Impulsive pinning synchronization of stochastic discrete-time networks

ARTICLE IN PRESS Neurocomputing 73 (2010) 2132–2139

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Neurocomputing journal homepage: www.elsevier.com/locate/neucom

Impulsive pinning synchronization of stochastic discrete-time networks$ Yang Tang a,b,, S.Y.S. Leung b, W.K. Wong b, Jian-an Fang a a b

College of Information Science and Technology, Donghua University, Shanghai 201620, PR China Institute of Textiles and Clothing, Hong Kong Polytechnic University, Hong Kong, PR China

a r t i c l e in f o

a b s t r a c t

Article history: Received 24 October 2009 Received in revised form 1 February 2010 Accepted 8 February 2010 Communicated by Z. Wang Available online 1 March 2010

The pinning synchronization problem of stochastic impulsive networks (SIN) is investigated. Using Lyapunov stability theory and pinning method, some sufficient criteria are derived for asymptotical synchronization and exponential synchronization of such dynamical networks in mean square. Illustrative examples are provided to verify the effectiveness of the proposed approach. & 2010 Elsevier B.V. All rights reserved.

Keywords: Pinning synchronization Stochastic neural networks Impulsive system Exponential synchronization

1. Introduction Synchronization is a very important characteristic of complex networks, which can be seen in many fields, and has been found applications in everywhere of real world [1–26]. Itˆo-type stochastic systems are well known for their significant effect on practical applications such as chemistry, biology, control, and information systems. In real complex networks, the signal transmission could be a noisy process brought on by random fluctuations from the release of probabilistic causes such as neurotransmitters. On the other hand, in today’s digital world, when implementing continuous-time networks for the sake of computer-based simulation and computation, it is usual to discretize the continuous-time networks. In fact, discrete-time networks have already been applied in a wide range of areas, such as image processing, time series analysis, quadratic optimization problems, and system identification [8]. Recently, synchronization of stochastic discrete networks has drawn much interests since stochastic disturbances in networks are necessary to be taken into

$ This research was supported by the National Natural Science Foundation of P.R. China (No. 60874113), the Research Fund for the Doctoral Program of Higher Education (No. 200802550007), the Key Foundation Project of Shanghai (Grant No. 09JC1400700) and the Key Creative Project of Shanghai Education Community (No. 09ZZ66), the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. PolyU 5212/07E).  Corresponding author at: College of Information Science and Technology, Donghua University, Shanghai 201620, PR China. E-mail addresses: [email protected] (Y. Tang), [email protected] (S.Y.S. Leung), [email protected] (W.K. Wong), [email protected] (J.-a. Fang).

0925-2312/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.neucom.2010.02.010

consideration for constructing a realistic network model, see [8,22] and references therein. Recently, various synchronization schemes have been proposed for synchronization of networks, see [2–16] for example. It has been revealed that, in the process of controlling various networks, feedback control serves as a simple and effective method for stabilization and synchronization. However, it is widely believed that it is impossible to add controllers to all nodes. To reduce the control cost, some feedback injections may be added to a fraction of network nodes, which is known as pinning control [12–15,17]. However, up to now, few works have been reported on the pinning problem of stochastic discrete-time networks. On the other hand, impulsive dynamical systems have attracted renewing interest in science and engineering during the past decades because they provide a natural framework for mathematical modeling of many real world evolutionary processes where the states undergo abrupt changes at certain instants in the fields such as medicine and biology, economics, electronics and telecommunications. Impulsive control methods have been widely used to synchronize coupled dynamical systems due to its potential advantages over general continuous control schemes [4–6]. Moreover, impulsive controller usually has a relatively simple structure [27–30]. In this sense, impulsive synchronization strategy is applicable in real-world applications. It is worth pointing out that most recent results appeared in the literature for stabilization or synchronization for networks by impulsive control have only considered the networks without stochastic disturbances. Due to the difficulty in investigating stochastic impulsive discrete-time system, synchronization

ARTICLE IN PRESS Y. Tang et al. / Neurocomputing 73 (2010) 2132–2139

problem of stochastic impulsive discrete-time networks has not been addressed so far. The contribution of this paper can be listed as follows: (1) a stochastic impulsive discrete-time networks is considered and its synchronization is investigated; (2) a pinning controller is employed to realize synchronization of stochastic impulsive discrete-time networks; (3) the average dwell-time is taken into account to derive the exponential synchronization criteria; (4) both asymptotical synchronization and exponential synchronization of stochastic impulsive networks are investigated in this paper. Motivated by the above discussions, the aim of this paper is to study the pinning synchronization of stochastic impulsive discrete-time networks in mean square. Via Lyapunov functionals and some analysis techniques, some sufficient conditions for pinning synchronization are derived for such networks. In the end, the developed approaches are applied to a scale-free network composing of discrete-time neural networks. The simulation results are given to show the effectiveness of the proposed results.

Notations: Throughout this paper, Rn and Rnm denote, respectively, the n-dimensional Euclidean space and the set of all real matrices. The superscript ‘T’ denotes matrix transposition and the notation X Z Y (respectively, X 4 Y) where X and Y are symmetric matrices, means that X Y is positive semi-definite (respectively, positive definite). In symmetric block matrices, the symbol  is used as an ellipsis for terms induced by symmetry. Z denotes the set including positive integers and zero. ðO; F ; PÞ is a complete probability space with a natural filtration F t Z 0 . Efxg and Efxjyg denote the expectation of x and the expectation of x conditional on y. J  J stands for the Euclidean vector norm in Rn . The Kronecker product of matrices Q A Rmn and R A Rpq is a matrix in Rmpnq and denoted as Q  R. lmin ðÞ and lmax ðÞ represents the maximum and minimum eigenvalue of a matrix. In denotes the identity matrix with dimension n  n. sðkÞ A Rn is a solution of an isolated node sðkÞ ¼ AsðkÞ þ f~ ðsðkÞÞ;

ð1Þ

n

with x0 A R . We consider an impulsive discrete-time networks consisting of N coupled identical neural networks, where each node of the network is an n-dimensional system: 8 PN ~ > > < xi ðk þ 1Þ ¼ Axi ðkÞ þ f ðxi ðkÞÞ þ ui ðkÞ þ c j ¼ 1 hij Gxj ðkÞ; ka Nl ;

Dxi ðk þ 1Þ ¼ xi ðkþ 1Þxi ðkÞ ¼ Uik ðk;xi ðkÞsðkÞÞ;

> > : i ¼ 1;2; . . . ; N;

k ¼ Nl ;

ð2Þ

where xi ðkÞ ¼ ½xi1 ðkÞ; xi2 ðkÞ; . . . ; xin ðkÞT A Rn is the state vector of the i th node; ui = [u1,u2,y,uN]T is the pinning control input and Uik is the impulsive control input; A ¼ diagða1 ; . . . ; an Þ 4 0 represents the connection weight matrix; G ¼ diagðg1 ; . . . ; gn Þ 40 represents the inner coupling matrix between the subsystems; matrix H ¼ ðhij ÞNN is the coupling configuration matrix representing the coupling strength and the topological structure of the networks which satisfies N X

hij ;

or without stochastic disturbances and impulsive effects has not been addressed in the literature. Here, let ei(k) =xi(k)  s(k) be the error states. It follows from (1), (2), the condition (2) and stochastic disturbances are considered that 8 PN > > < ei ðkþ 1Þ ¼ Aei ðkÞ þ f ðei ðkÞÞ þui ðkÞ þ c j ¼ 1 hij Gej ðkÞ þ kðk;ei ðkÞÞoðkÞ; Dei ðk þ 1Þ ¼ ei ðkþ 1Þei ðkÞ ¼ Uik ðk;ei ðkÞÞ; > > : i ¼ 1;2; . . . ;N;

ka Nl ;

ð4Þ k ¼ Nl ;

where f ðei ðkÞ; xi ðkÞÞ ¼ f~ ðxi ðkÞÞf~ ðsðkÞÞ. oðkÞ is a scalar Wiener process on a probability space ðO; F ; PÞ with

EfoðkÞg ¼ 0; Efo2 ðkÞg ¼ 1; EfoðiÞoðjÞg ¼ 0 ði a jÞ:

ð5Þ

The controller ui(k) is a pinning feedback controller described by ui ðkÞ ¼ cdi Gei ðkÞ; ui ðkÞ ¼ 0;

i ¼ 1;

i ¼ 2;3; . . . ;N;

ð6Þ

where di 4 0 is the feedback gain. In the following, we choose a linear impulsive controller, i.e., Uik(Nl, ei(k)) = Bikei(k), where each Bik is an n  n constant matrix. Thus, we have the following hybrid controlled stochastic impulsive discrete-time complex networks:

2. Preliminaries

hii ¼ 

2133

hij ¼ hji ;

hij Z 0ði ajÞ; i; j ¼ 1;2; . . . ; N:

ð3Þ

j ¼ 1;j a i

Remark 1. The complex network model (2) is general, which contains the nonlinear function, impulsive effects and stochastic disturbances. It should be pointed out that the pinning synchronization problem of discrete-time complex networks with

8 PN > < ei ðkþ 1Þ ¼ Aei ðkÞ þ f ðei ðkÞÞ þ c j ¼ 1 gij Gej ðkÞ þ kðk;ei ðkÞÞoðkÞ; Dei ðkþ 1Þ ¼ ei ðkþ 1Þei ðkÞ ¼ Bik ei ðkÞ; > : i ¼ 1;2; . . . ; N;

k a Nl ; k ¼ Nl ;

ð7Þ where G ¼ ðgij ÞNN and ( hij di ; i ¼ j ¼ 1; gij ¼ otherwise: hij

ð8Þ

Throughout this paper, the following assumptions are needed: (A1) For i A f1; 2; . . . ; ng, the nonlinear function in (1) satisfies H1 r

f~ i ðxÞf~ i ðyÞ r H2 ; xy

i ¼ 1;2; . . . ;n;

ð9Þ

where H1, H2 are constants. It can be verified from (9) that H1 r

fi ðxÞfi ðyÞ r H2 ; fi ð0Þ ¼ 0; xy

i ¼ 1;2; . . . ;n;

ð10Þ

where H1, H2 are constants. (A2) The stochastic intensity kðk; ei ðkÞÞ satisfies the following inequality.

kðk;ei ðkÞÞ r eTi ðkÞST Sei ðkÞ;

ð11Þ

where S is a matrix with appropriate dimension. (A3). The sequences {Nl} satisfies: Nl A Z and N0 ¼ 0 o N1 o N2 o    o Nl o    ; with Nl þ 1 Nl 4 1, l A Z. (A4). The function f~ i ðxÞ satisfies the Lipschitz condition, i.e., for all i=1,2,y,n, there exist constants li 4 0 such that jf~ i ðxÞf~ i ðyÞj r li jxyj;

8x; y A Rn :

The following definitions are needed in this paper: Definition 1. The complex networks is said to have average dwell time Ta if there exist two positive numbers N0 and Ta such that Nðk; k0 Þ r N0 þ

ðkk0 Þ ; Ta

k Zk0 ;

holds for N0 Z 0 and Ta 40, then Ta is called the average dwell time and N0 the chatter bound. Definition 2. Let xi ðk; k0 ; X0 Þ ð1r i rNÞ be a solution of the array of stochastic impulsive discrete-time complex networks (1), where X0 = (x01, x02,y, x0N). If there is a nonempty subset O D Rn , with x0i A O ð1 ri rNÞ, such that xi ðk; k0 ; X0 Þ A Rn for 1r i rN,

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Y. Tang et al. / Neurocomputing 73 (2010) 2132–2139

and lim E

k-1

N X

Jei ðkÞJ2 ¼ 0;

i ¼ 1;2; . . . ;N:

ð12Þ

i¼1

Then the stochastic impulsive complex networks (1) which is said to achieve synchronization in mean square and O      O is called the region of synchrony for networks (1). Definition 3. Let xi ðk; k0 ; X0 Þ ð1 ri r NÞ be a solution of the array of stochastic impulsive discrete-time complex networks (1), where X0 = (x01, x02,y, xN0). If there is a nonempty subset O D Rn , with x0i A Oð1 ri r NÞ, such that xi ðk; k0 ; X0 Þ A Rn for all k Zk0 ; 1 r ir N, and lim E

k-1

b ¼ maxfb1 ; . . . ; bl g; bl ¼ lmax ðP1 ðI þBiNl ÞT PðI þBiNl ÞÞ, c ¼ O2 , 1 lmax ðjP1 Þ, O ¼ ½O O1 ¼ AT PA þ c2 slmax ðaÞ þ 2cmlmax ðGÞ OT O3 where

N X

Jei ðkÞJ2 r M0 d

eðkk0 Þ

;

i ¼ 1;2; . . . ;N;

ð13Þ

i¼1

where e 40, M0 4 0 and d 4 1. Then the stochastic impulsive complex networks (1) which is said to achieve exponential synchronization in mean square and O      O is called the region of synchrony for networks (1). Remark 2. Inspired by [27], the average dwell time is considered in this paper for deriving exponential synchronization criteria. Usually, N0 is taken as N0 = 0. On the other hand, recently, the control and stability analysis of stochastic impulsive system have drawn initial attention [29,30]. However, in these well-studied works, stochastic discrete-time impulsive system has not been investigated.

2

^  O3 ¼ P þcRlmax ðaÞU, L ¼ O2 ¼ AT P þ U H, þ cR þ xST SU HP, 2

maxflj : j ¼ 1; 2; . . . ; ng and ( lmax ðOÞ þ Llmax ðOÞ if lmax ðOÞ Z 0; j¼ lmax ðOÞ if lmax ðOÞ o 0: Then, the system (1) is said to be synchronized in mean square, that is the trivial solution of the error system (7) is stable in mean square. Proof. 8 PN > > < ei ðk þ1Þ ¼ Aei ðkÞ þ f ðei ðkÞÞ þc j ¼ 1 gij Gej ðkÞ þ kðk;ei ðkÞÞoðkÞ; ei ðk þ1Þ ¼ ðI þBik Þei ðkÞ; > > : i ¼ 1;2; . . . ;N;

N X

VðkÞ ¼ eT ðkÞðIN  PÞeðkÞ ¼

eTi ðkÞPei ðkÞ;

ð18Þ

i¼1

where e(k) =[eT1(k),y,eTN(k)]T. Calculating the difference of V(k) along the system (17) for any ka Nl , we have

EfDVðkÞg ¼ EfVðkþ 1ÞVðkÞg

82 3T N < N X X 4Aei ðkÞ þ f ðei ðkÞÞ þ c ¼E gij Gej ðkÞ5 P : i¼1 j¼1 2 3 N X 4Aei ðkÞ þ f ðei ðkÞÞ þc gij Gej ðkÞ5 j¼1



eTi ðkÞPei ðkÞ þ½ ðk;ei ðkÞÞT P½ ðk;ei ðkÞÞ N  X ¼E eTi ðkÞAT PAei ðkÞ þf T ðei ðkÞÞPf ðei ðkÞÞ i¼1 0 1T 0 1 N N X X

k

3. Main results

þ @c

The following lemmas will be essential in achieving the main results.

þ 2eTi ðkÞAT P @c

k

gij Gej ðkÞA P @c

j¼1

gij Gej ðkÞA þ2eTi ðkÞAT Pf ðei ðkÞÞ

j¼1

0 0

Lemma 1 (See Ref. [27]). Let O1 ; O2 ; O3 be real matrices of appropriate dimensions with O3 4 0. Given any vector x,y of appropriate dimensions, then the following inequality holds:

N X

1 gij Gej ðkÞA

j¼1

þ 2f T ðei ðkÞÞP@c

N X

13 gij Gej ðkÞA5

j¼1

 eTi ðkÞPei ðkÞ þ½kðk;ei ðkÞÞT P½kðk;ei ðkÞÞ :

2xT OT1 O2 yr xT OT1 O3 O1 x þ yT OT2 O1 3 O2 y:

lmin ðH1 MÞxT Hx r xT Mx r lmax ðH1 MÞxT Hx:

ð17Þ k ¼ Nl :

Choose a Lyapunov function for (4) and (17) as follows:

Remark 3. Assumption (A1) was first introduced in Ref. [31] and has been subsequently studied in some complex network papers (see, e.g. [7,8]). The conditions in (A1) are more general than the usual sigmoid functions and the recently commonly used Lipschitz conditions. Such a description is very precise in quantifying the lower and upper bounds of the nonlinear functions, therefore it is beneficial for employing LMI-based method to reduce the possible conservatism.

Lemma 2 (See Ref. [28]). Let HA Rnn be a positive definite matrix and MA Rnn a symmetric matrix. Then, for any x A Rn , the following inequality holds:

k a Nl ;

Denote a ¼ ðaij ÞNN ¼ GT G, one has 0 1T 0 1 N N N X X X @c gij Gej ðkÞA P @c gij Gej ðkÞA i¼1

j¼1

j¼1 T

As discussed in Ref. [31], for brevity of the following analysis,  ¼ diagððH11 ^ ¼ diagðH11 H12 ; H21 H22 ; . . . ; Hn1 Hn2 Þ, H we denote H þH12 Þ=2; ðH21 þ H22 Þ=2; . . . ; ðHn1 þ Hn2 Þ=2Þ. Theorem 1. Under Assumptions ðA1 Þ2ðA4 Þ, if there exist positive constants x, R, s, m, y, positive matrices P and U such that P r xIn ;

P G r RIn ;

¼ ½cðG  GÞeðkÞ ðIN  PÞ½cðG  GÞeðkÞ ¼ ½ceT ðkÞðGT  GÞðIN  PÞ½cðG  GÞeðkÞ N N X X eTi ðkÞ aij GPGej ðkÞ ¼ c2 eT ðkÞðGT G  GP GÞeðkÞ ¼ c2 ¼ c2

i¼1j¼1

ð14Þ ¼ c2

P G2 r sIn ;

AP G Z mIn ;

ð15Þ

lnð1 þ cÞ þ

llnb r y o0; kl

ð16Þ

N X N X

n X l¼1

¼ c2

n X l¼1

"

aij 2

pl g2l 4

i¼1 n X

#

j¼1

eil ðkÞpl g2l ejl ðkÞ

l¼1 N X N X

3

aij eil ðkÞejl ðkÞ5

i¼1j¼1

pl g2l ½eTl ðkÞael ðkÞ r c2

n X l¼1

pl g2l lmax ðaÞeTl ðkÞel ðkÞ

ð19Þ

ARTICLE IN PRESS Y. Tang et al. / Neurocomputing 73 (2010) 2132–2139 N X

r c2 slmax ðaÞ

eTi ðkÞei ðkÞ;

ð20Þ

i¼1

It is easy to check that 0 1 N N X X T T @ 2ei ðkÞA P c gij Gej ðkÞA i¼1

r 2c

"

gij

i¼1j¼1

¼ 2c

n X

2

al pl g l 4

¼ 2c

eil ðkÞal pl gl ejl ðkÞ

l¼1 N X

¼E

3

al pl gl ½eTl ðkÞGel ðkÞ r 2c

n X

ZTi ðkÞlmax ðOÞZi ðkÞ ½eTi ðkÞlmax ðOÞei ðkÞ þf T ðei ðkÞÞlmax ðOÞf ðei ðkÞÞ

jeTi ðkÞei ðkÞ r E

N X

ceTi ðkÞPei ðkÞ ¼ EcVðkÞ;

ð27Þ

i¼1

^ O1 ¼ AT PA þc2 slmax ðaÞ þ2cmlmax ðGÞ þ cR þ xST SU HP,

 O3 ¼ P þ cRlmax ðaÞU, c ¼ lmax ðjP1 Þ, L ¼ max O2 ¼ AT P þ U H,

al pl gl lmax ðGÞeTl ðkÞel ðkÞ

2

eTi ðkÞei ðkÞ;

ð21Þ

i¼1

flj : j ¼ 1; 2; . . . ; ng, ZTi ðkÞ ¼ ½eTi ðkÞ; f T ðei ðkÞÞ, and ( lmax ðOÞ þ Llmax ðOÞ if lmax ðOÞ Z 0; j¼ lmax ðOÞ if lmax ðOÞ o 0: Using (27), one has

and N X

i¼1 N X

where

l¼1

r 2cmlmax ðGÞ

i¼1 N X

gij eil ðkÞejl ðkÞ5

l¼1 N X

i¼1 N X

i¼1

i¼1j¼1

l¼1 n X

¼E

#

n X

N X

f T ðei ðkÞÞUf ðei ðkÞÞeTi ðkÞPei ðkÞ þ xeTi ðkÞST Sei ðkÞg #" " #T " # N X O1 O2 ei ðkÞ ei ðkÞ ¼E T O2 O3 f ðei ðkÞÞ f ðei ðkÞÞ rE

j¼1 N X N X

2135

0 2f T ðei ðkÞÞP @c

i¼1

N X

1

EfVðkþ 1Þg ¼ Efð1 þ cÞVðkÞg:

gij Gej ðkÞA

When k= Nl, we have from Lemma 2 that ( ) N X T T EfVðNl þ1Þg ¼ E ei ðkÞðI þ BiNl Þ PðI þ BiNl Þei ðkÞ r Efbl VðNl Þg;

j¼1

¼ 2c rc rc

n X

pl gl ½f T ðel ðkÞÞGel ðkÞ

i¼1

l¼1 n X

T

pl gl ½f ðel ðkÞÞG

l¼1 n X

T

ð29Þ

Gf ðel ðkÞÞ þ eTl ðkÞel ðkÞ

where bl ¼ lmax ðP1 ðI þBiNl ÞT PðI þBiNl ÞÞ. It follows from (28) and (29) that for any k A ðNl1 ; Nl 

R½f T ðel ðkÞÞaf ðel ðkÞÞþ eTl ðkÞel ðkÞ

l¼1 N X

r cR

ð28Þ

T

lmax ðaÞf ðei ðkÞÞf ðei ðkÞÞ þ cR

i¼1

N X

EfVðkÞg r Efð1 þ cÞVðk1Þg r    r ð1 þ cÞkNl1 1 EfVðNl1 þ 1Þg eTi ðkÞei ðkÞ:

ð22Þ

i¼1

r Efð1þ cÞkNl1 1 bl1 VðNl1 Þg r Efð1þ cÞkNl1 1 ð1 þ cÞNl1 Nl2 1 bl1 bl2 VðNl2 Þg l

r    r Efð1þ cÞkl b Vð0Þg:

By (A2), one can get ½kðk;ei ðkÞÞT P½kðk;ei ðkÞÞ r xei ðkÞST Sei ðkÞ:

ð23Þ

From (9) and (10), we have ðfi ðei ðkÞÞHi1 ei ðkÞÞT ðfi ðei ðkÞÞHi2 ei ðkÞÞ r 0:

ð24Þ

It can be deduced from (24) that there exist U ¼ diagfu1 ; . . . ; un g 4 0, such that 3 2 Hi1 þHi2 T " " #T # T H H e e  e e n i1 i2 i i i 7 X i ei ðkÞ ei ðkÞ 6 2 7 6 ui 5 f ðei ðkÞÞ f ðei ðkÞÞ 4 Hi1 þHi2 T i¼1  ei ei ei eTi 2 #" " #T " # ^  ei ðkÞ ei ðkÞ UH U H ¼ r0; ð25Þ  TU f ðei ðkÞÞ f ðei ðkÞÞ U H where ei denotes a column vector having ‘1’ element on its i-th row and zeros elsewhere. Then, one can derive from (25) that #" " #T " # ^  ei ðkÞ ei ðkÞ U H UH Z0: ð26Þ  T U f ðei ðkÞÞ f ðei ðkÞÞ UH Combining (19)–(23), (26) and (A4), one has

EfDVðkÞg r E

N X

f½eTi ðkÞAT PAei ðkÞ þ f T ðei ðkÞÞPf ðei ðkÞÞ

i¼1 þ c2 lmax ð ÞeTi ðkÞei ðkÞ þ 2f T ðei ðkÞÞAT Pei ðkÞ þ 2c lmax ðGÞeTi ðkÞei ðkÞ þc lmax ð Þf T ðei ðkÞÞf ðei ðkÞÞ ^ T ðkÞ þ 2eT ðkÞU Hf  ðei ðkÞÞ þ c eTi ðkÞei ðkÞeTi ðkÞU He i i

s m R

a

R

a

ð30Þ

From the definition of Lyapunov function (18), it can be obtained that ( ) ( ) N N X X 2 2 ð31Þ E lmin ðPÞ Jei ðkÞJ r EfVðkÞg r E lmax ðPÞ Jei ðkÞJ : i¼1

i¼1

Therefore, one can easily have ( ) ( ) N N X X lmax ðPÞ l ð1 þ cÞkl b E Jei ðkÞJ2 r E Jei ð0ÞJ2 lmin ðPÞ i¼1 i¼1 ( ) N lmax ðPÞ ðklÞy X 2 e Jei ð0ÞJ : rE lmin ðPÞ i¼1

ð32Þ

It follows from (32) and (A3) that the system (17) is asymptotically stable, i.e. an array of discrete-time stochastic impulsive neural networks (1) is asymptotically synchronized. Thus, this completes the proof. & In order to prove the exponential pinning synchronized of an array of discrete-time stochastic impulsive neural networks (1), we have the following results. Theorem 2. Under Assumptions ðA1 Þ2ðA4 Þ, if there exist positive constants x, R, s, m, l 41, Ta 4 0, positive matrices P and U such that ^ o0; O P r xIn ;

ð33Þ P G r RIn ;

ð34Þ

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P G2 r sIn ;

Ta 4Ta ¼ where 2 ^ ¼4 O

AP G Z mIn ;

ð35Þ

lnb^ ; lnl

^1 O

^2 O

^T O 2

^3 O

ð36Þ

When k= Nl, we have ( ) N X kþ1 T T EfVðNl þ1Þg ¼ E l ei ðkÞðI þBiNl Þ PðI þBiNl Þei ðkÞ i¼1

r Efb^ l VðNl Þg; 3

ð41Þ

where b^ l ¼ lmax ðlP1 ðI þBiNl ÞT PðI þBiNl ÞÞ. It follows from (40), (41) and Definition 1 that for any kA ðNl1 ; Nl 

5;

^ 1 ¼ lAT PA þ lc2 slmax ðaÞ þ 2lcmlmax ðGÞ þ lcR þ xST SlU HP; ^ O

EfVðkÞg r EfVðk1Þg r    r EfVðNl1 þ 1Þg r Efb^ l1 VðNl1 Þg l k=Ta Vð0Þg r Efb^ l1 b^ l2 VðNl2 Þg r    r Efb^ Vð0Þg r Efb^   b^ k=Ta ln ln l Vð0Þ ; rE l

^ 3 ¼ lP þ lcRlmax ðaÞlU, b^ ¼ maxfb^ ; . . . ; b^ g, ^ 2 ¼ lAT P þ lU H,  O O 1 l b^ l ¼ lmax ðlP1 ðI þ BiNl ÞT PðI þ BiNl ÞÞ. Then, the system (1) is said to

ð42Þ

be exponentially synchronized in mean square, that is the trivial solution of the error system (7) is exponential stable in mean square. Proof. Define a Lyapunov function for (4) and (17) as follows: k

k

VðkÞ ¼ l eT ðkÞðIN  PÞeðkÞ ¼ l

N X

eTi ðkÞPei ðkÞ;

ð37Þ

where b^ ¼ maxfb^ 1 ; . . . ; b^ l g. From (37), the following inequality holds: ( ) ( ) N N X X E lmin ðPÞ lk Jei ðkÞJ2 r EfVðkÞg r E lmax ðPÞ lk Jei ðkÞJ2 : i¼1

i¼1

where l 41. Calculating the difference of V(k) along the system (7) for any k aNl , we have

EfDVðkÞg ¼ EfVðkþ 1ÞVðkÞg ¼E 0

N X

þ @c

N X

1T 0 gij Gej ðkÞA P@c

j¼1

N X j¼1

þ 2f T ðei ðkÞÞP @c

N X

E

gij Gej ðkÞA

(

^

0

N X

1

N X

(

kð1lnb^ =Ta lnlÞ

Jei ðkÞJ2 r E el i

N X

N X

) Jei ð0ÞJ2

i¼1

) Jei ð0ÞJ2 ;

ð45Þ

i¼1

gij Gej ðkÞA where

gij Gej ðkÞA5

e¼ ð38Þ

ð39Þ

N X

lk fl½eTi ðkÞAT PAei ðkÞ þf T ðei ðkÞÞPf ðei ðkÞÞ i¼1 þ c2 lmax ð ÞeTi ðkÞei ðkÞ þ 2f T ðei ðkÞÞAT Pei ðkÞ þ 2c lmax ðGÞeTi ðkÞei ðkÞ þ c lmax ð Þf T ðei ðkÞÞf ðei ðkÞÞ ^ T ðkÞ þ 2eT ðkÞU Hf  ðei ðkÞÞ þ c eTi ðkÞei ðkÞeTi ðkÞU He i i T T f ðei ðkÞÞUf ðei ðkÞÞei ðkÞPei ðkÞ þ xeTi ðkÞST Sei ðkÞg 3 " #T 2 ^ ^ 2 " e ðkÞ # N X O1 O ei ðkÞ i 4 T 5 ¼E lk ^ ^3 f ðei ðkÞÞ f ðei ðkÞÞ O O i¼1 2

lmax ðPÞ ; lmin ðPÞ

i ¼ 1

lnb^ 4 0: Ta lnl

Thus, this completes the proof.

&

Remark 4. Note that one can easily extend our results to the case of discrete-time complex networks without stochastic disturbances and discrete-time complex networks without impulsive effects. On the other hand, the asymptotical and exponential pinning problem of discrete-time complex networks has been addressed in this paper, which has not been investigated in the literature. Remark 5. It is worth noting that the results can be generalized to stability and control problem of stochastic impulsive discretetime system. The asymptotical and exponential stability criteria of such system can be obtained without any difficulty.

a

^ Z ðkÞ r 0; lk ZTi ðkÞO i

) Jei ð0ÞJ2 ;

j¼1

13

R

N X i¼1

r E eðl Þk

Thus, one has

N X

)

lk Jei ðkÞJ2 r E lðk=Ta Þlnb =lnl lmax ðPÞ )

(

The following inequality holds: #" " #T " # ^  ei ðkÞ ei ðkÞ U H UH kþ1 l Z 0:  T U f ðei ðkÞÞ f ðei ðkÞÞ UH

rE

N X

i¼1

 eTi ðkÞPei ðkÞ þ ½kðk;ei ðkÞÞT P½kðk;ei ðkÞÞ :

s m R

E lmin ðPÞ

ð44Þ

j¼1

EfDVðkÞg r E

Therefore (

(

1

þ 2eTi ðkÞAT Pf ðei ðkÞÞ þ2eTi ðkÞAT P @c 0

ð43Þ

i¼1

  lk l eTi ðkÞAT PAei ðkÞ þ f T ðei ðkÞÞPf ðei ðkÞÞ

i¼1

i¼1

a

ð40Þ

i¼1

where ^ 1 ¼ lAT PA þ lc2 slmax ðaÞ þ 2lcmlmax ðGÞ þ lcR þ xST SlU HP, ^ O T ^  O 2 ¼ lA P þ lU H, ^ 3 ¼ lP þ lcRlmax ðaÞlU and ZT ðkÞ ¼ ½ eT ðkÞ; f T ðe ðkÞÞ. O i i i

Remark 6. Generally, a complex networks is very large consisting of a number of nodes. It will spend much time when using the criteria to solve the LMIs in [7]. Different from the synchronization criteria in [7], the complexity of linear matrix inequalities is determined by a single node of complex networks instead of the scale of the complex networks.

4. Numerical simulation In this section, two numerical simulation examples are given to show the effectiveness and superiority of proposed method.

ARTICLE IN PRESS Y. Tang et al. / Neurocomputing 73 (2010) 2132–2139

Example 1. Let the number of nodes is N= 100 and assume that the network structure of (1) obeys the scale-free distribution of the Baraba´si–Albert model [32]. Consider the following complex networks composing by an array of neural networks: 8 PN ~ > < xi ðk þ 1Þ ¼ Axi ðkÞ þ f ðxi ðkÞÞ þ ui ðkÞ þc j ¼ 1 hij Gxj ðkÞ; k a Nl ; Dxi ðk þ 1Þ ¼ xi ðk þ1Þxi ðkÞ ¼ Uik ðk;xi ðkÞsðkÞÞ; k ¼ Nl ; : ð46Þ > : i ¼ 1;2; . . . ; N; with xi ðkÞ ¼ ðx1 ðkÞ; x2 ðkÞÞT ; f~ ðxðkÞÞ ¼ tanhðxðkÞÞ ¼ ðtanhð0:1x1 Þ; tanhð0:1x2 ÞÞT ; and A= diag(0.895,0.895), G ¼ diagð0:1; 0:1Þ and c= 0.01. The noise intensity is chosen as kðk; ei ðkÞÞ ¼ 0:05ei ðkÞ. ^ ¼ diagð0; 0Þ The l is taken as l ¼ 1:001. It is easy to check that H  and H ¼ diagð0:05; 0:05Þ. The equidistant impulsive interval Nl Nl1  Ta ¼ 3 and Bik is chosen as Bik =diag( 0.1,  0.1). Only one node is randomly pinned and d =10. We obtain a feasible solution by using LMI toolbox in Matlab [33] in the following:     319:6450 0 16:2188 0 P¼ ; U¼ ; 0 16:2188 0 319:6450

x ¼ 23:5466; R ¼ 2:2857; s ¼ 1:1752 and m ¼ 0:4860. Thus, one has Ta ¼ 34 Ta ¼ 210:3:

lnð1þ cÞ þ

0

1.5

−0.5

1

e (t)

2

−1

0

−2

−0.5

50

100 t

150

−1

200

1

3

0.5

2.5

0

2

−0.5

1

−1.5

0.5

0

50

100 t

150

200

0

50

100 t

150

200

0

50

100 t

150

200

1.5

−1

−2

ð48Þ

0.5

−1.5

0

llnb ¼ 0:0225 r y ¼ 0:02 o 0: kl

It can be checked that all the conditions in Theorem 1 are satisfied. Thus, pinning synchronization of discrete-time stochastic impulsive complex networks (1) can be realized asymptotically in mean square. Note that Theorem 2 cannot generate a feasible solution. However, using our criteria in Theorem 1,

x (t)

e (t)

Example 2. Consider the complex networks (46) with A= diag(1,1). The equidistant impulsive interval Nl =3l and Bik is chosen as Bik = diag( 0.2,  0.2). The other parameters are the same as Example 1. Theorem 2 is used to solve this model and cannot obtain a feasible solution using LMI toolbox in Matlab. We use P, U, x, R, s and m in Example 1. Let y ¼ 0:02 and one has

0.5

−2.5

x (t)

difficulty of mathematical derivation. On the other hand, the pinning synchronization of stochastic impulsive complex networks is investigated. The pinning synchronization conditions in well-studied works [13–15,17] cannot generate a feasible solution since the model is in the form of discrete-time and stochastic disturbances are also considered. However, using our criteria in Theorem 2, the stochastic impulsive complex networks (1) can be uniformly synchronized exponentially in mean square. Fig. 1(a) and (b) show the error states of synchronization errors when one node in the networks is pinned. The stochastic impulsive discrete-time complex networks can be synchronized exponentially in mean square, as depicted in Fig. 1(a) and (b). Fig. 1(c) and (d) show that the states of the stochastic impulsive discrete-time complex networks reach consensus.

ð47Þ

Obviously, the conditions of Theorem 2 are all satisfied. Therefore, the exponential pinning synchronization of discrete-time stochastic impulsive complex networks (1) can be realized in mean square. For this example, all the impulsive synchronization stability criteria in [4–6] cannot succeed to draw a conclusion since the model is in the discrete-time form, which results in the

2137

0

Fig. 1. (a) Synchronization errors of k ei 1(k). (b) k  ei 2(k). (c) k  xi 1(k). (d) k  xi 2(k).

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Y. Tang et al. / Neurocomputing 73 (2010) 2132–2139

15

0

10

−5

5 −10

e (t)

e (t)

0 −15 −20

−10

−25

−15

−30 −35

−5

−20

0

50

100 t

150

−25

200

200

150

150

100

100

50

100 t

150

200

0

50

100 t

150

200

x (t)

x (t)

200

0

50

50

0

0

−50

0

50

100 t

150

200

−50

Fig. 2. (a) Synchronization errors of k  ei 1(k). (b) k ei 2(k). (c) k  xi 1(k). (d) k  xi 2(k).

asymptotical pinning synchronization of stochastic impulsive complex networks (1) can be achieved in mean square. Fig. 2(a) and (b) illustrate the error states of synchronization errors when only one node in the networks is pinned. The stochastic impulsive discrete-time complex networks can be synchronized in mean square, as shown in Fig. 2(a) and (b). Fig. 2(c) and (d) depict that the states of the stochastic impulsive discrete-time complex networks achieve consensus in a few iterations. 5. Conclusion The asymptotical and exponential pinning synchronization problems of stochastic impulsive discrete-time networks are investigated in this paper. Using Lyapunov stability theory and pinning impulsive controller, several sufficient conditions are established for synchronization of stochastic impulsive discrete-time networks in mean square. Finally, two numerical simulation examples are offered to verify the effectiveness of the proposed approach. We would also like to extend our main results to genetic regulatory networks and neural networks using the methods presented in well-studied works [34–38].

Acknowledgments The authors are grateful to the Editor and anonymous reviewers for their careful reading and constructive comments.

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Yang Tang received the B.S. degree in electrical engineering from Donghua University, Shanghai, China, in 2006. Now he is a Ph.D. student with Department of Automation, Donghua University, Shanghai, China. Since January 2009, he has been a Research Associate in Hong Kong Polytechnic University, Hung Hom Kowloon, Hong Kong. His current research interests include stochastic systems, neural networks, image encryption, complex networks, data mining and evolutionary computation.

S.Y.S. Leung received both Ph.D. degree in supply chain management and M.Sc. degree in advanced manufacture (clothing) from the Manchester Metropolitan University, Manchester, UK. He is currently an Assistant Professor at Institute of Textiles and Clothing of The Hong Kong Polytechnic University, Hong Kong. His research interests include discrete event simulation for clothing manufacture, apparel supply chain management, lean and agile production, application of artificial intelligent techniques in fabric cutting, and utilization of RFID in fashion cross-selling and he has authored and co-authored conference and journal papers in these areas. Dr. Leung is the Scheme Leader of Fashion and Textile Studies.

W.K. Wong received his Ph.D. degree from The Hong Kong Polytechnic University, Hong Kong. He is currently an Assistant Professor with the Institute of Textiles and Clothing (Business Division), The Hong Kong Polytechnic University. He has published more than 40 scientific articles in refereed journals, including European Journal of Operational Research, International Journal of Production Economics, Expert Systems with Applications, International Journal of Production Research, Computers in Industry, among others. His main research interests include neural network, pattern recognition, time series forecasting as well as manufacturing planning and scheduling.

Jian-an Fang received the B.S., M.S., and Ph.D. degrees in electrical engineering from Donghua University (China Textile University), Shanghai, China, in 1988, 1991 and 1994 respectively. Subsequently, he joined the College of Information Science and Technology, Donghua University, Shanghai, China, where he became a Dean and Professor in 2001. During February 1998 and May 1998, he was the visiting scholar in the University of Michigan at Ann Arbor. During May 1998 and February 1999, he was the visiting scholar in the University of Maryland at College Park. During June 2005 and August 2005, he was the senior visiting scholar in the University of Southern California. In 2005 and 2006, Prof. Fang was elected as a Council Member of Shanghai Automation Association and a Council Member of Shanghai Microcomputer Applications, respectively. His research interests are mainly in complex system modeling and control, intelligent control systems, chaotic system control and synchronization, and digitalized technique for textile and fashion.