# Preserving Global Exponential Stability of Hybrid BAM Neural Networks with Reaction Diffusion Terms in the Presence of Stochastic Noise and Connection Weight Matrices Uncertainty

Authors: Yan Li; Yi Shen
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

We study the impact of stochastic noise and connection weight matrices uncertainty on global exponential stability of hybrid BAM neural networks with
reaction diffusion terms. Given globally exponentially stable hybrid BAM neural
networks with reaction diffusion terms, the question to be addressed here is how much
stochastic noise and connection weights matrices uncertainty the neural networks can
tolerate while maintaining global exponential stability. The upper threshold of stochastic noise and connection weights matrices uncertainty is defined by using the
transcendental equations. We find that the perturbed hybrid BAM neural networks
with reaction diffusion terms preserve global exponential stability if the intensity of
both stochastic noise and connection weights matrices uncertainty is smaller than the
defined upper threshold. A numerical example is also provided to illustrate the theoretical conclusion.


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## Body

## 1. Introduction

The bidirectional associative memory (BAM) neural networks were first introduced by Kosko in which the neurons in one layer are fully interconnected to the neurons in the other layer, while there are no interconnection among the neurons in the same layers [1–3]. The BAM neural networks widely have applications in pattern recognition, robot, signal processing, associative memory, solving optimization problems, and automatic control engineering. For most successful applications of BAM neural networks, the stability analysis on BAM neural networks is usually a prerequisite. The exponential stability and periodic oscillatory solution of BAM neural networks with delays were studied by Cao et al. [4, 5]. Moreover, in BAM neural networks, diffusion phenomena can hardly be avoided when electrons are moving in asymmetric electromagnetic fields. The BAM neural networks with reaction diffusion terms described by partial differential equations were investigated by many authors [6–11]. Sometimes, it is necessary to assess the parameters of the neural network that may experience abrupt changes caused by certain phenomena such as component failure or repair, change of subsystem interconnection, and environmental disturbance. The continuous-time Markov chains have been used to model these parameter jumps [12–14]. These neural networks with Markov chains are usually called hybrid neural networks. The almost surely exponential stability, moment exponential stability, and stabilization of hybrid neural networks were also researched; see, for example, [15–17]. By making use of impulsive control, Zhu and Cao [18] considered the stability of hybrid neural networks with mixed delay.For neural networks with stochastic noise, the system is usually described by stochastic differential equations. The stability of stochastic neural networks with delay or reaction diffusion terms was extensively analyzed by using the Itô formula and the linear matrix inequality (LMI) methods [18–22]. As is well known, stochastic noise is often the sources of instability and may destabilize the stable neural networks [23]. For stable hybrid BAM neural networks with reaction diffusion terms, it is interesting to determine how much noise the stochastic neural networks can tolerate while maintaining global exponential stability.Moreover, the connection weights of neurons depend on certain resistance and capacitance values which include uncertainty. The robust stability about parameter matrices uncertainty in neural networks was investigated by many authors [24, 25]. If the uncertainty in connection weights matrices is too large, the neural networks may be unstable. Therefore, for stable hybrid BAM neural networks with reaction diffusion terms, it is also interesting to determine how much connection weights matrices uncertainty the neural networks can also tolerate while maintaining global exponential stability.In this paper, we will study the impact of stochastic noise and connection weight matrices uncertainty of hybrid BAM neural networks with reaction diffusion terms. We give the upper threshold of stochastic noise and connection weights matrices uncertainty defined by using the transcendental equations. We find that the perturbed hybrid BAM neural networks with reaction diffusion terms preserve global exponential stability if the intensity of both stochastic noise and connection weights matrices uncertainty is smaller than the defined upper threshold.The remainder of this paper is organized as follows. Some preliminaries are given in Section2. Section 3 discusses the impact of the stochastic noise on global exponential stability of these neural networks. Section 4 discusses the impact of the connection weight matrices uncertainty and stochastic noise on global exponential stability of these neural networks. Finally, an example with numerical simulation is given to illustrate the effectiveness of the obtained results in Section 5.


## 2. Preliminaries

Throughout this paper, unless otherwise specified, let(Ω,ℱ,{ℱt}t≥0,ℙ) be complete probability space with a filtration {ℱt}t≥0 satisfying the usual conditions (i.e., it is increasing and right continuous while ℱ0 contains all ℙ-null sets). Let W(t) be a scalar Brownian motion (Wiener process) defined on the probability space. Let AT denote the transpose of A. If A is a matrix, its operator norm is denoted by ∥A∥=sup⁡{|Ax|:|x|=1}, where |·| is the Euclidean norm. Let r(t), t≥0, be a right-continuous Markov chain on the probability space taking values in a finite state space 𝕊={1,2,…,N} with the generator Γ=(γpq)N×N given by
(1)ℙ{r(t+Δ)=q∣r(t)=p}={γpqΔ+o(Δ)ifp≠q1+γppΔ+o(Δ)ifp=q,
where Δ>0. Here, γpq>0 is the transition rate from p to q if p≠q while
(2)γpp=-∑q≠p‍γpq.
We assume that the Markov chain r(·) is independent of the Brownian motion W(·). It is well known that almost every sample path of r(·) is a right-continuous step function with finite number of simple jumps in any finite subinterval of ℝ+:=[0,+∞).In this paper, we will consider the following hybrid BAM neural networks with reaction diffusion terms:(3)∂u~i(t,x)∂t=∑k=1l‍∂∂xk(D¯ik(r(t))∂u~i(t,x)∂xk)-ai(r(t))u~i(t,x)+∑j=1n‍cji(r(t))f~j(v~j(t,x))+Ii,∂v~j(t,x)∂t=∑k=1l‍∂∂xk(D¯jk*(r(t))∂v~j(t,x)∂xk)-bj(r(t))v~j(t,x)+∑i=1m‍eij(r(t))g~i(u~i(t,x))+Jj,
where i=1,2,…,m, j=1,2,…,n, t≥t0≥0, t0∈ℝ+, and the initial value r(t0)=i0∈𝕊. Consider x=(x1,x2,…,xl)∈Ω0⊂ℝl; Ω0 is a compact set with smooth boundary ∂Ω0 in space ℝl, and 0<mesΩ0<+∞. u~(t,x)   =  (u~1(t,x),…,u~m(t,x))∈ℝm and v~(t,x)   =  (v~1(t,x),…,v~n(t,x))∈ℝnu~i(t,x), v~j(t,x), are the state of the ith neurons and the jth neurons at times t and in space x, respectively. f~j and g~i denote the signal functions on the jth neurons and the ith neurons at times t and in space x, respectively. Ii and Jj denote the external input on the ith neurons and the jth neurons, respectively. ai(r(t))>0 and bj(r(t))>0 denote the rates with which the ith neurons and the jth neurons will reset its potential to the resting state in isolation when disconnected from the networks and external inputs, respectively. cji(r(t)) and eij(r(t)) denote the strength of the jth neurons on the ith neurons and the ith neurons on the jth neurons, respectively. Smooth functions D¯ik(r(t)):=D¯ik(r(t),x,u)≥0 and D¯jk*(r(t)):=D¯jk*(r(t),x,u)≥0 correspond to the transmission diffusion operator along the ith neurons and the jth neurons, respectively.The initial conditions and boundary conditions are given by(4)u~i(t0,x)=ϕ¯i(x),x∈Ω0,t0∈ℝ+,i=1,2,…,m,v~j(t0,x)=ψ¯j(x),x∈Ω0,t0∈ℝ+,j=1,2,…,n,∂u~i(t,x)∂n→|∂Ω0=(∂u~i(t,x)∂x1,…,∂u~i(t,x)∂xl)T=0,bbbb(t,x)∈[t0,+∞)×∂Ω0,i=1,2,…,m,∂v~j(t,x)∂n→|∂Ω0=(∂v~j(t,x)∂x1,…,∂v~j(t,x)∂xl)T=0,bbbbi(t,x)∈[t0,+∞)×∂Ω0,j=1,2,…,n.The neuron activation functionsf~ and g~ are global Lipschitz continuous; that is, there exist constants K>0 and L>0, such that
(5)|f~(v~)-f~(v~*)|≤K|v~-v~*|,∀v~,v~*∈ℝn|g~(u~)-g~(u~*)|≤L|u~-u~*|,∀u~,u~*∈ℝm.Then, the neural networks (3) have a unique state (u~(t,x;t0,ϕ¯(x)) and v~(t,x;t0,ψ¯(x))) for any initial values (ϕ¯(x),ψ¯(x)) (see [26, 27]).In addition, we assume that the neural networks (3) have an equilibrium point u*=(u1*,…,um*)∈ℝm, v*=(v1*,…,vn*)∈ℝn.Letu(t,x)=u~(t,x)-u*, v(t,x)=v~(t,x)-v*, f(v(t,x)) = f~(v(t,x)+v*)-f~(v*), g(u(t,x)) = g~(u(t,x)+u*)-f~(u*), Dik(r(t))=D¯ik(r(t),x,u(t,x)+u*), and Dik*(r(t))=D¯ik*(r(t),x,v(t,x)+v*), and then (3) can be rewritten as
(6)∂ui(t,x)∂t=∑k=1l‍∂∂xk(Dik(r(t))∂ui(t,x)∂xk)-ai(r(t))ui(t,x)+∑j=1n‍cji(r(t))fj(vj(t,x))∂vj(t,x)∂t=∑k=1l‍∂∂xk(Djk*(r(t))∂vj(t,x)∂xk)-bj(r(t))vj(t,x)+∑i=1m‍eij(r(t))gi(ui(t,x)).The initial conditions and boundary conditions are given by(7)iui(t0,x)=ϕi(x)=ϕ¯i(x)-ui*,x∈Ω0,t0∈ℝ+,i=1,2,…,m,ivj(t0,x)=ψj(x)=ψ¯j(x)-vj*,x∈Ω0,t0∈ℝ+,j=1,2,…,n,∂ui(t,x)∂n→|∂Ω0=(∂ui(t,x)∂x1,…,∂ui(t,x)∂xl)T=0,bbbi(t,x)∈[t0,+∞)×∂Ω0,i=1,2,…,m,∂vj(t,x)∂n→|∂Ω0=(∂vj(t,x)∂x1,…,∂vj(t,x)∂xl)T=0,bbbb(t,x)∈[t0,+∞)×∂Ω0,j=1,2,…,n.
Hence, the origin is an equilibrium point of (6). The stability of the equilibrium point of (3) is equivalent to the stability of the origin of the state space of (6).From (5), we give the assumption about activations functions f and g.Assumption (H1). The neuron activation functions f and g are global Lipschitz continuous; that is, there exist constants K>0 and L>0, such that
(8)|f(v)-f(v*)|≤K|v-v*|,∀v,v*∈ℝn,f(0)=0,|g(u)-g(u*)|≤L|u-u*|,∀u,u*∈ℝm,g(0)=0.We consider the following function vector space:(9)U={v(t,x):[t0,+∞)×Ω0⟶ℝn,v(t,x)iscontinuouson⁡tandtwicecontinuousdifferentiableon⁡x.For every pair of(v,z) in U and every given t∈ℝ+, define inner product for v and z with
(10)〈v,z〉=∫Ω0‍(v(·,x))Tz(·,x)dx∈ℝ+.
Obviously, it satisfies inner product axiom, and the norm can be deduced by
(11)∥v(·,x)∥2=〈v(·,x),v(·,x)〉=∫Ω0‍|v(·,x)|2dx=∑i=1n‍∫Ω0‍|vi(·,x)|2dx.Definition 1.
The neural networks (6) are said to be global exponentially stable if for any ϕ,ψ, there exist α>0 and β>0, such that
(12)∥u(t,x;t0,ϕ)∥22+∥v(t,x;t0,ψ)∥22≤α(∥ϕ∥22+∥ψ∥22)exp⁡(-β(t-t0)),∀t≥t0.
For the purpose of simplicity, we rewrite (6) as follows:
(13)∂u∂t=∇·(D(r(t))∘∇u)-A(r(t))u(t,x)+C(r(t))f(v(t,x))∂v∂t=∇·(D*(r(t))∘∇v)-B(r(t))v(t,x)+E(r(t))g(u(t,x)).
The initial conditions and boundary conditions are given by(14)u(t0,x)=ϕ(x)=ϕ¯(x)-u*,x∈Ω0,t0∈ℝ+,v(t0,x)=ψ(x)=ψ¯(x)-v*,x∈Ω0,t0∈ℝ+,∂u(t,x)∂n→|∂Ω0=(∂u(t,x)∂x1,…,∂u(t,x)∂xl)T=0,bbbbbi(t,x)∈[t0,+∞)×∂Ω0,∂v(t,x)∂n→|∂Ω0=(∂v(t,x)∂x1,…,∂v(t,x)∂xl)T=0,bbbbbi(t,x)∈[t0,+∞)×∂Ω0,
where
(15)D(r(t))=(Dik(r(t),x,u))m×l,D*(r(t))=(Djk*(r(t),x,v))n×l,u(t,x)=(u1(t,x),…,um(t,x))T,v(t,x)=(v1(t,x),…,vn(t,x))T,∇u=(∇u1,…,∇um)T,∇v=(∇v1,…,∇vn)T,∇ui=(∂ui∂x1,…,∂ui∂xl)T,∇vj=(∂vj∂x1,…,∂vj∂xl)T,A(r(t))=diag⁡(a1(r(t)),…,am(r(t))),B(r(t))=diag⁡(b1(r(t)),…,bn(r(t))),C(r(t))=(cji(r(t)))n×m,E(r(t))=(eij(r(t)))m×n,f(v)=(f1(v1),…,fn(vn))T,g(u)=(g1(u1),…,gm(um))T,(D(r(t))∘∇u)=(Dik(r(t))∂ui∂xk),(D*(r(t))∘∇v)=(Djk*(r(t))∂vj∂xk).
Here, ∘ denotes Hadamard product of matrix D and ∇u and D* and ∇v.


## 3. Noise Impact on Stability

In this section, we consider the noise-induced neural networks (6) described by the stochastic partial differential equations
(16)du-i(t,x)={∑j=1n‍∑k=1l‍∂∂xk(Dik(r(t))∂u-i(t,x)∂xk)m-ai(r(t))u-i(t,x)m+∑j=1n‍cji(r(t))fj(v-j(t,x))}dt+σu-i(t,x)dW(t),dv-j(t,x)={∑k=1l‍∂∂xk(Djk*(r(t))∂v-j(t,x)∂xk)m-bj(r(t))v-j(t,x)m+∑i=1m‍eij(r(t))gi(u-i(t,x))}dt+σv-j(t,x)dW(t).The initial conditions and boundary conditions are given by(17)u-i(t0,x)=ϕi(x),x∈Ω0,t0∈ℝ+,i=1,2,…,m,v-j(t0,x)=ψj(x),x∈Ω0,t0∈ℝ+,j=1,2,…,n,∂u-i(t,x)∂n→|∂Ω0=(∂u-i(t,x)∂x1,…,∂u-i(t,x)∂xl)T=0,bbbb(t,x)∈[t0,+∞)×∂Ω0,i=1,2,…,m,∂v-j(t,x)∂n→|∂Ω0=(∂v-j(t,x)∂x1,…,∂v-j(t,x)∂xl)T=0,bbbbi(t,x)∈[t0,+∞)×∂Ω0,j=1,2,…,n,
where σ is the noise intensity.We rewrite (16) as follows:
(18)du-(t,x)={∇·(D(r(t))∘∇u-)-A(r(t))u-(t,x)m+C(r(t))f(v-(t,x))}dt+σu-(t,x)dW(t),dv-(t,x)={∇·(D*(r(t))∘∇v-)-B(r(t))v-(t,x)m+E(r(t))g(u-(t,x))}dt+σv-(t,x)dW(t).
For the globally exponentially stable neural networks (6), we will characterize how much stochastic noise the neural networks (16) can tolerate while maintaining global exponential stability.Definition 2.
The neural networks (16) are said to be almost surely globally exponentially stable, if for any ϕandψ the Lyapunov exponent
(19)limsup⁡t→∞log⁡(∥u-(t,x;t0,ϕ)∥2+∥v-(t,x;t0,ψ)∥2)t<0,a.s.Definition 3.
The neural networks (16) are said to be mean square globally exponentially stable, if, for any ϕandψ, the Lyapunov exponent
(20)limsup⁡t→∞log⁡𝔼{∥u-(t,x;t0,ϕ)∥22+∥v-(t,x;t0,ψ)∥22}t<0,
where (u-(t,x;t0,ϕ), v-(t,x;t0,ψ)) is the state of neural networks (16).From the above definitions, it is clear that the almost sure global exponential stability of the neural networks (16) implies the mean square global exponential stability of the neural networks (16) (see [26, 27]) but not vice versa.Theorem 4.
Under Assumption (H1), the mean square global exponential stability of neural networks (16) implies the almost sure global exponential stability of the neural networks (16).Proof.
For any(ϕ(x),ψ(x))≢(0,0), we denote the state (u-(t,x;t0,ϕ), v-(t,x;t0,ψ)) of (16) as (u-(t,x),v-(t,x)). By Definition 3, there exist λ>0 and C>0, such that
(21)𝔼{∥u-(t,x)∥22+∥v-(t,x)∥22}≤C(∥ϕ∥22+∥ψ∥22)e-λ(t-t0),t≥t0.
Let r(t)=p∈𝕊. Construct average Lyapunov functional
(22)V(u-(t,x),v-(t,x),p)=∫Ω0‍|u-(t,x)|2dx+∫Ω0‍|v-(t,x)|2dx=∫Ω0‍∑i=1m‍u-i2(t,x)dx+∫Ω0‍∑j=1n‍v-j2(t,x)dx.
Let n=1,2,…, by Itô formula and Assumption (H1), for t0+n-1≤t≤t0+n,
(23)V(u-(t,x),v-(t,x),p)=V(u-(t0+n-1,x),v-(t0+n-1,x),p)+∫t0+n-1t‍∫Ω0‍2u-T(s,x)[+C(r(s))f(v-)∇·(D(r(s))∘∇u-)-A(r(s))u-mmmmmmmmmmmmm+C(r(s))f(v-)]dxds+σ2∫t0+n-1t‍∫Ω0‍|u-(s,x)|2dxds+∫t0+n-1t‍∫Ω0‍2v-T(s,x)[∇·(D*(r(s))∘∇v-)-B(r(s))v-mmmmmmmmmmmmm+E(r(s))g(u-)]dxds+σ2∫t0+n-1t‍∫Ω0‍|v-(s,x)|2dxds+2σ∫t0+n-1t‍∫Ω0‍|u-(s,x)|2dxdW(s)+2σ∫t0+n-1t‍∫Ω0‍|v-(s,x)|2dxdW(s)+∑q=1N‍γpq∫t0+n-1t‍∫Ω0‍(|u-(s,x)|2+|v-(s,x)|2)dxds.
By boundary condition and Gauss formula, we get
(24)2∫Ω0‍u-T(s,x)[∇·(D(r(s))∘∇u-)]dx=2∑i=1m‍∑k=1l‍∫Ω0‍u-i∂∂xk(Dik(r(s))∂u-i∂xk)dx=2∑i=1m‍∫Ω0‍∇·(u-iDik(r(s))∂u-i∂xk)k=1ldx-2∑i=1m‍∫Ω0‍(Dik(r(s))∂u-i∂xk)k=1l·∇u-idx=2∑i=1m‍∫∂Ω0‍(u-iDik(r(s))∂u-i∂xk)k=1ldx-2∑i=1m‍∑k=1l‍∫Ω0‍Dik(r(s))(∂u-i∂xk)2dx=-2∑i=1m‍∑k=1l‍∫Ω0‍Dik(r(s))(∂u-i∂xk)2dx,(25)2∫Ω0‍v-T(s,x)[∇·(D*(r(s))∘∇v-)]dx=2∑j=1n‍∑k=1l‍∫Ω0‍v-j∂∂xk(Djk*(r(s))∂v-j∂xk)dx=2∑j=1n‍∫Ω0‍∇·(v-jDjk*(r(s))∂v-j∂xk)k=1ldx-2∑j=1n‍∫Ω0‍(Djk*(r(s))∂v-j∂xk)k=1l·∇v-jdx=2∑j=1n‍∫∂Ω0‍(v-jDjk*(r(s))∂v-j∂xk)k=1ldx-2∑j=1n‍∑k=1l‍∫Ω0‍Djk*(r(s))(∂v-j∂xk)2dx=-2∑j=1n‍∑k=1l‍∫Ω0‍Djk*(r(s))(∂v-j∂xk)2dx.
By Hölder’s inequality, we have
(26)∫Ω0‍2u-(t,x)TC(r(s))f(v-(t,x))dx≤max⁡p∈𝕊∥C(p)∥[∫Ω0‍|u-(t,x)|2dx+K2∫Ω0‍|v-(t,x)|2dx],(27)∫Ω0‍2v-(t,x)TE(r(s))g(u-(t,x))dx≤max⁡p∈𝕊∥E(p)∥[∫Ω0‍|v-(t,x)|2dx+L2∫Ω0‍|u-(t,x)|2dx].
Substituting (24)–(27) into (23), we get
(28)V(u-(t,x),v-(t,x),p)=V(u-(t0+n-1,x),v-(t0+n-1,x),p)+max⁡p∈𝕊[2∥A(p)∥+∥C(p)∥+∥E(p)∥L2+σ2]×∫t0+n-1t‍∫Ω0‍|u-(s,x)|2dxds+max⁡p∈𝕊[2∥B(p)∥+∥E(p)∥+∥C(p)∥K2+σ2]×∫t0+n-1t‍∫Ω0‍|v-(s,x)|2dxds+2|σ|∫t0+n-1t‍∫Ω0‍(|u-(s,x)|2+|v-(s,x)|2)dxdW(s),
where we use ∑q=1N‍γpq=0.
From (28), we have
(29)𝔼(sup⁡t0+n-1≤t≤t0+nV(u-(t,x),v-(t,x),p))≤V(u-(t0+n-1,x),v-(t0+n-1,x),p)+C1∫t0+n-1t0+n‍𝔼V(u-(s,x),v-(s,x),r(s))ds+2|σ|𝔼(sup⁡t0+n-1≤t≤t0+n∫t0+n-1t‍V(u-(s,x),v-(s,x),mmmmmmmmmmmmmmmmr(s))dW(s)sup⁡t0+n-1≤t≤t0+n∫t0+n-1t‍),
where C1=[2∥A^∥+2∥B^∥+∥C^∥+∥C^∥K2+∥E^∥+∥E^∥L2+2σ2] and ∥A^∥=max⁡p∈𝕊∥A(p)∥.
On the other hand, by the Burkholder-Davis-Gundy inequality [27] and 2ab≤(a/ε)+εb(a>0,b>0,ε>0), we have
(30)2|σ|𝔼(sup⁡t0+n-1≤t≤t0+n∫t0+n-1t‍V(u-(s,x),v-(s,x),r(s))dW(s))≤42𝔼(∫t0+n-1t0+n‍4σ2V2(u-(s,x),v-(s,x),r(s))ds)1/2≤42𝔼(sup⁡t0+n-1≤t≤t0+nV(u-(s,x),v-(s,x),r(s))mmmmmmm×∫t0+n-1t0+n‍4σ2V(u-(s,x),v-(s,x),r(s))dssup⁡t0+n-1≤t≤t0+n)1/2≤12𝔼(sup⁡t0+n-1≤t≤t0+nV(u-(t,x),v-(t,x),p))+64σ2∫t0+n-1t0+n‍𝔼V(u-(s,x),v-(s,x),r(s))ds.
Substituting the above inequality into (29), we get
(31)𝔼(sup⁡t0+n-1≤t≤t0+nV(u-(t,x),v-(t,x),p))≤2𝔼V(u-(t0+n-1,x),v-(t0+n-1,x),p)+2[C1+64σ2]∫t0+n-1t0+n‍𝔼V(s)ds.
By induction and the mean square global exponential stability of neural networks (16),
(32)𝔼(sup⁡t0+n-1≤t≤t0+nV(u-(t,x),v-(t,x),p))≤C(∥ϕ∥22+∥ψ∥22)(2+2[C1+64σ2])e-λ(n-1).
Let ε∈(0,λ), by Chebyshev’s inequality [27], it follows from (32) that
(33)ℙ{sup⁡t0+n-1≤t≤t0+nV(u-(t,x),v-(t,x),p)>e-(λ-ε)(n-1)}≤e-(λ-ε)(n-1)𝔼(sup⁡t0+n-1≤t≤t0+nV(u-(t,x),v-(t,x),p))≤C(∥ϕ∥22+∥ψ∥22)(2+2[C1+64σ2])e-ε(n-1).
By Borel-Cantelli Lemma [27], for almost all ω∈Ω,
(34)sup⁡t0+n-1≤t≤t0+n2V(u-(t,x),v-(t,x),p)≤2e-(λ-ε)(n-1)
holds for all but finitely many n. Hence, there exists an n0=n0(ω), for all ω∈Ω, excluding a ℙ-null set, for the above inequality that holds whenever n≥n0. Consequently, for almost all ω∈Ω,
(35)log⁡2V(u-(t,x),v-(t,x),p)t≤-(λ-ε)(n-1)t0+n-1+2t0+n-1,
if t0+n-1≤t≤t0+n. Therefore,
(36)limsup⁡t→∞log⁡(∥u-(t,x)∥2+∥v-(t,x)∥2)t≤-(λ-ε)2a.s.Theorem 5.
Let Assumption (H1) hold and the neural networks (6) be globally exponentially stable. Then, the neural networks (16) is mean square globally exponentially stable and also almost surely globally exponentially stable, if there exist μq>0,(q∈𝕊) and |σ|<σ-, where σ- is a unique positive solution of the transcendental equation
(37)4σ-2αμ^βexp⁡{2Δ(μ^C2+max⁡p∈𝕊∑q=1N‍γpqμq)μ˘}+2αexp⁡{-βΔ}=1,(38)Δ>ln⁡(2α)β>0,
where C2=[2∥A^∥+2∥B^∥+(1+K2)∥C^∥+(1+L2)∥E^∥+2σ-2], ∥A^∥=max⁡p∈𝕊∥A(p)∥, and so forth and μ^=max⁡p∈𝕊μp and μ˘=min⁡p∈𝕊μp.Proof.
For any(ϕ(x),ψ(x)), we denote the state (u-(t,x;t0,ϕ),v-(t,x;t0,ψ)) of (16) as (u-(t,x),v-(t,x)) and the state (u(t,x;t0,ϕ),v(t,x;t0,ψ)) of (6) as (u(t,x),v(t,x)).
From (6) and (18) and stochastic Fubini’s Theorem, we have
(39)∫Ω0‍(u(t,x)-u-(t,x))dx+∫Ω0‍(v(t,x)-v-(t,x))dx=∫t0t‍∫Ω0‍∇·(D(r(s))∘∇(u-u-))dxds+∫t0t‍∫Ω0‍[(f(v(s,x))-f(v-(s,x)))-A(r(s))(u(s,x)-u-(s,x))mmmmmmmm+C(r(s))(f(v(s,x))-f(v-(s,x)))]dxds-∫t0t‍∫Ω0‍σu-(s,x)dxdW(s)+∫t0t‍∫Ω0‍∇·(D*(r(s))∘∇(v-v-))dxds+∫t0t‍∫Ω0‍[(g(u(s,x))-g(u-(s,x)))-B(r(s))(v(s,x)-v-(s,x))mmmmm+E(r(s))(g(u(s,x))-g(u-(s,x)))]dxds-∫t0t‍∫Ω0‍σv-(s,x)dxdW(s).
Construct average Lyapunov functional
(40)V(u(t,x),v(t,x),u-(t,x),v-(t,x),r(t))=∫Ω0‍μr(t)[|u(t,x)-u-(t,x)|2+|v(t,x)-v-(t,x)|2]dx,
where μr(t)>0.
By applying generalized Itô formula [27], we have
(41)dV(u(t,x),v(t,x),u-(t,x),v-(t,x),p)=∫Ω0‍2μp(u(t,x)-u-(t,x))T(∇·(D(p)∘∇(u-u-)))dxdt+∫Ω0‍2μp(u(t,x)-u-(t,x))Tmml×[(f(v(t,x))-f(v-(t,x)))-A(p)(u(t,x)-u-(t,x))mmmm+C(p)(f(v(t,x))-f(v-(t,x)))]dxdt+∫Ω0‍σ2μp|u-(t,x)|2dxdt-2∫Ω0‍σ(u(t,x)-u-(t,x))Tu-(t,x)dxdW(t)+∫Ω0‍2μp(v(t,x)-v-(t,x))Tmmmm×(∇·(D*(p)∘∇(v-v-)))dxdt+∫Ω0‍2μp(v(t,x)-v-(t,x))Tmml×[(g(u(t,x))-g(u-(t,x)))-B(p)(v(t,x)-v-(t,x))mmml+E(p)(g(u(t,x))-g(u-(t,x)))]dxdt+∫Ω0‍σ2μp|v-(t,x)|2dxdt-2∫Ω0‍σ(v(t,x)-v-(t,x))Tv-(t,x)dxdW(t)+∑q=1N‍γpqμq∫Ω0‍[|u(t,x)-u-(t,x)|2mmmmmmmmmmm+|v(t,x)-v-(t,x)|2]dx.
By boundary condition and (24), we have
(42)2μp∫Ω0‍(u(t,x)-u-(t,x))T(∇·(D(p)∘∇(u-u-)))dxdt=-2μp∑i=1m‍∑k=1l‍∫Ω0‍Dik(p)(∂(ui-u-i)∂xk)2dx.
By boundary condition and (25), we have
(43)2μp∫Ω0‍(v(t,x)-v-(t,x))T(∇·(D*(p)∘∇(v-v-)))dxdt=-2μp∑j=1n‍∑k=1l‍∫Ω0‍Djk*(p)(∂(vj-v-j)∂xk)2dx.
By Hölder’s inequality, we get
(44)2μp∫Ω0‍(u(t,x)-u-(t,x))TC(p)mmm×(f(v(t,x))-f(v-(t,x)))dx≤max⁡p∈𝕊(μp∥C(p)∥)[∫Ω0‍|u(t,x)-u-(t,x)|2dxmmmmmmmmmmmmn+K2∫Ω0‍|v(t,x)-v-(t,x)|2dx],(45)2μp∫Ω0‍(v(t,x)-v-(t,x))TE(p)mmmm×(g(u(t,x))-g(u-(t,x)))dx≤max⁡p∈𝕊(μp∥E(p)∥)[∫Ω0‍|v(t,x)-v-(t,x)|2dxmmmmmmmmmmm+L2∫Ω0‍|u(t,x)-u-(t,x)|2dx].
From (42)–(45) and Assumption (H1), we obtain that
(46)dV(u(t,x),v(t,x),u-(t,x),v-(t,x),p)≤(μ^C1+max⁡p∈𝕊∑q=1N‍γpqμq)×∫Ω0‍(|u(t,x)-u-(t,x)|2+|v(t,x)-v-(t,x)|2)dxdt+2σ2μ^∫Ω0‍(|u(t,x)|2+|v(t,x)|2)dxdt-2∫Ω0‍σ(u(t,x)-u-(t,x))Tu-(t,x)dxdW(t)-2∫Ω0‍σ(v(t,x)-v-(t,x))Tv-(t,x)dxdW(t).
When t≤t0+2Δ, we have
(47)𝔼V(u(t,x),v(t,x),u-(t,x),v-(t,x),r(t))≤(μ^C1+max⁡p∈𝕊∑q=1N‍γpqμq)×∫t0t‍𝔼∫Ω0‍(|u(s,x)-u-(s,x)|2mmmmmmmm+|v(s,x)-v-(s,x)|2)dxds+2σ2μ^∫t0t‍α(∥ϕ∥22+∥ψ∥22)exp⁡(-β(s-t0))ds-2σ𝔼∫t0t‍∫Ω0‍(u(s,x)-u-(s,x))Tu-(s,x)dxdW(s)-2σ𝔼∫t0t‍∫Ω0‍(v(s,x)-v-(s,x))Tv-(s,x)dxdW(s).
By stochastic Fubini’s Theorem, we have
(48)𝔼∫t0t‍∫Ω0‍(u(s,x)-u-(s,x))Tu-(s,x)dxdW(s)=0,𝔼∫t0t‍∫Ω0‍(v(s,x)-v-(s,x))Tv-(s,x)dxdW(s)=0.
By (47), one get
(49)𝔼V(u(t,x),v(t,x),u-(t,x),v-(t,x),r(t))≤(μ^C1+max⁡p∈𝕊∑q=1N‍γpqμq)μ˘×∫t0t‍𝔼V(u(s,x),v(s,x),u-(s,x),v-(s,x),r(s))ds+2σ2αμ^(∥ϕ∥22+∥ψ∥22)β.
When t0+Δ≤t≤t0+2Δ, by applying Gronwall’s inequality, we have
(50)𝔼(∥u(t,x)-u-(t,x)∥22+∥v(t,x)-v-(t,x)∥22)=𝔼V(u(t,x),v(t,x),u-(t,x),v-(t,x),r(t))≤2σ2αμ^(∥ϕ∥22+∥ψ∥22)β×exp⁡(μ^C1+max⁡p∈𝕊∑q=1N‍γpqμq)μ˘(t-t0)≤sup⁡t0≤t≤t0+Δ𝔼(∥u-(t,x)∥2+∥v-(t,x)∥2)×2σ2αμ^βexp⁡2Δ(μ^C1+max⁡p∈𝕊∑q=1N‍γpqμq)μ˘.
By the global exponential stability of (6), we have
(51)𝔼(∥u-(t,x)∥22+∥v-(t,x)∥22)≤2𝔼(∥u(t,x)-u-(t,x)∥22+∥v(t,x)-v-(t,x)∥22)+2𝔼(∥u(t,x)∥22+∥v(t,x)∥22)≤sup⁡t0≤t≤t0+Δ𝔼(∥u-(t,x)∥2+∥v-(t,x)∥2)×4σ2αμ^βexp⁡2Δ(μ^C1+max⁡p∈𝕊∑q=1N‍γpqμq)μ˘+2α(∥ϕ∥22+∥ψ∥22)exp⁡{-β(t-t0)}.
Moreover,
(52)𝔼(∥u-(t,x)∥22+∥v-(t,x)∥22)≤{exp⁡{2Δ(μ^C1+max⁡p∈𝕊∑q=1N‍γpqμq)μ˘}4σ2αμ^βexp⁡{2Δ(μ^C1+max⁡p∈𝕊∑q=1N‍γpqμq)μ˘}mmexp⁡{2Δ(μ^C1+max⁡p∈𝕊∑q=1N‍γpqμq)μ˘}+2αexp⁡{-βΔ}}×sup⁡t0≤t≤t0+Δ𝔼(∥u-(t,x)∥2+∥v-(t,x)∥2).
From (37), when |σ|<σ-, we have
(53)4σ2αμ^βexp⁡{2Δ(μ^C1+max⁡p∈𝕊∑q=1N‍γpqμq)μ˘}+2αexp⁡{-βΔ}<1.
Let
(54)γ=(-log⁡{{2Δ(μ^C1+max⁡p∈𝕊∑q=1N‍γpqμq)μ˘}4σ2αμ^βmmmmm×exp⁡{2Δ(μ^C1+max⁡p∈𝕊∑q=1N‍γpqμq)μ˘}ll{2Δ(μ^C1+max⁡p∈𝕊∑q=1N‍γpqμq)μ˘}mmm+2αexp⁡{-βΔ}})×(Δ)-1>0.
By (52), we have
(55)sup⁡t0+Δ≤t≤t0+2Δ𝔼(∥u-(t,x)∥22+∥v-(t,x)∥22)≤exp⁡(-γΔ)(sup⁡t0≤t≤t0+Δ𝔼(∥u-(t,x)∥22+∥v-(t,x)∥22)).
For any positive integer m=1,2,…, from the existence and uniqueness of the flow of (16) (see [28]), when t≥t0+(m-1)Δ, we have
(56)(u-(t,x;t0,ϕ),v-(t,x;t0,ψ))=(v-(t,x;t0+(m-1)Δ,v-(t0+(m-1)Δ,x;t0,ψ))u-(t,x;t0+(m-1)Δ,u-(t0+(m-1)Δ,x;t0,ϕ)),v-(t,x;t0+(m-1)Δ,v-(t0+(m-1)Δ,x;t0,ψ))).
From (55) and (56),
(57)sup⁡t0+mΔ≤t≤t0+(m+1)Δ𝔼(∥u-(t,x;t0,ϕ)∥22+∥v-(t,x;t0,ψ)∥22)=sup⁡t0+(m-1)Δ+Δ≤t≤t0+(m-1)Δ+2Δ𝔼∥u-(t,x;t0+(m-1)Δ,mmmmmmmmmmmmmmmnu-(t0+(m-1)Δ,x;t0,ϕ))∥22+sup⁡t0+(m-1)Δ+Δ≤t≤t0+(m-1)Δ+2Δ𝔼∥v-(t,x;t0+(m-1)Δ,mmmmmmmmmmmmmmmmv-(t0+(m-1)Δ,x;t0,ψ))∥22≤exp⁡(-γΔ)(sup⁡t0+(m-1)Δ≤t≤t0+mΔ𝔼(∥u-(t,x;t0,ϕ)∥22mmmmmmmmmmmmmmmmmm+∥v-(t,x;t0,ψ)∥22)sup⁡t0+(m-1)Δ≤t≤t0+mΔ)mmmmmmmmmmmmml⋮≤exp⁡(-γmΔ)(sup⁡t0≤t≤t0+Δ𝔼(∥u-(t,x;t0,ϕ)∥22mmmmmmmmmmmmmm+∥v-(t,x;t0,ψ)∥22)sup⁡t0+(m-1)Δ≤t≤t0+mΔ).
Hence, for any t≥t0+Δ, there exists a positive integer m, such that t0+mΔ≤t≤t0+(m+1)Δ, and we have
(58)𝔼(∥u-(t,x;t0,ϕ)∥22+∥v-(t,x;t0,ψ)∥22)≤exp⁡(-γmΔ)×(sup⁡t0≤t≤t0+Δ𝔼(∥u-(t,x;t0,ϕ)∥22+∥v-(t,x;t0,ψ)∥22))≤exp⁡{-γt+γt0+γΔ}×(sup⁡t0≤t≤t0+Δ𝔼(∥u-(t,x;t0,ϕ)∥22+∥v-(t,x;t0,ψ)∥22))≤C3exp⁡{γΔ}exp⁡{-γ(t-t0)},
where C3=sup⁡t0≤t≤t0+Δ𝔼(∥u-(t,x;t0,ϕ)∥22+∥v-(t,x;t0,ψ)∥22). The above inequality also holds for t0≤t≤t0+Δ.
Therefore, the neural networks (16) are mean square globally exponentially stable, and by Theorem 4, the neural networks (16) are also almost surely globally exponentially stable.


## 4. Connection Weight Matrices Uncertainty and Noise Impact on Stability

In this section, we first consider the parameter uncertainty intensity which is added to the self-feedback matrix(A,B)T of the neural networks (16). Then, the neural networks (16) are changed as
(59)du-i(t,x)={∑k=1l‍∂∂xk(Dik(r(t))∂u-i(t,x)∂xk)m-(1+λ)ai(r(t))u-i(t,x)m+∑j=1n‍cji(r(t))fj(v-j(t,x))}dt+σu-i(t,x)dW(t)dv-j(t,x)={∑k=1l‍∂∂xk(Djk*(r(t))∂v-j(t,x)∂xk)m-(1+λ)bj(r(t))v-j(t,x)m+∑i=1m‍eij(r(t))gi(u-i(t,x))}dt+σv-j(t,x)dW(t).The initial conditions and boundary conditions are given by(60)u-i(t0,x)=ϕi(x),x∈Ω0,t0∈ℝ+,i=1,2,…,m,v-j(t0,x)=ψj(x),x∈Ω0,t0∈ℝ+,j=1,2,…,n,∂u-i(t,x)∂n→|∂Ω0=(∂u-i(t,x)∂x1,…,∂u-i(t,x)∂xl)T=0,bbbi(t,x)∈[t0,+∞)×∂Ω0,i=1,2,…,m,∂v-j(t,x)∂n→|∂Ω0=(∂v-j(t,x)∂x1,…,∂v-j(t,x)∂xl)T=0,bbbb(t,x)∈[t0,+∞)×∂Ω0,j=1,2,…,n,
where λ is the self-feedback matrix (A,B)T uncertainty intensity and σ is the noise intensity.We rewrite (59) as follows:
(61)du-(t,x)={∇·(D(r(t))∘∇u-)-(1+λ)A(r(t))u-(t,x)m+C(r(t))f(v-(t,x))}dt+σu-(t,x)dW(t)dv-(t,x)={∇·(D*(r(t))∘∇v-)-(1+λ)B(r(t))v-(t,x)m+E(r(t))g(u-(t,x))}dt+σv-(t,x)dW(t).For the global exponential stability of neural networks (6), we will characterize how much the intensity of both the self-feedback matrix (A,B)T uncertainty and stochastic noise the stochastic neural networks (59) can tolerate while maintaining global exponential stability.Theorem 6.
Let Assumption (H1) hold and let the neural networks (6) be globally exponentially stable. Then, the neural networks (59) are mean square globally exponential stability and also almost sure globally exponential stability, if there exists μq>0,(q∈𝕊), and (λ,σ) is in the inner of the closed curve described by the following transcendental equation:
(62)4μ^(σ2+λ2(∥A^∥+∥B^∥))αβ×exp⁡{2Δ(μ^C4+max⁡p∈𝕊∑q=1N‍γpqμq)μ˘},+2αexp⁡{-βΔ}=1,(63)Δ>ln⁡(2α)β>0,
where C4=[(3+2λ2)(∥A^∥+∥B^∥)+(1+K2)∥C^∥+(1+L2)∥E^∥+2σ2], ∥A^∥=max⁡p∈𝕊∥A(p)∥, and so forth and μ^=max⁡p∈𝕊μp and μ˘=min⁡p∈𝕊μp.Proof.
For any(ϕ(x),ψ(x)), we denote the state (u-(t,x;t0,ϕ),v-(t,x;t0,ψ)) of (59) as (u-(t,x),v-(t,x)) and the state (u(t,x;t0,ϕ),v(t,x;t0,ψ)) of (6) as (u(t,x),v(t,x)).
From (6) and (61) and stochastic Fubini’s Theorem, we have
(64)∫Ω0‍(u(t,x)-u-(t,x))dx+∫Ω0‍(v(t,x)-v-(t,x))dx=∫t0t‍∫Ω0‍∇·(D(r(s))∘∇(u-u-))dxds+∫t0t‍∫Ω0‍[(f(v(s,x))-f(v-(s,x)))-A(r(s))(u(s,x)-u-(s,x))mmmmm+C(r(s))(f(v(s,x))-f(v-(s,x)))]dxds-∫t0t‍∫Ω0‍σu-(s,x)dxdW(s)+∫t0t‍∫Ω0‍λA(r(s))u-(s,x)dxds+∫t0t‍∫Ω0‍∇·(D*(r(s))∘∇(v-v-))dxds+∫t0t‍∫Ω0‍[(g(u(s,x))-g(u-(s,x)))-B(r(s))(v(s,x)-v-(s,x))l+E(r(s))(g(u(s,x))-g(u-(s,x)))]dxds-∫t0t‍∫Ω0‍σv-(s,x)dxdW(s)+∫t0t‍∫Ω0‍λB(r(s))v-(s,x)dxds.
Construct the average Lyapunov functional
(65)V(u(t,x),v(t,x),u-(t,x),v-(t,x),r(t))=∫Ω0‍μr(t)[|u(t,x)-u-(t,x)|2+|v(t,x)-v-(t,x)|2]dx,
where μr(t)>0.
By applying generalized Itô formula [27], we have
(66)dV(u(t,x),v(t,x),u-(t,x),v-(t,x),p)|(21)=∫Ω0‍2μp(u(t,x)-u-(t,x))T(∇·(D(p)∘∇(u-u-)))dxdt+∫Ω0‍2μp(u(t,x)-u-(t,x))Tm×[-A(p)(u(t,x)-u-(t,x))mmmmml+C(p)(f(v(t,x))-f(v-(t,x)))mmmmm+λA(p)u-(t,x)]dxdt+∫Ω0‍σ2μp|u-(t,x)|2dxdt-2∫Ω0‍σ(u(t,x)-u-(t,x))Tu-(t,x)dxdW(t)+∫Ω0‍2μp(v(t,x)-v-(t,x))Tmmmmm×(∇·(D*(p)∘∇(v-v-)))dxdt+∫Ω0‍2μp(v(t,x)-v-(t,x))Tmm×[-B(p)(v(t,x)-v-(t,x))mmmm+E(p)(g(u(t,x))-g(u-(t,x)))mmmm+λB(p)v-(t,x)]dxdt+∫Ω0‍σ2μp|v-(t,x)|2dxdt-2∫Ω0‍σ(v(t,x)-v-(t,x))Tv-(t,x)dxdW(t)+∑q=1N‍γpqμq∫Ω0‍[|u(t,x)-u-(t,x)|2mmmmmmmmm+|v(t,x)-v-(t,x)|2]dx.
By Hölder’s inequality, we get
(67)∫Ω0‍2μp(u(t,x)-u-(t,x))TλA(p)u-(t,x)dx≤max⁡p∈𝕊∥μpA(p)∥[∫Ω0‍|u(t,x)-u-(t,x)|2dxmmmmmmmmmm+λ2∫Ω0‍|u-(t,x)|2dx]=max⁡p∈𝕊∥μpA(p)∥[∫Ω0‍|u(t,x)-u-(t,x)|2dxmmmmmmmmlm+λ2∫Ω0‍|u(t,x)-u-(t,x)-u(t,x)|2dx]≤max⁡p∈𝕊∥μpA(p)∥[(1+2λ2)∫Ω0‍|u(t,x)-u-(t,x)|2dxmmmmmmmmmmm+2λ2∫Ω0‍|u(t,x)|2dx],∫Ω0‍2μp(v(t,x)-v-(t,x))TλB(p)v-(t,x)dx≤max⁡p∈𝕊∥μpB(p)∥[∫Ω0‍|v(t,x)-v-(t,x)|2dxmmmmmmmmm+λ2∫Ω0‍|v-(t,x)|2dx]=max⁡p∈𝕊∥μpB(p)∥[∫Ω0‍|v(t,x)-v-(t,x)|2dxmmmmmmmmml+λ2∫Ω0‍|v(t,x)-v-(t,x)-v(t,x)|2dx]≤max⁡p∈𝕊∥μpB(p)∥[(1+2λ2)∫Ω0‍|v(t,x)-v-(t,x)|2dxmmmmmmmmmm+2λ2∫Ω0‍|v(t,x)|2dx].
From (42), (43), and (67) and Assumption (H1), we obtain that
(68)dV(u(t,x),v(t,x),u-(t,x),v-(t,x),p)≤(μ^C4+max⁡p∈𝕊∑q=1N‍γpqμq)×∫Ω0‍(|u(t,x)-u-(t,x)|2mmmmm+|v(t,x)-v-(t,x)|2)dxdt+2μ^(σ2+λ2(∥A^∥+∥B^∥))×∫Ω0‍(|u(t,x)|2+|v(t,x)|2)dxdt-2∫Ω0‍σ(u(t,x)-u-(t,x))Tu-(t,x)dxdW(t)-2∫Ω0‍σ(v(t,x)-v-(t,x))Tv-(t,x)dxdW(t).
When t≤t0+2Δ, we have
(69)𝔼V(u(t,x),v(t,x),u-(t,x),v-(t,x),r(t))≤(μ^C4+max⁡p∈𝕊∑q=1N‍γpqμq)×∫t0t‍𝔼∫Ω0‍(|u(s,x)-u-(s,x)|2mmmmmmmm+|v(s,x)-v-(s,x)|2)dxds+2μ^(σ2+λ2(∥A^∥+∥B^∥))×∫t0t‍α(∥ϕ∥22+∥ψ∥22)exp⁡(-β(s-t0))ds-2σ𝔼∫t0t‍∫Ω0‍(u(s,x)-u-(s,x))Tu-(s,x)dxdW(s)-2σ𝔼∫t0t‍∫Ω0‍(v(s,x)-v-(s,x))Tv-(s,x)dxdW(s).
By stochastic Fubini’s Theorem, we have
(70)𝔼∫t0t‍∫Ω0‍(u(s,x)-u-(s,x))Tu-(s,x)dxdW(s)=0,𝔼∫t0t‍∫Ω0‍(v(s,x)-v-(s,x))Tv-(s,x)dxdW(s)=0.
By (69), one get
(71)𝔼V(u(t,x),v(t,x),u-(t,x),v-(t,x),r(t))≤(μ^C4+max⁡p∈𝕊∑q=1N‍γpqμq)μ˘×∫t0t‍𝔼V(u(s,x),v(s,x),u-(s,x),v-(s,x),r(s))ds+2μ^(σ2+λ2(∥A^∥+∥B^∥))α(∥ϕ∥22+∥ψ∥22)β.
When t0+Δ≤t≤t0+2Δ, by applying Gronwall’s inequality, we have
(72)𝔼(∥u(t,x)-u-(t,x)∥22+∥v(t,x)-v-(t,x)∥22)=𝔼V(u(t,x),v(t,x),u-(t,x),v-(t,x),r(t))≤2μ^(σ2+λ2(∥A^∥+∥B^∥))α(∥ϕ∥22+∥ψ∥22)β×exp⁡(μ^C4+max⁡p∈𝕊∑q=1N‍γpqμq)μ˘(t-t0)≤sup⁡t0≤t≤t0+Δ𝔼(∥u-(t,x)∥2+∥v-(t,x)∥2)×2μ^(σ2+λ2(∥A^∥+∥B^∥))αβ×exp⁡2Δ(μ^C4+max⁡p∈𝕊∑q=1N‍γpqμq)μ˘.
By the global exponential stability of (6), we have
(73)𝔼(∥u-(t,x)∥22+∥v-(t,x)∥22)≤2𝔼(∥u(t,x)-u-(t,x)∥22+∥v(t,x)-v-(t,x)∥22)+2𝔼(∥u(t,x)∥22+∥v(t,x)∥22)≤sup⁡t0≤t≤t0+Δ𝔼(∥u-(t,x)∥2+∥v-(t,x)∥2)×4μ^(σ2+λ2(∥A^∥+∥B^∥))αβ×exp⁡2Δ(μ^C4+max⁡p∈𝕊∑q=1N‍γpqμq)μ˘+2α(∥ϕ∥22+∥ψ∥22)exp⁡{-β(t-t0)}.
Moreover,
(74)𝔼(∥u-(t,x)∥22+∥v-(t,x)∥22)≤{4μ^(σ2+λ2(∥A^∥+∥B^∥))αβ{2Δ(μ^C4+max⁡p∈𝕊∑q=1N‍γpqμq)μ˘}mm×exp⁡{2Δ(μ^C4+max⁡p∈𝕊∑q=1N‍γpqμq)μ˘}lmm{2Δ(μ^C4+max⁡p∈𝕊∑q=1N‍γpqμq)μ˘}+2αexp⁡{-βΔ}}×sup⁡t0≤t≤t0+Δ𝔼(∥u-(t,x)∥2+∥v-(t,x)∥2).
From (62), when (λ,σ) is in the inner of the closed curve described by the transcendental equation, we have
(75)4μ^(σ2+λ2(∥A^∥+∥B^∥))αβ×exp⁡{2Δ(μ^C4+max⁡p∈𝕊∑q=1N‍γpqμq)μ˘}+2αexp⁡{-βΔ}<1.
Let
(76)γ=(-log⁡{{2Δ(μ^C4+max⁡p∈𝕊∑q=1N‍γpqμq)μ˘}+2αexp⁡{-βΔ}{2Δ(μ^C4+max⁡p∈𝕊∑q=1N‍γpqμq)μ˘}})4μ^(σ2+λ2(∥A^∥+∥B^∥))αβmmmmm×exp⁡{2Δ(μ^C4+max⁡p∈𝕊∑q=1N‍γpqμq)μ˘}mmmmm{2Δ(μ^C4+max⁡p∈𝕊∑q=1N‍γpqμq)μ˘}+2αexp⁡{-βΔ}{2Δ(μ^C4+max⁡p∈𝕊∑q=1N‍γpqμq)μ˘}})(Δ)-1>0.
By (74), we have
(77)sup⁡t0+Δ≤t≤t0+2Δ𝔼(∥u-(t,x)∥22+∥v-(t,x)∥22)≤exp⁡(-γΔ)(sup⁡t0≤t≤t0+Δ𝔼(∥u-(t,x)∥22+∥v-(t,x)∥22)).
Similar to the proof of Theorem 5, we can prove that the neural networks (59) are mean square globally exponentially stable and also almost surely globally exponentially stable.To continue, we consider the parameter uncertainty intensity which is added to the connection weight matrix(C,E)T of the neural networks (16). Then, the neural networks (16) are changed as
(78)du-i(t,x)={∑j=1n‍∑k=1l‍∂∂xk(Dik(r(t))∂u-i(t,x)∂xk)m-ai(r(t))u-i(t,x)m+∑j=1n‍(1+δ)cji(r(t))fj(v-j(t,x))}dt+σu-i(t,x)dW(t),dv-j(t,x)={∑k=1l‍∂∂xk(Djk*(r(t))∂v-j(t,x)∂xk)m-bj(r(t))v-j(t,x)m+∑i=1m‍(1+δ)eij(r(t))gi(u-i(t,x))}dt+σv-j(t,x)dW(t).The initial conditions and boundary conditions are given by(79)u-i(t0,x)=ϕi(x),x∈Ω0,t0∈ℝ+,i=1,2,…,m,v-j(t0,x)=ψj(x),x∈Ω0,t0∈ℝ+,j=1,2,…,n,∂u-i(t,x)∂n→|∂Ω0=(∂u-i(t,x)∂x1,…,∂u-i(t,x)∂xl)T=0,bbbi(t,x)∈[t0,+∞)×∂Ω0,i=1,2,…,m,∂v-j(t,x)∂n→|∂Ω0=(∂v-j(t,x)∂x1,…,∂v-j(t,x)∂xl)T=0,bbbb(t,x)∈[t0,+∞)×∂Ω0,j=1,2,…,n,
where δ is the connection weight matrix (C,E)T uncertainty intensity and σ is the noise intensity.We rewrite (78) as follows:
(80)du-(t,x)={C(r(t))f(v-(t,x))∇·(D(r(t))∘∇u-)-A(r(t))u-(t,x)l+(1+δ)C(r(t))f(v-(t,x))}dt+σu-(t,x)dW(t),dv-(t,x)={∇·(D*(r(t))∘∇v-)-B(r(t))v-(t,x)l+(1+δ)E(r(t))g(u-(t,x))}dt+σv-(t,x)dW(t).
For the global exponential stability of neural networks (6), we will characterize how much the intensity of both the connection weight matrix (C,E)T uncertainty and stochastic noise the stochastic neural networks (78) can tolerate while maintaining global exponential stability.Theorem 7.
Let Assumption (H1) hold and let the neural networks (6) be global exponential stability. Then, the neural networks (78) are mean square globally exponentially stable and also almost surely globally exponentially stable, if there exists μq>0, (q∈𝕊), and (δ,σ) is in the inner of the closed curve described by the following transcendental equation:
(81)4μ^(σ2+δ2(K2∥C^∥+L2∥E^∥))αβ×exp⁡{2Δ(μ^C5+max⁡p∈𝕊∑q=1N‍γpqμq)μ˘}+2αexp⁡{-βΔ}=1,Δ>ln⁡(2α)β>0,
where C5=[2(∥A^∥+∥B^∥)+(2+(1+2δ2)K2)∥C^∥+(2+(1+2δ2)L2)∥E^∥+2σ2], ∥A^∥=max⁡p∈𝕊∥A(p)∥, and so forth and μ^=max⁡p∈𝕊μp and μ˘=min⁡p∈𝕊μp.The proof is similar to the proof of Theorem6.


## 5. Illustrate Example

Example 1.
Consider hybrid BAM neural networks with reaction diffusion terms(82)∂u(t,x)∂t=D(r(t))∂2u(t,x)∂x2-a(r(t))u(t,x)+c(r(t))f(v(t,x)),∂v(t,x)∂t=Dk*(r(t))∂2v(t,x)∂x2-b(r(t))v(t,x)+e(r(t))g(u(t,x)).
The initial conditions and boundary conditions are given by(83)u(0,x)=sin(x),x∈[-5,5],v(0,x)=cos⁡(x)-1,x∈[-5,5],∂u(t,x)∂n→|∂Ω0=∂u(t,x)∂x=0,(t,x)∈[t0,+∞)×∂Ω0,∂v(t,x)∂n→|∂Ω0=∂v(t,x)∂x=0,(t,x)∈[t0,+∞)×∂Ω0,
where
(84)Γ=(-221-1),(D(1)00D*(1))=(0.002000.003),(D(2)00D*(2))=(0.001000.002),(a(1)00b(1))=(0.2000.3),(0c(1)e(1)0)=(00.20.20),(a(2)00b(2))=(0.3000.2),(0c(2)e(2)0)=(00.20.20),
and f(v)=sin(v), g(u)=(1/2)(|u+1|-|u-1|), and K=L=1. According to Theorem  1 in [9] and Theorem 1 in [29], the neural networks (82) are global exponential stability with α=1 and β=1.
In the presence of stochastic noise and self-feedback matrix(A,B)T uncertainty, the neural networks (82) become
(85)du(t,x)={D(r(t))∂2u(t,x)∂x2m-(a(r(t))+λ)u(t,x)m∂2u(t,x)∂x2+c(r(t))f(v(t,x))}dt+σu(t,x)dW(t),dv(t,x)={D*(r(t))∂2v(t,x)∂x2m-(b(r(t))+λ)v(t,x)mm∂2v(t,x)∂x2m+e(r(t))g(u(t,x))}dt+σv(t,x)dW(t).
According to Theorem 6, let Δ=0.7>log⁡(2α)/β=0.6931 and μ1=1andμ2=2. From (62), we have
(86)8(σ2+0.6λ2)exp⁡{10.08+3.36λ2+5.6σ2}+2exp⁡{-0.7}=1.
Then, we can obtain its closed curve for (λ,σ). Figure 1 depicts the stability region for (λ,σ) in (85).Figure 1
The stability region with(λ,σ) of the neural networks (85).Figures2, 3, 4, and 5 depict the surface curves of the neural networks (85) with (λ,σ)=(-10-4,10-4). It shows that the state of the neural networks (85) is mean square globally exponential stability and almost surely globally exponential stable, as the parameter (λ,σ) in the inner of the curve of Figure 1.Figure 2
Surface curve ofu(t,x) of the neural networks (85) in model 1.Figure 3
Surface curve ofv(t,x) of the neural networks (85) in model 1.Figure 4
Surface curve ofu(t,x) of the neural networks (85) in model 2.Figure 5
Surface curve ofv(t,x) of the neural networks (85) in model 2.Figures6, 7, 8 and 9 show the surface curves of the neural networks (85) with (λ,σ)=(-0.1,1.5). It shows that when the conditions in Theorem 6 do not hold, the neural networks (85) become unstable.Figure 6
Surface curve ofu(t,x) of the neural networks (82) in model 1.Figure 7
Surface curve ofv(t,x) of the neural networks (82) in model 1.Figure 8
Surface curve ofu(t,x) of the neural networks (82) in model 2.Figure 9
Surface curve ofv(t,x) of the neural networks (82) in model 2.



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*Source: 486052-2014-04-17.xml*