stability analysis of stochastic fuzzy cellular neural networks with delays

ARTICLE IN PRESS

0925-2312/$ - se

doi:10.1016/j.ne

$This resear

Project of Jian

University of A�CorrespondE-mail addr

(H. Zhao).

Neurocomputing 72 (2008) 436–444

www.elsevier.com/locate/neucom

Stability analysis of stochastic fuzzy cellular neural networkswith delays$

Ling Chena,b, Hongyong Zhaoa,�

aDepartment of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR ChinabDepartment of Basic Science, Jinling Institute of Technology, Nanjing 210001, PR China

Received 23 April 2007; received in revised form 16 November 2007; accepted 10 December 2007

Communicated by J. Cao

Available online 12 February 2008

Abstract

In this paper, stochastic fuzzy cellular neural networks with delays are considered. By constructing suitable Lyapunov functionals and

using stochastic analysis we give a family of sufficient conditions ensuring the almost sure exponential stability of the networks. These

results obtained are helpful to design stability of networks when stochastic noise is taken into consideration.

r 2008 Elsevier B.V. All rights reserved.

Keywords: Almost sure exponential stability; Stochastic; Lyapunov functional; Fuzzy cellular neural networks

1. Introduction

In the past few decades, neural networks such as cellularneural networks, Cohen–Grossberg neural networks,and Cohen–Grossberg-type bidirectional associativememory neural networks have drawn much attention, andmany important results have been reported, see [7,14,12,27,3,20,5,28,17,6,19,4,1,2,18,23,15,8] for some recentpublications. The another fundamental neural networks,Yang et al. in [25] have introduced a new class of neuralnetworks, namely fuzzy cellular neural networks (FCNNs).This class of networks integrate fuzzy logic into the structureof traditional cellular neural networks and maintain localconnectedness among cells. Unlike the previous cellularneural network structures, FCNNs have fuzzy logic betweentheir template input and/or output besides the sum ofproduct operation. Studies [22,24,26,9] have shown thepotential of FCNNs in image processing and patternrecognition. Such applications heavily depend on the

e front matter r 2008 Elsevier B.V. All rights reserved.

ucom.2007.12.005

ch was supported by the Grant of ‘‘Qing-Lan Engineering’’

gsu Province, and the Science Foundation of Nanjing

eronautics and Astronautics.

ing author. Tel.: +8625 84489977.

esses: [email protected], [email protected]

dynamical behaviors. Thus, the analysis of the dynamicalbehaviors such as stability is a necessary step for practicaldesign of FCNNs. Recently, many scientific and technicalworks have been joining the study fields with great interest,and various interesting results for FCNN models have beenobtained, see e.g. [23,15,8] and references therein. MostFCNN models proposed and discussed in existing literatureare deterministic. However, a real system is usually affectedby external perturbations which in many cases are of greatuncertainty and hence may be treated as random, as pointedout by Haykin [11] that in real nervous systems, the synaptictransmission is a noisy process brought on by randomfluctuations from the release of neurotransmitters and otherprobabilistic causes. Under the effect of the noise, thetrajectory of system becomes a stochastic process. There arevarious kinds stability concepts to describe limiting beha-viors of stochastic processes, see, for example [10]. Thealmost sure exponential stability is the most useful because itis closer to the real situation during computation than otherforms of convergence (see [21,13] for the detailed discus-sions). Therefore, it is of great significance to study thealmost sure exponential stability for stochastic FCNNmodels. To the best our knowledge, few authors investigatethe almost sure exponential stability for stochastic FCNNswith delays [29], which is still open.

www.elsevier.com/locate/neucom

dx.doi.org/10.1016/j.neucom.2007.12.005

mailto:[email protected],

mailto:[email protected]

mailto:[email protected]

ARTICLE IN PRESSL. Chen, H. Zhao / Neurocomputing 72 (2008) 436–444 437

Based on the above discussion, our objective in thispaper is to study stochastic FCNNs with delays, and give afamily of sufficient conditions ensuring the almost sureexponential stability by constructing suitable Lyapunovfunctionals and applying stochastic analysis. It is easy toapply these conditions to the real networks.

2. Preliminary

Rn and C½X ;Y � denote the n-dimensional Euclideanspace and a continuous mapping set from the topologicalspace X to the topological space Y, respectively. Especially,C9C½½�t; 0�;Rn�, where t40.

Consider the following stochastic FCNNs with delays:

dxiðtÞ ¼ �cixiðtÞ þPnj¼1

aijf jðxjðtÞÞ þPnj¼1

bijuj þ I i

"

þVnj¼1

aijf jðxjðt� tÞÞ þWnj¼1

bij f jðxjðt� tÞÞ

þVnj¼1

Tijuj þWnj¼1

Hijuj

#dt

þPnj¼1

sijðxjðtÞ;xjðt� tÞÞdojðtÞ; tX0;

xiðtÞ ¼ fiðtÞ; �tptp0;

8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:

(1)

where i ¼ 1; . . . ; n. aij ; bij ;Tij and Hij are elements of fuzzyfeedback MIN template, fuzzy feedback MAX template,fuzzy feedforward MIN template and fuzzy feedforwardMAX template, respectively. aij and bij are elements offeedback template and feedforward template.

Vand

Wdenote the fuzzy AND and fuzzy OR operation, respec-tively. xi, ui and I i denote state, input and bias of theith neuron, respectively. ci40 is the neuron firing rate. f jð�Þ

is the activation function. t represents transmissiondelay. sð�; �Þ ¼ ðsijð�; �ÞÞn�n is the diffusion coefficientmatrix and oð�Þ ¼ ðo1ð�Þ; . . . ;onð�ÞÞ

T is an n-dimensionalBrownian motion defined on a complete probabilityspace (O;F ;P) with a natural filtration fFtgtX0 (i.e.Ft ¼ sfoðsÞ : 0psptg). fiðtÞ is the initial functionwhere fiðtÞ 2 L2

F0ð½t; 0�;RnÞ, here L2

F0ð½t; 0�;RnÞ denotes

the family of all C-valued random processes xðsÞ such thatxðsÞ is F 0-measurable and

R 0�t EkxðsÞk2 dso1. Assume,

throughout this paper, that f jð�Þ and sijð�; �Þ are locallyLipschitz continuous and satisfy the linear growth condi-tion as well. So it is known that Eq. (1) has a unique globalsolution on tX0, which is denoted by xðtÞ, wherexðtÞ ¼ ðx1ðtÞ; . . . ;xnðtÞÞ

T.Assume that the nonlinear functions f jð�Þ and sijð�; �Þ

satisfy the following condition:

(H1)
There exist positive constants Lj and Zij such that
jf jðuÞ � f jðvÞjpLjju� vj,

jsijðu; uÞ � sijðv; vÞj2pZijðju� vj2 þ ju� vj2Þ,

for any u; v; u; v 2 R, i; j ¼ 1; . . . ; n.

We first give the following lemmas that are useful inderiving our stability conditions.

Lemma 1 (Semi-martingale Convergence Theorem, Mao

[16]). Let AðtÞ and UðtÞ be two continuous adapted

increasing processes on tX0 with Að0Þ ¼ Uð0Þ ¼ 0, a.s.Let MðtÞ be a real-valued continuous local martingale with

Mð0Þ ¼ 0, a.s. Let z be a nonnegative F 0-measurable random

variable with Ezo1. Define

X ðtÞ ¼ zþ AðtÞ �UðtÞ þMðtÞ for tX0.

If X ðtÞ is nonnegative, then

limt!1

AðtÞo1n o

� limt!1

X ðtÞo1n o

\ limt!1

UðtÞo1n o

a:s:;

where B � D a.s. means PðB \DcÞ ¼ 0. In particular, if

limt!1AðtÞo1 a.s., then for almost all o 2 O

limt!1

X ðtÞo1 and limt!1

UðtÞo1,

that is both X ðtÞ and UðtÞ converge to finite random

variables.

Lemma 2 (Yang and Yang [23]). Suppose x ¼ ðx1; . . . ;xnÞT

and y ¼ ðy1; . . . ; ynÞT are two states of system (1), then

we have

(1)

n

j¼1

aij f jðxjÞ �

n

j¼1

aij f jðyjÞ

��pXn

j¼1

Ljjaijjjxj � yjj,

(2)

_nj¼1

bij f jðxjÞ �_nj¼1

bijf jðyjÞ

��pXn

j¼1

Ljjbijjjxj � yjj,

where Lj is given as (H1).

Lemma 3 (Chen and Liao [8]). Suppose x ¼ ðx1; . . . ;xnÞT

and y ¼ ðy1; . . . ; ynÞT are two states of system (1), then

we have

(1)

n

j¼1

aij f jðxjÞ �

n

j¼1

aijf jðyjÞ

��

p max1pjpn

fLjjaijjjxj � yjjg,

(2)

_nj¼1

bij f jðxjÞ �_nj¼1

bij f jðyjÞ

��

p max1pjpn

fLjjbijjjxj � yjjg,

where Lj is given as (H1).

ARTICLE IN PRESSL. Chen, H. Zhao / Neurocomputing 72 (2008) 436–444438

Throughout the paper, we suppose that

(H2)
There are a set of positive constants d1; . . . ; dn, suchthat
� dici þXn

j¼1

jajijdjLi þXn

j¼1

jajijdjLi þXn

j¼1

jbjijdjLio0,

i ¼ 1; . . . ; n.

(H3)
There are a set of positive constants p1; . . . ; pn, suchthat
� ci þXn

j¼1

jaijjLjp�1j pi þ max

1pjpnfLjjaijjp

�1j gpi

þ max1pjpn

fLjjbijjp�1j gpio0; i ¼ 1; . . . ; n.

(H4)
There are a set of positive constants q1; . . . ; qn, suchthat
� 2qici þXn

j¼1

jaijjqiLj þXn

j¼1

jajijqjLi þXn

j¼1

jaijjqiLj

þXn

j¼1

jajijqjLi þXn

j¼1

jbijjqiLj þXn

j¼1

jbjijqjLi

þ 2Xn

j¼1

Zjiqjo0; i ¼ 1; . . . ; n.

(H5)
There are a set of positive constants w1; . . . ;wn, suchthat
� 2ci þXn

j¼1

jaijjLj þXn

j¼1

jaijjLjw�1j wi þ max

1pjpnfLjjaijjg

þ max1pjpn

fLjjaijjw�1j gwi þ max

1pjpnfLjjbijjg

þ max1pjpn

fLjjbijjw�1j gwi þ 2

Xn

j¼1

Zijw�1j wio0,

i ¼ 1; . . . ; n.

For any x ¼ ðx1; . . . ;xnÞT2 Rn, we define the vector

norm k � k1, k � k2, k � k1, k � k, respectively, by kxk1 ¼Pni¼1 dijxij, kxk2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPni¼1 qijxij

2q

, kxk1 ¼ maxifpijxijg,

kxk ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimaxifwijxij

2p

g.For any fðtÞ ¼ ðf1ðtÞ; . . . ;fnðtÞÞ

T2 L2

F0ð½�t; 0�;RnÞ,

we define kfk1 ¼ sup�tpsp0

kfðsÞk1; kfk2 ¼ sup�tpsp0

kfðsÞk2;

kfk1 ¼ sup�tpsp0

kfðsÞk1; kfk ¼ sup�tpsp0

kfðsÞk, where

kfðsÞk1 ¼Pni¼1

dijfiðsÞj; kfðsÞk2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPni¼1

qijfiðsÞj2

s; kfðsÞk1 ¼

maxifpijfiðsÞjg; kfðsÞk ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimax

ifwijfiðsÞj

2gq

.

3. Main results

For the deterministic system

dxiðtÞ ¼ �cixiðtÞ þPnj¼1

ðaijf jðxjðtÞÞ þ bijujÞ þ I i

"

þVnj¼1

aij f jðxjðt� tÞÞ þWnj¼1

bijf jðxjðt� tÞÞ

þVnj¼1

Tijuj þWnj¼1

Hijuj

#dt; tX0;

xiðtÞ ¼ fiðtÞ; �tptp0;

8>>>>>>>>>>>><>>>>>>>>>>>>:

(2)

we have the following results.

Theorem 1. If (H1) holds. Assume furthermore that one of

(H2)–(H5) with Zij ¼ 0 holds. Then system (2) has a unique

equilibrium point x� ¼ ðx�1; . . . ;x�nÞ

T.

Proof. The proof is similar to that of [20,28]. So we omitit here. &

In the paper, we assume that

(H6)
sijðx�j ;x�j Þ ¼ 0; i; j ¼ 1; . . . ; n.
Thus, system (1) admits an equilibrium point x� ¼ ðx�1; . . . ;x�nÞ

T. Let yiðtÞ ¼ xiðtÞ � x�i ;jiðtÞ ¼ fiðtÞ � x�i , yðtÞ ¼ ðy1ðtÞ;. . . ; ynðtÞÞ

T, jðtÞ ¼ ðj1ðtÞ; . . . ;jnðtÞÞT, then system (1)

becomes

dyiðtÞ ¼ �ciyiðtÞ þPnj¼1

aijðf jðyjðtÞ þ x�j Þ � f jðx�j ÞÞ

"

þVnj¼1

aij f jðyjðt� tÞ þ x�j Þ �Vnj¼1

aij f jðx�j Þ

þWnj¼1

bij f jðyjðt� tÞ þ x�j Þ �Vnj¼1

bij f jðx�j Þ

#dt

þPnj¼1

sijðyjðtÞ þ x�j ; yjðt� tÞ þ x�j ÞdojðtÞ; tX0;

yiðtÞ ¼ jiðtÞ; �tptp0:

8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:

(3)

Clearly, the equilibrium point x� of (1) is almost surelyexponentially stable if and only if the equilibrium point O

of system (3) is almost surely exponentially stable.Thus in the following, we only consider the almostsure exponential stability of the equilibrium point O forsystem (3).

Theorem 2. Suppose that (H1), (H2), and (H6) hold. Then

system (3) has an equilibrium point O which is almost surely

exponentially stable.


Proof. It follows from (H2) that there exists a sufficientlysmall constant 0olomin1pipn fcig such that

diðl� ciÞ þXn

j¼1

jajijdjLi þ eltXn

j¼1

jajijdjLi

þ eltXn

j¼1

jbjijdjLip0; i ¼ 1; . . . ; n. (4)

Taking V ðyðtÞ; tÞ ¼ eltPn

i¼1 dijyiðtÞj, and applying Ito’sformula to V ðyðtÞ; tÞ, we have

V ðyðtÞ; tÞpV ðyð0Þ; 0Þ þ

Z t

0

lelsXn

i¼1

dijyiðsÞjds

þ

Z t

0

elsXn

i¼1

di � cijyiðsÞj

"

þXn

j¼1

aij½f jðyjðsÞ þ x�j Þ � f jðx�j Þ�

��

þ

n

j¼1

aij f jðyjðs� tÞ þ x�j Þ �n

j¼1

aij f jðx�j Þ

��

þ_nj¼1

bij f jðyjðs� tÞ þ x�j Þ �n

j¼1

bij f jðx�j Þ

��#dsþMðvÞ,

where

MðvÞ ¼

Z t

0

elsXn

i¼1

disgnðyiðsÞÞXn

j¼1

sijðyjðsÞ þ x�j ; yjðs� tÞ þ x�j ÞdojðsÞ

By using Lemma 2, we obtain

V ðyðtÞ; tÞpkjk1 þZ t

0

lelsXn

i¼1

dijyiðsÞjds

þ

Z t

0

elsXn

i¼1

di �cijyiðsÞj þXn

j¼1

jaijjLjjyjðsÞj

"

þXn

j¼1

jaijjLjjyjðs� tÞj

þXn

j¼1

jbijjLjjyjðs� tÞj

#dsþMðvÞ

¼ kjk1 þZ t

0

lelsXn

i¼1

dijyiðsÞjds

þ

Z t

0

elsXn

i¼1

�dicijyiðsÞj þXn

j¼1

jajijLidjjyiðsÞj

"

þXn

j¼1

jajijLidjjyiðs� tÞj

þXn

j¼1

jbjijLidjjyiðs� tÞj

#dsþMðvÞ.

Note that

Z t

t�telsjyiðsÞjds ¼

Z t

�telsjyiðsÞjds

�

Z t

0

elðs�tÞjyiðs� tÞjds.

So

Z t

0

elsjyiðs� tÞjds ¼ eltZ t

�telsjyiðsÞjds

� eltZ t

t�telsjyiðsÞjds,

that is

Z t

0

elsjyiðs� tÞjdspeltZ t

�telsjyiðsÞjds.

Following from (4) we have

V ðyðtÞ; tÞpkjk1 þZ 0

�telselt

Xn

i¼1

Xn

j¼1

jajijLidjjyiðsÞjds

þ

Z 0

�telselt

Xn

i¼1

Xn

j¼1

jbjijLidjjyiðsÞjdsþMðvÞ.

(5)

It is obvious that the right-hand side of (5) is a non-negative semi-martingale. From Lemma 1, it can be easilyseen that

lim supt!1

V ðyðtÞ; tÞoþ1; PFa:s.

Since

V ðyðtÞ; tÞ ¼ eltXn

i¼1

dijyiðtÞj ¼ eltkyðtÞk1,

we obtain

lim supt!1

1

tlogðkyðtÞk1Þp� l; PFa:s.

According to [26,9], the equilibrium point O of (3) is almostsurely exponentially stable. This completes the proof. &

Theorem 3. Assume that (H1), (H3), and (H6) hold, then



Proof. From (H3), there exists a sufficiently small constant0olomin1pipn fcig such that

l� ci þXn

j¼1

jaijjLjjp�1j pi þ elt max

1pjpnfLjjaijjp

�1j gpi

þ elt max1pjpn

fLjjbijjp�1j gpip0; i ¼ 1; . . . ; n.


The Lyapunov functional is defined as V ðyðtÞ; tÞ ¼eltmax1pipn fpijyiðtÞjg. Suppose pkjykðtÞj ¼ max1pipn

fpijyiðtÞjg, where k 2 f1; . . . ; ng. Applying Ito’s formula toV ðyðtÞ; tÞ, we have

V ðyðtÞ; tÞ

pV ðyð0Þ; 0Þ þ

Z t

0

lelspkjykðsÞjds

þ

Z t

0

elspk � ckjykðsÞj

"

þXn

j¼1

akj ½f jðyjðsÞ þ x�j Þ � f jðx�j Þ�

��

þ

n

j¼1

akjf jðyjðs� tÞ þ x�j Þ �n

j¼1

akjf jðx�j Þ

��

þ_nj¼1

bkjf jðyjðs� tÞ þ x�j Þ �_nj¼1

bkjf jðx�j Þ

��#ds

þ

Z t

0

elspksgnðykðsÞÞXn

j¼1

skjðyjðsÞ þ x�j ; yjðs� tÞ þ x�j Þ dojðsÞ.

By using Lemma 3, we obtain

V ðyðtÞ; tÞ

pkjk1 þZ t

0

lelspkjykðsÞjds

þ

Z t

0

elspk �ckjykðsÞj þXn

j¼1

jakjjLjjyjðsÞj

"

þ max1pjpn

fLjjakjjp�1j gpkjykðs� tÞj

þ max1pjpn

fLjjbkjjp�1j gpkjykðs� tÞj

#ds

þ

Z t

0


j¼1


Similar to the discussion of Theorem 2, we have

V ðyðtÞ; tÞ

pkjk1 þZ 0

�telseltp2

k

� max1pjpn

fLjjakjjp�1j gjykðsÞjds

þ

Z 0

�telseltp2

k max1pjpn

fLjjbkjjp�1j gjykðsÞjds

þ

Z t

0


j¼1


From Lemma 1, we obtain

lim supt!1

1

tlogðkyðtÞk1Þp� l: PFa:s.

According to [26,9], the equilibrium point O of (3) is almostsurely exponentially stable. We complete the proof. &

Theorem 4. Suppose that (H1), (H4), and (H6) hold. Then



Proof. It follows from (H4) that there exists a sufficientlysmall constant 0olo2min1pipn fcig such that

lqi � 2qici þXn

j¼1

jaijjqiLj þXn

j¼1

jajijqjLi

þXn

j¼1

jaijjqiLj þ eltXn

j¼1

jajijqjLi

þXn

j¼1

jbijjqiLj þ eltXn

j¼1

jbjijqjLi þXn

j¼1

Zjiqj

þ eltXn

j¼1

Zjiqjp0; i ¼ 1; . . . ; n.

Taking V ðyðtÞ; tÞ ¼ eltPn

i¼1 qijyiðtÞj2, and applying Ito’s

formula to V ðyðtÞ; tÞ, we have


Z t

0

lelsXn

i¼1

qijyiðsÞj2 ds

þ

Z t

0

2elsXn

i¼1

qijyiðsÞj � cijyiðsÞj

"

þXn

j¼1

aij½f jðyjðsÞ þ x�j Þ � f jðx�j Þ�

��

þ

n

j¼1

aijf jðyjðs� tÞ þ x�j Þ �n

j¼1

aijf jðx�j Þ

��

þ_nj¼1

bijf jðyjðs� tÞ þ x�j Þ �_nj¼1

bijf jðx�j Þ

��#ds

þ

Z t

0

2elsXn

i¼1

qiyiðsÞXn

j¼1

sijðyjðsÞ

þ x�j ; yjðs� tÞ þ x�j ÞdojðsÞ

þ

Z t

0

elsXn

i¼1

qi

Xn

j¼1

jsijðyjðsÞ

þ x�j ; yjðs� tÞ þ x�j Þj2 ds.

By using Lemma 2 and inequality 2abpa2 þ b2, we obtain


0

lelsXn

i¼1

qijyiðsÞj2 ds

þ

Z t

0

elsXn

i¼1

qi �2cijyiðsÞj2 þ

Xn

j¼1

jaijjLjjyiðsÞj2

"

þXn

j¼1

jaijjLjjyjðsÞj2 þ

Xn

j¼1

jaijjLjjyiðsÞj2

þXn

j¼1

jaijjLjjyjðs� tÞj2 þXn

j¼1

jbijjLjjyiðsÞj2


þXn

j¼1

jbijjLjjyjðs� tÞj2#ds

þ

Z t

0

elsXn

i¼1

qi

Xn

j¼1

ZijðjyjðsÞj2 þ jyjðs� tÞj2Þds

þ

Z t

0

2elsXn

i¼1

qiyiðsÞXn

j¼1

sijðyjðsÞ

þ x�j ; yjðs� tÞ þ x�j ÞdojðsÞ.



�t

Xn

i¼1

qi

Xn

j¼1

elselt½ðjaijj

þ jbijjÞLj þ Zij �jyjðsÞj2 ds

þ

Z t

0

2elsXn

i¼1

qiyiðsÞXn

j¼1

sijðyjðsÞ



lim supt!1

1

tlogðkyðtÞk2Þp�

l2; PFa:s.

According to [26,9], the equilibrium point O of (3) is almostsurely exponentially stable. This completes the proof. &

Theorem 5. Assume that (H1), (H5), and (H6) hold, then



Proof. From (H5), there exists a sufficiently small constant0olo2min1pipn fcig such that

l� 2ci þXn

j¼1

jaijjLj þXn

j¼1

jaijjLjw�1j wi þ max

1pjpnfLjjaijjg

þ elt max1pjpn

fLjjaijjw�1j gwi þ max

1pjpnfLjjbijjg

þ elt max1pjpn

fLjjbijjw�1j gwi þ

Xn

j¼1

Zijw�1j wi

þ eltXn

j¼1

Zijw�1j wip0; i ¼ 1; . . . ; n.

The Lyapunov functional is defined as V ðyðtÞ; tÞ ¼

eltmax1pipnfwijyiðtÞj2g. Suppose wkjykðtÞj

2 ¼ max1pipn

fwijyiðtÞj2g, where k 2 f1; . . . ; ng. Applying Ito’s formula

to V ðyðtÞ; tÞ, we have


Z t

0

lelswkjykðsÞj2 ds

þ

Z t

0

2elswkjykðsÞj � ckjykðsÞj

"

þXn

j¼1

akjf jðyjðsÞ þ x�j Þ �Xn

j¼1

akjf jðx�j Þ

��

þ

n

j¼1

akjf jðyjðs� tÞ þ x�j Þ �n

j¼1

akjf jðx�j Þ

��

þ_nj¼1

bkjf jðyjðs� tÞ þ x�j Þ �_nj¼1

bkjf jðx�j Þ

��#ds

þ

Z t

0

2elswkykðsÞXn

j¼1

skjðyjðsÞ

þ x�j ; yjðs� tÞ þ x�j ÞdojðsÞ

þ

Z t

0

elswk

Xn

j¼1

s2kjðyjðsÞ þ x�j ; yjðs� tÞ þ x�j Þds.

By using inequality 2abpa2 þ b2 and Lemma 3, we obtain


0

lelswkjykðsÞj2 ds

þ

Z t

0

elswk �2ckjykðsÞj2 þ

Xn

j¼1

jakjjLjjykðsÞj2

"

þXn

j¼1

jakjjLjjyjðsÞj2 þ max

1pjpnfLjjakjjgjykðsÞj

2

þ max1pjpn

fLjjakjjw�1j gwkjykðs� tÞj2

þ max1pjpn

fLjjbkjjgjykðsÞj2

þ max1pjpn

fLjjbkjjw�1j gwkjykðs� tÞj2

#ds

þ

Z t

0

elswk

Xn

j¼1

ZkjðjyjðsÞj2 þ jyjðs� tÞj2Þds

þ

Z t

0

2elswkykðsÞXn

j¼1

skjðyjðsÞ




�telselt

� max1pjpn

fLjjakjjw�1j gw

2kjykðsÞj

2 ds

þ

Z 0

�telselt max

1pjpnfLjjbkjjw

�1j gw

2kjykðsÞj

2 ds

þ

Z 0

�telseltwk

Xn

j¼1

ZkjjyjðsÞj2 ds


þ

Z t

0

2elswkykðsÞXn

j¼1

skjðyjðsÞ



lim supt!1

1

tlogðkyðtÞkÞp�

l2; PFa:s.

According to [26,9], the equilibrium point O of (3) is almostsurely exponentially stable. We complete the proof. &

When di ¼ pi ¼ qi ¼ wi ¼ 1 ði ¼ 1; . . . ; nÞ in Theorems1–5, we can obtain the following result.

Corollary 1. System (3) has an equilibrium point O which is

almost surely exponentially stable if the nonlinear functions

f jð�Þ and sijð�; �Þ ði; j ¼ 1; . . . ; nÞ satisfy hypothesis (H1).Assume furthermore that one of the following conditions

holds:

(A1)

� ci þXn

j¼1

jajijLi þXn

j¼1

jajijLi

þXn

j¼1

jbjijLio0; i ¼ 1; . . . ; n.

(A2)

� ci þXn

j¼1

jaijjLj þ max1pjpn

fLjjaijjg

þ max1pjpn

fLjjbijjgo0; i ¼ 1; . . . ; n.

(A3)

� 2ci þXn

j¼1

jaijjLj þXn

j¼1

jajijLi þXn

j¼1

jaijjLj

þXn

j¼1

jajijLi þXn

j¼1

jbijjLj þXn

j¼1

jbjijLi

þ 2Xn

j¼1

Zjio0; i ¼ 1; . . . ; n.

(A4)

� ci þXn

j¼1

jaijjLj þ max1pjpn

fLjjaijjg

þ max1pjpn

fLjjbijjg þXn

j¼1

Zijo0; i ¼ 1; . . . ; n.

mples
4. Exa
Example 1. Consider the following stochastic FCNNs
with delays:
dx1ðtÞ ¼ �c1x1ðtÞ þP2j¼1

ða1j f jðxjðtÞÞ þ b1jujÞ þ I1

"

þV2j¼1

a1j f jðxjðt� tÞÞ þW2j¼1

b1j f jðxjðt� tÞÞ

þV2j¼1

T1juj þW2j¼1

H1juj

#dt

þP2j¼1

s1jðxjðtÞ;xjðt� tÞÞdojðtÞ

dx2ðtÞ ¼ �c2x2ðtÞ þP2j¼1

ða2j f jðxjðtÞÞ þ b2jujÞ þ I2

"

þV2j¼1

a2j f jðxjðt� tÞÞ þW2j¼1

b2j f jðxjðt� tÞÞ

þV2j¼1

T2juj þW2j¼1

H2juj

#dt

þP2j¼1

s2jðxjðtÞ;xjðt� tÞÞdojðtÞ:

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

(6)

Let f 1ðx1Þ ¼12 ðjx1 þ 1j þ jx1 � 1jÞ, f 2ðx2Þ ¼ x2, sijðxjðtÞ;

xjðt� tÞÞ ¼ffiffi5p

10xjðtÞ; i; j ¼ 1; 2. Obviously, L1 ¼ L2 ¼ 1,

Zij ¼120; i; j ¼ 1; 2. Choose c1 ¼ 3; c2 ¼ 2; a11 ¼ a12 ¼

12; a21 ¼

16; a22 ¼

12; aij ¼

14ði; j ¼ 1; 2Þ;b11 ¼ b12 ¼ b22 ¼

14;

b21 ¼13: It can easily see that condition (A3) is satisfied.

Thus, system (6) has an equilibrium point which is almostsurely exponentially stable. However,

�c2 þX2j¼1

jaj2jL2 þX2j¼1

jaj2jL2 þX2j¼1

jbj2jL2 ¼ 0,

that is condition (A1) does not hold.On the other hand, we choose f 1ðx1Þ ¼ x1, f 2ðx2Þ ¼

e�x2�1, s11ðx1ðtÞ; x1ðt� tÞÞ ¼2ffiffiffi2p

x1ðtÞ, s21ðx1ðtÞ;x1ðt� tÞÞ¼

ffiffiffi7p

x1 ðtÞ, s12 ðx2ðtÞ; x2 ðt � tÞÞ ¼ s22 ðx2ðtÞ; x2 ðt � tÞÞ ¼x2ðtÞ. Obviously, L1 ¼ L2 ¼ 1, Z11 ¼ 8; Z21 ¼ 7; Z12 ¼Z22 ¼ 1. Taking c1 ¼ 36; c2 ¼ 26; a11 ¼ a21 ¼ 5; a12 ¼ 8;a22 ¼ 6; a11 ¼ 3; a12 ¼ 1

4; a21 ¼ 9; a22 ¼ 2;b11 ¼ 2;b12¼

14;

b21 ¼ 10;b22 ¼ 1. By simple calculation, we obtain

� 2c1 þX2j¼1

ja1jjLj þXn

j¼1

jaj1jL1 þX2j¼1

ja1jjLj

þX2j¼1

jaj1jL1 þX2j¼1

jb1jjLj þX2j¼1

jbj1jL1

þ 2X2j¼1

Zj1 ¼ 10:540.


This shows that condition (A3) does not hold.However, it can be easily verified that condition (A1) issatisfied.

Example 2. For system (6), choose f 1ðx1Þ ¼ sin x1,f 2ðx2Þ ¼ x2, sijðxjðtÞ; xjðt� tÞÞ ¼

ffiffiffiffi10p

10xjðtÞ; i; j ¼ 1; 2. Ob-

viously, L1 ¼ L2 ¼ 1, Zij ¼110; i; j ¼ 1; 2. Let c1 ¼ 1; c2 ¼

20; a11 ¼ a21 ¼ a22 ¼120; a12 ¼

110; aij ¼

14ði; j ¼ 1; 2Þ; b11 ¼

b12 ¼ b22 ¼14;b21 ¼

13: It can easily see that condition (A4)

is satisfied. Thus, system (6) has an equilibrium point whichis almost surely exponentially stable. However,

� 2c1 þX2j¼1

ja1jjLj þX2j¼1

jaj1jL1 þX2j¼1

ja1jjLj

þX2j¼1

jaj1jL1 þX2j¼1

jb1jjLj þX2j¼1

jbj1jL1

þ 2X2j¼1

Zj1 ¼44

6040,

that is condition (A3) does not hold.On the other hand, we choose f 1ðx1Þ ¼

12ðjx1 þ 1j

þjx1 � 1jÞ, f 2 ðx2 Þ ¼ x2, s11 ðx1 ðtÞ; x1 ðt � tÞÞ ¼ x1 ðtÞ;s21ðx1ðtÞ;x1ðt � tÞÞ ¼

ffiffiffi3p

x1ðtÞ;s22ðx2ðtÞ; x2ðt � tÞÞ ¼ x2ðtÞ;s12ðx2ðtÞ;x2ðt� tÞÞ ¼

ffiffi2p

2x2ðtÞ. Obviously, L1 ¼ L2 ¼ 1,

Z11 ¼ Z22 ¼ 1; Z21 ¼ 3; Z12 ¼12. Taking c1 ¼ 22; c2 ¼ 26; a11

¼ 4; a21¼1; a22 ¼ 2; a12 ¼14; a11 ¼ 3; a12 ¼ 1

4; a21 ¼ 9; a22 ¼

2;b11 ¼ 2;b12 ¼14; b21 ¼ 5;b22 ¼ 1. By simple calculation,

we obtain

� c2 þX2j¼1

ja2jjLj þ max1pjp2

fLjja2jjg

þ max1pjp2

fLjjb2jjg þX2j¼1

Z2j ¼ 440.

This shows that condition (A4) does not hold. However, itcan be easily verified that condition (A3) is satisfied.

5. Conclusions

In this paper, stochastic FCNNs model with delays hasbeen investigated. We have derived a family of sufficientconditions ensuring the almost sure exponential stabilityfor the networks by constructing suitable Lyapunovfunctionals and applying stochastic analysis. Moreover,two examples are given to demonstrate the advantages ofour method.

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[29] H. Zhao, N. Ding, L. Chen, Almost sure exponential stability of

stochastic fuzzy cellular neural networks with delays, Chaos, Solitons

and Fractals, in press, doi:10.1016/j.chaos.2007.09.044.

Ling Chen is now pursuing the M.S. degree in the

Department of Mathematics at Nanjing Univer-

sity of Aeronautics and Astronautics, Nanjing,

China.

From August 2000 until now, she is with the

Department of Basic Science at Jinling Institute

of Technology, Nanjing, China. Her research

interests include neural networks and stability

theory.

Hongyong Zhao received the Ph.D. from Sichuan

University, Chendu, China, and the Post-Doctoral

Fellow in the Department of Mathematics at

Nanjin University, Nanjin, China.

He was with Department of Mathematics at

Nanjing University of Aeronautics and Astronau-

tics, Nanjin, China. He is currently a Professor of

Nanjin University of Aeronautics and Astronau-

tics, Nanjin, China. He is also the author or

coauthor of more than 60 journal papers. His

research interests include nonlinear dynamic systems, neural networks,

control theory.

http://dx.doi.org/10.1016/j.chaos.2007.09.044

stability analysis of stochastic fuzzy cellular neural networks with delays

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