New Results on Stability and Stabilization of Markovian Jump Systems with Time Delay

Abstract

This technical paper deals with the problem of stochastic stability and stabilization for a class of linear Markovian jumping systems with discrete time-varying delay. A novel delay-dependent stochastic stability criterion for Markovian delay systems is established based on new augmented Lyapunov-Krasovskii functional and delay fractioning techniques. Then a state feedback controller is designed to guarantee the stochastic stability of the resulting closed-loop system. Numerical examples are provided to illustrate the effectiveness of the proposed design approach in this paper.

1. Introduction

Markovian jump systems are a special class of hybrid systems with both modes and state variables. Since many dynamical systems such as the chemical processes, transportation systems, and manufacturing systems can be modeled as Markovian jump systems, considerable attention has been devoted to the problem of analysis and synthesis for Markovian jump systems over the past decades; see, for example, [1–4] and the references therein.

On the other hand, time delays are often encountered in various practical systems such as biological, electronics, and economical systems, which are frequently one of the main causes of poor performance, oscillation, or even instability [5–10]. Thus the analysis and synthesis of Markovian jump systems with time delay are of theoretical and practical importance and have received considerable research interest in the past few decades (e.g., [1–4, 10–13]).

In the literature, different design approaches have been proposed to deal with the delays. Generally, to get stability criteria, a Lyapunov function (or functional) is involved. For more details, one can see [10–14] and references therein. In those results, the information of delay is included in some criteria, which is so-called delay dependent, while others are irrelevant to the delay [15–17], which is called delay independent. It is well known that the former ones are generally much less conservative than the latter ones due to using the information of delay such as its length or rules of changes [18–20]. Therefore, how to choose a Lyapunov function (or functional) and derive a less conservative condition from the time derivative of such a function (or functional) is one of the most important techniques in the analysis and synthesis process.

Since cross products between two vectors are commonly encountered, different bound techniques are introduced. For example, Park’s inequality and Moon’s inequality were frequently used for bounding some cross terms. Moreover, many researchers tried to construct special forms of Lyapunov functional to obtain a feasible result. Some delay-dependent conditions were obtained through model transformation approaches; more details can be found in [18–22] and references therein. But these choices of specific functional and overbounding techniques are the origin of conservatism [20–23].

Slack matrices technique has been proven to be an effective approach to establish an improved result. To mention a few, by introducing two auxiliary matrices a new LMI representation of H∞ performance is presented for the linear systems with a single time-varying state delay in [19]. Wu et al. [20] and He et al. [21] tried to seek the relationship between the terms in the Newton-Leibniz formula and to introduce some free weighting matrices to express this relationship and some delay-dependent stability conditions with less conservativeness have been established. These methods are quite different from the conventional one of using model transformation which may introduce some superfluous dynamics into the original system. Additional studies can be found in [22–26] and references cited therein.

To improve the delay-dependent conditions, some efforts on constructing new Lyapunov functional were made by employing the delay partitioning method [21, 23–26]. To mention a few, a delay partitioning method has been developed for the delay-dependent stability analysis of fuzzy time-varying state delay systems based on a novel fuzzy Lyapunov-Krasovskii functional in [23]. In [24], by choosing an augmented Lyapunov-Krasovskii functional and utilizing the delay-partitioning method, novel delay-dependent mean square asymptotic stability conditions are derived in terms of LMIs. With the idea of delay partitioning, in [25] a new delay-dependent stochastic stability criterion on the stochastic stability of Markovian jump systems with time-delay has been proposed, which is less conservative than most existing results. However, when revisiting those problems, we find that there is still much room left for improvement, which motivates the present study.

In this context, we are interested in investigating the delay-dependent stability analysis and stabilization problem for a class of Markovian jump systems with time-varying delay. By employing the delay partitioning method and introducing some slack matrix variables based on the Newton-Leibniz formula, a less conservative delay-dependent stability condition is proposed. Then a state feedback controller, which guarantees the stochastic stability of the corresponding closed-loop system, is designed.

The rest of the paper is organized as follows. Section 2 gives problem formulation and some necessary lemmas. In Section 3, we present our main results of stability analysis and stabilization problem. Numerical examples are given in Section 4, followed by the conclusions, which are presented in Section 5.

Notations. The notations used throughout the paper are standard except where otherwise stated. The superscript “𝑇” stands for matrix transposition. “𝐼” and “0” denote the identity matrix and a zero matrix with compatible dimensions, respectively. The notation “𝑃>0(𝑃≥0)” means that 𝑃 is real symmetric and positive definite (positive semidefinite); diag{𝑀₁,𝑀₂,…,𝑀_𝑛} stands for a block-diagonal matrix with matrices 𝑀₁,𝑀₂,…,𝑀_𝑛 on the diagonal. “∗” represents the elements below the main diagonal of a symmetric matrix. ‖⋅‖ denotes the Euclidean norm of a vector or the spectral norm of a matrix. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

2. Problem Formulations and Preliminaries

Consider the following stochastic linear state-delay systems with Markovian jumping parameters: 𝑥̇(𝑡)=𝐴(𝑟(𝑡))𝑥(𝑡)+𝐴_𝑑(𝑟(𝑡𝑥((𝑡(𝑟(𝑡𝑢(𝑡)𝑥(𝑡)(𝑡)[−))𝑡−𝜏))+𝐵)),𝑡>0,=𝜑,𝑡∈𝜏,0],𝑟(0)=0, (1) where 𝑥(𝑡)∈𝑅^𝑛 is the system state vector, 𝑢(𝑡)∈𝑅^𝑚 is the controlled input vector, and 𝐴(𝑟(𝑡)),𝐴_𝑑(𝑟(𝑡)), and 𝐵(𝑟(𝑡)) are matrix functions of the random jumping process {𝑟(𝑡)}. 𝑟(𝑡) is a continuous-time Markovian jump process with right continuous trajectories taking discrete values in a given finite set 𝑆={1,2,…,𝑁} with transition probability matrix Γ≜{𝜋_𝑖𝑗} given by 𝑟()(𝑡)=𝜋Pr𝑡+Δ=𝑗∣𝑟=𝑖^𝑖𝑗Δ+𝑜(Δ),𝑖≠𝑗1+𝜋_𝑖𝑗Δ+𝑜(Δ),𝑖=𝑗, (2) where Δ>0 and 𝜋_𝑖𝑗≥0, for 𝑗≠𝑖, is the transition rate from mode 𝑖 at time 𝑡 to mode 𝑗 at time 𝑡+Δ and 𝜋_𝑖𝑖=−^𝑁_{𝑗=1,𝑗≠𝑖}𝜋_𝑖𝑗, (3) for each mode 𝑖∈𝑆, lim_ℎ→0(𝑜(Δ)/Δ)=0. 𝜑(𝑡)(𝑡∈[−𝜏,0]) is the system’s initial function which is continuous differentiable on [−𝜏,0] and 𝑟₀∈𝑆 are the initial conditions of the continuous state and mode. 𝜏(𝑡) is a continuously differentiable function, which denotes the time-varying delay when the mode is in 𝑟(𝑡)=𝑖, satisfying the conditions as follows: 0≤𝜏(𝑡)≤𝜏,𝜏̇(𝑡)≤𝜇∀𝑖∈𝑆, (4) where 𝜏 and 𝜇 are known positive real constants.

Remark 1. It should be pointed out that, when a time-varying delay 𝜏(𝑡) is considered, it is usually assumed that 𝜏̇(𝑡)≤𝜇<1, while we only require 𝜏̇(𝑡)≤𝜇 in this paper, which is more general, as discussed by Xu et al. [22].

For notational simplicity, in the sequel, for each possible 𝑟(𝑡)=𝑖, 𝑖∈𝑆, a matrix ℵ(𝑟(𝑡)) will be denoted as ℵ_𝑖; for example, 𝐴(𝑟(𝑡)) and 𝐴_𝑑(𝑟(𝑡)) are denoted as 𝐴_𝑖 and 𝐴_𝑑𝑖, respectively, and so on.

To facilitate further developments, we introduce the following lemma and definition that will be frequently used in deriving the main results.

Lemma 2 (Schur complement). Given constant matrices 𝑆₁, 𝑆₂, and 𝑆₃ with appropriate dimensions, where 𝑆₁=𝑆^𝑇₁ and 𝑆₂=𝑆^𝑇₂>0, then 𝑆₁+𝑆^𝑇₃𝑆₂⁻¹𝑆₃<0 if and only if 𝑆₁𝑆^𝑇₃∗−𝑆₂<0𝑜𝑟−𝑆₂𝑆^𝑇₃∗𝑆₁<0. (5)

Definition 3. The Markovian jump system 𝑥̇(𝑡)=𝐴(𝑟(𝑡))𝑥(𝑡)+𝐴_𝑑(𝑟(𝑡𝑥((𝑡,𝑥(𝑡)(𝑡)[−))𝑡−𝜏))𝑡>0,=𝜑,∀𝑡∈𝜏,0],𝑟(0)=𝑟₀ (6) is said to be stochastically stable, if for any finite 𝜑(𝑡)∈𝑅^𝑛 defined on [−𝜏,0] and 𝑟₀∈𝑆 the following condition is satisfied: lim_𝑡→∞𝐸^𝑡₀𝑥^𝑇(𝑡)𝑥(𝑡)𝑑𝑡∣𝜑,𝑟₀<∞. (7)

3. Main Results

In this section, we will consider the stability and stabilization problem of the system (1). First of all, we introduce the following results which will be used in deriving the main results.

By the Leibniz-Newton formula, we have 𝑥(𝑡−𝜏(𝑡))=𝑥(𝑡)−^𝑡_{(𝑡)𝑡−𝜏}𝑥̇(𝑠)𝑑𝑠; (8) and then for any appropriate dimensional matrices 𝑁_1𝑖, 𝑁_2𝑖, 𝑁_3𝑖, 𝑁_4𝑖, 𝑁_5𝑖, (𝑖∈𝑆), we have that the following equation holds: 2𝑥^𝑇(𝑡)𝑁_1𝑖+𝑥^𝑇𝜏(𝑡)𝑡−2𝑁_2𝑖+𝑥^𝑇(𝑡−𝜏(𝑡))𝑁_3𝑖+𝑥^𝑇𝜏(𝑡)+𝑡−𝜏2𝑁_4𝑖+𝑥^𝑇(𝑡−𝜏)𝑁_5𝑖×𝑥(𝑡)−𝑥(𝑡−𝜏(𝑡))−^𝑡_{(𝑡)𝑡−𝜏}𝑥̇(𝑠)𝑑𝑠=0. (9)

Moreover, since 0≤𝜏(𝑡)≤𝜏, 𝜏̇(𝑡)≤𝜇  ∀𝑖∈𝑆, for any appropriate dimensional matrix 𝑋≥0, 𝜏𝜂^𝑇(𝑡)𝑋𝜂(𝑡)−^𝑡_{(𝑡)𝑡−𝜏}𝜂^𝑇(𝑡)𝑋𝜂(𝑡)𝑑𝑠≥0, (10) where𝜂(𝑡)=𝑥^Τ(𝑡)𝑥^𝑇𝜏(𝑡)𝑡−2𝑥^𝑇(𝑡−𝜏(𝑡))𝑥^𝑇𝜏(𝑡)+𝑡−𝜏2𝑥^𝑇(𝑡−𝜏)^𝑇. (11)

3.1. Stability Analysis

Combining delay partitioning approach and free-weighting matrices technique, a new delay-dependent stochastic stability condition is established in the theorem as follows.

Theorem 4. For given positive scalars 𝜏>0 and 𝜇>0, the autonomous Markovian jump system (1) with 𝑢(𝑡)≡0 is stochastically stable if there exist matrices 𝑃_𝑖=𝑃^𝑇_𝑖>0, 𝑄_1𝑖=𝑄^𝑇_1𝑖≥0, 𝑄_2𝑖=𝑄^𝑇_2𝑖≥0, 𝑄_3𝑖=𝑄^𝑇_3𝑖≥0, 𝑄_4𝑖=𝑄^𝑇_4𝑖≥0, (𝑖∈𝑆), 𝑍=𝑍^𝑇>0, 𝑋𝑋=¹¹𝑋₁₂𝑋₁₃𝑋₁₄𝑋₁₅∗𝑋₂₂𝑋₂₃𝑋₂₄𝑋₂₅∗∗𝑋₃₃𝑋₃₄𝑋₃₅∗∗∗𝑋₄₄𝑋₄₅∗∗∗∗𝑋₅₅≥0, (12) and free-weighting matrices 𝑁_1𝑖,𝑁_2𝑖,𝑁_3𝑖,𝑁_4𝑖,𝑁_5𝑖,(𝑖∈𝑆) with appropriate dimensions, such that the following LMIs hold: ΦΦ=¹¹Φ₁₂Φ₁₃Φ₁₄Φ₁₅𝜏𝐴^𝑇_𝑖𝑍∗Φ₂₂Φ₂₃Φ₂₄Φ₂₅0∗∗Φ₃₃Φ₃₄Φ₃₅𝜏𝐴^𝑇_𝑑𝑖𝑍∗∗∗Φ₄₄Φ₄₅0∗∗∗∗Φ₅₅0∗∗∗∗∗−𝜏𝑍<0,𝑋Ψ=¹¹𝑋₁₂𝑋₁₃𝑋₁₄𝑋₁₅𝑁_1𝑖∗𝑋₂₂𝑋₂₃𝑋₂₄𝑋₂₅𝑁_2𝑖∗∗𝑋₃₃𝑋₃₄𝑋₃₅𝑁_3𝑖∗∗∗𝑋₄₄𝑋₄₅𝑁_4𝑖∗∗∗∗𝑋₅₅𝑁_5𝑖∗∗∗∗∗𝑍≥0,_𝑁𝑗=1𝜋_𝑖𝑗𝑄_1𝑗≤0,_𝑁𝑗=1𝜋_𝑖𝑗𝑄_2𝑗≤0,_𝑁𝑗=1𝜋_𝑖𝑗𝑄_3𝑗≤0,_𝑁𝑗=1𝜋_𝑖𝑗𝑄_4𝑗≤0, (13)(14)(15) where Φ₁₁=𝑃_𝑖𝐴_𝑖+𝐴^𝑇_𝑖𝑃_𝑖+^𝑁_𝑗=1𝜋_𝑖𝑗𝑃_𝑗+𝑄_1𝑖+𝑁_1𝑖+𝑁^𝑇_1𝑖+𝜏𝑋₁₁,Φ₁₂=𝑁^𝑇_2𝑖+𝜏𝑋₁₂,Φ₁₃=𝑃_𝑖𝐴_𝑑𝑖−𝑁_1𝑖+𝑁^𝑇_3𝑖+𝜏𝑋₁₃,Φ₁₄=𝑁^𝑇_4𝑖+𝜏𝑋₁₄,Φ₁₅=𝑁^𝑇_5𝑖+𝜏𝑋₁₅,Φ₂₂𝜇=−1−2𝑄_1𝑖+𝑄_2𝑖+𝜏𝑋₂₂,Φ₂₃=−𝑁_2𝑖+𝜏𝑋₂₃,Φ₂₄=𝜏𝑋₂₄,Φ₂₅=𝜏𝑋₂₅,Φ₃₃=−1−𝜇𝑄_2𝑖+𝑄_3𝑖−𝑁_3𝑖−𝑁^𝑇_3𝑖+𝜏𝑋₃₃,Φ₃₄=−𝑁^𝑇_4𝑖+𝜏𝑋₃₄,Φ₃₅=−𝑁^𝑇_5𝑖+𝜏𝑋₃₅,Φ₄₄𝜇=−1−2𝑄_3𝑖+𝑄_4𝑖+𝜏𝑋₄₄,Φ₄₅=𝜏𝑋₄₅,Φ₅₅=−𝑄_4𝑖+𝜏𝑋₅₅. (16)

Proof. We define the following Lyapunov-Krasovskii functional candidate: 𝑉(𝑥(𝑡),𝑟(𝑡))=𝑉₁(𝑥(𝑡),𝑟(𝑡))+𝑉₂(𝑥(𝑡),𝑟(𝑡))+𝑉₃(𝑥(𝑡),𝑟(𝑡))+𝑉₄(𝑥(𝑡),𝑟(𝑡))+𝑉₅(𝑥(𝑡),𝑟(𝑡))+𝑉₆(𝑥(𝑡),𝑟(𝑡)), (17) with 𝑉₁(𝑥(𝑡),𝑟(𝑡))=𝑥^𝑇(𝑡)𝑃_𝑖𝑥(𝑡),𝑉₂(𝑥(𝑡),𝑟(𝑡))=^𝑡_{(𝜏(𝑡))𝑡−/2}𝑥^𝑇(𝑠)𝑄_1𝑖𝑥(𝑠)𝑉𝑑𝑠,₃(𝑥(𝑡),𝑟(𝑡))=_(𝑡)^{𝑡−(𝜏(𝑡)/2)}_𝑡−𝜏𝑥^𝑇(𝑠)𝑄_2𝑖𝑥(𝑠)𝑉𝑑𝑠,₄(𝑥(𝑡),𝑟(𝑡))=_𝜏(𝑡)+^{𝑡−𝜏(𝑡)}_{𝑡−((𝜏))/2}𝑥^𝑇(𝑠)𝑄_3𝑖𝑥(𝑠)𝑉𝑑𝑠,₅(𝑥(𝑡),𝑟(𝑡))=^{𝑡−((𝜏(𝑡)+𝜏)/2)}_𝑡−𝜏𝑥^𝑇(𝑠)𝑄_4𝑖𝑥(𝑠)𝑉𝑑𝑠,₆(𝑥(𝑡),𝑟(𝑡))=⁰_−𝜏^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠)𝑍𝑥̇(𝑠)𝑑𝑠𝑑𝜃, (18) where 𝑃_𝑖=𝑃^𝑇_𝑖>0, 𝑄_1𝑖=𝑄^𝑇_1𝑖≥0, 𝑄_2𝑖=𝑄^𝑇_2𝑖≥0, 𝑄_3𝑖=𝑄^𝑇_3𝑖≥0, 𝑄_4𝑖=𝑄^𝑇_4𝑖≥0, (𝑖∈𝑆), and 𝑍=𝑍^𝑇>0 are matrices to be determined.
Let ℘ be the weak infinitesimal generator of the random process {𝑥(𝑡),𝑟(𝑡)}. Then, for each 𝑟(𝑡)=𝑖, 𝑖∈𝑆, we have ℘𝑉₁(𝑥(𝑡),𝑖)=2𝑥^𝑇(𝑡)𝑃_𝑖𝑥̇(𝑡)+𝑥^𝑇(𝑡)^𝑁_𝑗=1𝜋_𝑖𝑗𝑃_𝑗𝑥(𝑡)=2𝑥^𝑇(𝑡)𝑃_𝑖𝐴_𝑖𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑃_𝑖𝐴_𝑑𝑖×𝑥(𝑡−𝜏(𝑡))+𝑥^𝑇(𝑡)^𝑁_𝑗=1𝜋_𝑖𝑗𝑃_𝑗𝑥(𝑡),℘𝑉₂(𝑥(𝑡),𝑖)=𝑥^𝑇(𝑡)𝑄_1𝑖𝑥(𝑡)−𝜏̇(𝑡)1−2𝑥^𝑇𝜏(𝑡)𝑡−2𝑄_1𝑖𝑥𝜏(𝑡)𝑡−2+^𝑡_{(𝜏(𝑡))𝑡−/2}𝑥^𝑇(𝑠)^𝑁_𝑗=1𝜋_𝑖𝑗𝑄_1𝑗𝑥(𝑠)𝑑𝑠,℘𝑉₃(𝑥(𝑡))=,𝑖_{(𝑡)𝑡−(𝜏(𝑡)/2)𝑡−𝜏}𝑥^𝑇(𝑠)^𝑁_𝑗=1𝜋_𝑖𝑗𝑄_2𝑗𝑥(𝑠)+𝜏̇(𝑡)𝑑𝑠1−2𝑥^𝑇𝜏(𝑡)𝑡−2𝑄_2𝑖𝑥𝜏(𝑡)𝑡−2−(1−𝜏̇(𝑡))𝑥^𝑇(𝑡−𝜏(𝑡))𝑄_2𝑖𝑥(𝑡−𝜏(𝑡)),℘𝑉₄(𝑥(𝑡))=,𝑖_{𝜏(𝑡)+𝑡−𝜏(𝑡)𝑡−((𝜏))/2}𝑥^𝑇(𝑠)^𝑁_𝑗=1𝜋_𝑖𝑗𝑄_3𝑗𝑥(𝑠)+(𝜏̇(𝑡𝑥𝑑𝑠1−))^𝑇(𝑡−𝜏(𝑡))𝑄_3𝑖𝑥((𝑡−𝜏̇(𝑡)𝑡−𝜏))1−2𝑥^𝑇𝜏(𝑡)+𝑡−𝜏2𝑄_3𝑖𝑥𝜏(𝑡)+𝑡−𝜏2,℘𝑉₅(𝑥(𝑡))=𝜏̇(𝑡),𝑖1−2𝑥^𝑇𝜏(𝑡)+𝑡−𝜏2𝑄_4𝑖𝑥𝜏(𝑡)+𝑡−𝜏2−𝑥^𝑇(𝑡−𝜏)𝑄_4𝑖𝑥(𝑡−𝜏)+^{𝑡−((𝜏(𝑡)+𝜏)/2)}_𝑡−𝜏𝑥^𝑇(𝑠)^𝑁_𝑗=1𝜋_𝑖𝑗𝑄_4𝑗𝑥(𝑠)𝑑𝑠,℘𝑉₆(𝑥(𝑡),𝑖)=𝜏𝑥̇^𝑇(𝑡)𝑍𝑥̇(𝑡)−^𝑡_𝑡−𝜏𝑥̇^𝑇(𝑠)𝑍𝑥̇(𝑠)𝑑𝑠. (19)
According to (8)–(11) and considering 0≤𝜏(𝑡)≤𝜏, 𝜏̇(𝑡)≤𝜇∀𝑖∈𝑆 in (4), we have ℘𝑉(𝑥(𝑡),𝑖)≤2𝑥^𝑇(𝑡)𝑃_𝑖𝐴_𝑖𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑃_𝑖𝐴_𝑑𝑖𝑥(𝑡−𝜏(𝑡))+𝑥^𝑇(𝑡)^𝑁_𝑗=1𝜋_𝑖𝑗𝑃_𝑗𝑥(𝑡)+𝑥^𝑇(𝑡)𝑄_1𝑖𝑥(𝑡)−1−𝜏(𝑡)2𝑥^𝑇𝜏(𝑡)𝑡−2𝑄_1𝑖𝑥𝜏(𝑡)𝑡−2+^𝑡_{(𝜏(𝑡))𝑡−/2}𝑥^𝑇(𝑠)^𝑁_𝑗=1𝜋_𝑖𝑗𝑄_1𝑗𝑥(𝑠)+𝑑𝑠_{(𝑡)𝑡−(𝜏(𝑡)/2)𝑡−𝜏}𝑥^𝑇(𝑠)^𝑁_𝑗=1𝜋_𝑖𝑗𝑄_2𝑗𝑥(𝑠)+𝜏̇(𝑡)𝑑𝑠1−2𝑥^𝑇𝜏(𝑡)𝑡−2𝑄_2𝑖𝑥𝜏(𝑡)𝑡−2−(1−𝜏(𝑡))𝑥^𝑇(𝑡−𝜏(𝑡))𝑄_2𝑖𝑥((𝑡+𝑡−𝜏))_{𝜏(𝑡)+𝑡−𝜏(𝑡)𝑡−((𝜏))/2}𝑥^𝑇(𝑠)^𝑁_𝑗=1𝜋_𝑖𝑗𝑄_3𝑗𝑥(𝑠)+(𝜏̇(𝑡𝑥𝑑𝑠1−))^𝑇(𝑡−𝜏(𝑡))𝑄_3𝑖𝑥((𝑡−𝑡−𝜏))1−𝜏(𝑡)2𝑥^𝑇𝜏(𝑡)+𝑡−𝜏2𝑄_3𝑖𝑥𝜏(𝑡)+𝑡−𝜏2+𝜏̇(𝑡)1−2𝑥^𝑇𝜏(𝑡)+𝑡−𝜏2𝑄_4𝑖𝑥𝜏(𝑡)+𝑡−𝜏2−𝑥^𝑇(𝑡−𝜏)𝑄_4𝑖𝑥(𝑡−𝜏)+^{𝑡−((𝜏(𝑡)+𝜏)/2)}_𝑡−𝜏𝑥^𝑇(𝑠)^𝑁_𝑗=1𝜋_𝑖𝑗𝑄_4𝑗𝑥(𝑠)+𝑑𝑠𝜏𝑥̇^𝑇(𝑡)𝑍𝑥̇(𝑡)−^𝑡_𝑡−𝜏𝑥̇^𝑇(𝑠)𝑍𝑥̇(𝑠)𝑥𝑑𝑠+2^𝑇(𝑡)𝑁_1𝑖+𝑥^𝑇𝜏(𝑡)𝑡−2𝑁_2𝑖+𝑥^𝑇(𝑡−𝜏(𝑡))𝑁_3𝑖+𝑥^𝑇𝜏(𝑡)+𝑡−𝜏2𝑁_4𝑖+𝑥^𝑇(𝑡−𝜏)𝑁_5𝑖×𝑥(𝑡)−𝑥(𝑡−𝜏(𝑡))−^𝑡_{(𝑡)𝑡−𝜏}𝑥̇(𝑠)+𝑑𝑠𝜏𝜂^𝑇(𝑡)𝑋𝜂(𝑡)−^𝑡_{(𝑡)𝑡−𝜏}𝜂^𝑇(𝑡)𝑋𝜂(𝑡)𝑑𝑠=𝜂^𝑇(𝑡)Λ₁𝜂(𝑡)−^𝑡_{(𝑡)𝑡−𝜏}𝜉^𝑇(𝑡,𝑠)Ψ𝜉(𝑡,𝑠)𝑑𝑠+Λ₂+Λ₃, (20) where 𝜉(𝑡)=𝜂^𝑇(𝑡)𝑥̇^𝑇(𝑠)^𝑇,Λ₁=Φ₁₁Φ₁₂Φ₁₃Φ₁₄Φ₁₅∗Φ₂₂Φ₂₃Φ₂₄Φ₂₅∗∗Φ₃₃Φ₃₄Φ₃₅∗∗∗Φ₄₄Φ₄₅∗∗∗∗Φ₅₅,Φ₁₁=Φ₁₁+𝐴^𝑇_𝑖𝐻𝐴_𝑖,Φ₁₃=Φ₁₃+𝐴^𝑇_𝑖𝐻𝐴_𝑑𝑖,Φ₃₃=Φ₃₃+𝐴^𝑇_𝑑𝑖𝐻𝐴_𝑑𝑖.Λ₂=^𝑡_{(𝜏(𝑡))𝑡−/2}𝑥^𝑇(𝑠)^𝑁_𝑗=1𝜋_𝑖𝑗𝑄_1𝑗𝑥(𝑠)+𝑑𝑠_{(𝑡)𝑡−(𝜏(𝑡)/2)𝑡−𝜏}𝑥^𝑇(𝑠)^𝑁_𝑗=1𝜋_𝑖𝑗𝑄_2𝑗𝑥(𝑠)+𝑑𝑠_{𝜏(𝑡)+𝑡−𝜏(𝑡)𝑡−((𝜏))/2}𝑥^𝑇(𝑠)^𝑁_𝑗=1𝜋_𝑖𝑗𝑄_3𝑗𝑥(𝑠)+𝑑𝑠_{𝑡−((𝜏(𝑡)+𝜏)/2)𝑡−𝜏}𝑥^𝑇(𝑠)^𝑁_𝑗=1𝜋_𝑖𝑗𝑄_4𝑗𝑥(𝑠)Λ𝑑𝑠,₃𝜏̇(𝑡)=−2𝑥^𝑇𝜏(𝑡)𝑡−2𝑄_2𝑖𝑥𝜏(𝑡)𝑡−2−𝜏̇(𝑡)𝑥^𝑇(𝑡−𝜏(𝑡))𝑄_3𝑖𝑥((𝑡−𝜏̇(𝑡)𝑡−𝜏))2𝑥^𝑇𝜏(𝑡)+𝑡−𝜏2𝑄_4𝑖𝑥𝜏(𝑡)+𝑡−𝜏2, (21)(22)(23) and Ψ is denoted in (14).
By Schur complement (Lemma 2) and some matrices primary manipulations, it is easy to see that the abovementioned matrix Λ₁<0 in (22) is equivalent to (13).
Since 𝑄_2𝑖≥0, 𝑄_3𝑖≥0, 𝑄_4𝑖≥0, we have Λ₃≤0.
It is easy to see that ℘𝑉(𝑥(𝑡),𝑖)<0 if LMIs (13)–(15) hold.
Set 𝜆₁=min{𝜆_min(−Λ₁),𝑖∈𝑆}; then 𝜆₁>0. For any 𝑡≥𝜏, we have ℘𝑉(𝑥(𝑡),𝑖)≤−𝜆₁𝜂^𝑇(𝑡)𝜂(𝑡)≤−𝜆₁𝑥^𝑇(𝑡)𝑥(𝑡). (24)
By Dynkin’s formula, we have 𝐸{𝑉(𝑥(𝑡),𝑖)}−𝐸𝑉𝜑(𝑡),𝑟₀=𝐸^𝑡₀℘𝑉(𝑥(𝑠),𝑟(𝑠))𝑑𝑠≤−𝜆₁^𝑡₀𝐸𝑥^𝑇(𝑠)𝑥(𝑠)𝑑𝑠. (25)
Hence ^𝑡₀𝐸𝑥^𝑇(𝑠)𝑥(𝑠)𝑑𝑠≤𝜆₁⁻¹𝑉𝜑(𝑡),𝑟₀. (26)
Set 𝜆₂=min{𝜆_min(𝑃_𝑖),𝑖∈𝑆}; then 𝜆₂>0. From (17), it is easy to see that for any 𝑡≥0𝐸{𝑉(𝑥(𝑡)𝑉,𝑖)}=𝐸₁(𝑥(𝑡),𝑖)+𝐸𝑉₂(𝑥(𝑡),𝑖)+𝐸𝑉₃(𝑥(𝑡)𝑉,𝑖)+𝐸₄(𝑥(𝑡),𝑖)+𝐸𝑉₅(𝑥(𝑡),𝑖)+𝐸𝑉₆(𝑥(𝑡),𝑖)≥𝜆₂𝐸𝑥^𝑇(𝑡)𝑥(𝑡). (27)
From inequalities (26) and (27), we know that 𝐸𝑥^𝑇(𝑡)𝑥(𝑡)≤−𝜒₁𝐸^𝑡₀𝑥^𝑇(𝑠)𝑥(𝑠)𝑑𝑠+𝜒₂𝑉𝑥₀,𝑟₀, (28) where 𝜒₁=𝜆₁𝜆₂⁻¹, 𝜒₂=𝜆₂⁻¹.
By Gronwell-Bellman lemma, we have 𝐸𝑥^𝑇(𝑡)𝑥(𝑡)≤exp−𝜒₁𝑡𝜒₂𝑉𝑥₀,𝑟₀. (29)
Thus, according to (28), we have 𝐸^𝑡₀𝑥^𝑇(𝑠)𝑥(𝑠)𝑑𝑠∣𝜑,𝑟₀≤𝜒₁⁻¹𝜒₂1−exp−𝜒₁𝑡𝑉𝑥₀,𝑟₀. (30)
As 𝑡→∞, we have lim_𝑡→∞𝐸^𝑡₀𝑥^𝑇(𝑠)𝑥(𝑠)𝑑𝑠∣𝜑,𝑟₀≤𝜒₁⁻¹𝜒₂𝑉𝑥₀,𝑟₀. (31)
For there always exists a scalar 𝜀>0, such that lim_𝑡→∞𝐸^𝑡₀𝑥^𝑇(𝑠)𝑥(𝑠)𝑑𝑠∣𝜑,𝑟₀≤𝜒₁⁻¹𝜒₂𝑉𝑥₀,𝑟₀≤𝜀sup_{−𝜏≤𝑠≤0}𝜑(𝑠)². (32)
From Definition 3, we know that the Markovian jump system in (1) with 𝑢(𝑡)≡0 is stochastically stable.
The proof is thus completed.

Remark 5. Benefiting from a novel LKF construction, Theorem 4 presents a delay-dependent stochastic stability criterion which is much less conservative than most existing results in the literature. The idea of delay portioning used here is different from those approaches in the literature. For example, assuming the time delay can be denoted as a constant part and time-varying part, the idea of delay partitioning is employed to the constant part and some LKF are constructed in [21, 23–26]. As a further improved result, an appropriate LKF is established with the idea of partitioning the lower bound of time delay in [26]. However, in this brief, we divided the whole time varying delay interval [0,𝜏] into multi-parts, with which a novel LKF is constructed.

If the system mode set 𝑆={1}, the jump linear system (1) is simplified into a general linear system. Then we have the following simplified result.

Corollary 6. For given positive scalars 𝜏>0 and 𝜇>0, the autonomous Markovian jump system (1) with 𝑖∈𝑆={1} is stochastically stable if there exist matrices 𝑃=𝑃^𝑇>0, 𝑄_𝑘=𝑄^𝑇_𝑘≥0, (𝑘=1,2,3,4), 𝑍=𝑍^𝑇>0, 𝑋𝑋=¹¹𝑋₁₂𝑋₁₃𝑋₁₄𝑋₁₅∗𝑋₂₂𝑋₂₃𝑋₂₄𝑋₂₅∗∗𝑋₃₃𝑋₃₄𝑋₃₅∗∗∗𝑋₄₄𝑋₄₅∗∗∗∗𝑋₅₅≥0, (33) and free-weighting matrices 𝑁_𝑗, (𝑗=1,2,…,5) with appropriate dimensions, such that the following LMIs hold: ΦΦ=¹¹Φ₁₂Φ₁₃Φ₁₄Φ₁₅𝜏𝐴^𝑇𝑍∗Φ₂₂Φ₂₃Φ₂₄Φ₂₅0∗∗Φ₃₃Φ₃₄Φ₃₅𝜏𝐴^𝑇_𝑑𝑍∗∗∗Φ₄₄Φ₄₅0∗∗∗∗Φ₅₅0∗∗∗∗∗−𝑋𝜏𝑍<0,Ψ=¹¹𝑋₁₂𝑋₁₃𝑋₁₄𝑋₁₅𝑁₁∗𝑋₂₂𝑋₂₃𝑋₂₄𝑋₂₅𝑁₂∗∗𝑋₃₃𝑋₃₄𝑋₃₅𝑁₃∗∗∗𝑋₄₄𝑋₄₅𝑁₄∗∗∗∗𝑋₅₅𝑁₅∗∗∗∗∗𝑍≥0,_𝑁𝑗=1𝜋_𝑖𝑗𝑄₁≤0,_𝑁𝑗=1𝜋_𝑖𝑗𝑄₂≤0,_𝑁𝑗=1𝜋_𝑖𝑗𝑄₃≤0,_𝑁𝑗=1𝜋_𝑖𝑗𝑄₄≤0, (34) where Φ₁₁=𝑃𝐴+𝐴^𝑇𝑃+𝑄₁+𝑁₁+𝑁^𝑇₁+𝜏𝑋₁₁,Φ₁₂=𝑁^𝑇₂+𝜏𝑋₁₂,Φ₁₃=𝑃𝐴_𝑑−𝑁₁+𝑁^𝑇₃+𝜏𝑋₁₃,Φ₁₄=𝑁^𝑇₄+𝜏𝑋₁₄,Φ₁₅=𝑁^𝑇₅+𝜏𝑋₁₅,Φ₂₂𝜇=−1−2𝑄₁+𝑄₂+𝜏𝑋₂₂,Φ₂₃=−𝑁₂+𝜏𝑋₂₃,Φ₂₄=𝜏𝑋₂₄,Φ₂₅=𝜏𝑋₂₅,Φ₃₃=−1−𝜇𝑄₂+𝑄₃−𝑁₃−𝑁^𝑇₃+𝜏𝑋₃₃,Φ₃₄=−𝑁^𝑇₄+𝜏𝑋₃₄,Φ₃₅=−𝑁^𝑇₅+𝜏𝑋₃₅,Φ₄₄𝜇=−1−2𝑄₃+𝑄₄+𝜏𝑋₄₄,Φ₄₅=𝜏𝑋₄₅,Φ₅₅=−𝑄₄+𝜏𝑋₅₅. (35)

3.2. Stabilization Controller Design

Now we are in a position to present a solution to the stabilization problem for system (1). We assume that the state feedback controller is in the general formula as follows: 𝑢_𝑖=𝐾_𝑖𝑥(𝑡)+𝐾_𝑑𝑖𝑥(𝑡−𝜏(𝑡)), (36) where 𝐾_𝑖 and 𝐾_𝑑𝑖 are the gain matrices of the state feedback controller to be designed.

Then, for each 𝑟(𝑡)=𝑖∈𝑆, the resulting close-loop system of (1) with (36) becomes 𝑥̇(𝑡)=𝐴_𝑖𝑥(𝑡)+𝐴_𝑑𝑖𝑥(𝑡−𝜏(𝑡)), (37) where 𝐴_𝑖=𝐴_𝑖+𝐵_𝑖𝐾_𝑖, 𝐴_𝑑𝑖=𝐴_𝑑𝑖+𝐵_𝑖𝐾_𝑑𝑖. Thus, representing 𝐴_𝑖 by 𝐴_𝑖, 𝐴_𝑑𝑖 by 𝐴_𝑑𝑖 in Theorem 4, and by some matrices primary manipulations, we can obtain the following delay-dependent theorem.

Theorem 7. For given positive scalars 𝜏>0 and 𝜇>0, the autonomous Markovian jump system (1) is stochastically stable if there exist matrices 𝑃_𝑖=𝑃^𝑇_𝑖>0, 𝑄_1𝑖=𝑄^𝑇_1𝑖≥0, 𝑄_2𝑖=𝑄^𝑇_2𝑖≥0, 𝑄_3𝑖=𝑄^𝑇_3𝑖≥0, 𝑄_4𝑖=𝑄^𝑇_4𝑖≥0, (𝑖∈𝑆), 𝑍=𝑍^𝑇>0, 𝑋𝑋=¹¹𝑋₁₂𝑋₁₃𝑋₁₄𝑋₁₅∗𝑋₂₂𝑋₂₃𝑋₂₄𝑋₂₅∗∗𝑋₃₃𝑋₃₄𝑋₃₅∗∗∗𝑋₄₄𝑋₄₅∗∗∗∗𝑋₅₅≥0, (38)𝐾_𝑖,𝐾_𝑑𝑖 and free-weighting matrices 𝑁_1𝑖,𝑁_2𝑖,𝑁_3𝑖,𝑁_4𝑖,𝑁_5𝑖,(𝑖∈𝑆) with appropriate dimensions, such that the following LMIs hold: ΦΦ=¹¹Φ₁₂Φ₁₃Φ₁₄Φ₁₅Φ₁₆∗Φ₂₂Φ₂₃Φ₂₄Φ₂₅0∗∗Φ₃₃Φ₃₄Φ₃₅Φ₃₆∗∗∗Φ₄₄Φ₄₅0∗∗∗∗Φ₅₅0∗∗∗∗∗Φ₆₆<0,𝑋Ψ=¹¹𝑋₁₂𝑋₁₃𝑋₁₄𝑋₁₅𝑁_1𝑖∗𝑋₂₂𝑋₂₃𝑋₂₄𝑋₂₅𝑁_2𝑖∗∗𝑋₃₃𝑋₃₄𝑋₃₅𝑁_3𝑖∗∗∗𝑋₄₄𝑋₄₅𝑁_4𝑖∗∗∗∗𝑋₅₅𝑁_5𝑖∗∗∗∗∗𝑍≥0,_𝑁𝑗=1𝜋_𝑖𝑗𝑄_1𝑗≤0,_𝑁𝑗=1𝜋_𝑖𝑗𝑄_2𝑗≤0,_𝑁𝑗=1𝜋_𝑖𝑗𝑄_3𝑗≤0,_𝑁𝑗=1𝜋_𝑖𝑗𝑄_4𝑗≤0, (39)(40)(41) where Φ₁₁=𝐴_𝑖𝑃_𝑖+𝑃_𝑖𝐴^𝑇_𝑖+𝐵_𝑖𝐾_𝑖+𝐾^𝑇_𝑖𝐵^𝑇_𝑖+𝑄_1𝑖+𝑁_1𝑖+𝑁^𝑇_1𝑖+𝜏𝑋₁₁,Φ₁₂=𝑁^𝑇_2𝑖+𝜏𝑋₁₂,Φ₁₃=𝐴_𝑑𝑖𝑃_𝑖+𝐵_𝑖𝐾_𝑑𝑖−𝑁_1𝑖+𝑁^𝑇_3𝑖+𝜏𝑋₁₃,Φ₁₄=𝑁^𝑇_4𝑖+𝜏𝑋₁₄,Φ₁₅=𝑁^𝑇_5𝑖+𝜏𝑋₁₅,Φ₁₆=𝑃𝐴^𝑇_𝑖+𝐾^𝑇_𝑖𝐵^𝑇_𝑖,Φ₂₂𝜇=−1−2𝑄_1𝑖+𝑄_2𝑖+𝜏𝑋₂₂,Φ₂₃=−𝑁_2𝑖+𝜏𝑋₂₃,Φ₂₄=𝜏𝑋₂₄,Φ₂₅=𝜏𝑋₂₅,Φ₃₃=−1−𝜇𝑄_2𝑖+𝑄_3𝑖−𝑁_3𝑖−𝑁^𝑇_3𝑖+𝜏𝑋₃₃,Φ₃₄=−𝑁^𝑇_4𝑖+𝜏𝑋₃₄,Φ₃₅=−𝑁^𝑇_5𝑖+𝜏𝑋₃₅,Φ₃₆=𝑃𝐴^𝑇_𝑑𝑖+𝐾^𝑇_𝑑𝑖𝐵^𝑇_𝑖,Φ₄₄𝜇=−1−2𝑄_3𝑖+𝑄_4𝑖+𝜏𝑋₄₄,Φ₄₅=𝜏𝑋₄₅,Φ₅₅=−𝑄_4𝑖+𝜏𝑋₅₅,Φ₆₆=𝜏⁻¹𝑍−2𝑃_𝑖. (42)

In this case, if the conditions abovementioned are feasible, the matrices 𝐾_𝑖,𝐾_𝑑𝑖 in (36) are given by 𝐾_𝑖=𝐾_𝑖𝑃_𝑖⁻¹ and 𝐾_𝑑𝑖=𝐾_𝑑𝑖𝑃_𝑖⁻¹. That is, the state feedback control law (36) can be designed as 𝑢_𝑖=𝐾_𝑖𝑃_𝑖⁻¹𝑥(𝑡)+𝐾_𝑑𝑖𝑃_𝑖⁻¹𝑥(𝑡−𝜏(𝑡)). (43)

Proof. By performing a congruence transformation diag{𝑃_𝑖⁻¹,𝑃_𝑖⁻¹,𝑃_𝑖⁻¹,𝑃_𝑖⁻¹,𝑃_𝑖⁻¹} to both sides of (12), and defining matrices 𝑋_𝑚𝑛≜𝑃_𝑖⁻¹𝑋_𝑚𝑛𝑃_𝑖⁻¹, (𝑚,𝑛=1,2,3,4,5,𝑖∈𝑆), we can obtain (38).
Similarly, by performing a congruence transformation to (14) by diag{𝑃_𝑖⁻¹,𝑃_𝑖⁻¹,𝑃_𝑖⁻¹,𝑃_𝑖⁻¹,𝑃_𝑖⁻¹,𝑃_𝑖⁻¹} and defining matrices 𝑍≜𝑃_𝑖⁻¹𝑍𝑃_𝑖⁻¹, 𝑁_𝑘𝑖≜𝑃_𝑖⁻¹𝑁_𝑘𝑖𝑃_𝑖⁻¹, (𝑘=1,2,3,4,5,𝑖∈𝑆), we have (40).
By performing a congruence transformation to (13) with diag{𝑃_𝑖⁻¹,𝑃_𝑖⁻¹,𝑃_𝑖⁻¹,𝑃_𝑖⁻¹,𝑃_𝑖⁻¹,𝑍⁻¹} and defining matrices 𝑃_𝑖≜𝑃_𝑖⁻¹, 𝑄_𝑘𝑖≜𝑃_𝑖⁻¹𝑄_𝑘𝑖𝑃_𝑖⁻¹, (𝑘=1,2,3,4), 𝐾_𝑖=𝐾_𝑖𝑃_𝑖⁻¹, 𝐾_𝑑𝑖=𝐾_𝑑𝑖𝑃_𝑖⁻¹, we have ΦΦ=¹¹Φ₁₂Φ₁₃Φ₁₄Φ₁₅𝐴_𝑖𝑃_𝑖+𝐵_𝑖𝐾_𝑖^𝑇∗Φ₂₂Φ₂₃Φ₂₄Φ₂₅0∗∗Φ₃₃Φ₃₄Φ₃₅𝐴_𝑑𝑖𝑃_𝑖+𝐵_𝑖𝐾_𝑑𝑖^𝑇∗∗∗Φ₄₄Φ₄₅0∗∗∗∗Φ₅₅0∗∗∗∗∗−𝜏⁻¹𝑃_𝑖𝑍⁻¹𝑃_𝑖<0. (44)
According to the definition of 𝑃_𝑖 and 𝑍, the following equation always holds: 𝑃_𝑖−𝑍^𝑇𝑍⁻¹𝑃_𝑖−𝑍≥0, (45) which implies that −𝑃_𝑖𝑍⁻¹𝑃_𝑖≤𝑍−2𝑃_𝑖, (46)
This completes the proof.

Remark 8. The stabilization controller design problem is solved in Theorem 7, which guarantees the addressed Markovian jump system to be stochastically stable. Sometimes, we would like to design a memory-less state feedback control law 𝑢_𝑖=𝐾_𝑖𝑥(𝑡); in this case, just set 𝐾_𝑑𝑖=0 in Theorem 7.

Remark 9. It is worth pointing out that our main results are based on the LMI conditions, which can be easily obtained using LMI Control Toolbox.

4. Numerical Examples

In this section, some examples are presented to demonstrate the effectiveness of our proposed theoretical results.

Example 1. Consider a Markovian jump system in (1) with two modes and the following parameters [27]: 𝐴₁=0.5−10−3,𝐴_𝑑1=,𝐴0.5−0.20.20.3₂=−5110.2,𝐴_𝑑2=.−0.30.50.4−0.5 (47)
The generator matrix of the stochastic process Γ is 𝜋Γ=¹¹−𝜋₁₁−𝜋₂₂𝜋₂₂. (48)
To compare our stability result in Theorem 4 with those in [27–30], we assume 𝜇=0. When 𝜋₁₁=−7 and 𝜋₂₂≥−6, the result of [28] cannot be applied to stability. When 𝜋₁₁=−7 and 𝜋₂₂=−6, based on the result of [29], the system is found to be delay-independent stable. If we increase 𝜋₂₂, for example, 𝜋₁₁=−7 and 𝜋₂₂=−3, the result of [29] cannot guarantee system stability. By the results in [27, 30], we can obtain a feasible solution with 𝜏≤𝜏=0.404 and 𝜏≤𝜏=0.7316, respectively. By Theorem 4 of this paper, we can obtain a feasible solution with 𝜏≤𝜏=1.3384, which is much larger than that of [27, 30]. We can see that our criterion gives a less conservative result than those in [27–30].

Example 2. Consider the Markovian jump system (1) as Example 1. For 𝜋₁₁=−7 and 𝜋₂₂=−3 and various 𝜇, the results obtained by the proposed method in Theorem 4 for the admissible upper bound value of 𝜏 for which the system (1) remains stochastic stable are shown in Table 1.

Table 1 

Allowable upper bound of 𝜏 for various 𝜇.

Example 3. Consider the Markovian jump system (1) as Example 1. Suppose 𝜋₁₁=−7, 𝜋₂₂=−3, 𝜏=1.32, and 𝜇=0.1; according to Theorem 7, one can obtain a state feedback controller designed as 𝑢_𝑖=𝑥(𝑡)+𝑥((𝑡𝑥(𝑡)+𝑥((𝑡−3.70840.9821−0.3845−0.0491𝑡−𝑑)),𝑖=1−1.1560−0.6338−0.83940.5502𝑡−𝑑)),𝑖=2. (49)
The simulation results are illustrated in Figures 1–3 with the initial condition 𝜑(𝑡)=11^𝑇(𝑡∈−𝜏0). Among them, Figure 1 shows the Markovian jumping signal. The control input signal 𝑢(𝑡) is given in Figure 2, while Figure 3 plots the state responses of system (1) under the presented control law 𝑢(𝑡). From the simulation, it is shown that the designed control scheme is feasible and effective since the resulting closed-loop system is stable.

Figure 1 

Markovian jumping signal.

Figure 2 

Control input.

Figure 3 

States of the closed-loop system.

5. Conclusion

In this paper, new delay-dependent conditions for stochastic stability and stabilization of a class of Markovian jump systems with time delay are presented. By the idea of delay fractioning approach and introducing some zero equations, a sufficient condition is given to guarantee the stochastic stability of the Markovian jump systems, which is less conservative than most existing results. Furthermore, based on the new criterion, an approach has been shown to design a state feedback controller, which guarantees the resulting closed-loop system to be stochastically stable. Finally, numerical examples have been provided to demonstrate the effectiveness of the proposed methods in this paper.

Conflict of Interests

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

Acknowledgments

This work is supported by the National Natural Science Foundation of China (nos. 61304108, 61273094, and 51307035) and the Project for Distinguished Young Scholars of the Basic Research Plan in Shenzhen City under Contract no. JCJ201110001. The authors are very thankful to the reviewers for their valuable suggestions and comments.