Stochastic Stability Criteria for Neutral Distributed Parameter Systems with Markovian Jump

Li, Yanbo; Chen, Chao-Yang; Li, Chengqun

doi:https://doi.org/10.1155/2020/9450786

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Abstract Introduction Preliminaries Examples Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 9450786 | https://doi.org/10.1155/2020/9450786

Stochastic Stability Criteria for Neutral Distributed Parameter Systems with Markovian Jump

Yanbo Li,¹Chao-Yang Chen,^2,3and Chengqun Li¹

Academic Editor: Mohamed Boutayeb

Received16 Jun 2019

Revised16 Jan 2020

Accepted20 Jan 2020

Published11 Feb 2020

Abstract

This paper deals with the problem of stochastic stability for a class of neutral distributed parameter systems with Markovian jump. In this model, we only need to know the absolute maximum of the state transition probability on the principal diagonal line; other transition rates can be completely unknown. Based on calculating the weak infinitesimal generator and combining Poincare inequality and Green formula, a stochastic stability criterion is given in terms of a set of linear matrix inequalities (LMIs) by the Schur complement lemma. Because of the existence of the neutral term, we need to construct Lyapunov functionals showing more complexity to handle the cross terms involving the Laplace operator. Finally, a numerical example is provided to support the validity of the mathematical results.

1. Introduction

Time-delay models are popular in all kinds of fields such as demography, biology, economics, and chemistry. Neutral systems as a special type of time-delay systems are often encountered because these systems have a wider application value than the general time-delay systems in many dynamical systems such as bioengineering systems, dynamic systems of offshore platform, and dynamic economic models. Hence, there are so many investigations about time-delay systems [1–7]. As we all know, the systems inevitably receive the impact of sudden changes in the environment, abrupt failure of components, unexpected changes in system parameters, and so on. These random diversifications usually follow the law of Markov jump. These systems are called Markovian jump systems. Markovian jump systems spur investigators’ consuming interest [8, 9].

The stability and performance of stochastic systems are quite different from those of deterministic systems [10–12]. More recently, neutral-type Markovian jump systems have aspirated considerable attention. All kinds of analysis methods have been used to discover the stochastic stability criteria of neutral-type Markovian jump systems such as Lyapunov–Krasovskii functional approach, reciprocally convex combination inequality method, and stochastic analysis theory in [13–16]. For less conservative results, the delay-dependent stability has been discussed in [17–20]. Other control methods have also been extensively and thoroughly studied such as robust delay-dependent H_∞ control [21, 22], nonfragile control [23–25], sliding mode control [26, 27], H_∞ sliding mode control [28], and the references therein.

The transition rates in many references mentioned above have been supposed to be completely known. But it is very hard to acquire the accurate transfer probability, and even if the exact transfer probability can be obtained, the cost is also very huge. So the study of Markovian jump systems with general unknown transition rates [29–50] has appealed to a great many scholars. Stability, stabilization, and robust control of Markovian jump systems with partially unknown transition have been reported in [29–33]. Stability analysis for neutral Markovian jump systems with partially unknown transition probabilities has been proposed in [34, 35]. Kao et al. and Yang et al. [36, 37] have settled the delay-dependent stability for Markovian jump systems with partially unknown transition probabilities and Markovian jump neutral stochastic systems with general unknown transition rates, respectively. Singular Markovian jump systems with general incomplete transition probabilities have been presented in [38–40]. Finite-time stochastic stability and control of Markovian jump systems with general incomplete transition probabilities have been discussed in [41–44]. Stabilization of discrete-time Markovian jump systems with partially unknown transition probabilities was probed in [45].

In parallel, many researchers have extensively studied the time-delay distributed parameter systems [46–54]. Three main approaches of time-delay distributed parameter systems are semigroup theory [47], matrix norm theory [48], and linear matrix inequality theory (LMI) [49]. The semigroup method cannot guarantee the system to be of a fine dynamic character and performance index undergoing two transformations. It is not easy to apply the results to practical problems by matrix norm. The problem of exponential stability and stabilization [50], sliding mode control [51], and feedback control [52] has been proposed in terms of the linear matrix inequality approach. However, less attention has been paid to the distributed parameter systems with Markovian jump, especially neutral distributed parameter systems with Markovian jump which requires a lot of research to be performed.

Based on previous discussions, we are considering the problem of stochastic stability of a class of Markovian jumping neutral distributed parameter systems in this paper. The linear matrix inequality approach together with the Lyapunov functional method is employed to develop stochastic stability criteria for the described systems. The results are given in a group of linear matrix inequalities (LMIs).

2. Problem Formulation and Preliminaries

Consider the neutral distributed parameter systems with Markovian jump of the following form:where , is the bounded domain with smooth boundary , and . Also,where is the gradient operator. is the state function, and is the Laplace operator on .

The initial value and boundary value conditions satisfywhere and and are constants. is the unit outward normal vector of , and is the smooth function. , and are constants; , , and are constant matrices.

Let be a right-continuous Markov process and take values in a finite set with transition probability matrix ; the mode transition probabilities are defined as follows:where and . donates the transition rate from mode to mode in the time interval and . For each , let , , , and . Then, we can represent system (1) in the following form:

Lemma 1 (see [53] (Green formula)). Let be the bounded domain with smooth boundary , is the unit outward normal vector of , and is the smooth subdomain. If , thenwhere is the Hamilton operator and is the area element over the boundary region.

Lemma 2 (see [54] (Friedrichs’s inequality)). Let and be included in the closed region . Then,where .

Lemma 3 (see [55]). Let be the real matrices and ; then, for an arbitrary given scalar , the following inequality holds:

3. Main Results

Theorem 1. Given matrices , time-delay constants and and constant , the neutral distributed parameter systems with Markovian jump (5) is stochastically stable. If there exist positive symmetric matrices , , , , , and and positive scalars , such that for any Markovian jump mode , the following linear matrix inequalities (LMIs) hold:where , , ,

Proof. For system (5), we construct the following stochastic Lyapunov functional:whereLet be the weak infinitesimal generator; then, we calculatewhereApplying Lemma 1 and Lemma 2,Substituting (16) and (17) into (15), we haveNoticing and combining and , we derive the following inequality:Then,For the same reason, we can also obtainSoThrough applying Lemma 3, the following inequality holds:Taking advantage of Lemma 1 and Lemma 2, we obtainCombining (26)–(31) together yieldsSynthesizing (18), (21), (23)–(25), and (32)–(35), the following inequality holds:Select appropriate such that , then we transform it into (9) by the Schur complement lemma.
SetThen,The proof is concluded.

Remark 1. Because of the simultaneous existence of neutral term and Markov jump, the stochastic stability of neutral distributed parameter systems with jump is given by finding the maximum value of absolute value on the main diagonal of the state transition probability matrix. Compared with the general time-delay system, the derivation process of the system is complex, but the conclusion is simple.

4. Examples

Consider a neutral distributed parameter system (5) with two Markovian jump modes and the parameters as follows:

The transition rate matrix is defined by

Obviously, the absolute maximum of the principle diagonal line is 1. We get feasible solutions by solving linear matrix inequalities (9)–(12) in Theorem 1 and obtain the following parameters:

Obviously matrices are positive. The effectiveness of the method of Theorem 1 is illustrated.

5. Conclusion

First, we choose a set of appropriate Lyapunov stochastic functionals; some of them show more complexity in dealing with the neutral cross terms containing the Laplace operator. Then, by taking advantage of boundary conditions, Green formula, and Schur complement lemma, we obtain a sufficient condition for stochastic stability of the studied model in this paper via linear matrix inequalities. At last, a numerical example is given to prove the effectiveness of the proposed method.

Data Availability

The simulation data used to support the findings of this study are included within the article.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported in part by the Guangxi Natural Science Foundation Program under Grant nos. 2016GXNSFAA380157 and 2019JJA110076 and the National Natural Science Foundation of China (61973110, 71961002, and 61503133).

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Copyright © 2020 Yanbo Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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