] , f ( t ,0 ) = 0, and, for ( t , x) E J x S,, ,


0, t, E J , there exist positive functions 8, = %Po 4, 8, = s2(t0, E ~ ) ,and i , = to , it reaches the boundary of S(el) n Z(e2). This means that either 11 x(t1 , t o ,xo)ll = or I/ x(t, , to , x,)lI = e2 . Also, I! x ( t , to , xg)lI -: p, t E [t,,, t l ] , and therefore, for t E [to, tl], we can apply Theorem 4.1.3 to obtain

4.7.

STABILITY OF ASYMPTOTICALLY SELF-INVARIANT SETS

297

where r ( t , t o ,u,,),p(t, t, , vo) are the maximal, minimal solutions of (4.6.1), (4.6.2), respectively, such that no = V(to, xo) = u, . This implies that

c Pi(4 to

N

N

i=l

* uo)

< i=l1 Vdt, 4 4 t o ,

.ON

(4.6.5)

for t E [ t o ,t J . I n the first instance, if 11 x( t, , t o ,xo)lI = e l , using the right side inequality in (4.6.3) and (4.6.5), we arrive at the contradiction N

b(4

< 1 Vztt, i=l

7

X(t,

7

to xo>> Y

because of the left side inequalities in (4.6.3) and (4.6.5). This shows that (CS,) follows from (CST), and the proof of the theorem is complete.

4.7. Stability of asymptotically self-invariant sets One has to consider, in many concrete problems like adaptive control systems, the stability of sets that are not self-invariant; this rules out Lyapunov stability, because those definitions of stability imply the existence of a self-invariant set. T o describe such situations, the notion of eventual stability has been introduced in Sect. 3.14. It is easy to observe that, although such sets are not self-invariant in the usual sense, they are so in the asymptotic sense. This leads us to a new concept of asymptotically self-invariant sets. Evidently, asymptotically self-invariant sets form a special subclass of self-invariant sets, and therefore it is natural to expect that their stability properties closely resemble those of invariant sets.

298

CHAPTER

4

Let zu E C[Rn,R"]. Define (4.7.1) We shall denote the sets

[x E R" : I/ 4411

< €1

by G, S(G, E ) , S(G, E ) , respectively. Suppose that x ( t ) = x ( t , to , xo) is any solution of (4.1.1). DEFINITION 4.7.1. A set G is said to be asymptotically self-invariant with respect to the system (4.1.1) if, given any monotonic decreasing sequence { ep} , e p ---f 0 as p + CO, there exists a monotonic increasing sequence {tJE)}, t , ( ~ -+ ) 00 as p + 00, such that xo E G, to 3 t p ( E ) , implies x(t)

C S(G, cD),

t

2 to,

p

=

1, 2,... .

be an (n - k) dimensional manifold containing the set G. Let We shall assume that G is an asymptotically self-invariant set with respect to the system (4.1.1). DEFINITION 4.7.2. T h e asymptotically self-invariant set G of the system (4.1.1) is said to be (AS,) conditionally equistable if, for each E > 0, there exists a tl(e), tl(e) + 00 as E + 0, and a S = S(t, , E ) , to 2 t , ( e ) , which is continuous in to for each e such that x(t)

provided

c S(G, €1, xo E

t

3 to 2 tl(.),

S(G, 6) n M n - - k )

*

On the basis of this definition, it is easy to formulate the remaining notions (AS,)-(AS,) corresponding to (Cl)-(C8) of Sect. 4.4. The following theorem gives sufficient conditions for the set G to be asymptotically self-invariant with respect to the system (4.1.1).

THEOREM 4.7.1.

Assume that

(i) g E C[J x R+N,R N ] , and decreasing in u for each t E J ;

g ( t , u ) is

quasi-monotone

non-

4.7.

STABILITY OF ASYMPTOTICALLY SELF-INVARIANT SETS

299

V I EC [ J x S(G,p), R+N],V ( t ,x) is locally Lipschitzian in x,

(ii)

< c Vi(t, x), N

b(ll w(x)ll)

( t , x)

E

i=l

and

c Vi(t,x) N

=

J

a(t)

X

S(G,p),

if

x E G,

6 E .X,

(4.7.2)

(4.7.3)

2-1

where u E 9; (iii) Vi(t,x) = 0 (i = 1, 2,..., k), k < n, if x E M(n--li), where M(n-k) is an (n - k) dimensional manifold containing the set G; (iv) f E C [ J x S(G,p), R"],and

o+v(t, x) < g(t, v(t,x)),

( t , x) E

J x S(G,p ) ;

(v) for any function P(t, u), which is continuous for t 2 0, u 2 0, decreasing in t for each fixed u, increasing in u for each fixed t such that lim lim/3(t, u ) t-m

u-0

=

0,

(4.7.4)

we have

provided uiO = 0 (i = 1, 2, ..., k), where u(t, t o , uo) is any solution of (4.1.4). Then, the set G = [x E Rn : 11 w(x)lI invariant with respect to (4.1.1).

=

01 is asymptotically self-

Proof. Let x0 E G. Since G C M(n-,c. , it follows that x0 E M(n--li). As a consequence, we have, by (iii), Vi(to, XJ = 0 (i = 1, 2,..., k), k < n. We choose uiO= V i ( t o xO) , (i = 1, 2 ,..., N ) . Then, because of (4.7.3), we obtain N

1

c N

ui0

i=l

=

i=l

Vi(t0 9

(4.7.6)

xo) = O(t0).

Consider the function y ( t ) = p(t, u(t)), which decreases to zero as t+ co because of the assured monotonic properties of the functions /3 and u. Let now {E,} be a decreasing sequence such that E , + 0 a s p -+ 00. Then, the sequence {b(e,)} is a similar sequence. Since y(to)+ 0 as to + m, it is possible to find an increasing sequence { t p ( e ) } ,t p ( e ) + co as p + 00, such that y(t0)

, t , ( ~ )for a certain p , (1 w(x(t,))I(= c p €or some t = t, > to >, t , , ( ~ )and ,

II 44t))ll G ~p < P , For t

E

t

E

[to 9 t i l e

[t,, , t J , we obtain, on account of Theorem 4.1.1, the inequality

where r ( t , t, , uo) is the maximal solution of (4.1.4). At t = t, , we arrive at the contradiction

< c Vdtl N

b(%)

N

7

+l))

i=l

< 1Y d t l < i=l

>

to

> %I)

P(t0 1 4 t O ) )

= At,)

0, there exists a t , ( ~ ) t, , ( ~-+ ) 00 as E 4 0, and a 6 == S ( t o , E ) , to >, t l ( c ) , which is continuous in to for each E , such that

provided N

1uio < 8, i=l

ui0 = 0

(i = 1 , 2 )...,k).

T h e following theorem assures the conditional equistability of the asymptotically self-invariant set G.

4.7.

STABILITY OF ASYMPTOTICALLY SELF-INVARIANT SETS

301

THEOREM 4.7.2.

Suppose that hypotheses (i), (ii), (iii), and (iv) of Theorem 4.7.1 hold, except (4.7.3). Assume further that the set G is asymptotically self-invariant and

c N

i=l

< a(t, I1 w(x)II),

Vi(t,x)

(4 "4E J x S(G, P),

(4.7.9)

where the function a(t, r ) is defined and continuous for t >, 0, r 2 0, monotonic decreasing in t for each fixed r, monotonic increasing in r for each fixed t, and lim lim a(t, T ) = 0. t-m

r-0

Then (ACT)implies (AC,). Proof. Let 0 < E < p be given. Assume that the definition (ACT)holds. Then, given b ( ~ > ) 0, there exist a t , ( ~ ) t,l ( e ) -+ co as E -+ 0, and a 6 = S ( t o , E ) , to >, t , ( ~ such ) that N

1 udt, t o , uo)
, t Z ( e ) ,such that 4to

9

ll w(x0)ll)

< 8,

I/ w(x0)lI G 6,

9

(4.7.12)

provided to 3 t2(e). Let t3(c) = max[t,(E), t 2 ( c ) ] .I t can then be claimed that, if xo E S(G, S,) n M(n--k), we have x(t, to , xo)C S(G, E ) for t >, to >, t3(c), where x(t, to , xo) is any solution of (4.1.1). Let us assume that this is not true. Then, there exists a solution x ( t ) of (4.1.1) such that, , x(t) C S(G, c) for t E [ t o , t,], whenever xo E S(G, 6,) n t, > to 2 t 3 ( e ) , and x(tl) lies on the boundary of S(G, c). This implies that

I/ w(x(t))ll

and


II)

< i=l1 vi(t, < 44 I1 w(x)II),

t

3 0,

x E R",

where a(t, r ) is continuous for t >, 0, r 2 0, montonic decreasing in t for each Y, monotonic increasing in r for each t , and lim lim a(t, r )

2-02

7-0

= 0,


0. Let M(npk)denote, as before, an ( n - k) dimensional manifold containing the set G. We define S(B, C)

=

S(G, 01

+ c),

E

> 0.

DEFINITION 4.8.2. T h e conditionally invariant set B with respect to the set G and the system (4.1.1) is said to be (CC,) conditionally equistable if, for each E > 0 and to E 1, there exists a positive function S = S ( t o , E ) , which is continuous in to for each E , such that

Evidently, on the strength of (CC,), we can define (CC,)-(CC,) corresponding to (Cz)-(C8).

REMARK4.8.1. We observe that the set B need not be self-invariant. If 01 = 0, these definitions coincide with (C,)-(C,), that is, the conditional stability concepts of the self-invariant G.

306

4

CHAPTER

T o define the corresponding definitions (CCF)-( CC:) for the auxiliary system (4.1.4), let us define the set, for some p > 0, N

1 ui < 81,

u E R+N:

(4.8.1)

iL1

and assume that B* is conditionally invariant with respect to the set 0 and the system (4.1.4).

zi =

DEFIKITION 4.8.3. T h e conditionally invariant set B* with respect to the set 11 - 0 and the system (4.1.4) is said to be (CCF) conditionally equistahle if, for each E > 0 and t,, E J , there exists a positive function S = S ( t , , E ) , which is continuous in t, for each E , such that N

1 ~ t ( ttn,

7

un)

z=l

provided N

1

Ui"

1-1

THPORERI 4.8.1.

< 6,

ui,

B

=

+ 0

t

E,

2 to

7

(i = 1 , 2 )...,K).

Assume that

, = 0, and g(t, a) is quasi-monotone (i) g E C [ J x R+N,R N ] , g ( t0) nondecreasing in u for each t E J ; (ii) V E C [ J x R", R . , N ] ,V ( t ,x) is locally Lipschitzian in x, and N

b(Il 4.z)lI)

< 1 Vdt, x) < 4w(x)II),

( t , x) E

J x R",

2-1

where a , h E f on the interval [0, a)and h(r) + co

(iii) f

E

as

Y + co;

C [ J Y R", R"], and D - l ' ( t , x)

< g ( t , V ( t ,x)),

( t ,x) E J x R".

Then, if the set B* is conditionally invariant with respect to the set 0 and the system (4.1.4), the set B = S(G, a ) , where N = b-l(P), is conditionally invariant with respect to the set C and the system (4.1.1).

zi =

PFoof. Assume that the set B* defined by (4.8.1) is a conditionally invariant set. This implies that, if ui0 = 0 (i = 1, 2 ,..., N ) , N

1 ~ , ( tt ,o , 0 ) < P,

7=1

t

2 t o 3 0.

(4.8.2)

4.8.

STABILITY OF CONDITIONALLY INVARIANT SETS

307

Let us choose uin = V i ( t o x ,,,) (i = 1, 2, ..., N ) . Then, it follows that xo E G and Vi(to, xo) == 0 (i = 1, 2, ..., N ) hold simultaneously. By Theorem 4.1.1, we obtain

where r ( t , to , u,,) is the maximal solution of (4.1.4) through ( t o ,u,,). Since b(ll w(x)ll) CC, Vi(t,x), we readily get the inequality


0, y > 0, and to E J be given. Suppose that

V i ( t o xo) , = 0 (i = 1, 2, ..., k). Choose uio = Vi(t, , x,,)(i = 1, 2,..., N ) . Then, we have by Theorem 4.1.2 that every solution x(t, t,, , x,,) of (4.1.1) exists for t 3 to and satisfies

so that we can infer that

"(4 x ( t , t o , xo))

< Y ( t , t" , 4,

t

2 to,

where r ( t , t o ,u,,) is the maximal solution of (4.1.4). Define y1

(4.8.4)

= a(y),

308

CHAPTER

4

+

and assume that (CC,") holds. Let 01 = h-'(P). Then, given b(oc C) > 0, > 0, and to E 1, there exists a positive number T = T ( y , e ) such that

y,

V

1

U , ( f , f,,

, UO)

t=1

< b(a

+

+ T,

(4.8.5)

(i = I , 2,..., k).

(4.8.6)

t

c),

2 4,

provided V

2 U," < y1 ,

/

u,,, = 0

I

Clearly, by the choice of y1 and u i O , the condition (4.8.6) is satisfied. Hence, wc obtain, using (4.8.4), (4.8.5), and the fact that

WI w(4Il)

c V,(f, N

d

4

7

2-1

the relation b(ll zu(.v(t, t o , ~ 0 ) ) l l )

< b(a

+

e),

t

3 f0

+ T,

whenever xo E S(G, y ) n M(,L--I,) . Evidently, this implies that the conditionally invariant set 13 is conditionally quasi-uniform asymptotically stable. T h e proof of the theorem is thus complete.

4.9. Existence and stability of stationary points This section is concerned with the conditions sufficient to assure the existence of yo satisfying (4.9.1)

f(Yo) = 0

and the stability of the solution x ( t ) = y o of the autonomous differential system x' = f ( x ) , x(0) = x0 , (4.9.2) where f E C[R",R"].

TIIEORENI 4.9. I . (i)

Assume that

I/ E C[R",K + N ] , V(x)is locally Lipschitzian in N

C

V ( ( x )4

as

11 x 11

---f

x,and

co;

t=1

, is quasi-monotone nondecreasing in u , and (ii) g E C [ R ,N , R N ] g(u) D+V(x) g( V ( x ) ) ,x E RtI;


t o ,


t o ,

'rHEORER.1

x

-

Ks < y

-

Ky,

(5.1.5)

5.1. where x, y

E

INTEGRAL INEQUALITIES

317

C [ ] ,R]. Then ~ ( t , < ) y(t,) implies x(t) , I, 0, 0 < t a(t) f 4 0 ) ; (ii) g E C[R,R ] , xg(x) > 0, x # 0, and G(x) = l z g ( f )d f

as I x

co

-+

0

< co, i

I

+

(iii) u ( t ) is any solution of (5.5.1) existing on 0 Under these assumptions, lim ui(t) = 0

Proof.

Differentiating (5.5.1), we get x"(t)

+ a(0)g(x(t)) =

= 0,

(i

t-m

-

=

0, 1, 2, 3, and

a;

< t < co.

1 , 2).

(5.5.7)

it 0

a'(t - s) g(x(s)) ds.

(5.5.8)

Whenever we refer to (5.5.1) and (5.5.8), we mean the identities that result from substituting u(t) into them. The possibility of none of a'(O),a"(O),a"'(0) being finite necessitates that a little care be exercised in handling certain integrals that arise. In all the cases, the arguments used in the preceding lemmas supply the rigor, and hence, in this proof, we tacitly assume such considerations whenever they are relevant. Consider the function

-

$

st 0

a'(t - s)

[It

g(U(T))

dT]' ds

S

Using (5.5.1) and integrating by parts, we obtain

0.

(5.5.9)

332

CHAPTER

5

which implies that

< v(t)< v(0)= G(un),

G(u(t))

where u,, = u(0). I t then follows from assumption (ii) that


0 arbitrarily if the first alternative of Lemma 5.5.2 holds, and choose 0 < T , < to if the second one does. Then, clearly (5.5.1 4)

00

I4tn)i 3 A. This, together with the relations (5.5.8), (5.5.13), and the mean value theorem, implies the existence of a 6 > 0 and a p > 0, where 0 < 6 min( To , t l ) , such that


0

(12

=

1, 2,...),

which contradicts (5.5.14). Thus, limt+mu(t) = 0 is established. Formula (5.5.7), i = 1, follows from (5.5.7), i = 0, (5.5.12), and the mean value theorem by employing an argument similar to the proof of Lemma 5.5.1. Similarly, formula (5.5.7), i = 2, follows from (5.5.7), i = 0, assumption (ii), (5.5.8), and the fact that a'(t) E L ~ ( O a , ).This completes the proof of the theorem.

5.6. Perturbed integral equations Corresponding to the integral equation (5.5. I), let us first consider the perturbed equation x'(t)

=

-Jt

n

a(t

-

s ) g ( x ( s ) )ds

-

b(t) +f(t).

(5.6.1)

As in the previous section, the letter ,l3 denotes a finite a priori bound that may vary from time to time. Concerning Eq. (5.6.1), we have the following result.

334

CHAPTER

5

THEORIN 5.6.1. Assume that t -< m, i = 0, 1, 2; (i) a E C [ J ,121, (- I)”Q(Q(t)2 0 for 0 (ii) g E C [ R ,K], xg(x) 3 0, G(x) = J:g([) d [ -+ co as I x I --t m, and 1 g(s)/ Kl(l G(x)) for some K , > 0;

+


0 being some n um b e r ; and bounded. (viii) f(t) is continuously differentiable on p t \ ‘I‘hen, ~


0 with the property If(t,

u)IB