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Global robust exponential synchronization of BAM recurrent FNNs with infinite distributed delays and diffusion terms on time scales

Kaihong Zhao*

Correspondence: zhaokaihongs@126.com Department of Applied Mathematics, Kunming University of Science and Technology, Kunming, Yunnan 650093, People's Republic of China

Abstract

In this article, the global robust exponential synchronization of reaction-diffusion BAM recurrent fuzzy neural networks (FNNs) with infinite distributed delays on time scales is investigated. Applied Lyapunov functional and inequality skills, some sufficient criteria are established to guarantee the global robust exponential synchronization of reaction-diffusion BAM recurrent FNNs with infinite distributed delays on time scales. One example is given to illustrate the effectiveness of our results.

Keywords: globally robust exponential synchronization; reaction-diffusion BAM recurrent FNNs; infinite distributed delays; Lyapunov functional; time scales

ringer

1 Introduction

The study on the artificial neural networks has attracted much attention because of their potential applications such as signal processing, image processing, pattern classification, quadratic optimization, associative memory, moving object speed detection, etc. Many kinds of models of neural networks have been proposed by some famous scholars. One of these important neural network models is the bidirectional associative memory (BAM) neural network models, which were first introduced by Kosko [1-3]. It is a special class of recurrent neural networks that can store bipolar vector pairs. The BAM neural network is composed of neurons arranged in two layers, the X-layer and the Y-layer. The neurons in one layer are fully interconnected to the neurons in the other layer. Through iterations of forward and backward information flows between the two layers, it performs a two-way associative search for stored bipolar vector pairs and generalize the single-layer autoassociative Hebbian correlation to a two-layer pattern-matched heteroassociative circuits. Therefore, this class of networks possesses good application prospects in some fields such as pattern recognition, signal and image process, artificial intelligence [4]. In general, artificial neural networks have complex dynamical behaviors such as stability, synchronization, periodic or almost periodic solutions, invariant sets and attractors, and so forth. We can refer to [5-27] and the references cited therein. Therefore, the analysis of dynamical behaviors for neural networks is a necessary step for practical design of neural networks. As one of the famous neural network models, it has attracted many attention in the past two decades [28-48] since the BAM model was proposed by Kosko. The dynamical behaviors such as uniqueness, global asymptotic stability, exponential stability and invariant

© 2014 Zhao; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the originalworkis properly cited.

sets and attractors of the equilibrium point or periodic solutions were investigated for BAM neural networks with different types of time delays (see [28-44, 48]).

Synchronization has attracted much attention after it was proposed by Carrol et al. [49, 50]. The principle of drive-response synchronization is this: the driver system sends a signal through a channel to the responder system, which uses this signal to synchronize itself with the driver. Namely, the response system is influenced by the behavior of the drive system, but the drive system is independent of the response one. In recent years, many results concerning a synchronization problem of time lag neural networks have been investigated in the literature [5, 6, 8-15, 27, 36, 49, 50].

As is well known, both in biological and man-made neural networks, strictly speaking, diffusion effects cannot be avoided when electrons are moving in asymmetric electromagnetic fields, so we must consider that the activations vary in space as well as in time. Many researchers have studied the dynamical properties of continuous time reaction-diffusion neural networks (see, for example, [8,11,17,18, 24, 25, 27, 32, 48]).

However, in mathematical modeling of real world problems, we will encounter some other inconveniences such as complexity and uncertainty or vagueness. Fuzzy theory is considered as a more suitable setting for the sake of taking vagueness into consideration. Based on traditional cellular neural networks (CNNs), T Yang and LB Yang proposed the fuzzy CNNs (FCNNs) [23] which integrate fuzzy logic into the structure of traditional CNNs and maintain local connectedness among cells. Unlike previous CNNs structures, FCNNs have fuzzy logic between their template input and/or output besides the sum of product operation. FCNNs are very a useful paradigm for image processing problems, which is a cornerstone in image processing and pattern recognition. Therefore, it is necessary to consider both the fuzzy logic and delay effect on dynamical behaviors of neural networks. To the best of our knowledge, few authors have considered the synchronization of reaction-diffusion recurrent fuzzy neural networks with delays and Dirichlet boundary conditions on time scales which is a challenging and important problem in theory and application. Therefore, in this paper, we will investigate the global robust exponential synchronization of delayed reaction-diffusion BAM recurrent fuzzy neural networks (FNNs) on time scales as follows:

u?(t,X) = £h £-k (aikg) - biui(t,x) + jjj - r,x)) + I

+ A"=1 PijFj(uj(t - r, x)) + AJ=1 rij /0+TO kij(s)Fj (uj(t - s, x)) As + V/U qijFj(uj(t - r, x)) + V"=1 Wij fo™ kij(s)Fj(uj (t - s, x))As

+ £/=1 dij lj + A/=1 Sij lj + V/=1 jj,

vA(t,x) = £k=1 4k(j§) - jj(t,x) + En=1 zjigi(ui(t - r,x)) + Jj

+ AxjiGi(vi(t- r,x)) +/\Pji /0+TOKji(s)Gi(vi(t -s,x))As + V *i=1 njiGi(vi(t - r, x)) + V m=1 j /0+TO Kji(s)Gi(vi(t - s, x))As + £ i=1 hjiVi + A i=1 MjiVi + V i=1 NjVi,

subject to the following initial conditions

|ui(s,x) = tyi(s,x), (s,x) e [-r,0]T x (12)

I v;(s, x) = js, x), (s, x) e [-r,0]T x

and Dirichlet boundary conditions

Ui(t,x) = 0, (t,x) e [0, to)t x dO,

Vj(t ,x) = 0, (t,x) e [0, to)t x 9O,

where i = 1,2,..., n; j = 1,2,..., m. T c R is a time scale and T n [0, +to) = [0, +to)t is unbounded and T n [-t,0] = [t,0]t = <. t > 0 is constant time delay. x = (x1,x2,...,xl)T e O c Rl and O = {x = (x1,x2,...,xl)T : |xi| < li,i = 1,2,...,/} is a bounded compact set with smooth boundary dO in space Rl. u = (u1,u2,...,un)T e Rn, v = (v1,v2,...,vm)T e Rm. ui(t,x) and v,(t,x) are the state of the ith neurons and the jth neurons at time t and in space x, respectively. I = (I1,12,.. .,In)T e Rn and J = (J1, J2,... ,Jm)T e Rm are constant input vectors. The smooth functions aik > 0 and j > 0 correspond to the transmission diffusion operators along with the ith neurons and the jth neurons, respectively. bi > 0, n > 0, fij, Vi, Cij, pij, Tijt qij, Wij, dij, Sij, Tij, j, kji, pji, ^ji, oji, hji, Mji, Mji are constants. bi and nj denote the rate with which the ith neurons and jth neurons will reset their potential to the resting state in isolation when disconnected from the network and external inputs, respectively. Cij, pij, t^, qij, wv, dj, Sij, Tij, j j pn, j Oji, hji, Mji, Mji denote the connection weights. fj(■) (j = 1,2,..., m) and $(•) (i = 1,2,..., n) denote the activation function of the jth neurons of Y-layer on the ith neurons of X-layer and the ith neurons of X-layer on the jth neurons of Y-layer at time t and in space x, respectively. Fj(■) (j = 1,2,..., n) denotes the fuzzy activation function of the jth neurons on the ith neurons inside of X-layer. Gi(■) (i = 1,2,...,m) denotes the fuzzy activation function of the ith neurons on the jth neurons inside of Y-layer. fj (j = 1,2,..., n) denotes the bias of the jth neurons on the ith neurons inside of X-layer. vi (i = 1,2,..., m) denotes the bias of the ith neurons on the jth neurons inside of Y-layer. /\, V denote the fuzzy AND and fuzzy OR operations, respectively. <(t,x) = (</>i(t,x),<2(t,x), ...,<n(t,x))T : [-t,0]t x O ^ Rn, p(t, x) = (<p1(t, x), p2(t, x),...,pm(t, x))T : [-t ,0]t x O ^ Rm are rd-continuous with respect to t e [-t, 0]T and continuous with respect to x e O.

In order to investigate the global robust exponential synchronization for system (1.1)-(1.3), the quantities bi, aik, cij, pij, Tij, qij, wij, nj, j, Zji, j Pji, nji and Oji may be considered as intervals as follows: 0 < bi < bi < to, a^k < aik < aik, lcij| < 1%| < |Cij|, p. | < pji < |pij|, iTij| < T| < |Tij|, q| < |qij| < q|, |Wij| < |Wij| < |Wij|, 0 < n < nj < to, j < j < j |j < | Zji | < j lkjil < | j < j |Pji| < | Pji | < |Pji|, ^ji | < |nji | < j |Oji | < |Oji| <

| Oji |.

Take the time scale T = R (real number set), then system (1.1)-(1.3) can be changed into the following continuous case (1.4)-(1.6):

^ = Ek=1 £(aikH) - biUi(t,x) + ££ Cijfj(vj(t - T,x)) +1

+ A;n=1 pijFj(uj(t - T,x)) + AL Tij /0+to kij(s)Fj(uj(t - s,x)) ds

\j=1F - /\j=1' j j0

qijFj(uj(t - t , x)) + \ZJ=1 wij /0+to kij(s)Fj (uj(t - s, x)) ds

+ £1=1 dijfj ^m Sijfj W1=1 Tijfj, dvtf) ^ £k=1 (j^) - njvj(t,x) ^n=1 zjigi(ui(t - T,x)) + Jj

^ Am=1 xjiGi(vi(t- t,x)) +/\m=1 Pji /0+to Kji(s)Gi(vi(t -s,x))ds Wm=1 nji Gi (vi(t - t, x)) + Vm=1 Oji ¡r Kji (s) Gi (vi(t - s, x)) ds ^m=1 hjiVi ^m=1 MjiVi + Vm=1 NjV,

subject to the following initial conditions

i Ui(s, x) = & (s, x), (s, x) e [-T, 0] x Q, ^^ ^

I Vj(s,x) = js,x), (s,x) e [-T,0] x Q,

and Dirichlet boundary conditions

Iui(t,x) = 0, (t,x) e [0, to) x dQ, vj(t,x) = 0, (t,x) e [0, to) x aa

Take the time scale T = Z (integer number set), then system (1.1)-(1.3) can be changed into the following discrete case (1.7)-(1.9):

AtUi(t,x) = £k=1 dk(aiklu ) - biUi(t,x) + °iifj(vj(t - t,x)) + Ii + AJ=1 PiFj(uj(t - t,x)) + A"=1 rij Es°=0 kij(s)Fj(uj(t - s,x))

+ V"=1 qijFj(uj(t - t,x)) + \/n=1 Wij kij(s)Fj(uj(t - s,x)) + E"=1 jj + A"=1 jj + V"=1

^tVj(t,x) = £k=1 ^(ljk5) - jjfex) + £"=1 t;i&(Mi(t - T,x)) + /

+ Am=1 hGi(Vi(t- T,x)) + Am=1 ^i(s)Gi(vi(t -s,x))

+ vr=1 njiGi(Vi(t - T,x)) + vr=1 j ETOc Kji(s)Gi(Vi(t - s,x)) + EÎ=1 hjVi + Ai=1 M.jiVi +\Jm1

subject to the following initial conditions

|ui(s,x) = (s,x), (s,x) e {-t,-t + 1,...,-2,-1,0} x Q, Vj(s,x) = js,x), (s,x) e {-t, -t + 1,..., -2, -1,0} x Q,

and Dirichlet boundary conditions

Jui(t,x) = 0, (t,x) e Z+ x dQ, I Vj(t,x) = 0, (t,x) e Z+ x 9Q,

where t e Z, r is a positive integer, Z+ = {0,1,2,...}, Atui(t,x) = ui(t + 1,x) - ui(t,x), Atv,(t,x) = v,(t + 1,x) - v,(t,x).

If we choose T = R, then a (t) = t, |(t) = 0. In this case, system (1.1)-(1.3) is the continuous reaction-diffusion BAM recurrent FNNs (1.4)-(1.6). If T = Z, then |(t) = 1, system (1.1)-(1.3) is the discrete difference reaction-diffusion BAM recurrent FNNs (1.7)-(1.9). In this paper, we study the global robust exponential synchronization of reaction-diffusion BAM recurrent FNNs (1.1)-(1.3), which unify both the continuous case and the discrete difference case. What is more, system (1.1)-(1.3) is a good model for handling many problems such as predator-prey forecast or optimizing of goods output.

The rest of this paper is organized as follows. In Section 2, some notations and basic theorems or lemmas on time scales are given. In Section 3, the main results of global robust exponential synchronization are obtained by constructing the appropriate Lyapunov functional and applying inequality skills. In Section 4, one example is given to illustrate the effectiveness of our results.

2 Preliminaries

In this section, we first recall some basic definitions and lemmas on time scales which are used in what follows.

Let T be a nonempty closed subset (time scale) of R. The forward and backward jump operators p, a : T ^ T and the graininess i R+ are defined, respectively, by

a (t) = inf{s e T : s > t}, p (t) = sup{s e T: s < t}, l(t) = a (t)-t.

A point t e T is called left-dense if t > inf T and p (t) = t, left-scattered if p (t) < t, right-dense if t < sup T and a (t) = t, and right-scattered if a (t) > t. If T has a left-scattered maximum m, then Tk = T \ {m}, otherwise Tk = T.If T has a right-scattered minimum m, then Tk = T \ {m}, otherwise Tk = T.

Definition 2.1 ([51]) A function/: T ^ R is called regulated provided its right-hand side limits exist (finite) at all right-hand side points in T and its left-hand side limits exist (finite) at all left-hand side points in T.

Definition 2.2 ([51]) A function/ : T ^ R is called rd-continuous provided it is continuous at right-dense point in T and its left-hand side limits exist (finite) at left-dense points in T. The set of rd-continuous function/: T ^ R will be denoted by Crd = Crd(T) = Crd(T, R).

Definition 2.3 ([51]) Assume/: T ^ R and t e Tk. Then we define/A(t) to be the number (if it exists) with the property that given any e >0 there exists a neighborhood U of t (i.e., U = (t - A, t + A) n T for some A >0) such that

for all s e U. We call/A(t) the delta (or Hilger) derivative of/ at t. The set of functions /: T ^ R that is a differentiable and whose derivative is rd-continuous is denoted by Cjd =

If/ is continuous, then/ is rd-continuous. If/ is rd-continuous, then/ is regulated. If/ is delta differentiable at t, then/ is continuous at t.

Lemma 2.1 ([51]) Let/ be regulated, then there exists a/unction F which is delta differentiable with region o/differentiation D such that FA (t) =/(t) /or all t e D.

Definition 2.4 ([51]) Assume that/: T ^ R is a regulated function. Any function F as in Lemma 2.1 is called a A-antiderivative of/. We define the indefinite integral of a regulated function / by

|/(a(t)) -/(s)] -/A(t)[a(t) -s]| < e|a(t)-s

Cjd(T) = Cjd(R, T).

where C is an arbitrary constant and F is a A-antiderivative off. We define the Cauchy integral by f^f (s)As = F(b) - F(a) for all a, b e T.

A function F: T ^ R is called an antiderivative off: T ^ R provided FA(t) =f (t) for all t e T*.

Lemma 2.2 ([51]) If a, b e T, a, p e R andf,g e C(T, R), then

(i) /ab[af (t) + Pg(t)] At = a fbf (t) At + p tfg(t)At,

(ii) iff (t) > 0 for all a < t < b, then fhaf (t)At > 0,

(iii) f f (t)| < g(t) on [a, b) 4 {t e T: a < t < b}, then | fabf (t)At| < fbag(t) At.

A function p : T ^ R is called regressive if 1 + f (t)p(t) =0 for all t e Tk. The set of all regressive and rd-continuous functions f: T ^ R will be denoted by R = R(T) = R(T, R). We define the set R+ of all positively regressive elements of R by R+ = R+(T, R) = {p e R: 1 + f (t)p(t) > 0 for all t e T}. If p is a regressive function, then the generalized exponential function ep(t,s) is defined by ep(t,s) = exp{/ji ff(T)(p(T))At} for all s, t e T, withthe cylinder transformation

fLoga+fe) if h =0

&(*)= h , ifhn

I z, if h = 0.

Let p, q: T ^ R be two regressive functions, we define

p ® q = p + q + fpq, p

Op = ,

1 + fp

p © q = p

Ifp e R+, then ©p e R+. The generalized exponential function has the following properties.

Lemma 2.3 ([51]) Assume that p, q: T ^ R are two regressive functions, then

(i) ep(O(t),s) = (1 + fi(t)p(t))ep(t,s);

(ii) 1/ep (t, s) = e©p(t, s);

(iii) ep(t,s) = 1/ep(s, t) = e©p(s, t);

(iv) ep(t,s)ep(s, t) = ep(t, t);

(v) [ep(t,s)]A = p(t)ep(t,s);

(vi) [ep{c, -)]A = -p[ep(c, -)]° for all c e T;

(vii) (d/dz)[ez(t,s)] = [ft 1/(1 + f(T)z)AT]ez(t,s).

Lemma 2.4 ([51]) Assume thatf,g: T ^ R are delta differentiable att e Tk. Then

(fg)A(t) =f A(t)g (t) +f (o (t))gA(t) =gA (t)f (t) +g(O (t))fA (t).

Lemma 2.5 ([52]) For each t e T, let N be a neighborhood oft. Then, for V e Crd(T, R+), define D+ VA (t) to mean that, given e >0, there exists a right neighborhood Ne n N oft such

[V(a(t)) - V(t) - i(t)/(t)] < D+VA(t) + e /or each s e N, s > t,

where i(t) = a(t)- s. I/1 is right-scattered and V(t) is continuous at t, this reduces to D+V A(t) = .

Next, we introduce the Banach space which is suitable for system (1.1)-(1.3). Let fi = {x = (xi, x2,...,xi)T: |xi| < li, i = 1,2,..., l} be an open bounded domain in Rl with smooth boundary d fi. Let Crd(T x fi, Rn+m) be the set consisting of all the vector function y(t,x) = (y1(t,x),y2(t,x),...,yn+m(t,x))T which is rd-continuous with respect to t e T and continuous with respect to x e fi. For every t e T and x e fi,we define the set CT = {y(t, •): y e C(fi, Rn+m)}. Then CT is a Banach space with the norm ||y(t, •) || = (£n=+1m llyi(t, •) ||2)1/2, where ||y (t, •) N2 = (ffi |yi(t,x)|2dx)1/2. Let Crd([-r,0]t x fi, Rn+m) consist of all functions /(t,x) which map [-r,0]T x fi into Rn+m and /(t,x) is rd-continuous with respect to t e [-r,0]T and continuous with respect to x e fi. For every t e [-r,0]T and x e fi, we define the set C- 0]T = {u(t, •): u e C(fi, Rn+m)}. Then C- 0]T is a Banach space equipped with the norm ||f || 0 = (£ Zi || ji||2)1/2,where f (t, x) = (f1(t, x), x),...,fn+m(t, x))T e Ct_T,0]T, Hfi(t, •)|1 = (fi x) 12 dx)1/2, x)|r = sup,ehr ,„]T fi(s, x)|.

In order to achieve the global robust exponential synchronization, the following system (2.1)-(2.3) is the controlled slave system corresponding to the master system (1.1)-(1.3):

uA(t,x) = £k=1 axk(aik^u) - bui(t,x) + c/jj - r,x)) + I

+ Am PijFj(uj(t - r, x)) + A^ rij /0+~ kij(s)Fj(uj(t - s, x)) As + Vm qijFj(uj(t - r,x)) + Vm WiJ+m kij(s)Fj(uj(t - s,x))As

+ E^ dij lj + A/=1 Sij ij + \Z/=1 Tijlj + m^ x), vA(t,x) = Ek=1 4(&S) - jj(t,x) + £n=1 ?jigi(ui(t - r,x)) + Jj

+ Am=1 XjiGi(vi(t - r,x)) + Am=1 pa f<T Kji(s)Gi(vi(t - s,x)) As + Vm1 njiGi(\>i(t - r,x)) + Vm1 aji /0+TO Kji(s)Gi(v>i(t - s,x))As + Ei=1 hjiVi + Am1 MjiVi + \Jm1 NjiVi + mn+jEn+j(t,x),

subject to the following initial conditions

Iui(s,x) = jji(s,x), (s,x) e [-r, 0]T x fi, js,x) = ipj(s,x), (s,x) e [-r, 0]T x fi,

and Dirichlet boundary conditions

(ui(t, x) = 0, (t, x) e [0, to)t x d fi, v,(t,x) = 0, (t,x) e [0, to)t x dfi,

where Ei(t,x) = ûi(t,x) - ui(t,x) (i = 1,2, ...,n) and En+j(t,x) = Vn+j(t,x) - Vn+j(t,x) (j = 1,2,..., m) are error functions. mk >0(k = 1,2,..., n + m) is a constant error weighting coefficient. U(t,x) = (iû1(t,x),U2(t,x),...,Un(t,x))T e Crd(T x Q,Rn), V(t,x) = (i>1(t,x),V2(t,x), ...,Vm(t,x))T e Crd(TxQ,Rm),&(t,x) = (<Mt,x),<fc(t,x),...,4>n(t,x))T e C([-t,0]x Q,Rn), ((t,x) = (<^1(t,x), (f>2(t,x),..., <Pm(t,x))T e C([-t,0] x Q, Rm).

From (1.1)-(1.3) and (2.1)-(2.3), we obtain the error system (2.4)-(2.6) as follows:

E,A(t,x) = EL wk («'kH) + (mi - bi)E(i'x)

+ E^ Cijfj(Vj(t - T,x)) -fj(Vj(t - t,x))] + A"=iPiAFj(ûi(t - T,x)) - Fj(uj(t - t,x))] + A"=1 rnHœ kij(s)[Fj(Uj(t - s,x)) - Fj(uj(t - s,x))] Hs + Vn=1 ?ii'[F;(ti;(t - t,x)) - Fj(uj(t - t,x))]

+ VL Wj !<++ kij(s)[Fj(ùj(t - s,x)) - Fj(uj(t - s,x))] Hs,

EH+j(t,x) = Ek=1 4 Hk-) + (mn+j - nj)En+j(t,x) + £n=1 Zji[gi(Ui(t- T,x)) -gi(ui(t- T,x))] + Am=1 i(t - T,x)) - Gi(vi(t - T,x))]

+ Am P-i /o+œ K-ï(s)[Gi(Vi(t - s,x)) - Gi(vi(t - s,x))]Hs

+ Vm=1 n-i[Gi(Vi(t- t,x)) - Gi(Vi(t- t,x))]

+ Vm=1 i r K-i(s)[Gi(Vi(t -s,x)) - Gi(Vi(t- s,x))]Hs, subject to the following initial conditions

\Ei(s,x)=(f>i(s,x)-<fri(s,x), (s,x) e [-t,0]t x O, )

I En+j(s,x) = tp-(s,x) - js,x), (s,x) e [-t, 0]T x O,

and Dirichlet boundary conditions

y d o,

{Ei (t, x) = 0, (t, x) e [0, to)t x d O, En+j(t, x) = 0, (t, x) e [0, to)t x d O.

The following definition is significant to study the global robust exponential synchronization of coupled neural networks (1.1)-(1.3) and (2.1)-(2.3).

Definition 2.5 Let y(t,x) = (u1(t,x),u2(t,x),...,un(t,x),v1(t,x),v2(t,x),...,vm(t,x))T e Rn+m andy(t,x) = (ui1(t,x),u2(t,x),...,un(t,x),v1(t,x),v2(t,x),...,vm(t,x))T e Rn+m be the solution vectors of system (1.1)-(1.3) and its controlled slave system (2.1)-(2.3), respectively. E(t,x) = (E1(t,x),E2(t,x),...,En+m(t,x))T e Rn+m is the error vector. Then the coupled systems (1.1)-(1.3) and (2.1)-(2.3) are said to be globally exponentially synchronized if there exists a controlled input vector z(t,x) = (m1E1(t,x),m2E2(t,x),...,mn+mEn+m(t,x))T and a positive constant a e R+ and M > 1 such that

||E(t, -)|| = ||y(t, ■)-y(t, -)|| < Me©a(t, 0), t e [0, to)t,

where a is called the degree of exponential synchronization on time scales.

3 Main results

In this section, we will consider the global robust exponential synchronization of coupled systems (1.1)-(1.3) and (2.1)-(2.3). At first, we need to introduce some useful lemmas.

Lemma 3.1 ([53]) Let O be a cube |xi| < li (i = 1,2,...,l) and assume that h(x) is a real-valued function belonging to C1(O) which vanishes on the boundary d O of O, i.e.,

h(x)\aq = 0. Then

i h2(x)dx < lf i Jq Jq

WW / \vijj\yj) j=1 ;=i

ww V 'WW'

j=1 j=i

Lemma 3.2 ([23]) Suppose that y = (y1, y2,...,yn+m)T and y = y2,...,yn+m)T are the solutions to systems (1.1)-(1.3) and (2.1)-(2.3), respectiVely, then

<Т,\рчШ)-№\>

^\qij\\fj(yj)-fj(yj)\. y=i

<J2\pj\\&(yj)-&(yM>

<E\Pij\\F/(y/)—

<£pj\fj(yj) —Gj(yj)\,

4&>\/>) / \FiJ&J \yj) j=1 j=1

4&>\y>) V hj&j\yj)

j=1 j=i

/\PijF0,) — /\PijPj(yj) j=1 ;=i nn

\j qiFi(y i)-\j qiFj(y j) y=i j=i

/\PiiGj(y j) — ДPij'GJ'(y;)

V ii'^-V ^ВД j=i j'=i

j'=i m

< £\qij \\Gj(yj)-Gj(yj)\.

Throughout this paper, we always assume that:

(Hi) The neurons activationfj, Fi, gi and Gj are Lipschitz continuous, that is, there exist positive constants aj, ft, ^ and 5j such that fj(f ) —fj(^) \ < a/\f - n\, \Fi(f )—Fi(n)\ < A\f -n\, gi(f) — gi(n)\< — n\, \Gj(f) — Gj(n)\ < 5j\f — n\ for any f,n e R, i = i,2,...,n; j = 1,2, ...,m.

(H2) The delay kernels kij, j : [0, +œ)T ^ [0, +œ) (i = 1, 2, ..., n; j = 1, 2, ..., m) are real-valued non-negative rd-continuous functions and satisfy the following conditions:

/> œ p œ

I kij (s)As = 1, I skij(s)As < œ, Jo Jo

p œ /> œ

I Kji(s)As = 1, I sKji(s)As < œ, Jo Jo

and there exist constants w1 > 0, w2 > 0 such that

kij (s)e^(s, 0) As < œ,

Kji(s)em2(s,0)As < œ.

(H3) The following conditions are always satisfied:

l 2a m n

-J2 ~JT + 2(mi -bi) + J2aj^^ + J2Pv{\piv| + |qiv| + |Tiv | + |wiv|)

k=1 k j=1 v=1

n n p+to

+ m Pi( pj + qj) e1®1(T ,0) + Y^ Pi( |Tvi| + |wvi|) / kvi(s)e1®1(s,0)As

V=1 V=1

+ m Yi |e1®1(T, 0) < 0, i = 1,2,..., n;

l 2f n m

- "TT + 2(mn+j - Vj) + J] Yi |Zji | + Se(| + |nje | + |Pje | + |Oje |)

k=1 k i=1 e=1

m m />+TO

+ mjM + |ne/0e1®1(T,0) + ^j|pej| + |Oe/|) / Kej(s)e1©1(s,0)As

e=1 e=1 Jq

+ ^aj\cij|e1®1(T,0) <0, j = 1,2,...,m.

Theorem 3.1 Assume that (H1)-(H3) hold. Then the controlled slave system (2.1)-(2.3) is globally robustly exponentially synchronous with the master system (1.1)-(1.3).

Proof Calculating the delta derivation of ||Ei(t,-)H2 (i = 1,2,...,n) and ||En+j(t,-)||2 (j = 1,2,..., m) along the solution of (2.1), we can obtain

(l|Ei(t, -)||2)A

t, x))2)A dx

(o (t), x)) (Ei (t, x))A dx

= f (2Ei(t,x) + f(t)(Ei(t,x))A)(Ei(t,x))A dx Jq

■ 2 I Ei(t, x)(Ei(t, x))A dx + f(t) j ((Ei(t, x))A)2dx Jq Jq

■■2 [ Ei(t,x)-^- (aik^) dx + 2 / (mi - bi)(Ei(t,x))2dx k=1 Jq dxk \ dxk / Jq

+ 2 1 Ei(t,x)J2cij\fj(vj(t - T,x)) -fj(vj(t - T,x))] dx

Jq j=1

+ 2 1 Ei(t,x) yypij[Fj(u7(t - t,x)) - Fj(uj(t - t,x))]dx

+ 2 1 Ei(t, x)\J qi^F^uij(t - t , x)) - Fj(uj(t - t , x))] dx

JQ j=1

p n p + to

+ 2 1 Ei(t,x) A Tij / kij(s) [Fj(tij(t - s,x))- Fj(uj(t - s,x))] As jq jq

p n p+œ

+ 2 1 Ei(t, x) \J Wi- kij (s) [Fj(Uj(t - s, x))- Fj(uj(t - s, x))] Hs Jo Jo

+ Mt)||(Ei(t, •))

Hi 2 2

I En+j(t, • / 11 2/

= ((En+j(t, x))2) H dx JQ

= (En+j(t, x) + En+j( a (t), x))(En+j(t, x)) H dx JQ

= (2En+j(t, x) + Mt)(En+j(t, x)) A) (En+j(t, x)) H dx JQ

= 2 I En+j(t, x){Ei(t,x)) dx + /(t) ((En+j(t,x)) ) dx Jq JQ

= 2^ i En+j(t, x)^— (^jkdEn+) dx + 2 f (mn+j - m)(En+j(t, x))2dx k=1 JQ 9xA 9xW JQ

+ 2 1 En+j(t, x)V" Zji[gi(Ui(t - t , x))- gi(ui(t - t , x))]dx

,/q i=1

+ 2 1 En+j(t, x) f\\ji[Gi(yi(t - T, x)) - Gi(Vi(t - T, x))]dx

+ 2 / En+j(t, x)\J nji[Gi( Vi(t - t , x))- Gi(Vi(t - t, x))]dx JQ ,.=1

Kji(s)[Gi(Vi(t - s,x)) - G^Vi(t - s,x))]Hs

m p+œ

\Jaji Kji(s) [Gi(î>i(t - s,x)) - Gi(Vi(t - s,x))]Hs

i=1 o H2

+ 2 I En+j(t,x)

+2 En+j(t,x)

■/(t)||(En+j(t, •))H|2.

Employing Green's formula [17], Dirichlet boundary condition (2.6) and Lemma 3.1, we have

V i Ei(t,x)^~(«k^)

f-fJo dxk\ dxkJ

ST- i rt+ \QEi(t,x)

«ikEi(t, x)—;-dS -

YfaJ'EtxX

tlJo \ dxk )

k=1J q

/BEi(t, x)V

V dxk J

£ f ~JT (Ei(t,x))2dx

k=1 jq lk

8 ( BE,

H L En+'{,,x) dx

tf.jnj j dS - É fj'kj'd'

É /a* k (

k=1 ' 0 Q k k=1 l

k=1 „ Q X dxk

En+ (t, x)

, ia ik

f(En+j(t,x))2 dx. (.4)

By applying Lemma 3.2, (3.1)-(3.4), conditions (H1)-(H3) and the Holder inequality, and noting the robustness of parameter intervals, we get

l|Ei(t, -)||2)A < - É ^ ||Ei(t, .)|2 + 2(mi -bi)|Ei(t, .)|2 k=1 lk

+ 2^a'Cj||En+j(t- t, •)||21Ei(t, -)|2

j=1 2 2

+ 2^ IPv I + Iqiv 0 |Ev (t - t , OU^t, •) 12

n £ + TO

+ 2£ Pv( Wiv I + W I)^ kiv (s)|Ev (t - s, •) || 21 Ei (t, •) || 2 As

Mt)||(Ei(t, •))

< |Ei(t, ^)|2 + 2(mi - bi)|Ei(t, •)£

k=1 lk

+ > aj\Cij\u\tn+i(t - T, •)|2 +

J2a'I~CijI [|En+j(t - T, 0||2+ |Ei(t, ^)|2]

+ £ M IPv I + Iqiv 0 [|Ev (t - T, •)|2+ |Ei(t, •)|2]

n r p+to

Iriv I + W I) / kiv (s)||Ev (t - s, -)|2as + |Ei(t, •)|2 Ljû

+ > Iriv I + W I) fn kiv (s)|Ev (t - s, •) || 2 As + |Ei(t, ^)|2

+ ^(t)Q(t)|(Ei(t, 0)||2

-Ys-OT + 2(mi - bi ) + £ a'IC' I

k=1 lk j=1

+ £ ^v (IPiv I + Iqiv I + Iriv I + Iwiv I) + fi(t)Q(t)

+ £a'ICijI x |En+j(t - T, •)|2

llEi(t, 0||

+ E M Pv i + i«iv Ox ||Ev (t - t , o|2

n p+œ

+ V Pv(\riv | + |Wiv \)x/ kiv(s)|Ev(t - s, 0||2As, (3.5)

v=1 Jo

where || (Ei(t, 0)H ||2 = Q(t) ||Ei(t, •) ||2, Q(t) > 0, i = 1,2,..., n. Similar to the arguments of (3.5), we obtain

|En+j(t, o||2)H <

- E ^T + 2(mn+j - j + E Yi1Zj1

k=1 k ' i=1

+ Y^H\ + \nje \ + \Pje\ + \ajeO + /(t)R(t)

+ E Yi\Zji\ x ||Ei(t - t, 0||2

+ Y &e{ \j \ + \nje 0 x |En+e (t - t , •)|2

| En + j ( t,

n+j(t, ) || 2

m ( ) +

+ Z! \ pje \ + \ aje \ )x

Kje(s) |En+e(t s, O^s,

where || (En+j(t, 0)H ||2 = R(t) ||En+j(t, •) ||2, R(t) > 0, j = 1,2,..., m.

If the first inequality of condition (H3) holds, there exists one positive number ^ >0 (may be sufficiently small) such that

l 2« m n

-J2 ~ir + 2(mi -bi) + J2— + M\Piv\ + \qiv\ + \riv \ + \Wiv\)

k=1 lk

n n p+œ

+ m M \Pvi\ + \qviO 61®1(t ,0) + Y M\?vi\ + M / kUi(s)el©l(s,0)As

v=1 v=1

+ Yi\Zji\61®1(T ,0) + £■ <0, i = 1,2, ...,n.

Now we consider the functions

hi(zi) =Zi © Zi + 2(mi - bi) + J2 —

k=1 lk j=1

+ m Pv( \Piv \ + \qiv \ + \riv \ + \Wiv \) + J2 Pi( \Pvi\ + \iviO e1©1(T,°)

n p+œ m _

+ Pi(\rvi\ + \Wvi \) / kvi(s)e1©1(s,0)Hs+ ^Yi\Zji\e1©1(T,0)

v=1 j=1

max{ezi©zi(a (t),0), e,e K

(0 (Zi-HMtQWUEi-MU

(t, °)}e (zi)/(t)Q(t)

eZi©Zi (a (t),0)

where Q (z) = f^' (eZi-sl{zi — s)2) ds, i = 1,2,... , n. From (3.7) we achieve hi(Q) <—g < 0 and hi(zi) is continuous for zi e [0, Moreover, hi(zi) ^ as zi ^ therebythere exist constants e e (0, such that hi(e*) = 0 and hi(ei) < 0 for I' e (0,I*) n (0,1). Choosing I = min1<i<n I', obviously 1 > I > 0, we have, for i = 1,2,..., n,

i 2a m hi(I)=I e I — +2(mi — ki) + J2 a I Cij |

k=1 j=i n n

-V \ iKni I T ^,'n I T |/ iV I T I ^ iy I / T / |p„; I +

+ J2 M Pv l + ^ I + \riv I + \Wiv 0 + J2 M IPvil + liv/0 ei®l(T ,0)

v=i v=l

« /.+œ m _

+ £M\rvil + |Wwl) / kw(s)ei®i(s,0)As + £ y lZjilei®i(r,0) J 0

v=i v=i

v=i "0 j=i

max{e?œ? (a (t),0), e,e (-)_i)„(t)0(t), Ei(t,)ii2(t, 0)}e (-)Mt)Q(t)

(e®-i)v.(t)Q(t)\\Ei(t,-)\\i ■ - ..........„ ,„

+--'/—71—\- — 0. (3.9)

e-®- (a (t),0)

Similar to the above arguments of (3.7)-(3.9), we can always choose 0 < - < i such that for j = i, 2, ..., m,

l 2|.k « _ - ® - - £ - j + 2(m«+j - n) + £yilj k=i ' ;=i

+ £ l^je l + lnie l + lPje l + lOré 0 + £ j M + l^ejO ei®i(T ,0)

e=i e=i

m f+œ «

Ke/(s)eMi(s,0)As + > ail

+ £ jl^ejl + l°ejl) Kej(s)ei®i(s, 0) As + ^ a/lc,jlei®i(r, 0)

e=i 0 ,=i

max{e-®-(a (t),0), e(e(-)-i)^(t)«(t)NEf(t,.)N2(t,0)}e (-)^(t)R(t)

e=œ= (a (t),0)

— 0. (3.i0)

Thus, taking e = min{— e}, we derive, for i = i, 2,..., «; j = i, 2,..., m,

i 2a m

- ® - -J2-TT + 2(m, - b,) + £ aj Ic. l

k=i lk j=i

+ £ M lPiv l + l?iv l + lr,v l + lwiv l) + £ lpVil + l^viO ei®i(r ,0)

v=i v=i

« ç+œ m _

+ £Mlrvil + lwvil) / kvi(s)ei®i(s,0)As+ ^Kilfj,lei®i(r,0) 0

(e(e)-i)^t)Q(t)\\Ei(tr)\\22('-,0)Se (e )^(t)Q(t) — 0 (3 ii)

e-®e (a (t),0) and

l 2|'k « -e ® - -^-j + 2(m«+j - j + Yh yilZjil

k=i k ' i=i mm

+ £ l^je l + l^je l + lPje l + laje 0 + £ j M + l^e/l) ei®i(r ,0)

e=i e=i

m pn + Sj(lPejl + M / Kej(s)e191(s, 0) As + ^ a/|ci/|e1®1(T, 0)

e=1 Jq i=1

max{eeee(o(t),0),e(0(e)-1)f(t)i(t)|£i(t,-)|2(t,O)}0(e)f(t)R(t) <

ee®e (o (t), 0) " .

Take the Lyapunov functional V(t) as follows: V(t) = V (t, E(t)) = V1(t) + V2(t),

(3.12)

(3.13)

V1(t) = Ie*®* (t,°)||Ei(t, 0|2 + e(e(e)_1)/(t)Q(t)||Ei(t,•)||2 (t, 0)

+ Yaj\Cij\ ee©e(a(s + t),°)|En+j(s, •)|2hs

j=1 j*_t

n p t + ePv{\Pv\ + \qiv\) I ee©e(a(s + t),°) ||EV(s, •)|2hs

v=1 Jt_T

n p r p t

+ V Pv(\Tiv \ + \Wiv\) / kiV (s) ee©e(a (s + r),°) ||EV (r, -)|2Hr

v=1 Jo Ut_s

V2 (t) = ee©e (t, 0) ||En+j(t, •)|2 + e(e (6)_1)/(t)R(t)|£n+i'(t,•)l2 (t, 0)

n _ f t + e Yi\Zi\ ee©e(a(s + t),0) |Ei(s, ■)|2hs i=1 Jt_T m _ f t + Y se(\kie \ + \wie\) ee©e(a(s + t),°) |E n+e (s, •) 12 Hs e=1 Jt_T

m t+œ T 11

+ X]se(\Pje\ + \ âje\)/ Ke(s) ee©e(a(s + r),°) |En+e(r, •)|2Hr

Calculating D+V1A(t) along (2.1) associated with (3.5) and noting that (d/dz)[ez(t, s)] = [/; 1/(1 + f(T )z)AT ]ez(t, s) > 0 if and only if z e R+ (that is, ez(t, s) is increasing with respect to z if and only if z e R+), we have

D+ VH (t) = £ (e © e)ec©c (t, 0) |Ei(t, •) || 2 + e6©^a(t), °) ( ||Ei(t, •) 12)

i=1 2 2

+ (e (e) - 1 /(t)Q(t) pi^, •) || 2e(e(e)_1)/(t)Q(t)|£i(t,•)12 (t, 0)

+ J2 aj\Cij\eeee{a (t + T ),°) |En+j (t, •)|2

- Y a \ cij \ ee©^a (t), °) |En+j(t - t, •

— L^T + 2 (mi — bi)

+ £Pv ( IPiv I + I qiv I)eeee (a(t + r), 0) \\EV(t, ■) ||2

— £IPiv I + I qy IHee(a(t),0) |Ey(t — r, -)|2

n p + TO

+ £Pv(ITivI + WivI) kiv(s)eeee(a(s + t),0) |E„(t, .)| 2 As

v=1 J0

n p + TO

— £IrivI + WI ) kiv(s)eeee(a(t),0) \\Ev(t — s, O^As

v=1 j0

< £ (e e e)eeee (t, 0) |Ei(t, ■) |2 + eee (a(t), 0)

+ £ a I Cij I + £ P^ IPiv I + I qiv I + I riv I + I Wi„ I) + fi(t)Q(t)\ WEi(t, -)|

j=1 v=1 /

+ £ a; I % I x |En+j (t — r, -)|2^£ Pv( IPiv I + I qiv I )x |Ev (t — r, •)|2

j=1 v=1

n p + TO

+ £ Pv{ Wiv I + I Wiv I )x / kv (s)|Ev (t — s, .)|2As,

v=1 J°

+ (Q (e) — ^Mt)Q(t) |Ei(t, ■) W 2e(Q(e)—1)^(t)Q(t)yEi"(t,-)y2 (t, 0) m

+ £ajI CijI eee^a(t + r),0) |En+j(t, ■)|2

— £ aI I eee^a(t), 0 |En+j(t — r, ■) |2

+ £ P^IPiv I + I qiv I)e6ee (a(t + r), 0) |Ev(t, ■) 12

— £ P^ IPiv I + I qv I)eeee(a (t),0) ||EV(t — r, .)||2

n p + TO

+ £ Pv(Wiv I + I Wiv I) kiv (s)eeee(a (s + t),0) |Ev (t, -)|2 As

v=1 J°

n p + TO

— £ Pv( Wiv I + IWiv I ) kiv (s)eeee(a (t),0) ||EV (t — s, O^As

v=1 J°

< £ (e e e)eeee(a(t), 0) |Ei(t, ■) |2 + ee(S)e(a(t), 0)

— £ ¥ + 2(mi — bi)

+ £ aj I Cij I + £ Pv{ IPiv I + I qiv I + I Wiv I + I Wiv I )

+ fi(t)Q(t) ||Ei(t, -)|2 x max{e6e^a(t),0), e, + (Q(e) —l)^(t)Q(t)|Ei(t, -)|2

l^t' -)|2

(Q (e)-1)Mt)Q(t)l|£l■(t,.)ll2(t,0)}

x max{ee©e(a(t),°), e.

(e(e)_1)/(t)Q(t)HEi(t,)H

• ) ||- (t, 0) }

+ j2 aj\Cij\ee®e (t ,°K©e(a (t),°) |En+j (t, •)|2

+ EP^\Piv \ + \qiv 0ee©e(t, 0)e6©6 (a(t), °) ||EV(t, • ) |2

n p+œ 1

+ J2Pv(\Tiv\ + \Wivkiv(s)ee©e(s,0)ee©e(a(t),°) |Ev(&• )|2Hs

< ee©e (a(t),°)£ |Ei(t, •

© e-Y^-Ht +2(mi _ bi)

k=1 lk

+ aj\Cij\ + Y Pv( \Piv \ + \qiv \ + \ r iv \ + \Wiv\)

j=1 v=1

max{ee©e (a (t),0), e(e (e)_1)/(t)Q(t)|£i(t,) ||- (t, 0)}e (e)/(t)Q(tY

ee©e (a (t),0) YaCM©(t, 0) ||En+j(t, •) |2

J2Pv( \Piv \ + \qiv 0 ee©e (T ,0)||Ev (t,

n p+œ "I

+ Pv(\riv \ + \Wiv\)Jo kiv (s)ee©e (s,0)|Ev (t, •) || 2Hs |

< ee©e (a(t),0)£ |Ei(t, •

© e -Y^TT + 2(mi - bi) + ^ a \ Cij \

k=1 lk j=1

+ J2 Pv( \Piv \ + \ qiv \ + \ riv \ + \ Wiv \ ) + ^ Pi( \Pvi \ + \ qvi \ )e1©1(T ,0)

n p+œ

+ V Pi( Wvi \ + \ Wvi \ ) kvi(s)e1©1(s,0)Hs

max{ee©e (a (t),0), e(e (e)_1/(t)Q(t)||£i(t/)||-(t,0)}e (e)/(t)Q(t)'

ee©e (a (t),0)

+ ee©e (a (t), 0 Y m a \ Cij \ e1©1(T, °) ||En+j(t, •)|-.

j=1 i=1

By applying (3.6), we can similarly calculate D+ V2H(t) along (2.1) as follows:

(3.14)

D+V-H (t) < ee©e (a (t),°)£ ||En+j(t, •

e © e -Y^T + -(mn+j _ j + J2 Yi \ Zji \

k=1 lk m

+ Y Se( \ \ + \ n je \ + \ Pje \ + \ a je \ ) + M \ Àej \ + \n ej \ )e1©1(T ,0)

e=1 e=1

m p+œ

+ £ j l Pej l + l Oej l ) Kej (s)ei®i(s,0) As

max{ec®c (a (t),0), e,

(e (c)-i)^(t)R(t)\Ei(t,-)i

:(t,0)}e (f)Mt)R(t)

e-®- (a (t),0)

+ e-®- (a(t), 0) £ £ ylZa l ei®i(r, 0) ||Ei(t, •) ||2.

i=i j=i

(3.i5)

From (3.ii)-(3.i5), we get

D+V (t) = D+V (t, E(t))

= D+Vi(t)+D+Vz(t)

— e-®- (a(t),0) ¿j |Ei(t, •

® - -J2 2jr + 2(mi - bi) + £ ajl cij l

k=i ik j=i ««

+ £ M lpiv l + l qiv l + l riv l + l wiv l ) + £ Pi( lpvi l + l qvi l )ei®i(T ,0)

v=i «

+ £Ml?vil + l Wvil ) kvi(s)ei®i(s,0)As + £ yl Zjil ei®i(r,0)

v=i 0 j=i

max{e-®-(a (t), 0), e(e (e)-l)^(t)Q(t)\Ei•(t,•)\2(t, 0)}e (-)Mt)Q(t)

e-®- (a (t),0)

+ e-®e(a(t),0)£ | |E«+j(t, •

l 2ijk

® - -£ -l-i- + 2(m«+j - j

k=i ik

+ £ yi l Zji l + £ se( l j l + l n je l + l Pje l + l a je l )

i=i e=i

+ £ j l V l + l wej l )ei®i(r ,0)

m p+œ «

+ £ SA lPej l + lâe; l ) Kej (s)ei®i(s,0) As + £ aj lcj l ei®i(r ,0) 0

— 0. (3.i6)

e=i 0 i=i

max{e-®- (a (t), 0) e(e (e)-l)^t)R(t)\\E¡(t,•)\\22(t, 0)}e (-)Mt)R(t) '

eeee (a (t),0)

Note that (3.16) means that the Lyapunov functional V(t,E(t)) is monotone decreasing with respect to t e [0, +to)t. Therefore, in the light of (3.13) we get, for t e [0, ,

eeee (t,0)|E(t, OH2

= eeee(t,0)£|Ei(t, -)|2 + eeee(t,0)£|En+j(t, -)|2 < V(t,e(t)) < V(0,e(0))

« ( m p 0 £ lE^ O^ + l + Eaj l c ij11 e-®4a(s+T),0 |E«+j s ol2As i=i[ j=i J-r

" />0 + E PA Ipv l + lqiv l) ee(Be(o (s + t),Q ||Ev (s, -)|2as

v=1 J-T

n tr t o

+ E PA ITiv I + Iwiv l) / kiv (s) / ee©e(o (s + r),0) ||Ey (r, O^Ar

V=1 ./Q Li-s

m i n _ /. 0 + E |En+j(0, -)|2 + 1^ YilTjil I e6®6(o(s + t),0) |Ei(s, -)|2as

j=1 I i=1 J-T

m _ ^ Q

+ E l^jel + Inie I) / ee©e(o (s + t ),0) |En+e (s, •)|2As

e=1 J-T

m tr t Q

+ V $e( IPje I + |Oje I) Kje (s) ee©e(o (s + r),0) |En+e (r, •)|2at

e=1 0 -s 2

n n m n q

<£||* - + n + E 5^aj|cij|H<Pn+j - Vn+j ||2 ee©e(o (s + t),0) As

i=1 i=1 j=1

n n „ Q

+ £ Ipiv I + Iqiv OH'^v - fa ||2 ee©e(o(s + t),0) As

i=1 v=1 J~T

n n p+TO r p 0

+ Y EPA. ITiv I + Iwiv l)||fa - fa ||2 kiv (s) ee©e(o (s + r),0) Ar

i=1 v=1 •/o Li-

m m n n q

+ J2 HVj - Vj|2+ m + YilZji l||fa - fai |2 ee©e(o (s + t),0) As

j=1 j=1 i=1 J~T

mm . q

+ Y Hsa. &je I + Inje l)HVe - Ve ||2 ee©e(o (s + t ),0) As

j=1 e=1 J-T

m m p +TO r /. 0

+ £ IPje I + lOje 0 HVe - Ve ||2 j (s) ee©e(o (s + r),0) Ar

± M2,

which implies that

|| E(t, 0|| < Meee (t, 0).

(3.17)

Obviously, M >1. According to Definition 2.5, we conclude that the controlled slave system (2.1)-(2.3) is globally robustly exponentially synchronous with the master system (1.1)-(1.3) on the time scale [0, The proof is complete. □

When the time scale T = R and T = Z, we will obtain the following two important corollaries.

Corollary 3.1 Assume that the following (H4)-(H6) hold. Then the master system (1.4)-(1.6) and its controlled slave system are globally robustly exponentially synchronous.

(H4) The neurons activation fj, Fi, gi and Gj are Lipschitz continuous, that is, there exist positive constants aj, Pi, Yi and 5j such that f (f )-f(n)l < a/|f - nl, lFi(f )-Fi(n)| < PiiH -

nI, Igi(f) -gi(n) I <YiI f - nI, I j) - Gj(n) | <Sj| f - nI for any f,n e R, i = 1, 2, ...,n; j = 1, 2, ..., m.

(H5) The delay kernels kij, j : [0, +to) ^ [0, +to) (i = 1, 2, ..., n; j =1, 2, ..., m) are real-valued non-negative continuous functions and satisfy the following conditions:

/> TO /> TO /> TO /> TO

I kij(s) ds = 1, I skij(s)ds < to, I Kji(s)ds = 1, I sKji(s)ds < to Jo Jo Jo Jo

and there exist constants w\ > 0, «2 > 0 such that

/> TO /> TO

/ kij(s)esmi ds < TO, / K;,(s)esffl2ds < to. Jo Jo

(H6) The following conditions are always satisfied:

1 2a;t

+ 2(mi - ¿i) + £oj|Cj| + £|pi„| + | + |Fi„ | + \wiv|)

k=l k j=l v=l

« « +TO

+ £ PJ + |ij)e-T + £M|Fvi| + w^) kvi(s)e2s ds

v=l v=l

+ £Xi|fj|e-T <0, i = l,-,...,«;

l —f « m

- £ ~Î-T + 2(m«+/- ^) + £ Yi |Fji| + £ ^| + Wjg| + [pje | + ^ ^

k=l k i=l e=l

m _ m ç +TO

+ £ Sj( + |ne/ 0 e2T + £ j |pe/| + ^j) Kej(s)e2s ds

e=l e=l

YajC<0, j = l,—,...,m.

Corollary 3.2 Assume that the followi«g (H7)-(H9) hold. The« the master system (l.7)-(l.9) a«d its co«trolled slave system are globally robustly expo«e«tially sy«chro«ous.

(H7) The neurons activationf, Fi, gi and Gj are Lipschitz continuous, that is, there exist positive constants a/, Pi, Yi and 5j such that f-(f )-f(n)| < aj|f - n|, )-Fi(n)| < Pi|f -n|, |gi(f) - gi(n)|< Yi|f - n|, |Gj(f) - G/(n)| < 5/- n| for any f,n e R, i = l,—,...,«; j = l,—,..., m.

(H8) The delay kernels kij, : Z+ ^ [0, +to) (i = l, —, ..., «; j = l, —, ..., m) are real-valued non-negative rd-continuous functions and satisfy the following conditions:

TO TO TO TO

Y kij (s) = l, Y skij(s) < to, £ K/i (s) = l, Y sK/i (s) < to,

s=0 s=0 s=0 s=0

and there exist constants wl > 0, w— > 0 such that

Ykij(s)(l + «l)s < to, Y,K/i(s)(l + 0)—)s < to.

(H9) The following conditions are always satisfied:

1 2aiV

-Y~7T + 2(mi - bi) + J2 " + ßv( IPiv I + Iqiv I + \riv I + Wiv i)

k=l k j=l v=l

+ EIPvil + IqviO4T + EMIrvil + I Wvi I )Ekvi(s)4;

v=l v=l s=0

+ E Yi IZfi I 4T <0, i = l, 2, ..., n;

I 2f n m

- "TT + 2(mn+j - n) + Y Yi I Zji I + E Se(I I + I ne I + I Pje I + I wie I )

k=l k i=l e=l

m m +œ

+ EjI kejI + I nejI )4T + EjI p^ I + I ö«j I ) E^K

e=l e=l

+ E a I Cij I 4T <0, j = l, 2, ..., m.

4 Illustrative example

Consider the following reaction-diffusion BAM recurrent FNNs on time scales:

UA(t, X) = Ek=l -irk (aikîrt ) - biUi(t, X) + Eml Cijfj(Vj(t - t, x)) + /i

+ AJ=lPjFj(uj(t - T,x)) + /\n=l rj /0+œ kij(s)Fj(Uj(t - s,x)) As

+ V"=l qiiFi(uj(t - t , x)) + Vj=l Wj /0+œ kj(s)Fj(uj (t - s, x))As + £J=l ¿j'My + AJ=l My + j ji vA(t,x) = £k=l £-k idVk) - jj(t,x) + £n=l Zjigi(Ui(t - t,x)) + Ji

+ Am=l >knGi(vi(t- t,x)) + Am=l Pji /¡TKß(s)Gi(Vi(t -s,x))As + V ml njiGi(vi(t - t , x)) + \J m=l oji f+œ Kji(s)Gi(vi(t - s, x))As

+ £ ml hiivi + A "U Miivi + V ml

subject to the following initial conditions

J Ui(s, x) = & (s, x), (s, x) e [-t,0]T x Q, I js,x) = js,x), (s,x) e [-t, 0]T x Q,

and Dirichlet boundary conditions

Iui(t,x) = 0, (t,x) e [0, x dQ, Vj(t,x) = 0, (t,x) e [0, œ)T x dQ,

where n = m = l = 2,f(v) = Fi(v) = gi(v) = G,(v) = fv+f-^ (i,j = 1,2), %(t) = ^,(t) = ff(|)t (i,j = 1,2), T = {3n : n = 0,±1,±2,...}, Q = {x : |xi| <1,i = 1,2}, t = 1.1 =(Ii,I2) and J = (A, J2) are the constant input vectors. ¡x = (¡1, ¡2) and v = (v1, v2) are the constant bias vectors. Obviously,f (v), Fi(v),gi(v) and Gi(v) satisfy the Lipschitz condition with aj = fa =

Yi = Sj = 1. Let (b1, b2) = (9.5,10.5), (n1, n2) = (8.5,9),

/«11 au\ /0.7 0.4\ /cu C1A /0.4 0.5\ \a2i a22/ \0.2 0.8/, \C21 C22) \0.6 0.1/,

/pn A2W0.1 0.^ iqn qu\_(0.2 0.A VP21 P22/ \0.3 0.5^ ^21 q-j \0.7 0.8^

/rn F1A = / 0.4 0.3\ (wn wn\ /0.2 0.A \r21 r—y \0.6 0.9/, \W21 W22J \0.8 0.3/,

(in I12 \ = /°.6 04\ (Z11 Z12 \ = ^ \l21 i22) (0.2 0.7/ , (z21 Z^ (0.1 O.^

/X11 X1A = /0.2 0.A /n11 S12\/0.3 0.2\ ^21 X2^ = \0.3 0.4/, yn21 W22) = \0.6 0.4/,

(pn /0.5 0.2\ 11 ^1^/0.3 0.A

V21 P22) \0.7 0.8/, \W21 022) \0.9 0.2^

Take the controlled input vector z(t,x) = (m1E1(t,x),m2E2(t,x),m3E3(t,x),m4E4(t,x)) here (m1, m2, m3, m4) = (5,3,2,3). By a simple calculation, we have

a (t) = t + 3, fi(t) = 3, e1®1(t, 0) = (e1(t, 0))2 = 4t,

f+TO p+TO +TO 26 / 1 \ 3s

I k^As=1 "ms=£57(3) =',

f+TO f+TO +TO 78 /1 \ 3s 3

I skij(s)As = I sk«(s)As = 7 —s| - i = — < +to, J0 ' J0 ' 27 V3) 26 ,

/+to p+to +TO 78 /1 \ 3s

se«(s, 0)kij (s)As = J se«(s, 0)km(s)As = £ — s(1 + 3«)s 3 j

27(1 + 3«)

(26-^<+TO (0< « <13)

+ 2(m1 - ¿1) + £ o/ Ic1j I + £ IP1„ I + Iq^ I + I r^ I + I W1„ I)

k=1 k j_1 v=1

+ £ Pi( IPv1 I + M 4T + £ M Irv1I + IWv1^ k„1(3s)4

+ £ Y1IZ/1I4t ^ -0.24 <0,

2 2a 22

-E"«^ + 2(m2 - ¿2) + £ 0/IC2/ I + J2 Pv{ Ip2v I + Iq2v I + I^y I + fe I)

k=1 k j=1 v=1

2 2 +TO

+ £ P^ IPv2 I + I qv2 I )4T + £ P2( I rv2 I + I Wv2 I ) £M3s)4s

v=1 v=1 s=0

+ E Y21 Zj2 I 4T w -0.78 <0,

2 2| 2 2 -e ""ТТ +2(w3-П^ + щ yi IZu I + щ i^e I + I^ie I + I P I + I^ie I )

k=i Lk i=i e=i

2 2 +TO

+ £ 5i(I lei I + I ^iI )4T + £ 5i(I Pel I + I ^ I ) £>ei(3s)4s

e=i e=i s=o

+ £ ai I сц I 4T w -0.i9 < 0,

2 2f 2 2 - J2 ~TT + 2(m4 - П2) + E Yi IZ2i I + E I ^2e I + I I + I P2e I + I °2в I )

k=i k i=i e=i

2 2 +TO

+ £I Xe2 I + I We2I )4T + £I Pe2 I + I ^ I ) £>e2(3s)4s

e=i e=i s=o

+ J2 a21 ca I 4T w -i.03<0.

Thus, conditions (Hi)-(H3) are satisfied. It follows from Theorem 3.i that the master system (4.i)-(4.3) and its controlled slave system are globally robustly exponentially synchronized.

Competing interests

The author declares to have no competing Interests. Author's contributions

The author read and approved the finalmanuscript. Acknowledgements

The author would like to thank the anonymous referees for their usefuland valuable suggestions. This work is supported by the NationalNaturalSciences Foundation of Peoples Republic of China under Grant (No. 11161025; No. 11326101), Yunnan Province naturalscientific research fund project (No. 2011FZ058).

Received: 3 September 2014 Accepted: 28 November 2014 Published: 16 December 2014

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doi:10.1186/1687-1847-2014-317

Cite this article as: Zhao: Global robust exponential synchronization of BAM recurrent FNNs with infinite distributed delays and diffusion terms on time scales. Advances in Difference Equations 2014 2014:317.

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