On Delay-Range-Dependent Stochastic Stability Conditions of Uncertain Neutral Delay Markovian Jump Systems

Abstract

The delay-range-dependent stochastic stability for uncertain neutral Markovian jump systems with interval time-varying delays is studied in this paper. The uncertainties under consideration are assumed to be time varying but norm bounded. To begin with the nominal systems, a novel augmented Lyapunov functional which contains some triple-integral terms is introduced. Then, by employing some integral inequalities and the nature of convex combination, some less conservative stochastic stability conditions are presented in terms of linear matrix inequalities without introducing any free-weighting matrices. Finally, numerical examples are provided to demonstrate the effectiveness and to show that the proposed results significantly improve the allowed upper bounds of the delay size over some existing ones in the literature.

1. Introduction

Time delays are frequently encountered in various engineering systems, such as chemical or process control systems, networked control systems, and manufacturing systems. To sum up, delays can appear in the state, input, or output variables (retarded systems), as well as in the state derivative (neutral systems). In fact, neutral delay systems constitute a more general class than those of the retarded type because such systems can be found in places such as population ecology [1], distributed neural networks [2], heat exchangers, and robots in contact with rigid environments [3]. Since it is shown that the existence of delays in a dynamic system may result in instability, oscillations, or poor performances [3–5], the stability of time-delay systems has been an important problem of recurring interest for many years. Existing results on this topic can be roughly classified into two categories, namely, delay-independent criteria [6] and delay-dependent criteria, and it is generally recognized that the latter cases are less conservative. Actually, the stability of neutral time-delay systems proves to be a more complex issue as well as singular systems [7–9] because the systems involve the derivative of the delayed state. So considerable attention has been devoted to the problem of robust delay-independent stability or delay-dependent stability and stabilization via different approaches for linear neutral systems with delayed state input and parameter uncertainties. Results are mainly presented based on Lyapunov-Krasovskii (L-K) method; see, for example, [10–16] and the references therein. However, there is room for further investigation because the conservativeness of the neutral systems can be further reduced by a better technique.

On the other hand, with the development of science and technology, many practical dynamics, for example, solar thermal central receivers, robotic manipulator systems, aircraft control systems, economic systems, and so on, experience abrupt changes in their structures, whose parameters are caused by phenomena such as component failures or repairs, changes in subsystem interconnections, and sudden environmental changes. This class of systems is more appropriate to be described as Markovian jump systems (MJSs), which can be regarded as a special class of hybrid systems with finite operation modes. The system parameters jump among finite modes, and the mode switching is governed by a Markov process to represent the abrupt variation in their structures and parameters. With so many applications in engineering systems, a great deal of attention has been paid to the stability analysis and controller synthesis for Markovian jump systems (MJSs) in recent years. Many researchers have made a lot of progress on Markovian jump delay systems and Markovian jump control theory; see, for example, [17–23] and references therein for more details. However, a few of these papers have considered the effect of delay on the stability or stabilization for the corresponding neutral systems. Besides, to the best of the authors’ knowledge, it seems that the problem of stochastic stability for neutral Markovian jumping systems with interval time-varying delays has not been fully investigated and it is very challenging. Motivated by the previous description, this paper investigates the stochastic stability of neutral Markovian jumping systems with interval time-varying delays to seek less conservative stochastic stability conditions than some previous ones.

In order to simplify the treatment of the problem, in this paper, we first investigate the nominal systems and construct a new augmented Lyapunov functional containing some triple-integral terms to reduce conservativeness. By some integral inequalities and the nature of convex combination, the delay-range-dependent stochastic stability conditions are derived for the nominal neutral systems with Markovian jump parameters and interval time-varying delays. Then, the results are extended to the corresponding uncertain case on the basis of obtained conditions. In addition, these conditions are expressed in linear matrix inequalities (LMIs), which can be easily checked by utilizing the LMI Toolbox in MATLAB. Numerical examples are given to show the effectiveness and reduced conservativeness over some previous references.

The main contributions of this paper can be summarized as follows: (1) the proposed Lyapunov functional contains some triple-integral terms which is very effective in the reduction of conservativeness, and has not been used in any of the existing literatures in the same context before; (2) the delay-range-dependent stability conditions are obtained in terms of LMIs without introducing any free-weighting matrices besides the Lyapunov matrices, which will reduce the number of variables and decrease the complexity of computation; (3) the proposed results are expressed in a new representation and proved to be less conservative than some existing ones.

The remainder of this paper is organized as follows: Section 2 contains the problem statement and preliminaries; Section 3 presents the main results; Section 4 provides a numerical example to verify the effectiveness of the results; Section 5 draws a brief conclusion.

1.1. Notations

In this paper, ℝ^𝑛 denotes the 𝑛 dimensional Euclidean space and ℝ^𝑚×𝑛 is for the set of all 𝑚×𝑛 matrices. The notation 𝑋<𝑌 (𝑋>𝑌), where 𝑋 and 𝑌 are both symmetric matrices, means that 𝑋−𝑌 is negative (positive) definite. 𝐼 denotes the identity matrix with proper dimensions. 𝜆_max(min)(𝐴) is the eigenvalue of matrix 𝐴 with maximum (minimum) real part. For a symmetric block matrix, we use the sign ∗ to denote the terms introduced by symmetry. ℰ stands for the mathematical expectation, and ‖𝑣‖ is the Euclidean norm of vector 𝑣, ‖𝑣‖=(𝑣^𝑇𝑣)^1/2, while ‖𝐴‖ is spectral norm of matrix 𝐴, ‖𝐴‖=[𝜆_max(𝐴^𝑇𝐴)]^1/2. 𝐶([−𝜌,0],ℝ^𝑛) is the space of continuous function from [−𝜌,0] to ℝ^𝑛. In addition, if not explicitly stated, matrices are assumed to have compatible dimensions.

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. {𝑟_𝑡,𝑡≥0} is a homogeneous, finite-state Markovian process with right continuous trajectories and taking values in a finite set 𝑆={1,2,3,…,𝑁}, with the mode transition probability matrix 𝑃𝑟_𝑡+Δ𝑡=𝑗∣𝑟_𝑡=𝜋=𝑖^𝑖𝑗Δ𝑡+𝑜(Δ𝑡)𝑖≠𝑗,1+𝜋_𝑖𝑖Δ𝑡+𝑜(Δ𝑡)𝑖=𝑗, (1) where Δ𝑡>0, lim_Δ𝑡→0(𝑜(Δ𝑡)/Δ𝑡)=0, and 𝜋_𝑖𝑗≥0(𝑖,𝑗∈𝑆,𝑖≠𝑗) denote the transition rate from mode 𝑖 to 𝑗. For any state or mode 𝑖∈𝑆, we have 𝜋_𝑖𝑖=−^𝑁_{𝑗=1,𝑗≠𝑖}𝜋_𝑖𝑗. (2)

In this paper, we consider the following uncertain neutral systems with Markovian jump parameters and time-varying delay over the space {Ω,ℱ,𝐏} as follows: 𝑥̇(𝑡)−𝐶𝑟_𝑡𝑥̇(𝑡−𝜏)=𝐴𝑟_𝑡+Δ𝐴𝑟_𝑡𝑥(𝑡)+𝐵𝑟_𝑡+Δ𝐵𝑟_𝑡𝑥((𝑡,𝑥(𝑠)(𝑠)𝑡−𝑑))=𝜑,𝑟_𝑠=𝑟₀,𝑠∈−𝜌,0, (3)(4) where 𝑥(𝑡)∈ℝ^𝑛 is the system state and 𝜏>0 is a constant neutral delay. It is assumed that the time-varying delay 𝑑(𝑡) satisfies 0<𝑑₁≤𝑑(𝑡)≤𝑑₂,𝑑̇(𝑡)≤𝜇, (5) where 𝑑₁<𝑑₂ and 𝜇≥0 are constant real values. The initial condition 𝜑(𝑠) is a continuously differentiable vector-valued function. The continuous norm of 𝜑(𝑠) is defined as 𝜑_𝑐=max_{𝑠∈[−𝜌,0]}𝜑(𝑠),𝜌=max𝜏,𝑑₂; (6)𝐴(𝑟_𝑡)∈ℝ^𝑛×𝑛, 𝐵(𝑟_𝑡)∈ℝ^𝑛×𝑛, and 𝐶(𝑟_𝑡)∈ℝ^𝑛×𝑛 are known mode-dependent constant matrices, while Δ𝐴(𝑟_𝑡)∈ℝ^𝑛×𝑛 and Δ𝐵(𝑟_𝑡)∈ℝ^𝑛×𝑛 are uncertainties. For notational simplicity, when 𝑟_𝑡=𝑖∈𝑆, 𝐴(𝑟_𝑡), Δ𝐴(𝑟_𝑡), 𝐵(𝑟_𝑡), Δ𝐵(𝑟_𝑡), and 𝐶(𝑟_𝑡) are, respectively, denoted as 𝐴_𝑖, Δ𝐴_𝑖, 𝐵_𝑖, Δ𝐵_𝑖, and 𝐶_𝑖. Throughout this paper, the parametric matrix ‖𝐶_𝑖‖<1 and the admissible parametric uncertainties are assumed to satisfy the following condition: Δ𝐴_𝑖(𝑡)Δ𝐵_𝑖(𝑡)=𝐻_𝑖𝐹_𝑖(𝑡)𝐸_𝐴𝑖𝐸_𝐵𝑖, (7) where 𝐻_𝑖, 𝐸_𝐴𝑖, and 𝐸_𝐵𝑖 are known mode-dependent constant matrices with appropriate dimensions and 𝐹_𝑖(𝑡) is an unknown and time-varying matrix satisfying 𝐹^𝑇_𝑖(𝑡)𝐹_𝑖(𝑡)≤𝐼,∀𝑡. (8) Particularly, when we consider 𝐹_𝑖(𝑡)=0, we get the nominal systems which can be described as 𝑥̇(𝑡)−𝐶_𝑖𝑥̇(𝑡−𝜏)=𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑥(𝑡−𝑑(𝑡)). (9)

Before proceeding further, the following assumptions, definitions, and lemmas need to be introduced.

Assumption 1. The system matrix 𝐴_𝑖(forall𝑖∈𝑆) is Hurwitz matrix with all the eigenvalues having negative real parts for each mode. The matrix 𝐻_𝑖(forall𝑖∈𝑆) is chosen as a full row rank matrix.

Assumption 2. The Markov process is irreducible, and the system mode 𝑟_𝑡 is available at time 𝑡.

With regard to neutral systems, the operator 𝔇∶𝐶([−𝜌,0],ℝ^𝑛)→ℝ^𝑛 is defined to be 𝔇𝑥_𝑡=𝑥(𝑡)−𝐶𝑥(𝑡−𝜏). (10) Then, the stability of operator 𝔇 is defined as follows.

Definition 3 (see [4]). The operator 𝔇 is said to be stable if the homogeneous difference equation 𝔇𝑥_𝑡𝑥=0,𝑡≥0,₀=𝜓∈𝜙∈𝐶−𝜌,0,ℝ^𝑛∶𝔇𝜙=0 (11) is uniformly asymptotically stable. In order to guarantee the stability of the operator 𝔇, one has assumed that ‖𝐶_𝑖‖<1 as previosuly mentioned, which was introduced in [24].

Definition 4 (see [25]). The systems which are described in (3) are said to be stochastically stable if there exists a positive constant Υ such that ℰ^∞₀𝑥𝑟_𝑡,𝑡²𝑑𝑡∣𝜑(𝑠),𝑠∈−𝜌,0,𝑟₀<Υ. (12)

Definition 5 (see [26]). In the Euclidean space {ℝ^𝑛×𝑆×𝑅⁺}, where 𝑥(𝑡)∈ℝ^𝑛, 𝑟_𝑡∈𝑆, and 𝑡∈𝑅⁺, one introduces the stochastic Lyapunov-Krasovskii function of system (3) as 𝑉(𝑥(𝑡),𝑟_𝑡=𝑖,𝑡>0)=𝑉(𝑥_𝑡,𝑖,𝑡), the infinitesimal generator satisfying 𝔏𝑉(𝑥(𝑡),𝑖,𝑡)=lim_Δ𝑡→01ℰ𝑉𝑥()Δ𝑡𝑡+Δ𝑡,𝑟_𝑡+Δ𝑡,𝑡+Δ𝑡∣𝑥(𝑡)=𝑥,𝑟_𝑡(𝑥(𝑡)=𝜕=𝑖−𝑉,𝑖,𝑡)𝑉(𝑥(𝑡))+𝜕𝜕𝑡,𝑖,𝑡𝑉(𝑥(𝑡))𝑥̇(𝑡)+𝜕𝑥,𝑖,𝑡_𝑁𝑗=1𝜋_𝑖𝑗𝑉𝑥(𝑡),𝑗,𝑡. (13)

Lemma 6 (see [27, 28]). For any constant matrix 𝐻=𝐻^𝑇>0 and scalars 𝜏₂>𝜏₁>0 such that the following integrations are well defined, then (a)−𝜏₂−𝜏₁^𝑡−𝜏1_𝑡−𝜏2𝑥^𝑇(𝑠)𝐻𝑥(𝑠)𝑑𝑠≤−^𝑡−𝜏1_𝑡−𝜏2𝑥^𝑇(𝑠)𝑑𝑠𝐻^𝑡−𝜏1_𝑡−𝜏2𝑥(𝑠),(b)−1𝑑𝑠2𝜏²₂−𝜏²₁^−𝜏1_−𝜏2^𝑡_𝑡+𝜃𝑥^𝑇(𝑠)𝐻𝑥(𝑠)𝑑𝑠𝑑𝜃≤−^−𝜏1_−𝜏2^𝑡_𝑡+𝜃𝑥^𝑇(𝑠)𝑑𝑠𝑑𝜃𝐻^−𝜏1_−𝜏2^𝑡_𝑡+𝜃𝑥(𝑠)𝑑𝑠𝑑𝜃. (14)

Lemma 7 (see [19]). Suppose that  0≤𝜏_𝑚≤𝜏(𝑡)≤𝜏_𝑀, Ξ₁, Ξ₂, and Ω are constant matrices of appropriate dimensions, then 𝜏(𝑡)−𝜏_𝑚Ξ₁+𝜏_𝑀−𝜏(𝑡)Ξ₂+Ω<0 (15) if and only if (𝜏_𝑀−𝜏_𝑚)Ξ₁+Ω<0 and (𝜏_𝑀−𝜏_𝑚)Ξ₂+Ω<0 hold.

Lemma 8 (see [29]). For given matrices 𝑄=𝑄^𝑇, 𝑀, and 𝑁 with appropriate dimensions, 𝑄+𝑀𝐹(𝑡)𝑁+𝑁^𝑇𝐹^𝑇(𝑡)𝑀^𝑇<0 (16) for all 𝐹(𝑡) satisfying 𝐹^𝑇(𝑡)𝐹(𝑡)≤𝐼 if and only if there exists a scalar 𝛿>0 such that 𝑄+𝛿⁻¹𝑀𝑀^𝑇+𝛿𝑁𝑁^𝑇<0. (17)

Lemma 9 (see [30]). Given constant matrices Ω₁, Ω₂, and Ω₃, where Ω₁=Ω^𝑇₁ and Ω₂=Ω^𝑇₂>0, then Ω₁+Ω^𝑇₃Ω₂⁻¹Ω₃<0 if and only if Ω₁Ω^𝑇₃∗−Ω₂<0𝑜𝑟−Ω₂Ω^𝑇₃∗Ω₁<0. (18)

3. Main Results

In this section, we first consider the nominal systems described by (9) and extend to the uncertain case. The following theorems present sufficient conditions to guarantee the stochastic stability for the neutral systems with Markovian jump parameters and time-varying delays.

3.1. Stochastic Stability for the Nominal Systems

Theorem 10. For the given finite set 𝑆 of modes with transition rates matrix, scalars 𝑑₁, 𝑑₂, 𝜏, and 𝜇, the neutral systems with Markovian jump parameters and time-varying delays as described by (9) are stochastically stable if the operator 𝔇 is stable, and there exist symmetric positive matrices 𝑃_𝑖>0(𝑖∈𝑆), 𝑄_𝑗>0(𝑗=1,2,3,4,5), and 𝑅_𝑘>0(𝑘=1,2,…,11) such that the following linear matrix inequalities hold: Π_𝑖1+Γ^𝑇Π𝑀Γ<0,_𝑖2+Γ^𝑇𝑀Γ<0, (19) where Π_𝑖1=Π_𝑖0−2𝑒₂−𝑒₄𝑅₇𝑒^𝑇₂−𝑒^𝑇₄−𝑒₃−𝑒₂𝑅₇𝑒^𝑇₃−𝑒^𝑇₂−𝑒₈𝑅₉𝑒^𝑇₈−2𝑒₉𝑅₉𝑒^𝑇₉,Π_𝑖2=Π_𝑖0−𝑒₂−𝑒₄𝑅₇𝑒^𝑇₂−𝑒^𝑇₄−2𝑒₃−𝑒₂𝑅₇𝑒^𝑇₃−𝑒^𝑇₂−2𝑒₈𝑅₉𝑒^𝑇₈−𝑒₉𝑅₉𝑒^𝑇₉, (20) where Π_𝑖0=𝑒₁𝑌𝑒^𝑇₁−𝑒₂1−𝜇𝑅₃𝑒^𝑇₂+𝑒₁𝑃_𝑖𝐵_𝑖𝑒^𝑇₂+𝑒₂𝐵^𝑇_𝑖𝑃_𝑖𝑒^𝑇₁+𝑒₁𝑃_𝑖𝐶_𝑖𝑒^𝑇₁₁+𝑒₁₁𝐶^𝑇_𝑖𝑃_𝑖𝑒^𝑇₁+𝑒₃𝑅₂+𝑅₃−𝑅₁𝑒^𝑇₃−𝑒₄𝑅₂𝑒^𝑇₄+𝑒₅𝑅₅−𝑅₄𝑒^𝑇₅−𝑒₆𝑅₅𝑒^𝑇₆−𝑒₇𝑅₈𝑒^𝑇₇−𝑒₁₀𝑄₁𝑒^𝑇₁₀−𝑒₁₁𝑄₂𝑒^𝑇₁₁−𝑒₁₂𝑄₃𝑒^𝑇₁₂−𝑒₁−𝑒₁₀𝑄₄𝑒^𝑇₁−𝑒^𝑇₁₀−𝑒₁−𝑒₃𝑅₆𝑒^𝑇₁−𝑒^𝑇₃−𝜏𝑒₁−𝑒₁₂𝑄₅𝜏𝑒^𝑇₁−𝑒^𝑇₁₂−𝑑₁𝑒₁−𝑒₇𝑅₁₀𝑑₁𝑒^𝑇₁−𝑒^𝑇₇−𝑑₁₂𝑒₁−𝑒₈−𝑒₉𝑅₁₁𝑑₁₂𝑒^𝑇₁−𝑒^𝑇₈−𝑒^𝑇₉, (21) where 𝑒_𝑖{𝑖=1,2,…,12} are block entry matrices; for instance, 𝑒^𝑇₂=0𝐼0000000000,𝑌=𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+^𝑁_𝑗=1𝜋_𝑖𝑗𝑝_𝑗+𝑄₁+𝜏²𝑄₃+𝑅₁+𝑑²₁𝑅₈+𝑑²₁₂𝑅₉,𝑀=𝑄₂+𝜏²𝑄₄+𝜏⁴4𝑄₅+𝑅₄+𝑑²₁𝑅₆+𝑑²₁₂𝑅₇+𝑑⁴₁4𝑅₁₀+𝑑²_𝑚𝑅₁₁,Γ=𝐴_𝑖𝐵_𝑖00000000𝐶_𝑖0,𝑑₁₂=𝑑₂−𝑑₁,𝑑_𝑚=12𝑑²₂−𝑑²₁. (22)

Proof. Construct the novel Lyapunov functional as follows: 𝑉(𝑥(𝑡),𝑖,𝑡)=𝑉_𝑟𝑥_𝑡,𝑖+𝑉_𝜏𝑥_𝑡,𝑖+𝑉_𝑑1𝑥_𝑡,𝑖+𝑉_𝑑2𝑥_𝑡,𝑖+𝑉_𝑑3𝑥_𝑡,𝑖, (23) where 𝑉_𝑟𝑥_𝑡,𝑖=𝑥^𝑇(𝑡)𝑃_𝑖𝑥(𝑡),𝑉_𝜏𝑥_𝑡,𝑖=^𝑡_𝑡−𝜏𝑥^𝑇(𝑠)𝑄₁𝑥(𝑠)𝑑𝑠+^𝑡_𝑡−𝜏𝑥̇^𝑇(𝑠)𝑄₂𝑥̇(𝑠)+𝑑𝑠_{0−𝜏𝑡𝑡+𝜃}𝑥^𝑇(𝑠)𝜏𝑄₃𝑥(𝑠)𝑑𝑠𝑑𝜃+⁰_−𝜏^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠)𝜏𝑄₄𝑥̇(𝑠)+𝑑𝑠𝑑𝜃_{0−𝜏0𝜃𝑡𝑡+𝜆}𝑥̇^𝑇(𝑠𝜏)²2𝑄₅𝑥̇(𝑠)𝑉𝑑𝑠𝑑𝜆𝑑𝜃,_𝑑1𝑥_𝑡,𝑖=^𝑡_𝑡−𝑑1𝑥^𝑇(𝑠)𝑅₁𝑥(𝑠)𝑑𝑠+^𝑡−𝑑1_𝑡−𝑑2𝑥^𝑇(𝑠)𝑅₂𝑥(𝑠)+𝑑𝑠_{𝑡−𝑑1(𝑡)𝑡−𝑑}𝑥^𝑇(𝑠)𝑅₃𝑥(𝑠)+𝑑𝑠_{𝑡𝑡−𝑑1}𝑥̇^𝑇(𝑠)𝑅₄𝑥̇(𝑠)𝑑𝑠+^𝑡−𝑑1_𝑡−𝑑2𝑥̇^𝑇(𝑠)𝑅₅𝑥̇(𝑠)𝑉𝑑𝑠,_𝑑2𝑥_𝑡,𝑖=⁰_−𝑑1^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠)𝑑₁𝑅₆𝑥̇(𝑠)+𝑑𝑠𝑑𝜃_{−𝑑1−𝑑2𝑡𝑡+𝜃}𝑥̇^𝑇(𝑠)𝑑₁₂𝑅₇𝑥̇(𝑠)+𝑑𝑠𝑑𝜃_{0−𝑑1𝑡𝑡+𝜃}𝑥^𝑇(𝑠)𝑑₁𝑅₈𝑥(𝑠)𝑑𝑠𝑑𝜃+^−𝑑1_−𝑑2^𝑡_𝑡+𝜃𝑥^𝑇(𝑠)𝑑₁₂𝑅₉𝑥(𝑠)𝑉𝑑𝑠𝑑𝜃,_𝑑3𝑥_𝑡,𝑖=⁰_−𝑑1⁰_𝜃^𝑡_𝑡+𝜆𝑥̇^𝑇(𝑠𝑑)²¹2𝑅₁₀𝑥̇(𝑠)+𝑑𝑠𝑑𝜆𝑑𝜃_{−𝑑1−𝑑20𝜃𝑡𝑡+𝜆}𝑥̇^𝑇(𝑠)𝑑_𝑚𝑅₁₁𝑥̇(𝑠)𝑑𝑠𝑑𝜆𝑑𝜃. (24)
From Definition 5, taking 𝔏 as its infinitesimal generator along the trajectory of system (9), then from (23) and (24) we get the following equalities and inequalities: 𝔏𝑉(𝑥(𝑡),𝑖,𝑡)=𝔏𝑉_𝑟𝑥_𝑡,𝑖+𝔏𝑉_𝜏𝑥_𝑡,𝑖+𝔏𝑉_𝑑1𝑥_𝑡,𝑖+𝔏𝑉_𝑑2𝑥_𝑡,𝑖+𝔏𝑉_𝑑3𝑥_𝑡,𝑖,𝔏𝑉_𝑟𝑥_𝑡,𝑖=2𝑥^𝑇(𝑡)𝐴^𝑇_𝑖+𝑥^𝑇(𝑡−𝑑(𝑡))𝐵^𝑇_𝑖+𝑥̇^𝑇(𝑡−𝜏)𝐶^𝑇_𝑖𝑃_𝑖𝑥(𝑡)+^𝑁_𝑗=1𝜋_𝑖𝑗𝑥^𝑇(𝑡)𝑃_𝑗𝑥(𝑡),𝔏𝑉_𝜏𝑥_𝑡,𝑖=𝑥^𝑇(𝑡)𝑄₁+𝜏²𝑄₃𝑥(𝑡)+𝑥̇^𝑇(𝑡)𝑄₂+𝜏²𝑄₄+𝜏⁴4𝑄₅𝑥̇(𝑡)−𝑥^𝑇(𝑡−𝜏)𝑄₁𝑥()−𝑥̇𝑡−𝜏_𝑇(𝑡−𝜏)𝑄₂𝑥̇()−𝑡−𝜏_{𝑡𝑡−𝜏}𝑥^𝑇(𝑠)𝜏𝑄₃𝑥(𝑠)𝑑𝑠−^𝑡_𝑡−𝜏𝑥̇^𝑇(𝑠)𝜏𝑄₄𝑥̇(𝑠)−𝑑𝑠_{0−𝜏𝑡𝑡+𝜃}𝑥̇(𝑠𝜏)²2𝑄₅𝑥̇(𝑠)𝑑𝑠𝑑𝜃,𝔏𝑉_𝑑1𝑥_𝑡,𝑖=𝑥^𝑇(𝑡)𝑅₁𝑥(𝑡)+𝑥̇^𝑇(𝑡)𝑅₄𝑥̇(𝑡)+𝑥^𝑇𝑡−𝑑₁𝑅₂+𝑅₃−𝑅₁𝑥𝑡−𝑑₁+𝑥̇^𝑇𝑡−𝑑₁𝑅₅−𝑅₄𝑥̇𝑡−𝑑₁−𝑥̇^𝑇𝑡−𝑑₂𝑅₅𝑥̇𝑡−𝑑₂−𝑥^𝑇𝑡−𝑑₂𝑅₂𝑥𝑡−𝑑₂−1−𝑑̇(𝑡)𝑥^𝑇(𝑡−𝑑(𝑡))𝑅₃𝑥(𝑡−𝑑(𝑡)),𝔏𝑉_𝑑2𝑥_𝑡,𝑖=𝑥^𝑇(𝑡)𝑑²₁𝑅₈+𝑑²₁₂𝑅₉𝑥(𝑡)+𝑥̇^𝑇(𝑡)𝑑²₁𝑅₆+𝑑²₁₂𝑅₇𝑥̇(𝑡)−^𝑡_𝑡−𝑑1𝑥̇^𝑇(𝑠)𝑑₁𝑅₆𝑥̇(𝑠)𝑑𝑠−^𝑡−𝑑1_𝑡−𝑑2𝑥̇^𝑇(𝑠)𝑑₁₂𝑅₇𝑥̇(𝑠)−𝑑𝑠_{𝑡𝑡−𝑑1}𝑥^𝑇(𝑠)𝑑₁𝑅₈𝑥(𝑠)−𝑑𝑠_{𝑡−𝑑1𝑡−𝑑2}𝑥^𝑇(𝑠)𝑑₁₂𝑅₉𝑥(𝑠)𝑑𝑠,𝔏𝑉_𝑑3𝑥_𝑡,𝑖=𝑥̇^𝑇(𝑡𝑑)⁴¹4𝑅₁₀+𝑑²_𝑚𝑅₁₁𝑥̇(𝑡)−⁰_−𝑑1^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠𝑑)²¹2𝑅₁₀𝑥̇(𝑠)−𝑑𝑠𝑑𝜃_{−𝑑1−𝑑2𝑡𝑡+𝜃}𝑥̇^𝑇(𝑠)𝑑_𝑚𝑅₁₁𝑥̇(𝑠)𝑑𝑠𝑑𝜃. (25)(26)(27)
Let us define 𝜉(𝑡)=col𝑥(𝑡)𝑥(𝑡−𝑑(𝑡))𝑥𝑡−𝑑₁𝑥𝑡−𝑑₂𝑥̇𝑡−𝑑₁𝑥̇𝑡−𝑑₂^𝑡_𝑡−𝑑1𝑥(𝑠)𝑑𝑠^𝑡−𝑑1_{(𝑡)𝑡−𝑑}𝑥(𝑠)𝑑𝑠_{𝑡−𝑑(𝑡)𝑡−𝑑2}𝑥(𝑠)𝑑𝑠𝑥(𝑡−𝜏)𝑥̇(𝑡−𝜏)^𝑡_𝑡−𝜏𝑥(𝑠)𝑑𝑠. (28)
Applying (a) of Lemma 6, we obtain −^𝑡_𝑡−𝜏𝑥^𝑇(𝑠)𝜏𝑄₃𝑥(𝑠)𝑑𝑠≤−^𝑡_𝑡−𝜏𝑥^𝑇(𝑠)𝑑𝑠𝑄₃^𝑡_𝑡−𝜏𝑥(𝑠)𝑑𝑠=−𝜉^𝑇(𝑡)𝑒₁₂𝑄₃𝑒^𝑇₁₂𝜉(𝑡). (29) Following the same procedure, we also obtain the inequalities as follows: −^𝑡_𝑡−𝜏𝑥̇^𝑇(𝑠)𝜏𝑄₄𝑥̇(𝑠)𝑑𝑠≤−𝜉^𝑇(𝑡)𝑒₁−𝑒₁₀𝑄₄𝑒^𝑇₁−𝑒^𝑇₁₀𝜉(𝑡),−^𝑡_𝑡−𝑑1𝑥̇^𝑇(𝑠)𝑑₁𝑅₆𝑥̇(𝑠)𝑑𝑠≤−𝜉^𝑇(𝑡)𝑒₁−𝑒₃𝑅₆𝑒^𝑇₁−𝑒^𝑇₃𝜉(𝑡),−^𝑡_𝑡−𝑑1𝑥^𝑇(𝑠)𝑑₁𝑅₈𝑥(𝑠)𝑑𝑠≤−^𝑡_𝑡−𝑑1𝑥^𝑇(s)𝑑𝑠𝑅₈^𝑡_𝑡−𝑑1𝑥(𝑠)𝑑𝑠. (30)
Applying (b) of Lemma 6, we have −⁰_−𝜏^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠𝜏)²2𝑄₅𝑥̇(𝑠)𝑑𝑠𝑑𝜃≤−⁰_−𝜏^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠)𝑑𝑠𝑑𝜃𝑄₅⁰_−𝜏^𝑡_𝑡+𝜃𝑥̇(𝑠)𝑑𝑠𝑑𝜃=−𝜏𝑥^𝑇(𝑡)−^𝑡_𝑡−𝜏𝑥^𝑇(𝑠)𝑑𝑠𝑄₅𝜏𝑥(𝑡)−^𝑡_𝑡−𝜏𝑥(𝑠)𝑑𝑠=−𝜉^𝑇(𝑡)𝜏𝑒₁−𝑒₁₂𝑄₅𝜏𝑒^𝑇₁−𝑒^𝑇₁₂𝜉(𝑡). (31) Then the following inequalities are obtained by the same technique: −⁰_−𝑑1^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠𝑑)²¹2𝑅₁₀𝑥̇(𝑠)𝑑𝑠𝑑𝜃≤−𝜉^𝑇(𝑡)𝑑₁𝑒₁−𝑒₇𝑅₁₀𝑑₁𝑒^𝑇₁−𝑒^𝑇₇𝜉(𝑡),−^−𝑑1_−𝑑2^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠)𝑑_𝑚𝑅₁₁𝑥̇(𝑠)𝑑𝑠𝑑𝜃≤−𝜉^𝑇(𝑡)𝑑₁₂𝑒₁−𝑒₈−𝑒₉𝑅₁₁𝑑₁₂𝑒^𝑇₁−𝑒^𝑇₈−𝑒^𝑇₉𝜉(𝑡). (32)
Let 𝜆(𝑡)=(𝑑(𝑡)−𝑑₁)/𝑑₁₂, then we have −^𝑡−𝑑1_𝑡−𝑑2𝑥̇^𝑇(𝑠)𝑑₁₂𝑅₇𝑥̇(𝑠)𝑑𝑠=−𝑑₁₂^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑥̇^𝑇(𝑠)𝑅₇𝑥̇(𝑠)𝑑𝑠−𝑑₁₂^𝑡−𝑑1_{(𝑡)𝑡−𝑑}𝑥̇^𝑇(𝑠)𝑅₇𝑥̇(𝑠)𝑑𝑑𝑠=−₂−𝑑(𝑡)^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑥̇^𝑇(𝑠)𝑅₇𝑥̇(𝑠)−𝑑(𝑡)𝑑𝑠−𝑑₁^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑥̇^𝑇(𝑠)𝑅₇𝑥̇(𝑠)−𝑑(𝑡)𝑑𝑠−𝑑₁^𝑡−𝑑1_{(𝑡)𝑡−𝑑}𝑥̇^𝑇(𝑠)𝑅₇𝑥̇(𝑠)−𝑑𝑑𝑠₂−𝑑(𝑡)^𝑡−𝑑1_{(𝑡)𝑡−𝑑}𝑥̇^𝑇(𝑠)𝑅₇𝑥̇(𝑠)𝑑𝑠=−𝜉^𝑇(𝑡)𝑒₂−𝑒₄𝑅₇𝑒^𝑇₂−𝑒^𝑇₄𝜉(𝑡)−𝑑(𝑡)−𝑑₁𝑑₂(𝑡)𝜉−𝑑^𝑇(𝑡)𝑒₂−𝑒₄𝑅₇𝑒^𝑇₂−𝑒^𝑇₄𝜉(𝑡)−𝜉^𝑇(𝑡)𝑒₃−𝑒₂𝑅₇𝑒^𝑇₃−𝑒^𝑇₂𝜉(𝑡)−𝑑₂−𝑑(𝑡)𝑑(𝑡)−𝑑₁𝜉^𝑇(𝑡)𝑒₃−𝑒₂𝑅₇𝑒^𝑇₃−𝑒^𝑇₂𝜉(𝑡)≤−𝜉^𝑇(𝑡)𝑒₂−𝑒₄𝑅₇𝑒^𝑇₂−𝑒^𝑇₄𝜉(𝑡)−𝜆(𝑡)𝜉^𝑇(𝑡)𝑒₂−𝑒₄𝑅₇𝑒^𝑇₂−𝑒^𝑇₄𝜉(𝑡)−𝜉^𝑇(𝑡)𝑒₃−𝑒₂𝑅₇𝑒^𝑇₃−𝑒^𝑇₂𝜉(𝑡)−(1−𝜆(𝑡))𝜉^𝑇(𝑡)𝑒₃−𝑒₂𝑅₇𝑒^𝑇₃−𝑒^𝑇₂𝜉(𝑡)=−(1+𝜆(𝑡))𝜉^𝑇(𝑡)𝑒₂−𝑒₄𝑅₇𝑒^𝑇₂−𝑒^𝑇₄𝜉(𝑡)−(2−𝜆(𝑡))𝜉^𝑇(𝑡)𝑒₃−𝑒₂𝑅₇𝑒^𝑇₃−𝑒^𝑇₂𝜉(𝑡). (33) Consistent with the technique of (33), we obtain −^𝑡−𝑑1_𝑡−𝑑2𝑥^𝑇(𝑠)𝑑₁₂𝑅₉𝑥(𝑠)((𝑡𝜉𝑑𝑠≤−2−𝜆))^𝑇(𝑡)𝑒₈𝑅₉𝑒^𝑇₈𝜉(𝑡)−(1+𝜆(𝑡))𝜉^𝑇(𝑡)𝑒₉𝑅₉𝑒^𝑇₉𝜉(𝑡), (34) considering 𝑥̇^𝑇(𝑡)𝑀𝑥̇(𝑡)=𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑥(𝑡−𝑑(𝑡))+𝐶_𝑖𝑥̇(𝑡−𝜏)^𝑇×𝑀𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑥(𝑡−𝑑(𝑡))+𝐶_𝑖𝑥̇(𝑡−𝜏)=𝜉^𝑇(𝑡)Γ^𝑇𝑀Γ𝜉(𝑡), (35) where 𝑀, Γ have been defined as before.
We take the previous equalities and inequalities (26)–(35) into (25); thus, we finally get 𝔏𝑉(𝑥(𝑡),𝑖,𝑡)≤𝜉^𝑇(𝑡)𝜆(𝑡)Π_𝑖1+(1−𝜆(𝑡))Π_𝑖2+Γ^𝑇(𝑡).𝑀Γ×𝜉 (36)
Since 0≤𝜆(𝑡)≤1, by utilizing Lemma 7, we know that 𝜆(𝑡)Π_𝑖1+(1−𝜆(𝑡))Π_𝑖2+Γ^𝑇𝑀Γ<0 is equivalent to (19). So we choose 𝛽=max_{𝑖∈𝑆,𝜆(𝑡)∈[0,1]}𝜆_max𝜆(𝑡)Π_𝑖1+(1−𝜆(𝑡))Π_𝑖2+Γ^𝑇𝑀Γ. (37) Then 𝛽<0 and 𝔏𝑉(𝑥(𝑡),𝑖,𝑡)≤𝛽𝜉(𝑡)²≤𝛽‖𝑥(𝑡)‖². (38) According to (38), from Dynkin's formula [31], we obtain ℰ{𝑉(𝑥(𝑡),𝑖,𝑡)}−𝑉𝑥₀,𝑟₀≤𝛽ℰ^𝑡₀‖𝑥(𝑠)‖²𝑑𝑠. (39) Let 𝑡→∞, then we have lim_𝑡→∞ℰ^𝑡₀‖𝑥(𝑠)‖²𝑑𝑠≤−𝛽⁻¹𝑉𝑥₀,𝑟₀. (40) From Definition 4, we know that the systems described by (9) are stochastically stable. This completes the proof.

Remark 11. Theorem 10 provides a delay-range-dependent stochastic stability criterion for nominal neutral systems with interval time-varying delays and Markovian jump parameters as described by (9). By utilizing a new Lyapunov functional, the less conservative criterion is obtained in terms of LMIs and it can be verified in Section 4.

Remark 12. In the same context of the stochastic stability for neutral systems with Markovian jumping parameters and time-varying delays, the type of augmented Lyapunov functional has not been used in any of the existing literatures. Compared with the existing Lyapunov functional, the proposed one (23) contains some triple-integral terms, which is very effective in the reduction of conservativeness in [28]. Besides, the information on the lower bound of the delay is sufficiently used in the Lyapunov functional by introducing the terms, such as ∫^𝑡−𝑑1_𝑡−𝑑2𝑥^𝑇(𝑠)𝑅₂𝑥(𝑠)𝑑𝑠 and ∫^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑥^𝑇(𝑠)𝑅₃𝑥(𝑠)𝑑𝑠.

In many circumstances, the information on the delay derivative may not be available. That is, 𝜇 is usually unknown in the real systems. So we give the following result as a corollary which can be obtained from Theorem 10 by setting 𝑅₃=0.

Corollary 13. For the given finite set 𝑆 of modes with transition rates matrix, scalars 𝑑₁, 𝑑₂, and 𝜏, the neutral systems with Markovian jump parameters and time-varying delays as described by (9) are stochastically stable if the operator 𝔇 is stable, and there exist symmetric positive matrices 𝑃_𝑖>0(𝑖∈𝑆), 𝑄_𝑗>0(𝑗=1,2,3,4,5), and 𝑅_𝑘>0(𝑘=1,2,4,5,…,11) such that the following linear matrix inequalities hold: Π_𝑖1+Γ^𝑇Π𝑀Γ<0,_𝑖2+Γ^𝑇𝑀Γ<0, (41) where Π_𝑖1=Π_𝑖0−2𝑒₂−𝑒₄𝑅₇𝑒^𝑇₂−𝑒^𝑇₄−𝑒₃−𝑒₂𝑅₇𝑒^𝑇₃−𝑒^𝑇₂−𝑒₈𝑅₉𝑒^𝑇₈−2𝑒₉𝑅₉𝑒^𝑇₉,Π_𝑖2=Π_𝑖0−𝑒₂−𝑒₄𝑅₇𝑒^𝑇₂−𝑒^𝑇₄−2𝑒₃−𝑒₂𝑅₇𝑒^𝑇₃−𝑒^𝑇₂−2𝑒₈𝑅₉𝑒^𝑇₈−𝑒₉𝑅₉𝑒^𝑇₉, (42) where Π_𝑖0=𝑒₁𝑌𝑒^𝑇₁+𝑒₁𝑃_𝑖𝐵_𝑖𝑒^𝑇₂+𝑒₂𝐵^𝑇_𝑖𝑃_𝑖𝑒^𝑇₁+𝑒₁𝑃_𝑖𝐶_𝑖𝑒^𝑇₁₁+𝑒₁₁𝐶^𝑇_𝑖𝑃_𝑖𝑒^𝑇₁+𝑒₃𝑅₂−𝑅₁𝑒^𝑇₃−𝑒₄𝑅₂𝑒^𝑇₄+𝑒₅𝑅₅−𝑅₄𝑒^𝑇₅−𝑒₆𝑅₅𝑒^𝑇₆−𝑒₇𝑅₈𝑒^𝑇₇−𝑒₁₀𝑄₁𝑒^𝑇₁₀−𝑒₁₁𝑄₂𝑒^𝑇₁₁−𝑒₁₂𝑄₃𝑒^𝑇₁₂−𝑒₁−𝑒₁₀𝑄₄𝑒^𝑇₁−𝑒^𝑇₁₀−𝑒₁−𝑒₃𝑅₆𝑒^𝑇₁−𝑒^𝑇₃−𝜏𝑒₁−𝑒₁₂𝑄₅𝜏𝑒^𝑇₁−𝑒^𝑇₁₂−𝑑₁𝑒₁−𝑒₇𝑅₁₀𝑑₁𝑒^𝑇₁−𝑒^𝑇₇−𝑑₁₂𝑒₁−𝑒₈−𝑒₉𝑅₁₁𝑑₁₂𝑒^𝑇₁−𝑒^𝑇₈−𝑒^𝑇₉, (43) and other notations are the same as Theorem 10.

3.2. Stochastic Stability for the Uncertain Neutral Markovian Jump Systems

In this subsection, we consider the uncertain case which can be described by (3). Based on Theorem 10, we obtain the following theorem to guarantee the stochastic stability for the uncertain neutral systems with interval time-varying delays and Markovian jump parameters.

Theorem 14. For the given finite set 𝑆 of modes with transition rates matrix, scalars 𝑑₁, 𝑑₂, 𝜏, and 𝜇, the uncertain neutral systems with Markovian jump parameters and time-varying delays as described by (3) are stochastically stable if the operator 𝔇 is stable, and there exist scalars 𝛿₁>0, 𝛿₂>0, symmetric positive matrices 𝑃_𝑖>0(𝑖∈𝑆), 𝑄_𝑗>0(𝑗=1,2,3,4,5), and 𝑅_𝑘>0(𝑘=1,2,…,11) such that the following matrix inequalities hold: Π_𝑖1+1𝛿₁𝑒₁𝑃_𝑖𝐻_𝑖𝐻^𝑇_𝑖𝑃_𝑖𝑒^𝑇₁+𝛿₁𝜀𝜀^𝑇Γ^𝑇+1𝛿₁𝑒₁𝑃_𝑖𝐻_𝑖𝐻^𝑇_𝑖𝑀∗1𝛿₁𝑀𝐻_𝑖𝐻^𝑇_𝑖Π𝑀−𝑀<0,_𝑖2+1𝛿₂𝑒₁𝑃_𝑖𝐻_𝑖𝐻^𝑇_𝑖𝑃_𝑖𝑒^𝑇₁+𝛿₂𝜀𝜀^𝑇Γ^𝑇+1𝛿₂𝑒₁𝑃_𝑖𝐻_𝑖𝐻^𝑇_𝑖𝑀∗1𝛿₂𝑀𝐻_𝑖𝐻^𝑇_𝑖𝑀−𝑀<0, (44)(45) where 𝜀^𝑇=𝐸_𝐴𝑖𝑒^𝑇₁+𝐸_𝐵𝑖𝑒^𝑇₂, Π_𝑖1, Π_𝑖2, Γ, and 𝑀 have been defined in Theorem 10.

Proof. On the basis of Theorem 10, we directly replace 𝐴_𝑖 and 𝐵_𝑖 with 𝐴_𝑖+Δ𝐴_𝑖(𝑡), 𝐵_𝑖+Δ𝐵_𝑖(𝑡) and obtain Π_𝑖1(𝑡)+Γ^𝑇(𝑡)(𝑡)Π𝑀Γ<0,_𝑖2(𝑡)+Γ^𝑇(𝑡)𝑀Γ(𝑡)<0, (46)(47) where Π_𝑖1(𝑡)=Π_𝑖1+𝑒₁Δ𝐴^𝑇_𝑖(𝑡)𝑃_𝑖+𝑃_𝑖Δ𝐴_𝑖(𝑡)𝑒^𝑇₁+𝑒₁𝑃_𝑖Δ𝐵_𝑖(𝑡)𝑒^𝑇₂+𝑒₂Δ𝐵^𝑇_𝑖(𝑡)𝑃_𝑖𝑒^𝑇₁,Π_𝑖2(𝑡)=Π_𝑖2+𝑒₁Δ𝐴^𝑇_𝑖(𝑡)𝑃_𝑖+𝑃_𝑖Δ𝐴_𝑖(𝑡)𝑒^𝑇₁+𝑒₁𝑃_𝑖Δ𝐵_𝑖(𝑡)𝑒^𝑇₂+𝑒₂Δ𝐵^𝑇_𝑖(𝑡)𝑃_𝑖𝑒^𝑇₁. (48) Considering (46) and combining the uncertainties condition (7) by Lemma 9, we have Π_𝑖1Γ^𝑇𝑀+𝑒𝑀Γ−𝑀¹𝑃_𝑖0𝐻𝑀0^𝑖𝐹_𝑖(𝑡)𝜀^𝑇+𝜀𝐹^𝑇_𝑖(𝑡)𝐻^𝑇_𝑖𝑃_𝑖𝑒^𝑇₁𝑀00<0. (49) With (8), by Lemma 8 from (49), we obtain Π_𝑖1Γ^𝑇𝑀+1𝑀Γ−𝑀𝛿₁𝑒₁𝑃_𝑖0𝐻𝑀0^𝑖𝐻^𝑇_𝑖𝑃_𝑖𝑒^𝑇₁𝑀00+𝛿₁𝜀𝜀^𝑇000<0. (50) Obviously, (50) is equivalent to (44). Similarly, considering (46) and following the same procedure, we can get (45). Finally, following from the latter proof of Theorem 10, we know that the uncertain neutral systems with Markovian jump parameters and time-varying delay as described by (3) are stochastically stable. This completes the proof.

Remark 15. It should be noted that (44) and (45) can be viewed as linear matrix inequalities by introducing new variables. That is, define matrices 𝑃_𝑖⁽¹⁾=1𝛿₁𝑃_𝑖𝐻_𝑖𝐻^𝑇_𝑖𝑃_𝑖,𝑃_𝑖⁽²⁾=1𝛿₂𝑃_𝑖𝐻_𝑖𝐻^𝑇_𝑖𝑃_𝑖,𝑃⁽¹⁾_𝑖𝑀=1𝛿₁𝑃_𝑖𝐻_𝑖𝐻^𝑇_𝑖𝑀,𝑃⁽²⁾_𝑖𝑀=1𝛿₂𝑃_𝑖𝐻_𝑖𝐻^𝑇_𝑖𝑀𝑀,₍₁₎=1𝛿₁𝑀𝐻_𝑖𝐻^𝑇_𝑖𝑀,𝑀⁽²⁾=1𝛿₂𝑀𝐻_𝑖𝐻^𝑇_𝑖𝑀, (51) where 𝐻_𝑖, 𝑖∈𝑆, are known constant matrices and have been defined in (7). Then (44) and (45) can be easily solved by the LMI Toolbox in MATLAB.

Remark 16. It should be mentioned that Theorem 14 is an extension of Theorem 10 to uncertain neutral Markovian jump systems with interval time-varying delays. It provides a stochastic delay-range-dependent stability criterion for (3) and it will be verified to be less conservative than some existing ones in Section 4.

Consistent with the nominal systems, in the uncertain case, we have the following result as a corollary if the information on the delay derivative 𝜇 may not be available. The corollary is also obtained by setting 𝑅₃=0 in Theorem 14.

Corollary 17. For the given finite set 𝑆 of modes with transition rates matrix, scalars 𝑑₁, 𝑑₂, and 𝜏, the uncertain neutral systems with Markovian jump parameters and time-varying delays as described by (3) are stochastically stable if the operator 𝔇 is stable, and there exist scalars 𝛿₁>0, 𝛿₂>0, symmetric positive matrices 𝑃_𝑖>0(𝑖∈𝑆), 𝑄_𝑗>0(𝑗=1,2,3,4,5), and 𝑅_𝑘>0(𝑘=1,2,4,5,…,11) such that the following symmetric matrix inequalities hold: Π_𝑖1+1𝛿₁𝑒₁𝑃_𝑖𝐻_𝑖𝐻^𝑇_𝑖𝑃_𝑖𝑒^𝑇₁+𝛿₁𝜀𝜀^𝑇Γ^𝑇+1𝛿₁𝑒₁𝑃_𝑖𝐻_𝑖𝐻^𝑇_𝑖𝑀∗1𝛿₁𝑀𝐻_𝑖𝐻^𝑇_𝑖Π𝑀−𝑀<0,_𝑖2+1𝛿₂𝑒₁𝑃_𝑖𝐻_𝑖𝐻^𝑇_𝑖𝑃_𝑖𝑒^𝑇₁+𝛿₂𝜀𝜀^𝑇Γ^𝑇+1𝛿₂𝑒₁𝑃_𝑖𝐻_𝑖𝐻^𝑇_𝑖𝑀∗1𝛿₂𝑀𝐻_𝑖𝐻^𝑇_𝑖𝑀−𝑀<0, (52) where Π_𝑖1 and Π_𝑖2 have been defined in Corollary 13, the remaining notations are the same as Theorem 14.

4. Numerical Examples

In this section, numerical examples are given to show that the proposed theoretical results in this paper are effective and less conservative than some previous ones in the literature.

Example 1. Consider the nominal system in the form of (9) described as follows: 𝑥̇(𝑡)−𝐶_𝑖𝑥̇(𝑡−0.1)=𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑥(𝑡−𝑑(𝑡)), (53) where 𝑖∈𝑆={1,2} and the mode switching is governed by the rate matrix [𝜋_𝑖𝑗]_2×2=⁻⁵⁵₄₋₄, which is described by Figure 1𝐴₁=25−2−3,𝐵₁=,𝐶−0.30.5−0.2−0.3₁=−0.200.1−0.2,𝐴₂=,𝐵−5−1.62−4₂=−0.30.5−0.2−0.3,𝐶₂=.−0.10.20−0.2 (54) Given the time-varying delay 𝑑(𝑡)=0.5(2+sin³𝑡), from the graph of 𝑑(𝑡) with 𝑡∈[0,2𝜋] in Figure 2, we easily obtain 𝑑₁=0.5 and 𝑑₂=1.5. In addition, we have 𝑑(̇𝑡)=1.5sin²𝑡cos𝑡 and its maximum 𝜇=√3/3.

Figure 1 

Operation modes of Example 1.

Figure 2 

Interval time-varying delay 𝑑(𝑡) of Example 1.

By Theorem 10, with the help of LMI toolbox in MATLAB, we solve (19) and get a group of matrices for the solution to guarantee the stochastic stability for the system (53) as follows: for simplicity, we only list the matrices for 𝑃_𝑖, 𝑖∈𝑆={1,2}, 𝑄_𝑗, 𝑗=1,2,…,5. 𝑃₁=1.4857−0.3614∗0.6329,𝑃₂=,𝑄2.0645−0.3761∗0.5086₁=0.7342−0.1546∗0.2978,𝑄₂=,𝑄0.4083−0.1873∗0.3652₃=0.4165−0.2137∗0.3056,𝑄₄=,𝑄0.4576−0.2539∗0.3684₅=.0.2673−0.0845∗0.1766 (55) Therefore, it can be seen that the system (9) is determined to be stochastically stable by Theorem 10.

Example 2. As said in the literature [32], with the abrupt variation in its structures and parameters, we can present the partial element equivalent circuit (PEEC) model as a stochastic jump one. Then, a general form of PEEC model is given by (3), where we assume that the neutral delay of the PEEC model is constant. Consider the stochastic neutral partial element equivalent circuit (PEEC) model described by the following equation: 𝑥̇(𝑡)−𝐶_𝑖𝑥̇(𝑡−0.3)=𝐴_𝑖(𝑡)𝑥(𝑡)+𝐵+Δ𝐴_𝑖+Δ𝐵(𝑡)𝑥(𝑡−𝑑(𝑡)), (56) where 𝑖∈𝑆={1,2} and the mode switching is governed by the rate matrix [𝜋_𝑖𝑗]_2×2=⁻⁴⁴₃₋₃, which is described by Figure 3𝐴₁=−500−6,𝐵₁=,𝐶−1.60−1.8−1.5₁=0.5𝐼,𝐻₁=,𝐴0.20.2₂=−400−5,𝐵₂=,𝐶−20−0.9−1.2₂=0.3𝐼,𝐻₂=0,𝐸−0.3_𝐴1=0.20,𝐸_𝐴2=,𝐸00.2_𝐵1=−0.30.3,𝐸_𝐵2=0.20.2. (57) Given ‖𝐹_𝑖(𝑡)‖<1 and the time-varying delay 𝑑(𝑡)=0.5(2+cos³𝑡), from the graph of 𝑑(𝑡) with 𝑡∈[0,2𝜋] in Figure 4, we easily obtain 𝑑₁=0.5 and 𝑑₂=1.5. In addition, we have 𝑑(̇𝑡)=−1.5cos²𝑡sin𝑡 and its maximum 𝜇=√3/3.

Figure 3 

Operation modes of Example 2.

Figure 4 

Interval time-varying delay 𝑑(𝑡) of Example 2.

By Theorem 14, with the help of LMI toolbox in MATLAB, we solve (44) and (45) and get a group of matrices for the solution to guarantee the stochastic stability for the system (56) as follows: for simplicity, we only list the matrices for 𝑃_𝑖, 𝑖∈𝑆={1,2}, 𝑄_𝑗, 𝑗=1,2,…,5. 𝑃₁=2.4327−0.4713∗0.7846,𝑃₂=,𝑄2.7685−0.4617∗0.7432₁=0.5122−0.1558∗0.3976,𝑄₂=,𝑄0.6083−0.1898∗0.3976₃=0.3164−0.2058∗0.4751,𝑄₄=,𝑄0.2563−0.1584∗0.3476₅=.0.1574−0.0713∗0.1798 (58) Therefore, according to Theorem 14, the uncertain neutral PEEC system presented by (3) is stochastically stable.

Example 3. In the study of practical electrical circuit systems, a small test circuit which consists of a partial element equivalent circuit (PEEC) was considered in [33], which can be described as the following form: 𝑥̇(𝑡)−𝐶𝑥̇(𝑡−𝜏)=𝐴𝑥(𝑡)+𝐵𝑥(𝑡−𝑑). (59) Compared with (9), (59) can be regarded as 𝑖∈𝑆={1} and 𝑑(𝑡)=𝑑. So we have 𝑑₁=𝑑₂=𝑑, 𝜇=0 and utilize Theorem 10 to compute the maximum discrete delay for system stability.

Remark 18. It should be pointed out that we require 𝑑₁≠𝑑₂ and 𝑑₁≠0 in order to conveniently organize this paper. But from the results of the theorems and corollaries in this paper, we know that they are applicable to many special cases, such as 𝑑(𝑡)≡𝑑, 𝑑₁=𝑑₂, or 𝑑₁=0, 𝜏=0. Actually, we just need to delete the corresponding integral terms in the Lyapunov functional and obtain homologous results.
Consider (59) with the following parameters: ,.𝐴=−0.90.20.1−0.9,𝐵=−1.1−0.2−0.1−1.1𝐶=−0.200.2−0.1 (60) For given 𝜏, by Theorem 10, the maximum 𝑑, which satisfies the LMIs in (19), can be calculated by solving a quasiconvex optimization problem. This neutral system was considered in references [34–36]. The results on the maximum upper bound of 𝑑 are compared in Table 1.
From Table 1, we know that the maximum upper bound of delay 𝑑=2.3026 in this paper by setting 𝜏=0.1, while the maximum upper bound of delay 𝑑=1.7100 for [36], 𝑑=2.1229 for [34], and 𝑑=2.2951 for [35]. The results are also given by setting 𝜏=0.5 and 𝜏=1, and it is found that the maximum upper bound in this paper is larger than those in [34–36]. So it can be demonstrated that the stability condition Theorem 10 in this paper yields less conservative results than the previous ones.

Table 1 

Maximum upper bound of 𝑑 with different neutral delay 𝜏.

Example 4. Further consideration on Example 3, if we take the parameter uncertainties commonly existing in the modeling of a real system into account, a general form of PEEC model is given by 𝑥̇(𝑡)𝑥̇()=((𝑡𝑥(𝑡)+((𝑡𝑥(),−𝐶𝑡−𝜏𝐴+Δ𝐴))𝐵+Δ𝐵))𝑡−𝑑 (61) where 𝐴, 𝐵, and 𝐶 are given in Example 3 and the uncertain matrices Δ𝐴(𝑡) and Δ𝐵(𝑡) satisfy ‖Δ𝐴(𝑡)‖≤𝜅,‖Δ𝐵(𝑡)‖≤𝜅,𝜅≥0. (62) Moreover, in the form of (7) and (8), we assume that 𝐻=𝜅𝐼,𝐸_𝐴=𝐸_𝐵=I,0≤𝜅≤1. (63) Consider (61), for given 𝜏 and 𝜅, by Theorem 14, the maximum upper bound of 𝑑, which satisfies the LMIs in (44) and (45), can be calculated by solving a quasiconvex optimization problem. When 𝜏=1.0, Table 2 gives the comparisons of the maximum allowed delay of 𝑑 for various parameters 𝜅 in different methods.
From Table 2, provided that 𝜏=1.0, we know that the maximum upper bound of delay 𝑑=1.5316 in this paper by setting 𝜅=0.10, while the maximum upper bound of delay 𝑑=1.3864 for [37], 𝑑=1.4385 for [36], and 𝑑=1.5047 for [38]. The results are also given by setting 𝜅=0.15, 𝜅=0.20, and 𝜅=0.25, and it is found that the maximum upper bound in this paper is larger than those in [36–38]. So it can be seen that the delay-range-dependent stability condition Theorem 14 in this paper is less conservative than some earlier reported ones in the literature.

Table 2 

Maximum upper bound of 𝑑 with 𝜏=1.0 and different parameter 𝜅.

Example 5. In this example, to compare the stochastic stability result in Theorem 10 with those in [23, 39, 40], we consider the nominal system (9) with 𝐶_𝑟𝑡=0 and 𝑑₁=0. In fact, we consider here that there are no longer neutral delay systems, for given (9) with the following parameters: 𝐴₁=−3.490.81−0.65−3.27,𝐴₂=,𝐵−2.490.291.34−0.02₁=−0.86−1.29−0.68−2.07,𝐵₂=,𝐶−2.830.50−0.84−1.01₁=𝐶₂𝑃=0,_𝑖𝑗=𝜋_𝑖𝑗2×2,𝑖,𝑗∈𝑆={1,2}. (64) As described previously, for given 𝜋₂₂=−0.8, different values of 𝜋₁₁ and different values of 𝜇, by Theorem 10 and the maximum 𝑑₂, which satisfies the LMIs in (19), can be calculated by solving a quasiconvex optimization problem. Tables 3 and 4 give the contrastive results.
From Table 3, provided that 𝜇=0, we know that the maximum upper bound of delay 𝑑₂=0.6853 in this paper by setting 𝜋₁₁=−0.10, while the maximum upper bound of delay 𝑑₂=0.5012 for [40], 𝑑₂=0.5012 for [39], and 𝑑₂=0.6797 for [23]. The results are also given by setting 𝜋₁₁=−0.50, 𝜋₁₁=−0.80, and 𝜋₁₁=−1.00, and it is found that the maximum upper bound of delay 𝑑₂ in this paper is larger than those in [23, 39, 40]. So it also can be shown that the stochastic stability result in Theorem 10 is less conservative than those results in [23, 39, 40].
From Table 4, provided that 𝜇=1.5, we know that the maximum upper bound of delay 𝑑₂=0.3953 in this paper by setting 𝜋₁₁=−0.10, while the methods in [39, 40] cannot be applicable to the case 𝜇≥1, and the maximum upper bound of delay 𝑑₂=0.3860 for [23]. So it can be shown that Theorem 10 in this paper is less conservative and can be applied to the time-varying delay without the requirement on 𝜇<1.

Table 3 

Maximum upper bound of 𝑑₂ with 𝜇=0 and different parameter 𝜋₁₁.

Table 4 

Maximum upper bound of 𝑑₂ with 𝜇=1.5 and different parameter 𝜋₁₁.

5. Conclusions

In this paper, some new delay-range-dependent conditions have been provided to guarantee the stochastic stability of the neutral systems with Markovian jumping parameters and interval time-varying delays. A novel augmented Lyapunov-Krasovskii functional which contains some triple-integral terms is constructed. By some integral inequalities and the nature of convex combination, some less conservative delay-range-dependent stochastic stability criteria are obtained. Numerical examples are given to demonstrate the effectiveness and less conservativeness of our result.

Acknowledgments

This work was supported in part by the National Key Scientific Research Project (61233003), the National Natural Science Foundation of China (60935001, 61174061, 61074033, and 60934006), and the Fundamental Research Funds for the Central Universities.