Delay-Dependent 𝐻_∞ Control for Descriptor Markovian Jump Systems with Time-Varying Delay

Abstract

This paper is concerned with the delay-dependent 𝐻_∞ control problem for continuous-time descriptor Markovian jump systems with time-varying delay. By constructing various Lyapunov-Krasovskii functionals for different subsystems, together with delay decomposition method, a new delay-dependent bounded real lemma (BRL) is derived, under which descriptor Markovian jump time-delay systems are regular, impulse-free, and stochastically stable and satisfy a prescribed 𝐻_∞ performance level. Since the reciprocally convex combination approach is adopted to estimate the upper bound of the integral terms, the BRL obtained in this paper is less conservative than some existing ones. Based on the proposed BRL, a sufficient condition for the existence of state feedback controller is provided. Finally, three numerical examples are provided to demonstrate the validity of the proposed methods.

1. Introduction

Over the past decades, stability problems for time-delay systems have received considerable attention. Many methods have been presented to determine the stability. In the published results, two types of criteria have been widely investigated (see [1]), that is, delay-independent one and delay-dependent one. As we know, the delay-dependent criteria are less conservative than delay-independent ones, especially when the time delay is fairly small. Therefore, in order to reduce the conservatism of stability conditions, a great number of delay-dependent results have been reported, including, descriptor modeled transformation method (e.g., [2]), parameterized modeled transformation one (e.g., [3]), free-weighting matrix one (e.g., [4]), discretized Lyapunov functional one (e.g., [5]), delay decomposition one (e.g., [6]) and so on (e.g., [7, 8]). Among the aforementioned methods, since discretized Lyapunov functional method is complicated and it is difficult to apply this method to design controller, the delay decomposition method is less conservative than others except the discretized Lyapunov functional method.

Markovian jump systems are modeled by many stochastic systems, which arise abrupt random changes in their structure, such as component failures or repairs and unexpected environmental changes. A great deal of attention has been devoted to Markovian jump systems (see [9, 10]). Recently, many efforts have been made to study the stability analysis and controller synthesis problems for Markovian jump systems with time-varying and time-invariant delay. Reference [11] divides the time-invariant delay into 𝑚 segments and uses the delay division method to obtain a less conservative criterion than [12]. In [13], stability and stabilization are discussed for Markovian jump systems with time-varying delay by selecting distinct Lyapunov matrices for different subsystems and introducing a triple-integral term. Reference [14] studies the 𝐻_∞ analysis for Markovian jump stochastic systems with time-varying delay. By dividing the delay interval into two parts, two improved delay-dependent conditions have been presented to guarantee the stochastic stability.

Descriptor systems are also referred to as singular systems, semistate systems, which are a natural representation of dynamical systems and describe a much wider range of systems than standard state-space systems (see, e.g., [10, 15]). When descriptor systems are subject to time delay and stochastic disturbance in their structure, it is very natural to cast this model into descriptor Markovian jump time-delay systems. Very recently, a large number of researches have been published to study descriptor Markovian jump time-delay systems. By constructing more complicated Lyapunov-Krasovskii functionals, [16] studies the passivity and stochastic stability of descriptor Markovian jump systems with time-varying delay. Based on the delay decomposition method, [17] investigates the 𝐻_∞ control problem for descriptor Markovian jump time-delay systems. To the best of our knowledge, the time-varying delay in most published results has not been tackled adequately, which leaves us much room to improve. All of these motivate the current study.

In this paper, we investigate 𝐻_∞ control problem for continuous-time descriptor Markovian jump systems with time-varying delay. Motivated by delay decomposition method, through designing various Lyapunov-Krasovskii functionals for different subsystems, a delay-dependent BRL is established, which guarantees that the considered system is stochastically admissible and satisfies a prescribed 𝐻_∞ performance level. The reciprocally convex combination method is used in the proof of the proposed results, which is one of the contributions to this paper. The other is the employed Lyapunov-Krasovskii functionals. Thanks to all of these, the obtained results are less conservative than some existing results.

The rest of this paper is organized as follows. Section 2 gives problem description and some essential lemmas. Section 3 focuses on 𝐻_∞ control problem for continuous-time descriptor Markovian jump time-delay systems. A new delay-dependent BRL is proposed, and the design method for the state feedback controller, which guarantees that the resultant closed-loop system is stochastically admissible with a prescribed 𝐻_∞ performance level, is provided. Section 4 provides numerical examples to illustrate the effectiveness of our methods, and Section 5 concludes the paper.

Notation. The notation used throughout this paper is quite standard. ℝ^𝑛 stands for the 𝑛-dimensional Euclidean space, and ℝ^𝑚×𝑛 represents the set of all 𝑚×𝑛 real matrices. The superscripts 𝑇 and −1 stand for matrix transposition and matrix inverse, respectively. ℒ₂[0,∞) represents the space of square integrable vector functions over the interval [0,∞). (Ω,ℱ,𝒫) is the probability space, Ω is the sample space, ℱ is the 𝜎-algebra of subsets of the sample space, and 𝒫 is the probability measure on ℱ. ℰ{⋅} is the expectation operator with respect to some probability measure 𝒫. ℕ⁺ represents the set of positive integers. The notation 𝑃>0 (𝑃≥0) implies that 𝑃 is a real symmetric and positive definite (semipositive definite) matrix. For simplicity, 𝑀_𝑖 is often adopted to denote 𝑀(𝑖). The asterisk ∗ in matrix block implies that it can be induced by symmetric position. A block diagonal matrix with diagonal blocks 𝑁₁,𝑁₂,…,𝑁_𝑚 will be denoted by diag{𝑁₁,𝑁₂,…,𝑁_𝑚}. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

2. Problem Formulation and Preliminaries

Fix the probability space (Ω,ℱ,𝒫) and consider the following continuous-time descriptor Markovian jump systems with time-varying delay: 𝐸𝑥̇(𝑡)=𝐴𝑟_𝑡𝑥(𝑡)+𝐴_𝑑𝑟_𝑡𝑥((𝑡𝑟𝑡−ℎ))+𝐵_𝑡𝑢(𝑡)+𝐵_𝑤𝑟_𝑡𝑤(𝑡),𝑧(𝑡)=𝐶𝑟_𝑡𝑥(𝑡)+𝐶_𝑑𝑟_𝑡𝑥((𝑡𝑟𝑡−ℎ))+𝐷_𝑡𝑢(𝑡)+𝐷_𝑤𝑟_𝑡𝑤(𝑡),𝑥(𝜃)=𝜙(𝜃),∀𝜃∈[−ℎ,0], (1) where 𝑥(𝑡)∈ℝ^𝑛 is the state vector, 𝑢(𝑡)∈ℝ^𝑚 is the input vector, 𝑧(𝑡)∈ℝ^𝑞 is the controlled output vector, and 𝑤(𝑡)∈ℝ^𝑝 is the disturbance input vector which belongs to ℒ₂[0,∞). 𝐸 is a known real matrix which may be singular, and it is generally assumed that rank (𝐸)=𝑟≤𝑛. The jumping mode {𝑟_𝑡,𝑡≥0}, which takes value in a finite set ℓ={1,2,…,𝑁}, is the continuous-time Markov process with the following mode transition probabilities: 𝒫𝑟_𝑡+Δ=𝑗∣𝑟_𝑡=𝛾=𝑖^𝑖𝑗Δ+𝑜(Δ),if𝑗≠𝑖1+𝛾_𝑖𝑖Δ+𝑜(Δ),if𝑗=𝑖, (2) where Δ>0, lim_𝑡→0(𝑜(Δ)/Δ)=0, and 𝛾_𝑖𝑗≥0(𝑖,𝑗∈ℓ,𝑗≠𝑖), satisfying 𝛾_𝑖𝑖=−∑_{𝑗∈ℓ,𝑗≠𝑖}𝛾_𝑖𝑗, represent the switching rate from mode 𝑖 at time 𝑡 to mode 𝑗 at time 𝑡+Δ for all 𝑖∈ℓ. 𝐴(𝑟_𝑡), 𝐴_𝑑(𝑟_𝑡), 𝐵(𝑟_𝑡), 𝐵_𝑤(𝑟_𝑡), 𝐶(𝑟_𝑡), 𝐶_𝑑(𝑟_𝑡), 𝐷(𝑟_𝑡), and 𝐷_𝑤(𝑟_𝑡) are known real constant matrices representing the deterministic systems for each 𝑟_𝑡∈ℓ. The considered time-varying delay ℎ(𝑡) satisfies 0≤ℎ(𝑡)≤ℎ,ℎ̇(𝑡)≤ℎ_𝑑, (3) where ℎ and ℎ_𝑑 are known constants. Throughout this paper, for each 𝑟_𝑡=𝑖∈ℓ, we use 𝑀_𝑖(𝑡) and 𝑁_𝑖 to denote matrices 𝑀(𝑟_𝑡,𝑡) and 𝑁(𝑟_𝑡).

To facilitate exposition, the unforced descriptor Markovian jump time-delay systems are given as follows: 𝐸𝑥̇(𝑡)=𝐴𝑟_𝑡𝑥(𝑡)+𝐴_𝑑𝑟_𝑡𝑥(𝑡−ℎ(𝑡))+𝐵_𝑤𝑟_𝑡𝑤(𝑡)𝑧(𝑡)=𝐶𝑟_𝑡𝑥(𝑡)+𝐶_𝑑𝑟_𝑡𝑥(𝑡−ℎ(𝑡))+𝐷_𝑤𝑟_𝑡𝑤(𝑡)𝑥(𝜃)=𝜙(𝜃),∀𝜃∈[−ℎ,0]. (4)

In the sequel, we recall some indispensable definitions.

Definition 1 (see [10]). When the disturbance signal 𝑤(𝑡)≡0, for any given scalars ℎ>0 and ℎ_𝑑, consider the following statements.(1)The continuous-time descriptor Markovian jump time-delay systems (4) are said to be regular and impulse-free for any time-varying delay satisfying (3) if the pairs (𝐸,𝐴_𝑖) are regular and impulse-free for each 𝑟_𝑡∈ℓ.(2)The continuous-time descriptor Markovian jump time-delay systems (4) are said to be stochastically stable if there exists a scalar 𝑀(𝜙(⋅),𝑟₀)>0 such that lim_𝑡→∞ℰ^𝑡₀𝑥^𝑇𝑠,𝑥_𝑡,𝑟₀𝑥𝑠,𝑥_𝑡,𝑟₀𝑑𝑠∣𝑥_𝑡(𝑠)(𝑠),[]=𝑥=𝜙𝑠∈−ℎ,0,𝑟₀∈ℓ≤𝑀𝜙(⋅),𝑟₀, (5) where 𝑥(𝑡,𝑥_𝑡,𝑟₀) denotes the solution to systems (4) at time 𝑡 under the given initial conditions.(3)The continuous-time descriptor Markovian jump time-delay systems (4) are said to be stochastically admissible if they are regular, impulse-free, and stochastically stable.

Definition 2 (see [17]). For a given scalar 𝛾>0, the descriptor Markovian jump time-delay systems (4) are said to be stochastically admissible with 𝐻_∞ performance level 𝛾; if the system with 𝑤(𝑡)≡0 is stochastically admissible and under zero initial condition, it satisfies ℰ^∞₀𝑧^𝑇(𝑡)𝑧(𝑡)𝑑𝑡≤𝛾^2∞₀𝑤^𝑇(𝑡)𝑤(𝑡)𝑑𝑡 (6) for any nonzero 𝑤(𝑡)∈ℒ[0,∞).

In this paper, the purpose is to design a mode-dependent state feedback controller for the continuous-time descriptor Markovian jump time-delay systems (1) with a prescribed scalar 𝛾>0 such that the resultant closed-loop systems are stochastically admissible with 𝐻_∞ performance level 𝛾. Specifically, the desired state feedback controller can be expressed as 𝑢(𝑡)=𝐾𝑟_𝑡𝑥(𝑡)+𝐾_𝑑𝑟_𝑡𝑥(𝑡−ℎ(𝑡)). (7)

To this end, some essential lemmas are introduced to expedite the following discussion.

Lemma 3 (see [18]). For any constant matrix 𝑊∈ℝ^𝑛×𝑛, 𝑊>0, scalar 𝛾>0, and vector function 𝑥∶[−𝛾,0]→ℝ^𝑛 such that the following integration is well defined, then −𝛾^𝑡_𝑡−𝛾𝑥^𝑇(𝑠)𝑊𝑥(𝑠)𝑑𝑠≤−^𝑡_𝑡−𝛾𝑥^𝑇(𝑠)𝑑𝑠𝑊^𝑡_𝑡−𝛾𝑥(𝑠)𝑑𝑠. (8)

Lemma 4 (see [8]). Let 𝑓₁,𝑓₂,…,𝑓_𝑁∶ℝ^𝑚↦ℝ have positive values in an open subset 𝔻 of ℝ^𝑚. Then the reciprocally convex combination of 𝑓_𝑖 over 𝔻 satisfies min_{𝛼𝑖|𝛼𝑖>0,∑𝑖𝛼𝑖=1𝑖}1𝛼_𝑖𝑓_𝑖(𝑡)=_𝑖𝑓_𝑖(𝑡)+max_{𝑔𝑖,𝑗(𝑡)𝑖≠𝑗}𝑔_𝑖,𝑗(𝑡), (9) subject to 𝑔_𝑖,𝑗∶ℝ^𝑚↦ℝ,𝑔_𝑗,𝑖(𝑡)=𝑔_𝑖,𝑗(𝑡),𝑓_𝑖(𝑡)𝑔_𝑖,𝑗(𝑡)𝑔_𝑖,𝑗(𝑡)𝑓_𝑖(𝑡)≥0. (10)

Based on Lemma 4, we present the following lemma, which can be regarded as an alternative form of the reciprocally convex combination.

Lemma 5. For any constant matrix 𝑊∈ℝ^𝑛×𝑛, 𝑊≥0, and a variable 𝛽 satisfying 𝑎≤𝛽≤𝑏, if there exists a matrix 𝑊 satisfying _𝑊𝑊𝑊^𝑇_𝑊≥0, then −1𝑒𝛽−𝑎¹(𝑡)^𝑇𝑊𝑒₁(𝑡)−1𝑒𝑏−𝛽^𝑇2(𝑡)𝑊𝑒₂(𝑡)1≤−𝑒𝑏−𝑎¹(𝑡)𝑒₂(𝑡)^𝑇𝑊𝑊𝑊^𝑇𝑊𝑒₁(𝑡)𝑒₂(𝑡), (11) where 𝑒₁(𝑡) and 𝑒₂(𝑡) are vector functions with compatible dimensions.

Proof. Taking 𝑓₁(𝑡)=𝑒^𝑇₁(𝑡)𝑊𝑒₁(𝑡),𝑓₂(𝑡)=𝑒^𝑇₂(𝑡)𝑊𝑒₂(𝑡),𝑔_1,2(𝑡)=𝑒^𝑇₁(𝑡)𝑊𝑒₂(𝑡),𝑔_2,1(𝑡)=𝑒^𝑇₂(𝑡)𝑊^𝑇𝑒₁(𝑡),𝛼₁=𝛽−𝑎𝑏−𝑎,𝛼₂=𝑏−𝛽,𝑏−𝑎 (12) it can be easily verified that the above choice satisfies constraint (10) in Lemma 4. Note that with the equality (9), we have min_{𝛼𝑖|𝛼𝑖>0,∑𝑖𝛼𝑖=1𝑖}1𝛼_𝑖𝑓_𝑖(𝑡)≤𝑏−𝑎𝑒𝛽−𝑎^𝑇1(𝑡)𝑊𝑒₁(𝑡)+𝑏−𝑎𝑒𝑏−𝛽^𝑇2(𝑡)𝑊𝑒₂(𝑡),_𝑖𝑓_𝑖(𝑡)+max_{𝑔𝑖,𝑗(𝑡)𝑖≠𝑗}𝑔_𝑖,𝑗(𝑡)≥𝑒₁(𝑡)𝑒₂(𝑡)^𝑇𝑊𝑊𝑊^𝑇𝑊𝑒₁(𝑡)𝑒₂(𝑡). (13)(14) By Lemma 4, (13) and (14) yield that −1𝑒𝛽−𝑎^𝑇1(𝑡)𝑊𝑒₁(𝑡)−1𝑒𝑏−𝛽^𝑇2(𝑡)𝑊𝑒₂(𝑡)1≤−𝑒𝑏−𝑎¹(𝑡)𝑒₂(𝑡)^𝑇𝑊𝑊𝑊^𝑇𝑊𝑒₁(𝑡)𝑒₂(𝑡). (15) This completes the proof.

Lemma 6 (see [10]). Given any real matrices 𝑆₁, 𝑆₂, and 𝑆₃ with 𝑆^𝑇₁=𝑆₁ and 𝑆₃>0, then one has 𝑆₁+𝑆₂𝑆₃⁻¹𝑆^𝑇₂<0 (16) if and only if 𝑆₁𝑆₂𝑆^T₂−𝑆₃<0, (17) or equivalently −𝑆₃𝑆^𝑇₂𝑆₂𝑆₁<0. (18)

Lemma 7 (see [10]). Given matrices Φ=Φ^𝑇<0, 𝐷 and 𝐸 with appropriate dimensions Φ+𝐷𝐹(𝑡)𝐸+𝐸^𝑇𝐹(𝑡)^𝑇𝐷^𝑇<0 (19) for all 𝐹(𝑡) satisfying 𝐹^𝑇(𝑡)𝐹(𝑡)≤𝐼 if and only if there exists scalar 𝜖>0 such that Φ+𝜀𝐷𝐷^𝑇+𝜀⁻¹𝐸^𝑇𝐸<0. (20)

Lemma 8 (see [19]). Let 𝑃 be symmetric such that 𝐸^𝑇_𝑅𝑃𝐸_𝑅>0, and let Φ be nonsingular. Then, 𝑃𝐸+𝑈^𝑇Φ𝑉^𝑇 is nonsingular and its inverse is expressed as 𝑃𝐸+𝑈^𝑇Φ𝑉^𝑇−1=𝑃𝐸^𝑇+𝑉Φ𝑈, (21) where 𝐸_𝐿 and 𝐸_𝑅 are full rank with 𝐸=𝐸_𝐿𝐸^𝑇_𝑅 and 𝑈^𝑇 and 𝑉 are of full column rank and composed of bases of null space of 𝐸^𝑇 and 𝐸, respectively. 𝑃 is symmetric and Φ is nonsingular such that 𝐸^𝑇_𝑅𝑃𝐸_𝑅=(𝐸^𝑇_𝐿𝑃𝐸_𝐿)⁻¹, Φ=(𝑉^𝑇𝑉)⁻¹Φ⁻¹(𝑈𝑈^𝑇)⁻¹.

3. Main Results

In this section, the delay-dependent 𝐻_∞ control problem will be studied for the descriptor Markovian jump time-delay system (1) by virtue of the delay decomposition method. To this end, by dividing the delay interval (0,ℎ] into two disjoint subintervals (0,𝛼ℎ] and (𝛼ℎ,ℎ], (0<𝛼<1), the following delay-dependent bounded real lemma (BRL) for the system (4) is firstly proposed, which will play an indispensable role in the derivation of our main results.

Theorem 9. For given scalars 𝛾>0, ℎ>0, and 0≤ℎ_𝑑<1, the descriptor Markovian jump time-delay system (4) is stochastically admissible with 𝐻_∞ performance level 𝛾 for any ℎ(𝑡) satisfying (3), if there exist a set of positive definite matrices 𝑃_𝑖, 𝑄_1𝑖, 𝑄_2𝑖, 𝑄_3𝑖, 𝑅_2𝑖, 𝑅_3𝑖, 𝑈₁, 𝑈₂, 𝑊, and matrices Φ_𝑖, 𝑆_𝑖, 𝑇_𝑖, ∀𝑖∈ℓ, satisfying the following LMIs for each 𝑖∈ℓ:Θ_𝑖11Θ_𝑖12Θ_𝑖130Θ_𝑖15𝐴^𝑇_𝑖𝑅_2𝑖𝐴^𝑇_𝑖𝑅_3𝑖12ℎ²𝐴^𝑇_𝑖𝑊𝐶^𝑇_𝑖∗Θ_𝑖22Θ_𝑖2300𝐴^𝑇_𝑑𝑖𝑅_2𝑖𝐴^𝑇_𝑑𝑖𝑅_3𝑖12ℎ²𝐴^𝑇_𝑑𝑖𝑊𝐶^𝑇_𝑑𝑖∗∗Θ_𝑖33Θ_𝑖3400000∗∗∗Θ_𝑖4400000∗∗∗∗−𝛾²𝐼𝐵^𝑇_𝑤𝑖𝑅_2𝑖𝐵^𝑇_𝑤𝑖𝑅_3𝑖12ℎ²𝐵^𝑇_𝑤𝑖𝑊𝐷^𝑇_𝑤𝑖∗∗∗∗∗−𝛼₁𝑅_2𝑖000∗∗∗∗∗∗−𝛼₂𝑅_3𝑖100∗∗∗∗∗∗∗−2ℎ²(𝑡)∈(]𝑊0∗∗∗∗∗∗∗∗−𝐼<0,ℎ0,𝛼ℎΘ_𝑖11Θ_𝑖12Θ_𝑖130Θ_𝑖15𝐴^𝑇_𝑖𝑅_2𝑖𝐴^𝑇_𝑖𝑅_3𝑖12ℎ²𝐴^𝑇_𝑖𝑊𝐶^𝑇_𝑖∗Θ_𝑖22Θ_𝑖23Θ_𝑖240𝐴^𝑇_𝑑𝑖𝑅_2𝑖𝐴^𝑇_𝑑𝑖𝑅_3𝑖12ℎ²𝐴^𝑇_𝑑𝑖𝑊𝐶^𝑇_𝑑𝑖∗∗Θ_𝑖33Θ_𝑖3400000∗∗∗Θ_𝑖4400000∗∗∗∗−𝛾²𝐼𝐵^𝑇_𝑤𝑖𝑅_2𝑖𝐵^𝑇_𝑤𝑖𝑅_3𝑖12ℎ²𝐵^𝑇_𝑤𝑖𝑊𝐷^𝑇_𝑤𝑖∗∗∗∗∗−𝛼₁𝑅_2𝑖000∗∗∗∗∗∗−𝛼₂𝑅_3𝑖100∗∗∗∗∗∗∗−2ℎ²(𝑡)∈(]𝑊0∗∗∗∗∗∗∗∗−𝐼<0,ℎ𝛼ℎ,ℎ_𝑁𝑗=1𝛾_𝑖𝑗𝑅_2𝑗−𝑊<0,^𝑁_𝑗=1𝛾_𝑖𝑗𝑅_3𝑗−𝑊<0_𝑁𝑗=1𝛾_𝑖𝑗𝑄_1𝑗+^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_2𝑗−𝑈₁<0,^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_2𝑗−𝑈₁<0_𝑁𝑗=1𝛾_𝑖𝑗𝑄_1𝑗+^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_3𝑗−𝑈₂<0,^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_3𝑗−𝑈₂<0𝑍_1𝑖=𝑅_2𝑖𝑆_𝑖𝑆^𝑇_𝑖𝑅_2𝑖>0,𝑍_2𝑖=𝑅_3𝑖𝑇_𝑖𝑇^𝑇_𝑖𝑅_3𝑖>0, (22)(23)(24)(25)(26)(27)where 𝛼₁=1/𝛼ℎ, 𝛼₂=1/(ℎ−𝛼ℎ), and Θ_𝑖11=𝐴^𝑇_𝑖𝑃_𝑖𝐸+𝑈^𝑇Φ_𝑖𝑉^𝑇+𝑃_𝑖𝐸+𝑈^𝑇Φ_𝑖𝑉^𝑇𝑇𝐴_𝑖+^𝑁_𝑗=1𝛾_𝑖𝑗𝐸^𝑇𝑃_𝑗𝐸+𝑄_1𝑖+𝑄_2𝑖+𝛼ℎ𝑈₁+(1−𝛼)ℎ𝑈₂−𝛼₁𝐸^𝑇𝑅_2𝑖Θ𝐸,_𝑖12=𝑃_𝑖𝐸+𝑈^𝑇Φ_𝑖𝑉^𝑇𝑇𝐴_𝑑𝑖+𝛼₁𝐸^𝑇𝑅_2𝑖𝐸−𝛼₁𝐸^𝑇𝑆_𝑖Θ𝐸,_𝑖13=𝛼₁𝐸^𝑇𝑆_𝑖Θ𝐸,_𝑖15=𝑃_𝑖𝐸+𝑈^𝑇Φ_𝑖𝑉^𝑇𝑇𝐵_𝑤𝑖,Θ_𝑖22=−1−ℎ_𝑑𝑄_1𝑖−2𝛼₁𝐸^𝑇𝑅_2𝑖𝐸+𝛼₁𝐸^𝑇𝑆_𝑖𝐸+𝛼₁𝐸^𝑇𝑆^𝑇_𝑖Θ𝐸,_𝑖23=𝛼₁𝐸^𝑇𝑅_2𝑖𝐸−𝛼₁𝐸^𝑇𝑆_𝑖Θ𝐸,_𝑖33=𝑄_3𝑖−𝑄_2𝑖−𝛼₁𝐸^𝑇𝑅_2𝑖𝐸−𝛼₂𝐸^𝑇𝑅_3𝑖Θ𝐸,_𝑖34=𝛼₂𝐸^𝑇𝑅_3𝑖Θ𝐸,_𝑖44=−𝑄_3𝑖−𝛼₂𝐸^𝑇𝑅_3𝑖Θ𝐸,_𝑖12=𝑃_𝑖𝐸+𝑈^𝑇Φ_𝑖𝑉^𝑇𝑇𝐴_𝑑𝑖,Θ_𝑖13=𝛼₁𝐸^𝑇𝑅_2𝑖Θ𝐸,_𝑖22=−1−ℎ_𝑑𝑄_1𝑖−2𝛼₂𝐸^𝑇𝑅_3𝑖𝐸+𝛼₂𝐸^𝑇𝑇_𝑖𝐸+𝛼₂𝐸^𝑇𝑇^𝑇_𝑖Θ𝐸,_𝑖23=𝛼₂𝐸^𝑇𝑅_3𝑖𝐸−𝛼₂𝐸^𝑇𝑇^𝑇_𝑖Θ𝐸,_𝑖24=𝛼₂𝐸^𝑇𝑅_3𝑖𝐸−𝛼₂𝐸^𝑇𝑇_𝑖Θ𝐸,_𝑖34=𝛼₂𝐸^𝑇𝑇_𝑖𝐸. (28)

Proof. Assume that there exist matrices such that LMIs (22)–(27) hold. With these conditions in hand, we firstly show the regularity and absence of impulse of the system (4). To this end, it is from [10] that there exist a couple of nonsingular matrices 𝑀 and 𝑁 such that for each 𝑖∈ℓ𝐼𝑀𝐸𝑁=^𝑟000,𝑀𝐴_𝑖𝐴𝑁=^𝑖1𝐴_𝑖2𝐴_𝑖3𝐴_𝑖4,𝑀𝐴_𝑑𝑖𝐴𝑁=^𝑑𝑖1𝐴_𝑑𝑖2𝐴_𝑑𝑖3𝐴_𝑑𝑖4. (29)
Decomposing the associated matrices with a compatible dimension, we have 𝑀^−𝑇𝑃_𝑖𝑀⁻¹=𝑃_𝑖1𝑃_𝑖2𝑃_𝑖3𝑃_𝑖4,𝑀^−𝑇𝑈^𝑇=0𝐼𝑍₁,𝑉^𝑇𝑁=𝑍₂0𝐼, (30) where 𝑍₁∈ℝ^{(𝑛−𝑟)×(𝑛−𝑟)} and 𝑍₂∈ℝ^{(𝑛−𝑟)×(𝑛−𝑟)} are arbitrarily nonsingular matrices.
From the (1,1) block in the left-hand matrices of (22) or (23), and observing that 𝑄_1𝑖, 𝑄_2𝑖, 𝑈₁, and 𝑈₂ are positive definite, it can be easily seen that 𝐴^𝑇_𝑖𝑃_𝑖𝐸+𝑈^𝑇Φ_𝑖𝑉^𝑇+𝑃_𝑖𝐸+𝑈^𝑇Φ_𝑖𝑉^𝑇𝑇𝐴_𝑖+𝐸^𝑇𝑓𝑃_𝑗,𝑅_2𝑖𝐸<0, (31) where 𝑓(𝑃_𝑗,𝑅_2𝑖) is the function of 𝑃_𝑗 and 𝑅_2𝑖 and makes no difference to the subsequent proof.
Pre- and postmultiplying (31) by 𝑁^𝑇 and 𝑁, respectively, together with (29) and (30), we get ∗∗∗𝐴^𝑇_𝑖4𝑍₁Φ_𝑖𝑍₂+𝑍^𝑇₂Φ^𝑇_𝑖𝑍^𝑇₁𝐴_𝑖4<0. (32) According to [10], we can know that the pairs (𝐸,𝐴_𝑖) are regular and impulse-free for each 𝑖∈ℓ. Therefore, by Definition 1, the system (4) is regular and impulse-free.
Next, we will show the stochastic stability of the system (4). To this end, we define a new process {(𝑥_𝑡,𝑟_𝑡),𝑡≥0} by {𝑥_𝑡(𝑠)=𝑥(𝑡+𝑠),−2ℎ≤𝑠≤0}; then from [12], we can know {(𝑥_𝑡,𝑟_𝑡),𝑡≥ℎ} is a Markov process with initial state (𝜙(⋅),𝑟₀). Now, construct the stochastic Lyapunov functional candidate to be 𝑉𝑥_𝑡,𝑟_𝑡,𝑡=𝑉₁𝑥_𝑡,𝑟_𝑡,𝑡+𝑉₂𝑥_𝑡,𝑟_𝑡,𝑡+𝑉₃𝑥_𝑡,𝑟_𝑡,𝑡, (33) where 𝑉₁𝑥_𝑡,𝑟_𝑡,𝑡=𝑥^𝑇(𝑡)𝐸^𝑇𝑃𝑟_𝑡(𝑡),𝑉𝐸𝑥₂𝑥_𝑡,𝑟_𝑡,𝑡=^𝑡_{𝑡−ℎ(𝑡)}𝑥^𝑇(𝑠)𝑄₁𝑟_𝑡𝑥(𝑠)𝑑𝑠+^𝑡_{𝑡−𝛼ℎ}𝑥^𝑇(𝑠)𝑄₂𝑟_𝑡𝑥(𝑠)+𝑑𝑠_{𝑡−𝛼ℎ𝑡−ℎ}𝑥^𝑇(𝑠)𝑄₃𝑟_𝑡𝑥(𝑠)+𝑑𝑠_{0−𝛼ℎ𝑡𝑡+𝜃}𝑥^𝑇(𝑠)𝑈₁𝑥(𝑠)+𝑑𝑠𝑑𝜃_{−𝛼ℎ−ℎ𝑡𝑡+𝜃}𝑥^𝑇(𝑠)𝑈₂𝑥(𝑠)𝑉𝑑𝑠𝑑𝜃,₃𝑥_𝑡,𝑟_𝑡,𝑡=⁰_−𝛼ℎ^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠)𝐸^𝑇𝑅₂𝑟_𝑡𝐸𝑥̇(𝑠)+𝑑𝑠𝑑𝜃_{−𝛼ℎ−ℎ𝑡𝑡+𝜃}𝑥̇^𝑇(𝑠)𝐸^𝑇𝑅₃𝑟_𝑡𝐸𝑥̇(𝑠)+𝑑𝑠𝑑𝜃_{0−ℎ0𝛽𝑡𝑡+𝜃}𝑥̇^𝑇(𝑠)𝐸^𝑇𝑊𝐸𝑥̇(𝑠)𝑑𝑠𝑑𝜃𝑑𝛽. (34)
Let ℒ be the weak infinitesimal generator of the random process {𝑥_𝑡,𝑡} acting on 𝑉(𝑥_𝑡,𝑟_𝑡,𝑡) [12]. Then for each 𝑟_𝑡=𝑖∈ℓ, it can be shown that ℒ𝑉₁𝑥_𝑡=𝑥̇,𝑖,𝑡_𝑇(𝑡)𝐸^𝑇𝑃_𝑖𝐸𝑥(𝑡)+𝑥^𝑇(𝑡)𝐸^𝑇𝑃_𝑖𝐸𝑥̇(𝑡)+𝑥^𝑇(𝑡)^𝑁_𝑗=1𝛾_𝑖𝑗𝐸^𝑇𝑃_𝑗𝐸𝑥(𝑡)=𝑥̇^𝑇(𝑡)𝐸^𝑇𝑃_𝑖𝐸+𝑈^𝑇Φ_𝑖𝑉^𝑇𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃_𝑖𝐸+𝑈^𝑇Φ_𝑖𝑉^𝑇𝑇𝐸𝑥̇(𝑡)+𝑥^𝑇(𝑡)^𝑁_𝑗=1𝛾_𝑖𝑗𝐸^𝑇𝑃_𝑗𝐸𝑥(𝑡)=𝑥^𝑇(𝑡)𝐴^𝑇_𝑖𝑃_𝑖𝐸+𝑈^𝑇Φ_𝑖𝑉^𝑇𝑥(𝑡)+𝑥^𝑇(𝑡−ℎ(𝑡))𝐴^𝑇_𝑑𝑖𝑃_𝑖𝐸+𝑈^𝑇Φ_𝑖𝑉^𝑇𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃_𝑖𝐸+𝑈^𝑇Φ_𝑖𝑉^𝑇𝑇𝐴_𝑖𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃_𝑖𝐸+𝑈^𝑇Φ_𝑖𝑉^𝑇𝑇𝐴_𝑑𝑖𝑥(𝑡−ℎ(𝑡))+𝑥^𝑇(𝑡)^𝑁_𝑗=1𝛾_𝑖𝑗𝐸^𝑇𝑃_𝑗𝐸𝑥(𝑡)=𝑥^𝑇(𝑡)𝐴^𝑇_𝑖𝑃_𝑖𝐸+𝑈^𝑇Φ_𝑖𝑉^𝑇+𝑃_𝑖𝐸+𝑈^𝑇Φ_𝑖𝑉^𝑇𝑇𝐴_𝑖+^𝑁_𝑗=1𝛾_𝑖𝑗𝐸^𝑇𝑃_𝑗𝐸𝑥(𝑡)+𝑥^𝑇(𝑡−ℎ(𝑡))𝐴^𝑇_𝑑𝑖𝑃_𝑖𝐸+𝑈^𝑇Φ_𝑖𝑉^𝑇𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃_𝑖𝐸+𝑈^𝑇Φ_𝑖𝑉^𝑇𝑇𝐴_𝑑𝑖𝑥(𝑡−ℎ(𝑡)),ℒ𝑉₂𝑥_𝑡,𝑖,𝑡=𝑥^𝑇(𝑡)𝑄_1𝑖+𝑄_2𝑖+𝛼ℎ𝑈₁+(1−𝛼)ℎ𝑈₂𝑥(𝑡)−1−ℎ̇(𝑡)𝑥^𝑇(𝑡−ℎ(𝑡))𝑄_1𝑖𝑥(𝑡−ℎ(𝑡))+𝑥^𝑇(𝑡−𝛼ℎ)𝑄_3𝑖−𝑄_2𝑖𝑥(𝑡−𝛼ℎ)−𝑥^𝑇(𝑡−ℎ)𝑄_3𝑖𝑥()+𝑡−ℎ_{𝑡(𝑡)𝑡−ℎ}𝑥^𝑇(𝑠)^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_1𝑗𝑥(𝑠)+𝑑𝑠_{𝑡𝑡−𝛼ℎ}𝑥^𝑇(𝑠)^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_2𝑗𝑥(𝑠)+𝑑𝑠_{𝑡−𝛼ℎ𝑡−ℎ}𝑥^𝑇(𝑠)^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_3𝑗𝑥(𝑠)𝑑𝑠−^𝑡_{𝑡−𝛼ℎ}𝑥^𝑇(𝑠)𝑈₁𝑥(𝑠)−𝑑𝑠_{𝑡−𝛼ℎ𝑡−ℎ}𝑥^𝑇(𝑠)𝑈₂𝑥(𝑠)𝑑𝑠≤𝑥^𝑇(𝑡)𝑄_1𝑖+𝑄_2𝑖+𝛼ℎ𝑈₁+(1−𝛼)ℎ𝑈₂𝑥(𝑡)−1−ℎ_𝑑𝑥^𝑇(𝑡−ℎ(𝑡))𝑄_1𝑖𝑥(𝑡−ℎ(𝑡))+𝑥^𝑇(𝑡−𝛼ℎ)𝑄_3𝑖−𝑄_2𝑖𝑥(𝑡−𝛼ℎ)−𝑥^𝑇(𝑡−ℎ)𝑄_3𝑖𝑥()+𝑡−ℎ_{𝑡(𝑡)𝑡−ℎ}𝑥^𝑇(𝑠)^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_1𝑗𝑥(𝑠)𝑑s+^𝑡_{𝑡−𝛼ℎ}𝑥^𝑇(𝑠)^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_2𝑗𝑥(𝑠)+𝑑𝑠_{𝑡−𝛼ℎ𝑡−ℎ}𝑥^𝑇(𝑠)^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_3𝑗𝑥(𝑠)−𝑑𝑠_{𝑡𝑡−𝛼ℎ}𝑥^𝑇(𝑠)𝑈₁𝑥(𝑠)−𝑑𝑠_{𝑡−𝛼ℎ𝑡−ℎ}𝑥^𝑇(𝑠)𝑈₂𝑥(𝑠)𝑑𝑠,ℒ𝑉₃𝑥_𝑡=𝑥̇,𝑖,𝑡_𝑇(𝑡)𝐸^𝑇𝛼ℎ𝑅_2𝑖+(1−𝛼)ℎ𝑅_3𝑖+12ℎ²𝑊𝐸𝑥̇(𝑡)−^𝑡_{𝑡−𝛼ℎ}𝑥̇^𝑇(𝑠)𝐸^𝑇𝑅_2𝑖𝐸𝑥̇(𝑠)−𝑑𝑠_{𝑡−𝛼ℎ𝑡−ℎ}𝑥̇^𝑇(𝑠)𝐸^𝑇𝑅_3𝑖𝐸𝑥̇(𝑠)+𝑑𝑠_{0−𝛼ℎ𝑡𝑡+𝜃}𝑥̇^𝑇(𝑠)^𝑁_𝑗=1𝛾_𝑖𝑗𝐸^𝑇𝑅_2𝑗𝐸𝑥̇(𝑠)−𝑑𝑠𝑑𝜃_{0−𝛼ℎ𝑡𝑡+𝜃}𝑥̇^𝑇(𝑠)𝐸^𝑇𝑊𝐸𝑥̇(𝑠)𝑑𝑠𝑑𝜃+^−𝛼ℎ_−ℎ^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠)^𝑁_𝑗=1𝛾_𝑖𝑗𝐸^𝑇𝑅_3𝑗𝐸𝑥̇(𝑠)−𝑑𝑠𝑑𝜃_{−𝛼ℎ−ℎ𝑡𝑡+𝜃}𝑥̇^𝑇(𝑠)𝐸^𝑇𝑥̇(𝑠)=𝑥̇𝑊𝐸𝑑𝑠𝑑𝜃_𝑇(𝑡)𝐸^𝑇𝛼ℎ𝑅_2𝑖+(1−𝛼)ℎ𝑅_3𝑖+12ℎ²𝑊𝐸𝑥̇(𝑡)−^𝑡_{𝑡−𝛼ℎ}𝑥̇^𝑇(𝑠)𝐸^𝑇𝑅_2𝑖𝐸𝑥̇(𝑠)+𝑑𝑠_{0−𝛼ℎ𝑡𝑡+𝜃}𝑥̇^𝑇(𝑠)𝐸^𝑇𝑁_𝑗=1𝛾_𝑖𝑗𝑅_2𝑗𝐸𝑥̇(𝑠)−−𝑊𝑑𝑠𝑑𝜃_{𝑡−𝛼ℎ𝑡−ℎ}𝑥̇^𝑇(𝑠)𝐸^𝑇𝑅_3𝑖𝐸𝑥̇(𝑠)+𝑑𝑠_{−𝛼ℎ−ℎ𝑡𝑡+𝜃}𝑥̇^𝑇(𝑠)𝐸^𝑇𝑁_𝑗=1𝛾_𝑖𝑗𝑅_3𝑗−𝑊𝐸𝑥̇(𝑠)𝑑s𝑑𝜃. (35)(36)(37)
Next, we will estimate the upper bound of the integral terms in ℒ𝑉₂(𝑥_𝑡,𝑖,𝑡) and ℒ𝑉₃(𝑥_𝑡,𝑖,𝑡).
Case 1 (0<ℎ(𝑡)≤𝛼ℎ). The upper bound can be evaluated for the last five integral terms in ℒ𝑉₂(𝑥_𝑡,𝑖,𝑡), which can be shown as ^𝑡_{(𝑡)𝑡−ℎ}𝑥^𝑇(𝑠)^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_1𝑗𝑥(𝑠)+𝑑𝑠_{𝑡𝑡−𝛼ℎ}𝑥^𝑇(𝑠)^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_2𝑗𝑥(𝑠)+𝑑𝑠_{𝑡−𝛼ℎ𝑡−ℎ}𝑥^𝑇(𝑠)^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_3𝑗𝑥(𝑠)−𝑑𝑠_{𝑡𝑡−𝛼ℎ}𝑥^𝑇(𝑠)𝑈₁𝑥(𝑠)𝑑𝑠−^{𝑡−𝛼ℎ}_𝑡−ℎ𝑥^𝑇(𝑠)𝑈₂𝑥(𝑠)=𝑑𝑠_{𝑡(𝑡)𝑡−ℎ}𝑥^𝑇(𝑠)^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_1𝑗+^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_2𝑗−𝑈₁𝑥(𝑠)𝑑𝑠+^{𝑡−ℎ(𝑡)}_{𝑡−𝛼ℎ}𝑥^𝑇(𝑠)^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_2𝑗−𝑈₁𝑥(𝑠)+𝑑𝑠_{𝑡−𝛼ℎ𝑡−ℎ}𝑥^𝑇(𝑠)^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_3𝑗−𝑈₂𝑥(𝑠)𝑑𝑠. (38)
Likewise, by applying Jensen inequality of Lemma 3 and reciprocally convex combination method of Lemma 5, the upper bound for the two definite integrals in ℒ𝑉₃(𝑥_𝑡,𝑖,𝑡) can be shown as −^𝑡_{𝑡−𝛼ℎ}𝑥̇^𝑇(𝑠)𝐸^𝑇𝑅_2𝑖𝐸𝑥̇(𝑠)𝑑𝑠−^{𝑡−𝛼ℎ}_𝑡−ℎ𝑥̇^𝑇(𝑠)𝐸^𝑇𝑅_3𝑖𝐸𝑥̇(𝑠)𝑑𝑠=−^𝑡_{(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝑠)𝐸^𝑇𝑅_2𝑖𝐸𝑥̇(𝑠)−𝑑𝑠_{𝑡−ℎ(𝑡)𝑡−𝛼ℎ}𝑥̇^𝑇(𝑠)𝐸^𝑇𝑅_2𝑖𝐸𝑥̇(𝑠)−𝑑𝑠_{𝑡−𝛼ℎ𝑡−ℎ}𝑥̇^𝑇(𝑠)𝐸^𝑇𝑅_3𝑖𝐸𝑥̇(𝑠)1𝑑𝑠≤−ℎ(𝑡)(𝑥(𝑡)−𝑥(𝑡−ℎ(𝑡)))^𝑇×𝐸^𝑇𝑅_2𝑖𝐸(𝑥(𝑡)((𝑡−1−𝑥𝑡−ℎ)))(𝑡)(𝑥((𝑡(𝛼ℎ−ℎ𝑡−ℎ))−𝑥𝑡−𝛼ℎ))^𝑇×𝐸^𝑇𝑅_2𝑖𝐸(𝑥((𝑡(−1𝑡−ℎ))−𝑥𝑡−𝛼ℎ))(𝑥()(ℎ−𝛼ℎ𝑡−𝛼ℎ−𝑥𝑡−ℎ))^𝑇×𝐸^𝑇𝑅_3𝑖𝐸(𝑥()(1𝑡−𝛼ℎ−𝑥𝑡−ℎ))≤−𝑥(𝑡)((𝑡𝑥((𝑡()𝛼ℎ−𝑥𝑡−ℎ))𝑡−ℎ))−𝑥𝑡−𝛼ℎ^𝑇×𝐸^𝑇𝑅_2𝑖𝐸𝐸^𝑇𝑆_𝑖𝐸𝐸^𝑇𝑆^𝑇_𝑖𝐸𝐸^𝑇𝑅_2𝑖𝐸𝑥(𝑡)((𝑡𝑥((𝑡()−1−𝑥𝑡−ℎ))𝑡−ℎ))−𝑥𝑡−𝛼ℎ(𝑥()(ℎ−𝛼ℎ𝑡−𝛼ℎ−𝑥𝑡−ℎ))^𝑇×𝐸^𝑇𝑅_3𝑖𝐸(𝑥()(𝑥(𝑡)((𝑡𝑥((𝑡()𝑥()()𝑡−𝛼ℎ−𝑥𝑡−ℎ))=−−𝑥𝑡−ℎ))𝑡−ℎ))−𝑥𝑡−𝛼ℎ𝑡−𝛼ℎ−𝑥𝑡−ℎ^𝑇×𝛼₁𝐸^𝑇𝑍_1𝑖𝐸00𝛼₂𝐸^𝑇𝑅_3𝑖𝐸𝑥(𝑡)((𝑡𝑥((𝑡()𝑥()().−𝑥𝑡−ℎ))𝑡−ℎ))−𝑥𝑡−𝛼ℎ𝑡−𝛼ℎ−𝑥𝑡−ℎ (39)
Combining (35)–(39), it yields that ℒ𝑉𝑥_𝑡,𝑖,𝑡≤𝜂^𝑇(𝑡)Ψ_1𝑖𝜂(𝑡)+^𝑡_{𝑡−ℎ(𝑡)}𝑥^𝑇(𝑠)^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_1𝑗+^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_2𝑗−𝑈₁𝑥(𝑠)+𝑑𝑠_{0−𝛼ℎ𝑡𝑡+𝜃}𝑥̇^𝑇(𝑠)𝐸^𝑇𝑁_𝑗=1𝛾_𝑖𝑗𝑅_2𝑗𝐸𝑥̇(𝑠)+−𝑊𝑑𝑠𝑑𝜃_{𝑡−ℎ(𝑡)𝑡−𝛼ℎ}𝑥^𝑇(𝑠)^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_2𝑗−𝑈₁𝑥(𝑠)+𝑑𝑠_{−𝛼ℎ−ℎ𝑡𝑡+𝜃}𝑥̇^𝑇(𝑠)𝐸^𝑇𝑁_𝑗=1𝛾_𝑖𝑗𝑅_3𝑗−𝑊𝐸𝑥̇(𝑠)𝑑𝑠𝑑𝜃+^{𝑡−𝛼ℎ}_𝑡−ℎ𝑥^𝑇(𝑠)^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_3𝑗−𝑈₂𝑥(𝑠)𝑑𝑠, (40) where 𝜂^𝑇(𝑡)=[𝑥^𝑇(𝑡)𝑥^𝑇(𝑡−ℎ(𝑡))𝑥^𝑇(𝑡−𝛼ℎ)𝑥^𝑇(𝑡−ℎ)], and Ψ_1𝑖=Ψ_𝑖−𝐼−𝐼000𝐼−𝐼000𝐼−𝐼^𝑇𝛼₁𝐸^𝑇𝑍_1𝑖𝐸00𝛼₂𝐸^𝑇𝑅_3𝑖𝐸×,𝐼−𝐼000𝐼−𝐼000𝐼−𝐼 (41) where 𝐸=diag{𝐸,𝐸} and Ψ_𝑖=Ψ_𝑖11Ψ_𝑖1200∗Ψ_𝑖2200∗∗Ψ_𝑖330∗∗∗Ψ_𝑖44,Ψ_𝑖11=𝐴^𝑇_𝑖𝑃_𝑖𝐸+𝑈^𝑇Φ_𝑖𝑉^𝑇+𝑃_𝑖𝐸+𝑈^𝑇Φ_𝑖𝑉^𝑇𝑇𝐴_𝑖+^𝑁_𝑗=1𝛾_𝑖𝑗𝐸^𝑇𝑃_𝑗𝐸+𝑄_1𝑖+𝑄_2𝑖+𝛼ℎ𝑈₁+(1−𝛼)ℎ𝑈₂+𝐴^𝑇_𝑖𝛼ℎ𝑅_2𝑖+(1−𝛼)ℎ𝑅_3𝑖+12ℎ²𝑊𝐴_𝑖,Ψ_𝑖12=𝑃_𝑖𝐸+𝑈^𝑇Φ_𝑖𝑉^𝑇𝑇𝐴_𝑑𝑖+𝐴^𝑇_𝑖𝛼ℎ𝑅_2𝑖+(1−𝛼)ℎ𝑅_3𝑖+12ℎ²𝑊𝐴_𝑑𝑖,Ψ_𝑖22=−1−ℎ_𝑑𝑄_1𝑖+𝐴^𝑇_𝑑𝑖𝛼ℎ𝑅_2𝑖+(1−𝛼)ℎ𝑅_3𝑖+12ℎ²𝑊𝐴_𝑑𝑖,Ψ_𝑖33=𝑄_3𝑖−𝑄_2𝑖,Ψ_𝑖44=−𝑄_3𝑖. (42)
From the Schur complement lemma of Lemma 5, it can be easily verified that Ψ_1𝑖<0 holds if (22) holds for 0<ℎ(𝑡)≤𝛼ℎ.
Case 2 (𝛼ℎ<ℎ(𝑡)≤ℎ). The upper bound can be evaluated for the last five integral terms in ℒ𝑉₂(𝑥_𝑡,𝑖,𝑡), which can be shown as ^𝑡_{𝑡−ℎ(𝑡)}𝑥^𝑇(𝑠)^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_1𝑗𝑥(𝑠)+𝑑𝑠_{𝑡𝑡−𝛼ℎ}𝑥^𝑇(𝑠)^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_2𝑗𝑥(𝑠)+𝑑𝑠_{𝑡−𝛼ℎ𝑡−ℎ}𝑥^𝑇(𝑠)^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_3𝑗𝑥(𝑠)−𝑑𝑠_{𝑡𝑡−𝛼ℎ}𝑥^𝑇(𝑠)𝑈₁𝑥(𝑠)𝑑𝑠−^{𝑡−𝛼ℎ}_𝑡−ℎ𝑥^𝑇(𝑠)𝑈₂𝑥(𝑠)=𝑑𝑠_{𝑡𝑡−𝛼ℎ}𝑥^𝑇(𝑠)^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_1𝑗+^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_2𝑗−𝑈₁𝑥(𝑠)𝑑𝑠+^{𝑡−ℎ(𝑡)}_𝑡−ℎ𝑥^𝑇(𝑠)^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_3𝑗−𝑈₂𝑥(𝑠)+𝑑𝑠_{(𝑡)𝑡−𝛼ℎ𝑡−ℎ}𝑥^𝑇(𝑠)^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_1𝑗+^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_3𝑗−𝑈₂𝑥(𝑠)𝑑𝑠. (43)
Likewise, by applying Jensen inequality of Lemma 3 and reciprocally convex combination method of Lemma 5, the upper bound for the two definite integrals in ℒ𝑉₃(𝑥_𝑡,𝑖,𝑡) can be shown as −^𝑡_{𝑡−𝛼ℎ}𝑥̇^𝑇(𝑠)𝐸^𝑇𝑅_2𝑖𝐸𝑥̇(𝑠)𝑑𝑠−^{𝑡−𝛼ℎ}_𝑡−ℎ𝑥̇^𝑇(𝑠)𝐸^𝑇𝑅_3𝑖𝐸𝑥̇(𝑠)𝑑𝑠=−^𝑡_{𝑡−𝛼ℎ}𝑥̇^𝑇(𝑠)𝐸^𝑇𝑅_2𝑖𝐸𝑥̇(𝑠)−𝑑𝑠_{(𝑡)𝑡−𝛼ℎ𝑡−ℎ}𝑥̇^𝑇(𝑠)𝐸^𝑇𝑅_3𝑖𝐸𝑥̇(𝑠)−𝑑𝑠_{𝑡−ℎ(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝑠)𝐸^𝑇𝑅_3𝑖𝐸𝑥̇(𝑠)1𝑑𝑠≤−(𝑥(𝑡)(𝛼ℎ−𝑥𝑡−𝛼ℎ))^𝑇×𝐸^𝑇𝑅_2𝑖𝐸(𝑥(𝑡)(−1−𝑥𝑡−𝛼ℎ))ℎ(𝑡)(𝑥()((𝑡−𝛼ℎ𝑡−𝛼ℎ−𝑥𝑡−ℎ)))^𝑇×𝐸^𝑇𝑅_3𝑖𝐸(𝑥()((𝑡−1𝑡−𝛼ℎ−𝑥𝑡−ℎ)))(𝑡)(𝑥((𝑡(ℎ−ℎ𝑡−ℎ))−𝑥𝑡−ℎ))^𝑇×𝐸^𝑇𝑅_3𝑖𝐸(𝑥((𝑡(1𝑡−ℎ))−𝑥𝑡−ℎ))≤−(𝑥(𝑡)(𝛼ℎ−𝑥𝑡−𝛼ℎ))^𝑇×𝐸^𝑇𝑅_2𝑖𝐸(𝑥(𝑡)(−1−𝑥𝑡−𝛼ℎ))𝑥()((𝑡𝑥((𝑡()ℎ−𝛼ℎ𝑡−𝛼ℎ−𝑥𝑡−ℎ))𝑡−ℎ))−𝑥𝑡−ℎ^𝑇×𝐸^𝑇𝑅_3𝑖𝐸𝐸^𝑇𝑇_𝑖𝐸𝐸^𝑇𝑇^𝑇_𝑖𝐸𝐸^𝑇𝑅_3𝑖𝐸𝑥()((𝑡𝑥((𝑡()𝑥(𝑡)()𝑥()((𝑡𝑥((𝑡()𝑡−𝛼ℎ−𝑥𝑡−ℎ))𝑡−ℎ))−𝑥𝑡−ℎ=−−𝑥𝑡−𝛼ℎ𝑡−𝛼ℎ−𝑥𝑡−ℎ))𝑡−ℎ))−𝑥𝑡−ℎ^𝑇𝛼₁𝐸^𝑇𝑅_2𝑖𝐸00𝛼₂𝐸^𝑇𝑍_2𝑖𝐸×𝑥(𝑡)()𝑥()((𝑡𝑥((𝑡().−𝑥𝑡−𝛼ℎ𝑡−𝛼ℎ−𝑥𝑡−ℎ))𝑡−ℎ))−𝑥𝑡−ℎ (44)
Combining (35)–(37) and (43)-(44), it yields that ℒ𝑉𝑥_𝑡,𝑖,𝑡≤𝜂^𝑇(𝑡)Ψ_2𝑖𝜂(𝑡)+^𝑡_{𝑡−𝛼ℎ}𝑥^𝑇(𝑠)^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_1𝑗+^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_2𝑗−𝑈₁𝑥(𝑠)+𝑑𝑠_{(𝑡)𝑡−𝛼ℎ𝑡−ℎ}𝑥^𝑇(𝑠)^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_1𝑗+^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_3𝑗−𝑈₂𝑥(𝑠)+𝑑𝑠_{𝑡−ℎ(𝑡)𝑡−ℎ}𝑥^𝑇(𝑠)^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_3𝑗−𝑈₂𝑥(𝑠)+𝑑𝑠_{0−𝛼ℎ𝑡𝑡+𝜃}𝑥̇^𝑇(𝑠)𝐸^𝑇𝑁_𝑗=1𝛾_𝑖𝑗𝑅_2𝑗−𝑊𝐸𝑥̇(𝑠)𝑑𝑠𝑑𝜃+^−𝛼ℎ_−ℎ^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠)𝐸^𝑇𝑁_𝑗=1𝛾_𝑖𝑗𝑅_3𝑗−𝑊𝐸𝑥̇(𝑠)𝑑𝑠𝑑𝜃, (45) where 𝜂^𝑇(𝑡)=[𝑥^𝑇(𝑡)𝑥^𝑇(𝑡−ℎ(𝑡))𝑥^𝑇(𝑡−𝛼ℎ)𝑥^𝑇(𝑡−ℎ)], and Ψ_2𝑖=Ψ_𝑖−𝐼0−𝐼00−𝐼𝐼00𝐼0−𝐼^𝑇𝛼₁𝐸^𝑇𝑅_2𝑖𝐸00𝛼₂𝐸^𝑇𝑍_2𝑖𝐸×.𝐼0−𝐼00−𝐼𝐼00𝐼0−𝐼 (46)
From the Schur complement lemma of Lemma 5, it can be easily verified that Ψ_2𝑖<0 holds if (23) holds for 𝛼ℎ<ℎ(𝑡)≤ℎ.
Based on the above analysis, we can know that there exists a scalar 𝜆>0 such that for each 𝑖∈ℓℒ𝑉𝑥_𝑡,𝑖,𝑡≤−𝜆‖𝑥(𝑡)‖². (47) Then, by Dynkin’s formula, we can have for each 𝑖∈ℓ and 𝑡≥−ℎℰ𝑉𝑥_𝑡,𝑖,𝑡≤ℰ𝑉𝑥₀,𝑖,0−𝜆ℰ‖𝑥(𝑡)‖², (48) which implies that ℰ‖𝑥(𝑡)‖²≤𝜆⁻¹ℰ𝑉𝑥₀,𝑖,0. (49)
Following a similar line in [12], we can find that the system (4) is stochastically stable under the condition that 𝑤(𝑡)=0. By Definition 1, the system (4) is stochastically admissible.
In the following, we will establish the 𝐻_∞ performance index of the system (4). Using the stochastic Lyapunov functional (33), under the zero initial condition, it is from (22)–(27) that 𝐽_𝑧𝑤(𝑡)=ℰ^𝑡₀𝑧^𝑇(𝑠)𝑧(𝑠)−𝛾²𝑤^𝑇(𝑠)𝑤(𝑠)𝑑𝑠=ℰ^𝑡₀𝑧^𝑇(𝑠)𝑧(𝑠)−𝛾²𝑤^𝑇(𝑠)𝑤(𝑠)+ℒ𝑉𝑥_𝑠𝑥,𝑖,𝑠𝑑𝑠−ℰ𝑉_𝑡,𝑖,𝑡≤ℰ^𝑡₀𝑧^𝑇(𝑠)𝑧(𝑠)−𝛾²𝑤^𝑇(𝑠)𝑤(𝑠)+ℒ𝑉𝑥_𝑠,𝑖,𝑠𝑑𝑠≤ℰ^𝑡₀𝜂^𝑇(𝑠)Θ_𝑖𝜂(𝑠)𝑑𝑠, (50) where 𝜂^𝑇(𝑡)=[𝜂^𝑇(𝑡)𝑤^𝑇(𝑡)]; for ℎ(𝑡)∈(0,𝛼ℎ], Θ_𝑖=Θ_𝑖11+𝐶^𝑇_𝑖𝐶_𝑖Θ_𝑖12+𝐶^𝑇_𝑖𝐶_𝑑𝑖Θ_𝑖130Θ_𝑖15+𝐶^𝑇_𝑖𝐷_𝑤𝑖∗Θ_𝑖22+𝐶^T_𝑑𝑖𝐶_𝑑𝑖Θ_𝑖230𝐶^𝑇_𝑑𝑖𝐷_𝑤𝑖∗∗Θ_𝑖33Θ_𝑖340∗∗∗Θ_𝑖440∗∗∗∗𝐷^𝑇_𝑤𝑖𝐷_𝑤𝑖−𝛾²𝐼+𝐴^𝑇_𝑖𝐴^𝑇_𝑑𝑖00𝐵^𝑇_𝑤𝑖𝛼ℎ𝑅_2𝑖+(1−𝛼)ℎ𝑅_3𝑖+12ℎ²𝑊𝐴^𝑇_𝑖𝐴^𝑇_𝑑𝑖00𝐵^𝑇_𝑤i^𝑇, (51) and for ℎ(𝑡)∈(𝛼ℎ,ℎ], Θ_𝑖=Θ_𝑖11+𝐶^𝑇_𝑖𝐶_𝑖Θ_𝑖12+𝐶^𝑇_𝑖𝐶_𝑑𝑖Θ_𝑖130Θ_𝑖15+𝐶^𝑇_𝑖𝐷_𝑤𝑖∗Θ_𝑖22+𝐶^𝑇_𝑑𝑖𝐶_𝑑𝑖Θ_𝑖23Θ_𝑖24𝐶^𝑇_𝑑𝑖𝐷_𝑤𝑖∗∗Θ_𝑖33Θ_𝑖340∗∗∗Θ_𝑖440∗∗∗∗𝐷^𝑇_𝑤𝑖𝐷_𝑤𝑖−𝛾²𝐼+𝐴^𝑇_𝑖𝐴^𝑇_𝑑𝑖00𝐵^𝑇_𝑤𝑖𝛼ℎ𝑅_2𝑖+(1−𝛼)ℎ𝑅_3𝑖+12ℎ²𝑊𝐴^𝑇_𝑖𝐴^𝑇_𝑑𝑖00𝐵^𝑇_𝑤𝑖^𝑇. (52)
By Schur complement lemma of Lemma 6, together with (22)–(27), we have that 𝐽_𝑧𝑤(𝑡)<0 for all 𝑡>0. Consequently, under the zero initial condition, we have that (6) holds for any nonzero 𝑤(𝑡)∈ℒ₂[0,∞). Therefore, by Definition 2, the descriptor Markovian jump time-delay system (4) is stochastically admissible with 𝐻_∞ performance level 𝛾 for any time delay ℎ(𝑡) satisfying (3). This completes the proof.

Remark 10. By virtue of the novel Lyapunov-Krasovskii functional, Theorem 9 provides a delay-dependent and delay-derivative-dependent 𝐻_∞ criterion to guarantee the system (4) to be stochastically admissible with 𝐻_∞ performance level. The novelty of Theorem 9 lies in the new Lyapunov-Krasovskii functional and the reciprocally convex combination method. By dividing the delay interval (0,ℎ] into two subintervals (0,𝛼ℎ] and (𝛼ℎ,ℎ] the following mode-dependent definite integrals ^𝑡_{𝑡−𝛼ℎ}𝑥^𝑇(𝑠)𝑄₂𝑟_𝑡𝑥(𝑠)𝑑𝑠+^{𝑡−𝛼ℎ}_𝑡−ℎ𝑥^𝑇(𝑠)𝑄₃𝑟_𝑡𝑥(𝑠)𝑑𝑠 (53) and mode-dependent double integrals 𝑥̇⁰_−𝛼ℎ^𝑡_𝑡+𝜃^𝑇(𝑠)𝐸^𝑇𝑅₂𝑟_𝑡𝐸𝑥̇(𝑠)+𝑑𝑠𝑑𝜃_{−𝛼ℎ−ℎ𝑡𝑡+𝜃}𝑥̇^𝑇(𝑠)𝐸^𝑇𝑅₃𝑟_𝑡𝐸𝑥̇(𝑠)𝑑𝑠𝑑𝜃 (54) are introduced. Obviously, the above integral terms take the mode 𝑟_𝑡 into full consideration and contain the mode-independent one [12] as a special case. Since ℎ(𝑡) is restricted to the subinterval (0,𝛼ℎ] or (𝛼ℎ,ℎ] and various Lyapunov-Krasovskii functionals are selected for different subsystems, our method can lead to reduction of conservatism.

For the unforced system (4), if disturbance signal 𝑤(𝑡)≡0, then we have the following time-delay systems: 𝐸𝑥̇(𝑡)=𝐴𝑟_𝑡𝑥(𝑡)+𝐴_𝑑𝑟_𝑡𝑥((𝑡𝑥(𝜃)(𝜃)[].𝑡−ℎ))=𝜙,∀𝜃∈−ℎ,0 (55)

Based on Theorem 9, the following stochastic admissibility result can be given.

Corollary 11. For given scalars ℎ>0, 0≤ℎ_𝑑<1, the system (55) is stochastically admissible if there exist a set of positive definite matrices 𝑃_𝑖, 𝑄_1𝑖, 𝑄_2𝑖, 𝑄_3𝑖, 𝑅_2𝑖, 𝑅_3𝑖, 𝑈₁, 𝑈₂, 𝑊 and matrices Φ_𝑖, 𝑆_𝑖, 𝑇_𝑖, ∀𝑖∈ℓ, satisfying the following LMIs for each 𝑖∈ℓ: Θ_1𝑖=Θ_𝑖−𝐼−𝐼0000𝐼−𝐼0000𝐼−𝐼00000𝐼^𝑇×𝛼₁𝐸^𝑇𝑍_1𝑖𝐸000𝛼₂𝐸^𝑇𝑅_3𝑖Θ𝐸0000𝐼−𝐼0000𝐼−𝐼0000𝐼−𝐼00000𝐼<0,_2𝑖=Θ_𝑖−𝐼0−𝐼000−𝐼𝐼000𝐼0−𝐼00000𝐼^𝑇×𝛼₁𝐸^𝑇𝑅_2𝑖𝐸000𝛼₂𝐸^𝑇𝑍_2𝑖𝐸0000𝐼0−𝐼000−𝐼𝐼000𝐼0−𝐼00000𝐼<0,_𝑁𝑗=1𝛾_𝑖𝑗𝑅_2𝑗−𝑊<0,^𝑁_𝑗=1𝛾_𝑖𝑗𝑅_3𝑗−𝑊<0,_𝑁𝑗=1𝛾_𝑖𝑗𝑄_1𝑗+^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_2𝑗−𝑈₁<0,^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_2𝑗−𝑈₁<0,_𝑁𝑗=1𝛾_𝑖𝑗𝑄_1𝑗+^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_3𝑗−𝑈₂<0,^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_3𝑗−𝑈₂<0,𝑍_1𝑖=𝑅_2𝑖𝑆_𝑖𝑆^𝑇_𝑖𝑅_2𝑖>0,𝑍_2𝑖=𝑅_3𝑖𝑇_𝑖𝑇^𝑇_𝑖𝑅_3𝑖>0, (56)(57) where 𝛼₁=1/𝛼ℎ, 𝛼₂=1/(ℎ−𝛼ℎ), 𝐸=diag{𝐸,𝐸}, and Θ_𝑖=Θ_𝑖11Θ_𝑖1200Θ_𝑖15∗Θ_𝑖2200Θ_𝑖25∗∗Θ_𝑖3300∗∗∗Θ_𝑖440∗∗∗∗Θ_𝑖55,Θ_𝑖11=𝐴^𝑇_𝑖𝑃_𝑖𝐸+𝑈^𝑇Φ_𝑖𝑉^𝑇+𝑃_𝑖𝐸+𝑈^𝑇Φ_𝑖𝑉^𝑇𝑇𝐴_𝑖+^𝑁_𝑗=1𝛾_𝑖𝑗𝐸^𝑇𝑃_𝑗𝐸+𝑄_1𝑖+𝑄_2𝑖+𝛼ℎ𝑈₁+(1−𝛼)ℎ𝑈₂,Θ_𝑖12=𝑃_𝑖𝐸+𝑈^𝑇Φ_𝑖𝑉^𝑇𝑇𝐴_𝑑𝑖,Θ_𝑖15=𝐴^𝑇_𝑖𝛼ℎ𝑅_2𝑖+(1−𝛼)ℎ𝑅_3𝑖+12ℎ²𝑊,Θ_𝑖22=−1−ℎ_𝑑𝑄_1𝑖,Θ_𝑖25=𝐴^𝑇_𝑑𝑖𝛼ℎ𝑅_2𝑖+(1−𝛼)ℎ𝑅_3𝑖+12ℎ²𝑊,Θ_𝑖33=𝑄_3𝑖−𝑄_2𝑖,Θ_𝑖44=−𝑄_3𝑖,Θ_𝑖55=−𝛼ℎ𝑅_2𝑖+(1−𝛼)ℎ𝑅_3𝑖+12ℎ²𝑊. (58)

When the time delay is time-invariant, the system (55) reduces to the one 𝐸𝑥̇(𝑡)=𝐴𝑟_𝑡𝑥(𝑡)+𝐴_𝑑𝑟_𝑡𝑥()𝑥(𝜃)(𝜃)[].𝑡−ℎ=𝜙,∀𝜃∈−ℎ,0 (59) Using the similar method of Corollary 11, we have the stochastic admissibility condition for (59).

Corollary 12. For given scalars ℎ>0, the system (59) is stochastically admissible if there exist a set of positive definite matrices 𝑃_𝑖, 𝑄_2𝑖, 𝑄_3𝑖, 𝑅_2𝑖, 𝑅_3𝑖, 𝑈₁, 𝑈₂, 𝑊 and matrices Φ_𝑖, ∀𝑖∈ℓ, satisfying the following LMIs for each 𝑖∈ℓ: Θ_𝑖=Θ_𝑖−𝐼−𝐼000𝐼−𝐼0000𝐼^𝑇𝛼₁𝐸^𝑇𝑅_2𝑖𝐸000𝛼₂𝐸^𝑇𝑅_3𝑖×𝐸0000𝐼−𝐼000𝐼−𝐼0000𝐼<0,_𝑁𝑗=1𝛾_𝑖𝑗𝑅_2𝑗−𝑊<0,^𝑁_𝑗=1𝛾_𝑖𝑗𝑅_3𝑗−𝑊<0,_𝑁𝑗=1𝛾_𝑖𝑗𝑄_2𝑗−𝑈₁<0,^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_3𝑗−𝑈₂<0, (60) where 𝛼₁=1/𝛼ℎ, 𝛼₂=1/(ℎ−𝛼ℎ); and Θ_𝑖=Θ_𝑖110Θ_𝑖13Θ_𝑖14∗Θ_𝑖2200∗∗Θ_𝑖33Θ_𝑖34∗∗∗Θ_𝑖44,Θ_𝑖11=𝐴^𝑇_𝑖𝑃_𝑖𝐸+𝑈^𝑇Φ_𝑖𝑉^𝑇+𝑃_𝑖𝐸+𝑈^𝑇Φ_𝑖𝑉^𝑇𝑇𝐴_𝑖+^𝑁_𝑗=1𝛾_𝑖𝑗𝐸^𝑇𝑃_𝑗𝐸+𝑄_2𝑖+𝛼ℎ𝑈₁+(1−𝛼)ℎ𝑈₂,Θ_𝑖13=𝑃_𝑖𝐸+𝑈^𝑇Φ_𝑖𝑉^𝑇𝑇𝐴_𝑑𝑖,Θ_𝑖14=𝐴^𝑇_𝑖𝛼ℎ𝑅_2𝑖+(1−𝛼)ℎ𝑅_3𝑖+12ℎ²𝑊,Θ_𝑖22=𝑄_3𝑖−𝑄_2𝑖,Θ_𝑖33=−𝑄_3𝑖,Θ_𝑖34=𝐴^𝑇_𝑑𝑖𝛼ℎ𝑅_2𝑖+(1−𝛼)ℎ𝑅_3𝑖+12ℎ²𝑊,Θ_𝑖44=−𝛼ℎ𝑅_2𝑖+(1−𝛼)ℎ𝑅_3𝑖+12ℎ²𝑊. (61)

Remark 13. By decomposing the delay interval (0,ℎ] into two segments (0,𝛼ℎ] and (𝛼ℎ,ℎ], Theorem 9, Corollary 11, and Corollary 12 give delay-dependent conditions for the underlying systems with time-varying and time-invariant delay, respectively. The algorithm about how to select tuning parameter 𝛼 optimally can be referred to the algorithm in the end of this section.

Now, we are ready to deal with the delay-dependent 𝐻_∞ control problem for the descriptor Markovian jump systems (1). For this purpose, applying the controller in (7) to the system (1), then we can have the following closed-loop systems: 𝐸𝑥̇(𝑡)=𝐴𝑟_𝑡+𝐵𝑟_𝑡𝐾𝑟_𝑡𝑥(𝑡)+𝐴_𝑑𝑟_𝑡+𝐵𝑟_𝑡𝐾_𝑑𝑟_𝑡𝑥(𝑡−ℎ(𝑡))+𝐵_𝑤𝑟_𝑡𝑤(𝑡),𝑧(𝑡)=𝐶𝑟_𝑡+𝐷𝑟_𝑡𝐾𝑟_𝑡𝑥(𝑡)+𝐶_𝑑𝑟_𝑡+𝐷𝑟_𝑡𝐾_𝑑𝑟_𝑡𝑥(𝑡−ℎ(𝑡))+𝐷_𝑤𝑟_𝑡𝑤(𝑡),𝑥(𝜃)=𝜙(𝜃),∀𝜃∈[−ℎ,0]. (62)

On the basis of Theorem 9, the following result, which provides the design method of state feedback controller and ensures the closed-loop systems (62) to be stochastically admissible with 𝐻_∞ performance level 𝛾, can be obtained.

Theorem 14. For given scalars 𝛾>0, 𝜀_𝑖>0, ℎ>0, and 0≤ℎ_𝑑<1, the descriptor Markovian jump time-delay system (1) is stochastically admissible with 𝐻_∞ performance level 𝛾 for any ℎ(𝑡) satisfying (3), if there exist a set of positive definite matrices 𝑃_𝑖, 𝑄_1𝑖𝑗, 𝑄_2𝑖𝑗, 𝑄_3𝑖𝑗, 𝑅_2𝑖, 𝑅_3𝑖, 𝑈_1𝑖, 𝑈_2𝑖, 𝑊 and matrices Φ_𝑖, 𝐻_𝑖, 𝐿_𝑖, ∀𝑖∈ℓ,𝑗∈ℓ, satisfying the following LMIs for each 𝑖∈ℓ:Υ_𝑖11Ῠ_𝑖1200𝐵_𝑤𝑖Υ_𝑖1612ℎ²𝐴_𝑖Λ_𝑖+𝐵_𝑖𝐻_𝑖^𝑇𝐶_𝑖Λ_𝑖+𝐷_𝑖𝐻_𝑖^𝑇Υ_𝑖19∗Ῠ_𝑖22Ῠ_𝑖2300Υ_𝑖2612ℎ²𝐴_𝑑𝑖Λ_𝑖+𝐵_𝑖𝐿_𝑖^𝑇𝐶_𝑑𝑖Λ_𝑖+𝐷_𝑖𝐿_𝑖^𝑇0∗∗Υ_𝑖33Ῠ_𝑖3400000∗∗∗Υ_𝑖4400000∗∗∗∗−𝛾²𝐼Υ_𝑖5612ℎ²𝐵^𝑇_𝑤𝑖𝐷^𝑇_𝑤𝑖0∗∗∗∗∗Υ_𝑖661000∗∗∗∗∗∗−2ℎ²𝑊00∗∗∗∗∗∗∗−𝐼0∗∗∗∗∗∗∗∗Υ_𝑖99<0,ℎ(𝑡)∈(0,𝛼ℎ],Υ_𝑖11Υ_𝑖12Υ_𝑖130𝐵_𝑤𝑖Υ_𝑖1612ℎ²𝐴_𝑖Λ_𝑖+𝐵_𝑖𝐻_𝑖^𝑇𝐶_𝑖Λ_𝑖+𝐷_𝑖𝐻_𝑖^𝑇Υ_𝑖19∗Υ_𝑖22Υ_𝑖23Υ_𝑖240Υ_𝑖2612ℎ²𝐴_𝑑𝑖Λ_𝑖+𝐵_𝑖𝐿_𝑖^𝑇𝐶_𝑑𝑖Λ_𝑖+𝐷_𝑖𝐿_𝑖^𝑇0∗∗Υ_𝑖33000000∗∗∗Υ_𝑖4400000∗∗∗∗−𝛾²𝐼Υ_𝑖5612ℎ²𝐵^𝑇_𝑤𝑖𝐷^𝑇_𝑤𝑖0∗∗∗∗∗Υ_𝑖661000∗∗∗∗∗∗−2ℎ²𝑊00∗∗∗∗∗∗∗−𝐼0∗∗∗∗∗∗∗∗Υ_𝑖99<0,ℎ(𝑡)∈(𝛼ℎ,ℎ],𝛾_𝑖𝑗(𝑁−1)𝑊<𝑅_2𝑗,𝛾_𝑖𝑗(𝑁−1)𝑊<𝑅_3𝑗,∀𝑗≠𝑖,𝛾_𝑖𝑗≠0,_𝑁𝑗=1𝛾_𝑖𝑗𝑄_1𝑖𝑗+^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_2𝑖𝑗−𝑈_1𝑖<0,^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_2𝑖𝑗−𝑈_1𝑖<0,_𝑁𝑗=1𝛾_𝑖𝑗𝑄_1𝑖𝑗+^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_3𝑖𝑗−𝑈_2𝑖<0,^𝑁_𝑗=1𝛾_𝑖𝑗𝑄_3𝑖𝑗−𝑈_2𝑖<0, (63)(64)(65)(66)(67)where 𝛼₁=1/𝛼ℎ, 𝛼₂=1/(ℎ−𝛼ℎ), Λ_𝑖=𝑃_𝑖𝐸^T+𝑉Φ_𝑖𝑈, and Υ_𝑖11=Λ^𝑇_𝑖𝐴^𝑇_𝑖+𝐻^𝑇_𝑖𝐵^𝑇_𝑖+𝐴_𝑖Λ_𝑖+𝐵_𝑖𝐻_𝑖+𝛾_𝑖𝑖𝜖_𝑖Λ^𝑇_𝑖𝐸^𝑇+𝜖_𝑖𝐸Λ_𝑖−𝜖²_𝑖𝐸𝑃_𝑖𝐸^𝑇+𝑄_1𝑖𝑖+𝑄_2𝑖𝑖+𝛼ℎ𝑈_1𝑖+(1−𝛼)ℎ𝑈_2𝑖−Λ^𝑇_𝑖𝐸^𝑇−𝐸Λ_𝑖+𝛼ℎ𝑅_2𝑖,Ῠ_𝑖12=𝐴_𝑑𝑖Λ_𝑖+𝐵_𝑖𝐿_𝑖+Λ^𝑇_𝑖𝐸^𝑇+𝐸Λ_𝑖−𝛼ℎ𝑅_2𝑖,Υ_𝑖16=𝐴_𝑖Λ_𝑖+𝐵_𝑖𝐻_𝑖^𝑇𝐴_𝑖Λ_𝑖+𝐵_𝑖𝐻_𝑖^𝑇,Υ_𝑖19=√𝛾_𝑖1Λ^𝑇_𝑖𝐸_𝑅⋯𝛾_{𝑖(𝑖−1)}Λ^𝑇_𝑖𝐸_𝑅𝛾_𝑖(𝑖+1)Λ^𝑇_𝑖𝐸_𝑅⋯√𝛾_𝑖𝑁Λ^𝑇_𝑖𝐸_𝑅,Ῠ_𝑖22=−1−ℎ_𝑑𝑄_1𝑖𝑖−2Λ^𝑇_𝑖𝐸^𝑇−2𝐸Λ_𝑖+2𝛼ℎ𝑅_2𝑖,Ῠ_𝑖23=Λ^𝑇_𝑖𝐸^𝑇+𝐸Λ_𝑖−𝛼ℎ𝑅_2𝑖,Υ_𝑖26=𝐴_𝑑𝑖Λ_𝑖+𝐵_𝑖𝐿_𝑖^𝑇𝐴_𝑑𝑖Λ_𝑖+𝐵_𝑖𝐿_𝑖^𝑇,Υ_𝑖33=𝑄_3𝑖𝑖−𝑄_2𝑖𝑖−2Λ^𝑇_𝑖𝐸^𝑇−2𝐸Λ_𝑖+𝛼ℎ𝑅_2𝑖+(ℎ−𝛼ℎ)𝑅_3𝑖,Ῠ_𝑖34=Λ^𝑇_𝑖𝐸^𝑇+𝐸Λ_𝑖−(ℎ−𝛼ℎ)𝑅_3𝑖,Υ_𝑖44=−𝑄_3𝑖𝑖−Λ^𝑇_𝑖𝐸^𝑇−𝐸Λ_𝑖+(ℎ−𝛼ℎ)𝑅_3𝑖,Υ_𝑖56=𝐵^𝑇_𝑤𝑖𝐵^𝑇_𝑤𝑖,Υ_𝑖66=diag−𝛼₁𝑅_2𝑖,−𝛼₂𝑅_3𝑖,Υ_𝑖99=−diag𝐸^𝑇_𝑅𝑃₁𝐸_𝑅,…,𝐸^𝑇_𝑅𝑃_𝑖−1𝐸_𝑅,𝐸^𝑇_𝑅𝑃_𝑖+1𝐸_𝑅,…,𝐸^𝑇_𝑅𝑃_𝑁𝐸_𝑅,Υ_𝑖12=𝐴_𝑑𝑖Λ_𝑖+𝐵_𝑖𝐿_𝑖,Υ_𝑖13=Λ^𝑇_𝑖𝐸^𝑇+𝐸Λ_𝑖−𝛼ℎ𝑅_2𝑖,Υ_𝑖22=−1−ℎ_𝑑𝑄_1𝑖𝑖−2Λ^𝑇_𝑖𝐸^𝑇−2𝐸Λ_𝑖+2(ℎ−𝛼ℎ)𝑅_3𝑖,Υ_𝑖23=Λ^𝑇_𝑖𝐸^𝑇+𝐸Λ_𝑖−(ℎ−𝛼ℎ)𝑅_3𝑖,Υ_𝑖24=Υ_𝑖23. (68) Furthermore, if the above LMIs have feasible solutions, then the desired state feedback controller can be solved as 𝐾_𝑖=𝐻_𝑖Λ_𝑖⁻¹, 𝐾_𝑑𝑖=𝐿_𝑖Λ_𝑖⁻¹.

Proof. It follows from Theorem 9 that the closed-loop system (62) is stochastically admissible with 𝐻_∞ performance level 𝛾 if there exist a set of matrices such that (22)–(27) hold, where 𝐴_𝑖, 𝐴_𝑑𝑖, 𝐶_𝑖, and 𝐶_𝑑𝑖 are replaced by 𝐴_𝑖+𝐵_𝑖𝐾_𝑖, 𝐴_𝑑𝑖+𝐵_𝑖𝐾_𝑑𝑖, 𝐶_𝑖+𝐷_𝑖𝐾_𝑖, and 𝐶_𝑑𝑖+𝐷_𝑖𝐾_𝑑𝑖, respectively.
According to Lemma 8, we can set Λ_𝑖=(𝑃_𝑖𝐸+𝑈^𝑇Φ_𝑖𝑉^𝑇)⁻¹=𝑃_𝑖𝐸^𝑇+𝑉Φ_𝑖𝑈; then pre- and postmultiplying (22) and (23) by diag{Λ^𝑇_𝑖,Λ^𝑇_𝑖,Λ^𝑇_𝑖,Λ^𝑇_𝑖,𝐼,𝑅^−𝑇_2𝑖,𝑅^−𝑇_3𝑖,𝑊^−𝑇,𝐼} and its transpose and leting 𝐻_𝑖=𝐾_𝑖(𝑃_𝑖𝐸^𝑇+𝑉Φ_𝑖𝑈), 𝐿_𝑖=𝐾_𝑑𝑖(𝑃_𝑖𝐸^𝑇+𝑉Φ_𝑖𝑈), we have Θ̆_𝑖11Θ̆_𝑖12Θ̆_𝑖130𝐵_𝑤𝑖Θ̆_𝑖1612ℎ²𝐴_𝑖Λ_𝑖+𝐵_𝑖𝐻_𝑖^𝑇𝐶_𝑖Λ_𝑖+𝐷_𝑖𝐻_𝑖^𝑇∗Θ̆_𝑖22Θ̆_𝑖2300Θ̆_𝑖2612ℎ²𝐴_𝑑𝑖Λ_𝑖+𝐵_𝑖𝐿_𝑖^𝑇𝐶_𝑑𝑖Λ_𝑖+𝐷_𝑖𝐿_𝑖^𝑇∗∗Θ̆_𝑖33Θ̆_𝑖34Θ̆0000∗∗∗_𝑖440000∗∗∗∗−𝛾²𝐼Θ̆_𝑖5612ℎ²𝐵^𝑇_𝑤𝑖𝐷^𝑇_𝑤𝑖∗∗∗∗∗Θ̆_𝑖66100∗∗∗∗∗∗−2ℎ²𝑊⁻¹0(𝑡)∈(],∗∗∗∗∗∗∗−𝐼<0,ℎ0,𝛼ℎΘ̆_𝑖11Θ_𝑖12Θ_𝑖130𝐵_𝑤𝑖Θ̆_𝑖1612ℎ²𝐴_𝑖Λ_𝑖+𝐵_𝑖𝐻_𝑖^𝑇𝐶_𝑖Λ_𝑖+𝐷_𝑖𝐻_𝑖^𝑇∗Θ_𝑖22Θ_𝑖23Θ_𝑖240Θ̆_𝑖2612ℎ²𝐴_𝑑𝑖Λ_𝑖+𝐵_𝑖𝐿_𝑖^𝑇𝐶_𝑑𝑖Λ_𝑖+𝐷_𝑖𝐿_𝑖^𝑇∗∗Θ̆_𝑖33Θ_𝑖34Θ̆0000∗∗∗_𝑖440000∗∗∗∗−𝛾²𝐼Θ̆_𝑖5612ℎ²𝐵^𝑇_𝑤𝑖𝐷^𝑇_𝑤𝑖∗∗∗∗∗Θ̆_𝑖66100∗∗∗∗∗∗−2ℎ²𝑊⁻¹0(𝑡)∈(],∗∗∗∗∗∗∗−𝐼<0,ℎ𝛼ℎ,ℎ (69)(70)where Θ̆_𝑖11=Λ^𝑇_𝑖𝐴^𝑇_𝑖+𝐻^𝑇_𝑖𝐵^𝑇_𝑖+𝐴_𝑖Λ_𝑖+𝐵_𝑖𝐻_𝑖+Λ^𝑇_𝑖^𝑁_𝑗=1𝛾_𝑖𝑗𝐸_𝑅𝐸^𝑇_𝑅𝑃_𝑗𝐸_𝑅⁻¹𝐸^𝑇_𝑅Λ_𝑖+Λ^𝑇_𝑖𝑄_1𝑖Λ_𝑖+Λ^𝑇_𝑖𝑄_2𝑖Λ_𝑖+𝛼ℎΛ^𝑇_𝑖𝑈₁Λ_𝑖+(1−𝛼)ℎΛ^𝑇_𝑖𝑈₂Λ_𝑖−𝛼₁Λ^𝑇_𝑖𝐸^𝑇𝑅_2𝑖𝐸Λ_𝑖,Θ̆_𝑖12=𝐴_𝑑𝑖Λ_𝑖+𝐵_𝑖𝐿_𝑖+𝛼₁Λ^𝑇_𝑖𝐸^𝑇𝑅_2𝑖𝐸Λ_𝑖−𝛼₁Λ^𝑇_𝑖𝐸^𝑇𝑆_𝑖𝐸Λ_𝑖,Θ̆_𝑖13=𝛼₁Λ^𝑇_𝑖𝐸^𝑇𝑆_𝑖𝐸Λ_𝑖,Θ̆_𝑖16=𝐴_𝑖Λ_𝑖+𝐵_𝑖𝐻_𝑖^𝑇𝐴_𝑖Λ_𝑖+𝐵_𝑖𝐻_𝑖^𝑇,Θ̆_𝑖22=−1−ℎ_𝑑Λ^𝑇_𝑖𝑄_1𝑖Λ_𝑖−2𝛼₁Λ^𝑇_𝑖𝐸^𝑇𝑅_2𝑖𝐸Λ_𝑖,+𝛼₁Λ^𝑇_𝑖𝐸^𝑇𝑆_𝑖𝐸Λ_𝑖+𝛼₁Λ^𝑇_𝑖𝐸^𝑇𝑆^𝑇_𝑖𝐸Λ_𝑖,Θ̆_𝑖23=𝛼₁Λ^𝑇_𝑖𝐸^𝑇𝑅_2𝑖𝐸Λ^𝑇_𝑖−𝛼₁Λ^𝑇_𝑖𝐸^𝑇𝑆_𝑖𝐸Λ^𝑇_𝑖,Θ̆_𝑖26=𝐴_𝑑𝑖Λ_𝑖+𝐵_𝑖𝐿_𝑖^𝑇𝐴_𝑑𝑖Λ_𝑖+𝐵_𝑖𝐿_𝑖^𝑇,Θ̆_𝑖33=Λ^𝑇_𝑖𝑄_3𝑖Λ_𝑖−Λ^𝑇_𝑖𝑄_2𝑖Λ_𝑖−𝛼₁Λ^𝑇_𝑖𝐸^𝑇𝑅_2𝑖𝐸Λ_𝑖−Λ^𝑇_𝑖𝛼₂𝐸^𝑇𝑅_3𝑖𝐸Λ_𝑖,Θ̆_𝑖34=𝛼₂Λ^𝑇_𝑖𝐸^𝑇𝑅_3𝑖𝐸Λ_𝑖,Θ̆_𝑖44=−Λ^𝑇_𝑖𝑄_3𝑖Λ_𝑖−𝛼₂Λ^𝑇_𝑖𝐸^𝑇𝑅_3𝑖𝐸Λ_𝑖,Θ̆_𝑖56=𝐵^𝑇_𝑤𝑖𝐵^𝑇_𝑤𝑖,Θ̆_𝑖66=diag−𝛼₁𝑅⁻¹_2𝑖,−𝛼₂𝑅⁻¹_3𝑖,Θ_𝑖12=𝐴_𝑑𝑖Λ_𝑖+𝐵_𝑖𝐿_𝑖,Θ_𝑖13=𝛼₁Λ^𝑇_𝑖𝐸^𝑇𝑅_2𝑖𝐸Λ_𝑖,Θ_𝑖22=−1−ℎ_𝑑Λ^𝑇_𝑖𝑄_1𝑖Λ_𝑖−2𝛼₂Λ^𝑇_𝑖𝐸^𝑇𝑅_3𝑖𝐸Λ_𝑖+𝛼₂Λ^𝑇_𝑖𝐸^𝑇𝑇_𝑖𝐸Λ_𝑖+𝛼₁Λ^𝑇_𝑖𝐸^𝑇𝑇^𝑇_𝑖𝐸Λ_𝑖,Θ_𝑖23=𝛼₂Λ^𝑇_𝑖𝐸^𝑇𝑅_3𝑖𝐸Λ_𝑖−𝛼₂Λ^𝑇_𝑖𝐸^𝑇𝑇^𝑇_𝑖𝐸Λ_𝑖,Θ_𝑖24=𝛼₂Λ^𝑇_𝑖𝐸^𝑇𝑅_3𝑖𝐸Λ_𝑖−𝛼₂Λ^𝑇_𝑖𝐸^𝑇𝑇_𝑖𝐸Λ_𝑖,Θ_𝑖34=𝛼₂Λ^𝑇_𝑖𝐸^𝑇𝑇_3𝑖𝐸Λ_𝑖. (71)
Note that 𝛾_𝑖𝑖≤0, we have 𝛾_𝑖𝑖Λ^𝑇_𝑖𝐸_𝑅𝐸^𝑇_𝑅𝑃_𝑖𝐸_𝑅⁻¹𝐸^𝑇_𝑅Λ_𝑖≤𝛾_𝑖𝑖𝜀_𝑖Λ^𝑇_𝑖𝐸^𝑇+𝜀_𝑖𝐸Λ_𝑖−𝜀²_𝑖𝐸𝑃_𝑖𝐸^𝑇. (72) If we choose 𝑆_𝑖=0 and 𝑇_𝑖=0, we have −𝛼₁Λ^𝑇_𝑖𝐸^𝑇𝑅_2𝑖𝐸Λ_𝑖0−𝛼₁Λ^𝑇_𝑖𝐸^𝑇𝑅_2𝑖𝐸Λ_𝑖002𝛼₂Λ^𝑇_𝑖𝐸^𝑇𝑅_3𝑖𝐸Λ_𝑖−𝛼₂Λ^𝑇_𝑖𝐸^𝑇𝑅_3𝑖𝐸Λ_𝑖−𝛼₂Λ^𝑇_𝑖𝐸^𝑇𝑅_3𝑖𝐸Λ_𝑖−𝛼₁Λ^𝑇_𝑖𝐸^𝑇𝑅_2𝑖𝐸Λ_𝑖−𝛼₂Λ^𝑇_𝑖𝐸^𝑇𝑅_3𝑖𝐸Λ_𝑖𝛼₁Λ^𝑇_𝑖𝐸^𝑇𝑅_2𝑖𝐸Λ_𝑖+𝛼₂Λ^𝑇_𝑖𝐸^𝑇𝑅_3𝑖𝐸Λ_𝑖00−𝛼₂Λ^𝑇_𝑖𝐸^𝑇𝑅_3𝑖𝐸Λ_𝑖0𝛼₂Λ^𝑇_𝑖𝐸^𝑇𝑅_3𝑖𝐸Λ_𝑖=𝐼00−𝐼−𝛼₁Λ^𝑇_𝑖𝐸^𝑇𝑅_2𝑖𝐸Λ_𝑖+0𝐼0𝐼0−𝐼0−𝐼−𝛼₂Λ^𝑇_𝑖𝐸^𝑇𝑅_3𝑖𝐸Λ_𝑖+0𝐼00𝐼−𝐼0−𝐼−𝛼₂Λ^𝑇_𝑖𝐸^𝑇𝑅_3𝑖𝐸Λ_𝑖≤𝐼000𝐼0−𝐼−𝐼−Λ^𝑇_𝑖𝐸^𝑇−𝐸Λ_𝑖+𝛼ℎ𝑅⁻¹_2𝑖+0𝐼0𝐼0−𝐼0−𝐼−Λ^𝑇_𝑖𝐸^𝑇−𝐸Λ_𝑖+(ℎ−𝛼ℎ)𝑅⁻¹_3𝑖+0𝐼00𝐼−𝐼0−𝐼−Λ^𝑇_𝑖𝐸^𝑇−𝐸Λ_𝑖+(ℎ−𝛼ℎ)𝑅⁻¹_3𝑖,−𝛼0𝐼0−𝐼₁Λ^𝑇_𝑖𝐸^𝑇𝑅_2𝑖𝐸Λ_𝑖−𝛼₁Λ^𝑇_𝑖𝐸^𝑇𝑅_2𝑖𝐸Λ_𝑖00−𝛼₁Λ^𝑇_𝑖𝐸^𝑇𝑅_2𝑖𝐸Λ_𝑖2𝛼₁Λ^𝑇_𝑖𝐸^𝑇𝑅_2𝑖𝐸Λ_𝑖−𝛼₁Λ^𝑇_𝑖𝐸^𝑇𝑅_2𝑖𝐸Λ_𝑖00−𝛼₁Λ^𝑇_𝑖𝐸^𝑇𝑅_2𝑖𝐸Λ_𝑖𝛼₁Λ^𝑇_𝑖𝐸^𝑇𝑅_2𝑖𝐸Λ_𝑖+𝛼₂Λ^𝑇_𝑖𝐸^𝑇𝑅_3𝑖𝐸Λ_𝑖−𝛼₂Λ^𝑇_𝑖𝐸^𝑇𝑅_3𝑖𝐸Λ_𝑖00−𝛼₂Λ^𝑇_𝑖𝐸^𝑇𝑅_3𝑖𝐸Λ_𝑖𝛼₂Λ^𝑇_𝑖𝐸^𝑇𝑅_3𝑖𝐸Λ_𝑖=𝐼00−𝐼−𝛼₁Λ^𝑇_𝑖𝐸^𝑇𝑅_2𝑖𝐸Λ_𝑖+0𝐼0𝐼−𝐼00−𝐼−𝛼₁Λ^𝑇_𝑖𝐸^𝑇𝑅_2𝑖𝐸Λ_𝑖+00𝐼0𝐼−𝐼0−𝐼−𝛼₂Λ^𝑇_𝑖𝐸^𝑇𝑅_3𝑖𝐸Λ_𝑖≤𝐼0000𝐼−𝐼−𝐼−Λ^𝑇_𝑖𝐸^𝑇−𝐸Λ_𝑖+𝛼ℎ𝑅⁻¹_2𝑖+0𝐼0𝐼−𝐼00−𝐼−Λ^𝑇_𝑖𝐸^𝑇−𝐸Λ_𝑖+𝛼ℎ𝑅⁻¹_2𝑖+00𝐼0𝐼−𝐼0−𝐼−Λ^𝑇_𝑖𝐸^𝑇−𝐸Λ_𝑖+(ℎ−𝛼ℎ)𝑅⁻¹_3𝑖00𝐼−𝐼. (73) Let 𝑄_1𝑖𝑗=Λ^𝑇_𝑖𝑄_1𝑗Λ_𝑖, 𝑄_2𝑖𝑗=Λ^𝑇_𝑖𝑄_2𝑗Λ_𝑖, 𝑄_3𝑖𝑗=Λ^𝑇_𝑖𝑄_3𝑖Λ_𝑖, 𝑅_2𝑖=𝑅⁻¹_2𝑖, 𝑅_3𝑖=𝑅⁻¹_3𝑖, 𝑈_1𝑖=Λ^𝑇_𝑖𝑈₁Λ^𝑇_𝑖, 𝑈_2𝑖=Λ^𝑇_𝑖𝑈₂Λ^𝑇_𝑖, and 𝑊=𝑊⁻¹; then by using Schur complement of Lemma 6, we have that (69) and (70) hold if (63) and (64) hold. Intuitively, we can verify that (66) and (67) are equivalent to (25) and (26).
On the other hand, it is from (65) that for ∀𝑗≠𝑖,𝛾_𝑖𝑗≠0, 𝑊𝑁−1>𝛾_𝑖𝑗𝑅_2𝑗. (74) It should also be noted that if 𝛾_𝑖𝑗=0, (74) always holds. Thus, summing (74) for all 𝑗∈ℓ, 𝑗≠𝑖, implies that ^𝑁_{𝑗=1,𝑗≠𝑖}𝛾_𝑖𝑗𝑅_2𝑗−𝑊<0. (75) Similarly, we can also obtain ^𝑁_{𝑗=1,𝑗≠𝑖}𝛾_𝑖𝑗𝑅_3𝑗−𝑊<0. (76) Since 𝛾_𝑖𝑖<0, 𝑅_2𝑖>0 and 𝑅_3𝑖>0, it can be verified that (65) guarantees (24). This completes the proof.

To show how to decide the parameter 𝛼 optimally, we construct the following iterative algorithm for Theorem 9. Following a similar way, we can obtain the iterative algorithms for other theorems and corollaries.

Algorithm 15. Select optimal parameter 𝛼 for maximum delay ℎ. Step 1.For a prescribed scalar ℎ_𝑑, choose appropriate step lengths Δℎ and Δ𝛼 for ℎ and 𝛼, such that (22)–(27) in Theorem 9 hold. Set 𝑘 as the iterative times, and select the initial values 𝑘=1, ℎ₀=Δℎ, and 𝛼₀=Δ𝛼. Let the upper bound ℎ=ℎ₀+Δℎ. Step 2. Let 𝛼=𝑘Δ𝛼; if LMIs (22)–(27) are feasible, go to Step 3; otherwise, go to Step 4. Step 3. Update ℎ₀, 𝛼₀, 𝑘 and ℎ as ℎ₀=ℎ, 𝛼₀=𝛼, 𝑘=1, and ℎ=ℎ₀+Δℎ, go to Step 2.  Step 4. Set 𝑘=𝑘+1. If 𝑘Δ𝛼<1; then go to Step 2; otherwise, stop.

From the above algorithm, the output 𝛼₀ is the optimal 𝛼 to ensure the delay upper bound ℎ to achieve its maximum value ℎ₀.

4. Numerical Examples

In this section, we provide three examples to clarify the applicability and less conservatism of the results proposed in this paper.

Example 1. Consider the nominal Markovian jump systems (55) with two modes and the following parameters: 𝐴₁=−3.48880.8057−0.6451−3.2684,𝐴₂=,𝐴−2.48980.28951.3396−0.0211_𝑑1=−0.8620−1.2919−0.6841−2.0729,𝐴_𝑑2.−2.83060.4978−0.8436−1.0115 (77) The transition rate matrix is given as 𝛾Γ=¹¹𝛾₁₂.0.8−0.8 (78)
It is noted that if 𝐸=𝐼, descriptor Markovian jump time-delay systems (55) reduce to the standard state-space Markovian jump time-delay systems. For the time-invariant delay, the stochastic stability problem can be solved by Corollary 12. Table 1 shows the compared results on the maximum allowable upper bound for different 𝛾₁₁ with the recent results [12, 14]. For ℎ_𝑑=0.5 and ℎ_𝑑=1.5, the maximum upper bound on allowable time delay is shown with given 𝛾₁₁ in Tables 2 and 3, respectively.

Table 1 

The maximum allowable upper bound for different 𝛾₁₁ and time-invariant delay.

Table 2 

The maximum allowable upper bound for different 𝛾₁₁ and ℎ_𝑑=0.5.

Table 3 

The maximum allowable upper bound for different 𝛾₁₁ and ℎ_𝑑=1.5.

As shown in the above tables, the methods in this paper have advantages over some previous ones in that the obtained maximum upper bound for allowable time delay is much larger. Therefore, the developed results have less conservatism than references [12, 14].

Example 2. Consider the continuous-time descriptor Markovian jump time-delay systems (59) with two modes and the following parameters: 𝐸=1000,𝐴₁=,𝐴0.497200−0.9541₂=,𝐴0.512100−0.7215_𝑑1=−1.0101.541500.5449,𝐴_𝑑2=.−0.85211.972100.4321 (79) The transition rate matrix is given as 𝛾Γ=¹¹𝛾₁₂.0.3−0.3 (80)

Compared with the existing results [17, 20], we find that Theorem 14 in this paper reaches the much larger bounds than the aforementioned results. Therefore, we can conclude that Theorem 14 in this paper is less conservative than the results in references [17, 20]. The compared results can be seen in Table 4.

Table 4 

The maximum allowable upper bound for different 𝛾₁₁ and time-invariant delay.

Example 3. Consider a Markovian jump system (1) with two modes and the following parameters: 𝐴₁=−2.2460−1.4410−1.5937−2.9289,𝐴₂=,𝐴−1.89990.81560.6900−0.7881_𝑑1=−0.70981.19080.6686−3.2025,𝐴_𝑑2=,𝐶−1.5198−1.6041−0.1567−1.2427₁=−0.3775−0.2959,𝐶₂=,𝐶−1.4751−0.2340_𝑑1=00,𝐶_𝑑2=,𝐵00_𝑤1=0.04030.6771,𝐵_𝑤2=,𝐷0.5689−0.2556_𝑤1=0.1184,𝐷_𝑤2=0.3148. (81) The transition matrix is given as Γ=.−330.6−0.6 (82)
When 𝐸=𝐼, we assume that 𝐵₁=𝐵₂=0, 𝐷₁=𝐷₂=0, and ℎ_𝑑=0; the compared results for different 𝛾>0 are given in Table 5. Obviously, the BRL obtained in this paper is less conservative than that in [12].
When we take 𝐸=¹⁰₀₀, then 𝐸_𝐿=𝐸_𝑅=10 satisfies 𝐸=𝐸_𝐿𝐸^𝑇_𝑅. The coefficient matrices of the input vector can be given as 𝐵₁=1.50.99,𝐵₂=𝐷1.20.99₁=0.5,𝐷₂=−0.3. (83) In this case, we take 𝑉=01, 𝑈=⁰₁, 𝜖₁=1.5, and 𝜖₂=1.8.

Table 5 

The maximum allowable upper bound for different 𝛾 with ℎ_𝑑=0.

By applying the LMIs conditions in Theorem 14, the minimum attenuation level is 𝛾_min=1.15 under the assumption ℎ=0.35 and ℎ_𝑑=0.3, and the mode-dependent state feedback controller gain can be obtained. Consider 𝐾₁=2.81740,𝐾₂=,𝐾−63.9993−0.0246_𝑑1=−0.69960,𝐾_𝑑2=−50.39730.0141. (84)

Therefore, it is easy to see that our results are effective.

5. Conclusions

This paper has studied the 𝐻_∞ control problem for continuous-time descriptor Markovian jump systems with time-varying delay. By constructing novel Lyapunov-Krasovskii functionals, delay-dependent stability criteria and BRL have been developed under which the underlying systems are regular, impulse-free, and stochastically stable with a prescribed 𝐻_∞ performance level. Compared with other time-varying delay results based on Jensen inequality, the obtained results in this paper have advantages over some previous results, which is partially attributed to the reciprocally convex combination method. The illustrated examples show that the maximum allowable upper bound of time delay is much larger than some latest results.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant no. 61273008 and the Natural Science Foundation of Liaoning Province under Grant no. 201202063, respectively.