0 Advances in Difference Equations

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On pth moment exponential stability of stochastic fuzzy cellular neural networks with time-varying delays and impulses

Peiying Xiong1,2* and Lihong Huang1

Correspondence: xiongpy68@163.com College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, P.R. China 2College ofMathematics and Computer Science, Hunan City University, Yiyang, Hunan 413000, PR. China

Abstract

This paper is concerned with the pth moment exponential stability of fuzzy cellular neural networks with time-varying delays under impulsive perturbations and stochastic noises. Based on the Lyapunov function, stochastic analysis and differential inequality technique, a set of novel sufficient conditions on pth moment exponential stability of the system are derived. These results generalize and improve some of the existing ones. Moreover, an illustrative example is given to demonstrate the effectiveness of the results obtained.

ft Springer

1 Introduction

In the last decades, cellular neural networks [1, 2] have been extensively studied and applied in many different fields such as associative memory, signal processing and some optimization problems. In such applications, it is of prime importance to ensure that the designed neural networks are stable. In practice, due to the finite speeds of the switching and transmission of signals, time delays do exist in a working network and thus should be incorporated into the model equation. In recent years, the dynamical behaviors of cellular neural networks with constant delays or time-varying delays or distributed delays have been studied; see, for example, [3-11] and the references therein.

In addition to the delay effects, recently, studies have been intensively focused on stochastic models. It has been realized that the synaptic transmission is a noisy process brought on by random fluctuations from the release of neurotransmitters and other probabilistic causes, and it is of great significance to consider stochastic effects on the stability of neural networks or dynamical system described by stochastic functional differential equations (see [12-23]). On the other hand, most neural networks can be classified as either continuous or discrete. Therefore most of the investigations focused on the continuous or discrete systems, respectively. However, there are many real-world systems and neural processes that behave in piecewise continuous style interlaced with instantaneous and abrupt change (impulses). Motivated by this fact, several new neural networks with impulses have been recently proposed and studied (see [24-33]).

In this paper, we would like to integrate fuzzy operations into cellular neural networks. Speaking of fuzzy operations, Yang and Yang [34] first introduced fuzzy cellular neural networks (FCNNs) combining those operations with cellular neural networks. So far researchers have found that FCNNs are useful in image processing, and some results have

© 2013 Xiong and Huang; 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 originalwork is properly cited.

been reported on stability and periodicity of FCNNs [34-40]. However, to the best of our knowledge, few author investigated the stability of stochastic fuzzy cellular neural networks with time-varying delays and impulses.

Motivated by the above discussions, in this paper, we consider the following stochastic fuzzy cellular neural networks with time-varying delays and impulses

' dxi(t) = [-CiXi(t) + £;=1 Uijfj(xj(t)) + A;=1 aijgj(xj(t - Tj(t)))

+ vn=i Pijgj(Xj(t - Tj(t))) + li] dt (1)

+ E;=1 aij(t,Xi(t),Xi(t - Ti(t))) dtoj(t), t = kk, Axi(tk) = Iik(x(tk)) = Xi(tk) - Xi(t-), k e Z+, i = 1,2,...,n,

where n corresponds to the number of units in the neural networks. For i = 1,2,..., n, Xi(t) corresponds to the state of the ith neuron. f(■), gj(■) are signal transmission functions. ci >0 denotes the rate at which a cell i resets its potential to the resting state when isolated from other cells and inputs; Tj(t) corresponds to the transmission delay. atj represents the elements of the feedback template. Ii = Ii + /\ TjUj + \J HijUj, aij, fy, Tij and Hij are elements of fuzzy feedback MIN template and fuzzy feedback MAX template, fuzzy feed-forward MIN template and fuzzy feed-forward MAX template, respectively; /\ and V denote the fuzzy AND and fuzzy OR operation, respectively; Uj denotes the external input of the ith neurons. Ii is the external bias of the ith unit. Ti(t) is a transmission delay satisfying 0 < Ti(t) < t; a(t,X,y) = (a^t, Xi,yi))nxn e Rnxn is the diffusion coefficient matrix and ai(t,Xi,yi) = (afl(t,Xi,yi),ai2(t,Xi,yi),...,a;(t,Xi,y) is the ith row vector of a(t,X,y): a>(t) = (<y1(t),a)2(t),...,a)n(t))T is an n-dimensional Brownian motion defined on a complete probability space (fi, F, {Ft}t>0, P) with a filtration {Ft}t>0 satisfying the usual conditions (i.e., it is right continuous and F0 contains all P-null sets). AXi(tk) = Xi(t+) -Xi(t-) is the impulses at moment tk, the fixed moments of time tk satisfy 0 = t0 < t1 < t2 < ..., limk^+TO tk = +(»; Xt(s) = X(t + s), s e [-t,0].

System (1) is supplemented with the initial condition given by

where <p(s) is F0-measurable and continuous everywhere except at a finite number of points tk, at which <p(t+) and ^(t-) exist and ^(t+) = q>(tk).

Let PC1,2([tk, tk+1) x Rn;R+) denote the family of all nonnegative functions V(t,x) on [tk, tk+1) x Rn which are continuous once differentiable in t and twice differentiable in x. If V(t,x) e PC1,2([tk, tk+1) x Rn;R+), define an operator LV associated with (1) as

Xt0(s)=^(s), s e [-t,0],

LV (t, x) = Vt (t, x) + Vxi (t, x) -CiXi(t) + aijxj (t)) + /\ ajxjit - T;(t)))

For convenience, we introduce several notations. x = (x1,x2,...,xn)T e Rn denotes a column vector. ||x|| denotes a vector norm defined by ||x|| = (^n=i Ixi\p)llp; C[X, Y] denotes the space of continuous mappings from topological space X to topological space Y. Denoted by CbFo[[ -t,0),Rn] is the family of all bounded F0-measurable, C[[-t,0),Rn]-valued random variables $, satisfying ||$||lp = sup-T<g<0E||$(0)|| < +cc, where E(-) denotes the expectation of a stochastic process. Throughout the paper, we give the following assumptions. (Al) The signal transmission functionsf(-), g(-) (j = 1,2,..., n) are Lipschitz

continuous on R with Lipschitz constants f and Vj, namely, for any u, v e R,

f (u) -f(v) | < fj\u - v|, \gj(u) -gj(v) | < Vj|u - v|, f(0) = gj(0) = 0.

(A2) There exist non-negative numbers si, wi such that for all x,y,x',y! e R, i = 1,2, ...,n,

[o;(t,x',/) - o(t,x,y)][ffi(t,x',y) -o(t,x,y)]T < si\x' -x\2 + wi\y -y\2.

(A3) Iik(xi(tk)) = -yik(xi(tk) - x*), where x* is the equilibrium point of (l) with the initial condition (2), yik satisfies 0 < yik < 2.

Definition 1.1 The equilibrium point x* = (x*,x*,...,xn)T of system (1) is said to be pth moment exponentially stable if there exist positive constants M >0,X >0 such that

E( ||x(t) - x* ||p) < M|| V - x* |pe-A(t-t0), t > to,

where x(t) = (x1(t), x2(t),...,xn (t))T is any solution of system (1) with initial value xi(t + s) = Vi(s) e PC([-t,0],R), i = 1,2,...,n.

When p = 2, it is usually said to be exponentially stable in mean square.

Lemma 1.1 [34] Suppose x andy are two states of system (1), then we have

f\aijgj(x)- Aaijgj(y) j=i j=i

V ßijgj(x)-\J ßijgj(y) j=i j=i

<J2aij \\gj(x)-gj (y)|

<J2ßij\\gj(x)-gj(y)\-

Lemma 1.2 Ifai > 0 (i = 1,2,..., m) denote p nonnegative real numbers, then

ai + a2 + ••• + am , *

aia2 ■■■am <-, (4)

where p > 1 denotes an integer. A particular form of (4), namely

p-i ^ (p - 1)a? ap

ai a2 <-+ —.

2 Main results

In this section, we consider the existence and global pth moment exponential stability of system (1).

Lemma 2.1 For two positive real numbers a and b, assume that there exists a constant number 0 < n <1 such that 0 < b < na. Then the equation

X = a - be" (5)

has a unique solution k > 0.

Proof Let F (k) = k - a + bekT. It is easy to see F (0) = -a + b <0, F (a) = beaT > 0, F'(k) = 1 + brekT > 0. Thus, F(k) is strictly increasing on [0, to). Therefore, Eq. (5) has a unique positive solution k >0. □

Lemma 2.2 [18] For two positive real numbers a and b, assume that there exists a constant number 0 < n <1 such that 0 < b < na. Assume that z(t) is a nonnegative continuous function on [t0 - t , t0] and satisfies the following inequality:

D+z(t) < -az(t) + b||zt ||, fort >0, (6)

then z(t) < ||zt0 ||e-k(t-t0), where k is a solution of (5) and the upper right Dini derivative of z(t) is defined as

n+ (a r z(t + S)-z(t) D z(t) = lim sup-.

Theorem 2.1 Under conditions (A1)-(A3), if there exist constants Ki > 0 (i = 1,2), 0 < n <1 such that

0<K2< nK1, (7)

Kl = min |pci -(p - 1) V(mja| + Vj(\aij\ + |))

1<i<n I ' '

" " I j=1

A p(p -1) (p-1)(p-2) \

\aji\im--2— si--2-wi r

j=1 2 2

K2 = maJ ViY] (\aji \ + \Pjii) + (p - 1)si 1.

1< i< n

I j=1 I

Then x* = (xj, x2,..., x*)T is a unique equilibrium which is globally pth moment exponential stable.

Proof The proof of existence and uniqueness of equilibrium for the system is similar to that of [39]. So we omit it. Suppose that x* = (x^,x^,...,x;)t is the unique equilibrium of system (1).

yi(t)=Xi(t)-X*,

j,yi(t),yi(t - T(f})) = Oij(t,yi(t)+x*i,yi(t - r;(i)+X*)) - Oij(t,X*,X*), then system (1) can be transformed into the following equation, for i = 1,2,..., n:

dyi(t) = [-Ciyi(t) + EJ=1 aij(fj(yj(t - Tij(t)) + xJ) -fj(Xj))

+ (A;=i aijgj(yj(s) + Xj) ^n=i ajj))

+ (Vn=i Pjigj(yj(s~) + Xj) -j ^jigj(Xj))] dt + En=1 j, yi (t), yi (t - Ti (t))) dwj(t), t = tk, k = 1,2,..., Ayi(tk) = ilk(y(tk)) = Iik(y(tk) + Xj) - Iik(Xj).

We define a Lyapunov function V(t,y(t)) = En=1 iyKt)ip. Let t > to, t e [tk-1, tk), then we can get the operator LV(t, y(t)) associated with system (8) of the following form:

LV(t,y(t)) = pJ2\yi(t)P sg^yi(t))

-Ciyi(t) + J2 jjj - Ti/(t)) + j-fj(Xj))

Aaijgj(yj(s)+j - A «jj

I V^jigj(yj(s)+X;j) -\/ Pjigj (Xj

V=1 P(P -1)

EM E j, yi(t), - Ti(t)))

i=1 j=1 nn

< -pEci\yi(t)\P + pEEiaiji^\yi(t)\p-1\yj(t)\

+(ai + ijyj\yi(t)\p \yj(t-\

i=1 j=1

+ E\yi(t)\p-2(si\yi(t)\2 + Wi\yi(t- Ti(t))\2)

< -pJ2a \yi(t) \p + E E ia^j((p -1) \yi(t) \p + \yj(t) \p)

i=1 j=1

+ E E (a i + ij j (p -1) \yi(t) \p + \yj(t - Tj(t)) \p)

i=1 j=1

+ E> \yi(t) \p + ^ E Wi((p -1) \yi(t) \p + 2\yi(t- Ti(t)) \p)

pci - (p - 1) Y^{lxjiaiji + jiaiji + iAji))

A, , p(p -1) (p-1)(p-2) Iaji \fi--2— si--2-wi

\yi(t)\p

(|«jiI + |j8/i|) + (p -1)si

\yi(t - Ti(t)) \p

< -K1 V(t,y(t) + K2 sup V(s,y(s)), (9)

t-T <s<t

K1 = min jpCi -(p - 1) ^ f a| + Vj(|«ij| + |Aj|))

" p(p -1) (p-1)(p-2) - |aji |fi--2—si--9-wi ,

K2 = max j ViY (|aji| + + (p - 1)si |.

1 < i < n I ' ■ I

1< i< n

" " I j=1

Firstly, for t e [t0, t1), applying the Ito formula, we obtain that V(t + 5,y(t + 5)) - V(t,y(t))

p t+5 p t+5

= LV(s,y(s)) ds + / Vy(s,y(sj)o(s,y(s),y(s - t(s))) dw(s). (10)

Since E[ Vy (s, y(s))o (s, y(s), y(s - t (s))) d«(s)] = 0, taking expectations on both sides of equality (10) and applying the inequality (9) yields

E(V(t + 5, y(t + 5)))- E(V(t, y(t)))

i f-K1E(V(s,y(s))) + Kz^ sup V(0,y(0)))

Jt ^ ^s-T<0<s ' J

ds. (11)

Since the Dini derivative D+ is

D+E(V (t, y(t))) = lirn sup E(V (t + 5, y(t + 5)- E(V (t, y(t))), (12)

denote z(t) = E(V(t,y(t))), the preceding result (11) leads directly to

D+z(t) < -K1z(t)+K2||zt ||p. (13)

Hence, from Lemma 2.2, we have

z(t) < ||z(to) ||pe-A(t-to). Namely,

E[|x(t) -x*|p] < M¡v -x*|pe-A(t-to),

where M = 1, X is the unique positive solution of the following equation:

X = Ki- K2ekt. Next, suppose that for k = 1,2,..., m, the inequality

E(V(t,y(t})) < M\H\pe-X(t-to), t e [tk-i, tk),k =1,2,..., (14)

holds. From (A3), we get

^V(tm, y(tm))) = £ E\y^tm)+ T^t-,) + x*) \p = E £ |1 - Yim |p \yi(tm)\p

i=1 i=1 n

< E^\y^tm) \p = E(V(tm),y(tm)) < M|v|pe-A(tm-t0). i=1

This, together with (14), leads to

E(V(t,y(t))) < M||v||pe-A(t-to), t e [-t, tm]. (15)

On the other hand, for t e [tm, tm+1), applying the Ito formula, we get

V(t, y(t) - V(tm, y(tm))

= I LV(s,y(s)) ds + / Vy(s,y(s))o(s,y(s),y(s - t(s))) da(s).

v tm v tm

Then, we have

EV(t,y(t)) - EV(tm,y(tm)) =i ELV(s,y(s)) ds. (16)

So, for small enough 5 ^ 0+, we have

EV(t + 5, y(t + 5))- EV(tm, y(tm))= ELV(s, y(s)) ds. (17)

From (16) and (17), we have

EV(t + 5,y(t + 5)) - EV(t,y(t))

ELV(s, y(s)) ds

ds. (18)

< i [-K1E(V(s,y(s))) + K2^ sup V(0,y(0)))

Jt L Vs-r<0<s ' J

Similarly, we obtain

E( V(t,y(t))) < M\\v ype-X(t-to), t e [tm, tm+1). (9)

Hence, by the mathematical induction, for any k = 1,2,..., we conclude that

E( V(t,y(t))) < M\W\\pre-x(t-to), t > t0,

which implies that the equilibrium point of the impulsive system (1) is pth moment exponentially stable. This completes the proof of the theorem. □

3 Comparisons and remarks

It can be easily seen that many neural networks are special cases of system (1). Thus, in this section, we give some comparisons and remarks.

Suppose that Iik(x) = x e Rn, system (1) becomes the stochastic fuzzy cellular neural networks with time-varying delays.

dxi(t) = [-CiXi(t) + £;=. aijfj(xj(t)) + A;=i aijgj(xj(t - tj(t)))

+ V;=i Pijgj(xj(t - Tj(t)))+ Ii] dt

+ Xj=i °ij(t,Xi(t),Xi(t- Ti(t))) dwj(t), t > to, xi(t0 + 5) = Wi(s), -to < s < 0, i = 1,2,..., n.

For (20), we have the following corollary by Theorem 2.1.

Corollary 3.1 If (A1)-(A2) hold, if there exist constants Ki > 0 (i = 1,2), 0 < n <1 such that

0 < K2 < nKi, where

Kl = min Ipci -(p - 1) V(pja| + vA\aij\ + \faj|))

1<i<n I

~ ~ y j=i

A, , p(p -1) p-1)(p-2) \

- \aji\Pi--2— si--2-Wi I > 0,

j=1 2 2

K2 = maJ Viy(\a.ji \ + \^ji\) + (p- 1)si |,

1< i< n 1 i n j=1

then the unique equilibrium x* = (x* , x2,..., xn )T of system (20) is globally pth moment exponential stable.

If we do not consider fuzzy AND and fuzzy OR operations and when Iik(x) = x e Rn in system (1), then system (1) becomes impulsive stochastic cellular neural networks with time-varying delays

dxi(t) = [-cixi(t) + £n=1 aijfj(xj(t)) + Xj=i aijgj (jt - jt)))

+ j j(xj(t - Tj (t)))+ Ii] dt

+ Xj=i °ij(t,xi(t),xi(t - Ti(t))) d«j(t), t > t0, xi(t0 + s) = Wi(s), -to < s < 0, i = 1,2,..., n.

Remark 3.1 The stability of system (21) has been investigated in [31]. In [31], authors required the differentiability and monotonicity of a time delays function which satisfied Tj(t) < f <1. Hence, this assumption may impose a very strict constraint on the model because time delays sometimes vary dramatically with time in real circuits. Obviously, Theorem 2.1 does not require these conditions.

Remark 3.2 In Theorem 2.1, if we do not consider fuzzy AND and OR operation, it becomes traditional cellular neural networks. The results in [32] are the corollary of Theorem 2.1. Therefore the results of this paper extend the previous known publication.

4 An example

Example 4.1 Consider the following stochastic fuzzy neural networks with time-varying delays and time-varying delays and impulses

' dX1(t) = [-11X1 (t) + 0.1/1 (X1(t)) + 0fX2(t)) + Aj=1 ayfj(Xj(t - Tj (t))) + Vj=1 Pjj (Xj(t - Tj(t))) + Aj=1 TjUj + j j] dt

+ on(t,Xi(t),Xi(t - Ti(t))) d«i + O12(t, Xz(t), X1(t - T1(t))) d«2, t = tk, dX2(t) = [-17X2(t) - 0.6/2(X1(t)) + 0.3f2(X2(t)) + /\j2=1 a2jfj(Xj(t - Tj(t))) + V=1 j(Xj(t - Tj(t))) + Aj=1 T2jUj + V=1 HjUj] dt

+ O21(t,X2(t),X2(t - T2(t))) d«1 + O22(t,X2(t),X2(t - T2(t))) d«2, t = tk,

AX1(tk) = -(1 + 0.3 sin(1 + k2)X1(t-)), AX2(tk) = -(1 + 0.6 sin(1 + k)X2(t-)),

where t0 = 0, tk = tk-1 + 0.2k, k = 1,2,..., and

fi(r)=gi(r)^1(ir + 1i-ir-1i), Tj(t) = 0.3 i sin ti +0.1< 0.4, i,j = 1,2, 5 1 13

a11 = -, a21 = —, ai9 =—, am = —;

o11(X, y) = 0.2x - 0.1y, o12(x, y) = 0.3x + 0.1y, o21(X, y) = 0.1x + 0.2y, o22(x, y) = 0.2x + 0.1y,

and Tij = Hij = Sij = Lij = Ui = Uj = 1 (i, j = 1,2).

Let p = 2, obviously, ii = ii = 1, i = 1,2. By simple computation, we can easily get that K1 = min{19.4,31.6} = 19.4, K2 = max{6,2} = 6. Letting n = 0.4, we have

nK1 > K2 = 6 > 0. (23)

Thus, system (22) satisfies assumptions (A1)-(A3). It follows from Theorem 2.1 that system (22) is exponentially stable in mean square.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

The authors indicated in parentheses made substantialcontributions to the following tasks of research: drafting the manuscript (PX);participating in the design of the study (LH);writing of paper (PX).

Acknowledgements

The authors would like to thank the editor and the referees for their helpfulcomments and valuable suggestions regarding this paper. This work is partially supported by 973 program (2009CB326202), the NationalNaturalScience Foundation of China (11071060,60835004,11101133), FundamentalResearch Fund for the Central Universities of China (531107040222) and Scientific Research Fund of Hunan ProvincialEducation Department (Grant No. 12A206).

Received: 13 March 2013 Accepted: 28 May 2013 Published: 17 June 2013

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doi:10.1186/1687-1847-2013-172

Cite this article as: Xiong and Huang: On pth moment exponential stability of stochastic fuzzy cellular neural networks with time-varying delays and impulses. Advances in Difference Equations 2013 2013:172.

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