Structural stability of linear difference equations in Hilbert space.PDF

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Pergamon Computers Math Applic. Vol. 36, No. 10-12, pp. 71-76, 1998 1998 Elsevier Science Ltd. All rights reservedPrinted in Great Britain0898-1221/98 19.00 + 0.00P I I : S 0 8 9 8 - 1 2 2 1 ( 9 8 ) 0 0 2 0 1 - 6

S t r u c t u r a l S t a b i l i t y o f L i n e a ri f f e r e n c e E q u a t i o n s i nH i l b e r t S p a c eB. AULBACHDepartment of Mathematics, University of AugsburgD-86135 Augsburg, Germany

N. VAN MINHDepartment of Mathematics, University of HanoiKhoa Toan Dai hoc Tong hop, Hanoi, VietnamP. P. ZABREIKODepartment of Mathematics and MechanicsByelorussian State University, Minsk 220080, Byelorussia

Abs t rac t - - In this note, we prove that linear differenceequations in Hilbert space are structurallystable if and only if they have an exponential dichotomy. 1998 Elsevier Science Ltd. All rightsreserved.Keywordsmstr uc tu ra l stability, Linear differenceequations, Exponential dichotomy.

1 . I N T R O D U C T I O NIn the Qualitative Theory of Differential Equations and Difference Equations, the notion ofStructural Stability plays a vital role. This notion, introduced by Pontrjagin and Andronovfor differential equations, has been developed for dynamical systems of different types, such asdifferentiable dynamical systems, nonautonomons differential equations, difference equations etc.In this note, we consider the structural stability problem for linear difference equations, a problemwhich has been investigated before see [1] and the references therein) . In [1], it is proved that inthe finite dimensional space R n a linear difference equation is s tructu rally stable if and only if ithas an exponential dichotomy. The main result is proved by referring to the differential equationcase [2,3]. The proof of the main result in [1], however, is not complete. This is due to thefact th at in thi s proof the given difference equation is converted into a differential equation withpiecewise constant , discontinuous coefficients. The result applied from [2,3], however, requiresthe cont inui ty of those coefficients.In this note, we give a direct proof of the main result in [1] for difference equations in Hilbertspace. Based on the main result in [4], we then show the st ructural stability of linear differenceequations having an exponential dichotomy.

2 P R E L I M I N A R I E SWe consider linear difference equations of the form

x ~ l = A ~ x ~ , 1 )Typeset by .A~I,q-TEX

71


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72 B . AULBAOH e t al.w h e r e n i s a n i n t e g e r a n d e a c h An : X --* X i s a n i n v e r t i b l e li n e a r o p e r a t o r o n a H i l b e r t s p a c e X .T h r o u g h o u t t h i s p a p e r , w e a s s u m e t h a t

s u p [ [Anl l < o o a n d s u p I I A f f l ll oo.n e ZDEFINITION 1. E q u a t i o n 1 ) i s s a i d to h a v e a n e x p on e n t ia l d i c h o t o m y f f t h e r e e x i s t s a p r o j e c t i o nP : X - 4 X a n d p o s i t i v e c o n s t a n t s D , a s u c h t h a t

I I X ( n ) P X - ' ( m ) l l < _ D e - ' ~ ( n - O ,I I X n ) / - P ) X - ( m ) l l < - D e - C ( m - n) ,

f o r n ~ m ,f o r n _~ m ,

wh e r e X n ) i s th e s o l u ti o n o p e r a t o r o f e q u a t i o n 1 ) s a t i s f y in g X O ) = I : = i d x .A p a r t f r o m e q u a t i o n 1 ) , w h i c h w e k e e p fi x e d i n t h e s e q u e l , w e d e a l w i t h f u r t h e r e q u a t i o n s o f

t h e t y p eY n l = B n Y n , 2 )

w h e r e f o r e a c h n E Z , t h e B n : 3 --* Y i s a l i n e a r o p e r a t o r o n a H i l b e r t sp a c e y .DEFINITION 2. Eq uat io ns 1 ) and 2 ) are sa id to be topolog ica ll y equ iva len t f f t h e r e ex i s t s as e q ue n c e h n ) n e z o f h o m e o m o r p h i s m s hn : X --~ Y h a v i n g t h e p r o p e r t i e s

i) hn+l o An = Bn o hn , for M1 n E Z ,ii) limllzll_~oo I h . z ) l l = l i m l l x l l - ~ I I h v , X x ) l l oo un i formly for n E Z .

F r o m w e l l - k n o w n r e s u l t s o n e c a n d e r i v e t h e f o l lo w i n g l e m m a .L E M M A 1 . S u p p o s e e q u a t i o n 1 ) h a s a n e x p o n e n t i a l d i c h o t o m y . T h e n t h e r e ex i s t s a sequenceTn ) n e z o f b o u n d e d l i n e a r o p e r a t o r s Tn : X ~ X h a v i n g t h e f o l lo w i n g p r o p e r t ie s :i ) s u p n e z I I T . I I c ~ , s u P n e z I I T ~ X l l o o ,

i i) T n + I A n T ~ 1 c o m m u t e s f o r each n E Z w i th t h e p r o j e c t i o n c o r r e s p o n d i n g t o t h e e x p o n e n t i a ld i c h o t o m y o f 1 ) .

R EM A RK . F r o m L e m m a 1 it f ol lo w s t h a t i f e q u a t i o n 1 ) h a s a n e x p o n e n t i a l d i c h o t o m y w i t hp r o j e c t i o n P , t h e n i t i s t o p o l o g i c a ll y e q u i v a le n t t o t h e d e c o u p l e d e q u a t i o n

U n l = P A n u n ,v n + l = I - P ) A n v n ,

un E I m Pvn E K e r P ,

w h e r e - 4n : = Tn + x A n T ~ a w i t h Tn as i n L e m m a 1 .THEOREM 1. S e e , e .g ., [ 4 ]. ) A s s u m e t h a t e q u a t i o n 1 ) h a s a n e x p o n e n t i a l d i c h o to m y . Th e nf o r ~ s u t~c i e n t l y s m a l l , a n y p e r t u r b e d e q u a t i o n o f t h e f o r m

x . + 1 = A . + C . ) z . , 3 )w i t h s u p n e z l [C n[ l < _ 6 a s o h a s a n e x p o n e n t i a l i c h o t o m y . I n a d d i t i o n, e n o t i n g b y P a n d P 6 t h ep r o j e c t i o n s o r r e s p o n d i n g t o t h e e x p o n e n t i a l i c h o t o m i e s o f e q u a t i o n s i ) a n d 3 ) , e sp e c t iv e l y,t h e m a p p i n g

P : I m P 6 --* I m Ps a n e a r h o m e o m o r p h i s m .


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E q u a t i o n s i n H i l b e r t S p ace 73

3 S T R U C T U R A L S T A B I L I T YDEFINITION 3 . Equ at ion 1) i s said to be s t ruc tural ly s table i f t h e r e ex is ts a posi t ive 60 sucht h a t any pe r turb e d e qua t ion o f t he f orm 3) w i th s u p n e z [ [C , [[ < 60 i s topological ly equivalentto equat ion 1) .DEFINITION 4 . E qua t ion 1 ) i s sa id t o b e k inemat ical ly s imi lar to equat ion 2) , i f t h e r e ex is ts ase que nc e T . ) . ~ z of l inear operators Tn : X --* X such tha t

i ) s u p . e z I I T . ]i < o 0 , s u p . e z I l T ; l i i < ~ o ,ii) T , + I A n T ~ 1 = B , , f o r a ll n Z .PROPOSITION 1 . Sup pose equat ion 1) i s s t ruc tura l ly s table a n d k ine mat i c a l l y s im i lar t o equa-t ion 2) . Th en also equat ion 2) i s s t ruc tu ral ly s t a b l e .P R O O F . S u p p o s e t h a t 6 o i s a p o s i t iv e c o n s t a n t s u c h t h a t f o r a l l 6 < 6 o e q u a t i o n 3 ) w i t hs u p n e z [ [ C , [[ < 6 o i s t o p o l o g i c a l l y e q u i v a l e n t t o e q u a t i o n 1 ). C o n s i d e r a n y p e r t u r b a t i o n

x . + l = B . + D . ) x n , 4 )o f e q u a t i o n 2 ) w i t h s u p n e z l I D . l l < 6 0 / s u p . I I T .I I s u p . l I T ; I l l . W e a r e g o i n g to s h o w t h a te q u a t i o n 4 ) i s t o p o l o g i c a l l y e q u i v a l e n t t o e q u a t i o n 2 ) w h i c h m e a n s e q u a t i o n 2 ) i s s t r u c t u r a l l ys t a b l e . I n f a c t , s e t t i n g C n := T ~ . I D n T , w e h a v e

sup HC , II < sup I IT ,~ l l ] . sup I ITnl l. sup [ ID,[ I < 60 .. . ? .T h e n , t h e e q u a t i o n

X . + l = A . + C . ) x . , 5 )i s t o p o l o g i c a l l y e q u i v a l e n t t o e q u a t i o n 1 ) . B u t 5 ) i s k i n e m a t i c a l l y s i m i l a r t o 4 ) , a n d 1 ) i sk i n e m a t i c a l l y s i m i l a r t o 2 ) . H e n c e , e q u a t i o n 5 ) i s t o p o l o g i c a l ly e q u i v a l e n t t o e q u a t i o n 2 ) . T h i sc o m p l e t e s t h e p r o o f o f th e p r o p o s it i o n.R E M A R K . I t f o l l o w s f r o m L e m m a 1 a n d P r o p o s i t i o n 1 t h a t w h e n d e a l i n g w i t h t h e s t r u c t u r a ls t a b i li t y o f e q u a t i o n s h a v i n g a n e x p o n e n t i a l d i c h o t o m y a n d s a ti s fy i n g s o m e n o , d e g e n e r a c y c o n-d i t io n s i n L e m m a 1 , w e m a y a s s u m e w i t h o u t l o ss o f g e n e r a li ty t h a t t h e u n d e r l y i n g e q u a t io n i sd e c o u p l e d , i .e . , t h a t t h e o p e r a t o r - c o e ff i c i e n t c o m m u t e s w i t h t h e p r o j e c t i o n P c o r r e s p o n d i n g t ot h e e x p o n e n t i a l d i c h o t o m y o f t h e e q u a t i o n .L E M M A 2 . Le t e qua t ion 1 ) sa t i s f y a ll as sumpt ions o f Le m ma 1 and suppose P = O. The n ,equat ion 1) i s topological ly equivalent to the equat ion

= . + 1 = e = . . 6 )

P R O O F . F i r s t , w e d e fi n e a n o p e r a t o r X t , s ) , t , s E R b y t h e r e l a t i o nA[, ]_ I .A [ t ]_ 2 . . - A [ , ] , fo r [t] _> [s] + 1 ,

X t , s ) = I , for It] = [s],X s , t ) ) - 1 , fo r [ t ] < [s ] ,

w h e r e [ t] d e n o t e s t h e i n t e g e r p a r t o f t h e r e a l n u m b e r t . N o t e t h a t X n , m ) , n , m Z is th eC a u c h y o p e r a t o r o f e q u a t i o n 1 ). N e x t , w e s e t

V t , x ) : = S t ) x , x ) , f o r a l l x X , 7 )wh r t

s t ) : = / X * s, t ) X s, t ) d s.--00

8 )


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74 B. AULBACHat aLH e r e , X * s , t) d e n o t e s t h e c o n j u g a t e o p e r a t o r o f X s , t ). T h e n , V t , x ) h a s t h e f o l l o w i n g p r o p e r -t i e s :

i) V t , n ) i s p i e c e w i s e c o n t i n u o u s w i t h r e s p e c t t o t ,i i) i f x n ) i s a s o l u t i o n o f e q u a t i o n 1 ) , t h e n V t , x [t ] ) ) i s c o n t i n u o u s w i t h r e s p e c t t o t E R ,

i ii ) f o r a l l t ~ Z , w e h a v ed v t , z [ t ] ) ) : x c t , 0 ) z o = z [ t ] ) , 9 )

w h e r e x n ) i s s o m e s o l u t i o n o f e q u a t i o n 1 ) .i v ) T h e r e e x i s t p o s i t i v e c o n s t a n t s a > b s u c h t h a t

b l lz l l 2 - 1 / v ~ , w e h a v e S s ) z , z) >_ 1 . So t s , z ) > _I,I I X t , s ) x l l 2 a

h o l d t r u e f o r a ll ~ Z . T h u s , w e g e to _ < s - t s , ~ ) _ < a C a I 1 ~ 1 1 2 - 1 ) . 1 4 )

F o r z s a t is f y i n g S s ) z , z ) < I , s a y HzH


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E q u a t i o n s i n H i l b e rt p a c e 7 5w h e r e f o r t E N , w e p u t ~ , ( u , s , x ) = O . T h u s , w e o b t a i n

t(s,z)0


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76 B. AULBACHet al.F r o m t h e d e f i n i t i o n o f h a , h~ 1, it i s e a s i l y c h e c k e d t h a t

h , o x t ) = x s ) o h . , 2 0 )f o r a l l t , s E R , w h e r e X t ) d e n o t e s X t , 0 ) , t E R . N o w it s u ff i ce s o c h o o s e t , s E Z t o ge t t h ea s se r ti o n o f L e m m a 2 .T H E O R E M 2 . L e t e q u a t i o n 1 ) h a v e a n e x p o n e n t i a l d i c h o t o m y . T h e n , e q u a t i o n 1 ) i s s t r u c t u r a l l ys t a b l e .P R O O F . I n f ac t , f r o m L e r n m a 2 i t f o l l o w s t h a t e q u a t i o n 1 ) i s t o p o l o gi c a l l y q u i v a l e n t t o

~ ) n + l

F r o m T h e o r e m 1 i t fo ll ow s t h a t s i n c e f o ra l e n t t o

U n + l = e t tn ,1~)n +l - --~ --~)n,ei t i s t o p o l o g i c a l l y e q u i v a l e n t t o e q u a t i o n 21) .c o m p l e t e s t h e p r o o f o f T h e o r e m 2 .

e u n , u n E K e r P ,1 2 1 )- v ~ , v . E I m P .e

6 su f f ic i e n t ly sm a l l ) e qua t ion 3 ) i s t opo lo g i c a l l y e qu iv -u n E K e r P 6 ,v n E I m P 6 , 2 2 )S o e q u a t i o n 1 ) i s s t r u c t u r a l l y s t a b l e . T h i s

R E M A R K . I n t h i s p a p e r , w e h a v e c o n s i d e r e d o n l y t h e c a s e w h e r e n E 7 . . T h e r e i s n o d i f f i c u l t y ,h o w e v e r , t o e x t e n d a l l t h e r e s u l ts t o t h e c a s e w h e r e t h e d i ff e re n c e e q u a t i o n s u n d e r c o n s i d e r a t io na r e o n l y d e f i n e d f o r n E N .

R E F E R E N C E S1 . J . K urzw ef l and G . P apasch inop ou loe , S t ruc tu ra l s t ab i l i t y o f l i nea r d i sc r e t e sys t em s v i a t he expone n t i a ld i c h o t o m y , Czech. Math. J . $8, 280-284 (1988) .2 . K . P a lm er , A ch a rac t e r i z a t i on o f expo nen t i a l d i cho tom y in t e rm s o f t opo log i ca l equ iva l ence , J. Math. Anal.Appl. 69, 8--16 (1979).3 . K . P a l m e r , T h e s t r u c t u r a l l y s t a b l e l i n e a r s y st e m s o n t h e h a l f -l in e a r e t h o s e w i t h e x p o n e n t i a l d ic h o t o m ie s ,J . D /~ . E q~ . aS , 16 -25 (1979) .4 . B . A u lbac h , N . V an M inh a nd P . P . Z sb re iko , T he concep t o f spec t r a l d i cho tom y fo r l i nea r d i f f e rence equa -t i ons , ,7. Math. Anal. Appl. 185, 276-287 (1994) .

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Structural stability of linear difference equations in Hilbert space.PDF