Global exponential stability of cycle associative neural network with constant delays

SHI Renxiang

( School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China )

Globalexponentialstabilityofcycleassociativeneuralnetworkwithconstantdelays

SHI Renxiang*

( School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China )

The global exponential stability of cycle associative neural network with constant delays is discussed. During the discussion, by constructing homeomorphism mapping, it is demonstrated that there exists an equilibrium point which is unique for this system, then the global exponential stability of the unique equilibrium point is testified by constructing proper Lyapunov function. Similar to previous work about neural network stability, under the assumption that the activation function about neuron satisfies Lipschitz condition and the matrix constructed by correlation coefficient satisfies given condition, the dynamics of global exponential stability forn-layer neural network with constant delays are obtained. The results contain that when the passive rate of neuron is sufficiently large, the neural network is global exponential stable.

exponential stability; equilibrium point; neural network; Lyapunov function

0 Introduction

The dynamical behaviors of delayed neural networks have attracted increasing interest for their intense application. Especially, there are many works about stability of neural network[1-8]. In Lits. [2-3], the authors discussed the static network with S-type distributed delays. In Lit. [4], the author discussed the global exponential stability of a class of neural networks with delays by natureM-matrix. In Lits. [3-5], the authors discussed the global exponential stability of the one-layer neural network. At the same time, the stability of bidirectional associative memory neural networks of the two-layers with delays has also been studied by many researchers[6-8]. In Lits. [5-6] the authors discussed the existence of equilibrium point and the global exponential stability by homeomorphism and constructing proper Lyapunov function. Inspired by above work, we should discuss the exponential stability ofn-layers neural networks with constant delays, which should be taken as general form for work[6].

1 Model and preliminaries

In this paper, we should discuss the cycle associative neural network of then-layers with constant delays:

u.

u.

u.

(1)

(2)

Letτ=max(τ1,τ2,…,τn), initial conditions for network (1) are of the form

φ=(φ1…φl1φl1+1…φl1+l2…φl1+l2+…+ln-1+1…φl1+l2+…+ln)∈C=C([-τ,0],Rl1+l2+…+ln)

(u1(t,φ),u2(t,φ),…,un(t,φ))= (u1,1(t,φ) …ul1,1(t,φ)u1,2(t,φ) …ul2,2(t,φ) …u1,n(t,φ) …uln,n(t,φ))

Denotex=(u1u2…un)=(u1,1…ul1,1u1,2…ul2,2…u1,n…uln,n). Hence,we write network (1) as

u.T2=-A2uT2+W2S(2)(u3(t-τ2))+J(2)

…

u.Tn=-AnuTn+WnS(n)(u1(t-τn))+J(n)

whereA1=diag{a1,1,…,al1,1},A2=diag{a1,2,…,al2,2}, …,An=diag{a1,n,…,aln,n}, andW1=(w1,i1,i2)l1×l2,W2=(w2,i2,i3)l2×l3, …,Wn=(wn,in,i1)ln×l1.

Theorem1For network (1), the assumption (2) and condition (T) hold. Then the neural network (1) has a unique equilibrium point.

Theorem2For network (1), the assumption (2) and condition (T) hold. Then the equilibrium point of neural network (1) is global exponential stable.

We should discuss the existence and uniqueness of the equilibrium point, the global exponential stability, and compare our result with previous results and give an example.

2 The existence and uniqueness of equilibrium point

For convenience we state the following lemma,which is special case of lemma (2.1) in Lit. [6].

Lemma1Given any real vectorsX,Yof appropriate dimensions, then the following inequality holds

…

Let

(3)

LetS(x)=(S(n)(x)S(1)(x) …S(n-1)(x))T,xandybe two vectors such thatx≠y. Under the assumption (2) on the activation functionsx≠yimply two cases: (i)x≠yandS(x)-S(y)≠0; (ii)x≠yandS(x)-S(y)=0, now we write

H1(x)-H1(y)=-A1u1x+A1u1y+W1(S(1)(x)-S(1)(y))

H2(x)-H2(y)=-A2u2x+A2u2y+W2(S(2)(x)-S(2)(y)) …

Hn(x)-Hn(y)=-Anunx+Anuny+Wn(S(n)(x)-S(n)(y))

(4)

whereu1x=(u1,1x…ul1,1x)T,u1y=(u1,1y…

ul1,1y)T,u2x=(u1,2x…ul2,2x)T,u2y=(u1,2y…

ul2,2y)T, …,unx=(u1,nx…uln,nx)T,uny=(u1,ny…uln,ny)T.

First, we consider the case (i). In this case,there existsk∈(1,2,…,n) such thatS(k)(x)≠S(k)(y). Multiplying both sides of the first equation in Eq. (4) by 2(S(n)(x)-S(n)(y))TP1, results in

2(S(n)(x)-S(n)(y))TP1(H1(x)-H1(y))= -2(S(n)(x)-S(n)(y))TP1(A1u1x-A1u1y)+ 2(S(n)(x)-S(n)(y))TP1W1(S(1)(x)-S(1)(y))

we have

(S(n)(x)-S(n)(y))TP1(A1u1x-A1u1y)≥ (S(n)(x)-S(n)(y))TP1A1(α(n))-1(S(n)(x)-S(n)(y))

It follows from Lemma 1

(5)

Similarly

(6)

…

(7)

which imply that

Ψ(x,y)=(2(S(n)(x)-S(n)(y))T2(S(1)(x)-S(1)(y))T… 2(S(n-1)(x)-S(n-1)(y))T)diag{P1,P2,…,Pn}× (H(x)-H(y))

Ψ(x,y)≤-(S(1)(x)-S(1)(y))TΩ1(S(1)(x)-S(1)(y))-(S(2)(x)-S(2)(y))T×Ω2(S(2)(x)-S(2)(y))-…- (S(n)(x)-S(n)(y))TΩn(S(n)(x)-S(n)(y))<0

(8)

That isH(x)≠H(y). Sincediag{P1,P2,…,Pn} is a positive diagonal matrix,we prove thatH(x)-H(y)≠0whenx≠yandS(x)≠S(y).

Now we consider the case (ii). In view ofx≠yandS(x)-S(y)=0,we have

which implies thatH(x)≠H(y) forx≠y.

Ψ(x,0)≤-λmin[(S(x)-S(0))T(S(x)-S(0))]

whereλmindenotes the minimum eigenvalue of the positive definite matricesΩ1,Ω2, …,Ωn. Similar to Lemma 2.2 in Lit. [6], we obtain

Hence

3 The global exponential stability of the equilibrium point

v.

1(t)=-A1v1(t)+W1f(1)(v2(t-τ1))

v.

2(t)=-A2v2(t)+W2f(2)(v3(t-τ2))

…

v.

n(t)=-Anvn(t)+Wnf(n)(v1(t-τn))

(9)

i1=1,2,…,l1

The Lipschitz condition implies that

ProofofTheorem2We employ the following Lyapunov function

V(v1(t),v2(t),…,vn(t),t)=ε1V1(v1(t),v2(t),…,vn(t))+V2(v1(t),v2(t),…,vn(t),t)

(10)

where

First we compute the derivative ofValong trajectories of Eq. (9), then determine positive constantε1and positive definite matricesR1,R2, …,Rn.

V.

(v1(t),v2(t),…,vn(t),t)=ε1

V.

1(v1(t),v2(t), …,vn(t))+

V.

2(v1(t),v2(t), …,vn(t),t)

where

V.

and

V.

2(v1(t),v2(t),…,vn(t),t)= 2f(1)T(v2(t))P2[-A2v2(t)+W2f(2)(v3(t-τ2))]+2f(2)T(v3(t))P3[-A3v3(t)+W3f(3)(v4(t-τ3))]+…+ 2f(n)T(v1(t))P1[-A1v1(t)+W1f(1)(v2(t-τ1))]+f(1)T(v2(t))R1f(1)(v2(t))-f(1)T(v2(t-τ1))×R1f(1)(v2(t-τ1))+f(2)T(v3(t))R2f(2)(v3(t))-f(2)T(v3(t-τ2))R2f(2)(v3(t-τ2))+…+f(n)T(v1(t))Rnf(n)(v1(t))-f(n)T(v1(t-τn))×Rnf(n)(v1(t-τn))

Rewriting

V.

1as

V.

It follows from Lemma 1 that

V.

we get

-f(1)T(v2(t))P2A2v2(t)≤ -f(1)T(v2(t))P2A2(α(1))-1f(1)(v2(t)) -f(2)T(v3(t))P3A3v3(t)≤ -f(2)T(v3(t))P3A3(α(2))-1f(2)(v3(t)) … -f(n)T(v1(t))P1A1v1(t)≤ -f(n)T(v1(t))P1A1(α(n))-1f(n)(v1(t))

V.

2≤-f(1)T(v2(t))2P2A2(α(1))-1f(1)(v2(t))-f(2)T(v3(t))2P3A3(α(2))-1f(2)(v3(t))-…-f(n)T(v1(t))2P1A1(α(n))-1f(n)(v1(t))+ 2(f(1)T(v2(t))P2(W21K2-1)(K2W22)×f(2)(v3(t-τ2)))+2(f(2)T(v3(t))×P3(W31K3-1)(K3W32)f(3)(v4(t-τ3)))+…+ 2(f(n)T(v1(t))P1(W11K1-1)(K1W12)×f(1)(v2(t-τ1)))+f(1)T(v2(t))×R1f(1)(v2(t))-f(1)T(v2(t-τ1))×R1f(1)(v2(t-τ1))+f(2)T(v3(t))×R2f(2)(v3(t))-f(2)T(v3(t-τ2))×R2f(2)(v3(t-τ2))+…+f(n)T(v1(t))×Rnf(1)(v1(t))-f(n)T(v1(t-τn))×Rnf(n)(v1(t-τn))

That

V.

2is bounded by Lemma 1.

V.

V.

2(v1(t),v2(t),…,vn(t),t)≤ -f(1)T(v2(t))(Ω1-2ε2Il2+ε2Il2)f(1)(v2(t))-f(2)T(v3(t))(Ω2-2ε2Il3+ε2Il3)f(2)(v3(t))-…-f(n)T(v1(t))(Ωn-2ε2Il1+ε2Il1)f(n)(v1(t))-ε2f(1)T(v2(t-τ1))f(1)(v2(t-τ1))-ε2f(2)T(v3(t-τ2))f(2)(v3(t-τ2))-…-ε2f(n)T(v1(t-τn))f(n)(v1(t-τn))≤ -ε2f(1)T(v2(t))f(1)(v2(t))-ε2f(2)T(v3(t))f(2)(v3(t))-…-ε2f(n)T(v1(t))f(n)(v1(t))-ε2f(1)T(v2(t-τ1))f(1)(v2(t-τ1))-ε2f(2)T(v3(t-τ2))f(2)(v3(t-τ2))-…-ε2f(n)T(v1(t-τn))f(n)(v1(t-τn))

Chooseε1>0 such thatMε1≤ε2, we have

V.

εε1+εpθ-ε1a+rθ2ετeετ<0

(11)

We obtain

Noting that

(12)

Integrating both sides of Eq. (12) from 0 tos, concerned with Eq. (11), similar to Theorem 2.3 in Lit. [6], we obtain

Therefore

(13)

According to Eq. (13) and the above inequality

that is,

(14)

Inequality (14) implies that the origin of system (9) is global exponential stable.

4 Comparison with previous results

Now we compare our results with the previous result in Lit. [6], where authors gave a new sufficient condition for the existence, uniqueness and global stability of the equilibrium point for BAM neural network with constant delays:

a.

i=1,2,…,n

z.

j=1,2,…,m

(15)

We could obtain the result in Lit. [6] from our work,whenn=2, network (1) is similar to Eq. (15), Theorems (1), (2) became Lemma (2.2), Theorem (2.3) in Lit. [6].

Example1Assume the parameters in Eq. (9) are given as follows:

andA1=A2=…=An=aIn,Q1=Q2=…=Qn=rIn, (α(1))-1=(α(2))-1=…=(α(n))-1=P1=P2=…=Pn=W11=W21=…=Wn1=In, whereInisn×nidentity matrix. Hence, we have

5 Conclusion

We study a class of neural networks with constant delays in this paper, comparing with previous work[6], we expand the result of neural network from 2-layer ton-layer by constructing Lyapunov function. Our result includes the result of work in Lit. [6].

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1000-8608(2017)05-0537-08

带有常时滞循环耦合神经网络的全局指数稳定性

石 仁 祥*

( 上海交通大学 数学科学学院, 上海 200240 )

讨论了带有常时滞循环耦合神经网络的全局指数稳定性，在讨论过程中通过构造同胚映射论证了该系统平衡点的存在性与唯一性，再通过构造合适的Lyapunov函数论证唯一平衡点是全局指数稳定的．类似于已有的神经网络稳定性方面工作，在神经元的激励函数满足Lipschitz条件且相关系数构成矩阵也满足给定条件下，得到n层带有常时滞的神经网络全局指数稳定的动力学性质．所得结果同时也蕴含当神经元的衰减速率足够大时，神经网络是全局指数稳定的．

指数稳定性；平衡点；神经网络；Lyapunov函数

O175.13；TP183

A

2016-10-07；

2017-06-20.

江苏省自然科学基金资助项目(BK20131285).

石仁祥*(1983-)，男，博士生，E-mail:srxahu@aliyun.com.

SHI Renxiang*(1983-), Male, Doc., E-mail:srxahu@aliyun.com.

10.7511/dllgxb201705015

Receivedby2016-10-07；Revisedby2017-06-20.

SupportedbyNatural Science Foundation of Jiangsu (BK20131285).