Stochastic Stability for Time-Delay Markovian Jump Systems with Sector-Bounded Nonlinearities and More General Transition Probabilities

Abstract

This paper is concerned with delay-dependent stochastic stability for time-delay Markovian jump systems (MJSs) with sector-bounded nonlinearities and more general transition probabilities. Different from the previous results where the transition probability matrix is completely known, a more general transition probability matrix is considered which includes completely known elements, boundary known elements, and completely unknown ones. In order to get less conservative criterion, the state and transition probability information is used as much as possible to construct the Lyapunov-Krasovskii functional and deal with stability analysis. The delay-dependent sufficient conditions are derived in terms of linear matrix inequalities to guarantee the stability of systems. Finally, numerical examples are exploited to demonstrate the effectiveness of the proposed method.

1. Introduction

During the past decades, much attention has been devoted to the stochastic systems since stochastic modeling has an important science and engineering application [1, 2]. As an important class of stochastic systems, MJSs have attracted a lot of interest, since they can be used to model many practical dynamical systems, such as power systems, manufacturing systems, and economic systems in which they may experience failure or repairs, abrupt environmental disturbances, and abrupt changes in the operating point of a nonlinear plant (see [3, 4]). Theoretical results of this class of systems have been applied in many processes, such as target tracking, manufactory processes, solar thermal receivers, and fault-detection and fault-tolerant systems. Some examples can be found in [5–7].

For MJSs, the transition probabilities of the jumping process are important, but most of the previous issues on this kind of systems usually assumed that the elements of the transition probability matrix are completely known. Based on this condition, a lot of research results have been worked out in the literature, such as [8–13]. However, in many practical engineering applications, we cannot get all of the transition probabilities elements, and some elements might be time variable in some cases. Therefore, the study of Markov jump systems with partly known transition probabilities becomes necessary and some well-known results have been published, see [11–13]. In some cases, the exact value of some elements of the transition probability matrix can not be obtained, but their lower and upper bounds can be determined. The case has been addressed by [8], but its method cannot be used to deal with the case where there is no information available for transition probabilities. In [9], the more general partly known transition probabilities are investigated, which covers the cases that the transition probabilities are exactly known, unknown, and unknown but with known bounds. However, a conservative condition to relax inequality is used to deal with this kind of transition probabilities, and the time delay is not considered.

It is well known that time delay is very common in variously practical systems such as manufacturing systems, chemical processes, network control systems (NCSs), and telecommunication and economic systems. They are often the causes of oscillation, instability, and poor performance in many control systems. In the past years, time-delay system has been widely studied, and many analysis results have been reported (see [14, 15]). However, many approaches for time-delay problems only employed partial information of the delay-related terms. The quadratic integral terms of the form ∫^𝑡_{𝑡−ℎ(𝑡)}𝑓(𝛼)𝑑(𝛼) have been the major choices to construct the Lyapunov-Krasovskii functional, ignoring the employment of the delay upper bound ℎ (see [16]). Recently, Markovian systems with time delay have been widely studied (see [17–19]). But the time delays in [17, 19] are invariant, which is limited in the practical application. Although [18] considers the systems with time-varying delay, some useful terms are ignored to construct the Lyapunov-Krasovskii functional, which may lead to considerable conservativeness. Unfortunately, in the aforementioned papers about time delay, the elements of the considered transition probability matrix are completely known. To the best of the authors’ knowledge, up to now, there are few papers concerning both time delay and more general transition probability to deal with stochastic stability for nonlinear Markov jump systems.

Motivated by the previous points, in this paper, we are concerned with the stochastic stability analysis for a class of nonlinear MJSs with time-varying delay and general transition probabilities. The considered nonlinearity is the so-called sector-bounded nonlinearity [20], which is quite more general than the usual Lipschitz conditions. By exploiting full information of the delay-related terms including ℎ(𝑡) and ℎ and inherent relationship between the elements in transition probabilities, less conservative stability criterions are obtained in the framework of LMIs. Numerical examples are given to demonstrate the superiorities and effectiveness of the proposed method.

Notations. The superscript 𝑇 stands for matrix transposition, and 𝑅^𝑛 denotes the 𝑛 dimensional Euclidean space. 𝐸[⋅] stands for the 𝑛 mathematical expectation; 𝐻𝑒(𝑋) represents the sum of 𝑋 and 𝑋^𝑇. In addition, in symmetric block matrices or long matrix expressions, ∗ is used as an ellipsis for the terms that are introduced by symmetry. The notation 𝑃>0 (≥0) means that 𝑃 is real symmetric positive (semipositive) definite. 𝜀[⋅] stands for the expectation operator with respect to the given probability measure 𝑃. 𝐿²_ℱ0([−ℎ,0];ℝ^𝑛) denotes the family of ℝ^𝑛-valued stochastic process 𝑥(𝑡),−ℎ≤𝑡≤0, such that 𝑥(𝑡) is ℱ₀-measurable and ∫⁰_−ℎ𝐸{||𝑥(𝑡)||²}𝑑𝑡<∞. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

2. Problem Statement

Consider a class of stochastic nonlinear MJSs with sector-bounded nonlinearity and time-varying delay as follows: 𝑥̇(𝑡)=𝐴(𝑟(𝑡))𝑥(𝑡)+𝐴_𝑑(𝑟(𝑡𝑥((𝑡((𝑡)(𝑡))𝑡−ℎ))+𝑓𝑡,𝑥,𝑟))+𝑓_𝑑(((𝑡(𝑡,𝑥(𝑡)(𝑡)𝑡,𝑥𝑡−ℎ)),𝑟))=𝜓,−ℎ≤𝑡≤0, (1) where 𝑥(𝑡)∈𝑅^𝑛 is the state variable, and 𝐴(𝑟(𝑡))∈𝑅^𝑛×𝑛 and 𝐴_𝑑(𝑟(𝑡))∈𝑅^𝑛×𝑛 are known matrix functions of the Markov jump process 𝑟(𝑡). 𝑓(∗) and 𝑓_𝑑(∗) are the vector-valued nonlinear functions. Also ℎ(𝑡)∈[0,ℎ], ℎ̇(𝑡)≤𝜇, and 𝜓(𝑡)∈𝐿²_ℱ0([−ℎ,0];ℝ^𝑛). 𝑟(𝑡) is a time-homogeneous Markov process with right continuous trajectories and takes values on the finite set 𝕃={1,2,…,ℕ} with the following mode transition probabilities: 𝑟()(𝑡)=𝜋𝑃𝑟𝑡+𝑑𝑡=𝑗∣𝑟=𝑖^𝑖𝑗𝑑𝑡+𝑜(𝑑𝑡),𝑖≠𝑗,1+𝜋_𝑖𝑖𝑑𝑡+𝑜(𝑑𝑡),𝑖=𝑗, (2) where 𝑑𝑡>0, where lim_𝑑𝑡→0(𝑜(𝑑𝑡)/𝑑𝑡)=0. 𝜋_𝑖𝑗≥0, for 𝑖≠𝑗, is the transition rate from mode 𝑖 to mode 𝑗, and ^ℕ_{𝑗=1,𝑖≠𝑗}𝜋_𝑖𝑗=−𝜋_𝑖𝑖,𝑖=(1,…,ℕ). (3)

In this paper, more general transition probabilities of the jumping process in [9] are considered. Namely, some elements in matrix have been exactly known, some ones have been merely known with lower and upper bounds, and others may have no information to use. For instance, the transition probability matrix might be described by matrix 𝜋₁₁?𝜋₁₃⋯𝜋_1ℕ??𝛼⋯𝜋_2ℕ𝜋⋮⋯⋯⋱⋮_ℕ1𝛽⋯⋯𝜋_ℕℕ, (4) where "?" denotes the inaccessible elements, 𝛼 and 𝛽 have known lower and upper bounds (𝛼≤𝛼≤𝛼 and 𝛽≤𝛽≤𝛽) and 𝜋_𝑖𝑗 is exactly known.

For notation clarity, for all 𝑖,𝑗∈𝕃, we further rewrite 𝕃=𝕃^𝑖_𝑘∪𝕃^𝑖_𝑢𝑘 with 𝕃^𝑖_𝑘≜𝑗∶𝜋_𝑖𝑗≤𝜋_𝑖𝑗≤𝜋_𝑖𝑗,𝕃^𝑖_𝑢𝑘≜𝑗∶𝑗∉𝕃^𝑖_𝑘,ℒ^𝑖_𝑘≜𝑚∣𝑚∈𝕃^𝑖_𝑘,ℒ,𝑚≠𝑖_𝑖𝑢𝑘≜𝑚∣𝑚∈𝕃^𝑖_𝑢𝑘,𝑚≠𝑖, (5) where 𝜋_𝑖𝑗 and 𝜋_𝑖𝑗 are known lower and upper bounds of 𝜋_𝑖𝑗, respectively.

Remark 1. In order to get more general result, this paper considers that 𝕃^𝑖_𝑘 includes two cases. One is that 𝜋_𝑖𝑗 is exactly known, and the other is that it is unknown but with upper and lower known bounds, which has been considered in [8]. If 𝜋_𝑖𝑗 is exactly known, it also can be seen as a rate with equal bounds.
The set 𝕃 comprises the various operation modes of system (1), and for each possible value of 𝑟(𝑡)=𝑖(𝑖∈𝕃), the matrices and the vector-valued nonlinear functions associated with the 𝑖th mode will be denoted by 𝐴_𝑖=𝐴(𝑟(𝑡)=𝑖),𝐴_𝑑𝑖=𝐴_𝑑(𝑟(𝑡)),𝑓=𝑖_𝑖(𝑡)((𝑡)(𝑡)),𝑓=𝑓𝑡,𝑥,𝑟=𝑖_𝑑𝑖(𝑡)=𝑓_𝑑(𝑡,𝑥(𝑡−ℎ(𝑡)),𝑟(𝑡)=𝑖). (6)
In system (1), the vector-valued nonlinear functions 𝑓(∗) and 𝑓_𝑑(∗) are assumed to satisfy the following sector-bounded conditions: 𝑓(𝑥)−𝑓𝑦−𝑀₁𝑥−𝑦^𝑇×𝑓(𝑥)−𝑓𝑦−𝑀₂𝑓𝑥−𝑦≤0,_𝑑(𝑥)−𝑓_𝑑𝑦−𝑁₁𝑥−𝑦^𝑇×𝑓_𝑑(𝑥)−𝑓_𝑑𝑦−𝑁₂𝑥−𝑦≤0, (7) where for all 𝑥,𝑦∈𝑅^𝑛, 𝑀₁,𝑀₂,𝑁₁,𝑁₂∈𝑅_𝑛×𝑛 are known constant matrices. Without loss of generality, the following equations are always assumed: 𝑓(0)=0,𝑓_𝑑(0)=0. (8)

Remark 2. As in [20], the nonlinear functions 𝑓(∗), 𝑓_𝑑(∗) are said to belong to sectors. In other words, the nonlinearities are bounded by sectors. Further, both the filter design problems and control analysis have been investigated; see [21, 22]. The nonlinear descriptions in (7) are more general than the usual sigmoid functions and the recently commonly used Lipschitz conditions [21], and 𝑀₁, 𝑀₂, 𝑁₁, 𝑁₂ are lower and upper slope bounds, respectively.

Lemma 3. Assume that 𝑓(∗) is a vector-valued nonlinear function and 𝑀₁, 𝑀₂ are known constant matrices. If there exists 𝑓(𝑥)−𝑀₁𝑥^𝑇𝑓(𝑥)−𝑀₂𝑥≤0, (9) it can be obtained that 𝑥𝑓(𝑥)^𝑇𝑀₁𝑀₂𝑥∗𝐼𝑓(𝑥)≤0, (10) with 𝑀₁=(𝑀^𝑇₁𝑀₂+𝑀^𝑇₂𝑀₁)/2, 𝑀₂=−(𝑀^𝑇₁+𝑀^𝑇₂)/2.

Proof. Notice the fact that 𝑓(𝑥)−𝑀₁𝑥^𝑇𝑓(𝑥)−𝑀₂𝑥𝑓(𝑥)≤0,−𝑀₂𝑥^𝑇𝑓(𝑥)−𝑀₁𝑥≤0; (11) so, 0.5𝑓(𝑥)−𝑀₁𝑥^𝑇𝑓(𝑥)−𝑀₂𝑥+𝑓(𝑥)−𝑀₂𝑥^𝑇𝑓(𝑥)−𝑀₁𝑥≤0. (12) Then, (10) can be obtained. Thus, the proof is complete.

For the sake of simplicity, the solution 𝑥(𝑡,𝑥₀,𝑟₀) of system (1) with 𝑟₀∈𝕃 is denoted by 𝑥(𝑡). It is known that {𝑥(𝑡),𝑟(𝑡)} is a Markov process with an initial state (𝑥₀,𝑟₀), and its weak infinitesimal generator acting on function 𝑉 is defined as follows: 𝜉𝑉(𝑥(𝑡),𝑟(𝑡),𝑡)=lim_Δ→0+1Δ[𝜀(𝑉(𝑥()((𝑡)(𝑡))(𝑥(𝑡)(𝑡).𝑡+Δ,𝑡+Δ,𝑟𝑡+Δ))∣𝑥,𝑟=𝑖−𝑉,𝑟,𝑡)] (13)

Throughout this paper, the following definition is used.

Definition 4. System (1) is said to be stochastically stable if for finite 𝜓(𝑡)∈𝑅^𝑛 defined on [−ℎ,0] and 𝑟(0)∈𝕃, the following is satisfied: lim_𝑡→∞𝜀^𝑡₀𝑥^𝑇𝑠,𝜓,𝑟(0)𝑥𝑠,𝜓,𝑟(0)𝑑𝑠<∞, (14) where 𝑥(𝑠,𝜓,𝑟(0)) denotes the solution to system at time 𝑡 under the initial conditions 𝜓(𝑡) and 𝑟(0).

3. Stochastic Stability Analysis

In this section, new LMI-based sufficient conditions for system (1) will be presented based on a special Lyapunov-Krasovskii functional and Lemma 3.

For presentation convenience, it is necessary to define 𝜒_𝑖(𝑡)=𝑥^𝑇(𝑡),𝑥^𝑇(𝑡−ℎ(𝑡)),𝑥^𝑇𝑡−ℎ,𝑥̇^𝑇(𝑡),𝑓^𝑇_𝑖(𝑡),𝑓^𝑇_𝑑𝑖(𝑡)^𝑇=𝑒₁𝑥(𝑡)+𝑒₂𝑥(𝑡−ℎ(𝑡))+𝑒₃𝑥𝑡−ℎ+𝑒₄𝑥̇(𝑡)+𝑒₅𝑓_𝑖(𝑡)+𝑒₆𝑓_𝑑𝑖(𝑡), (15) and the corresponding block entry matrices are 𝑒_𝑗, 𝑗∈{1,…,6}. In addition, there are some other definitions. Consider 𝜆^𝑖_𝑘=_{𝑚∈ℒ𝑖𝑘}𝜋_𝑖𝑚,𝜆^𝑖_𝑘=_{𝑚∈ℒ𝑖𝑘}𝜋_𝑖𝑚,𝜆^𝑖_𝑢𝑘=_{𝑚∈ℒ𝑖𝑢𝑘}𝜋_𝑖𝑚,𝒫^𝑖_𝑘=_{𝑚∈ℒ𝑖𝑘}𝜋_𝑖𝑗𝑃_𝑚,𝒫^𝑖_𝑘=_{𝑚∈ℒ𝑖𝑘}𝜋_𝑖𝑗𝑃_𝑚, (16)

Theorem 5. For given scalars ℎ>0 and 𝜇, the delayed system (1) with ℎ̇(𝑡)≤𝜇 is stochastically stable if there exist scalars 𝜂_1𝑖>0 and 𝜂_2𝑖>0 and matrices 𝑃_𝑖>0, 𝑄_0𝑖>0, 𝑄_1𝑖>0, 𝑆₀>0, 𝑆₁>0, 𝑅>0, 𝑌^𝑖₁₁,𝑌^𝑖₁₂,𝑌^𝑖₂₂,𝑍^𝑖₁₁,𝑍^𝑖₁₂,𝑍^𝑖₂₂, Φ_𝑖, 𝐻_1𝑖, and 𝐻_2𝑖 such that the following conditions hold: 𝐻𝑒Φ_𝑖𝐴_𝑖𝑒^𝑇₁+𝐴_𝑑𝑖𝑒^𝑇₂−𝑒^𝑇₄+𝑒^𝑇₅+𝑒^𝑇₆+𝑍^𝑖₁₂𝑒₂−𝑒₃^𝑇+𝑌^𝑖₁₂𝑒₁−𝑒₂^𝑇+𝐻𝑒𝑒₄𝑃_𝑖𝑒^𝑇₁−𝑒₃𝑄_0𝑖𝑒^𝑇₃+ℎ𝑒₄𝑆₀+𝑆₁𝑒^𝑇₄−1−𝜇𝑒₂𝑄_1𝑖𝑒^𝑇₂+ℎ𝑌^𝑖₁₁+𝑒₁ℎ𝑅+𝑄_0𝑖+𝑄_1𝑖+Ψ_𝑖𝑒^𝑇₁−𝜂_1𝑖𝑒₁𝑀_1𝑖𝑒^𝑇₁+𝑒₅𝑀^𝑇_2𝑖𝑒^𝑇₁+𝑒₁𝑀_2𝑖𝑒^𝑇₅+𝑒₅𝑒^𝑇₅−𝜂_2𝑖𝑒₂𝑁_1𝑖𝑒^𝑇₂+𝑒₆𝑁^𝑇_2𝑖𝑒^𝑇₂+𝑒₂𝑁_2𝑖𝑒^𝑇₆+𝑒₆𝑒^𝑇₆Φ<0,𝐻𝑒_𝑖𝐴_𝑖𝑒^𝑇₁+𝐴_𝑑𝑖𝑒^𝑇₂−𝑒^𝑇₄+𝑒^𝑇₅+𝑒^𝑇₆+𝑍^𝑖₁₂𝑒₂−𝑒₃^𝑇+𝑌^𝑖₁₂𝑒₁−𝑒₂^𝑇+𝐻𝑒𝑒₄𝑃_𝑖𝑒^𝑇₁−𝑒₃𝑄_0𝑖𝑒^𝑇₃+ℎ𝑒₄𝑆₀𝑒^𝑇₄−1−𝜇𝑒₂𝑄_1𝑖𝑒^𝑇₂+ℎ𝑍^𝑖₁₁+𝑒₁ℎ𝑅+𝑄_0𝑖+𝑄_1𝑖+Ψ_𝑖𝑒^𝑇₁−𝜂_1𝑖𝑒₁𝑀_1𝑖𝑒^𝑇₁+𝑒₅𝑀^𝑇_2𝑖𝑒^𝑇₁+𝑒₁𝑀_2𝑖𝑒^𝑇₅+𝑒₅𝑒^𝑇₅−𝜂_2𝑖𝑒₂𝑁_1𝑖𝑒^𝑇₂+𝑒₆𝑁^𝑇_2𝑖𝑒^𝑇₂+𝑒₂𝑁_2𝑖𝑒^𝑇₆+𝑒₆𝑒^𝑇₆𝑌<0,_𝑖11𝑌^𝑖₁₂∗𝑌^𝑖₂₂𝑍≥0,^𝑖11𝑍^𝑖₁₂∗𝑍^𝑖₂₂𝑆≥0,₀≥𝑍^𝑖₂₂,𝑆₀+1−𝜇𝑆₁≥𝑌^𝑖₂₂,𝑖∈𝕃^𝑖_𝑘𝑚∈∑_ℒ𝑖𝑘𝜋_𝑖𝑚𝑄_0𝑚+𝑄_1𝑚+𝜋_𝑖𝑖𝑄_0𝑖+𝑄_1𝑖−𝜆^𝑖_𝑘+𝜋_𝑖𝑖𝑄_0𝑗+𝑄_1𝑗−𝑅<0,𝑗∈𝕃^𝑖_𝑢𝑘,_𝑚∈∑_ℒ𝑖𝑘𝜋_𝑖𝑚𝑄_0𝑚+𝜋_𝑖𝑖𝑄_0𝑖−𝜆^𝑖_𝑘+𝜋_𝑖𝑖𝑄_0𝑗−𝑅<0,𝑗∈𝕃^𝑖_𝑢𝑘,𝑖∈𝕃^𝑖_𝑢𝑘𝑚∑_{∈𝕃𝑖𝑘}𝜋_𝑖𝑚𝑄_0𝑚+𝑄_1𝑚−∑_{𝑚∈𝕃𝑖𝑘}𝜋_𝑖𝑚𝐻_2𝑖∑−𝑅<0,_{𝑚∈𝕃𝑖𝑘}𝜋_𝑖𝑚𝑄_0𝑚−∑_{𝑚∈𝕃𝑖𝑘}𝜋_𝑖𝑚𝐻_2𝑖𝑄−𝑅<0,_0𝑗+𝑄_1𝑗−𝐻_2𝑖≤0,𝑗∈𝕃^𝑖_𝑢𝑘𝑄,𝑗≠𝑖,_0𝑗−𝐻_2𝑖≥0,𝑗∈𝕃^𝑖_𝑢𝑘𝑃,𝑗=𝑖,_𝑗−𝐻_1𝑖≤0,𝑗∈𝕃^𝑖_𝑢𝑘𝑃,𝑗≠𝑖,_𝑗−𝐻_1𝑖≥0,𝑗∈𝕃^𝑖_𝑢𝑘,𝑗=𝑖, (17)(18)(19)(20)(21)(22) where Ψ_𝑖=𝒫^𝑖_𝑘+𝜋_𝑖𝑖𝑃_𝑖−𝜋_𝑖𝑖+𝜆^𝑖_𝑘𝑃_𝑙,𝑖∈𝕃^𝑖_𝑘,𝑙∈𝕃^𝑖_𝑢𝑘,𝒫^𝑖_𝑘−∑_{𝑚∈𝕃𝑖𝑘}𝜋_𝑖𝑚𝐻_1𝑖,𝑖∈𝕃^𝑖_𝑢𝑘,𝑀₁=𝑀^𝑇₁𝑀₂+𝑀^𝑇₂𝑀₁2,𝑀₂𝑀=−^𝑇1+𝑀^𝑇₂2,𝑁₁=𝑁^𝑇₁𝑁₂+𝑁^𝑇₂𝑁₁2,𝑁₂𝑁=−^𝑇1+𝑁^𝑇₂2, (23)

Proof. First, in order to cast the model involved into the framework of the Markov processes, a new process is defined as {(𝑥_𝑡,𝑟(𝑡)),𝑡≥0} by 𝑥_𝑡(𝑠)=𝑥(𝑡+𝑠),−2ℎ≤𝑠≤0; (24) so, {(𝑥_𝑡,𝑟(𝑡)),𝑡≥ℎ} is a Markov process with initial state (𝜓(⋅),𝑟(0)).
Then, the stochastic Lyapunov-Krasovskii functional is chosen as follows: 𝑉𝑥_𝑡,𝑟(𝑡),𝑡=⁶_𝑗=1𝑉_𝑗𝑥_𝑡,𝑟(𝑡),𝑡, (25) where 𝑉₁𝑥_𝑡,𝑟(𝑡),𝑡=𝑥^𝑇(𝑡)𝑃(𝑟(𝑡𝑥(𝑡),𝑉))₂𝑥_𝑡,𝑟(𝑡),𝑡=^𝑡_𝑡−ℎ𝑥^𝑇(𝛼)𝑄₀(𝑟(𝑡𝑥(𝛼)𝑉))𝑑𝛼,₃𝑥_𝑡,𝑟(𝑡),𝑡=^𝑡_{(𝑡)𝑡−ℎ}𝑥^𝑇(𝛼)𝑄₁(𝑟(𝑡𝑥(𝛼)𝑉))𝑑𝛼,₄𝑥_𝑡,𝑟(𝑡),𝑡=⁰_−ℎ^𝑡_𝑡+𝛽𝑥̇^𝑇(𝛼)𝑆₀𝑥̇(𝛼)𝑉𝑑𝛼𝑑𝛽,₅𝑥_𝑡,𝑟(𝑡),𝑡=⁰_(𝑡)−ℎ^𝑡_𝑡+𝛽𝑥̇^𝑇(𝛼)𝑆₁𝑥̇(𝛼)𝑉𝑑𝛼𝑑𝛽,₆𝑥_𝑡,𝑟(𝑡),𝑡=⁰_−ℎ^𝑡_𝑡+𝛽𝑥^𝑇(𝛼)(𝛼)𝑃(𝑟(𝑡𝑅𝑥𝑑𝛼𝑑𝛽,))>0,𝑄₀(𝑟(𝑡))>0,𝑄₁(𝑟(𝑡𝑆))>0,₀>0,𝑆₁>0,𝑅>0. (26)
The infinitesimal generator of the Markov process acting on 𝑉(⋅) and emanating from the point 𝑟(𝑡)=𝑖(𝑖∈𝕃) is given as follows: 𝜉𝑉𝑥_𝑡,𝑟(𝑡),𝑡=2𝜒^𝑇_𝑖(𝑡)𝑒₄𝑃_𝑖𝑒^𝑇₁𝜒_𝑖(𝑡)+𝜒^𝑇_𝑖(𝑡)𝑒₁^ℕ_𝑗=1𝜋_𝑖𝑗𝑃_𝑗𝑒^𝑇₁+𝑒₁𝑄_0𝑖𝑒^𝑇₁−𝑒₃𝑄_0𝑖𝑒^𝑇₃𝜒_𝑖(𝑡)+^𝑡_𝑡−ℎ𝑥^𝑇(𝛼)^ℕ_𝑗=1𝜋_𝑖𝑗𝑄_0𝑗𝑥(𝛼)𝑑𝛼+𝜒^𝑇_𝑖(𝑡)𝑒₁𝑄_1𝑖𝑒^𝑇₁−1−ℎ̇(𝑡)𝑒₂𝑄_1𝑖𝑒^𝑇₂𝜒_𝑖(𝑡)+^𝑡_{(𝑡)𝑡−ℎ}𝑥^𝑇(𝛼)^ℕ_𝑗=1𝜋_𝑖𝑗𝑄_1𝑗𝑥(𝛼)+𝑑𝛼ℎ𝜒^𝑇_𝑖(𝑡)𝑒₄𝑆₀𝑒^𝑇₄𝜒_𝑖(𝑡)−^𝑡_𝑡−ℎ𝑥̇^𝑇(𝛼)𝑆₀𝑥̇(𝛼)𝑑𝛼+ℎ(𝑡)𝜒^𝑇_𝑖(𝑡)𝑒₄𝑆₁𝑒^𝑇₄𝜒_𝑖(𝑡)−1−ℎ̇(𝑡)^𝑡_{(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝛼)𝑆₁𝑥̇(𝛼)+𝑑𝛼ℎ𝜒^𝑇_𝑖(𝑡)𝑒₁𝑅𝑒^𝑇₁𝜒_𝑖(𝑡)−^𝑡_𝑡−ℎ𝑥^𝑇(𝛼)𝑅𝑥(𝛼)𝑑𝛼. (27) Then, choose 𝑌^𝑖₁₁𝑌^𝑖₁₂∗𝑌^𝑖₂₂𝑍≥0,^𝑖11𝑍^𝑖₁₂∗𝑍^𝑖₂₂≥0; (28) it follows that _{𝑡(𝑡)𝑡−ℎ}𝜒_𝑖(𝑡)𝑥̇(𝛼)^𝑇𝑌^𝑖₁₁𝑌^𝑖₁₂∗𝑌^𝑖₂₂𝜒_𝑖(𝑡)𝑥̇(𝛼)𝑑𝛼=𝜒^𝑇_𝑖(𝑡)𝐻𝑒𝑌^𝑖₁₂𝑒₁−𝑒₂^𝑇+ℎ(𝑡)𝑌^𝑖₁₁𝜒_𝑖(𝑡)+^𝑡_{(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝛼)𝑌₂₂𝑥̇(𝛼)𝑑𝛼≥0,_{𝑡−ℎ(𝑡)𝑡−ℎ}𝜒_𝑖(𝑡)𝑥̇(𝛼)^𝑇𝑍^𝑖₁₁𝑍^𝑖₁₂∗𝑍^𝑖₂₂𝜒_𝑖(𝑡)𝑥̇(𝛼)𝑑𝛼=𝜒^𝑇_𝑖(𝑡)𝐻𝑒𝑍^𝑖₁₂𝑒₂−𝑒₃^𝑇+ℎ−ℎ(𝑡)𝑍^𝑖₁₁𝜒_𝑖(𝑡)+^{𝑡−ℎ(𝑡)}_𝑡−ℎ𝑥̇^𝑇(𝛼)𝑍₂₂𝑥̇(𝛼)𝑑𝛼≥0. (29) And the constraints of the model dynamics can be written as 𝜒^𝑇_𝑖(𝑡)Φ_𝑖[𝐴_𝑖𝑒^𝑇₁+𝐴_𝑑𝑖𝑒^𝑇₂−𝑒^𝑇₄+𝑒^𝑇₅+𝑒^𝑇₆]𝜒_𝑖(𝑡)≡0 by introducing free variables Φ_𝑖. In addition, according to Lemma 3 and choosing scalar 𝜂_1𝑖>0, 𝜂_2𝑖>0, the following inequalities hold: −𝜂_1𝑖𝜒^𝑇_𝑖(𝑡)𝑒₁𝑀_1𝑖𝑒^𝑇₁+𝑒₅𝑀^𝑇_2𝑖𝑒^𝑇₁+𝑒₁𝑀_2𝑖𝑒^𝑇₅+𝑒₅𝑒^𝑇₅𝜒_𝑖(𝑡)≥0,−𝜂_2𝑖𝜒^𝑇_𝑖(𝑡)𝑒₂𝑁_1𝑖𝑒^𝑇₂+𝑒₆𝑁^𝑇_2𝑖𝑒^𝑇₂+𝑒₂𝑁_2𝑖𝑒^𝑇₆+𝑒₆𝑒^𝑇₆𝜒_𝑖(𝑡)≥0. (30)
So, 𝜉𝑉(𝑥(𝑡),𝑟(𝑡),𝑡) can be upper bounded as 𝜉𝑉𝑥_𝑡,𝑟(𝑡),𝑡≤𝜒^𝑇_𝑖(𝑡)Ω₁(ℎ(𝑡))+𝑒₁^ℕ_𝑗=1𝜋_𝑖𝑗𝑃_𝑗𝑒^𝑇₁𝜒_𝑖(𝑡)+Ω₂+Ω₃, (31) where Ω₁(ℎ(𝑡))=ℎ(𝑡)𝑌^𝑖₁₁+𝑒₄𝑆₁𝑒^𝑇₄+ℎ−ℎ(𝑡)𝑍^𝑖₁₁+𝐻𝑒Φ_𝑖𝐴_𝑖𝑒^𝑇₁+𝐴_𝑑𝑖𝑒^𝑇₂−𝑒^𝑇₄+𝑒^𝑇₅+𝑒^𝑇₆+𝑒₄𝑃_𝑖𝑒^𝑇₁+𝑌^𝑖₁₂𝑒₁−𝑒₂^𝑇+𝑍^𝑖₁₂𝑒₂−𝑒₃^𝑇+ℎ𝑒₄𝑆₀𝑒^𝑇₄−𝑒₃𝑄_0𝑖𝑒^𝑇₃−1−𝜇𝑒₂𝑄_1𝑖𝑒^𝑇₂+𝑒₁𝑄_0𝑖+𝑄_1𝑖+ℎ𝑅𝑒^𝑇₁−𝜂_1𝑖𝑒₁𝑀_1𝑖𝑒^𝑇₁+𝑒₅𝑀^𝑇_2𝑖𝑒^𝑇₁+𝑒₁𝑀_2𝑖𝑒^𝑇₅+𝑒₅𝑒^𝑇₅−𝜂_2𝑖𝑒₂𝑁_1𝑖𝑒^𝑇₂+𝑒₆𝑁^𝑇_2𝑖𝑒^𝑇₂+𝑒₂𝑁_2𝑖𝑒^𝑇₆+𝑒₆𝑒^𝑇₆,Ω₂=−^{𝑡−ℎ(𝑡)}_𝑡−ℎ𝑥̇^𝑇(𝛼)𝑆₀−𝑍^𝑖₂₂𝑥̇(𝛼)𝑑𝛼−^𝑡_{(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝛼)𝑆₀−𝑌^𝑖₂₂+1−𝜇𝑆₁𝑥̇(𝛼)Ω𝑑𝛼,₃=^𝑡_{(𝑡)𝑡−ℎ}𝑥^𝑇(𝛼)^ℕ_𝑗=1𝜋_𝑖𝑗𝑄_0𝑗+𝑄_1𝑗𝑥(𝛼)+−𝑅𝑑𝛼_{𝑡−ℎ(𝑡)𝑡−ℎ}𝑥^𝑇(𝛼)^ℕ_𝑗=1𝜋_𝑖𝑗𝑄_0𝑗−𝑅𝑥(𝛼)𝑑𝛼. (32)
Since Ω₁(ℎ(𝑡)) is a convex combination of the matrices 𝑌^𝑖₁₁+𝑒₄𝑆₁𝑒^𝑇₄ and 𝑍^𝑖₁₁ on ℎ(𝑡), it can be handled nonconservatively by two corresponding boundary LMIs: one for ℎ(𝑡)=ℎ and the other for ℎ(𝑡)=0.
So, Ω₁(ℎ(𝑡))<0 is equivalent to Ω₁(0)Ω<0,₁ℎ<0. (33) Then, the proof is separated into two cases, 𝑖∈𝕃^𝑖_𝑘 and 𝑖∈𝕃^𝑖_𝑢𝑘.
Case 1 (𝑖∈𝕃^𝑖_𝑘). In this case, 𝜋_𝑖𝑖 is bounded or exactly known. Based on ∑^ℕ_𝑗=1𝜋_𝑖𝑗=𝜆^𝑖_𝑘+𝜋_𝑖𝑖+𝜆^𝑖_𝑢𝑘=0 and −∑_{𝑙∈ℒ𝑖𝑢𝑘}𝜋_𝑖𝑙/(𝜋_𝑖𝑖+𝜆^𝑖_𝑘)=1, it is easy to obtain that Ω₁(ℎ(𝑡))+𝑒₁^ℕ_𝑗=1𝜋_𝑖𝑗𝑃_𝑗𝑒^𝑇₁=Ω₁(ℎ(𝑡))+𝑒_{1𝑗∈ℒ𝑖𝑘}𝜋_𝑖𝑗𝑃_𝑗+𝜋_𝑖𝑖𝑃_𝑖+_{𝑙∈ℒ𝑖𝑢𝑘}𝜋_𝑖𝑙𝑃_𝑙𝑒^𝑇₁∑=−^{𝑙∈ℒ𝑖𝑢𝑘}𝜋_𝑖𝑙𝜋_𝑖𝑖+𝜆^𝑖_𝑘Ω₁(ℎ(𝑡))+𝑒_{1𝑗∈ℒ𝑖𝑘}𝜋_𝑖𝑗𝑃_𝑗+𝜋_𝑖𝑖𝑃_𝑖−𝜋_𝑖𝑖+𝜆^𝑖_𝑘𝑃_𝑙𝑒^𝑇₁. (34)
Considering the fact that 0≤−𝜋_𝑖𝑙/(𝜋_𝑖𝑖+𝜆^𝑖_𝑘)≤1, it is easy to obtain that Ω₁(ℎ(𝑡)) + 𝑒₁[∑_{𝑗∈ℒ𝑖𝑘}𝜋_𝑖𝑗𝑃_𝑗 + 𝜋_𝑖𝑖𝑃_𝑖−(𝜋_𝑖𝑖 + 𝜆^𝑖_𝑘)𝑃_𝑙]𝑒^𝑇₁<0, 𝑙∈𝕃^𝑖_𝑢𝑘 can guarantee Ω₁(ℎ(𝑡))+𝑒₁∑^ℕ_𝑗=1𝜋_𝑖𝑗𝑃_𝑗𝑒^𝑇₁<0.
Since 𝜋_𝑖𝑗≤𝜋_𝑖𝑗≤𝜋_𝑖𝑗, the following inequality can be obtained: Ω₁(ℎ(𝑡))+𝑒_{1𝑗∈ℒ𝑖𝑘}𝜋_𝑖𝑗𝑃_𝑗+𝜋_𝑖𝑖𝑃_𝑖−𝜋_𝑖𝑖+𝜆^𝑖_𝑘𝑃_𝑙𝑒^𝑇₁≤Ω₁(ℎ(𝑡))+𝑒₁𝒫^𝑖_𝑘+𝜋_𝑖𝑖𝑃_𝑖−𝜋_𝑖𝑖+𝜆^𝑖_𝑘𝑃_𝑙𝑒^𝑇₁. (35) So, if (17) and (18) hold, Ω₁(ℎ(𝑡))+𝑒₁∑^ℕ_𝑗=1𝜋_𝑖𝑗𝑃_𝑗𝑒^𝑇₁<0 can be obtained.
Similarly, (21) ⇒Ω₃<0.
And (20) ⇒Ω₂<0.
Therefore, when (17)–(21) are satisfied, 𝜉𝑉(𝑥_𝑡,𝑟(𝑡),𝑡)<0.
Case 2 (𝑖∈𝕃^𝑖_𝑢𝑘). Because ∑^ℕ_𝑗=1𝜋_𝑖𝑗=0, there exist −𝑥^𝑇(𝑡)^ℕ_𝑗=1𝜋_𝑖𝑗𝐻_1𝑖𝑥(𝑡)−=0,_{𝑡𝑡−ℎ}𝑥^𝑇(𝛼)^ℕ_𝑗=1𝜋_𝑖𝑗𝐻_2𝑖𝑥(𝛼)𝑑𝛼=0. (36) Adding the left sides of (36) into (31), the inequality can be written as 𝜉𝑉𝑥_𝑡,𝑟(𝑡),𝑡≤𝜒^𝑇(𝑡)Γ₁(ℎ(𝑡))𝜒(𝑡)+Γ₂+Γ₃, (37) where Γ₁(ℎ(𝑡))=ℎ(𝑡)𝑌^𝑖₁₁+𝑒₄𝑆₁𝑒^𝑇₄+ℎ−ℎ(𝑡)𝑍^𝑖₁₁+𝐻𝑒Φ_𝑖𝐴_𝑖𝑒^𝑇₁+𝐴_𝑑𝑖𝑒^𝑇₂−𝑒^𝑇₄+𝑒^𝑇₅+𝑒^𝑇₆+𝑒₄𝑃_𝑖𝑒^𝑇₁+𝑌^𝑖₁₂𝑒₁−𝑒₂^𝑇+𝑍^𝑖₁₂𝑒₂−𝑒₃^𝑇+ℎ𝑒₄𝑆₀𝑒^𝑇₄−𝑒₃𝑄_0𝑖𝑒^𝑇₃−1−𝜇𝑒₂𝑄_1𝑖𝑒^𝑇₂+𝑒₁𝑄_0𝑖+𝑄_1𝑖+ℎ𝑅+_{𝑗∈𝕃𝑖𝑘}𝜋_𝑖𝑗𝑃_𝑗−𝐻_1𝑖𝑒^𝑇₁+𝑒_{1𝑗∈𝕃𝑖𝑢𝑘}𝜋_𝑖𝑗𝑃_𝑗−𝐻_1𝑖𝑒^𝑇₁−𝜂_1𝑖𝑒₁𝑀_1𝑖𝑒^𝑇₁+𝑒₅𝑀^𝑇_2𝑖𝑒^𝑇₁+𝑒₁𝑀_2𝑖𝑒^𝑇₅+𝑒₅𝑒^𝑇₅−𝜂_2𝑖𝑒₂𝑁_1𝑖𝑒^𝑇₂+𝑒₆𝑁^𝑇_2𝑖𝑒^𝑇₂+𝑒₂𝑁_2𝑖𝑒^𝑇₆+𝑒₆𝑒^𝑇₆,Γ₂=Ω₂,Γ₃=^𝑡_{(𝑡)𝑡−ℎ}𝑥^𝑇(𝛼)_{𝑗∈𝕃𝑖𝑘}𝜋_𝑖𝑗𝑄_0𝑗+𝑄_1𝑗−𝐻_2𝑖−𝑅+_{𝑗∈𝕃𝑖𝑢𝑘}𝜋_𝑖𝑗𝑄_0𝑗+𝑄_1𝑗−𝐻_2𝑖𝑥(𝛼)+𝑑𝛼_{𝑡−ℎ(𝑡)𝑡−ℎ}𝑥^𝑇(𝛼)^ℕ_{𝑗∈𝕃𝑖𝑘}𝜋_𝑖𝑗𝑄_0𝑗−𝐻_2𝑖+−𝑅_{ℕ𝑗∈𝕃𝑖𝑢𝑘}𝜋_𝑖𝑗𝑄_0𝑗−𝐻_2𝑖𝑥(𝛼)𝑑𝛼. (38)
So, Γ₁(ℎ(𝑡)) is also a convex combination of the matrices 𝑌^𝑖₁₁+𝑒₄𝑆₁𝑒^𝑇₄ and 𝑍^𝑖₁₁ on ℎ(𝑡).
It is known that 𝜋_𝑖𝑖<0. At the same time, the boundary information 𝜋_𝑖𝑗≤𝜋_𝑖𝑗≤𝜋_𝑖𝑗 of the transition probabilities need to be used. It is clear that if (17)–(20) and (22) hold, 𝜉𝑉(𝑥_𝑡,𝑟(𝑡),𝑡)<0 can be achieved.
Taking into account (17)–(22), there exists a scalar 𝑐>0 for each 𝑖∈𝕃 such that 𝜉𝑉(𝑥(𝑡),𝑟(𝑡),𝑡)≤−𝑐‖𝑥(𝑡)‖², (39) which implies that system (1) is stochastically stable.
Therefore, the proof is completed.

If 𝑓(𝑡,𝑥(𝑡),𝑟(𝑡))=Δ𝐴(𝑡,𝑟(𝑡))𝑥(𝑡), 𝑓_𝑑(𝑡,𝑥(𝑡−ℎ(𝑡)), 𝑟(𝑡))=Δ𝐴_𝑑(𝑡,𝑟(𝑡))𝑥(𝑡−ℎ(𝑡)), system (1) becomes 𝑥̇(𝑡)=(𝐴(𝑟(𝑡((𝑡𝑥(𝑡)+𝐴))+Δ𝐴𝑡,𝑟)))_𝑑(𝑟(𝑡))+Δ𝐴_𝑑((𝑡𝑥((𝑡,𝑥(𝑡)(𝑡)𝑡,𝑟))𝑡−ℎ))=𝜓,−ℎ≤𝑡≤0, (40) where Δ𝐴(𝑡,𝑟(𝑡)), Δ𝐴_𝑑(𝑡,𝑟(𝑡)) are unknown time-varying matrices with appropriate dimensions and are assumed to be of the following form: Δ𝐴(𝑡,𝑟(𝑡))Δ𝐴_𝑑((𝑡(𝑟(𝑡𝐸((𝑡𝐹(𝑟(𝑡𝐹𝑡,𝑟))=𝐷))𝑡,𝑟))))_𝑑(𝑟(𝑡)), (41) where 𝐷(𝑟(𝑡)), 𝐹(𝑟(𝑡)), 𝐹_𝑑(𝑟(𝑡)) are known real constant matrices for all 𝑟(𝑡)∈𝕃, and 𝐸(𝑡,𝑟(𝑡)), for all 𝑟(𝑡)∈𝕃, are the uncertain time-varying matrices satisfying 𝐸(𝑡,𝑟(𝑡))^𝑇𝐸(𝑡,𝑟(𝑡))≤𝐼,∀𝑟(𝑡)∈𝕃. (42)

It can be easily seen that system (40) is just the MJSs with norm bounded uncertainty (41).

Choosing 𝑝(𝑡,𝑟(𝑡))=𝐸(𝑡,𝑟(𝑡))𝑞(𝑡,𝑟(𝑡)) and 𝑞(𝑡,𝑟(𝑡))=𝐹(𝑟(𝑡))𝑥(𝑡)+𝐹_𝑑(𝑟(𝑡))𝑥(𝑡−ℎ(𝑡)), system (40) is equivalent to the following system: 𝑥̇(𝑡)=𝐴(𝑟(𝑡))𝑥(𝑡)+𝐴_𝑑(𝑟(𝑡𝑥((𝑡(𝑟(𝑡𝑝((𝑡,𝑥(𝑡)(𝑡)))𝑡−ℎ))+𝐷))𝑡,𝑟))=𝜓,−ℎ≤𝑡≤0. (43)

For each possible value of 𝑟(𝑡)=𝑖  (𝑖∈𝕃), 𝑝(𝑡,𝑟(𝑡)), 𝐷(𝑟(𝑡)), 𝐹(𝑟(𝑡)), and 𝐹_𝑑(𝑟(𝑡)) can be denoted as 𝑝_𝑖(𝑡), 𝐷_𝑖, 𝐹_𝑖, and 𝐹_𝑑𝑖. Then, it is necessary to choose 𝜒_𝑖(𝑡)=[𝑥^𝑇(𝑡),𝑥^𝑇(𝑡−ℎ(𝑡)),𝑥^𝑇(𝑡−ℎ),𝑥̇^𝑇(𝑡),𝑝^𝑇_𝑖(𝑡)]^𝑇 and the corresponding block entry matrices as 𝑒_𝑗,𝑗∈{1,…,5}.

Then, the following corollary is obtained.

Corollary 6. If there exist matrices 𝑃_𝑖>0, 𝑄_0𝑖>0, 𝑄_1𝑖>0, 𝑆₀>0, 𝑆₁>0, 𝑅>0, 𝑌^𝑖₁₁, 𝑌^𝑖₁₂, 𝑌^𝑖₂₂, 𝑍^𝑖₁₁, 𝑍^𝑖₁₂, 𝑍^𝑖₂₂, Φ_𝑖, 𝐻_1𝑖, and 𝐻_2𝑖 such that (19)–(22) and the following LMIs are feasible for 𝑖=1,2…,ℕ: 𝐻𝑒Φ_𝑖𝐴_𝑖𝑒^𝑇₁+𝐴_𝑑𝑖𝑒^𝑇₂−𝑒^𝑇₄+𝐷_𝑖𝑒^𝑇₅+𝑍₁₂𝑒₂−𝑒₃^𝑇+𝑌₁₂𝑒₁−𝑒₂^𝑇+𝐻𝑒𝑒₄𝑃_𝑖𝑒^𝑇₁−𝑒₃𝑄_0𝑖𝑒^𝑇₃+ℎ𝑒₄𝑆₀+𝑆₁𝑒^𝑇₄−1−𝜇𝑒₂𝑄_1𝑖𝑒^𝑇₂+ℎ𝑌₁₁+𝑒₁ℎ𝑅+𝑄_0𝑖+𝑄_1𝑖+Ψ_𝑖𝑒^𝑇₁+𝑒₁𝐹^𝑇_𝑖+𝑒₂𝐹^𝑇_𝑑𝑖𝐹_𝑖𝑒^𝑇₁+𝐹_𝑑𝑖𝑒^𝑇₂−𝑒₅𝑒^𝑇₅Φ<0,𝐻𝑒_𝑖𝐴_𝑖𝑒^𝑇₁+𝐴_𝑑𝑖𝑒^𝑇₂−𝑒^𝑇₄+𝐷_𝑖𝑒^𝑇₅+𝑍₁₂𝑒₂−𝑒₃^𝑇+𝑌₁₂𝑒₁−𝑒₂^𝑇+𝐻𝑒𝑒₄𝑃_𝑖𝑒^𝑇₁−𝑒₃𝑄_0𝑖𝑒^𝑇₃+ℎ𝑒₄𝑆₀𝑒^𝑇₄−1−𝜇𝑒₂𝑄_1𝑖𝑒^𝑇₂+ℎ𝑍₁₁+𝑒₁ℎ𝑅+𝑄_0𝑖+𝑄_1𝑖+Ψ_𝑖𝑒^𝑇₁+𝑒₁𝐹^𝑇_𝑖+𝑒₂𝐹^𝑇_𝑑𝑖𝐹_𝑖𝑒^𝑇₁+𝐹_𝑑𝑖𝑒^𝑇₂−𝑒₅𝑒^𝑇₅<0, (44) then the delayed uncertain system (40) with partly known transition probabilities is stochastically stable.

Proof. Here, the constraints of the model dynamics can be written as 𝜒^𝑇_𝑖(𝑡)Φ_𝑖[𝐴_𝑖𝑒^𝑇₁+𝐴_𝑑𝑖𝑒^𝑇₂−𝑒^𝑇₄+𝐷_𝑖𝑒^𝑇₅]𝜒_𝑖(𝑡)≡0. And the additional uncertainty constraint (42) is given as follows: 0≤𝑞^𝑇(𝑡)𝑞(𝑡)−𝑝^𝑇(𝑡)𝑝(𝑡)=𝜒^𝑇_𝑖(𝑡)𝑒₁𝐹^𝑇_𝑖+𝑒₂𝐹^𝑇_𝑑𝑖𝐹_𝑖𝑒^𝑇₁+𝐹_𝑑𝑖𝑒^𝑇₂−𝑒₅𝑒^𝑇₅𝜒_𝑖(𝑡). (45)
Choose the Lyapunov functional as (25); so, 𝜉𝑉(𝑥(𝑡),𝑟(𝑡),𝑡)≤𝜉𝑉(𝑥(𝑡),𝑟(𝑡),𝑡)+𝑞^𝑇(𝑡)𝑞(𝑡)−𝑝^𝑇(𝑡)𝑝(𝑡). (46)
Use the same method as the proof of Theorem 5 to make 𝜉𝑉(𝑥(𝑡),𝑟(𝑡),𝑡)+𝑞^𝑇(𝑡)𝑞(𝑡)−𝑝^𝑇(𝑡)𝑝(𝑡)<0, and then Corollary 6 can be obtained.

4. Numerical Examples

In this section, numerical examples are given to demonstrate the benefits and effectiveness of the proposed methods.

First, Example 1 in [16] is given to show the superiority of Theorem 5 for MJS (1) with no nonlinear terms and completely known transition probability matrix.

Example 7. Consider MJS (1) with 𝑓(𝑡,𝑥(𝑡),𝑟(𝑡))=0 and 𝑓_𝑑(𝑡,𝑥(𝑡−ℎ(𝑡)),𝑟(𝑡))=0. The two modes are given as 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 (47) Supposing that 𝜇=0.8 and 𝜋₂₂=−0.8, compare the stochastic stability result in Theorem 5 with the stability theorem in [16, 18]. For given 𝜋₁₁, the maximum ℎ, which satisfies the LMIs in (17)–(22), can be calculated.

Table 1 presents the comparison results, which shows that the stochastic stability result in Theorem 5 is less conservative than those in [16, 18] when no nonlinear terms exist and transition probabilities are completely known.

Table 1 

Comparison of maximum allowed ℎ.

Next numerical example is for uncertain MJS (43) with more general transition probability.

Example 8. Consider the following MJS (43) with four modes, whose system matrices are 𝐴₁=−2.2460−1.4410−1.5937−2.9289,𝐴₂=,𝐴−1.89990.8156−0.6900−0.7881₃=−0.75230.75001.5630−2.3540,𝐴₄=,𝐴−1.78400.36401.3670−1.5640_𝑑1=−0.5−0.50.050.01,𝐴_𝑑2=,𝐴0.0100.05−0.01_𝑑3=−0.01−0.020.050.01,𝐴_𝑑4=,𝐹0.030.01−0.2−0.1₁=𝐹₂=𝐹₃=𝐹₄=,𝐹10_𝑑1=𝐹_𝑑2=𝐹_𝑑3=𝐹_𝑑4=,𝐷−50.5₁=𝐷₂=𝐷₃=𝐷₄=.0.10.1 (48)
The more general transition probability matrix is chosen as follows, which includes completely known elements, boundary known elements, and completely unknown elements: −1.30.2????0.30.30.6?−1.5????𝛼. (49) Choose −1.5≤𝛼≤−0.9. According to Corollary 6, Table 2 can be obtained for different values of 𝜇. Also, the uncertain MJS (43) is stochastically stable. Moreover, as the upper bound of ℎ̇(𝑡), that is, 𝜇, increases, the maximum allowed upper bound of ℎ(𝑡) may be reduced.
Furthermore, a nonlinear MJS with sector-bounded nonlinearity is given.

Table 2 

Maximum allowed ℎ by Corollary 6.

Example 9. Consider MJS (1) with four models, whose 𝐴_𝑖, 𝐴_𝑑𝑖 and the transition probability matrix are the same matrices as the ones of Example 8. The nonlinear functions are given as follows:𝑓₁(𝑡)=0.1𝑥₁(𝑡)cos²𝑥₁(𝑡)−0.05𝑥₁(𝑡)−𝑥₂(𝑡)−0.05𝑥₁(𝑡)−0.1𝑥₂(𝑡),𝑓₂(𝑡)=2𝑓₁(𝑡),𝑓₃(𝑡)=0.15𝑥₂(𝑡)cos²𝑥₁(𝑡)+𝑥₂(𝑡)+0.1𝑥₁(𝑡)+0.25𝑥₂(𝑡)0.15𝑥₁(𝑡)+0.05𝑥₂(𝑡),𝑓₄(𝑡)=2𝑓₃(𝑡),𝑓_𝑑1(𝑡)=𝑓_𝑑2(𝑡)=𝑓_𝑑3(𝑡)=𝑓_𝑑4(𝑡)=0.12𝑥₁(𝑡−ℎ(𝑡))sin²𝑥₂(𝑡−ℎ(𝑡))+0.04𝑥₁(𝑡−ℎ(𝑡))+0.08𝑥₂(𝑡−ℎ(𝑡))0.04𝑥₁(𝑡−ℎ(𝑡))+0.04𝑥₂((𝑡.𝑡−ℎ)) (50)According to (7), the corresponding matrices, which describe the sector bound, can be chosen as 𝑀₁₁=0.050.050.050.05,𝑀₁₂=2𝑀₁₁,𝑀₁₃=0.10.10.1−0.05,𝑀₁₄=2𝑀₁₃,𝑀₂₁=−0.050.05−0.15−0.25,𝑀₂₂=2𝑀₂₁,𝑀₂₃=0.10.250.20.15,𝑀₂₄=2𝑀₂₃,𝑁₁₁=𝑁₁₂=𝑁₁₃=𝑁₁₄=,𝑁0.040.08−0.040.12₂₁=𝑁₂₂=𝑁₂₃=𝑁₂₄=.0.160.080.12−0.04 (51) Choose −1.5≤𝛼≤−1. Using Matlab LMI toolbox and Theorem 5, the maximum allowed upper bound of ℎ(𝑡) can be obtained for different values of 𝜇.
From Table 3, it can be seen that the stochastic stability can be guaranteed for nonlinear MJS (1) when more general transition probabilities (4) are considered.

Table 3 

Maximum allowed ℎ by Theorem 5.

5. Conclusion

This paper has presented a new method for solving the stochastic stability analysis problem for MJS with time-varying delay and sector-bounded nonlinearity. A general transition probability matrix is adopted, which includes completely known elements, boundary known elements, and completely unknown elements. Based on a new mode-dependent stochastic Lyapunov-Krasovskii functional and some new techniques to deal with transition probabilities, sufficient conditions with less conservativeness are derived in terms of LMIs to obtain stochastic stability of systems. Numerical examples have also demonstrated the effectiveness of the results.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (nos. 61273155, 61273355, 61273356, 61035005), New Century Excellent Talents in University (no. NCET-11-0083), Foundation for the Author of National Excellent Doctoral Dissertation of China (no. 201157), the Foundation of State Key Laboratory of Robotics (no. 2012-001).