# Existence and Stability of Almost Periodic Solution for a Stochastic Cellular Neural Network with External Perturbation

Authors: Hui Zhou; Zongfu Zhou; Wei Jiang
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

A class of stochastic cellular neural networks with external perturbation is investigated. By employing fixed points principle and some stochastic analysis techniques, we establish some sufficient conditions for existence and exponential stability of a quadratic mean almost periodic solution of the model. The new criteria not only improve some classical results but also are applied in real problems due to the changes of external input.


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## Body

## 1. Introduction

Cellular neural networks (CNNs) and the various generalizations have attracted many scientists' attention due to their important applications, such as associative memory, optimization problems, parallel computation, and so on [1–7]. Huang et al. [7] studied almost periodic solutions of a delayed cellular neural networks as follows:
(1)dxi(t)dt=-ci(t)xi(t)+∑j=1N‍aij(t)gj(xj(t-τ(t)))+Ii(t),hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhi=1,2,…,N.
The authors obtained some good criteria ensuring exponential global attractivity of almost periodic solution to (1).The concept of almost periodic stochastic process is of great importance in probability for investigating stochastic process [8]. Recently, the existence and stability of almost periodic solution to stochastic cellular neural networks were considered [9]. To the best of our knowledge, there are few works about the quadratic mean almost periodic solution for stochastic cellular neural networks. Motivated by [7–13], in this paper, we will consider the existence and exponential stability of quadratic mean almost periodic for stochastic cellular neural networks with distributed delay as follows:
(2)dxi(t)=[-ci(t)xi(t)+∑j=1n‍aij(t)fj(xi(t-τij(t)))+∑j=1n‍bij(t)gj(∫-∞0‍kj(s)xi(s+t)ds)]dt+Ii(t)dt+σi(t,xi(t))dωi(t),i=1,2,…,n,
where t∈[0,+∞) and x(t)=(x1(t),x2(t),…,xn(t))∈Rn,n is the number of neurons in the network; xi denotes the state variable of the ith neuron; fj and gj denote the activation function of the jth neuron; the feedback function aij and bij indicate the strength of the neuron interconnections within the network; ci represents an amplification function; Ii represents external input; σi can be viewed as a stochastic perturbation on the neuron states and ωi is a Brownian motion; τij is a variable delay function of the neuron xi and the kernel function kj satisfies ∫-∞0‍kj(t)=1. Some sufficient conditions ensuring the existence and stability of square mean almost periodic solutions are shown. The results in this paper improve some previous results and are applied in real problems such as signal processing and the design of networks and secure communication.The rest of the paper is organized as follows. In Section2, we introduce some definitions, lemmas, and some notations which would be useful to get the main results. In Section 3, the main results of existence and stability to (2) are obtained.


## 2. Preliminaries

Now let us state the following definitions and lemmas, which will be used to prove our main results.Let(𝔹,∥·∥) be a Banach space and let (Ω,𝔽,ℙ) be a probability space. Define Lp(ℙ;𝔹) for p≥1 to be the space of all 𝔹-value random variable V such that
(3)E∥V∥p=∫Ω‍∥V∥pdℙ<∞.
It is easy to find that Lp(ℙ,𝔹) is a Banach space when it is equipped with its natural norm ∥·∥p defined by
(4)∥V∥p:=(∫Ω‍∥V∥pdℙ)1/p.Definition 1.
A stochastic processX:R→Lp(ℙ;𝔹) is said to be continuous whenever
(5)lim⁡t→sE∥X(t)-X(s)∥p=0.Definition 2.
A stochastic processX:R→Lp(ℙ;𝔹) is said to be stochastically bounded whenever
(6)lim⁡N→∞sup⁡t∈Rℙ{∥X(t)-X(s)∥>N}=0.Definition 3.
A stochastic processX:R→Lp(ℙ;𝔹) is said to be p-mean almost periodic if for each ε>0 there exists l(ε)>0 such that any interval of length l(ε) contains at least a number τ for which
(7)sup⁡t∈RE∥X(t+τ)-X(t)∥p<ε.
The number τ will be called an ε-translation of X and the set of all ε-translation of X is denoted by ℵ(ε,X).The collection of all stochastic processesX:R→Lp(ℙ;𝔹) which are p-mean almost periodic is denoted by AP(R;Lp(ℙ;𝔹)). Let CUB(R;Lp(ℙ;𝔹)) denote the collection of all stochastic processes X:R→Lp(ℙ;𝔹), which are continuous and uniformly bounded. Obviously, CUB(R;Lp(ℙ,𝔹)) is a Banach space when it is equipped with the norm
(8)∥X∥∞=sup⁡t∈R(E∥X(t)∥p)1/p.Lemma 4 (see [8]).
A
P
(
R
;
L
p
(
ℙ
;
𝔹
)
)
⊂
C
U
B
(
R
;
L
p
(
ℙ
;
𝔹
)
) is a closed subspace.Lemma 5 (see [8]).
IfX belongs to AP(R;Lp(ℙ;𝔹)), then (i)
the mappingt→E∥X(t)∥p is uniformly continuous;(ii)
there exists a constantM>0 such that E∥X(t)∥p≤M, for each t∈R;(iii)
X is stochastically bounded.Let(𝔹1,∥·∥1) and (𝔹2,∥·∥2) be Banach spaces and let Lp(ℙ;𝔹1) and Lp(ℙ;𝔹2) be their corresponding Lp-spaces, respectively.Definition 6.
A functionf:R×Lp(ℙ;𝔹1)→Lp(ℙ;𝔹2),(t,y)→f(t,y), which is jointly continuous, is said to be p-mean almost periodic in t∈R uniformly in y∈𝕂 where 𝕂⊂Lp(ℙ;𝔹1) is compact if, for any ε>0, there exists l(ε,𝕂)>0 such that any interval of length l(ε,𝕂) constants at least a number τ for which
(9)sup⁡t∈R(E∥f(t+τ,y)-f(t,y)∥p)1/p<ε,
for each stochastic process y:R→𝕂. The number τ will be called an ε-translation of f and the set of all ε-translation of f is denoted by ℵ(ε,f,𝕂).Lemma 7 (see [8, 9]).
Letf:R×Lp(ℙ;𝔹1)→Lp(ℙ;𝔹2), (t,x)→f(t,x) be a p-mean almost periodic process in t∈R uniformly in x∈𝕂, where 𝕂⊂Lp(ℙ;𝔹1) is compact. Suppose that f is Lipschitzian in the following sense:
(10)E∥f(t,x)-f(t,y)∥2p≤ME∥x-y∥1p
for all x,y∈Lp(ℙ;𝔹1) and for each t∈R, where M>0. Then for any p-mean almost periodic process ϕ:R→Lp(ℙ;𝔹1)); then stochastic process t→f(t,ϕ(t)) is p-mean almost periodic.We need to introduce the following notations. For every real sequenceα=(αn) and a continuous stochastic process f:R→Lp(ℙ;𝔹), if lim⁡n→∞E∥f(t+αn)∥p exists, we define TαE∥f∥p=lim⁡n→∞E∥f(t+αn)∥p. Like the proof of Fink [14], we have the following lemma.Lemma 8.
f
:
R
→
L
p
(
ℙ
;
𝔹
) is p-mean almost periodic if and only if f is continuous and, for each α=(αn), there exists a subsequence α′ of (αn) such that Tα′E∥f∥p=E∥g∥p uniformly on R.Lemma 9.
Ifu(t),g(t):R→L2(ℙ;𝔹) are square almost periodic stochastic process, then u(t-g(t)) is square mean almost periodic.Proof.
It is obvious thatu(t-g(t)) is continuous for t∈R; that is, lim⁡t→sE∥u(t-g(t))-u(s-g(s))∥2=0. For any sequence α′=(αn′), since u(t),g(t):R→L2(ℙ;𝔹) are square almost periodic, we have
(11)TαE∥u(t)∥2=E∥u¯(t)∥2,TαE∥g(t)∥2=E∥g¯(t)∥2
uniformly for t∈R. On the other hand, since u(t) is almost periodic, it is uniformly continuous on R. For any ε>0, there exists a positive number δ(ε), such that |t1-t2|<δ implies that u(t1)-u(t2)<ε. From (11), there exists a positive integer N, when n>N, we have
(12)E∥u(t+αn)-u¯(t)∥2<ε4,E∥g(t+αn)-g¯(t)∥2<min⁡{ε4,δ},t∈R.
Since (a+b)2≤2a2+2b2, we have
(13)E∥u(t+αn-g(t+αn))-u¯(t-g¯(t))∥2=E∥u(t+αn-g(t+αn))-u(t+αn-g¯(t))+u(t+αn-g¯(t))-u¯(t-g¯(t))∥2≤2E∥u(t+αn-g(t+αn))-u(t+αn-g¯(t))∥2+2E∥u(t+αn-g¯(t))-u¯(t-g¯(t))∥2≤2·ε4+2·ε4=ε,
when n>N. Thus, u(t-g(t)) is square mean almost periodic.


## 3. Main Results

In this section, we state and prove our main results concerning the existence and stability of square mean almost solutions of (2). Throughout the rest of the paper, the following assumptions are satisfied.(H1)
The functionsaij,bij,ci,Ii:R→L2(ℙ;𝔹)(i,j=1,2,…,n) are square mean almost periodic functions, where inf⁡t∈Rci(t)>0(i=1,2,…,n).(H2)
The activation functionsfj,gj:R→L2(ℙ;𝔹) are square mean almost periodic, and fj and gj are Lipschitz in the following sense: there exist Lfj>0 and Lgj>0 for which
(14)E∥fj(u)-fj(v)∥2≤Lfj∥u-v∥2,hhhhhhhhhhhhhhu,v∈L2(ℙ;𝔹),hhhhhhhhhhhhhhhhj=1,2,…,n,E∥gj(u)-gj(v)∥2≤Lgj∥u-v∥2,hhhhhhhhhhhhhhhu,v∈L2(ℙ;𝔹),hhhhhhhhhhhhhhhhhj=1,2,…,n.(H3)
The functionsσi(t,xi(t)):R×L2(ℙ;𝔹)→L2(ℙ;𝔹) are square mean almost periodic in t∈R uniformly in x∈S(S⊂L2(ℙ;𝔹) being a compact subspace). Moreover, σi is Lipschitz in the following sense: there exists Lσi>0 for which
(15)E∥σi(u)-σi(v)∥2≤Lσi∥u-v∥2,hhhhhhhhhhhhhhhu,v∈L2(ℙ,𝔹),hhhhhhhhhhhht∈R,i=1,2,…,n.Let denote the signs h*:=sup⁡t∈R⁡h(t), h*:=inf⁡t∈Rh(t) and z*:=sup⁡(t,x)∈R×L2(ℙ;𝔹)z(t,x).Theorem 10.
Assume that conditions (H1)–(H3) are satisfied and3(a*2n2Lf*+b*2n2Lg*+Lσ*)<c*2; then (2) has a unique square mean almost periodic solution.Proof.
By (2), we can obtain that
(16)xi(t)=∫-∞t‍e-ci(s)(t-s)×∑j=1n[(∫-∞0‍kj(θ)xi(s+θ)dθ)aij(s)fj(xi(s-τij(s)))‍+bij(s)gj(∫-∞0‍kj(θ)xi(s+θ)dθ)]ds+∫-∞t‍e-ci(s)(t-s)Ii(s)ds+∫-∞t‍e-ci(s)(t-s)σi(s,xi(s))dωi(s),i=1,2,…,n,
for all t∈R, xi(t)(i=1,2,…,n) given by (16) is the solution to (2).
Define(𝕃xi)(t)=∑m=14(ϕm)xi(t), where
(17)(ϕ1xi)(t)=∫-∞t‍e-ci(s)(t-s)×∑j=1n‍aij(s)fj(xi(s-τij(s)))ds,(ϕ2xi)(t)=∫-∞t‍e-ci(s)(t-s)×∑j=1n‍bij(s)gj(∫-∞0‍kj(θ)xi(s+θ)dθ)ds,(ϕ3xi)(t)=∫-∞t‍e-ci(s)(t-s)Ii(s)ds,(ϕ4xi)(t)=∫-∞t‍e-ci(s)(t-s)σi(s,xi(s))dωi(s).
For i=1,2,…,n, we show that (𝕃xi)(·) is square mean almost periodic if xi is. Assuming that xi(i=1,2,…,n) is square mean almost periodic and applying conditions (H1) and (H2) and Lemmas 7 and 9, one can easily obtain that aij(s)fj(xi(s-τij(s)), (i,j=1,2,…,n;s∈R) is square mean almost periodic. Therefore, for each ε>0, there exists l(ε)>0 such that any interval of length l(ε) contains at least τ for which
(18)E∥aij(s+τ)fjxi(s+τ-τij(s+τ))-aij(s)fjxi(s-τij(s))∥2≤(c*n)2ε,
for each s∈R.
Now, by using Cauchy-Schwarz inequality, we can write(19)E∥(ϕ1xi)(t+τ)-(ϕ1xi)(t)∥2=E∥∫-∞t+τ‍e-ci(s)(t+τ-s)∑j=1n‍aij(s)fj(xi(s-τij(s)))ds-∫-∞t‍e-ci(s)(t-s)∑j=1n‍aij(s)fj(xi(s-τij(s)))ds∥2≤E∥∫-∞t‍e-c*(t-s)∑j=1n‍[aij(s+τ)fj(xi(s+τ-τij(s+τ)))-aij(s)fj(xi(s-τij(s)))]ds∑j=1n‍∥2≤(∫-∞t‍e-c*(t-s)ds)×(∫-∞t‍e-c*(t-s)E∥∑j=1n‍[aij(s+τ)fj(xi(s+τ-τij(s+τ)))-aij(s)fj(xi(s-τij(s)))]∑j=1n‍∥ds)2≤(∫-∞t‍e-c*(t-s))2×sup⁡s∈R{n∑j=1n‍E∥aij(s+τ)fj(xi(s+τ-τij(s+τ)))-aij(s)fj(xi(s-τij(s)))∥2∑j=1n‍}≤nc*2·nE∥aij(s+τ)fjxi(s+τ-τij(s+τ))-aij(s)fjxi(s-τij(s))∥2≤nc*2·n(c*n)2ε=ε.
According to the above, for i=1,2,…,n, E∥(ϕ1xi)(t+τ)-(ϕ1xi)(t)∥2<ε for each t∈R; that is, (ϕ1xi)(·) is square mean almost periodic.
Next, fori=1,2,…,n, we show that (ϕ2xi)(·) is square mean almost periodic if xi is. Since xi(i=1,2,…,n) is square mean almost periodic, for each ε>0 there exists l(ε)>0 such that any interval of length l(ε) contains at least τ for which
(20)E∥xi(s+τ+θ)-xi(s+τ)∥2≤c*24n2b*2Lg*ε,(21)E∥bij(s+τ)-bij(s)∥2≤c*24n2g*2ε,
for each s∈R.
So, we have(22)∥(ϕ2xi)(t+τ)-(ϕ2xi)(t)∥=∥∫-∞t+τ‍e-ci(s)(t+τ-s)×∑j=1n‍bij(s)gj(∫-∞0‍kj(θ)xi(s+θ)dθ)ds-∫-∞t‍e-ci(s)(t-s)∑j=1n‍bij(s)gj×(∫-∞0‍kj(θ)xi(s+θ)dθ)ds∥=∥∫-∞t‍e-ci(s)(t-s)×∑j=1n‍[(∫-∞0‍kj(θ)xi(s+τ+θ)dθ)bij(s+τ)gj(∫-∞0‍kj(θ)xi(s+τ+θ)dθ)-bij(s)gj(∫-∞0‍kj(θ)xi(s+θ)dθ)]ds∥=∥∑j=1n‍}∫-∞t‍e-ci(s)(t-s)×{∑j=1n‍[(∫-∞0‍kj(θ)xi(s+τ+θ)dθ)bij(s+τ)gj(∫-∞0‍kj(θ)xi(s+τ+θ)dθ)-bij(s)gj(∫-∞0‍kj(θ)xi(s+τ+θ)dθ)]+∑j=1n‍[(∫-∞0‍kj(θ)xi(s+τ+θ)dθ)bij(s)gj(∫-∞0‍kj(θ)xi(s+τ+θ)dθ)-bij(s)gj(∫-∞0‍kj(θ)xi(s+θ)dθ)]∑j=1n‍}ds∥=∥∑j=1n‍}∫-∞t‍e-ci(s)(t-s)×{∑j=1n‍[bij(s+τ)-bij(s)]gj×(∫-∞0‍kj(θ)xi(s+τ+θ)dθ)+∑j=1n‍bij(s)[gj(∫-∞0‍kj(θ)xi(s+τ+θ)dθ)-gj(∫-∞0‍kj(θ)xi(s+θ)dθ)]∑j=1n‍}ds∥.
Similarly, from (20)–(22), by using Cauchy-Schwarz inequality, we obtain that
(23)E∥(ϕ2xi)(t+τ)-(ϕ2xi)(t)∥2≤2E∥∫-∞t‍e-ci(s)(t-s)×∑j=1n‍[bij(s+τ)-bij(s)]gj×(∫-∞0‍kj(θ)xi(s+τ+θ)dθ)ds∥2+2E∥∑j=1n‍∫-∞t‍e-ci(s)(t-s)×∑j=1n‍bij(s)×[gj(∫-∞0‍kj(θ)xi(s+τ+θ)dθ)-gj(∫-∞0‍kj(θ)xi(s+θ)dθ)]ds∑j=1n‍∥2≤2(∫-∞t‍e-ci(s)(t-s)ds)×∫-∞t‍e-ci(s)(t-s)×E∥∑j=1n‍[bij(s+τ)-bij(s)]gj×(∫-∞0‍kj(θ)xi(s+τ+θ)dθ)∑j=1n‍∥2ds+2(∫-∞t‍e-ci(s)(t-s)ds)×∫-∞t‍e-ci(s)(t-s)×E∥∑j=1n‍bij(s)×[gj(∫-∞0‍kj(θ)xi(s+τ+θ)dθ)-gj(∫-∞0‍kj(θ)xi(s+θ)dθ)]∑j=1n‍∥2ds≤2g*2c*∫-∞t‍e-ci(s)(t-s)n∑j=1n‍E∥bij(s+τ)-bij(s)∥2ds+2b*2c*×∫-∞t‍e-ci(s)(t-s)×n∑j=1n‍E∥gj(∫-∞0‍kj(θ)xi(s+τ+θ)dθ)-gj(∫-∞0‍kj(θ)xi(s+θ)dθ)∥2ds≤n22g*2c*2·c*24n2g*2ε+n22b*2c*2·Lg*∫-∞0‍kj(θ)E∥xi(s+τ+θ)-xi(s+θ)∥2dθ≤ε2+ε2=ε.
So, for i=1,2,…,n, E∥(ϕ2xi)(t+τ)-(ϕ2xi)(t)∥2<ε for each t∈R; that is, (ϕ2xi)(·) is square mean almost periodic. From (H1), it is easy to see that (ϕ3xi)(·)∈AP(R;L2(ℙ;𝔹)).
Now, we prove that(ϕ4xi)(·)∈AP(R;L2(ℙ;𝔹)) provided xi∈AP(R;L2(ℙ;𝔹))(i=1,2,…,n). From (H3) and Lemma 7, one can see that σi(s,xi(s))(i=1,2,…,n) is square mean almost periodic for s∈R; for each ε>0 there exists l(ε)>0 such that any interval of length l(ε) contains at least τ for which
(24)E∥σi(s+τ,xi(s+τ))-σi(s,xi(s))∥2≤c*2ε,
for each s∈R. We make extensive use of the Ito's isometry identity and the properties of W¯ defined by ωi¯:=ωi(s+τ)-ωi(τ) for each s∈R. And ωi¯ is also a Brownian motion and has the same distribution as ωi. Now,
(25)(ϕ4xi)(t+τ)-(ϕ4xi)(t)=∫-∞t‍e-ci(s)(t-s)[σi(s+τ,xi(s+τ))-σi(s,xi(s))]dωi(s).
By making a change of variables s=ν-τ, we get
(26)E∥(ϕ4xi)(t+τ)-(ϕ4xi)(t)∥2=E∥∫-∞t‍e-ci(s)(t-s)×[σi(s+τ,xi(s+τ))-σi(s,xi(s))]dωi¯(s)∫-∞t‍∥2.
Thus, applying Ito's isometry identity, we have
(27)E∥(ϕ4xi)(t+τ)-(ϕ4xi)(t)∥2=∫-∞t‍E∥e-ci(s)(t-s)[σi(s+τ,xi(s+τ))-σi(s,xi(s))]∥2ds≤(∫-∞t‍e-ci(s)(t-s)ds)×E(∫-∞t‍e-ci(s)(t-s)∥σi(s+τ,xi(s+τ))-σi(s,xi(s))∥2∫-∞t‍)ds≤1c*2E∥σi(s+τ,xi(s+τ))-σi(s,xi(s))∥2≤1c*2·c*2ε=ε,
which implies that (ϕ4xi)(·)∈AP(R;L2(ℙ;𝔹))(i=1,2,…,n).
From the above, fori=1,2,…,n, it is clear that 𝕃 maps AP(R;Lp(ℙ,𝔹)) into itself. In the next section, we show that 𝕃 is a contraction mapping. Actually, for i=1,2,…,n, we have
(28)∥(𝕃xi)(t)-(𝕃yi)(t)∥=∥∑j=1n‍∫-∞t‍e-ci(s)(t-s)×∑j=1n‍aij(s)[fj(xi(s-τij(s)))-fj(yi(s-τij(s)))]ds+∫-∞t‍e-ci(s)(t-s)×∑j=1n‍bij(s)[gj(∫-∞0‍kj(θ)xi(s+θ)dθ)-gj(∫-∞0‍kj(θ)yi(s+θ)dθ)]ds+∫-∞t‍e-ci(s)(t-s)×[σi(s,xi(s))-σi(s,yi(s))]dωi(s)∑j=1n‍∥≤a*∫-∞t‍e-ci(s)(t-s)×∑j=1n‍∥fj(xi(s-τij(s)))-fj(yi(s-τij(s)))∥ds+b*∫-∞t‍e-ci(s)(t-s)×∑j=1n‍∥gj(∫-∞0‍kj(θ)xi(s+θ)dθ)-gj(∫-∞0‍kj(θ)yi(s+θ)dθ)∥ds+∫-∞t‍e-ci(s)(t-s)∥σi(s,xi(s))-σi(s,yi(s))∥dωi(s).
Since (a+b+c)2≤3a2+3b2+3c2, we can get
(29)E∥(𝕃xi)(t)-(𝕃yi)(t)∥2≤3a*2E(∑j=1n‍∫-∞t‍e-ci(s)(t-s)×∑j=1n‍∥fj(xi(s-τij(s)))-fj(yi(s-τij(s)))∥ds∑j=1n‍)2+3b*2E×(∑j=1n‍∫-∞t‍e-ci(s)(t-s)×∑j=1n‍∥gj(∫-∞0‍kj(θ)xi(s+θ)dθ)-gj(∫-∞0‍kj(θ)yi(s+θ)dθ)∥ds∑j=1n‍)2+3E(∫-∞t‍e-ci(s)(t-s)∥σi(s,xi(s))-σi(s,yi(s))∥dωi(s)∫-∞t‍)2:=F(t)+G(t)+Φ(t).
Firstly, we evaluate the first term of the right hand side as follows:
(30)F(t)≤3a*2E(∫-∞t‍e-ci(s)(t-s)ds)×(-fj(yi(s-τij(s)))∥∑j=1n‍)2∫-∞t‍e-ci(s)(t-s)×(∑j=1n‍∥fj(xi(s-τij(s)))-fj(yi(s-τij(s)))∥∑j=1n‍)2ds)≤3a*2(∫-∞t‍e-ci(s)(t-s)ds)×n(∑j=1n‍∫-∞t‍e-ci(s)(t-s)×∑j=1n‍E∥fj(xi(s-τij(s)))-fj(yi(s-τij(s)))∥2ds∑j=1n‍)≤3a*2n2(∫-∞t‍e-ci(s)(t-s)ds)2×Lf*sup⁡t∈RE∥xi(t)-yi(t)∥2≤3a*2n2c*2Lf*∥x-y∥∞.
Secondly, we evaluate the second term of the right hand side as follows:
(31)G(t)≤3b*2E(∫-∞t‍e-ci(s)(t-s)ds)×(∫-∞t‍e-ci(s)(t-s)×∑j=1n‍∥gj(∫-∞0‍kj(θ)xi(s+θ)dθ)-gj(∫-∞0‍kj(θ)yi(s+θ)dθ)∥ds)≤3b*2c*n×∫-∞t‍e-ci(s)(t-s)×(∑j=1n‍E∥gj(∫-∞0‍kj(θ)xi(s+θ)dθ)-gj(∫-∞0‍kj(θ)yi(s+θ)dθ)∥2)ds≤3b*2c*n2(∫-∞t‍e-ci(s)(t-s)ds)×Lg*sup⁡t∈RE∥∫-∞0‍kj(θ)[xi(t)-yi(t)]dθ∥2≤3b*2n2c*2Lg*sup⁡t∈RE∥xi(t)-yi(t)∥2≤3b*2n2c*2Lg*∥x-y∥∞.
As to the third term, we apply isometry identity and have
(32)Φ(t)≤3(∫-∞t‍e-ci(s)(t-s)ds)×∫-∞t‍e-ci(s)(t-s)×E∥σi(s,xi(s))-σi(s,yi(s))∥2dωi(s)≤3c*Lσ*∫-∞t‍e-ci(s)(t-s)E∥xi(s)-yi(s)∥2ds≤3c*2Lσ*sup⁡t∈RE∥xi(t)-yi(t)∥2≤3c*2Lσ*∥x-y∥∞.
Thus, for i=1,2,…,n, by combining (30), (31), and (32), we can obtain that
(33)E∥(𝕃xi)(t)-(𝕃yi)(t)∥2≤(3a*2n2c*2Lf*+3b*2n2c*2Lg*+3c*2Lσ*)∥x-y∥∞≤3c*2(a*2n2Lf*+b*2n2Lg*+Lσ*)∥x-y∥∞,
and it follows that
(34)∥(𝕃xi)(t)-(𝕃yi)(t)∥∞≤3c*2(a*2n2Lf*+b*2n2Lg*+Lσ*)∥x-y∥∞.
From (3/c*2)(a*2n2Lf*+b*2n2Lg*+Lσ*)<1, it suffices to show that 𝕃 has a unique fixed point, which is clearly the unique square mean almost periodic solution to (2). The proof is completed.Remark 11.
According to the conditions of Theorem10, we can find that the activation function of the neuron, the feedback function, the neuron interconnections, amplification function, external input, and stochastic perturbation have key effect on the unique of square mean almost periodic solution.Theorem 12.
Assume that the conditions of Theorem10 are held; then the square mean almost periodic solution of system (2) is global exponential stability.Proof.
Take a positive constantα such that 0<α<c*. For i=1,2,…,n, it follows from (16) that
(35)eαtE∥xi(t)∥2≤6eαtE∥∑j=1n‍∫-∞t‍e-ci(s)(t-s)×∑j=1n‍aij(s)fj(xi(s-τij(s)))ds∥2+6eαtE∥∑j=1n‍∫-∞t‍e-ci(s)(t-s)×∑j=1n‍bij(s)gj×(∫-∞0‍kj(θ)xi(s+θ)dθ)ds∑j=1n‍∥2+3eαtE∥∫-∞t‍e-ci(s)(t-s)Ii(s)ds∥2+3eαtE∥∫-∞t‍e-ci(s)(t-s)σi(s,xi(s))dωi(s)∥2.
Now, we evaluate the first term on the right hand side of (35); we obtain
(36)6eαtE∥∑j=1n‍∫-∞t‍e-ci(s)(t-s)×∑j=1n‍aij(s)fj(xi(s-τij(s)))ds∥2=6eαt(∫-∞t‍e-ci(s)(t-s)ds)×∫-∞t‍e-ci(s)(t-s)E∥∑j=1n‍aij(s)fj(xi(s-τij(s)))∥2ds≤6c*eαt×∫-∞t‍e-c*(t-s)E∥∑j=1n‍aij(s)fj(xi(s-τij(s)))∥2ds≤6n2c*e-(c*-α)ta*2Lf*×∫-∞t‍e(c*-α)seαsE∥xi(s)∥2ds.
For any xi(t)∈R(i=1,2,…,n) and any ε>0, there exists t1∈R such that eαsE∥xi(s)∥2<ε for t≥t1. Therefore
(37)6eαtE∥∑j=1n‍∫-∞t‍e-ci(s)(t-s)×∑j=1n‍aij(s)fj(xi(s-τij(s)))ds∥2≤6n2c*e-(c*-α)ta*2Lf*×∫-∞t‍e(c*-α)seαsE∥xi(s)∥2ds=6n2c*e-(c*-α)ta*2Lf*×∫-∞t1‍e(c*-α)seαsE∥xi(s)∥2ds+6n2c*e-(c*-α)ta*2Lf*×∫t1t‍e(c*-α)seαsE∥xi(s)∥2ds≤6n2c*e-(c*-α)ta*2Lf*×∫-∞t1‍e(c*-α)seαsE∥xi(s)∥2ds+6n2c*a*2Lf*1c*(c*-α)ε.
When t→+∞, e-(c*-α)t→0, that is, there exists t2≥t1 such that for any t≥t2, one can get
(38)6n2c*e-(c*-α)ta*2Lf*∫-∞t1‍e(c*-α)seαsE∥xi(s)∥2ds≤ε-6n2c*a*2Lf*1c*(c*-α)ε.
Thus, from (37) and (38), we have for any t2≤t, when t→+∞,
(39)6eαtE∥∫-∞t‍e-ci(s)(t-s)∑j=1n‍aij(s)fj(xi(s-τij(s)))ds∥2⟶0.
Like the discussion of other terms on the right hand side of (35), as t→+∞, we have
(40)6eαtE∥∑j=1n‍∫-∞t‍e-ci(s)(t-s)×∑j=1n‍bij(s)gj×(∫-∞0‍kj(θ)xi(s+θ)dθ)ds∑j=1n‍∥2⟶0,3eαtE∥∫-∞t‍e-ci(s)(t-s)Ii(s)ds∥2⟶0,3eαtE∥∫-∞t‍e-ci(s)(t-s)σi(s,xi(s))dωi(s)∥2⟶0.
Therefore, from the above, we get that eαtE∥xi(t)∥2→0 (i=1,2,…,n) as t→+∞. Thus we know that (2) has a unique square mean almost periodic solution, which is an exponentially stable.Remark 13.
From the assumptions of Theorem10, we can see that stochastic perturbations on the neuron states also have key effect on the exponentially stability of square mean almost periodic.



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*Source: 905415-2014-04-09.xml*