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fference EquationsVolume 2007, Article ID 78160, 18 pages doi:10.1155/2007/78160 Research Article Exponential Stability for Impulsive BAM Neural Networks with Time-Varying Delays and Reac

fference Equations

Volume 2007, Article ID 78160, 18 pages

doi:10.1155/2007/78160

Research Article

Exponential Stability for Impulsive BAM Neural Networks with Time-Varying Delays and Reaction-Diffusion Terms

Qiankun Song and Jinde Cao

Received 9 March 2007; Accepted 16 May 2007

Recommended by Ulrich Krause

Impulsive bidirectional associative memory neural network model with time-varying de-lays and reaction-diffusion terms is considered Several sufficient conditions ensuring the existence, uniqueness, and global exponential stability of equilibrium point for the ad-dressed neural network are derived byM-matrix theory, analytic methods, and

inequal-ity techniques Moreover, the exponential convergence rate index is estimated, which de-pends on the system parameters The obtained results in this paper are less restrictive than previously known criteria Two examples are given to show the effectiveness of the obtained results

Copyright © 2007 Q Song and J Cao 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

The bidirectional associative memory (BAM) neural network model was first introduced

by Kosko [1] This class of neural networks has been successfully applied to pattern recog-nition, signal and image processing, artificial intelligence due to its generalization of the single-layer auto-associative Hebbian correlation to two-layer pattern-matched heteroas-sociative circuits Some of these applications require that the designed network has a unique stable equilibrium point

In hardware implementation, time delays occur due to finite switching speed of the amplifiers and communication time [2] Time delays will affect the stability of designed neural networks and may lead to some complex dynamic behaviors such as periodic oscil-lation, bifurcation, or chaos [3] Therefore, study of neural dynamics with consideration

of the delayed problem becomes extremely important to manufacture high-quality neural networks Some results concerning the dynamical behavior of BAM neural networks with

delays have been reported, for example, see [2–12] and references therein The circuits di-agram and connection pattern implementing for the delayed BAM neural networks can

be found in [8]

Most widely studied and used neural networks can be classified as either continuous

or discrete Recently, there has been a somewhat new category of neural networks which are neither purely continuous-time nor purely discrete-time ones, these are called im-pulsive neural networks This third category of neural networks displays a combination

of characteristics of both the continuous-time and the discrete systems [13] Impulses can make unstable systems stable, so they have been widely used in many fields such as physics, chemistry, biology, population dynamics, and industrial robotics Some results for impulsive neural networks have been given, for example, see [13–22] and references therein

It is well known that diffusion effect cannot be avoided in the neural networks when electrons are moving in asymmetric electromagnetic fields [23], so we must consider that the activations vary in space as well as in time There have been some works devoted to the investigation of the stability of neural networks with reaction-diffusion terms, which are expressed by partial differential equations, for example, see [23–26] and references therein To the best of our knowledge, few authors have studied the stability of impulsive BAM neural network model with both time-varying delays and reaction-diffusion terms Motivated by the above discussions, the objective of this paper is to give some sufficient conditions ensuring the existence, uniqueness, and global exponential stability of equilib-rium point for impulsive BAM neural networks with time-varying delays and reaction-diffusion terms, without assuming the boundedness, monotonicity, and differentiability

on these activation functions Our methods, which do not make use of Lyapunov func-tional, are simple and valid for the stability analysis of impulsive BAM neural networks with time-varying or constant delays

2 Model description and preliminaries

In this paper, we consider the following model:

∂u i(t, x)

l



k =1

∂

∂x k



D ik ∂u i(t, x)

∂x k



− a i u i(t, x)

+

m



j =1

c i j f j



v j



t − τ i j(t), x

+α i, t = t k, =1, , n,

Δu i

t k,x

= I k

u i

t k,x

, i =1, , n, k =1, 2, ,

∂v j(t, x)

l



k =1

∂

∂x k



D ∗ jk ∂v j(t, x)

∂x k



− b j v j(t, x)

+

n



i =1

d ji g i

u i

t − σ ji(t), x

+β j, t = t k, j =1, , m,

Δv j



t k,x

= J k

v i

t k,x

, j =1, , m, k =1, 2, .

(2.1)

fort > 0, where x =(x1,x2, , x l)T ∈Ω⊂ R l,Ω is a bounded compact set with smooth boundary∂ Ω and mesΩ > 0 in space R l;u =(u1,u2, , u n)T ∈ R n;v =(v1,v2, , v m)T ∈

Rm;u i(t, x) and v j(t, x) are the state of the ith neurons from the neural field F Uand the

jth neurons from the neural field F Vat timet and in space x, respectively; f jandg idenote the activation functions of thejth neurons from F V and theith neurons from F Uat time

t and in space x, respectively; α i andβ j are constants, and denote the external inputs

on theith neurons from F Uand the jth neurons from F V, respectively;τ i j(t) and σ ji(t)

correspond to the transmission delays and satisfy 0≤ τ i j(t) ≤ τ i jand 0≤ σ ji(t) ≤ σ ji(τ i j

andσ jiare constants);a iandb jare positive constants, and denote the rates with which the

ith neurons from F Uand thejth neurons from F Vwill reset their potentials to the resting state in isolation when disconnected from the networks and external inputs, respectively;

c i j andd jiare constants, and denote the connection strengths; smooth functionsD ik =

D ik(t, x) ≥0 andD ∗ jk = D ∗ jk(t, x) ≥0 correspond to the transmission diffusion operator along theith neurons from F U and the jth neurons from F V, respectively.Δu i(t k,x) =

u i(t+

k,x) − u i(t k −,x) and Δv j(t k,x) = v j(t+

k,x) − v j(t − k,x) are the impulses at moments t k

and in spacex, and t1< t2< ··· is a strictly increasing sequence such that limk →∞ t k =

+∞ The boundary conditions and initial conditions are given by

∂u i

∂n :=



∂u i

∂x1

,∂u i

∂x2

, , ∂u i

∂x l

=0, i =1, 2, , n,

∂v j

∂n :=

j

∂x1,∂v j

∂x2, , ∂v j

∂x l

=0, j =1, 2, , m,

(2.2)

u i(s, x) = φ u i(s, x), s ∈− σ, 0

1≤ i ≤ n, 1 ≤ j ≤ m

σ ji , i =1, 2, , n,

v j(s, x) = φ v j(s, x), s ∈[− τ, 0], τ = max

1≤ i ≤ n, 1 ≤ j ≤ m

τ i j , j =1, 2, , m, (2.3)

whereφ u i(s, x), φ v j(s, x) (i =1, 2, , n, j =1, 2, , m) denote real-valued continuous

func-tions defined on [− σ, 0] ×Ω and [− τ, 0] ×Ω, respectively

Since the solution (u1(t, x), , u n(t, x), v1(t, x), , v m(t, x)) T of model (2.1) is discon-tinuous at the pointt k, by theory of impulsive differential equations, we assume that (u1(t k,x), , u n(t k,x), v1(t k,x), , v m(t k,x)) ≡(u1(t k −0,x), , u n(t k −0,x), v1(t k −0,x), , v m(t k −0,x)) T It is clear that, in general, the partial derivatives ∂u i(t k,x)/∂t and

∂v j(t k,x)/∂t do not exist On the other hand, according to the first and the third

equa-tions of model (2.1), there exist the limits∂u i(t k ∓0,x)/∂t and ∂v j(t k ∓0,x)/∂t According

to the above convention, we assume∂u i(t k,x)/∂t = ∂u i(t k −0,x)/∂t and ∂v j(t k,x)/∂t =

∂v j(t k −0,x)/∂t.

Throughout this paper, we make the following assumption

(H) There exist two positive diagonal matricesG =diag(G1,G2, , G n) andF =diag (F1,F2, , F m) such that

g i

u1 

− g i

v1 

− f j

for allu1,u2,v1,v2∈ R, =1, 2, , n, j =1, 2, , m.

For convenience, we introduce two notations For anyu(t, x) =(u1(t, x), u2(t, x), ,

u k(t, x)) T ∈ R k, define

u i(t, x)

2=

Ω

u i(t, x) 2

dx

, i =1, 2, , k. (2.5)

For anyu(t) =(u1(t), u2(t), , u k(t)) T ∈ R k, define u(t) =[k

i =1| u i(t) | r]1/r,r > 1 Definition 2.1 A constant vector (u ∗1, , u ∗ n,v1∗, , v ∗ m)T is said to be an equilibrium of model (2.1) if

− a i u ∗ i +

m



j =1

c i j f j

v ∗ j

+α i =0, i =1, 2, , n,

I k



u ∗ i

=0, i =1, 2, , n, k ∈ Z+,

− b j v ∗ j +

n



i =1

d ji g i

u ∗ i

+β j =0, j =1, 2, , m,

J k

v ∗ j

=0, j =1, 2, , m, k ∈ Z+,

(2.6)

whereZ +denotes the set of all positive integers

Definition 2.2 (see [3]) A real matrixA =(a i j)n × nis said to be anM-matrix if a i j ≤0 (i,

j =1, 2, , n, i = j) and successive principle minors of A are positive.

Definition 2.3 (see [27]) A mapH :Rn → R n is a homomorphism ofRn onto itself if

H ∈ C0,H is one-to-one, H is onto, and the inverse map H −1∈ C0

To prove our result, the following four lemmas are necessary

Lemma 2.4 (see [3]) Let Q be n × n matrix with nonpositive o ff-diagonal elements, then Q

is an M-matrix if and only if one of the following conditions holds.

(i) There exists a vector ξ > 0 such that Qξ > 0.

(ii) There exists a vector ξ > 0 such that ξ T Q > 0.

Lemma 2.5 (see [27]) If H(x) ∈ C0satisfies the following conditions:

(i)H(x) is injective onRn ,

(ii) H(x) →+∞ as  x →+∞ ,

then H(x) is homomorphism ofRn

Lemma 2.6 (see [28]) Let a, b ≥ 0, p > 1, then

a p −1b ≤ p −1

p a

p+1

p b

Lemma 2.7 (see [29]) (C p inequality) Let a ≥ 0, b ≥ 0, p > 1, then

(a + b)1/ p ≤ a1/ p+b1/ p (2.8)

3 Existence and uniqueness of equilibria

Theorem 3.1 Under assumption (H), if there exist real constants α i j , β i j , α ∗ ji , β ∗ ji (i =

1, 2, , n, j =1, 2, , m), and r > 1 such that

W =



A −  C − C ∗

− D ∗ B −  D



(3.1)

is an M-matrix, and

I k

u ∗ i

=0, i =1, 2, , n, k ∈ Z+,

J k

v ∗ j

then model ( 2.1 ) has a unique equilibrium point (u ∗1, , u ∗ n,v1∗, , v ∗ m)T , where

A =diag

a1,a2, , a n

,



C =diag



c1, , cn

with ci =

m



j =1

r −1

r c i j (r − α i j)/(r −1)

F(r − β i j)/(r −1)

B =diag

b1,b2, , b m

,



D =diag d1, , dm with dj =n

i =1

r −1

r d ji (r − α ∗ ji)/(r −1)

G(r − β

∗

ji)/(r −1)

C ∗ =c ∗ i j

n × m with c i j ∗ =1

r c i j α i j F β i j

j ,

D ∗ =d ∗ ji

m × n with d ∗ ji =1

r d ji α ∗ ji

G β

∗ ji

i

(3.3)

Proof Define the following map associated with model (2.1):

H(x, y) =



− A 0



x y



+





g(x)

f (y)



+



α β



where

C =c i j

n × m, D =d ji

m × n,

g(x) =g1(x1 

,g2(x2), , g n

x n)T

,

f (y) =f1



y1



, 2



y2



, , f m

y mT

,

α =α1,α2, , α nT

, β =β1,β2, , β mT

.

(3.5)

In the following, we will prove thatH(x, y) is a homomorphism.

First, we prove thatH(x, y) is an injective map onRn+m.

In fact, if there exist (x, y) T, (x, y) T ∈ R n+mand (x, y) T =(x, y) T such thatH(x, y) =

H(x, y), then

a i

x i − x i

=

m



j =1

c i j

f j

y j

− f j

y j

, i =1, 2, , n, (3.6)

b j

y j − y j

=

n



i =1

d ji

g i

x i

− g i

x i

, j =1, 2, , m. (3.7)

Multiply both sides of (3.6) by| x i − x i | r −1, it follows from assumption (H) andLemma 2.6that

a i x i − x i r

≤

m



j =1

j

≤

m



j =1

r −1

r c i j (r − α i j)/(r −1)

F(r − β i j)/(r −1)

j x i − x i r

+1

r

m



j =1

c i j α i j

F β i j

j y j − y

j r

.

(3.8)

Similarly, we have

b j y j − y

j r

≤

n



i =1

r −1

r d ji (r − α ∗ ji)/(r −1)

G(r − β

∗

ji)/(r −1)

j r

+1

r

n



i =1

d ji α ∗ ji g β

∗ ji

i x i − x i r

.

(3.9)

From (3.8) and (3.9) we get

W x1− x1 r

, , x n − x n r

, y1− y

1 r

, , y m − y

m rT

SinceW is an M-matrix, we get x i = x i,y j = y j, =1, 2, , n, j =1, 2, , m, which is a

contradiction So,H(x, y) is an injective map onRn+m

Second, we prove that H(x, y) →+∞as(x, y) T →+∞

Since W is an M-matrix, fromLemma 2.4, we know that there exists a vectorγ =

(λ1, , λ n,λ n+1, , λ n+m)T > 0 such that γ T W > 0, that is,

λ i

a i −  c i

−

m



j =1

λ n+ j d ∗ ji > 0, i =1, 2, , n,

λ n+ j

b j −  d j

−

n



i =1

λ i c i j ∗ > 0, j =1, 2, , m.

(3.11)

We can choose a small numberδ such that

λ i

a i −  c i

−

m



j =1

λ n+ j d ∗ ji ≥ δ > 0, i =1, 2, , n,

λ n+ j

b j −  d j

−

n



i =1

λ i c ∗ i j ≥ δ > 0, j =1, 2, , m.

(3.12)

LetH(x, y) = H(x, y) − H(0, 0), and sgn(θ) is the signum function defined as 1 if θ > 0, 0

ifθ =0,−1 ifθ < 0 From assumption (H),Lemma 2.6, and (3.12) we have

n



i =1

λ i x i r −1

sgn

x i H i(x, y) +m

j =1

λ n+ j y j r −1

sgn

y j H n+ j(x, y)

≤ −

n



i =1

λ i a i x i r

+

n



i =1

λ i

m



j =1

c i j F j y j x i r −1

−

m



j =1

λ n+ j b j y j r

+

m



j =1

λ n+ j

n



i =1

d ji G i x i y j r −1

≤

n



i =1

λ i



− a i+

m



j =1

r −1

r c i j (r − α i j)/(r −1)

F(r − β i j)/(r −1)

j



x i r

+

m



j =1

1

r c i j α i j

F β j i j y j r



+

m



j =1

λ n+ j



− b j+

n



i =1

r −1

r d ji (r − α ∗ ji)/(r −1)

G(r − β

∗

ji)/(r −1)

i



y j r

+

n



i =1

1

r d ji α ∗ ji

G β

∗ ji

i x i r



= −

n



i =1



λ i

a i −  c i

−

m



j =1

λ n+ j d ∗ ji



x i r

−

m



j =1



λ n+ j

b j −  d j

−

n



i =1

λ i c ∗ i j



y j r

≤ − δ (x, y) T r

.

(3.13) From (3.13) we have

δ (x, y) T r

≤ −



i =1

λ i x i r −1

sgn

x i H i(x, y) +m

j =1

λ n+ j y j r −1

sgn

y j H n+ j(x, y)

≤ max

1≤ i ≤ n+m

λ i



i =1

x i r −1 H i(x, y) +m

j =1

y j r −1 H n+ j(x, y) .

(3.14)

By using H¨older inequality we get

(x, y) T r

≤max1≤ i ≤ n+m

λ i

δ

i =1

x i r

+

m



j =1

y j r

×

i =1

H i(x, y) r

+

m



j =1

H n+ j(x, y) r

,

(3.15)

that is,

(x, y) T ≤max1≤ i ≤ n+m

λ i

δ H(x, y) . (3.16)

Therefore,  H(x, y) ∞ → +∞ as (x, y) T ∞ → +∞, which directly implies that

 H(x, y) →+∞ as(x, y) T →+∞ FromLemma 2.5 we know thatH(x, y) is a

ho-momorphism onRn+m Thus, equation

− a i u i+

m



j =1

c i j f j

v j

+α i =0, i =1, 2, , n,

− b j v j+

n



i =1

d ji g i

u i

+β j =0, j =1, 2, , m

(3.17)

has unique solution (u ∗1, , u ∗ n,v ∗1, , v m ∗)T, which is one unique equilibrium point of

4 Global exponential stability

Theorem 4.1 Under assumption (H), if W in Theorem 3.1 is an M-matrix, and

I k(u i(t k,x)) and J k(v j(t k,x)) satisfy

I k



u i



t k,x

= − γ ik



u i



t k,x

− u ∗ i

, 0< γ ik < 2, i =1, 2, , n, k ∈ Z+,

J k

v j

t k,x

= − δ jk

v j

t k,x

− v ∗ j

, 0< δ ik < 2, j =1, 2, , m, k ∈ Z+, (4.1)

then model ( 2.1 ) has a unique point (u ∗1, , u ∗ n,v ∗1, , v m ∗)T , which is globally exponentially stable.

Proof From (4.1) we know thatI k(u ∗ i )=0 andJ k(v ∗ j)=0 (i =1, 2, , n, j =1, 2, , m,

k ∈ Z+), so the existence and uniqueness of equilibrium point of (2.1) follow from

Theorem 3.1

Let (u1(t, x), , u n(t, x), v1(t, x), , v m(t, x)) Tbe any solution of model (2.1), then

∂

u i(t, x) − u ∗ i

∂t

=

l



k =1

∂

∂x k



D ik ∂



u i(t, x) − u ∗ i 

∂x k



− a i

u i(t, x) − u ∗ i 

+

m



j =1

c i j



f j



v j



t − τ i j(t), x

− f j



v ∗ j

, t > 0, t = t k, =1, , n, k ∈ Z+,

(4.2)

∂

v j(t, x) − v ∗ j

∂t

=

l



k =1

∂

∂x k



D ∗ jk ∂



v j(t, x) − v ∗ j

∂x k



− b j

v j(t, x) − v ∗ j

+

n



i =1

d ji



g i



u i



t − σ ji(t), x

− g i



u ∗ i

, t > 0, t = t k, j =1, , m, k ∈ Z+.

(4.3) Multiply both sides of (4.2) byu i(t, x) − u ∗ i, and integrate, then we have

1

2

d

dt



Ω



u i(t, x) − u ∗ i 2

dx =

l



k =1



Ω



u i(t, x) − u ∗ i ∂

∂x k



D ik ∂

u i(t, x) − u ∗ i

∂x k



dx

− a i



Ω



u i(t, x) − u ∗ i 2

dx

+

m



j =1

c i j



Ω



u i(t, x) − u ∗ i 

f j



v j



t − τ i j(t), x

− f j



v ∗ j

dx.

(4.4) From the boundary condition (2.2) and the proof of [22, Theorem 1] we get

l



k =1



Ω



u i(t, x) − u ∗ i  ∂

∂x k



D ik ∂



u i(t, x) − u ∗ i 

∂x k



dx = −

l



k =1



ΩD ik

i(t, x) − u ∗ i

∂x k

dx.

(4.5) From (4.4), (4.5), assumption (H), and Cauchy integrate inequality we have

d u i(t, x) − u ∗

i 2 2

dt ≤ −2a i u i(t, x) − u ∗

i 2 2

+ 2

m



j =1

c i j F j u i(t, x) − u ∗

i

2 v j

t − τ i j(t), x

− v ∗ j

2.

(4.6)

D+ u i(t, x) − u ∗

i

2≤ − a i u i(t, x) − u ∗

i

2+

m



j =1

c i j F j v j

t − τ i j(t), x

− v ∗ j

fort > 0, t = t k, =1, , n, k ∈ Z+

Multiply both sides of (4.3) byv j(t, x) − v ∗ j, similarly, we can get

D+ v j(t, x) − v ∗

j

2≤ − b j v j(t, x) − v ∗

j

2+

n



i =1

t − σ ji(t), x

− u ∗ i

fort > 0, t = t k, =1, , m, k ∈ Z+

It follows from (4.1) that

t k+ 0,x

− u ∗ i

2= 1− γ ik u i

t k,x

− u ∗ i

2, i =1, , n, k ∈ Z+,

t k+ 0,x

− v ∗ j

2= 1− δ jk v j

t k,x

− v ∗ j

2, i = j, , m, k ∈ Z+. (4.9)

Let us consider functions

ρ i(θ) = λ i

r − a i+ci



+

m



j =1

λ n+ j c ∗ i j e τθ, i =1, 2, , n,

χ j(θ) = λ n+ j

r − b j+dj+n

i =1

λ i d ∗ ji e σθ, j =1, 2, , m.

(4.10)

Since W is an M-matrix, from Lemma 2.4, we know that there exists a vector γ =

(λ1, , λ n,λ n+1, , λ n+m)T > 0 such that Wγ > 0, that is,

λ i

a i −  c i

−

m



j =1

λ n+ j c ∗ i j > 0, i =1, 2, , n,

λ n+ j

b j −  d j

−

n



i =1

λ i d ∗ ji > 0, j =1, 2, , m.

(4.11)

From (4.11) and (4.10) we know thatρ i(0)< 0, χ j(0)< 0, and ρ i(θ) and χ j(θ) are

con-tinuous forθ ∈[0, +∞) Moreover,ρ i(θ), χ j(θ) →+∞asθ →+∞ Since dρ i(θ)/dθ > 0,

dχ j(θ)/dθ > 0, ρ i(θ) and χ j(θ) are strictly monotone increasing functions on [0, + ∞) Thus, there exist constantsz ∗ i andz∗ j ∈(0, +∞) such that

ρ i

z ∗ i

= λ i

i

r − a i+c i



+

m



j =1

λ n+ j c i j ∗ e z ∗ i τ =0, i =1, 2, , n,

χ j



z ∗ j

= λ n+ j z ∗ j

r − b j+dj+n

i =1

λ i d ∗ ji e z∗ j σ =0, j =1, 2, , m.

(4.12)

3 Existence and uniqueness of equilibria

Theorem... existence and uniqueness of equilibrium point of (2.1) follow from

Theorem 3.1