# Stability Analysis for Neutral Delay Markovian Jump Systems with Nonlinear Perturbations and Partially Unknown Transition Rates

Authors: Xinghua Liu; Hongsheng Xi
Journal: Advances in Mathematical Physics
 (2013)

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## Abstract

The problem of exponential stability for the uncertain neutral Markovian jump systems with interval time-varying delays and nonlinear perturbations is investigated in this paper. This study starts from the corresponding nominal systems with known and partially unknown transition rates, respectively. By constructing a novel augmented Lyapunov functional which contains triple-integral terms and fully utilizes the bound of the delay, the delay-range-dependent and rate-dependent exponential stability criteria are developed by the Lyapunov theory, reciprocally convex lemma, and free weighting matrices. Then, the results about nominal systems are extended to the uncertain case. Finally, numerical examples are given to demonstrate the effectiveness of the proposed methods.


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## Body

## 1. Introduction

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], and so forth. The existence of time delay may cause the instability of the systems, thus making the stability analysis of time-delay systems an interesting topic. 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. In addition, it should be pointed out that the stability of neutral time-delay systems is more difficult to tackle since the derivative of the delayed state is involved. The situation is similar as singular systems [3, 4], whose stability problem 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 [5–8]. Furthermore, when nonlinear perturbations or parameter uncertainties appear in neutral systems, some results on stability analysis have been also presented [9–14]. 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. [14] 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, many complex systems with uncertainties and neutral types as well as time-varying or state-dependent delays are still inviting further investigation.It is noted that many practical dynamics such as solar thermal central receivers, robotic manipulator systems, aircraft control systems, and economic systems, experience abrupt changes in their structures, caused by phenomena such as component failures or repairs, changes in subsystem interconnections, and sudden environmental changes. This class of systems can be described as Markovian jump systems (MJSs) where the abrupt variation in the structures and parameters can be naturally represented by the jumps in MJSs. Since its first introduction by Krasovskii and Lidskii in 1961, MJSs have received much attention, and considerable progress has been made; see, for example, [15–22] and references therein for more details. In view of these results, although related research has made good achievement, we have to admit that the transition probabilities in the jumping process determine the system behavior to a large extent. However, the likelihood of obtaining such available knowledge is actually questionable, and the cost is probably expensive. Thus, it is significant and necessary, from control perspectives, to further study more general jump systems with partly unknown transition probabilities. Recently, many results on the Markovian jump systems with partly unknown transition probabilities are obtained [23–27]. Most of these improved results just require some free matrices or the knowledge of the known elements in transition rate matrix, such as the bounds or structures of uncertainties, and some else of the unknown elements do not need to be considered. It is a great progress on the analysis of Markovian jump systems. However, few of these results are concerned with neutral delay systems. It is urgent and significant to consider the problem of delay-dependent exponential stability for Markovian jumping neutral systems with partially unknown transition rates. Besides, to the best of the authors’ knowledge, the neutral Markovian jump systems have not been fully investigated, and it is very challenging, especially when nonlinear perturbations exist. These facts thus motivate our study.In this paper, the exponential stability problem of neutral Markovian jump systems with mixed interval time-varying delay, nonlinear perturbations, and partially unknown transition rates is investigated. A new augmented 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. Moreover, in contrast with the recent research on uncertain transition rates, our proposed concept of the partly unknown transition rates does not require any knowledge of the unknown elements, such as the bounds or structures of uncertainties. On the basis of the obtained results about nominal systems, 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 main contributions of this paper can be summarized as follows:(1) the bound of the delay is fully utilized in this paper; that is, improved bounding technique is used to reduce the conservativeness. (2) The constructed Lyapunov functional contains some triple-integral terms which is very effective in the reduction of conservativeness and has not appeared in the context of neutral Markovian jump systems with nonlinear perturbations before. (3) The reciprocally convex lemma is used to derive the delay-range-dependent stability conditions, which can reduce the conservativeness of the investigated systems. (4) The proposed results are applicable to the uncertain transition rates and expressed in a new representation, which are proved to be less conservative than some existing ones.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.Notation 1.
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 is defined to be the expectation operator with respect to the probability measure. ∥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, and 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.The following uncertain neutral Markovian jump systems with interval time-varying delays and nonlinear perturbations over the space{Ω,ℱ,P} are considered:
(3)x˙(t)-C(rt)x˙(t-τ(t))=[A(rt)+ΔA(rt)]x(t)+[B(rt)+ΔB(rt)]x(t-d(t))+f1(x(t),t)+f2(x(t-d(t)),t)+f3(x˙(t-τ(t)),t),(4)x(s)=φ(s),rs=r0,s∈[-ρ,0],
where x(t)∈ℝn is the system state and τ(t) is time-varying neutral delay which satisfies 0≤τ(t)≤τ, τ˙(t)≤τd. The time-varying retarded delay d(t) is such that
(5)0<d1≤d(t)≤d2,d˙(t)≤μ,
where d1<d2, μ≥0, and ρ=max{τ,d2} are constant real values. The initial condition φ(s) is a continuously differentiable vector-valued function. f1(x(t),t)∈ℝn, f2(x(t-d(t)),t)∈ℝn, and f3(x˙(t-τ(t)),t)∈ℝn are unknown nonlinear perturbations which are, with respect to the current state x(t), the delayed state x(t-d(t)) and the neutral delay state x˙(t-τ(t)), respectively. For all t, they are assumed to be bounded in magnitude as
(6)∥f1(x(t),t)∥≤α∥x(t)∥,∥f2(x(t-d(t)),t)∥≤β∥x(t-d(t))∥,∥f3(x˙(t-τ(t)),t)∥≤γ∥x˙(t-τ(t))∥,
where α≥0, β≥0, and γ≥0 are given constants, for simplicity, f1≜f1(x(t),t), f2≜f2(x(t-d(t)),t), and f3≜f3(x˙(t-τ(t)),t).A
(
r
t
)
∈
ℝ
n
×
n, B(rt)∈ℝn×n, and C(rt)∈ℝn×n are known mode-dependent constant matrices, and ΔA(rt)∈ℝn×n, ΔB(rt)∈ℝn×n are uncertainties. For simplicity of notations, we denote A(rt), ΔA(rt), B(rt), ΔB(rt), C(rt) by Ai, ΔAi, Bi, ΔBi, Ci for rt=i∈S. The parametric matrix ∥Ci∥<1 and the admissible parametric uncertainties satisfy the following condition:
(7)[ΔAi(t)ΔBi(t)]=HiFi(t)[EAiEBi],
where Hi, EAi, and EBi are known mode-dependent constant matrices with appropriate dimensions and Fi(t) is an unknown and time-varying matrix satisfying
(8)FiT(t)Fi(t)≤I,∀t.
Particularly, the following nominal systems can be obtained for Fi(t)=0:
(9)x˙(t)-Cix˙(t-τ(t))=Aix(t)+Bix(t-d(t))+f1+f2+f3.Before proceeding with the main results, we present the following definitions, assumptions, and lemmas.Assumption 1.
System matricesAi, (∀i∈S) are Hurwitz, and all the eigenvalues have negative real parts for each mode. Hi, (∀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 [28]).
Define operator𝔇:C([-ρ,0],ℝn)→ℝn as 𝔇(xt)=x(t)-Cx(t-τ). 𝔇 is said to be stable if the homogeneous difference equation
(10)𝔇(xt)=0,t≥0,x0=ψ∈{ϕ∈C([-ρ,0],ℝn):𝔇ϕ=0}
is uniformly asymptotically stable. In this paper, that is, ∥Ci∥+γ<1.Definition 4 (see [29]).
The system in (3) is exponentially stable with a decay rate ε for all rt=i∈S, if there exist scalars ε>0 and σ≥1 such that for all x(t),
(11)∥x(t)∥≤σexp{-ε(t-t0)}∥xt0∥ρ,
where ε is the exponential decay rate, ∥*∥ denotes the Euclidean norm, and
(12)∥xt0∥ρ=supθ∈[-ρ,0]{∥x(t0+θ)∥,∥x˙(t0+θ)∥}=sups∈[-ρ,t0]{∥φ(s)∥,∥φ˙(s)∥}.Definition 5 (see [30]).
Define the stochastic Lyapunov-Krasovskii function of system (3) as V(x(t),rt=i,t>0)=V(xt,i,t), where its infinitesimal generator is defined as
(13)𝔏V(x(t),i,t)=limΔt⟶01Δt[ℰ{V(x(t+Δt),rt+Δt,t+Δt)∣x(t)=x,𝔏V(x(t)fxx,it)rt=i}-V(x(t),i,t)rt+Δt]=∂∂tV(x(t),i,t)+∂∂xV(x(t),i,t)x˙(t)+∑j=1N‍πijV(x(t),j,t).Lemma 6 (see [31]).
For given real constant matricesΩ1, Ω2, and Ω3, with appropriate dimensions, where Ω1=Ω1T and Ω2=Ω2T>0. Ω1+Ω3TΩ2-1Ω3<0 if and only if
(14)[Ω1Ω3T*-Ω2]<0or[-Ω2Ω3T*Ω1]<0.Lemma 7.
For any constant matrixH=HT>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)
(15)exp{εh2(t)}-exp{εh1(t)}ε×∫t-h2(t)t-h1(t)‍exp{ε(s-t)}xT(s)Hx(s)ds≥[∫t-h2(t)t-h1(t)‍xT(s)ds]H[∫t-h2(t)t-h1(t)‍x(s)ds],(b)
(16)τ22-τ122∫-τ2-τ1‍∫t+θt‍exp{ε(s-t)}xT(s)Hx(s)dsdθ≥exp{-ετ2}[∫-τ2-τ1‍∫t+θt‍xT(s)dsdθ]×H[∫-τ2-τ1‍∫t+θt‍x(s)dsdθ].Proof.
(a) Is directly obtained from [32]. In addition, from -τ2≤θ≤-τ1 and t+θ≤s≤t, it is held that -τ2≤θ≤s-t≤0. Then,
(17)∫-τ2-τ1‍∫t+θt‍exp{ε(s-t)}xT(s)Hx(s)dsdθ≥exp{-ετ2}∫-τ2-τ1‍∫t+θt‍xT(s)Hx(s)dsdθ
(b) Is thus true by [33].Lemma 8 (see [34]).
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
(18)[PTTTQ]>0,
and the following inequality holds:
(19)1λ1(t)η1Pη1T+1λ2(t)η2Qη2T≥[η1η2][PTTTQ][η1Tη2T].Lemma 9 (see [35]).
For given matricesQ=QT, M, and N with appropriate dimensions:
(20)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
(21)Q+δ-1MMT+δNNT<0.


## 3. Main Results

This section will first state the exponential stability analysis for (9) with known and partly unknown transition rates, respectively. Then, the uncertain systems described by (3) are considered. 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 with Known Transition Rates

In this subsection, we consider the full information on the transition rates and give the following conditions to guarantee the exponential stability of the nominal systems.Theorem 10.
For the given finite setS of modes with transition rates matrix, scalars α, β, γ, ε, τ, τd, d1, d2, μ, and constant scalar dm satisfying d1<dm<d2, the systems described by (9) are exponentially stable with decay rate ε/2 and σ=λ¯/λ_ if the operator 𝔇 is stable, and there exist symmetric positive matrices Pi>0, (i∈S), Qj>0, (j=1,2,3,4,5), Rk>0, (k=1,2,3,4,5), Wl>0, (l=1,2,3,4), Ym>0, (m=1,2,3,4), Zn>0, (n=1,2,3,4) and matrices U1, U2, X1, X2, X3, X4  for any scalars ε1, ε2, ε3 and any matrices Tk, (k=1,2,…,23) with appropriate dimensions, such that the following linear matrix inequalities (22), (23), and (24) hold:
(22)[W1U1U1TW1]>0,[Y1U2U2TY1]>0,(23)(i)[W3X1X1TW3]>0,[Y3X2X2TY3]>0,(ii)Ξi0ε+Ξi1ε<0,(24)(iii)[W4X3X3TW4]>0,[Y4X4X4TY4]>0,(iv)Ξi0ε+Ξi2ε<0,
where
(25)Ξi0ε=∑m=123‍emGmemT+𝒜(✠)-εd1exp{εd1}-1(e1-e4)Y2(e1T-e4T)-exp{-ετ}(τe1-e20)Z1(τe1T-e20T)-exp{-εd1}(d1e1-e10)Z2(d1e1T-e10T)-exp{-εdm}(ϱ1e1-e13)Z3(ϱ1e1T-e13T)-exp{-εd2}(ϱ2e1-e14)Z4(ϱ2e1T-e14T)-ετexp{ετ}-1[e19e20-e19][W1U1U1TW1][e19Te20T-e19T]-ετexp{ετ}-1[e1-e17e17-e15]×[Y1U2U2TY1][e1T-e17Te17T-e15T],
where 𝒜 is a linear operator on ℝn×n by
(26)𝒜(F)=F+FT,∀F∈ℝn×n,✠=-e1T1e2T+e1(PiBi+T1Bi)e3T+e1(PiCi)e18T+∑m=223‍e1(AiTTmT)emT+∑m=2123‍e1(Pi)emT+e2(T2Bi)e3T-∑m=323‍e2(TmT)emT+∑m=423‍e3(BiTTmT)emT+∑m=117‍em(TmCi)e18T+∑m=1923‍e18(CiTTmT)emT+∑m=120‍∑n=2123‍emTmenT+e21T21(e22T+e23T)+e22T22e23T+∑m=2223‍e21TmTemT+e22T23Te23T,G1=AiTPi+PiAi+∑j=1N‍πijPj+εPi+Q1+Q3+τ2W1+d12W2+ϱ12W3+ϱ22W4+ε1α2I+T1Ai+AiTT1T,G2=R1+R2+R5+τ2Y1+d12Y2+ϱ12Y3+ϱ22Y4+τ44Z1+d144Z2+ϱ32Z3+ϱ42Z4-T2-T2T,G3=χ(ε,μ)Q2+T3Bi+BiTT3T+ε2β2I,G4=exp{-εd1}(Q2+Q4-Q3),G5=exp{-εdm}(Q5-Q4),G6=-exp{-εd2}Q5,G7=exp{-εd1}(R3-R2),G8=exp{-εdm}(R4-R3),G9=-exp{-εd2}R4,G10=-εd1exp{εd1}-1W2,Gm=0,(m=11,12,13,14,17,19,20),G15=-exp{-ετ}Q1,G16=-exp{-ετ}R1,G18=-(1-τd)exp{-ετ}R5+T18Ci+CiTT18T+ε3γ2I,G21=-ε1I+T21+T21T,G22=-ε2I+T22+T22T,G23=-ε3I+T23+T23T,Ξi1ε=-εϱ2exp{εd2}-exp{εdm}e14W4e14T-εϱ2exp{εd2}-exp{εdm}(e5-e6)Y4(e5T-e6T)-εϱ1exp{εdm}-exp{εd1}×[e11e13-e11][W3X1X1TW3][e11Te13T-e11T]-εϱ1exp{εdm}-exp{εd1}×[e4-e3e3-e5][Y3X2X2TY3][e4T-e3Te3T-e5T],Ξi2ε=-εϱ1exp{εdm}-exp{εd1}e13W3e13T-εϱ1exp{εdm}-exp{εd1}(e4-e5)Y3(e4T-e5T)-εϱ2exp{εd2}-exp{εdm}×[e12e14-e12][W4X3X3TW4][e12Te14T-e12T]-εϱ2exp{εd2}-exp{εdm}×[e3-e6e5-e3][Y4X4X4TY4][e3T-e6Te5T-e3T],
where ei{i=1,2,…,23} are block entry matrices; that is,
(27)e2T=[0I000000000000000000000],χ(ε,μ)=(μ-1)exp{-εd1}withμ≥1,(μ-1)exp{-εd2}withμ<1,λ¯=maxi∈S{λmax(Pi)}+1-exp{-ερ}ε{∑j=15‍λmax(Qj)+∑k=15‍λmax(Rk)}+exp{-ερ}+ερ-1ε2×{∑l=14‍λmax(Wl)+∑m=14‍λmax(Ym)}+2-2exp{-ερ}+ρ2ε2-2ερ2ε3{∑n=14‍λmax(Zn)},λ_=mini∈S{λmin(Pi)},ϱ1=dm-d1,ϱ2=d2-dm,ϱ3=12(dm2-d12),ϱ4=12(d22-dm2).Proof.
Construct the following Lyapunov functional:(28)V(x(t),i,t)=∑k=16‍Vk(xt,i),
where
(29)V1(xt,i)=xT(t)Pix(t),(30)V2(xt,i)=∫t-τt‍exp{ε(s-t)}xT(s)Q1x(s)ds+∫t-d(t)t-d1‍exp{ε(s-t)}xT(s)Q2x(s)ds+∫t-d1t‍exp{ε(s-t)}xT(s)Q3x(s)ds+∫t-dmt-d1‍exp{ε(s-t)}xT(s)Q4x(s)ds+∫t-d2t-dm‍exp{ε(s-t)}xT(s)Q5x(s)ds,(31)V3(xt,i)=∫t-τt‍exp{ε(s-t)}x˙T(s)R1x˙(s)ds+∫t-d1t‍exp{ε(s-t)}x˙T(s)R2x˙(s)ds+∫t-dmt-d1‍exp{ε(s-t)}x˙T(s)R3x˙(s)ds+∫t-d2t-dm‍exp{ε(s-t)}x˙T(s)R4x˙(s)ds+∫t-τ(t)t‍exp{ε(s-t)}x˙T(s)R5x˙(s)ds,(32)V4(xt,i)=∫-τ0‍∫t+θt‍τexp{ε(s-t)}xT(s)W1x(s)dsdθ+∫-d10‍∫t+θt‍d1exp{ε(s-t)}xT(s)W2x(s)dsdθ+∫-dm-d1‍∫t+θt‍ϱ1exp{ε(s-t)}xT(s)W3x(s)dsdθ+∫-d2-dm‍∫t+θt‍ϱ2exp{ε(s-t)}xT(s)W4x(s)dsdθ,(33)V5(xt,i)=∫-τ0‍∫t+θt‍τexp{ε(s-t)}x˙T(s)Y1x˙(s)dsdθ+∫-d10‍∫t+θt‍d1exp{ε(s-t)}x˙T(s)Y2x˙(s)dsdθ+∫-dm-d1‍∫t+θt‍ϱ1exp{ε(s-t)}x˙T(s)Y3x˙(s)dsdθ+∫-d2-dm‍∫t+θt‍ϱ2exp{ε(s-t)}x˙T(s)Y4x˙(s)dsdθ,(34)V6(xt,i)=∫-τ0‍∫θ0‍∫t+λt‍τ22exp{ε(s-t)}x˙T(s)Z1x˙(s)dsdλdθ+∫-d10‍∫θ0‍∫t+λt‍d122exp{ε(s-t)}x˙T(s)Z2x˙(s)dsdλdθ+∫-dm-d1‍∫θ0‍∫t+λt‍ϱ3exp{ε(s-t)}x˙T(s)Z3x˙(s)dsdλdθ+∫-d2-dm‍∫θ0‍∫t+λt‍ϱ4exp{ε(s-t)}x˙T(s)Z4x˙(s)dsdλdθ.
Taking𝔏 as its infinitesimal generator along the trajectory of system (9), we obtain the following from Definition 5 and (28)–(34):
(35)𝔏V(x(t),i,t)=𝔏V1(xt,i)+𝔏V2(xt,i)+𝔏V3(xt,i)+𝔏V4(xt,i)+𝔏V5(xt,i)+𝔏V6(xt,i),(36)𝔏V1(xt,i)=2[xT(t)AiT+xT(t-d(t))BiT+x˙T(t-τ(t))CiT+f1T+f2T+f3T]×Pix(t)+∑j=1N‍πijxT(t)Pjx(t)+εxT(t)Pix(t)-εV1(xt,i),(37)𝔏V2(xt,i)=xT(t)[Q1+Q3]x(t)+exp{-εd1}xT×(t-d1)[Q2+Q4-Q3]x(t-d1)-εV2(xt,i)-(1-d˙(t))exp{-εd(t)}xT(t-d(t))×Q2x(t-d(t))-exp{-ετ}xT(t-τ)×Q1x(t-τ)+exp{-εdm}×xT(t-dm)[Q5-Q4]×x(t-dm)-exp{-εd2}xT(t-d2)Q5x(t-d2),(38)𝔏V3(xt,i)=x˙T(t)[R1+R2+R5]x˙(t)+exp{-εd1}x˙T(t-d1)[R3-R2]×x˙(t-d1)-εV3(xt,i)+exp{-εdm}x˙T(t-dm)[R4-R3]×x˙(t-dm)-exp{-ετ}×x˙T(t-τ)R1x˙(t-τ)-(1-τ˙(t))exp{-ετ(t)}×x˙T(t-τ(t))R5x˙(t-τ(t))-exp{-εd2}x˙T(t-d2)R4x˙(t-d2),(39)𝔏V4(xt,i)=xT(t)[τ2W1+d12W2+ϱ12W3+ϱ22W4]×x(t)-εV4(xt,i)-∫t-τt‍τexp{ε(s-t)}xT(s)W1x(s)ds-∫t-d1t‍d1exp{ε(s-t)}xT(s)W2x(s)ds-∫t-dmt-d1‍ϱ1exp{ε(s-t)}xT(s)W3x(s)ds-∫t-d2t-dm‍ϱ2exp{ε(s-t)}xT(s)W4x(s)ds,(40)𝔏V5(xt,i)=x˙T(t)[τ2Y1+d12Y2+ϱ12Y3+ϱ22Y4]x˙(t)-εV5(xt,i)-∫t-τt‍τexp{ε(s-t)}x˙T(s)Y1x˙(s)ds-∫t-d1t‍d1exp{ε(s-t)}x˙T(s)Y2x˙(s)ds-∫t-dmt-d1‍ϱ1exp{ε(s-t)}x˙T(s)Y3x˙(s)ds-∫t-d2t-dm‍ϱ2exp{ε(s-t)}x˙T(s)Y4x˙(s)ds,(41)𝔏V6(xt,i)=x˙T(t)×[τ44Z1+d144Z2+ϱ32Z3+ϱ42Z4]x˙(t)-εV6(xt,i)-∫-τ0‍∫t+θt‍τ22exp{ε(s-t)}x˙T(s)εV6(xt,id)×Z1x˙(s)dsdθ-∫-d10‍∫t+θt‍d122exp{ε(s-t)}x˙T(s)εV6(xt,di)×Z2x˙(s)dsdθ-∫-dm-d1‍∫t+θt‍ϱ3exp{ε(s-t)}x˙T(s)εV6(xtd,i)×Z3x˙(s)dsdθ-∫-d2-dm‍∫t+θt‍ϱ4exp{ε(s-t)}x˙T(s)εV6(xt,id)×Z4x˙(s)dsdθ.
In view of (6), the following inequalities hold for any scalars ε1>0, ε2>0, and ε3>0:
(42)ε1[α2xT(t)x(t)-f1T(x(t),t)f1(x(t),t)]≥0,ε2[β2xT(t-d(t))x(t-d(t))-f2T(x(t-d(t)),t)f2(x(t-d(t)),t)]≥0,ε3[γ2x˙T(t-τ(t))x˙(t-τ(t))-f3T(x˙(t-τ(t)),t)f3(x˙(t-τ(t)),t)]≥0.
From (35) and (42), we have
(43)𝔏V(x(t),i,t)≤𝔏V1(xt,i)+𝔏V2(xt,i)+𝔏V3(xt,i)+𝔏V4(xt,i)+𝔏V5(xt,i)+𝔏V6(xt,i)+ε1[α2xT(t)x(t)-f1Tf1]+ε2[β2xT(t-d(t))x(t-d(t))-f2Tf2]+ε3[γ2x˙T(t-τ(t))x˙(t-τ(t))-f3Tf3].
Define(44)ξ(t)=col{∫t-d(t)t-d1‍x(t)x˙(t)x(t-d(t))x(t-d1)x(t-dm)x(t-d2)x˙(t-d1)x˙(t-dm)x˙(t-d2)∫t-d1t‍x(s)ds∫t-d(t)t-d1‍x(s)ds∫t-d2t-d(t)x(s)ds‍∫t-dmt-d1‍x(s)ds∫t-d2t-dm‍x(s)dsx(t-τ)x˙(t-τ)x(t-τ(t))x˙(t-τ(t))∫t-τ(t)t‍x(s)ds∫t-τt‍x(s)dsf1f2f3}.
Since it is easy to see-(1-τ˙(t))exp{-ετ(t)}≤-(1-τd)exp{-ετ}; then from (5) we also obtain that
(45)-(1-d˙(t))exp{-εd(t)}≤χ(ε,μ),
where χ(ε,μ) is defined in Theorem 10. Notice (a) of Lemma 7; then
(46)-∫t-d1t‍d1exp{ε(s-t)}xT(s)W2x(s)ds≤-εd1exp{εd1}-1ξT(t)e10W2e10Tξ(t),-∫t-d1t‍d1exp{ε(s-t)}x˙T(s)Y2x˙(s)ds≤-εd1exp{εd1}-1ξT(t)(e1-e4)Y2(e1T-e4T)ξ(t).
Notice (b) of Lemma7; then
(47)-∫-τ0‍∫t+θt‍τ22exp{ε(s-t)}x˙T(s)Z1x˙(s)dsdθ≤-exp{-ετ}ξT(t)(τe1-e20)Z1(τe1T-e20T)ξ(t),-∫-d10‍∫t+θt‍d122exp{ε(s-t)}x˙T(s)Z2x˙(s)dsdθ≤-exp{-εd1}ξT(t)(d1e1-e10)Z2(d1e1T-e10T)ξ(t),-∫-dm-d1‍∫t+θt‍ϱ3exp{ε(s-t)}x˙T(s)Z3x˙(s)dsdθ≤-exp{-εdm}ξT(t)(ϱ1e1-e13)Z3(ϱ1e1T-e13T)ξ(t),-∫-d2-dm‍∫t+θt‍ϱ4exp{ε(s-t)}x˙T(s)Z4x˙(s)dsdθ≤-exp{-εd2}ξT(t)(ϱ2e1-e14)Z4(ϱ2e1T-e14T)ξ(t).
Ford(t)∈[d1,dm], the following is held from (a) of Lemma 7:
(48)-∫t-dmt-d1‍ϱ1exp{ε(s-t)}x˙T(s)Y3x˙(s)ds=-{∫t-d(t)t-d1‍+∫t-dmt-d(t)‍}ϱ1exp{ε(s-t)}x˙T(s)Y3x˙(s)ds≤-εϱ1exp{εd(t)}-exp{εd1}[∫t-d(t)t-d1‍x˙T(s)ds]×Y3[∫t-d(t)t-d1‍x˙(s)ds]-εϱ1exp{εdm}-exp{εd(t)}[∫t-dmt-d(t)‍x˙T(s)ds]×Y3[∫t-dmt-d(t)‍x˙(s)ds]=-εϱ1exp{εdm}-exp{εd1}ξT(t)×[1λ1(t)(e4-e3)Y3(e4T-e3T)(e4Tff)+1λ2(t)(e3-e5)Y3(e3T-e5T)]ξ(t),
where
(49)λ1(t)=exp{εd(t)}-exp{εd1}exp{εdm}-exp{εd1},λ2(t)=exp{εdm}-exp{εd(t)}exp{εdm}-exp{εd1}.
By Lemma8, there exists matrix X2 with appropriate dimensions such that
(50)-∫t-dmt-d1‍ϱ1exp{ε(s-t)}x˙T(s)Y3x˙(s)ds≤-εϱ1exp{εdm}-exp{εd1}ξT(t)×[e4-e3e3-e5][Y3X2X2TY3][e4T-e3Te3T-e5T]ξ(t),[Y3X2X2TY3]>0.
Similarly, considering-∫t-dmt-d1‍ϱ1exp{ε(s-t)}xT(s)W3x(s)ds and following the same procedure, there exists matrix X1 with appropriate dimensions such that
(51)-∫t-dmt-d1‍ϱ1exp{ε(s-t)}xT(s)W3x(s)ds≤-εϱ1exp{εdm}-exp{εd1}ξT(t)×[e11e13-e11][W3X1X1TW3][e11Te13T-e11T]ξ(t),[W3X1X1TW3]>0.
Forτ(t)∈[0,τ], with the same matrix inequalities technique, we obtain the following:
(52)-∫t-τt‍τexp{ε(s-t)}xT(s)W1x(s)ds≤-ετexp{ετ}-1ξT(t)×[e19e20-e19][W1U1U1TW1][e19Te20T-e19T]ξ(t),[W1U1U1TW1]>0,-∫t-τt‍τexp{ε(s-t)}x˙T(s)Y1x˙(s)ds≤-ετexp{ετ}-1ξT(t)×[e1-e17e17-e15][Y1U2U2TY1][e1T-e17Te17T-e15T]ξ(t),[Y1U2U2TY1]>0.
Consider-∫t-d2t-dm‍ϱ2exp{ε(s-t)}xT(s)W4x(s)ds  and  -∫t-d2t-dm‍ϱ2exp{ε(s-t)}x˙T(s)Y4x˙(s)ds, which are directly estimated by (a) of Lemma 7; that is,
(53)-∫t-d2t-dm‍ϱ2exp{ε(s-t)}xT(s)W4x(s)ds≤-εϱ2exp{εd2}-exp{εdm}ξT(t)e14W4e14Tξ(t),-∫t-d2t-dm‍ϱ2exp{ε(s-t)}x˙T(s)Y4x˙(s)ds≤-εϱ2exp{εd2}-exp{εdm}ξT(t)×(e5-e6)Y4(e5T-e6T)ξ(t).
In addition, there exist matricesT=col{Tk,(k=1,2,…,23)} with appropriate dimensions, such that the following equality holds according to (9):
(54)2ξT(t)T[-x˙(t)+Aix(t)+Bix(t-d(t))+Cix˙(t-τ(t))+f1+f2+f3]=0.
Substituting (36)–(42) and (45)–(54) into (43), we obtain
(55)𝔏V(x(t),i,t)+εV(x(t),i,t)≤ξT(t)(Ξi0ε+Ξi1ε)ξ(t).
On the other hand, ford(t)∈[dm,d2], the integral terms
(56)-∫t-d2t-dm‍ϱ2exp{ε(s-t)}xT(s)W4x(s)ds,-∫t-d2t-dm‍ϱ2exp{ε(s-t)}x˙T(s)Y4x˙(s)ds.
The above equations: are disposed and estimated by Lemma 8. 									(57)-∫t-dmt-d1‍ϱ1exp{ε(s-t)}xT(s)W3x(s)ds,-∫t-dmt-d1‍ϱ1exp{ε(s-t)}x˙T(s)Y3x˙(s)ds
are directly estimated by (a) of Lemma 7. Therefore,
(58)𝔏V(x(t),i,t)+εV(x(t),i,t)≤ξT(t)(Ξi0ε+Ξi2ε)ξ(t).
With (55) and (58), the following inequality (59) is held for d(t)∈[d1,d2] if (22), (23), and (24) are satisfied
(59)𝔏V(x(t),i,t)+εV(x(t),i,t)<0.
From the Lyapunov functional (28) and (59), it is held that
(60)V(x(t),i,t)≥mini∈S{λmin(Pi)}∥x(t)∥2=λ_∥x(t)∥2,V(x(t),i,t)<exp{-ε(t-t0)}V(xt0,rt0,t0).
Moreover, we have(61)V(xt0,rt0,t0)=∑k=16‍Vk(xt0,rt0)≤λ¯∥xt0∥ρ2.
Then, from (60) and (61), it is readily seen that
(62)∥x(t)∥≤λ¯λ_exp{-ε2(t-t0)}∥xt0∥ρ,
where σ=λ¯/λ_≥1.
Therefore, by Definition4, the system (9) is exponentially stable with a decay rate ε/2. This completes the proof.Remark 11.
It is noted that the integral intervals in (61) are enlarged as follows:
(63)∑k=16‍Vk(xt0,rt0)≤xT(t0)Pix(t0)+∫t0-ρt0‍exp{ε(s-t0)}xT(s)(∑j=15‍Qj)x(s)ds+∫t0-ρt0‍exp{ε(s-t0)}x˙T(s)(∑k=15‍Rk)x˙(s)ds+∫t-ρ0‍∫t0+θt0‍ρexp{ε(s-t0)}xT(t0)Pixf(t0)×xT(s)(∑l=14‍Wl)x(s)dsdθ+∫-ρ0‍∫t0+θt0‍ρexp{ε(s-t0)}xT(t0)Pixf(t0)×x˙T(s)(∑m=14‍Ym)x˙(s)dsdθ+∫-ρ0‍∫θ0‍∫t0+λt0‍ρ22exp{ε(s-t0)}xT(t0)Pixf(t0)×x˙T(s)(∑n=14‍Zn)x˙(s)dsdλdθ.
Equation (61) can be obtained by letting λ¯ be defined as previously.Remark 12.
In Theorem10, the factors
(64)exp{εd2}-exp{εd(t)}ε,exp{εd(t)}-exp{εd1}ε
may be enlarged as (exp{εd2}-exp{εd1})/ε. This will lead conservative results due to the fact that d(t) cannot achieve d1 and d2 at the same time. While we apply Lemma 7 to these terms, the method by using reciprocally convex lemma [34] can achieve less conservative results. Moreover, for ε>0, the factor (d˙(t)-1)exp{-εd(t)} that appeared in the derivative of Lyapunov functional may be directly enlarged as μexp{-εd1}-exp{-εd2}. In this paper, we enlarge it as χ(ε,μ) to reduce the conservativeness of the obtained criteria. In the literature [32, 36, 37], this factor is enlarged as (μ-1)exp{-εd2}, which only holds for μ<1.Remark 13.
The proposed Lyapunov functional (28) contains some triple-integral terms, which has not been used in any of the existing literatures in the same context before. Compared with the existing ones, [33] has shown that such a Lyapunov functional type is very effective in the reduction of conservatism. Besides, the information on the lower bound of the delay is sufficiently used in the Lyapunov functional by introducing the terms such as ∫t-dmt-d1‍exp{ε(s-t)}xT(s)Q4x(s)ds, ∫t-d2t-dm‍exp{ε(s-t)}xT(s)Q5x(s)ds, and ∫t-d(t)t-d1‍exp{ε(s-t)}xT(s)Q2x(s)ds.Remark 14.
It should be also mentioned that the result obtained in Theorem10 is delay-range-dependent and decay rate-dependent stability condition for (9), 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 τ(t). That is, τ1≤τ(t)≤τ2. The results can be obtained by using the similar methods.The information on the delay derivative may not be available in many cases. The following corollary is therefore given, which can be obtained from Theorem10 by setting Q2=0 and R5=0.Corollary 15.
For the given finite setS of modes with transition rates matrix, scalars α, β, γ, ε, τ, d1, d2, and constant scalar dm satisfying d1<dm<d2, the systems described by (9) are exponentially stable with decay rate ε/2 and σ=λ¯/λ_ if the operator 𝔇 is stable, and there exist symmetric positive matrices Pi>0, (i∈S), Qj>0, (j=1,3,4,5), Rk>0, (k=1,2,3,4), Wl>0, (l=1,2,3,4), Ym>0, (m=1,2,3,4), Zn>0, (n=1,2,3,4) and matrices U1, U2, X1, X2, X3, X4 for any scalars ε1, ε2, ε3 and any matrices Tk, (k=1,2,…,23) with appropriate dimensions, such that linear matrix inequalities (22) and (65) hold:
(65)(i)[W3X1X1TW3]>0,[Y3X2X2TY3]>0,(ii)Ξ~i0ε+Ξi1ε<0,(iii)[W4X3X3TW4]>0,[Y4X4X4TY4]>0,(iv)Ξ~i0ε+Ξi2ε<0,
where
(66)Ξ~i0ε=∑m=123‍emG~memT+𝒜(✠)-εd1exp{εd1}-1(e1-e4)Y2(e1T-e4T)-exp{-ετ}(τe1-e20)Z1(τe1T-e20T)-exp{-εd1}(d1e1-e10)Z2(d1e1T-e10T)-exp{-εdm}(ϱ1e1-e13)Z3(ϱ1e1T-e13T)-exp{-εd2}(ϱ2e1-e14)Z4(ϱ2e1T-e14T)-ετexp{ετ}-1[e19e20-e19]×[W1U1U1TW1][e19Te20T-e19T]-ετexp{ετ}-1[e1-e17e17-e15]×[Y1U2U2TY1][e1T-e17Te17T-e15T],
where
(67)G~2=R1+R2+τ2Y1+d12Y2+ϱ12Y3+ϱ22Y4+τ44Z1+d144Z2+ϱ32Z3+ϱ42Z4-T2-T2T,G~3=T3Bi+BiTT3T+ε2β2I,G~4=exp{-εd1}(Q4-Q3),G~18=T18Ci+CiTT18T+ε3γ2I.
Other G~m=Gm, (m≠2,3,4,18), which have been defined previously. 									(68)λ¯=maxi∈S{λmax(Pi)}+1-exp{-ερ}ε{∑j=1,j≠25‍λmax(Qj)+∑k=14‍λmax(Rk)}+exp{-ερ}+ερ-1ε2{∑l=14‍λmax(Wl)+∑m=14‍λmax(Ym)}+2-2exp{-ερ}+ρ2ε2-2ερ2ε3{∑n=14‍λmax(Zn)},
and the remaining notations are the same as Theorem 10.In Theorem10, it is assumed that ε≠0. For ε=0, by L'Hospital’s rule, the following asymptotic stability criterion can be obtained.Corollary 16.
For the given finite setS of modes with transition rates matrix, scalars α, β, γ, τ, τd, d1, d2, μ and constant scalar dm satisfying d1<dm<d2, the systems described by (9) are asymptotically stable if the operator 𝔇 is stable, and there exist symmetric positive matrices Pi>0, (i∈S), Qj>0, (j=1,2,3,4,5), Rk>0, (k=1,2,3,4,5), Wl>0, (l=1,2,3,4), Ym>0, (m=1,2,3,4), Zn>0, (n=1,2,3,4) and matrices U1, U2, X1, X2, X3, X4 for any scalars ε1, ε2, ε3 and any matrices Tk, (k=1,2,…,23) with appropriate dimensions, such that linear matrix inequalities (22) and (69) hold:
(69)(i)[W3X1X1TW3]>0,[Y3X2X2TY3]>0,(ii)Ξi0+Ξi1<0,(iii)[W4X3X3TW4]>0,[Y4X4X4TY4]>0,(iv)Ξi0+Ξi2<0,
where
(70)Ξi0=∑m=123‍emG¯memT+𝒜(✠)-(e1-e4)Y2(e1T-e4T)-(τe1-e20)Z1(τe1T-e20T)-(d1e1-e10)Z2(d1e1T-e10T)-(ϱ1e1-e13)Z3(ϱ1e1T-e13T)-(ϱ2e1-e14)Z4(ϱ2e1T-e14T)-[e19e20-e19][W1U1U1TW1][e19Te20T-e19T]-[e1-e17e17-e15][Y1U2U2TY1][e1T-e17Te17T-e15T],G¯1=AiTPi+PiAi+∑j=1N‍πijPj+Q1+Q3+τ2W1+d12W2+ϱ12W3+ϱ22W4+ε1α2I+T1Ai+AiTT1T,G¯3=(μ-1)Q2+T3Bi+BiTT3T+ε2β2I,G¯4=Q2+Q4-Q3,G¯5=Q5-Q4,G¯6=-Q5,G¯7=R3-R2,G¯8=R4-R3,G¯9=-R4,G¯10=-W2,G¯15=-Q1,G¯16=-R1,G¯18=(τd-1)R5+T18Ci+CiTT18T+ε3γ2I.
Other G¯m=Gm, (m=2,11,12,13,14,17,19,…,23), which have been defined in Theorem 10:
(71)Ξi1=-e14W4e14T-(e5-e6)Y4(e5T-e6T)-[e11e13-e11][W3X1X1TW3][e11Te13T-e11T]-[e4-e3e3-e5][Y3X2X2TY3][e4T-e3Te3T-e5T],Ξi2=-e13W3e13T-(e4-e5)Y3(e4T-e5T)-[e12e14-e12][W4X3X3TW4][e12Te14T-e12T]-[e3-e6e5-e3][Y4X4X4TY4][e3T-e6Te5T-e3T],
and the remaining notations are the same as Theorem 10.


### 3.2. Exponential Stability for the Nominal Systems with Partially Unknown Transition Rates

In this subsection, we take into account the situation that the information of transition rates is not accessible completely and propose the following conditions to guarantee the exponential stability of system (9) with partially unknown transition rates.Since the transition rates of the Markov chain are partially unknown, then some elements in matrixΠ=[πij]N×N are inaccessible. For instance, the system (3) with five operation modes, the jump rates matrix Π may be viewed as
(72)[?π12??π15π21??π24???π33π34??π42???π51???π55],
where ? represents the unknown element. For notation clarity, we denote that 𝒮i=𝒮ki⋃‍𝒮uki,foralli∈S and
(73)𝒮ki≜{j:πijisknownforj∈S},𝒮uki≜{j:πijisunknownforj∈S}.
If 𝒮ki≠∅, it is further described as
(74)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. Specially, when 𝒮uki=∅, 𝒮i=𝒮ki, the full information about Π is obtained, and it becomes the case of Section 3.1.Theorem 17.
For the given finite setS of modes with partly unknown transition rates matrix, scalars α, β, γ, ε, τ, τd, d1, d2, and μ, and constant scalar dm satisfying d1<dm<d2, the systems described by (9) with partly unknown transition rates are exponentially stable with decay rate ε/2 and σ=λ¯/λ_ if the operator 𝔇 is stable, and there exist symmetric positive matrices Pi>0, (i∈S), Qj>0, (j=1,2,3,4,5), Rk>0, (k=1,2,3,4,5), Wl>0, (l=1,2,3,4), Ym>0, (m=1,2,3,4), Zn>0, (n=1,2,3,4) and matrices U1, U2, X1, X2, X3, X4  for any scalars ε1, ε2, ε3 and any symmetric matrices Vi, (i∈S) and any matrices Tk, (k=1,2,…,23) with appropriate dimensions, such that linear matrix inequalities (22), (75), (76), and (77) hold:
(75)Pj-Vi≤0,j∈𝒮uki,j≠i,Pj-Vi≥0,j∈𝒮uki,j=i,(76)(i)[W3X1X1TW3]>0,[Y3X2X2TY3]>0,(ii)Ξ𝒮i0ε+Ξ𝒮i1ε<0,(77)(iii)[W4X3X3TW4]>0,[Y4X4X4TY4]>0,(iv)Ξ𝒮i0ε+Ξ𝒮i2ε<0,
where
(78)Ξ𝒮i0ε=∑m=123‍emG𝒮memT+𝒜(✠)-εd1exp{εd1}-1(e1-e4)Y2(e1T-e4T)-exp{-ετ}(τe1-e20)Z1(τe1T-e20T)-exp{-εd1}(d1e1-e10)Z2(d1e1T-e10T)-exp{-εdm}(ϱ1e1-e13)Z3(ϱ1e1T-e13T)-exp{-εd2}(ϱ2e1-e14)Z4(ϱ2e1T-e14T)-ετexp{ετ}-1[e19e20-e19]×[W1U1U1TW1][e19Te20T-e19T]-ετexp{ετ}-1[e1-e17e17-e15]×[Y1U2U2TY1][e1T-e17Te17T-e15T],G𝒮1=AiTPi+PiAi+∑j∈𝒮kiN‍πij(Pj-Vi)+εPi+Q1+Q3+τ2W1+d12W2+ϱ12W3+ϱ22W4+ε1α2I+T1Ai+AiTT1T,G𝒮m=Gm,(m≠1),Ξ𝒮i1ε=Ξi1ε,Ξ𝒮i2ε=Ξi2ε,
and the remaining notations are the same as Theorem 10.Proof.
Choose the same Lyapunov functional as (28), and V2(xt,i), V3(xt,i), V4(xt,i), V5(xt,i), and V6(xt,i) are disposed with the identical method in Theorem 10. Then, giving consideration on ∑j=1N‍πij=0, the following equation holds for arbitrary matrices Vi=ViT, i∈S; that is,
(79)-xT(t)(∑j=1N‍πijVi)x(t)=0.
With (36) and (79), we obtain the following:
(80)𝔏V1(xt,i)=2[xT(t)AiT+xT(t-d(t))BiT+x˙T(t-τ(t))CiT+f1T+f2T+f3T]Pix(t)+xT(t)[∑j∈𝒮kiN‍πij(Pj-Vi)]x(t)+xT(t)[∑j∈𝒮ukiN‍πij(Pj-Vi)]x(t)+εxT(t)Pix(t)-εV1(xt,i).
Ford(t)∈[d1,dm] and d(t)∈[dm,d2], with (80) and on the basis of the result in Theorem 10, we have the following equations, respectively:
(81)𝔏V(x(t),i,t)+εV(x(t),i,t)≤ξT(t)(Ξ𝒮i0ε+Ξ𝒮i1ε)ξ(t)+xT(t)[∑j∈𝒮ukiN‍πij(Pj-Vi)]x(t),𝔏V(x(t),i,t)+εV(x(t),i,t)≤ξT(t)(Ξ𝒮i0ε+Ξ𝒮i2ε)ξ(t)+xT(t)[∑j∈𝒮ukiN‍πij(Pj-Vi)]x(t).
With (81), due to λii=-∑j=1,j≠iN‍λij and λij>0, for all i≠j, (59) is also held for d(t)∈[d1,d2] if linear matrix inequalities (22), (75), (76), and (77) are satisfied. With the same procedure in the latter proof of Theorem 10, we draw the conclusion that the system (9) with partially unknown transition rates is exponentially stable with a decay rate ε/2. This completes the proof.Consider the system (9) with partially unknown transition rates, and the following corollaries are given, for unknown information on the delay derivative and ε=0, respectively.Corollary 18.
For the given finite setS of modes with partly unknown transition rates matrix, scalars α, β, γ, ε, τ, d1, and d2, and constant scalar dm satisfying d1<dm<d2, the systems described by (9) with partly unknown transition rates are exponentially stable with decay rate ε/2 and σ=λ¯/λ_ if the operator 𝔇 is stable, and there exist symmetric positive matrices Pi>0, (i∈S), Qj>0, (j=1,3,4,5), Rk>0, (k=1,2,3,4), Wl>0, (l=1,2,3,4), Ym>0, (m=1,2,3,4), Zn>0, (n=1,2,3,4) and matrices U1, U2, X1, X2, X3, X4   for any scalars ε1, ε2, ε3 and any symmetric matrices Vi, (i∈S) and any matrices Tk, (k=1,2,…,23) with appropriate dimensions, such that linear matrix inequalities (22), (75), and (82) hold:
(82)(i)[W3X1X1TW3]>0,[Y3X2X2TY3]>0,(ii)Ξ~𝒮i0ε+Ξ𝒮i1ε<0,(iii)[W4X3X3TW4]>0,[Y4X4X4TY4]>0,(iv)Ξ~𝒮i0ε+Ξ𝒮i2ε<0,
where
(83)Ξ~𝒮i0ε=∑m=123‍emG~𝒮memT+𝒜(✠)-εd1exp{εd1}-1(e1-e4)Y2(e1T-e4T)-exp{-ετ}(τe1-e20)Z1(τe1T-e20T)-exp{-εd1}(d1e1-e10)Z2(d1e1T-e10T)-exp{-εdm}(ϱ1e1-e13)Z3(ϱ1e1T-e13T)-exp{-εd2}(ϱ2e1-e14)Z4(ϱ2e1T-e14T)-ετexp{ετ}-1[e19e20-e19]×[W1U1U1TW1][e19Te20T-e19T]-ετexp{ετ}-1[e1-e17e17-e15]×[Y1U2U2TY1][e1T-e17Te17T-e15T],G~𝒮1=G𝒮1,G~𝒮m=G~m,(m≠1),Ξ𝒮i1ε=Ξi1ε,Ξ𝒮i2ε=Ξi2ε,
and the remaining notations are the same as Corollary 15.Corollary 19.
For the given finite setS of modes with partly unknown transition rates matrix, scalars α, β, γ, τ, τd, d1, d2, μ, and constant scalar dm satisfying d1<dm<d2, the systems described by (9) with partly unknown transition rates are exponentially stable with decay rate ε/2 and σ=λ¯/λ_ if the operator 𝔇 is stable, and there exist symmetric positive matrices Pi>0, (i∈S), Qj>0, (j=1,2,3,4,5), Rk>0, (k=1,2,3,4,5), Wl>0, (l=1,2,3,4), Ym>0, (m=1,2,3,4), Zn>0, (n=1,2,3,4) and matrices U1, U2, X1, X2, X3, X4  for any scalars ε1, ε2, ε3 and any symmetric matrices Vi, (i∈S) and any matrices Tk, (k=1,2,…,23) with appropriate dimensions, such that linear matrix inequalities (22), (75), and (84) hold:
(84)(i)[W3X1X1TW3]>0,[Y3X2X2TY3]>0,(ii)Ξ𝒮i0+Ξ𝒮i1<0,(iii)[W4X3X3TW4]>0,[Y4X4X4TY4]>0,(iv)Ξ𝒮i0+Ξ𝒮i2<0,
where
(85)Ξ𝒮i0=∑m=123‍emG¯𝒮memT+𝒜(✠)-(e1-e4)Y2(e1T-e4T)-(τe1-e20)Z1(τe1T-e20T)-(d1e1-e10)Z2(d1e1T-e10T)-(ϱ1e1-e13)Z3(ϱ1e1T-e13T)-(ϱ2e1-e14)Z4(ϱ2e1T-e14T)-[e19e20-e19][W1U1U1TW1][e19Te20T-e19T]-[e1-e17e17-e15]×[Y1U2U2TY1][e1T-e17Te17T-e15T],G¯𝒮1=AiTPi+PiAi+∑j∈𝒮kiN‍πij(Pj-Vi)+Q1+Q3+τ2W1+d12W2+ϱ12W3+ϱ22W4+ε1α2I+T1Ai+AiTT1T,G¯𝒮m=G¯m,(m≠1),Ξ𝒮i1=Ξi1,Ξ𝒮i2=Ξi2,
and the remaining notations are the same as Corollary 16.Remark 20.
Corollaries16 and 19 provide new delay-range-dependent asymptotic stability conditions for the systems described by (9) with known and partially unknown transition rates. Though reciprocally convex lemma has not helped here, the results are still less conservative than some previous ones because of some triple-integral terms, which will be verified in Section 4.


### 3.3. Extension to the Uncertain Case

In this subsection, the uncertain neutral delay Markovian jump systems with nonlinear perturbations described by (3) with known and partially unknown transition rates are, respectively, considered. The corresponding conditions are presented in the following theorems and corollaries.Theorem 21.
For the given finite setS of modes with transition rates matrix, scalars α, β, γ, ε, τ, τd, d1, d2, μ, and constant scalar dm satisfying d1<dm<d2, the uncertain neutral systems described by (3) are exponentially stable with decay rate ε/2 and σ=λ¯/λ_ if the operator 𝔇 is stable, and there exist scalars δ1>0, δ2>0, symmetric positive matrices Pi>0, (i∈S), Qj>0, (j=1,2,3,4,5), Rk>0, (k=1,2,3,4,5), Wl>0, (l=1,2,3,4), Ym>0, (m=1,2,3,4), Zn>0, (n=1,2,3,4), and matrices U1, U2, X1, X2, X3, X4   for any scalars ε1, ε2, ε3 and any matrices Tk, (k=1,2,…,23) with appropriate dimensions, such that (22), (86), and (87) hold:
(86)(i)[W3X1X1TW3]>0,[Y3X2X2TY3]>0,(ii)[δ1Ξi0ε+δ1Ξi1ε+δ12MTMNT*-I]<0,(87)(iii)[W4X3X3TW4]>0,[Y4X4X4TY4]>0,(iv)[δ2Ξi0ε+δ2Ξi2ε+δ22MTMNT*-I]<0,
where
(88)M=e1EAiT+e3EBiT,N=HiTPiTe1+∑m=123‍HiTTmTem,
and other notations are the same as Theorem 10.Proof.
DefineE=col{em,(m=1,2,…,23)} and T^=col{Pi+T1,Tm(m=2,…,23)}, and we replace Ai, Bi with Ai+ΔAi(t), Bi+ΔBi(t) on the basis of Theorem 10; that is,
(89)(i)[W3X1X1TW3]>0,[Y3X2X2TY3]>0,(ii)Ξi0ε+Ξi1ε+𝒜[e1ΔAiT(t)T^TE+e3ΔBiT(t)T^TE]<0,(90)(iii)[W4X3X3TW4]>0,[Y4X4X4TY4]>0,(iv)Ξi0ε+Ξi2ε+𝒜[e1ΔAiT(t)T^TE+e3ΔBiT(t)T^TE]<0.
Considering (ii) of (89) and combining the uncertainties condition (7), we have
(91)Ξi0ε+Ξi1ε+𝒜{[e1e3][EAiTEBiT]FiT(t)HiTT^TE}<0.
By the definition of 𝒜, we obtain
(92)Ξi0ε+Ξi1ε+MFiT(t)N+NTFi(t)MT<0.
According to (8), by Lemmas 9 and 6, with (92), we obtain (ii) of (86). Following the same procedure, (90) is considered and (iv) of (87) can be obtained. Finally, following the proof of Theorem 10, the systems described by (3) are exponentially stable with a decay rate ε/2. This completes the proof.Theorem 22.
For the given finite setS of modes with partly unknown transition rates matrix, scalars α, β, γ, ε, τ, τd, d1, d2, μ, and constant scalar dm satisfying d1<dm<d2, the uncertain neutral systems described by (3) with partly unknown transition rates are exponentially stable with decay rate ε/2 and σ=λ¯/λ_ if the operator 𝔇 is stable, and there exist scalars δ1>0, δ2>0, symmetric positive matrices Pi>0, (i∈S), Qj>0, (j=1,2,3,4,5), Rk>0, (k=1,2,3,4,5), Wl>0, (l=1,2,3,4), Ym>0, (m=1,2,3,4), Zn>0, (n=1,2,3,4), and matrices U1, U2, X1, X2, X3, X4   for any scalars ε1, ε2, ε3 and any symmetric matrices Vi, (i∈S) and any matrices Tk, (k=1,2,…,23) with appropriate dimensions, such that (22), (75), and (93) hold:
(93)(i)[W3X1X1TW3]>0,[Y3X2X2TY3]>0,(ii)[δ1Ξℐi0ε+δ1Ξℐi1ε+δ12MTMNT*-I]<0,(iii)[W4X3X3TW4]>0,[Y4X4X4TY4]>0,(iv)[δ2Ξℐi0ε+δ2Ξℐi2ε+δ22MTMNT*-I]<0,
where
(94)M=e1EAiT+e3EBiT,N=HiTPiTe1+∑m=123‍HiTTmTem
and other notations are the same as Theorem 17.The proof of Theorem22 is omitted here because it is identical with the proof of Theorem 21. In addition, considering the system (3) with fully or partly known transition rates, the corollaries for unknown information on the delay derivative and ε=0 are also omitted here because they are uniform results as the nominal case.


## 3.1. Exponential Stability for the Nominal Systems with Known Transition Rates

In this subsection, we consider the full information on the transition rates and give the following conditions to guarantee the exponential stability of the nominal systems.Theorem 10.
For the given finite setS of modes with transition rates matrix, scalars α, β, γ, ε, τ, τd, d1, d2, μ, and constant scalar dm satisfying d1<dm<d2, the systems described by (9) are exponentially stable with decay rate ε/2 and σ=λ¯/λ_ if the operator 𝔇 is stable, and there exist symmetric positive matrices Pi>0, (i∈S), Qj>0, (j=1,2,3,4,5), Rk>0, (k=1,2,3,4,5), Wl>0, (l=1,2,3,4), Ym>0, (m=1,2,3,4), Zn>0, (n=1,2,3,4) and matrices U1, U2, X1, X2, X3, X4  for any scalars ε1, ε2, ε3 and any matrices Tk, (k=1,2,…,23) with appropriate dimensions, such that the following linear matrix inequalities (22), (23), and (24) hold:
(22)[W1U1U1TW1]>0,[Y1U2U2TY1]>0,(23)(i)[W3X1X1TW3]>0,[Y3X2X2TY3]>0,(ii)Ξi0ε+Ξi1ε<0,(24)(iii)[W4X3X3TW4]>0,[Y4X4X4TY4]>0,(iv)Ξi0ε+Ξi2ε<0,
where
(25)Ξi0ε=∑m=123‍emGmemT+𝒜(✠)-εd1exp{εd1}-1(e1-e4)Y2(e1T-e4T)-exp{-ετ}(τe1-e20)Z1(τe1T-e20T)-exp{-εd1}(d1e1-e10)Z2(d1e1T-e10T)-exp{-εdm}(ϱ1e1-e13)Z3(ϱ1e1T-e13T)-exp{-εd2}(ϱ2e1-e14)Z4(ϱ2e1T-e14T)-ετexp{ετ}-1[e19e20-e19][W1U1U1TW1][e19Te20T-e19T]-ετexp{ετ}-1[e1-e17e17-e15]×[Y1U2U2TY1][e1T-e17Te17T-e15T],
where 𝒜 is a linear operator on ℝn×n by
(26)𝒜(F)=F+FT,∀F∈ℝn×n,✠=-e1T1e2T+e1(PiBi+T1Bi)e3T+e1(PiCi)e18T+∑m=223‍e1(AiTTmT)emT+∑m=2123‍e1(Pi)emT+e2(T2Bi)e3T-∑m=323‍e2(TmT)emT+∑m=423‍e3(BiTTmT)emT+∑m=117‍em(TmCi)e18T+∑m=1923‍e18(CiTTmT)emT+∑m=120‍∑n=2123‍emTmenT+e21T21(e22T+e23T)+e22T22e23T+∑m=2223‍e21TmTemT+e22T23Te23T,G1=AiTPi+PiAi+∑j=1N‍πijPj+εPi+Q1+Q3+τ2W1+d12W2+ϱ12W3+ϱ22W4+ε1α2I+T1Ai+AiTT1T,G2=R1+R2+R5+τ2Y1+d12Y2+ϱ12Y3+ϱ22Y4+τ44Z1+d144Z2+ϱ32Z3+ϱ42Z4-T2-T2T,G3=χ(ε,μ)Q2+T3Bi+BiTT3T+ε2β2I,G4=exp{-εd1}(Q2+Q4-Q3),G5=exp{-εdm}(Q5-Q4),G6=-exp{-εd2}Q5,G7=exp{-εd1}(R3-R2),G8=exp{-εdm}(R4-R3),G9=-exp{-εd2}R4,G10=-εd1exp{εd1}-1W2,Gm=0,(m=11,12,13,14,17,19,20),G15=-exp{-ετ}Q1,G16=-exp{-ετ}R1,G18=-(1-τd)exp{-ετ}R5+T18Ci+CiTT18T+ε3γ2I,G21=-ε1I+T21+T21T,G22=-ε2I+T22+T22T,G23=-ε3I+T23+T23T,Ξi1ε=-εϱ2exp{εd2}-exp{εdm}e14W4e14T-εϱ2exp{εd2}-exp{εdm}(e5-e6)Y4(e5T-e6T)-εϱ1exp{εdm}-exp{εd1}×[e11e13-e11][W3X1X1TW3][e11Te13T-e11T]-εϱ1exp{εdm}-exp{εd1}×[e4-e3e3-e5][Y3X2X2TY3][e4T-e3Te3T-e5T],Ξi2ε=-εϱ1exp{εdm}-exp{εd1}e13W3e13T-εϱ1exp{εdm}-exp{εd1}(e4-e5)Y3(e4T-e5T)-εϱ2exp{εd2}-exp{εdm}×[e12e14-e12][W4X3X3TW4][e12Te14T-e12T]-εϱ2exp{εd2}-exp{εdm}×[e3-e6e5-e3][Y4X4X4TY4][e3T-e6Te5T-e3T],
where ei{i=1,2,…,23} are block entry matrices; that is,
(27)e2T=[0I000000000000000000000],χ(ε,μ)=(μ-1)exp{-εd1}withμ≥1,(μ-1)exp{-εd2}withμ<1,λ¯=maxi∈S{λmax(Pi)}+1-exp{-ερ}ε{∑j=15‍λmax(Qj)+∑k=15‍λmax(Rk)}+exp{-ερ}+ερ-1ε2×{∑l=14‍λmax(Wl)+∑m=14‍λmax(Ym)}+2-2exp{-ερ}+ρ2ε2-2ερ2ε3{∑n=14‍λmax(Zn)},λ_=mini∈S{λmin(Pi)},ϱ1=dm-d1,ϱ2=d2-dm,ϱ3=12(dm2-d12),ϱ4=12(d22-dm2).Proof.
Construct the following Lyapunov functional:(28)V(x(t),i,t)=∑k=16‍Vk(xt,i),
where
(29)V1(xt,i)=xT(t)Pix(t),(30)V2(xt,i)=∫t-τt‍exp{ε(s-t)}xT(s)Q1x(s)ds+∫t-d(t)t-d1‍exp{ε(s-t)}xT(s)Q2x(s)ds+∫t-d1t‍exp{ε(s-t)}xT(s)Q3x(s)ds+∫t-dmt-d1‍exp{ε(s-t)}xT(s)Q4x(s)ds+∫t-d2t-dm‍exp{ε(s-t)}xT(s)Q5x(s)ds,(31)V3(xt,i)=∫t-τt‍exp{ε(s-t)}x˙T(s)R1x˙(s)ds+∫t-d1t‍exp{ε(s-t)}x˙T(s)R2x˙(s)ds+∫t-dmt-d1‍exp{ε(s-t)}x˙T(s)R3x˙(s)ds+∫t-d2t-dm‍exp{ε(s-t)}x˙T(s)R4x˙(s)ds+∫t-τ(t)t‍exp{ε(s-t)}x˙T(s)R5x˙(s)ds,(32)V4(xt,i)=∫-τ0‍∫t+θt‍τexp{ε(s-t)}xT(s)W1x(s)dsdθ+∫-d10‍∫t+θt‍d1exp{ε(s-t)}xT(s)W2x(s)dsdθ+∫-dm-d1‍∫t+θt‍ϱ1exp{ε(s-t)}xT(s)W3x(s)dsdθ+∫-d2-dm‍∫t+θt‍ϱ2exp{ε(s-t)}xT(s)W4x(s)dsdθ,(33)V5(xt,i)=∫-τ0‍∫t+θt‍τexp{ε(s-t)}x˙T(s)Y1x˙(s)dsdθ+∫-d10‍∫t+θt‍d1exp{ε(s-t)}x˙T(s)Y2x˙(s)dsdθ+∫-dm-d1‍∫t+θt‍ϱ1exp{ε(s-t)}x˙T(s)Y3x˙(s)dsdθ+∫-d2-dm‍∫t+θt‍ϱ2exp{ε(s-t)}x˙T(s)Y4x˙(s)dsdθ,(34)V6(xt,i)=∫-τ0‍∫θ0‍∫t+λt‍τ22exp{ε(s-t)}x˙T(s)Z1x˙(s)dsdλdθ+∫-d10‍∫θ0‍∫t+λt‍d122exp{ε(s-t)}x˙T(s)Z2x˙(s)dsdλdθ+∫-dm-d1‍∫θ0‍∫t+λt‍ϱ3exp{ε(s-t)}x˙T(s)Z3x˙(s)dsdλdθ+∫-d2-dm‍∫θ0‍∫t+λt‍ϱ4exp{ε(s-t)}x˙T(s)Z4x˙(s)dsdλdθ.
Taking𝔏 as its infinitesimal generator along the trajectory of system (9), we obtain the following from Definition 5 and (28)–(34):
(35)𝔏V(x(t),i,t)=𝔏V1(xt,i)+𝔏V2(xt,i)+𝔏V3(xt,i)+𝔏V4(xt,i)+𝔏V5(xt,i)+𝔏V6(xt,i),(36)𝔏V1(xt,i)=2[xT(t)AiT+xT(t-d(t))BiT+x˙T(t-τ(t))CiT+f1T+f2T+f3T]×Pix(t)+∑j=1N‍πijxT(t)Pjx(t)+εxT(t)Pix(t)-εV1(xt,i),(37)𝔏V2(xt,i)=xT(t)[Q1+Q3]x(t)+exp{-εd1}xT×(t-d1)[Q2+Q4-Q3]x(t-d1)-εV2(xt,i)-(1-d˙(t))exp{-εd(t)}xT(t-d(t))×Q2x(t-d(t))-exp{-ετ}xT(t-τ)×Q1x(t-τ)+exp{-εdm}×xT(t-dm)[Q5-Q4]×x(t-dm)-exp{-εd2}xT(t-d2)Q5x(t-d2),(38)𝔏V3(xt,i)=x˙T(t)[R1+R2+R5]x˙(t)+exp{-εd1}x˙T(t-d1)[R3-R2]×x˙(t-d1)-εV3(xt,i)+exp{-εdm}x˙T(t-dm)[R4-R3]×x˙(t-dm)-exp{-ετ}×x˙T(t-τ)R1x˙(t-τ)-(1-τ˙(t))exp{-ετ(t)}×x˙T(t-τ(t))R5x˙(t-τ(t))-exp{-εd2}x˙T(t-d2)R4x˙(t-d2),(39)𝔏V4(xt,i)=xT(t)[τ2W1+d12W2+ϱ12W3+ϱ22W4]×x(t)-εV4(xt,i)-∫t-τt‍τexp{ε(s-t)}xT(s)W1x(s)ds-∫t-d1t‍d1exp{ε(s-t)}xT(s)W2x(s)ds-∫t-dmt-d1‍ϱ1exp{ε(s-t)}xT(s)W3x(s)ds-∫t-d2t-dm‍ϱ2exp{ε(s-t)}xT(s)W4x(s)ds,(40)𝔏V5(xt,i)=x˙T(t)[τ2Y1+d12Y2+ϱ12Y3+ϱ22Y4]x˙(t)-εV5(xt,i)-∫t-τt‍τexp{ε(s-t)}x˙T(s)Y1x˙(s)ds-∫t-d1t‍d1exp{ε(s-t)}x˙T(s)Y2x˙(s)ds-∫t-dmt-d1‍ϱ1exp{ε(s-t)}x˙T(s)Y3x˙(s)ds-∫t-d2t-dm‍ϱ2exp{ε(s-t)}x˙T(s)Y4x˙(s)ds,(41)𝔏V6(xt,i)=x˙T(t)×[τ44Z1+d144Z2+ϱ32Z3+ϱ42Z4]x˙(t)-εV6(xt,i)-∫-τ0‍∫t+θt‍τ22exp{ε(s-t)}x˙T(s)εV6(xt,id)×Z1x˙(s)dsdθ-∫-d10‍∫t+θt‍d122exp{ε(s-t)}x˙T(s)εV6(xt,di)×Z2x˙(s)dsdθ-∫-dm-d1‍∫t+θt‍ϱ3exp{ε(s-t)}x˙T(s)εV6(xtd,i)×Z3x˙(s)dsdθ-∫-d2-dm‍∫t+θt‍ϱ4exp{ε(s-t)}x˙T(s)εV6(xt,id)×Z4x˙(s)dsdθ.
In view of (6), the following inequalities hold for any scalars ε1>0, ε2>0, and ε3>0:
(42)ε1[α2xT(t)x(t)-f1T(x(t),t)f1(x(t),t)]≥0,ε2[β2xT(t-d(t))x(t-d(t))-f2T(x(t-d(t)),t)f2(x(t-d(t)),t)]≥0,ε3[γ2x˙T(t-τ(t))x˙(t-τ(t))-f3T(x˙(t-τ(t)),t)f3(x˙(t-τ(t)),t)]≥0.
From (35) and (42), we have
(43)𝔏V(x(t),i,t)≤𝔏V1(xt,i)+𝔏V2(xt,i)+𝔏V3(xt,i)+𝔏V4(xt,i)+𝔏V5(xt,i)+𝔏V6(xt,i)+ε1[α2xT(t)x(t)-f1Tf1]+ε2[β2xT(t-d(t))x(t-d(t))-f2Tf2]+ε3[γ2x˙T(t-τ(t))x˙(t-τ(t))-f3Tf3].
Define(44)ξ(t)=col{∫t-d(t)t-d1‍x(t)x˙(t)x(t-d(t))x(t-d1)x(t-dm)x(t-d2)x˙(t-d1)x˙(t-dm)x˙(t-d2)∫t-d1t‍x(s)ds∫t-d(t)t-d1‍x(s)ds∫t-d2t-d(t)x(s)ds‍∫t-dmt-d1‍x(s)ds∫t-d2t-dm‍x(s)dsx(t-τ)x˙(t-τ)x(t-τ(t))x˙(t-τ(t))∫t-τ(t)t‍x(s)ds∫t-τt‍x(s)dsf1f2f3}.
Since it is easy to see-(1-τ˙(t))exp{-ετ(t)}≤-(1-τd)exp{-ετ}; then from (5) we also obtain that
(45)-(1-d˙(t))exp{-εd(t)}≤χ(ε,μ),
where χ(ε,μ) is defined in Theorem 10. Notice (a) of Lemma 7; then
(46)-∫t-d1t‍d1exp{ε(s-t)}xT(s)W2x(s)ds≤-εd1exp{εd1}-1ξT(t)e10W2e10Tξ(t),-∫t-d1t‍d1exp{ε(s-t)}x˙T(s)Y2x˙(s)ds≤-εd1exp{εd1}-1ξT(t)(e1-e4)Y2(e1T-e4T)ξ(t).
Notice (b) of Lemma7; then
(47)-∫-τ0‍∫t+θt‍τ22exp{ε(s-t)}x˙T(s)Z1x˙(s)dsdθ≤-exp{-ετ}ξT(t)(τe1-e20)Z1(τe1T-e20T)ξ(t),-∫-d10‍∫t+θt‍d122exp{ε(s-t)}x˙T(s)Z2x˙(s)dsdθ≤-exp{-εd1}ξT(t)(d1e1-e10)Z2(d1e1T-e10T)ξ(t),-∫-dm-d1‍∫t+θt‍ϱ3exp{ε(s-t)}x˙T(s)Z3x˙(s)dsdθ≤-exp{-εdm}ξT(t)(ϱ1e1-e13)Z3(ϱ1e1T-e13T)ξ(t),-∫-d2-dm‍∫t+θt‍ϱ4exp{ε(s-t)}x˙T(s)Z4x˙(s)dsdθ≤-exp{-εd2}ξT(t)(ϱ2e1-e14)Z4(ϱ2e1T-e14T)ξ(t).
Ford(t)∈[d1,dm], the following is held from (a) of Lemma 7:
(48)-∫t-dmt-d1‍ϱ1exp{ε(s-t)}x˙T(s)Y3x˙(s)ds=-{∫t-d(t)t-d1‍+∫t-dmt-d(t)‍}ϱ1exp{ε(s-t)}x˙T(s)Y3x˙(s)ds≤-εϱ1exp{εd(t)}-exp{εd1}[∫t-d(t)t-d1‍x˙T(s)ds]×Y3[∫t-d(t)t-d1‍x˙(s)ds]-εϱ1exp{εdm}-exp{εd(t)}[∫t-dmt-d(t)‍x˙T(s)ds]×Y3[∫t-dmt-d(t)‍x˙(s)ds]=-εϱ1exp{εdm}-exp{εd1}ξT(t)×[1λ1(t)(e4-e3)Y3(e4T-e3T)(e4Tff)+1λ2(t)(e3-e5)Y3(e3T-e5T)]ξ(t),
where
(49)λ1(t)=exp{εd(t)}-exp{εd1}exp{εdm}-exp{εd1},λ2(t)=exp{εdm}-exp{εd(t)}exp{εdm}-exp{εd1}.
By Lemma8, there exists matrix X2 with appropriate dimensions such that
(50)-∫t-dmt-d1‍ϱ1exp{ε(s-t)}x˙T(s)Y3x˙(s)ds≤-εϱ1exp{εdm}-exp{εd1}ξT(t)×[e4-e3e3-e5][Y3X2X2TY3][e4T-e3Te3T-e5T]ξ(t),[Y3X2X2TY3]>0.
Similarly, considering-∫t-dmt-d1‍ϱ1exp{ε(s-t)}xT(s)W3x(s)ds and following the same procedure, there exists matrix X1 with appropriate dimensions such that
(51)-∫t-dmt-d1‍ϱ1exp{ε(s-t)}xT(s)W3x(s)ds≤-εϱ1exp{εdm}-exp{εd1}ξT(t)×[e11e13-e11][W3X1X1TW3][e11Te13T-e11T]ξ(t),[W3X1X1TW3]>0.
Forτ(t)∈[0,τ], with the same matrix inequalities technique, we obtain the following:
(52)-∫t-τt‍τexp{ε(s-t)}xT(s)W1x(s)ds≤-ετexp{ετ}-1ξT(t)×[e19e20-e19][W1U1U1TW1][e19Te20T-e19T]ξ(t),[W1U1U1TW1]>0,-∫t-τt‍τexp{ε(s-t)}x˙T(s)Y1x˙(s)ds≤-ετexp{ετ}-1ξT(t)×[e1-e17e17-e15][Y1U2U2TY1][e1T-e17Te17T-e15T]ξ(t),[Y1U2U2TY1]>0.
Consider-∫t-d2t-dm‍ϱ2exp{ε(s-t)}xT(s)W4x(s)ds  and  -∫t-d2t-dm‍ϱ2exp{ε(s-t)}x˙T(s)Y4x˙(s)ds, which are directly estimated by (a) of Lemma 7; that is,
(53)-∫t-d2t-dm‍ϱ2exp{ε(s-t)}xT(s)W4x(s)ds≤-εϱ2exp{εd2}-exp{εdm}ξT(t)e14W4e14Tξ(t),-∫t-d2t-dm‍ϱ2exp{ε(s-t)}x˙T(s)Y4x˙(s)ds≤-εϱ2exp{εd2}-exp{εdm}ξT(t)×(e5-e6)Y4(e5T-e6T)ξ(t).
In addition, there exist matricesT=col{Tk,(k=1,2,…,23)} with appropriate dimensions, such that the following equality holds according to (9):
(54)2ξT(t)T[-x˙(t)+Aix(t)+Bix(t-d(t))+Cix˙(t-τ(t))+f1+f2+f3]=0.
Substituting (36)–(42) and (45)–(54) into (43), we obtain
(55)𝔏V(x(t),i,t)+εV(x(t),i,t)≤ξT(t)(Ξi0ε+Ξi1ε)ξ(t).
On the other hand, ford(t)∈[dm,d2], the integral terms
(56)-∫t-d2t-dm‍ϱ2exp{ε(s-t)}xT(s)W4x(s)ds,-∫t-d2t-dm‍ϱ2exp{ε(s-t)}x˙T(s)Y4x˙(s)ds.
The above equations: are disposed and estimated by Lemma 8. 									(57)-∫t-dmt-d1‍ϱ1exp{ε(s-t)}xT(s)W3x(s)ds,-∫t-dmt-d1‍ϱ1exp{ε(s-t)}x˙T(s)Y3x˙(s)ds
are directly estimated by (a) of Lemma 7. Therefore,
(58)𝔏V(x(t),i,t)+εV(x(t),i,t)≤ξT(t)(Ξi0ε+Ξi2ε)ξ(t).
With (55) and (58), the following inequality (59) is held for d(t)∈[d1,d2] if (22), (23), and (24) are satisfied
(59)𝔏V(x(t),i,t)+εV(x(t),i,t)<0.
From the Lyapunov functional (28) and (59), it is held that
(60)V(x(t),i,t)≥mini∈S{λmin(Pi)}∥x(t)∥2=λ_∥x(t)∥2,V(x(t),i,t)<exp{-ε(t-t0)}V(xt0,rt0,t0).
Moreover, we have(61)V(xt0,rt0,t0)=∑k=16‍Vk(xt0,rt0)≤λ¯∥xt0∥ρ2.
Then, from (60) and (61), it is readily seen that
(62)∥x(t)∥≤λ¯λ_exp{-ε2(t-t0)}∥xt0∥ρ,
where σ=λ¯/λ_≥1.
Therefore, by Definition4, the system (9) is exponentially stable with a decay rate ε/2. This completes the proof.Remark 11.
It is noted that the integral intervals in (61) are enlarged as follows:
(63)∑k=16‍Vk(xt0,rt0)≤xT(t0)Pix(t0)+∫t0-ρt0‍exp{ε(s-t0)}xT(s)(∑j=15‍Qj)x(s)ds+∫t0-ρt0‍exp{ε(s-t0)}x˙T(s)(∑k=15‍Rk)x˙(s)ds+∫t-ρ0‍∫t0+θt0‍ρexp{ε(s-t0)}xT(t0)Pixf(t0)×xT(s)(∑l=14‍Wl)x(s)dsdθ+∫-ρ0‍∫t0+θt0‍ρexp{ε(s-t0)}xT(t0)Pixf(t0)×x˙T(s)(∑m=14‍Ym)x˙(s)dsdθ+∫-ρ0‍∫θ0‍∫t0+λt0‍ρ22exp{ε(s-t0)}xT(t0)Pixf(t0)×x˙T(s)(∑n=14‍Zn)x˙(s)dsdλdθ.
Equation (61) can be obtained by letting λ¯ be defined as previously.Remark 12.
In Theorem10, the factors
(64)exp{εd2}-exp{εd(t)}ε,exp{εd(t)}-exp{εd1}ε
may be enlarged as (exp{εd2}-exp{εd1})/ε. This will lead conservative results due to the fact that d(t) cannot achieve d1 and d2 at the same time. While we apply Lemma 7 to these terms, the method by using reciprocally convex lemma [34] can achieve less conservative results. Moreover, for ε>0, the factor (d˙(t)-1)exp{-εd(t)} that appeared in the derivative of Lyapunov functional may be directly enlarged as μexp{-εd1}-exp{-εd2}. In this paper, we enlarge it as χ(ε,μ) to reduce the conservativeness of the obtained criteria. In the literature [32, 36, 37], this factor is enlarged as (μ-1)exp{-εd2}, which only holds for μ<1.Remark 13.
The proposed Lyapunov functional (28) contains some triple-integral terms, which has not been used in any of the existing literatures in the same context before. Compared with the existing ones, [33] has shown that such a Lyapunov functional type is very effective in the reduction of conservatism. Besides, the information on the lower bound of the delay is sufficiently used in the Lyapunov functional by introducing the terms such as ∫t-dmt-d1‍exp{ε(s-t)}xT(s)Q4x(s)ds, ∫t-d2t-dm‍exp{ε(s-t)}xT(s)Q5x(s)ds, and ∫t-d(t)t-d1‍exp{ε(s-t)}xT(s)Q2x(s)ds.Remark 14.
It should be also mentioned that the result obtained in Theorem10 is delay-range-dependent and decay rate-dependent stability condition for (9), 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 τ(t). That is, τ1≤τ(t)≤τ2. The results can be obtained by using the similar methods.The information on the delay derivative may not be available in many cases. The following corollary is therefore given, which can be obtained from Theorem10 by setting Q2=0 and R5=0.Corollary 15.
For the given finite setS of modes with transition rates matrix, scalars α, β, γ, ε, τ, d1, d2, and constant scalar dm satisfying d1<dm<d2, the systems described by (9) are exponentially stable with decay rate ε/2 and σ=λ¯/λ_ if the operator 𝔇 is stable, and there exist symmetric positive matrices Pi>0, (i∈S), Qj>0, (j=1,3,4,5), Rk>0, (k=1,2,3,4), Wl>0, (l=1,2,3,4), Ym>0, (m=1,2,3,4), Zn>0, (n=1,2,3,4) and matrices U1, U2, X1, X2, X3, X4 for any scalars ε1, ε2, ε3 and any matrices Tk, (k=1,2,…,23) with appropriate dimensions, such that linear matrix inequalities (22) and (65) hold:
(65)(i)[W3X1X1TW3]>0,[Y3X2X2TY3]>0,(ii)Ξ~i0ε+Ξi1ε<0,(iii)[W4X3X3TW4]>0,[Y4X4X4TY4]>0,(iv)Ξ~i0ε+Ξi2ε<0,
where
(66)Ξ~i0ε=∑m=123‍emG~memT+𝒜(✠)-εd1exp{εd1}-1(e1-e4)Y2(e1T-e4T)-exp{-ετ}(τe1-e20)Z1(τe1T-e20T)-exp{-εd1}(d1e1-e10)Z2(d1e1T-e10T)-exp{-εdm}(ϱ1e1-e13)Z3(ϱ1e1T-e13T)-exp{-εd2}(ϱ2e1-e14)Z4(ϱ2e1T-e14T)-ετexp{ετ}-1[e19e20-e19]×[W1U1U1TW1][e19Te20T-e19T]-ετexp{ετ}-1[e1-e17e17-e15]×[Y1U2U2TY1][e1T-e17Te17T-e15T],
where
(67)G~2=R1+R2+τ2Y1+d12Y2+ϱ12Y3+ϱ22Y4+τ44Z1+d144Z2+ϱ32Z3+ϱ42Z4-T2-T2T,G~3=T3Bi+BiTT3T+ε2β2I,G~4=exp{-εd1}(Q4-Q3),G~18=T18Ci+CiTT18T+ε3γ2I.
Other G~m=Gm, (m≠2,3,4,18), which have been defined previously. 									(68)λ¯=maxi∈S{λmax(Pi)}+1-exp{-ερ}ε{∑j=1,j≠25‍λmax(Qj)+∑k=14‍λmax(Rk)}+exp{-ερ}+ερ-1ε2{∑l=14‍λmax(Wl)+∑m=14‍λmax(Ym)}+2-2exp{-ερ}+ρ2ε2-2ερ2ε3{∑n=14‍λmax(Zn)},
and the remaining notations are the same as Theorem 10.In Theorem10, it is assumed that ε≠0. For ε=0, by L'Hospital’s rule, the following asymptotic stability criterion can be obtained.Corollary 16.
For the given finite setS of modes with transition rates matrix, scalars α, β, γ, τ, τd, d1, d2, μ and constant scalar dm satisfying d1<dm<d2, the systems described by (9) are asymptotically stable if the operator 𝔇 is stable, and there exist symmetric positive matrices Pi>0, (i∈S), Qj>0, (j=1,2,3,4,5), Rk>0, (k=1,2,3,4,5), Wl>0, (l=1,2,3,4), Ym>0, (m=1,2,3,4), Zn>0, (n=1,2,3,4) and matrices U1, U2, X1, X2, X3, X4 for any scalars ε1, ε2, ε3 and any matrices Tk, (k=1,2,…,23) with appropriate dimensions, such that linear matrix inequalities (22) and (69) hold:
(69)(i)[W3X1X1TW3]>0,[Y3X2X2TY3]>0,(ii)Ξi0+Ξi1<0,(iii)[W4X3X3TW4]>0,[Y4X4X4TY4]>0,(iv)Ξi0+Ξi2<0,
where
(70)Ξi0=∑m=123‍emG¯memT+𝒜(✠)-(e1-e4)Y2(e1T-e4T)-(τe1-e20)Z1(τe1T-e20T)-(d1e1-e10)Z2(d1e1T-e10T)-(ϱ1e1-e13)Z3(ϱ1e1T-e13T)-(ϱ2e1-e14)Z4(ϱ2e1T-e14T)-[e19e20-e19][W1U1U1TW1][e19Te20T-e19T]-[e1-e17e17-e15][Y1U2U2TY1][e1T-e17Te17T-e15T],G¯1=AiTPi+PiAi+∑j=1N‍πijPj+Q1+Q3+τ2W1+d12W2+ϱ12W3+ϱ22W4+ε1α2I+T1Ai+AiTT1T,G¯3=(μ-1)Q2+T3Bi+BiTT3T+ε2β2I,G¯4=Q2+Q4-Q3,G¯5=Q5-Q4,G¯6=-Q5,G¯7=R3-R2,G¯8=R4-R3,G¯9=-R4,G¯10=-W2,G¯15=-Q1,G¯16=-R1,G¯18=(τd-1)R5+T18Ci+CiTT18T+ε3γ2I.
Other G¯m=Gm, (m=2,11,12,13,14,17,19,…,23), which have been defined in Theorem 10:
(71)Ξi1=-e14W4e14T-(e5-e6)Y4(e5T-e6T)-[e11e13-e11][W3X1X1TW3][e11Te13T-e11T]-[e4-e3e3-e5][Y3X2X2TY3][e4T-e3Te3T-e5T],Ξi2=-e13W3e13T-(e4-e5)Y3(e4T-e5T)-[e12e14-e12][W4X3X3TW4][e12Te14T-e12T]-[e3-e6e5-e3][Y4X4X4TY4][e3T-e6Te5T-e3T],
and the remaining notations are the same as Theorem 10.


## 3.2. Exponential Stability for the Nominal Systems with Partially Unknown Transition Rates

In this subsection, we take into account the situation that the information of transition rates is not accessible completely and propose the following conditions to guarantee the exponential stability of system (9) with partially unknown transition rates.Since the transition rates of the Markov chain are partially unknown, then some elements in matrixΠ=[πij]N×N are inaccessible. For instance, the system (3) with five operation modes, the jump rates matrix Π may be viewed as
(72)[?π12??π15π21??π24???π33π34??π42???π51???π55],
where ? represents the unknown element. For notation clarity, we denote that 𝒮i=𝒮ki⋃‍𝒮uki,foralli∈S and
(73)𝒮ki≜{j:πijisknownforj∈S},𝒮uki≜{j:πijisunknownforj∈S}.
If 𝒮ki≠∅, it is further described as
(74)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. Specially, when 𝒮uki=∅, 𝒮i=𝒮ki, the full information about Π is obtained, and it becomes the case of Section 3.1.Theorem 17.
For the given finite setS of modes with partly unknown transition rates matrix, scalars α, β, γ, ε, τ, τd, d1, d2, and μ, and constant scalar dm satisfying d1<dm<d2, the systems described by (9) with partly unknown transition rates are exponentially stable with decay rate ε/2 and σ=λ¯/λ_ if the operator 𝔇 is stable, and there exist symmetric positive matrices Pi>0, (i∈S), Qj>0, (j=1,2,3,4,5), Rk>0, (k=1,2,3,4,5), Wl>0, (l=1,2,3,4), Ym>0, (m=1,2,3,4), Zn>0, (n=1,2,3,4) and matrices U1, U2, X1, X2, X3, X4  for any scalars ε1, ε2, ε3 and any symmetric matrices Vi, (i∈S) and any matrices Tk, (k=1,2,…,23) with appropriate dimensions, such that linear matrix inequalities (22), (75), (76), and (77) hold:
(75)Pj-Vi≤0,j∈𝒮uki,j≠i,Pj-Vi≥0,j∈𝒮uki,j=i,(76)(i)[W3X1X1TW3]>0,[Y3X2X2TY3]>0,(ii)Ξ𝒮i0ε+Ξ𝒮i1ε<0,(77)(iii)[W4X3X3TW4]>0,[Y4X4X4TY4]>0,(iv)Ξ𝒮i0ε+Ξ𝒮i2ε<0,
where
(78)Ξ𝒮i0ε=∑m=123‍emG𝒮memT+𝒜(✠)-εd1exp{εd1}-1(e1-e4)Y2(e1T-e4T)-exp{-ετ}(τe1-e20)Z1(τe1T-e20T)-exp{-εd1}(d1e1-e10)Z2(d1e1T-e10T)-exp{-εdm}(ϱ1e1-e13)Z3(ϱ1e1T-e13T)-exp{-εd2}(ϱ2e1-e14)Z4(ϱ2e1T-e14T)-ετexp{ετ}-1[e19e20-e19]×[W1U1U1TW1][e19Te20T-e19T]-ετexp{ετ}-1[e1-e17e17-e15]×[Y1U2U2TY1][e1T-e17Te17T-e15T],G𝒮1=AiTPi+PiAi+∑j∈𝒮kiN‍πij(Pj-Vi)+εPi+Q1+Q3+τ2W1+d12W2+ϱ12W3+ϱ22W4+ε1α2I+T1Ai+AiTT1T,G𝒮m=Gm,(m≠1),Ξ𝒮i1ε=Ξi1ε,Ξ𝒮i2ε=Ξi2ε,
and the remaining notations are the same as Theorem 10.Proof.
Choose the same Lyapunov functional as (28), and V2(xt,i), V3(xt,i), V4(xt,i), V5(xt,i), and V6(xt,i) are disposed with the identical method in Theorem 10. Then, giving consideration on ∑j=1N‍πij=0, the following equation holds for arbitrary matrices Vi=ViT, i∈S; that is,
(79)-xT(t)(∑j=1N‍πijVi)x(t)=0.
With (36) and (79), we obtain the following:
(80)𝔏V1(xt,i)=2[xT(t)AiT+xT(t-d(t))BiT+x˙T(t-τ(t))CiT+f1T+f2T+f3T]Pix(t)+xT(t)[∑j∈𝒮kiN‍πij(Pj-Vi)]x(t)+xT(t)[∑j∈𝒮ukiN‍πij(Pj-Vi)]x(t)+εxT(t)Pix(t)-εV1(xt,i).
Ford(t)∈[d1,dm] and d(t)∈[dm,d2], with (80) and on the basis of the result in Theorem 10, we have the following equations, respectively:
(81)𝔏V(x(t),i,t)+εV(x(t),i,t)≤ξT(t)(Ξ𝒮i0ε+Ξ𝒮i1ε)ξ(t)+xT(t)[∑j∈𝒮ukiN‍πij(Pj-Vi)]x(t),𝔏V(x(t),i,t)+εV(x(t),i,t)≤ξT(t)(Ξ𝒮i0ε+Ξ𝒮i2ε)ξ(t)+xT(t)[∑j∈𝒮ukiN‍πij(Pj-Vi)]x(t).
With (81), due to λii=-∑j=1,j≠iN‍λij and λij>0, for all i≠j, (59) is also held for d(t)∈[d1,d2] if linear matrix inequalities (22), (75), (76), and (77) are satisfied. With the same procedure in the latter proof of Theorem 10, we draw the conclusion that the system (9) with partially unknown transition rates is exponentially stable with a decay rate ε/2. This completes the proof.Consider the system (9) with partially unknown transition rates, and the following corollaries are given, for unknown information on the delay derivative and ε=0, respectively.Corollary 18.
For the given finite setS of modes with partly unknown transition rates matrix, scalars α, β, γ, ε, τ, d1, and d2, and constant scalar dm satisfying d1<dm<d2, the systems described by (9) with partly unknown transition rates are exponentially stable with decay rate ε/2 and σ=λ¯/λ_ if the operator 𝔇 is stable, and there exist symmetric positive matrices Pi>0, (i∈S), Qj>0, (j=1,3,4,5), Rk>0, (k=1,2,3,4), Wl>0, (l=1,2,3,4), Ym>0, (m=1,2,3,4), Zn>0, (n=1,2,3,4) and matrices U1, U2, X1, X2, X3, X4   for any scalars ε1, ε2, ε3 and any symmetric matrices Vi, (i∈S) and any matrices Tk, (k=1,2,…,23) with appropriate dimensions, such that linear matrix inequalities (22), (75), and (82) hold:
(82)(i)[W3X1X1TW3]>0,[Y3X2X2TY3]>0,(ii)Ξ~𝒮i0ε+Ξ𝒮i1ε<0,(iii)[W4X3X3TW4]>0,[Y4X4X4TY4]>0,(iv)Ξ~𝒮i0ε+Ξ𝒮i2ε<0,
where
(83)Ξ~𝒮i0ε=∑m=123‍emG~𝒮memT+𝒜(✠)-εd1exp{εd1}-1(e1-e4)Y2(e1T-e4T)-exp{-ετ}(τe1-e20)Z1(τe1T-e20T)-exp{-εd1}(d1e1-e10)Z2(d1e1T-e10T)-exp{-εdm}(ϱ1e1-e13)Z3(ϱ1e1T-e13T)-exp{-εd2}(ϱ2e1-e14)Z4(ϱ2e1T-e14T)-ετexp{ετ}-1[e19e20-e19]×[W1U1U1TW1][e19Te20T-e19T]-ετexp{ετ}-1[e1-e17e17-e15]×[Y1U2U2TY1][e1T-e17Te17T-e15T],G~𝒮1=G𝒮1,G~𝒮m=G~m,(m≠1),Ξ𝒮i1ε=Ξi1ε,Ξ𝒮i2ε=Ξi2ε,
and the remaining notations are the same as Corollary 15.Corollary 19.
For the given finite setS of modes with partly unknown transition rates matrix, scalars α, β, γ, τ, τd, d1, d2, μ, and constant scalar dm satisfying d1<dm<d2, the systems described by (9) with partly unknown transition rates are exponentially stable with decay rate ε/2 and σ=λ¯/λ_ if the operator 𝔇 is stable, and there exist symmetric positive matrices Pi>0, (i∈S), Qj>0, (j=1,2,3,4,5), Rk>0, (k=1,2,3,4,5), Wl>0, (l=1,2,3,4), Ym>0, (m=1,2,3,4), Zn>0, (n=1,2,3,4) and matrices U1, U2, X1, X2, X3, X4  for any scalars ε1, ε2, ε3 and any symmetric matrices Vi, (i∈S) and any matrices Tk, (k=1,2,…,23) with appropriate dimensions, such that linear matrix inequalities (22), (75), and (84) hold:
(84)(i)[W3X1X1TW3]>0,[Y3X2X2TY3]>0,(ii)Ξ𝒮i0+Ξ𝒮i1<0,(iii)[W4X3X3TW4]>0,[Y4X4X4TY4]>0,(iv)Ξ𝒮i0+Ξ𝒮i2<0,
where
(85)Ξ𝒮i0=∑m=123‍emG¯𝒮memT+𝒜(✠)-(e1-e4)Y2(e1T-e4T)-(τe1-e20)Z1(τe1T-e20T)-(d1e1-e10)Z2(d1e1T-e10T)-(ϱ1e1-e13)Z3(ϱ1e1T-e13T)-(ϱ2e1-e14)Z4(ϱ2e1T-e14T)-[e19e20-e19][W1U1U1TW1][e19Te20T-e19T]-[e1-e17e17-e15]×[Y1U2U2TY1][e1T-e17Te17T-e15T],G¯𝒮1=AiTPi+PiAi+∑j∈𝒮kiN‍πij(Pj-Vi)+Q1+Q3+τ2W1+d12W2+ϱ12W3+ϱ22W4+ε1α2I+T1Ai+AiTT1T,G¯𝒮m=G¯m,(m≠1),Ξ𝒮i1=Ξi1,Ξ𝒮i2=Ξi2,
and the remaining notations are the same as Corollary 16.Remark 20.
Corollaries16 and 19 provide new delay-range-dependent asymptotic stability conditions for the systems described by (9) with known and partially unknown transition rates. Though reciprocally convex lemma has not helped here, the results are still less conservative than some previous ones because of some triple-integral terms, which will be verified in Section 4.


## 3.3. Extension to the Uncertain Case

In this subsection, the uncertain neutral delay Markovian jump systems with nonlinear perturbations described by (3) with known and partially unknown transition rates are, respectively, considered. The corresponding conditions are presented in the following theorems and corollaries.Theorem 21.
For the given finite setS of modes with transition rates matrix, scalars α, β, γ, ε, τ, τd, d1, d2, μ, and constant scalar dm satisfying d1<dm<d2, the uncertain neutral systems described by (3) are exponentially stable with decay rate ε/2 and σ=λ¯/λ_ if the operator 𝔇 is stable, and there exist scalars δ1>0, δ2>0, symmetric positive matrices Pi>0, (i∈S), Qj>0, (j=1,2,3,4,5), Rk>0, (k=1,2,3,4,5), Wl>0, (l=1,2,3,4), Ym>0, (m=1,2,3,4), Zn>0, (n=1,2,3,4), and matrices U1, U2, X1, X2, X3, X4   for any scalars ε1, ε2, ε3 and any matrices Tk, (k=1,2,…,23) with appropriate dimensions, such that (22), (86), and (87) hold:
(86)(i)[W3X1X1TW3]>0,[Y3X2X2TY3]>0,(ii)[δ1Ξi0ε+δ1Ξi1ε+δ12MTMNT*-I]<0,(87)(iii)[W4X3X3TW4]>0,[Y4X4X4TY4]>0,(iv)[δ2Ξi0ε+δ2Ξi2ε+δ22MTMNT*-I]<0,
where
(88)M=e1EAiT+e3EBiT,N=HiTPiTe1+∑m=123‍HiTTmTem,
and other notations are the same as Theorem 10.Proof.
DefineE=col{em,(m=1,2,…,23)} and T^=col{Pi+T1,Tm(m=2,…,23)}, and we replace Ai, Bi with Ai+ΔAi(t), Bi+ΔBi(t) on the basis of Theorem 10; that is,
(89)(i)[W3X1X1TW3]>0,[Y3X2X2TY3]>0,(ii)Ξi0ε+Ξi1ε+𝒜[e1ΔAiT(t)T^TE+e3ΔBiT(t)T^TE]<0,(90)(iii)[W4X3X3TW4]>0,[Y4X4X4TY4]>0,(iv)Ξi0ε+Ξi2ε+𝒜[e1ΔAiT(t)T^TE+e3ΔBiT(t)T^TE]<0.
Considering (ii) of (89) and combining the uncertainties condition (7), we have
(91)Ξi0ε+Ξi1ε+𝒜{[e1e3][EAiTEBiT]FiT(t)HiTT^TE}<0.
By the definition of 𝒜, we obtain
(92)Ξi0ε+Ξi1ε+MFiT(t)N+NTFi(t)MT<0.
According to (8), by Lemmas 9 and 6, with (92), we obtain (ii) of (86). Following the same procedure, (90) is considered and (iv) of (87) can be obtained. Finally, following the proof of Theorem 10, the systems described by (3) are exponentially stable with a decay rate ε/2. This completes the proof.Theorem 22.
For the given finite setS of modes with partly unknown transition rates matrix, scalars α, β, γ, ε, τ, τd, d1, d2, μ, and constant scalar dm satisfying d1<dm<d2, the uncertain neutral systems described by (3) with partly unknown transition rates are exponentially stable with decay rate ε/2 and σ=λ¯/λ_ if the operator 𝔇 is stable, and there exist scalars δ1>0, δ2>0, symmetric positive matrices Pi>0, (i∈S), Qj>0, (j=1,2,3,4,5), Rk>0, (k=1,2,3,4,5), Wl>0, (l=1,2,3,4), Ym>0, (m=1,2,3,4), Zn>0, (n=1,2,3,4), and matrices U1, U2, X1, X2, X3, X4   for any scalars ε1, ε2, ε3 and any symmetric matrices Vi, (i∈S) and any matrices Tk, (k=1,2,…,23) with appropriate dimensions, such that (22), (75), and (93) hold:
(93)(i)[W3X1X1TW3]>0,[Y3X2X2TY3]>0,(ii)[δ1Ξℐi0ε+δ1Ξℐi1ε+δ12MTMNT*-I]<0,(iii)[W4X3X3TW4]>0,[Y4X4X4TY4]>0,(iv)[δ2Ξℐi0ε+δ2Ξℐi2ε+δ22MTMNT*-I]<0,
where
(94)M=e1EAiT+e3EBiT,N=HiTPiTe1+∑m=123‍HiTTmTem
and other notations are the same as Theorem 17.The proof of Theorem22 is omitted here because it is identical with the proof of Theorem 21. In addition, considering the system (3) with fully or partly known transition rates, the corollaries for unknown information on the delay derivative and ε=0 are also omitted here because they are uniform results as the nominal case.


## 4. Numerical Examples

In this section, numerical examples are provided with known and partially unknown transition rates, respectively, which demonstrate that the proposed theoretical results in this paper are effectiveness.


### 4.1. Systems with Known Transition Rates

In this subsection, numerical systems with full information on transition rates are given to show the effectiveness of ours.Example 1.
Partial element equivalent circuit (PEEC) model can be represented as a stochastic jump system as in (3) with the abrupt variation in structures and parameters [38]. The practical system is given as follows:
(95)x˙(t)-Cix˙(t-τ(t))=(Ai+ΔA(t))x(t)+(Bi+ΔB(t))x(t-d(t)),
where α=β=γ=0, i∈S={1,2}, and the mode switching is governed by the rate matrix Π=[-332-2] as shown in Figure 1, and the state matrices
(96)A1=[-400-3],A2=[-200-3],B1=[-10-1-1],B2=[-10-0.9-1.2],C1=0.5I,C2=0.3I,H1=[0.20.2],H2=[0-0.3],EA1=[0.20],EA2=[00.2],EB1=[-0.30.3],EB2=[0.20.2].Figure 1
The mode switching of Example1.Given the decay rateε/2=0.4, the time-varying neutral delay τ(t)=0.1sin2t, and retarded delay d(t)=0.1(1+cos2t), from τ(t), d(t) with t∈[0,2π] in Figures 2 and 3, it is readily obtained that τ=0.1, d1=0.1, and d2=0.2. Without loss of generality, we choose dm=0.12. In addition, we have τ˙(t)=0.1sin2t, τd=0.1 and d˙(t)=-0.1sin2t, μ=0.1. Thus, the exponential stability can be readily established by Theorem 21.Figure 2
Interval time-varying neutral delayτ(t) of Example 1.Figure 3
Interval time-varying retarded delayd(t) of Example 1.Example 2.
To show the reduced conservativeness of the exponential stability condition in Theorem10, consider the time-delay system in the form of (9) with Ci=0, α=β=γ=0, d(t)=d, and
(97)Ai=[-3-210],Bi=[-0.50.10.30],i∈S={1}.
For givend, by Theorem 10, the maximum exponential decay rate ε^, which satisfies the LMIs in (22), (23), and (24), can be calculated by solving a quasi-convex optimization problem. The results are presented in Table 1.Table 1
Maximum exponential decay rateε^ with different parameter d.
Methods
d
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Mondie and Kharitonov [39]
ε
^
0.7344
0.6715
0.6145
0.5642
0.5202
0.4818
0.4481
Liu [40]
ε
^
0.9367
0.5903
0.3400
0.1813
0.0752
0.0014
0
Xu et al. [41]
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 Table1, 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 [39], ε^=0.9367 for [40], and ε^=0.9366 for [41]. 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 [39–41]. So, it can be demonstrated that Theorem 10 in this paper yields less conservative results than [39–41].Consider the previous system again but with parameter uncertainties as follows:(98)x˙(t)=(Ai+ΔAi(t))x(t)+(Bi+ΔBi(t))x(t-d),
where
(99)Ai=[-410-4],Bi=[0.1040.1],i∈S={1},
and the uncertain matrices ΔAi(t) and ΔBi(t) satisfy
(100)Hi=0.2I,EAi=EBi=I,∥Fi(t)∥<1,i∈S={1}.For givend, by Theorem 21, the maximum exponential decay rate ε^, which satisfies the LMIs in (22), (86) and (87), can be calculated by solving a quasi-convex optimization problem. The results are presented in Table 2, where the method of [40] is no longer applicable.Table 2
Maximum exponential decay rateε^ with different parameter d.
Methods
d
0.3
0.5
0.7
0.9
1.1
1.3
1.5
Mondie and Kharitonov [39]
ε
^
0.6255
0.4760
0.3825
0.3191
0.2735
0.2392
0.2125
Xu et al. [41]
ε
^
1.0108
0.8366
0.7103
0.6156
0.5425
0.4845
0.4375
Theorem21
ε
^
1.0419
0.9207
0.8035
0.7048
0.6234
0.5876
0.5043From Table2, we know that the maximum exponential decay rate ε^=1.0419 in this paper by setting d=0.3, while the maximum exponential decay rate ε^=0.6255 for [39] and ε^=1.0108 for [41]. The results are also given by setting d=0.5,  d=0.7,  d=0.9,  d=1.1,  d=1.3, and d=1.5, and it is found that the maximum exponential decay rate in this paper is larger than those in [3, 22, 26]. So, it can be seen that the delay-range-dependent and rate-dependent exponential stability conditions in Theorem 21 in this paper are less conservative than previous results in [39, 41].Example 3.
Consider the system (3) with τ(t)=d(t), and the parameters are listed in the following:
(101)Ai=[-200-0.9],Bi=[-10-1-1],Ci=[0.1000.1],i∈S={1},α=0.05,β=0.1,γ=0,d1=0,d2=τ,μ=τd.
For given ε and  μ, choose dm=d2/2 and utilize Theorem 10, and the maximum upper bound of d2, which satisfies the LMIs in (22), (23), and (24), can be obtained by solving a quasi-convex optimization problem. The results are presented in Tables 3, 4, and 5.Table 3
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. [42]
d
2
1.2999
0.8781
0.6917
0.5792
0.5015
Qiu and Cui [10]
d
2
1.4008
1.0199
0.8457
0.7395
0.6667
Theorem10 with ε=2ε^
d
2
1.5167
1.1365
1.0674
0.9677
0.7558Table 4
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. [42]
d
2
0.9442
0.7275
0.6096
0.5321
0.4761
Qiu and Cui [10]
d
2
1.0120
0.8324
0.7311
0.6941
0.6063
Theorem10 with ε=2ε^
d
2
1.0643
0.9156
0.7985
0.7368
0.6849Table 5
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. [42]
d
2
0.5471
0.5015
0.4650
0.4350
0.4089
Qiu and Cui [10]
d
2
0.6438
0.5789
0.5214
0.4954
0.4231
Theorem10 with ε=2ε^
d
2
0.7136
0.6344
0.5987
0.5473
0.4963From Tables3, 4, and 5, 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=0.713, 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 [42], and the maximum upper bound of delay d2=1.4008, d2=1.0120, d2=0.6438, respectively, for [10]. 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 [10, 42]. So, it also can be seen that Theorem 10 in this paper yields less conservative results than existing results in [10, 42].


### 4.2. Systems with Partially Unknown Transition Rates

In this subsection, numerical systems with partly unknown transition rates are given to show the effectiveness of the approaches presented in this paper.Example 4.
Regard Example1 again, and we consider the practical system of PEEC model with partially unknown transition rates Π=[??2-2] and nonlinear perturbations
(102)∥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))∥.
Following the same procedure in Example 1, we solve the LMIs (22), (75), and (93) in Theorem 22 and obtain the feasible solutions to guarantee the exponential stability of the uncertain neutral delay Markovian jump system with nonlinear perturbations and partially unknown transition rates.Example 5.
Consider the nominal system (9) with four operation modes S={1,2,3,4}, α=β=γ=0, and the following parameters:
(103)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 unknown transition rate matrix is considered as the following two cases:
(104)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???].
For given ε=0 and d(t)=τ(t), we have d1=0, d2=τ, and μ=τd. Set dm=d2/2 and employ Corollary 19, and the maximum upper bound of the time delay d2, which satisfies LMIs (22), (75), and (84), can be calculated by solving a quasi-convex optimization problem. This neutral Markovian jump system with partially unknown transition rates was also considered in reference [43]. The results on the maximum upper bound of d2 are compared in Tables 6 and 7.Table 6
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. [43]
d
2
0.4489
0.3317
0.2242
0.1814
0.1486
0.1220
0.0983
0.0781
Corollary19
d
2
0.4763
0.3652
0.2573
0.2154
0.1891
0.1466
0.1134
0.0946Table 7
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. [43]
d
2
12.0750
0.9924
0.5147
0.3749
0.2742
0.2139
0.1737
0.1433
Corollary19
d
2
12.4382
1.2574
0.6253
0.4026
0.2997
0.2475
0.2015
0.1789From Tables6 and 7, we consider the previous system with Π1, Π2 and obtain the maximum upper bound of delay d2=0.4763, d2=12.4382, respectively, in this paper by setting μ=0.18, while the maximum upper bound of delay d2=0.4489, d2=12.0750, respectively for [43]. 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 [43]. So, it can be seen that our proposed method is less conservative than the result in [43]. Besides, we know that the maximum upper bound of delay to guarantee stability is dependent on transition rates knowledge.


## 4.1. Systems with Known Transition Rates

In this subsection, numerical systems with full information on transition rates are given to show the effectiveness of ours.Example 1.
Partial element equivalent circuit (PEEC) model can be represented as a stochastic jump system as in (3) with the abrupt variation in structures and parameters [38]. The practical system is given as follows:
(95)x˙(t)-Cix˙(t-τ(t))=(Ai+ΔA(t))x(t)+(Bi+ΔB(t))x(t-d(t)),
where α=β=γ=0, i∈S={1,2}, and the mode switching is governed by the rate matrix Π=[-332-2] as shown in Figure 1, and the state matrices
(96)A1=[-400-3],A2=[-200-3],B1=[-10-1-1],B2=[-10-0.9-1.2],C1=0.5I,C2=0.3I,H1=[0.20.2],H2=[0-0.3],EA1=[0.20],EA2=[00.2],EB1=[-0.30.3],EB2=[0.20.2].Figure 1
The mode switching of Example1.Given the decay rateε/2=0.4, the time-varying neutral delay τ(t)=0.1sin2t, and retarded delay d(t)=0.1(1+cos2t), from τ(t), d(t) with t∈[0,2π] in Figures 2 and 3, it is readily obtained that τ=0.1, d1=0.1, and d2=0.2. Without loss of generality, we choose dm=0.12. In addition, we have τ˙(t)=0.1sin2t, τd=0.1 and d˙(t)=-0.1sin2t, μ=0.1. Thus, the exponential stability can be readily established by Theorem 21.Figure 2
Interval time-varying neutral delayτ(t) of Example 1.Figure 3
Interval time-varying retarded delayd(t) of Example 1.Example 2.
To show the reduced conservativeness of the exponential stability condition in Theorem10, consider the time-delay system in the form of (9) with Ci=0, α=β=γ=0, d(t)=d, and
(97)Ai=[-3-210],Bi=[-0.50.10.30],i∈S={1}.
For givend, by Theorem 10, the maximum exponential decay rate ε^, which satisfies the LMIs in (22), (23), and (24), can be calculated by solving a quasi-convex optimization problem. The results are presented in Table 1.Table 1
Maximum exponential decay rateε^ with different parameter d.
Methods
d
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Mondie and Kharitonov [39]
ε
^
0.7344
0.6715
0.6145
0.5642
0.5202
0.4818
0.4481
Liu [40]
ε
^
0.9367
0.5903
0.3400
0.1813
0.0752
0.0014
0
Xu et al. [41]
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 Table1, 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 [39], ε^=0.9367 for [40], and ε^=0.9366 for [41]. 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 [39–41]. So, it can be demonstrated that Theorem 10 in this paper yields less conservative results than [39–41].Consider the previous system again but with parameter uncertainties as follows:(98)x˙(t)=(Ai+ΔAi(t))x(t)+(Bi+ΔBi(t))x(t-d),
where
(99)Ai=[-410-4],Bi=[0.1040.1],i∈S={1},
and the uncertain matrices ΔAi(t) and ΔBi(t) satisfy
(100)Hi=0.2I,EAi=EBi=I,∥Fi(t)∥<1,i∈S={1}.For givend, by Theorem 21, the maximum exponential decay rate ε^, which satisfies the LMIs in (22), (86) and (87), can be calculated by solving a quasi-convex optimization problem. The results are presented in Table 2, where the method of [40] is no longer applicable.Table 2
Maximum exponential decay rateε^ with different parameter d.
Methods
d
0.3
0.5
0.7
0.9
1.1
1.3
1.5
Mondie and Kharitonov [39]
ε
^
0.6255
0.4760
0.3825
0.3191
0.2735
0.2392
0.2125
Xu et al. [41]
ε
^
1.0108
0.8366
0.7103
0.6156
0.5425
0.4845
0.4375
Theorem21
ε
^
1.0419
0.9207
0.8035
0.7048
0.6234
0.5876
0.5043From Table2, we know that the maximum exponential decay rate ε^=1.0419 in this paper by setting d=0.3, while the maximum exponential decay rate ε^=0.6255 for [39] and ε^=1.0108 for [41]. The results are also given by setting d=0.5,  d=0.7,  d=0.9,  d=1.1,  d=1.3, and d=1.5, and it is found that the maximum exponential decay rate in this paper is larger than those in [3, 22, 26]. So, it can be seen that the delay-range-dependent and rate-dependent exponential stability conditions in Theorem 21 in this paper are less conservative than previous results in [39, 41].Example 3.
Consider the system (3) with τ(t)=d(t), and the parameters are listed in the following:
(101)Ai=[-200-0.9],Bi=[-10-1-1],Ci=[0.1000.1],i∈S={1},α=0.05,β=0.1,γ=0,d1=0,d2=τ,μ=τd.
For given ε and  μ, choose dm=d2/2 and utilize Theorem 10, and the maximum upper bound of d2, which satisfies the LMIs in (22), (23), and (24), can be obtained by solving a quasi-convex optimization problem. The results are presented in Tables 3, 4, and 5.Table 3
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. [42]
d
2
1.2999
0.8781
0.6917
0.5792
0.5015
Qiu and Cui [10]
d
2
1.4008
1.0199
0.8457
0.7395
0.6667
Theorem10 with ε=2ε^
d
2
1.5167
1.1365
1.0674
0.9677
0.7558Table 4
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. [42]
d
2
0.9442
0.7275
0.6096
0.5321
0.4761
Qiu and Cui [10]
d
2
1.0120
0.8324
0.7311
0.6941
0.6063
Theorem10 with ε=2ε^
d
2
1.0643
0.9156
0.7985
0.7368
0.6849Table 5
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. [42]
d
2
0.5471
0.5015
0.4650
0.4350
0.4089
Qiu and Cui [10]
d
2
0.6438
0.5789
0.5214
0.4954
0.4231
Theorem10 with ε=2ε^
d
2
0.7136
0.6344
0.5987
0.5473
0.4963From Tables3, 4, and 5, 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=0.713, 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 [42], and the maximum upper bound of delay d2=1.4008, d2=1.0120, d2=0.6438, respectively, for [10]. 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 [10, 42]. So, it also can be seen that Theorem 10 in this paper yields less conservative results than existing results in [10, 42].


## 4.2. Systems with Partially Unknown Transition Rates

In this subsection, numerical systems with partly unknown transition rates are given to show the effectiveness of the approaches presented in this paper.Example 4.
Regard Example1 again, and we consider the practical system of PEEC model with partially unknown transition rates Π=[??2-2] and nonlinear perturbations
(102)∥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))∥.
Following the same procedure in Example 1, we solve the LMIs (22), (75), and (93) in Theorem 22 and obtain the feasible solutions to guarantee the exponential stability of the uncertain neutral delay Markovian jump system with nonlinear perturbations and partially unknown transition rates.Example 5.
Consider the nominal system (9) with four operation modes S={1,2,3,4}, α=β=γ=0, and the following parameters:
(103)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 unknown transition rate matrix is considered as the following two cases:
(104)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???].
For given ε=0 and d(t)=τ(t), we have d1=0, d2=τ, and μ=τd. Set dm=d2/2 and employ Corollary 19, and the maximum upper bound of the time delay d2, which satisfies LMIs (22), (75), and (84), can be calculated by solving a quasi-convex optimization problem. This neutral Markovian jump system with partially unknown transition rates was also considered in reference [43]. The results on the maximum upper bound of d2 are compared in Tables 6 and 7.Table 6
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. [43]
d
2
0.4489
0.3317
0.2242
0.1814
0.1486
0.1220
0.0983
0.0781
Corollary19
d
2
0.4763
0.3652
0.2573
0.2154
0.1891
0.1466
0.1134
0.0946Table 7
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. [43]
d
2
12.0750
0.9924
0.5147
0.3749
0.2742
0.2139
0.1737
0.1433
Corollary19
d
2
12.4382
1.2574
0.6253
0.4026
0.2997
0.2475
0.2015
0.1789From Tables6 and 7, we consider the previous system with Π1, Π2 and obtain the maximum upper bound of delay d2=0.4763, d2=12.4382, respectively, in this paper by setting μ=0.18, while the maximum upper bound of delay d2=0.4489, d2=12.0750, respectively for [43]. 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 [43]. So, it can be seen that our proposed method is less conservative than the result in [43]. Besides, we know that the maximum upper bound of delay to guarantee stability is dependent on transition rates knowledge.


## 5. Conclusions

This paper addresses the exponential stability for neutral Markovian jumping systems with interval time-varying delays, nonlinear perturbations, and partially unknown transition rates. According to the Lyapunov theory, a novel augmented Lyapunov-Krasovskii functional, which contains some triple-integral terms and sufficiently takes advantage of the delay bound, is constructed by dividing the time-varying delay interval into two subintervals. Then, less conservative delay-range-dependent and rate-dependent exponential stability criteria are obtained by the reciprocally convex lemma and free weighting matrices, both of which can be used to reduce the conservativeness. Furthermore, these theoretical results are successfully verified through some numerical examples.



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*Source: 592483-2013-09-11.xml*