## Recent Developments on the Stability and Control of Stochastic Systems

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# Further Results on Stability Analysis for Markovian Jump Systems with Time-Varying Delays

**Academic Editor:**Leonid Shaikhet

#### Abstract

This paper is concerned with the problem of stability analysis for Markovian jump systems with time-varying delays. By constructing a newly augmented Lyapunov-Krasovskii functional and combining Wirtinger-based integral inequality, an improved delay-dependent stability criterion within the framework of linear matrix inequalities (LMIs) is introduced. Based on the result of delay-dependent stability criterion, when linear systems have fast time-varying delays, a corresponding stability condition is given. Via three numerical examples, the improvements of the proposed criteria are shown by comparing maximum delay bounds provided by our theorems with the recent results.

#### 1. Introduction

Stability analysis of dynamic systems is a prerequisite and essential job before designing a controller to achieve the prescribed specifications. In particular, a great concern of stability analysis for systems with time-delays has been received due to the fact that time-delay naturally occurs in many practical systems such as networked control system, chemical processing, hot rolling mill, synchronization between chaotic systems, neural networks, and multiagent systems. For instance, see [1, 2] and references therein.

The main issue in delay-dependent stability analysis for time-delay systems with the framework of LMIs is how to increase maximum delay bounds for guaranteeing the asymptotic stability of systems. Thus, the choosing of Lyapunov-Krasovskii functional (LKF) and some techniques in estimating an upper bound of time-derivative value of the constructed LKF are the most important factors in enhancing the stability feasible region. In the LKF aspect, quadratic form, single integral, and double integral of quadratic form are the most utilized functionals. Recently, since the triple integral form of LKF was introduced in [3], this form of LKF has been utilized in many works such as [4–6]. Moreover, in [4, 5], it was shown that some augmented LKFs can increase the feasible region of stability criteria. In estimating an upper bound of time-derivative value of LKF, Jensen’s inequality [7], free-weighting matrix technique [8], and reciprocally convex optimization theory [9] make big impacts on the enhancement of delay-dependent stability and stabilization. Seuret and Gouaisbaut [10] proposed the Wirtinger-based integral inequality which provides more tight lower bounds than Jensen’s inequality and showed that the utilization of Wiritinger-based integral inequality can improve maximum delay bounds in many systems such as systems with constant and known delay, systems with a time-varying delay, systems with a constant distributed delay, and sampled-date systems. Cheng and Xiong [11] reduced conservative condition of stabilization criteria for continuous-time systems with time-varying input by introducing a new integral inequality. Recently, in [12, 13], for neural network with time-varying delay, it can be confirmed that the utilization of Wirtinger-based integral inequality in obtaining an upper bound of time-derivative values of some augmented LKFs can provide larger delay bounds than some other literatures. Very recently, in [14], it was shown that the results obtained by [10] can be further improved by choosing some new augmented LKFs. From the statements mentioned above, one can see that the choosing of LKF and some techniques play key roles to reduce the conservatism of stability criteria.

On the other hand, increasing attention has been paid to Markovian jumping systems (MJSs) which are a special sort of hybrid systems and driven by Markov chain. MJSs may undergo unexpected changes in their structure and parameters including economic systems, aerospace systems, power systems, and networked control system [15, 16]. Very recently, a survey on recent developments of modeling, analysis, and design of MJSs was reported in Shi and Li [17].

In this regard, many researchers put their times and efforts into stability and stabilization of Markovian jumping systems with time-delays. In [18], the problems of robust control and filtering for uncertain MJSs with time-varying delays were investigated by utilizing bounded real lemma. In [19], some new results on stabilization of MJSs with time-delays were proposed based on a delay-partitioning approach. Wu et al. [20] investigated the problem of stability and filtering for singular Markovian jump systems with time-delay via a delay-dependent bounded real lemma. Li et al. [21] utilized an input-output approach to stability and stabilization of MJSs with time-varying delays and showed the reduction of conservatism of the concerned criteria by a precise approximation of time-varying delay. By constructing new LKFs having distinct Lyapunov matrices for different modes, the mean square exponential stability and stabilization problems were studied in [22] for MJSs with constant time-delays. In [23], improved delay-dependent stability and control for singular Markovian jump systems with time-delay by utilizing delay-partitioning technique with a tuning parameter. Zhu [24] derived some new conditions for ensuring the asymptotic stability of singular nonlinear MJSs with unknown parameters and continuously distributed delays. Recently, some new augmented LKFs and techniques in estimating upper bounds of time-derivative of LKFs were introduced in [25] in studying stability and performance analysis of MJSs with time-varying delays. Very recently, in [26], an input-output approach to the delay-dependent stability analysis and control for MJSs with time-varying delays and deficient transition descriptions. The problem of finite-time estimation for a class of discrete-time Markov jump systems with time-varying transition probabilities subject to average dwell time switching was investigated in [27]. However, as mentioned in [17], the results on stability have still some conservativeness. Thus, there are rooms for further reduction of conservativeness caused by time-delays with the construction of a newly augmented Lyapunov-Krasovskii functional and utilization of a Wirtinger-based integral inequality [10].

Motivated by [17] and based on the result of [25], the goal of this paper is to propose a further improved result of delay-dependent stability for MJSs with time-varying delays. In Theorem 5, a new and improved stability criterion will be proposed based on the results of [25]. To derive less conservative results, Wirtinger-based integral inequality is applied to the augmented LKFs and some new techniques are introduced. When an upper bound of time-derivative value of time-varying delay is larger than one or unknown, a corresponding result will be presented in Corollary 6 by constructing some part of LKF utilized in Theorem 5. Comparing with the result of [25], the constructed Lyapunov-Krasovskii functionals in Theorem 5 and Corollary 6 are simple since the triple and quadruple integral form of Lyapunov-Krasovskii functionals will not be utilized. Via three numerical examples, the advantage and effectiveness of the proposed results will be explained by comparing maximum delay bounds with some recent results presented in other literatures.

*Notation*. Throughout this paper, the following notations will be used. means that is a real symmetric positive definitive matrix (positive semidefinite). The subscript “” represents the transpose. denotes a basis for the null-space of . denotes the -dimensional Euclidean space and is the set of all real matrix. denotes the Banach space of continuous functions mapping the interval into with the topology of uniform convergence. means the space of square-integrable vector functions over . denotes the expectation operator with respect to some measure . , , and denote identity matrix and and zero matrices, respectively. refers to the induced matrix 2-norm. denotes the block diagonal matrix. means that the elements of matrix include the scalar value of . For any matrix , means . means .

#### 2. Problem Statement and Preliminaries

Consider the Markovian jump system with time-varying delays: where is the state vector, which belongs to means the initial function, and are known system matrices with appropriate dimensions, and denotes a finite state Markovian jump process representing the system mode. That is, takes values in the finite discrete set with transition probability matrix .

The transition probability is described as where , , for and .

The delay in states, , is a time-varying and continuous function satisfying where is a known positive scalar and is any constant one.

For simplicity, a matrix of th node is denoted by for each possible in the rest of this paper. For example, and of th node will be represented as and , respectively. Let for . From [28], it should be noted that is a Markov process for . Then, its weak infinitesimal operator acting on a functional is defined by In stability analysis of system (1), the following definition will be utilized.

*Definition 1 (see [29]). *For any finite , and the initial condition of the mode , the system is said to (a)be stochastically stable if there exists a constant such that (b)be mean square stable if hold for any initial condition ,(c)be mean exponentially stable if there exist constants and such that the following holds for any initial condition :

Based on the results of [25], the objective of this paper is to develop further improved delay-dependent stability criteria of system (1) which will be conducted in next section.

The following lemmas will be utilized in deriving main results.

Lemma 2. *Consider a given matrix . Then, for all continuous function in , the following inequality holds: *

*Proof. *From the original Wirtinger-based integral inequality [10], since inequality (8) holds.

Lemma 3 (see [30]). *Let , , and such that . Then, the following two statements are equivalent:*(a)*, ,*(b)*, where is a right orthogonal complement of .*

Lemma 4 (see [31]). *For the symmetric appropriately dimensional matrices , , an any matrix , the following two statements are equivalent:*(a)*,*(b)*there exists a matrix of appropriate dimension such that *

#### 3. Main Results

In this section, improved delay-dependent stability criteria for MJSs (1) will be proposed. To express vectors and matrices in simple forms, block entry matrices will be used. For example, means . And some of scalars, vectors, and matrices are defined asNow, we have the following theorem.

Theorem 5. *For given scalars and , system (1) is stochastically stable for and if there exist positive definite matrices , , , and , any matrices and , and any symmetric matrices and satisfying the following LMIs for all , : **where means the two vertices of with the bounds of . That is, and .*

*Proof. *For each , , let us consider the Lyapunov-Krasovskii functional candidate: whereFrom the following relationship: can be represented as where is defined in (11).

Note that From (17), calculation of leads to An upper bound of can be obtained as Inspired by the work of [32], for any symmetric matrices , the following two zero equalities are satisfied:By summing the two zero equalities in (20), we have Let . By using the similar methods presented in (18) to (19), the calculation of can be represented as Here, the following equations are utilized in (22): With the consideration of in (18), in (19), and the two integral terms and in (21), the last integral term at (22) with the addition of integral terms mentioned above can be estimated by the use of (a) in Lemma 2 and reciprocally convex optimization approach [9] as where which were defined in (11).

With the use of Lemma 2, the integral term can be bounded as where The other integral term can be estimated as Sincefrom (28) to (29), we have where From (22) to (30), by utilizing reciprocally convex optimization approach [9], it can be confirmed that