# Globalμ-Stability Analysis for Impulsive Stochastic Neural Networks with Unbounded Mixed Delays

Authors: Lizi Yin; Xinchun Wang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We investigate the globalμ-stability in the mean square of impulsive stochastic neural networks with unbounded time-varying delays and continuous distributed delays. By choosing an appropriate Lyapunov-Krasovskii functional, a novel robust stability condition, in the form of linear matrix inequalities, is derived. These sufficient conditions can be tested by MATLAB LMI software packages. The results extend and improve the earlier publication. Two numerical examples are provided to illustrate the effectiveness of the obtained theoretical results.


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## Body

## 1. Introduction

Since 1982 when American California Engineering institute physicist Hopfield proposed Hopfield neural networks models [1, 2], artificial neural networks theory and applied research have attracted more and more attentions. The application of artificial neural networks is very extensive. It can process a variety of information, such as artificial intelligence, secure communications, network optimization, military information, pattern recognition, and so forth. Information processing of neural networks greatly depends on the system's dynamic characteristics. The stability is one of very important dynamic characteristics of neural network.External perturbations can affect the stability of a real system, so it is necessary to consider the stochastic effects for the stability property of neural networks. Recently, some results to guarantee the global asymptotic stability or exponential stability for stochastic neural networks are obtained, see [3–8]. For example, in [4],  Blythe et al. investigated stochastic stability of neural networks with constant delay. In [6], Wang et al. analysed the stability in the mean square for stochastic Cohen-Grossberg neural networks with time-varying delays and continuous distributed delays.As we known, artificial neural networks are often subject to impulsive perturbations which can affect systems’ stability, see [9–11]. Impulsive perturbations often make stable systems unstable. Therefore, we should consider systems’ stability in impulsive effects. Very recently, some results on stability of stochastic neural networks with impulses have been proposed and studied, see [12, 13].The neural network may be disturbed by environmental noises, which cause the uncertainty of the connection weights of the neurons and affect the dynamical behaviors of system. Therefore, parameter uncertainties should be taken into account in system model. There are some global stability results of neural networks under the influence of parameter uncertainties [14–20]. In [14], the authors kept a watchful eye on robust exponential stability for uncertain stochastic neural networks with discrete interval and distributed time-varying delays by Lyapunov-Krasovski functional method and the random analysis theory.The time delays happen frequently in the neurotransmission. Many researchers have a large amount interest in time delay neural networks. There are many sufficient conditions have been proposed to guarantee the stability of neural networks with various type of time delays [21–26]. For example, in [22], Lou and Cui investigated the global asymptotic stability of the Hopfield neural networks with bounded time delay by constructing suitable Lyapunov-krasovski functional and employing LMI method. In [24], Raja et al. further obtained some results of stochastic neural networks with mixed time delays by using Lyapunov stability theory and LMI technology. However, most of the results have only focused on bounded delays. As we know, in many engineering time delays depend on the histories heavily and may be unbounded [27, 28]. In this case, the conditions on delays of those existing results are too restrictive.Recently, Chen and Wang [29], Liu and Chen [30] proposed a new concept of μ-stability, which can be applied to neural networks with unbounded time-varying delay. Moreover, few results have been reported in the literature concerning the problem of μ-stability for parameter impulsive stochastic neural networks with unbounded time-varying delays and continuously distributed delays, which inspire our interests.This paper is concerned with the global stability analysis in the mean square for impulsive stochastic neural networks with unbounded time-varying delays and continuously distributed delays. By choosing an appropriate Lyapunov-Krasovskii functional, a novel robust stability condition, in the form of linear matrix inequalities, is derived. These sufficient conditions can be tested by Matlab LMI software packages. Our results extend and improve the some results [29, 30].The paper is organized as following. In Section2, the basic definitions and assumptions are given together with the statement of the problem. In Section 3, the sufficient conditions of μ-stability in the mean square of impulsive stochastic neural networks with unbounded time-varying delays and continuously distributed delays are obtained. In Section 4, global robust μ-stability criteria in the mean square are derived for uncertain neural networks model. Then, two numerical examples are provided to demonstrate the effectiveness of our results in Section 5. Finally, concluding remarks are given in Section 6.Notations. Throughout this paper, R and Rn denote, respectively, the set of real numbers and n-dimensional real spaces equipped with the Euclidean norm |·|; Z+ denotes the set of positive integers; Rn×m denotes the set of all n×m. Let A≥0 (A≤0) denote that the matrix A is a symmetric and positive semidefinite (negative semidefinite) matrix; AT denotes the transpose of the matrix A; λmax(·) (λmin) denotes the maximum eigenvalue (minimum eigenvalue) of matrix A. ℒ2[0,∞) is the space of square integrable vector. Moreover, let (Ω,ℱ,(ℱt)t≥0,𝒫) be a probability space with a filtration (ℱt)t≥0 satisfying the usual conditions (It is right continuous and ℱ0 contains all 𝒫-null sets). Lℱ02(-∞,0];Rn denotes the family of all ℱ0-measurable. E(·) stands for the mathematical expectation operator with respect to the given probability measure 𝒫.


## 2. Model Description and Some Assumptions

Consider the following neural network model:(1)x˙(t)=-A0x(t)+A1f(x(t))+A2f(x(t-τ(t)))+A3∫0∞‍h(s)f(x(t-s))ds+J,t≠tk,t>0,Δx(tk)=x(tk)-x(tk-)=Jk(x(tk-)),t=tk,k∈Z+,
where x=(x1(t),x2(t),…,xn(t))T is the neuron state vector of the neural network; A0=diag(a1(0),a2(0),…,an(0)),ai(0)>0,i=1,2,…,n, is decay rate; A1=(aij(1))n×n, A2=(aij(2))n×n and A3=(aij(3))n×n are connection weight matrix; f(·)=(f1(·),f2(·),…,fn(·))T is the neuron activation function; τ(t) represents the transmission delay of neural networks; h(·)=diag(h1(·),h2(·),…,hn(·)) is the delay kernel function; J is an input constant matrix, and Jk is the impulsive function.Throughout the paper, we make the following assumptions.Assumption 1.
The neuron activation functionsfj(·),j=1,2,…,n, are bounded, continuous differentiable, and satisfy
(2)0≤fj(u)-fj(v)u-v≤lj,u≠v,u,v∈R,
where lj,j=1,2,…,n are some positive constants.Assumption 2.
τ
(
t
) is a nonnegative and continuous differentiable time-varying delay and satisfies τ˙(t)≤ρ<1,  ρ is a positive constant.Assumption 3.
The delay kernelshi,i=1,2,…,n, are some real nonnegative continuous functions defined in [0,∞) and satisfy
(3)∫0∞‍hi(s)ds=1,i=1,2,…,n.Assumption 4.
The impulse timestk satisfy 0=t0<t1<⋯<tk<⋯,limt→∞tk=+∞.If the functionsfj(·),j=1,2,…,n satisfy Assumption 1, the system (1) exists an equilibrium point, see [14]. Assume that x*=(x1*,x2*,…,xn*)T is an equilibrium point of system (1), and the impulsive function of system (1) characterized by Jk(x(tk-))=-Mk(x(tk-)-x*), where Jk(x(tk-))=(J1k(x(tk-)),J2k(x(tk-)),…,Jnk(x(tk-)))T, Mk is a real matrix.Lettingyi=xi-x*, one can switch the equilibrium point of system (1) to the origin and have
(4)y˙(t)=-A0y(t)+A1g(y(t))+A2g(y(t-τ(t)))+A3∫0∞‍h(s)g(y(t-s))ds,t≠tk,t>0,Δy(tk)=y(tk)-y(tk-)=-Mk(y(tk-)),t=tk,k∈Z+,
where y(t)=(y1(t),y2(t),…,yn(t))T, g(y(s))=(g1(y1(s)),g2(y2(s)),…,gn(yn(s)))T, and gi(yi(s))=fi(xi(s)+xi*)-fi(xi*). By the definition of gi(·),i=1,2,…,n and Assumption 1, gi satisfies the sector condition
(5)0≤gi(u)u≤li,u∈R
and gi(0)=0,i=1,2,…,n, where lj,j=1,2,…,n are some positive constants.In the practical application, a neural system is affected by external perturbations. It is necessary to consider the effect of randomness to neural networks. We obtain stochastic impulsive neural networks with delays:(6)dy(t)=[∫0∞‍-A0y(t)+A1g(y(t))+A2g(y(t-τ(t)))+A3∫0∞‍h(s)g(y(t-s))ds]dt+σ(t,y(t),y(t-τ(t)))dω(t),t≠tk,t>0,Δy(tk)=y(tk)-y(tk-)=-Mk(y(tk-)),t=tk,k∈Z+,
where ω(t)=(ω1(t),ω2(t),…,ωn(t))T is n dimensional Brownian motions defined on a complete probability space (Ω,ℱ,(ℱt)t≥0,𝒫).Assumption 5.
Assume thatσ:R+×Rn×Rn→Rn×n is local Lipschitz continuous and satisfies the linear growth condition, that is, σ(t,0,0)=0 for all t∈R+. Moreover, there exist n×n dimension matrices Γ0>0,Γ1>0, such that
(7)trace[σ(t,y(t),y(t-τ(t)))Tσ(t,y(t),y(t-τ(t)))]≤yT(t)Γ0y(t)+yT(t-τ(t))Γ1y(t-τ(t)).The initial conditions for system (6) are y(s)=φ(s),s∈(-∞,0],φ∈Cℱ02((-∞,0],Rn), where Cℱ02 denotes the family of all bounded ℱ0-measurable, C((-∞,0],Rn)-valued variables, satisfying ∥φ∥=sups≤0E|φ(s)|2<∞.For completeness, we first give the following definition and lemmas.Definition 1 (see [29]).
Suppose thatμ(t) is a nonnegative continuous function and satisfieslimt→∞μ(t)=∞. If there exists a scalar M>0 such that
(8)E∥y(t)∥≤Mμ(t),t≥0,
then the system (6) is said to be globally stochastically μ-stable in the mean square.Obviously, the definition of global stochasticμ-stability in the mean square includes the globally stochastically asymptotical stability and exponential stability in the mean square.Lemma 2 (see [31]).
LetW1,W2,Q=QT>0 be real matrices of appropriate dimensions, and ε>0 be a scalar, then one has
(9)W1TW2+W2TW1≤εW1TQW1+ε-1W2TQ-1W2.Lemma 3 (see [32]).
LetU,V,W and F(t) be the real matrices of appropriate dimensions with W satisfying WT=W, then
(10)W+UF(t)V+VTFT(t)U<0,∀FT(t)F(t)≤I,
if and only if there exists a scalar ε>0, such that
(11)W+εUUT+ε-1VTV<0.Lemma 4 (see [32]).
For a given matrix(12)S=(S11S12S21S22)>0,
where S11T=S11, S22T=S22 is equivalent to any one of the following conditions:(1)
S22>0,S11-S12S22-1S12T>0;(2)
S11>0,S22-S12TS11-1S12>0.


## 3. Stochastic Stability Analysis of  Neural Networks

LetC2,1(Rn×R+→R+) denote the family of all nonnegative functions V(y,t) on Rn×R+, which are continuous one differentiable in t and twice differentiable in x. For every such V, we define an operator LV associated with system (6),
(13)LV=Vt(y,t)+Vy(y,t)[∫0∞‍-A0y(t)+A1g(y(t))Vt(y,t)+Vy(y,t)+A2g(y(t-τ(t)))Vt(y,t)+Vy(y,t)+A3∫0∞‍h(s)g(y(t-s))ds]+12trace[σTVyy(y,t)σ],
where
(14)Vt(y,t)=∂V(y,t)∂t,Vy(y,t)=(∂V(y,t)∂y1,…,∂V(y,t)∂yn)T,Vyy(y,t)=(∂V(y,t)∂yiyj)n×n,j=1,2,…,n.Theorem 5.
Assume that Assumptions1–5 hold. Then, the zero solution of system (6) is globally stochastically μ-stable in the mean sense if there exist diagonal matrices P>0, Qi>0,i=1,2 and Q3=diag(q1(3),…,qn(3)), some constants α1≥0, α2>0, α3>0, δj>0,j=0,1,2, mik∈[0,2],i=1,2,…,n,k∈Z+, a nonnegative continuous differential function μ(t) defined on [0,∞), and a constant T>0 such that, for t≥T,
(15)μ˙(t)μ(t)≤α1,μ(t-τ(t))μ(t)≥α2,∫0∞‍hi(s)μ(t+s)dsμ(t)≤α3,i=1,2,…,n,
and the following LMI hold:
(16)Ξ*=[Π0δ0PA1δ11-ρPA2δ2PA3*-Q100**-Q20***-Q3]<0,
where Π0=α1P-PA0-A0TP+λΓ0+(1/(1-ρ))α2-1λΓ1+L[δ0-1Q1+δ1-1α2-1Q2+δ2-1α3Q3]L
L
=
diag
(
l
1
,
l
2
,
…
,
l
n
), λ=λmax(P), and impulsive operator Mk=diag(m1k,…,mnk).Proof.
By Lemma4, (16) is equal to the following inequality:
(17)α1P-PA0-A0TP+δ0PA1Q1-1A1TP+δ111-ρPA2Q2-1A2TP+δ2PA3Q3-1A3TP+λΓ0+11-ρα2-1λΓ1+L[δ0-1Q1+δ1-1α2-1Q2+δ2-1α3Q3]L<0.
Based on the system (6), we construct the following Lyapunov-Krasovski functional:
(18)V(y(t),t)=V1(y(t),t)+V2(y(t),t)+V3(y(t),t),
where
(19)V1(y(t),t)=μ(t)yT(t)Py(t),V2(y(t),t)=11-ρα2-1λ∫t-τ(t)t‍μ(s)yT(s)Γ1y(s)ds+δ1-1α2-1∫t-τ(t)t‍μ(s)gT(y(s))Q2g(y(s))ds,V3(y(t),t)=∑i=1n‍qi(3)δ2-1∫0∞‍hi(σ)∫t-σt‍μ(s+σ)gi2×(yi(s))dsdσ.
By Itô's formula, we can obtain the following stochastic differential:
(20)dV(y(t),t)=LV(y(t),t)dt+Vy(y,t)σ(t,y(t),y(t-τ(t)))dω(t),
where
(21)LV(y(t),t)=LV1(y(t),t)+LV2(y(t),t)+LV3(y(t),t).
Along the trajectories of system (6), we have that
(22)LV1(y(t),t)=μ˙yT(t)Py(t)+2μ(t)yT(t)P×[∫0∞‍-A0y(t)+A1g(y(t))+A2g(y(t-τ(t)))+A3∫0∞h(s)g(y(t-s))ds]+12trace[σTVyy(y,t)σ].
By Lemma 2, we get that
(23)2yT(t)PA1g(y(t))≤δ0yT(t)PA1Q1-1A1TPy(t)+δ0-1gT(y(t))Q1g(y(t)),2yT(t)PA2g(y(t-τ(t)))≤δ1(1-ρ)-1yT(t)PA2Q2-1A2TPy(t)+δ1-1(1-ρ)gT(y(t-τ(t)))Q2g(y(t-τ(t))),2yT(t)PA3∫0∞‍h(s)g(y(t-s))ds≤δ2yT(t)PA3Q3-1A3TPy(t)+δ2-1∫0∞‍h(s)g(y(t-s))dsQ3×∫0∞‍h(s)g(y(t-s))ds.
From Assumption 5, we obtain that
(24)12trace[σTVyy(y,t)σ]=μ(t)trace[σTPσ]≤μ(t)λmax(P)trace[σTσ]≤μ(t)λ[yT(t)Γ0y(t)≤μ(t)λ+yT(t-τ(t))Γ1y(t-τ(t))].
Therefore,
(25)LV1(y(t),t)≤μ(t)yT(t)[α1P-PA0-A0TPμ(t)yT(t)+δ0PA1Q1-1A1TPμ(t)yT(t)+δ1(1-ρ)-1PA2Q2-1A2TP+δ2PA3Q3-1A3TP+λΓ0]y(t)+μ(t)δ0-1gT(y(t))Q1g(y(t))+μ(t)λyT(t-τ(t))Γ1y(t-τ(t))+μ(t)δ1-1(1-ρ)gTy×(t-τ(t))Q2g(y(t-τ(t)))+μ(t)δ2-1∫0∞‍h(s)g(y(t-s))dsQ3×∫0∞h(s)g(y(t-s))ds,LV2(y(t),t)≤11-ρα2-1λ[yTμ(t)yT(t)Γ1y(t)11-ρα2-1λ-μ(t-τ(t))yT(t-τ(t))11-ρα2-1λ×Γ1yT(t-τ(t))1-τ˙(t)]+δ1-1α2-1[μ(t)gT(y(t))Q2g(y(t))+δ1-1α2-1-μ(t-τ(t))gT(y(t-τ(t)))+δ1-1α2-1×Q2g(y(t-τ(t)))1-τ˙(t)]≤11-ρα2-1λμ(t)yT(t)Γ1y(t)-α2-1λμ(t-τ(t))yT(t-τ(t))×Γ1yT(t-τ(t))+δ1-1α2-1×[μ(t)gT(y(t))Q2g(y(t))-μ(t-τ(t))gT×(y(t-τ(t)))Q2g×(y(t-τ(t)))(1-ρ)gT].
From (15), we have
(26)LV2(y(t),t)≤11-ρα2-1λμ(t)yT(t)Γ1y(t)-λμ(t)yT(t-τ(t))Γ1yT(t-τ(t))+δ1-1α2-1μ(t)gT(y(t))Q2g(y(t))-δ1-1α2-1μ(t-τ(t))gT(y(t-τ(t)))×Q2g(y(t-τ(t)))(1-ρ),LV3(y(t),t)=δ2-1∑i=1n‍qi(3)gi2(yi(t))∫0∞‍hi(σ)μ(t+σ)dσ-μ(t)∑i=1n‍qi(3)δ2-1∫0∞‍hi(σ)gi2(yi(t-σ))dσ≤δ2-1μ(t)∫0∞‍hi(σ)μ(t+σ)dσμ(t)∑i=1n‍qi(3)gi2(yi(t))-μ(t)∑i=1n‍qi(3)δ2-1∫0∞‍hi(σ)gi2(yi(t-σ))dσ≤δ2-1μ(t)α3∑i=1n‍qi(3)gi2(yi(t))-μ(t)∑i=1n‍qi(3)δ2-1∫0∞‍hi(σ)gi2(yi(t-σ))dσ.
We use Cauchy's inequality (∫‍p(s)q(s))2≤(∫‍p2(s)ds)(∫‍q2(s)ds) and get
(27)μ(t)∑i=1n‍qi(3)δ2-1∫0∞‍hi(σ)dσ∫0∞‍hi(σ)gi2(yi(t-σ))dσ≥μ(t)δ2-1[∫0∞‍h(σ)g(y(t-σ))dσ]T×Q3[∫0∞‍h(σ)g(y(t-σ))dσ].
Thus,
(28)LV3(y(t),t)≤δ2-1μ(t)α3gT(y(t))Q3g(y(t))-μ(t)δ2-1[∫0∞‍h(σ)g(y(t-σ))dσ]T,Q3[∫0∞‍h(σ)g(y(t-σ))dσ].
Substituting (25), (26), and (28) to (21), we get
(29)LV(y(t),t)≤μ(t)yT(t)[α1P-PA0-A0TP+δ0PA1Q1-1A1TP≤μ(t)yT(t)+δ1(1-ρ)-1PA2Q2-1A2TP≤μ(t)yT(t)+δ2PA3Q3-1A3TP≤μ(t)yT(t)+λΓ0+(1-ρ)-1α2-1λΓ1]y(t)+μ(t)gT(y(t))×[δ0-1Q1+δ1-1α2-1Q2+δ2-1α3Q3]g(y(t))≤μ(t)yT(t)[α1P-PA0-A0TP+δ0PA1Q1-1A1TPμ(t)yT(t)+δ1(1-ρ)-1PA2Q2-1A2TPμ(t)yT(t)+δ2PA3Q3-1A3TP+λΓ0+(1-ρ)-1α2-1λΓ1μ(t)yT(t)+L[δ0-1Q1+δ1-1α2-1Q2μ(t)yT(t)+L+δ2-1α3Q3]L]y(t)=μ(t)yT(t)Ξy(t),
where
(30)Ξ=α1P-PA0-A0TP+δ0PA1Q1-1A1TP+δ1(1-ρ)-1PA2Q2-1A2TP+δ2PA3Q3-1A3TP+λΓ0+(1-ρ)-1α2-1λΓ1+L[δ0-1Q1+δ1-1α2-1Q2+δ2-1α3Q3]L.
So, we have
(31)dV(y(t),t)≤μ(t)yT(t)Ξy(t)+Vy(y(t),t)σ(t,y(t),y(t-τ(t)))dω(t).
Taking the mathematical expectation, we get
(32)dEV(y(t),t)dt≤μ(t)yT(t)Ξy(t).
By Ξ<0, we get
(33)dEV(y(t),t)dt≤0,t∈[tk-1,tk)∩[T,+∞),k∈Z+.
In addition,
(34)V1(tk)=μ(tk)yT(tk)Py(tk)=μ(tk-)[y(tk-)-Mky(tk-)]T×P[yT(tk-)-MkyT(tk-)]=μ(tk-)[(I-Mk)y(tk-)]TP[(I-Mk)y(tk-)]=μ(tk-)yT(tk-)(I-Mk)TP(I-Mk)y(tk-).
Noting Mk=diag(m1k,m2k,…,mnk),mik∈[0,2],i=1,2,…,n,k∈Z+, we have
(35)V1(tk)≤μ(tk-)yT(tk-)Py(tk-)=V1(tk-).
It is obvious that V2(tk)=V2(tk-), V3(tk)=V3(tk-), therefore
(36)V(tk)≤V(tk-),k∈Z+.
By (33) and (36), we know that V is monotonically nonincreasing for t∈[T,∞), which implies that
(37)EV(y(t),t)≤EV(y(T),T),t≥T.
From the definition of V, we can deduce that
(38)μ(t)λmin(P)E∥y(t)∥2≤μ(t)EyT(t)PEy(t)≤EV(y(t),t)≤M<∞,t≥0,
where M=max0≤s≤TEV(y,s). It implies that
(39)E∥y(t)∥2≤Mμ(t)λmin(P),t≥0.
This completes the proof of Theorem 5.Remark 6.
Theorem5 provides a global stochastic μ-stability in the mean sense for a impulsive stochastic differential system (6). It should be noted that our results depend on the upper bound of the derivative of time-varying delay, the influence of randomness, the delay kernels of continuous distribution, and have nothing to do with the range of time-varying delay. Therefore, our results can be applied to the impulsive stochastic neural networks with unbounded time-varying delay and continuous distributed delay.Remark 7.
In [29, 30], different approaches were used to study μ-stability of neural networks with unbounded time-varying. However, the impulsive effect and unbounded continuous distributed delay are not considered in their models. It is known that neural networks with unbounded continuous distributed delays in impulsive effect are of great importance in many practically problems. Hence, our results in this paper are more general than those reported in [29, 30],If the diffusion coefficientσ(t,y(t),y(t-τ(t)))=0, the model (6) becomes the model (4), then the following result can be obtained.Corollary 8.
Assume that Assumptions1–4 hold. Then, the zero solution of system (4) is globally μ-stable if there exist diagonal matrices P>0, Qi>0,i=1,2 and Q3=diag(q1(3),…,qn(3)), some constants α1≥0,  α2>0,  α3>0,  δj>0,j=0,1,2, mik∈[0,2],i=1,2,…,n,k∈Z+, a nonnegative continuous differential function μ(t) defined on [0,∞), and a constant T>0 such that, for t≥T,
(40)μ˙(t)μ(t)≤α1,μ(t-τ(t))μ(t)≥α2,∫0∞‍hi(s)μ(t+s)dsμ(t)≤α3,i=1,2,…,n,
and the following LMI hold:
(41)Ξ*=[Π0δ0PA1δ11-ρPA2δ2PA3*-Q100**-Q20***-Q3]<0,
where Π0=α1P-PA0-A0TP+L[δ0-1Q1+δ1-1α2-1Q2+δ2-1α3Q3]L, L=diag(l1,l2,…,ln),  λ=λmax(P), and impulsive operator Mk=diag(m1k,…,mnk).If we takeP=Q1=Q2=Q3=I in Theorem 5, then we obtain the following testable condition.Corollary 9.
Assume that Assumptions1–5 hold. Then, the zero solution of system (6) is globally stochastically μ-stable in the mean square if there exist some constants α1≥0,α2>0,α3>0,δj>0,j=0,1,2,mik∈[0,2],i=1,2,…,n,k∈Z+, a nonnegative continuous differential function μ(t) defined on [0,∞), and a constant T>0 such that, for t≥T,
(42)μ˙(t)μ(t)≤α1,μ(t-τ(t))μ(t)≥α2,∫0∞‍hi(s)μ(t+s)dsμ(t)≤α3,i=1,2,…,n,
and the following LMI hold:
(43)Ξ*=[Π0δ0A1δ11-ρA2δ2A3*-I00**-I0***-I]<0,
where Π0=α1-A0-A0T+λΓ0+(1/(1-ρ))α2-1λΓ1+L[δ0-1+δ1-1α2-1+δ2-1α3]L, L=diag(l1,l2,…,ln), λ=λmax(P), and impulsive operator Mk=diag(m1k,…,mnk).


## 4. Robust Stochastic Stability Analysis of Neural Networks

Consider the following uncertain stochastic neural networks:(44)y˙(t)=-A0(t)y(t)+A1(t)g(y(t))+A2(t)g(y(t-τ(t)))+A3(t)∫0∞‍h(s)g(y(t-s))ds+σ(t,y(t),y(t-τ(t)))dω(t),t≠tk,t>0,Δy(tk)=y(tk)-y(tk-)=-Mk(y(tk-)),t=tk,k∈Z+,
where some parameters and variables are introduced in Section 2, the uncertainties are described as follows:
(45)Ai(t)=Ai+ΔAi(t),i=1,2,3,
where ΔAi(t),i=1,2,3 are perturbed matrices satisfying
(46)[ΔA0(t),ΔA1(t),ΔA2(t),ΔA3(t)]=HF(t)[T0,T1,T2,T3],
where H,Ti,i=1,2,3 are some appropriate dimensions constant matrices, F(t) is an unknown real time-varying function with appropriate dimensions and satisfies
(47)FT(t)F(t)≤I,
where I is appropriate dimensions unit matrix.Theorem 10.
Assume that Assumptions1–5 hold. Then, the zero solution of system (44) is globally robustly stochastically μ-stable in the mean square if there exist diagonal matrices P>0,Qi>0,  i=1,2 and Q3=diag(q1(3),…,qn(3)), some constants α1≥0,  α2>0,  α3>0, δj>0,  j=0,1,2,εj>0, j=0,1,2,3,mik∈[0,2],   i=1,2,…,n,  k∈Z+, a nonnegative continuous differential function μ(t) defined on [0,∞), and a constant T>0 such that, for t≥T,
(48)μ˙(t)μ(t)≤α1,μ(t-τ(t))μ(t)≥α2,∫0∞‍hi(s)μ(t+s)dsμ(t)≤α3,i=1,2,…n,
and the following LMI hold:
(49)Ξ**=[Π1δ0PA1δ11-ρPA2δ2PA3*R100**R20***R3]<0,
where Π1=α1P-PA0-A0TP+λΓ0+(1/(1-ρ))α2-1λΓ1+L[δ0-1Q1+δ1-1α2-1Q2+δ2-1α3Q3]L+ε0-1PHHTP+ε0T0TT0+ε1δ0T1TT1+ε2(δ1/(1-ρ))T2TT2+ε3δ2T3TT3, L=diag(l1,l2,…,ln), λ=λmax(P), R1=-Q1+ε1-1PHHTP, R2=-Q2+ε2-1PHHTP, R3=-Q3+ε3-1PHHTP, and impulsive operator Mk=diag(m1k,…,mnk).Proof.
Let the Lyapunov-Krasovskii functional as defined in Theorem5, then by Itô's formula, we have(50)Ξ**=Ξ*+(-PΔA0(t)-ΔA0T(t)Pδ0PΔA1(t)δ11-ρPΔA2(t)δ2PΔA3(t)*000**00***0)=Ξ*+(-PHF(t)T0-T0TFT(t)HTPδ0PHF(t)T1δ11-ρPHF(t)T2δ2PHF(t)T3*000**00***0)=Ξ*+(-T0T000)FT(t)(HTP000)+(HTP000)TF(t)(-T0T000)T+(δ0T1T000)FT(t)(0HTP00)+(0HTP00)TF(t)(δ0T1T000)T+(δ11-ρT2T000)FT(t)(00HTP0)+(00HTP0)TF(t)(δ11-ρT2T000)T+(δ2T3T000)FT(t)(000HTP)+(000HTP)TF(t)(δ2T3T000)T.
By Lemma3, we get
(51)Ξ**≤Ξ*+ε0-1(PH000)(HTP000)+ε0(-T0T000)(-T0000)+ε1-1(0PH00)(0HTP00)+ε1(δ0T1T000)(δ0T1000)+ε2-1(00PH0)(00HTP0)+ε2(δ11-ρT2T000)(δ11-ρT2000)+ε3-1(000PH)(000HTP)+ε3(δ2T3T000)(δ2T3000)=Ξ*+(Π2000*ε1-1PHHTP00**ε2-1PHHTP0***ε3-1PHHTP),whereΠ2=ε0-1PHHTP+ε0T0TT0+ε1δ0T1TT1+ε2(δ1/(1-ρ))T2TT2+ε3δ2T3TT3.
This completes the proof of Theorem10.Remark 11.
The neural network can be disturbed by environmental noises, which cause system parameters' uncertainty. It can lead to system instability. As far as we know, there is no literature that is published on robustμ-stability on uncertain neural networks with unbounded continuous distributed delays in impulsive perturbations.There exist only parameter uncertainties and no stochastic perturbations. So the uncertain neural networks can be described as(52)y˙(t)=-A0(t)y(t)+A1(t)g(y(t))+A2(t)g(y(t-τ(t)))+A3(t)∫0∞‍h(s)g(y(t-s))ds,+A3(t)∫0∞‍h(s)gt≠tk,t>0,Δy(tk)=y(tk)-y(tk-)=-Mk(y(tk-)),t=tk,k∈Z+.Corollary 12.
Assume that Assumptions1–5 hold. Then, the zero solution of system (52) is globally robustly μ-stable if there exist diagonal matrices P>0, Qi>0, i=1,2 and Q3=diag(q1(3),…,qn(3)), some constants α1≥0, α2>0, α3>0, δj>0,  j=0,1,2,  εj>0,j=0,1,2,3, mik∈[0,2],i=1,2,…,n,k∈Z+, a nonnegative continuous differential function μ(t) defined on [0,∞], and a constant T>0 such that, for t≥T,
(53)μ˙(t)μ(t)≤α1,μ(t-τ(t))μ(t)≥α2,∫0∞‍hi(s)μ(t+s)dsμ(t)≤α3,i=1,2,…n,
and the following LMI hold:
(54)Ξ**=[Π1δ0PA1δ11-ρPA2δ2PA3*R100**R20***R3]<0,
where Π1=α1P-PA0-A0TP+L[δ0-1Q1+δ1-1α2-1Q2+δ2-1α3Q3]L+ε0-1PHHTP+ε0T0TT0+ε1δ0T1TT1+ε2(δ1/(1-ρ))T2TT2+ε3δ2T3TT3, L=diag(l1,l2,…,ln), λ=λmax(P), R1=-Q1+ε1-1PHHTP, R2=-Q2+ε2-1PHHTP, R3=-Q3+ε3-1PHHTP, and impulsive operator Mk=diag(m1k,…,mnk).


## 5. Illustrative Examples

In this section, we will give two examples to show the validity of the results obtained.Example 13.
Consider the stochastic impulsive neural network with two neurons:(55)dx(t)=[∫0∞‍-A0x(t)+A1g(x(t))+A2g(x(t-τ(t)))+A3∫0∞‍h(s)g(x(t-s))ds]dt+σ(t,x(t),x(t-τ(t)))dω(t),t≠tk,t>0,Δx(tk)=x(tk)-x(tk-)=-Mk(x(tk-)),t=tk,k∈Z+,
where the activation function is described by gi(s)=tanh(s), i=1,2,…,n, τ(t)=0.4t, hi(s)=e-s. It is obvious that (0,0)T is an equilibrium point of system (55). Let μ(t)=t and choose α1=0.1, α2=0.6, α3=1.2,  Mk=diag(0.5,0.5), and the parameter matrices are, respectively, given by
(56)A0=(2.8003.2),A1=(0.2-0.10.10.1),A2=(0.060.07-0.070.03),A3=(0.02-0.030.010.01),Γ0=Γ1=(1.1001.1).In this case, we getL=diag(1,1),ρ=0.4. Let Q1=Q2=Q3=I, δ0=δ1=δ2=1, and P=diag(10,9). We can get λmax(Ξ)=-0.9590<0 via MATLAB LMI toolbox. By Theorem 5, the equilibrium point of model (55) with unbounded time-varying delay and continuously distributed delay is globally stochastically μ-stable in the mean square. The numerical simulation is shown in Figure 1.(a) Time-series of theE(x1)2 of model (55) without impulsive effects for t∈[0,5]. (b) Time-series of the E(x2)2 of model (55) without impulsive effects for t∈[0,5]. (c) Time-series of the E(x1)2 of model (55) with impulsive effects for t∈[0,5]. (d) Time-series of the E(x2)2 of model (55) with impulsive effects for t∈[0,5].
(a)
(b)
(c)
(d)Example 14.
Consider the uncertain stochastic impulsive neural network with two neurons:(57)dx(t)=[-A0(t)x(t)+A1g(x(t))+A2(t)g(x(t-τ(t)))+A3(t)∫0∞‍h(s)g(x(t-s))ds]dt+σ(t,x(t),x(t-τ(t)))dω(t),t≠tk,t>0,Δx(tk)=x(tk)-x(tk-)=-Mk(x(tk-)),t=tk,k∈Z+,
where the activation function is described by gi(s)=tanh(s),i=1,2,…,n, τ(t)=0.5t, hi(s)=e-s. It is obvious that (0,0)T is an equilibrium point of system (57). Let μ(t)=t and choose α1=0.1, α2=0.5, α3=1.2, Mk=diag(1.5,1.5), and the parameter matrices are, respectively, given by
(58)A0=(9.80011.2),A1=(0.2-0.20.40.3),A2=(0.60.7-0.10.3),A3=(0.12000.21),Γ0=Γ1=(0.5000.5).
In this case, we get L=diag(1,1),ρ=0.5. Let H=T0=T1=T2=T3=0.2I,  δ0=δ1=δ2=ε0=ε1=ε2=ε3=1, we can get the following feasible solution via MATLAB LMI toolbox:
(59)P=(0.0863000.0863),Q1=(0.1416000.1416),Q2=(0.0527000.0527),Q3=(0.1461000.1461),
and obtain λmax(Ξ**)=-1.0865<0. By Theorem 10, the equilibrium point of model (24) is globally robustly stochastically μ-stable in the mean square.


## 6. Concluding Remarks

This paper firstly investigated the global stochasticμ-stability in the mean square of a class of impulsive stochastic neural network with unbounded mixed delays. We obtained some sufficient conditions by using Lyapunov-Krasovskii functional method, the random analysis theory, and the linear matrix inequality (LMI) technique. Secondly, we researched global robust μ-stability in the mean square for a class of uncertain impulsive neural network with unbounded mixed delays. Our results improve and generalize some earlier works reported in the literature. Finally, two examples and their numerical simulations are given to illustrate the effectiveness of obtained results.



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*Source: 746241-2013-03-28.xml*