Applied Mathematics and Computation 206 (2008) 818–824

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

Exponential stability of stochastic reaction–diffusion Cohen–Grossbergneural networks with delays

Li Wan a,*, Qinghua Zhou b

a Department of Mathematics and Physics, Wuhan University of Science and Engineering, Wuhan Fangzhi Road No. 1, Wuhan 430073, Chinab Department of Mathematics, Zhaoqing University, Zhaoqing 526061, China

a r t i c l e i n f o a b s t r a c t

Keywords:Cohen–Grossberg neural networksReaction–diffusionMean value exponential stabilityTime-varying delays

0096-3003/$ - see front matter � 2008 Elsevier Incdoi:10.1016/j.amc.2008.10.002

* Corresponding author.E-mail addresses: wanlinju@yahoo.com.cn (L. Wa

In this paper, stochastic Cohen–Grossberg neural network with time-varying delays andreaction–diffusion terms is considered. The sufficient conditions are obtained to guaranteethe mean value exponential stability of an equilibrium solution.

� 2008 Elsevier Inc. All rights reserved.

1. Introduction

The stability analysis problem for Cohen–Grossberg neural network (CGNN) has gained much research attention, and alarge amount of results related to this problem have been published, see e.g. [2,4,5,7,8,17,21,23,28,29,33,38]. However,strictly speaking, diffusion effects cannot be avoided in the neural networks when electrons are moving in asymmetric elec-tromagnetic fields. Therefore we must consider that the activations vary in space as well as in time. In [9–11,15,18,19,26,27,31,35,36,39], the authors have considered the stability of neural networks with reaction–diffusion terms.

On the other hand, a real system is usually affected by external perturbations which in many cases are of great uncer-tainty and hence may be treated as random, as pointed out by Haykin [3] that in real nervous systems, the synaptic trans-mission is a noisy process brought on by random fluctuations from the release of neurotransmitters and other probabilisticcauses. It has also been known that a neural network could be stabilized or destabilized by certain stochastic inputs [1].Hence, the stability analysis problem for stochastic neural network becomes increasingly significant, and some results onstability have been derived, see e.g. [13,14,16,20,22,24,25,30,32,34,37].

To the best of our knowledge, however, few authors have considered the exponential stability for stochastic CGNN withdelays and diffusion terms. Motivated by the above discussions, our objective in this paper is to present the sufficient con-ditions ensuring the mean value exponential stability of stochastic CGNN.

2. Model description and assumptions

Consider the following stochastic CGNN with delays and diffusion terms

dyiðt; xÞ ¼Xm

k¼1

o

oxkDik

oyiðt; xÞoxk

� �dt � diðyiðt; xÞÞ ciðyiðt; xÞÞ �

Xn

j¼1

aijfjðyjðt; xÞÞ �Xn

j¼1

bijgjðyjðt � sijðtÞ; xÞÞ þ Ji

!dt

þXn

j¼1

rijðyiðt; xÞÞdwjðtÞ; x 2 X;oyi

on:¼ oyi

ox1; . . . ;

oyi

oxm

� �T

¼ 0; x 2 oX; ð2:1Þ

yiðs; xÞ ¼ niðs; xÞ;�s 6 s 6 0; x 2 X;

. All rights reserved.

n), zqhmath@yahoo.com.cn (Q. Zhou).

L. Wan, Q. Zhou / Applied Mathematics and Computation 206 (2008) 818–824 819

for 1 6 i 6 n and t P 0. In the above model, n P 2 is the number of neurons in the network; xi is space variable; yiðt; xÞ is thestate variable of the ith neuron at time t and in space x; fjðyjðt; xÞÞ and gjðyjðt; xÞÞ denote the activation functions of the jth unitat time t and in space x; smooth function Dik ¼ Dikðt; x; yÞP 0 is diffusion operator; diðyiðt; xÞÞ presents an amplification func-tion; ciðyiðt; xÞÞ is an appropriately behavior function; aij and bij denote the connection strengths of the jth unit on the ith unit,respectively; sijðtÞ corresponds to the transmission delay and satisfies 0 6 sijðtÞ 6 sij 6 sðsij and s are constants); Ji denotesthe external bias on the ith unit; X is a compact set with smooth boundary oX and measure mes X > 0 in Rm; niðs; xÞ is theinitial boundary value; wðtÞ ¼ ðw1ðtÞ; . . . ;wnðtÞÞT is n-dimensional Brownian motion defined on a complete probability spaceðX;F;PÞ with a natural filtration fFtgtP0 generated by fwðsÞ : 0 6 s 6 tg, where we associate X with the canonical spacegenerated by all fwiðtÞg, and denote by F the associated r-algebra generated by fwðtÞg with the probability measure P.

Let L2ðXÞ be the space of real Lebesgue measurable functions on X and be a Banach space for the L2-norm

kuðtÞk22 ¼

ZX

u2ðt; xÞdx:

Note that, n ¼ fðn1ðs; xÞ; . . . ; nnðs; xÞÞT : �s 6 s 6 0g is Cð½�s;0� � Rm; RnÞ-valued function and F0-measurable Rn-valued ran-dom variable, where F0 ¼Fs on ½�s;0�, Cð½�s;0� � Rm; RnÞ is the space of all continuous Rn-valued functions defined on½�s;0� � Rm with a norm kniðtÞk2

2 ¼R

X n2i ðt; xÞdx.

Throughout this paper, for system (2.1), we have the following assumptions:

(A1) fj; gj and rij are Lipschitz continuous with Lipschitz constant aj > 0; bj > 0 and Lij > 0, respectively, andjfjj 6 Mf

j ; jgjj 6 Mgj ; for i; j ¼ 1;2; . . . ;n.

(A2) ci 2 C1ðR;RÞ and ci ¼ infu2Rc0iðuÞ > 0; for i ¼ 1;2; . . . ;n.(A3) di is bounded, positive and continuous, i.e. there exist constants di; di such that 0 < di 6 diðuÞ 6 di; for

u 2 R; i ¼ 1;2; . . . ;n.

When rij � 0; i; j ¼ 1;2; . . . ;n; system (2.1) becomes the deterministic system

dyiðtÞ ¼Xm

k¼1

o

oxkDik

oyi

oxk

� �dt � diðyiðtÞÞ � fciðyiðtÞÞ �

Xn

j¼1

aijfjðyjðt; xÞÞ �Xn

j¼1

bijgjðyjðt � sijðtÞ; xÞÞ þ Jigdt; x 2 X: ð2:2Þ

Furthermore when Dik � 0; i;¼ 1;2; . . . ;n, k ¼ 1;2; . . . ;m, system (2.2) becomes

dyiðtÞ ¼ �diðyiðtÞÞ � ciðyiðtÞÞ �Xn

j¼1

aijfjðyjðt; xÞÞ �Xn

j¼1

bijgjðyjðt � sijðtÞ; xÞÞ þ Ji

( )dt; x 2 X: ð2:3Þ

From [28], we know that under the assumptions (A1)–(A3), system (2.3) has an equilibrium solution y� ¼ ðy�1; . . . ; y�nÞT: It is

easy to see that y� ¼ ðy�1; . . . ; y�nÞT is also the equilibrium solution of system (2.2).

Now we assume that

(A4) rijðy�i Þ ¼ 0; for i; j ¼ 1;2; . . . ;n.

Then system (2.1) admits an equilibrium solution y� ¼ ðy�1; . . . ; y�nÞT and can be described by

dðyiðtÞ � y�i Þ ¼Xm

k¼1

o

oxkDik

oðyiðtÞ � y�i Þoxk

� �dt þ diðyiðtÞÞf�½ciðyiðtÞ � ciðy�i Þ� þ

Xn

j¼1

aij½fjðyjðt; xÞÞ � fjðy�j Þ�

þXn

j¼1

bij½gjðyjðt � sijðtÞ; xÞÞ � gjðy�j Þ�gdt þXn

j¼1

rijðyiðt; xÞÞdwjðtÞ; x 2 X;

since �Ji ¼ ciðy�i Þ �Pn

j¼1aijfjðy�j Þ �Pn

j¼1bijgjðy�j Þ.It follows from [12] that under the conditions (A1)–(A4), system (2.1) has the global solution on t P 0, which is denoted

by yðt; nÞ; or, yðtÞ, if no confusion occurs.

Definition 2.1. The equilibrium solution y� of system (2.1) is said to be pth moment exponentially stable if there exists a pairof positive constants k and C such that for any n

Ejyðt; nÞ � y�jp 6 CEjn� y�jpe�kt; t P 0;

where jyðtÞj2 ¼Pn

i¼1kyiðtÞk22: When p ¼ 1 it is usually called the mean value exponential stability.

For the convenience, we introduce some notations. For n� n matrix A, A�1 denotes the inverse of A, A P 0 means A is anonnegative matrix.

Lemma 2.2 [6]. If M P 0 and qðMÞ < 1, then ðI �MÞ�1 P 0; where I denotes the identity matrix and qðMÞ denotes the spectralradius of a square matrix M.

820 L. Wan, Q. Zhou / Applied Mathematics and Computation 206 (2008) 818–824

3. Mean value exponential stability

For system (2.1), we have the following result.

Theorem 3.1. Suppose that system (2.1) satisfies the assumptions ðA1Þ—ðA4Þ and

(A5) C > 0; qðC�1ðD1Aþaþ D1BþbÞ < 1;

whereC ¼ diag ðd1; d2; . . . ; dnÞ; di ¼ dici � 1

2

Pnj¼1L2

ij; i ¼ 1;2; . . . ;n,D1 ¼ diag ðd1; d2; . . . ; dnÞ;Aþ ¼ ðjaijjÞn�n;Bþ ¼ ðjbijjÞn�n,a ¼ diag ða1;a2;

. . . ;anÞ; b ¼ diag ðb1; b2; . . . ; bnÞ.

Then the equilibrium solution y� of system (2.1) is mean value exponentially stable.

Proof. Let ziðt; xÞ ¼ yiðt; xÞ � y�i ; i ¼ 1;2; . . . ;n: Applying Ito formula to z2i ðt; xÞ and integrating with respect to x, we have

dkziðtÞk22 ¼ 2

ZX

ziðtÞXm

k¼1

o

oxkDik

ozi

oxk

� �dxdt þ 2

ZX

ziðtÞdiðyiðtÞÞf�½ciðyiðtÞ � ciðy�i Þ� þXn

j¼1

aij½fjðyjðt; xÞÞ � fjðy�j Þ�

þXn

j¼1

bij½gjðyjðt � sijðtÞ; xÞÞ � gjðy�j Þ�gdxdt þZ

X

Xn

j¼1

r2ijðyiðt; xÞÞdxdt þ 2

ZX

Xn

j¼1

ziðtÞrijðyiðt; xÞÞdxdwjðtÞ:

Applying Ito formula to ðkziðtÞk22Þ

1=2, we have

dkziðtÞk2 ¼ �dikziðtÞk2 þ dikziðtÞk2 þ1

kziðtÞk2

ZX

zi

Xm

k¼1

o

oxkDik

oziðtÞoxk

� �dxdt þ 1

kziðtÞk2

ZX

ziðtÞdiðyiðtÞÞf�½ciðyiðtÞ � ciðy�i Þ�

þXn

j¼1

aij½fjðyjðt; xÞÞ � fjðy�j Þ� þXn

j¼1

bij½gjðyjðt � sijðtÞ; xÞÞ � gjðy�j Þ�gdxdt þ 12kziðtÞk2

ZX

Xn

j¼1

r2ijðyiðt; xÞÞdxdt

þ 1kziðtÞk2

ZX

Xn

j¼1

ziðtÞrijðyiðt; xÞÞdxdwjðtÞ �1

2kziðtÞk32

Xn

j¼1

ZX

ziðtÞrijðyiðt; xÞÞdx� �2

dt:

Notice that, it follows from the boundary condition that

Xm

k¼1

ZX

zio

oxkDik

ozi

oxk

� �dx ¼

ZX

zir � Dikozi

oxk

� �m

k¼1dx ¼

ZXr � ziDik

ozi

oxk

� �m

k¼1dx�

ZX

Dikozi

oxk

� �m

k¼1� rzidx

¼Xm

k¼1

ZoX

ziDikozi

oxk

� �m

k¼1� ds�

Xm

k¼1

ZX

Dikozi

oxk

� �2

dx ¼ �Xm

k¼1

ZX

Dikozi

oxk

� �2

dx;

where ‘‘�” is inner product, r ¼ ð oox1; . . . ; o

oxmÞ is the gradient operator, and

Dikozi

oxk

� �m

k¼1¼ Di1

ozi

ox1

� �; . . . ; Dim

ozi

oxm

� �� �T

:

Then, by using the method of variation parameter, Hölder inequality and (A1)–(A4), we obtain

kziðtÞk2 ¼ e�di tkzið0Þk2 þZ t

0e�diðt�sÞ dikziðsÞk2 �

1kziðsÞk2

Xm

k¼1

ZX

DikoziðsÞoxk

� �2

dx

(

þ 1kziðsÞk2

ZX

ziðtÞdiðyiðsÞÞf�½ciðyiðsÞ � ciðy�i Þ� þXn

j¼1

aij½fjðyjðs; xÞÞ � fjðy�j Þ�

þXn

j¼1

bij½gjðyjðs� sijðsÞ; xÞÞ � gjðy�j Þ�gdxþ 12kziðsÞk2

ZX

Xn

j¼1

r2ijðyiðs; xÞÞdx

)ds

þ NiðtÞ �Z t

0e�diðt�sÞ 1

2kziðsÞk32

Xn

j¼1

ZX

ziðsÞrijðyiðs; xÞÞdx� �2

ds 6 e�di tkzið0Þk2

þZ t

0e�diðt�sÞ dikziðsÞk2 þ

1kziðsÞk2

f�dicikziðsÞk22 þ di

Xn

j¼1

jaijjajkziðsÞk2kzjðsÞk2

(

L. Wan, Q. Zhou / Applied Mathematics and Computation 206 (2008) 818–824 821

þ di

Xn

j¼1

jbijjbjkziðsÞk2kzjðs� sijðsÞÞk2 þ12

Xn

j¼1

L2ijkziðsÞk2

2g)

dsþ NiðtÞ ¼ e�di tkzið0Þk2

þ di

Z t

0e�diðt�sÞ

Xn

j¼1

jaijjajkzjðsÞk2 þXn

j¼1

jbijjbjkzjðs� sijðsÞÞk2

( )dsþ NiðtÞ;

where

NiðtÞ ¼Z t

0e�diðt�sÞ 1

kziðsÞk2

ZX

Xn

j¼1

ziðsÞrijðyjðs; xÞÞdxdwjðsÞ:

Therefore, for a sufficiently small positive constant k < minfdi : 1 6 i 6 ng; we have

EkziðtÞk2ekt6 eðk�diÞtEkzið0Þk2 þ

Z t

0eðk�diÞðt�sÞdi

Xn

j¼1

jaijjaj½EkzjðsÞk2eks� þXn

j¼1

jbijjeksijðsÞbj½Ekzjðs� sijðsÞÞk2ekðs�sijðsÞÞ�( )

ds

6 Ekzið0Þk2 þZ t

0eðk�diÞðt�sÞdsdi

Xn

j¼1

jaijjaj þXn

j¼1

jbijjbjeks

( )GjðtÞ

6 Ekzið0Þk2 þ1

di � kdi

Xn

j¼1

jaijjaj þXn

j¼1

jbijjbjeks

( )GjðtÞ;

which yields

GiðtÞ 6 Ekzið0Þk2 þ1

di � kdi

Xn

j¼1

jaijjaj þXn

j¼1

jbijjbjeks

( )GjðtÞ;

that is,

GðtÞ 6 Ekzð0Þk þ ðC � kIÞ�1ðD1Aþaþ D1BþbeksÞGðtÞ;

where

GjðtÞ ¼ sup�s6h6t

EkzjðhÞk2ekh; j ¼ 1;2; . . . ; n;

GðtÞ ¼ðG1ðtÞ;G2ðtÞ; . . . ;GnðtÞÞT ;Ekzð0Þk ¼ðEkz1ð0Þk2;Ekz2ð0Þk2; . . . ;Ekznð0Þk2Þ

T:

Since

qðC�1ðD1Aþaþ D1BþbÞÞ < 1; C�1ðD1Aþaþ D1BþbÞP 0;

from Lemma 2.2, it deduces

ðI � C�1ðD1Aþaþ D1BþbÞÞ�1 P 0:

Then there exists a sufficiently small positive constant c < k such that

MðcÞ :¼ ðI � ðC � cIÞ�1ðD1Aþaþ D1BþbecsÞÞ�1 P 0:

Therefore, one can derive that

EkzðtÞk 6 MðcÞEkzð0Þke�kt6 MðcÞEkzð0Þke�ct ;

in which

EkzðtÞk ¼ ðEkz1ðtÞk2;Ekz2ðtÞk2; . . . ;EkznðtÞk2ÞT:

From

jzðtÞj2 ¼Xn

i¼1

kziðtÞk22 6

Xn

i¼1

kziðtÞk2

!2

;

it follows

EjzðtÞj 6Xn

i¼1

EkziðtÞk2 6 e�ctXn

i¼1

Xn

j¼1

MijðcÞEkzjð0Þk2:

The proof is completed. h

822 L. Wan, Q. Zhou / Applied Mathematics and Computation 206 (2008) 818–824

Notice that qðAÞ 6 kAk for any A 2 Rn�n, in which k�k is an arbitrary matrix norm. Moreover, for any matrix norm and anynonsingular matrix S, a matrix norm kAkS can be given by kAkS ¼ kS

�1ASk. For the convenience of calculation, in general, tak-ing S ¼ diag fs1; . . . ; sng > 0: Therefore, corresponding to the matrix norm widely applied— the row norm, column norm andFrobenius norm, we can obtain the following sufficient conditions to guarantee kAkS < 1, respectively.

(1)Pn

j¼1ðsisjjaijjÞ < 1, for 1 6 i 6 n;

(2)Pn

i¼1ðsisjjaijjÞ < 1, for 1 6 j 6 n;

(3)Pn

i¼1

Pnj¼1ð

sisjjaijjÞ2 < 1.

From the above statement, we have

Corollary 3.2. The equilibrium solution y�of system (2.1) is mean value exponentially stable provided that one of the followingconditions holds: there exist positive real numbers s1; . . . ; sn such that

(1)

Xn

j¼1

si

sjdi

Pnj¼1jaijjaj þ

Pnj¼1jbijjbj

di

0BBB@

1CCCA

26664

37775 < 1; for 1 6 i 6 n;

(2)

Xn

i¼1

si

sjdi

Pnj¼1jaijjaj þ

Pnj¼1jbijjbj

di

0BBB@

1CCCA

26664

37775 < 1; for 1 6 j 6 n;

(3)

Xn

i¼1

Xn

j¼1

si

sjdi

Pnj¼1jaijjaj þ

Pnj¼1jbijjbj

di

0BBB@

1CCCA

26664

37775

2

< 1:

From Theorem 3.1, we obtain the following results discussed in [28].

Corollary 3.3. Suppose that system (2.2) (or (2.3)) satisfies the assumptions (A1)–(A3). Then the equilibrium solution y� of system(2.2) (or (2.3)) is unique and globally exponentially stable provided that the following condition holds:

(A6) qðC�11 ðD1Aþaþ D1BþbÞ < 1, where C1 ¼ diag ðd1c1; d2c2; . . . ; dncnÞ.

4. One example

The following illustrative example will demonstrate the effectiveness of our results.

Example 1. Consider the following stochastic Cohen–Grossberg neural networks

dy1ðtÞy2ðtÞ

� �¼

2þ sin y1 00 2þ cos y2

� �� �

3 00 3

� �y1ðtÞy2ðtÞ

� �dt

�

þ0:2 0:10:3 0:1

� �sinðy1ðtÞÞcosðy2ðtÞÞ

� �dt þ

�0:3 0:20:1 0:3

� �

�tanhðy1ðt � s1ðtÞÞÞtanhðy2ðt � s2ðtÞÞÞ

� �dt�þ

L11y1ðtÞ L12y1ðtÞL21y2ðtÞ L22y2ðtÞ

� �dwðtÞ

þD11

oy1ox1

D12oy1ox2

D21oy2ox1

D22oy2ox2

!o

ox1

oox2

!dt

oyi

on:¼ oyi

ox1; � � � ; oyi

oxm

� �T

¼ 0; t P 0; x 2 oX;

yiðs; xÞ ¼ niðs; xÞ;�sij 6 s 6 0; x 2 X;

L. Wan, Q. Zhou / Applied Mathematics and Computation 206 (2008) 818–824 823

where tanhðxÞ ¼ ex�e�x

exþe�x ;Dik ¼ xk or constant, i; k ¼ 1;2. It is clear that dj ¼ 3; dj ¼ 1; cj ¼ 3;aj ¼ bj ¼ 1; j ¼ 1;2, and

C ¼3�

P2j¼1

L21j 0

0 3�P2j¼1

L22j

0BBBB@

1CCCCA; D1 ¼

3 00 3

� �;

Aþ ¼0:2 0:10:3 0:1

� �; Bþ ¼

0:3 0:20:1 0:3

� �;

Therefore,

C�1ðD1Aþaþ D1BþbÞ ¼

3

3�P2

j¼1

L21j

0

0 3

3�P2

j¼1

L22j

0BBBB@

1CCCCA

0:5 0:30:4 0:4

� �:

Taking

L ¼0 0ffiffiffiffiffiffiffi0:4p ffiffiffiffiffiffiffi

0:2p

� �;

we have

C�1ðD1Aþaþ D1BþbÞ ¼0:5 0:30:5 0:5

� �;

qðC�1ðD1Aþaþ D1BþbÞ ¼ 0:5þffiffiffiffiffiffiffiffiffiffi0:15p

< 1:

It follows from Theorem 3.1 that the equilibrium solution of such system is mean value exponentially stable.

5. Conclusions

The results in this paper show that, the sufficient criteria on the mean value exponential stability of stochastic CGNN withdelays and diffusion terms are independent of the magnitude of the delays and diffusion effect, but are dependent of themagnitude of noise, and therefore, in the above content, diffusion and delays are harmless, but noisy fluctuations areimportant.

Acknowledgements

The authors would like to thank the referee and the editor for their helpful comments and suggestions. This research waspartly supported by the National Natural Science Foundation of China (10801109) and the Natural Science Foundation ofWuhan University of Science and Engineering (2008Z25).

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