LINEAR
SYSTEMS
OF
ORDINARY
DIFFERENTIAL
EQUATIONS N.
A.
Izobov
UDC 517.941.92
The p r e s e n t s u r v e y dea...

Author:
Izobov

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LINEAR

SYSTEMS

OF

ORDINARY

DIFFERENTIAL

EQUATIONS N.

A.

Izobov

UDC 517.941.92

The p r e s e n t s u r v e y deals chiefly with the Lyapunov t h e o r y of c h a r a c t e r i s t i c exponents of the g e n e r a l f o r m of linear differential s y s t e m s , x=A(t)x

with bounded p i e c e w i s e continuous coefficients and p a r t i c u l a r l y with its m o s t r e c e n t advances. A sufficiently c o m p l e t e bibliography on such other independent b r a n c h e s of the t h e o r y of linear s y s t e m s as " L i n e a r S y s t e m s with P e r i o d i c Coefficients" (already s u r v e y e d in p a r t [469] in the s e r i e s G e n e r a l Mechanics) and the "Analytical T h e o r y of L i n e a r S y s t e m s " can be found in Sec. 15. The bibliography p r e s e n t e d in our s u r v e y b a s i c a l l y e n c o m p a s s e s all works on l i n e a r s y s t e m s a b s t r a c t e d in R e f e r a t i v n y i Zhurnal (Matematika). 1.

DEFINITION

AND

BASIC

PROPERTIES

OF

DIFFERENT

EXPONENTS

The Lyapunov c h a r a c t e r i s t i c exponent ([349], p. 27) of the solution x (t) # 0 of a linear s y s t e m is the number

-- ~

11 )It

A linear s y s t e m cannot have m o r e than n nonzero solutions with p a i r w i s e different exponents (Lyapunov [349], p. 34). A fundamental s y s t e m of solutions of a linear s y s t e m is called n o r m a l (Lyapunov [349], p. 34) if the sum of the exponents of its solutions is a m i n i m u m in the set of all fundamental s y s t e m s and is said to be b i n o r m a l (R. l~. Vinograd [126]) if in addition X-1 (t) is n o r m a l r e l a t i v e to the adjoint s y s t e m ~ = - A T ( t ) y . The exponents ~I - ~2 - 9 - ~n of a n o r m a l o r d e r e d (i.e., a r r a n g e d in n o n d e c r e a s i n g o r d e r of the exponents of the solution) s y s t e m will be called the exponents of the l i n e a r s y s t e m while the e x t r e m e exponents, Xi and 7~n, will be called its lowest and highest exponents. The solutions a r e distributed with r e s p e c t to t h e i r exponents in the following way (A. M. Lyapunov [349], p.34, and Yu. S. Bogdanov [48]): the e n t i r e s p a c e of initial values is divided into m l i n e a r s p a c e s L k embedded in each other, such that e v e r y nonzero solution outside it h a s an exponent l e s s than ~k (~'l < 9 . . < ~'m). Stability of the Exponents: The exponents of a linear s y s t e m a r e said to be stable if for any e > 0, a 6 > 0 can be found such that e v e r y exponent Xy of any p e r t u r b e d s y s t e m

~=A(Oy+Q(t)y, IIO(t)ll--0,

rain Ikr-- kil ~< .~. i

T h i s concept a p p e a r e d in P e r r o n [808] w h e r e it was e s t a b l i s h e d for the f i r s t t i m e that exponents can be unstable. T r a n s l a t e d f r o m Itogi Nauki i Tekhniki ( M a t e m a t i c h e s k i i Analiz), Vol. 12, P a r t 1, pp. 71-146, 1974,

9 Plenum P~+blishing Corporation, 22 7 West 17th Street, New York, N. ~: 10011. No part o f this publication may be reproduced, stored ht a retrieval system, or transmitted, in an), form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of tile publisher. A copy o f this article is aPailable from the publisher for $15.00.

46

The lower exponent ( P e r r o n [808]) is given by

~_[xl ~ li__~m+ in Ilx (011. Unlike the c h a r a c t e r i s t i c exponents of a linear s y s t e m , t h e r e m a y e x i s t m o r e than n different lower e x p o nents ( P e r r o n [808]). M o r e o v e r , a s y s t e m has been c o n s t r u c t e d whose set of lower exponents was the segment [0, 1] (N. A. Izobov [251]). In addition, the lower exponents a l s o obey a p r o p e r t y s i m i l a r to that obeyed by the c h a r a c t e r i s t i c exponents (N. A. Izobov [248]). T h i s p r o p e r t y m a y be stated a s follows: A l m o s t nil solutions beginning in a k - d i m e n s i o n a l (k = 2, . . . . n) s u b s p a c e of the s p a c e of initial data have a 2 o w e r exponent which is the g r e a t e s t exponent for the given solutions. The exact exponent o c c u r s in those solutions in which the c h a r a c t e r i s t i c and lower exponents coincide. t

The g r e a t e s t lower bound of the c o n s t a n t s R satisfying

llX(t, s)fI..- . . . ~ d n [X] a r e positive s q u a r e r o o t s of the eigenvalues of the m a t r i x X* X, and the set of a u x i l i a r y exponents will be called its a u x i l i a r y s p e c t r u m (v i coincides with the c e n t r a l exponent ~2). The Lyapunov n o r m X (Yu. S. Bogdanbv [ 51]) is defined on an a r b i t r a r y linear s e t A, h a s its own v a l ues X(a) of an e l e m e n t of a given o r d e r e d s e t A , and is such that: l) X(ca) - A(a), a E A and c is a r e a l n u m b e r , and 2) k (a + a ' ) ~ m a x {~.(a), ~.(a')}, w h e r e a, a v • A. The c h a r a c t e r i s t i c d e g r e e (B. P. Demidovs v Ix], such that * [xl ~- lira

[187]) of a solution x (t) with exponent X is that n u m b e r

* in Ilx (t)e-~qE.

We r e f e r the r e a d e r also to Refs. 57, 103, 122, 125, 219, 247, 389, 399, 400, 436, 456, 465, 690, 791, and 827. 2.

REDUCIBLE

AND ALMOST

REDUCIBLE

SYSTEMS

The Lyapunov t r a n s f o r m a t i o n s ([349], p. 42) keep the: a s y m p t o t i c c h a r a c t e r i s t i c s of s y s t e m s invariant and a r e therefDre used for c l a s s i f y i n g these c h a r a c t e r i s t i c s . The s i m p l e s t and m o s t i m p o r t a n t c l a s s of s y s t e m s singled out by Lyapunov include the following. 1_~, Reducible S y s t e m s ([349], p. 43). A l i n e a r s y s t e m is called r e d u c i b l e if t h e r e exists a Lyapunov t r a n s f o r m a t i o n that c o n v e r t s it into a s y s t e m with constant coefficients.

47

Lyapunov ([349], p. 195) indicated one i m p o r t a n t c l a s s of reducible s y s t e m s , n a m e l y s y s t e m s with periodic coefficient m a t r i c e s . F u r t h e r investigations of reducible s y s t e m s a r e a s s o c i a t e d with the n a m e N. P. Erugin and deal with the g e n e r a l theory he c o n s t r u c t e d of such s y s t e m s in the monograph "Reducible S y s t e m s . ~ In this monograph N. P. Erugin p r o p o s e d two methods for investigating r e d u c i b i l i t y p r o b l e m s . The f i r s t method is that of s u c c e s s i v e a p p r o x i m a t i o n s (fromwhich, incidentally, follow well-known a s y m p totic decompositions and which p e r m i t e s t i m a t e s to be obtained of the r e m a i n d e r t e r m in a s y m p t o t i c f o r m u las for the c a s e 6f an i r r e g u l a r singular.point, something that had not p r e v i o u s l y been done) (cf. V. V. Khoroshilov [556], L. I. Donskaya [192, 193], and I. N. Zboichik [232]). The second method c o n s i s t s in r e duction to s y s t e m s of a special type suitable for investigations (cf. A. E. G e l ' m a n [143; 144], L. Ya. A d r i a n o va [3], and L. V. T r i g u b o v i c h [525]). Let us d i s c u s s c e r t a i n c o n c r e t e r e s u l t s of N. P. E r u g i n ' s monograph. Reduciblity C r i t e r i o n ([199], p. 9). A l i n e a r s y s t e m is reducible if and only i f its fundamental s y s t e m of solutions can be r e p r e s e n t e d in the f o r m X(t) = S(t) e x p A t with Lyapunov m a t r i x S (t). R e a l n e s s of the T r a n s f o r m a t i o n ([199], p. 17; [213], p. 119). A r e d u c i b l e s y s t e m of a r e a l Lyapunov t r a n s f o r m a t i o n can be c o n v e r t e d into a s y s t e m with constant r e a l canonical coefficients m a t r i c e s . N e c e s s a r y Reducibility Condition ([199], p. 17). The integral of the t r a c e of the coefficient m a t r i x of t

a r e d u c i b l e s y s t e m can be r e p r e s e n t e d in the f o r m f SPA (Q d~=at-b~

(t)

with bounded function ~0(t).

0

Reducibility of an A n t i s y m m e t r i c System ([199], p. 20). A l i n e a r s y s t e m with an anti,symmetric m a t r i x (ajk = - akj for all j and k) is reducible, I. A. L a p p o - D a n i l e v s k i i [3051 C a s e and Reducibility ([1991, p. 24). If the m a t r i x A(t) c o m m u t e s with its integral ~ A (~)dr

for e v e r y s --- t and if we have the r e p r e s e n t a t i o n

s

fA (~)de =A + B(t)

with bounded

0

m a t r i x B (t) the linear s y s t e m can be r e d u c e d to the s y s t e m ~r = Ay. T r i a n g u l a r S y s t e m s ([199], p. 36). A t w o - d i m e n s i o n a l upper t r i a n g u l a r linear s y s t e m is r e d u c i b l e to a diagonal f o r m if and only if its diagonal coefficients have the r e p r e s e n t a t i o n aii (t) = a i + 9 i (t) with bounded t

integrals

I ~'(:)d:

t

and, if at = a2 = a, t h e integral

](t)=fax~(~)ex p I(y2--*t)d~d*

0

0

sional upper t r i a n g u l a r linear s y s t e m is r e d u c i b l e to the s y s t e m

(~

is bounded. A t w o - d i m e n -

0

y = Oa

y. if the function Cvl (t) + C2t,

w h e r e C 2 ~ 0, is bounded. In p a r t i c u l a r ([199], pp. 28-32) the reducibility p r o b l e m for the t w o - d i m e n s i o n a l s y s t e m (1) with upper t r i a n g u l a r m a t r i x

A(O= ~ A~t-.,.

(*)

nt~O

has been exhaustively investigated. The nonanalytic c a s e for a m a t r i x A (t) as well a s the n - d i m e n s i o n a l c a s e have a l s o b e e n examined. R i c c a t i Equation and Reducibility. N. P. Erugin ([199], p. 68) e s t a b l i s h e d a r e l a t i o n between r e d u c i b i l ity of a t w o - d i m e n s i o n a l s y s t e m to t r i a n g u l a r f o r m and the existence of a bounded solution or a solution that tends to +oo a s t --~ + ~ for the equation u' = - al2 u2 + ( a 2 2 - - all) U + a 2 1 . New reducibility t e s t s w e r e obtained using it, for example, a reducibility c r i t e r i o n for a s y s t e m with a nontriangular m a t r i x of the f o r m (*) as well as ([199], p. 82) a linear s y s t e m i n w h i c h aj2, a~2--aH>~>O, a~t>O, akk~ak+~k(t), at-~a~, t >0,

48

is reducible to the s y s t e m ~ --- diag[a I, a2]y if the i n t e g r a l s

t

t

l?k(~.)d~, and

~a~x(Qd~ a r e

0

-0

bounded.

R e m a r k . In r e d u c i n g a linear s y s t e m to t r i a n g u l a r f o r m it is n e c e s s a r y to find a given bounded solution. But s o m e t i m e s such a r e a l bounded solution does not exist while a complex bounded solution does exist, which also leads to a solution of the p r o b l e m (see e x a m p l e [199], p. 80). P e r t u r b a t i o n s that P r e s e r v e the Reducibility P r o p e r t y . N. P. Erugin ([199], p. 63) set forth the p r o b l e m a s t o how s m a l l Q (t) m u s t be such that the p e r t u r b e d s y s t e m ~r = [A (t) + Q (t)]y r e m a i n s r e d u c i b l e to the s a m e s y s t e m a s initially. Its solution was given in the f o r m of u n i m p r o v a b l e (differential) e s t i m a t e s or an a l g o r i t h m for obtaining t h e m by m e a n s of the f i r s t method. N. P. E r u g i n ' s m o n o g r a p h [213] in which his r e s u l t s w e r e set forth and studies of A. M. Lyapunov, I. A. L a p p o - D a n i l e v s k i i , N. N. Bogulyubov, and L Z. Shtokalo on linear s y s t e m s w e r e brought t o g e t h e r , also c o n s t i t u t e s an i m p o r t a n t contribution to the l i t e r a t u r e on t h e s e p r o b l e m s . T h i s book h a s e x e r t e d a significant influence on the f u r t h e r development of different lines of r e s e a r c h into l i n e a r s y s t e m s . We r e f e r the r e a d e r also to Secs. 6 and 12 and to Refs. 24, 27, 54, 103, 137, !41, 207, 203, 209, 211, 218-220, 228, 271, 319, 332, 333, 361,362. 370, 373, 442, 463, 600, 605, 757, and 826. 2 ~ A l m o s t Reducible S y s t e m s (B. F. Bylov [85, 88] and Lillo [766]). A linear s y s t e m is said to be almosY-reducible to the s y s t e m

:~=B (t) y.

(2.1)

if for any 6 > 0 t h e r e e x i s t s a Lyapunov t r a n s f o r m a t i o n y = Ss(t)x that t r a n s f o r m s it into the s y s t e m ~ = [B (t) + Q6(t)]z if the inequality IIQ6 (0tl 0 there, e x i s t s a Lyapunov t r a n s f o r m a t i o n y = S6(t) x that r e d u c e s it to the f o r m = diag [xi . . . . . x,,] y ~- B6 (t) y ~u Q6 (t) y, w h e r e Q5 (t) is a t r i a n g u l a r m a t r i x with z e r o diagonal, IlQ~ (t)I[ a ( L k l i + . . . + l k m l ) -~, a, d > 0 , where k is any integral-valued vector, the inner Lebesgue m e a s u r e of the set M (Q) of constant m a t r i c e s A f r o m the unit sphere $1, such that the s y s t e m is reducible, satisfies the condition rues M (Q)-* mes S 1 when IIQtt -~ 0. Other proofs of the Adrianova-Samoilenko t h e o r e m can be obtained by means of the Lille t h e o r e m (cf. 2~ below). Yu. A. Mitropol'skii and A. M. Samoilenko's joint paper [405], which preceded [473], constructed reducibility t e s t s for such quasi-periodic s y s t e m s by t r a n s f o r m a t i o n with a quasi-periodic maVrix, while Yu. A. Mit-ropol'skii and E. A. Belan [401] used an a c c e l e r a t e d convergence method to c o n s t r u c t a uniform13, a l m o s t periodic t r a n s f o r m a t i o n of an a l m o s t diagonal (with s e p a r a b l e diagonal) uniformly a l m o s t periodic s y s t e m to a diagonal uniformly a l m o s t periodic f o r m whose e x i s t e n c e had been proved by B. F, Bylov. (cf. 2~

53

V. Kh. Kharasakhal and his colleagues studied the quasi-periodic s y s t e m ~ = A (t) x with an m - d i m e n sional basis by means of the s y s t e m of linear equations and partial derivatives with coefficients of different periods, Ox/Ou=F(ul . . . . . urn)x, F ( t . . . . . t)==A(t).

An analog of the Floquet t h e o r e m was successfully proved for the l a t t e r s y s t e m under additional a s s u m p tions on the p r o p e r t i e s of the initial s y s t e m , which p e r m i t t e d reducibility and r e g u l a r i t y t e s t s for quasiperiodic s y s t e m s to be derived. We r e f e r the r e a d e r also to Refs. 295, 296, 467, 528, 544-551, 593, 594, and 601. The f i r s t set also includes V. G. Sprindzhuk [487] in which r e p r e s e n t a b i l i t y conditions for an integral of a function f i x , r . . . . . Wmx) one-periodic in each independent variable in the f o r m at + O (1) w e r e indicated and as a c o r o l l a r y reducibility t e s t s w e r e obtained for two-dimensional quasi--periodic s y s t e m s . T h e s e p a p e r s (cf. also K. G. Valeev [110]) also contain stability t e s t s for quasi-periodic and uniformly a l m o s t periodic s y s t e m s . 2 ~ a) General Study of Uniformly Almost P e r i o d i c S y s t e m s : Lillo [766], in which the a l m o s t p e r i o d i c ity of the P e r r o n t r a n s f o r m a t i o n m a t r i x for a uniformly a l m o s t periodic system with separable solutions (cf. Sec. 7) to triangular f o r m was proved by means of a set of displacements (cf. Sec. 1), was one of the f i r s t works along these lines. In his p r e c e d i n g work [764], it was proved that the set of uniformly a l m o s t periodic s y s t e m s with separable solutions is open in the space of all uniformly a l m o s t periodic s y s t e m s . B. F. Bylov [91] constructed a reducibility test for the uniformly a l m o s t periodic s y s t e m ~ = A (t) x using a l m o s t periodic Lyapunov t r a n s f o r m a t i o n s , to a diagonal s y s t e m with uniformly a l m o s t periodic coefficients (and consequently (B. F. Bylov [88]), a l m o s t reducibility) in the f o r m of c e r t a i n constraints on e v e r y s y s t e m ~ = ~(t) y. In p a r t i c u l a r such a reduction is possible if a uniformly a l m o s t periodic s y s t e m p o s s e s s e s a s y s t e m of separable solutions, which is a strengthening of the above r e s u l t of LiUo. V. M. Millionshchikov [380] examined r e c u r s i o n m a t r i c e s A (t) (bourided and uniformly continuous on a line, such that e v e r y path of the s y s t e m DA is e v e r y w h e r e dense in RA; for notation see Sec. 1) and for these m a t r i c e s proved the following a s s e r t i o n : 1. A linear s y s t e m with a r e c u r s i o n coefficient m a t r i x can be r e d u c e d by a P e r r o n t r a n s f o r m a t i o n with a r e c u r s i o n m a t r i x to triangular f o r m also with a r e c u r s i o n matrix. 2. If the s y s t e m ~ = A (t)x with r e c u r s i o n m a t r i x is not a l m o s t r e d u c i b l e , an~(t) E R A can be found, such that ~ = ~(t) x is i r r e g u l a r . V. M. Millionshchikov [381, 397] also proved that for e v e r y uniformly a l m o s t periodic m a t r i x ACt) and for e v e r y e > 0, t h e r e exists a uniformly a l m o s t periodic (with the same frequency moduli) m a t r i x B(t), such that lIB ( t ) - A (t)[] ~ s and such that the s y s t e m ~ = B (t)x is statistically a l m o s t reducible (cf. Sec. 3). b) Exponents of Uniformly Alrnost P e r i o d i c Systems. This section and the preceding one join together in the-following a s s e r t i o n , which is fundamental in the general t h e o r y of uniformly almost periodic s y s t e m s . Millionshchikov T h e o r e m [380, 382, 393]. A uniformly a l m o s t periodic s y s t e m is almost reducible if and only if its exponents a r e stable. A r e f i n e m e n t of this t h e o r e m can be found in V. L. Novikov [420]. We note other r e s u l t s of the t h e o r y of exponents of uniformly a l m o s t periodic s y s t e m s . Bylov T h e o r e m s ([87] and [103], p. 190). 1. The upper (lower) singular and central exponents (cf. Sec. 1) of uniformly a l m o s t periodic s y s t e m s coincide. 2. The highest (lowest) exponents of a uniformly a l m o s t periodic s y s t e m is upper (lower) stable if and only if it coincides with the upper (lower) singular exponent. Millionshchikov's T h e o r e m s [381, 393,398]. 1. The highest (lowest) exponent for an absolutely r e g u l a r uniformly a l m o s t periodic s y s t e m coincides with the upper (lower) singular exponent ~o(~o).

54

2. The exponents of a uniformly a l m o s t periodic s y s t e m a r e stable if and only if the exponents of e v e r y s y s t e m ~/= :~(t) y, where ~(t) E R A , a r e stable. 3. F o r a l m o s t e v e r y ~(t) E R A (in the sense of any invariant m e a s u r e on DA), the highest (lowest) exponent of the s y s t e m ~ = ~(t) y is equal to its upper (respectively, lower) singular exponent. 4. We have for a two-dimensional uniformly a l m o s t periodic s y s t e m , a) its probable s p e c t r u m Ap(A) Consists of two n u m b e r s $t0 and COo;b) its P e r r o n t r a n s f o r m a t i o n can be r e d u c e d to triangular f o r m for int e g r a l mean diagonal coefficients Pi (t) under the equalities p~ = Pi and P2 = P2 (and the s y s t e m is almost reducible) or Pl = P2 and Pl = P2. 5. The probable exponents for e v e r y uniformly a l m o s t periodic s y s t e m coincide with the auxiliary exponents (cf. Sec. 1). 3 ~ Examples. N. P. E r u g i n ' s ([199], p. 88; [2i3], p. 137) c l a s s i c a l problem is considered one of the most important theoretical p r o b l e m s in the t h e o r y of linear s y s t e m s with a l m o s t periodic coefficients, This problem c o n s i s t s in determining the existence of i r r e g u l a r s y s t e m s with a l m o s t periodic and with quasiperiodic coefficients. V. M. Millionshchikov ([387]; cf. also Differents.Uravnen., 10, No. 3, 569 (1974)) constructed an e x a m ple of a not a l m o s t reducible s y s t e m with a l m o s t periodic coefficients (the question as to the existence of such s y s t e m s also r e m a i n s open); he also derived in this paper (proceeding on the basis of his r e s u l t s [380] p r e s e n t e d above) the existence of i r r e g u l a r s y s t e m s with almost periodic coefficients. Subsequently V. M. Millionshchikov [397] proved (using a proof principle identical to that of [387]) the existence as well of i r r e g u l a r s y s t e m s with quasi-periodic coefficients. V. L. Novikov, complicating the construction of Millionshchikov's example [387], constructed examples that proved: 1) T h e r e exists a uniformly a l m o s t periodic s y s t e m with highest exponent less than the upper singular exponent [420]; 2) T h e r e exists a not almost reducible uniformly a l m o s t periodic s y s t e m ~ = A (t)x, such that the highest exponent in e v e r y s y s t e m ~r = ~ ( t ) y , where ~(t) E RA, is equal to the upper singular exponent [421]. We r e f e r the r e a d e r also to Refs, 33, 41-47, 74, 83, 108, 114, 142, 180, 184, 188,222, 225-227, 241-245, 286,325, 352, 353,367, 378, 384, 386, 389, 390,399, 400, 403, 404, 425, 595, 632, 683, 672, 707, 708, 710, 762, 781, 786, 787, and 846. 7,

STABILITY

OF EXPONENTS

AND INTEGRAL

SEPARABILITY

OF SOLUTIONS P e r r o n [811], in which was proved (cf. also V. V. Nemytskii and V. V. Stepanov [416], p, 193) that the exponents of the perturbed ~ = [P (t) + Q (t)]y and initial diagonal of ~ = P (t)x of s y s t e m s when Q (t) "" 0 coincide as well as the separability condition Pi+t ( t ) - P i ( t ) >- a > 0, t >- 0 on the diagonal, was the f i r s t work on s y s t e m s with integral separability of solutions. 1

B. F. Bylov [85] weakened the separability condition to the r e q u i r e m e n t that i [P~+l(=)--Pl(~)]d:>~ s

> a (t -- s)-- d, a, d > 0, t > s, be i n t e g r a l - s e p a r a b l e and proved the stability of the exponents (of. Sec. 1) of a diagonal s y s t e m , under the assumption that it was r e g u l a r . The l a t t e r r e q u i r e m e n t was dispensed with by R. E. Vinograd [127]. Lillo [764] defined for a r b i t r a r y linear s y s t e m s the concept of separability of some of its solutions x 1 (t) . . . . . x n (t), such that the set of inequalities 1 gt(t--a)--~ fl--sl / ]] xi (t) [J / p ~.~(t-*)+v.lt-,f ~-e

-~ ~

with given r e a l n u m b e r s Xi and p =

~,~

, i=1

.....

n,

!

~ min[kj-),~l=#O , holds for all s and t.

Finally, B. F. Bylov [92] provided the following definition. 9Definition. A linear s y s t e m is said to be a s y s t e m with integral separability if it has solutions xl (t), . . . . xn(t), such that for all t > - s,

55

l[xt+, (t) li. If xi (t) II IIx~+,(s) II" Ii xt (s)It > dea(t-*)'

i = 1, 9

n - - 1,

(7,1)

with given c o n s t a n t s a, d > 0, and proved the reducibility of such s y s t e m s to P e r r o n - s e p a r a b l e diagonal s y s t e m s , m o r e o v e r e s t a b l i s h i n g the stability of the exponents of a linear s y s t e m with integral s e p a r a b i l i t y . The definition of integral s e p a r a b i l i t y is c l o s e l y r e l a t e d to the concept of exponential dichotomy; an i m p o r t a n t use and the h i s t o r y of the l a t t e r concept can be found in D. V. Anosov [16]. The theory of s y s t e m s with exponential dichotomy has been set forth in M a s s e r a and Shaeffer [364], Coppel [680], and Yu. L. Daletskii and M. G. Krein [181]. S

Subsequently B. F. Bylov and R. E. Vinograd {[96, 97]; [103], p. 207) proved a g e n e r a l t h e o r e m for the g e o m e t r i c location and e s t i m a t e of the growth of solutions of p e r t u r b e d s y s t e m s with s m a l l p e r t u r b a t i o n s , a s s u m i n g that the initial s y s t e m is b l o c k - t r i a n g u l a r with blocks of dimension n k, and such that we have the following integral s e p a r a b i l i t y conditions holding on the diagonal: t

t

t

t

J" r~d, 4 J Pi .(') d , 4 j' R~d,, ien~; ~ (rt+, -- Ok) d , > a (t -- s) -- d.

$

$

,$

$

The a s s e r t i o n that the set of s y s t e m s with integral s e p a r a b i l i t y is open in the m e t r i c s p a c e M n of n - d i m e n sional linear s y s t e m s with distance p[A it), B it) = sup ][ A ( t ) - B (t)[I i s implicit in this t h e o r e m and can be found in Bylov [98]. V. M. Millionshchikov [385] e s t a b l i s h e d that the integral s e p a r a b i l i t y condition (7.1) of a linear s y s t e m is equivalent to the p r o p e r t y that for e v e r y s > 0, a 6 > 0 can be found, such that if p[A (t), B (t)] O, such that [IV;L 1 (t, , ) p ' > c U a " - ' [ [ V ~

(t, ")If, t > * .

k = 1. . . . . m -

1,

w h e r e U k It, T) is the Cauchy m a t r i x of the s y s t e m hk = P k it) u k. 2. The upper ~ k and lower c0k c e n t r a l exponents for e v e r y block coincide. [

The sufficiency of this c r i t e r i o n had been p r e v i o u s l y proved in I3. F. Bylov, R. E. Vinograd, et al. ([103], p. 208) in T h e o r e m 15.2.1, which is a g e n e r a l i z a t i o n of the P e r r o n t h e o r e m given above. To conclude this section we p r e s e n t a stability coefficient t e s t for exponents (N. A. Izobov [258]). The exponents of a t w o - d i m e n s i o n a l l i n e a r s y s t e m a r e stable if the g r e a t e s t and l e a s t c h a r a c t e r i s t i c n u m b e r s of the coefficient m a t r i x a r e s e p a r a b l e and if the c h a r a c t e r i s t i c v e c t o r s c o r r e s p o n d i n g to t h e m a r e c o m p l e t e l y s e p a r a b l e . T h e s i m p l e s t e x a m p l e of such a s y s t e m is one with positive (negative) nonzero coefficientm not *The union of all oPen s e t s contained in this set. *Cf. Diba [190] for n-th 9 linear equations.

56

on the diagonals. It is wellknown (K. P. P e r s i d s k i i [436]) that only i n t e g r a l s e p a r a b i l i t y of the c h a r a c t e r i s tic n u m b e r s is sufficient for the stability of the exponents of a l i n e a r s y s t e m with P e r s i d s k i i weakly v a r y ing functional coefficients. We r e f e r the r e a d e r also to Secs. 4, 6, and 8-11 and to Refs. 50, 85, 86, 100, 101, 105, 129, 149, 167, 190, 358, 377, 384, 386, 389, 393, 398-400, 504, 505, 516-518, 667, 682, 683, 742, 744, 751, 760, 761, 763, 765, 766, 783, 858, 859, and 878. 8.

UNIFORMLY

COARSE

SEQUENCES

OF

PERIODIC

SYSTEMS

The following r e s u l t s of V. A. P l i s s a r e r e l a t e d to the p r e c e d i n g section. Definition [444, 449]. A sequence of n - d i m e n s i o n a l s y s t e m s

~=Am(t)x, m =

1, 2. . . . .

(8.1)

with p i e c e w i s e continuous u n i f o r m l y bounded w i n - p e r i o d i c m a t r i c e s , where com --- oo when t >- 0, is said to be u n i f o r m l y c o a r s e if for s o m e e > 0 e v e r y s y s t e m y = [A m (t) + Qm (t)]y, w h e r e lIQm (t)l] 1. We s e t Be (t) =- 0 on the i n t e r v a l [0, T] ( o n ( T - l , T), Be(t) m a y possibly vary) and w e t a k e as the solution y (t) of the s y s t e m (9.1) whose exponent we wish to be at l e a s t ~ - e , a r a p i d solution of the l i n e a r s y s t e m on [0, T]. If x it, y (T)) is a r a p i d solution on IT, 2T] of the l i n e a r s y s t e m , B e (t) -- 0 a l s o when t E [T, 2T]. But if it is a slow solution, then, i n a c c o r d a n c e with L e m m a 1, a v e c t o r a T can be found i n a n e - n e i g h b o r h o o d of the v e c t o r y iT), which we denote as b e f o r e by y (T), such that the solution x (t, y (T)) will turn out to be a r a p i d solutior/on [T, 2T]. We continuously connect the two values y ( T - 1) and y (T) of the solution y (t) of the s y s t e m (9.1) by m e a n s of the turning (9.2) indicated in L e m m a 2, which is not d e r i v e d f r o m the c l a s s of r e a c h a b l e p e r t u r b a t i o n s for 5 = e/(2M + 1). Thus in this c a s e B(t) = Q (t) when t E [ T - 1, T) and Be (t) - 0 when t E [T, 2T). By extending this c o n s t r u c t i o n to the entire h a l f : axis we will have a p e r t u r b e d s y s t e m (9.1) with solution y (t) whose n o r m s a t i s f i e s the r e q u i r e d bound k--1

[ly(kT)ll>~llY(0)tt H llX((Z+ 0T, ir)l[, ~ = ~ - sin~. i~l

The f i r s t p a r t of the t h e o r e m is proved. The r e a s o n why j u m p s a p p e a r in the exponents of linear s y s t e m s with s m a l l p e r t u r b a t i o n s was a l s o indicated in Yu. S. Bogdanov [60]. I

R. E. Vinograd ([103], p. 180), for diagonal s y s t e m s of even o r d e r , and T. E. Nuzhdova [423], for twodimensional s y s t e m s of g e n e r a l f o r m , each proved the s i m u l t a n e o u s r e a c h a b i l i t y of the c e n t r a l exponents, i.e., that t h e r e e x i s t s a m a t r i x Qe (t), such that both a s s e r t i o n s of the Millionshchikov t h e o r e m a r e t r u e . The C e n t r a l and Singular Exponents a r e Unstable iV. M. Millionshchikov [394]). T h e r e e x i s t s a l i n e a r s y s t e m of any o r d e r with nonzero singular exponents, such that for e v e r y natural m t h e r e exists a s y s t e m ~r = [A (t) + Qm (t)]y with z e r o singular exponents in which supllQm (t)ll --~ 0 as m--* o% However, t h e r e e x i s t v a r i o u s s e m i s t a b l e exponents (t3. F. Bylov, R. ~.. Vinograd, et al. [103] (p. 166). An r e a c h a b l e bound f o r the highest exponent of a s y s t e m with p e r t u r b a t i o n s IIQ (t)lI 0, x-p~r.pjj=x~._pi~ = 0

Millionshchikov [386, 389] also in the c a s e of a s t r i c t l y ergodic ([416], Chap. 6) dynamic s y s t e m D A proved that the probable s p e c t r u m of a linear s y s t e m is stable if and only it it is a l m o s t r e d u c i b l e (cf. also Secs. 4 and 6). 12.

ASYMPTOTICALLY

EQUIVALENT

SYSTEMS

Two linear s y s t e m s a r e said to be a s y m p t o t i c a l l y equivalent if/u. S. Bogdanov [56, 62, 66] or k i n e m a t ically s i m i l a r ( C e s a r i [562], p. 93) if t h e r e e x i s t s a Lyapunov t r a n s f o r m a t i o n that c o n v e r t s one into the other. T h i s concept lies at the b a s i s of the Lyapunov t h e o r y of linear differential s y s t e m s . An i m p o r t a n t r e s u l t in the g e n e r a l t h e o r y of a s y m p t o t i c a l l y equivalent s y s t e m s is the following a s s e r t i o n . P e r r o n Triangulation T h e o r e m [809]. F o r e v e r y l i n e a r s y s t e m t h e r e e x i s t s a unitary Lyapunov t r a n s f o r m a t i o n that c o n v e r t s it into a linear s y s t e m with a t r i a n g u l a r bounded and piecewise continuous coefficient m a t r i x .

60

New proofs of this t h e o r e m have been given by P. A. Kuz'min [297], R. E. Vinograd [118]~ and Dilib e r t o [695]. Hardly e v e r y linear s y s t e m can be r e d u c e d by a Lyapunov t r a n s f o r m a t i o n to diagonal or to even blockt r i a n g u l a r f o r m . The c o r r e s p o n d i n g c r i t e r i a of this reducibility in t e r m s of the Gramian were given by B. F. Bylov [92, 94]. In p a r t i c u l a r to r e d u c e a linear s y s t e m to diagonal f o r m it is n e c e s s a r y and sufficient that det X(t)[ > 7 H [[xi(t)[I, where T > 0 for the fundamental s y s t e m of solutions X = [xl . . . . .

Xn]. However,

it is t r u e that e v e r y linear s y s t e m with "slowly varying" coefficients is a l m o s t reducible to diagonal form, namely, we have the following important a s s e r t i o n . P e r s i d s k i i T h e o r e m [436]. For e v e r y s > 0, a Lyapunov t r a n s f o r m a t i o n y = Ss (t) x can be found that t r a n s f o r m s a l i n e a r s y s t e m with a w e a k l y v a r y i n g m a t r i x A (t) and having c h a r a c t e r i s t i c numbers Xi (t) into the almost diagonal s y s t e m y = diag [kt(t) . . . . .

~At)l y + Q (t) y

(12~

with components satisfying the inequality [!Q (t)][ a M. M o r e o v e r t h e r e e x i s t s a p e r t u r bation (5.1) with a < aM, such that the p e r t u r b e d system is no longer asymptotically equivalent to the initial system. The equivalence of a linear s y s t e m r e l a t i v e to the t r a n s f ~ Y (t) = X (t) S (t) has been studied by I. N. Zboichik [231, 234], O. A. K a s t r i t s a , and G. N. P e t r o v s k i i [268, 269]. The investigations of Yu. S. Bogdanov [58] and V. I. Arnol'd [20] on the analytic s t r u c t u r e of a m a t r i x that t r a n s f o r m s a variable m a t r i x to normal Jordan f o r m turns out to be useful in c o n s t r u c t i n g a triangulation t r a n s f o r m a t i o n . We r e f e r the r e a d e r also to Secs. 2, 4-7, and 11 and~to Refs. 1, 2, 52, 54, 93, 95, 100, 106, 188, 1 9 9 , 2 1 3 , 2 3 9 , 252, 258-260, 274, 280, 281,310, 338, 349,351, 356, 396, 402, 409, 424, 426, 427, 432, 442, 472, 482, 499, 500, 519, 521-523, 535, 603, 673, 676, 678, 692, 696, 724, 725, 746, 756, 757, 768, 775, 780, 790, 801, 828, 848, 853, 864,and 869. 13.

LINEAR

APPROXIMATION

STABILITY

Together with the initial linear s y s t e m with highest exponent X, we consider the s y s t e m

~=A(t)y+f(t,

y)

(13.1)

with m - p e r t u r b a t i o n s f , [If(t, Y)[I -~ N [I yll m, where m > I and t ~- 0 n e a r y = 0. In f i r s t approximation stability t h e o r y we have the following initial and fundamental a s s e r t i o n .

61

Lyapunov T h e o r e m (A. M. Lyapunov [349], pp.52-55, and M a s s e r a [363]). The zero solution of the s y s t e m (13.1) is asymptotically stable when

(m-- I) k-saL< 0.

(13.2)

This theorem was subsequentlyrefined by D. M. Grobman [168], who asserted that the variable (rL in Eq. (13.2) can be replaced by a smaller variable ~G (cf. Sec. 5). As regards the replacement of crL by ap in Eq. (13.2) we refer the reader to N. E. Bol'shakov [72]. Necessary and sufficient conditions on the diagonal coefficients, under which no given perturbations violate the stability or asymptotic stability of the zero solution, have been indicated for the case of a first approximation triangular system by Perron [810]. K. P. Persidskii [435] obtained sufficient conditions for such stability in the ease of a regular first approximation system and for unbounded coefficients in the expansions of f(t, y) in series. I. G. Malkin ([358], p. 379) used the bound 1]X(t.s)If< D exp[a(t--s) +[3s] of the Cauchymatrix of a linear system to prove the asymptotic stability of the zero solution of the system (13.1) under the condition ( m - l ) ~ +fl < 0. First approximation instabilitytests were constructed by N. G. Chataev [564] and by N. P. Erugin (Prikl. Matem. i Mekh., 16, No. 3 (1952)).

The Lyapunov t h e o r e m and its r e f i n e m e n t s p e r m i t bounds to be constructed of the value of m0 (though not to calculate this value exactly), beginning at which stability is observed, a s well as to obtain not always r e a c h a b l e upper bounds on the highest exponent A(f) (cf. [253, 134] for its definition) of the s y s t e m (13.1) withan m - p e r t u r b a t i o n f . The construction of these exact c h a r a c t e r i s t i c s was c a r r i e d out by means of R. E. Vinograd's central m-exponents ([103], p. 233). We have the following a s s e r t i o n . T h e o r e m {R. E . Vinograd [132, 134] and N. A. Izobov [253]). The highest exponent A(f) of e v e r y s y s tem (1)with an m - p e r t u r b a t i o n f does not exceed the c e n t r a l m-exponent ~2m that can be calculated by the Cauchy m a t r i x of the linear s y s t e m and its zero solution is asymptotically stable when 12m < 0. T h e r e exist m - p e r t u r b a t i o n s f that r e a l i z e t2 m (A(f) = ~m). t

The f i r s t a s s e r t i o n of the t h e o r e m was proved by means of an evaluating exponent (R. E. Vinograd ([103], p. 233; [132]) ~m----infl~m -1 R(t), where inf is taken over all pairs (W, R) of numbers W < 0 and #-+oo

t

piecewise continuous functions R (t) that obey the bound in iIX (t, s)H ~< W (t-- s)-5 R (t)-- mR (s), where s

SYSTEMS

OF

ORDINARY

DIFFERENTIAL

EQUATIONS N.

A.

Izobov

UDC 517.941.92

The p r e s e n t s u r v e y deals chiefly with the Lyapunov t h e o r y of c h a r a c t e r i s t i c exponents of the g e n e r a l f o r m of linear differential s y s t e m s , x=A(t)x

with bounded p i e c e w i s e continuous coefficients and p a r t i c u l a r l y with its m o s t r e c e n t advances. A sufficiently c o m p l e t e bibliography on such other independent b r a n c h e s of the t h e o r y of linear s y s t e m s as " L i n e a r S y s t e m s with P e r i o d i c Coefficients" (already s u r v e y e d in p a r t [469] in the s e r i e s G e n e r a l Mechanics) and the "Analytical T h e o r y of L i n e a r S y s t e m s " can be found in Sec. 15. The bibliography p r e s e n t e d in our s u r v e y b a s i c a l l y e n c o m p a s s e s all works on l i n e a r s y s t e m s a b s t r a c t e d in R e f e r a t i v n y i Zhurnal (Matematika). 1.

DEFINITION

AND

BASIC

PROPERTIES

OF

DIFFERENT

EXPONENTS

The Lyapunov c h a r a c t e r i s t i c exponent ([349], p. 27) of the solution x (t) # 0 of a linear s y s t e m is the number

-- ~

11 )It

A linear s y s t e m cannot have m o r e than n nonzero solutions with p a i r w i s e different exponents (Lyapunov [349], p. 34). A fundamental s y s t e m of solutions of a linear s y s t e m is called n o r m a l (Lyapunov [349], p. 34) if the sum of the exponents of its solutions is a m i n i m u m in the set of all fundamental s y s t e m s and is said to be b i n o r m a l (R. l~. Vinograd [126]) if in addition X-1 (t) is n o r m a l r e l a t i v e to the adjoint s y s t e m ~ = - A T ( t ) y . The exponents ~I - ~2 - 9 - ~n of a n o r m a l o r d e r e d (i.e., a r r a n g e d in n o n d e c r e a s i n g o r d e r of the exponents of the solution) s y s t e m will be called the exponents of the l i n e a r s y s t e m while the e x t r e m e exponents, Xi and 7~n, will be called its lowest and highest exponents. The solutions a r e distributed with r e s p e c t to t h e i r exponents in the following way (A. M. Lyapunov [349], p.34, and Yu. S. Bogdanov [48]): the e n t i r e s p a c e of initial values is divided into m l i n e a r s p a c e s L k embedded in each other, such that e v e r y nonzero solution outside it h a s an exponent l e s s than ~k (~'l < 9 . . < ~'m). Stability of the Exponents: The exponents of a linear s y s t e m a r e said to be stable if for any e > 0, a 6 > 0 can be found such that e v e r y exponent Xy of any p e r t u r b e d s y s t e m

~=A(Oy+Q(t)y, IIO(t)ll--0,

rain Ikr-- kil ~< .~. i

T h i s concept a p p e a r e d in P e r r o n [808] w h e r e it was e s t a b l i s h e d for the f i r s t t i m e that exponents can be unstable. T r a n s l a t e d f r o m Itogi Nauki i Tekhniki ( M a t e m a t i c h e s k i i Analiz), Vol. 12, P a r t 1, pp. 71-146, 1974,

9 Plenum P~+blishing Corporation, 22 7 West 17th Street, New York, N. ~: 10011. No part o f this publication may be reproduced, stored ht a retrieval system, or transmitted, in an), form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of tile publisher. A copy o f this article is aPailable from the publisher for $15.00.

46

The lower exponent ( P e r r o n [808]) is given by

~_[xl ~ li__~m+ in Ilx (011. Unlike the c h a r a c t e r i s t i c exponents of a linear s y s t e m , t h e r e m a y e x i s t m o r e than n different lower e x p o nents ( P e r r o n [808]). M o r e o v e r , a s y s t e m has been c o n s t r u c t e d whose set of lower exponents was the segment [0, 1] (N. A. Izobov [251]). In addition, the lower exponents a l s o obey a p r o p e r t y s i m i l a r to that obeyed by the c h a r a c t e r i s t i c exponents (N. A. Izobov [248]). T h i s p r o p e r t y m a y be stated a s follows: A l m o s t nil solutions beginning in a k - d i m e n s i o n a l (k = 2, . . . . n) s u b s p a c e of the s p a c e of initial data have a 2 o w e r exponent which is the g r e a t e s t exponent for the given solutions. The exact exponent o c c u r s in those solutions in which the c h a r a c t e r i s t i c and lower exponents coincide. t

The g r e a t e s t lower bound of the c o n s t a n t s R satisfying

llX(t, s)fI..- . . . ~ d n [X] a r e positive s q u a r e r o o t s of the eigenvalues of the m a t r i x X* X, and the set of a u x i l i a r y exponents will be called its a u x i l i a r y s p e c t r u m (v i coincides with the c e n t r a l exponent ~2). The Lyapunov n o r m X (Yu. S. Bogdanbv [ 51]) is defined on an a r b i t r a r y linear s e t A, h a s its own v a l ues X(a) of an e l e m e n t of a given o r d e r e d s e t A , and is such that: l) X(ca) - A(a), a E A and c is a r e a l n u m b e r , and 2) k (a + a ' ) ~ m a x {~.(a), ~.(a')}, w h e r e a, a v • A. The c h a r a c t e r i s t i c d e g r e e (B. P. Demidovs v Ix], such that * [xl ~- lira

[187]) of a solution x (t) with exponent X is that n u m b e r

* in Ilx (t)e-~qE.

We r e f e r the r e a d e r also to Refs. 57, 103, 122, 125, 219, 247, 389, 399, 400, 436, 456, 465, 690, 791, and 827. 2.

REDUCIBLE

AND ALMOST

REDUCIBLE

SYSTEMS

The Lyapunov t r a n s f o r m a t i o n s ([349], p. 42) keep the: a s y m p t o t i c c h a r a c t e r i s t i c s of s y s t e m s invariant and a r e therefDre used for c l a s s i f y i n g these c h a r a c t e r i s t i c s . The s i m p l e s t and m o s t i m p o r t a n t c l a s s of s y s t e m s singled out by Lyapunov include the following. 1_~, Reducible S y s t e m s ([349], p. 43). A l i n e a r s y s t e m is called r e d u c i b l e if t h e r e exists a Lyapunov t r a n s f o r m a t i o n that c o n v e r t s it into a s y s t e m with constant coefficients.

47

Lyapunov ([349], p. 195) indicated one i m p o r t a n t c l a s s of reducible s y s t e m s , n a m e l y s y s t e m s with periodic coefficient m a t r i c e s . F u r t h e r investigations of reducible s y s t e m s a r e a s s o c i a t e d with the n a m e N. P. Erugin and deal with the g e n e r a l theory he c o n s t r u c t e d of such s y s t e m s in the monograph "Reducible S y s t e m s . ~ In this monograph N. P. Erugin p r o p o s e d two methods for investigating r e d u c i b i l i t y p r o b l e m s . The f i r s t method is that of s u c c e s s i v e a p p r o x i m a t i o n s (fromwhich, incidentally, follow well-known a s y m p totic decompositions and which p e r m i t e s t i m a t e s to be obtained of the r e m a i n d e r t e r m in a s y m p t o t i c f o r m u las for the c a s e 6f an i r r e g u l a r singular.point, something that had not p r e v i o u s l y been done) (cf. V. V. Khoroshilov [556], L. I. Donskaya [192, 193], and I. N. Zboichik [232]). The second method c o n s i s t s in r e duction to s y s t e m s of a special type suitable for investigations (cf. A. E. G e l ' m a n [143; 144], L. Ya. A d r i a n o va [3], and L. V. T r i g u b o v i c h [525]). Let us d i s c u s s c e r t a i n c o n c r e t e r e s u l t s of N. P. E r u g i n ' s monograph. Reduciblity C r i t e r i o n ([199], p. 9). A l i n e a r s y s t e m is reducible if and only i f its fundamental s y s t e m of solutions can be r e p r e s e n t e d in the f o r m X(t) = S(t) e x p A t with Lyapunov m a t r i x S (t). R e a l n e s s of the T r a n s f o r m a t i o n ([199], p. 17; [213], p. 119). A r e d u c i b l e s y s t e m of a r e a l Lyapunov t r a n s f o r m a t i o n can be c o n v e r t e d into a s y s t e m with constant r e a l canonical coefficients m a t r i c e s . N e c e s s a r y Reducibility Condition ([199], p. 17). The integral of the t r a c e of the coefficient m a t r i x of t

a r e d u c i b l e s y s t e m can be r e p r e s e n t e d in the f o r m f SPA (Q d~=at-b~

(t)

with bounded function ~0(t).

0

Reducibility of an A n t i s y m m e t r i c System ([199], p. 20). A l i n e a r s y s t e m with an anti,symmetric m a t r i x (ajk = - akj for all j and k) is reducible, I. A. L a p p o - D a n i l e v s k i i [3051 C a s e and Reducibility ([1991, p. 24). If the m a t r i x A(t) c o m m u t e s with its integral ~ A (~)dr

for e v e r y s --- t and if we have the r e p r e s e n t a t i o n

s

fA (~)de =A + B(t)

with bounded

0

m a t r i x B (t) the linear s y s t e m can be r e d u c e d to the s y s t e m ~r = Ay. T r i a n g u l a r S y s t e m s ([199], p. 36). A t w o - d i m e n s i o n a l upper t r i a n g u l a r linear s y s t e m is r e d u c i b l e to a diagonal f o r m if and only if its diagonal coefficients have the r e p r e s e n t a t i o n aii (t) = a i + 9 i (t) with bounded t

integrals

I ~'(:)d:

t

and, if at = a2 = a, t h e integral

](t)=fax~(~)ex p I(y2--*t)d~d*

0

0

sional upper t r i a n g u l a r linear s y s t e m is r e d u c i b l e to the s y s t e m

(~

is bounded. A t w o - d i m e n -

0

y = Oa

y. if the function Cvl (t) + C2t,

w h e r e C 2 ~ 0, is bounded. In p a r t i c u l a r ([199], pp. 28-32) the reducibility p r o b l e m for the t w o - d i m e n s i o n a l s y s t e m (1) with upper t r i a n g u l a r m a t r i x

A(O= ~ A~t-.,.

(*)

nt~O

has been exhaustively investigated. The nonanalytic c a s e for a m a t r i x A (t) as well a s the n - d i m e n s i o n a l c a s e have a l s o b e e n examined. R i c c a t i Equation and Reducibility. N. P. Erugin ([199], p. 68) e s t a b l i s h e d a r e l a t i o n between r e d u c i b i l ity of a t w o - d i m e n s i o n a l s y s t e m to t r i a n g u l a r f o r m and the existence of a bounded solution or a solution that tends to +oo a s t --~ + ~ for the equation u' = - al2 u2 + ( a 2 2 - - all) U + a 2 1 . New reducibility t e s t s w e r e obtained using it, for example, a reducibility c r i t e r i o n for a s y s t e m with a nontriangular m a t r i x of the f o r m (*) as well as ([199], p. 82) a linear s y s t e m i n w h i c h aj2, a~2--aH>~>O, a~t>O, akk~ak+~k(t), at-~a~, t >0,

48

is reducible to the s y s t e m ~ --- diag[a I, a2]y if the i n t e g r a l s

t

t

l?k(~.)d~, and

~a~x(Qd~ a r e

0

-0

bounded.

R e m a r k . In r e d u c i n g a linear s y s t e m to t r i a n g u l a r f o r m it is n e c e s s a r y to find a given bounded solution. But s o m e t i m e s such a r e a l bounded solution does not exist while a complex bounded solution does exist, which also leads to a solution of the p r o b l e m (see e x a m p l e [199], p. 80). P e r t u r b a t i o n s that P r e s e r v e the Reducibility P r o p e r t y . N. P. Erugin ([199], p. 63) set forth the p r o b l e m a s t o how s m a l l Q (t) m u s t be such that the p e r t u r b e d s y s t e m ~r = [A (t) + Q (t)]y r e m a i n s r e d u c i b l e to the s a m e s y s t e m a s initially. Its solution was given in the f o r m of u n i m p r o v a b l e (differential) e s t i m a t e s or an a l g o r i t h m for obtaining t h e m by m e a n s of the f i r s t method. N. P. E r u g i n ' s m o n o g r a p h [213] in which his r e s u l t s w e r e set forth and studies of A. M. Lyapunov, I. A. L a p p o - D a n i l e v s k i i , N. N. Bogulyubov, and L Z. Shtokalo on linear s y s t e m s w e r e brought t o g e t h e r , also c o n s t i t u t e s an i m p o r t a n t contribution to the l i t e r a t u r e on t h e s e p r o b l e m s . T h i s book h a s e x e r t e d a significant influence on the f u r t h e r development of different lines of r e s e a r c h into l i n e a r s y s t e m s . We r e f e r the r e a d e r also to Secs. 6 and 12 and to Refs. 24, 27, 54, 103, 137, !41, 207, 203, 209, 211, 218-220, 228, 271, 319, 332, 333, 361,362. 370, 373, 442, 463, 600, 605, 757, and 826. 2 ~ A l m o s t Reducible S y s t e m s (B. F. Bylov [85, 88] and Lillo [766]). A linear s y s t e m is said to be almosY-reducible to the s y s t e m

:~=B (t) y.

(2.1)

if for any 6 > 0 t h e r e e x i s t s a Lyapunov t r a n s f o r m a t i o n y = Ss(t)x that t r a n s f o r m s it into the s y s t e m ~ = [B (t) + Q6(t)]z if the inequality IIQ6 (0tl 0 there, e x i s t s a Lyapunov t r a n s f o r m a t i o n y = S6(t) x that r e d u c e s it to the f o r m = diag [xi . . . . . x,,] y ~- B6 (t) y ~u Q6 (t) y, w h e r e Q5 (t) is a t r i a n g u l a r m a t r i x with z e r o diagonal, IlQ~ (t)I[ a ( L k l i + . . . + l k m l ) -~, a, d > 0 , where k is any integral-valued vector, the inner Lebesgue m e a s u r e of the set M (Q) of constant m a t r i c e s A f r o m the unit sphere $1, such that the s y s t e m is reducible, satisfies the condition rues M (Q)-* mes S 1 when IIQtt -~ 0. Other proofs of the Adrianova-Samoilenko t h e o r e m can be obtained by means of the Lille t h e o r e m (cf. 2~ below). Yu. A. Mitropol'skii and A. M. Samoilenko's joint paper [405], which preceded [473], constructed reducibility t e s t s for such quasi-periodic s y s t e m s by t r a n s f o r m a t i o n with a quasi-periodic maVrix, while Yu. A. Mit-ropol'skii and E. A. Belan [401] used an a c c e l e r a t e d convergence method to c o n s t r u c t a uniform13, a l m o s t periodic t r a n s f o r m a t i o n of an a l m o s t diagonal (with s e p a r a b l e diagonal) uniformly a l m o s t periodic s y s t e m to a diagonal uniformly a l m o s t periodic f o r m whose e x i s t e n c e had been proved by B. F, Bylov. (cf. 2~

53

V. Kh. Kharasakhal and his colleagues studied the quasi-periodic s y s t e m ~ = A (t) x with an m - d i m e n sional basis by means of the s y s t e m of linear equations and partial derivatives with coefficients of different periods, Ox/Ou=F(ul . . . . . urn)x, F ( t . . . . . t)==A(t).

An analog of the Floquet t h e o r e m was successfully proved for the l a t t e r s y s t e m under additional a s s u m p tions on the p r o p e r t i e s of the initial s y s t e m , which p e r m i t t e d reducibility and r e g u l a r i t y t e s t s for quasiperiodic s y s t e m s to be derived. We r e f e r the r e a d e r also to Refs. 295, 296, 467, 528, 544-551, 593, 594, and 601. The f i r s t set also includes V. G. Sprindzhuk [487] in which r e p r e s e n t a b i l i t y conditions for an integral of a function f i x , r . . . . . Wmx) one-periodic in each independent variable in the f o r m at + O (1) w e r e indicated and as a c o r o l l a r y reducibility t e s t s w e r e obtained for two-dimensional quasi--periodic s y s t e m s . T h e s e p a p e r s (cf. also K. G. Valeev [110]) also contain stability t e s t s for quasi-periodic and uniformly a l m o s t periodic s y s t e m s . 2 ~ a) General Study of Uniformly Almost P e r i o d i c S y s t e m s : Lillo [766], in which the a l m o s t p e r i o d i c ity of the P e r r o n t r a n s f o r m a t i o n m a t r i x for a uniformly a l m o s t periodic system with separable solutions (cf. Sec. 7) to triangular f o r m was proved by means of a set of displacements (cf. Sec. 1), was one of the f i r s t works along these lines. In his p r e c e d i n g work [764], it was proved that the set of uniformly a l m o s t periodic s y s t e m s with separable solutions is open in the space of all uniformly a l m o s t periodic s y s t e m s . B. F. Bylov [91] constructed a reducibility test for the uniformly a l m o s t periodic s y s t e m ~ = A (t) x using a l m o s t periodic Lyapunov t r a n s f o r m a t i o n s , to a diagonal s y s t e m with uniformly a l m o s t periodic coefficients (and consequently (B. F. Bylov [88]), a l m o s t reducibility) in the f o r m of c e r t a i n constraints on e v e r y s y s t e m ~ = ~(t) y. In p a r t i c u l a r such a reduction is possible if a uniformly a l m o s t periodic s y s t e m p o s s e s s e s a s y s t e m of separable solutions, which is a strengthening of the above r e s u l t of LiUo. V. M. Millionshchikov [380] examined r e c u r s i o n m a t r i c e s A (t) (bourided and uniformly continuous on a line, such that e v e r y path of the s y s t e m DA is e v e r y w h e r e dense in RA; for notation see Sec. 1) and for these m a t r i c e s proved the following a s s e r t i o n : 1. A linear s y s t e m with a r e c u r s i o n coefficient m a t r i x can be r e d u c e d by a P e r r o n t r a n s f o r m a t i o n with a r e c u r s i o n m a t r i x to triangular f o r m also with a r e c u r s i o n matrix. 2. If the s y s t e m ~ = A (t)x with r e c u r s i o n m a t r i x is not a l m o s t r e d u c i b l e , an~(t) E R A can be found, such that ~ = ~(t) x is i r r e g u l a r . V. M. Millionshchikov [381, 397] also proved that for e v e r y uniformly a l m o s t periodic m a t r i x ACt) and for e v e r y e > 0, t h e r e exists a uniformly a l m o s t periodic (with the same frequency moduli) m a t r i x B(t), such that lIB ( t ) - A (t)[] ~ s and such that the s y s t e m ~ = B (t)x is statistically a l m o s t reducible (cf. Sec. 3). b) Exponents of Uniformly Alrnost P e r i o d i c Systems. This section and the preceding one join together in the-following a s s e r t i o n , which is fundamental in the general t h e o r y of uniformly almost periodic s y s t e m s . Millionshchikov T h e o r e m [380, 382, 393]. A uniformly a l m o s t periodic s y s t e m is almost reducible if and only if its exponents a r e stable. A r e f i n e m e n t of this t h e o r e m can be found in V. L. Novikov [420]. We note other r e s u l t s of the t h e o r y of exponents of uniformly a l m o s t periodic s y s t e m s . Bylov T h e o r e m s ([87] and [103], p. 190). 1. The upper (lower) singular and central exponents (cf. Sec. 1) of uniformly a l m o s t periodic s y s t e m s coincide. 2. The highest (lowest) exponents of a uniformly a l m o s t periodic s y s t e m is upper (lower) stable if and only if it coincides with the upper (lower) singular exponent. Millionshchikov's T h e o r e m s [381, 393,398]. 1. The highest (lowest) exponent for an absolutely r e g u l a r uniformly a l m o s t periodic s y s t e m coincides with the upper (lower) singular exponent ~o(~o).

54

2. The exponents of a uniformly a l m o s t periodic s y s t e m a r e stable if and only if the exponents of e v e r y s y s t e m ~/= :~(t) y, where ~(t) E R A , a r e stable. 3. F o r a l m o s t e v e r y ~(t) E R A (in the sense of any invariant m e a s u r e on DA), the highest (lowest) exponent of the s y s t e m ~ = ~(t) y is equal to its upper (respectively, lower) singular exponent. 4. We have for a two-dimensional uniformly a l m o s t periodic s y s t e m , a) its probable s p e c t r u m Ap(A) Consists of two n u m b e r s $t0 and COo;b) its P e r r o n t r a n s f o r m a t i o n can be r e d u c e d to triangular f o r m for int e g r a l mean diagonal coefficients Pi (t) under the equalities p~ = Pi and P2 = P2 (and the s y s t e m is almost reducible) or Pl = P2 and Pl = P2. 5. The probable exponents for e v e r y uniformly a l m o s t periodic s y s t e m coincide with the auxiliary exponents (cf. Sec. 1). 3 ~ Examples. N. P. E r u g i n ' s ([199], p. 88; [2i3], p. 137) c l a s s i c a l problem is considered one of the most important theoretical p r o b l e m s in the t h e o r y of linear s y s t e m s with a l m o s t periodic coefficients, This problem c o n s i s t s in determining the existence of i r r e g u l a r s y s t e m s with a l m o s t periodic and with quasiperiodic coefficients. V. M. Millionshchikov ([387]; cf. also Differents.Uravnen., 10, No. 3, 569 (1974)) constructed an e x a m ple of a not a l m o s t reducible s y s t e m with a l m o s t periodic coefficients (the question as to the existence of such s y s t e m s also r e m a i n s open); he also derived in this paper (proceeding on the basis of his r e s u l t s [380] p r e s e n t e d above) the existence of i r r e g u l a r s y s t e m s with almost periodic coefficients. Subsequently V. M. Millionshchikov [397] proved (using a proof principle identical to that of [387]) the existence as well of i r r e g u l a r s y s t e m s with quasi-periodic coefficients. V. L. Novikov, complicating the construction of Millionshchikov's example [387], constructed examples that proved: 1) T h e r e exists a uniformly a l m o s t periodic s y s t e m with highest exponent less than the upper singular exponent [420]; 2) T h e r e exists a not almost reducible uniformly a l m o s t periodic s y s t e m ~ = A (t)x, such that the highest exponent in e v e r y s y s t e m ~r = ~ ( t ) y , where ~(t) E RA, is equal to the upper singular exponent [421]. We r e f e r the r e a d e r also to Refs, 33, 41-47, 74, 83, 108, 114, 142, 180, 184, 188,222, 225-227, 241-245, 286,325, 352, 353,367, 378, 384, 386, 389, 390,399, 400, 403, 404, 425, 595, 632, 683, 672, 707, 708, 710, 762, 781, 786, 787, and 846. 7,

STABILITY

OF EXPONENTS

AND INTEGRAL

SEPARABILITY

OF SOLUTIONS P e r r o n [811], in which was proved (cf. also V. V. Nemytskii and V. V. Stepanov [416], p, 193) that the exponents of the perturbed ~ = [P (t) + Q (t)]y and initial diagonal of ~ = P (t)x of s y s t e m s when Q (t) "" 0 coincide as well as the separability condition Pi+t ( t ) - P i ( t ) >- a > 0, t >- 0 on the diagonal, was the f i r s t work on s y s t e m s with integral separability of solutions. 1

B. F. Bylov [85] weakened the separability condition to the r e q u i r e m e n t that i [P~+l(=)--Pl(~)]d:>~ s

> a (t -- s)-- d, a, d > 0, t > s, be i n t e g r a l - s e p a r a b l e and proved the stability of the exponents (of. Sec. 1) of a diagonal s y s t e m , under the assumption that it was r e g u l a r . The l a t t e r r e q u i r e m e n t was dispensed with by R. E. Vinograd [127]. Lillo [764] defined for a r b i t r a r y linear s y s t e m s the concept of separability of some of its solutions x 1 (t) . . . . . x n (t), such that the set of inequalities 1 gt(t--a)--~ fl--sl / ]] xi (t) [J / p ~.~(t-*)+v.lt-,f ~-e

-~ ~

with given r e a l n u m b e r s Xi and p =

~,~

, i=1

.....

n,

!

~ min[kj-),~l=#O , holds for all s and t.

Finally, B. F. Bylov [92] provided the following definition. 9Definition. A linear s y s t e m is said to be a s y s t e m with integral separability if it has solutions xl (t), . . . . xn(t), such that for all t > - s,

55

l[xt+, (t) li. If xi (t) II IIx~+,(s) II" Ii xt (s)It > dea(t-*)'

i = 1, 9

n - - 1,

(7,1)

with given c o n s t a n t s a, d > 0, and proved the reducibility of such s y s t e m s to P e r r o n - s e p a r a b l e diagonal s y s t e m s , m o r e o v e r e s t a b l i s h i n g the stability of the exponents of a linear s y s t e m with integral s e p a r a b i l i t y . The definition of integral s e p a r a b i l i t y is c l o s e l y r e l a t e d to the concept of exponential dichotomy; an i m p o r t a n t use and the h i s t o r y of the l a t t e r concept can be found in D. V. Anosov [16]. The theory of s y s t e m s with exponential dichotomy has been set forth in M a s s e r a and Shaeffer [364], Coppel [680], and Yu. L. Daletskii and M. G. Krein [181]. S

Subsequently B. F. Bylov and R. E. Vinograd {[96, 97]; [103], p. 207) proved a g e n e r a l t h e o r e m for the g e o m e t r i c location and e s t i m a t e of the growth of solutions of p e r t u r b e d s y s t e m s with s m a l l p e r t u r b a t i o n s , a s s u m i n g that the initial s y s t e m is b l o c k - t r i a n g u l a r with blocks of dimension n k, and such that we have the following integral s e p a r a b i l i t y conditions holding on the diagonal: t

t

t

t

J" r~d, 4 J Pi .(') d , 4 j' R~d,, ien~; ~ (rt+, -- Ok) d , > a (t -- s) -- d.

$

$

,$

$

The a s s e r t i o n that the set of s y s t e m s with integral s e p a r a b i l i t y is open in the m e t r i c s p a c e M n of n - d i m e n sional linear s y s t e m s with distance p[A it), B it) = sup ][ A ( t ) - B (t)[I i s implicit in this t h e o r e m and can be found in Bylov [98]. V. M. Millionshchikov [385] e s t a b l i s h e d that the integral s e p a r a b i l i t y condition (7.1) of a linear s y s t e m is equivalent to the p r o p e r t y that for e v e r y s > 0, a 6 > 0 can be found, such that if p[A (t), B (t)] O, such that [IV;L 1 (t, , ) p ' > c U a " - ' [ [ V ~

(t, ")If, t > * .

k = 1. . . . . m -

1,

w h e r e U k It, T) is the Cauchy m a t r i x of the s y s t e m hk = P k it) u k. 2. The upper ~ k and lower c0k c e n t r a l exponents for e v e r y block coincide. [

The sufficiency of this c r i t e r i o n had been p r e v i o u s l y proved in I3. F. Bylov, R. E. Vinograd, et al. ([103], p. 208) in T h e o r e m 15.2.1, which is a g e n e r a l i z a t i o n of the P e r r o n t h e o r e m given above. To conclude this section we p r e s e n t a stability coefficient t e s t for exponents (N. A. Izobov [258]). The exponents of a t w o - d i m e n s i o n a l l i n e a r s y s t e m a r e stable if the g r e a t e s t and l e a s t c h a r a c t e r i s t i c n u m b e r s of the coefficient m a t r i x a r e s e p a r a b l e and if the c h a r a c t e r i s t i c v e c t o r s c o r r e s p o n d i n g to t h e m a r e c o m p l e t e l y s e p a r a b l e . T h e s i m p l e s t e x a m p l e of such a s y s t e m is one with positive (negative) nonzero coefficientm not *The union of all oPen s e t s contained in this set. *Cf. Diba [190] for n-th 9 linear equations.

56

on the diagonals. It is wellknown (K. P. P e r s i d s k i i [436]) that only i n t e g r a l s e p a r a b i l i t y of the c h a r a c t e r i s tic n u m b e r s is sufficient for the stability of the exponents of a l i n e a r s y s t e m with P e r s i d s k i i weakly v a r y ing functional coefficients. We r e f e r the r e a d e r also to Secs. 4, 6, and 8-11 and to Refs. 50, 85, 86, 100, 101, 105, 129, 149, 167, 190, 358, 377, 384, 386, 389, 393, 398-400, 504, 505, 516-518, 667, 682, 683, 742, 744, 751, 760, 761, 763, 765, 766, 783, 858, 859, and 878. 8.

UNIFORMLY

COARSE

SEQUENCES

OF

PERIODIC

SYSTEMS

The following r e s u l t s of V. A. P l i s s a r e r e l a t e d to the p r e c e d i n g section. Definition [444, 449]. A sequence of n - d i m e n s i o n a l s y s t e m s

~=Am(t)x, m =

1, 2. . . . .

(8.1)

with p i e c e w i s e continuous u n i f o r m l y bounded w i n - p e r i o d i c m a t r i c e s , where com --- oo when t >- 0, is said to be u n i f o r m l y c o a r s e if for s o m e e > 0 e v e r y s y s t e m y = [A m (t) + Qm (t)]y, w h e r e lIQm (t)l] 1. We s e t Be (t) =- 0 on the i n t e r v a l [0, T] ( o n ( T - l , T), Be(t) m a y possibly vary) and w e t a k e as the solution y (t) of the s y s t e m (9.1) whose exponent we wish to be at l e a s t ~ - e , a r a p i d solution of the l i n e a r s y s t e m on [0, T]. If x it, y (T)) is a r a p i d solution on IT, 2T] of the l i n e a r s y s t e m , B e (t) -- 0 a l s o when t E [T, 2T]. But if it is a slow solution, then, i n a c c o r d a n c e with L e m m a 1, a v e c t o r a T can be found i n a n e - n e i g h b o r h o o d of the v e c t o r y iT), which we denote as b e f o r e by y (T), such that the solution x (t, y (T)) will turn out to be a r a p i d solutior/on [T, 2T]. We continuously connect the two values y ( T - 1) and y (T) of the solution y (t) of the s y s t e m (9.1) by m e a n s of the turning (9.2) indicated in L e m m a 2, which is not d e r i v e d f r o m the c l a s s of r e a c h a b l e p e r t u r b a t i o n s for 5 = e/(2M + 1). Thus in this c a s e B(t) = Q (t) when t E [ T - 1, T) and Be (t) - 0 when t E [T, 2T). By extending this c o n s t r u c t i o n to the entire h a l f : axis we will have a p e r t u r b e d s y s t e m (9.1) with solution y (t) whose n o r m s a t i s f i e s the r e q u i r e d bound k--1

[ly(kT)ll>~llY(0)tt H llX((Z+ 0T, ir)l[, ~ = ~ - sin~. i~l

The f i r s t p a r t of the t h e o r e m is proved. The r e a s o n why j u m p s a p p e a r in the exponents of linear s y s t e m s with s m a l l p e r t u r b a t i o n s was a l s o indicated in Yu. S. Bogdanov [60]. I

R. E. Vinograd ([103], p. 180), for diagonal s y s t e m s of even o r d e r , and T. E. Nuzhdova [423], for twodimensional s y s t e m s of g e n e r a l f o r m , each proved the s i m u l t a n e o u s r e a c h a b i l i t y of the c e n t r a l exponents, i.e., that t h e r e e x i s t s a m a t r i x Qe (t), such that both a s s e r t i o n s of the Millionshchikov t h e o r e m a r e t r u e . The C e n t r a l and Singular Exponents a r e Unstable iV. M. Millionshchikov [394]). T h e r e e x i s t s a l i n e a r s y s t e m of any o r d e r with nonzero singular exponents, such that for e v e r y natural m t h e r e exists a s y s t e m ~r = [A (t) + Qm (t)]y with z e r o singular exponents in which supllQm (t)ll --~ 0 as m--* o% However, t h e r e e x i s t v a r i o u s s e m i s t a b l e exponents (t3. F. Bylov, R. ~.. Vinograd, et al. [103] (p. 166). An r e a c h a b l e bound f o r the highest exponent of a s y s t e m with p e r t u r b a t i o n s IIQ (t)lI 0, x-p~r.pjj=x~._pi~ = 0

Millionshchikov [386, 389] also in the c a s e of a s t r i c t l y ergodic ([416], Chap. 6) dynamic s y s t e m D A proved that the probable s p e c t r u m of a linear s y s t e m is stable if and only it it is a l m o s t r e d u c i b l e (cf. also Secs. 4 and 6). 12.

ASYMPTOTICALLY

EQUIVALENT

SYSTEMS

Two linear s y s t e m s a r e said to be a s y m p t o t i c a l l y equivalent if/u. S. Bogdanov [56, 62, 66] or k i n e m a t ically s i m i l a r ( C e s a r i [562], p. 93) if t h e r e e x i s t s a Lyapunov t r a n s f o r m a t i o n that c o n v e r t s one into the other. T h i s concept lies at the b a s i s of the Lyapunov t h e o r y of linear differential s y s t e m s . An i m p o r t a n t r e s u l t in the g e n e r a l t h e o r y of a s y m p t o t i c a l l y equivalent s y s t e m s is the following a s s e r t i o n . P e r r o n Triangulation T h e o r e m [809]. F o r e v e r y l i n e a r s y s t e m t h e r e e x i s t s a unitary Lyapunov t r a n s f o r m a t i o n that c o n v e r t s it into a linear s y s t e m with a t r i a n g u l a r bounded and piecewise continuous coefficient m a t r i x .

60

New proofs of this t h e o r e m have been given by P. A. Kuz'min [297], R. E. Vinograd [118]~ and Dilib e r t o [695]. Hardly e v e r y linear s y s t e m can be r e d u c e d by a Lyapunov t r a n s f o r m a t i o n to diagonal or to even blockt r i a n g u l a r f o r m . The c o r r e s p o n d i n g c r i t e r i a of this reducibility in t e r m s of the Gramian were given by B. F. Bylov [92, 94]. In p a r t i c u l a r to r e d u c e a linear s y s t e m to diagonal f o r m it is n e c e s s a r y and sufficient that det X(t)[ > 7 H [[xi(t)[I, where T > 0 for the fundamental s y s t e m of solutions X = [xl . . . . .

Xn]. However,

it is t r u e that e v e r y linear s y s t e m with "slowly varying" coefficients is a l m o s t reducible to diagonal form, namely, we have the following important a s s e r t i o n . P e r s i d s k i i T h e o r e m [436]. For e v e r y s > 0, a Lyapunov t r a n s f o r m a t i o n y = Ss (t) x can be found that t r a n s f o r m s a l i n e a r s y s t e m with a w e a k l y v a r y i n g m a t r i x A (t) and having c h a r a c t e r i s t i c numbers Xi (t) into the almost diagonal s y s t e m y = diag [kt(t) . . . . .

~At)l y + Q (t) y

(12~

with components satisfying the inequality [!Q (t)][ a M. M o r e o v e r t h e r e e x i s t s a p e r t u r bation (5.1) with a < aM, such that the p e r t u r b e d system is no longer asymptotically equivalent to the initial system. The equivalence of a linear s y s t e m r e l a t i v e to the t r a n s f ~ Y (t) = X (t) S (t) has been studied by I. N. Zboichik [231, 234], O. A. K a s t r i t s a , and G. N. P e t r o v s k i i [268, 269]. The investigations of Yu. S. Bogdanov [58] and V. I. Arnol'd [20] on the analytic s t r u c t u r e of a m a t r i x that t r a n s f o r m s a variable m a t r i x to normal Jordan f o r m turns out to be useful in c o n s t r u c t i n g a triangulation t r a n s f o r m a t i o n . We r e f e r the r e a d e r also to Secs. 2, 4-7, and 11 and~to Refs. 1, 2, 52, 54, 93, 95, 100, 106, 188, 1 9 9 , 2 1 3 , 2 3 9 , 252, 258-260, 274, 280, 281,310, 338, 349,351, 356, 396, 402, 409, 424, 426, 427, 432, 442, 472, 482, 499, 500, 519, 521-523, 535, 603, 673, 676, 678, 692, 696, 724, 725, 746, 756, 757, 768, 775, 780, 790, 801, 828, 848, 853, 864,and 869. 13.

LINEAR

APPROXIMATION

STABILITY

Together with the initial linear s y s t e m with highest exponent X, we consider the s y s t e m

~=A(t)y+f(t,

y)

(13.1)

with m - p e r t u r b a t i o n s f , [If(t, Y)[I -~ N [I yll m, where m > I and t ~- 0 n e a r y = 0. In f i r s t approximation stability t h e o r y we have the following initial and fundamental a s s e r t i o n .

61

Lyapunov T h e o r e m (A. M. Lyapunov [349], pp.52-55, and M a s s e r a [363]). The zero solution of the s y s t e m (13.1) is asymptotically stable when

(m-- I) k-saL< 0.

(13.2)

This theorem was subsequentlyrefined by D. M. Grobman [168], who asserted that the variable (rL in Eq. (13.2) can be replaced by a smaller variable ~G (cf. Sec. 5). As regards the replacement of crL by ap in Eq. (13.2) we refer the reader to N. E. Bol'shakov [72]. Necessary and sufficient conditions on the diagonal coefficients, under which no given perturbations violate the stability or asymptotic stability of the zero solution, have been indicated for the case of a first approximation triangular system by Perron [810]. K. P. Persidskii [435] obtained sufficient conditions for such stability in the ease of a regular first approximation system and for unbounded coefficients in the expansions of f(t, y) in series. I. G. Malkin ([358], p. 379) used the bound 1]X(t.s)If< D exp[a(t--s) +[3s] of the Cauchymatrix of a linear system to prove the asymptotic stability of the zero solution of the system (13.1) under the condition ( m - l ) ~ +fl < 0. First approximation instabilitytests were constructed by N. G. Chataev [564] and by N. P. Erugin (Prikl. Matem. i Mekh., 16, No. 3 (1952)).

The Lyapunov t h e o r e m and its r e f i n e m e n t s p e r m i t bounds to be constructed of the value of m0 (though not to calculate this value exactly), beginning at which stability is observed, a s well as to obtain not always r e a c h a b l e upper bounds on the highest exponent A(f) (cf. [253, 134] for its definition) of the s y s t e m (13.1) withan m - p e r t u r b a t i o n f . The construction of these exact c h a r a c t e r i s t i c s was c a r r i e d out by means of R. E. Vinograd's central m-exponents ([103], p. 233). We have the following a s s e r t i o n . T h e o r e m {R. E . Vinograd [132, 134] and N. A. Izobov [253]). The highest exponent A(f) of e v e r y s y s tem (1)with an m - p e r t u r b a t i o n f does not exceed the c e n t r a l m-exponent ~2m that can be calculated by the Cauchy m a t r i x of the linear s y s t e m and its zero solution is asymptotically stable when 12m < 0. T h e r e exist m - p e r t u r b a t i o n s f that r e a l i z e t2 m (A(f) = ~m). t

The f i r s t a s s e r t i o n of the t h e o r e m was proved by means of an evaluating exponent (R. E. Vinograd ([103], p. 233; [132]) ~m----infl~m -1 R(t), where inf is taken over all pairs (W, R) of numbers W < 0 and #-+oo

t

piecewise continuous functions R (t) that obey the bound in iIX (t, s)H ~< W (t-- s)-5 R (t)-- mR (s), where s

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