# On Exponential Stability for a Class of Uncertain Neutral Markovian Jump Systems with Mode-Dependent Delays

Authors: Xinghua Liu; Hongsheng Xi
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The exponential stability of neutral Markovian jump systems with interval mode-dependent time-varying delays, nonlinear perturbations, and partially known transition rates is investigated. A novel augmented stochastic Lyapunov functional is constructed, which employs the improved bounding technique and contains triple-integral terms to reduce conservativeness; then the delay-range-dependent and rate-dependent exponential stability criteria are developed by Lyapunov stability theory, reciprocally convex lemma, and free-weighting matrices. The corresponding results are extended to the uncertain case. Finally, numerical examples are given to illustrate the effectiveness of the proposed methods.


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## Body

## 1. Introduction

Delay differential equations or systems are assuming an increasingly important role in many disciplines like mathematics, science, and engineering. In particular, the stability and stabilization problem for neutral delay differential dynamic systems have received considerable attention during the decades and neutral time-delay systems have been the focus of the research community, which are often encountered in such practical situations as distributed networks, population ecology, processes including steam or heat exchanges [1], and robots in contact with rigid environments [2]. Existing results can be roughly classified into two categories, delay-independent criteria and delay-dependent criteria, where the latter is generally regarded as less conservative. Moreover, since the derivative of the delayed state is involved, it should be pointed out that the stability of neutral time-delay systems is more difficult to tackle, which is identical with singular systems [3, 4]. The stability problem of them is more complicated than that for regular systems because more factors need to be considered. In the past decades, considerable attention has been devoted to the robust delay-independent stability and delay-dependent stability of linear neutral systems, which are mainly obtained based on the Lyapunov-Krasovskii (L-K) method [5–11], and references therein. It should be noted that the delay-partitioning approach is used in [6–8]. Furthermore, when nonlinear perturbations or parameter uncertainties appear in neutral systems, some results on stability analysis have been also presented [12–18]. Various techniques have been proposed in these papers, for example, model transformation techniques, the improved bounding techniques, and matrix decomposition approaches. In particular, He et al. [18] propose a new method for dealing with time-delay systems, which employs free weighting matrices to express the relationships between the terms in the Newton-Leibniz formula and has brought novel results. However, these results have conservativeness to some extent, which exist room for further improvement.In another line, Markovian jump systems (MJSs) have attracted much attention during the past few decades since its first introduction by Krasovskii and Lidskii in 1961, which can be regarded as a special class of hybrid systems with finite operation modes whose structures are subject to random abrupt changes. The system parameters usually jump among finite modes, and the mode switching is governed by a Markov process. MJSs have many applications, such as failure prone manufacturing systems, power systems, solar thermal central receivers, robotic manipulator systems, aircraft control systems, and economic systems. A large number of results on estimation and control problems related to such systems have been reported in the literature; see, for example, [19–25] and references therein for more details. However, these lines of literature about the transition probabilities in the jumping process have been assumed to be completely accessible. The ideal assumption on the transition probabilities inevitably limits the application of the traditional Markovian jump system theory. Actually, the likelihood of obtaining such available knowledge is questionable, and the cost may be very expensive. Thus, it is really significant and meaningful, from control perspectives, to further study more general jump systems with partially known transition rates. Recently, many results on the Markovian jump systems with partially known transition rates are obtained [26–31]. Most of these improved results just require some free matrices or the knowledge of the known elements in transition rate matrix, such as the structures of uncertainties, and some else of the unknown elements need not be considered. It is a great progress on the analysis of Markovian jump systems. However, few of these results are concerned with neutral Markovian jump systems with mode-dependent time-varying delays and perturbations. To the best of the authors' knowledge, neutral Markovian jump systems with mode-dependent time-varying delays and partially known transition rates have not been fully investigated, and it is very challenging, especially when nonlinear perturbations exist. Besides, seeking and proposing less conservative delay-range-dependent criterion for uncertain neutral MJSs with nonlinear perturbations and partially known transition rates to desired performance are still open problems. These facts thus motivate our study.In this paper, the investigated neutral Markovian jump systems are more general than the neutral MJSs with completely known or completely unknown transition rates, which can be viewed as two special cases of the ones tackled here. Specifically, a new augmented stochastic Lyapunov functional containing triple-integral terms is constructed by dividing the delay interval into two subintervals, and then the delay-range-dependent and rate-dependent exponential stability criteria are obtained by reciprocally convex lemma and free weighting matrices. We further extend the criteria to the uncertain case. All the obtained results are presented in terms of LMIs that can be solved numerically. The remainder of the paper is organized as follows. Section2 presents the problem and preliminaries. Section 3 gives the main results, which are then verified by numerical examples in Section 4. Section 5 concludes the paper.Notations. The following notations are used throughout the paper. ℝn denotes the n dimensional Euclidean space and ℝm×n is the set of all m×n matrices. X<Y(X>Y), where X and Y are both symmetric matrices, means that X-Y is negative (positive) definite. I is the identity matrix with proper dimensions. For a symmetric block matrix, we use * to denote the terms introduced by symmetry. ℰ stands for the mathematical expectation, ∥v∥ is the Euclidean norm of vector v, ∥v∥=(vTv)1/2, while ∥A∥ is spectral norm of matrix A, ∥A∥=[λmax(ATA)]1/2. λmax(min)(A) is the eigenvalue of matrix A with maximum(minimum) real part. Matrices, if their dimensions are not explicitly stated, are assumed to have compatible dimensions for algebraic operations.


## 2. Problem Statement and Preliminaries

Given a probability space{Ω,ℱ,P} where Ω is the sample space, ℱ is the algebra of events and P is the probability measure defined on ℱ. {rt,t≥0} is a homogeneous, finite-state Markovian process with right continuous trajectories taking values in a finite set S={1,2,3,…,N}, with the mode transition probability matrix being
(1)P(rt+Δt=j∣rt=i)={πijΔt+o(Δt),i≠j,1+πiiΔt+o(Δt),i=j,
where Δt>0, limΔt→0(o(Δt)/Δt)=0, πij≥0(i,j∈S,i≠j) is the transition rate from mode i to j and for any state or mode i∈S; it satisfies
(2)πii=-∑j=1,j≠iN‍πij.
Since the transition rates of the Markov chain are partially known in this paper, some elements in matrix Π=[πij]N×N are inaccessible. For instance, the system with five operation modes, the jump rates matrix Π may be viewed as
(3)[?π12??π15π21??π24???π33π34??π42???π51???π55],
where ? represents the unknown element. For notation clarity, we denote 𝒮i=𝒮ki∪𝒮uki,  for all i∈S and
(4)𝒮ki≜{j:πijisknownforj∈S},𝒮uki≜{j:πijisunknownforj∈S}.
If 𝒮ki≠∅, it is further described as
(5)𝒮ki={k1i,k2i,…,kmi},1≤m≤N,
where kji, (j=1,2,…,m) represent the jth known element of the set 𝒮ki in the ith row of the transition rate matrix Π. Furthermore, let πim and πiM be the lower and upper bound for the diagonal elements of the jump rates matrix Π.In this paper, the following uncertain neutral Markovian jump systems with mode-dependent interval time-varying delays, nonlinear perturbations, and partially known transition rates over the space{Ω,ℱ,P} are considered:
(6)x˙(t)-C(t,rt)x˙(t-τ(t,rt))=A(t,rt)x(t)+B(t,rt)x(t-d(t,rt))+D(t,rt)f1(x(t),t)+E(t,rt)f2(x(t-d(t,rt)),t)+F(t,rt)f3(x˙(t-τ(t,rt)),t),(7)x(s)=φ(s),rs=r0,s∈[-ς,0],
where x(t)∈ℝn is the system state and τ(t,rt) is mode-dependent interval time-varying neutral delay which satisfies 0≤τi(t)≤τi¯, τ˙i(t)≤νi when rt=i∈S. The mode-dependent interval time-varying retarded delay d(t,rt) is assumed that
(8)0≤d1i≤di(t)≤d2i,maxi∈S{d1i}≤minj∈S{d2j},d˙i(t)≤μi,ifrt=i∈S,
where τi¯, νi≥0, d1i, d2i, μi≥0, and ς=maxi∈S{τi¯,d2i} are real constant scalars. The initial condition φ(s) is a continuously differentiable vector-valued function. f1(x(t),t)∈ℝn, f2(x(t-d(t,rt)),t)∈ℝn, and f3(x˙(t-τ(t,rt)),t)∈ℝn are unknown nonlinear perturbations which are, with respect to the current state x(t), the delayed state x(t-d(t,rt)) and the neutral delay state x˙(t-τ(t,rt)), respectively. For all t and rt=i∈S, they are assumed to be bounded in magnitude as
(9)∥f1(x(t),t)∥≤α∥x(t)∥,∥f2(x(t-di(t)),t)∥≤β∥x(t-di(t))∥,∥f3(x˙(t-τi(t)),t)∥≤γ∥x˙(t-τi(t))∥,
where α≥0, β≥0, and γ≥0 are given constants, for simplicity, f1≜f1(x(t),t), f2≜f2(x(t-di(t)),t), and f3≜f3(x˙(t-τi(t)),t).For notational simplicity further, wherert=i∈S, the parametric matrices A(t,rt)∈ℝn×n, B(t,rt)∈ℝn×n, C(t,rt)∈ℝn×n, D(t,rt)∈ℝn×n, E(t,rt)∈ℝn×n, and F(t,rt)∈ℝn×n are denoted by Ai(t), Bi(t), Ci(t), Di(t), Ei(t), and Fi(t), which can be described as
(10)Ai(t)=Ai+ΔAi(t),Bi(t)=Bi+ΔBi(t),Ci(t)=Ci,Di(t)=Di+ΔDi(t),Ei(t)=Ei+ΔEi(t),Fi(t)=Fi+ΔFi(t),
where Ai, Bi, Ci, Di, Ei, and Fi are known constant matrices with appropriate dimensions. ΔAi(t)∈ℝn×n, ΔBi(t)∈ℝn×n, ΔDi(t)∈ℝn×n, ΔEi(t)∈ℝn×n, and ΔFi(t)∈ℝn×n are uncertainties. The parametric matrix ∥Ci∥<1 and the admissible parametric uncertainties satisfy the following condition:
(11)[ΔAi(t)ΔBi(t)ΔDi(t)ΔEi(t)ΔFi(t)]=LiHi(t)[NAiNBiNDiNEiNFi],
where Li, NAi, NBi, NDi, NEi, and NFi are known mode-dependent constant matrices with appropriate dimensions and Hi(t) is an unknown and time-varying matrix satisfying
(12)HiT(t)Hi(t)≤I,∀t.Particularly, the following nominal systems can be obtained forHi(t)=0:
(13)x˙(t)-Cix˙(t-τi(t))=Aix(t)+Bix(t-di(t))+Dif1+Eif2+Fif3.Before proceeding with the main results, we present the following assumptions, definitions, and lemmas.Assumption 1.
System matricesAi, (for all i∈S), are Hurwitz and all the eigenvalues have negative real parts for each mode. Li, (for all i∈S), is full rank in row.Assumption 2.
The Markov process is irreducible and the system modert is available at time t.Definition 3 (see [32]).
Define operator𝔇:C([-ς,0],ℝn)→ℝn as 𝔇(xt)=x(t)-Cx(t-τ). 𝔇 is said to be stable if the homogeneous difference equation
(14)𝔇(xt)=0,t≥0,x0=ψ∈{ϕ∈C([-ς,0],ℝn):𝔇ϕ=0}is uniformly asymptotically stable. In this paper, that is,∥Ci∥+γ<1.Definition 4 (see [33]).
The system in (6) is exponentially stable with a decay rate ε for all rt=i∈S, if there exist scalars ε>0 and κ≥1 such that for all x(t),
(15)∥x(t)∥≤κexp{-ε(t-t0)}∥xt0∥ς,
where ε is the exponential decay rate, ∥*∥ denotes the Euclidean norm, and
(16)∥xt0∥ς=supθ∈[-ς,0]{∥x(t0+θ)∥,∥x˙(t0+θ)∥}=sups∈[-ς,t0]{∥φ(s)∥,∥φ˙(s)∥}.Definition 5 (see [34]).
Define the stochastic Lyapunov-Krasovskii function of system (6) as V(x(t),rt=i,t>0)=V(xt,i,t), where its infinitesimal generator is defined as
(17)ΓV(x(t),i,t)=limΔt→01Δt[ℰ{V(x(t+Δt),rt+Δt,t+Δt)∣x(t)=x,=limΔt→01Δt(hrt=i}-V(x(t),i,t)]=∂∂tV(x(t),i,t)+∂∂xV(x(t),i,t)x˙(t)+∑j=1N‍πijV(x(t),j,t).Lemma 6 (see [35]).
Given constant matricesΩ1, Ω2, and Ω3, where Ω1=Ω1T and Ω2=Ω2T>0. Ω1+Ω3TΩ2-1Ω3<0 if and only if
(18)[Ω1Ω3T*-Ω2]<0,or[-Ω2Ω3T*Ω1]<0.Lemma 7.
For any constant matrixQ=QT>0, continuous functions 0≤h1(t)≤h2(t), constant scalars 0≤τ1<τ2, and constant ε>0 such that the following integrations are well defined,(a)
(19)exp{2εh2(t)}-exp{2εh1(t)}2ε×∫t-h2(t)t-h1(t)‍exp{2ε(s-t)}xT(s)Qx(s)ds≥[∫t-h2(t)t-h1(t)‍xT(s)ds]Q[∫t-h2(t)t-h1(t)‍x(s)ds].(b)
(20)τ22-τ122∫-τ2-τ1‍∫t+θt‍exp{2ε(s-t)}xT(s)Qx(s)dsdθ≥exp{-2ετ2}[∫-τ2-τ1‍∫t+θt‍xT(s)dsdθ]×Q[∫-τ2-τ1‍∫t+θt‍x(s)dsdθ].Proof.
(a) Is directly obtained from [36]. In addition, from -τ2≤θ≤-τ1 and t+θ≤s≤t, it is held that -τ2≤θ≤s-t≤0. Then
(21)∫-τ2-τ1‍∫t+θt‍exp{2ε(s-t)}xT(s)Qx(s)dsdθ≥exp{-2ετ2}∫-τ2-τ1‍∫t+θt‍xT(s)Qx(s)dsdθ.
(b) Is thus true by [37].Lemma 8 (see [38]).
For functionsλ1(t), λ2(t)∈[0,1], λ1(t)+λ2(t)=1, and η1=0 with λ1(t)=0 and η2=0 with λ2(t)=0, matrices P>0, Q>0, then there exists matrix T such that
(22)[PTTTQ]>0
and the following inequality holds:
(23)1λ1(t)η1Pη1T+1λ2(t)η2Qη2T≥[η1η2][PTTTQ][η1Tη2T].Lemma 9 (see [39]).
For given matricesQ=QT, M, and N with appropriate dimensions,
(24)Q+MF(t)N+NTFT(t)MT<0
for all F(t) satisfying FT(t)F(t)≤I, if and only if there exists a scalar δ>0, such that
(25)Q+δ-1MMT+δNNT<0.


## 3. Main Results

This section will state the exponential stability analysis for neutral Markovian jump systems with mode-dependent interval time-varying delays, nonlinear perturbations, and partially known transition rates. With creative Lyapunov functional and novel matrix inequalities analysis, delay-range-dependent and rate-dependent exponential stability conditions are presented.


### 3.1. Exponential Stability for the Nominal Systems

Theorem 10.
For given scalarsπim, πiM, α, β, γ, ε, τi¯, νi, d1i, d2i, μi and constant scalar dmi satisfying d1i<dmi<d2i, the systems described by (13) with partially known transition rates are exponentially stable with decay rate ε and κ=λ¯/λ_ if ∥Ci∥+γ<1 and there exist symmetric positive matrices Pi>0, Q1i>0, Q2i>0, R1i>0, T1i>0, (i∈S), Qj>0, (j=3,4), Rk>0, Tl>0, (k,l=2,3,4), Um>0, Vn>0, Ws>0, (m,n,s=1,2,3,4) and matrices M1, M2, N1, N2, N3, N4  for any scalars ε1, ε2, ε3, any symmetric matrices Xi, Y1i, Y2i, Z1i, Z2i, (i∈S) and any matrices Jk, (k=1,2,…,24) with appropriate dimensions, such that the following linear matrix inequalities hold.
Wheni∈𝒮ki(26)∑j∈𝒮ki,j≠i‍πij(Q1j-Y1i)-πiiY1i≤0,∑j∈𝒮ki,j≠i‍πij(Q2j-Y2i)-πiiY2i≤0,∑j∈𝒮ki,j≠i‍πij(R1j-Z1i)-πiiZ1i≤0,∑j∈𝒮ki,j≠i‍πij(T1j-Z2i)-πiiZ2i≤0,Pj-Xi≤0,j∈𝒮uki,Q1j-Y1i≤0,j∈𝒮uki,Q2j-Y2i≤0,j∈𝒮uki,R1j-Z1i≤0,j∈𝒮uki,T1j-Z2i≤0,j∈𝒮uki.
When i∈𝒮uki(27)∑j∈𝒮ki‍πij(Q1j-Y1i)≤0;∑j∈𝒮ki‍πij(Q2j-Y2i)≤0,(28)∑j∈𝒮ki‍πij(R1j-Z1i)≤0;∑j∈𝒮ki‍πij(T1j-Z2i)≤0,(29)Pj-Xi≤0,j∈𝒮uki,j≠i,Pj-Xi≥0,j∈𝒮uki,j=i,(30)-∑j∈𝒮uki,j≠i‍πiM(Q1j-Y1i)-πimY1i≤0,-∑j∈𝒮uki,j≠i‍πiM(Q2j-Y2i)-πimY2i≤0,(31)-∑j∈𝒮uki,j≠i‍πiM(R1j-Z1i)-πimZ1i≤0,-∑j∈𝒮uki,j≠i‍πiM(T1j-Z2i)-πimZ2i≤0,(32)[U1M1M1TU1]>0,[V1M2M2TV1]>0,(33)(i)[U3N1N1TU3]>0,[V3N2N2TV3]>0,(ii)Ωi0ε+Ωi1ε<0,(34)(iii)[U4N3N3TU4]>0,[V4N4N4TV4]>0,(iv)Ωi0ε+Ωi2ε<0,
where
(35)Ωi0ε=∑m=124‍emΥmemT+ℒ(Ξ)-2εd1iexp{2εd1i}-1(e1-e5)V2(e1T-e5T)-exp{-2ετi¯}(τi¯e1-e21)W1(τi¯e1T-e21T)-exp{-2εd1i}(d1ie1-e11)W2(d1ie1T-e11T)-exp{-2εdmi}(ϱ1ie1-e14)W3(ϱ1ie1T-e14T)-exp{-2εd2i}(ϱ2ie1-e15)W4(ϱ2ie1T-e15T)-2ετi¯exp{2ετi¯}-1[e20e21-e20]×[U1M1M1TU1][e20Te21T-e20T]-2ετi¯exp{2ετi¯}-1[e1-e18e18-e16]×[V1M2M2TV1][e1T-e18Te18T-e16T],
where ℒ is a linear operator on ℝn×n by
(36)ℒ(G)=G+GT,∀G∈ℝn×n,Ξ=-e1J1e2T+e1(PiBi+J1Bi)e3T+e1(PiCi)e19T+∑m=224‍e1(AiTJmT)emT+e1Pi(Die22T+Eie23T+Fie24T)+e2(J2Bi)e3T-∑m=324‍e2(JmT)emT+∑m=424‍e3(BiTJmT)emT+∑m=118‍em(JmCi)e19T+∑m=2024‍e19(CiTJmT)emT+∑m=121‍emJm(Die22T+Eie23T+Fie24T)+e22J22(Eie23T+Fie24T)+∑m=2324‍e22(DiTJmT)emT+e23(J23Fi)e24T+e23(EiTJ24T)e24T,Υ1=AiTPi+PiAi+∑j∈𝒮ki‍πij(Pj-Xi)+2εPi+Q1i+Q3+R2+τi¯2U1+d1i2U2+ϱ1i2U3+ϱ2i2U4+ε1α2I+J1Ai+AiTJ1T,Υ2=Q2i+Q4+T2+τi¯2V1+d1i2V2+ϱ1i2V3+ϱ2i2V4+τi¯44W1+d1i44W2+ϱ3i2W3+ϱ4i2W4-J2-J2T,Υ3=ϖ(ε,μi)R1i+ε2β2I+J3Bi+BiTJ3T,Υ4=ϖ(ε,μi)T1i,Υ5=exp{-2εd1i}(R1i+R3-R2),Υ6=exp{-2εdmi}(R4-R3),Υ7=-exp{-2εd2i}R4,Υ8=exp{-2εd1i}(T1i+T3-T2),Υ9=exp{-2εdmi}(T4-T3),Υ10=-exp{-2εd2i}T4,Υ11=-2εd1iexp{2εd1i}-1U2,Υ16=-exp{-2ετi¯}Q3,Υ17=-exp{-2ετi¯}Q4,Υ18=-(1-νi)exp{-2ετi¯}Q1i,Υ19=-(1-νi)exp{-2ετi¯}Q2i+ε3γ2I+J19Ci+CiTJ19T,Υ22=-ε1I+J22Di+DiTJ22T,Υ23=-ε2I+J23Ei+EiTJ23T,Υ24=-ε3I+J24Fi+FiTJ24T,Υm=0,(m=12,13,14,15,20,21),Ωi1ε=-2εϱ2iexp{2εd2i}-exp{2εdmi}e15U4e15T-2εϱ2iexp{2εd2i}-exp{2εdmi}×(e6-e7)V4(e6T-e7T)-2εϱ1iexp{2εdmi}-exp{2εd1i}[e12e14-e12]×[U3N1N1TU3][e12Te14T-e12T]-2εϱ1iexp{2εdmi}-exp{2εd1i}[e5-e3e3-e6]×[V3N2N2TV3][e5T-e3Te3T-e6T],Ωi2ε=-2εϱ1iexp{2εdmi}-exp{2εd1i}e14U3e14T-2εϱ1iexp{2εdmi}-exp{2εd1i}×(e5-e6)V3(e5T-e6T)-2εϱ2iexp{2εd2i}-exp{2εdmi}[e13e15-e13]×[U4N3N3TU4][e13Te15T-e13T]-2εϱ2iexp{2εd2i}-exp{2εdmi}[e3-e7e6-e3]×[V4N4N4TV4][e3T-e7Te6T-e3T],
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1
2
(
d
2
i
2
-
d
m
i
2
)
.Proof.
Construct the following stochastic Lyapunov functional:(38)V(x(t),i,t)=∑k=17‍Vk(xt,i),
where
(39)V1(xt,i)=xT(t)Pix(t),(40)V2(xt,i)=∫t-τi(t)t‍exp{2ε(s-t)}xT(s)Q1ix(s)ds+∫t-τi(t)t‍exp{2ε(s-t)}x˙T(s)Q2ix˙(s)ds+∫t-τi¯t‍exp{2ε(s-t)}xT(s)Q3x(s)ds+∫t-τi¯t‍exp{2ε(s-t)}x˙T(s)Q4x˙(s)ds,(41)V3(xt,i)=∫t-di(t)t-d1i‍exp{2ε(s-t)}xT(s)R1ix(s)ds+∫t-d1it‍exp{2ε(s-t)}xT(s)R2x(s)ds+∫t-dmit-d1i‍exp{2ε(s-t)}xT(s)R3x(s)ds+∫t-d2it-dmi‍exp{2ε(s-t)}xT(s)R4x(s)ds,(42)V4(xt,i)=∫t-di(t)t-d1i‍exp{2ε(s-t)}x˙T(s)T1ix˙(s)ds+∫t-d1it‍exp{2ε(s-t)}x˙T(s)T2x˙(s)ds+∫t-dmit-d1i‍exp{2ε(s-t)}x˙T(s)T3x˙(s)ds+∫t-d2it-dmi‍exp{2ε(s-t)}x˙T(s)T4x˙(s)ds,(43)V5(xt,i)=∫-τi¯0‍∫t+θt‍τi¯exp{2ε(s-t)}xT(s)U1x(s)dsdθ+∫-d1i0‍∫t+θt‍d1iexp{2ε(s-t)}xThhhhhhhhhhhhhh×(s)U2x(s)dsdθ+∫-dmi-d1i‍∫t+θt‍ϱ1iexp{2ε(s-t)}xThhhhhhhhhhhhhh×(s)U3x(s)dsdθ+∫-d2i-dmi‍∫t+θt‍ϱ2iexp{2ε(s-t)}xThhhhhhhhhhhhhh×(s)U4x(s)dsdθ,(44)V6(xt,i)=∫-τi¯0‍∫t+θt‍τi¯exp{2ε(s-t)}x˙T(s)V1x˙(s)dsdθ+∫-d1i0‍∫t+θt‍d1iexp{2ε(s-t)}x˙Thhhhhhhhhhhhhh×(s)V2x˙(s)dsdθ+∫-dmi-d1i‍∫t+θt‍ϱ1iexp{2ε(s-t)}x˙Thhhhhhhhhhhhhhh×(s)V3x˙(s)dsdθ+∫-d2i-dmi‍∫t+θt‍ϱ2iexp{2ε(s-t)}x˙Thhhhhhhhhhhhhhh×(s)V4x˙(s)dsdθ,(45)V7(xt,i)=∫-τi¯0‍∫θ0‍∫t+λt‍τi¯22exp{2ε(s-t)}x˙Thhhhhhhhhh×(s)W1x˙(s)dsdλdθ+∫-d1i0‍∫θ0‍∫t+λt‍d1i22exp{2ε(s-t)}hhhhhhhhhh×x˙T(s)W2x˙(s)dsdλdθ+∫-dmi-d1i‍∫θ0‍∫t+λt‍ϱ3iexp{2ε(s-t)}hhhhhhhhhhhh×x˙T(s)W3x˙(s)dsdλdθ+∫-d2i-dmi‍∫θ0‍∫t+λt‍ϱ4iexp{2ε(s-t)}hhhhhhhhhhhh×x˙T(s)W4x˙(s)dsdλdθ.Remark  11. It should be pointed out that the proposed stochastic augmented Lyapunov functional (38) contains some triple-integral terms, which has not been used in of the existing literature in the same context before. Compared with the existing ones, [37] has shown that such triple-integral terms are very effective in the reduction of conservativeness.
TakingΓ as its infinitesimal generator along the trajectory of system (13), we obtain the following from Definition 5 and (38)–(45):
(46)ΓV(x(t),i,t)=∑k=17‍ΓVk(xt,i),ΓV1(xt,i) can be easily obtained by the following equation:
(47)ΓV1(xt,i)=2[xT(t)AiT+xT(t-di(t))BiT=2Hh+x˙T(t-τi(t))CiT+f1TDiT=2Hh+f2TEiT+f3TFiT]×Pix(t)+xT(t)[∑j∈𝒮ki‍πijPj+∑j∈𝒮uki‍πijPj]x(t)+2εxT(t)Pix(t)-2εV1(xt,i).
With regard toΓV2(xt,i), the detailed procedures are given as follows. 									(48)DefineV2(xt,i)=V21(xt,i)+V22(xt,i)+V23(xt,i)+V24(xt,i),
where
(49)V21(xt,i)=∫t-τi(t)t‍exp{2ε(s-t)}xT(s)Q1ix(s)ds,V22(xt,i)=∫t-τi(t)t‍exp{2ε(s-t)}x˙T(s)Q2ix˙(s)ds,V23(xt,i)=∫t-τi¯t‍exp{2ε(s-t)}xT(s)Q3x(s)ds,V24(xt,i)=∫t-τi¯t‍exp{2ε(s-t)}x˙T(s)Q4x˙(s)ds.
By the infinitesimal generatorΓ, we obtain
(50)ΓV21(xt,i)=exp{-2εt}×Γ(∫t-τi(t)t‍exp{2εs}xT(s)Q1ix(s)ds)-2εV21(xt,i),
where
(51)Γ(∫t-τi(t)t‍exp{2εs}xT(s)Q1ix(s)ds)=limΔ→0+1Δ{ℰ[∫t+Δ-τ(t+Δ,rt+Δ)t+Δ‍exp{2εs}xT(s)Q1(rt+Δ)=limΔ→0+1Δ(hhhhhh×x(s)ds∣x(t),rt=i∫t+Δ-τ(t+Δ,rt+Δ)t+Δ‍]=limΔ→0+1Δ(hhhhhh-∫t-τi(t)t‍exp{2εs}xT(s)Q1ix(s)ds∫t+Δ-τ(t+Δ,rt+Δ)t+Δ‍}=limΔ→0+1Δ{∑j=1N‍P(rt+Δ=j∣rt=i)ggggggggg×[∫t+Δ-τj(t+Δ)t+Δ‍exp{2εs}xT(s)Q1jx(s)ds]ggggggggg-∫t-τi(t)t‍exp{2εs}xT(s)Q1ix(s)ds∑j=1N‍}=limΔ→0+1Δ[(∑j=1N‍πijΔ)∫t+Δ-τj(t+Δ)t+Δ‍exp{2εs}xT(s)jjjjjjjjjjjjjjjjjjjj=limΔ→0+1Δ×Q1jx(s)dsjjjjjjjjjjjjjjjjjjjj+∫t+Δ-τi(t+Δ)t+Δ‍exp{2εs}xT(s)Q1ix(s)dsjjjjjjjjjjjjjjjjjjjj-∫t-τi(t)t‍exp{2εs}xT(s)Q1ix(s)ds∑j=1N‍πijΔ]=∑j=1N‍πij∫t-τj(t)t‍exp{2εs}xT(s)Q1jx(s)ds+exp{2ε(t)}xT(t)Q1ix(t)-(1-τ˙i(t))exp{2ε(t-τi(t))}xT×(t-τi(t))Q1ix(t-τi(t)).
Following the same procedure,ΓV22(xt,i) is also obtained:
(52)ΓV22(xt,i)=x˙T(t)Q2ix˙(t)-(1-τ˙i(t))exp{2ε(t-τi(t))}×x˙T(t-τi(t))Q2ix˙(t-τi(t))-∑j=1N‍πij∫t-τj(t)t‍exp{2εs}x˙T(s)Q2jx˙(s)ds-2εV22(xt,i).
Moreover,ΓV23(xt,i), ΓV24(xt,i) are easily calculated as shown in the following:
(53)ΓV23(xt,i)=xT(t)Q3x(t)-exp{-2ετi¯}×xT(t-τi¯)Q3x(t-τi¯)-2εV23(xt,i),ΓV24(xt,i)=x˙T(t)Q4x˙(t)-exp{-2ετi¯}×x˙T(t-τi¯)Q4x˙(t-τi¯)-2εV24(xt,i).
According to (50), (51), (52), and (53), we can easily obtain the following (54):
(54)ΓV2(xt,i)=xT(t)[Q1i+Q3]x(t)+x˙T(t)[Q2i+Q4]x˙(t)-2εV2(xt,i)-(1-τ˙i(t))exp{-2ετi(t)}xT(t-τi(t))Q1ix×(t-τi(t))-(1-τ˙i(t))exp{-2ετi(t)}x˙T×(t-τi(t))Q2ix˙(t-τi(t))-exp{-2ετi¯}xT(t-τi¯)Q3x(t-τi¯)-exp{-2ετi¯}x˙T(t-τi¯)Q4x˙(t-τi¯)+∑j∈S‍πij∫t-τj(t)t‍exp{2ε(s-t)}+∑j∈S‍πij×[xT(s)Q1jx(s)+x˙T(s)Q2jx˙(s)]ds≤xT(t)[Q1i+Q3]x(t)+x˙T(t)[Q2i+Q4]x˙(t)-2εV2(xt,i)-(1-τ˙i(t))exp{-2ετi(t)}xT(t-τi(t))Q1ix×(t-τi(t))-(1-τ˙i(t))exp{-2ετi(t)}x˙T×(t-τi(t))Q2ix˙(t-τi(t))-exp{-2ετi¯}xT(t-τi¯)Q3x(t-τi¯)-exp{-2ετi¯}x˙T(t-τi¯)Q4x˙(t-τi¯)+∑j≠i‍πij∫t-τj¯t‍exp{2ε(s-t)}+∑j≠i‍πij×[xT(s)Q1jx(s)+x˙T(s)Q2jx˙(s)]ds.
Then, in the same method,ΓV3(xt,i), ΓV4(xt,i) can be calculated and the results are given by the following, respectively:
(55)ΓV3(xt,i)=xT(t)R2x(t)+exp{-2εd1i}xT×(t-d1i)[R1i+R3-R2]x(t-d1i)+exp{-2εdmi}xT(t-dmi)[R4-R3]x(t-dmi)-exp{-2εd2i}xT(t-d2i)R4x(t-d2i)-2εV3(xt,i)-(1-d˙i(t))×exp{-2εdi(t)}xT(t-di(t))R1ix(t-di(t))+∑j∈S‍πij∫t-dj(t)t-d1i‍exp{2ε(s-t)}xT(s)R1jx(s)ds≤xT(t)R2x(t)+exp{-2εd1i}xT(t-d1i)×[R1i+R3-R2]x(t-d1i)+exp{-2εdmi}xT(t-dmi)[R4-R3]x×(t-dmi)-exp{-2εd2i}xT×(t-d2i)R4x(t-d2i)-2εV3(xt,i)-(1-d˙i(t))exp{-2εdi(t)}xT×(t-di(t))R1ix(t-di(t))+∑j≠i‍πij∫t-d2jt-d1i‍exp{2ε(s-t)}xT(s)R1jx(s)ds,(56)ΓV4(xt,i)=x˙T(t)T2x˙(t)+exp{-2εd1i}x˙T×(t-d1i)[T1i+T3-T2]x˙(t-d1i)+exp{-2εdmi}x˙T(t-dmi)×[T4-T3]x˙(t-dmi)-exp{-2εd2i}x˙T(t-d2i)T4x˙(t-d2i)-2εV4(xt,i)-(1-d˙i(t))exp{-2εdi(t)}x˙T×(t-di(t))T1ix˙(t-di(t))+∑j∈S‍πij∫t-dj(t)t-d1i‍exp{2ε(s-t)}x˙T(s)T1jx˙(s)ds≤x˙T(t)T2x˙(t)+exp{-2εd1i}x˙T(t-d1i)×[T1i+T3-T2]x˙(t-d1i)+exp{-2εdmi}x˙T(t-dmi)[T4-T3]x˙(t-dmi)-exp{-2εd2i}x˙T(t-d2i)T4x˙(t-d2i)-2εV4(xt,i)-(1-d˙i(t))×exp{-2εdi(t)}x˙T(t-di(t))T1ix˙(t-di(t))+∑j≠i‍πij∫t-d2jt-d1i‍exp{2ε(s-t)}x˙T(s)T1jx˙(s)ds.
Moreover,ΓV5(xt,i), ΓV6(xt,i), and ΓV7(xt,i) can be directly obtained as follows:
(57)ΓV5(xt,i)=xT(t)[τi¯2U1+d1i2U2+ϱ1i2U3+ϱ2i2U4]x(t)-2εV5(xt,i)-∫t-τi¯t‍τi¯exp{2ε(s-t)}xT(s)U1x(s)ds-∫t-d1it‍d1iexp{2ε(s-t)}xT(s)U2x(s)ds-∫t-dmit-d1i‍ϱ1iexp{2ε(s-t)}xT(s)U3x(s)ds-∫t-d2it-dmi‍ϱ2iexp{2ε(s-t)}xT(s)U4x(s)ds,ΓV6(xt,i)=x˙T(t)[τi¯2V1+d1i2V2+ϱ1i2V3+ϱ2i2V4]x˙(t)-2εV6(xt,i)-∫t-τi¯t‍τi¯exp{2ε(s-t)}x˙T(s)V1x˙(s)ds-∫t-d1it‍d1iexp{2ε(s-t)}x˙T(s)V2x˙(s)ds-∫t-dmit-d1i‍ϱ1iexp{2ε(s-t)}x˙T(s)V3x˙(s)ds-∫t-d2it-dmi‍ϱ2iexp{2ε(s-t)}x˙T(s)V4x˙(s)ds,ΓV7(xt,i)=x˙T(t)[τi¯44W1+d1i44W2+ϱ3i2W3+ϱ4i2W4]x˙(t)-2εV7(xt,i)-∫-τi¯0‍∫t+θt‍τi¯22exp{2ε(s-t)}x˙T(s)W1x˙(s)dsdθ-∫-d1i0‍∫t+θt‍d1i22exp{2ε(s-t)}x˙T(s)W2x˙(s)dsdθ-∫-dmi-d1i‍∫t+θt‍ϱ3iexp{2ε(s-t)}x˙T(s)W3x˙(s)dsdθ-∫-d2i-dmi‍∫t+θt‍ϱ4iexp{2ε(s-t)}x˙T(s)W4x˙(s)dsdθ.
Define(58)ξ(t)=col{∫t-τi(t)t‍x(t)x˙(t)x(t-di(t))x˙(t-di(t))x(t-d1i)x(t-dmi)x(t-d2i)x˙(t-d1i)x˙(t-dmi)x˙(t-d2i)∫t-d1it‍x(s)ds∫t-di(t)t-d1i‍x(s)ds∫t-d2it-di(t)‍x(s)ds∫t-dmit-d1i‍x(s)ds∫t-d2it-dmi‍x(s)dsx(t-τi¯)x˙(t-τi¯)x(t-τi(t))x˙(t-τi(t))∫t-τi(t)t‍x(s)ds∫t-τi¯t‍x(s)dsf1f2f3}.
Then, there exist matricesJ=col{Jk,(k=1,2,…,24)} with appropriate dimensions, such that the following equality holds according to (13):
(59)2ξT(t)J[-x˙(t)+Aix(t)+Bix(t-di(t))hhhHhhhj+Cix˙(t-τi(t))+Dif1+Eif2+Fif3]=0.
Due to∑j=1N‍πij=0, the following zero equations hold for arbitrary matrices Xi=XiT, Y1i=Y1iT, Y2i=Y2iT, Z1i=Z1iT, Z2i=Z2iT,i∈S; that is,
(60)-xT(t)[∑j∈𝒮ki‍πijXi+∑j∈𝒮uki‍πijXi]x(t)=0,-∫t-τj¯t‍exp{2ε(s-t)}xT(s)×[∑j∈𝒮ki‍πijY1i+∑j∈𝒮uki‍πijY1i]x(s)ds=0,-∫t-τj¯t‍exp{2ε(s-t)}x˙T(s)×[∑j∈𝒮ki‍πijY2i+∑j∈𝒮uki‍πijY2i]x˙(s)ds=0,-∫t-d2jt-d1i‍exp{2ε(s-t)}xT(s)×[∑j∈𝒮ki‍πijZ1i+∑j∈𝒮uki‍πijZ1i]x(s)ds=0,-∫t-d2jt-d1i‍exp{2ε(s-t)}x˙T(s)×[∑j∈𝒮ki‍πijZ2i+∑j∈𝒮uki‍πijZ2i]x˙(s)ds=0.
In view of (9), the following inequalities hold for any scalars ε1>0, ε2>0, and ε3>0:
(61)ε1[α2xT(t)x(t)-f1T(x(t),t)f1(x(t),t)]≥0,ε2[β2xT(t-di(t))x(t-di(t))-f2T(x(t-di(t)),t)f2(x(t-di(t)),t)]≥0,ε3[γ2x˙T(t-τi(t))x˙(t-τi(t))-f3T(x˙(t-τi(t)),t)f3(x˙(t-τi(t)),t)]≥0.
From (46) and (61), we have
(62)ΓV(x(t),i,t)≤∑k=17‍ΓVk(xt,i)+ε1[α2xT(t)x(t)-f1Tf1]+ε2[β2xT(t-di(t))x(t-di(t))-f2Tf2]+ε3[γ2x˙T(t-τi(t))x˙(t-τi(t))-f3Tf3].
Since it is easy to see-(1-τ˙i(t))exp{-2ετi(t)}≤-(1-νi)exp{-2ετi¯}, then from (8) we also obtain that
(63)-(1-d˙i(t))exp{-2εdi(t)}≤ϖ(ε,μi),
where ϖ(ε,μi) is defined in Theorem 10. Notice (a) of Lemma 7, then
(64)-∫t-d1it‍d1iexp{2ε(s-t)}xT(s)U2x(s)ds≤-2εd1iexp{2εd1i}-1ξT(t)e11U2e11Tξ(t),-∫t-d1it‍d1iexp{2ε(s-t)}x˙T(s)V2x˙(s)ds≤-2εd1iexp{2εd1i}-1ξT(t)(e1-e5)V2(e1T-e5T)ξ(t).
Notice (b) of Lemma7, then
(65)-∫-τi¯0‍∫t+θt‍τi¯22exp{2ε(s-t)}x˙T(s)W1x˙(s)dsdθ≤-exp{-2ετi¯}ξT(t)(τi¯e1-e21)×W1(τi¯e1T-e21T)ξ(t),-∫-d1i0‍∫t+θt‍d1i22exp{2ε(s-t)}x˙T(s)W2x˙(s)dsdθ≤-exp{-2εd1i}ξT(t)(d1ie1-e11)×W2(d1ie1T-e11T)ξ(t),-∫-dmi-d1i‍∫t+θt‍ϱ3iexp{2ε(s-t)}x˙T(s)W3x˙(s)dsdθ≤-exp{-2εdmi}ξT(t)(ϱ1ie1-e14)×W3(ϱ1ie1T-e14T)ξ(t),-∫-d2i-dmi‍∫t+θt‍ϱ4iexp{2ε(s-t)}x˙T(s)W4x˙(s)dsdθ≤-exp{-2εd2i}ξT(t)(ϱ2ie1-e15)×W4(ϱ2ie1T-e15T)ξ(t).
Fordi(t)∈[d1i,dmi], the following is held from (a) of Lemma 7:
(66)-∫t-dmit-d1i‍ϱ1iexp{2ε(s-t)}x˙T(s)V3x˙(s)ds=-{∫t-di(t)t-d1i‍+∫t-dmit-di(t)‍}ϱ1iexp{2ε(s-t)}×x˙T(s)V3x˙(s)ds≤-2εϱ1iexp{2εdi(t)}-exp{2εd1i}×[∫t-di(t)t-d1i‍x˙T(s)ds]V3[∫t-di(t)t-d1i‍x˙(s)ds]-2εϱ1iexp{2εdmi}-exp{2εd(t)}×[∫t-dmit-di(t)‍x˙T(s)ds]V3[∫t-dmit-di(t)‍x˙(s)ds]=-2εϱ1iexp{2εdmi}-exp{2εd1i}ξT(t)×[1λ1i(t)(e5-e3)V3(e5T-e3T)h+1λ2i(t)(e3-e6)V3(e3T-e6T)]ξ(t),
where
(67)λ1i(t)=exp{2εdi(t)}-exp{2εd1i}exp{2εdmi}-exp{2εd1i},λ2i(t)=exp{2εdmi}-exp{2εdi(t)}exp{2εdmi}-exp{2εd1i}.
By Lemma8, there exists matrix N2 with appropriate dimensions such that
(68)-∫t-dmit-d1i‍ϱ1iexp{2ε(s-t)}x˙T(s)V3x˙(s)ds≤-2εϱ1iexp{2εdmi}-exp{2εd1i}ξT(t)×[e5-e3e3-e6][V3N2N2TV3][e5T-e3Te3T-e6T]ξ(t),[V3N2N2TV3]>0.
Similarly, considering-∫t-dmit-d1i‍ϱ1iexp{2ε(s-t)}xT(s)U3x(s)ds and following the same procedure, there exists matrix N1 with appropriate dimensions such that
(69)-∫t-dmit-d1i‍ϱ1iexp{2ε(s-t)}xT(s)U3x(s)ds≤-2εϱ1iexp{2εdmi}-exp{2εd1i}ξT(t)×[e12e14-e12][U3N1N1TU3][e12Te14T-e12T]ξ(t),[U3N1N1TU3]>0.
Forτi(t)∈[0,τi¯], with the same matrix inequalities technique, we obtain the following:
(70)-∫t-τi¯t‍τi¯exp{2ε(s-t)}xT(s)U1x(s)ds≤-2ετi¯exp{2ετi¯}-1ξT(t)[e20e21-e20]×[U1M1M1TU1][e20Te21T-e20T]ξ(t),[U1M1M1TU1]>0,-∫t-τi¯t‍τi¯exp{2ε(s-t)}x˙T(s)V1x˙(s)ds≤-2ετi¯exp{2ετi¯}-1ξT(t)[e1-e18e18-e16]×[V1M2M2TV1][e1T-e18Te18T-e16T]ξ(t),[V1M2M2TV1]>0.
Consider-∫t-d2it-dmi‍ϱ2iexp{2ε(s-t)}xT(s)U4x(s)ds and -∫t-d2it-dmi‍ϱ2iexp{2ε(s-t)}x˙T(s)V4x˙(s)ds, which are directly estimated by (a) of Lemma 7; that is,
(71)-∫t-d2it-dmi‍ϱ2iexp{2ε(s-t)}xT(s)U4x(s)ds≤-2εϱ2iexp{2εd2i}-exp{2εdmi}ξT(t)e15U4e15Tξ(t),-∫t-d2it-dmi‍ϱ2iexp{2ε(s-t)}x˙T(s)V4x˙(s)ds≤-2εϱ2iexp{2εd2i}-exp{2εdmi}ξT(t)×(e6-e7)V4(e6T-e7T)ξ(t).
Substituting (47), (54)–(60), and (63)–(71) into (62) we obtain
(72)ΓV(x(t),i,t)+2εV(x(t),i,t)≤ξT(t)(Ωi0ε+Ωi1ε)ξ(t)+℧,
where
(73)℧=∫t-τj¯t‍exp{2ε(s-t)}xT(s){∑j≠i‍πijQ1j-∑j∈𝒮ki‍πijY1ihhhhhhhhhhhhhhhhhhhhhh-∑j∈𝒮uki‍πijY1i}x(s)ds+∫t-τj¯t‍exp{2ε(s-t)}x˙T(s){∑j≠i‍πijQ2j-∑j∈𝒮ki‍πijY2ihhhhhhhhhhhhhhhhhhhhhh-∑j∈𝒮uki‍πijY2i}x˙(s)ds+∫t-d2jt-d1i‍exp{2ε(s-t)}xT(s){∑j≠i‍πijR1j-∑j∈𝒮ki‍πijZ1immmmmmmmmmmmmmm-∑j∈𝒮uki‍πijZ1i}x(s)ds+∫t-d2jt-d1i‍exp{2ε(s-t)}x˙T(s){∑j≠i‍πijT1j-∑j∈𝒮ki‍πijZ2ihhhhhhhhhhhhhhhhhhhhhhh-∑j∈𝒮uki‍πijZ2i}x(s)ds+xT(t){∑j∈𝒮uki‍πij(Pj-Xi)}x(t).
On the other hand, fordi(t)∈[dmi,d2i], the integral terms
(74)-∫t-d2it-dmi‍ϱ2iexp{2ε(s-t)}xT(s)U4x(s)ds,-∫t-d2it-dmi‍ϱ2iexp{2ε(s-t)}x˙T(s)V4x˙(s)ds
are disposed and estimated by Lemma8, and
(75)-∫t-dmit-d1i‍ϱ1iexp{2ε(s-t)}xT(s)U3x(s)ds,-∫t-dmit-d1i‍ϱ1iexp{2ε(s-t)}x˙T(s)V3x˙(s)ds
are directly estimated by (a) of Lemma7. Therefore,
(76)ΓV(x(t),i,t)+2εV(x(t),i,t)≤ξT(t)(Ωi0ε+Ωi2ε)ξ(t)+℧.
With (72) and (76), the following inequality (77) is held for di(t)∈[d1i,d2i] if (26)–(34) are satisfied
(77)ΓV(x(t),i,t)+2εV(x(t),i,t)<0.
From the stochastic Lyapunov functional (38) and (77), it is held that
(78)V(x(t),i,t)≥mini∈S{λmin(Pi)}∥x(t)∥2=λ_∥x(t)∥2,V(x(t),i,t)<exp{-2ε(t-t0)}×V(xt0,rt0,t0).
Moreover, we have(79)V(xt0,rt0,t0)=∑k=16‍Vk(xt0,rt0)≤λ¯∥xt0∥ς2.
Then from (78) and (79), it is readily seen that
(80)∥x(t)∥≤λ¯λ_exp{-ε(t-t0)}∥xt0∥ς,
where κ=λ¯/λ_≥1.
Therefore, by Definition4, the system (13) is exponentially stable with a decay rate ε. This completes the proof.Remark 12.
It is noted that the integral intervals in (79) are enlarged as follows:
(81)∑k=16‍Vk(xt0,rt0)≤∫t0-ςt0‍exp{2ε(s-t0)}xT(s)×{Q1i+Q3+R1i+∑k=24‍Rk}x(s)ds+∫t0-ςt0‍exp{2ε(s-t0)}x˙T(s)×{Q2i+Q4+T1i+∑l=24‍Tl}x˙(s)ds+∫-ς0‍∫t0+θt0‍ςexp{2ε(s-t0)}xT(s)×(∑m=14‍Um)x(s)dsdθ+∫-ς0‍∫t0+θt0‍ςexp{2ε(s-t0)}x˙T(s)×(∑n=14‍Vn)x˙(s)dsdθ+∫-ς0‍∫θ0‍∫t0+λt0‍ς22exp{2ε(s-t0)}x˙T(s)×(∑s=14‍Ws)x˙(s)dsdλdθ+xT(t0)Pix(t0)(79) can be obtained by letting λ¯ be defined as previously mentioned.Remark 13.
In Theorem10, the factors
(82)exp{2εd2i}-exp{2εdi(t)}2ε,exp{2εdi(t)}-exp{2εd1i}2ε
may be enlarged as(exp{2εd2i}-exp{2εd1i})/2ε. This will lead conservative results due to the fact that di(t) cannot achieve d1i and d2i at the same time. While we apply Lemma 7 to these terms, the method by using reciprocally convex lemma [38] can achieve less conservative results. Moreover, for ε>0, the factor (d˙i(t)-1)exp{-2εdi(t)} that appeared in the derivative of Lyapunov functional may be directly enlarged as μiexp{-2εd1i}-exp{-2εd2i}. In this paper, we enlarge it as ϖ(ε,μi) to reduce the conservativeness of the obtained criteria. In the literature [36, 45, 46], this factor is enlarged as (μi-1)exp{-2εd2i}, which only holds for μi<1.Remark 14.
The information on the lower bound of the delay is sufficiently used in the Lyapunov functional by introducing the terms such as∫t-dmit-d1i‍exp{2ε(s-t)}xT(s)R3x(s)ds, ∫t-d2it-dmi‍exp{2ε(s-t)}xT(s)R4x(s)ds, and ∫t-di(t)t-d1i‍exp{2ε(s-t)}xT(s)R1ix(s)ds, which is equivalent to the improved bounding technique.Remark 15.
It should be also mentioned that the result obtained in Theorem10 is delay-range-dependent and decay rate-dependent stability condition for (13), which is less conservative than the previous ones and will be verified in Section 4. Although the large number of introduced free weighting matrices may increase the complexity of computation, utilizing the technique of free weighting matrices would reduce the conservativeness. In addition, the given results can be extended to more general systems with neutral delay τi(t). That is, τ1i≤τi(t)≤τ2i. The corresponding results can be obtained by using the similar methods.In Theorem10, it is assumed that ε≠0. For ε=0, by L'Hospital rule, the following asymptotic stability criterion can be obtained.Corollary 16.
For given scalarsπim, πiM, α, β, γ, τi¯, νi, d1i, d2i, μi and constant scalar dmi satisfying d1i<dmi<d2i, the systems described by (13) with partially known transition rates are asymptotically stable if ∥Ci∥+γ<1 and there exist symmetric positive matrices Pi>0, Q1i>0, Q2i>0, R1i>0, T1i>0, (i∈S), Qj>0, (j=3,4), Rk>0, Tl>0, (k,l=2,3,4), Um>0, Vn>0, Ws>0, (m,n,s=1,2,3,4) and matrices M1, M2, N1, N2, N3, N4 for any scalars ε1, ε2, ε3, any symmetric matrices Xi, Y1i, Y2i, Z1i, Z2i, (i∈S) and any matrices Jk, (k=1,2,…,24) with appropriate dimensions, such that (26)–(32) and the following linear matrix inequalities hold:
(83)(i)[U3N1N1TU3]>0,[V3N2N2TV3]>0,(ii)Ωi0+Ωi1<0,(iii)[U4N3N3TU4]>0,[V4N4N4TV4]>0,(iv)Ωi0+Ωi2<0,
where
(84)Ωi0=∑m=124‍emΥ^memT+ℒ(Ξ)-(e1-e5)V2(e1T-e5T)-(τi¯e1-e21)W1(τi¯e1T-e21T)-(d1ie1-e11)W2(d1ie1T-e11T)-(ϱ1ie1-e14)W3(ϱ1ie1T-e14T)-(ϱ2ie1-e15)W4(ϱ2ie1T-e15T)-[e20e21-e20][U1M1M1TU1][e20Te21T-e20T]-[e1-e18e18-e16][V1M2M2TV1][e1T-e18Te18T-e16T],Υ^1=AiTPi+PiAi+∑j∈𝒮ki‍πij(Pj-Xi)+Q1i+Q3+R2+τi¯2U1+d1i2U2+ϱ1i2U3+ϱ2i2U4+ε1α2I+J1Ai+AiTJ1T,Υ^3=-(1-μi)R1i+ε2β2I+J3Bi+BiTJ3T,Υ^4=-(1-μi)T1i,Υ^5=R1i+R3-R2,Υ^6=R4-R3,Υ^7=-R4,Υ^8=T1i+T3-T2,Υ^9=T4-T3,Υ^10=-T4,Υ^11=-U2,Υ^16=-Q3,Υ^17=-Q4,Υ^18=-(1-νi)Q1i,Υ^19=-(1-νi)Q2i+ε3γ2I+J19Ci+CiTJ19T.
Other Υ^m=Υm, (m=2,12,13,14,15,20,…,24), which have been defined in Theorem 10,
(85)Ωi1=-e15U4e15T-(e6-e7)V4(e6T-e7T)-[e12e14-e12][U3N1N1TU3][e12Te14T-e12T]-[e5-e3e3-e6][V3N2N2TV3][e5T-e3Te3T-e6T],Ωi2=-e14U3e14T-(e5-e6)V3(e5T-e6T)-[e13e15-e13][U4N3N3TU4][e13Te15T-e13T]-[e3-e7e6-e3][V4N4N4TV4][e3T-e7Te6T-e3T]
and the remaining notations are the same as Theorem 10.Further, the information on the transition rates may be completely unknown in some circumstances, which viewed the systems as switched systems with arbitrary switching. The following corollary is therefore given to guarantee the exponential stability for this case.Corollary 17.
For given scalarsπim, πiM, α, β, γ, ε, τi¯, νi, d1i, d2i, μi, and constant scalar dmi satisfying d1i<dmi<d2i, the systems described by (13) with completely unknown transition rates are exponentially stable with decay rate ε and κ=λ¯/λ_ if ∥Ci∥+γ<1 and there exist symmetric positive matrices P>0, Qj>0, Rk>0, Tl>0, Um>0, Vn>0, Ws>0, (j,k,l,m,n,s=1,2,3,4) and matrices M1, M2, N1, N2, N3, N4  for any scalars ε1, ε2, ε3, any symmetric matrices Y1, Y2, Z1, Z2, and any matrices Jk, (k=1,2,…,24) with appropriate dimensions, such that the following linear matrix inequalities hold:
(86)-∑j∈𝒮uki,j≠i‍πiM(Q1-Y1)-πimY1≤0;-∑j∈𝒮uki,j≠i‍πiM(Q2-Y2)-πimY2≤0,(87)-∑j∈𝒮uki,j≠i‍πiM(R1-Z1)-πimZ1≤0;-∑j∈𝒮uki,j≠i‍πiM(T1-Z2)-πimZ2≤0,(88)(i)[U3N1N1TU3]>0,[V3N2N2TV3]>0,(ii)Ω~i0ε+Ω~i1ε<0,(89)(iii)[U4N3N3TU4]>0,[V4N4N4TV4]>0,(iv)Ω~i0ε+Ω~i2ε<0,
where
(90)Ω~i0ε=∑m=124‍emΥ~memT+ℒ(Ξ)-2εd1iexp{2εd1i}-1(e1-e5)V2(e1T-e5T)-exp{-2ετi¯}(τi¯e1-e21)W1(τi¯e1T-e21T)-exp{-2εd1i}(d1ie1-e11)W2(d1ie1T-e11T)-exp{-2εdmi}(ϱ1ie1-e14)W3(ϱ1ie1T-e14T)-exp{-2εd2i}(ϱ2ie1-e15)W4(ϱ2ie1T-e15T)-2ετi¯exp{2ετi¯}-1[e20e21-e20]×[U1M1M1TU1][e20Te21T-e20T]-2ετi¯exp{2ετi¯}-1[e1-e18e18-e16]×[V1M2M2TV1][e1T-e18Te18T-e16T],Υ~1=AiTP+PAi+2εP+Q1+Q3+R2+τi¯2U1+d1i2U2+ϱ1i2U3+ϱ2i2U4+ε1α2I+J1Ai+AiTJ1T,Υ~2=Q2+Q4+T2+τi¯2V1+d1i2V2+ϱ1i2V3+ϱ2i2V4+τi¯44W1+d1i44W2+ϱ3i2W3+ϱ4i2W4-J2-J2T,Υ~3=ϖ(ε,μi)R1+ε2β2I+J3Bi+BiTJ3T,Υ~4=ϖ(ε,μi)T1,Υ~5=exp{-2εd1i}(R1+R3-R2),Υ~6=exp{-2εdmi}(R4-R3),Υ~7=-exp{-2εd2i}R4,Υ~8=exp{-2εd1i}(T1+T3-T2),Υ~9=exp{-2εdmi}(T4-T3),Υ~10=-exp{-2εd2i}T4,Υ~11=-2εd1iexp{2εd1i}-1U2,Υ~16=-exp{-2ετi¯}Q3,Υ~17=-exp{-2ετi¯}Q4,Υ~18=-(1-νi)exp{-2ετi¯}Q1,Υ~19=-(1-νi)exp{-2ετi¯}Q2+ε3γ2I+J19Ci+CiTJ19T,Υ~22=-ε1I+J22Di+DiTJ22T,Υ~23=-ε2I+J23Ei+EiTJ23T,Υ~24=-ε3I+J24Fi+FiTJ24T,Υ~m=0,(m=12,13,14,15,20,21),
where
(91)Ω~i1ε=Ωi1ε,Ω~i2ε=Ωi2ε,λ¯=λmax(P)+1-exp{-2ες}2ε×{∑j=14‍λmax(Qj)+∑k=14‍λmax(Rk)+∑l=14‍λmax(Tl)}+exp{-2ες}+2ες-14ε2×{∑m=14‍λmax(Um)+∑n=14‍λmax(Vn)}+1-exp{-2ες}+2ε2ς2-2ες8ε3{∑s=14‍λmax(Ws)}
and the remaining notations are the same as Theorem 10.Proof.
Since the elements of transition rates are entirely unknown, we choose the Lyapunov functional as follows:(92)V(x(t),i,t)=xT(t)Px(t)+∑k=27‍Vk(xt,i),
where
(93)V2(xt,i)=∫t-τi(t)t‍exp{2ε(s-t)}xT(s)Q1x(s)ds+∫t-τi(t)t‍exp{2ε(s-t)}x˙T(s)Q2x˙(s)ds+∫t-τi¯t‍exp{2ε(s-t)}xT(s)Q3x(s)ds+∫t-τi¯t‍exp{2ε(s-t)}x˙T(s)Q4x˙(s)ds,(94)V3(xt,i)=∫t-di(t)t-d1i‍exp{2ε(s-t)}xT(s)R1x(s)ds+∫t-d1it‍exp{2ε(s-t)}xT(s)R2x(s)ds+∫t-dmit-d1i‍exp{2ε(s-t)}xT(s)R3x(s)ds+∫t-d2it-dmi‍exp{2ε(s-t)}xT(s)R4x(s)ds,(95)V4(xt,i)=∫t-di(t)t-d1i‍exp{2ε(s-t)}x˙T(s)T1x˙(s)ds+∫t-d1it‍exp{2ε(s-t)}x˙T(s)T2x˙(s)ds+∫t-dmit-d1i‍exp{2ε(s-t)}x˙T(s)T3x˙(s)ds+∫t-d2it-dmi‍exp{2ε(s-t)}x˙T(s)T4x˙(s)ds
andV5(xt,i), V6(xt,i), V7(xt,i) are the same as (43), (44), and (45).
Then, we follow a similar line as in proof of Theorem10 and obtain the result.


### 3.2. Extension to the Uncertain Case

In this subsection, the uncertain neutral Markovian jump systems described by (6) with partially unknown transition rates are considered. The delay-range-dependent and rate-dependent exponential stability conditions are presented in the following theorems and corollaries.Theorem 18.
For given scalarsπim, πiM, α, β, γ, ε, τi¯, νi, d1i, d2i, μi and constant scalar dmi satisfying d1i<dmi<d2i, the systems described by (6) with partially known transition rates are exponentially stable with decay rate ε and κ=λ¯/λ_ if ∥Ci∥+γ<1 and there exist scalars δ1>0, δ2>0, symmetric positive matrices Pi>0, Q1i>0, Q2i>0, R1i>0, T1i>0, (i∈S), Qj>0, (j=3,4), Rk>0, Tl>0, (k,l=2,3,4), Um>0, Vn>0, Ws>0, (m,n,s=1,2,3,4) and matrices M1, M2, N1, N2, N3, N4 for any scalars ε1, ε2, ε3, any symmetric matrices Xi, Y1i, Y2i, Z1i, Z2i, (i∈S), and any matrices Jk, (k=1,2,…,24) with appropriate dimensions, such that (26)-(32) and the following inequalities hold:
(96)(i)[U3N1N1TU3]>0,[V3N2N2TV3]>0,(ii)[δ1Ωi0ε+δ1Ωi1ε+δ12𝕄T𝕄ℕT*-I]<0,(97)(iii)[U4N3N3TU4]>0,[V4N4N4TV4]>0,(iv)[δ2Ωi0ε+δ2Ωi2ε+δ22𝕄T𝕄ℕT*-I]<0,
where
(98)𝕄=e1NAiT+e3NBiT+e22NDiT+e23NEiT+e24NFiT,ℕ=LiTPiTe1+∑m=124‍LiTJmTem
and other notations are the same as Theorem10.Proof.
Defining𝔼=col{em,(m=1,2,…,24)} and 𝕁=col{Pi+J1,Jm,(m=2,…,24)}, we replace Ai, Bi, Di, Ei, and Fi with Ai+ΔAi(t), Bi+ΔBi(t), Di+ΔDi(t), Ei+ΔEi(t), and Fi+ΔFi(t) on the basis of Theorem 10. That is,
(99)(i)[U3N1N1TU3]>0,[V3N2N2TV3]>0,(ii)Ωi0ε+Ωi1ε+ℒ[e1ΔAiT(t)𝕁T𝔼+e3ΔBiT(t)𝕁T𝔼Ωi0ε+Ωi1ε+ℒhh+e22ΔDiT(t)𝕁T𝔼+e23ΔEiT(t)𝕁T𝔼Ωi0ε+Ωi1ε+ℒhh+e24ΔFiT(t)𝕁T𝔼]<0,(100)(iii)[U4N3N3TU4]>0,[V4N4N4TV4]>0,(iv)Ωi0ε+Ωi2ε+ℒ[e1ΔAiT(t)𝕁T𝔼+e3ΔBiT(t)𝕁T𝔼Ωi0ε+Ωi2ε+ℒhhhhhh+e22ΔDiT(t)𝕁T𝔼+e23ΔEiT(t)𝕁T𝔼Ωi0ε+Ωi2ε+ℒhhhhhh+e24ΔFiT(t)𝕁T𝔼]<0.
Considering (ii) of (99) and combining the uncertainties condition (11), we have
(101)Ωi0ε+Ωi1ε+ℒ×{[e1e3e22e23e24][NAiTNBiTNDiTNEiTNFiT]HiT(t)LiT𝕁T𝔼}<0.
By the definition of ℒ, we obtain
(102)Ωi0ε+Ωi1ε+𝕄HiT(t)ℕ+ℕTHi(t)𝕄T<0.
According to (12), by Lemmas 9 and 6, with (102) we obtain (ii) of (96). Following the same procedure, (iv) of (100) is considered and (iv) of (97) can be obtained. Finally, following the proof of Theorem 10, the systems described by (6) are exponentially stable with a decay rate ε. This completes the proof.Considering the uncertain case, the following corollaries are given, forε=0 and entirely unknown transition rates, respectively.Corollary 19.
For given scalarsπim, πiM, α, β, γ, τi¯, νi, d1i, d2i, μi and constant scalar dmi satisfying d1i<dmi<d2i, the systems described by (6) with partially known transition rates are asymptotically stable if ∥Ci∥+γ<1 and there exist scalars δ1>0, δ2>0, symmetric positive matrices Pi>0, Q1i>0, Q2i>0, R1i>0, T1i>0, (i∈S), Qj>0, (j=3,4), Rk>0, Tl>0, (k,l=2,3,4), Um>0, Vn>0, and Ws>0, (m,n,s=1,2,3,4) and matrices M1, M2, N1, N2, N3, N4 for any scalars ε1, ε2, ε3, any symmetric matrices Xi, Y1i, Y2i, Z1i, Z2i, (i∈S) and any matrices Jk, (k=1,2,…,24) with appropriate dimensions, such that (26)–(32), and the following inequalities hold:
(103)(i)[U3N1N1TU3]>0,[V3N2N2TV3]>0,(ii)[δ1Ωi0+δ1Ωi1+δ12𝕄T𝕄ℕT*-I]<0,(iii)[U4N3N3TU4]>0,[V4N4N4TV4]>0,(iv)[δ2Ωi0+δ2Ωi2+δ22𝕄T𝕄ℕT*-I]<0,
where
(104)𝕄=e1NAiT+e3NBiT+e22NDiT+e23NEiT+e24NFiT,ℕ=LiTPiTe1+∑m=124‍LiTJmTem
and other notations are the same as Corollary16.Corollary 20.
For given scalarsπim, πiM, α, β, γ, ε, τi¯, νi, d1i, d2i, μi and constant scalar dmi satisfying d1i<dmi<d2i, the systems described by (6) with completely unknown transition rates are exponentially stable with decay rate ε and κ=λ¯/λ_ if  ∥Ci∥+γ<1 and there exist scalars δ1>0, δ2>0, symmetric positive matrices P>0, Qj>0, Rk>0, Tl>0, Um>0, Vn>0, Ws>0, (j,k,l,m,n,s=1,2,3,4) and matrices M1, M2, N1, N2, N3, N4 for any scalars ε1, ε2, ε3, any symmetric matrices Y1, Y2, Z1, Z2, and any matrices Jk, (k=1,2,…,24) with appropriate dimensions, such that (86), (87), and the following inequalities hold:
(105)(i)[U3N1N1TU3]>0,[V3N2N2TV3]>0,(ii)[δ1Ω~i0ε+δ1Ω~i1ε+δ12𝕄T𝕄ℕT*-I]<0,(iii)[U4N3N3TU4]>0,[V4N4N4TV4]>0,(iv)[δ2Ω~i0ε+δ2Ω~i2ε+δ22𝕄T𝕄ℕT*-I]<0,
where
(106)𝕄=e1NAiT+e3NBiT+e22NDiT+e23NEiT+e24NFiT,ℕ=LiTPiTe1+∑m=124‍LiTJmTem
and other notations are the same as Corollary17.


## 3.1. Exponential Stability for the Nominal Systems

Theorem 10.
For given scalarsπim, πiM, α, β, γ, ε, τi¯, νi, d1i, d2i, μi and constant scalar dmi satisfying d1i<dmi<d2i, the systems described by (13) with partially known transition rates are exponentially stable with decay rate ε and κ=λ¯/λ_ if ∥Ci∥+γ<1 and there exist symmetric positive matrices Pi>0, Q1i>0, Q2i>0, R1i>0, T1i>0, (i∈S), Qj>0, (j=3,4), Rk>0, Tl>0, (k,l=2,3,4), Um>0, Vn>0, Ws>0, (m,n,s=1,2,3,4) and matrices M1, M2, N1, N2, N3, N4  for any scalars ε1, ε2, ε3, any symmetric matrices Xi, Y1i, Y2i, Z1i, Z2i, (i∈S) and any matrices Jk, (k=1,2,…,24) with appropriate dimensions, such that the following linear matrix inequalities hold.
Wheni∈𝒮ki(26)∑j∈𝒮ki,j≠i‍πij(Q1j-Y1i)-πiiY1i≤0,∑j∈𝒮ki,j≠i‍πij(Q2j-Y2i)-πiiY2i≤0,∑j∈𝒮ki,j≠i‍πij(R1j-Z1i)-πiiZ1i≤0,∑j∈𝒮ki,j≠i‍πij(T1j-Z2i)-πiiZ2i≤0,Pj-Xi≤0,j∈𝒮uki,Q1j-Y1i≤0,j∈𝒮uki,Q2j-Y2i≤0,j∈𝒮uki,R1j-Z1i≤0,j∈𝒮uki,T1j-Z2i≤0,j∈𝒮uki.
When i∈𝒮uki(27)∑j∈𝒮ki‍πij(Q1j-Y1i)≤0;∑j∈𝒮ki‍πij(Q2j-Y2i)≤0,(28)∑j∈𝒮ki‍πij(R1j-Z1i)≤0;∑j∈𝒮ki‍πij(T1j-Z2i)≤0,(29)Pj-Xi≤0,j∈𝒮uki,j≠i,Pj-Xi≥0,j∈𝒮uki,j=i,(30)-∑j∈𝒮uki,j≠i‍πiM(Q1j-Y1i)-πimY1i≤0,-∑j∈𝒮uki,j≠i‍πiM(Q2j-Y2i)-πimY2i≤0,(31)-∑j∈𝒮uki,j≠i‍πiM(R1j-Z1i)-πimZ1i≤0,-∑j∈𝒮uki,j≠i‍πiM(T1j-Z2i)-πimZ2i≤0,(32)[U1M1M1TU1]>0,[V1M2M2TV1]>0,(33)(i)[U3N1N1TU3]>0,[V3N2N2TV3]>0,(ii)Ωi0ε+Ωi1ε<0,(34)(iii)[U4N3N3TU4]>0,[V4N4N4TV4]>0,(iv)Ωi0ε+Ωi2ε<0,
where
(35)Ωi0ε=∑m=124‍emΥmemT+ℒ(Ξ)-2εd1iexp{2εd1i}-1(e1-e5)V2(e1T-e5T)-exp{-2ετi¯}(τi¯e1-e21)W1(τi¯e1T-e21T)-exp{-2εd1i}(d1ie1-e11)W2(d1ie1T-e11T)-exp{-2εdmi}(ϱ1ie1-e14)W3(ϱ1ie1T-e14T)-exp{-2εd2i}(ϱ2ie1-e15)W4(ϱ2ie1T-e15T)-2ετi¯exp{2ετi¯}-1[e20e21-e20]×[U1M1M1TU1][e20Te21T-e20T]-2ετi¯exp{2ετi¯}-1[e1-e18e18-e16]×[V1M2M2TV1][e1T-e18Te18T-e16T],
where ℒ is a linear operator on ℝn×n by
(36)ℒ(G)=G+GT,∀G∈ℝn×n,Ξ=-e1J1e2T+e1(PiBi+J1Bi)e3T+e1(PiCi)e19T+∑m=224‍e1(AiTJmT)emT+e1Pi(Die22T+Eie23T+Fie24T)+e2(J2Bi)e3T-∑m=324‍e2(JmT)emT+∑m=424‍e3(BiTJmT)emT+∑m=118‍em(JmCi)e19T+∑m=2024‍e19(CiTJmT)emT+∑m=121‍emJm(Die22T+Eie23T+Fie24T)+e22J22(Eie23T+Fie24T)+∑m=2324‍e22(DiTJmT)emT+e23(J23Fi)e24T+e23(EiTJ24T)e24T,Υ1=AiTPi+PiAi+∑j∈𝒮ki‍πij(Pj-Xi)+2εPi+Q1i+Q3+R2+τi¯2U1+d1i2U2+ϱ1i2U3+ϱ2i2U4+ε1α2I+J1Ai+AiTJ1T,Υ2=Q2i+Q4+T2+τi¯2V1+d1i2V2+ϱ1i2V3+ϱ2i2V4+τi¯44W1+d1i44W2+ϱ3i2W3+ϱ4i2W4-J2-J2T,Υ3=ϖ(ε,μi)R1i+ε2β2I+J3Bi+BiTJ3T,Υ4=ϖ(ε,μi)T1i,Υ5=exp{-2εd1i}(R1i+R3-R2),Υ6=exp{-2εdmi}(R4-R3),Υ7=-exp{-2εd2i}R4,Υ8=exp{-2εd1i}(T1i+T3-T2),Υ9=exp{-2εdmi}(T4-T3),Υ10=-exp{-2εd2i}T4,Υ11=-2εd1iexp{2εd1i}-1U2,Υ16=-exp{-2ετi¯}Q3,Υ17=-exp{-2ετi¯}Q4,Υ18=-(1-νi)exp{-2ετi¯}Q1i,Υ19=-(1-νi)exp{-2ετi¯}Q2i+ε3γ2I+J19Ci+CiTJ19T,Υ22=-ε1I+J22Di+DiTJ22T,Υ23=-ε2I+J23Ei+EiTJ23T,Υ24=-ε3I+J24Fi+FiTJ24T,Υm=0,(m=12,13,14,15,20,21),Ωi1ε=-2εϱ2iexp{2εd2i}-exp{2εdmi}e15U4e15T-2εϱ2iexp{2εd2i}-exp{2εdmi}×(e6-e7)V4(e6T-e7T)-2εϱ1iexp{2εdmi}-exp{2εd1i}[e12e14-e12]×[U3N1N1TU3][e12Te14T-e12T]-2εϱ1iexp{2εdmi}-exp{2εd1i}[e5-e3e3-e6]×[V3N2N2TV3][e5T-e3Te3T-e6T],Ωi2ε=-2εϱ1iexp{2εdmi}-exp{2εd1i}e14U3e14T-2εϱ1iexp{2εdmi}-exp{2εd1i}×(e5-e6)V3(e5T-e6T)-2εϱ2iexp{2εd2i}-exp{2εdmi}[e13e15-e13]×[U4N3N3TU4][e13Te15T-e13T]-2εϱ2iexp{2εd2i}-exp{2εdmi}[e3-e7e6-e3]×[V4N4N4TV4][e3T-e7Te6T-e3T],
where ei{i=1,2,…,24} are block entry matrices; that is,(37)
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.Proof.
Construct the following stochastic Lyapunov functional:(38)V(x(t),i,t)=∑k=17‍Vk(xt,i),
where
(39)V1(xt,i)=xT(t)Pix(t),(40)V2(xt,i)=∫t-τi(t)t‍exp{2ε(s-t)}xT(s)Q1ix(s)ds+∫t-τi(t)t‍exp{2ε(s-t)}x˙T(s)Q2ix˙(s)ds+∫t-τi¯t‍exp{2ε(s-t)}xT(s)Q3x(s)ds+∫t-τi¯t‍exp{2ε(s-t)}x˙T(s)Q4x˙(s)ds,(41)V3(xt,i)=∫t-di(t)t-d1i‍exp{2ε(s-t)}xT(s)R1ix(s)ds+∫t-d1it‍exp{2ε(s-t)}xT(s)R2x(s)ds+∫t-dmit-d1i‍exp{2ε(s-t)}xT(s)R3x(s)ds+∫t-d2it-dmi‍exp{2ε(s-t)}xT(s)R4x(s)ds,(42)V4(xt,i)=∫t-di(t)t-d1i‍exp{2ε(s-t)}x˙T(s)T1ix˙(s)ds+∫t-d1it‍exp{2ε(s-t)}x˙T(s)T2x˙(s)ds+∫t-dmit-d1i‍exp{2ε(s-t)}x˙T(s)T3x˙(s)ds+∫t-d2it-dmi‍exp{2ε(s-t)}x˙T(s)T4x˙(s)ds,(43)V5(xt,i)=∫-τi¯0‍∫t+θt‍τi¯exp{2ε(s-t)}xT(s)U1x(s)dsdθ+∫-d1i0‍∫t+θt‍d1iexp{2ε(s-t)}xThhhhhhhhhhhhhh×(s)U2x(s)dsdθ+∫-dmi-d1i‍∫t+θt‍ϱ1iexp{2ε(s-t)}xThhhhhhhhhhhhhh×(s)U3x(s)dsdθ+∫-d2i-dmi‍∫t+θt‍ϱ2iexp{2ε(s-t)}xThhhhhhhhhhhhhh×(s)U4x(s)dsdθ,(44)V6(xt,i)=∫-τi¯0‍∫t+θt‍τi¯exp{2ε(s-t)}x˙T(s)V1x˙(s)dsdθ+∫-d1i0‍∫t+θt‍d1iexp{2ε(s-t)}x˙Thhhhhhhhhhhhhh×(s)V2x˙(s)dsdθ+∫-dmi-d1i‍∫t+θt‍ϱ1iexp{2ε(s-t)}x˙Thhhhhhhhhhhhhhh×(s)V3x˙(s)dsdθ+∫-d2i-dmi‍∫t+θt‍ϱ2iexp{2ε(s-t)}x˙Thhhhhhhhhhhhhhh×(s)V4x˙(s)dsdθ,(45)V7(xt,i)=∫-τi¯0‍∫θ0‍∫t+λt‍τi¯22exp{2ε(s-t)}x˙Thhhhhhhhhh×(s)W1x˙(s)dsdλdθ+∫-d1i0‍∫θ0‍∫t+λt‍d1i22exp{2ε(s-t)}hhhhhhhhhh×x˙T(s)W2x˙(s)dsdλdθ+∫-dmi-d1i‍∫θ0‍∫t+λt‍ϱ3iexp{2ε(s-t)}hhhhhhhhhhhh×x˙T(s)W3x˙(s)dsdλdθ+∫-d2i-dmi‍∫θ0‍∫t+λt‍ϱ4iexp{2ε(s-t)}hhhhhhhhhhhh×x˙T(s)W4x˙(s)dsdλdθ.Remark  11. It should be pointed out that the proposed stochastic augmented Lyapunov functional (38) contains some triple-integral terms, which has not been used in of the existing literature in the same context before. Compared with the existing ones, [37] has shown that such triple-integral terms are very effective in the reduction of conservativeness.
TakingΓ as its infinitesimal generator along the trajectory of system (13), we obtain the following from Definition 5 and (38)–(45):
(46)ΓV(x(t),i,t)=∑k=17‍ΓVk(xt,i),ΓV1(xt,i) can be easily obtained by the following equation:
(47)ΓV1(xt,i)=2[xT(t)AiT+xT(t-di(t))BiT=2Hh+x˙T(t-τi(t))CiT+f1TDiT=2Hh+f2TEiT+f3TFiT]×Pix(t)+xT(t)[∑j∈𝒮ki‍πijPj+∑j∈𝒮uki‍πijPj]x(t)+2εxT(t)Pix(t)-2εV1(xt,i).
With regard toΓV2(xt,i), the detailed procedures are given as follows. 									(48)DefineV2(xt,i)=V21(xt,i)+V22(xt,i)+V23(xt,i)+V24(xt,i),
where
(49)V21(xt,i)=∫t-τi(t)t‍exp{2ε(s-t)}xT(s)Q1ix(s)ds,V22(xt,i)=∫t-τi(t)t‍exp{2ε(s-t)}x˙T(s)Q2ix˙(s)ds,V23(xt,i)=∫t-τi¯t‍exp{2ε(s-t)}xT(s)Q3x(s)ds,V24(xt,i)=∫t-τi¯t‍exp{2ε(s-t)}x˙T(s)Q4x˙(s)ds.
By the infinitesimal generatorΓ, we obtain
(50)ΓV21(xt,i)=exp{-2εt}×Γ(∫t-τi(t)t‍exp{2εs}xT(s)Q1ix(s)ds)-2εV21(xt,i),
where
(51)Γ(∫t-τi(t)t‍exp{2εs}xT(s)Q1ix(s)ds)=limΔ→0+1Δ{ℰ[∫t+Δ-τ(t+Δ,rt+Δ)t+Δ‍exp{2εs}xT(s)Q1(rt+Δ)=limΔ→0+1Δ(hhhhhh×x(s)ds∣x(t),rt=i∫t+Δ-τ(t+Δ,rt+Δ)t+Δ‍]=limΔ→0+1Δ(hhhhhh-∫t-τi(t)t‍exp{2εs}xT(s)Q1ix(s)ds∫t+Δ-τ(t+Δ,rt+Δ)t+Δ‍}=limΔ→0+1Δ{∑j=1N‍P(rt+Δ=j∣rt=i)ggggggggg×[∫t+Δ-τj(t+Δ)t+Δ‍exp{2εs}xT(s)Q1jx(s)ds]ggggggggg-∫t-τi(t)t‍exp{2εs}xT(s)Q1ix(s)ds∑j=1N‍}=limΔ→0+1Δ[(∑j=1N‍πijΔ)∫t+Δ-τj(t+Δ)t+Δ‍exp{2εs}xT(s)jjjjjjjjjjjjjjjjjjjj=limΔ→0+1Δ×Q1jx(s)dsjjjjjjjjjjjjjjjjjjjj+∫t+Δ-τi(t+Δ)t+Δ‍exp{2εs}xT(s)Q1ix(s)dsjjjjjjjjjjjjjjjjjjjj-∫t-τi(t)t‍exp{2εs}xT(s)Q1ix(s)ds∑j=1N‍πijΔ]=∑j=1N‍πij∫t-τj(t)t‍exp{2εs}xT(s)Q1jx(s)ds+exp{2ε(t)}xT(t)Q1ix(t)-(1-τ˙i(t))exp{2ε(t-τi(t))}xT×(t-τi(t))Q1ix(t-τi(t)).
Following the same procedure,ΓV22(xt,i) is also obtained:
(52)ΓV22(xt,i)=x˙T(t)Q2ix˙(t)-(1-τ˙i(t))exp{2ε(t-τi(t))}×x˙T(t-τi(t))Q2ix˙(t-τi(t))-∑j=1N‍πij∫t-τj(t)t‍exp{2εs}x˙T(s)Q2jx˙(s)ds-2εV22(xt,i).
Moreover,ΓV23(xt,i), ΓV24(xt,i) are easily calculated as shown in the following:
(53)ΓV23(xt,i)=xT(t)Q3x(t)-exp{-2ετi¯}×xT(t-τi¯)Q3x(t-τi¯)-2εV23(xt,i),ΓV24(xt,i)=x˙T(t)Q4x˙(t)-exp{-2ετi¯}×x˙T(t-τi¯)Q4x˙(t-τi¯)-2εV24(xt,i).
According to (50), (51), (52), and (53), we can easily obtain the following (54):
(54)ΓV2(xt,i)=xT(t)[Q1i+Q3]x(t)+x˙T(t)[Q2i+Q4]x˙(t)-2εV2(xt,i)-(1-τ˙i(t))exp{-2ετi(t)}xT(t-τi(t))Q1ix×(t-τi(t))-(1-τ˙i(t))exp{-2ετi(t)}x˙T×(t-τi(t))Q2ix˙(t-τi(t))-exp{-2ετi¯}xT(t-τi¯)Q3x(t-τi¯)-exp{-2ετi¯}x˙T(t-τi¯)Q4x˙(t-τi¯)+∑j∈S‍πij∫t-τj(t)t‍exp{2ε(s-t)}+∑j∈S‍πij×[xT(s)Q1jx(s)+x˙T(s)Q2jx˙(s)]ds≤xT(t)[Q1i+Q3]x(t)+x˙T(t)[Q2i+Q4]x˙(t)-2εV2(xt,i)-(1-τ˙i(t))exp{-2ετi(t)}xT(t-τi(t))Q1ix×(t-τi(t))-(1-τ˙i(t))exp{-2ετi(t)}x˙T×(t-τi(t))Q2ix˙(t-τi(t))-exp{-2ετi¯}xT(t-τi¯)Q3x(t-τi¯)-exp{-2ετi¯}x˙T(t-τi¯)Q4x˙(t-τi¯)+∑j≠i‍πij∫t-τj¯t‍exp{2ε(s-t)}+∑j≠i‍πij×[xT(s)Q1jx(s)+x˙T(s)Q2jx˙(s)]ds.
Then, in the same method,ΓV3(xt,i), ΓV4(xt,i) can be calculated and the results are given by the following, respectively:
(55)ΓV3(xt,i)=xT(t)R2x(t)+exp{-2εd1i}xT×(t-d1i)[R1i+R3-R2]x(t-d1i)+exp{-2εdmi}xT(t-dmi)[R4-R3]x(t-dmi)-exp{-2εd2i}xT(t-d2i)R4x(t-d2i)-2εV3(xt,i)-(1-d˙i(t))×exp{-2εdi(t)}xT(t-di(t))R1ix(t-di(t))+∑j∈S‍πij∫t-dj(t)t-d1i‍exp{2ε(s-t)}xT(s)R1jx(s)ds≤xT(t)R2x(t)+exp{-2εd1i}xT(t-d1i)×[R1i+R3-R2]x(t-d1i)+exp{-2εdmi}xT(t-dmi)[R4-R3]x×(t-dmi)-exp{-2εd2i}xT×(t-d2i)R4x(t-d2i)-2εV3(xt,i)-(1-d˙i(t))exp{-2εdi(t)}xT×(t-di(t))R1ix(t-di(t))+∑j≠i‍πij∫t-d2jt-d1i‍exp{2ε(s-t)}xT(s)R1jx(s)ds,(56)ΓV4(xt,i)=x˙T(t)T2x˙(t)+exp{-2εd1i}x˙T×(t-d1i)[T1i+T3-T2]x˙(t-d1i)+exp{-2εdmi}x˙T(t-dmi)×[T4-T3]x˙(t-dmi)-exp{-2εd2i}x˙T(t-d2i)T4x˙(t-d2i)-2εV4(xt,i)-(1-d˙i(t))exp{-2εdi(t)}x˙T×(t-di(t))T1ix˙(t-di(t))+∑j∈S‍πij∫t-dj(t)t-d1i‍exp{2ε(s-t)}x˙T(s)T1jx˙(s)ds≤x˙T(t)T2x˙(t)+exp{-2εd1i}x˙T(t-d1i)×[T1i+T3-T2]x˙(t-d1i)+exp{-2εdmi}x˙T(t-dmi)[T4-T3]x˙(t-dmi)-exp{-2εd2i}x˙T(t-d2i)T4x˙(t-d2i)-2εV4(xt,i)-(1-d˙i(t))×exp{-2εdi(t)}x˙T(t-di(t))T1ix˙(t-di(t))+∑j≠i‍πij∫t-d2jt-d1i‍exp{2ε(s-t)}x˙T(s)T1jx˙(s)ds.
Moreover,ΓV5(xt,i), ΓV6(xt,i), and ΓV7(xt,i) can be directly obtained as follows:
(57)ΓV5(xt,i)=xT(t)[τi¯2U1+d1i2U2+ϱ1i2U3+ϱ2i2U4]x(t)-2εV5(xt,i)-∫t-τi¯t‍τi¯exp{2ε(s-t)}xT(s)U1x(s)ds-∫t-d1it‍d1iexp{2ε(s-t)}xT(s)U2x(s)ds-∫t-dmit-d1i‍ϱ1iexp{2ε(s-t)}xT(s)U3x(s)ds-∫t-d2it-dmi‍ϱ2iexp{2ε(s-t)}xT(s)U4x(s)ds,ΓV6(xt,i)=x˙T(t)[τi¯2V1+d1i2V2+ϱ1i2V3+ϱ2i2V4]x˙(t)-2εV6(xt,i)-∫t-τi¯t‍τi¯exp{2ε(s-t)}x˙T(s)V1x˙(s)ds-∫t-d1it‍d1iexp{2ε(s-t)}x˙T(s)V2x˙(s)ds-∫t-dmit-d1i‍ϱ1iexp{2ε(s-t)}x˙T(s)V3x˙(s)ds-∫t-d2it-dmi‍ϱ2iexp{2ε(s-t)}x˙T(s)V4x˙(s)ds,ΓV7(xt,i)=x˙T(t)[τi¯44W1+d1i44W2+ϱ3i2W3+ϱ4i2W4]x˙(t)-2εV7(xt,i)-∫-τi¯0‍∫t+θt‍τi¯22exp{2ε(s-t)}x˙T(s)W1x˙(s)dsdθ-∫-d1i0‍∫t+θt‍d1i22exp{2ε(s-t)}x˙T(s)W2x˙(s)dsdθ-∫-dmi-d1i‍∫t+θt‍ϱ3iexp{2ε(s-t)}x˙T(s)W3x˙(s)dsdθ-∫-d2i-dmi‍∫t+θt‍ϱ4iexp{2ε(s-t)}x˙T(s)W4x˙(s)dsdθ.
Define(58)ξ(t)=col{∫t-τi(t)t‍x(t)x˙(t)x(t-di(t))x˙(t-di(t))x(t-d1i)x(t-dmi)x(t-d2i)x˙(t-d1i)x˙(t-dmi)x˙(t-d2i)∫t-d1it‍x(s)ds∫t-di(t)t-d1i‍x(s)ds∫t-d2it-di(t)‍x(s)ds∫t-dmit-d1i‍x(s)ds∫t-d2it-dmi‍x(s)dsx(t-τi¯)x˙(t-τi¯)x(t-τi(t))x˙(t-τi(t))∫t-τi(t)t‍x(s)ds∫t-τi¯t‍x(s)dsf1f2f3}.
Then, there exist matricesJ=col{Jk,(k=1,2,…,24)} with appropriate dimensions, such that the following equality holds according to (13):
(59)2ξT(t)J[-x˙(t)+Aix(t)+Bix(t-di(t))hhhHhhhj+Cix˙(t-τi(t))+Dif1+Eif2+Fif3]=0.
Due to∑j=1N‍πij=0, the following zero equations hold for arbitrary matrices Xi=XiT, Y1i=Y1iT, Y2i=Y2iT, Z1i=Z1iT, Z2i=Z2iT,i∈S; that is,
(60)-xT(t)[∑j∈𝒮ki‍πijXi+∑j∈𝒮uki‍πijXi]x(t)=0,-∫t-τj¯t‍exp{2ε(s-t)}xT(s)×[∑j∈𝒮ki‍πijY1i+∑j∈𝒮uki‍πijY1i]x(s)ds=0,-∫t-τj¯t‍exp{2ε(s-t)}x˙T(s)×[∑j∈𝒮ki‍πijY2i+∑j∈𝒮uki‍πijY2i]x˙(s)ds=0,-∫t-d2jt-d1i‍exp{2ε(s-t)}xT(s)×[∑j∈𝒮ki‍πijZ1i+∑j∈𝒮uki‍πijZ1i]x(s)ds=0,-∫t-d2jt-d1i‍exp{2ε(s-t)}x˙T(s)×[∑j∈𝒮ki‍πijZ2i+∑j∈𝒮uki‍πijZ2i]x˙(s)ds=0.
In view of (9), the following inequalities hold for any scalars ε1>0, ε2>0, and ε3>0:
(61)ε1[α2xT(t)x(t)-f1T(x(t),t)f1(x(t),t)]≥0,ε2[β2xT(t-di(t))x(t-di(t))-f2T(x(t-di(t)),t)f2(x(t-di(t)),t)]≥0,ε3[γ2x˙T(t-τi(t))x˙(t-τi(t))-f3T(x˙(t-τi(t)),t)f3(x˙(t-τi(t)),t)]≥0.
From (46) and (61), we have
(62)ΓV(x(t),i,t)≤∑k=17‍ΓVk(xt,i)+ε1[α2xT(t)x(t)-f1Tf1]+ε2[β2xT(t-di(t))x(t-di(t))-f2Tf2]+ε3[γ2x˙T(t-τi(t))x˙(t-τi(t))-f3Tf3].
Since it is easy to see-(1-τ˙i(t))exp{-2ετi(t)}≤-(1-νi)exp{-2ετi¯}, then from (8) we also obtain that
(63)-(1-d˙i(t))exp{-2εdi(t)}≤ϖ(ε,μi),
where ϖ(ε,μi) is defined in Theorem 10. Notice (a) of Lemma 7, then
(64)-∫t-d1it‍d1iexp{2ε(s-t)}xT(s)U2x(s)ds≤-2εd1iexp{2εd1i}-1ξT(t)e11U2e11Tξ(t),-∫t-d1it‍d1iexp{2ε(s-t)}x˙T(s)V2x˙(s)ds≤-2εd1iexp{2εd1i}-1ξT(t)(e1-e5)V2(e1T-e5T)ξ(t).
Notice (b) of Lemma7, then
(65)-∫-τi¯0‍∫t+θt‍τi¯22exp{2ε(s-t)}x˙T(s)W1x˙(s)dsdθ≤-exp{-2ετi¯}ξT(t)(τi¯e1-e21)×W1(τi¯e1T-e21T)ξ(t),-∫-d1i0‍∫t+θt‍d1i22exp{2ε(s-t)}x˙T(s)W2x˙(s)dsdθ≤-exp{-2εd1i}ξT(t)(d1ie1-e11)×W2(d1ie1T-e11T)ξ(t),-∫-dmi-d1i‍∫t+θt‍ϱ3iexp{2ε(s-t)}x˙T(s)W3x˙(s)dsdθ≤-exp{-2εdmi}ξT(t)(ϱ1ie1-e14)×W3(ϱ1ie1T-e14T)ξ(t),-∫-d2i-dmi‍∫t+θt‍ϱ4iexp{2ε(s-t)}x˙T(s)W4x˙(s)dsdθ≤-exp{-2εd2i}ξT(t)(ϱ2ie1-e15)×W4(ϱ2ie1T-e15T)ξ(t).
Fordi(t)∈[d1i,dmi], the following is held from (a) of Lemma 7:
(66)-∫t-dmit-d1i‍ϱ1iexp{2ε(s-t)}x˙T(s)V3x˙(s)ds=-{∫t-di(t)t-d1i‍+∫t-dmit-di(t)‍}ϱ1iexp{2ε(s-t)}×x˙T(s)V3x˙(s)ds≤-2εϱ1iexp{2εdi(t)}-exp{2εd1i}×[∫t-di(t)t-d1i‍x˙T(s)ds]V3[∫t-di(t)t-d1i‍x˙(s)ds]-2εϱ1iexp{2εdmi}-exp{2εd(t)}×[∫t-dmit-di(t)‍x˙T(s)ds]V3[∫t-dmit-di(t)‍x˙(s)ds]=-2εϱ1iexp{2εdmi}-exp{2εd1i}ξT(t)×[1λ1i(t)(e5-e3)V3(e5T-e3T)h+1λ2i(t)(e3-e6)V3(e3T-e6T)]ξ(t),
where
(67)λ1i(t)=exp{2εdi(t)}-exp{2εd1i}exp{2εdmi}-exp{2εd1i},λ2i(t)=exp{2εdmi}-exp{2εdi(t)}exp{2εdmi}-exp{2εd1i}.
By Lemma8, there exists matrix N2 with appropriate dimensions such that
(68)-∫t-dmit-d1i‍ϱ1iexp{2ε(s-t)}x˙T(s)V3x˙(s)ds≤-2εϱ1iexp{2εdmi}-exp{2εd1i}ξT(t)×[e5-e3e3-e6][V3N2N2TV3][e5T-e3Te3T-e6T]ξ(t),[V3N2N2TV3]>0.
Similarly, considering-∫t-dmit-d1i‍ϱ1iexp{2ε(s-t)}xT(s)U3x(s)ds and following the same procedure, there exists matrix N1 with appropriate dimensions such that
(69)-∫t-dmit-d1i‍ϱ1iexp{2ε(s-t)}xT(s)U3x(s)ds≤-2εϱ1iexp{2εdmi}-exp{2εd1i}ξT(t)×[e12e14-e12][U3N1N1TU3][e12Te14T-e12T]ξ(t),[U3N1N1TU3]>0.
Forτi(t)∈[0,τi¯], with the same matrix inequalities technique, we obtain the following:
(70)-∫t-τi¯t‍τi¯exp{2ε(s-t)}xT(s)U1x(s)ds≤-2ετi¯exp{2ετi¯}-1ξT(t)[e20e21-e20]×[U1M1M1TU1][e20Te21T-e20T]ξ(t),[U1M1M1TU1]>0,-∫t-τi¯t‍τi¯exp{2ε(s-t)}x˙T(s)V1x˙(s)ds≤-2ετi¯exp{2ετi¯}-1ξT(t)[e1-e18e18-e16]×[V1M2M2TV1][e1T-e18Te18T-e16T]ξ(t),[V1M2M2TV1]>0.
Consider-∫t-d2it-dmi‍ϱ2iexp{2ε(s-t)}xT(s)U4x(s)ds and -∫t-d2it-dmi‍ϱ2iexp{2ε(s-t)}x˙T(s)V4x˙(s)ds, which are directly estimated by (a) of Lemma 7; that is,
(71)-∫t-d2it-dmi‍ϱ2iexp{2ε(s-t)}xT(s)U4x(s)ds≤-2εϱ2iexp{2εd2i}-exp{2εdmi}ξT(t)e15U4e15Tξ(t),-∫t-d2it-dmi‍ϱ2iexp{2ε(s-t)}x˙T(s)V4x˙(s)ds≤-2εϱ2iexp{2εd2i}-exp{2εdmi}ξT(t)×(e6-e7)V4(e6T-e7T)ξ(t).
Substituting (47), (54)–(60), and (63)–(71) into (62) we obtain
(72)ΓV(x(t),i,t)+2εV(x(t),i,t)≤ξT(t)(Ωi0ε+Ωi1ε)ξ(t)+℧,
where
(73)℧=∫t-τj¯t‍exp{2ε(s-t)}xT(s){∑j≠i‍πijQ1j-∑j∈𝒮ki‍πijY1ihhhhhhhhhhhhhhhhhhhhhh-∑j∈𝒮uki‍πijY1i}x(s)ds+∫t-τj¯t‍exp{2ε(s-t)}x˙T(s){∑j≠i‍πijQ2j-∑j∈𝒮ki‍πijY2ihhhhhhhhhhhhhhhhhhhhhh-∑j∈𝒮uki‍πijY2i}x˙(s)ds+∫t-d2jt-d1i‍exp{2ε(s-t)}xT(s){∑j≠i‍πijR1j-∑j∈𝒮ki‍πijZ1immmmmmmmmmmmmmm-∑j∈𝒮uki‍πijZ1i}x(s)ds+∫t-d2jt-d1i‍exp{2ε(s-t)}x˙T(s){∑j≠i‍πijT1j-∑j∈𝒮ki‍πijZ2ihhhhhhhhhhhhhhhhhhhhhhh-∑j∈𝒮uki‍πijZ2i}x(s)ds+xT(t){∑j∈𝒮uki‍πij(Pj-Xi)}x(t).
On the other hand, fordi(t)∈[dmi,d2i], the integral terms
(74)-∫t-d2it-dmi‍ϱ2iexp{2ε(s-t)}xT(s)U4x(s)ds,-∫t-d2it-dmi‍ϱ2iexp{2ε(s-t)}x˙T(s)V4x˙(s)ds
are disposed and estimated by Lemma8, and
(75)-∫t-dmit-d1i‍ϱ1iexp{2ε(s-t)}xT(s)U3x(s)ds,-∫t-dmit-d1i‍ϱ1iexp{2ε(s-t)}x˙T(s)V3x˙(s)ds
are directly estimated by (a) of Lemma7. Therefore,
(76)ΓV(x(t),i,t)+2εV(x(t),i,t)≤ξT(t)(Ωi0ε+Ωi2ε)ξ(t)+℧.
With (72) and (76), the following inequality (77) is held for di(t)∈[d1i,d2i] if (26)–(34) are satisfied
(77)ΓV(x(t),i,t)+2εV(x(t),i,t)<0.
From the stochastic Lyapunov functional (38) and (77), it is held that
(78)V(x(t),i,t)≥mini∈S{λmin(Pi)}∥x(t)∥2=λ_∥x(t)∥2,V(x(t),i,t)<exp{-2ε(t-t0)}×V(xt0,rt0,t0).
Moreover, we have(79)V(xt0,rt0,t0)=∑k=16‍Vk(xt0,rt0)≤λ¯∥xt0∥ς2.
Then from (78) and (79), it is readily seen that
(80)∥x(t)∥≤λ¯λ_exp{-ε(t-t0)}∥xt0∥ς,
where κ=λ¯/λ_≥1.
Therefore, by Definition4, the system (13) is exponentially stable with a decay rate ε. This completes the proof.Remark 12.
It is noted that the integral intervals in (79) are enlarged as follows:
(81)∑k=16‍Vk(xt0,rt0)≤∫t0-ςt0‍exp{2ε(s-t0)}xT(s)×{Q1i+Q3+R1i+∑k=24‍Rk}x(s)ds+∫t0-ςt0‍exp{2ε(s-t0)}x˙T(s)×{Q2i+Q4+T1i+∑l=24‍Tl}x˙(s)ds+∫-ς0‍∫t0+θt0‍ςexp{2ε(s-t0)}xT(s)×(∑m=14‍Um)x(s)dsdθ+∫-ς0‍∫t0+θt0‍ςexp{2ε(s-t0)}x˙T(s)×(∑n=14‍Vn)x˙(s)dsdθ+∫-ς0‍∫θ0‍∫t0+λt0‍ς22exp{2ε(s-t0)}x˙T(s)×(∑s=14‍Ws)x˙(s)dsdλdθ+xT(t0)Pix(t0)(79) can be obtained by letting λ¯ be defined as previously mentioned.Remark 13.
In Theorem10, the factors
(82)exp{2εd2i}-exp{2εdi(t)}2ε,exp{2εdi(t)}-exp{2εd1i}2ε
may be enlarged as(exp{2εd2i}-exp{2εd1i})/2ε. This will lead conservative results due to the fact that di(t) cannot achieve d1i and d2i at the same time. While we apply Lemma 7 to these terms, the method by using reciprocally convex lemma [38] can achieve less conservative results. Moreover, for ε>0, the factor (d˙i(t)-1)exp{-2εdi(t)} that appeared in the derivative of Lyapunov functional may be directly enlarged as μiexp{-2εd1i}-exp{-2εd2i}. In this paper, we enlarge it as ϖ(ε,μi) to reduce the conservativeness of the obtained criteria. In the literature [36, 45, 46], this factor is enlarged as (μi-1)exp{-2εd2i}, which only holds for μi<1.Remark 14.
The information on the lower bound of the delay is sufficiently used in the Lyapunov functional by introducing the terms such as∫t-dmit-d1i‍exp{2ε(s-t)}xT(s)R3x(s)ds, ∫t-d2it-dmi‍exp{2ε(s-t)}xT(s)R4x(s)ds, and ∫t-di(t)t-d1i‍exp{2ε(s-t)}xT(s)R1ix(s)ds, which is equivalent to the improved bounding technique.Remark 15.
It should be also mentioned that the result obtained in Theorem10 is delay-range-dependent and decay rate-dependent stability condition for (13), which is less conservative than the previous ones and will be verified in Section 4. Although the large number of introduced free weighting matrices may increase the complexity of computation, utilizing the technique of free weighting matrices would reduce the conservativeness. In addition, the given results can be extended to more general systems with neutral delay τi(t). That is, τ1i≤τi(t)≤τ2i. The corresponding results can be obtained by using the similar methods.In Theorem10, it is assumed that ε≠0. For ε=0, by L'Hospital rule, the following asymptotic stability criterion can be obtained.Corollary 16.
For given scalarsπim, πiM, α, β, γ, τi¯, νi, d1i, d2i, μi and constant scalar dmi satisfying d1i<dmi<d2i, the systems described by (13) with partially known transition rates are asymptotically stable if ∥Ci∥+γ<1 and there exist symmetric positive matrices Pi>0, Q1i>0, Q2i>0, R1i>0, T1i>0, (i∈S), Qj>0, (j=3,4), Rk>0, Tl>0, (k,l=2,3,4), Um>0, Vn>0, Ws>0, (m,n,s=1,2,3,4) and matrices M1, M2, N1, N2, N3, N4 for any scalars ε1, ε2, ε3, any symmetric matrices Xi, Y1i, Y2i, Z1i, Z2i, (i∈S) and any matrices Jk, (k=1,2,…,24) with appropriate dimensions, such that (26)–(32) and the following linear matrix inequalities hold:
(83)(i)[U3N1N1TU3]>0,[V3N2N2TV3]>0,(ii)Ωi0+Ωi1<0,(iii)[U4N3N3TU4]>0,[V4N4N4TV4]>0,(iv)Ωi0+Ωi2<0,
where
(84)Ωi0=∑m=124‍emΥ^memT+ℒ(Ξ)-(e1-e5)V2(e1T-e5T)-(τi¯e1-e21)W1(τi¯e1T-e21T)-(d1ie1-e11)W2(d1ie1T-e11T)-(ϱ1ie1-e14)W3(ϱ1ie1T-e14T)-(ϱ2ie1-e15)W4(ϱ2ie1T-e15T)-[e20e21-e20][U1M1M1TU1][e20Te21T-e20T]-[e1-e18e18-e16][V1M2M2TV1][e1T-e18Te18T-e16T],Υ^1=AiTPi+PiAi+∑j∈𝒮ki‍πij(Pj-Xi)+Q1i+Q3+R2+τi¯2U1+d1i2U2+ϱ1i2U3+ϱ2i2U4+ε1α2I+J1Ai+AiTJ1T,Υ^3=-(1-μi)R1i+ε2β2I+J3Bi+BiTJ3T,Υ^4=-(1-μi)T1i,Υ^5=R1i+R3-R2,Υ^6=R4-R3,Υ^7=-R4,Υ^8=T1i+T3-T2,Υ^9=T4-T3,Υ^10=-T4,Υ^11=-U2,Υ^16=-Q3,Υ^17=-Q4,Υ^18=-(1-νi)Q1i,Υ^19=-(1-νi)Q2i+ε3γ2I+J19Ci+CiTJ19T.
Other Υ^m=Υm, (m=2,12,13,14,15,20,…,24), which have been defined in Theorem 10,
(85)Ωi1=-e15U4e15T-(e6-e7)V4(e6T-e7T)-[e12e14-e12][U3N1N1TU3][e12Te14T-e12T]-[e5-e3e3-e6][V3N2N2TV3][e5T-e3Te3T-e6T],Ωi2=-e14U3e14T-(e5-e6)V3(e5T-e6T)-[e13e15-e13][U4N3N3TU4][e13Te15T-e13T]-[e3-e7e6-e3][V4N4N4TV4][e3T-e7Te6T-e3T]
and the remaining notations are the same as Theorem 10.Further, the information on the transition rates may be completely unknown in some circumstances, which viewed the systems as switched systems with arbitrary switching. The following corollary is therefore given to guarantee the exponential stability for this case.Corollary 17.
For given scalarsπim, πiM, α, β, γ, ε, τi¯, νi, d1i, d2i, μi, and constant scalar dmi satisfying d1i<dmi<d2i, the systems described by (13) with completely unknown transition rates are exponentially stable with decay rate ε and κ=λ¯/λ_ if ∥Ci∥+γ<1 and there exist symmetric positive matrices P>0, Qj>0, Rk>0, Tl>0, Um>0, Vn>0, Ws>0, (j,k,l,m,n,s=1,2,3,4) and matrices M1, M2, N1, N2, N3, N4  for any scalars ε1, ε2, ε3, any symmetric matrices Y1, Y2, Z1, Z2, and any matrices Jk, (k=1,2,…,24) with appropriate dimensions, such that the following linear matrix inequalities hold:
(86)-∑j∈𝒮uki,j≠i‍πiM(Q1-Y1)-πimY1≤0;-∑j∈𝒮uki,j≠i‍πiM(Q2-Y2)-πimY2≤0,(87)-∑j∈𝒮uki,j≠i‍πiM(R1-Z1)-πimZ1≤0;-∑j∈𝒮uki,j≠i‍πiM(T1-Z2)-πimZ2≤0,(88)(i)[U3N1N1TU3]>0,[V3N2N2TV3]>0,(ii)Ω~i0ε+Ω~i1ε<0,(89)(iii)[U4N3N3TU4]>0,[V4N4N4TV4]>0,(iv)Ω~i0ε+Ω~i2ε<0,
where
(90)Ω~i0ε=∑m=124‍emΥ~memT+ℒ(Ξ)-2εd1iexp{2εd1i}-1(e1-e5)V2(e1T-e5T)-exp{-2ετi¯}(τi¯e1-e21)W1(τi¯e1T-e21T)-exp{-2εd1i}(d1ie1-e11)W2(d1ie1T-e11T)-exp{-2εdmi}(ϱ1ie1-e14)W3(ϱ1ie1T-e14T)-exp{-2εd2i}(ϱ2ie1-e15)W4(ϱ2ie1T-e15T)-2ετi¯exp{2ετi¯}-1[e20e21-e20]×[U1M1M1TU1][e20Te21T-e20T]-2ετi¯exp{2ετi¯}-1[e1-e18e18-e16]×[V1M2M2TV1][e1T-e18Te18T-e16T],Υ~1=AiTP+PAi+2εP+Q1+Q3+R2+τi¯2U1+d1i2U2+ϱ1i2U3+ϱ2i2U4+ε1α2I+J1Ai+AiTJ1T,Υ~2=Q2+Q4+T2+τi¯2V1+d1i2V2+ϱ1i2V3+ϱ2i2V4+τi¯44W1+d1i44W2+ϱ3i2W3+ϱ4i2W4-J2-J2T,Υ~3=ϖ(ε,μi)R1+ε2β2I+J3Bi+BiTJ3T,Υ~4=ϖ(ε,μi)T1,Υ~5=exp{-2εd1i}(R1+R3-R2),Υ~6=exp{-2εdmi}(R4-R3),Υ~7=-exp{-2εd2i}R4,Υ~8=exp{-2εd1i}(T1+T3-T2),Υ~9=exp{-2εdmi}(T4-T3),Υ~10=-exp{-2εd2i}T4,Υ~11=-2εd1iexp{2εd1i}-1U2,Υ~16=-exp{-2ετi¯}Q3,Υ~17=-exp{-2ετi¯}Q4,Υ~18=-(1-νi)exp{-2ετi¯}Q1,Υ~19=-(1-νi)exp{-2ετi¯}Q2+ε3γ2I+J19Ci+CiTJ19T,Υ~22=-ε1I+J22Di+DiTJ22T,Υ~23=-ε2I+J23Ei+EiTJ23T,Υ~24=-ε3I+J24Fi+FiTJ24T,Υ~m=0,(m=12,13,14,15,20,21),
where
(91)Ω~i1ε=Ωi1ε,Ω~i2ε=Ωi2ε,λ¯=λmax(P)+1-exp{-2ες}2ε×{∑j=14‍λmax(Qj)+∑k=14‍λmax(Rk)+∑l=14‍λmax(Tl)}+exp{-2ες}+2ες-14ε2×{∑m=14‍λmax(Um)+∑n=14‍λmax(Vn)}+1-exp{-2ες}+2ε2ς2-2ες8ε3{∑s=14‍λmax(Ws)}
and the remaining notations are the same as Theorem 10.Proof.
Since the elements of transition rates are entirely unknown, we choose the Lyapunov functional as follows:(92)V(x(t),i,t)=xT(t)Px(t)+∑k=27‍Vk(xt,i),
where
(93)V2(xt,i)=∫t-τi(t)t‍exp{2ε(s-t)}xT(s)Q1x(s)ds+∫t-τi(t)t‍exp{2ε(s-t)}x˙T(s)Q2x˙(s)ds+∫t-τi¯t‍exp{2ε(s-t)}xT(s)Q3x(s)ds+∫t-τi¯t‍exp{2ε(s-t)}x˙T(s)Q4x˙(s)ds,(94)V3(xt,i)=∫t-di(t)t-d1i‍exp{2ε(s-t)}xT(s)R1x(s)ds+∫t-d1it‍exp{2ε(s-t)}xT(s)R2x(s)ds+∫t-dmit-d1i‍exp{2ε(s-t)}xT(s)R3x(s)ds+∫t-d2it-dmi‍exp{2ε(s-t)}xT(s)R4x(s)ds,(95)V4(xt,i)=∫t-di(t)t-d1i‍exp{2ε(s-t)}x˙T(s)T1x˙(s)ds+∫t-d1it‍exp{2ε(s-t)}x˙T(s)T2x˙(s)ds+∫t-dmit-d1i‍exp{2ε(s-t)}x˙T(s)T3x˙(s)ds+∫t-d2it-dmi‍exp{2ε(s-t)}x˙T(s)T4x˙(s)ds
andV5(xt,i), V6(xt,i), V7(xt,i) are the same as (43), (44), and (45).
Then, we follow a similar line as in proof of Theorem10 and obtain the result.


## 3.2. Extension to the Uncertain Case

In this subsection, the uncertain neutral Markovian jump systems described by (6) with partially unknown transition rates are considered. The delay-range-dependent and rate-dependent exponential stability conditions are presented in the following theorems and corollaries.Theorem 18.
For given scalarsπim, πiM, α, β, γ, ε, τi¯, νi, d1i, d2i, μi and constant scalar dmi satisfying d1i<dmi<d2i, the systems described by (6) with partially known transition rates are exponentially stable with decay rate ε and κ=λ¯/λ_ if ∥Ci∥+γ<1 and there exist scalars δ1>0, δ2>0, symmetric positive matrices Pi>0, Q1i>0, Q2i>0, R1i>0, T1i>0, (i∈S), Qj>0, (j=3,4), Rk>0, Tl>0, (k,l=2,3,4), Um>0, Vn>0, Ws>0, (m,n,s=1,2,3,4) and matrices M1, M2, N1, N2, N3, N4 for any scalars ε1, ε2, ε3, any symmetric matrices Xi, Y1i, Y2i, Z1i, Z2i, (i∈S), and any matrices Jk, (k=1,2,…,24) with appropriate dimensions, such that (26)-(32) and the following inequalities hold:
(96)(i)[U3N1N1TU3]>0,[V3N2N2TV3]>0,(ii)[δ1Ωi0ε+δ1Ωi1ε+δ12𝕄T𝕄ℕT*-I]<0,(97)(iii)[U4N3N3TU4]>0,[V4N4N4TV4]>0,(iv)[δ2Ωi0ε+δ2Ωi2ε+δ22𝕄T𝕄ℕT*-I]<0,
where
(98)𝕄=e1NAiT+e3NBiT+e22NDiT+e23NEiT+e24NFiT,ℕ=LiTPiTe1+∑m=124‍LiTJmTem
and other notations are the same as Theorem10.Proof.
Defining𝔼=col{em,(m=1,2,…,24)} and 𝕁=col{Pi+J1,Jm,(m=2,…,24)}, we replace Ai, Bi, Di, Ei, and Fi with Ai+ΔAi(t), Bi+ΔBi(t), Di+ΔDi(t), Ei+ΔEi(t), and Fi+ΔFi(t) on the basis of Theorem 10. That is,
(99)(i)[U3N1N1TU3]>0,[V3N2N2TV3]>0,(ii)Ωi0ε+Ωi1ε+ℒ[e1ΔAiT(t)𝕁T𝔼+e3ΔBiT(t)𝕁T𝔼Ωi0ε+Ωi1ε+ℒhh+e22ΔDiT(t)𝕁T𝔼+e23ΔEiT(t)𝕁T𝔼Ωi0ε+Ωi1ε+ℒhh+e24ΔFiT(t)𝕁T𝔼]<0,(100)(iii)[U4N3N3TU4]>0,[V4N4N4TV4]>0,(iv)Ωi0ε+Ωi2ε+ℒ[e1ΔAiT(t)𝕁T𝔼+e3ΔBiT(t)𝕁T𝔼Ωi0ε+Ωi2ε+ℒhhhhhh+e22ΔDiT(t)𝕁T𝔼+e23ΔEiT(t)𝕁T𝔼Ωi0ε+Ωi2ε+ℒhhhhhh+e24ΔFiT(t)𝕁T𝔼]<0.
Considering (ii) of (99) and combining the uncertainties condition (11), we have
(101)Ωi0ε+Ωi1ε+ℒ×{[e1e3e22e23e24][NAiTNBiTNDiTNEiTNFiT]HiT(t)LiT𝕁T𝔼}<0.
By the definition of ℒ, we obtain
(102)Ωi0ε+Ωi1ε+𝕄HiT(t)ℕ+ℕTHi(t)𝕄T<0.
According to (12), by Lemmas 9 and 6, with (102) we obtain (ii) of (96). Following the same procedure, (iv) of (100) is considered and (iv) of (97) can be obtained. Finally, following the proof of Theorem 10, the systems described by (6) are exponentially stable with a decay rate ε. This completes the proof.Considering the uncertain case, the following corollaries are given, forε=0 and entirely unknown transition rates, respectively.Corollary 19.
For given scalarsπim, πiM, α, β, γ, τi¯, νi, d1i, d2i, μi and constant scalar dmi satisfying d1i<dmi<d2i, the systems described by (6) with partially known transition rates are asymptotically stable if ∥Ci∥+γ<1 and there exist scalars δ1>0, δ2>0, symmetric positive matrices Pi>0, Q1i>0, Q2i>0, R1i>0, T1i>0, (i∈S), Qj>0, (j=3,4), Rk>0, Tl>0, (k,l=2,3,4), Um>0, Vn>0, and Ws>0, (m,n,s=1,2,3,4) and matrices M1, M2, N1, N2, N3, N4 for any scalars ε1, ε2, ε3, any symmetric matrices Xi, Y1i, Y2i, Z1i, Z2i, (i∈S) and any matrices Jk, (k=1,2,…,24) with appropriate dimensions, such that (26)–(32), and the following inequalities hold:
(103)(i)[U3N1N1TU3]>0,[V3N2N2TV3]>0,(ii)[δ1Ωi0+δ1Ωi1+δ12𝕄T𝕄ℕT*-I]<0,(iii)[U4N3N3TU4]>0,[V4N4N4TV4]>0,(iv)[δ2Ωi0+δ2Ωi2+δ22𝕄T𝕄ℕT*-I]<0,
where
(104)𝕄=e1NAiT+e3NBiT+e22NDiT+e23NEiT+e24NFiT,ℕ=LiTPiTe1+∑m=124‍LiTJmTem
and other notations are the same as Corollary16.Corollary 20.
For given scalarsπim, πiM, α, β, γ, ε, τi¯, νi, d1i, d2i, μi and constant scalar dmi satisfying d1i<dmi<d2i, the systems described by (6) with completely unknown transition rates are exponentially stable with decay rate ε and κ=λ¯/λ_ if  ∥Ci∥+γ<1 and there exist scalars δ1>0, δ2>0, symmetric positive matrices P>0, Qj>0, Rk>0, Tl>0, Um>0, Vn>0, Ws>0, (j,k,l,m,n,s=1,2,3,4) and matrices M1, M2, N1, N2, N3, N4 for any scalars ε1, ε2, ε3, any symmetric matrices Y1, Y2, Z1, Z2, and any matrices Jk, (k=1,2,…,24) with appropriate dimensions, such that (86), (87), and the following inequalities hold:
(105)(i)[U3N1N1TU3]>0,[V3N2N2TV3]>0,(ii)[δ1Ω~i0ε+δ1Ω~i1ε+δ12𝕄T𝕄ℕT*-I]<0,(iii)[U4N3N3TU4]>0,[V4N4N4TV4]>0,(iv)[δ2Ω~i0ε+δ2Ω~i2ε+δ22𝕄T𝕄ℕT*-I]<0,
where
(106)𝕄=e1NAiT+e3NBiT+e22NDiT+e23NEiT+e24NFiT,ℕ=LiTPiTe1+∑m=124‍LiTJmTem
and other notations are the same as Corollary17.


## 4. Numerical Examples

In this section, numerical examples are given to demonstrate that the proposed theoretical results in this paper are effectiveness. In particular, wheni∈S={1}, di(t), d1i, dmi, and d2i are expressed into d(t), d1, dm, and d2.Example 1.
We consider an interval mode-dependent time-varying delay system in the form of (6) with three modes and nonlinear perturbations:
(107)∥f1(x(t),t)∥≤0.1∥x(t)∥,∥f2(x(t-d(t)),t)∥≤0.2∥x(t-d(t))∥,∥f3(x˙(t-τ(t)),t)∥≤0.1∥x˙(t-τ(t))∥;
that is, α=0.1, β=0.2, γ=0.1. The parametric matrices of the system are given as follows:
(108)A1=[-0.75-0.751.50-1.50],A2=[-0.15-0.091.50-0.10],A3=[-0.30-0.150.50-0.50],B1=[0.110.23-0.52-0.37],B2=[-0.590.02-0.06-0.61],B3=[0.510.240.02-0.44],C1=[0.30000.20],C2=[0.10000.20],C3=[0.20000.40],D1=[-0.70-0.340.50-0.65],D2=[-0.67-0.500.390.23],D3=[-0.230.160.02-0.57],E1=[-0.30-0.440.50-0.65],E2=[-0.40-0.350.390.28],E3=[-0.240.190.12-0.38],F1=[-0.50-0.360.26-0.75],F2=[-0.48-0.330.270.22],F3=[-0.450.230.12-0.35],L1=L2=L3=[0.10.2],NA1=NA2=NA3=[0.10.1],NB1=NB2=NB3=[0.10.2],ND1=ND2=ND3=[0.20.2],NE1=NE2=NE3=[0.20.3],NF1=NF2=NF3=[0.10.3].
The partially known transition rates matrix is described by
(109)[-0.8???-0.9?0.70.4-1.1],
where πim=-1.1, πiM=-0.8.
The interval mode-dependent time-varying neutral delays are assumed to be(110)τ1(t)=0.3(1+cost),τ2(t)=0.8(1+sint),τ3(t)=2+sint+cost.
The interval mode-dependent time-varying retarded delays are assumed to be(111)d1(t)=0.4(1+sin2(3t)),d2(t)=0.5(1+cos2(4t)),d3(t)=0.6sin4t.
They are governed by the Markov process {rt,t≥0} and shown in Figures 1 and 2. It can be readily obtain that
(112)τ1¯=0.6,τ2¯=1.6,τ3¯=22;ν1=0.3,ν2=0.8,ν3=2;d11=0.4,d21=0.8;d12=0.5,d22=1;d13=0,d23=0.6;μ1=1.2,μ2=2,μ3=9320.
Providing that decay rateε=0.5, we choose qualified values of dm1, dm2, dm3, for example, dm1=0.5, dm2=0.7, dm3=0.3, and then we solve (26)–(32), (96), and (97) in Theorem 18. Thus, it is facile to establish the exponential stability of this uncertain neutral system described by (6), which shows that the approach presented in this paper is effectiveness.Figure 1
Neutral time-varying delayτi(t) at Mode 1, Mode 2, and Mode 3.Figure 2
Retarded time-varying delaydi(t) at Mode 1, Mode 2, and Mode 3.Example 2.
Consider the nominal system (13) with four operation modes S={1,2,3,4}, α=β=γ=0, and the following parameters:
(113)A1=[-1.15-0.751.50-1.50],B1=[-1.200.120.24-0.25],C1=[-0.15-0.060.50-0.50],A2=[-2.15-0.491.50-2.10],B2=[-1.45-0.960.47-1.57],C2=[0.25-0.350.05-0.65],A3=[-1.30-0.151.50-1.80],B3=[0.58-0.68-0.130.96],C3=[0.330.070.19-0.36],A4=[-1.90-0.341.50-1.65],B4=[-0.67-1.501.391.23],C4=[-0.230.160.02-0.57].
The partially known transition rate matrix is considered as the following two cases:
(114)Π1=[-1.30.2????0.30.30.6?-1.5?0.4???],Π2=[-1.30.20.40.7??0.30.30.6?-1.5?0.4???],
where πim=-1.8 and πiM=-1.3.
Providing thatε=0 and τi(t)=di(t)=d(t), i∈S={1,2,3,4}, we have d1=0,τ¯=d2,ν=μ. Set dm=d2/2 and employ Corollary 16; the maximum upper bound of the time delay d2 which satisfies LMIs (26)–(32), and (83), can be calculated by solving a quasi-convex optimization problem. This neutral Markovian jump system with partially unknown transition rates was also considered in [40]. The results on the maximum upper bound of d2 are compared in Tables 1 and 2.Table 1
Maximum upper bound ofd2 with Π1 and different parameter μ.
Methods
μ
0.18
0.19
0.20
0.21
0.22
0.23
0.24
0.25
Xiong et al. [40]
d
2
0.4489
0.3317
0.2242
0.1814
0.1486
0.1220
0.0983
0.0781
Corollary16
d
2
0.4771
0.3664
0.2583
0.2165
0.1897
0.1473
0.1144
0.0956Table 2
Maximum upper bound ofd2 with Π2 and different parameter μ.
Methods
μ
0.18
0.19
0.20
0.21
0.22
0.23
0.24
0.25
Xiong et al. [40]
d
2
12.0750
0.9924
0.5147
0.3749
0.2742
0.2139
0.1737
0.1433
Corollary16
d
2
12.4394
1.2587
0.6263
0.4037
0.3003
0.2485
0.2018
0.1796From Tables1 and 2, we consider the previous system with Π1, Π2 and obtain the maximum upper bound of delay d2=0.4771, d2=12.4394, respectively, in this paper by setting μ=0.18, while the maximum upper bound of delay d2=0.4489, d2=12.0750, respectively, for [40]. The results are also given by setting μ=0.19, μ=0.20, μ=0.21, μ=0.22, μ=0.23, μ=0.24, and μ=0.25, and it is found that the maximum upper bound of delay in this paper is larger than that of [40]. So it can be seen that our proposed method is less conservative than the result in [40]. Besides, according to the data in Tables 1 and 2, we know that the maximum upper bound of delay to guarantee stability is dependent on transition rates knowledge.Example 3.
To show the reduced conservativeness of the exponential stability condition in Theorem10, consider the time-delay system in the form of (13) with Ci=0 and α=β=γ=0, di(t)=d and
(115)Ai=[-3-210],Bi=[-0.50.10.30],i∈S={1}.
For givend, the maximum exponential decay rate ε, which satisfies the LMIs (32), (33), and (34) in Theorem 10, can be calculated by solving a quasi-convex optimization problem. The results are presented in Table 3.Table 3
Maximum upper bound ofε with different parameter d.
Methods
d
0.8
1
1.2
1.4
1.6
1.8
2.0
Mondié and Kharitonov [41]
ε
0.7344
0.6715
0.6145
0.5642
0.5202
0.4818
0.4481
Liu [42]
ε
0.9367
0.5903
0.3400
0.1813
0.0752
0.0014
0
Xu et al. [43]
ε
0.9366
0.9192
0.8991
0.8115
0.6990
0.6148
0.5494
Theorem10
ε
0.9569
0.9407
0.9218
0.9024
0.8154
0.7394
0.6933From Table3, we know that the maximum exponential decay rate ε=0.9569 in this paper by setting d=0.8, while the maximum exponential decay rate ε=0.7344 for [41], ε=0.9367 for [42], and ε=0.9366 for [43]. The results are also given by setting d=1.0, d=1.2, d=1.4, d=1.6, d=1.8, and d=2.0, and it is found that the maximum exponential decay rate in this paper is larger than those in [41–43]. So it can be demonstrated that Theorem 10 in this paper yields less conservative results than [41–43].Consider the previous system again, but with parametric matrices and parameter uncertainties as follows:(116)Ai=[-410-4],Bi=[0.1040.1],i∈S={1}
and the uncertain matrices ΔAi(t) and ΔBi(t) satisfy
(117)Li=0.2I,NAi=NBi=I,∥Hi(t)∥<1,hhhhhhhhhhhhhhhhhhhhhhhhi∈S={1}.For givend, the maximum exponential decay rate ε, which satisfies the LMIs in (32), (96), and (97) in Theorem 18, can be calculated by solving a quasi-convex optimization problem. The results are presented in Table 4 and it also can be seen that the delay-range-dependent and rate-dependent exponential stability conditions in this paper are less conservative than previous results in [41, 43].Table 4
Maximum upper bound ofε with different parameter d.
Methods
d
0.3
0.5
0.7
0.9
1.1
1.3
1.5
Mondié and Kharitonov [41]
ε
0.6255
0.4760
0.3825
0.3191
0.2735
0.2392
0.2125
Xu et al. [43]
ε
1.0108
0.8366
0.7103
0.6156
0.5425
0.4845
0.4375
Theorem18
ε
1.0419
0.9207
0.8035
0.7048
0.6234
0.5876
0.5043Example 4.
Consider the system (6) with τi(t)=di(t)=d(t) and the parameters are listed in the following:
(118)Ai=[-200-0.9],Bi=[-10-1-1],Ci=[0.1000.1],Di=Ei=Fi=I,i∈S={1},α=0.05,β=0.1,γ=0,d1=0,d2=τ¯,μ=ν.
For given ε and μ, choose dm=d2/2 and utilize Theorem 10, the maximum upper bound of d2, which satisfies the LMIs in (32), (33), and (34), can be obtained by solving a quasi-convex optimization problem. The results are presented in Tables 5, 6, and 7.Table 5
Maximum upper bound ofd2 with different ε and parameter μ=0.
Methods
ε
0.1
0.3
0.5
0.7
0.9
Chen et al. [44]
d
2
1.2999
0.8781
0.6917
0.5792
0.5015
Qiu and Cui [13]
d
2
1.4008
1.0199
0.8457
0.7395
0.6667
Theorem10
d
2
1.5167
1.1365
1.0674
0.9677
0.7558Table 6
Maximum upper bound ofd2 with different ε and parameter μ=0.5.
Methods
ε
0.1
0.3
0.5
0.7
0.9
Chen et al. [44]
d
2
0.9442
0.7275
0.6096
0.5321
0.4761
Qiu and Cui [13]
d
2
1.0120
0.8324
0.7311
0.6941
0.6063
Theorem10
d
2
1.0643
0.9156
0.7985
0.7368
0.6849Table 7
Maximum upper bound ofd2 with different ε and parameter μ=0.9.
Methods
ε
0.1
0.3
0.5
0.7
0.9
Chen et al. [44]
d
2
0.5471
0.5015
0.4650
0.4350
0.4089
Qiu and Cui [13]
d
2
0.6438
0.5789
0.5214
0.4954
0.4231
Theorem10
d
2
0.7136
0.6344
0.5987
0.5473
0.4963From Tables5, 6, and 7, we consider μ=0, μ=0.5, and μ=0.9 and obtain the maximum upper bound of delay d2=1.5167, d2=1.0643, and d2=0.7136, respectively, in this paper by setting ε=0.1, while the maximum upper bound of delay d2=1.2999, d2=0.9442, and d2=0.5471, respectively, for [44], the maximum upper bound of delay d2=1.4008, d2=1.0120, and d2=0.6438, respectively, for [13]. The results are also given by setting ε=0.3, ε=0.5, ε=0.7, and ε=0.9, and it is found that the maximum upper bound of delay in this paper is larger than those in [13, 44]. According to the comparative result, it can be seen that our proposed method is less conservative than those in [13, 44].Example 5.
Partial element equivalent circuit (PEEC) model can be represented as a stochastic jump system as in (6) with the abrupt variation in structures and parameters [47]. Consider the practical PEEC system described by (6) with α=β=γ=0 and completely unknown transition rates
(119)Π=[????],i∈S={1,2},
where πim=-1.0 and πiM=-0.6.
The parametric matrices of the system are given as follows:(120)A1=[-400-3],A2=[-200-3],B1=[-10-1-1],B2=[-10-0.9-1.2],C1=0.5I,C2=0.3I,L1=[0.20.2],L2=[0-0.3],NA1=[0.20],NA2=[00.2],NB1=[-0.30.3],NB2=[0.20.2].
Given the decay rate ε=0.4, the bound of mode-dependent time-varying neutral delay 0≤τ1(t)≤0.5, 0≤τ2(t)≤0.6, and the bound of mode-dependent time-varying retarded delay 0.1≤d1(t)≤0.2, 0.3≤d2(t)≤0.4. Without loss of generality, we choose dm1=0.12 and dm2=0.33. Since the information on the delay derivative is not available, by setting Q1=Q2=0, R1=0, T1=0 in the Lyapunov functional (92), we follow the same procedure as in the proof of Theorem 10 and obtain the corresponding result. Thus, the exponential stability of the practical PEEC system can be readily established, which shows the effectiveness of the approach presented in this paper.


## 5. Conclusions

The problem of exponential stability for neutral Markovian jumping systems with interval mode-dependent time-varying delays, nonlinear perturbations, and partially unknown transition rates is investigated in this paper. A novel augmented stochastic Lyapunov-Krasovskii functional is constructed, which contains some triple-integral terms and sufficiently takes advantage of the delay bound. Then, less conservative delay-range-dependent and rate-dependent exponential stability criteria are obtained by novel technique of matrix inequalities and free weighting matrices. These theoretical results are successfully verified through some numerical examples. Finally, the main contributions of this paper can be summarized as follows:(1) the constructed stochastic Lyapunov functional contains some triple-integral terms which are very effective in the reduction of conservativeness and has not appeared in the context of neutral Markovian jump systems with partially known transition rates and nonlinear perturbations before; (2) the bound of the delay is fully utilized in this paper; that is, improved bounding technique is used to reduce the conservativeness; (3) the reciprocally convex lemma is used to derive the delay-range-dependent and rate-dependent stability conditions, which can well reduce the conservativeness of the investigated systems; and (4) the proposed results are applicable to the partially known transition rates and expressed in a new representation, which are proved to be less conservative than some existing ones.



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*Source: 691650-2013-12-01.xml*