Global exponential stability of impulsive discrete-time neural networks with time-varying delays

Applied Mathematics and Computation 217 (2010) 537–544

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Global exponential stability of impulsive discrete-time neural networks with time-varying delays Honglei Xu a,b,*, Yuanqiang Chen c, Kok Lay Teo b a

Department of Mathematics, Guizhou University, Guiyang 550025, China Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia c Department of Mathematics, National Minorities College of Guizhou, Guiyang 550025, China b

a r t i c l e

i n f o

Keywords: Impulsive discrete-time neural networks Global exponential stability Exponential convergence rate Halanay inequality

a b s t r a c t This paper studies the problem of global exponential stability and exponential convergence rate for a class of impulsive discrete-time neural networks with time-varying delays. Firstly, by means of the Lyapunov stability theory, some inequality analysis techniques and a discrete-time Halanay-type inequality technique, sufficient conditions for ensuring global exponential stability of discrete-time neural networks are derived, and the estimated exponential convergence rate is provided as well. The obtained results are then applied to derive global exponential stability criteria and exponential convergence rate of impulsive discrete-time neural networks with time-varying delays. Finally, numerical examples are provided to illustrate the effectiveness and usefulness of the obtained criteria.  2010 Elsevier Inc. All rights reserved.

1. Introduction Neural networks have received extensive interests in recent years and have witnessed many promising potential applications in different areas such as signal processing, content addressable memory, pattern recognition and combinatorial optimization (see e.g. [2–4,6–8] and the references therein). It is well known that the existence of delays in neural networks causes undesirable complex dynamical behaviors such as instability, oscillation and chaotic phenomena. Stability problems of time-delayed neural networks have attracted much attention in view of their theoretical as well as practical importance. There are now many results being reported in the literature on neural networks with time delays, see, for example, [1–3,5,6,8–17], and references therein. In practice, for computation convenience, continuous-time neural networks are often discretized to generate discrete-time neural networks. Thus, the study of discrete-time neural networks attracts more and more interests. In recent years, asymptotic behaviors of discrete-time neural networks have been investigated, and many results have been reported (see [14–18] and the references therein). Complex dynamical systems usually undergo abrupt changes of their states at certain moments due to unexpected internal or external effects. There is no exception for discrete neural networks. These impulsive perturbations can also cause undesirable dynamical behaviors leading to poor performance. Therefore, it is necessary to take into account both impulsive effects and delay effects in the stability analysis of discrete neural networks. Impulsive differential equations, which are mathematical models for continuous-time dynamical systems with impulsive perturbations, have been successfully applied to many practical problems, see, e.g. Refs. [19–23]. Similarly, impulsive difference equations are suitable mathematical tools to model impulsive discrete-time neural networks. Continuous-time neural networks with impulsive perturbations have

* Corresponding author at: Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia. E-mail address: [email protected] (H. Xu). 0096-3003/$ - see front matter  2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.05.087

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H. Xu et al. / Applied Mathematics and Computation 217 (2010) 537–544

been reported (see e.g. [24–27]). In [24], global stability properties have been analyzed for impulsive Hopfield-type neural networks whose impulses contain both the functional term and its integral. Sufficient conditions are derived in [25] based on vector Lyapunov functions and the M-matrix theory for ensuring global exponential stability of the neural networks with impulsive effects. In addition, the estimated exponential convergence rate is given as well. Exponential stability analysis of impulsive delay neural networks has been investigated in [26,27] based on the M-matrix theory and an impulsive delay differential inequality. However, it appears that little attention is devoted to the investigation of stability for discrete-time neural networks with time delays subject to impulsive perturbations, although such neural networks are important in the fields of natural sciences and applied technology. This motivates our study. In this paper, we shall deal with a class of discrete-time neural networks with time-varying delays subject to impulsive perturbations. Firstly, by utilizing the Lyapunov stability theory and discrete-time Halanay-type inequality, we shall establish some sufficient conditions for global exponential stability of discrete-time neural networks with time-varying delays. Secondly, we shall apply the results obtained to impulsive discrete-time neural networks and obtain new global exponential stability criteria and new estimated exponential convergence rate for impulsive discrete-time neural networks with timevarying delays. The main contributions of the current paper include: (i) some new global exponential stability criteria are derived by means of the discrete-time Halanay-type inequality; (ii) the form of impulsive perturbations is more general than the existing ones in the literature which are described by either a linear matrix function or a simple nonlinear matrix function; and (iii) new global exponential stability criteria and convergence rate of impulsive neural networks with time-varying delays are obtained. The rest of this paper is organized as follows. In Section 2, impulsive discrete-time neural networks with time-varying delays are introduced and some preliminary lemmas are presented. In Section 3, based on the Lyapunov stability theory and the discrete-time Halanay-type inequality, global exponential stability criteria are derived for discrete-time neural networks with time-varying delays for the case when the neural networks are free of impulsive perturbations as well as the case when they are subject to impulsive perturbations. Moreover, numerical examples are presented in Section 4. Section 5 concludes the paper. 2. Preliminaries and problem statement Consider the following impulsive neural network with time-varying delays:

8 n P > > > ui ðm þ 1Þ ¼ ai ui ðmÞ þ T ij ^f j ðuj ðm  sij ðmÞÞÞ þ Ii ; m – nk ; > > j¼1 < > Dui ðmÞ ¼ g^iðkÞ ðm; u1 ðmÞ; . . . ; un ðmÞÞ; > > > > : ui ðmÞ ¼ /i ðmÞ;

m ¼ nk ;

m 2 Nð1Þ;

ð1Þ

m 2 Nðs; 0Þ;

where ui(t) denotes the state of the ith neuron at time t; ai 2 [0, 1), i 2 N(1, n), represents the passive decay rate, where NðkÞ ¼ fk; k þ 1; k þ 2; . . .g; Nðk; lÞ ¼ fk; k þ 1; k þ 2; . . . ; lg; ^f j is the neuron output signal function which is a continuous function; T ij ; sij ðmÞ P 0 denote, respectively, the connection weight and the transmission delay from the neuron j to the neuron i with s = maxi,j2N(1, n) {sij(m)} and m  sij(m) ? 1 as m ? 1; fj: R ? R is the neuron activation function with fj(0) = 0; and Ii is the exogenous input, where i 2 N(1, n)  ui(m) = /i(m), m 2 N(s, 0), is the initial condition for (1), where /i: N(s, 0) ? R,  T i 2 N(1, n), is a continuous function. A vector u ¼ u1 ; u2 ; . . . ; un is said to be an equilibrium point of the impulsive discrete-time neural network (1) if it satisfies

ui ¼ ai ui þ

n X

  T ij ^f j uj þ Ii :

j¼1

ui

Let be an equilibrium point of (1). For the purpose of brevity, we can shift the equilibrium ui to the origin by setting xi ðmÞ ¼ ui ðmÞ  ui ; i 2 Nð1; nÞ. Then, the neural network (1) is transformed into

8 n P > > xi ðm þ 1Þ ¼ ai xi ðmÞ þ T ij fj ðxj ðm  sij ðmÞÞ; m – nk ; > > > j¼1 < > Dxi ðmÞ ¼ g ðkÞ > i ðm; x1 ðmÞ; . . . ; xn ðmÞÞ; > > > : xi ðmÞ ¼ /i ðmÞ;

m ¼ nk ;

m 2 Nð1Þ;

m 2 Nðs; 0Þ;

where

  fj ðxj ðm  sij ðmÞÞÞ ¼ ^f j ðuj ðm  sij ðmÞÞÞ  ^f j uj ðm  sij ðmÞÞ and

 ðkÞ ðkÞ ðkÞ  g i ðm; x1 ðmÞ; . . . ; xn ðmÞÞ ¼ g^i ðm; u1 ðmÞ; . . . ; un ðmÞÞ  g^i m; u1 ðmÞ; . . . ; un ðmÞ :

ð2Þ

539

H. Xu et al. / Applied Mathematics and Computation 217 (2010) 537–544 ðkÞ

Without loss of generality, we may assume that the sequence fnk ; g k g of the impulsive effects satisfies the following assumptions. Assumption 1. The sequence {nk} of the impulsive time points satisfies nk 2 N(1), nk + 2 6 nk+1 and limk?1nk = 1 with k 2 N(0). ðkÞ

ðkÞ

Assumption 2. For the impulsive increment function sequence g k , there exists xij P 0 such that "(x1(t), x2(t), . . . , xn(t)) 2 Rn, t 2 2 N(0), the following condition is satisfied,

  X n   ðkÞ xðkÞ xi ðtÞ þ g i ðt; x1 ðtÞ; . . . ; xn ðtÞÞ 6 ij jxj ðtÞj;

j 2 Nð1; nÞ:

ð3Þ

j¼1

Clearly, the stability properties of the impulsive neural network (1) are equivalent to the stability properties of the impulsive neural network (2). Furthermore, we need the following definitions and lemmas. Definition 1. For the impulsive discrete-time neural network (2), the trivial equilibrium point is uniformly stable if there exists a positive constant e > 0, and maxs2N(s, 0) {kx(s)k} 6 d(e), then, for any given d(e) > 0, kx(m)k < e, "m 2 N(1). Definition 2. For the impulsive discrete-time neural network (2), the trivial equilibrium point is asymptotically stable if the neural network (2) is uniformly stable and the following condition is satisfied,

lim kxðmÞk ¼ 0:

ð4Þ

m!þ1

Definition 3. For the impulsive discrete-time neural network (2), the trivial equilibrium point is exponentially stable if there exist positive constants j > 0 and r 2 (0, 1) such that

kxðmÞk 6 jrm ;

8m 2 Nð1Þ;

ð5Þ n

where r is called the exponential convergence rate. If (5) is satisfied for any initial condition x(m) 2 R , m 2 N(s, 0), the trivial equilibrium point is globally exponentially stable for the impulsive neural network (2). Lemma 1 ([28] Discrete-time Halanay-type inequality). Consider the following discrete-time system

DxðmÞ ¼ f ðm; xðmÞ; xðm  1Þ; . . . ; xðm  sÞÞ; where Dx(m) = x(m + 1)  x(m) and N  Rs+1 ? R. The initial condition is given that /i: N(s, 0) ? R, i 2 N(1, n). Suppose that the real numbers sequence {bn}nPs is such that

Dxn ¼ axn þ gðn; xn ; xn1 ; . . . ;ns Þ;

n 2 Nð0Þ;

a 2 ð0; 1;

and that there exists a b 2 (0, a) such that

Dbn 6 abn þ b max fbi g; i2Nðns;nÞ

8n 2 Nð0Þ:

Then, there exists a k 2 (0, 1) such that

bn 6 kn max fbi g; i2Nðs;0Þ

8n 2 Nð0Þ;

where Dbn = bn+1  bn, s 2 N(0), is a constant, g : Nð0Þ  Rsþ1 ! R; ðbs ; bsþ1 ; . . . ; b0 Þ is the initial condition and k is the smallest root in the interval (0, 1) of the following equation

ksþ1 þ ða  1Þks  b ¼ 0: 3. Main results In this section, we shall firstly establish sufficient conditions for global exponential stability of discrete-time neural networks with time-varying delays. On the basis of the obtained results, we shall investigate the global exponential stability criteria and the estimated exponential convergence rate of impulsive discrete-time neural networks. 3.1. Discrete-time neural networks Consider a discrete-time neural network described by

xi ðm þ 1Þ ¼ ai xi ðmÞ þ

n X

T ij fj ðxj ðm  sij ðmÞÞÞ;

m 2 Nð1Þ;

j¼1

xi ðmÞ ¼ /i ðmÞ;

m 2 Nðs; 0Þ;

i 2 Nð1; nÞ:

In the following theorem, the results on the global exponential stability of the neural network (6) are presented.

ð6Þ

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H. Xu et al. / Applied Mathematics and Computation 217 (2010) 537–544

Theorem 1. Suppose that Assumptions 1, 2 and the following conditions are satisfied. (i) There exists a constant dj > 0 such that "t1, t2 2 N(1), the neuron activation function fj(m) in (1) is bounded and satisfies the following Lipschitz condition

jfj ðt 1 Þ  fj ðt 2 Þj 6 dj jt 1  t 2 j;

j 2 Nð1; nÞ:

ð7Þ

(ii)

a þ d < 1; where a = maxi2N(1, n){ai} and d ¼ maxi2Nð1;nÞ

nP

n j¼1 jT ij jdj

o

ð8Þ .

Then, the trivial equilibrium point of (6) is globally exponentially stable with the convergence rate k which is the smallest root in the interval (0, 1) of the following equation

ksþ1  aks  d ¼ 0:

ð9Þ

Proof. From the trajectory {x(m)}, m 2 N(1), of system (6), we have

xi ðmÞ ¼ am i xi ð0Þ þ

m1 X

n X

am1s i

s¼0

T ij fj ðxj ðs  sij ðsÞÞÞ;

m 2 Nð1Þ;

i 2 Nð1; nÞ:

j¼1

Clearly,

jxi ðmÞj 6 am i jxi ð0Þj þ

m1 X

am1s i

s¼0

n X

jT ij jjfj ðxj ðs  sij ðsÞÞÞj:

ð10Þ

j¼1

Then, by the Lipschitz condition (7), it follows from (10) that

jxi ðmÞj 6 am i max fjxi ð0Þjg þ i2Nð1;nÞ

m 1 X

n X

s¼0

6 am i max fjxi ð0Þjg þ i2Nð1;nÞ

m1 X

i2Nð1;nÞ

m1 X

aim1s

n X

8 max fjxi ðmÞjg; > > < i2Nð1;nÞ

jT ij jdj max

aim1s max

( n X

i2Nð1;nÞ

nP n

j¼1 jT ij jdj



j2Nð1;nÞ

j¼1

s¼0

For any m 2 N(s), let d ¼ maxi2Nð1;nÞ

jT ij jdj jxj ðs  sij ðsÞÞj

j¼1

s¼0

6 am i max fjxi ð0Þjg þ

nm ¼

aim1s

o



max fjxj ðtÞjg

t2Nðss;sÞ

) jT ij jdj

j¼1

max

t2Nðss;sÞ

max fjxj ðtÞjg :

ð11Þ

j2Nð1;nÞ

and

m 2 Nðs; 0Þ;

m1 P m1s > m > max fjxi ð0Þjg þ d ai max : ai i2Nð1;nÞ t2Nðss;sÞ



s¼0

max fjxj ðtÞjg ;

j2Nð1;nÞ

m 2 Nð1Þ:

Then, (11) is reduced to

jxi ðmÞj 6 nm s:

ð12Þ

Since



Dnm ¼ nmþ1  nm ¼ ð1  aÞnm þ d

max

t2Nðms;mÞ

max fjxj ðtÞjg 6 ð1  aÞnm þ d

j2Nð1;nÞ

max fnt g;

t2Nðms;mÞ

8m 2 Nð1Þ;

it follows from Lemma 1 that there exists a k 2 (0, 1) such as

nm 6 km max fnt g; t2Nðs;0Þ

8m 2 Nð0Þ:

ð13Þ

Furthermore, by (12), we have

kxðmÞk1 ¼ max fjxi ðmÞjg 6 nm ; i2Nð1;nÞ

8m 2 NðsÞ:

Thus,

kxðmÞk1 6 max fnt gkm ; t2Nðs;0Þ

8m 2 Nð0Þ:

ð14Þ

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H. Xu et al. / Applied Mathematics and Computation 217 (2010) 537–544

Therefore, by virtue of Definition 3, the trivial equilibrium point of (6) is globally exponentially stable with the convergence rate k which is the smallest root in the interval (0, 1) of the following equation

ksþ1  aks  d ¼ 0: This completes the proof. h 3.2. Impulsive discrete-time neural networks We now give sufficient conditions for global exponential stability of impulsive discrete-time neural network (2). In addition, its convergence rate will also be presented. Theorem 2. Suppose that Assumption 1, 2 and the following conditions are satisfied. (i) There exists a constant dj > 0 such that "t1, t2 2 N(1), the neuron activation function fj(m) in (1) is bounded and satisfies the following Lipschitz condition

jfj ðt 1 Þ  fj ðt 2 Þj 6 dj jt 1  t 2 j;

j 2 Nð1; nÞ;

ð15Þ

k 2 Nð0Þ;

ð16Þ

(ii) k X

ln lj  ðk þ 1Þ ln a 6 0;

j¼0

(iii)

a þ d < 1; where a ¼ maxi2Nð1;nÞ fai g; d ¼ maxi2Nð1;nÞ

nP n

j¼1 jT ij jdj

o

and lk ¼ maxi2Nð1;nÞ

nP

n j¼1

ðkÞ ij

x

o

ð17Þ .

Then, the equilibrium point ui of (1) is globally exponentially stable with the convergence rate k, which is the smallest root in the interval (0, 1) of the following equation

ksþ1  aks  d ¼ 0:

ð18Þ

Proof. For any m 2 (Nk, Nk+1], we have mN k 1

jxi ðmÞj 6 ai

m 1 X

jxi ðNk þ 1Þj þ

am1s i

n X

s¼Nk þ1

jT ij jjfj ðxj ðs  sij ðsÞÞÞj:

ð19Þ

j¼1

Applying (3) and (15) to (19) yields mN k 1

jxi ðmÞj 6 ai

n X

mN k 1

Let d ¼ maxi2Nð1;nÞ

nP n

n X

am1s i

n X

s¼N k þ1

j¼1

6 ai

m 1 X

xijðkÞ jxj ðNk Þj þ

m1 X

xijðkÞ max fjxj ðNk Þjg þ j2Nð1;nÞ

j¼1

j¼1 jT ij jdj

o

jT ij jdj jxj ðs  sij ðsÞÞj

j¼1

am1s i

s¼N k þ1

and lk ¼ maxi2Nð1;nÞ m1 X

jxi ðmÞj 6 amNk 1 lk max fjxj ðNk Þjg þ d j2Nð1;nÞ

nP

n j¼1

o

n X

ð20Þ

j2Nð1;nÞ

. Then, it follows from (20) that xðkÞ ij

am1s max



t2Nðss;sÞ

s¼N k þ1

jT ij jdj max fjxj ðs  sij ðsÞÞjg:

j¼1

max fjxj ðtÞjg :

j2Nð1;nÞ

Thus, NX kþ1 1

jxi ðNkþ1 Þj 6 aNkþ1 Nk 1 lk max fjxj ðNk Þjg þ d j2Nð1;nÞ

aNkþ1 1s max

s¼Nk þ1



t2Nðss;sÞ

max fjxj ðtÞjg :

j2Nð1;nÞ

By induction, we obtain that

jxi ðNk Þj 6 aNk N0 k

k1 Y

lj max fjxj ðN0 Þjg þ d

j¼0

þd

k1 Y j¼2

þd

j2Nð1;nÞ

NX 2 1

lj

s¼N 1 þ1

N k 1 X s¼N k1 þ1

k1 Y

aNk ksþ1 max

t2Nðss;sÞ

aNk s1 max

t2Nðss;sÞ





j¼1

lj

NX 1 1 s¼N0 þ1

t2Nðss;sÞ



max fjxj ðtÞjg :

max fjxj ðtÞjg

j2Nð1;nÞ

max fjxj ðtÞjg þ    þ dlk1

j2Nð1;nÞ

j2Nð1;nÞ

aNk ks max

NX k1 1 s¼Nk2 þ1

aNk s2 max

t2Nðss;sÞ



max fjxj ðtÞjg

j2Nð1;nÞ

ð21Þ

542

H. Xu et al. / Applied Mathematics and Computation 217 (2010) 537–544

Since (21), we have

jxi ðNk Þj 6 aNk k

k1 Y

lj max fjxj ð0Þjg þ d j2Nð1;nÞ

j¼0

þd

k1 Y

NX 1 1

lj

aN1 ks max

NX k1 1

þ dlk1



t2Nðss;sÞ

aNk s2 max

j2Nð1;ntÞ



NX k1 2 1 Y max fjxj ðtÞjg þ d lj aNk ksþ1 max max fjxj ðtÞjg þ   

j2Nð1;nÞ



t2Nðss;sÞ

s¼Nk2 þ1

t2Nðss;sÞ

s¼0

j¼0

s¼N 0 þ1

j¼1



k1 NX 0 1 Y lj aNk ks1 max max fjxj ðtÞjg

j¼2

t2Nðss;sÞ

s¼N 1 þ1

j2Nð1;nÞ



N k 1 X max fjxj ðtÞjg þ d aNk s1 max max fjxj ðtÞtjg :

j2Nð1;nÞ

t2Nðss;sÞ

s¼Nk1 þ1

j2Nð1;nÞ

ð22Þ

Thus, it follows from (21) and (22) that

jxi ðmÞj 6 amk1

k Y

lj max fjxj ð0Þjg þ d j2Nð1;nÞ

j¼0

þd

k Y

NX 1 1

lj

j¼1

þd

j¼0

amks1 max



t2Nðss;sÞ

s¼N0 þ1

m1 X



NX k 0 1 Y lj amks2 max max fjxj ðtÞjg

am1s max



t2Nðss;sÞ

s¼Nk þ1

t2Nðss;sÞ

s¼0

j2Nð1;nÞ

max fjxj ðtÞjg þ    þ dlk

j2Nð1;nÞ

N k 1 X s¼Nk1 þ1

max fjxj ðtÞjg :



ams2 max

t2Nðss;sÞ

max fjxj ðtÞjg

j2Nð1;nÞ

ð23Þ

j2Nð1;nÞ

By (21) and (23), we obtain

jxi ðmÞj 6 am max fjxj ð0Þjg þ d j2Nð1;nÞ

NX 0 1

ams1 max

t2Nðss;sÞ

s¼0 N k 1 X

þ  þ d

ams1 max

t2Nðss;sÞ

s¼Nk1 þ1

8i 2 Nð1; nÞ;







NX 1 1 max fjxj ðtÞjg þ d ams1 max max fjxj ðtÞtjg

j2Nð1;nÞ

s¼N 0 þ1

t2Nðss;sÞ

j2Nð1;nÞ



m1 X max fjxj ðtÞjg þ d am1s max max fjxj ðtÞjg ¼ gm ;

j2Nð1;nÞ

s¼Nk þ1

t2Nðss;sÞ

j2Nð1;nÞ

m 2 Nð1Þ:

Let

( nm ¼

max fjxi ðmÞjg;

i2Nð1;nÞ

m 2 Nðs; 0Þ;

gm ; m 2 Nð1Þ:

Then, the rest of the proof follows readily from similar arguments as those given for the proof of Theorem 1. This completes the proof. h Corollary 1. Suppose that Assumptions 1, 2 and the following conditions are satisfied. (i) There exists a constant dj > 0 such that "t1, t2 2 N(1), the neuron activation function fj(m) in (1) is bounded and satisfies the following Lipschitz condition

jfj ðt 1 Þ  fj ðt 2 Þj 6 dj jt 1  t 2 j;

j 2 Nð1; nÞ;

(ii)

g þ d < 1; where g ¼ supk2Nð0Þ fa; lk g; a ¼ maxi2Nð1;nÞ fai g; d ¼ maxi2Nð1;nÞ

nP n

j¼1 jT ij jdj

o

and lk ¼ maxi2Nð1;nÞ

nP

n j¼1

o

. xðkÞ ij

Then, the equilibrium point ui of (1) is globally exponentially stable with the convergence rate k, which is the smallest root in the interval (0, 1) of the following equation

ksþ1  gks  d ¼ 0: Corollary 2. Suppose that Assumptions 1, 2 and the following conditions are satisfied. (i) There exists a constant dj > 0 such that "t1, t2 2 N(1), the neuron activation function fj(m) in (1) is bounded and satisfies the following Lipschitz condition

jfj ðt 1 Þ  fj ðt 2 j 6 dj jt1  t2 j;

j 2 Nð1; nÞ;

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H. Xu et al. / Applied Mathematics and Computation 217 (2010) 537–544

(ii)

b<

1 ; 2

where b ¼ supk2Nð0Þ fa; d; lk g; a ¼ maxi2Nð1;nÞ fai g; d ¼ maxi2Nð1;nÞ

nP

n j¼1 jT ij jdj

o

and lk ¼ maxi2Nð1;nÞ

nP n

j¼1

o

. xðkÞ ij

Then, the equilibrium point ui of (1) is globally exponentially stable with the convergence rate k, which is the smallest root in the interval (0, 1) of the following equation

ksþ1  gks  d ¼ 0: 4. Numerical examples In this section, two numerical examples are presented to verify and illustrate the usefulness of our main results. Example 1. In this example, we consider a two-neuron discrete-time neural network with time delays

(

x1 ðm þ 1Þ ¼ 12 x1 ðmÞ þ 18 f1 ðx1 ðm  1ÞÞ  14 f2 ðx2 ðm  1ÞÞ; x2 ðm þ 1Þ ¼ 13 x2 ðmÞ þ 18 f1 ðx1 ðm  1ÞÞ þ 13 f2 ðx2 ðm  1ÞÞ; x1 ðmÞ ¼ /1 ðmÞ; x2 ðmÞ ¼ /2 ðmÞ;

ðm 2 Nð1ÞÞ

ðm 2 Nð1; 0ÞÞ

where f1(t) = sint, f2(t) = t, /1(t) = t2, /2(t) = t3. It is easy to verify that

jf1 ðsÞ  f1 ðtÞj 6 js  tj;

8s; t 2 R;

jf2 ðsÞ  f2 ðtÞj 6 js  tj; 8s; t 2 R; ( ) 2 X max fai g þ max jT ij jdj ¼ 0:9583 < 1: i2Nð1;2Þ

i2Nð1;2Þ

j¼1

Thus, all the conditions of Theorem 1 are satisfied. Therefore, the trivial equilibrium point of (2) is globally exponentially stable with the convergence rate k = 0.9717. Example 2. In this example, we consider a three-neuron impulsive discrete-time neural network with time delays

8 1 1 1 1 > < x1 ðm þ 1Þ ¼ 2 x1 ðmÞ þ 8 f1 ðx1 ðm  2ÞÞ  4 f2 ðx2 ðm  1ÞÞ þ 16 f3 ðx3 ðm  1ÞÞ þ 1; 1 1 1 ðm – 4k; m 2 Nð1Þ; k 2 Nð1ÞÞ x2 ðm þ 1Þ ¼ 3 x2 ðmÞ þ 4 f1 ðx1 ðm  2ÞÞ þ 8 f2 ðx2 ðm  1ÞÞ þ 2; > : 1 1 f1 ðx1 ðm  2ÞÞ  18 f2 ðx2 ðm  1ÞÞ þ 16 f3 ðx3 ðm  1ÞÞ þ 1; x3 ðm þ 1Þ ¼ 14 x3 ðmÞ þ 16 8 2 > < Dx1 ðmÞ ¼  3 x1 ðmÞ; Dx2 ðmÞ ¼  23 x2 ðmÞ; ðm ¼ 4k; m 2 Nð1Þ; k 2 Nð1ÞÞ > : Dx3 ðmÞ ¼  23 x3 ðmÞ; 8 > < x1 ðmÞ ¼ /1 ðmÞ; > :

x2 ðmÞ ¼ /2 ðmÞ;

ðm 2 Nð2; 0ÞÞ

x3 ðmÞ ¼ /3 ðmÞ;

where f1(t) = f3(t) = sint, f2(t) = t, /1(t) = /2(t) = /3(t) = t3 + 2. Here, for computational convenience, we assume that the neural network is only subject to linear impulsive perturbations. It can be shown that the equilibrium point of the impulsive discrete-time neural network (1) is

x ¼ ðx1 ; x22 ; x3 ÞT ¼ ð1:5786; 1:8355; 1:2322ÞT : In addition, one can easily check that

jf1 ðsÞ  f1 ðtÞj 6 js  tj; jf2 ðsÞ  f2 ðtÞj 6 js  tj; jf3 ðsÞ  f3 ðtÞj 6 js  tj; k X

8s; t 2 R; 8s; t 2 R; 8s; t 2 R;

ln lj  ðk þ 1Þ ln a ¼ 0:4055 6 0;

j¼0

max fai g þ max

i2Nð1;2Þ

i2Nð1;2Þ

( 2 X j¼1

k 2 Nð0Þ;

) jT ij jdj

¼ 0:8750 < 1:

544

H. Xu et al. / Applied Mathematics and Computation 217 (2010) 537–544

Thus, all the conditions of Theorem 2 are satisfied. Therefore, the equilibrium point of (1) is globally exponentially stable with the convergence rate k = 0.9319. In conclusion, it is clear that all the state variables in both examples globally exponentially converge to their equilibrium points.

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