Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2012, Article ID 175934,20 pages doi:10.1155/2012/175934

Research Article

Pth Moment Exponential Stability of Impulsive Stochastic Neural Networks with Mixed Delays

Xiaoai Li,1 Jiezhong Zou,1 and Enwen Zhu2

1 School of Mathematics and Statistics, Central South University, Hunan, Changsha 410083, China

2 School of Mathematics and Computational Science, Changsha University of Science and Technology, Hunan, Changsha 410004, China

Correspondence should be addressed to Enwen Zhu, engwenzhu@126.com

Received 29 June 2012; Revised 15 October 2012; Accepted 2 November 2012

Academic Editor: Dane Quinn

Copyright © 2012 Xiaoai Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper investigates the problem of pth moment exponential stability for a class of stochastic neural networks with time-varying delays and distributed delays under nonlinear impulsive perturbations. By means of Lyapunov functionals, stochastic analysis and differential inequality technique, criteria on pth moment exponential stability of this model are derived. The results of this paper are completely new and complement and improve some of the previously known results (Stamova and Ilarionov (2010), Zhang et al. (2005), Li (2010), Ahmed and Stamova (2008), Huang et al. (2008), Huang et al. (2008), and Stamova (2009)). An example is employed to illustrate our feasible results.

1. Introduction

The dynamics of neural networks have drawn considerable attention in recent years due to their extensive applications in many fields such as image processing, associative memories, classification of patters, and optimization. Since the integration and communication delays are unavoidably encountered in biological and artificial neural systems, it may result in oscillation and instability. The stability analysis of delayed neural networks has been extensively investigated by many researchers, for instance, see [1-30].

In real nervous systems, there are many stochastic perturbations that affect the stability of neural networks. The result in Mao [24] suggested that one neural network could be stabilized or destabilized by certain stochastic inputs. It implies that the stability analysis of stochastic neural networks has primary significance in the design and applications of neural networks, such as [7,12-16,18, 20, 22-24, 26, 27, 30].

On the other hand, it is noteworthy that the state of electronic networks is often subjected to some phenomenon or other sudden noises. On that account, the electronic networks will experience some abrupt changes at certain instants that in turn affect dynamical behaviors of the systems [5, 6, 17-23, 28, 29]. Therefore, it is necessary to take both stochastic effects and impulsive perturbations into account on dynamical behaviors of delayed neural networks [18, 20, 22, 23].

Very recently, Li et al. [22] have employed the properties of M-cone and inequality technique to investigate the mean square exponential stability of impulsive stochastic neural networks with bounded delays. Wu et al. [23] studied the exponential stability of the equilibrium point of bounded discrete-time delayed dynamic systems with linear impulsive effects by using Razumikhin theorems. To the best of authors' knowledge, however, few authors have considered the pth moment exponential stability of impulsive stochastic neural networks with mixed delays.

Motivated by the discussions above, our object in this paper is to present the sufficient conditions ensuring pth moment exponential stability for a class of stochastic neural networks with time-varying delays and distributed delays under nonlinear impulsive perturbations by virtue of Lyapunov method, inequality technique and Ito formula. The results obtained in this paper generalize and improve some of the existing results [5, 8, 18, 19, 26-28]. The effectiveness and feasibility of the developed results have been shown by a numerical example.

2. Model Description and Preliminaries

Let R denote the set of real numbers, Rn the n-dimensional real space equipped with the Euclidean norm | ■ |, Z+ the set of nonnegative integral numbers. E(-) stands for the mathematical expectation operator. L denotes the well-known L-operator given by the Ito formula. w(t) = (w1(t),.. .,wm(t)) is m-dimensional Brownian motion defined on a complete probability space (Q, F,P) with a natural filtration (Ft}t>0 generated by {w(s) : 0 < s < t}, where we associate Q with the canonical space generated by w(t) and denote by F the associated a-algebra generated by w(t) with the probability measure P. Let a(t,x,y) = (an(t,xi,yi))nxm € Rnxm, and ai(t,xi,yi) be ith row vector of a(t,x,y).

In [5, 6], the researchers investigated the following impulsive neural networks with time-varying delays:

Xi(t) = -ai(xi(t)) + Vbijfj (xj(t)) + VCijgj (xj (t - Tj(t))) + Ii, t / tk,

j=1 j=1 (2.1)

xi(tk) = pik(x(tk)), k € Z+, i € A.

The authors in [7, 26, 27] studied the stochastic recurrent neural networks with time-varying delays:

( -Uiixm + 2 bijfj(xj (t))+£ Cijgj(xj(t - Tj (t)))+ Ii \dt

\ ^ ^ / (2.2)

+ y,Oil(t,xi(t),xi(t - Ti(t)))dWl(t).

In this paper, we will study the generalized stochastically perturbed neural network model with time-varying delays and distributed delays under nonlinear impulses defined by the state equations:

dxi(t) =

-ai(xi(t)) + ^bijfj(xj(t)) + ^Cijgj(xj(t - Tj(t))) j=i j=i

^dij Kij(t - s) ■ hj(xj(s))ds + Ii

7 = 1 J-CO

ij ij j = 1 J-X

+ y,oil(t,xi(t),xi(t - Ti(t)))dwi(t), t = tk,

xi(tk) = Pik(x(t-)), k e Z+, i e A,

where A = {\,2,...,n}, the time sequence {tk} satisfies 0 = t0 < t\ < t2 < ••• < tk < tk+\ •••, limk^TOtk = to; x(t) = (x\(t),X2(t),...,xn(t))T and Xi(t) corresponds to the state of the ith unit at time t; bj, cij, and dij denote the constant connection weight; Tj (t) is the time-varying transmission delay and satisfies 0 < Tj(t) < Tj, 0 < n = infie^+ {1 - Tj (t)}, for j e A. fj (•), gj (•), hj (•) denote the activation functions of the jth neuron; the delay kernel Kij(•) is the real-valued nonnegative piecewise continuous functions defined on [0, to); n corresponds to the numbers of units in a neural network; Ii denotes the external bias on the ith unit; pik(x(tk)) represents the abrupt change of the state xi(t) at the impulsive moment tk.

System (2.3) is supplemented with initial condition given by

xi(s) = yi(s), s e (-to,0], i e A, (2.4)

where y(s) = (<p\(s),yi(s),.. .,yn(s))T e PCB^((-to,0],Rn) = BPCB^. Denote by PCB^ the family of all bounded Fo-measurable, PC((-to,0],Rn)-value random variables y, satisfying supee^0]E\y(9)\p < to, where PC((-to,0],Rn) = {y : [-to,0] ^ Rn} is continuous everywhere except at finite number of points tk, at which y(t£) and y(t-) exist and y(t^) =

y(tk).

The norms are defined by the following norms, respectively:

/ n \ \/P / n \ \/P

\W\\P = suW ZWi(s)\P) , ||x||p =(Bxi\P) . (25)

se(-TO,0]\ i=\ / \i=\ /

Throughout this paper, the following standard hypothesis are needed. (H\) Functions ai(•) : R ^ R are continuous and monotone increasing, that is, there exist real numbers ai> 0 such that

ai(u) - ai(v) > a, (2.6)

for all u,v e R, u = v, i e A.

(H2) Functions fj, gj, and hj are Lipschitz-continuous on R with Lipschitz constants Lf > 0, Lg > 0, and Lh > 0, respectively. That is,

Ifj(u) - fj(v)| < Lf \u - v\, \gj(u) - gj(v)| < Lg\u - v\, \hj(u) - hj(v)| < Lh\u - v\,

for all u,v e R, i e A.

(H3) The delay kernels Kj : [0, to) ^ R+ satisfy

¡•CO

Kij (s) <K(s) e A, s e [0, to), K(s)e^°sds

< <x>, (2.8)

where K(s) : [0, to) ^ R+ is continuous and integrable, and the constant fi0 denotes some positive number.

(H4) There exist nonnegative constants ei, li such that

[oi(t,u',v') - Oi(t,u,v)] [oi(t,u',v') - Oi(t,u,v)]T < ei\u' - u\2 + k\v' - v\2, (2.9)

for all u, v, u', v' e R, i e A.

nnegative matrixes P>_ = (i,„

(H5) There exist nonnegative matrixes Pk = (pk )„x„ such that

pifc(u1,u2,..,un) - Pik (V1,V2,...,V„)|P <YjP,ij\uj - Vj\P, (2.10)

for any (u1,u2, ...,un )T, (v1, v2,..., vn)T e Rn, where p > 1 is an integer. We end this section by introducing three definitions.

Definition 2.1. A constant vector x* = (x*, x*,..., x*n)T e Rn is said to be an equilibrium point of system (2.3) if x* is governed by the algebraic system

„ „ „ft ai(x*i) = X bijfj(xj) + X cijgj(x*j) + X dij\ Kij (t - s)hj(Xj) ds + ^ (2.11)

j=1 j=1 j=1 J

where it is assumed that impulse functions pik(■) satisfy pik(x\,x2,,...,x*n) = x* for all i e A and a(t,x*,x*) = 0.

Definition 2.2. The equilibrium x* = (x*1,x2l,...,x'n)T e Rn of system (2.3) is said to be pth moment exponentially stable if there exist X> 0 and M > 1 such that

E||x(t) - x*\\p < E\\y - x*\\pMe-u for t > 0, (2.12)

where x(t) is an any solution of system (2.3) with initial value y e PCB^o((-to,0],Rn).

Definition 2.3 (Forti and Tesi, 1995 [25]). A map H : R" ^ R" is a homeomorphism of R" onto itself if H is continuous and one-to-one, and its inverse map H-1 is also continuous.

3. Main Result

For convenience, we denote that

" P 1 / (X \

^ = - " E fa Ff + \dj\mi>lfu' + \djK(s)ds)

j=i Hi j=i Hi (3.!)

= * - ^ei - (p - ^ - 2) lit = + (p - l)/i,

ti-j = H11 dijl^Lf»', H = max{Hi}, H = min{H },

where h- are positive constant, ai-j, fim, y^, Si^, h^, and niij are real numbers and satisfy

p p p p p p

Yaui =1 =1 , Yjnj =1 , YSnj =1 , Yjiiij =1 , Yniij =1

i=i i=1 i=1 i=1 i=1 i=1

Lemma 3.1. If ai (i = 1,2,..., p) denote p nonnegative real numbers, then

a + a + ••• + ap p

| "2 1 1 "p /o a1a2 ■ ■ ■ ap <-, (3.3)

where p > 1 denotes an integer.

A particular form of (3.3), namely,

p-1 (p - 1) ap ap .„ ..

ap 1a2 < —-a + —, for p = 1,2,3,.... (3.4)

Lemma 3.2. If H : R" — R" is a continuous function and satisfies the following conditions.

(1) H (x) is injective on R", that is, H(x) f H (y) for all x / y.

(2) ||H(x)|| —> go as ||x|| — go.

Then, H(x) is homeomorphism of R".

Theorem 3.3. System (2.3) exists a unique equilibrium x* under the assumptions (H1)-(H3) if the following condition is also satisfied:

-in > |*to

JJ=1 W' J 0

(H6) + £?=1 Zjt J0° K(s)ds.

Proof. Defining a map H(x) = (hi(x)rh2(x)r...rhn(x))1 e C0(Rn,Rn),where

n n n ft

hi(x) = -äi(xi) + ^bijfj (x-j) + ^Cijgj (xj) + ^dj Kij(t - s)hj (xj)ds + Ii, (3.5) j=1 j=i j=i

the map H is a homeomorphism on Rn if it is injective on Rn and satisfies ||H(x)|| ^ to as ||x^ ^ to.

In the following, we will prove that H (x) is a homeomorphism. Firstly, we claim that H(x) is an injective map on Rn. Otherwise, there exist xT,yT e Rn, and xT / yT such that H(x) = H(y), then

ai(xi) - ai(yi) = £ bij[fj{xj) - fj{Vj)] + £ cij[gj{xj) - gjj j=1 j=1

J]dij Kij (t - s) [hj(xj) - hj(yj)]ds.

j=1 -°

It follows from (H1)-(H3) that

i\xi - yi| < ^|bij\Lf\xj - yj\ \cij\Lg\xj - yj\ j=1 j=1

^|dj \Lh\xj - yj | K(t - s)ds.

j=1 -°

Therefore,

Yp^iai\xi - yi|

j^iai I xi yi

< Xbij\Lf \xi-yi\p-1\xj-yj\ + XYp^i\cij\Lg\xi-yi\p-1\xj-yj\ i=1 j=1 i=1 j=1

\Lh\xi - yi\v'l\xj - yj| K(t - s)ds i=1 j=1 -°

n n /p-1

< X XM X \b'j r'LfPßU,\x - v\ + Ibijr^^'lxj - '

j ^ j rt y M ^rvl ^ j

i=1 j=1 \'=1 n n /p-1 i=1 j=1 \l=1

n n ft p-1

n n ft p-1

X|> K(t - s)dsZ\dj f'^Lf^' X - y\

i=1 j=1 '=1

XXM K(t- s)ds\dij\pêp,i'LhPnp,i'\Xj -y,\p i=1 '=1 J

ni n />œ

= Xpi ( pai - <Pi + Vi + X'

i=1 \ j=1 J(

j=l J0

K(s)ds ) \ xi - yi|

From (H6), it leads to a contradiction with our assumption. Therefore, H(x) is an injective map on Rn.

To demonstrate the property ||H(x)|| ^ œ as ||x|| ^ œ, we have

Xsgn(Xi)ppi(hi(x) - hi(0))\xi\p i=1

= Xsgn(xi)ppi\xi\p 1 ( -ai(xi) + Xbijfj(xj) + Xcijgj(xj) i=1 \ j=1 j=1

X sgn(xi)ppi|xi\p-1Xdij Kij(t - s)hj(x,)ds i=1 j=1 -'-œ

n n / n />œ \

< -Xpaipi\xi\p + XPi \ pai - fri + Vi + XZji\ K(s)ds j \xi\p i=1 i=1 \ ,=1 J0 y

ni n />œ

= -XH-i{ fr - Vi - XZji\ K(s)ds ) \xi\p i=0 \ j=1 J0

< -wHxfp,

where w = min1<i<n{pi(fri - vi -Y!j=1 j Jo°° K(s)ds)}. Then, we have

œnxfp < \h,(x) - hi(0)\\xi\p-1. (3.10)

Using the Holder inequality, we obtain

1-1/p / n \!/P

xfp 1 Xx>n (X\h>x - h«(0)iO , (311)

which leads to

1 / \t=i

||x||p < \\H(x)\\p + ||H(0)||p). (3.12)

From (3.12), we see that ||H(x)|| ^ œ as ||x|| ^ œ. Thus, the map H(x) is a homeomorphism on R" under the sufficient condition (H6), and hence it has a unique fixed point x*. This fixed point is the unique solution of the system (2.3). The proof is now complete. □

To establish some sufficient conditions ensuring the pth moment exponential stability of the equilibrium point of x* of system (2.3), we transform x* to the origin by using the transformation yi(t) = xi(t) - x* for i e A. Then system (2.3) can be rewritten as the following form:

dyi(t) =

-ai{vi(t)) + X bijfi(yj (t)) + X cijgj( - Tj(t)) j=i j=i

Kii(t - s)h Kyj(s))ds

(3.13)

+ ^da( t,yi(t),yi(t - Ti(t))dwi(t), t = tk,

yi(tk) = pik(y(tk)), k e Z+, i e A,

äi(yi(t)) = ai(yi(t) + x*) - ai(x*), f (y,(t)) = /¿(y](t) + ] - fj(],

gj(yj(t - tj(t))) = - tj(t)) + ] - gj(xj),

hj (yj (s)) = hj(yj(s) + xj) - h^xj), (3.14)

ëijityj (t),y^t - Tj(t))) = Oij^yj (t)+ x*,y^t - Tj(t)) + ] - Oij(t,x*,j, pik(y(tk)) = Pik(y(tk) + x*) - Pik(x*).

In order to obtain our results, the following assumptions are necessary.

(H7) When m > 1, Oi> % + — Zji Jo°° K(s)ds for any i e A.

When 0 <m < 1, ®i> (%i/m) + — 0ji J0°° K(s)ds for any i e A.

(H8) There exist constant X e (0,p°) and a e [0,1), pk = maxieA{1^"=1 (^-/^i)plki} <

ea(tk-tk-i) sUch that

m n <•<»

X< Oi--l-eXT - Y Zji] K(s)eXsds

Hi j=\ J o

/"TO /"TO

K(s)eXssds < K(s)e^°sds.

(3.15)

Theorem 3.4. Assume that (H1)-(H5) and (H7)-(H8) hold, then system (2.3) exists a unique equilibrium x*, and the equilibrium point is pth moment exponentially stable.

Proof. From (H7), if 0 <n < 1, then

% n fTO m- n fTO n fTO

<i> O-> — + Y ZjA K(s)ds > — + Y iji] K(s)ds >fi + Y iji\ K(s)ds, (3.16)

H- j=1 JO Hi j=1 JO j=i Jo

if ni ^ 1,

n /"TO n /"TO

<i> Oi> % + ^Zji\ K(s)ds >fi + ^Zji\ K(s)ds, (3.17)

j=1 Jo j=1 Jo

for any i e A.

Then regardless of cases, (H6) and

% n *to

®i> — + Y iji\ K(s)ds (3.18)

ni j=1 JO

are satisfied. Therefore, system (2.3) exists a unique equilibrium point x*.

Now, we define

l(t,y(t)) = e1i^l\yl(t)\p = ut(i), U(t,y(t))=^ Ui(t).

(3.19)

For t / tk, k e Z+, we obtain

Ut + 1 trace [a T(t,yi(t),yi(t - n(t))) Uym(t,y)ö (t,yl(t),yl(t - Ti(t)))]

n / n n \

+Xuvi (-Hym+xbijJj(yj(t))+xcnzj - Tjm ) i=1 \ j=1 j=1 /

+ XTUyidin Kij(t - s)hj (yj(s))ds i=1 j=1

< X( 1 - pai + 1 ei )ui(t)

+ peXt{xZ V\bij\Lf\yi(t)\p-1\yj (t)\

I i=1 j=1

i=1 j=1

\Cjj\Lg \yi (t)\p-1\yj(t - Tj (t)) \ + ¡i\yi(t)\p-2\yi(t - Ti(t))\2

n n /"TO

yi\dl]\Lh\yi(t)\p-1 K(s)\yj(t - s)\ds I i=1 j=1 0 j

< X ( 1 - pai

p(p - 1)

ei ui(t)

p(p - 1)

ext^Myi(t)\p-2\yi(t - Ti(t))\2

+ eXti ±J X ^r'Lf^'y (t)\p + \btj\pap^0p"\yj (t)\p i=1 j=1 \i=1

n n /p-1

+ eXtZMBcj\mi'Lfli'\ymp + \cij\PJF,i'Lj j i=1 j=1 \i=1

n n fca /p-1 \

+ e^XXM K(s)(^j\dij\pki'Lfnii'\yl (t)\p + dj^'Lf^'y (t - s)\p)ds i=1 j=1 Jo \i=1 /

Z|V\bij\r^Lf11' + Ic^'Lf6'-" + j^L*™" f K(s)ds

Z (x - p»i + PpjO + h+¿t ¡bj'\(t)

i=i\ 2 2 j=1 ri J

j=ir i

n n .. ATO

+ V V \dj' fvt'Lf^' K(s)eXsuj(t - s)ds i=i j=i Hi ' Jo

+ Ze^ (^jif^ + (p - 1)li\ui(t - Ti(t))

i=i \j=1/

n n n /"TO

Y((l - Oi)Ui(t) +%eXTi(Vui(t - Ti(t))) + Yt y]ZiA K(s)eXsUj (t - s)ds. i=1 X 7 i=1 i=1 Jo

while Lemma 3.1 is used in the second inequality. Let

(3.20)

V(t,y(t)) = U(t,y(t)) + £WieXTi f 1 ,u;(S-)1( ds

i=1 Jt-Ti(t) 1 - TiWi(s))

n n /"TO ft

+ Z S K(s)eXs\ uj (r)drds,

i=1 i=1 J0 Jt-s

(3.21)

for t > 0, where fi(s) = s - Ti(s), applying Ito formula to (3.21), we can get

LV = LU + Z t-TU§W) - ^(t - T (0)'

n n /"to

EX' K(s)eXs(uj(t) - uj(t - s))ds (322)

i=1 j=1 o

n ( m n fTO

< -^l O - X - -eXT-Ztji\ K(s)eXsdAul (t). i=1 I ni j=1 Jo j

From (H8), we have

LV < 0. (3 . 23)

On the other hand, we have

EU(tk,y(tk)) = Eextk^\yi(tk )\p < Ee^^fpkMj (tk)\P < PkEU (tk,y (tk)), (3.24)

¿=1 ¿=1 j=i

EV(tk,y(tk)) = EU(tk,y(tk)) + e(WielTi C ufs) ds

Jh-Ti(h) 1- TilW; L(s))

¿=1 Jtk-ndk) 1 kfi(yk (s))

(3.25)

Inn fca ftk \

+ K(s)eXs\ u, (r )drds)

\i=1 j=1 Jtk-s J

< pkEV(tk,y(tk))

for t e [tkk 1,tk), k e Z+, by (3.23), (3.25), and (H8), we have

EU(t,y(t)) < EV (t,y(t)) < EV (tk- 1,y(tk-1)) < pk- 1EV (t^ vy(t~k-1))

< P0P1 •••pk- 1EV(to,y(to)) < eat1 ■ ■ ■ ea(ikk 1 -tkk2)EV(to,y(to)) < eatEV(0,y(0)),

(3.26)

where p0 = 1.

On the other hand, we observe that

V(0,y(0)) < ¿^(0)1' + jt^ f 1 U;(s)1(), ds 1=1 i=1 J-T 1 1(s))

n n /-to />0

XX&i K(s)eXs uj(r )drds (327)

i=1 j=1 •'0 j-s \ • J

n f n fTO ^ — \ <XM 1 + XZji\ K(s)eXssds + ^-eX-i sup |yi(s)|

i=1 \ ,=1 J0 Hi I -00<s<0

Zu^i i 1 T ZjZji Ksds ^ —e

i=1 \ j=1 J0 ni J

It follows that

E( £\y>(t)\p) < Me-(X-a)t sup X\y(s)\p, (3.28)

i=1 / -to<s<0 i=1

for t > 0, where

1 < M = £ ( 1 + maxi K(s)eXssds + —eX-i L ) < to, (3.29)

' L j=1 j° n J /

1 + maxi Vf« f K(s)eXssds + — "X-i

A ie^ 1 ^J 0 " ni

Mathematical Problems in Engineering which means that

E\\x(t) - x*f < - x*||pMe-(X-a)t for t > to. (3.30)

Therefore, the equilibrium point of system (2.3) is pth moment exponentially stable. □

Corollary 3.5. If p > 2, under the assumptions (H1)-(H5), system (2.3) exists a unique equilibrium point x*, and x* is pth moment exponentially stable if the following two conditions are satisfied: (H9)

n / /"TO \ n

Pa - (p - H) X(¡bij\Lf + \cij\Lg + \dtj|lH X(s)ds) - £\bn\Lf j=1 V J0 / j=i

(3.31)

p(p - 1)

ji\' +.........".....+

n / /"TO

> X(\cji\Lg + \dji\Lh}0 K(s)ds) + + ei).

(H10) There exist constant 1 e (0,^0) and a e [0,1), pk = maxi eA[1^n=1(^j /¡i)p1)i) -£a(tk-tk-1) such that

n , r~ \ n - p(p - 1)

n / /"to \ n

\<püi - (p- 1) £(\bij\Lf + \Cij\Lg + \dij¡LU K(s)ds) - X\bji\Lf - ei

j=l\ J0 ' j=1

(p - 2)2p - 1 k - (t\Cji\Lf + (p - 1)lXlT - ±\dji\Lh Cx(s)eXsds 2 \j=i / ,=1 h

(3.32)

Proof. In Theorem 3.4, let airij = ¡3lrij = jlrij = Sirij = h^ = n,lrij = 1/p, ¡i = 1 for all i,j,l e A. The result is obtained directly. □

Remark 3.6. In Corollary 3.5, if p = 2, the conditions (H9) and (H10) are less weak than the following conditions:

/ n \ / n \ I n fTO \

2min ai - maxi "Y\bij\Lf j - max i YClLf J - maxi "VldiALhl K(s)ds J

feA feA y" ' J feA y" ' J feA y" ' Jo J

n n n / /"TO \

> Xmax (\bji\Lf)+Xmax (\'L)+Xmax(*(s)dsj + maxk (3.33)

j=i j=i j=i n

+ Y max ei.

r~f ieA

(H10) There exist (

ea(tk-tk-i) such that

(H10) There exist constant X e (0,p0) and a e [0,1), pk = max [1,2j=1(pj/pdpk} <

ieA j j

/ n \ / n \ / n

X< 2minai - maxi V\bi^Lf ) - maxi VIc^U ) - maxi V||I K(s)ds

ieA ieA \j=1 J ieA \j=i J ieA \j=i j ^0

- V max I bjA Lf - max ei - ( V max I Cji I Lg + max li ) e

rr1 i eA ieA \ rr1 ieA 1 ieA /

j=1 i i j=1 i i

n /"TO

Vmax |dj^LM K(s)eXsds, ,=1 1 e A J0

while (H9) and (H10) were asked in Theorem 3.2 in [18].

(3.34)

Corollary 3.7. Under the assumptions (H2) and (H8), system (2.1) exists a unique equilibrium point x*, and x* is globally exponentially stable if 0 <n < 1, and the following condition is also satisfied:

pai - (p - (Lf\blj\pai'/(p-1) + Lg\cj\pjp-i)) j=1

(3.35)

- \bjr-a^>n cr-1^-

j=1 ri Hi j=1 ri

Proof. In Theorem 3.4, when p > 1, choosing aliij = aij/(p - 1), jliij = Jij/(p - 1) for l = 1,2,...,p - 1, and dij = 0, ¡5liij = Sliij = 1/p for all l e A, then the result is obtained. □

Remark 3.8. If 0 <n < 1, l{ = Lg = Li, it follows from (3.35) that

n n u n r

pai - (P - 1)n(Lj\bljP>/(p-1) + Lj\Cijr/(p-V)) - i~~\bji\pPl~a'iiLi > ijc^*U, j=1 j=1 ri j=1 ri

(3.36)

this is less conservative than the following inequality:

mAn [p* - (p - 1)ii (Lj\bjP'/(P-1) + L1\c^^) - t^b^U} ieA [ j=1 j=1ri j

> mJi^j^LX,

ie-A \ Pi ^ j

(3.37)

while (3.37) was required in Theorem 3.1 in [5] and in the only theorem in [8].

When p = 2, aij = j/ = 1/2, ¡i, = 1, (3.37) is equal to

n n I I n

min -< 2ai -J] (Lj\bij\ + Lj\dj\) -2\bji\Li\ > ma^\cn\Li L (3.38) I j=1 j=1 j j

while (3.38) was required in Theorem 3.2 in [19].

Similar to Corollaries 3.5 and 3.7, the following results are directly gained from Theorem 3.4.

Corollary 3.9. Under the assumptions (H2), (H4), and (H8), system (2.2) exists a unique equilibrium point x*, and x* is pth moment exponentially stable if 0 < n < 1, and the following condition is satisfied:

pai - (p - 1)£(\bij\L

1 / ¡j

j=1 ¡i

p(p- ^ (p- ^ (p- 2).

(3.39)

ir.j \cji\Lg + (p - 1)k

Especially, if p = 2, then the equilibrium x* is exponentially stable in mean square if the following condition is also satisfied:

2ai - £ (\b/, j^ + \c//12^/ - \b,raL j=1 j=1 ¡i

. (3.40)

1 (¡¡j 2-2S'i+h)-

Remark 3.10. If 0 <n < 1, it follows from (3.39) that

pa. - (p - (|b„ L ♦ |c | Lf) - ^jlf - ^ e, - fclfc2.,,

j=1 j=1 ¡i

(3.41)

j£\c/i\Lf + (p - 1)l,

¡i j=1

This is less conservative than the following inequality:

(3.42)

mini pa - (p - 1)£ (+ jg) - - i{p -1}2( p - 2 (ei + li)

I j=1 j=1 1i j=1

n I ^ f II n II

- Z j - 0 > mg{ jML> + jp - !) l,J,

while (3.42) was required in Theorem 3.3 in [26]. Remark 3.11. If 0 <n < 1, it follows from (3.40) that

2fli - f (\bj j ^ + \cj ) - jjf ^ - e>

>I f jj2-2j+^ 1 j=1

this is less conservative than the following inequality:

minLi - f(\btjj^ + \ctjj26-) - fH\b}ifj{2-2j - ^^^e\ 1<i<n ^ ptv " ' " ' / pt Ii ' i Ii I

(3.43)

{ I f \C\ \ 2-2JH Lg 2-2j + max1<i<n{li} l l

{lijri ji i Ii l)

(3.44)

while (3.44) was required in Theoerm 3.3 in [27].

4. Illustrative Example

In this section, we will give an example to show that the conditions given in the previous sections are less weak than those given in some of the earlier literatures, such as [18]. Consider the following stochastic neural networks with mixed time delays

2 2 2 dxi(t) = I -aiXi(t) + Zbijfj(xj(0) + XCi'gAxAt - T'(t))) + Xdij \ j=1 j=1 j=1

^ (4.1)

• Kij(t - s)hj(xj(s))ds + Ii ldt + Oi(t/Xi(t)/Xi(t - Ti(t)))dwi(t),

J -co /

where t /tk, i e A = (1,2}, tk = k, f (s) = gj(s) = f(s) = (1/2)[|x + 1| - |x - 1|],and Kj(s) = e-s.

(fli)2x1 = ( ¡3 )' ^^2x2 = (q.4 0.3) , (Cij) 2x2 = (0,2 Q.4/

,d , = /0.1 0.2N ( ) = /0.94\ ) = /0.1(x1(t - n(t)) -1.2A

2x2 = yo.2 0.3J, (ii)2xi = ^ 2.5 ' (ai)2x1 = \0.1(x2(t - T2(t)) - 1.4) J'

In the following, we introduce the following nonlinear impulsive controllers:

xi(tk) = 0.05sin(xi(t-) - 1.2) - 0.02x2(tk) + 1.48, k e Z+, x2(tk)= - 0.03x1(tk)+ 0.04cos(x2(t-) - 1.4) + 1.72, k e Z+

In this case, we have if = Lg = Lf = 1, K(s) = e-s, ey = 0, lj = 0.01 for j = 1,2, and po = 0.9. For p = 2, we can compute that

2fl1 - £ \b1j\Lf + |c1j|Lg + |d1j|Lf K(s)ds - £|bj 11if = 1.8

j=1 L JQ j=1

2 r /"to

X |Lg + |d711Ll\ K(s)ds

+11 + e1 = 0.61,

2 /«to 2

2fl2 - £ \b2j\Lf + \C2j \Lg + \d2j\Lf\ K(s)ds - £\bj2\Lf = 3.8

j=1 0 j=1

2 r /"TO

X jLg + |dj21Ll\ K(s)ds

= / 0.2 0.08X Pk \0.12 0.16/.

+ l2 + e2 = 1.21,

Choosing fii = 1 for i = 1,2, we have pk = maxieA P^} = 1 and ^ = 0-6 and a =

0.5, then

2 T fTO 1 2

0.6 = 1< 2fli |biy|Lf + |ciy\Lg + \dij|LH K(s)ds - £- ei

/=1 L Jo ;=i

2 /"to

V|dji|lH K(s)eXsds = 1.05 - 0.31e012 = 0.7, j=i J0

2 T /*TO 1 2

0.6 = 1 < 2a2 - £ |b2jL + |c2;|Lg + |d2;|Lh K(s)ds - £|bj21L^ - e2

;=i L J0 =

XMLg + /1

X|c;2 |Lg +12

E|d;2|Lh

j=1 ^0

K(s)eds = 2.55 - 0.71e012 = 1.749

/"TO /"TO

K(s)e1ssds = 6.25 < K(s)e^sds = 10.

Thus, all conditions of Theorem 3.4 in this paper are satisfied; the equilibrium solution is exponentially stable in mean square. From above discussion, it is easy to see that

Xmf (|ML0 = ^

\ /2 \ /

j- mmasx^ ^|Lg)- max^

2min ai - maxi Vlb«ILf ) - max( Vlc/lLf )- ma4 Y!|dj/|Lh f K(s)ds

i e A i e A \ ^ J j / i e A \ ^ J j / i e A \ ■j=11 1 ' J 0

2 2 / /"to \ 2

< X mAx^ |Cji|Lf ^ + ^ max

/=1 ieA j=1 j eA v J° ' j eA j=1 j eA

which implies that the condition (H9) in [18] do not hold for this example. So our results are less weaker than some previous results.

5. Conclusion

In this paper, we investigate the pth moment exponential stability for stochastic neural networks with mixed delays under nonlinear impulsive effects. By means of Lyapunov functionals, stochastic analysis, and differential inequality technique, some sufficient conditions for the pth moment exponential stability of this system are derived. The results of this paper are new, and they supplement and improve some of the previously known results [5, 8,18,19, 26, 27]. Moreover, examples are given to illustrate the effectiveness of our results. Furthermore, the method given in this paper may be extended to study other neural networks, such as the model in [29] and stochastic Cohen-Grossberg neural networks in [30], and we can get improved results too.

Acknowledgments

The authors would like to thank the editor and the anonymous referees for their detailed comments and valuable suggestions which considerably improved the presentation of this paper. This work was supported in part by the National Natural Science Foundation of China under Grants no. 11101054, Hunan Provincial Natural Science Foundation of China under Grant no. 12jj4005, the Open Fund Project of Key Research Institute of Philosophies and Social Sciences in Hunan Universities under Grants no. 11FEFM11 and the Scientific Research Funds of Hunan Provincial Science and Technology Department of China under Grants no. 2012SK3096.

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