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

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.

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. Section 2 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. ℝ^𝑛 denotes the 𝑛 dimensional Euclidean space, and ℝ^𝑚×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‖=(𝑣^𝑇𝑣)^1/2, while ‖𝐴‖ is spectral norm of matrix 𝐴, ‖𝐴‖=[𝜆_max(𝐴^𝑇𝐴)]^1/2. 𝜆_max(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. {𝑟_𝑡,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 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) is the transition rate from mode 𝑖 to 𝑗, and for any state or mode 𝑖∈𝑆, it satisfies 𝜋_𝑖𝑖=−^𝑁_{𝑗=1,𝑗≠𝑖}𝜋_𝑖𝑗. (2)

The following uncertain neutral Markovian jump systems with interval time-varying delays and nonlinear perturbations over the space {Ω,ℱ,𝐏} are considered: 𝑥̇(𝑡)−𝐶𝑟_𝑡𝑥̇((𝑡=𝐴𝑟𝑡−𝜏))_𝑡+Δ𝐴𝑟_𝑡𝑥(𝑡)+𝐵𝑟_𝑡+Δ𝐵𝑟_𝑡𝑥(𝑡−𝑑(𝑡))+𝑓₁(𝑥(𝑡),𝑡)+𝑓₂(𝑥(𝑡−𝑑(𝑡)),𝑡)+𝑓₃(𝑥̇((𝑡),𝑥(𝑠)(𝑠)𝑡−𝜏)),𝑡=𝜑,𝑟_𝑠=𝑟₀,𝑠∈−𝜌,0, (3)(4) where 𝑥(𝑡)∈ℝ^𝑛 is the system state and 𝜏(𝑡) is time-varying neutral delay which satisfies 0≤𝜏(𝑡)≤𝜏, 𝜏̇(𝑡)≤𝜏_𝑑. The time-varying retarded delay 𝑑(𝑡) is such that 0<𝑑₁≤𝑑(𝑡)≤𝑑₂,𝑑̇(𝑡)≤𝜇, (5) 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)∈ℝ^𝑛, 𝑓₂(𝑥(𝑡−𝑑(𝑡)),𝑡)∈ℝ^𝑛, and 𝑓₃(𝑥̇(𝑡−𝜏(𝑡)),𝑡)∈ℝ^𝑛 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 𝑥̇(𝑡−𝜏(𝑡)), respectively. For all 𝑡, they are assumed to be bounded in magnitude as 𝑓₁(𝑥(𝑡)‖𝑥(𝑡,𝑓,𝑡)≤𝛼)‖₂(𝑥((𝑡‖𝑥((𝑡,𝑓𝑡−𝑑)),𝑡)≤𝛽𝑡−𝑑))‖₃(𝑥̇(𝑡−𝜏(𝑡)),𝑡)≤𝛾‖𝑥̇(𝑡−𝜏(𝑡))‖, (6) 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(𝑥̇(𝑡−𝜏(𝑡)),𝑡).

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, and Δ𝐴(𝑟_𝑡)∈ℝ^𝑛×𝑛, Δ𝐵(𝑟_𝑡)∈ℝ^𝑛×𝑛 are uncertainties. For simplicity of notations, we denote 𝐴(𝑟_𝑡), Δ𝐴(𝑟_𝑡), 𝐵(𝑟_𝑡), Δ𝐵(𝑟_𝑡), 𝐶(𝑟_𝑡) by 𝐴_𝑖, Δ𝐴_𝑖, 𝐵_𝑖, Δ𝐵_𝑖, 𝐶_𝑖 for 𝑟_𝑡=𝑖∈𝑆. The parametric matrix ‖𝐶_𝑖‖<1 and the admissible parametric uncertainties 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, the following nominal systems can be obtained for 𝐹_𝑖(𝑡)=0: 𝑥̇(𝑡)−𝐶_𝑖𝑥̇(𝑡−𝜏(𝑡))=𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑥(𝑡−𝑑(𝑡))+𝑓₁+𝑓₂+𝑓₃. (9)

Before proceeding with the main results, we present the following definitions, assumptions, and lemmas.

Assumption 1. System matrices 𝐴_𝑖, (∀𝑖∈𝑆) are Hurwitz, and all the eigenvalues have negative real parts for each mode. 𝐻_𝑖, (∀𝑖∈𝑆) is full rank in row.

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

Definition 3 (see [28]). Define operator 𝔇∶𝐶([−𝜌,0],ℝ^𝑛)→ℝ^𝑛 as 𝔇(𝑥_𝑡)=𝑥(𝑡)−𝐶𝑥(𝑡−𝜏). 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 (10) is uniformly asymptotically stable. In this paper, that is, ‖𝐶_𝑖‖+𝛾<1.

Definition 4 (see [29]). The system in (3) is exponentially stable with a decay rate 𝜀 for all 𝑟_𝑡=𝑖∈𝑆, if there exist scalars 𝜀>0 and 𝜎≥1 such that for all 𝑥(𝑡), ‖𝑥(𝑡)‖≤𝜎exp−𝜀𝑡−𝑡₀𝑥_𝑡0𝜌, (11) where 𝜀 is the exponential decay rate, ‖∗‖ denotes the Euclidean norm, and 𝑥_𝑡0𝜌=sup_{𝜃∈−𝜌,0}𝑥𝑡₀+𝜃,𝑥̇𝑡₀+𝜃=sup_{𝑠∈−𝜌,𝑡0}𝜑(𝑠),𝜑̇(𝑠). (12)

Definition 5 (see [30]). Define the stochastic Lyapunov-Krasovskii function of system (3) as 𝑉(𝑥(𝑡),𝑟_𝑡=𝑖,𝑡>0)=𝑉(𝑥_𝑡,𝑖,𝑡), where its infinitesimal generator is defined as 𝔏𝑉(𝑥(𝑡),𝑖,𝑡)=lim_Δ𝑡⟶01ℰ𝑉𝑥()Δ𝑡𝑡+Δ𝑡,𝑟_𝑡+Δ𝑡(𝑡)𝑟,𝑡+Δ𝑡∣𝑥=𝑥,_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 [31]). For given real constant matrices Ω₁, Ω₂, and Ω₃, with appropriate dimensions, where Ω₁=Ω^𝑇₁ and Ω₂=Ω^𝑇₂>0. Ω₁+Ω^𝑇₃Ω₂⁻¹Ω₃<0 if and only if Ω₁Ω^𝑇₃∗−Ω₂<0or−Ω₂Ω^𝑇₃∗Ω₁<0. (14)

Lemma 7. For any constant matrix 𝐻=𝐻^𝑇>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)exp𝜀ℎ₂(𝑡)−exp𝜀ℎ₁(𝑡)𝜀×^{𝑡−ℎ1(𝑡)}_{𝑡−ℎ2(𝑡)}exp{𝜀(𝑠−𝑡)}𝑥^𝑇(𝑠)(𝑠)≥𝐻𝑥𝑑𝑠_{𝑡−ℎ1(𝑡)𝑡−ℎ2(𝑡)}𝑥^𝑇(𝑠)𝑑𝑠𝐻^{𝑡−ℎ1(𝑡)}_{𝑡−ℎ2(𝑡)}𝑥(𝑠)𝑑𝑠, (15)(b)𝜏²₂−𝜏²₁2^−𝜏1_−𝜏2^𝑡_𝑡+𝜃exp{𝜀(𝑠−𝑡)}𝑥^𝑇(𝑠)𝐻𝑥(𝑠)𝑑𝑠𝑑𝜃≥exp−𝜀𝜏₂^−𝜏1_−𝜏2^𝑡_𝑡+𝜃𝑥^𝑇(𝑠)𝑑𝑠𝑑𝜃×𝐻^−𝜏1_−𝜏2^𝑡_𝑡+𝜃𝑥(𝑠)𝑑𝑠𝑑𝜃. (16)

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, exp{^−𝜏1_−𝜏2^𝑡_𝑡+𝜃𝜀(𝑠−𝑡)}𝑥^𝑇(𝑠)𝐻𝑥(𝑠)𝑑𝑠𝑑𝜃≥exp−𝜀𝜏₂^−𝜏1_−𝜏2^𝑡_𝑡+𝜃𝑥^𝑇(𝑠)𝐻𝑥(𝑠)𝑑𝑠𝑑𝜃 (17)
(b) Is thus true by [33].

Lemma 8 (see [34]). For functions 𝜆₁(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 𝑇𝑃𝑇_𝑇𝑄>0, (18) and the following inequality holds: 1𝜆₁(𝑡)𝜂₁𝑃𝜂^𝑇₁+1𝜆₂(𝑡)𝜂₂𝑄𝜂^𝑇₂≥𝜂₁𝜂₂𝑇𝑃𝑇_𝑇𝑄𝜂^𝑇₁𝜂^𝑇₂. (19)

Lemma 9 (see [35]). For given matrices 𝑄=𝑄^𝑇, 𝑀, and 𝑁 with appropriate dimensions: 𝑄+𝑀𝐹(𝑡)𝑁+𝑁^𝑇𝐹^𝑇(𝑡)𝑀^𝑇<0, (20) 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. (21)

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 set 𝑆 of modes with transition rates matrix, scalars 𝛼, 𝛽, 𝛾, 𝜀, 𝜏, 𝜏_𝑑, 𝑑₁, 𝑑₂, 𝜇, and constant scalar 𝑑_𝑚 satisfying 𝑑₁<𝑑_𝑚<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 𝑃_𝑖>0, (𝑖∈𝑆), 𝑄_𝑗>0, (𝑗=1,2,3,4,5), 𝑅_𝑘>0, (𝑘=1,2,3,4,5), 𝑊_𝑙>0, (𝑙=1,2,3,4), 𝑌_𝑚>0, (𝑚=1,2,3,4), 𝑍_𝑛>0, (𝑛=1,2,3,4) and matrices 𝑈₁, 𝑈₂, 𝑋₁, 𝑋₂, 𝑋₃, 𝑋₄  for any scalars 𝜀₁, 𝜀₂, 𝜀₃ and any matrices 𝑇_𝑘, (𝑘=1,2,…,23) with appropriate dimensions, such that the following linear matrix inequalities (22), (23), and (24) hold: 𝑊₁𝑈₁𝑈^𝑇₁𝑊₁𝑌>0,¹𝑈₂𝑈^𝑇₂𝑌₁𝑊>0,(i)³𝑋₁𝑋^𝑇₁𝑊₃𝑌>0,³𝑋₂𝑋^𝑇₂𝑌₃>0,(ii)Ξ^𝜀_𝑖0+Ξ^𝜀_𝑖1𝑊<0,(iii)⁴𝑋₃𝑋^𝑇₃𝑊₄𝑌>0,⁴𝑋₄𝑋^𝑇₄𝑌₄>0,(iv)Ξ^𝜀_𝑖0+Ξ^𝜀_𝑖2<0, (22)(23)(24) where Ξ^𝜀_𝑖0=²³_𝑚=1𝑒_𝑚𝐺_𝑚𝑒^𝑇_𝑚(✠)−+𝒜𝜀𝑑₁exp𝜀𝑑₁𝑒−1¹−𝑒₄𝑌₂𝑒^𝑇₁−𝑒^𝑇₄−exp{−𝜀𝜏}𝜏𝑒₁−𝑒₂₀𝑍₁𝜏𝑒^𝑇₁−𝑒^𝑇₂₀−exp−𝜀d₁𝑑₁𝑒₁−𝑒₁₀𝑍₂𝑑₁𝑒^𝑇₁−𝑒^𝑇₁₀−exp−𝜀𝑑_𝑚𝜚₁𝑒₁−𝑒₁₃𝑍₃𝜚₁𝑒^𝑇₁−𝑒^𝑇₁₃−exp−𝜀𝑑₂𝜚₂𝑒₁−𝑒₁₄𝑍₄𝜚₂𝑒^𝑇₁−𝑒^𝑇₁₄−𝜀𝜏{}𝑒exp𝜀𝜏−1¹⁹𝑒₂₀−𝑒₁₉𝑊₁𝑈₁𝑈^𝑇₁𝑊₁𝑒^𝑇₁₉𝑒^𝑇₂₀−𝑒^𝑇₁₉−𝜀𝜏{}𝑒exp𝜀𝜏−1¹−𝑒₁₇𝑒₁₇−𝑒₁₅×𝑌₁𝑈₂𝑈^𝑇₂𝑌₁𝑒^𝑇₁−𝑒^𝑇₁₇𝑒^𝑇₁₇−𝑒^𝑇₁₅, (25) where 𝒜 is a linear operator on ℝ^𝑛×𝑛 by 𝒜(𝐹)=𝐹+𝐹^𝑇,∀𝐹∈ℝ^𝑛×𝑛,✠=−𝑒₁𝑇₁𝑒^𝑇₂+𝑒₁𝑃_𝑖𝐵_𝑖+𝑇₁𝐵_𝑖𝑒^𝑇₃+𝑒₁𝑃_𝑖𝐶_𝑖𝑒^𝑇₁₈+²³_𝑚=2𝑒₁𝐴^𝑇_𝑖𝑇^𝑇_𝑚𝑒^𝑇_𝑚+²³_𝑚=21𝑒₁𝑃_𝑖𝑒^𝑇_𝑚+𝑒₂𝑇₂𝐵_𝑖𝑒^𝑇₃−²³_𝑚=3𝑒₂𝑇^𝑇_𝑚𝑒^𝑇_𝑚+²³_𝑚=4𝑒₃𝐵^𝑇_𝑖𝑇^𝑇_𝑚𝑒^𝑇_𝑚+¹⁷_𝑚=1𝑒_𝑚𝑇_𝑚𝐶_𝑖𝑒^𝑇₁₈+²³_𝑚=19𝑒₁₈𝐶^𝑇_𝑖𝑇^𝑇_𝑚𝑒^𝑇_𝑚+²⁰_𝑚=1²³_𝑛=21𝑒_𝑚𝑇_𝑚𝑒^𝑇_𝑛+𝑒₂₁𝑇₂₁𝑒^𝑇₂₂+𝑒^𝑇₂₃+𝑒₂₂𝑇₂₂𝑒^𝑇₂₃+²³_𝑚=22𝑒₂₁𝑇^𝑇_𝑚𝑒^𝑇_𝑚+𝑒₂₂𝑇^𝑇₂₃𝑒^𝑇₂₃,𝐺₁=𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+^𝑁_𝑗=1𝜋_𝑖𝑗𝑃_𝑗+𝜀𝑃_𝑖+𝑄₁+𝑄₃+𝜏²𝑊₁+𝑑²₁𝑊₂+𝜚²₁𝑊₃+𝜚²₂𝑊₄+𝜀₁𝛼²𝐼+𝑇₁𝐴_𝑖+𝐴^𝑇_𝑖𝑇^𝑇₁,𝐺₂=𝑅₁+𝑅₂+𝑅₅+𝜏²𝑌₁+𝑑²₁𝑌₂+𝜚²₁𝑌₃+𝜚²₂𝑌₄+𝜏⁴4𝑍₁+𝑑⁴₁4𝑍₂+𝜚²₃𝑍₃+𝜚²₄𝑍₄−𝑇₂−𝑇^𝑇₂,𝐺₃=𝜒𝜀,𝜇𝑄₂+𝑇₃𝐵_𝑖+𝐵^𝑇_𝑖𝑇^𝑇₃+𝜀₂𝛽²𝐺𝐼,₄=exp−𝜀𝑑₁𝑄₂+𝑄₄−𝑄₃,𝐺₅=exp−𝜀𝑑_𝑚𝑄₅−𝑄₄,𝐺₆=−exp−𝜀𝑑₂𝑄₅,𝐺₇=exp−𝜀𝑑₁𝑅₃−𝑅₂,𝐺₈=exp−𝜀𝑑_𝑚𝑅₄−𝑅₃,𝐺₉=−exp−𝜀𝑑₂𝑅₄,𝐺₁₀=−𝜀𝑑₁exp𝜀𝑑₁𝑊−1²,𝐺_𝑚(),𝐺=0,𝑚=11,12,13,14,17,19,20₁₅=−exp{−𝜀𝜏}𝑄₁,𝐺₁₆=−exp{−𝜀𝜏}𝑅₁,𝐺₁₈=−1−𝜏_𝑑exp{−𝜀𝜏}𝑅₅+𝑇₁₈𝐶_𝑖+𝐶^𝑇_𝑖𝑇^𝑇₁₈+𝜀₃𝛾²𝐺𝐼,₂₁=−𝜀₁𝐼+𝑇₂₁+𝑇^𝑇₂₁,𝐺₂₂=−𝜀₂𝐼+𝑇₂₂+𝑇^𝑇₂₂,𝐺₂₃=−𝜀₃𝐼+𝑇₂₃+𝑇^𝑇₂₃,Ξ^𝜀_𝑖1=−𝜀𝜚₂exp𝜀𝑑₂−exp𝜀𝑑_𝑚𝑒₁₄𝑊₄𝑒^𝑇₁₄−𝜀𝜚₂exp𝜀𝑑₂−exp𝜀𝑑_𝑚𝑒₅−𝑒₆𝑌₄𝑒^𝑇₅−𝑒^𝑇₆−𝜀𝜚₁exp𝜀𝑑_𝑚−exp𝜀𝑑₁×𝑒₁₁𝑒₁₃−𝑒₁₁𝑊₃𝑋₁𝑋^𝑇₁𝑊₃𝑒^𝑇₁₁𝑒^𝑇₁₃−𝑒^𝑇₁₁−𝜀𝜚₁exp𝜀𝑑_𝑚−exp𝜀𝑑₁×𝑒₄−𝑒₃𝑒₃−𝑒₅𝑌₃𝑋₂𝑋^𝑇₂𝑌₃𝑒^𝑇₄−𝑒^𝑇₃𝑒^𝑇₃−𝑒^𝑇₅,Ξ^𝜀_𝑖2=−𝜀𝜚₁exp𝜀𝑑_𝑚−exp𝜀𝑑₁𝑒₁₃𝑊₃𝑒^𝑇₁₃−𝜀𝜚₁exp𝜀𝑑_𝑚−exp𝜀𝑑₁𝑒₄−𝑒₅𝑌₃𝑒^𝑇₄−𝑒^𝑇₅−𝜀𝜚₂exp𝜀𝑑₂−exp𝜀𝑑_𝑚×𝑒₁₂𝑒₁₄−𝑒₁₂𝑊₄𝑋₃𝑋^𝑇₃𝑊₄𝑒^𝑇₁₂𝑒^𝑇₁₄−𝑒^𝑇₁₂−𝜀𝜚₂exp𝜀𝑑₂−exp𝜀𝑑_𝑚×𝑒₃−𝑒₆𝑒₅−𝑒₃𝑌₄𝑋₄𝑋^𝑇₄𝑌₄𝑒^𝑇₃−𝑒^𝑇₆𝑒^𝑇₅−𝑒^𝑇₃, (26) where 𝑒_𝑖{𝑖=1,2,…,23} are block entry matrices; that is, 𝑒^𝑇₂=_{0𝐼000000000000000000000},𝜒𝜀,𝜇=𝜇−1exp−𝜀𝑑₁𝑤𝑖𝑡ℎ𝜇≥1,𝜇−1exp−𝜀𝑑₂𝑤𝑖𝑡ℎ𝜇<1,𝜆=max_𝑖∈𝑆𝜆_max𝑃_𝑖+1−exp−𝜀𝜌𝜀⁵_𝑗=1𝜆_max𝑄_𝑗+⁵_𝑘=1𝜆_max𝑅_𝑘+exp−𝜀𝜌+𝜀𝜌−1𝜀²×⁴_𝑙=1𝜆_max𝑊_𝑙+⁴_𝑚=1𝜆_max𝑌_𝑚+2−2exp−𝜀𝜌+𝜌²𝜀²−2𝜀𝜌2𝜀³⁴_𝑛=1𝜆_max𝑍_𝑛,𝜆=min_𝑖∈𝑆𝜆_min𝑃_𝑖,𝜚₁=𝑑_𝑚−𝑑₁,𝜚₂=𝑑₂−𝑑_𝑚,𝜚₃=12𝑑²_𝑚−𝑑²₁,𝜚₄=12𝑑²₂−𝑑²_𝑚. (27)

Proof. Construct the following Lyapunov functional: 𝑉(𝑥(𝑡),𝑖,𝑡)=⁶_𝑘=1𝑉_𝑘𝑥_𝑡,𝑖, (28) where 𝑉₁𝑥_𝑡,𝑖=𝑥^𝑇(𝑡)𝑃_𝑖𝑥(𝑡),𝑉₂𝑥_𝑡,𝑖=^𝑡_𝑡−𝜏exp{𝜀(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑄₁𝑥(𝑠)+𝑑𝑠_{𝑡−𝑑1(𝑡)𝑡−𝑑}exp{𝜀(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑄₂𝑥(𝑠)+𝑑𝑠_{𝑡𝑡−𝑑1}exp{𝜀(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑄₃𝑥(𝑠)𝑑𝑠+^𝑡−𝑑1_{𝑡−𝑑𝑚}exp{𝜀(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑄₄𝑥(𝑠)𝑑𝑠+^{𝑡−𝑑𝑚}_𝑡−𝑑2exp{𝜀(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑄₅𝑥(𝑠)𝑉𝑑𝑠,₃𝑥_𝑡,𝑖=^𝑡_𝑡−𝜏exp{𝜀(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑅₁𝑥̇(𝑠)+𝑑𝑠_{𝑡𝑡−𝑑1}exp{𝜀(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑅₂𝑥̇(𝑠)+𝑑𝑠_{𝑡−𝑑1𝑡−𝑑𝑚}exp{𝜀(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑅₃𝑥̇(𝑠)+𝑑𝑠_{𝑡−𝑑𝑚𝑡−𝑑2}exp{𝜀(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑅₄𝑥̇(𝑠)+𝑑𝑠_{𝑡(𝑡)𝑡−𝜏}exp{𝜀(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑅₅𝑥̇(𝑠)𝑉𝑑𝑠,₄𝑥_𝑡,𝑖=⁰_−𝜏^𝑡_𝑡+𝜃𝜏exp{𝜀(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑊₁𝑥(𝑠)+𝑑𝑠𝑑𝜃_{0−𝑑1𝑡𝑡+𝜃}𝑑₁exp{𝜀(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑊₂𝑥(𝑠)+𝑑𝑠𝑑𝜃_{−𝑑1−𝑑𝑚𝑡𝑡+𝜃}𝜚₁exp{𝜀(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑊₃𝑥(𝑠)+𝑑𝑠𝑑𝜃_{−𝑑𝑚−𝑑2𝑡𝑡+𝜃}𝜚₂exp{𝜀(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑊₄𝑥(𝑠)𝑉𝑑𝑠𝑑𝜃,₅𝑥_𝑡,𝑖=⁰_−𝜏^𝑡_𝑡+𝜃𝜏exp{𝜀(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑌₁𝑥̇(𝑠)+𝑑𝑠𝑑𝜃_{0−𝑑1𝑡𝑡+𝜃}𝑑₁exp{𝜀(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑌₂𝑥̇(𝑠)+𝑑𝑠𝑑𝜃_{−𝑑1−𝑑𝑚𝑡𝑡+𝜃}𝜚₁exp{𝜀(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑌₃𝑥̇(𝑠)+𝑑𝑠𝑑𝜃_{−𝑑𝑚−𝑑2𝑡𝑡+𝜃}𝜚₂exp{𝜀(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑌₄𝑥̇(𝑠)𝑉𝑑𝑠𝑑𝜃,₆𝑥_𝑡=,𝑖_{0−𝜏0𝜃𝑡𝑡+𝜆}𝜏²2exp{𝜀(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑍₁𝑥̇(𝑠)+𝑑𝑠𝑑𝜆𝑑𝜃_{0−𝑑10𝜃𝑡𝑡+𝜆}𝑑²₁2exp{𝜀(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑍₂𝑥̇(𝑠)+𝑑𝑠𝑑𝜆𝑑𝜃_{−𝑑1−𝑑𝑚0𝜃𝑡𝑡+𝜆}𝜚₃exp{𝜀(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑍₃𝑥̇(𝑠)+𝑑𝑠𝑑𝜆𝑑𝜃_{−𝑑𝑚−𝑑20𝜃𝑡𝑡+𝜆}𝜚₄exp{𝜀(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑍₄𝑥̇(𝑠)𝑑𝑠𝑑𝜆𝑑𝜃. (29)(30)(31)(32)(33)(34)
Taking 𝔏 as its infinitesimal generator along the trajectory of system (9), we obtain the following from Definition 5 and (28)–(34): 𝔏𝑉(𝑥(𝑡),𝑖,𝑡)=𝔏𝑉₁𝑥_𝑡,𝑖+𝔏𝑉₂𝑥_𝑡,𝑖+𝔏𝑉₃𝑥_𝑡,𝑖+𝔏𝑉₄𝑥_𝑡,𝑖+𝔏𝑉₅𝑥_𝑡,𝑖+𝔏𝑉₆𝑥_𝑡,𝑖, (35)𝔏𝑉₁𝑥_𝑡,𝑖=2𝑥^𝑇(𝑡)𝐴^𝑇_𝑖+𝑥^𝑇(𝑡−𝑑(𝑡))𝐵^𝑇_𝑖+𝑥̇^𝑇(𝑡−𝜏(𝑡))𝐶^𝑇_𝑖+𝑓^𝑇₁+𝑓^𝑇₂+𝑓^𝑇₃×𝑃_𝑖𝑥(𝑡)+^𝑁_𝑗=1𝜋_𝑖𝑗𝑥^𝑇(𝑡)𝑃_𝑗𝑥(𝑡)+𝜀𝑥^𝑇(𝑡)𝑃_𝑖𝑥(𝑡)−𝜀𝑉₁𝑥_𝑡,𝑖, (36)𝔏𝑉₂𝑥_𝑡,𝑖=𝑥^𝑇(𝑡)𝑄₁+𝑄₃𝑥(𝑡)+exp−𝜀𝑑₁𝑥^𝑇×𝑡−𝑑₁𝑄₂+𝑄₄−𝑄₃𝑥𝑡−𝑑₁−𝜀𝑉₂𝑥_𝑡−𝑑̇(𝑡){(𝑡𝑥,𝑖1−exp−𝜀𝑑)}^𝑇(𝑡−𝑑(𝑡))×𝑄₂𝑥(𝑡−𝑑(𝑡))−exp{−𝜀𝜏}𝑥^𝑇(𝑡−𝜏)×𝑄₁𝑥(𝑡−𝜏)+exp−𝜀𝑑_𝑚×𝑥^𝑇𝑡−𝑑_𝑚𝑄₅−𝑄₄×𝑥𝑡−𝑑_𝑚−exp−𝜀𝑑₂𝑥^𝑇𝑡−𝑑₂𝑄₅𝑥𝑡−𝑑₂,𝔏𝑉₃𝑥_𝑡,𝑖=𝑥̇^𝑇(𝑡)𝑅₁+𝑅₂+𝑅₅𝑥̇(𝑡)+exp−𝜀𝑑₁𝑥̇^𝑇𝑡−𝑑₁𝑅₃−𝑅₂×𝑥̇𝑡−𝑑₁−𝜀𝑉₃𝑥_𝑡,𝑖+exp−𝜀𝑑_𝑚𝑥̇^𝑇𝑡−𝑑_𝑚𝑅₄−𝑅₃×𝑥̇𝑡−𝑑_𝑚{}×𝑥̇−exp−𝜀𝜏_𝑇(𝑡−𝜏)𝑅₁𝑥̇()−(𝜏̇(𝑡{(𝑡×𝑥̇𝑡−𝜏1−))exp−𝜀𝜏)}_𝑇(𝑡−𝜏(𝑡))𝑅₅𝑥̇(𝑡−𝜏(𝑡))−exp−𝜀𝑑₂𝑥̇^𝑇𝑡−𝑑₂𝑅₄𝑥̇𝑡−𝑑₂, (37)(38)𝔏𝑉₄𝑥_𝑡,𝑖=𝑥^𝑇(𝑡)𝜏²𝑊₁+𝑑²₁𝑊₂+𝜚²₁𝑊₃+𝜚²₂𝑊₄×𝑥(𝑡)−𝜀𝑉₄𝑥_𝑡−,𝑖_{𝑡𝑡−𝜏}𝜏exp{𝜀(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑊₁𝑥(𝑠)𝑑𝑠−^𝑡_𝑡−𝑑1𝑑₁exp{𝜀(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑊₂𝑥(𝑠)𝑑𝑠−^𝑡−𝑑1_{𝑡−𝑑𝑚}𝜚₁exp{𝜀(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑊₃𝑥(𝑠)𝑑𝑠−^{𝑡−𝑑𝑚}_𝑡−𝑑2𝜚₂exp{𝜀(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑊₄𝑥(𝑠)𝑑𝑠,𝔏𝑉₅𝑥_𝑡,𝑖=𝑥̇^𝑇(𝑡)𝜏²𝑌₁+𝑑²₁𝑌₂+𝜚²₁𝑌₃+𝜚²₂𝑌₄𝑥̇(𝑡)−𝜀𝑉₅𝑥_𝑡,𝑖−^𝑡_𝑡−𝜏𝜏exp{𝜀(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑌₁𝑥̇(𝑠)−𝑑𝑠_{𝑡𝑡−𝑑1}𝑑₁exp{𝜀(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑌₂𝑥̇(𝑠)−𝑑𝑠_{𝑡−𝑑1𝑡−𝑑𝑚}𝜚₁exp{𝜀(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑌₃𝑥̇(𝑠)−𝑑𝑠_{𝑡−𝑑𝑚𝑡−𝑑2}𝜚₂exp{𝜀(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑌₄𝑥̇(𝑠)𝑑𝑠,𝔏𝑉₆𝑥_𝑡,𝑖=𝑥̇^𝑇(𝑡)×𝜏⁴4𝑍₁+𝑑⁴₁4𝑍₂+𝜚²₃𝑍₃+𝜚²₄𝑍₄𝑥̇(𝑡)−𝜀𝑉₆𝑥_𝑡−,𝑖_{0−𝜏𝑡𝑡+𝜃}𝜏²2exp{𝜀(𝑠−𝑡)}𝑥̇^𝑇(𝑠)×𝑍₁𝑥̇(𝑠)−𝑑𝑠𝑑𝜃_{0−𝑑1𝑡𝑡+𝜃}𝑑²₁2exp{𝜀(𝑠−𝑡)}𝑥̇^𝑇(𝑠)×𝑍₂𝑥̇(𝑠)−𝑑𝑠𝑑𝜃_{−𝑑1−𝑑𝑚𝑡𝑡+𝜃}𝜚₃exp{𝜀(𝑠−𝑡)}𝑥̇^𝑇(𝑠)×𝑍₃𝑥̇(𝑠)−𝑑𝑠𝑑𝜃_{−𝑑𝑚−𝑑2𝑡𝑡+𝜃}𝜚₄exp{𝜀(𝑠−𝑡)}𝑥̇^𝑇(𝑠)×𝑍₄𝑥̇(𝑠)𝑑𝑠𝑑𝜃. (39)(40)(41) In view of (6), the following inequalities hold for any scalars 𝜀₁>0, 𝜀₂>0, and 𝜀₃>0: 𝜀₁𝛼²𝑥^𝑇(𝑡)𝑥(𝑡)−𝑓^𝑇₁(𝑥(𝑡),𝑡)𝑓₁(𝑥(𝑡)𝜀,𝑡)≥0,₂𝛽²𝑥^𝑇(𝑡−𝑑(𝑡))𝑥(𝑡−𝑑(𝑡))−𝑓^𝑇₂(𝑥(𝑡−𝑑(𝑡)),𝑡)𝑓₂(𝑥((𝑡𝜀𝑡−𝑑)),𝑡)≥0,₃𝛾²𝑥̇^𝑇(𝑡−𝜏(𝑡))𝑥̇(𝑡−𝜏(𝑡))−𝑓^𝑇₃(𝑥̇(𝑡−𝜏(𝑡)),𝑡)𝑓₃(𝑥̇(𝑡−𝜏(𝑡)),𝑡)≥0. (42) From (35) and (42), we have 𝔏𝑉(𝑥(𝑡),𝑖,𝑡)≤𝔏𝑉₁𝑥_𝑡,𝑖+𝔏𝑉₂𝑥_𝑡,𝑖+𝔏𝑉₃𝑥_𝑡,𝑖+𝔏𝑉₄𝑥_𝑡,𝑖+𝔏𝑉₅𝑥_𝑡,𝑖+𝔏𝑉₆𝑥_𝑡,𝑖+𝜀₁𝛼²𝑥^𝑇(𝑡)𝑥(𝑡)−𝑓^𝑇₁𝑓₁+𝜀₂𝛽²𝑥^𝑇(𝑡−𝑑(𝑡))𝑥(𝑡−𝑑(𝑡))−𝑓^𝑇₂𝑓₂+𝜀₃𝛾²𝑥̇^𝑇(𝑡−𝜏(𝑡))𝑥̇(𝑡−𝜏(𝑡))−𝑓^𝑇₃𝑓₃. (43)
Define𝜉(𝑡)=col𝑥(𝑡)𝑥̇(𝑡)𝑥(𝑡−𝑑(𝑡))𝑥𝑡−𝑑₁𝑥𝑡−𝑑_𝑚𝑥𝑡−𝑑₂𝑥̇𝑡−𝑑₁𝑥̇𝑡−𝑑_𝑚𝑥̇𝑡−𝑑₂∫^𝑡_𝑡−𝑑1𝑥(𝑠)𝑑𝑠∫^𝑡−𝑑1_{(𝑡)𝑡−𝑑}𝑥(𝑠)𝑑𝑠∫^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑥(𝑠)𝑑𝑠∫^𝑡−𝑑1_{𝑡−𝑑𝑚}𝑥(𝑠)𝑑𝑠∫^{𝑡−𝑑𝑚}_𝑡−𝑑2𝑥(𝑠)𝑥()𝑥̇()𝑥((𝑡𝑥̇((𝑡∫𝑑𝑠𝑡−𝜏𝑡−𝜏𝑡−𝜏))𝑡−𝜏))^𝑡_{(𝑡)𝑡−𝜏}𝑥(𝑠)𝑑𝑠∫^𝑡_𝑡−𝜏𝑥(𝑠)𝑑𝑠𝑓₁𝑓₂𝑓₃. (44)
Since it is easy to see −(1−𝜏̇(𝑡))exp{−𝜀𝜏(𝑡)}≤−(1−𝜏_𝑑)exp{−𝜀𝜏}; then from (5) we also obtain that −1−𝑑̇(𝑡)exp{−𝜀𝑑(𝑡)}≤𝜒𝜀,𝜇, (45) where 𝜒(𝜀,𝜇) is defined in Theorem 10. Notice (a) of Lemma 7; then −^𝑡_𝑡−𝑑1𝑑₁exp{𝜀(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑊₂𝑥(𝑠)𝑑𝑠≤−𝜀𝑑₁exp𝜀𝑑₁𝜉−1^𝑇(𝑡)𝑒₁₀𝑊₂𝑒^𝑇₁₀𝜉(𝑡),−^𝑡_𝑡−𝑑1𝑑₁exp{𝜀(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑌₂𝑥̇(𝑠)𝑑𝑠≤−𝜀𝑑₁exp𝜀𝑑₁𝜉−1^𝑇(𝑡)𝑒₁−𝑒₄𝑌₂𝑒^𝑇₁−𝑒^𝑇₄𝜉(𝑡). (46)
Notice (b) of Lemma 7; then −⁰_−𝜏^𝑡_𝑡+𝜃𝜏²2exp{𝜀(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑍₁𝑥̇(𝑠){}𝜉𝑑𝑠𝑑𝜃≤−exp−𝜀𝜏^𝑇(𝑡)𝜏𝑒₁−𝑒₂₀𝑍₁𝜏𝑒^𝑇₁−𝑒^𝑇₂₀𝜉(𝑡),−⁰_−𝑑1^𝑡_𝑡+𝜃𝑑²₁2exp{𝜀(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑍₂𝑥̇(𝑠)𝑑𝑠𝑑𝜃≤−exp−𝜀𝑑₁𝜉^𝑇(𝑡)𝑑₁𝑒₁−𝑒₁₀𝑍₂𝑑₁𝑒^𝑇₁−𝑒^𝑇₁₀𝜉(𝑡),−^−𝑑1_−𝑑𝑚^𝑡_𝑡+𝜃𝜚₃exp{𝜀(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑍₃𝑥̇(𝑠)𝑑𝑠𝑑𝜃≤−exp−𝜀𝑑_𝑚𝜉^𝑇(𝑡)𝜚₁𝑒₁−𝑒₁₃𝑍₃𝜚₁𝑒^𝑇₁−𝑒^𝑇₁₃𝜉(𝑡),−^−𝑑𝑚_−𝑑2^𝑡_𝑡+𝜃𝜚₄exp{𝜀(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑍₄𝑥̇(𝑠)𝑑𝑠𝑑𝜃≤−exp−𝜀𝑑₂𝜉^𝑇(𝑡)𝜚₂𝑒₁−𝑒₁₄𝑍₄𝜚₂𝑒^𝑇₁−𝑒^𝑇₁₄𝜉(𝑡). (47)
For 𝑑(𝑡)∈[𝑑₁,𝑑_𝑚], the following is held from (a) of Lemma 7: −^𝑡−𝑑1_{𝑡−𝑑𝑚}𝜚₁exp{𝜀(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑌₃𝑥̇(𝑠)𝑑𝑠=−^𝑡−𝑑1_{(𝑡)𝑡−𝑑}+^{𝑡−𝑑(𝑡)}_{𝑡−𝑑𝑚}𝜚₁exp{𝜀(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑌₃𝑥̇(𝑠)𝑑𝑠≤−𝜀𝜚₁exp{𝜀𝑑(𝑡)}−exp𝜀𝑑₁^𝑡−𝑑1_{(𝑡)𝑡−𝑑}𝑥̇^𝑇(𝑠)𝑑𝑠×𝑌₃^𝑡−𝑑1_{(𝑡)𝑡−𝑑}𝑥̇(𝑠)−𝑑𝑠𝜀𝜚₁exp𝜀𝑑_𝑚−exp{𝜀𝑑(𝑡)}^{𝑡−𝑑(𝑡)}_{𝑡−𝑑𝑚}𝑥̇^𝑇(𝑠)𝑑𝑠×𝑌₃^{𝑡−𝑑(𝑡)}_{𝑡−𝑑𝑚}𝑥̇(𝑠)𝑑𝑠=−𝜀𝜚₁exp𝜀𝑑_𝑚−exp𝜀𝑑₁𝜉^𝑇(𝑡)×1𝜆₁(𝑡)𝑒₄−𝑒₃𝑌₃𝑒^𝑇₄−𝑒^𝑇₃+1𝜆₂(𝑡)𝑒₃−𝑒₅𝑌₃𝑒^𝑇₃−𝑒^𝑇₅𝜉(𝑡), (48) where 𝜆₁(𝑡)=exp{𝜀𝑑(𝑡)}−exp𝜀𝑑₁exp𝜀𝑑_𝑚−exp𝜀𝑑₁,𝜆₂(𝑡)=exp𝜀𝑑_𝑚−exp{𝜀𝑑(𝑡)}exp𝜀𝑑_𝑚−exp𝜀𝑑₁. (49)
By Lemma 8, there exists matrix 𝑋₂ with appropriate dimensions such that −^𝑡−𝑑1_{𝑡−𝑑𝑚}𝜚₁exp{𝜀(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑌₃𝑥̇(𝑠)𝑑𝑠≤−𝜀𝜚₁exp𝜀𝑑_𝑚−exp𝜀𝑑₁𝜉^𝑇(𝑡)×𝑒₄−𝑒₃𝑒₃−𝑒₅𝑌₃𝑋₂𝑋^𝑇₂𝑌₃𝑒^𝑇₄−𝑒^𝑇₃𝑒^𝑇₃−𝑒^𝑇₅𝜉(𝑡),𝑌₃𝑋₂𝑋^𝑇₂𝑌₃>0. (50)
Similarly, considering −∫^𝑡−𝑑1_{𝑡−𝑑𝑚}𝜚₁exp{𝜀(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑊₃𝑥(𝑠)𝑑𝑠 and following the same procedure, there exists matrix 𝑋₁ with appropriate dimensions such that −^𝑡−𝑑1_{𝑡−𝑑𝑚}𝜚₁exp{𝜀(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑊₃𝑥(𝑠)𝑑𝑠≤−𝜀𝜚₁exp𝜀𝑑_𝑚−exp𝜀𝑑₁𝜉^𝑇(𝑡)×𝑒₁₁𝑒₁₃−𝑒₁₁𝑊₃𝑋₁𝑋^𝑇₁𝑊₃𝑒^𝑇₁₁𝑒^𝑇₁₃−𝑒^𝑇₁₁𝜉(𝑡),𝑊₃𝑋₁𝑋^𝑇₁𝑊₃>0. (51)
For 𝜏(𝑡)∈[0,𝜏], with the same matrix inequalities technique, we obtain the following: −^𝑡_𝑡−𝜏𝜏exp{𝜀(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑊₁𝑥(𝑠)𝑑𝑠≤−𝜀𝜏{}𝜉exp𝜀𝜏−1^𝑇(𝑡)×𝑒₁₉𝑒₂₀−𝑒₁₉𝑊₁𝑈₁𝑈^𝑇₁𝑊₁𝑒^𝑇₁₉𝑒^𝑇₂₀−𝑒^𝑇₁₉𝜉(𝑡),𝑊₁𝑈₁𝑈^𝑇₁𝑊₁−>0,_{𝑡𝑡−𝜏}𝜏exp{𝜀(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑌₁𝑥̇(𝑠)𝑑𝑠≤−𝜀𝜏{}𝜉exp𝜀𝜏−1^𝑇(𝑡)×𝑒₁−𝑒₁₇𝑒₁₇−𝑒₁₅𝑌₁𝑈₂𝑈^𝑇₂𝑌₁𝑒^𝑇₁−𝑒^𝑇₁₇𝑒^𝑇₁₇−𝑒^𝑇₁₅𝜉(𝑡),𝑌₁𝑈₂𝑈^𝑇₂𝑌₁>0. (52)
Consider  −∫^{𝑡−𝑑𝑚}_𝑡−𝑑2𝜚₂exp{𝜀(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑊₄𝑥(𝑠)𝑑𝑠  and  −∫^{𝑡−𝑑𝑚}_𝑡−𝑑2𝜚₂exp{𝜀(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑌₄𝑥̇(𝑠)𝑑𝑠, which are directly estimated by (a) of Lemma 7; that is, −^{𝑡−𝑑𝑚}_𝑡−𝑑2𝜚₂exp{𝜀(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑊₄𝑥(𝑠)𝑑𝑠≤−𝜀𝜚₂exp𝜀𝑑₂−exp𝜀𝑑_𝑚𝜉^𝑇(𝑡)𝑒₁₄𝑊₄𝑒^𝑇₁₄𝜉(𝑡),−^{𝑡−𝑑𝑚}_𝑡−𝑑2𝜚₂exp{𝜀(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑌₄𝑥̇(𝑠)𝑑𝑠≤−𝜀𝜚₂exp𝜀𝑑₂−exp𝜀𝑑_𝑚𝜉^𝑇(𝑡)×𝑒₅−𝑒₆𝑌₄𝑒^𝑇₅−𝑒^𝑇₆𝜉(𝑡). (53)
In addition, there exist matrices 𝑇=col{𝑇_𝑘,(𝑘=1,2,…,23)} with appropriate dimensions, such that the following equality holds according to (9): 2𝜉^𝑇(𝑡)𝑇−𝑥̇(𝑡)+𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑥(𝑡−𝑑(𝑡))+𝐶_𝑖𝑥̇(𝑡−𝜏(𝑡))+𝑓₁+𝑓₂+𝑓₃=0. (54)
Substituting (36)–(42) and (45)–(54) into (43), we obtain 𝔏𝑉(𝑥(𝑡),𝑖,𝑡)+𝜀𝑉(𝑥(𝑡),𝑖,𝑡)≤𝜉^𝑇(𝑡)Ξ^𝜀_𝑖0+Ξ^𝜀_𝑖1𝜉(𝑡). (55)
On the other hand, for 𝑑(𝑡)∈[𝑑_𝑚,𝑑₂], the integral terms −^{𝑡−𝑑𝑚}_𝑡−𝑑2𝜚₂exp{𝜀(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑊₄𝑥(𝑠)−𝑑𝑠,_{𝑡−𝑑𝑚𝑡−𝑑2}𝜚₂exp{𝜀(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑌₄𝑥̇(𝑠)𝑑𝑠. (56) The above equations: are disposed and estimated by Lemma 8. −^𝑡−𝑑1_{𝑡−𝑑𝑚}𝜚₁exp{𝜀(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑊₃𝑥(𝑠)−𝑑𝑠,_{𝑡−𝑑1𝑡−𝑑𝑚}𝜚₁exp{𝜀(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝑌₃𝑥̇(𝑠)𝑑𝑠 (57) are directly estimated by (a) of Lemma 7. Therefore, 𝔏𝑉(𝑥(𝑡),𝑖,𝑡)+𝜀𝑉(𝑥(𝑡),𝑖,𝑡)≤𝜉^𝑇(𝑡)Ξ^𝜀_𝑖0+Ξ^𝜀_𝑖2𝜉(𝑡). (58)
With (55) and (58), the following inequality (59) is held for 𝑑(𝑡)∈[𝑑₁,𝑑₂] if (22), (23), and (24) are satisfied 𝔏𝑉(𝑥(𝑡),𝑖,𝑡)+𝜀𝑉(𝑥(𝑡),𝑖,𝑡)<0. (59)
From the Lyapunov functional (28) and (59), it is held that 𝑉(𝑥(𝑡),𝑖,𝑡)≥min_𝑖∈𝑆𝜆_min𝑃_𝑖‖𝑥(𝑡)‖²=𝜆‖𝑥(𝑡)‖²,𝑉(𝑥(𝑡),𝑖,𝑡)<exp−𝜀𝑡−𝑡₀𝑉𝑥_𝑡0,𝑟_𝑡0,𝑡₀. (60)
Moreover, we have 𝑉𝑥_𝑡0,𝑟_𝑡0,𝑡₀=⁶_𝑘=1𝑉_𝑘𝑥_𝑡0,𝑟_𝑡0≤𝜆𝑥_𝑡0²_𝜌. (61)
Then, from (60) and (61), it is readily seen that ‖𝑥(𝑡)‖≤𝜆𝜆−𝜀exp2𝑡−𝑡₀𝑥_𝑡0𝜌, (62) where 𝜎=𝜆/𝜆≥1.
Therefore, by Definition 4, 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: ⁶_𝑘=1𝑉_𝑘𝑥_𝑡0,𝑟_𝑡0≤𝑥^𝑇𝑡₀𝑃_𝑖𝑥𝑡₀+^𝑡0_𝑡0−𝜌exp𝜀𝑠−𝑡₀𝑥^𝑇(𝑠)⁵_𝑗=1𝑄_𝑗𝑥(𝑠)+𝑑𝑠_{𝑡0𝑡0−𝜌}exp𝜀𝑠−𝑡₀𝑥̇^𝑇(𝑠)⁵_𝑘=1𝑅_𝑘𝑥̇(𝑠)𝑑𝑠+⁰_𝑡−𝜌^𝑡0_𝑡0+𝜃𝜌exp𝜀𝑠−𝑡₀×𝑥^𝑇(𝑠)⁴_𝑙=1𝑊_𝑙𝑥(𝑠)+𝑑𝑠𝑑𝜃_{0−𝜌𝑡0𝑡0+𝜃}𝜌exp𝜀𝑠−𝑡₀×𝑥̇^𝑇(𝑠)⁴_𝑚=1𝑌_𝑚𝑥̇(𝑠)+𝑑𝑠𝑑𝜃_{0−𝜌0𝜃𝑡0𝑡0+𝜆}𝜌²2exp𝜀𝑠−𝑡₀×𝑥̇^𝑇(𝑠)⁴_𝑛=1𝑍_𝑛𝑥̇(𝑠)𝑑𝑠𝑑𝜆𝑑𝜃. (63)  Equation (61) can be obtained by letting 𝜆 be defined as previously.

Remark 12. In Theorem 10, the factors exp𝜀𝑑₂−exp{𝜀𝑑(𝑡)}𝜀,exp{𝜀𝑑(𝑡)}−exp𝜀𝑑₁𝜀 (64) 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−𝑑1_{𝑡−𝑑𝑚}exp{𝜀(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑄₄𝑥(𝑠)𝑑𝑠, ∫^{𝑡−𝑑𝑚}_𝑡−𝑑2exp{𝜀(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑄₅𝑥(𝑠)𝑑𝑠, and ∫^𝑡−𝑑1_{𝑡−𝑑(𝑡)}exp{𝜀(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑄₂𝑥(𝑠)𝑑𝑠.

Remark 14. It should be also mentioned that the result obtained in Theorem 10 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 Theorem 10 by setting 𝑄₂=0 and 𝑅₅=0.

Corollary 15. For the given finite set 𝑆 of modes with transition rates matrix, scalars 𝛼, 𝛽, 𝛾, 𝜀, 𝜏, 𝑑₁, 𝑑₂, and constant scalar 𝑑_𝑚 satisfying 𝑑₁<𝑑_𝑚<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 𝑃_𝑖>0, (𝑖∈𝑆), 𝑄_𝑗>0, (𝑗=1,3,4,5), 𝑅_𝑘>0, (𝑘=1,2,3,4), 𝑊_𝑙>0, (𝑙=1,2,3,4), 𝑌_𝑚>0, (𝑚=1,2,3,4), 𝑍_𝑛>0, (𝑛=1,2,3,4) and matrices 𝑈₁, 𝑈₂, 𝑋₁, 𝑋₂, 𝑋₃, 𝑋₄ for any scalars 𝜀₁, 𝜀₂, 𝜀₃ and any matrices 𝑇_𝑘, (𝑘=1,2,…,23) with appropriate dimensions, such that linear matrix inequalities (22) and (65) hold: 𝑊(i)³𝑋₁𝑋^𝑇₁𝑊₃𝑌>0,³𝑋₂𝑋^𝑇₂𝑌₃Ξ>0,(ii)^𝜀_𝑖0+Ξ^𝜀_𝑖1𝑊<0,(iii)⁴𝑋₃𝑋^𝑇₃𝑊₄𝑌>0,⁴𝑋₄𝑋^𝑇₄𝑌₄Ξ>0,(iv)^𝜀_𝑖0+Ξ^𝜀_𝑖2<0, (65) where Ξ^𝜀_𝑖0=²³_𝑚=1𝑒_𝑚𝐺_𝑚𝑒^𝑇_𝑚(✠)−+𝒜𝜀𝑑₁exp𝜀𝑑₁𝑒−1¹−𝑒₄𝑌₂𝑒^𝑇₁−𝑒^𝑇₄−exp{−𝜀𝜏}𝜏𝑒₁−𝑒₂₀𝑍₁𝜏𝑒^𝑇₁−𝑒^𝑇₂₀−exp−𝜀𝑑₁𝑑₁𝑒₁−𝑒₁₀𝑍₂𝑑₁𝑒^𝑇₁−𝑒^𝑇₁₀−exp−𝜀𝑑_𝑚𝜚₁𝑒₁−𝑒₁₃𝑍₃𝜚₁𝑒^𝑇₁−𝑒^𝑇₁₃−exp−𝜀𝑑₂𝜚₂𝑒₁−𝑒₁₄𝑍₄𝜚₂𝑒^𝑇₁−𝑒^𝑇₁₄−𝜀𝜏{}𝑒exp𝜀𝜏−1¹⁹𝑒₂₀−𝑒₁₉×𝑊₁𝑈₁𝑈^𝑇₁𝑊₁𝑒^𝑇₁₉𝑒^𝑇₂₀−𝑒^𝑇₁₉−𝜀𝜏{}𝑒exp𝜀𝜏−1¹−𝑒₁₇𝑒₁₇−𝑒₁₅×𝑌₁𝑈₂𝑈^𝑇₂𝑌₁𝑒^𝑇₁−𝑒^𝑇₁₇𝑒^𝑇₁₇−𝑒^𝑇₁₅, (66) where 𝐺₂=𝑅₁+𝑅₂+𝜏²𝑌₁+𝑑²₁𝑌₂+𝜚²₁𝑌₃+𝜚²₂𝑌₄+𝜏⁴4𝑍₁+𝑑⁴₁4𝑍₂+𝜚²₃𝑍₃+𝜚²₄𝑍₄−𝑇₂−𝑇^𝑇₂,𝐺₃=𝑇₃𝐵_𝑖+𝐵^𝑇_𝑖𝑇^𝑇₃+𝜀₂𝛽²𝐺𝐼,₄=exp−𝜀𝑑₁𝑄₄−𝑄₃,𝐺₁₈=𝑇₁₈𝐶_𝑖+𝐶^𝑇_𝑖𝑇^𝑇₁₈+𝜀₃𝛾²𝐼. (67) Other 𝐺_𝑚=𝐺_𝑚, (𝑚≠2,3,4,18), which have been defined previously. 𝜆=max_𝑖∈𝑆𝜆_max𝑃_𝑖+1−exp−𝜀𝜌𝜀⁵_{𝑗=1,𝑗≠2}𝜆_max𝑄_𝑗+⁴_𝑘=1𝜆_max𝑅_𝑘+exp−𝜀𝜌+𝜀𝜌−1𝜀²⁴_𝑙=1𝜆_max𝑊_𝑙+⁴_𝑚=1𝜆_max𝑌_𝑚+2−2exp−𝜀𝜌+𝜌²𝜀²−2𝜀𝜌2𝜀³⁴_𝑛=1𝜆_max𝑍_𝑛, (68) and the remaining notations are the same as Theorem 10.

In Theorem 10, 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 set 𝑆 of modes with transition rates matrix, scalars 𝛼, 𝛽, 𝛾, 𝜏, 𝜏_𝑑, 𝑑₁, 𝑑₂, 𝜇 and constant scalar 𝑑_𝑚 satisfying 𝑑₁<𝑑_𝑚<𝑑₂, the systems described by (9) are asymptotically 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), 𝑅_𝑘>0, (𝑘=1,2,3,4,5), 𝑊_𝑙>0, (𝑙=1,2,3,4), 𝑌_𝑚>0, (𝑚=1,2,3,4), 𝑍_𝑛>0, (𝑛=1,2,3,4) and matrices 𝑈₁, 𝑈₂, 𝑋₁, 𝑋₂, 𝑋₃, 𝑋₄ for any scalars 𝜀₁, 𝜀₂, 𝜀₃ and any matrices 𝑇_𝑘, (𝑘=1,2,…,23) with appropriate dimensions, such that linear matrix inequalities (22) and (69) hold: 𝑊(i)³𝑋₁𝑋^𝑇₁𝑊₃𝑌>0,³𝑋₂𝑋^𝑇₂𝑌₃>0,(ii)Ξ_𝑖0+Ξ_𝑖1𝑊<0,(iii)⁴𝑋₃𝑋^𝑇₃𝑊₄𝑌>0,⁴𝑋₄𝑋^𝑇₄𝑌₄>0,(iv)Ξ_𝑖0+Ξ_𝑖2<0, (69) where Ξ_𝑖0=²³_𝑚=1𝑒_𝑚𝐺_𝑚𝑒^𝑇_𝑚+𝒜(✠)−𝑒₁−𝑒₄𝑌₂𝑒^𝑇₁−𝑒^𝑇₄−𝜏𝑒₁−𝑒₂₀𝑍₁𝜏𝑒^𝑇₁−𝑒^𝑇₂₀−𝑑₁𝑒₁−𝑒₁₀𝑍₂𝑑₁𝑒^𝑇₁−𝑒^𝑇₁₀−𝜚₁𝑒₁−𝑒₁₃𝑍₃𝜚₁𝑒^𝑇₁−𝑒^𝑇₁₃−𝜚₂𝑒₁−𝑒₁₄𝑍₄𝜚₂𝑒^𝑇₁−𝑒^𝑇₁₄−𝑒₁₉𝑒₂₀−𝑒₁₉𝑊₁𝑈₁𝑈^𝑇₁𝑊₁𝑒^𝑇₁₉𝑒^𝑇₂₀−𝑒^𝑇₁₉−𝑒₁−𝑒₁₇𝑒₁₇−𝑒₁₅𝑌₁𝑈₂𝑈^𝑇₂𝑌₁𝑒^𝑇₁−𝑒^𝑇₁₇𝑒^𝑇₁₇−𝑒^𝑇₁₅,𝐺₁=𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+^𝑁_𝑗=1𝜋_𝑖𝑗𝑃_𝑗+𝑄₁+𝑄₃+𝜏²𝑊₁+𝑑²₁𝑊₂+𝜚²₁𝑊₃+𝜚²₂𝑊₄+𝜀₁𝛼²𝐼+𝑇₁𝐴_𝑖+𝐴^𝑇_𝑖𝑇^𝑇₁,𝐺₃=𝜇−1𝑄₂+𝑇₃𝐵_𝑖+𝐵^𝑇_𝑖𝑇^𝑇₃+𝜀₂𝛽²𝐼,𝐺₄=𝑄₂+𝑄₄−𝑄₃,𝐺₅=𝑄₅−𝑄₄,𝐺₆=−𝑄₅,𝐺₇=𝑅₃−𝑅₂,𝐺₈=𝑅₄−𝑅₃,𝐺₉=−𝑅₄,𝐺₁₀=−𝑊₂,𝐺₁₅=−𝑄₁,𝐺₁₆=−𝑅₁,𝐺₁₈=𝜏_𝑑−1𝑅₅+𝑇₁₈𝐶_𝑖+𝐶^𝑇_𝑖𝑇^𝑇₁₈+𝜀₃𝛾²𝐼. (70) Other 𝐺_𝑚=𝐺_𝑚, (𝑚=2,11,12,13,14,17,19,…,23), which have been defined in Theorem 10: Ξ_𝑖1=−𝑒₁₄𝑊₄𝑒^𝑇₁₄−𝑒₅−𝑒₆𝑌₄𝑒^𝑇₅−𝑒^𝑇₆−𝑒₁₁𝑒₁₃−𝑒₁₁𝑊₃𝑋₁𝑋^𝑇₁𝑊₃𝑒^𝑇₁₁𝑒^𝑇₁₃−𝑒^𝑇₁₁−𝑒₄−𝑒₃𝑒₃−𝑒₅𝑌₃𝑋₂𝑋^𝑇₂𝑌₃𝑒^𝑇₄−𝑒^𝑇₃𝑒^𝑇₃−𝑒^𝑇₅,Ξ_𝑖2=−𝑒₁₃𝑊₃𝑒^𝑇₁₃−𝑒₄−𝑒₅𝑌₃𝑒^𝑇₄−𝑒^𝑇₅−𝑒₁₂𝑒₁₄−𝑒₁₂𝑊₄𝑋₃𝑋^𝑇₃𝑊₄𝑒^𝑇₁₂𝑒^𝑇₁₄−𝑒^𝑇₁₂−𝑒₃−𝑒₆𝑒₅−𝑒₃𝑌₄𝑋₄𝑋^𝑇₄𝑌₄𝑒^𝑇₃−𝑒^𝑇₆𝑒^𝑇₅−𝑒^𝑇₃, (71) 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 matrix Π=[𝜋_𝑖𝑗]_𝑁×𝑁 are inaccessible. For instance, the system (3) with five operation modes, the jump rates matrix Π may be viewed as ?𝜋₁₂??𝜋₁₅𝜋₂₁??𝜋₂₄???𝜋₃₃𝜋₃₄??𝜋₄₂𝜋???₅₁???𝜋₅₅, (72) where ? represents the unknown element. For notation clarity, we denote that 𝒮^𝑖=𝒮^𝑖_𝑘⋃𝒮^𝑖_𝑢𝑘,forall𝑖∈𝑆 and 𝒮^𝑖_𝑘≜𝑗∶𝜋_𝑖𝑗,𝒮isknownfor𝑗∈𝑆_𝑖𝑢𝑘≜𝑗∶𝜋_𝑖𝑗isunknownfor𝑗∈𝑆. (73) If 𝒮^𝑖_𝑘≠∅, it is further described as 𝒮^𝑖_𝑘=𝑘^𝑖₁,𝑘^𝑖₂,…,𝑘^𝑖_𝑚,1≤𝑚≤𝑁, (74) where 𝑘^𝑖_𝑗, (𝑗=1,2,…,𝑚) represent the 𝑗th known element of the set 𝒮^𝑖_𝑘 in the 𝑖th row of the transition rate matrix Π. Specially, when 𝒮^𝑖_𝑢𝑘=∅, 𝒮^𝑖=𝒮^𝑖_𝑘, the full information about Π is obtained, and it becomes the case of Section 3.1.

Theorem 17. For the given finite set 𝑆 of modes with partly unknown transition rates matrix, scalars 𝛼, 𝛽, 𝛾, 𝜀, 𝜏, 𝜏_𝑑, 𝑑₁, 𝑑₂, and 𝜇, and constant scalar 𝑑_𝑚 satisfying 𝑑₁<𝑑_𝑚<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 𝑃_𝑖>0, (𝑖∈𝑆), 𝑄_𝑗>0, (𝑗=1,2,3,4,5), 𝑅_𝑘>0, (𝑘=1,2,3,4,5), 𝑊_𝑙>0, (𝑙=1,2,3,4), 𝑌_𝑚>0, (𝑚=1,2,3,4), 𝑍_𝑛>0, (𝑛=1,2,3,4) and matrices 𝑈₁, 𝑈₂, 𝑋₁, 𝑋₂, 𝑋₃, 𝑋₄  for any scalars 𝜀₁, 𝜀₂, 𝜀₃ and any symmetric matrices 𝑉_𝑖, (𝑖∈𝑆) and any matrices 𝑇_𝑘, (𝑘=1,2,…,23) with appropriate dimensions, such that linear matrix inequalities (22), (75), (76), and (77) hold: 𝑃_𝑗−𝑉_𝑖≤0,𝑗∈𝒮^𝑖_𝑢𝑘𝑃,𝑗≠𝑖,_𝑗−𝑉_𝑖≥0,𝑗∈𝒮^𝑖_𝑢𝑘𝑊,𝑗=𝑖,(i)³𝑋₁𝑋^𝑇₁𝑊₃𝑌>0,³𝑋₂𝑋^𝑇₂𝑌₃>0,(ii)Ξ^𝜀_𝒮𝑖0+Ξ^𝜀_𝒮𝑖1𝑊<0,(iii)⁴𝑋₃𝑋^𝑇₃𝑊₄𝑌>0,⁴𝑋₄𝑋^𝑇₄𝑌₄>0,(iv)Ξ^𝜀_𝒮𝑖0+Ξ^𝜀_𝒮𝑖2<0, (75)(76)(77) where Ξ^𝜀_𝒮𝑖0=²³_𝑚=1𝑒_𝑚𝐺_𝒮𝑚𝑒^𝑇_𝑚(✠)−+𝒜𝜀𝑑₁exp𝜀𝑑₁𝑒−1¹−𝑒₄𝑌₂𝑒^𝑇₁−𝑒^𝑇₄−exp{−𝜀𝜏}𝜏𝑒₁−𝑒₂₀𝑍₁𝜏𝑒^𝑇₁−𝑒^𝑇₂₀−exp−𝜀𝑑₁𝑑₁𝑒₁−𝑒₁₀𝑍₂𝑑₁𝑒^𝑇₁−𝑒^𝑇₁₀−exp−𝜀𝑑_𝑚𝜚₁𝑒₁−𝑒₁₃𝑍₃𝜚₁𝑒^𝑇₁−𝑒^𝑇₁₃−exp−𝜀𝑑₂𝜚₂𝑒₁−𝑒₁₄𝑍₄𝜚₂𝑒^𝑇₁−𝑒^𝑇₁₄−𝜀𝜏{}𝑒exp𝜀𝜏−1¹⁹𝑒₂₀−𝑒₁₉×𝑊₁𝑈₁𝑈^𝑇₁𝑊₁𝑒^𝑇₁₉𝑒^𝑇₂₀−𝑒^𝑇₁₉−𝜀𝜏{}𝑒exp𝜀𝜏−1¹−𝑒₁₇𝑒₁₇−𝑒₁₅×𝑌₁𝑈₂𝑈^𝑇₂𝑌₁𝑒^𝑇₁−𝑒^𝑇₁₇𝑒^𝑇₁₇−𝑒^𝑇₁₅,𝐺_𝒮1=𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+^𝑁_{𝑗∈𝒮𝑖𝑘}𝜋_𝑖𝑗𝑃_𝑗−𝑉_𝑖+𝜀𝑃_𝑖+𝑄₁+𝑄₃+𝜏²𝑊₁+𝑑²₁𝑊₂+𝜚²₁𝑊₃+𝜚²₂𝑊₄+𝜀₁𝛼²𝐼+𝑇₁𝐴_𝑖+𝐴^𝑇_𝑖𝑇^𝑇₁,𝐺_𝒮𝑚=𝐺_𝑚,(𝑚≠1),Ξ^𝜀_𝒮𝑖1=Ξ^𝜀_𝑖1,Ξ^𝜀_𝒮𝑖2=Ξ^𝜀_𝑖2, (78) and the remaining notations are the same as Theorem 10.

Proof. Choose the same Lyapunov functional as (28), and 𝑉₂(𝑥_𝑡,𝑖), 𝑉₃(𝑥_𝑡,𝑖), 𝑉₄(𝑥_𝑡,𝑖), 𝑉₅(𝑥_𝑡,𝑖), and 𝑉₆(𝑥_𝑡,𝑖) are disposed with the identical method in Theorem 10. Then, giving consideration on ∑^𝑁_𝑗=1𝜋_𝑖𝑗=0, the following equation holds for arbitrary matrices 𝑉_𝑖=𝑉^𝑇_𝑖, 𝑖∈𝑆; that is, −𝑥^𝑇(𝑡)^𝑁_𝑗=1𝜋_𝑖𝑗𝑉_𝑖𝑥(𝑡)=0. (79) With (36) and (79), we obtain the following: 𝔏𝑉₁𝑥_𝑡,𝑖=2𝑥^𝑇(𝑡)𝐴^𝑇_𝑖+𝑥^𝑇(𝑡−𝑑(𝑡))𝐵^𝑇_𝑖+𝑥̇^𝑇(𝑡−𝜏(𝑡))𝐶^𝑇_𝑖+𝑓^𝑇₁+𝑓^𝑇₂+𝑓^𝑇₃𝑃_𝑖𝑥(𝑡)+𝑥^𝑇(𝑡)^𝑁_{𝑗∈𝒮𝑖𝑘}𝜋_𝑖𝑗𝑃_𝑗−𝑉_𝑖𝑥(𝑡)+𝑥^𝑇(𝑡)^𝑁_{𝑗∈𝒮𝑖𝑢𝑘}𝜋_𝑖𝑗𝑃_𝑗−𝑉_𝑖𝑥(𝑡)+𝜀𝑥^𝑇(𝑡)𝑃_𝑖𝑥(𝑡)−𝜀𝑉₁𝑥_𝑡,𝑖. (80)
For 𝑑(𝑡)∈[𝑑₁,𝑑_𝑚] and 𝑑(𝑡)∈[𝑑_𝑚,𝑑₂], with (80) and on the basis of the result in Theorem 10, we have the following equations, respectively: 𝔏𝑉(𝑥(𝑡),𝑖,𝑡)+𝜀𝑉(𝑥(𝑡),𝑖,𝑡)≤𝜉^𝑇(𝑡)Ξ^𝜀_𝒮𝑖0+Ξ^𝜀_𝒮𝑖1𝜉(𝑡)+𝑥^𝑇(𝑡)^𝑁_{𝑗∈𝒮𝑖𝑢𝑘}𝜋_𝑖𝑗𝑃_𝑗−𝑉_𝑖𝑥(𝑡),𝔏𝑉(𝑥(𝑡),𝑖,𝑡)+𝜀𝑉(𝑥(𝑡),𝑖,𝑡)≤𝜉^𝑇(𝑡)Ξ^𝜀_𝒮𝑖0+Ξ^𝜀_𝒮𝑖2𝜉(𝑡)+𝑥^𝑇(𝑡)^𝑁_{𝑗∈𝒮𝑖𝑢𝑘}𝜋_𝑖𝑗𝑃_𝑗−𝑉_𝑖𝑥(𝑡). (81) With (81), due to 𝜆_𝑖𝑖=−∑^𝑁_{𝑗=1,𝑗≠𝑖}𝜆_𝑖𝑗 and 𝜆_𝑖𝑗>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 set 𝑆 of modes with partly unknown transition rates matrix, scalars 𝛼, 𝛽, 𝛾, 𝜀, 𝜏, 𝑑₁, and 𝑑₂, and constant scalar 𝑑_𝑚 satisfying 𝑑₁<𝑑_𝑚<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 𝑃_𝑖>0, (𝑖∈𝑆), 𝑄_𝑗>0, (𝑗=1,3,4,5), 𝑅_𝑘>0, (𝑘=1,2,3,4), 𝑊_𝑙>0, (𝑙=1,2,3,4), 𝑌_𝑚>0, (𝑚=1,2,3,4), 𝑍_𝑛>0, (𝑛=1,2,3,4) and matrices 𝑈₁, 𝑈₂, 𝑋₁, 𝑋₂, 𝑋₃, 𝑋₄   for any scalars 𝜀₁, 𝜀₂, 𝜀₃ and any symmetric matrices 𝑉_𝑖, (𝑖∈𝑆) and any matrices 𝑇_𝑘, (𝑘=1,2,…,23) with appropriate dimensions, such that linear matrix inequalities (22), (75), and (82) hold: 𝑊(i)³𝑋₁𝑋^𝑇₁𝑊₃𝑌>0,³𝑋₂𝑋^𝑇₂𝑌₃Ξ>0,(ii)^𝜀_𝒮𝑖0+Ξ^𝜀_𝒮𝑖1𝑊<0,(iii)⁴𝑋₃𝑋^𝑇₃𝑊₄𝑌>0,⁴𝑋₄𝑋^𝑇₄𝑌₄Ξ>0,(iv)^𝜀_𝒮𝑖0+Ξ^𝜀_𝒮𝑖2<0, (82) where Ξ^𝜀_𝒮𝑖0=²³_𝑚=1𝑒_𝑚𝐺_𝒮𝑚𝑒^𝑇_𝑚(✠)−+𝒜𝜀𝑑₁exp𝜀𝑑₁𝑒−1¹−𝑒₄𝑌₂𝑒^𝑇₁−𝑒^𝑇₄−exp{−𝜀𝜏}𝜏𝑒₁−𝑒₂₀𝑍₁𝜏𝑒^𝑇₁−𝑒^𝑇₂₀−exp−𝜀𝑑₁𝑑₁𝑒₁−𝑒₁₀𝑍₂𝑑₁𝑒^𝑇₁−𝑒^𝑇₁₀−exp−𝜀𝑑_𝑚𝜚₁𝑒₁−𝑒₁₃𝑍₃𝜚₁𝑒^𝑇₁−𝑒^𝑇₁₃−exp−𝜀𝑑₂𝜚₂𝑒₁−𝑒₁₄𝑍₄𝜚₂𝑒^𝑇₁−𝑒^𝑇₁₄−𝜀𝜏{}𝑒exp𝜀𝜏−1¹⁹𝑒₂₀−𝑒₁₉×𝑊₁𝑈₁𝑈^𝑇₁𝑊₁𝑒^𝑇₁₉𝑒^𝑇₂₀−𝑒^𝑇₁₉−𝜀𝜏{}𝑒exp𝜀𝜏−1¹−𝑒₁₇𝑒₁₇−𝑒₁₅×𝑌₁𝑈₂𝑈^𝑇₂𝑌₁𝑒^𝑇₁−𝑒^𝑇₁₇𝑒^𝑇₁₇−𝑒^𝑇₁₅,𝐺_𝒮1=𝐺_𝒮1,𝐺_𝒮𝑚=𝐺_𝑚,(),Ξ𝑚≠1_{𝜀𝒮𝑖1}=Ξ^𝜀_𝑖1,Ξ^𝜀_𝒮𝑖2=Ξ^𝜀_𝑖2, (83) and the remaining notations are the same as Corollary 15.

Corollary 19. For the given finite set 𝑆 of modes with partly unknown transition rates matrix, scalars 𝛼, 𝛽, 𝛾, 𝜏, 𝜏_𝑑, 𝑑₁, 𝑑₂, 𝜇, and constant scalar 𝑑_𝑚 satisfying 𝑑₁<𝑑_𝑚<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 𝑃_𝑖>0, (𝑖∈𝑆), 𝑄_𝑗>0, (𝑗=1,2,3,4,5), 𝑅_𝑘>0, (𝑘=1,2,3,4,5), 𝑊_𝑙>0, (𝑙=1,2,3,4), 𝑌_𝑚>0, (𝑚=1,2,3,4), 𝑍_𝑛>0, (𝑛=1,2,3,4) and matrices 𝑈₁, 𝑈₂, 𝑋₁, 𝑋₂, 𝑋₃, 𝑋₄  for any scalars 𝜀₁, 𝜀₂, 𝜀₃ and any symmetric matrices 𝑉_𝑖, (𝑖∈𝑆) and any matrices 𝑇_𝑘, (𝑘=1,2,…,23) with appropriate dimensions, such that linear matrix inequalities (22), (75), and (84) hold: 𝑊(i)³𝑋₁𝑋^𝑇₁𝑊₃𝑌>0,³𝑋₂𝑋^𝑇₂𝑌₃>0,(ii)Ξ_𝒮𝑖0+Ξ_𝒮𝑖1𝑊<0,(iii)⁴𝑋₃𝑋^𝑇₃𝑊₄𝑌>0,⁴𝑋₄𝑋^𝑇₄𝑌₄>0,(iv)Ξ_𝒮𝑖0+Ξ_𝒮𝑖2<0, (84) where Ξ_𝒮𝑖0=²³_𝑚=1𝑒_𝑚𝐺_𝒮𝑚𝑒^𝑇_𝑚(✠)−𝑒+𝒜₁−𝑒₄𝑌₂𝑒^𝑇₁−𝑒^𝑇₄−𝜏𝑒₁−𝑒₂₀𝑍₁𝜏𝑒^𝑇₁−𝑒^𝑇₂₀−𝑑₁𝑒₁−𝑒₁₀𝑍₂𝑑₁𝑒^𝑇₁−𝑒^𝑇₁₀−𝜚₁𝑒₁−𝑒₁₃𝑍₃𝜚₁𝑒^𝑇₁−𝑒^𝑇₁₃−𝜚₂𝑒₁−𝑒₁₄𝑍₄𝜚₂𝑒^𝑇₁−𝑒^𝑇₁₄−𝑒₁₉𝑒₂₀−𝑒₁₉𝑊₁𝑈₁𝑈^𝑇₁𝑊₁𝑒^𝑇₁₉𝑒^𝑇₂₀−𝑒^𝑇₁₉−𝑒₁−𝑒₁₇𝑒₁₇−𝑒₁₅×𝑌₁𝑈₂𝑈^𝑇₂𝑌₁𝑒^𝑇₁−𝑒^𝑇₁₇𝑒^𝑇₁₇−𝑒^𝑇₁₅,𝐺_𝒮1=𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+^𝑁_{𝑗∈𝒮𝑖𝑘}𝜋_𝑖𝑗P_𝑗−𝑉_𝑖+𝑄₁+𝑄₃+𝜏²𝑊₁+𝑑²₁𝑊₂+𝜚²₁𝑊₃+𝜚²₂𝑊₄+𝜀₁𝛼²𝐼+𝑇₁𝐴_𝑖+𝐴^𝑇_𝑖𝑇^𝑇₁,𝐺_𝒮𝑚=𝐺_𝑚,(𝑚≠1),Ξ_𝒮𝑖1=Ξ_𝑖1,Ξ_𝒮𝑖2=Ξ_𝑖2, (85) and the remaining notations are the same as Corollary 16.

Remark 20. Corollaries 16 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 set 𝑆 of modes with transition rates matrix, scalars 𝛼, 𝛽, 𝛾, 𝜀, 𝜏, 𝜏_𝑑, 𝑑₁, 𝑑₂, 𝜇, and constant scalar 𝑑_𝑚 satisfying 𝑑₁<𝑑_𝑚<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 𝑃_𝑖>0, (𝑖∈𝑆), 𝑄_𝑗>0, (𝑗=1,2,3,4,5), 𝑅_𝑘>0, (𝑘=1,2,3,4,5), 𝑊_𝑙>0, (𝑙=1,2,3,4), 𝑌_𝑚>0, (𝑚=1,2,3,4), 𝑍_𝑛>0, (𝑛=1,2,3,4), and matrices 𝑈₁, 𝑈₂, 𝑋₁, 𝑋₂, 𝑋₃, 𝑋₄   for any scalars 𝜀₁, 𝜀₂, 𝜀₃ and any matrices 𝑇_𝑘, (𝑘=1,2,…,23) with appropriate dimensions, such that (22), (86), and (87) hold: 𝑊(i)³𝑋₁𝑋^𝑇₁𝑊₃𝑌>0,³𝑋₂𝑋^𝑇₂𝑌₃𝛿>0,(ii)¹Ξ^𝜀_𝑖0+𝛿₁Ξ^𝜀_𝑖1+𝛿²₁𝑀^𝑇𝑀𝑁^𝑇𝑊∗−𝐼<0,(iii)⁴𝑋₃𝑋^𝑇₃𝑊₄𝑌>0,⁴𝑋₄𝑋^𝑇₄𝑌₄𝛿>0,(iv)²Ξ^𝜀_𝑖0+𝛿₂Ξ^𝜀_𝑖2+𝛿²₂𝑀^𝑇𝑀𝑁^𝑇∗−𝐼<0, (86)(87) where 𝑀=𝑒₁𝐸^𝑇_𝐴𝑖+𝑒₃𝐸^𝑇_𝐵𝑖,𝑁=𝐻^𝑇_𝑖𝑃^𝑇_𝑖𝑒₁+²³_𝑚=1𝐻^𝑇_𝑖𝑇^𝑇_𝑚𝑒_𝑚, (88) and other notations are the same as Theorem 10.

Proof. Define 𝐸=col{𝑒_𝑚,(𝑚=1,2,…,23)} and 𝑇=col{𝑃_𝑖+𝑇₁,𝑇_𝑚(𝑚=2,…,23)}, and we replace 𝐴_𝑖, 𝐵_𝑖 with 𝐴_𝑖+Δ𝐴_𝑖(𝑡), 𝐵_𝑖+Δ𝐵_𝑖(𝑡) on the basis of Theorem 10; that is, 𝑊(i)³𝑋₁𝑋^𝑇₁𝑊₃𝑌>0,³𝑋₂𝑋^𝑇₂𝑌₃>0,(ii)Ξ^𝜀_𝑖0+Ξ^𝜀_𝑖1+𝒜𝑒₁Δ𝐴^𝑇_𝑖(𝑡)𝑇^𝑇𝐸+𝑒₃Δ𝐵^𝑇_𝑖(𝑡)𝑇^𝑇𝐸𝑊<0,(iii)⁴𝑋₃𝑋^𝑇₃𝑊₄𝑌>0,⁴𝑋₄𝑋^𝑇₄𝑌₄>0,(iv)Ξ^𝜀_𝑖0+Ξ^𝜀_𝑖2+𝒜𝑒₁Δ𝐴^𝑇_𝑖(𝑡)𝑇^𝑇𝐸+𝑒₃Δ𝐵^𝑇_𝑖(𝑡)𝑇^𝑇𝐸<0. (89)(90)
Considering (ii) of (89) and combining the uncertainties condition (7), we have Ξ^𝜀_𝑖0+Ξ^𝜀_𝑖1+𝒜𝑒₁𝑒₃𝐸^𝑇_𝐴𝑖𝐸^𝑇_𝐵𝑖𝐹^𝑇_𝑖(𝑡)𝐻^𝑇_𝑖𝑇^𝑇𝐸<0. (91) By the definition of 𝒜, we obtain Ξ^𝜀_𝑖0+Ξ^𝜀_𝑖1+𝑀𝐹^𝑇_𝑖(𝑡)𝑁+𝑁^𝑇𝐹_𝑖(𝑡)𝑀^𝑇<0. (92) 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 set 𝑆 of modes with partly unknown transition rates matrix, scalars 𝛼, 𝛽, 𝛾, 𝜀, 𝜏, 𝜏_𝑑, 𝑑₁, 𝑑₂, 𝜇, and constant scalar 𝑑_𝑚 satisfying 𝑑₁<𝑑_𝑚<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 𝑃_𝑖>0, (𝑖∈𝑆), 𝑄_𝑗>0, (𝑗=1,2,3,4,5), 𝑅_𝑘>0, (𝑘=1,2,3,4,5), 𝑊_𝑙>0, (𝑙=1,2,3,4), 𝑌_𝑚>0, (𝑚=1,2,3,4), 𝑍_𝑛>0, (𝑛=1,2,3,4), and matrices 𝑈₁, 𝑈₂, 𝑋₁, 𝑋₂, 𝑋₃, 𝑋₄   for any scalars 𝜀₁, 𝜀₂, 𝜀₃ and any symmetric matrices 𝑉_𝑖, (𝑖∈𝑆) and any matrices 𝑇_𝑘, (𝑘=1,2,…,23) with appropriate dimensions, such that (22), (75), and (93) hold: 𝑊(i)³𝑋₁𝑋^𝑇₁𝑊₃𝑌>0,³𝑋₂𝑋^𝑇₂𝑌₃𝛿>0,(ii)¹Ξ^𝜀_ℐ𝑖0+𝛿₁Ξ^𝜀_ℐ𝑖1+𝛿²₁𝑀^𝑇𝑀𝑁^𝑇𝑊∗−𝐼<0,(iii)⁴𝑋₃𝑋^𝑇₃𝑊₄𝑌>0,⁴𝑋₄𝑋^𝑇₄𝑌₄𝛿>0,(iv)²Ξ^𝜀_ℐ𝑖0+𝛿₂Ξ^𝜀_ℐ𝑖2+𝛿²₂𝑀^𝑇𝑀𝑁^𝑇∗−𝐼<0, (93) where 𝑀=𝑒₁𝐸^𝑇_𝐴𝑖+𝑒₃𝐸^𝑇_𝐵𝑖,𝑁=𝐻^𝑇_𝑖𝑃^𝑇_𝑖𝑒₁+²³_𝑚=1𝐻^𝑇_𝑖𝑇^𝑇_𝑚𝑒_𝑚 (94) and other notations are the same as Theorem 17.

The proof of Theorem 22 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: 𝑥̇(𝑡)−𝐶_𝑖𝑥̇((𝑡=𝐴𝑡−𝜏))_𝑖+Δ𝐴(𝑡)𝑥(𝑡)+𝐵_𝑖+Δ𝐵(𝑡)𝑥(𝑡−𝑑(𝑡)), (95) 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 𝐴₁=−400−3,𝐴₂=,𝐵−200−3₁=−10−1−1,𝐵₂=,𝐶−10−0.9−1.2₁=0.5𝐼,𝐶₂𝐻=0.3𝐼,₁=0.20.2,𝐻₂=0,𝐸−0.3_𝐴1=0.20,𝐸_𝐴2=,𝐸00.2_𝐵1=−0.30.3,𝐸_𝐵2=0.20.2. (96)

Figure 1 

The mode switching of Example 1.

Given the decay rate 𝜀/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 𝑑_𝑚=0.12. In addition, we have 𝜏̇(𝑡)=0.1sin2𝑡, 𝜏_𝑑=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 delay 𝜏(𝑡) of Example 1.

Figure 3 

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

Example 2. To show the reduced conservativeness of the exponential stability condition in Theorem 10, consider the time-delay system in the form of (9) with 𝐶_𝑖=0, 𝛼=𝛽=𝛾=0, 𝑑(𝑡)=𝑑, and 𝐴_𝑖=−3−210,𝐵_𝑖={1}−0.50.10.30,𝑖∈𝑆=. (97)
For given 𝑑, by Theorem 10, the maximum exponential decay rate 𝜀̂, 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 rate 𝜀̂ with different parameter 𝑑.

From Table 1, we know that the maximum exponential decay rate 𝜀̂=0.9569 in this paper by setting 𝑑=0.8, while the maximum exponential decay rate 𝜀̂=0.7344 for [39], 𝜀̂=0.9367 for [40], and 𝜀̂=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: 𝑥̇(𝑡)=𝐴_𝑖+Δ𝐴_𝑖(𝑡)𝑥(𝑡)+𝐵_𝑖+Δ𝐵_𝑖(𝑡)𝑥(𝑡−𝑑), (98) where 𝐴_𝑖=−410−4,𝐵_𝑖={1}0.1040.1,𝑖∈𝑆=, (99) and the uncertain matrices Δ𝐴_𝑖(𝑡) and Δ𝐵_𝑖(𝑡) satisfy 𝐻_𝑖=0.2𝐼,𝐸_𝐴𝑖=𝐸_𝐵𝑖=𝐼,𝐹_𝑖(𝑡)<1,𝑖∈𝑆={1}. (100)

For given 𝑑, by Theorem 21, the maximum exponential decay rate 𝜀̂, 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 rate 𝜀̂ with different parameter 𝑑.

From Table 2, we know that the maximum exponential decay rate 𝜀̂=1.0419 in this paper by setting 𝑑=0.3, while the maximum exponential decay rate 𝜀̂=0.6255 for [39] and 𝜀̂=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: 𝐴_𝑖=−200−0.9,𝐵_𝑖=−10−1−1,𝐶_𝑖=,{1},𝑑0.1000.1𝑖∈𝑆=𝛼=0.05,𝛽=0.1,𝛾=0,₁=0,𝑑₂=𝜏,𝜇=𝜏_𝑑. (101) For given 𝜀 and  𝜇, choose 𝑑_𝑚=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 of 𝑑₂ with different 𝜀 and parameter 𝜇=0.

Table 4 

Maximum upper bound of 𝑑₂ with different 𝜀 and parameter 𝜇=0.5.

Table 5 

Maximum upper bound of 𝑑₂ with different 𝜀 and parameter 𝜇=0.9.

From Tables 3, 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 𝜀̂=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 𝜀̂=0.3, 𝜀̂=0.5, 𝜀̂=0.7, and 𝜀̂=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 Example 1 again, and we consider the practical system of PEEC model with partially unknown transition rates Π=^??₂₋₂ and nonlinear perturbations 𝑓₁(𝑥(𝑡)‖𝑥(𝑡,𝑓,𝑡)≤0.1)‖₂(𝑥((𝑡‖𝑥((𝑡,𝑓𝑡−𝑑)),𝑡)≤0.2𝑡−𝑑))‖₃(𝑥̇(𝑡−𝜏(𝑡)),𝑡)≤0.1‖𝑥̇(𝑡−𝜏(𝑡))‖. (102) 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: 𝐴₁=−1.15−0.751.50−1.50,𝐵₁=,𝐶−1.200.120.24−0.25₁=,𝐴−0.15−0.060.50−0.50₂=−2.15−0.491.50−2.10,𝐵₂=,𝐶−1.45−0.960.47−1.57₂=,𝐴0.25−0.350.05−0.65₃=−1.30−0.151.50−1.80,𝐵₃=,𝐶0.58−0.68−0.130.96₃=,𝐴0.330.070.19−0.36₄=−1.90−0.341.50−1.65,𝐵₄=,𝐶−0.67−1.501.391.23₄=.−0.230.160.02−0.57 (103) The partially unknown transition rate matrix is considered as the following two cases: Π₁=,Π−1.30.2????0.30.30.6?−1.5?0.4???₂=.−1.30.20.40.7??0.30.30.6?−1.5?0.4??? (104) For given 𝜀=0 and 𝑑(𝑡)=𝜏(𝑡), we have 𝑑₁=0, 𝑑₂=𝜏, and 𝜇=𝜏_𝑑. Set 𝑑_𝑚=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 of 𝑑₂ with Π₁ and different parameter 𝜇.

Table 7 

Maximum upper bound of 𝑑₂ with Π₂ and different parameter 𝜇.

From Tables 6 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.

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), the Doctoral Fund of Ministry of Education of China (20093402110019), Anhui Provincial Natural Science Foundation (11040606M143), the Fundamental Research Funds for the Central Universities, and the Program for New Century Excellent Talents in University.