#### Abstract

The exponential stability of neutral Markovian jump systems with interval mode-dependent time-varying delays, nonlinear perturbations, and partially known transition rates is investigated. A novel augmented stochastic Lyapunov functional is constructed, which employs the improved bounding technique and contains triple-integral terms to reduce conservativeness; then the delay-range-dependent and rate-dependent exponential stability criteria are developed by Lyapunov stability theory, reciprocally convex lemma, and free-weighting matrices. The corresponding results are extended to the uncertain case. Finally, numerical examples are given to illustrate the effectiveness of the proposed methods.

#### 1. Introduction

Delay differential equations or systems are assuming an increasingly important role in many disciplines like mathematics, science, and engineering. In particular, the stability and stabilization problem for neutral delay differential dynamic systems have received considerable attention during the decades and neutral time-delay systems have been the focus of the research community, which are often encountered in such practical situations as distributed networks, population ecology, processes including steam or heat exchanges [1], and robots in contact with rigid environments [2]. Existing results can be roughly classified into two categories, delay-independent criteria and delay-dependent criteria, where the latter is generally regarded as less conservative. Moreover, since the derivative of the delayed state is involved, it should be pointed out that the stability of neutral time-delay systems is more difficult to tackle, which is identical with singular systems [3, 4]. The stability problem of them is more complicated than that for regular systems because more factors need to be considered. In the past decades, considerable attention has been devoted to the robust delay-independent stability and delay-dependent stability of linear neutral systems, which are mainly obtained based on the Lyapunov-Krasovskii (L-K) method [5–11], and references therein. It should be noted that the delay-partitioning approach is used in [6–8]. Furthermore, when nonlinear perturbations or parameter uncertainties appear in neutral systems, some results on stability analysis have been also presented [12–18]. Various techniques have been proposed in these papers, for example, model transformation techniques, the improved bounding techniques, and matrix decomposition approaches. In particular, He et al. [18] propose a new method for dealing with time-delay systems, which employs free weighting matrices to express the relationships between the terms in the Newton-Leibniz formula and has brought novel results. However, these results have conservativeness to some extent, which exist room for further improvement.

In another line, Markovian jump systems (MJSs) have attracted much attention during the past few decades since its first introduction by Krasovskii and Lidskii in 1961, which can be regarded as a special class of hybrid systems with finite operation modes whose structures are subject to random abrupt changes. The system parameters usually jump among finite modes, and the mode switching is governed by a Markov process. MJSs have many applications, such as failure prone manufacturing systems, power systems, solar thermal central receivers, robotic manipulator systems, aircraft control systems, and economic systems. A large number of results on estimation and control problems related to such systems have been reported in the literature; see, for example, [19–25] and references therein for more details. However, these lines of literature about the transition probabilities in the jumping process have been assumed to be completely accessible. The ideal assumption on the transition probabilities inevitably limits the application of the traditional Markovian jump system theory. Actually, the likelihood of obtaining such available knowledge is questionable, and the cost may be very expensive. Thus, it is really significant and meaningful, from control perspectives, to further study more general jump systems with partially known transition rates. Recently, many results on the Markovian jump systems with partially known transition rates are obtained [26–31]. Most of these improved results just require some free matrices or the knowledge of the known elements in transition rate matrix, such as the structures of uncertainties, and some else of the unknown elements need not be considered. It is a great progress on the analysis of Markovian jump systems. However, few of these results are concerned with neutral Markovian jump systems with mode-dependent time-varying delays and perturbations. To the best of the authors' knowledge, neutral Markovian jump systems with mode-dependent time-varying delays and partially known transition rates have not been fully investigated, and it is very challenging, especially when nonlinear perturbations exist. Besides, seeking and proposing less conservative delay-range-dependent criterion for uncertain neutral MJSs with nonlinear perturbations and partially known transition rates to desired performance are still open problems. These facts thus motivate our study.

In this paper, the investigated neutral Markovian jump systems are more general than the neutral MJSs with completely known or completely unknown transition rates, which can be viewed as two special cases of the ones tackled here. Specifically, a new augmented stochastic Lyapunov functional containing triple-integral terms is constructed by dividing the delay interval into two subintervals, and then the delay-range-dependent and rate-dependent exponential stability criteria are obtained by reciprocally convex lemma and free weighting matrices. We further extend the criteria to the uncertain case. All the obtained results are presented in terms of LMIs that can be solved numerically. The remainder of the paper is organized as follows. Section 2 presents the problem and preliminaries. Section 3 gives the main results, which are then verified by numerical examples in Section 4. Section 5 concludes the paper.

*Notations.* The following notations are used throughout the paper. denotes the dimensional Euclidean space and is the set of all matrices. , where and are both symmetric matrices, means that is negative (positive) definite. is the identity matrix with proper dimensions. For a symmetric block matrix, we use to denote the terms introduced by symmetry. stands for the mathematical expectation, is the Euclidean norm of vector , , while is spectral norm of matrix , . is the eigenvalue of matrix with maximum(minimum) real part. Matrices, if their dimensions are not explicitly stated, are assumed to have compatible dimensions for algebraic operations.

#### 2. Problem Statement and Preliminaries

Given a probability space where is the sample space, is the algebra of events and is the probability measure defined on . is a homogeneous, finite-state Markovian process with right continuous trajectories taking values in a finite set , with the mode transition probability matrix being where , , is the transition rate from mode to and for any state or mode ; it satisfies Since the transition rates of the Markov chain are partially known in this paper, some elements in matrix are inaccessible. For instance, the system with five operation modes, the jump rates matrix may be viewed as where represents the unknown element. For notation clarity, we denote , for all and If , it is further described as where , represent the th known element of the set in the th row of the transition rate matrix . Furthermore, let and be the lower and upper bound for the diagonal elements of the jump rates matrix .

In this paper, the following uncertain neutral Markovian jump systems with mode-dependent interval time-varying delays, nonlinear perturbations, and partially known transition rates over the space are considered: where is the system state and is mode-dependent interval time-varying neutral delay which satisfies , when . The mode-dependent interval time-varying retarded delay is assumed that where , , , , , and are real constant scalars. The initial condition is a continuously differentiable vector-valued function. , , and are unknown nonlinear perturbations which are, with respect to the current state , the delayed state and the neutral delay state , respectively. For all and , they are assumed to be bounded in magnitude as where , , and are given constants, for simplicity, , , and .

For notational simplicity further, where , the parametric matrices , , , , , and are denoted by , , , , , and , which can be described as where , , , , , and are known constant matrices with appropriate dimensions. , , , , and are uncertainties. The parametric matrix and the admissible parametric uncertainties satisfy the following condition: where , , , , , and are known mode-dependent constant matrices with appropriate dimensions and is an unknown and time-varying matrix satisfying

Particularly, the following nominal systems can be obtained for :

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

*Assumption 1. *System matrices , (for all ), are Hurwitz and all the eigenvalues have negative real parts for each mode. , (for all ), is full rank in row.

*Assumption 2. *The Markov process is irreducible and the system mode is available at time .

*Definition 3 (see [32]). *Define operator as . is said to be stable if the homogeneous difference equation

is uniformly asymptotically stable. In this paper, that is, .

*Definition 4 (see [33]). *The system in (6) is exponentially stable with a decay rate for all , if there exist scalars and such that for all ,
where is the exponential decay rate, denotes the Euclidean norm, and

*Definition 5 (see [34]). *Define the stochastic Lyapunov-Krasovskii function of system (6) as , where its infinitesimal generator is defined as

Lemma 6 (see [35]). *Given constant matrices , , and , where and . if and only if
*

Lemma 7. *For any constant matrix , continuous functions , constant scalars , and constant such that the following integrations are well defined,*(a)*(b)*

*Proof. *(a) Is directly obtained from [36]. In addition, from and , it is held that . Then

(b) Is thus true by [37].

Lemma 8 (see [38]). *For functions , , , and with and with , matrices , , then there exists matrix such that
**
and the following inequality holds:
*

Lemma 9 (see [39]). *For given matrices , , and with appropriate dimensions,
**
for all satisfying , if and only if there exists a scalar , such that
*

#### 3. Main Results

This section will state the exponential stability analysis for neutral Markovian jump systems with mode-dependent interval time-varying delays, nonlinear perturbations, and partially known transition rates. With creative Lyapunov functional and novel matrix inequalities analysis, delay-range-dependent and rate-dependent exponential stability conditions are presented.

##### 3.1. Exponential Stability for the Nominal Systems

Theorem 10. *For given scalars , , , , , , , , , , and constant scalar satisfying , the systems described by (13) with partially known transition rates are exponentially stable with decay rate and if and there exist symmetric positive matrices , , , , , , , , , , , , , , and matrices , , , , , for any scalars , , , any symmetric matrices , , , , , and any matrices , with appropriate dimensions, such that the following linear matrix inequalities hold.**When **
When **
where
**
where is a linear operator on by
**
where are block entry matrices; that is,*

*Proof. *Construct the following stochastic Lyapunov functional:
where
*Remark* *11*. It should be pointed out that the proposed stochastic augmented Lyapunov functional (38) contains some triple-integral terms, which has not been used in of the existing literature in the same context before. Compared with the existing ones, [37] has shown that such triple-integral terms are very effective in the reduction of conservativeness.

Taking as its infinitesimal generator along the trajectory of system (13), we obtain the following from Definition 5 and (38)–(45):
can be easily obtained by the following equation:

With regard to , the detailed procedures are given as follows.
where

By the infinitesimal generator , we obtain
where

Following the same procedure, is also obtained:

Moreover, , are easily calculated as shown in the following:

According to (50), (51), (52), and (53), we can easily obtain the following (54):

Then, in the same method, , can be calculated and the results are given by the following, respectively:

Moreover, , , and can be directly obtained as follows: