# Exponential Stability for a Class of Stochastic Reaction-Diffusion Hopfield Neural Networks with Delays

Authors: Xiao Liang; Linshan Wang
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

This paper studies the asymptotic behavior for a class of delayed reaction-diffusion Hopfield neural networks driven by finite-dimensional Wiener processes. Some new sufficient conditions are established to guarantee the mean square exponential stability of this system by using Poincaré’s inequality and stochastic analysis technique. The proof of the almost surely exponential stability for this system is carried out by using the Burkholder-Davis-Gundy inequality, the Chebyshev inequality and the Borel-Cantelli lemma. Finally, an example is given to illustrate the effectiveness of the proposed approach, and the simulation is also given by using the Matlab.


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## Body

## 1. Introduction

Recently, the dynamics of Hopfield neural networks with reaction-diffusion terms have been deeply investigated because their various generations have been widely used in some practical engineering problems such as pattern recognition, associate memory, and combinatorial optimization (see [1–3]). However, under closer scrutiny, that a more realistic model would include some of the past states of the system, and theory of functional differential equations systems has been extensively developed [4, 5], meanwhile many authors have considered the asymptotic behavior of the neural networks with delays [6–9]. In fact random perturbation is unavoidable in any situation [3, 10]; if we include some environment noise in these systems, we can obtain a more perfect model of this situation [3, 11–16]. So, this paper is devoted to the exponential stability of the following delayed reaction-diffusion Hopfield neural networks driven by finite-dimensional Wiener processes:(1.1)dui(t,x)=(∑j=1l∂∂xj(Dij(x)∂ui∂xj)-aiui+∑j=1ncijfj(uj(t-r,x)))dt+∑j=1mgij(ui(t-r,x))dWj,∂ui∂ν|∂O=0,t≥0,ui(θ,x)=ϕi(θ,x),x∈O∈Rl,θ∈[-r,0],i=1,2,…,n.There aren neural network units in this system and ui(t,x) denote the potential of the cell i at t and x. ai are positive constants and denote the rate with which the ith unit will reset its potential to the resting state in isolation when it is disconnected from the network and external inputs at t, and cij are the output connection weights from the jth neuron to the  ith neuron. fj are the active functions of the neural network. r is the time delay of a neuron. 𝒪 denotes an open bounded and connected subset of ℝl with a sufficient regular boundary ∂𝒪, ν is the unit outward normal on ∂𝒪, ∂ui/∂ν=(∇ui,ν)ℝl, and gij are noise intensities. Initial data ϕi are ℱ0-measurable and bounded functions, almost surely.We denote(Ω,ℱ,ℙ) a complete probability space with filtration {ℱt}t≥0 satisfying the usual conditions (see [10]). Wi(t),i=1,2,…,m, are scale standard Brownian motions defined on (Ω,ℱ,ℙ).For convenience, we rewrite system (1.1) in the vector form:
(1.2)du=(∇⋅(D(x)∘∇u)-Au+Cf(u(t-r)))dt+G(u(t-r))dW,∂u(t,x)∂ν|∂O=0,t≥0,u(0,x)=ϕ(x),
where C=(cij)n×n, u=(u1,u2,…,un)T, ∇u=(∇u1,…,∇un)T, W=(W1,W2,…,Wm)T, f(u)=(f1(u1),f2(u2),…,fn(un))T, A=Diag(a1,a2,…,an), ϕ=(ϕ1,ϕ2,…,ϕn)T, G(u)=(gij(ui))n×m, D=(Dij)n×l, and D∘∇u=(Dij∂ui/∂xj)n×l is the Hadamard product of matrix D and ∇u; for the definition of divergence operator ∇·u, we refer to [2, 3].


## 2. Preliminaries and Notations

In this paper, we introduce the following Hilbert spacesH≜L2(𝒪), V≜H1(𝒪), according to [17–19], V⊂H=H′⊂V′, where H′,V′ denote the dual of the space H,V, respectively, the injection is continuous, and the embedding is compact. ∥·∥,|∥·∥| represent the norm in H,V, respectively.U≜(L2(𝒪))n is the space of vector-valued Lebesgue measurable functions on 𝒪, which is a Banach space under the norm ∥u∥U=(∑i=1n∥ui(x)∥2)1/2.C≜C([-r,0],U) is the Banach space of all continuous functions from [-r,0] to U, when equipped with the sup-norm ∥ϕ∥C=sup-r≤s≤0∥ϕ∥U.With any continuousℱt-adapted U-valued stochastic process u(t):Ω→U, t≥-r, we associate a continuous ℱt-adapted C-valued stochastic process ut:Ω→C,t>0, by setting ut(s,x)(ω)=u(t+s,x)(ω), s∈[-r,0],x∈𝒪.Cℱ0b denote the space of all bounded continuous processes ϕ:[-r,0]×Ω→U such that ϕ(θ,·) is ℱ0-measurable for each θ∈[-r,0] and E∥ϕ∥C<∞.ℒ(K) is the set of all linear bounded operators from K into K; when equipped with the operator norm, it becomes a Banach space.In this paper, we assume the following.H1
fi and Gij are Lipschitz continuous with positive Lipschitz constants k1,k2 such that |fi(u)-fi(v)|≤k1|u-v| and |Gij(u)-Gij(v)|≤k2|u-v|,∀u,v∈ℝ, and fi(0)=0, gij(0)=0.H2
There existsα>0 such that Dij(x)≥α/l.H3
Letη=2αβ2+2k3-nk12σ2er-mk22er-2>0,k3=min{ai}, σ=max{|cij|}.Remark 2.1.
We can infer from H1 that system (1.1) has an equilibrium u(t,x,ω)=0.
Let us define the linear operator as follows:(2.1)A:Π(A)∈U⟶U,Au=∇⋅(D(x)∘∇u),
and Π(𝔄)={u∈H2(𝒪)n,∂u/∂ν|∂𝒪=0}.Lemma 2.2 (Poincaré’s inequality).
Let𝒪 be a bounded domain in Rl and ϕ belong to a collection of twice differentiable functions defined on 𝒪 into R; then
(2.2)‖ϕ‖≤β-1|‖ϕ‖|,
where the constant β depends on the size of 𝒪.Lemma 2.3.
Let us consider the equation(2.3)dudt=Au,t≥0,u(0)=ϕ.
For every ϕ∈U, let u(t)=S(t)ϕ denote the solution of (2.3); then S(t) is a contraction map in U.Proof.
Now we take the inner product of (2.3) with u(t) in U; by employing the Gaussian theorem and condition H2, we get that (𝔄u,u)≤-α∥|u|∥H1(𝒪)n2, (·,·) is the inner product in U, ∥|u|∥H1(𝒪)n2 denote the norm of H1(𝒪)n (see [3]), which means
(2.4)12ddt‖u(t)‖U2+α‖|u(t)|‖H1(O)n2≤0.
Thanks to the Poincaré inequality, one obtains
(2.5)ddt‖u(t)‖U2+2αβ2‖u(t)‖U2≤0.
Multiplying e2αβ2t in both sides of the inequality, we have
(2.6)ddt(e2αβ2t‖u(t)‖U2)≤0.
Integrating the above inequality from 0 to t, we obtain
(2.7)‖u(t)‖U2≤e-2αβ2t‖ϕ‖U2.
By the definition of ∥T(t)∥ℒ(U), we have ∥T(t)∥ℒ(U)≤1.Definition 2.4 (see [20–22]).
A stochastic processu(t):[-r,+∞)×Ω→U is called a global mild solution of (1.1) if(i)
u(t) is adapted to ℱt(ii)
u(t) is measurable with ∫0∞∥u(t)∥U2dt<∞ almost surely and(2.8)u(t)=S(t)ϕ-∫0tS(t-s)Ads+∫0tS(t-s)f(u(s-r))ds+∫0tS(t-s)G(u(s-r))dW,u(t)=ϕ∈CF0b,t∈[-r,0],
for all t∈[-r,+∞) with probability one.Definition 2.5.
Equation (1.1) is said to be almost surely exponentially stable if, for any solution u(t,x,ω) with initial data ϕ∈Cℱ0b, there exists a positive constant λ such that
(2.9)limsup⁡t⟶∞ln⁡‖ut‖C≤-λ,ut∈C,almostsurely.Definition 2.6.
System (1.1) is said to be exponentially stable in the mean square sense if there exist positive constants κ and α such that, for any solution u(t,x,ω) with the initial condition ϕ∈Cℱ0b, one has
(2.10)E‖u(t)‖C2≤κe-α(t-t0),t≥t0,ut∈C.


## 3. Main Result

Theorem 3.1.
Suppose conditions H1–H3 hold; then (1.1) is exponentially stable in the mean square sense.Proof.
Letu be the mild solution of (1.1); thanks to the Itô formula, we observe that
(3.1)d(eλtui2)=λeλtui2dt+eλt(2ui(∑j=1l∂∂xi(Dij∂ui∂xj)-aiui+∑j=1ncijfj(uj(t-r))))dt+eλt(GiGiT)dt+2eλtuiGidW,Gi=(Gi1,Gi2,…,Gim),
where λ is a positive constant that will be defined below. Then, by integration between 0 and t, we find that
(3.2)eλtui2(t)=ϕi(0)2+∫0tλeλsui2ds+2∫0teλs(ui∑j=1l∂∂xj(Dij∂ui∂xj))ds-2∫0teλsaiui2ds+2∫0teλsui∑j=1ncijfj(uj(s-r))ds+∫0teλsGiGiTds+2∫0teλsuiGidW.
Integrating the above equation over 𝒪, by virtue of Fubini’s theorem, we prove that
(3.3)eλt‖ui2‖2=‖ϕi(0)‖2+λ∫0teλs∫Oui2dxds+∫0teλs∫O2ui∑j=1l∂∂xj(Dij∂ui∂xj)dxds-2∫0teλs∫Oaiui2dxds+2∫0teλs∫Oui∑j=1ncijfj(uj(s-r))dxds+∫0teλs∫OGiGiTdxds+2∫0teλs∫OuiGidxdW.
Taking the expectation on both sides of the last equation, by means of [3, 10, 16]
(3.4)2E∫0t∫OeλsuiGidxdW=0.
Then, by Fubini’s theorem, we have
(3.5)eλtE‖ui2‖2=E‖ϕi(0)‖2+λ∫0teλs∫OEui2dxds+2E∫0teλs∫Oui∑j=1l∂∂xj(Dij∂ui∂xj)dxds-2∫0teλs∫OaiEui2dxds+2E∫0teλs∫Oui∑j=1ncijfj(uj(s-r))dxds+E∫0teλs∫OGiGiTdxds≜I1+I2+I3+I4+I5+I6.
We observe that
(3.6)I1≜E‖ϕi(0)‖2≤sup⁡θ∈[-r,0]E‖ϕi(θ)‖2,(3.7)I2≜λ∫0t∫OeλsEui2dxds=λ∫0teλsE‖ui‖2ds.
From the Neumann boundary condition, by means of Green’s formula and H2 (see [3, 6, 7]), we know
(3.8)I3≜2E∫0t∫Oeλs(ui∑j=1l∂∂xj(Dij∂ui∂xj))dxds=-2E∫0t∫Oeλs∑j=1lDij(∂ui∂xj)2dxds≤-2α∫0teλsE|‖ui‖|2ds≤-2αβ2∫0teλsE‖ui‖2ds.
Then, by using the positiveness of ai, one gets the relation
(3.9)I4≜-2∫0t∫OeλsaiEui2dxds≤-2k3∫0teλsE‖ui‖2ds,
where k3=min{a1,a2,…,an}>0. By using the Young inequality as well as condition H1, we have that
(3.10)I5≜2E∫0t∫Oeλsui∑j=1ncijfjdxds≤∫0t∫Oeλs(E|ui|2+E|∑j=1ncijfj|2)dxds≤∫0t∫Oeλs(E|ui|2+σ2∑j=1nE|fj(uj(s-r))|2)dxds≤∫0t∫Oeλs(E|ui|2+σ2k12∑j=1nE|uj(s-r)|2)dxds≤∫0teλs(E‖ui‖2+σ2k12E‖u(s-r)‖U2)ds,
where σ=max|cij|, and
(3.11)I6≜∫0t∫OeλsEGiGiTdxds≤mk22∫0teλsE‖ui(s-r)‖2ds.
We infer from (3.6)–(3.11) that
(3.12)eλtE‖ui(t)‖2≤sup⁡θ∈[-r,0]E‖ϕi(θ)‖2-(2αβ2+2k3-1-λ)∫0teλsE‖ui‖2ds+σ2k12∫0teλs‖u(t-r)‖U2ds+mk22∫0teλsE‖ui(s-r)‖2ds.
Adding (3.12) from i=1 to i=n, we obtain
(3.13)eλtE‖u‖U2≤E‖ϕ‖C2-(2αβ2+2k3-1-λ)∫0teλsE‖u‖U2ds+(nk12σ2+mk22)∫0teλsE‖u(s-r)‖U2ds,
due to
(3.14)∫0teλsE‖u(s-r)‖U2ds≤eλr∫-rteλsE‖u(s)‖U2ds≤e2λr∫-r0E‖ϕ(s)‖U2ds+eλr∫0teλsE‖u(s)‖U2ds≤re2λrE‖ϕ‖C2+eλr∫0teλsE‖u(s)‖U2ds;
we induce from the previous equations that
(3.15)eλtE‖u‖U2≤-c1∫0teλsE‖u‖U2ds+c2,
where c1=2αβ2+2k3-1-nk12σ2eλr-mk22eλr-λ and c2=(1+mk22re2λr+nk12σ2re2λr)E∥ϕ∥C2; so we choose λ=1 such that c1=η>0. By using the classical Gronwall inequality we see that
(3.16)eλtE‖u‖U2≤c2e-ηt;
in other words, we get
(3.17)E‖u‖U2≤c2e-(η+1)t.
So, for t+θ≥t/2≥0, we also have
(3.18)E‖u(t+θ)‖U2≤c2e-(η+1)(t+θ),≤c2e-κt,θ∈[-r,0],κ=(η+1)2,
and we can conclude that
(3.19)E‖ut‖C2≤c2e-κt.Theorem 3.2.
If the system (1.1) satisfies hypotheses H1–H3, then it is almost surely exponentially stable.Proof.
Letu(t) be the mild solution of (1.1). By Definition 2.4 as well as the inequality (∑i=1nai)2≤n∑i=1nai2,ai∈ℝ, we have
(3.20)Esup⁡N≤t≤N+1‖u(t)‖U2≤4sup⁡N≤t≤N+1‖S(t-N+1)u(N)‖U2+4sup⁡N≤t≤N+1‖∫N-1t-AS(t-s)uds‖U2+4sup⁡N≤t≤N+1‖∫N-1tS(t-s)Cf(u(s-r))ds‖U2+4sup⁡N≤t≤N+1‖∫N-1tS(t-s)G(u(s-r))dW‖U2≜I1+I2+I3+I4.
Using the contraction of the map S(t) and the result of Theorem 3.1, we find
(3.21)I1≜4sup⁡N≤t≤N+1E‖(S(t-N+1)u(N-1))‖U2≤4sup⁡N≤t≤N+1E‖uN-1‖C2≤4c2e-κ(N-1).
By the Hölder inequality, we obtain
(3.22)I2≜4sup⁡N≤t≤N+1E‖∫N-1t-AS(t-s)uds‖U2≤4sup⁡N≤t≤N+1(t-N+1)∫N-1tE‖-AS(t-s)u‖U2ds≤8sup⁡N≤t≤N+1∫N-1tE‖Au‖U2ds≤8k42∫N-1N+1E‖u‖U2ds≤8k42∫N-1N+1E‖us‖C2ds≤8k42c2∫N-1N+1e-κsds≤8k42ρ1e-κ(N-1),
where ρ1=c2/κ,k4=max{a1,a2,…,an}.
By virtue of Theorem3.1, Hölder inequality, and H1, we have
(3.23)I3≜4sup⁡N≤t≤N+1‖E∫N-1tS(t-s)Cf(u(s-r))ds‖U2≤4sup⁡N≤t≤N+1(t-N+1)E∫N-1t‖Cf(u(s-r))‖U2ds≤8σ2sup⁡N≤t≤N+1E∫N-1t‖f(u(s-r))‖U2ds≤8k12σ2∫N-1N+1E‖u(s-r)‖U2ds≤8k12σ2∫N-1N+1E‖us‖C2ds≤8k12c2∫N-1N+1e-κsds≤8k12ρ1e-κ(N-1).
Then, by the Burkholder-Davis-Gundy inequality (see [18, 22]), there exists c3 such that
(3.24)I4≜4sup⁡N≤t≤N+1E‖∫N-1tS(t-s)G(u(s-r))dW‖U2≤4c3sup⁡N≤t≤N+1E∫N-1t‖S(t-s)G(u(s-r))I‖U2ds≤4c3k22sup⁡N≤t≤N+1∫N-1tE‖u(s-r)‖U2ds≤4c3k22∫N-1N+1E‖us‖C2ds≤4c3k22c2∫N-1N+1e-κsds≤4c3k22ρ1e-κ(N-1),
where I=(1,1,…,1)T is an m-dimensional vector.
We can deduce from (3.21)–(3.24) that
(3.25)Esup⁡N≤t≤N+1‖u(t)‖U2≤ρ2e-κ(N-1),
where ρ2=4c2+(8k42+8k12+4c3k22)ρ1.
Thus, for any positive constantsɛN, thanks to the Chebyshev inequality we have that
(3.26)P(sup⁡N≤t≤N+1‖u(t)‖U>ɛN)≤1ɛN2sup⁡N≤t≤N+1E‖u(t)‖U2≤1ɛN2ρ2e-κ(N-1).
Due to the Borel-Cantelli lemma, we see that
(3.27)limsup⁡t⟶∞ln⁡‖u(t)‖U2t≤-κ,almostsurely.
This completes the proof of the theorem.


## 4. Simulation

Consider two-dimensional stochastic reaction-diffusion recurrent neural networks with delay as follows:(4.1)du1(t,x)=(10Δu1-7u1+1.3tanh(u1(t-1,x)))dt+u1(t-1,x)dW,du2(t,x)=(10Δu2-7u2+tanh(u1(t-1,x))-tanh(u2(t-1,x)))dt+u2(t-1,x)dW∂ui(t,0)∂x=∂ui(t,20)∂x=0,t≥0,u1(θ,x)=cos⁡(0.2πx),u2(θ,x)=cos⁡(0.1πx),x∈[0,20],θ∈[-1,0].Δ is the Laplace operator. We have β≥1/20, α≥10, k1=1, k2=1, k3=7, σ=1.3, n=2,  and η>0; by Theorems 3.1 and 3.2, this system is mean square exponentially stable as well as almost surely exponentially stable. The results can be shown in Figures 1, 2 and 3.(a)(b)Figure 2Figure 3We use the forward Euler method to simulate this example [23–25]. We choose the time step Δt=0.01 and space step Δx=1, and δ=Δt/Δx2=0.01.



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