# Delay-Dependent Robust Exponential Stability for Uncertain Neutral Stochastic Systems with Interval Time-Varying Delay

Authors: Weihua Mao; Feiqi Deng; Anhua Wan
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

This paper discusses the mean-square exponential stability of uncertain neutral linear stochastic systems with interval time-varying delays. A new augmented Lyapunov-Krasovskii functional (LKF) has been constructed to derive improved delay-dependent robust mean-square exponential stability criteria, which are forms of linear matrix inequalities (LMIs). By free-weight matrices method, the usual restriction that the stability conditions only bear slow-varying derivative of the delay is removed. Finally, numerical examples are provided to illustrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

Dynamical systems, such as the distributed networks, electric power systems, and communication systems, can be efficiently modeled by neutral stochastic functional differential equations, which have been extensively studied in recent years, see [1–8] and references therein. Basically, sufficient conditions on stability of time delay systems are divided into two categories: delay-dependent and delay-independent cases. It is well known that the latter is less conservative than the former, especially when the delay is very small. Therefore, increasing attention has been focused on delay-dependent stability analysis of delay systems in recent years, see [1–25].There are unstable systems without delay that can be stabilized with proper nonzero delay, see [26]. Therefore, it is of great significance to consider the stability of systems with interval time-varying delay, it is a time delay that varies in an interval, in which the lower bound is not restricted to 0. Recently, interval time-varying delay was introduced and investigated, see [11–17].In the past years, as an effective approach of improving the performance of delay-dependent stability criteria, free-weighting matrices method has attracted much attention, see [2–5, 10–12]. The mean-square exponential stability was studied for neutral stochastic systems with fixed delays in [2, 3]. The global asymptotic stability for a class of neutral stochastic neural networks was studied in [4]. Improved delay-dependent robust stability criteria of uncertain stochastic systems with interval time-varying delay were proposed in [12].The usual restriction on derivative of the time-varying delay thatτ˙(t)<1 in [5–7, 18] brings conservatism, which shows that the resulting conditions can only bear slow-varying delay. In order to remove this limitation, fast-varying rate condition was studied in [4, 12, 23].Stability of stochastic differential delay systems with nonlinear impulsive effects is studied in [24, 25], the equivalent relations are established between the stability of the stochastic differential delay equations with impulsive effects and that of a corresponding stochastic differential delay equations without impulses.To our best knowledge, few works on the robust delay-dependent exponential stability analysis have been reported for uncertain neutral stochastic distributed system with fast-varying interval time-varying delay. This paper focuses on the stability analysis of uncertain neutral stochastic distributed system with interval time-varying delay. By Lyapunov-Krasovskii functional theory and free-weighting matrices method, a new delay-dependent mean-square exponential stability criterion is formulated in terms of LMI, and the usual restriction ofτ˙(t)<1 is removed. Finally, three numerical example are given to illustrate the effectiveness of the proposed method.Notation 1.
Throughout this paper, the notations are standard. IfA is a vector or matrix, its transpose is denoted by AT. P>0 means that P is a symmetric positive definite matrix. * denotes the symmetry part of a symmetry matrix. ⋆ indicates terms that can be induced by symmetry, for example, A+AT=A+⋆ and MPMT=MP⋆. Denote by λM{·} and λm{·} the maximum and minimum eigenvalue of a matrix, respectively. Let ρ(·) denote the spectral norm of a matrix, while |·| refers to the Euclidean vector norm. Let (Ω,ℱ,𝒫) be a complete probability space relative to an increasing family {ℱ}t≥0 of σ algebras ℱt⊂ℱ and 𝔼{·} the mathematical expectation operator with respect to the probability measure 𝒫. ℒ2[0,∞) denotes the space of square integrable vector functions over [0,∞). Let τ>0 and C([-τ,0],Rn) denote the family of all continuous Rn-valued functions ϕ on [-τ,0] with the norm ∥ϕ∥=sup{|ϕ(θ)|:-τ⩽θ≤0}. Let ℒℱ02([-τ,0];Rn) be the family of all ℱ0-measurable bounded C([-τ,0],Rn)-valued random variables φ={φ(θ):-τ⩽θ⩽0}.


## 2.  Preliminaries

Consider the following uncertain Itô-type neutral stochastic system with both discrete and distributed interval time-varying delays(2.1)(Σ):d[x(t)-Dx(tτ)]={[A0+ΔA0(t)]x(t)+[A1+ΔA1(t)]x(tτ)+[A2+ΔA2(t)]∫tτt‍x(s)ds}+{[B0+ΔB0(t)]x(t)+[B1+ΔB1(t)]x(tτ)+[B2+ΔB2(t)]∫tτt‍x(s)ds}dB(t),(2.2)x(t)=ϕ(t),t∈[-τ2,0],
where x(t)∈Rn is the state vector; B(t) is a standard scalar Brownian motion defined on a complete probability space (Ω,ℱ,𝒫), we assume 𝔼{dB(t)}=0,𝔼{d2B(t)}=dt. ϕ(t) is any given initial data in ℒℱ02([-τ,0];Rn). Denote η(t)=x(t)-Dx(tτ),tτ=t-τ(t),tτ1=t-τ1,tτ2=t-τ2, τ(t) is time-varying delays, satisfying
(2.3)τ˙(t)⩽τ-<+∞,0⩽τ1⩽τ(t)⩽τ2<+∞,δ=τ2-τ1.Remark 2.1.
It should be noted that system (2.1)-(2.2) encompasses many state space models of neutral delay systems, which can be used to represent many important physical systems such as a large class of distributed networks containing lossless transmission lines, population ecology, heat exchangers, wind tunnel, and water resources systems (see [27–29] and the references therein). For example, signal transformation cannot be finished in time due to the finite signal propagation speed in distributed networks containing lossless transmission lines, or to the finite switching speed of amplifiers in electronic networks, so the models only depending on the discrete time delays are not complete, the more exact models should include the distributed time delays, and the delays are found both in the states and in the derivatives of the states.Remark 2.2.
It is worth pointing out that, when a time-varying delay appears, it is usually assumed thatτ˙(t)<1 is satisfied and the lower bound of the delay is restricted to be zero in the literature [5, 7, 8] and so forth. However, in this paper, we only require τ˙(t)⩽τ-, in addition, the range of delay may vary in a range for which the lower bound is not restricted to be zero. Therefore, the time-varying delay in this paper is more general.
A
i
,
B
j
,
i
,
j
=
0
,
1
,
2, are known real constant matrices of appropriate dimensions; ΔAi,ΔBj,i,j=0,1,2, are unknown matrices representing time-varying parameter uncertainties. For the sake of convenience, we assume that the uncertainties are norm-bounded and can be described as
(2.4)(ΔAi(t)ΔBi(t))=GF(t)(SiTi),
where Si,Ti,i=0,1,2, are known real constant matrices, F(t) is an unknown real time-varying function with appropriate dimension satisfying
(2.5)FT(t)F(t)⩽I.
It is assumed that the elements of F(t) are Lebesgue measurable. Throughout this paper, the following assumption, definitions, and lemmas will be used to develop our results.Assumption 2.3.
The difference operator matrixD satisfies ∥D∥<1.Definition 2.4 (see [1]).
The neutral stochastic delay system (2.1)-(2.2) is said to be mean-square exponentially stable if there is a pair of positive constants α,β such that
(2.6)𝔼{xT(t)x(t)}⩽αe-βtsup-τ2⩽θ⩽0𝔼{ϕ-(θ)}.Lemma 2.5 (see [30]).
If there exists a vector functionv(t):[0,r]→Rn such that ∫0r‍vT(s)Pv(s)ds and ∫0r‍v(s)ds are well defined, then the following inequality:
(2.7)[∫0r‍v(s)ds]TP[∫0r‍v(s)ds]⩽r∫0r‍vT(s)Pv(s)ds
holds for any pair of symmetric positive definite matrix P∈Rn×n and r>0.Lemma 2.6 (see [31]).
For any vectorsx,y∈Rn, matrices A,D,E,P, and F are real matrices of appropriate dimensions with P>0 and FTF⩽I, the following inequalities hold:(1)
2xTDFEy⩽ϵ-1xTDDTx+ϵyTEETy;(2)
for any scalarϵ>0 such that P-ϵDDT>0, then (A+DFE)TP-1(A+DFE)⩽ϵ-1ETE+AT(P-ϵDDT)-1A;(3)
2xTy⩽xTP-1x+yTPy.


## 3. Main Results

In this section, a robustly stochastically exponentially stable criterion for the uncertain linear neutral stochastic distributed delayed system(Σ) will be established by applying the Lyapunov-Krasovskii theory and free-weighting matrices method.For convenience, define a new state variable(3.1)f(t)=(A0+ΔA0(t))x(t)+(A1+ΔA1(t))x(tτ)+(A2+ΔA2(t))∫tτt‍x(s)ds
and a new perturbation variable
(3.2)g(t)=(B0+ΔB0(t))x(t)+(B1+ΔB1(t))x(tτ)+(B2+ΔB2(t))∫tτt‍x(s)ds.
Then, system (2.1) becomes
(3.3)d[x(t)-Dx(tτ)]=f(t)dt+g(t)dB(t).
Applying Leibniz-Newton formula to (2.1), it yields the following zero equations which will be used in our main result:
(3.4)2ξT(t)N[x(t)-Dx(tτ)-x(tτ)+Dx(tτ-τ(tτ))-∫tτt‍f(s)ds-∫tτt‍g(s)dB(s)]=0,2ξT(t)H[x(tτ1)-Dx(tτ1-τ(tτ1))-x(tτ)+Dx(tτ-τ(tτ))-∫tτtτ1‍f(s)ds-∫tτtτ1‍g(s)dB(s)]=0,2ξT(t)M[x(tτ)-Dx(tτ-τ(tτ))-x(tτ2)+Dx(tτ2-τ(tτ2))-∫tτ2tτ‍f(s)ds-∫tτ2tτ‍g(s)dB(s)]=0,
where N, H, and M are any matrices with appropriate dimensions, and
(3.5)ξT(t)=[xT(t),xT(tτ),1τ2∫tτt‍xT(s)ds,xT(tτ1),xT(tτ2),xT(tτ1-τ(tτ1)),xT(tτ2-τ(tτ2)),xT(tτ-τ(tτ))∫tτt‍].
With zero equations (3.4) above, we obtain the following theorem.Theorem 3.1.
For given scalars0⩽τ1⩽τ2 and τ-, system (Σ) is robustly stochastically exponentially stable for all time-varying delays satisfying (2.3) and for all admissible uncertainties satisfying (2.4) and (2.5), if there exist symmetric positive definite matrices P>0,Ri>0,i=0,1,2,…,10, scalars ϵj,j=1,2,3 and any matrices N,H,M with appropriate dimensions, such that the following LMI (3.6) holds:
(3.6)Φ=(Φ1Φ2*Φ3)<0,
where(3.7)Φ1=(ΘP-GV1TZ0V2TU0*-ϵ1I0000**-ZZG00***-ϵ2I00****-UUG*****-ϵ3I),Φ2=(N-H-M-N-H-M-),Φ3=diag={-R7-R8-R7-R8-R9-R10-R9-R10}.Θ=Γ+Ψ+ΨT,V1=[A0A1A200000],V2=[B0B1B200000],P-=[P-PD000000]T,Z=τ2R7+δR8,U=P+τ2R9+δR10,N-=[NT,0,0,0,0,0]T,H-=[HT,0,0,0,0,0]T,M-=[MT,0,0,0,0,0]T,
with
(3.8)Γ=(Γ11Γ12Γ1300000*Γ22Γ2300000**Γ3300000***-R10000****-R3000*****Γ6600******Γ770*******Γ88),Ψ=(N(M-H)-N(D+I)0H-M-HDMD(N+H-M)D),Γ11=PA0+A0TP+R1+R2+R3+τ2R0+(ϵ1+ϵ2)S0TS0+ϵ3T0TT0,Γ12=PA1-A0TPD+(ϵ1+ϵ2)S0TS1+ϵ3T0TT1,Γ13=PA2+(ϵ1+ϵ2)S0TS2+ϵ3T0TT2,Γ22=R4+R5+R6-(1-τ-)R2-DTPA1-A1TPD+(ϵ1+ϵ2)S1TS1+ϵ3T1TT1,Γ23=-DTPA2+(ϵ1+ϵ2)S1TS2+ϵ3T1TT2,Γ33=-τ2R0+(ϵ1+ϵ2)S2TS2+ϵ3T2TT2,Γ66=R4,Γ77=R6,Γ88=(1-τ-)R5.Proof.
Choose a Lyapunov-Krasovskii functional candidate as(3.9)V(t,ξt)=∑i=04‍Vi(t,ξt),
where
(3.10)V0(t,xt)=[x(t)-Dx(tτ)]TP[x(t)-Dx(tτ)];V1(t,xt)=∫tτ1t‍(x(s)x(s-τ(s)))T(R100R4)(x(s)x(s-τ(s)))ds+∫tτt‍(x(s)x(s-τ(s)))T(R200R5)(x(s)x(s-τ(s)))ds+∫tτ2t‍(x(s)x(s-τ(s)))T(R300R6)(x(s)x(s-τ(s)))ds;V2(t,xt)=∫tτ2t‍[s-(t-τ2)]xT(s)R0x(s)ds;V3(t,xt)=τ2∫-τ20‍∫t+θt‍fT(s)R7f(s)dsdθ+δ∫-τ2-τ1‍∫t+θt‍fT(s)R8f(s)dsdθ;V4(t,xt)=∫-τ20‍∫t+θt‍tr(gT(s)R9g(s))dsdθ+∫-τ2-τ1‍∫t+θt‍tr(gT(s)R10g(s))dsdθ.
According to Itô's differential formula [1], the stochastic differential is
(3.11)dV(t,xt)=LV(t,xt)dt+2[x(t)-Dx(tτ)]TPg(t)dB(t),
where
(3.12)LV(t,xt)dt=2[x(t)-Dx(tτ)]TPf(t)+g(t)TPg(t)+∑i=14‍V˙i(t,xt).
Direct computations give
(3.13)V˙1(t,xt)⩽xT(t)(R1+R2+R3)x(t)-xT(tτ1)R1x(tτ1)-xT(tτ2)R3x(tτ2)+xT(tτ)(R4+R5+R6-(1-τ-)R2)x(tτ)-xT(tτ1-τ(tτ1))R4x(tτ1-τ(tτ1))-(1-τ-)xT(tτ-τ(tτ))R5x(tτ-τ(tτ))-xT(tτ2-τ(tτ2))R6x(tτ2-τ(tτ2)){byτ˙(t)⩽τ-}V˙2(t,xt)⩽τ2xT(t)R0x(t)-∫tτt‍xT(s)R0x(s)ds≤τ2xT(t)R0x(t)-(1τ2∫tτt‍x(s)ds)T(τ2R0)(1τ2∫tτt‍x(s)ds);byLemma2.5,V˙3(t,xt)=fT(t)T(τ22R7+δ2R8)f(t)-τ2∫tτ2t‍fT(s)TR7f(s)ds-δ∫tτ2tτ1‍fT(s)TR8f(s)ds;V˙4(t,xt)=tr(gT(t)T(τ2R9+δR10)g(t))-∫tτ2t‍tr(gT(s)TR9g(s))ds-∫tτ2tτ1‍tr(gT(s)TR10g(s))ds.
By (1) of Lemma 2.6, for any scalar ϵ1>0, we have
(3.14)2[x(t)-Dx(tτ)]TPf(t)=2ξT(t)P-V1ξ(t)+2ξT(t)P-GF(t)V3ξ(t)≤2ξT(t)P-V1ξ(t)+ϵ1-1ξT(t)P-G⋆+ϵ1ξT(t)V3T⋆,
and by (2) of Lemma 2.6, for any ϵ2>0 satisfying [τ22R7+δ2R8]-1-ϵ2-1GGT>0, we have
(3.15)fT(t)[τ22R7+δ2R8]f(t)=ξT(t)(V1+GF(t)V3)TZ⋆≤ξT(t)V1T[Z-1-ϵ2-1GGT]-1⋆+ϵ2ξT(t)V3T⋆.
For any scalar ϵ3>0 satisfying [τ2R9+δR10]-1-ϵ3-1GGT>0, the following inequality holds:
(3.16)tr(gT(t)[P+τ2R9+δR10]g(t))=gT(t)[P+τ2R9+δR10]g(t)=ξT(t)(V2+GF(t)V4)TU⋆≤ξT(t)V2T[U-1-ϵ3-1GGT]-1⋆+ϵ3ξT(t)V4T⋆.
In addition, it follows from (3.4), (3)  of Lemma 2.6 and Lemma 2.5 that
(3.17)-2ξT(t)N∫tτt‍f(s)ds⩽ξT(t)NR7-1⋆+(∫tτt‍f(s)ds)TR7⋆⩽ξT(t)NR7-1⋆+τ2∫tτt‍fT(s)R7f(s)ds,-2ξT(t)H∫tτtτ1‍f(s)ds⩽ξT(t)HR8-1⋆+(∫tτtτ1‍f(s)ds)TR8⋆⩽ξT(t)HR8-1⋆+δ∫tτtτ1‍fT(s)R8f(s)ds,-2ξT(t)M∫tτ2tτ‍f(s)ds⩽ξT(t)M(R7+R8)-1⋆+(∫tτ2tτ‍f(s)ds)T(R7+R8)⋆≤ξT(t)M(R7+R8)-1⋆+∫tτ2tτ‍fT(s)(τ2R7+δR8)f(s)ds,-2ξT(t)N∫tτt‍g(s)dB(s)≤ξT(t)NR9-1⋆+(∫tτt‍g(s)dB(s))TR9⋆,-2ξT(t)H∫tτtτ1‍g(s)dB(s)≤ξT(t)HR10-1⋆+(∫tτtτ1‍g(s)dB(s))TR10⋆,-2ξT(t)M∫tτ2tτ‍g(s)dB(s)≤ξT(t)M(R9+R10)-1⋆+(∫tτ2tτ‍g(s)dB(s))T(R9+R10)⋆.
Note that
(3.18)𝔼{(∫tτt‍g(s)dB(s))TR9⋆}=∫tτt‍tr(gT(s)R9g(s))ds,𝔼{(∫tτtτ1‍g(s)dB(s))TR10⋆}=∫tτtτ1‍(gT(s)R10g(s))ds,(3.19)𝔼{(∫tτ2tτ‍g(s)dB(s))T(R9+R10)⋆}=∫tτ2tτ‍(gT(s)(R9+R10)g(s))ds.
Applying inequalities from (3.13) to (3.19) to (3.12), it yields
(3.20)ℒωV(t,ξt)⩽ξT(t)Λξ(t),
where
(3.21)Λ=Θ+ϵ1-1P-G(P-G)T+V1T[Z-1-ϵ2-1GGT]-1V1+V2T[U-1-ϵ3-1GGT]-1V2+N(R7-1+R9-1)NT+H(R8-1+R10-1)HT+M((R7+R8)-1+(R9+R10)-1)MT
with Θ being defined in Theorem 3.1.
By applying the Schur complement to (3.6) results in Λ<0, which implies
(3.22)LV(t,xt)⩽ξT(t)Λξ(t)<0
therefore,
(3.23)LV(t,xt)⩽-νξ(t)ξ(t)⩽-νxT(t)x(t),
where ν=λm{-Λ}. Now, integrating both sides of (3.23) from 0 to t>0, and then taking the expectation, we obtain
(3.24)𝔼{V(s,xs)|s=0s=t}=𝔼{∫0t‍LV(s,xs)ds}⩽-ν∫0t‍𝔼{xT(s)x(s)}ds.
On the other hand, it follows from (3.9) that
(3.25)𝔼{V(t,xt)}≥𝔼{ηT(t)Pη(t)}≥λm{P}𝔼{ηT(t)η(t)}.
Therefore, by (3.24) and (3.25)
(3.26)λm{P}𝔼{ηT(t)η(t)}⩽𝔼{V(0,x0)}-ν∫0t‍𝔼{xT(s)x(s)}ds.
Since ρ(D)<1, choose ϵ>0 such that eτ2ϵρ2(D)<1, which implies ρ2(D)<e-τ2ϵ<1, denote ϵ0=e-τ2ϵ. Let σ be any nonnegative real number. For all 0⩽t⩽σ, it follows from (1) of Lemma 2.6 that
(3.27)𝔼{ηT(t)η(t)}≥𝔼xT(t)x(t)-2𝔼{xT(t)Dx(tτ)}+𝔼xT(tτ)DTDx(tτ)≥(1-ϵ0)𝔼xT(t)x(t)-(ϵ0-1-1)𝔼xT(tτ)DTDx(tτ).
Hence
(3.28)𝔼xT(t)x(t)⩽11-ϵ0𝔼{ηT(t)η(t)}+1ϵ0𝔼{xT(tτ)DTDx(tτ)}.
By (3.25), (3.27), and (3.28), we then derive that for all 0⩽t⩽σ,
(3.29)𝔼xT(t)x(t)⩽11-ϵ0sup0⩽t⩽σ𝔼{ηT(t)η(t)}+1ϵ0sup0⩽t⩽σ𝔼{xT(tτ)DTDx(tτ)}⩽1(1-ϵ0)sup0⩽t⩽σ𝔼{ηT(t)η(t)}+λM{D′D}ϵ0sup-τ2⩽t⩽σ𝔼{xT(t)x(t)}=1(1-ϵ0)sup0⩽t⩽σ𝔼{ηT(t)η(t)}+∥D∥2ϵ0sup-τ2⩽t⩽σ𝔼{xT(t)x(t)}.
However, this holds for all -τ⩽t⩽0 as well. Therefore,
(3.30)sup-τ2⩽t⩽σ𝔼{xT(t)x(t)}⩽1(1-ϵ0)sup0⩽t⩽σ𝔼{ηT(t)η(t)}+ρ2(D)ϵ0sup-τ2⩽t⩽σ𝔼{xT(t)x(t)}.
Since ρ2(D)<ϵ0 and P>0, we obtain that
(3.31)(1-ϵ0)(ϵ0-ρ2(D))ϵ0sup-τ2⩽t⩽σ𝔼{xT(t)x(t)}⩽sup0⩽t⩽σ𝔼{ηT(t)η(t)}.
By supremum property,
(3.32)sup-τ2⩽t⩽σ𝔼{xT(t)x(t)}≥sup0⩽t⩽σ𝔼{xT(t)x(t)}.
By (3.31) and (3.32),
(3.33)(1-ϵ0)(ϵ0-ρ2(D))ϵ0sup0⩽t⩽σ𝔼{xT(t)x(t)}⩽sup0⩽t⩽σ𝔼{ηT(t)η(t)}.
For any t≥0, it follows from (3.26), (3.33) and σ being an arbitrary nonnegative real number that
(3.34)supt≥0𝔼{xT(t)x(t)}⩽μ𝔼{V(0,x0)}-νμ∫0t‍𝔼{xT(s)x(s)}ds.
Since σ is an arbitrary nonnegative number, we have
(3.35)𝔼{xT(t)x(t)}⩽μ𝔼{V(0,x0)}-νμ∫0t‍𝔼{xT(s)x(s)}ds,∀t≥0.
Then, applying Gronwall-Bellman lemma to (3.35), it yields
(3.36)𝔼{xT(t)x(t)}⩽μ𝔼{V(0,x0)}e-νμt.
Note that there exists a scalar α>0 such that
(3.37)μ𝔼{V(0,x0)}⩽αsup-τ2⩽θ⩽0𝔼{ϕ(θ)}.
Denote μ=ϵ0[(1-ϵ0)(ϵ0-ρ2(D))λm{P}]-1>0,β=νμ>0, then we have
(3.38)𝔼{xT(t)x(t)}⩽αe-βtsup-τ2⩽θ⩽0𝔼{ϕ(θ)}.
By Definition 2.4, system (Σ) is robustly stochastically mean-square exponentially stable.Remark 3.2.
Theorem3.1 provides a delay-dependent condition with a new Lyapunov-Krasovskii functional, which makes full use of the relationship among the time-varying delay, its upper and lower bounds, and their difference. It is noted that this condition is obtained by free-weighting matrices techniques; model transformation is not resorted to in the derivation of Theorem 3.1. Thus, the conservatism caused by model transformation will be reduced.
Ifτ1=τ2=τ0, then τ-=0,τ(t)=τ0. Applying Leibniz-Newton formula to (2.1), it yields the following zero equations:
(3.39)2ζT(t)L[x(t)-Dx(t-τ0)-x(t-τ0)+Dx(t-2τ0)-∫tτ0t‍f(s)ds-∫tτ0t‍g(s)dB(s)]=0,
where L is any matrices with appropriate dimension, and
(3.40)ζT(t)=[xT(t),xT(t-τ0),1τ0∫t-τ0t‍xT(s)ds,xT(t-2τ0)].
With zero equations (3.39) above, we obtain the following corollary.Corollary 3.3.
For given constant delayτ0, system (Σ) is robustly stochastically exponentially stable for all admissible uncertainties satisfying (2.4) and (2.5), if there exist symmetric positive definite matrices P0>0,Z0>0,Z1>0,Z2>0,Z3>0,Z4>0, scalars ϵi,i=1,2,3 and any matrices L with appropriate dimensions such that the following LMI (3.41) holds:
(3.41)Ω=(ΞP-0Gτ02W1TZ30τ0W2TZ40L-L-*-ϵ1I000000**-τ02Z3τ02Z3G0000***-ϵ2I0000****-τ0Z4τ0Z4G00*****-ϵ3I00******-Z30******I0-Z4)<0,
where
(3.42)Ξ=Π+Υ+ΥT,W1=[A0A1A20],W2=[B0B1B20],P-0=[P0-DP000]T,L-=[LT000]T,
with
(3.43)Π=(Π11Π12Π130*Π22Π230**Π330***-Z2),Υ=(L-L(D+I)0LD),Π11=P0A0+A0TP0+Z1+Z2+τ0Z0+(ϵ1+ϵ2)S0TS0+ϵ3T0TT0,Π12=P0A1-A0TDP0+(ϵ1+ϵ2)S0TS1+ϵ3T0TT1,Π13=P0A2+(ϵ1+ϵ2)S0TS2+ϵ3T0TT2,Π22=-Z1-P0DTA1-A1TDP0+(ϵ1+ϵ2)S1TS1+ϵ3T1TT1,Π23=-P0DTA2+(ϵ1+ϵ2)S1TS2+ϵ3T1TT2,Π33=-τ0Z0+(ϵ1+ϵ2)S2TS2+ϵ3T2TT2.IfA2=B2=0,ΔA2(t)=ΔB2(t)=0,D=0, then, system (2.1) becomes
(3.44)dx(t)=f-(t)dt+g-(t)dB(t),
where
(3.45)f-(t)=(A0+ΔA0(t))x(t)+(A1+ΔA1(t))x(tτ),g-(t)=(B0+ΔB0(t))x(t)+(B1+ΔB1(t))x(tτ).
Applying Leibniz-Newton formula to (3.44), it yields the following zero equations which will be used in our main result
(3.46)2ξ~T(t)N^[x(t)-x(tτ)-∫tτt‍f-(s)ds-∫tτt‍g-(s)dB(s)]=0,2ξ~T(t)H^[x(tτ1)-x(tτ)-∫tτtτ1‍f-(s)ds-∫tτtτ1‍g-(s)dB(s)]=0,2ξ~T(t)M^[x(tτ)-x(tτ2)-∫tτ2tτ‍f-(s)ds-∫tτ2tτ‍g-(s)dB(s)]=0,
where N,H, and M are any matrices with appropriate dimensions, and
(3.47)ξ~T(t)=[xT(t),xT(tτ),xT(tτ1),xT(tτ2)].
With zero equation (3.46) above, we obtain the following corollary.Corollary 3.4.
For given scalars0⩽τ1⩽τ2 and τ-, system (3.44) is robustly stochastically exponentially stable for all time-varying delays satisfying (2.3) and for all admissible uncertainties satisfying (2.4) and (2.5), if there exist symmetric positive definite matrices P>0,Ri>0,i=1,2,…,10, scalars ϵj,j=1,2,3 and any matrices N^,H^,M^ with appropriate dimensions, such that the following LMI (3.48) holds:
(3.48)Φ~=(Φ~1Φ~2*Φ~3)<0,
where
(3.49)Φ~1=(ΘP~GV~1TZ0V~2TU0*-ϵ1I0000**-ZZG00***-ϵ2I00****-UUG*****-ϵ3I),Φ~2=(N~H~M~N~H~M~),Φ~3=diag={-R7-R8-R7-R8-R9-R10-R9-R10},Θ~=Γ~+Ψ~+Ψ~T,V~1=[A0A100],V~2=[B0B100],P~=[P000]T,Z=τ2R7+δR8,U=P+τ2R9+δR10,N~=[N^T,0,0,0,0,0]T,H~=[H^T,0,0,0,0,0]T,M~=[M^T,0,0,0,0,0]T,
with
(3.50)Γ~=(Γ~11Γ~1200*Γ~2200**-R10***-R3),Ψ~=(N^M^-H^-N^H^-M^),Γ~11=PA0+A0TP+R1+R2+R3+(ϵ1+ϵ2)S0TS0+ϵ3T0TT0,Γ~12=PA1+(ϵ1+ϵ2)S0TS1+ϵ3T0TT1,Γ~22=-(1-τ-)R2+(ϵ1+ϵ2)S1TS1+ϵ3T1TT1.Ifτ1=τ2=τ0, A2=B2=0,ΔAi(t)=ΔBi(t)=0,i=0,1,2, then τ-=0,τ(t)=τ0 and (2.1) becomes
(3.51)d[x(t)-Dx(t-τ0)]=f^(t)dt+g^(t)dB(t),
where
(3.52)f^(t)=A0x(t)+A1x(t-τ0),g^(t)=B0x(t)+B1x(t-τ0).
Applying Leibniz-Newton formula to (3.51), it yields the following zero equations
(3.53)2ςT(t)L^[x(t)-Dx(t-τ0)-x(t-τ0)+Dx(t-2τ0)-∫tτ0t‍f^(s)ds-∫tτ0t‍g^(s)dB(s)]=0,
where L^ is any matrices with appropriate dimension, and
(3.54)ςT(t)=[xT(t),xT(t-τ0),xT(t-2τ0)].
With zero equations (3.53) above, we obtain the following corollary.Corollary 3.5.
For given constant delayτ0, system (3.51) is robustly stochastically exponentially stable, if there exist symmetric positive definite matrices P0>0,Z1>0,Z2>0,Z3>0,Z4>0, any matrices L^ with appropriate dimensions, such that the following LMI (3.41) holds:
(3.55)Ω~=(Ξ~τ02W~1TZ3τ0W~2TZ4L~L~*-τ02Z3000**-τ0Z400***-Z30****-Z4)<0,
where
(3.56)Ξ~=Π~+Υ~+Υ~T,W~1=[A0A10],W~2=[B0B10],L~=[L^T00]T,
with
(3.57)Π~=(Π~11Π~120*Π~220**-Z2),Υ~=(L^-L^(D+I)L^D),Π~11=P0A0+A0TP0+Z1+Z2,Π~12=P0A1-A0TDP0,Π~22=-Z1-P0DTA1-A1TDP0.Deterministic systems may be regarded as special class of stochastic systems, letBi=0,ΔBi=0,i=0,1,2, then system (2.1)-(2.2) becomes the following uncertain neutral system with both discrete and distributed interval time-varying delays
(3.58)(Σ0):x˙(t)-Dx˙(tτ)=[A0+ΔA0(t)]x(t)+[A1+ΔA1(t)]x(tτ)+[A2+ΔA2(t)]∫tτt‍x(s)ds,(3.59)x(t)=ϕ(t),t∈[-τ2,0].
Applying Leibniz-Newton formula to (3.58), it yields the following zero equations which will be used in our main result:
(3.60)2ξT(t)N[x(t)-Dx(tτ)-x(tτ)+Dx(tτ-τ(tτ))-∫tτt‍f(s)ds]=0,2ξT(t)H[x(tτ1)-Dx(tτ1-τ(tτ1))-x(tτ)+Dx(tτ-τ(tτ))-∫tτtτ1‍f(s)ds]=0,2ξT(t)M[x(tτ)-Dx(tτ-τ(tτ))-x(tτ2)+Dx(tτ2-τ(tτ2))-∫tτ2tτ‍f(s)ds]=0,
where N,  H, and M are any matrices with appropriate dimensions, and ξ(t) is defined in (3.5). With zero equations (3.60) above, we obtainCorollary 3.6.
For given scalars0⩽τ1⩽τ2 and τ-, system (Σ0) is robustly stochastically exponentially stable for all time-varying delays satisfying (2.3) and for all admissible uncertainties satisfying (2.4) and (2.5), if there exist symmetric positive definite matrices P>0,Ri>0,i=0,1,2,…,8, scalars ϵj,j=1,2,3 and any matrices N,H,M with appropriate dimensions, such that the following LMI (3.61) holds:
(3.61)Φ=(ΘP-GV1TZ0NHM*-ϵ1I00000**-ZZG000***-ϵ2I000****-R700*****-R80******-R7-R8)<0,
where
(3.62)Θ=Γ+Ψ+ΨT,V1=[A0A1A200000],V2=[B0B1B200000],P-=[P-PD000000]T,Z=τ2R7+δR8,U=P+τ2R9+δR10,
with
(3.63)Γ=(Γ11Γ12Γ1300000*Γ22Γ2300000**Γ3300000***-R10000****-R3000*****Γ6600******Γ770*******Γ88),Ψ=(N(M-H)-N(D+I)0H-M-HDMD(N+H-M)D),Γ11=PA0+A0TP+R1+R2+R3+τ2R0+(ϵ1+ϵ2)S0TS0+ϵ3T0TT0,Γ12=PA1-A0TPD+(ϵ1+ϵ2)S0TS1+ϵ3T0TT1,Γ13=PA2+(ϵ1+ϵ2)S0TS2+ϵ3T0TT2,Γ22=R4+R5+R6-(1-τ-)R2-DTPA1-A1TPD+(ϵ1+ϵ2)S1TS1+ϵ3T1TT1,Γ23=-DTPA2+(ϵ1+ϵ2)S1TS2+ϵ3T1TT2,Γ33=-τ2R0+(ϵ1+ϵ2)S2TS2+ϵ3T2TT2,Γ66=-R4,Γ77=-R6,Γ88=-(1-τ-)R5.Proof.
Choose a Lyapunov-Krasovskii functional candidate as(3.64)V(t,ξt)=∑i=03‍Vi(t,ξt),
where Vi(t,ξt),i=0,1,2,3 are defined in (3.9). It can be concluded from Theorem 3.1.Ifτ1=τ2=τ0, A2=0,Bi=ΔAi(t)=ΔBi(t)=0,i=0,1,2, then τ-=0,τ(t)=τ0 and (2.1) becomes
(3.65)x˙(t)-Dx˙(t-τ0)=A0x(t)+A1x(t-τ0).
Applying Leibniz-Newton formula to (3.65), it yields the following zero equations:
(3.66)2ςT(t)L^[x(t)-Dx(t-τ0)-x(t-τ0)+Dx(t-2τ0)-∫tτ0t‍f^(s)ds]=0,
where L^ is any matrices with appropriate dimension, f^(t)=A0x(t)+A1x(t-τ0), and
(3.67)ςT(t)=[xT(t),xT(t-τ0),xT(t-2τ0)].
With zero equations (3.66) above, we obtain the following corollary.Corollary 3.7.
For given constant delayτ0, system (3.65) is robustly stochastically exponentially stable, if there exist symmetric positive definite matrices P0>0,Z1>0,Z2>0,Z3>0,Z4>0, and any matrices L^ with appropriate dimensions, such that the following LMI (3.68) holds:
(3.68)Ω~=(Ξ~τ02W~1TZ3LˇLˇ*-τ02Z300**-Z30***-Z4)<0,
where
(3.69)Ξ~=Π~+Υ~+Υ~T,W~1=[A0A10],Lˇ=[L^T00]T,
with
(3.70)Π~=(Π~11Π~120*Π~220**-Z2),Υ~=(L^-L^(D+I)L^D),Π~11=P0A0+A0TP0+Z1+Z2,Π~12=P0A1-A0TDP0,Π~22=-Z1-P0DTA1-A1TDP0.


## 4. Numerical Examples

Example 4.1.
Consider the uncertain neutral linear stochastic time-varying delay system (Σ) with
(4.1)A0=[-2.00.00.0-1.9],A1=[-1.00.0-1.0-1.0],A2=-I,D=-0.5I,B0=-0.1I,B1=0.2I,B2=0.1I,G=0.1I,Si=Ti=0.1I,i=0,1,2,
and τ(t)=αexp(-1/(1+t)2),0⩽α⩽2.1315, set α=1, in this situation, τ1=0.3679,τ2=1,τ-=0.7358,δ=0.6321. Set ϕ(θ)=[x1T(θ),x2T(θ)]T, where x1(θ),x2(θ) are random initial values with zero mean and variance 1.0, and F(t)=sin(t), the trajectories of x1(t) and x2(t) are shown in Figure 1. According to Theorem 3.1, the lower bounds and the upper bounds on the time delay to guarantee the system is robustly stochastically mean-square exponentially stable are listed in Table 1.Table 1
Allowable lower bounds and upper bounds for differentτ-.
τ
-
0.3
0.6
1.0
1.2
1.5
1.52
τ
2
1.5970
1.6724
1.8719
1.9717
2.1215
2.1315
τ
1
0.1500
0.3000
0.5000
0.6000
0.7500
0.7600
δ
1.4470
1.3724
1.3719
1.3717
1.3715
1.3715Figure 1
Mean-square exponential stability of the neutral stochastic system with interval time-varying delay.Example 4.2.
Consider the uncertain linear stochastic system (3.44) with
(4.2)A0=[-2.00.00.0-0.9],A1=[-1.00.0-1.0-1.0],B0=B1=0,G=0.2I,Si=Ti=I,i=0,1,2.
According to Corollary 3.4, for τ1=0, the upper bounds on the time delay to guarantee the system is robustly stochastically mean-square exponentially stable are listed in Table 2. At the same time, Table 2 also lists the upper bounds obtained from the criterion in [15].Table 2
Allowable upper bounds ofτ2 for different  τ-.
τ
-
0.3
0.5
0.9
Yan et al. [15]
0.7288
0.5252
0.1489
Corollary3.4
0.9012
0.7498
0.5640Example 4.3.
Consider the uncertain neutral linear stochastic system (3.51) with
(4.3)A0=[0.50.00.00.3],A1=[-1.00.0-1.0-1.0],D=[-0.20.0-0.950.2],B0=0.2I,B1=0.3I.
If Corollary 3.5 in this paper is applied, the maximal admissible delay τM of this example is τM=0.6001.Example 4.4.
Consider the uncertain neutral linear stochastic system (3.65) with
(4.4)A0=[-2.00.00.0-0.9],A1=[-1.00.0-1.0-1.0],D=[c00c],0⩽c<1.
The maximum upper bound τM for this system in case of different cs is listed in Table 3, which shows that the results obtained by the methods proposed in this paper are less conservative than that in [19].Table 3
Allowable upper bounds ofτ2 for different c.
c
0
0.1
0.3
0.5
0.7
0.9
Fridman and Shaked [19]
4.47
3.49
2.06
1.14
0.54
0.13
Corollary3.7
4.47
4.05
3.29
2.58
1.85
0.92


## 5. Conclusion

The mean-square exponential stability for uncertain neutral stochastic system with both discrete and distributed time-varying delays has been investigated in this paper. Sufficient conditions have been established without ignoring any terms in the weak infinitesimal operator of Lyapunov-Krasovskii functional by considering the relationship among the time-varying delay, its upper bounds, and their difference. The usual restriction thatτ˙(t)<1 has been removed by free-weight matrices method. According to the numerical examples, it has been shown that the proposed results improve some existing ones.



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*Source: 593780-2012-10-02.xml*