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Volume 2009, Article ID 491268, 17 pagesdoi:10.1155/2009/491268 Research Article Global Exponential Stability of Delayed Cohen-Grossberg BAM Neural Networks with Impulses on Time Scales

Volume 2009, Article ID 491268, 17 pages

doi:10.1155/2009/491268

Research Article

Global Exponential Stability of

Delayed Cohen-Grossberg BAM Neural Networks with Impulses on Time Scales

Yongkun Li,1 Yuchun Hua,1 and Yu Fei2

1 Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China

2 School of Statistics and Mathematics, Yunnan University of Finance and Economics,

Kunming, Yunnan 650221, China

Correspondence should be addressed to Yongkun Li,yklie@ynu.edu.cn

Received 18 April 2009; Accepted 14 July 2009

Recommended by Patricia J Y Wong

Based on the theory of calculus on time scales, the homeomorphism theory, Lyapunov functional method, and some analysis techniques, sufficient conditions are obtained for the existence, uniqueness, and global exponential stability of the equilibrium point of Cohen-Grossberg bidirectional associative memoryBAM neural networks with distributed delays and impulses

on time scales This is the first time applying the time-scale calculus theory to unify the discrete-time and continuous-discrete-time Cohen-Grossberg BAM neural network with impulses under the same framework

Copyrightq 2009 Yongkun 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

1 Introduction

In the recent years, bidirectional associative memory BAM neural networks and Cohen-Grossberg neural networksCGNNs with their various generalizations have attracted the attention of many mathematicians, physicists, and computer scientistssee 1 17 due to their wide range of applications in, for example, pattern recognition, associative memory, and combinatorial optimization Particularly, as discussed in 18–20, in the hardware implementation of the neural networks, when communication and response of neurons happens time delays may occur Actually, time delays are known to be a possible source

of instability in many real-world systems in engineering, biology, and so forth see, e.g.,

21 and references therein However, besides delay effect, impulsive effect likewise exists

in a wide variety of evolutionary processes in which states are changed abruptly at certain moments of time, involving fields such as medicine and biology, economics, mechanics, electronics, and telecommunications As artificial electronic systems, neural networks such

as Hopfield neural networks, bidirectional neural networks, and recurrent neural networks

often are subject to impulsive perturbations which can affect dynamical behaviors of the systems just as time delays Therefore, it is necessary to consider both impulsive effect and delay effect on the stability of neural networks

As is well known, both continuous and discrete systems are very important in implementation and applications However, it is troublesome to study the stability for continuous and discrete systems, respectively Therefore, it is worth studying a new method, such as the time-scale theory, which can unify the continuous and discrete situations Motivated by the above discussions, the objective of this paper is to study the global exponential stability of the following Cohen-Grossberg bidirectional associative memory networks with impulses and time delays on time scales:

xΔi t −a i x i t

⎡

⎣b i x i t −m

j 1

p0ji f j



y j



t − τ ji



−m

j 1

p1

ji

∞ 0

h ij sf j



y j t − s i

⎤

⎦, t ≥ 0, t / t k , t ∈ T,

Δx i t k  I k x i t k , i 1, 2, , n, k 1, 2, ,

yΔj t −c j



y j t d j

y j t−n

i 1

q0ij g i

x i

t − σ ij



−n

i 1

q1ij

∞ 0

k ij sg i x i j

, t ≥ 0, t / t k , t ∈ T,

Δy j t k  J k



y j t k, j 1, 2, , m, k 1, 2, ,

1.1

where T is a time scale; I k , J k : R → R are continuous, x i t, y j t are the states of the

ith neuron from the neural field F X and the jth neuron from the neural field F Y at time

t, respectively; f j , g i denote the activation functions of the jth neuron from F Y and the ith neuron from F X , respectively; r i and s j are constants, which denote the external inputs on

the ith neuron from F X and the jth neuron from F Y , respectively; τ ji and σ ij correspond to

the transmission delays; a i x i t and c j y j t represent amplification functions; b i x i t and d j y j t are appropriately behaved functions such that the solutions of system 1.1

remain bounded; p0

ji , p1

ji , q0

ij , and q1

ijdenote the connection strengths which correspond to the

neuronal gains associated with the neuronal activations; I i and J jdenote the external inputs

For each interval I ofR, we denote that by IT IT, Δx i t k  x i t k −x i t−

k , Δy j t k  y j t k−

y j t−

k  are the impulses at moments t k , and x i t k , x i t−

k , y j t k , y j t−

k  i 1, 2, , n, j

1, 2, , m represent the right and left limits of x i t k  and y j t k in the sense of time scales;

0 < t1< t2< · · · < t k → ∞ is a strictly increasing sequence

The system1.1 is supplement with initial values given by

x i s ϕ i s, s ∈ −∞, 0T, i 1, 2, , n,

y j s ψ j s, s ∈ −∞, 0T, j 1, 2, , m, 1.2 where ϕ i , ψ jare continuous real-valued functions defined on their respective domains

As usual in the theory of impulsive differential equations, at the points of discontinuity

t k of the solution t → x1t, x2t, , x n t, y1t, y2t, , y m t Twe assume that

x i t k  x i



t−k

, y j t k  y j



t−k

, xΔi t k  xΔ

i



t−k

, y jΔt k  yΔ

j



t−k

, 1.3

for i 1, 2, , n, j 1, 2, , m.

The organization of the rest of this paper is as follows In Section 2, we introduce some notations and definitions, and state some preliminary results which are needed in later sections In Section 3, by means of homeomorphism theory, we study the existence and uniqueness of the equilibrium point of system 1.1 In Section 4, by constructing a suitable Lyapunov function, we establish the exponential stability of the equilibrium of1.1

obtained in previous sections

2 Preliminaries

In this section, we will cite some definitions and lemmas which will be used in the proofs of our main results

LetT be a nonempty closed subset time scale of R The forward and backward jump

operators σ, ρ : T → T and the graininess μ : T → R are defined, respectively, by

σ t inf{s ∈ T : s > t}, ρ t sup{s ∈ T : s < t}, μ t σt − t. 2.1

A point t ∈ T is called left dense if t > inf T and ρt t, left scattered if ρt < t, right dense if t < sup T and σt t, and right scattered if σt > t If T has a left-scattered maximum m, thenTk T \ {m}; otherwise T k T If T has a right-scattered minimum m,

thenTk T \ {m}; otherwise T k T

A function f :T → R is right dense continuous provided that it is continuous at right dense point inT and its left-side limits exist at left-dense points in T If f is continuous at each right dense point and each left-dense point, then f is said to be a continuous function

onT The set of continuous functions f : T → R will be denoted by CT.

For y : T → R and t ∈ T k , we define the delta derivative of yt, yΔt to be the

numberif it exists with the property that for a given ε > 0, there exists a neighborhood U

of t such that



y σt − ys− yΔtσt − s 2.2

for all s ∈ U.

If y is continuous, then y is right dense continuous, and y is delta differentiable at t, then y is continuous at t.

Let y be right dense continuous If yΔt yt, then we define the delta integral by

t

a

Definition 2.1see 22 For each t ∈ T, let N be a neighborhood of t, then, for V ∈ CrdT ×

Rn ,R , define D VΔt, xt to mean that, given ε > 0, there exists a right neighborhood

N ε ⊂ N of t such that



V σt, xσt − V s, xσt − μt, sft, xt

for each s ∈ N ε , s > t, where μt, s ≡ σt − s If t is rd and V t, xt is continuous at t, this

reduce to

D VΔt, xt V σt, xσt − V t, xσt

Definition 2.2see 23 If a ∈ T, sup T ∞, and f is right dense continuous on a, ∞, then

we define the improper integral by

∞

a

f tΔt lim

b→ ∞

b

a

provided that this limit exists, and we say that the improper integral converges in this case

If this limit does not exist, then we say that the improper integral diverges

A function r : T → R is called regressive if

for all t∈ Tk

If r is regressive function, then the generalized exponential function e r is defined by

e r t, s exp

t

s

ξ μ τ rτΔτ



with the cylinder transformation

ξ h z

⎧

⎪

⎪

Log

h , if h / 0,

Let p, q :T → R be two regressive functions, then we define

Then the generalized exponential function has the following properties

Lemma 2.3 see 24 Assume that p, q : T → R are two regressive functions, then

i e0t, s ≡ 1 and e p t, t ≡ 1

iii e p t, σs e p

iv 1/e p t, s e p t, s

v e p t, s 1/e p s, t e p s, t

vi e p t, se p s, r e p t, r

vii e p t, se q t, s e p ⊕q t, s

viii e p t, s/e q t, s e p q t, s.

Definition 2.4 The equilibrium point u∗ x∗

1, x2∗, , x∗n , y1∗, y∗2, , y∗mT of system

1.1 is said to be exponentially stable if there exists a positive constant α such that for every δ ∈ T, there exists N N δ ≥ 1 such that the

solu-tion ut x1t, x2t, , x n t, y1t, y2t, , y m t T of 1.1 with initial value

ϕ1s, ϕ2s, , ϕ n s, ψ1s, ψ2s, , ψ m s Tsatisfies

u − u∗ ≤ Ne −α t, δ

⎡

⎣n

i 1

max

δ ∈−∞,0T

ϕ i δ − x∗

i m



j 1

max

δ ∈−∞,0T j δ − y∗

j

⎤

Lemma 2.5 see 25 If Hx ∈ CR n ,Rn  satisfies the following conditions:

i Hx is injective on R n ,

then H x is a homeomorphism of R n onto itself.

For z x1, x2, , x n , y1, y2, , y mT∈ Rn , we define the norm as

z n

i 1

|x i m



j 1

Throughout this paper, we assume that

H1 a i , c j ∈ CT, R , and satisfy 0 < a i ≤ a i x ≤ a i , 0 < c j ≤ c j x ≤ c j , ∀x ∈ R, i

1, 2, , n, j 1, 2, , m;

H2 the activation functions f j , g i ∈ CR, R and there exist positive constants M j , N i

such that

f j x − f j



y j x − y , g i x − g i



for all x, y ∈ R, i 1, , n, j 1, , m;

H3 b i , d j ∈ CR, R, b i 0 d j 0 0, i 1, 2, , n, j 1, 2, , m, and there exist positive constants η i , ω jsuch that

b i x − b i



y

x − y ≥ η i , d j x − d j



y

x − y ≥ ω j , ∀x / y; 2.14

H4 the kernels h ji and k ijdefined on0, ∞Tare nonnegative continuous integral func-tions such that∞

0h ji sΔs 1,∞0 sh ji

∞

0 k ij sΔs 1,∞0 sk ij sΔs <

3 Existence and Uniqueness of the Equilibrium

In this section, using homeomorphism theory, we will study the existence and uniqueness of the equilibrium point of system1.1

An equilibrium point of1.1 is a constant vector x∗

1, x∗2, , x∗n , y∗1, y2∗, , y m∗T ∈ Rn

which satisfies the system

a i

x∗i⎡⎣b ix i∗

−m

j 1



p0

ji 1ji



f j

y j∗

i

⎤

⎦ 0, i 1, 2, , n,

c j



y∗j d j



y∗j

−n

i 1



q0ij 1ij

g i



x i∗

j

0, j 1, 2, , m,

3.1

where the impulsive jumps I k ·, J k· satisfy

I k

x i∗

0, i 1, 2, , n, J k



y∗j

0, j 1, 2, , m. 3.2 From the assumptionsH1 and H4, it follows that

b i



x∗i m

j 1



p0ji 1ji

f j



y∗j

i , i 1, 2, , n,

d j

y j∗ n

i 1



q0

ij 1ij



g i

x i∗

j , j 1, 2, , m.

3.3

Noting that if b−1i ·, d−1

j · exist and activation functions f j · and g j· are bounded, then the existence of an equilibrium point of system1.1 is easily obtained from Brouwer’s fixed point theorem We can refer to2 8

Theorem 3.1 Assume that H2 and H3 hold Suppose further that for each i 1, 2, , n, j

1, 2, , m, the following inequalities are satisfied:

η i >

m



j 1

0

ij 1ij i , ω j >

n



i 1

0

Then there exists a unique equilibrium point of system1.1.

Proof Consider a mappingΦ : Rn → Rn defined by

Φi z b i x i −m

j 1



p0ji 1ji

f j



y j



i , i 1, 2, , n,

Φi z d j



y j



−n

i 1



q0ij 1ij

g i x i j , j 1, 2, , m,

3.5

where z x1, x2, , x n , y1, y2, , y mT ∈ Rn , Φz Φ1z, , Φ n z, , Φ n z T ∈

Rn First, we want to show that Φ is an injective mapping on Rn By contradiction,

suppose that there exists a distinct z, z ∈ Rn such that Φz Φz, where z

x1, x2, , x n , y1, y2, , y mT ∈ Rn and z x1, x2 , x n , y1, y2, , y mT ∈ Rn Then

it follows from3.5 that

b i x i  − b i x i m

j 1



p0

ji 1ji



f j

y j

− f j



y j

, i 1, 2, , n,

d j

y j

− d j



y j n

i 1



q0

ij 1ij



g i x i  − g j x i, j 1, 2, , m.

3.6

In view ofH2-H3 and 3.6, we have

n



i 1

η i |x i − x i| ≤n

i 1

m



j 1

0

ji 1ji j j − y j

m



j 1

ω j j − y j m

j 1

n



i 1

0

ij 1ij i |x i − x i |.

3.7

Thus, we can obtain

n



i 1

⎡

⎣η i−m

j 1

0

ij 1ij i

⎤

⎦|x i − x i

m



j 1

ω j−n

i 1

0

ji 1ji j j − y j 3.8

It follows from3.4 and 3.8 that |x i − x i | 0 and |y j − y j | 0, i 1, 2, , n, j 1, 2, , m That is z z, which leads to a contradiction Therefore, Φ is an injective on R n

Then we will proveΦ is a homeomorphism on Rn For convenience, we let Φz Φz − Φ0, where

Φi z b i x i −m

j 1



p0ji 1ji

f j



y j



− f j0, i 1, 2, , n,

Φn z d j



y j



−n

i 1



q ij0 1ij

g i x i  − g i0, j 1, 2, , m.

3.9

We assert that Φ → ∞ as z → ∞ Otherwise there is a sequence {z v } such that z v →

∞ and Φz v  is bounded as v → ∞, where z v x v

1, x v

2, , x v , y v

1, y v

2, , y v

mT ∈ Rn Noting that

n



i 1

sgn

x v i

b i

x v i

− Φi z v n

i 1 sgn

x v im

j 1



p0ji ji1

f j

y j v

− f j0

≤n

i 1

m



j 1

0

ji ji1 j v j m



j 1

sgn

y v j

d j

y j v

− Φn z v m

j 1 sgn

y v jn

i 1



q0ij 1ij

g i

x v i

− g i0

≤m

j 1

n



i 1

0

ij ij1 i x v i ,

3.10

we have

n



i 1

sgn

x i v

b i

x i v

− Φi z v m

j 1 sgn

y v j

d j

y v j

− Φn z v

≤n

i 1

m



j 1

0

ji 1ji j j v

m



j 1

n



i 1

0

ij ij1 i x v i

3.11

On the other hand, we have

n



i 1

sgn

x i v

b i

x i v

− Φi z v m

j 1 sgn

y v j

d j

y v j

− Φn z v

≥n

i 1

η i x v i

n



i 1 i

z v m

j 1

ω j v j

m



j 1 n

z v

3.12

It follows from3.11 and 3.12 that

Θ

⎛

⎝n

i 1

x v i

m



j 1

v j

⎞

⎠ ≤n

i 1 i

z v m

j 1 n

where

Θ min

⎧

⎨

⎩min1≤i≤n

⎧

⎨

⎩η i−m

j 1

0

ij 1ij i

⎫

⎬

⎭, min1≤j≤m



ω j−n

i 1

0

ji ji1 j

⎫⎬

⎭> 0. 3.14 That is

z v ≤ Θ1%%% Φz v%%%, 3.15 which contradicts our choice of {z v } Hence, Φ satisfies Φ → ∞ as z → ∞.

x∗

1, x∗2, , x n∗, y1∗, y∗2, , y∗mT such thatΦz∗ 0 From the definition of Φ, we know that

z∗ x∗

1, x∗2, , x∗n , y∗1, y∗2, , y m∗Tis the unique equilibrium point of1.1

4 Global Exponential Stability of the Equilibrium

In this section, we will construct some suitable Lyapunov functions to derive the sufficient conditions which ensure that the equilibrium of1.1 is globally exponentially stable

Theorem 4.1 Assume that (H1)–( H4) hold, suppose further that

H5 for each i 1, 2, , n, j 1, 2, , m, the following inequalities are satisfied:

a i η i >

m



j 1

c j 0ij 1ij 

N i , c j ω j >

n



i 1

a i 0ji 1ji 

H6 the impulsive operators I ik x i t and J jk y j t satisfy

I ik x i t k  −γ ik



x i t k  − x∗

i



, 0 < γ ik < 2, i 1, , n, k ∈ Z ,

J jk

y j t k −γ jk



y j t k  − y∗

j



, 0 < γ jk < 2, j 1, , m, k ∈ Z 4.2

Then the unique equilibrium point of system1.1 is globally exponentially stable.

Proof According toTheorem 3.1, we know that 1.1 has a unique equilibrium point z∗

x∗

1, x∗2, , x n∗, y1∗, , y∗mT In view ofH6, it is easy to see that I i x∗

i  0 and J j y∗

j 0

Suppose that zt x1t, x2t, , x n t, y1t, y2t, , y m t T is an arbitrary solution of

1.1 Let u i t x i t − x∗

i , v j t y j t − y∗

j , t≥ 0, then system 1.1 can be rewritten as

uΔi t −a i u i t

⎡

⎣b i u i t −m

j 1

p0ji fjv jt − τ ji

−m

j 1

p ji1

∞ 0

h ji s  f j

v j t − sΔs − r i

⎤

⎦,

i 1, 2, , n, t > 0, t / t k , t ∈ T,

vΔj t −c j



v j t dj



v j t−n

i 1

q0ij g i



u i



t − σ ij



−n

i 1

q1ij

∞ 0

k ij sg i u i t − sΔs − s j

,

j 1, 2, , m, t > 0, t / t k , t ∈ T,

4.3

where, for i 1, 2, , n, j 1, 2, , m,

a i u i t a i



u i ∗i

, b i u i t b i



u i ∗i

− b i



x∗i

,

c j



v j t c j



v j ∗

j



, fi v i t f jv j ∗

j



− f j



y j∗

,



d j



v j t d j



v j ∗

j



− b i



y j∗

, g i u i t g i



u i ∗

i



− b i



x∗i

.

4.4

Also, for all t t k , k∈ Z , i 1, 2, , n, j 1, 2, , m,

u i

t k i

t k

− x∗

i i t k k x i t k  − x∗

i

1− γ ik



x i t k  − x∗

i i t k  − x∗

i i t k |,

v j

t k j

t k

− y∗



y j t k− y∗

j

1− γ jk



y j t k  − y∗

j j t k  − y∗

j v j t k .

4.5

4 Global Exponential Stability of the Equilibrium

In this section, we will construct some suitable Lyapunov functions to derive the sufficient conditions...

Then the generalized exponential function has the following properties

Lemma 2.3 see... which leads to a contradiction Therefore, Φ is an injective on R n