# Robust Stochastic Stability Analysis for Uncertain Neutral-Type Delayed Neural Networks Driven by Wiener Process

Authors: Weiwei Zhang; Linshan Wang
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

The robust stochastic stability for a class of uncertain neutral-type delayed neural networks driven by Wiener process is investigated. By utilizing the Lyapunov-Krasovskii functional and inequality technique, some sufficient criteria are presented in terms of linear matrix inequality (LMI) to ensure the stability of the system. A numerical example is given to illustrate the applicability of the result.


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## Body

## 1. Introduction

In the past few years, neural networks and their various generalizations have drawn much research attention owing to their promising potential applications in a variety of areas, such as robotics, aerospace, telecommunications, pattern recognition, image processing, associative memory, signal processing, and combinatorial optimization [1–3]. In such applications, it is of prime importance to ensure the asymptotic stability of the designed neural networks. Because of this, the stability of neural networks has been deeply investigated in the literature [4–14].It is known that time delays and stochastic perturbations are commonly encountered in the implementation of neural networks, and may result in instability or oscillation. So it is essential to investigate the stability of delayed stochastic neural networks [15, 16]. Moreover, uncertainties are unavoidable in practical implementation of neural networks due to modeling errors and parameter fluctuation, which also cause instability and poor performance [15, 17, 18]. Therefore, it is significant to introduce such uncertainties into delayed stochastic neural networks.On the other hand, because of the complicated dynamic properties of the neural cells in the real world, it is natural and important that systems will contain some information about the derivative of the past state. Practically, such phenomenon always appears in the study of automatic control, circuit analysis, chemical process simulation, and population dynamics, and so forth. Recently, there has been increasing interest in the study of delayed neural networks of neutral type, see [6–15, 18–24]. In [6, 8], the authors developed the global asymptotic stability of neutral-type neural networks with delays by utilizing the Lyapunov stability theory and LMI technique. In [9, 10], the global exponential stability of neutral-type neural networks with distributed delays is studied. However, the stochastic perturbations were not taken into account in those delayed neural networks [6–10].In [23, 24], the authors discussed the robust stability for uncertain stochastic neural networks of neutral-type with time-varying delays. However, the distributed delays were not taken into account in the models. So far, there are only a few papers that not only deal with the stochastic stability analysis for delayed neural networks of neutral-type, but also consider the parameter uncertainties.To the best of our knowledge, there are very few results on the stochastic stability analysis for uncertain neutral-type neural networks with both discrete and distributed delays driven by Wiener process. This motivates the research in this paper.In this paper, a class of uncertain neutral-type delayed neural networks driven by Wiener process is considered. By constructing a suitable Lyapunov functional, some new stability criteria to guarantee the system to be stochastically asymptotically stable in the mean square are given, which are less conservative than some existing reports. The structure of the addressed system is more general than in the other papers. The criteria can be checked easily by the LMI control toolbox in MATLAB. Moreover, a numerical example is given to illustrate the effectiveness and improvement over some existing results.


## 2. Preliminaries

Notations 2.
A<0 denotes that A is a negative definite matrix. The superscript “T” stands for the transpose of a matrix. (Ω,F,P) denotes a complete probability space, E(·) stands for the mathematical expectation operator. ∥·∥ stands for the Euclidean norm. I is the identity matrix of appropriate dimension, and the symmetric terms in a symmetric matrix are denoted by *.Consider the following class of uncertain neutral-type delayed neural networks driven by Wiener process:(2.1)d[x(t)-Cx(t-h(t))]=[-A(t)x(t)+B(t)f(x(t-τ(t)))+D(t)∫t-r(t)tf(x(s))ds]dt+[H0(t)x(t)+H1(t)x(t-τ(t))]dw(t),x(t0+s)=φ(s),s∈[t0-ρ,t0],
where x=(x1,x2,…,xn)T is the neuron state vector, A(t)=A̅+ΔA(t), B(t)=B̅+ΔB(t), D(t)=D̅+ΔD(t), H0(t)=H̅0+ΔH0(t), H1(t)=H̅1+ΔH1(t), A̅=diag(ai)n×n is a positive diagonal matrix, B̅,C,D̅∈Rn×n are the connection weight matrices, H̅0,H̅1∈Rn×n are known real constant matrices, ΔA(t),ΔB(t),ΔD(t),ΔH0(t),ΔH1(t) represent the time-varying parameter uncertain terms. f(x)=(f1(x1),f2(x2),…,fn(xn))T is the neuron activation function with f(0)=0. w(t)=(w1(t),w2(t),…,wn(t))T is an n-dimensional Wiener process defined on a complete probability space (Ω,F,P). r(t),τ(t),h(t) are nonnegative, bounded, and differentiable time varying delays satisfying(2.2)0<r(t)≤r¯<∞,0<τ(t)≤τ¯<∞,τ̇(t)≤η1<∞,0<h(t)≤h¯<∞,ḣ(t)≤η2<∞.The admissible parameter uncertain terms are assumed to be the following form:(2.3)[ΔA(t),ΔB(t),ΔD(t),ΔH0(t),ΔH1(t)]=UF(t)[M1,M2,M3,M4,M5],
where U,Mi, i=1,…,5 are known real constant matrices, F(t) is the time-varying uncertain matrix satisfying (2.4)FTF≤I.Suppose thatf(·) is bounded and satisfies the following condition:(2.5)‖f(x)‖≤‖Gx‖,
where G∈Rn×n is a known constant matrix.Assume that the initial valueφ:[-ρ,0]→Rn is F0-measurable and continuously differentiable, we introduce the following norm:(2.6)‖φ‖ρ2=max⁡{sup⁡-α≤s≤0E|φi(s)|2,sup⁡-h¯≤s≤0E|φi′(s)|2}<∞,
where ρ=max{τ¯,h¯,r¯}, α=max{τ¯,r¯}.Under the above assumptions, it is easy to verify that there exists a unique equilibrium point of system (2.1) (see [25]).Definition 2.1.
The equilibrium point of (2.1) is said to be globally robustly stochastically asymptotically stable in the mean square, if the following condition holds:
(2.7)lim⁡t→+∞E|x(t,t0,φ)|2=0,t≥t0,
where x(t,t0,φ) is any solution of model (2.1) with initial value φ.Lemma 2.2 (Schur complement [26]).
Given constant matricesΩ1,Ω2,Ω3 with appropriate dimensions, where Ω1T=Ω1 and Ω2T=Ω2>0, then
(2.8)Ω1+Ω3TΩ2-1Ω3<0,
if and only if
(2.9)(Ω1Ω3T*-Ω2)<0,or(-Ω2Ω3*Ω1)<0.Lemma 2.3 (see [26]).
Given matricesD,E, and F with FTF≤I and a scalar ε>0, then
(2.10)DFE+ETFTDT≤εDDT+ε-1ETE.Lemma 2.4 (see [27]).
For any constant matrixM∈Rn×n, M=MT>0, a scalar γ>0, vector function x(t):[0,γ]→Rn such that the integrations are well defined, then
(2.11)[∫0γx(s)ds]TM[∫0γx(s)ds]≤γ∫0γxT(s)Mx(s)ds.


## 3. Main Results

Theorem 3.1.
System (2.1) is globally robustly stochastically asymptotically stable in the mean square, if there exist symmetric positive definite matrices P,Q,R,S,U1,U2 and positive scalars δ,ε1,ε2>0 such that LMI holds:
(3.1)Λ=(Γ1A̅TPCPB̅-ε1M1TM2PD̅-ε1M1TM3ε2M4TM5H̅0TP-PU0*Γ2-CTPB̅-CTPD̅00-CTPU0**Γ3ε1M2TM30000***Γ40000****Γ5H̅1TP00*****-P0PU******-ε1I0*******-ε2I)<0,
where Γ1=-PA̅-A̅TP+Q+R+r¯GTSG+ε1M1TM1+ε2M4TM4, Γ2=-U1-(1-η2)R, Γ3=-δI+ε1M2TM2, Γ4=-r¯-1S+ε1M3TM3, Γ5=-U2-(1-η1)Q+δGTG+ε2M5TM5.Proof.
Using Lemma2.2, the matrix Λ<0 implies that
(3.2)(Γ1A̅TPCPB̅-ε1M1TM2PD̅-ε1M1TM3ε2M4TM5H̅0TP*Γ2-CTPB̅-CTPD̅00**Γ3ε1M2TM300***Γ400****Γ5H̅1TP*****-P)+(PU0-CTPU00000000PU)(ε1-1I0*ε2-1I)(UTP-UTPC000000000UTP)=(Φ1A̅TPCPB̅PD̅0H̅0TP*Γ2-CTPB̅-CTPD̅00**-δI000***-r¯-1S00****Φ2H̅1TP*****-P)+ε1(-M1T0M2TM3T00)(-M1T0M2TM3T00)T+ε2(M4T000M5T0)(M4T000M5T0)T+ε1-1(PU-CTPU0000)(PU-CTPU0000)T+ε2-1(00000PU)(00000PU)T<0,
where Φ1=-PA̅-A̅TP+Q+R+r¯GTSG,  Φ2=-U2-(1-η1)Q+δGTG.
From (2.3), (2.4), using Lemma 2.3, we have
(3.3)(-PΔA(t)-ΔAT(t)PΔAT(t)PCPΔB(t)PΔD(t)0ΔH0T(t)P*0-CTPΔB(t)-CTPΔD(t)00**0000***000****0ΔH1T(t)P*****0)=(-M1T0M2TM3T00)FT(t)(PU-CTPU0000)T+(PU-CTPU0000)F(t)(-M1T0M2TM3T00)T+(M4T000M5T0)FT(t)(00000PU)T+(00000PU)F(t)(M4T000M5T0)T≤ε1(-M1T0M2TM3T00)(-M1T0M2TM3T00)T+ε1-1(PU-CTPU0000)(PU-CTPU0000)T+ε2(M4T000M5T0)(M4T000M5T0)T+ε2-1(00000PU)(00000PU)T.
Together with (3.2), we get
(3.4)(ΨAT(t)PCPB(t)PD(t)0H0T(t)P*Γ2-CTPB(t)-CTPD(t)00**-δI000***-r¯-1S00****Φ2H1T(t)P*****-P)<0,
where Ψ=-PA(t)-AT(t)P+Q+R+r¯GTSG.
Utilizing Lemma2.2 again, we obtain
(3.5)Σ=(ΨAT(t)PCPB(t)PD(t)0*Γ2-CTPB(t)-CTPD(t)0**-δI00***-r¯-1S0****Φ2)+(H0T(t)000H1T(t))P(H0T(t)000H1T(t))T<0.
Constructing a positive definite Lyapunov-Krasovskii functional as follows:(3.6)V(t,x(t))=yT(t)Py(t)+∫t-τ(t)txT(s)Qx(s)ds+∫t-h(t)txT(s)Rx(s)ds+∫-r(t)0∫t+θtfT(x(s))Sf(x(s))dsdθ+∫tTxT(s-h(s))U1x(s-h(s))ds+∫tTxT(s-τ(s))U2x(s-τ(s))ds,
where y(t)=x(t)-Cx(t-h(t)), T>t is a constant.
By Ito’s differential formula, we get(3.7)dV(t,x(t))≤{2yT(t)P[-A(t)x(t)+B(t)f(x(t-τ(t)))+D(t)∫t-r(t)tf(x(s))ds]+xT(t)Qx(t)-(1-τ̇(t))xT(t-τ(t))Qx(t-τ(t))+xT(t)Rx(t)-(1-ḣ(t))xT(t-h(t))Rx(t-h(t))+r¯fT(x(t))Sf(x(t))-∫t-r(t)tfT(x(s))Sf(x(s))ds-xT(t-h(t))U1x(t-h(t))-xT(t-τ(t))U2x(t-τ(t))+[H0(t)x(t)+H1(t)x(t-τ(t))]TP[H0(t)x(t)+H1(t)x(t-τ(t))]2yT(t)P[-A(t)x(t)+B(t)f(x(t-τ(t)))+D(t)∫t-r(t)tf(x(s))ds]}dt+2yT(t)P[H0(t)x(t)+H1(t)x(t-τ(t))]dw(t)≤{∫t-r(t)tf(x(s))ds2[x(t)-Cx(t-h(t))]T×P[-A(t)x(t)+B(t)f(x(t-τ(t)))+D(t)∫t-r(t)tf(x(s))ds]+xT(t)Qx(t)-(1-η1)xT(t-τ(t))Qx(t-τ(t))+xT(t)Rx(t)-(1-η2)xT(t-h(t))Rx(t-h(t))+r¯fT(x(t))Sf(x(t))-∫t-r(t)tfT(x(s))Sf(x(s))ds-xT(t-h(t))U1x(t-h(t))-xT(t-τ(t))U2x(t-τ(t))+[H0(t)x(t)+H1(t)x(t-τ(t))]TP[H0(t)x(t)+H1(t)x(t-τ(t))][x(t)-Cx(t-h(t))]TP[-A(t)x(t)+B(t)f(x(t-τ(t)))+D(t)∫t-r(t)tf(x(s))ds]}dt+2yT(t)P[H0(t)x(t)+H1(t)x(t-τ(t))]dw(t).
From (2.5), for a scalar δ>0, we have
(3.8)-δ[fT(x(t-τ(t)))f(x(t-τ(t)))-xT(t-τ(t))GTGx(t-τ(t))]≥0.
Using Lemma 2.4, we have
(3.9)(∫t-r(t)tf(x(s))ds)Tr¯-1S(∫t-r(t)tf(x(s))ds)≤∫t-r(t)tfT(x(s))Sf(x(s))ds.
Together (3.8), (3.9) with dV(t,x(t)), we obtain
(3.10)dV(t,x(t))≤{xT(t)[-PA(t)-AT(t)P+Q+R+r¯GTSG]x(t)+xT(t)AT(t)PCx(t-h(t))+xT(t-h(t))CTPA(t)x(t)+xT(t)PB(t)f(x(t-τ(t)))+fT(x(t-τ(t)))BT(t)Px(t)+xT(t)PD(t)∫t-r(t)tf(x(s))ds+(∫t-r(t)tf(x(s))ds)TDT(t)Px(t)+xT(t-h(t))[-U1-(1-η2)R]x(t-h(t))-xT(t-h(t))CTPB(t)f(x(t-τ(t)))-fT(x(t-τ(t)))BT(t)PCx(t-h(t))-xT(t-h(t))CTPD(t)∫t-r(t)tf(x(s))ds-(∫t-r(t)tf(x(s))ds)TDT(t)PCx(t-h(t))+xT(t-τ(t))×[-U2-(1-η1)Q]x(t-τ(t))-(∫t-r(t)tf(x(s))ds)Tr¯-1S(∫t-r(t)tf(x(s))ds)+[H0(t)x(t)+H1(t)x(t-τ(t))]TP[H0(t)x(t)+H1(t)x(t-τ(t))]}dt+2yT(t)P[H0(t)x(t)+H1(t)x(t-τ(t))]dw(t).
That is,
(3.11)dV(t,x(t))≤ξT(t)Σξ(t)dt+2yT(t)P[H0(t)x(t)+H1(t)x(t-τ(t))]dw(t),
where ξT(t)=(xT(t),xT(t-h(t)),fT(x(t-τ(t))),(∫t-r(t)tf(x(s))ds)T,xT(t-τ(t))), and the matrix Σ is given in (3.5).
Taking the mathematical expectation, we get(3.12)E(dV(t,x(t))dt)≤E(ξT(t)Σξ(t))≤λmax⁡(Σ)E‖x(t)‖2.
From (3.5), we know Σ<0, that is, λmax(Σ)<0. By Lyapunov-Krasovskii stability theorems, the system (2.1) is globally robustly asymptotically stable. The proof is completed.Remark 3.2.
To the best of our knowledge, few authors have considered the stochastically asymptotic stability for uncertain neutral-type neural networks driven by Wiener process. We can find recent papers [18, 22–24]. However, it is assumed in [18] that the system is a linear model and all delays are constants. In [22], it is assumed that the time-varying delays satisfying τ̇(t)≤ρτ<1,  ḣ(t)≤ρh<1, in this paper, we relax it to τ̇(t)≤ρτ<∞,ḣ(t)≤ρh<∞. In [23, 24], the authors discussed the robust stability for uncertain stochastic neural networks of neutral-type with time-varying delays. However, the distributed delays were not taken into account in the models. Hence, our results in this paper have wider adaptive range.Remark 3.3.
Suppose thatC=0,D(t)=0 (i.e., without neutral-type and distributed delays), then the system (2.1) becomes the one investigated in [15].Remark 3.4.
In [17], the authors studied the global stability for uncertain stochastic neural networks with time-varying delay by Lyapunov functional method and LMI technique. However, the neutral term and distributed delays were not taken into account in the models. Therefore, our developed results in this paper are more general than those reported in [17].Remark 3.5.
It should be noted that the condition is given as linear matrix inequalities LMIs, therefore, by using the MATLAB LMI Toolbox, it is straightforward to check the feasibility of the condition.


## 4. Numerical Example

Consider the following uncertain neutral-type delayed neural networks:(4.1)d[x(t)-Cx(t-h(t))]=[(D+UF(t)M3)∫t-r(t)tf(x(s))ds-(A+UF(t)M1)x(t)+(B+UF(t)M2)f(x(t-τ(t)))+(D+UF(t)M3)∫t-r(t)tf(x(s))ds]dt+[UF(t)M4x(t)+UF(t)M5x(t-τ(t))]dw(t),
where n=2, fi(xi)=sinxi, i=1,2, η1=0.7, η2=0.5, 0<r(t)≤r¯=3, FT(t)F(t)≤I.The constant matrices are(4.2)A=(3003),B=(0.20.160.040.08),C=(0.2000.2),D=(0.040.03-0.020.05),U=(0.10.50.50.3),M1=(0.6000.6),M2=M3=M4=(0.2000.2),M5=(0.4000.4),G=(1001).
By using the MATLAB LMI Control Toolbox, we obtain the feasible solution as follows: δ=2.0876, ε1=5.0486,  ε2=8.0446,(4.3)P=(6.8465-0.7257-0.72576.6012),Q=(9.9371-0.2792-0.279210.4388),R=(8.0104-2.3936-2.39365.4991),S=(3.4984-0.8143-0.81431.9588),U1=(1.11110.38890.38891.1060),U2=(2.4193-0.6561-0.65612.6270).
That is the system (4.1) is globally robustly stochastically asymptotically stable in the mean square.


## 5. Conclusion

In this paper, the stochastically asymptotic stability problem has been studied for a class of uncertain neutral-type delayed neural networks driven by Wiener process by utilizing the Lyapunov-Krasovskii functional and linear matrix inequality (LMI) approach. A numerical example is given to illustrate the applicability of the result.



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