Lag stochastic synchronization of chaotic mixed time-delayed neural networks with uncertain parameters or perturbations

Neurocomputing 74 (2011) 1617–1625

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

Lag stochastic synchronization of chaotic mixed time-delayed neural networks with uncertain parameters or perturbations Xinsong Yang a,, Quanxin Zhu b, Chuangxia Huang c a

Department of Mathematics, Honghe University, Mengzi, Yunnan 661100, China Department of Mathematics, Ningbo University, Ningbo Zhejiang 315211, China c Department of Mathematics, College of Mathematics and Computing Science, Changsha University of Science and Technology, Changsha, Hunan 410076, China b

a r t i c l e i n f o

abstract

Article history: Received 20 October 2009 Received in revised form 3 December 2010 Accepted 19 January 2011 Communicated by J. Cao Available online 17 March 2011

This paper investigates the problem of lag synchronization for a kind of chaotic neural networks with discrete and distributed delays (mixed delays). The driver system has uncertain parameters and uncertain nonlinear external perturbations, while the response system has channel noises. A simple but all-powerful robust adaptive controller is designed to circumvent the effects of uncertain external perturbations such that the response system synchronize with the driver system. Based on the invariance principle of stochastic differential equations and some suitable Lyapunov functions, several sufficient conditions are developed to solve this problem. Moreover, under certain conditions, parameters of the uncertain master system can be estimated. Numerical simulations are exploited to show the effectiveness of the theoretical results. & 2011 Elsevier B.V. All rights reserved.

Keywords: Lag synchronization Mixed delay Nonlinear perturbations Parameter identification Vector-form noise

1. Introduction In the past decades, since the concept of drive-response synchronization for coupled chaotic systems was proposed in [1]. Much attention has been focused on control and chaos synchronization due to its potential applications such as secure communication, biological systems, information science, etc. [2,3]. In order to realize drive-response synchronization, a number of control schemes have been proposed such as linear and nonlinear feedback control [4], adaptive control [5], and so on. Recently, it has been revealed that the delayed neural networks (DNNs), such as delayed Hopfield neural network and delayed cellular network, can exhibit chaotic behaviors [6,7]. Hence, there has been a great deal of activities on investigating the synchronization problem of coupled chaotic DNNs. In the research area of neural network synchronization, several results have been appeared in the literature [5,8–22]. In [5], the authors investigated the adaptive synchronization of neural networks with or without time-varying delay. Two coupled identical neural networks are shown to achieve synchronization by enhancing the coupled strength. Yang and Cao [9] studied the exponential lag synchronization of a class of chaotic delayed neural networks with impulsive effects. Some sufficient conditions are

 Corresponding author.

E-mail addresses: [email protected] (X. Yang), [email protected] (Q. Zhu), [email protected] (C. Huang). 0925-2312/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.neucom.2011.01.010

established by the stability analysis of impulsive differential equations. The authors of [10] dealt with the exponential synchronization problem for a class of stochastic perturbed chaotic delayed neural networks. Based on the Lyapunov stability theory, stochastic analysis, Halanay inequality for stochastic differential equation, drive-response concept and time-delay feedback control techniques, several sufficient conditions were proposed to guarantee the exponential synchronization of two identical chaotic delayed neural networks with stochastic perturbation. In [11], Lou and Cui studied the synchronization of neural networks based on parameter identification via output or state coupling. In [14], He and Cao investigated the adaptive synchronization of a class of chaotic neural networks with known or unknown parameters. Time delays in [5,8–15] are constants or time-varying discrete delays. In practice, neural networks usually have a spatial extent due to the presence of an amount of parallel pathways with a variety of axon size and lengths [16–22], therefore, it is more practical to consider synchronization of coupled neural networks with both discrete and distributed delays, i.e., mixed delays. Effects of perturbations should be taken into account in researching synchronization of neural networks. White noises brought by some random fluctuations in the course of transmission and other probabilities causes have been received extensive attention in the literatures, for instance, see [17,20–22]. However, from practical point of view, neural networks are always in a changing environments and therefore maybe disturbed by some unknown factors from environments. Obviously, such kind of perturbations may be non-stochastic, i.e., the average values of

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X. Yang et al. / Neurocomputing 74 (2011) 1617–1625

them are not zero. A small perturbation to chaotic system will result in a drastic change in the chaotic behavior of the chaotic system [23]. One important reason for studying drive-response chaos synchronization is the application of chaos to secure communication [3]. Artificially adding some disturbances especially those nonlinear and non-stochastic perturbations can make the message transmission more complicated, and hence it is more difficult to realize the drive-response synchronization than that in the case without any perturbation, which in turn strengthens the security of communication. However, to the best of our knowledge, the synchronization problem of DNNs with mixed delays and both uncertain external and stochastic perturbations is still open. In this paper, we are particularly interested in the driveresponse synchronization of neural networks with mixed delays, uncertain parameters as well as uncertain external perturbations. Uncertain external perturbations include nonlinear non-stochastic perturbation and stochastic perturbation. In the driveresponse DNNs, not only the Lipschitz constants on neuron activation functions but also the bounds of external perturbations are unknown. A simple but robust adaptive controller is designed to circumvent these uncertainties and synchronize the coupled systems. Based on two new Lyapunov functions and the invariance principle of stochastic differential equations [24], several sufficient conditions are developed to solve this problem. Moreover, under certain conditions, parameters of the uncertain driver system can be estimated. Numerical simulations demonstrate the usefulness of the proposed design methods. The rest of this paper is organized as follows. In Section 2, new models of mixed delayed neural networks with external perturbations and stochastic distributions are presented. Some necessary assumptions, definitions and lemmas are also given in this section. Our main results and their rigorous proofs are described in Section 3. In Section 4, one example with numerical simulations is offered to show the effectiveness of our results. In Section 5, conclusions are given. At last, acknowledgments.

2. Preliminaries

þ

n X

dij

Z

t tZ

j¼1

fj ðxj ðsÞÞ ds

þ Ii þ si t,xðtÞ,xðtyÞ,

Z

!#

t

xðsÞ ds

dt,

or, in a compact form " dxðtÞ ¼ CxðtÞ þ Af ðxðtÞÞ þ Bf ðxðtyÞÞ þ D þ s t,xðtÞ,xðtyÞ,

i ¼ 1,2, . . . ,n,

ð3Þ

tZ

Z

t

Z

f ðxðsÞÞ ds þ I tZ

!# xðsÞ ds

t

dt,

ð4Þ

tZ

Rt Rt where sðt,xðtÞ,xðtyÞ, tZ xðsÞ dsÞ ¼ s1 ðt,xðtÞ,xðtyÞ, tZ xðsÞ dsÞ, Rt . . . , sn ðt,xðtÞ,xðtyÞ, tZ xðsÞ dsÞT represents the nonlinear vector that may include parametric perturbations and other external disturbances. The initial conditionals of (3) or (4) are given by xi ðtÞ ¼ fi ðtÞ A Ið½y,0, RÞ, where Ið½y,0, RÞ denotes the set of all continuous functions from ½y,0 to R. Suppose that some parameters C, A, B, D and external perturRt bations sðt,xðtÞ,xðtyÞ, tZ xðsÞ dsÞ are unknown priori, we design the following response system: 2 n n X X b~ ij fj ðyj ðtyÞÞ a~ ij fj ðyj ðtÞÞ þ dyi ðtÞ ¼ 4c~ i yi ðtÞ þ j¼1

þ

n X j¼1

Z

t

 tZ

d~ ij

Z

t

tZ

j¼1

3

fj ðxj ðsÞÞ ds þ Ii þui 5dt

n X

hij ðt,eðtÞ,eðtyÞ,

j¼1

! eðsÞ ds doi ðtÞ,

i ¼ 1,2, . . . ,n,

ð5Þ

or " ~ ðyðtÞÞ þ Bf ~ ~ j ðyðtyÞÞ þ D dyðtÞ ¼ C~ yðtÞ þ Af þ h t,eðtÞ,eðtyÞ,

Z

t

!

Z

#

t

f ðxðsÞÞ ds þI þ u dt tZ

eðsÞ ds doðtÞ,

ð6Þ

tZ

Usually, the chaotic mixed delayed neural networks with n neurons are described as [16] n X

x_ i ðtÞ ¼ ci xi ðtÞ þ

aij fj ðxj ðtÞÞ þ

j¼1

þ

n X j¼1

dij

Z

n X

bij fj ðxj ðtyÞÞ

j¼1

t tZ

fj ðxj ðsÞÞ dsþ Ii ,

i ¼ 1,2, . . . ,n,

ð1Þ

or, in a compact form x_ i ðtÞ ¼ CxðtÞ þ Af ðxðtÞÞ þ Bf ðxðtyÞÞ þD

Z

t

f ðxðsÞÞ ds þI,

ð2Þ

tZ

where xðtÞ ¼ ðx1 ðtÞ, . . . ,xn ðtÞÞn A Rn is the state vector of the neural network; C ¼ diagðc1 , . . . ,cn Þ is a diagonal matrix with ci 40,i ¼ 1,2, . . . ,n. A ¼ ðaij Þnn , B ¼ ðbij Þnn and D ¼ ðdij Þnn are the connection weight matrix, the connection discrete time-delayed weight matrix and the connection distributed time-delayed weight matrix, respectively. I ¼ ðI1 , . . . ,In ÞT is the constant input vector. Scalars y Z 0 and Z Z0 are the discrete time-delay and the distributed time-delay, respectively. f ðxðtÞÞ ¼ ðf1 ðx1 ðtÞÞ, . . . , fn ðxn ðtÞÞÞT denotes the neuron activation function. From practical point of view, nonlinear uncertain perturbations are always exist, hence, we propose the following chaotic delayed neural networks with external perturbations: 2 n n X X dxi ðtÞ ¼ 4ci xi ðtÞ þ aij fj ðxj ðtÞÞ þ bij fj ðxj ðtyÞÞ j¼1

j¼1

~ ¼ ðd~ ij Þ where C~ ¼ diagðc~ 1 , . . . , c~ n Þ, A~ ¼ ða~ ij Þnn , B~ ¼ ðb~ ij Þnn and D nn are parameters to be determined. u ¼ ðu1 , . . . ,un ÞT is the controller. eðtÞ ¼ yðtÞxðttÞ is the state error, the unknown constant t Z 0 is the transmission delay from neural network (3) to the response (5). oðtÞ ¼ ðo1 ðtÞ, . . . , on ðtÞÞT is a n-dimensional Brownian motion defined on a complete probability space ðO,F ,PÞ with a natural filtration fF t gt Z 0 generated by foðsÞ : 0 r s rtg, where we associate O with the canonical space generated by oðtÞ, and denote F the associated s-algebra generated by foðtÞg with the probability measure P. Here, the white noise doi ðtÞ is independent of doj ðtÞ for ia j, and h : R þ  Rn  Rn  Rn -Rnn is called the noise intensity function matrix. This type of stochastic perturbation can be regarded as a result from the occurrence of random uncertainties during the process of transmission. Also the initial conditionals of (6) are given by yi ðtÞ ¼ ji ðtÞ A Ið½y,0, RÞ. We assume that networks (2) and (4) exhibit chaotic behaviors and the output signals of the neural network (4) can be received by (6). Thought out this paper, we list assumptions as follows: (H1) There exist unknown positive constants ri, i ¼ 1,2, . . . ,n, such that jfi ðxÞfi ðyÞj r ri jxyj,

8x, y A R, x a y:

(H2) Networks (2) and (4) are chaotic, and if u,v,w A Rn are bounded, then jsi ðt,u,v,wÞj, i ¼ 1,2, . . . ,n are bounded.

X. Yang et al. / Neurocomputing 74 (2011) 1617–1625

(H3) h(t,0,0,0)¼0 and there exist unknown positive constants r3 , r2 and r3 such that "

trace hT t,eðtÞ,eðtyÞ,

Z

!

t

eðsÞ ds h t,eðtÞ,eðtyÞ, tZ

r r1 JeðtÞJ2 þ r2 JeðtyÞJ2 þ r3

Z

Z

!#

t

eðsÞ ds

tZ

t tZ

JeðsÞJ2 ds:

Remark 1. Note that condition (H2) is very mild. We do not impose the usual conditions such as Lipschitz condition and differentiability on the external perturbation functions. It can be discontinuous or even impulsive functions. Since chaotic neural networks have strange attractors, there exists a bounded region containing all attractors of (3) such that every orbit of the system (3) never leave it. Hence, condition (H2) can be easily satisfied. Remark 2. From the viewpoint of engineering applications and characteristics of channel, a time-delay always exists. Hence, we investigate lag synchronization with the lag time t Z0. Let eðtÞ ¼ yðtÞxðttÞ. Subtracting (4) from (6) yields the following error system: " Z t gðeðsÞÞ ds deðtÞ ¼ CeðtÞ þ AgðeðtÞÞ þ BgðeðtyÞÞ þ D

1619

3. Main results Our main objective in this subsection is to design an allpowerful adaptive feedback controller which is added to (6) such that states of (6) lag stochastic synchronize with (2) or (4), i.e., whether there are external perturbations or not for the driver network. Theorem 1. Suppose that the assumption conditions ðH1 ÞðH3 Þ hold. If some parameters C,A,B,D or external perturbations sðtÞ are unknown priori, then the driver network (4) can be synchronized with (6) under the controllers ui ¼ li ðtÞei ðtÞabi ðtÞsignðei ðtÞÞ,

i ¼ 1,2, . . . ,n,

ð8Þ

and the following adaptive laws: 8 _l ðtÞ ¼ e e2 ðtÞ, i ¼ 1,2, . . . ,l, > i i i > > > > _ > b ðtÞ ¼ p > i jei ðtÞj, i > > > > _ > ~ < c i ðtÞ ¼ gi ei ðtÞyi ðtÞ, a_~ ij ðtÞ ¼ dij ei ðtÞfj ðyj ðtÞÞ, > > > > _~ > > > > b ij ðtÞ ¼ Zij ei ðtÞfj ðyj ðtyÞÞ, > > R > > : d_~ ij ðtÞ ¼ zij ei ðtÞ t fj ðyj ðsÞÞ ds,

ð9Þ

tZ

tZ

~ ðC~ CÞyðtÞ þ ðAAÞf ðyðtÞÞ ~ ~ þðBBÞf ðyðtyÞÞ þ ðDDÞ

ði,j ¼ 1,2, . . . ,nÞ are arbitrary constants.

Z

t

f ðyðsÞÞ ds tZ

s tt,xðttÞ,xðtytÞ,

Z

!

t

#

xðstÞ ds þu dt

tZ

þh t,eðtÞ,eðtyÞ,

Z

t

!

eðsÞ ds doðtÞ,

ð7Þ

tZ

where gðeðtÞÞ ¼ ½g1 ðe1 ðtÞÞ, . . . ,gn ðen ðtÞÞT , gi ðei ðtÞÞ ¼ fi ðyi ðtÞÞ fi ðxi ðttÞÞ. For convenience of writing, we denote sðtt,xðttÞ,xðtytÞ, Rt Rt yÞ, t Z eðsÞ dsÞ as h(t). AccordtZ xðstÞ dsÞ as sðttÞ, hðt,eðtÞ,eðt Rt ingly, we denote sðt,xðtÞ,xðtyÞ, tZ xðsÞ dsÞ as sðtÞ. The initial condition of (7) is c ¼ jðtÞfðttÞ, where c A L2F 0 ð½ðy þ tÞ,0, Rn Þ, here L2F 0 ð½ðy þ tÞ,0, Rn Þ denotes the family of Rn -valued stochastic process xðsÞ, ðy þ tÞ rs r 0 such R0 that xðsÞ is F 0 -measurable and ðy þ tÞ E½JxðsÞJ2  ds o 1. Lemma 1 (Gu et al. [25]). For any constant matrix D A Rnn , DT ¼ D 4 0, scalar s 4 0 and vector function o : ½0, s-Rn , one has Z s T Z s Z s s oT ðsÞDoðsÞ ds Z oðsÞ ds D oðsÞ ds 0

0

0

provided that the integrals are all well defined, where T denotes transpose and D 4 0 denotes that D is positive definite matrix. n

Lemma 2 (Cohen and Grossberg [26]). For any vectors x,yA R and positive definite matrix G A Rn  Rn , the following matrix inequality holds: T

T

T

2x y rx Gx þ y G

1

y:

Definition 1. System (7) is said to be globally asymptotically stable in mean square if for any given condition such that

Proof. From assumption condition (H2) and Remark 1, it is reasonable that there exist positive constants Mi such that jsi ðt,u,v,wÞj r Mi , i ¼ 1,2, . . . ,n. Define the following Lyapunov functional candidate: " Z t Z t Z t 1 T e ðtÞeðtÞ þ VðtÞ ¼ eT ðsÞQeðsÞ ds þ eT ðsÞMeðsÞ ds dz 2 ty tZ z þ

þ

n X 1

e i¼1 i

g i¼1 i

ðc~ i ci Þ2 þ

n X n X 1 i¼1j¼1

dij

ða~ ij aij Þ2

where Q and M are positive definite matrices, K ¼ diagðk1 , k2 , . . . ,kn Þ, Q, M and K are to be determined. By Itˆo’s differential rule, the stochastic derivative of V(t) along trajectories of error system (7) can be obtained as follows: dVðtÞ ¼ LVðtÞ dt þ eT ðtÞhðt,eðtÞ,eðtyÞÞ doðtÞ,

ð10Þ

where the weak infinitesimal operator L is given by  LVðtÞ ¼ eT ðtÞ ðC þ LÞeðtÞ þ AgðeðtÞÞ þ BgðeðtyÞÞ Z t ~ ~ þD gðeðsÞÞ dsðC~ CÞyðtÞ þðAAÞf ðyðtÞÞ þðBBÞf ðyðtyÞÞ tZ

~ þ ðDDÞ

Z

t

f ðyðsÞÞ dssðttÞ tZ

 1 1 absignðeðtÞÞ þ trace½hT ðtÞhðtÞþ eT ðtÞQeðtÞ 2 2 Z 1 T 1 T 1 t T e ðsÞMeðsÞ ds  e ðtyÞQeðtyÞ þ Ze ðtÞMeðtÞ 2 2 2 tZ þ

lim EJeðtÞJ ¼ 0,

t- þ 1

Obviously, if system (7) realizes globally asymptotically stable in mean square for any given condition, then the stochastic synchronization between (4) and (6) is achieved.

n X 1

3 n X n n X n n X X X 1 ~ 1 ~ 1 ðb ij bij Þ2 þ ðd ij dij Þ2 þ ðMi bi Þ2 5, p Z z i ¼ 1 j ¼ 1 ij i ¼ 1 j ¼ 1 ij i¼1 i

2

where E½ is the mathematical expectation.

ðli ki Þ2 þ



n X

ðli ki Þe2i ðtÞ þ

i¼1 n X n X

n X

ðc~ i ci Þei ðtÞyi ðtÞ

i¼1

ða~ ij aij Þei ðtÞfj ðyj ðtÞÞ

i¼1j¼1



n X n X i¼1j¼1

n X n X

ðb~ ij bij Þei ðtÞfj ðyj ðtyÞÞ

i¼1j¼1

ðd~ ij dij Þei ðtÞ

Z

t tZ

fj ðyj ðsÞÞ ds

n X i¼1

ðMi bi Þjei ðtÞj

1620

X. Yang et al. / Neurocomputing 74 (2011) 1617–1625

¼ eT ðtÞCeðtÞ þ eT ðtÞAgðeðtÞÞ þ eT ðtÞBgðeðtyÞÞ Z t gðeðsÞÞ dseT ðtÞsðttÞ þ eT ðtÞD

Based on the LaSalle invariance principle of stochastic differential equation, which was developed in [24], we have eðtÞ-0, which in turn illustrates that E½JeðtÞJ2 -0. This completes the proof. &

tZ

1 1 1 þ trace½hT ðtÞhðtÞ þ eT ðtÞQeðtÞ eT ðtyÞQeðtyÞ 2 2 2 Z 1 T 1 t T þ Ze ðtÞMeðtÞ e ðsÞMeðsÞ dseT ðtÞKeðtÞ 2 2 tZ aeT ðtÞbsignðeðtÞÞ

n X

Remark 3. From the proof of Theorem 1 we know that LVðtÞ o0 as long as eðtÞ a0. Therefore, the synchronization error e(t) will converges to zero in mean square. On the other hand, from the _ _ adaptive law (9) we can see _l i ðtÞ, b_ i ðtÞ, c_~ i ðtÞ, a_~ ij ðtÞ, b~ ij ðtÞ and d~ ij ðtÞ turn to zero when e(t)¼0, which implies that li ðtÞ, bi ðtÞ, c~ i ðtÞ, a~ ij ðtÞ, b~ ij ðtÞ and dij ðtÞ approach to some constants as eðtÞ-0. However, this does not elaborate that bi ðtÞ-Mi , c~ i -ci , a~ ij - aij , b~ ij -bij , d~ ij -dij . This point is consistent with the results of [27].

ðMi bi Þjei ðtÞj:

i¼1

In view of Lemmas 1 and 2, we have 1 T 1 1 e ðtÞAAT eðtÞ þ g T ðeðtÞÞgðeðtÞÞ þ eT ðtÞBBT eðtÞ 2 2 2 Z t 1 1 1 þ g T ðeðtyÞÞgðeðtyÞÞþ eT ðtÞDDT eðtÞ þ Z g T ðeðsÞÞgðeðsÞÞ ds 2 2 2 tZ

LVðtÞ r eT ðtÞCeðtÞ þ

1 1 1 þ r1 eT ðtÞeðtÞ þ r2 eT ðtyÞeðtyÞ þ r3 2 2 2

Z

t

In fact, the unknown parameters C,A,B,D in (4) cannot identify ~ B, ~ D ~ in (6), respectively, when synchronization between with C~ , A, (4) and (6) is realized, as long as there are some iA f1,2, . . . ,ng such that si ðtÞ a 0. We offer the following proposition. Proposition 1. In Theorem1, if there are some i A f1,2, . . . ,ng such that si ðtÞ a 0, then the unknown parameters C,A,B,D in (4) cannot ~ B, ~ D ~ in (6), respectively, if synchronization between identify with C~ , A,

T

e ðsÞeðsÞ ds

tZ

1 T 1 1 e ðtÞQeðtÞ eT ðtyÞQeðtyÞ þ ZeT ðtÞMeðtÞ 2 2 2 Z 1 t T 1  e ðsÞMeðsÞ dseT ðtÞKeðtÞ r eT ðtÞCeðtÞ þ eT ðtÞAAT eðtÞ 2 tZ 2 þ

1 T 1 1 e ðtÞReðtÞ þ eT ðtÞBBT eðtÞ þ eT ðtÞDDT eðtÞ 2 2 2 Z t 1 T 1 T e ðsÞReðsÞ ds þ e ðtyÞReðtyÞ þ Z 2 2 tZ Z t 1 1 1 eT ðsÞeðsÞ ds þ r1 eT ðtÞeðtÞ þ r2 eT ðtyÞeðtyÞ þ r3 2 2 2 tZ þ

(4) and (6) is realized. Proof. We prove it by contrapositive. Suppose C,A,B,D can iden~ B, ~ D, ~ respectively, on the synchronization manifold. tify with C~ , A, From (7) we have 2 n n X X aij gj ðej ðtÞÞ þ bij gj ðej ðtyÞÞ dei ðtÞ ¼ 4ci ei ðtÞ þ j¼1

þ

n X

dij

Z

j¼1

1 T 1 1 e ðtÞQeðtÞ eT ðtyÞQeðtyÞ þ ZeT ðtÞMeðtÞ 2 2 2 Z 1 t T T e ðsÞMeðsÞ dse ðtÞKeðtÞ,  2 tZ

j¼1

t

tZ

gj ðej ðsÞÞ dsli ðtÞei ðtÞ

abi ðtÞsignðei ðtÞÞðc~ i ci Þyi ðtÞ þ

þ

þ

where b ¼ diagðb1 , b2 , . . . , bn Þ, signðeðtÞÞ ¼ ðsignðe1 ðtÞÞ,signðe2 ðtÞÞ, . . . ,signðen ðtÞÞÞT , R ¼ diagðr12 ,r22 , . . . ,rn2 Þ, and we have used the following deduction: n X

n X

ðb~ ij bij Þfj ðyj ðtyÞÞ þ

r

jei ðtÞjðMi bi Þða1Þ

i¼1

n X

jei ðtÞjbi 

i¼1

¼ ða1Þ

n X

n X

þ

ðd~ ij dij Þ

j¼1

hij ðtÞ doi ðtÞ,

Z

t tZ

3 fj ðyj ðsÞÞ dssi ðtÞ5dt

i ¼ 1,2, . . . ,n:

j¼1

0 ¼ ðc~ i ci Þyi ðtÞ þ

n X

ða~ ij aij Þfj ðyj ðtÞÞ þ

j¼1

ðMi bi Þjei ðtÞj

i¼1

þ jei ðtÞjbi r 0:

n X j¼1

i¼1

ðd~ ij dij Þ

Z

n X

ðb~ ij bij Þfj ðyj ðtyÞÞ

j¼1

t

tZ

fj ðyj ðsÞÞ dssi ðttÞ,

i ¼ 1,2, . . . ,n:

Since c~ i ¼ ci , a~ ij ¼ aij , b~ ij ¼ bij and d~ ij ¼ dij , we obtain ! Z t s tt,xðttÞ,xðtytÞ, xðstÞ ds ¼ 0, i ¼ 1,2, . . . ,n,

Let Q ¼ R þ r2 I, M ¼ ZR þ r3 I, we get

tZ

 1 1 1 LVðtÞ r e ðtÞ C þ AAT þ BBT þ DDT 2 2 2    1 1 þ 1 þ Z R þ ðr2 þ r3 ÞIK eðtÞ: 2 2 T

which is a contradiction. This completes the proof.

Take ki ¼ ci þ 12 JAJ2 þ 12 JBJ2 þ 12 JDJ2 þ ð1þ 12 ZÞri2 þ 12 ðr2 þ r3 Þ þ 1, then LVðtÞ r eT ðtÞeðtÞ:

n X

n X

Therefore, on the synchronization manifold yðtÞ ¼ xðttÞ, one has

ðMi bi Þjei ðtÞj

i¼1 n X

ða~ ij aij Þfj ðyj ðtÞÞ

j¼1

j¼1

eT ðtÞsðttÞaeT ðtÞbsignðeðtÞÞ

n X

ð11Þ

Taking mathematical expectation on both sides of (10), in view of (11) and the definition of V(t), we obtain Z t 1 EJeðtÞJ2 rEVðtÞ rEVð0Þ EJeðtÞJ2 dt: 2 0

&

If perturbations in (4) satisfy sðtÞ ¼ 0, then the network (4) turns to (2). The following theorem accounts for the validity of the controllers (8) and update laws (9) when they are added to (6) to synchronize (6) with (2). Theorem 2. Suppose that the assumption conditions (H1) and (H3) hold. Then, under the controllers (8) and update laws (9), the response system (6) can be synchronized with the driver system Rt (2). Moreover, if xi ðtÞ,fj ðxj ðtÞÞ,fj ðxj ðtyÞÞ, tZ fj ðyj ðsÞÞ ds, i,j ¼ 1, 2, . . . ,n are linearly independent on the synchronization manifold, then limt-1 ðc~ i ci Þ ¼ limt-1 ða~ ij aij Þ ¼ limt-1 ðb~ ij bij Þ ¼ limt-1 ðd~ ij dij Þ ¼ 0.

X. Yang et al. / Neurocomputing 74 (2011) 1617–1625

Proof. We can easily get the following error system from (2) and (4): "

þ

Z

t

~ ~ þðBBÞf ðyðtyÞÞ þ ðDDÞ

Z

#

t

f ðyðsÞÞ ds þ u dt þ hðtÞ doðtÞ: tZ

ð12Þ Define the following Lyapunov functional candidate:  Z t 1 T e ðtÞeðtÞ þ eT ðsÞQeðsÞ ds V1 ðtÞ ¼ 2 ty Z t Z t n X 1 eT ðsÞMeðsÞ ds dz þ ðli ki Þ2 þ tZ

þ

z

n X 1 i¼1

e i¼1 i

gi

ðc~ i ci Þ2 þ

n X n X 1 i¼1j¼1

dij

ða~ ij aij Þ2 þ

n X n X 1 ~ ðb ij bij Þ2 i¼1j¼1

Zij

3

þ

n X n X 1 ~ ðd ij dij Þ2 5,

z i ¼ 1 j ¼ 1 ij

where Q and M are positive definite matrices, K ¼ diagðk1 , k2 , . . . ,kn Þ. Q, M and K are to be determined. By Itˆo’s differential rule, the stochastic derivative of V1 ðtÞ along trajectories of error system (12) can be obtained as follows: dV1 ðtÞ ¼ LV1 ðtÞ dt þ eT ðtÞhðt,eðtÞ,eðtyÞÞ doðtÞ, where the weak infinitesimal operator L is given by  LV1 ðtÞ ¼ eT ðtÞ ðC þ LÞeðtÞ þ AgðeðtÞÞ þ BgðeðtyÞÞ Z t þD gðeðsÞÞ dsðC~ CÞyðtÞ tZ

~ ~ ~ þ ðAAÞf ðyðtÞÞ þ ðBBÞf ðyðtyÞÞ þðDDÞ

Z

t

f ðyðsÞÞ ds

Z

t

eT ðsÞeðsÞ ds

tZ

1 T 1 e ðtÞQeðtÞ eT ðtyÞQeðtyÞ 2 2 Z 1 T 1 t T e ðsÞMeðsÞ dseT ðtÞKeðtÞ, þ Ze ðtÞMeðtÞ 2 2 tZ

~ gðeðsÞÞ dsðC~ CÞyðtÞ þðAAÞf ðyðtÞÞ

tZ

1 1 r eT ðtyÞeðtyÞ þ r3 2 2 2

þ

deðtÞ ¼ CeðtÞ þ AgðeðtÞÞ þ BgðeðtyÞÞ þD

1621

where b ¼ diagðb1 , b2 , . . . , bn Þ, signðeðtÞÞ ¼ ðsignðe1 ðtÞÞ, signðe2 ðtÞÞ, . . . ,signðen ðtÞÞÞT , R ¼ diagðr12 ,r22 , . . . ,rn2 Þ. By the same procedure of the proof of Theorem 1, we also arrive at EJeðtÞJ2 -0. On the synchronization manifold yðtÞ ¼ xðttÞ, we have 0 ¼ ðc~ i ci Þyi ðtÞ þ

n X

ða~ ij aij Þfj ðyj ðtÞÞ þ

j¼1

þ

n X

ðd~ ij dij Þ

j¼1

Z

n X

ðb~ ij bij Þfj ðyj ðtyÞÞ

j¼1

t

tZ

fj ðyj ðsÞÞ ds,

i ¼ 1,2, . . . ,n:

Since xi ðtÞ,fj ðxj ðtÞÞ,fj ðxj ðtyÞÞ, i,j ¼ 1,2, . . . ,n are linearly independent on the synchronization manifold, yi ðtÞ,fj ðyj ðtÞÞ,fj ðyj ðtyÞÞ, i,j ¼ 1,2, . . . ,n are also linearly independent on the synchronization manifold, therefore, the above equality holds if and only if c~ i ¼ ci , a~ ij ¼ aij , b~ ij ¼ bij , d~ ij ¼ dij . This completes the proof. & Remark 4. To synchronize (6) and (2), the constant a in the controllers (8) can be relaxed to a Z0 and bi can be any positive constants. Certainly, without abi ðtÞsignðei ðtÞÞ and b_ i ðtÞ ¼ pi jei ðtÞj in (8) and (9), respectively, the controllers (8) and update laws (9) can also synchronize (6) and (2). However, it is more effective to synchronize (6) and (2) with these terms in (8) and (9). If yi ðtÞ 4 xi ðttÞ, then abi ðtÞsignðei ðtÞÞ will help to reduce yi ðtÞ, if yi ðtÞ o xi ðttÞ, then abi ðtÞsignðei ðtÞÞ will help to increase yi ðtÞ, hence, synchronization is realized faster than without these terms. When the response system (6) has no stochastic perturbation, we have the following corollaries from Theorems 1 and 2. We omit their proofs here.

tZ

 1 1 absignðeðtÞÞ þ trace½hT ðtÞhðtÞ þ eT ðtÞQeðtÞ 2 2 Z 1 1 1 t T e ðsÞMeðsÞ ds  eT ðtyÞQeðtyÞ þ ZeT ðtÞMeðtÞ 2 2 2 tZ n n X X þ ðli ki Þe2i ðtÞ þ ðc~ i ci Þei ðtÞyi ðtÞ 

i¼1 n X n X

i¼1

ða~ ij aij Þei ðtÞfj ðyj ðtÞÞ

i¼1j¼1



n X n X

n X n X

ðb~ ij bij Þei ðtÞfj ðyj ðtyÞÞ

i¼1j¼1

ðd~ ij dij Þei ðtÞ

i¼1j¼1

Z

t tZ

fj ðyj ðsÞÞ ds

¼ eT ðtÞCeðtÞ þ eT ðtÞAgðeðtÞÞ þ eT ðtÞBgðeðtyÞÞ Z t 1 1 gðeðsÞÞ ds þ trace½hT ðtÞhðtÞ þ eT ðtÞQeðtÞ þ eT ðtÞD 2 2 tZ 1 1  eT ðtyÞQeðtyÞ þ ZeT ðtÞMeðtÞ 2 2 Z 1 t T e ðsÞMeðsÞ dseT ðtÞKeðtÞaeT ðtÞbsignðeðtÞÞ:  2 tZ Since aeT ðtÞbsignðeðtÞÞ Z0, we get LV1 ðtÞ r eT ðtÞCeðtÞ þ

1 T 1 1 e ðtÞAAT eðtÞ þ g T ðeðtÞÞgðeðtÞÞ þ eT ðtÞBBT eðtÞ 2 2 2

1 T 1 g ðeðtyÞÞgðeðtyÞÞ þ eT ðtÞDDT eðtÞ 2 2 Z t 1 1 T g ðeðsÞÞgðeðsÞÞ ds þ r1 eT ðtÞeðtÞ þ Z 2 2 tZ þ

Corollary 1. Suppose (H1) and (H2) are satisfied. Then the driver network (4) can synchronize with " # Z t ~ ~ ~ ~ f ðxðsÞÞ ds þI þ u dt dyðtÞ ¼ C yðtÞ þ Af ðyðtÞÞ þ Bfj ðyðtyÞÞ þ D tZ

ð13Þ under the adaptive controllers (8) and update laws (9), where all the parameters are same as those in Theorem1. Corollary 2. Suppose (H1) is satisfied. Then the driver network (2) can synchronize with (13) under the adaptive controllers (8) and update laws (9), where a 40, the other parameters are same as those Rt in Theorem1. Moreover, if xi ðtÞ,fj ðxj ðtÞÞ,fj ðxj ðtyÞÞ, tZ fj ðyj ðsÞÞ ds, i,j ¼ 1,2, . . . ,n are linearly independent on the synchronization manifold, then limt-1 ðc~ i ci Þ ¼ limt-1 ða~ ij aij Þ ¼ limt-1 ðb~ ij bij Þ ¼ limt-1 ðd~ ij dij Þ ¼ 0. Remark 5. In this paper, suitable adaptive controllers are designed to guarantee lag synchronization of mixed delayed neural networks with unknown parameters, unknown external perturbations and stochastic perturbations. Sufficient conditions are derived to ensure parameters identification of the driver and response networks. While papers [8–15] only considered discrete time-delayed neural networks. Obviously, models in [8–15] are special cases of this paper. Moreover, conditions for parameter identification in [8–15] are not sufficient since they do not have linearly independent condition [28]. In [17,20], the authors only considered stochastic perturbations. One can easily extend our results to neural networks with mixed delays, Markovian

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X. Yang et al. / Neurocomputing 74 (2011) 1617–1625

jumpings and uncertain external perturbations. To the best of our knowledge, result on synchronization of mixed delayed neural networks with uncertain parameters, external perturbations as well as channel noises is few. In this sense, to some extent, results of this paper extend results of [8–15,17,20].

15

10

4. A numerical example In this section, we provide numerical simulations to illustrate the effectiveness of the theoretical results obtained above. Numerical simulations demonstrating that the part abi ðtÞ signðei ðtÞÞ in ui plays key role when synchronizing chaotic systems with non-stochastic perturbations are also given. Consider the following chaotic neural networks with mixed delays [16]: " # Z t dxðtÞ ¼ CxðtÞ þ Af ðxðtÞÞþ Bf ðxðtyÞÞ þD f ðxðsÞÞ ds þI dt, ð14Þ

x2 (t)

5

0

−5

−10 −1.2

tZ T

where xðtÞ ¼ ðx1 ðtÞ,x2 ðtÞÞ , f ðxðtÞÞ ¼ ðtanhðx1 ðtÞÞ,tanhðx2 ðtÞÞÞ , y ¼ 1, Z ¼ 0:3,       1:2 0 3 0:3 1:4 0:1 C¼ ,A ¼ ,B ¼ , 0 1 8 5 0:3 8     1:2 0:1 0 D¼ ,I ¼ : 2:8 1 2:1

tZ

−5

−10 −1.5

tZ

−1

−0.5

0 x1

ð16Þ

0.5

1

1.5

Fig. 2. Chaotic trajectory of model (15).

where transmission delay is t ¼ 0:8, the noise intensity function matrix is hðtÞ ¼ 0:5diagðe1 ðtÞ,e2 ðtÞÞ, e1 ðtÞ ¼ y1 ðtÞx1 ðt0:8Þ, e2 ðtÞ ¼ y2 ðtÞx2 ðt0:8Þ, u is the controller and " # a~ 11 0:3 ~ : A¼ 5 a~ 21

15

10

According to Theorem 1, networks (15) and (16) can be synchronized under the controllers

j

0.8

0

In the case that the where s initial condition is chosen as x1 ðtÞ ¼ 0:4,x2 ðtÞ ¼ 0:6, 8t A ½1,0, the chaotic trajectory of (15) can be seen in Fig. 2. Let system (15) is the driver network and suppose a11 ¼3 and a21 ¼ 8 in A are unknown, then the response network is " # Z t ~ ðxðtÞÞ þ Bf ðxðtyÞÞ þ D f ðxðsÞÞ ds þI þ u dt dxðtÞ ¼ CxðtÞ þ Af

5

ð17Þ

y2

and the following adaptive laws: 8 _l ðtÞ ¼ 2e2 ðtÞ, i ¼ 1,2, > > i < i b_ i ðtÞ ¼ jei ðtÞj, i ¼ 1,2, > > : a~_ ij ðtÞ ¼ e ðtÞf ðy ðtÞÞ, i ¼ 1,2, j ¼ 1:

0.6

5

ð15Þ

i ¼ 1,2,

0.4

10

ðtÞ ¼ ð0:01ðx1 ðtÞ þ x2 ðtÞÞ,0:01x22 ðtÞÞT .

þ hðtÞ doðtÞ,

0.2

x2

f ðxðsÞÞ ds þI þ sðtÞ dt,

0

15

t

dxðtÞ ¼ CxðtÞ þ Af ðxðtÞÞþ Bf ðxðtyÞÞ þD

i

−0.8 −0.6 −0.4 −0.2 x1 (t)

Fig. 1. Chaotic trajectory of model (14).

In the case that the initial condition is chosen as x1 ðtÞ ¼ 0:4, x2 ðtÞ ¼ 0:6, 8t A ½1,0, the chaotic attractor can be seen in Fig. 1. Now we suppose that (14) is perturbed, which is described by " # Z

ui ¼ li ðtÞei ðtÞ2bi ðtÞsignðei ðtÞÞ,

−1

T

0 ð18Þ

j

Taking the initial conditions of the numerical simulations as y1 ðtÞ ¼ 1, y2 ðtÞ ¼ 0:5, l1 ðtÞ ¼ l2 ðtÞ ¼ b1 ðtÞ ¼ b2 ðtÞ ¼ 1, 8t A ½1:8,0, a~ 11 ¼ 2, a~ 21 ¼ 5, we get the simulation results shown in Figs. 3–6. Fig. 3 describes the phase trajectory of the response network (16). Fig. 4 represents the state trajectories of drive (red) and response (black) networks. Fig. 5 shows the trajectories of the error states. Parameter trajectories of a~ 11 and a~ 21 of network (16) are shown in

−5

−10 −3

−2.5

−2

−1.5

−1

−0.5

0

0.5

y1 Fig. 3. Phase trajectory of the response network (16).

1

1.5

X. Yang et al. / Neurocomputing 74 (2011) 1617–1625

1.5

15

y1 x1

1

1623

y2 x2

10 x2 (t), y2 (t)

x1 (t), y1 (t)

0.5 0 −0.5

5 0

−1 −5

−1.5 −2

−10 0

10

20

30

40

50

60

0

10

20

30

40

50

60

t Fig. 4. State trajectories of drive (red) and response (black) networks. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

2.5

15

2 1.5

10

1 e2 (t)

e1 (t)

0.5 0

5

−0.5 −1

0

−1.5 −2 −2.5

−5 0

1

2

3 t

4

5

6

0

1

2

3 t

4

5

6

Fig. 5. Trajectories of the error states between (15) and (16) by using the controllers (17) and adaptive laws (18).

6

The above simulations demonstrate the designed controllers are effective. However, if the controllers ui are taken as the following form, which is often used to synchronize chaotic systems [21]:

a11 a21

5

ui ¼ li ðtÞei ðtÞ,

a11, a21

4

ð19Þ

and the adaptive laws are the same as those of (18), one can find from Fig. 8 that system (16) cannot be controlled synchronize with (15). Note that the perturbation sðtÞ in (15) do not disappear even the synchronization has been realized. Simulations in Figs. 5 and 8 demonstrate that the part abi ðtÞsignðei ðtÞÞ in ui plays key role when synchronizing chaotic systems with nonstochastic perturbations, which makes the controllers (17) have better robustness than (19). Therefore, results of this paper are new.

3

2

1

0

i ¼ 1,2,

0

1

2

3 t

4

5

6

Fig. 6. Parameter trajectories of a~ 11 (black) and a~ 21 (green) of network (16). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 6. Figs. 2–5 show that lag stochastic synchronization is realized quickly. Fig. 6 shows that a~ 11 and a~ 21 of network (16) approach to some constants but the corresponding values 3 and 8 in A. Fig. 7 describes the evolutions of control gains l(t) (left) and bðtÞ (right), which approach some constants when synchronization is realized. Simulations verify the theoretical results perfectly.

5. Conclusions In this paper, we investigated lag synchronization control for a kind of mixed delayed stochastic neural network with unknown external perturbations as well as unknown parameters. Based on the invariance principle of stochastic differential equations and two suitable Lyapunov functions, by designing a simple robust adaptive controller, sufficient conditions are derived to ensure the global asymptotic stochastic synchronization of the considered model. Sufficient conditions are also obtained to guarantee identification of the unknown parameters. The obtained results extend some existing results. Numerical simulations verify the effectiveness of the theoretical results. Models and results in this

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X. Yang et al. / Neurocomputing 74 (2011) 1617–1625

3

20

β1 β2

18 2.5

16 14

2 1, 2

l1, l2

12 10 8

1.5 1

6 4

0.5

2 0

0

1

2

3 t

4

5

0

6

0

1

2

3 t

4

5

6

3 t

4

5

6

Fig. 7. Evolutions of control gains lðtÞ (left) and bðtÞ (right).

2.5

15

2 1.5

10

1 e2 (t)

e1 (t)

0.5 0

5

−0.5 −1

0

−1.5 −2 −2.5

−5 0

1

2

3 t

4

5

6

0

1

2

Fig. 8. Trajectories of the error states between (15) and (16) by using the controllers (19) and adaptive laws (18).

paper provide possible new applications for neural network designers.

Acknowledgments This work was jointly supported by the Scientific Research Fund of Yunnan Province under Grant No. 2010ZC150, the National Natural Science Foundation of China under Grant No. 10801056, the Natural Science Foundation of Zhejiang Province, the Natural Science Foundation of Ningbo under Grant No. 2010A610094, the Foundation of Chinese Society for Electrical Engineering (2008), the Hunan Provincial Natural Science Foundation of China under Grant No. 07JJ4001, the Key Project of Chinese Ministry of Education (2011), the Excellent Youth Foundation of Educational Committee of Hunan Provincial (10B002). References [1] L. Pecora, T. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett. 64 (1990) 821–824. [2] S. Sundar, A. Minai, Synchronization of randomly multiplexed chaotic systems with application to communication, Phys. Rev. Lett. 85 (2000) 5456–5459. [3] S. Bowong, F.M. Moukam Kakmeni, H. Fotsin, A new adaptive observer-based synchronization scheme for private communication, Phys. Lett. A 355 (2006) 193–201. [4] G. Jiang, G. Chen, K. Tang, A new criterion for chaos synchronizaton using linear state feedback control, Int. J. Bifur. Chaos 13 (2003) 2343–2351. [5] J. Cao, J. Lu, Adaptive synchronization of neural networks with or without time-varying delay, Chaos 16 (2006) 013133. [6] M. Gilli, Strange attractors in delayed cellular neural networks, IEEE Trans. Circuits Syst.-I 40 (1993) 849–853.

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Quanxin Zhu received the B.S. and M.S. degree from Hunan Normal University, Changsha, China, in 1999 and 2002, respectively. In 2005, he received the Ph.D. degree from Sun Yatsen (Zhongshan) University, Guangzhou, China, in probability and statistic. From July 2005 to May 2009, he was with the South China Normal University. From February 2009 to June 2009, he was a Visiting Scholar in the Department of Mathematics, Southeast University, Nanjing, China. Dr. Zhu is a Reviewer of Mathematical Reviews and ZentralblattMath, and he is a Reviewer of more than 20 other journals. He is currently a professor of Ningbo University. He is the author or coauthor of more than 30 journal papers. His research interests include random processes, stochastic control, stochastic differential equation, stability theory, Markovian jump systems and stochastic neural networks.

Xinsong Yang received the B.S. degree in mathematics from Huaihua Normal University, Hunan, China, in 1992, and the M.S. degree in mathematics from Yunnan University, Yunnan, China, in 2006. He is currently an Associate Professor with the Department of Mathematics, Honghe University, Menfzi, Yunan 661100, China. From 2008 to 2009, he was a Visiting Scholar with the Department of Mathematics, Southeast University, China. His current research interests include collective behavior in complex dynamical networks, multi-agent system, chaos synchronization, control theory and neural networks. He is the author or coauthor of more than 20 papers in refereed international journals. Professor Yang serves as a reviewer of several international journals.

Chuangxia Huang received the B.S. degree in Mathematics in 1999 from National University of Defense Technology, Changsha, China. From September 2002, he began to pursue his M.S. degree in Applied Mathematics at Hunan University, Changsha, China, and from April 2004, he pursued his Ph.D. degree in Applied Mathematics in advance at Hunan University. He received the Ph.D. degree in June 2006. He is currently an Associate Professor of Changsha University of Science and Technology, Changsha, China. He is the author of more than 30 journal papers. His research interests include dynamics of neural networks, and stability theory of functional differential equations.