Continuity of numeric characteristics for asymptotic stability of solutions to linear difference equations with periodic coefficients

Applied Mathematics

Continuity of numeric characteristics for

asymptotic stability of solutions to linear

dierence equations with periodic coecients

Kemal Aydn, Haydar Bulgak

1

and Gennadi Demidenko

2? 1 Research Centre of Applied Mathematics, Selcuk University, Konya, Turkey

e-mail: kaydin@selcuk.edu.tr

e-mail: hbulgak@selcuk.edu.tr

2 Sobolev Institute of Mathematics, SB RAS, 630090 Novosibirsk, Russia

e-mail: demidenk@math.nsc.ru

Received: May 21, 2001

Summary.

We consider linear perturbation systems of dierence equations y(n+ 1) = (A(n) +B(n))y(n), n 0whereA(n), B(n) areNN periodic matrices with periodT. The spectrum of a

mon-odromy matrix of the systemx(n+ 1) = A(n)x(n),n 0 belongs to the unit diskfjj<1g. We indicate conditions on a perturbation

matrixB(n) for asymptotic stability of the zero solution to the per-turbation system and prove continuity one numeric characteristic of the asymptotic stability from 1].

Key words:

monodromy matrix, perturbation systems of dier-ence equations, asymptotic stability of solutions

Mathematics Subject Classication (1991): 39A11

1. Introduction

In this paper we consider linear perturbation systems of dierence equations with periodic coecients:

(1:1) y(n+ 1) = (A(n) + B(n))y(n) n 0

? The research was nancially supported by the Scientic and Technical

Re-search Council of Turkey (TUBITAK) in the framework of the NATO-PC Ad-vanced Fellowships Programme.

where A(n) and B(n) are N N periodic matrices with period T,

i. e.

A(n+T) =A(n) B(n+T) =B(n) n 0:

We suppose, that the zero solution of the system (1:2) x(n+ 1) = A(n)x(n) n 0

is asymptotically stable. By the Spectral Criterion, each eigenvalue of the monodromy matrix of the system

(1:3) X(T) =A(T;1):::A(1)A(0)

belongs to the unit diskfjj<1g. According to the Lyapunov

Crite-rion, the zero solution to system (1.2) is asymptotically stable if and only if the series

(1:4) F = 1

X k =0

(X(T))k(X(T))k

converges (see, for example, 2, 3]). Note that the matrix seriesF is a solution to the discrete matrix Lyapunov equation

(1:5) X(T)FX(T)

;F =;I:

In the paper 1] we indicated some numeric characteristics of the asymptotic stability of the zero solution to system (1.1) without ap-pealing to the spectrum of the monodromy matrix (1.3). Using these characteristics one can obtain various estimates for solutions fx(n)g

to system (1.2). In particular, we considered the following numeric characteristic for asymptotic stability of the zero solution to (1.2) (1:6) !1(AT) =

kFk

wherekFk is the spectral norm of the matrix series (1.4).

The following statement holds 1].

Theorem 1.

The solution to system (1.2) satises the estimates

kx(n)kkX(m)k  1; 1 !1(AT)  k =2 !1(AT) 1=2 kx(0)k n=kT+m k 0 0mT;1:

Our aim is to obtain conditions on a perturbation matrixB(n) for asymptotic stability of the zero solution to system (1.1) and prove continuity of the norm of series (1.4).

2. Conditions for asymptotic stability of solutions

In this section we obtain conditions on a perturbation matrix B(n) for asymptotic stability of the zero solution to system (1.1).

Let

Y(T) = (A(T ;1) +B(T;1)):::(A(1) +B(1))(A(0)+B(0))

be the monodromy matrix of system (1.1).

Theorem 2.

If the perturbation matrixB(n)such that the inequality holds (2:1) kY(T);X(T)k< s kX(T)k 2+ 1 !1(AT) ;kX(T)k

then the zero solution to system (1.1) is asymptotically stable. Proof. Note that inequality (2.1) is equivalent to

(2:2) = 1;(2kX(T)kkY(T);X(T)k+kY(T);X(T)k 2)

kFk>0:

ByT -periodicity,Y(kT) = (Y(T))k,k 0, therefore the sequence fY(kT)g is the solution to the problem

(2:3) Y((k+ 1)T) = (X(T)+ (Y(T);X(T))Y(kT) k 0

Y(0) =I:

Consider the form

hFY((k+ 1)T)v Y((k+ 1)T)vi v2E N: By (1.5) and (2.3), we have hFY((k+ 1)T)v Y((k+ 1)T)vi =hFY(kT)v Y(kT)vi;hY(kT)v Y(kT)vi +h(X (T)F(Y(T) ;X(T)) + (Y(T);X(T)) FX(T) +(Y(T);X(T)) F(Y(T) ;X(T)))Y(kT)v Y(kT)vi:

Using (2.2), we obtain the inequality

hFY((k+ 1)T)v Y((k+ 1)T)vi  1;  kFk  hFY(kT)v Y(kT)vi:

Consequently, for every k 0 we have

hFY(kT)v Y(kT)vi  1;  kFk  k hFv vi:

The matrixF is positive denite. Hence, for every vectorv2E N k(Y(T))

kv

k=kY(kT)vk!0 k!1:

Then the spectrum of the monodromy matrixY(T) must lie in the diskfjj<1g. According to the Spectral Criterion, the zero solution

to system (1.1) is asymptotically stable. ut

Corollary 1.

Let (2:4) a= max 0jT;1 fkA(j)kg b= max 0jT;1 fkB(j)kg:

If the perturbation matrix B(n) such that the inequality holds

(2:5) Tb(a+b)T;1< s kX(T)k 2+ 1 kFk ;kX(T)k

then the zero solution to system (1.1) is asymptotically stable. Proof. It is easy to verify that the inequality holds

kY(T);X(T)k(a+b) T ;a T Tb(a+b) T;1:

Therefore, asymptotic stability of the zero solution to system (1.1) is immediate from (2.1), (2.5). ut

3. Continuity of numeric characteristics for asymptotic

stability of solutions

We will suppose, that for the perturbation matrix B(n) the condi-tion of theorem 2 is true. Hence, the zero solucondi-tion to system (1.1) is asymptotically stable. Then one can consider the matrix series (3:1) Fe =

1 X k =0

(Y(T))k(Y(T))k:

The matrixFe is a solution to the discrete matrix Lyapunov equation

(3:2) Y(T) e

FY(T); e

F =;I:

By analogy with (1.6), we dene the numeric characteristic for asym-ptotic stability of the zero solution to (1.1) to be

!1(A+BT) = k

e

According to theorem 1, the estimates ky(n)kkY(m)k  1; 1 !1(A+BT)  k =2 !1(A+BT) 1=2 ky(0)k n=kT +m k 0 0mT;1

hold for solutions to system (1.1).

Theorem 3.

If the perturbation matrix B(n) such that inequality (2.1) holds, then (3:3) k e F ;Fk 1;  !1(AT)

where the constant  is dened by (2.2). Proof. By (1.5) and (3.2), we have (3:4) Y(T)( e F ;F)Y(T);( e F ;F) =;C where C=X(T)F(Y(T) ;X(T))+ (Y(T);X(T)) FX(T) +(Y(T);X(T)) F(Y(T) ;X(T)):

The discrete matrix Lyapunov equation (3.4) has a unique solution

e F ;F = 1 X k =0 (Y(T))kC(Y(T))k:

Hence, by (3.1), we have the inequality (3:5) k e F ;FkkCkk e FkkCkk e F ;Fk+kCkkFk: Obviously, kCk(2kX(T)kkY(T);X(T)k+kY(T);X(T)k 2) kFk= 1; and we obtain (3.3). ut

Corollary 2.

If the perturbation matrixB(n)such that the inequality holds (3:6) kY(T);X(T)k< s kX(T)k 2+ 1 2!1(AT) ;kX(T)k then (3:7) k e F ;Fk2kY(T);X(T)k! 1(AT)   kX(T)k+ q kX(T)k 2+ 1 2!1(AT)  :

Proof. By (3.5) and (3.6), one can obtain 1 2k e F ;Fk(2kX(T)k+kY(T);X(T)k)kY(T);X(T)kkFk   kX(T)k+ s kX(T)k 2+ 1 2!1(AT) ! kY(T);X(T)kkFk: Hence, we come to (3.7). ut

Remark 1.Some analogous results for dierence equations with con-stant coecients are given in 4{6].

Remark 2.In the paper 7] we consider another numeric characteris-tics for asymptotic stability of solutions to (1.1) from 1].

Remark 3.A survey of results on numeric characteristics for asymp-totic stability of solutions to linear dierential equations with con-stant coecients dy

dt = Ay is given in 6]. Results for the case of

linear dierential equations with periodic coecients dy

dt = A(t)y,

A(t+T) = A(t), were obtained in 8].

References

1. Aydn, K., Bulgak, H. and Demidenko, G. V. (2000): Numeric characteristics for asymptotic stability of solutions to linear dierence equations with periodic coecients, Siberian Math. J.,41, No. 6, 1005{1014.

2. Dalecki, Ju. L. and Kren M. G. (1974):StabilityofSolutionsofDierential Equations in BanachSpace, Translations of Mathematical Monographs. 43.

Providence, RI: American Mathematical Society.

3. Elaydi, S. N. (1996):AnIntroductiontoDierenceEquations, Springer-Verlag,

New York.

4. Godunov, S. K. (1998): Modern Aspects of Linear Algebra, Translations of

Mathematical Monographs. 175. Providence, RI: American Mathematical So-ciety.

5. Akn, O. and Bulgak, H. (1998):LinearDierenceEquationsandStability The-ory, in Turkish], Selcuk University, Research Centre of Applied Mathematics,

Konya.

6. Bulgak, H. (1999): Pseudoeigenvalues, spectral portrait of a matrix and their connections with dierent criteria of stability, in: Error Control and Adaptivity in Scientic Computing, Kluwer Acad. Publ., Dordrecht, pp. 95-124. (NATO Science Ser.)

7. Aydn, K., Bulgak, H. and Demidenko, G. V.: Asymptotic stability of solutions to perturbation linear dierence equations with periodic coecients, Siberian Math. J. (to appear).

8. Demidenko, G. V. and Matveeva, I. I. (2001): On stability of solutions to linear systems with periodic coecients, Siberian Math. J.,42, No. 2, 282{296.