Stability radii of delay difference systems under affine parameter perturbations in infinite dimensional spaces

Applied Mathematics and Computation 202 (2008) 562–570

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Stability radii of delay difference systems under affine parameter perturbations in infinite dimensional spaces Bui The Anh a,*, Nguyen Khoa Son b, Duong Dang Xuan Thanh c a b c

Department of Mathematics, University of Pedagogy, 280 An Duong Vuong Street, HoChiMinh City, Viet Nam Institute of Mathematics, 18 Hoang Quoc Viet Road, 10307 Hanoi, Viet Nam Department of Mathematics, Ton Duc Thang University, 98 Ngo Tat To Street, HoChiMinh City, Viet Nam

a r t i c l e

i n f o

a b s t r a c t In this paper, we study the stability radii of positive difference systems with delays under arbitrary affine parameter perturbations in infinite dimensional spaces. The obtained results are extensions of the recent results in [P.H.A. Ngoc, N.K. Son, Stability radii of positive linear difference equations under affine parameter perturbation, Appl. Math. Comput. 134 (2003) 577–594; A. Fischer, Stability radii of infinite-dimensional positive systems, Math. Control Signals Syst. 10 (1997) 223–236]. Ó 2008 Elsevier Inc. All rights reserved.

Keywords: Positive difference system Affine parameter perturbation Stability radius

1. Introduction In the last two decades, a considerable attention has been paid to problems of robust stability of dynamic systems in infinite dimensional spaces. The interested readers are referred to [1–6,10] and the biography therein for further references. One of the most important problems in study of robust stability is the calculation of the stability radius of a dynamic system subjected to various classes of parameter perturbations. Although there have been many works dedicated to stability radii problems of linear systems, however, so far the problem of computing the stability radii of delay difference systems under arbitrary affine perturbations has not yet been studied. In this paper, we study stability radii for delay difference systems in infinite dimensional spaces described by the linear difference equation of the form xðt þ K þ 1Þ ¼ A0 xðt þ KÞ þ A1 xðt þ K  1Þ þ    þ AK xðtÞ;

t; K 2 N

where Ai 2 LðXÞ; i 2 K ¼ f0; 1; . . . ; Kg, under arbitrary parameters of the form Ai ,! Ai þ

N X

Dij Dij Eij ;

i2K

j¼1

and Ai ,! Ai þ

N X

dij Aij ;

i 2 K;

j¼1

* Corresponding author. E-mail address: [email protected] (B.T. Anh). 0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.02.041

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where Dij 2 LðU ij ; XÞ; Eij 2 LðX; Y ij Þ, and Aij 2 LðXÞi 2 K; j 2 N ¼ f1; . . . ; Ng are given operators defining the scaling and structure of the parameter uncertainties, and Dij 2 LðY ij ; U ij Þ and dij are, respectively, unknown operators and scalars representing parameter uncertainties. The purpose of present paper is to generalize the results of [7] to the system in infinite dimensional spaces and these of [5] to linear system with delays. The organization of the paper is as follows. In the next section, we recall some notations and results on positive operators on Banach lattices. The main results will be addressed in Section 3. A formula for complex stability radius is given. Then we show that for positive systems the complex, real and positive stability radius coincide and can be computed by the simple formulas. Finally, a simple example is given to illustrate the obtained results. 2. Preliminaries In this section, we recall some useful results for later use. First, we recall some notations and definitions. Let X and Y be complex Banach lattices, and let X þ and Y þ denote positive cones of X and Y, respectively; LR ðX; YÞ and þ L ðX; YÞ, the sets of all the real and positive linear operators from X to Y, respectively. For A 2 LðXÞ let rðAÞ denote the spectrum of A, qðAÞ ¼ C n rðAÞ the resolvent set of A, and Rðk; AÞ ¼ ðkI  AÞ1 2 LðXÞ the resolvent of A defined on qðAÞ. The spectral radius rðAÞ is defined by rðAÞ ¼ maxfjkj : k 2 rðAÞg: The following result is known form the Perron–Frobenius theory of bounded positive operator, see [8]. Theorem 2.1. Suppose T 2 Lþ ðXÞ. Then (i) rðTÞ 2 rðTÞ. (ii) Rðk; TÞ P 0 if and only if k 2 R and k > rðTÞ. We end this section by the following proposition. Proposition 2.2. Suppose X; Y are Banach lattices and A 2 Lþ ðXÞ; B 2 Lþ ðX; YÞ. Then jBRðk; AÞxj 6 BRðjkj; AÞjxj;

jkj > rðAÞ;

x 2 X:

Proof. Because jkj > rðAÞ; k 2 qðAÞ and Rðjkj; AÞ is positive. Moreover, we have jBRðk; AÞxj 6

   1  1 1 X  BAn  X  x  X BAn   BAn  nþ1  6 jxj 6 BRðjkj; AÞjxj: knþ1 x 6 nþ1 j k n¼0 n¼0 n¼0 jk

The proof is complete. h

3. Stability radii of difference system with delays 3.1. Complex stability radius Now, we consider the difference system with delays of the form xðt þ K þ 1Þ ¼ A0 xðt þ KÞ þ A1 xðt þ K  1Þ þ    þ AK xðtÞ;

t; K 2 N;

ð1Þ

where Ai 2 LðXÞ; i 2 K ¼ f0; 1; . . . ; Kg. Set PðkÞ ¼ A0 þ k1 A1 þ    þ kK AK ;

k 6¼ 0:

We denote the resolvent set of PðÞ by qðPðÞÞ ¼ fk 2 C : ½kI  PðkÞ1 2 LðXÞg; and the spectrum set of PðÞ by rðPðÞÞ ¼ C n qðPðÞÞ. The spectral radius of PðÞ can be defined rðPðÞÞ ¼ maxfjkj : k 2 rðPðÞÞg: Now we will introduce to an yðtÞ ¼ ðxðt þ KÞ; xðt þ K  1Þ; . . . ; xðtÞÞ yðt þ 1Þ ¼ AyðtÞ; where A 2 LðX 0

1

Kþ1

equivalent

state-space

t 2 N;

system

easily

determined

by

the

state

vector ð2Þ

Þ defined by

Aðx ; x ; . . . ; xK Þ ¼ ðA0 x0 þ A1 x1 þ    þ AK xK ; x0 ; x1 ; . . . ; xK1 Þ:

ð3Þ

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With given k 2 C and ðy0 ; y1 ; . . . ; yK Þ 2 X Kþ1 we looking for ðx0 ; x1 ; . . . ; xK Þ such that ðkI  AÞðx0 ; x1 ; . . . ; xK Þ ¼ ðy0 ; y1 ; . . . ; yK Þ: Solving this equation for k 6¼ 0 we have 8 ½kI  PðkÞx0 ¼ y0 þ A1 ðk1 y1 Þ þ    þ AK ðk1 yK þ k2 yK1 þ    þ kK y1 Þ; > > > 1 < x ¼ k1 y1 þ k1 x0 ; >   > > : K x ¼ k1 yK þ k1 xK1 :

ð4Þ

From (4), we obtain that qðPðÞÞ ¼ qðAÞ n f0g:

ð5Þ

By direct calculation, we have Rðk; AÞðy; 0; . . . ; 0Þ ¼ ðRðk; PðkÞÞy; k1 Rðk; PðkÞÞy; . . . ; kK Rðk; PðkÞÞyÞ:

ð6Þ

Due to (5), it is easy to see that system (1) is stable if and only if system (3) is stable, i.e, rðAÞ  C1 ; C1 ¼ fk 2 C : jkj < 1g. Furthermore, this is equivalent rðPðÞÞ  C1 , or, rðPðÞÞ ¼ rðAÞ < 1. In this section, we always assume that system (1) is stable. Now, we suppose that all operators Ai ; i 2 K are subjected to perturbations of the form Ai ,! Ai þ

N X

Dij Dij Eij ;

i 2 K;

ð7Þ

j¼1

where Dij 2 LðU ij ; XÞ; Eij 2 LðX; Y ij Þ; i 2 K; j 2 N ¼ f1; . . . ; Ng are given operators defining the structure of perturbations, and Dij 2 LðY ij ; U ij Þ; i 2 K; j 2 N are unknown disturbance operators. For the perturbations determined by above way we set ! ! ! N N N X X X PD ðkÞ ¼ A0 þ D0j D0j E0j þ k1 A1 þ D1j D1j E1j þ    þ kK AK þ DKj DKj EKj : ð8Þ j¼1

j¼1

j¼1

The size of each perturbation D ¼ ðDij Þi2K;j2N is measured by cðDÞ ¼

P

kDij k. i2K;j2N

Definition 3.1. Let system (1) be stable. The complex, real and positive stability radius of system (1) under perturbations of the form (7) can be defined by rC ¼ inffcðDÞ : Dij 2 LðY ij ; U ij Þ; i 2 K; j 2 N; rðP D ðÞÞ 6 C1 g; rR ¼ inffcðDÞ : Dij 2 LR ðY ij ; U ij Þ; i 2 K; j 2 N; rðP D ðÞÞ 6 C1 g; rþ ¼ inffcðDÞ : Dij 2 Lþ ðY ij ; U ij Þ; i 2 K; j 2 N; rðPD ðÞÞ 6 C1 g; respectively, where we set inf Ø ¼ 1. For i; u 2 K; j; v 2 N let Gij;uv ðsÞ ¼ Eij Rðs; PðsÞÞDuv ;

s 2 qðPðÞÞ

be associated transfer function of system (1). The following result can be obtained by using arguments similar to those in Proposition 3.6 in [5]. Proposition 3.2. Let system (1) be stable. Then sup kGij;uv ðsÞk ¼ sup kGij;uv ðsÞk; jsjP1

i; u 2 K;

j; v 2 N:

jsj¼1

Before considering the stability radii of system (1), we need the following result. Proposition 3.3. Let k 2 qðPðÞÞ; jkj P 1 and D ¼ ðDij Þi2K;j2N . If cðDÞ <

1 ; maxi;u2K;j;v2N kGij;uv ðkÞk

ð9Þ

then k 2 qðP D ðÞÞ Proof. Let us consider the Banach spaces U ¼ X kuk ¼ kuij k; u ¼ ðuij Þi2k;j2N 2 U; i;j X kyk ¼ kyij k; y ¼ ðyij Þi2k;j2N 2 Y: i;j

Q

i;j U ij ;

Y¼

Q

i;j Y ij

with the norms

B.T. Anh et al. / Applied Mathematics and Computation 202 (2008) 562–570

565

Then we define the operators E 2 LðX; YÞ; D 2 LðU; XÞ by setting X Ex ¼ ðEij xÞi2k;j2N ; Du ¼ Dij uij for x 2 X; ðuij Þi2k;j2N 2 U: i;j

For Dij 2 LðY ij ; U ij Þ; i 2 k; j 2 N and k 2 C we also define a ‘‘block-diagonal” operator DðkÞ 2 LðY; UÞ by setting DðkÞy ¼ ðki Dij yij Þi2k;j2N ;

y ¼ ðyij Þi2k;j2N 2 Y:

Suppose that k 2 C; jkj P 1, and k satisfies (9). Then, by definition, we have, for each u ¼ ðuij Þi2k;j2N 2 Y, ! X u k Duv Guv;ij ðkÞuij : DðkÞERðk; PðkÞÞDu ¼ i;j

u2K;v2N

Therefore,    X X u  k Duv Guv;ij ðkÞuij  kDðkÞERðk; PðkÞÞDuk ¼    u;v i;j and hence, by (9), k½DðkÞERðk; PðkÞÞDk < 1. This follows that operator ½I  DðkÞERðk; PðkÞÞD is invertible. Consequently, ½I  DDðkÞERðk; PðkÞÞ is invertible, too. By a straightforward computation, we can verify that ½I  DDðkÞERðk; PðkÞÞ1 ¼ I þ D½I  DðkÞERðk; PðkÞÞD1 DðkÞERðk; PðkÞÞ: Moreover, we have ½kI  ðPðkÞ þ DDðkÞEÞ1 ¼ Rðk; PðkÞÞ½I  DDðkÞERðk; PðkÞÞ1 ; which implies that k 2 qðPðkÞ þ DDðkÞEÞ or k 2 qðPD ðÞÞ. The proof is complete. h The result concerning the complex stability radius of system (1) is given by the following theorem. Theorem 3.4. Let system (1) be stable. Then 1 1 6 rC 6 : maxjsj¼1 fkGij;uv ðsÞk : i; u 2 K; j; v 2 Ng maxjsj¼1 fkGij;ij ðsÞk : i 2 K; j 2 Ng

ð10Þ

In particular, if Dij ¼ Duv or Eij ¼ Euv for all i; u 2 K; j; v 2 N then rC ¼

1 maxjsj¼1 fkGij;ij ðsÞk : i 2 K; j 2 Ng

ð11Þ

:

Proof. We first will prove that rC P

1 : maxjsj¼1 fkGij;uv ðsÞk : i; u 2 K; j; v 2 Ng

ð12Þ

Indeed, we assume that this is not true. From Proposition 3.2, this means rC <

1 maxjsj¼1 fkGij;uv ðsÞk : i; u 2 K; j; v 2 Ng

¼

1 maxjsjP1 fkGij;uv ðsÞk : i; u 2 K; j; v 2 Ng

:

By the definition of rC , there exists D ¼ ðDi;j Þi2K;j2N ; Dij 2 LðY ij ; U ij Þ; i 2 K; j 2 N, such that cðDÞ <

1 maxjsjP1 fkGij;uv ðsÞk : i; u 2 K; j; v 2 Ng

ð13Þ

and rðP D ðÞÞ 6 C1 . Due to Proposition 3.3 and inequality (13), we have s 2 qðPD ðÞÞ for all s 2 C; jsj P 1. This contradicts to rðP D ðÞÞ 6 C1 . Thus we obtain rC P

1 : maxjsj¼1 fkGij;uv ðsÞk : i; u 2 K; j; v 2 Ng

It remains to show that rC 6

1 : maxjsj¼1 fkGij;ij ðsÞk : i 2 K; j 2 Ng

ð14Þ

Let e > 0; i0 2 K; j0 2 N and s0 ; js0 j ¼ 1 be arbitrarily given. Then, there exists ui0 j0 2 U i0 j0 satisfying kui0 j0 k ¼ 1; kGi0 j0 ;i0 j0 ðs0 Þk P kGi0 j0 ;i0 j0 ðs0 Þui0 j0 k P kGi0 j0 ;i0 j0 ðs0 Þk  e. By Hahn–Banach theorem there exists yi0 j0 2 Y i0 j0 such that kyi0 j0 k ¼ 1; yi0 j0 ðGi0 j0 ;i0 j0 ðs0 Þui0 j0 Þ ¼ kGi0 j0 ;i0 j0 ðs0 Þui0 j0 k. Let Di0 j0 : Y i0 j0 ! U i0 j0 be defined as

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B.T. Anh et al. / Applied Mathematics and Computation 202 (2008) 562–570

Di0 j0 yi0 j0 ¼

1 y ðy Þui j kGi0 j0 ;i0 j0 ðs0 Þuk i0 j0 i0 j0 0 0

8yi0 j0 2 Y i0 j0 :

It is clear that Di0 j0 2 LðY i0 j0 ; U i0 j0 Þ and kDi0 j0 k 6

1 1 6 : kGi0 j0 ;i0 j0 ðs0 Þui0 j0 k kGi0 j0 ;i0 j0 ðs0 Þk  e

Now we construct the disturbance D ¼ ðDij Þi2K;j2N defined as ( si00 D; i ¼ i0 ; j ¼ j0 ; i 2 N: Dij ¼ 0; i 6¼ i0 or j 6¼ j0 ; x ¼ Rðs0 ; Pðs0 ÞÞDi0 j0 ui0 j0 , then we can verify that ^ x 6¼ 0 and Obviously, cðDÞ ¼ kDk. Taking ^ implies s0 2 rðPD ðÞÞ and therefore, rðP D ðÞÞ 6 C1 . Consequently, by the definition of rC rC 6 cðDÞ ¼ kDk 6

PK

i i¼0 ½s0 ðAi

1 : kGi0 j0 ;i0 j0 ðs0 Þk  e

þ

PN

^ ¼ s0 ^ x. This

j¼1 Dij Dij Eij Þx

ð15Þ

Since the inequality (15) is hold for arbitrary s0 2 C; js0 j ¼ 1; i0 2 K; j0 2 N and e > 0, we have rC 6

1 maxjsj¼1 fkGij;ij ðsÞk : i 2 K; j 2 Ng

Hence, the second inequality in (10) is hold. Furthermore, if Dij ¼ Duv 8i; u 2 K; j; v 2 N (resp. Eij ¼ Euv 8i; u 2 K; j; v 2 N) then, by the definition Gij;uv ðsÞ ¼ Gij;ij ðsÞ 8i; u 2 K; j; v 2 N (resp. Guv;ij ðsÞ ¼ Gij;ij ðsÞ 8i; u 2 K; j; v 2 N). Thus, in this case, (10) implies (11). The proof is complete. h In general, the three stability radii: complex, real and positive radius are different. Theorem 3.4 reduces the computation of the complex stability radius to a global optimization problem over the unit circle in complex plane while the problem for the real stability radius is much more difficult even X is a infinite dimensional space, see [9]. It is therefore natural to investigate for which kind of systems these three radii coincide. The answer will be found in next section. 3.2. Positive stability radius We first address some results for the class of positive systems. System (1) is called positive system if and only if all operators Ai 2 Lþ ðXÞ; i 2 K. We are now in a position to introduce some properties on the operator polynomial PðÞ. Lemma 3.5. Let Ai 2 Lþ ðXÞ, for all i 2 K and k > 0. Then (i) Rðk; PðkÞÞ P 0 if and only if k > rðPðÞÞ. (ii) For D 2 Lþ ðX; YÞ and x 2 X, we have jDRðs; PðsÞÞxj 6 DRðjsj; PðjsjÞÞjxj

8jsj > rðPðÞÞ:

(iii) t1 P t2 ) Rðt2 ; Pðt2 ÞÞ P Rðt 1 ; Pðt 1 ÞÞ P 0. Proof (i) From (4) and (5), we obtain that the operator Rðk; PðkÞÞ P 0 if and only if Rðk; AÞ P 0. By invoking (ii) in Theorem 2.1 and following equation rðAÞ ¼ rðPðÞÞ, we obtain (i). (ii) Because Ai 2 Lþ ðXÞ; i 2 K, the operator A is positive. Using the Proposition 2.2 for the operator Aðx; 0; . . . ; 0Þ 2 X Kþ1 and the operator D 2 Lþ ðX Kþ1 ; YÞ defined by Dðx0 ; x1 ; . . . ; xK Þ ¼ Dx0 , gives jDRðs; AÞðx; 0; . . . ; 0Þj 6 DRðjsj; AÞjðx; 0; . . . ; 0Þj; which implies that jDRðs; PðsÞÞxj 6 DRðjsj; PðjsjÞÞjxj. (iii) Using the following equality: Rðt 2 ; Pðt 2 ÞÞ  Rðt1 ; Pðt1 ÞÞ ¼ Rðt2 ; Pðt2 ÞÞRðt 1 ; Pðt 1 ÞÞ

! N X i ðt i  t ÞA i ; 2 1 i¼0

we deduce (iii). The proof of Lemma 3.5 is complete. h Theorem 3.6. Let system (1) be stable and positive, and Dij 2 Lþ ðU ij ; XÞ; Eij 2 Lþ ðX; Y ij Þ; i 2 K; j 2 N. If Dij ¼ Duv or Eij ¼ Euv for all i; u 2 K; j; v 2 N then

B.T. Anh et al. / Applied Mathematics and Computation 202 (2008) 562–570

rC ¼ rR ¼ rþ ¼

567

1 maxfkGij;ij ð1Þk : i 2 K; j 2 Ng

Proof. First, due to Theorem 3.4 and (iii) of Lemma 3.5, we have rC ¼

1 : maxfkGij;ij ð1Þk : i 2 K; j 2 Ng

Finally, we have to show that rþ 6

1 maxfkGij;ij ð1Þk : i 2 K; j 2 Ng

:

Indeed, for arbitrary i 2 K; j 2 N and e > 0, using arguments similar to those in proof of Theorem 3.4 and the Hahn–Banach Theorem for positive operators, see [11], one can construct an one-rank positive destabilizing perturbation 1 D ¼ ðDij Þi2K;j2N ; Dij 2 Lþ ðY ij ; U ij Þ; i 2 K; j 2 N such that cðDÞ 6 kG ð1Þke . This implies ij;ij

rþ 6

1 maxfkGij;ij ð1Þk : i 2 K; j 2 Ng

;

concluding the proof. h Now assume that system (1) is stable and the coefficient operators of system (1) are subjected to perturbations of the following kind Ai ,! Ai þ

N X

dij Aij ;

i 2 K;

ð16Þ

j¼1

where Aij 2 Lþ ðXÞ; i 2 K; i 2 N are given operators and dij ; i 2 K; j 2 N are unknown scalar parameters. We can rewrite the perturbation of the form ! ! N N X X ð17Þ xðt þ K þ 1Þ ¼ A0 þ d0j A0j xðt þ KÞ þ    þ AK þ dKj AKj xðtÞ; t; K 2 N: j¼1

j¼1

Set Pd ðkÞ ¼

A0 þ

N X

! d0j A0j

þ k1 A1 þ

j¼1

N X

! d1j A1j

þ    þ kK AK þ

j¼1

N X

! dKj AKj :

ð18Þ

j¼1

Definition 3.7. Let system (1) be stable. The complex, real and positive radius of system (1) under perturbations of the form (16) can be defined by rdC ¼ inffkdk1 : d ¼ ðdij Þi2K;j2N 2 CKN ; rðP d ðÞÞ 6 C1 g; rdR ¼ inffkdk1 : d ¼ ðdij Þi2K;j2N 2 RKN ; rðP d ðÞÞ 6 C1 g; KN rdþ ¼ inffkdk1 : d ¼ ðdij Þi2K;j2N 2 Rþ ; rðP d ðÞÞ 6 C1 g;

respectively, where we set inf Ø ¼ 1 and kdk1 ¼ maxfjdij j : i 2 K; j 2 Ng. Theorem 3.8. Let system (1) be stable. Then we have rdC P

1 P : supjsjP1;jzij j61;i2K;j2N r½Rðs; PðsÞÞð i;j zij Aij Þ

Proof. First suppose that d ¼ ðdij Þi2K;j2N is a perturbation satisfying kdk1 <

1 P : supjsjP1;jzij j61;i2K;j2N r½Rðs; PðsÞÞð i;j zij Aij Þ

This implies that " r ðsI  PðsÞÞ1

X i;j

!# kdk1 zij Aij

<1

for all jsj P 1;

jzij j 6 1;

i 2 K;

j 2 N:

ð19Þ

568

B.T. Anh et al. / Applied Mathematics and Computation 202 (2008) 562–570 d

If we choose zij ¼ si jdjij ; i 2 K; j 2 N then jzij j 6 1; i 2 K; j 2 N. Due to (19), we have 1 " !# X 1 r ðsI  PðsÞÞ si dij Aij < 1 for all jsj P 1: i;j

P Therefore, the operator ½I  ðsI  PðsÞÞ1 ð i;j si dij Aij Þ1 exists for all jsj P 1. Moreover, by a straightforward computation, we have ! " !# X X 1 i i sI  PðsÞ þ s dij Aij ¼ ðsI  PðsÞÞ I  ðsI  PðsÞÞ s dij Aij : i;j

i;j

Thus, we conclude that the operator ½sI  ðPðsÞ þ hence we have rdC P

P

i;j s

i

dij Aij Þ is invertible for all jsj P 1. This means that rðPd ðÞÞ  C1 and

1 P : supjsjP1;jzij j61;i2K;j2N r½Rðs; PðsÞÞð zij Aij Þ i;j

The proof is complete.

h

Remark 3.9. In fact that we can prove that 1 P : supjsjP1;jzij j61;i2K;j2N r½Rðs; PðsÞÞð i;j zij si Aij Þ

rdC ¼

However, it is not necessary because in this section we only concern computing stability radii of positive systems. In following theorem, we will show that the three stability radii: complex, real and positive radius coincide and can be computed by a simple formula. Theorem 3.10. Assume that system (1) is positive and stable and all operators Aij are positive. Then rdC ¼ r dR ¼ r dþ ¼ where B ¼

P

i2K;j2N

1 ; r½Rð1; Pð1ÞÞB

ð20Þ

Aij .

Proof. Thanks to Theorem 3.8, we have rdC P

1 P : supjsjP1;jzij j61;i2K;j2N r½Rðs; PðsÞÞð i;j zij Aij Þ

For arbitrary x 2 X; n 2 N; jsj P 1; jzij j 6 1; i 2 K; j 2 N, due to Lemma 3.5, we have " !# n  " !#n   X X   zij Aij x 6 Rðjsj; PðjsjÞÞ jzij jAij jxj 6 ½Rð1; Pð1ÞÞBn jxj:  Rðs; PðsÞÞ   i;j i;j P This implies that k½Rðs; PðsÞÞð i;j zij Aij Þn k 6 k½Rð1; Pð1ÞÞBn k. This follows that: " !# X zij Aij 6 r½Rð1; Pð1ÞÞB: r Rðs; PðsÞÞ i;j

Therefore, rdC P

1 1 P : ¼ supjsjP1;jzij j61;i2K;j2N r½Rðs; PðsÞÞð i;j zij Aij Þ r½Rð1; Pð1ÞÞB

To complete the proof, we have to show that rdþ 6

1 : r½Rð1; Pð1ÞÞB

P P 1 Set s0 ¼ r½Rð1;Pð1ÞÞB , then r½Rð1; Pð1ÞÞð i;j s0 Aij Þ ¼ 1. Since Rð1; Pð1ÞÞð i;j s0 Aij Þ is positive, by invoking Theorem 2.1, we have P 1 2 r½Rð1; Pð1ÞÞð i;j s0 Aij Þ. On the other hand, by direct calculation, we have " !# " !# X X 1 I  Pð1Þ þ s0 Aij s0 Aij : ¼ ðI  Pð1ÞÞ I  ðI  Pð1ÞÞ i;j

i;j

P Therefore, since the operator ½I  ðI  Pð1ÞÞ ð i;j s0 Aij Þ is not invertible, operator ½I  ðPð1Þ þ i;j s0 Aij Þ is not invertible P either. Set dij ¼ s0 for all i 2 K; j 2 N, then we have 1 2 rðPð1Þ þ i;j dij Aij Þ and jdj1 ¼ s0 . By the definition of positive stability radius we obtain 1

P

B.T. Anh et al. / Applied Mathematics and Computation 202 (2008) 562–570

rdþ 6

1 r½Rð1; Pð1ÞÞ1 B

569

:

The proof is complete. h Now we will consider following example to illustrate the above results. 1

Example. Let X ¼ l ðCÞ be a space of all sequences ðxi Þi2N  C satisfying space if it is endowed with the norm kxk ¼

1 X

P1

i¼1 jxi j

< þ1. Then X is a complex Banach lattice

jxi j; x ¼ ðxi Þi2N 2 X

i¼1

and the module jxj ¼ ðjxi jÞi2N : Considering following system: xðt þ 2Þ ¼ A0 xðt þ 1Þ þ A1 xðtÞ where A0 ; A1 are positive operators on X defined by   1 1 1 1 1 x1 ; x1 þ x2 ; . . . ; xn þ xnþ1 ; . . . ; x ¼ ðxi Þi2N 2 X; A0 x ¼ 12 3 12 3 12   1 1 1 1 1 x1 ; x1 þ x2 ; . . . ; xn þ xnþ1 ; . . . ; x ¼ ðxi Þi2N 2 X: A1 x ¼ 4 6 4 6 4 It is easy to see that kA0 þ A1 k ¼ 12 þ 13 ¼ 56 < 1. It implies that system xðt þ 2Þ ¼ A0 xðt þ 1Þ þ A1 xðtÞ is stable. Now we assume that A0 ; A1 are subjected to perturbations of the form A0 ,! A0 þ D01 D01 E01 þ D02 D02 E02 ; A1 ,! A1 þ D11 D11 E11 þ D12 D12 E12 ; where Dij ¼ IX ; i ¼ 0; 1; j ¼ 1; 2 and Eij 2 Lþ ðXÞ; i ¼ 0; 1; j ¼ 1; 2 are defined by E01 x ¼ ð0; x2 ; x3 ; . . . ; xn ; . . .Þ; E02 x ¼ ðx1 ; 0; x3 ; . . . ; xn ; . . .Þ; E11 x ¼ ðx1 ; 0; 0; . . . ; 0; . . .Þ;

x ¼ ðxi Þi2N 2 X; x ¼ ðxi Þi2N 2 X; x ¼ ðxi Þi2N 2 X;

E12 x ¼ ðx1 ; x2 ; 0; 0; . . . ; 0; . . .Þ;

x ¼ ðxi Þi2N 2 X:

Thus, using the Theorem for stability radii for the positive systems, we have rC ¼ rR ¼ rþ ¼

1 : maxi2f0;1g;j2f1;2g kGij;ij ð1Þk

Now we will compute kG00;00 ð1Þk; kG01;01 ð1Þk; kG10;10 ð1Þk and kG11;11 ð1Þk. First one can check that ðI  A0  A1 Þ1 x ¼ ðy1 ; y2 ; . . . ; yn ; . . .Þ;

x ¼ ðxi Þi2N 2 X;

where y1 ¼

3 x1 ; 2

y2 ¼

3 32 x2 þ 3 x1 ; 2 2

 yn ¼

3 32 33 3n1 3n xn þ 3 xn1 þ 5 xn2 þ    þ 2n3 x2 þ 2n1 x1 ; 2 2 2 2 2

Thus, we can compute kG00;00 ð1Þk ¼ kE00 ðI  AÞ1 k ¼

1 X 3n

¼ 6; 22n1 1 3 X 3n 39 ; ¼ kG01;01 ð1Þk ¼ kE01 ðI  AÞ1 k ¼ þ 2n1 2 i¼3 2 8 i¼1

3 ; 2 3 32 21 kG11;11 ð1Þk ¼ kE11 ðI  AÞ1 k ¼ þ 3 ¼ : 2 2 8

kG10;10 ð1Þk ¼ kE10 ðI  AÞ1 k ¼

n P 3:

570

B.T. Anh et al. / Applied Mathematics and Computation 202 (2008) 562–570

Therefore, we obtain rC ¼ rR ¼ rþ ¼

1 : 6

Next we assume that operators Ai ; i ¼ 0; 1 are subjected by affine perturbations of the form A0 ,! A0 þ d01 A01 þ d02 A02 ; A1 ,! A1 þ d11 A11 þ d12 A12 ; where Aij 2 Lþ ðXÞ; i; j are defined by   1 1 A01 x ¼ x1 ; x2 ; . . . ; 0; . . . ; 2 3   1 A02 x ¼ 0; x2 ; 0; . . . ; 0; . . . ; 2   1 1 x1 ; x2 ; . . . ; 0; . . . ; A11 x ¼ 3 2   7 2 x1 ; x2 ; 0; . . . ; 0; . . . : A12 x ¼ 6 3 By invoking the theorem about positive stability radii under affine perturbations, we have rdC ¼ r dR ¼ r dþ ¼

1 r½ðI  A0  A1 Þ1 ðA01 þ A02 þ A11 þ A12 Þ

:

Now we compute r½ðI  A0  A1 Þ1 ðA01 þ A02 þ A11 þ A12 Þ. First we have ! 3 3 32 3n1 3n x1 ; x2 þ 3 x1 ; . . . ; 2n3 x2 þ 2n1 x1 ; . . . : ðI  A0  A1 Þ ðA01 þ A02 þ A11 þ A12 Þx ¼ 2 2 2 2 2 2 1

By a simple computation, one has r½ðI  A0  A1 Þ1 ðA01 þ A02 þ A11 þ A12 Þ ¼ lim

n!1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n kðI  A0  A1 Þ1 ðA01 þ A02 þ A11 þ A12 Þk ¼ 3:

Thus, we obtain rdC ¼ r dR ¼ r dþ ¼

1 : 3

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