# Delay-Dependent Exponential Stability for Uncertain Neutral Stochastic Systems with Mixed Delays and Markovian Jumping Parameters

Authors: Huabin Chen
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

This paper is mainly concerned with the globally exponential stability in mean square of uncertain neutral stochastic systems with mixed delays and Markovian jumping parameters. The mixed delays are comprised of the discrete interval time-varying delays and the distributed time delays. Taking the stochastic perturbation and Markovian jumping parameters into account, some delay-dependent sufficient conditions for the globally exponential stability in mean square of such systems can be obtained by constructing an appropriate Lyapunov-Krasovskii functional, which are given in the form of linear matrix inequalities (LMIs). The derived criteria are dependent on the upper bound and the lower bound of the time-varying delay and the distributed delay and are therefore less conservative. Two numerical examples are given to illustrate the effectiveness and applicability of our obtained results.


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## Body

## 1. Introduction

It is well known that many dynamical systems not only depend on the present and past states but also involve the derivative with delays as well as the functional of the past history. Neutral delay differential equations are often used to describe the following systems [1]:(1.1)d[x(t)-Dx(t-τ)]dt=f(t,x(t),x(t-τ)).
Many authors have considered the dynamical analysis of the neutral delay differential equations (see [2–7] and references therein). For example, Chen et al. in [3] and Wu et al. in [4, 5] have given some LMI-based conditions ensuring the stability analysis and the stabilization of neutral delay systems. Taking the environmental disturbances into account, the neutral stochastic delay differential equations can be given as follows:(1.2)d[x(t)-Dx(t-τ)]=f(t,x(t),x(t-τ))dt+g(t,x(t),x(t-τ))dB(t).
Some fundamental theories of neutral stochastic delay differential equations are introduced in [1, 8]. Since they can be extensively applied into many branches for the control field, the problem about the exponential stability and the asymptotical stability of the neutral stochastic delay systems has attracted many authors’ attention over the past few years, and many less conservative results of delay-dependent conditions ensuring the stabilization analysis and H∞ filtering design for such systems have been reported in many works, see, for example, [9–14] and references therein. The methods used include the Razumikhin-type theorems [10], the Lyapunov functional [13], the fixed point theorem [14], and the linear matrix inequality [9, 11, 12]. For example, Huang and Mao in [9] and Chen et al. in [11] have given the exponential stability criteria of neutral stochastic delay systems. Some LMI-based sufficient conditions for the mean-square exponential stability analysis of stochastic systems of neutral type have been obtained by introducing an auxiliary vector in [12]. In practice, the parameter uncertainty is considered as one of the main sources leading to undesirable behavior (e.g., instability) of dynamical systems, especially when implementing neural networks in applications. The stability analysis of the uncertain neutral stochastic system has received considerable research attention, see, for example, [15–18], and the problem of the H∞ filter design of the uncertain neutral stochastic delay systems has been discussed in [19, 20].On the other hand, Markovian jump systems introduced by [21] are the hybrid systems with two components in the state. The first one refers to the mode that is described by a continuous-time finite-state Markovian process, and the second one refers to the state that is represented by a system of differential equations. The jump systems have the advantage of modeling the dynamic systems subject to abrupt variation in their structures, such as component failures or repairs, sudden environmental disturbance, changing subsystem interconnections, and operating in different points of a nonlinear plant [22]. The stability analysis and H∞ filter design of stochastic delay systems with Markovian jumping parameters and delay systems with Markovian jumping parameters have been widely studied, see, for example, [23–38]. For example, in [24], Liu et al. have discussed the exponential stability of delayed recurrent neural networks with Markovian jumping parameters; Liu et al. in [24] and Wang et al. in [25] have considered some sufficient conditions for the exponential stability of stochastic neural networks with mixed time delays and Markovian switching; Mao in [13] has also given some sufficient conditions for the exponential stability of stochastic delay interval systems with Markovian switching. More recently, He and Liu in [39] and Balasubramaniam et al. in [40] have presented some LMI-based sufficient conditions for the exponential stability of uncertain neutral systems with Markovian jumping parameters. Although Kolmanovskii et al. in [17], and Mao et al. in [18] have derived the exponential stability of the neutral stochastic delay systems with Markovian jumping parameters, some sufficient conditions obtained by using the estimate method are not easily checked. Thus, the problem of the stability analysis of the uncertain neutral stochastic systems with mixed delays and Markovian jumping parameters has not been fully investigated and there is still much room left for further consideration, which constitutes the motivation for the present research.In this paper, the global exponential stability of a class of the uncertain neutral stochastic systems with mixed delays and Markovian jumping parameters is discussed. The delays include the discrete and distributed time delays, and the jumping parameters are generated from a finite state Markov chain. By constructing an appropriate Lyapunov functional, some LMIs-based sufficient conditions ensuring the exponential stability in mean square of the uncertain neutral stochastic systems with mixed delays and Markovian jumping parameters are obtained by using the stochastic analysis and some bounding technique. It is worth pointing out that compared with the earlier works in [17, 18], the obtained results given in the form of the linear matrix inequalities (LMIs) can be easily be solved by using the standard software packages. Two illustrative examples are exploited to demonstrate the effectiveness and applicability of the obtained results.The content of the paper is arranged as follows. In Section2, some necessary notations, definitions, and lemmas will be introduced. In Section 3, we mainly study the exponential stability in mean square of the uncertain neutral stochastic systems with mixed delays and Markovian jumping parameters. Two illustrative numerical examples are given to show the power of our obtained results in Section 4.Notations.
Unless otherwise specified, for a real square matrixA, the matrix A>0 (A≥0, A<0, A≤0) means that A is a positive definite (positive semidefinite, negative definite, and negative semidefinite, resp.); λmax(A) and λmin(A) denote the maximum and minimum eigenvalues of the square matrix A, respectively. Let (Ω,ℑ,{ℑt}t≥0,P) be a probability space with a natural {ℑt}t≥0 and let E{·} stand for the mathematical expectation operator with respect to this probability measure. If A is a vector or matrix, its transpose is denoted by AT. |B|=trace(BTB) denotes the Euclidean norm of a vector B and its induced norm of a matrix B. Unless explicitly stated, matrices are assumed to have real entries and compatible dimensions. Let τ>0 and C([-τ,0];Rn) be the family of all continuous Rn-valued functions ϕ on the interval [-τ,0] with the norm ∥ϕ∥=sup{|ϕ(θ)|:-τ≤θ≤0}. Denote by Lℑ02([-τ,0];Rn) the family of all ℑ0-measurable C([-τ,0];Rn)-valued random variables ξ={ξ(θ):-τ≤θ≤0} such that supθ∈[-τ,0]E|ξ(θ)|2<+∞. B(t)=[B1(t),B2(t),…,Bn(t)]T (t≥0) is an n-dimensional standard Brownian motion defined on the completed probability space (Ω,ℑ,P).


## 2. Problem Formulation

Letr(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 generator Γ={γij}N×N given by(2.1)P{r(t+Δ)=j∣r(t)=i}={γijΔ+o(Δ),ifi≠j,1+γiiΔ+o(Δ),ifi=j,
where Δ>0. Here, γij≥0 is the transition rate from i to j if i≠j while
(2.2)γii=-∑j≠iγij.
Now, we assume that the Markov chain r(·) is independent of the Brownian motion B(·). It is well known that almost every sample path of r(t) is a right-continuous step function with a finite number of simple jumps in any finite subinterval of R+.In this paper, we will consider the following uncertain neutral stochastic systems with mixed delays and Markovian switching:(2.3)d[x(t)-D(r(t))x(t-d(t))]=[∫t-τ(t)tA1(r(t),t)x(t)+A2(r(t),t)f(x(t))+A3(r(t),t)f(x(t-d(t)))+A4(r(t),t)∫t-σ(t)tf(x(s))ds]dt+σ(t,x(t),x(t-d(t)),∫t-σ(t)tx(s)ds,r(t))dB(t),t≥0,
with the initial value x0=φ∈Lℑ02([-τ,0],Rn) (τ=max{d2,σ}), where x(t)∈Rn is the system state vector associated with the neurons and d(t) and σ(t) are the time-varying delays. Here, we assume that the Markov chain r(·) is independent of the Brownian motion B(t)(t≥0). f:Rn→Rn is neuron activation function, and the noise perturbation σ:R+×Rn×Rn×Rn×S→Rn×n is the noise intensity matrix. When r(t)=i(i∈S), D(r(t)), A1(r(t),t), A2(r(t),t), A3(r(t),t), and A4(r(t),t) are, respectively, denoted as Di, A1i(t), A2i(t), A3i(t), and A4i(t), and Di (i∈S) are known matrices with |Di|<1(i∈S). A1i(t), A2i(t), A3i(t), and A4i(t) are matrix functions with time-varying uncertainties, that is,(2.4)A1i(t)=A1i+ΔA1i(t),A2i(t)=A2i+ΔA2i(t),A3i(t)=A3i+ΔA3i(t),A4i(t)=A4i+ΔA4i(t),
where A1i, A2i, A3i, and A4i (i∈S) are known real constant matrices and ΔA1i(t), ΔA2i(t), ΔA3i(t), and ΔA4i(t) (i∈S) are unknown matrices representing time-varying parameter uncertainties in system model. We assume that the uncertainties are norm-bounded and can be described as(2.5)[ΔA1i(t)ΔA2i(t)ΔA3i(t)ΔA4i(t)]=MiFi(t)[N1iN2iN3iN4i],FiT(t)Fi(t)≤I,i∈S,
where Mi,N1i,N2i,N3i,andN4i(i∈S) are known real matrices and Fi(t)(i∈S) is unknown real and possibly time-varying matrix for any given t. It is assumed that the elements of Fi(t)(i∈S) are Lebesgue measurable. When Fi(t)=0(i∈S), systems (2.3) have the following nominal case:(2.6)d[x(t)-Dix(t-d(t))]=[A1ix(t)+A2if(x(t))+A3if(x(t-d(t)))+A4i∫t-σ(t)tf(x(s))ds]dt+σ(t,x(t),x(t-d(t)),∫t-σ(t)tx(s)ds,i)dB(t),t≥0.In order to obtain our results, we need some assumptions as follows.Assumption 2.1.
The neuron activation functionsf(·) in (2.3) (or (2.6)) are bounded and satisfy the following Lipschitz condition:
(2.7)|f(x)-f(y)|≤|L(x-y)|,∀x,y∈Rn,
where L∈Rn×n is known constant matrix and f(0)=0.Assumption 2.2.
The noise perturbationσ satisfies the following condition:
(2.8)trace[σT(t,x(t),x(t-d(t)),∫t-σ(t)tx(s)ds,i)σ(t,x(t),x(t-d(t)),∫t-σ(t)tx(s)ds,i)]≤xT(t)R1iTR1ix(t)+xT(t-d(t))R2iTR2ix(t-d(t))+[∫t-σ(t)tx(s)ds]TR3iTR3i[∫t-σ(t)tx(s)ds],
where R1i, R2i, and R3i are known constant matrices with appropriate dimensions and σ(t,0,0,0,i)=0 (i∈S).Remark 2.3.
Under Assumptions2.1 and 2.2, it is easily shown that the system (2.3) with uncertainties (2.4) admits a unique trivial solution when the initial data ξ=0. The readers can refer to [41].Assumption 2.4.
d(t) and σ(t) are two time-varying continuous functions that satisfy
(2.9)0≤d1≤d(t)≤d2,ḋ(t)≤μ<1,0≤σ(t)≤σ,σ̇(t)≤ν<1,
where d1 and d2 are the lower and upper bounds of the time delay d(t), respectively.
We present the definitions and three useful lemmas as follows.Definition 2.5.
The neutral stochastic systems with mixed delays and Markovian jump parameters (2.6) is said to be exponentially stable in mean square if there exist a pair of positive scalar α>0 and l>0 such that every solution x(t,ξ,i) of systems (2.6) satisfies
(2.10)E|x(t,ξ,i)|2≤lsup⁡s∈[-τ,0]E|φ(s)|2e-αt,∀t≥0,
for any φ∈Lℑ02([-τ,0],Rn).Definition 2.6.
The uncertain neutral stochastic systems with mixed delays and Markovian jump parameters (2.3) are said to be exponentially stable in mean square if (2.10) holds for all admissible uncertainties (2.5).Lemma 2.7 (see [8]).
For any vectorsa,b∈Rn, the inequality
(2.11)±2aTb≤aTXa+bTX-1b
holds, in which X is any n×nmatrix with X>0.Lemma 2.8 (see [8]).
LetX∈Rn×n; then
(2.12)λmin⁡(X)aTb≤aTXb≤λmax⁡(X)aTb
for any a∈Rn if X is a symmetric matrix.Lemma 2.9 (see [42] Schur complement).
For a given matrix(2.13)S=(S11S12S12TS22)
with S11=S11T, S22=S22T, the following conditions are equivalent:(1)
S<0,(2)
S22<0, S11-S12S22-1S12T<0,(3)
S11<0, S22-S12TS11-1S12<0.Lemma 2.10 (see [42]).
LetU, V, W, and M be real matrices of appropriate dimensions with M satisfying M=MT; then
(2.14)M+UVW+WTVTUT<0,∀VTV≤I,
if and only if there exist a scalar ε>0 such that
(2.15)M+ε-1UUT+εWTW<0.Lemma 2.11 (see [43]).
For any positive symmetric constant matrixM∈Rn×n and a scalar γ>0, a vector function ω:[0,γ]→Rn such that the integrations concerned are well defined, and then the following inequality holds:
(2.16)[∫0γω(s)ds]TM[∫0γω(s)ds]≤γ∫0γωT(s)Mω(s)ds.


## 3.  Main Results

Theorem 3.1.
Suppose that Assumptions2.1–2.4 hold and for any given positive scalar κ∈(0,1), the neutral stochastic systems with mixed delays and Markovian switching (2.6) are exponentially stable in mean square if there exist λi>0 (i∈S) and some positive definite matrices Pi>0 (i∈S) and Ql>0 (l=1,2,…,10) such that the following linear matrix inequalities (LMIs) are satisfied: for i∈S,
(3.1)Pi≤λiI,(3.2)Ωi=[Ωi11Ωi120PiA2iPiA3iPiA4i000*Ωi220000Ωi27Ωi28Ωi29**Ωi33000000***-Q500000****-Q60000*****-Q7000******-Q800*******-Q90********-Q10]<0,
where * denotes the entries that are readily inferred bysymmetry of a symmetric matrix and
(3.3)Ωi11=PiA1i+A1iTPi+∑j=1NγijPj+λiR1iTR1i+LT(Q5+Q8)L+Q1+d2Q2+12(d22-d12)Q3+σ2Q4,Ωi12=A1iTPiDi-∑j=1NγijPjDi,Ωi22=LT(Q6+Q9)L+∑j=1NγijDiTPjDi+λiR2iTR2i-(1-u)Q1,Ωi33=LT(Q7+Q10)L+λiR3iTR3i-κ(1-v)Q5,Ωi27=DiTPiA2i,Ωi28=DiTPiA3i,Ωi29=DiTPiA4i.Proof.
Denote byC2,1(R+×Rn×S;Rn) the family of all nonnegative functions V(t,x,i) on R+×Rn×S that are once differentiable with respect to the first variable t and twice differentiable with respect to the second variable x. To obtain the stability conditions, we consider the following Lyapunov functional:
(3.4)V(t,x(t),i)=V1(t,x(t),i)+V2(t,x(t),i)+V3(t,x(t),i),
where
(3.5)V1(t,x(t),i)=[x(t)-Dix(t-d(t))]TPi[x(t)-Dix(t-d(t))],V2(t,x(t),i)=∫t-d(t)txT(s)Q1x(s)ds+∫t-d(t)t∫stxT(θ)Q2x(θ)dθds,V3(t,x(t),i)=∫-d2-d1∫t+st∫θtxT(u)Q3x(u)dudθds+σ∫-σ(t)0∫t+stxT(θ)Q4x(θ)dθds.
The weak infinitesimal operator𝔏V [17] along (2.6) from R+×Rn×S to R is given by(3.6)LV(t,x(t),i)=LV1(t,x(t),i)+LV2(t,x(t),i)+LV3(t,x(t),i),
where
(3.7)LV1(t,x(t),i)=xT(t)[PiA1i+A1iTPi+∑j=1NγijPj]x(t)+2xT(t)PiA2if(x(t))+2xT(t)PiA3if(x(t-d(t)))+2xT(t)PiA4i∫t-σ(t)tf(x(s))ds+2xT(t-d(t))DiTPiA1ix(t)-2xT(t-d(t))DiTPiA2if(x(t))-2xT(t-d(t))DiTPiA3if(x(t-d(t)))-2xT(t-d(t))DiTPiA4i∫t-σ(t)tf(x(s))ds+xT(t)[-∑j=1NγijPjDi]x(t-d(t))+xT(t-d(t))[-∑j=1NγijDiTPi]x(t)+xT(t-d(t))[∑j=1NDiTPjDi]x(t-d(t))+trace[σT(t,x(t),x(t-d(t)),∫t-σ(t)tx(s)ds)V1xxσ(t,x(t),x(t-d(t)),∫t-σ(t)tx(s)ds)].
By Lemma 2.7, we have
(3.8)xT(t)PiA2if(x(t),i)≤xT(t)PiA2iQ5-1A2iTPix(t)+fT(x(t))Q5f(x(t))≤xT(t)PiA2iQ5-1A2iTPix(t)+xT(t)LTQ5Lx(t),(3.9)2xT(t)PiA3if(x(t-d(t)),i)≤xT(t)PiA3iQ6-1A3iTPix(t)+fT(x(t-d(t)),i)Q6f(x(t-d(t)),i)≤xT(t)PiA3iQ6-1A3iTPix(t)+xT(t-d(t))LTQ6Lx(t-d(t)),(3.10)-2xT(t-d(t))DiTPiA2if(x(t),i)≤xT(t-d(t))DiTPiA2iQ8-1A2iTPiDix(t-d(t))+fT(x(t))Q8f(x(t))≤xT(t-d(t))DiTPiA2iQ8-1A2iTPiDix(t-d(t))+xT(t)LTQ8Lx(t),(3.11)-2xT(t-d(t))DiTPiA3if(x(t-d(t)),i)≤xT(t-d(t))DiTPiA3iQ9-1A3iTPiDix(t-d(t))+fT(x(t-d(t)))Q9f(x(t-d(t)))≤xT(t-d(t))DiTPiA3iQ9-1A3iTPiDix(t-d(t))+xT(t-d(t))LTQ9Lx(t-d(t)).
From Lemmas 2.7 and 2.11, it implies that
(3.12)2xT(t)PiA4i∫t-σ(t)tf(x(s))ds≤xT(t)PiA4iQ7-1A4iTPix(t)+[∫t-σ(t)tf(x(s))ds]TQ7[∫t-σ(t)tf(x(s))ds]≤xT(t)PiA4iQ7-1A4iTPix(t)+[∫t-σ(t)tx(s)ds]TLTQ7L[∫t-σ(t)tx(s)ds],(3.13)-2xT(t-d(t))DiTPiA4i∫t-σ(t)tf(x(s))ds≤xT(t-d(t))DiTPiA4iQ10-1A4iTPiDix(t-d(t))+[∫t-σ(t)tf(x(s))ds]TQ10[∫t-σ(t)tf(x(s))ds]≤xT(t-d(t))PiA4iQ10-1A4iTPix(t-d(t))+[∫t-σ(t)tx(s)ds]TLTQ10L[∫t-σ(t)tx(s)ds].
Substituting (3.8)–(3.13) into (3.7) and from Assumption 2.2, it follows that
(3.14)LV1(t,x(t),i)≤xT(t)[PiA1i+A1iTPi+∑j=1NγijPj+λiR1iTR1i+LT(Q5+Q8)L+PiA2iQ5-1A2iTPi+PiA3i(t)Q6-1A3i-TPi+PiA4iQ7-1A4iTPi∑j=1N]x(t)+xT(t)[A1iTPiDi-∑j=1NγijPjDi]x(t-d(t))+xT(t-d(t))×[DiTPiA1i-∑j=1NγijDiTPj]x(t)+x(t-d(t))×[∑j=1NDiTPiA2iQ8-1A2iTPiDi+DiTPiA3iQ9-1A3iTPiDi+DiTPiA4iQ10-1A4iTPiDi+LT(Q6+Q9)L+∑j=1NγijDiTPjDi+λiR2iTR2i]x(t-d(t))+[∫t-σ(t)tx(s)ds]T[LT(Q7+Q10)L+λiR3iTR3i][∫t-σ(t)tx(s)ds].
On the other hand, we can obtain
(3.15)LV2(t,x(t),i)≤xT[Q1+d2Q2]x(t)-(1-μ)xT(t-d(t))Q1x(t-d(t))-(1-μ)∫t-d(t)txT(s)Q2x(s)ds,(3.16)LV3(t,x(t),i)≤xT(t)[12(d22-d12)Q3+σ2Q4]x(t)-∫-d2-d1∫t+stxT(θ)Q3x(θ)dθds-σ(1-ν)∫t-σ(t)txT(s)Q5x(s)ds≤xT(t)[12(d22-d12)Q3+σ2Q4]x(t)-∫-d2-d1∫t+stxT(θ)Q3x(θ)dθds-σ(1-κ)(1-ν)∫t-σ(t)txT(s)Q5x(s)ds-κ(1-ν)[∫t-σ(t)tx(s)ds]TQ5[∫t-σ(t)tx(s)ds].
Substituting (3.14)–(3.16) into (3.6), we have
(3.17)LV(t,x(t),i)≤xT(t)[PiA1i+A1iTPi+∑j=1NγijPj+λiR1iTR1i+LT(Q5+Q8)L+PiA2iQ5-1A2iTPi+PiA3i(t)Q6-1A3iTPi+PiA4iQ7-1A4iTPi+Q1+d2Q2+12(d22-d12)Q3+σ2Q4∑j=1N]x(t)+xT(t)[A1iTPiDi-∑j=1NγijPjDi]x(t-d(t))+x(t-d(t))[DiTPiA1iT-∑j=1NγijDiTPj]x(t)+xT(t-d(t))[∑j=1NLT(Q6+Q9)L+DiTPiA2iQ8-1A2iTPiDi+DiTPiA3iQ9-1A3iTPiDi+DiTPiA4iQ10-1A4iTPiDi+λiR2iTR2i+∑j=1NγijDiTPjDi-(1-μ)Q1]x(t-d(t))+[∫t-σ(t)tx(s)ds]T[LT(Q7+Q10)L+λiR3iTR3i-κ(1-ν)Q4][∫t-σ(t)tx(s)ds]-(1-μ)∫t-d(t)txT(s)Q2x(s)ds-∫-d2-d1∫t+stxT(θ)Q3x(θ)dθds-σ(1-κ)(1-ν)∫t-σ(t)txT(s)Q5x(s)ds≤ξT(t)Πiξ(t)-(1-μ)∫t-d(t)txT(s)Q2x(s)ds-∫-d2-d1∫t+stxT(θ)Q3x(θ)dθds-σ(1-κ)(1-ν)∫t-σ(t)txT(s)Q4x(s)ds,
where ξ(t)=[xT(t)xT(t-d(t))∫t-σ(t)txT(s)ds]T and, for i∈S,
(3.18)Πi=[Π11Π120*Π220**Π33],Π11=PiA1i+A1iTPi+∑j=1NγijPj+λiR1iTR1i+LT(Q5+Q8)L+PiA2iQ5-1A2iTPi+PiA3i(t)Q6-1A3iTPi+PiA4iQ7-1A4iTPi+Q1+d2Q2+12(d22-d12)Q3+σ2Q4,Π12=A1iTPiDi-∑j=1NPjDi,Π22=LT(Q6+Q9)L+DiTPiA2iQ8-1A2iTPiDi+DiTPiA3iQ9-1A3iTPiDi+DiTPiA4iQ10-1A4iTPiDi+λiR2iTR2i+∑j=1NγijDiTPjDi-(1-μ)Q1,Π33=LT(Q7+Q10)L+λiR3iTR3i-κ(1-ν)Q4.
In view of (3.2), we have Πi<0, for i∈S. By the Lyapunov functional V(t,x(t),i),
(3.19)V(t,x(t),i)=[x(t)-Dix(t-d(t))]TPi[x(t)-Dix(t-d(t))]+∫t-d(t)txT(s)Q1x(s)ds+∫t-d(t)t∫stxT(θ)Q2x(θ)dθds+∫-d2-d1∫t+st∫θtxT(u)Q3x(u)dudθds+σ∫-σ(t)0∫t+stxT(θ)Q4x(θ)dθds≤ξT(t)Πi′ξ(t)+∫t-d(t)txT(s)[Q1+d2Q2]x(s)ds+σ2∫t-σ(t)txT(s)Q4x(s)ds+∫-d2-d1∫t+stxT(θ)d2Q3x(θ)dθds,
where
(3.20)Πi′=[Pi-PiDi0-DiTPiDiTPiDi0000].
Lettingα>0, for system (2.6), we can define another operator 𝔏[eαtV(t,x(t),i)]:R+×Rn×S→R as follows:(3.21)L[eαtV(t,x(t),i)]=eαt[αV(t,x(t),i)+LV(t,x(t),i)]≤eαtξT(t)[αΠi′+Πi]ξ(t)+∫t-d(t)txT(s)[α(Q1+d2Q2)-(1-μ)Q2]x(s)ds+∫-d2-d1∫t+stxT(θ)[αd2-1]Q3x(θ)dθds+∫t-σ(t)txT(s)[ασ2-σ(1-κ)(1-ν)]Q4ds.
Now, we can chooseα>0 sufficiently small such that(3.22)αλmax⁡(Πi′)+λmax⁡(Πi)<0,α(Q1+d2Q2)-(1-μ)Q2<0,αd2-1<0,ασ2-σ(1-κ)(1-ν)<0,
for i∈S.
By the weak infinitesimal operator along (2.6), it is obtained from (3.21) and (3.22) that(3.23)eαtEV(t,x(t),i)≤EV(0,x(0),i)+∫0tEL[eαsV(s,x(s),i)]ds≤EV(0,x(0),i).
Using the definition of the Lyapunov functional (3.4) again, we have(3.24)EV(0,x(0),i)=E[x(0)-Dix(-d(0))]TPi[x(0)-Dix(-d(0))]+∫-d(0)0ExT(s)Q1x(s)ds+∫-d(0)0∫s0ExT(θ)Q2x(θ)dθds+∫-d2-d1∫s0∫θ0ExT(u)Q3x(u)dudθds+σ∫-σ(0)0∫s0xT(θ)Q4x(θ)dθds≤[2λmax⁡(Πi)+d2λmax⁡(Q1)+τ2λmax⁡(Q2)+(d2-d1)τ2λmax⁡(Q3)+σ3λmax⁡(Q4)]×sup⁡θ∈[-τ,0]E|φ(θ)|2≜Mi,
for i∈S.
Thus, from (3.23) and (3.24), it follows that(3.25)E|x(t)-Dix(t-d(t))|2≤1ΘMe-αt,
where Θ=mini∈S{λmin(Pi)} and M=maxi∈S{Mi}.
From|Di|<1 (i∈S), we obtain that there exist a positive scalar l>0 such that l=maxi∈S|Di|<1. So, for all ε∈(0,min{α,-(2/τ)log|l|}) and any θ>0, by using the elementary inequality, we derive(3.26)eεtE|x(t)|2=eεtE|x(t)-Dix(t-d(t))+Dix(t-d(t))|2≤(1+θ)eεtE|x(t)-Dix(t-d(t))|2+(1+1θ)eεtE|Dix(t-d(t))|2≤1+θΘMe-(α-ε)t+(1+1θ)eεtE|Dix(t-d(t))|2≤1+θΘM+l2(1+1θ)eετeε(t-τ(t))E|x(t-d(t))|2.
From for all ε∈(0,min{α,-(2/τ)log|l|}), we have l2eετ<1. Thus, we can choose θ sufficiently large such that
(3.27)Δ≜l2(1+1θ)eετ<1.
So,(3.28)eεtE|x(t)|2≤1+θΘM+Δeε(t-d(t))E|x(t-d(t))|2.
For allT>0, from (3.28), it follows that(3.29)sup⁡0≤t≤T[eεtE|x(t)|2]≤1+θΘM+ΔE|φ|2+Δsup⁡0≤t≤T[eεtE|x(t)|2],
that is,(3.30)sup⁡0≤t≤T[eεtE|x(t)|2]≤((1+θ)/Θ)M+ΔE|φ|21-Δ.
WhenT→+∞, it follows from (3.30) that(3.31)sup⁡0≤t<∞[eεtE|x(t)|2]≤((1+θ)/Θ)M+ΔE|φ|21-Δ.
So, we can obtain
(3.32)E|x(t)|2≤((1+θ)/Θ)M+ΔE|φ|21-Δe-εt.
The proof of this theorem is completed.Theorem 3.2.
Suppose that Assumptions2.1–2.4 hold and for any given positive scalar κ∈(0,1), the uncertain neutral stochastic systems with mixed delays and Markovian switching (2.3) are robustly exponentially stable in mean square if there exist λi>0 (i∈S), ε1i>0, ε2i>0 (i∈S), and some positive definite matrices Pi>0 (i∈S) and Ql>0 (l=1,2,…,10) such that the following linear matrix inequalities (LMIs) are satisfied:
(3.33)Pi<λiI,i∈S,Ωi=[Ω11Ω120Ω14Ω15Ω16Ω17Ω18Ω19Ω1100*Ω220000Ω27Ω28Ω290Ω211**Ω3300000000***Ω44Ω45Ω4600000****Ω55Ω5600000*****Ω6600000******Ω77Ω78Ω7900*******Ω88Ω8900********Ω9900*********-ε1iI0**********-ε2iI]<0,i∈S,
where * denotes the entries that are readily inferred by symmetry of a symmetric matrix and
(3.34)Ω11=PiA1i+A1iTPi+∑j=1NγijPj+λiR1iTR1i+LT(Q5+Q8)L+Q1+d2Q2+12(d22-d12)Q3+σ2Q4+(ε1i+ε2i)N1iTN1i,Ω12=A1iTPiDi-∑j=1NγijPjDi,Ω14=PiA2i+ε1iN1iTN2i,Ω15=PiA3i+ε1iN1iTN3i,Ω110=PiMi,Ω16=PiA4i+ε1iN1iTN4i,Ω17=ε2iN1iTN2i,Ω18=ε2iN1iTN3i,Ω19=ε2iN1iTN4i,Ω22=LT(Q6+Q9)L+∑j=1NγijDiTPjDi+λiR2iTR2i-(1-μ)Q1,Ω27=DiTPiA2i,Ω28=DiTPiA3i,Ω29=DiTPiA4i,Ω211=DiTPiMi,Ω33=LT(Q7+Q10)L+λiR3iTR3i-κ(1-υ)Q4,Ω44=-Q5+ε1iN2iTN2i,Ω45=ε1iN2iTN3i,Ω46=ε1iN2iTN4i,Ω55=-Q6+ε1iN3iTN3i,Ω56=ε1iN3iTN4i,Ω66=-Q7+ε1iN4iTN4i,Ω77=-Q8+ε2iN2iTN2i,Ω78=ε2iN2iTN3i,Ω79=ε2iN2iTN4i,Ω88=-Q9+ε2iN3iTN3i,Ω89=ε2iN3iTN4i,Ω99=-Q10+ε2iN4iTN4i.Proof.
ReplacingA1i, A2i, A3i, and A4i in (3.2) with A1i+ΔA1i(t), A2i+ΔA2i(t), A3i+ΔA3i(t), and A4i+ΔA4i(t), ΔA1i(t), ΔA2i(t), ΔA3i(t), ΔA4i(t) are described in (2.4) and (2.5), in view of Lemma 2.9 and Lemma 2.10, we obtain
(3.35)Πi(t)=Πi+ΞiTFi(t)E1i+E1iTFiT(t)Ξi+ΨiTFi(t)E2i+E2iTFiT(t)Ψi=Πi+ε1i-1ΞiΞiT+ε1iE1iTE1i+ε2i-1ΨiΨiT+ε2iE2iTE2i=Ωi<0,
where
(3.36)Ξi=[MiTPi00000000]T,Ψi=[0MiTPiDi0000000]T,E1i=[N1iT00N2iTN3iTN4iT000]T,E2i=[N1iT00000N2iTN3iTN4iT]T,
where ε1i>0 and ε2i>0, for any i∈S. The proof of the remainder can be easily finished by following a similar line as in the proof of Theorem 3.1. The proof is completed.Remark 3.3.
The delay-dependent sufficient conditions ensuring the robustly exponential stability in mean square of the uncertain neutral stochastic systems with mixed delays and Markovian switching (2.3) are provided in Theorem 3.2. It should be pointed out that such conditions are given in the form of LMIs, which could be easily solved by using the standard software packages. Besides, the criteria derived are dependent upon both the upper and lower bound of the time-varying delay and the distributed delay, which are less conservative.Remark 3.4.
Besides, by the Borel-Cantelli Lemma, we can also obtain the almost surely exponential stability of systems (2.3). Here, for the sake of brevity, we omit it and the readers can refer to [17]. Besides, we can easily come to a conclusion that the uncertain neutral stochastic delay systems with mixed delays and Markovian jumping parameters (2.3) are asymptotically stable in mean square from the conditions |Di|<1 (i∈S). Thus, the results given in [15] are generalized.Case 1.
Consider the problem of delay-dependent robust exponential stability for a special case of the uncertain neutral stochastic systems with mixed delays and Markovian switching in (2.6), that is,
(3.37)d[x(t)-Dix(t-d(t))]=[A1i(t)x(t)+A2i(t)f(x(t))+A3i(t)f(x(t-d(t)))+A4i(t)∫t-σ(t)tf(x(s))ds]dt+[A5i(t)x(t)+A6i(t)x(t-d(t))+A7i(t)∫t-σ(t)tx(s)ds]dB(t),t≥0,
where A1i(t), A2i(t), A3i(t), and A4i(t) are given in (2.3), while A5i(t), A6i(t), and A7i(t) are of the following form:
(3.38)[A5i(t)A6i(t)A7i(t)]=[A5iA6iA7i]+MiFi(t)[N5iN6iN7i],
where A5i, A6i, A7i, Mi, N5i, N6i, and N7i (i∈S) are known constant matrices. For systems (3.37), we can also obtain the robust exponential stability. The proof can be easily established by following a similar line as in the proof of Theorem 3.1 and then is omitted here.Theorem 3.5.
Suppose that Assumptions2.1 and 2.2 hold and for any given positive scalar κ∈(0,1), the uncertain neutral stochastic systems with mixed delays and Markovian switching (3.37) are robustly exponentially stable in mean square if there exist λi>0, ε1i>0, ε2i>0, ε3i>0 (i∈S) and some positive definite matrices Pi>0 (i∈S) and Qi>0 (i=1,2,…,10) such that the following linear matrix inequalities (LMIs) are satisfied:
(3.39)Pi≤λiI,i∈S,[Γ11Γ12Γ13Γ14Γ15Γ16000A5iTPiPiMi00*Γ22Γ23000Γ27Γ28Γ29A6iTPi0DiTPiMi0**Γ33000000A7iTPi000***Γ44Γ45Γ460000000****Γ55Γ560000000*****Γ660000000******Γ77Γ78Γ790000*******Γ88Γ890000********Γ990000*********-Pi00PiMi**********-ε1iI00***********-ε2iI0************-ε3iI]<0,
where * denotes the entries that are readily inferred by symmetry of a symmetric matrix and
(3.40)Γ11=PiA1i+A1iTPi+∑j=1NγijPj+LT(Q5+Q8)L+Q1+d2Q2+12(d22-d12)Q3+σ2Q5+ε1iN1iTN1i+ε3iN5iTN5i,Γ12=A1iTPiDi-∑j=1NγijPjDiT+ε3iN5iTN6i,Γ13=ε3iN5iTN7i,Γ14=PiA2iT+ε1iN1iTN2i,Γ15=PiA3i+ε1iN1iTN3i,Γ16=PiA4i+ε1iN1iTN4i,Γ22=LT(Q6+Q9)L+∑j=1NγijDiTPjDi-(1-μ)Q1+ε3iN6iTN6i,Γ23=ε3iN6iTN7i,Γ27=DiTPiA2i,Γ28=DiTPiA3i,Γ29=DiTPiA3i,Γ33=LT(Q7+Q10)L-κ(1-ν)Q5+ε3iN7iTN7i,Γ44=-Q5+ε1iN2iTN2i,Γ45=ε1iN2iTN3i,Γ46=ε1iN2iTN4i,Γ55=-Q6+ε1iN3iTN3i,Γ56=ε1iN3iTN4i,Γ66=-Q7+ε1iN4iTN4i,Γ77=-Q8+ε2iN2iTN2i,Γ78=ε2iN2iTN3i,Γ79=ε2iN2iTN4i,Γ88=-Q9+ε2iN3iTN3i,Γ89=ε2iN3iTN4i,Γ99=ε2iN4iTN4i.Case 2.
Whenσ=0, the uncertain neutral stochastic systems (2.3) are described as(3.41)d[x(t)-Dix(t-d(t))]=[∫t-τ(t)tA1i(t)x(t)+A2i(t)f(x(t))+A3i(t)f(x(t-d(t)))+A4i(t)∫t-σ(t)tf(x(s))ds]dt,t≥0.Theorem 3.6.
Suppose that Assumptions2.1 and 2.2 hold and for any given positive scalar κ∈(0,1), the uncertain neutral stochastic systems with mixed delays and Markovian switching (3.41) are robustly exponentially stable in mean square if there exist λi>0 (i∈S), εi>0 (i∈S) and some positive definite matrices Pi>0 (i∈S) and Qi>0 (i=1,2,…,10) such that the following linear matrix inequalities (LMIs) are satisfied: for i∈S,
(3.42)Ωi=[Ω11Ω120Ω14Ω15Ω16Ω17Ω18Ω19Ω1100*Ω220000Ω27Ω28Ω290Ω211**Ω3300000000***Ω44Ω45Ω4600000****Ω55Ω5600000*****Ω6600000******Ω77Ω78Ω7900*******Ω88Ω8900********Ω9900*********-ε1iI0**********-ε2iI]<0,i∈S,
where * denotes the entries that are readily inferred by symmetry of a symmetric matrix and
(3.43)Ω11=PiA1i+A1iTPi+∑j=1NγijPj+LT(Q5+Q8)L+Q1+d2Q2+12(d22-d12)Q3+σ2Q4+(ε1i+ε2i)N1iTN1i,Ω22=LT(Q6+Q9)L+∑j=1NγijDiTPjDi-(1-μ)Q1,Ω27=DiTPiA2i,Ω33=LT(Q7+Q10)L-κ(1-υ)Q4,
and Ω12, Ω14, Ω15, Ω16, Ω17, Ω18, Ω19, Ω27, Ω28, Ω29, Ω33, Ω44, Ω45, Ω46, Ω55, Ω56, Ω66, Ω77, Ω78, Ω79, Ω88, Ω89, and Ω99 are given in Theorem 3.2.Remark 3.7.
The sufficient conditions ensuring the exponential stability in mean square of the uncertain neutral delay systems with Markovian jumping parameters in [39, 40] are dependent upon the upper bound of the exponential stability rate. But, Theorem 3.6 can remove this restrictive conditions. So, we can improve and generalize the results in [39, 40].


## 4.  Numerical Examples

In this section, two examples are provided to illustrate the feasibility and applicability of our obtained results.Example 4.1.
Consider the case of 2D Brownian motion, andr(t) is right-continuous Markovian chain taking values in S={1,2} with its generator Γ=[-334-4]. And the parameters in systems (2.3) are given as follows:
(4.1)A11=[-1.500-1.8],A12=[-1.800-1.8],A21=[0.30.2-0.6-0.8],A22=[0.6-0.5-0.4-0.3],A31=[-0.7-0.50.40.6],A32=[0.70.5-0.30.3],A41=[0.20.3-0.10.2],A42=[0.30.2-0.20.4],D1=[-0.20.01.00.1],D2=[0.10.00.70.2],M1=[0.10.20.00.1],M2=[0.20.10.10.2],N11=N12=N21=N22=N31=N32=N41=N42=[0.10.1].
The delays d(t)=0.45sin(t)+0.3, σ(t)=0.4cos(t)+0.4, and it is easily obtained that d1=0.15, d2=0.75, μ=0.45, ν=0.4, and σ=0.8. Assume that the activation function f satisfies Assumption 2.1 with the Lipschitz coefficient matrix L=[0.20.00.00.2] and the function σ satisfies Assumption 2.2 with the matrices Rij (i=1,2; j=1,2,3) given by
(4.2)R11=[0.150.00.200.35],R12=[0.50.10.280.20],R21=[0.30.00.20.1],R22=[0.10.00.20.1],R31=[0.10.20.10.2],R32=[0.10.10.20.1].
With the parameters above, when κ=0.9999, by using Matlab LMI Toolbox, according to Theorem 3.2, we solve LMIs (3.33) and obtain there feasible solutions as follows:
(4.3)P1=[70.093012.228512.228521.8248],P2=[74.154915.487315.487320.7284],Q1=[134.466225.422325.422320.5296],Q2=[1.8693-0.0065-0.00650.6213],Q3=[4.8828-0.0153-0.01531.6598],Q4=[16.7672-0.0153-0.01531.6598],Q5=[117.1703-17.8696-17.8696119.1560],Q6=[114.706167.043867.043891.0274],Q7=[58.328423.081123.081170.1310],Q8=[57.018112.728012.728058.0073],Q9=[41.692114.691914.691940.8478],Q10=[40.5029-0.2295-0.229533.2920],λ1=75.3750,λ2=80.6197,ε11=66.1002,ε12=97.9532,ε21=47.7433,ε22=56.9671.Example 4.2.
Letr(t) be right-continuous Markovian chain taking values in S={1,2} with its generator Γ=[-221-1]. Consider the following uncertain neutral systems with mixed delays and Markovian switching: for any i∈S,
(4.4)ddt[x(t)-Dix(t-d(t))]=A1i(t)x(t)+A2i(t)x(t-d(t)),t≥0,
where
(4.5)A11=A12=[-20.00.0-3],A21=A22=[-10-1-1],D1=[0.3000.3],D2=[0.10.00.00.1],M1=[0.10.10.20.1],M2=[0.2-0.10.10.1],N11=[0.10.0],N12=[0.00.1],N21=[-0.00.1],N22=[0.10.1].
Applying Matlab toolbox, by Theorem 3.6, for μ=0, it can be obtained that the uncertain neutral systems with delays and Markovian jumping parameters (4.4) are robustly exponentially stable in mean square with the delay d(t) satisfying 0≤d(t)≤6.0×1017.


## 5.  Conclusions

In this paper, the exponential stability in mean square of the uncertain neutral stochastic systems with mixed delays and Markovian jumping parameters has been considered. The mixed delays consist of the discrete delay and the distributed delay. The LMI-based conditions ensuring the exponential stability in mean square of such systems are obtained by constructing an appropriate Lyapunov functional, which are dependent upon the upper bound and lower bound of the discrete time delays and distributed delays. It is worth pointing out that, compared with the previous works [17, 18], the stability criteria in this paper can be easily checked by using some standard numerical packages. Two illustrative examples are provided to show the effectiveness and applicability of the proposed results.



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*Source: 748279-2012-05-27.xml*