Báo cáo hoa học: "Research Article Dynamic Analysis of Stochastic Reaction-Diffusion Cohen-Grossberg Neural Networks with Delays"

Volume 2009, Article ID 410823, 18 pagesdoi:10.1155/2009/410823 Research Article Dynamic Analysis of Stochastic Reaction-Diffusion Cohen-Grossberg Neural Networks with Delays 1 College o

Volume 2009, Article ID 410823, 18 pages

doi:10.1155/2009/410823

Research Article

Dynamic Analysis of Stochastic Reaction-Diffusion Cohen-Grossberg Neural Networks with Delays

1 College of Applied Mathematics, University of Electronic Science and Technology of China,

Chengdu, Sichuan 610054, China

2 Department of Mathematics, Sichuan Agricultural University, Yaan, Sichuan 625014, China

Correspondence should be addressed to Jie Pan,guangjiepan@163.com

Received 13 June 2009; Revised 20 August 2009; Accepted 2 September 2009

Recommended by Tocka Diagana

Stochastic effects on convergence dynamics of reaction-diffusion Cohen-Grossberg neural networks CGNNs with delays are studied By utilizing Poincar´e inequality, constructing suitable Lyapunov functionals, and employing the method of stochastic analysis and nonnegative semimartingale convergence theorem, some sufficient conditions ensuring almost sure exponential stability and mean square exponential stability are derived Diffusion term has played an important role in the sufficient conditions, which is a preeminent feature that distinguishes the present research from the previous Two numerical examples and comparison are given to illustrate our results

Copyrightq 2009 J Pan and S Zhong 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

1 Introduction

In the recent years, the problems of stability of delayed neural networks have received much attention due to its potential application in associative memories, pattern recognition and optimization A large number of results have appeared in literature, see, for example, 1

14 As is well known, a real system is usually affected by external perturbations which

in many cases are of great uncertainty and hence may be treated as random 15–17 As pointed out by Haykin 18 that in real nervous systems synaptic transmission is a noisy process brought on by random fluctuations from the release of neurotransmitters and other probabilistic causes, it is of significant importance to consider stochastic effects for neural networks In recent years, the dynamic behavior of stochastic neural networks, especially the stability of stochastic neural networks, has become a hot study topic Many interesting results on stochastic effects to the stability of delayed neural networks have been reported

see 16–23

In the factual operations, on other hand, diffusion phenomena could not be ignored in neural networks and electric circuits once electrons transport in a nonuniform electromagnetic field Thus, it is essential to consider state variables varying with time and space variables The delayed neural networks with diffusion terms can commonly be expressed by partial functional differential equation PFDE To study the stability of delayed reaction-diffusion neural networks, for instance, see 24–31, and references therein

Based on the above discussion, it is significant and of prime importance to consider the stochastic effects on the stability property of the delayed reaction-diffusion networks Recently, Sun et al 32, 33 have studied the problem of the almost sure exponential stability and the moment exponential stability of an equilibrium solution for stochastic reaction-diffusion recurrent neural networks with continuously distributed delays and constant delays, respectively Wan et al have derived the sufficient condition of exponential stability of stochastic reaction-diffusion CGNNs with delay 34, 35 In 36, the problem of stochastic exponential stability of the delayed reaction-diffusion recurrent neural networks with Markovian jumping parameters have been investigated In 32–

36, unfortunately, reaction-diffusion terms were omitted in the deductions, which result

in that the criteria of obtained stability do not contain the diffusion terms In other words, the diffusion terms do not take effect in their results The same cases appear also in other research literatures on the stability of reaction-diffusion neural network 24–

31

Motivated by the above discussions, in this paper, we will further investigate the convergence dynamics of stochastic reaction-diffusion CGNNs with delays, where the activation functions are not necessarily bounded, monotonic, and differentiable Utilizing Poincar´e inequality and constructing appropriate Lyapunov functionals, some sufficient conditions on the almost surely and mean square exponential stability for the equilibrium point are established The results show that diffusion terms have contributed to the almost surely and mean square exponential stability criteria Two examples have been provided to illustrate the effectiveness of the obtained results

The rest of this paper is organized as follows In Section 2, a stochastic delayed reaction-diffusion CGNNs model is described, and some preliminaries are given InSection 3, some sufficient conditions to guarantee the mean square and almost surely exponential stability of equilibrium point for the reaction-diffusion delayed CGNNs are derived Examples and comparisons are given in Section 4 Finally, in Section 5, conclusions are given

2 Model Description and Preliminaries

To begin with, we introduce some notations and recall some basic definitions and lemmas:

i X be an open bounded domain in R m with smooth boundary ∂X, and mesX > 0 denotes the measure of X X  X ∪ ∂x;

ii L2X is the space of real Lebesgue measurable functions on X which is a Banach space for the L2-normvx2 X |vx|2dx 1/2

, v ∈ L2X;

iii H1X  {w ∈ L2X, D i w ∈ L2X}, where D i w  ∂w/∂x i, 1≤ i ≤ m H1

0X  the closure of C0∞X in H1X;

iv C  CI × X, R n  is the space of continuous functions which map I × X into R nwith the normut, x2 n

i1u i t, x2

21/2 , for any ut, x  u1t, x, , u n t, x T ∈

C;

v ζ  {φ1s, x, , φ n s, x T : −τ, 0} ∈ BC−τ, 0 × X, R n and be an F0 -measurable R-valued random variable, where, for example, F0  Fs restricted

on −τ, 0, and BC be the Banach space of continuous and bounded functions

with the normφ τ  Σn

i1φ i2

τ1/2, whereφ iτ  sup−τ≤s≤0 φ i s, x2, for any

φs, x  φ1s, x, , φ n s, x T ∈ BC, i  1, , n;

vi ∇v  ∂v/∂x 1, , ∂v/∂x m  is the gradient operator, for v ∈ C1X |∇v|2 

m

l1|∂v/∂x m|2.Δu m

l1∂2u/∂x2

l  is the Laplace operator, for u ∈ C2X.

Consider the following stochastic reaction-diffusion CGNNs with constant delays

on X:

du i t, x  Σ m

l1

∂

∂x l



D il ∂u i t, x

∂x l



dt − a i u i t, x

×b i u i t, x − Σ n

j1w ij f j



u j t, x− Σn

j1v ij g j



u j



t − τ j , x

J i

dt

n

j1

σ ij u i t, xdw j t, t, x ∈ 0, ∞ × X, Bu i t, x  0, t, x ∈ 0, ∞ × ∂X,

u i t, x  φ i s, x, s, x ∈ −τ, 0 × X,

2.1

where i  1, , n, n ≥ 2 corresponds to the number of units in a neural network;

x  x1, , x mT ∈ X is a space variable, u i t, x corresponds to the state of the ith unit at time tand in space x; D il > 0 corresponds to the transmission diffusion

coefficient along the ith neuron; ai u i t, x represents an amplification function; b i u i t, x

is an appropriately behavior function; w ij , v ij denote the connection strengths of the

jth neuron on the ith neuron, respectively; g j u j t, x, f j u j t, x denote the activation functions of jth neuron at time t and in space x; τ j corresponds to the transmission delay and satisfies 0 ≤ τ j ≤ τ τ is a positive constant; J i is the constant input

from outside of the network Moreover, wt  w1t, , w n t T is an n-dimensional

Brownian motion defined on a complete probability spaceΩ, F, P with the natural filtration

{Ft}t≥0 generated by the process {ws : 0 ≤ s ≤ t}, where we associate Ω with

the canonical space generated by all {w i t}, and denote by F the associated σ-algebra generated by wt with the probability measure P The boundary condition is given by Bu i t, x  u i t, x Dirichlet type or Bu i t, x  ∂u i t, x/∂m Neumann type, where

∂u i t, x/∂m  ∂u i t, x/∂x1, , ∂u i t, x/∂x mT denotes the outward normal derivative

on ∂X.

Model 2.1 includes the following reaction-diffusion recurrent neural networks

RNNs as a special case:

du i t, x  Σ m

l1

∂

∂x l



D il ∂u i t, x

∂x l



dt

−b i u i t, x Σ n

j1w ij f j

u j t, x Σn

j1v ij g j

u j

t − τ j , x

J i

dt

n

j1

σ ij



u j t, xdw j t, t, x ∈ 0, ∞ × X, Bu i t, x  0, t, x ∈ 0, ∞ × ∂X,

u i t, x  φ i s, x, s, x ∈ −τ, 0 × X,

2.2

for i  1, , n.

When w i t  0 for any i  1, , n, model 2.1 also comprises the following reaction-diffusion CGNNs with no stochastic effects on space X:

∂u i t, x

∂t  Σm

l1

∂

∂x l



D il ∂u i t, x

∂x l



− a i u i t, x

×b i u i t, x − Σ n

j1w ij f j



u j t, x− Σn

j1v ij g j



u j



t − τ j , x

J i

,

t, x ∈ 0, ∞ × X, Bu i t, x  0, t, x ∈ 0, ∞ × ∂X,

u i t, x  φ i s, x, s, x ∈ −τ, 0 × X,

2.3

for i  1, , n.

Throughout this paper, we assume that

H1 each function a i ξ is bounded, positive and continuous, that is, there exist constants a i , a i such that 0 < a i ≤ a i ξ ≤ a i < ∞, for ξ ∈ R, i  1, , n,

H2 b i ξ ∈ C1R, R and b i infξ∈Rb i ξ > 0, for i  1, , n,

H3 f j , g j are bounded, and f j , g j , σ ij are Lipschitz continuous with Lipschitz constant

F j , G j , L ij > 0, for i, j  1, , n,

H4 σ ij u∗

i   0, for i, l  1, , n.

Using the similar method of25, it is easily to prove that under assumptions H1–

H3, model 2.3 has a unique equilibrium point u∗ u∗

1, , u∗nTwhich satisfies

b i

u∗i

− Σn

j1w ij f j

u∗j

− Σn

j1v ij g j

u∗j

J i  0, i  1, , n. 2.4 Suppose that system 2.1 satisfies assumptions H1–H4, then equilibrium point u∗ of model2.3 is also a unique equilibrium point of system 2.1

By the theory of stochastic differential equations, see 15,37, it is known that under the conditionsH1–H4, model 2.1 has a global solution denoted by ut, 0, x; φ or simply ut, φ, ut, x or ut, if no confusion should occur For the effects of stochastic forces to the

stability property of delayed CGNNs model2.1, we will study the almost sure exponential

stability and the mean square exponential stability of their equilibrium solution ut ≡ u∗in the following sections For completeness, we give the following definitions33, in which E denotes expectation with respect toP

Definition 2.1 The equilibrium solution u∗ of model 2.1 is said to be almost surely

exponentially stable, if there exists a positive constant μ such that for any φ there is a finite positive random variable M such that

ut, φ − u∗

2≤ Me −μt ∀t ≥ 0. 2.5

Definition 2.2 The equilibrium solution u∗ of model 2.1 is said to be pth moment exponentially stable, if there exists a pair of positive constants μ and M such that for any φ,

E ut, φ − u∗ p

τ e −μt ∀t ≥ 0. 2.6

When p  1 and 2, it is usually called the exponential stability in mean value and mean square, respectively

The following lemmas are important in our approach

Lemma 2.3 nonnegative semimartingale convergence theorem 16 Suppose At and Ut are two continuous adapted increasing processes on t ≥ 0 with A0  U0  0, a.s Let Mt be a real-valued continuous local martingale with M0  0, a.s and let ζ be a nonnegative F0-measurable random variable with Eζ < ∞ Define Xt  ζ At−UtMt for t ≥ 0 If Xt is nonnegative, then

 lim

t→ ∞At < ∞



⊂

 lim

t→ ∞Xt < ∞



∩

 lim

t→ ∞Ut < ∞



a.s., 2.7

where B ⊂ D a.s denotes PB ∪ D c   0 In particular, if lim t→ ∞At < ∞ a.s., then for almost all

w∈ Ωlimt→ ∞Xt, w < ∞ and lim t→ ∞Ut, w < ∞, that is, both Xt and Ut converge to finite random variables.

Lemma 2.4 Poincar´e inequality Let X be a bounded domain of R m with a smooth boundary ∂X

of classC2 by X vx is a real-valued function belonging to H1

0X and satisfies Bvx| ∂X  0 Then

λ1



X

|vx|2dx≤



X

|∇vx|2dx, 2.8

which λ1is the lowest positive eigenvalue of the boundary value problem

−Δψx  λψx, x ∈ X,

B

ψx 0, x ∈ ∂X. 2.9 Proof We just give a simple sketch here.

Case 1 Under the Neumann boundary condition, that is, Bvx  ∂vx/∂m According

to the eigenvalue theory of elliptic operators, the Laplacian −Δ on X with the Neumann

boundary conditions is a self-adjoint operator with compact inverse, so there exists a sequence of nonnegative eigenvalues going to ∞ and a sequence of corresponding eigenfunctions, which are denoted by 0  λ0 < λ1 < λ2 < · · · and ψ0x, ψ1x, ψ2x, ,

respectively In other words, we have

λ0  0, ψ0x  1,

−Δψ k x  λ k ψ k x, in X,

ψ k x  0, on ∂X,

2.10

where k ∈ N Multiply the second equation of 2.10 by ψ k x and integrate over X By

Green’s theorem, we obtain



X

∇ψ k x2

dx  λ k



X

ψ k2xdx, for k ∈ N. 2.11

Clearly,2.11 can also hold for k  0 The sequence of eigenfunctions {ψ k x}∞

k0defines an

orthonormal basis of L2X For any vx ∈ H1

0X, we have

vx  ∞

k0

c k ψ k 2.12

From2.11 and 2.12, we can obtain



X

|∇vx|2dx ≥ λ1



X

|vx|2dx. 2.13

Case 2 Under the Dirichlet boundary condition, that is, Bvx  vx By the same may, we

can obtained the inequality

This completes the proof

Remark 2.5 i The lowest positive eigenvalue λ1in the boundary problem2.9 is sometimes known as the first eigenvalue ii The magnitude of λ1 is determined by domain X For example, let Laplacian on X  {x1, x2T ∈ R2| 0 < x1< a, 0 < x2< b}, if Bvx  vx and Bvx  ∂vx/∂m, respectively, then λ1  π/a2 π/b2and λ1  min{π/a2, π/b2}

38, 39 iii Although the eigenvalue λ1 of the laplacian with the Dirichlet boundary

condition on a generally bounded domain X cannot be determined exactly, a lower bound

of it may nevertheless be estimated by λ1≥ m2/m 22π2/ω m−11/V  2/m , where ω m−1

is a surface area of the unit ball inRm , V is a volume of domain X40

InSection 4, we will compare the results between this paper and previous literatures

To this end, we recall some previous results as follows according to the symbols in this paper

In35, Wan and Zhou have studied the problem of convergence dynamics of model

2.1 with the Neumann boundary condition and obtained the following result see 35, Theorem 3.1

Proposition 2.6 Suppose that system 2.1 satisfies the assumptions (H1)–(H4) and

A C > 0, ρC−1A1WF A1VG < 1, where C  diagδ1, , δ n , δ i  a i b i −

1/2n

j1L2

ij , i  1, , n, A1  diaga1, , a n , W  |w ij|n ×n , V  |v ij|n ×n ,

F  diagF1, , F n , G  diagG1, , G n  Also, ρA denotes the spectral radius of a square matrix A.

Then model2.1 is mean value exponentially stable.

Remark 2.7 It should be noted that condition A in Proposition 2.6 is equivalent to C −

A1WF A1VG is a nonsingular M-matrix, where C > 0 Thus, the following result is

treated as a special case ofProposition 2.6

Proposition 2.8 see 33, Theorem 3.1 Suppose that model 2.2 satisfies the assumptions (H2)– (H4) and

B B − B − WF − VG is a nonsingular M-matrix, where B  diag{b1, , b n }, b i :

−b in

j1|w ij |F jn

j1|V ij |G jn

j1L2

ij ≥ 0, for 1 ≤ i ≤ n.

Then model2.2 is almost surely exponentially stable.

Remark 2.9 It is obvious that conditionsA and B are irrelevant to the diffusion term In other words, the diffusion term does not take effect in Propositions2.6and2.8

3 Main Results

Theorem 3.1 Under assumptions (H1)–(H4), if the following conditions hold:

H5 a  2λ1D i a i b i −n

j1|w ij |a i F j |w ji |a j F i |v ij |a i G j L2

ij  > b n

j1|v ji |a j G i , for any i  1, , n,

where λ1is the lowest positive eigenvalue of problem2.9, D i min1≤l≤m{D il }, i  1, , n Then model2.1 is almost surely exponentially stable.

Proof Let ut  u1t, , u n t T be an any solution of model2.1 and y i t  u i t − u∗

i Model2.1 is equivalent to

dy i t  Σ m

l1

∂

∂x l



D il

∂y i t

∂x l



dt − a i u i t

×b i



y i t− Σn

j1v ij g j



u j



t − τ j



− Σn

j1w ij fjy j t dt

n

j1

σ ij



y i tdw j t, t, x ∈ 0, ∞ × X,

3.1a

B

y i t 0, t, x ∈ 0, ∞ × ∂X, 3.1b

y i s, x  φ i s, x − u∗

i , s, x ∈ −τ, 0 × X, 3.1c

where

b iy i t b i



y i t u∗

i



− b i



u∗i

, fjy j t f jy i t u∗

j

− f j

u∗j

,

g j



y j t g j



y j t u∗

i



− g j

u∗j

, σ ij



y j t σ ij

y j t u∗

j

− σ ij

u∗j

,

3.2

for i, j  1, , n.

It follows fromH5 that there exists a sufficiently small constant μ > 0 such that

2

λ1D i a i b i

− μ − n

j1

w

ija i F jw jia j F iv ija i G j L2

ij

− n

j1

v jia j G i e μτ > 0, i  1, , n.

3.3

To derive the almost surely exponential stability result, we construct the following Lyapunov functional:

V zt, t  n

i1



Ωe μt

⎡

⎣y2

i t a i

n

j1

v ijG jt

t −τ j

e μs τ j y2

j sds

⎤

⎦dx. 3.4

By It ˆo’s formula to V zt, t along 3.1a, we obtain

V zt, t  V z0, 0

t

0

e μs

n

i1



Ω

⎧

⎨

⎩μy2i s 2y i s ∂

∂x l



D il ∂y i s

∂x l



− 2y i sa i u i s

×

⎡

⎣b i



y i s n

j1

w ij fjy j s Σn

j1v ij g j



y j



s − τ j

⎤⎦

a i n

j1

v ijG j e μτ j y j2s − a i

n

j1

v ijG j y2

j



s − τ j

⎫⎬

⎭ds dx

t

0



Ω

n

i1

n

j1

e μs σ ij2

y i sds dx

2 n

i1

t

0



j1y i sσ ij



y j sdw j sdx,

3.5

for t≥ 0

By the boundary condition, it is easy to calculate that



l1

∂

∂x l



D il ∂y i s

∂x l



dx

 −m

l1



∂y

i s

∂x l

2

dx ≤ −D i



Ω

m

l1

∂y

i s

∂x l

2

dx

 −D i



Ω∇y i s2

dx

≤ −λ1D i



Ωy2i sdx  −λ1D i y i s 2

2.

3.6

From assumptionsH1 and H2, we have



Ωy i sa i



y i sb i

y i sdx ≥ a i b i



Ωy2i tdx  a i b i y i s 2

2. 3.7

From assumptionsH1 and H3, we have

2



Ωy i sa i



y i s n

j1

w ij fiy j sdx

≤ 2



Ω

n

j1

a iw ijy i s f i



y j sdx

≤ 2



Ω

n

j1

a iw ijy i sF jy j sdx

≤ a i



Ω

n

j1

w ijF j y2

i tdx a i



Ω

n

j1

w ijF jy j s2

dx

≤ a i n

j1

w ijF j y i s 2

n

j1

w ijF j y j s 2

2.

3.8

By the same way, we can obtain

2



Ωy i sa i



y i sΣn

j1v ij g i



y j

s − τ j



dx

≤ a i n

j1

v ijG j y i s 2

n

j1

v ijG j y j

s − τ j 22.

3.9

Combining3.6–3.9 into 3.5, we get

V zt, t ≤ V z0, 0

t

0

e μs

⎧

⎨

⎩

n

i1

⎡

⎣−2λ1D i a i b i

μ n

j1

w ija i F j

n

j1

w jia j F i a i n

j1

v ijG j⎤ i s 2

2

a i n

j1

v ijG j e μτ j y j s 2

2

⎫

⎬

⎭ds

t

0



Ω

n

i1

n

j1

e μs σ2

ij



y i sdx ds

2

t

0

n

i1



j1y i sσ ij



y j sdw j sdx