delay range dependent global robust passivity analysis of discrete time uncertain recurrent neural networks with interval time varying delay

Volume 2009, Article ID 415786, 14 pagesdoi:10.1155/2009/415786 Research Article Periodic Solutions and Exponential Stability of a Class of Neural Networks with Time-Varying Delays Corre

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Volume 2009, Article ID 415786, 14 pages

doi:10.1155/2009/415786

Research Article

Periodic Solutions and Exponential

Stability of a Class of Neural Networks with

Time-Varying Delays

Correspondence should be addressed to Mingzhi Xue,whl2762@163.com

Received 8 March 2009; Accepted 17 June 2009

Recommended by Guang Zhang

Employing fixed point theorem, we make a further investigation of a class of neural networks with delays in this paper A family of suﬃcient conditions is given for checking global exponential stability These results have important leading significance in the design and applications of globally stable neural networks with delays Our results extend and improve some earlier publications

Copyrightq 2009 Y Guo and M Xue 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 stability of dynamical neural networks with time delay which have been used in many applications such as optimization, control, and image processing has received much attention recentlysee, e.g., 1 15 Particularly, the authors 3,8,9,14,16 have studied the stability

of neural networks with time-varying delays

As pointed out in8, Global dissipativity is also an important concept in dynamical neural networks The concept of global dissipativity in dynamical systems is a more general concept, and it has found applications in areas such as stability theory, chaos and synchronization theory, system norm estimation, and robust control 8 Global dissipativity of several classes of neural networks was discussed, and some suﬃcient conditions for the global dissipativity of neural networks with constant delays are derived in

8

In this paper, without assuming the boundedness, monotonicity, and diﬀerentiability

of activation functions, we consider the following delay diﬀerential equations:

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xi t −d i tx i t n

j1

aij tf j

xj t n

j1

bij tf j

xj

t − τ ij t

n

j1

c ij t

t

−∞H ij t − sf j

x j sds J i t, i 1, 2, , n,

1.1

where n denotes the number of the neurons in the network, x i t is the state of the ith neuron

at time t, xt x1t, x2t, , x n t T ∈ R n , f xt f1x1t, f2x2t, , f n x n t T ∈

R n denote the activation functions of the jth neuron at time t, and the kernels H ij:0, ∞ →

0, ∞ are piece continuous functions with ∞0 Hij sds h ij < ∞ for i, j 1, 2, , n.

Moreover, we consider model1.1 with τ ij t, d i t, a ij t, b ij t, c ij t, and J i t satisfying

the following assumptions:

A1 the time delays τ ij t ∈ CR, 0, ∞ are periodic functions with a common period

ω > 0 for i, j 1, 2, , n;

A2 c ij t ∈ CR, 0, ∞, a ij t, b ij t, c ij t, J i t ∈ CR, R are periodic functions with a common period ω> 0 and f i ∈ CR, R, i, j 1, 2, , n.

The organization of this paper is as follows InSection 2, problem formulation and preliminaries are given InSection 3, some new results are given to ascertain the global robust dissipativity of the neural networks with time-varying delays.Section 4gives an example to illustrate the eﬀectiveness of our results

2 Preliminaries and Lemmas

For the sake of convenience, two of the standing assumptions are formulated below as follows

A3 |f j u| ≤ p j |u| q j for all u ∈ R, j 1, 2, , n, where p j , q jare nonnegative constants

A4 There exist nonnegative constants p j , j 1, 2, , n, such that |f j u − f j v| ≤ p j |u −

v | for any u, v ∈ R.

Let

τ max

1≤i,j≤nsup

t≥0

The initial conditions associated with system1.1 are of the form

x i s φ i s, s ∈ −τ, 0, i 1, 2, , n, 2.2

in which φ i s is continuous for s ∈ −τ, 0.

For continuous functions φ i defined on −τ, 0, i 1, 2, , n, we set φ 

φ1, φ2, , φ nT If x0 x0

1, x0

1, , x0

nTis an equilibrium of system1.1, then we denote

φ − x0 n

i1

sup

−τ≤t≤0

φ i t − x0

i . 2.3

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Definition 2.1 The equilibrium x0  x0

1, x01, , x0

nT is said to be globally exponentially

stable, if there exist constants λ > 0 and m ≥ 1 such that for any solution xt 

x1t, x2t, , x n t T of1.1, we have

x i t − x0

i ≤ m φ − x0 e −λt 2.4

for t ≥ 0, where λ is called to be globally exponentially convergent rate.

Lemma 2.2 17 If ρK < 1 for matrix K k ijn ×n ≥ 0, then E − K−1≥ 0, where E denotes

the identity matrix of size n.

3 Periodic Solutions and Exponential Stability

We will use the coincidence degree theory to obtain the existence of a ω-periodic solution to

systems1.1 For the sake of convenience, we briefly summarize the theory as follows

Let X and Z be normed spaces, and let L : Dom L ⊂ X → Z be a linear mapping and

be a continuous mapping The mapping L will be called a Fredholm mapping of index zero

if dimKer L codimIm L < ∞ and Im L is closed in Z If L is a Fredholm mapping of index zero, then there exist continuous projectors P : X → X and Q : Z → Z such that Im P Ker L and Im L Ker Q ImI −Q It follows that L | Dom L∩Ker P : I −PX → Im L is invertible.

We denote the inverse of this map by K p If Ω is a bounded open subset of X, the mapping N

is called L-compact on Ω if QNΩ is bounded and K p I − QN : Ω → X is compact Because ImQ is isomorphic to Ker L, there exists an isomorphism J : ImQ → Ker L.

LetΩ ⊂ R n be open and bounded, f ∈ C1Ω, R n ∩ CΩ, R n and y ∈ R n \ f∂Ω ∪ S f , that is, y is a regular value of f Here, S f {x ∈ Ω : J f x 0}, the critical set of f, and J f is

the Jacobian of f at x Then the degree deg{f, Ω, y} is defined by

deg

f, Ω, y

x ∈f−1y

with the agreement that the above sum is zero if f−1y ∅ For more details about the degree

theory, we refer to the book of Deimling18

Lemma 3.1 continuation theorem 19, page 40 Let L be a Fredholm mapping of index zero, and

let N be L-compact on Ω Suppose that

a for each λ ∈ 0, 1, every solution x of Lx λNx is such that x ∈ ∂Ω;

b QNx / 0 for each x ∈ ∂Ω ∩ Ker L and

deg{JQN, Ω ∩ Ker L, 0} / 0. 3.2

Then the equation Lx Nx has at least one solution lying in Dom L ∩ Ω.

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For the simplicity of presentation, in the remaining part of this paper, for a continuous

function g : 0, ω → R, we denote

g∗ max

ω

0

g tdt. 3.3

ρ K < 1, then system 1.1 has at least a ω-periodic solution.

Proof Take X Z {xt x1t, x2t, , x n t T ∈ CR, R n : xt xt ω, for all t ∈ R},

and denote

|x i| max

t ∈0,ω |x i t|, i 1, 2, , n, x max

Equipped with the norms · , both X and Z are Banach spaces Denote

Δx i , t : −d i tx i t n

j1

a ij tf j

x j t n

j1

b ij tf j

x j

t − τ ij t

n

j1

cij t

t

−∞Hij t − sf j

xj sds J i t.

3.5

Since

n

j1

c ij t

t

−∞H ij t − sf j

x j sdsn

j1

c ij t

∞

0

H ij sf j

x j t − sds, 3.6

then, for any xt ∈ X, because of the periodicity, it is easy to check that

Δx i , t −d i tx i t n

j1

a ij tf j

x j t n

j1

b ij tf j

x j

t − τ ij t

n

j1

c ij t

∞

0

H ij sf j

x j t − sds J i t ∈ Z.

3.7

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L : Dom L x ∈ X : x ∈ C1R, R n x −→ x· ∈ Z,

P : X x −→ 1

ω

0

x tdt ∈ X,

Q : Z z −→ 1

ω

0

z tdt ∈ Z,

N : X x −→ Δx i, t ∈ Z.

3.8

Here, for any W w1, w2, , w nT ∈ R n , we identify it as the constant function in X or Z

with the value vector W w1, w2, , w nT Then system1.1 can be reduced to the operator

equation Lx Nx It is easy to see that

Ker L R n ,

Im L

z ∈ Z : 1

ω

0

z tdt 0

, which is closed in Z,

dimKer L codimIm L n < ∞,

3.9

and P , Q are continuous projectors such that

ImP ker L, Ker Q ImL ImI − Q. 3.10

It follows that L is a Fredholm mapping of index zero Furthermore, the generalized inverse

to L K p : ImL → Ker P ∩ Dom L is given by

K p z i t 

t

0

zi sds − 1

ω

0

s

0

zi vdv ds. 3.11

Then,

QNx i t  1

ω

0

Δx i, s ds,

K p I − QNx i t 

t

0

Δx i, s ds − 1

ω

0

t

0

Δx i, s ds dt

1

2− t

ω

0

Δx i, s ds.

3.12

Clearly, QN and K p I − QN are continuous For any bounded open subset Ω ⊂ X, QNΩ

is obviously bounded Moreover, applying the ArzelaCAscoli theorem, one can easily show

that K p I − QNΩ is compact Therefore, N is L-compact on with any bounded open subset

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Ω ∈ X Since ImQ Ker L, we take the isomorphism J of ImQ onto Ker L to be the identity

mapping

Now, we reach the point to search for an appropriate open bounded setΩ for the

application of the continuation theorem corresponding to the operator equation Lx λNx,

λ ∈ 0, 1, and we have

xi t λΔx i , t for 1 1, 2, , n. 3.13

Assume that x xt ∈ X is a solution of system 1.1 for some λ ∈ 0, 1 Integrating both

sides of3.13 over the interval 0, ω, we obtain

0

ω

0

xi tdt λ

ω

0Δx i, t dt. 3.14 Then

ω

0

d i tx i tdt 

ω

0

⎧

⎨

⎩

n

j1

a ij tf j

x j t n

j1

b ij tf j

x j

t − τ ij t

n

j1

c ij t

∞

0

H ij sf j

x j t − sds J i t

⎫

⎬

⎭dt.

3.15

Noting that

f j u ≤ p j |u| q j ∀u ∈ R, j 1, 2, , n, 3.16

we get

|x i|∗d i≤n

j1

|a ij | |b ij | |c ij h ij|p j x j ∗ n

j1

|a ij | |b ij | |c ij h ij|q j |J i |. 3.17

It follows that

|x i|∗≤ 1

di

n

j1

|a ij | |b ij | |c ij h ij|p j x j ∗ 1

di

⎧

⎨

⎩

n

j1

|a ij | |b ij | |c ij h ij|q j |J i|

⎫

⎬

⎭. 3.18

Note that each x i t is continuously diﬀerentiable for i 1, 2, , n, and it is certain that there exists t i ∈ 0, ω such that |x i t i | |x i t|∗ Set

D D1, D2, , D nT , D i 1

di ω

⎧⎨

⎩

n

j1

a

ij b ij c ij h ij q j |J i|⎫⎬

⎭. 3.19

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In view of ρK < 1 andLemma 2.2, we haveE − K−1D l l1, l2, , lnT ≥ 0, where l iis given by

lin

j1

kij lj D i, i 1, 2, , n. 3.20

Let

Ω x1, x2, , xnT ∈ R n;|x i | ≤ l i, i 1, 2, , n. 3.21

Then, for t ∈ t i , t i ω, we have

|x i t| ≤ |x i t i|

t

t i

D |x i t|dt

≤ |x i t|∗

t i ω

t i

D |x i t|dt

≤ 1

di

n

j1

|a ij | |b ij | |c ijhij|pj xj ∗

1

d i

⎧

⎨

⎩

n

j1

⎫

⎬

⎭

t i ω

t i

D |x i t|dt

≤ 1

di ω

n

j1

|a ij | |b ij | |c ij hij|pj xj ∗

1

d i ω

⎧⎨

⎩

n

j1

⎫

⎬

⎭

≤n

j1

kij lj D i

 l i,

3.22

where D denotes the right derivative Clearly, l i , i 1, 2, , n, are independent of λ Then there are no λ ∈ 0, 1 and x ∈ Ω such that Lx λNx When u x1, x2, , x nT ∈ ∂Ω ∩ Ker L ∂Ω ∩ R n , u is a constant vector in R nwith|x i | l i , i 1, 2, , n Note that QNu

JQNu; when u ∈ Ker L, it must be

QNu i −d i n

j1

a ij b ij c ij h ij

f j

x j

J i 3.23

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We claim that

|QNu i | > 0 for i 1, 2, , n. 3.24

On the contrary, suppose that there exists some i such that |QNu i | 0, that is,

dixin

j1

aij b ij c ijhij

fj

xj

Then, we have

li |x i|

1

d i

n

j1

f j

x j

J i

≤ 1

di

n

j1

|a ij | |b ij | |c ijhij|pjlj

1

d i

⎧

⎨

⎩

n

j1

⎫

⎬

⎭

≤ 1

di ω

n

j1

1

d i ω

⎧⎨

⎩

n

j1

⎫

⎬

⎭

n

j1

kijlj D i

 l i,

3.26

which is a contradiction Therefore,

QNu / 0 for any u ∈ ∂Ω ∩ Ker L ∂Ω ∩ R n 3.27

Consider the homotopy F : Ω ∩ Ker L × 0, 1 → Ω ∩ Ker L defined by

F

u, μ

 μ diag−d1, −d2, , −d n

u 1− μQNu, 3.28

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u, μ ∈ Ω ∩ Ker L × 0, 1 Note that F·, 0 JQN; if Fu, μ 0, then, as before, we have

|x i| 1− μ

di

n

j1

f j

x j

J i

≤ 1

d i

n

j1

|a ij | |b ij | |c ijhij|pj xj

1

di

⎧

⎨

⎩

n

j1

⎫

⎬

⎭

≤ 1

d i

n

j1

1

di

⎧

⎨

⎩

n

j1

⎫

⎬

⎭

<

n

j1

k ij l j D i

 l i ,

3.29

Hence

F

u, μ

/

 0, for u, μ

∈ ∂Ω ∩ Ker L × 0, 1. 3.30

It follows from the property of invariance under a homotopy that

deg{JQN, Ω ∩ Ker L, 0} deg{F·, 0, Ω ∩ Ker L, 0}

 deg{F·, 1, Ω ∩ Ker L, 0} deg diag

−d1, −d2, , −d n

/

 0.

3.31

Thus, we have shown thatΩ satisfies all the assumptions ofLemma 3.1 Hence, Lu Nu has

at least one ω-periodic solution on Dom L∩ Ω This completes the proof

When c ij 0, 1.1 turns into the following system:

xi t −d i tx i t n

j1

aij tf j

xj t n

j1

bij tf j

xj t − τt J i t, i 1, 2, , n P

Corollary 3.3 Let (A1)–(A3) hold, k ij 1/d i ω|a ij | |b ij |p j, and K k ijn ×n If ρ K < 1,

then systemP has at least a ω-periodic solution.

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k ijn ×n If ρ K < 1, and that

d i−n

j1

|a ij | |b ij | |c ij h ij|p j e d∗i τ > 0, 3.32

then system1.1 has exactly one ω-periodic solution Moreover, it is globally exponentially stable.

Proof Let C C−τ, 0, R n with the supnorm ϕ sup s ∈−τ,0;1≤i≤n |ϕ i s|, ϕ ∈ C As usual, if

−∞ ≤a ≤ b≤ ∞ and ψ ∈ C−τ a, b, R n , then for t ∈ a, b we define ψ t ∈ C by ψ t θ 

ψ t θ, θ ∈ −τ, 0 From A4, we can get |f j u| ≤ p j |u| |f j 0|, j 1, 2, , n Hence, all the

hypotheses inTheorem 3.2hold with q j |f j 0|, j 1, 2, , n Thus, system 1.1 has at least

one ω-periodic solution, say xt x1t, x2t, , x n t T Let x t x1t, x2t, , x n t T

be an arbitrary solution of system1.1 For t ≥ 0, a direct calculation of the right derivative

D |x i t − x i t| of |x i t − x i t| along the solutions of system 1.1 leads to

D |x i t − x i t| D

sgnxi t − x i t}x i t − x i t

≤ −d i t|x i t − x i t| n

j1

a ij t

f j

x j t− f j

x j t

n

j1

b ij t

f j

x j

t − τ ij t− f j

x j

t − τ ij t

n

j1

c ij t ∞

0

k ij sf j

x j t − s− f j

x j t − sds

≤ −d i t|x i t − x i t| n

j1

a ij t p j x j t − x j t

n

j1

b ij t p j x j

t − τ ij t− x j

t − τ ij t

n

j1

c ij th ij p jsup

−τ≤s≤t

x j s − x j s

≤ −d i t|x i t − x i t|

n

j1

aij t b ij t c ij th ij pj sup

−τ≤s≤t

xj s − x j t .

3.33

Let z i t |x i t − x i t| Then 3.33 can be transformed into

D z i t ≤ −d i tz i t n

j1

a ij t b ij t c ij th ij p jsup

the Jacobian of f at x Then the degree...

Assume that x xt ∈ X is a solution of system 1.1 for some λ ∈ 0, 1 Integrating both

sides of 3.13 over the interval 0, ω, we obtain

0

ω... has exactly one ω-periodic solution Moreover, it is globally exponentially stable.

Proof Let C C−τ, 0, R n with the supnorm ϕ sup s ∈−τ,0;1≤i≤n

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