# Stability of the Stochastic Reaction-Diffusion Neural Network with Time-Varying Delays andp-Laplacian

Authors: Pan Qingfei; Zhang Zifang; Huang Jingchang
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

The main aim of this paper is to discuss moment exponential stability for a stochastic reaction-diffusion neural network with time-varying delays andp-Laplacian. Using the Itô formula, a delay differential inequality and the characteristics of the neural network, the algebraic conditions for the moment exponential stability of the nonconstant equilibrium solution are derived. An example is also given for illustration.


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## Body

## 1. Introduction

In many neural networks, time delays cannot be avoided. For example, in electronic neural networks, time delays will be present due to the finite switching speed of amplifies. In fact, time delays are often encountered in various engineering, biological, and economical systems. On the other hand, when designing a neural network to solve a problem such as optimization or pattern recognition, we need foremost to guarantee that the neural networks model is globally asymptotically stable. However, the existence of time delay frequently causes oscillation, divergence, or instability in neural networks. In recent years, the stability of neural networks with delays or without delays has become a topic of great theoretical and practical importance (see [1–16]).The stability of neural networks which depicted by partial differential equations was studied in [6, 7]. Stochastic differential equations were employed to research the stability of neural networks in [8–11], while [12, 13] used stochastic partial differential equations to analysis this question. In [15], the authors studied almost exponential stability for a stochastic recurrent neural network with time-varying delays. In addition, moment exponential stability for a stochastic reaction-diffusion neural network with time-varying delays is discussed in [16].In this paper, we consider the stochastic reaction-diffusion neural network with time-varying delays andp-Laplacian as follows:
(1.1)dui(t,x)=[∑k=1m‍∂∂xk(|∇ui|p-2∂ui∂xk)-ai(u)ui+Ii+∑j=1n‍Tijgj(uj(t-τj(t),x))]dt+∑l=1m‍σil(ui(t,x))dWl(t),i=1,2,…,n,t>t0,x∈Ω,(1.2)∂ui∂n:=(∂ui∂x1,…,∂ui∂xm)T=0,i=1,2,…,n,t≥t0,x∈∂Ω,(1.3)ui(t0+s,x)=ϕi(s,x),-τi(t0)≤s≤0,0≤τi(t)≤τi,i=1,2,…,n,x∈Ω.
In (1.1), ∇ui=(∂ui/∂x1,…,∂ui/∂xm)T, p≥2 is a common number. Ω⊆Rm is a bounded convex domain with smooth boundary ∂Ω and measure mesΩ>0. n denotes the numbers of neurons in the neural network, ui(t,x) corresponds to the state of the ith neurons at time t and in space x, the ai(u) is an amplification function. Ii is output. gj(uj(t-τj,x)) denotes the output of the jth neuron at time t-τj and in space x, namely, activation function which shows how neurons respond to each other. W(t)=(W1(t),…,Wm(t))T is an m-dimensional Brownian motion which is defined on a complete probability space (𝒮,ℱ,𝒫) with a natural filtration {ℱt}t≥t0 (i.e., ℱt=σ{W(s):t0≤s≤t}). σ(u)=(σ1(u1),…,σn(un))T, σi(ui)=(σi1(ui),…,σim(ui)). σij(ui) denotes the intensity of the stochastic perturbation. Functions σij(ui) and gi are subject to certain conditions to be specified later. T:=(Tij)n×n is a real constant matrix and represents weight of the neuron interconnections, namely, Tij denotes the strength of jth neuron on the ith neuron at time t-τj and in space x, and τj∈[0,τ] corresponds to axonal signal transmission delay.


## 2. Definitions and Lemmas

Throughout this paper, unless otherwise specified, let|·| denote Euclidean norm. Define that |x|p=(∑i=1n‍|xi|2)p/2 and |x|p=∑i=1n‍|xi|p where x=(x1,…,xn)T∈Rn. Denote by C([-τ,0]×Ω;Rn) the family of continuous functions φ from [-τ,0]×Ω to Rn. For every t≥t0 and p≥2, denote by Lℱtp([-τ,0] × Ω;Rn) the family of all ℱt-measurable C([-τ,0] × Ω;Rn) valued random variables such that ∥ϕ∥Lpp=sup-τ≤θ≤0E(∥ϕ(θ)∥pp)<∞, where ∥ϕ(θ)∥p=(∫Ω‍|ϕ(θ,x)|pdx)1/p, E(ϕ) denotes the expectation of random variable ϕ.Definition 2.1.
Theu(t,x)=(u1(t,x),…,un(t,x))T is called a solution of problem (1.1)–(1.3) if it satisfies following conditions (1), (2), and (3):(1)
u(t,x) adapts {ℱt}t≥t0;(2)
forT∈Rt0+:={t∈R:t≥t0}, u(t,x)∈C([t0,T]×Ω,Rn), and E(maxx∈Ω∫t0T‍[|u(t,x)|p+|∇u(t,x)|p]dt)<+∞, where ∇u(x,t)=(∂u/∂x1,…,∂u/∂xn);(3)
forT∈Rt0+, t∈(t0,T], it holds that
(2.1)∫Ω‍ui(t,x)dx=∫Ω‍ϕi(t0,x)dx+∫Ω‍∫t0t‍{∑j=1n‍∑k=1m‍∂∂xk(|∇ui|p-2∂ui∂xk)-ai(u)ui+Ii+∑j=1n‍Tijgj(uj(ξ-τj(ξ),x))}dξdx+∑l=1m‍∫Ω‍∫t0t‍σil(ui(ξ,x))dWl(ξ)dx,i=1,…,n,(t,x)∈(t0,T]×Ω,Pa.s.,ui(t0+s,x)=ϕi(s,x),-τi≤s≤0,i=1,…,n,Pa.s.Definition 2.2.
Theu=u*(x) is called a nonconstant equilibrium solution of problem (1.1)–(1.3) if and only if u=u*(x) satisfies (1.1) and (1.2).Definition 2.3.
The nonconstant equilibrium solutionu*(x) of (1.1) about the given norm ∥·∥Ω is called exponential stability in pth moment, if there are constants M>0, δ>0 for every stochastic field solution u(t,x) of problem (1.1)–(1.3) such that
(2.2)E[∥u(t,x)-u*∥Ω]≤Me-δ(t-t0),
namely,
(2.3)limsupt→∞1tlogE[∥u(t,x)-u0∥Ω]≤-δ.
The constant -δ on the right hand side in (2.3) is called Lyapunov exponent of every solution of problem (1.1)–(1.3) converging on equilibrium about norm ∥·∥Ω.
In order to obtainpth moment exponential stability for a nonconstant equilibrium solution of problem (1.1)–(1.3), we need the following lemmas.Lemma 2.4 (see [17]).
LetP=(pij)n×n and Q=(qij)n×n be two real matrices. The continuous function ui(t)≥0 satisfies the delay differential inequalities
(2.4)D+ui(t)≤∑j=1n‍[pijuj(t)+qijuj(t-τj(t))],0≤τi(t)≤τ,i=1,…,n.
If pij>0 for i≠j and qij≥0(i,j=1,2,…,n) and -(P+Q) is an M-matrix, then there are constants ki>0, α>0 such that
(2.5)ui(t)≤ki(∑j=1n‍∥ϕj∥)exp(-α(t-t0)),i=1,…,n,t≥t0,
where ϕ=(ϕ1,…,ϕ1) are initial functions. D+ is right-hand upper derivate.∥·∥ represents a norm.Lemma 2.5 (see [10]).
Letp>2, then there are positive constants ep(n) and dp(n) for any x=(x1,…,xn)T∈Rn such that
(2.6)ep(n)(∑i=1n‍|xi|2)p/2≤∑i=1n‍|xi|p≤dp(n)(∑i=1n‍|xi|2)p/2.Remark 2.6.
Ifp=2, Lemma 2.5 also holds with ep(n)=dp(n)=1.
Suppose thatσij(ui), ai(u), and gi are Lipschitz continuous such that the following conditions hold: (H1)
|(σi(v1)-σi(v2))(σi(v1)-σi(v2))T|≤2λi|v1-v2|2, i=1,2,…,n,(H2)
|gi(v1)-gi(v2)|≤ci|v1-v2|, i=1,2,…,n,(H3)
(u-v)(ai(u)u-ai(v)v)≥ai|u-v|2, for all u,v∈R, i=1,2,…,n, where ci, ai, and λi(1≤i≤n) are positive constants.


## 3. Main Result

Setu(t,x)=(u1(t,x),…,un(t,x))T as a solution of the problem (1.1)–(1.3) and u*(x)=(u1*(x),…,un*(x))T as a nonconstant equilibrium solution of the problem (1.1)–(1.3).Theorem 3.1.
Letp≥2 and (H1)–(H3) hold. Assume that there are positive constants ε1,…,εn such that the matrix Mp=-(pji+qij)n×n:=PT+Q is an M-matrix, where
(3.1)pij=-pδijai+p(p-1)δijλi+(p-1)|Tij|cjεi,δii=1,δij=0(i≠j),i,j=1,…,n,qij=|Tij|cjεi1-p,i,j=1,…,n,
then the nonconstant equilibrium solution of problem (1.1)–(1.3) about Lp norm is exponential stability in pth moment, that is, there are constants C1>0 and α>0, for any t0∈R+ and any ϕ∈LFt0p such that
(3.2)E(∥u(t,x)-u*∥pp)≤C1e-α(t-t0),t≥t0.Proof.
SetVi(u(t,x))=|ui(t,x)-ui*(x)|p. For every t≥t0 and dt>0, by means of Itô formula and (H3), one has that
(3.3)dVi(u)=p|ui-ui*|p-2(ui-ui*){∑j=1n‍∑k=1m‍[∂∂xk(|∇ui|p-2∂ui∂xk)-∂∂xk(|∇ui*|p-2∂ui*∂xk)]-(ai(u)ui-ai(u*)ui*)+∑j=1n‍Tij(gj(uj(t-τj,x))-gj(uj*))}dt+12p(p-1)|ui-ui*|p-2sgn(ui-ui*)(σi(ui)-σi(ui*))(σi(ui)-σi(ui*))Tdt+p|ui-ui*|p-2(ui-ui*)(σi(ui)-σi(ui*))dW(t)≤p∑k=1m‍|ui-ui*|p-2(ui-ui*)[∂∂xk(|∇ui|p-2∂ui∂xk)-∂∂xk(|∇ui*|p-2∂ui*∂xk)]dt+(-pai+p(p-1)λi)|ui-ui*|pdt+p∑j=1n‍|Tij|cj|ui-ui*|p-2|ui-ui*||uj(t-τj,x)-uj*|dt+p|ui-ui*|p-2(ui-ui*)(σi(ui)-σi(ui*))dW(t).
Both sides of Inequality (3.3) are integrated about x over Ω. Set Vi(u)¯=∫ΩVi(u)dx=∥ui-ui*∥pp. One has that
(3.4)dVi(u)¯≤p∑k=1m‍∫Ω‍|ui-ui*|p-2(ui-ui*)[∂∂xk(|∇ui|p-2∂ui∂xk)-∂∂xk(|∇ui*|p-2∂ui*∂xk)]dxdt+(-pai+p(p-1)λi)∫Ω‍|ui-ui*|pdxdt+p∑j=1n‍∫Ω‍|ui-ui*|p-2|ui-ui*||Tij|cj|uj(t-τj,x)-uj*|dxdt+p∑l=1m‍∫Ω‍|ui-ui*|p-2(ui-ui*)(σil(ui)-σil(ui*))dxdWl(t).
Set (|∇ui|p-2(∂ui/∂xk)-|∇ui*|p-2(∂ui*/∂xk))k=1m:=(|∇ui|p-2(∂ui/∂x1)-|∇ui*|p-2(∂ui*/∂x1),…,|∇ui|p-2(∂ui/∂xm)-|∇ui*|p-2(∂ui*/∂xm)). By (1.2), one has that
(3.5)∑k=1m‍∫Ω‍|ui-ui*|p-2(ui-ui*)[∂∂xk(|∇ui|p-2∂ui∂xk)-∂∂xk(|∇ui*|p-2∂ui*∂xk)]dx=∫Ω‍|ui-ui*|p-2(ui-ui*)∇·(|∇ui|p-2∂ui∂xk-|∇ui*|p-2∂ui*∂xk)k=1mdx=∫Ω‍∇·(|ui-ui*|p-2(ui-ui*)(|∇ui|p-2∂ui∂xk-|∇ui*|p-2∂ui*∂xk)k=1m)dx-∫Ω‍(|∇ui|p-2∂ui∂xk-|∇ui*|p-2∂ui*∂xk)k=1m∇(|ui-ui*|p-2(ui-ui*))dx=∫∂Ω‍(|ui-ui*|p-2(ui-ui*)(|∇ui|p-2∂ui∂xk-|∇ui*|p-2∂ui*∂xk)k=1m)·n→ds-∫Ω‍(p-1)|ui-ui*|p-2∇(ui-ui*)·(|∇ui|p-2∂ui∂xk-|∇ui*|p-2∂ui*∂xk)k=1mdx=-∑k=1m‍∫Ω‍(p-1)|ui-ui*|p-2(|∇ui|p-2∂ui∂xk-|∇ui*|p-2∂ui*∂xk)·(∂ui∂xk-∂ui*∂xk)dx≤0,
where n→ is unit outer cotangent vector on ∂Ω. By (3.4), (3.5), (H1), and Young’s inequality, one has that
(3.6)dVi(u)¯≤-∑k=1m‍∫Ω‍(p-1)|ui-ui*|p-2(|∇ui|p-2∂ui∂xk-|∇ui*|p-2∂ui*∂xk)·(∂ui∂xk-∂ui*∂xk)dx+(-pai+p(p-1)λi)∥ui(t)-ui*∥ppdt+p∑j=1n‍∫G‍|ui-ui*|p-2|ui-ui*||Tij|cj|uj(t-τj,x)-uj*|dxdt+p∑l=1m‍∫G‍|ui-ui*|p-2(ui-ui*)(σil(ui)-σil(ui*))dxdWl(t)≤∑j=1n‍pij∥ui(t)-ui*∥ppdt+∑j=1n‍qij∥uj(t-τj,x)-uj*∥ppdt+p∑l=1m‍∫G‍|ui-ui*|p-2(ui-ui*)(σil(ui)-σil(ui*))dxdWl(t),
where pij and qij are defined by (3.1).
ForΔt>0, both sides of (3.6) are integrated about t from t to t+Δt, then both sides of (3.6) are calculated expectation. By the properties of Brownian motion, one has that
(3.7)E[∥ui(t+Δt)-ui*∥pp]-E[∥ui(t)-ui*∥pp]≤∑j=1n‍{pijE[∫tt+Δt‍∥ui(s)-ui*∥ppds]+qijE[∫tt+Δt‍∥uj(s-τj(s))-uj*∥ppds]}.
Since the integrals ∫tt+Δt‍E[∥ui(s)-ui*∥pp]ds and ∫tt+Δt‍E[∥uj(s-τj)-ui*∥pp]ds are finite, by Fubini theorem [18] and (3.7), one obtain that
(3.8)E[∥ui(t+Δt)-ui*∥pp]-E[∥ui(t)-ui*∥pp]≤∑j=1npij∫tt+Δt‍E[∥ui(s)-ui*∥pp]ds+∑j=1nqij∫tt+Δt‍E[∥uj(s-τj(s))-uj*∥pp]ds.
Setvi(t)=E[∥ui(t)-ui*∥pp]. Both sides of Inequality (3.8) are divided by Δt, let Δt→0, one has the following inequality:
(3.9)D+vi(t)≤∑j=1n‍[pijvi(t)+qijvj(t-τj(t))].
By Lemma 2.4, there are positive constants Ki, α such that
(3.10)E[∥ui(t)-ui*∥pp]≤Ki∑j=1n‍∥E[∥ϕj-uj*∥pp]∥exp(-α(t-t0)),i=1,…,n,t≥t0,
where ϕj is initial value. Set K=max{Ki:1≤i≤n}, then
(3.11)E[∑j=1n‍∥uj-uj*∥pp]≤nK∑j=1n‍∥E[∥ϕj-uj*∥pp]∥exp(-α(t-t0)),t≥t0.
By (3.11) and Lemma 2.5, one obtains that
(3.12)E[∥u-u*∥pp]≤ep-1(n)dp(n)nK∑j=1n‍∥E[∥ϕj-uj*∥pp]∥exp(-α(t-t0)),t≥t0.
In order to prove Theorem3.1, we need the following lemma.Lemma 3.2.
The nonconstant equilibrium solution of the problem (1.1)–(1.3), u*(x) satisfies E(∥u*∥pp)=0.Proof.
SetFi(u*(x))=|ui*(x)|p. Similar to (3.8) in proof of Theorem 3.1, one has that
(3.13)∑j=1npij∫tt+Δt‍E[∥ui*∥pp]ds+∑j=1nqij∫tt+Δt‍E[∥uj*∥pp]ds≥0.
By (3.13) and the assumption that -(pji+qij)n×n:=PT+Q is an M-matrix, one obtains that
(3.14)-∑j=1n‍(pji+qij)E[∥uj*∥pp]Δt≤0,i=1,2,…,n.
Because of E[∥uj*∥pp]≥0 and Δt>0, one has that E(∥u*∥pp)=0.
We continue the proof of Theorem3.1 as the following.
By Lemma3.2, one has that
(3.15)∥E[∥ϕj-uj*∥pp]∥≤M,
where M>0 is a common number. We derive every solution of problem (1.1)–(1.3) such that
(3.16)limsupt→∞1tlogE[∥u-u*∥pp]≤-α,
then a nonconstant equilibrium solution of problem (1.1)–(1.3) about Lp norm is exponential stability in pth moment. The proof of Theorem 3.1 is complete.In order to illustrate the application of the theorem, we give an example.Example 3.3.
Discuss the stochastic reaction-diffusion neural network with time-varying delays andp-Laplacian as the following:
(3.17)du1(t,x)=[∑k=12‍∂∂xk(|∇u1|p-2∂u1∂xk)-a1u1+I1+∑j=12‍T1jgj(uj(t-τj(t),x))]dt+σ1(u1(t,x))dW(t),(3.18)du2(t,x)=[∑k=12‍∂∂xk(|∇u2|p-2∂u2∂xk)-a2u2+I2+∑j=12‍T2jgj(uj(t-τj(t),x))]dt+σ2(u2(t,x))dW(t),(3.19)∂ui∂n:=(∂ui∂x1,∂ui∂x2)T=0,i=1,2,t≥t0≥0,x∈∂Ω,(3.20)ui(t0+s,x)=ϕi(s,x),-τi(t0)≤s≤0,0≤τi(t)≤τi,i=1,2,x∈Ω,
where
(3.21)a1=1.2,a2=1.8,σ1(u1)=0.2(u1-u1*),σ2(u2)=0.3(u2-u2*),T11=1.5,T12=0.5,T21=0.6,T22=1,g1(u1(t-τ1(t),x))=0.3u1(t-τ1(t),x),g2(u2(t-τ2(t),x))=0.2u2(t-τ2(t),x)cos(u2(t-τ2(t),x)).
Set u*(x)=(u1*(x),u1*(x))T as a nonconstant equilibrium solution of (3.17) and (3.18). One can derive that
(3.22)|g1(v1)-g1(v2)|≤0.3|v1-v2|,c1=0.3;|g2(v1)-g2(v2)|≤0.4|v1-v2|,c2=0.4;|(σ1(v1)-σ1(v2))(σ1(v1)-σ1(v2))T|≤0.04|v1-v2|,λ1=0.02;|(σ2(v1)-σ2(v2))(σ2(v1)-σ2(v2))T|≤0.09|v1-v2|,λ2=0.045.
Taking εi=1(i=1,2) and p=3, one has that
(3.23)M3=(2.58-0.56-0.581.53),
and M3 is an M-matrix. The nonconstant equilibrium solution of (3.17) and (3.18) about L3 norm is exponential stability in the 3rd moment.Remark 3.4.
The Theorem3.1 extends the correlative results in [12, 13, 16] to the situation related to the p-Laplacian.



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