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考察一类Markov切换时变时滞随机系统的均方指数稳定性. 利用基于Liapunov函数和线性矩阵不等式的方法, 给出了使状态反馈控制系统能克服不确定性和随机干扰, 在均方意义下达到指数稳定的充分条件. 当Markov链遍历所有模态时, 给出了一个独立于Markov链模态集的增益矩阵, 使得状态反馈控制系统均方指数稳定     

Exponential Stability in Mean Square for a General Class of Discrete-Time Linear Stochastic Systems

Abstract

The problem of the mean square exponential stability for a class of discrete-time linear stochastic systems subject to independent random perturbations and Markovian switching is investigated. The case of the linear systems whose coefficients depend both to present state and the previous state of the Markov chain is considered. Three different definitions of the concept of exponential stability in mean square are introduced and it is shown that they are not always equivalent. One definition of the concept of mean square exponential stability is done in terms of the exponential stability of the evolution defined by a sequence of linear positive operators on an ordered Hilbert space. The other two definitions are given in terms of different types of exponential behavior of the trajectories of the considered system. In our approach the Markov chain is not prefixed. The only available information about the Markov chain is the sequence of probability transition matrices and the set of its states. In this way one obtains that if the system is affected by Markovian jumping the property of exponential stability is independent of the initial distribution of the Markov chain.

The definition expressed in terms of exponential stability of the evolution generated by a sequence of linear positive operators, allows us to characterize the mean square exponential stability based on the existence of some quadratic Lyapunov functions.

The results developed in this article may be used to derive some procedures for designing stabilizing controllers for the considered class of discrete-time linear stochastic systems in the presence of a delay in the transmission of the data.     

An LMI approach to stability analysis of stochastic high-order Markovian jumping neural networks with mixed time delays

This paper deals with the problem of global exponential stability for a general class of stochastic high-order neural networks with mixed time delays and Markovian jumping parameters. The mixed time delays under consideration comprise both discrete time-varying delays and distributed time-delays. The main purpose of this paper is to establish easily verifiable conditions under which the delayed high-order stochastic jumping neural network is exponentially stable in the mean square in the presence of both mixed time delays and Markovian switching. By employing a new Lyapunov–Krasovskii functional and conducting stochastic analysis, a linear matrix inequality (LMI) approach is developed to derive the criteria ensuring exponential stability. Furthermore, the criteria are dependent on both the discrete time-delay and distributed time-delay, and hence less conservative. The proposed criteria can be readily checked by using some standard numerical packages such as the Matlab LMI Toolbox. A simple example is provided to demonstrate the effectiveness and applicability of the proposed testing criteria.     

Exponential stability of reaction-diffusion high-order Markovian jump Hopfield neural networks with time-varying delays

This paper studies the problems of global exponential stability of reaction-diffusion high-order Markovian jump Hopfield neural networks with time-varying delays. By employing a new Lyapunov-Krasovskii functional and linear matrix inequality, some criteria of global exponential stability in the mean square for the reaction-diffusion high-order neural networks are established, which are easily verifiable and have a wider adaptive. An example is also discussed to illustrate our results.     

Exponential estimates for stochastic delay hybrid systems with Markovian switching

This paper deals with the problem of norm bounds for the solutions of stochastic hybrid systems with Markovian switching and time delay.Based on Lyapunov-Krasovskii theory for functional differential equations and the linear matrix inequality(LMI)approach,mean square exponential estimates for the solutions of this class of linear stochastic hybrid systems are derived.Finally,An example is illustrated to show the applicability and effectiveness of our method.     

基于SLQ控制器的时滞微分系统的鲁棒镇定

该文基于随机线性二次控制问题, 讨论了多时滞、且具有马尔可夫跳变参数的微分系统的最优控制的鲁棒性及可镇定问题.应用了Lyapunov-Krasovskii型的泛函、伊藤(Ito)公式、及Schur补等工具, 分析了该随机多时滞、具有马尔可夫过程的微分系统的均方指数稳定性.得到了时滞相关与时滞无关的充分性的代数判据.     

Comparison principle and stability criteria for stochastic differential delay equations with Markovian switching

In the present paper we first obtain the comparison principle for the nonlinear stochastic differential delay equations with Markovian switching. Later, using this comparison principle, we obtain some stability criteria, including stability in probability, asymptotic stability in probability, stability in thepth mean, asymptotic stability in the pth mean and the pth moment exponential stability of such equations. Finally, an example is given to illustrate the effectiveness of our results.     

Stability of singularly perturbed nonlinear stochastic hybrid systems

The problem of exponential mean-square stability of nonlinear singularly perturbed, stochastic hybrid systems is studied in this article. Two groups of nonlinear systems are considered separately. To obtain the sufficient conditions of stability, two basic approaches of stability analysis for hybrid systems with a given Markovian switching rule and any Markovian switching rule and singularly perturbed non–hybrid systems were combined. The Lyapunov techniques were used in both approaches. The obtained results are illustrated by examples.     

Comparison principle and stability of Ito stochastic differential delay equations with Poisson jump and Markovian switching

In this paper the comparison principle for the nonlinear Itô stochastic differential delay equations with Poisson jump and Markovian switching is established. Later, using this comparison principle, we obtain some stability criteria, including stability in probability, asymptotic stability in probability, stability in the pth mean, asymptotic stability in the pth mean and the pth moment exponential stability of such equations. Some known results are generalized and improved.     

Impulsive stabilization of complex-valued stochastic complex networks via periodic self-triggered intermittent control

The aim of this article is to research the stabilization issue of complex-valued stochastic Markovian switching complex networks with time delays and time-varying multi-links (CSMCTM) via periodic self-triggered intermittent impulsive control (PSIIC). Thereinto, PSIIC is designed for the first time by combining intermittent impulsive control with periodic self-triggered control for intermittent control. It is worth emphasizing that the triggered protocol is designed to be more flexible, and easier to implement than previously reported triggered protocol. Then, by means of impulsive control, intermittent control, event-driven control theory and stability analysis, a stabilization criterion of CSMCTM in the sense of exponential stability in mean square is obtained. Whereafter, the stability of a class of complex-valued inertial neural networks is researched as a practical application of our theoretical results. Ultimately, a numerical example gives a corresponding verification.     

带Markov调制的随机微分延迟方程的稳定性

本文采用了一例特定的Lyapunov函数,来研究带Markov调制的随机微分延迟方程的p阶指数稳定性,并对其几乎必然指数稳定性也进行了探讨.     

Stability of neutral stochastic delayed systems with switching and distributed-delay dependent impulses

This paper is devoted to discuss the exponential stability in mean square of neutral stochastic delayed systems (NSDDs) with switching and distributed-delay dependent impulses. By using multiple Lyapunov functions and average dwell time (ADT), we provide some sufficient conditions for the exponential stability in mean square for NSDDs with switching and distributed-delay dependent impulses. Compared with the existing related works, we consider not only the influences of switches and neutral type on the stability of NSDDs with switching and distributed-delay dependent impulses but also the influences of both the stable continuous dynamics case and the stable discrete dynamics case. Finally, we provide two examples to illustrate the effectiveness of the theory.     

一类具马尔科夫链中立型随机神经网络的动力学性质（英文）

A generalized neutral stochastic functional differential equation(NSFDE) with Markovian switching is studied. We will discuss some important properties of the solutions including boundedness and exponential stability by using Lyapunov-Krasovskii functional,Matrix inequality and some analysis techniques. Finally, an numerical example for neutral stochastic neural networks with Markovian switching is given to show the effectiveness of the results in this paper.     

Stability of hybrid stochastic differential systems with switching and time delay

This article introduces a hybrid stochastic differential system with impulsive, switching and time-delay. Some stability criteria of p-moment global asymptotical stability, p-moment global exponential stability and mean square stability of this system are derived by using switching Lyapunov function approach, Itô formula, impulsive differential inequality method, and linear matrix equality techniques. Three examples are presented to demonstrate the efficiency of the obtained results.     

Almost sure and moment exponential stability of Euler–Maruyama discretizations for hybrid stochastic differential equations

Positive results are derived concerning the long time dynamics of numerical simulations of stochastic differential equation systems with Markovian switching. Euler–Maruyama discretizations are shown to capture almost sure and moment exponential stability for all sufficiently small timesteps under appropriate conditions.     

The asymptotic stability and exponential stability of nonlinear stochastic differential systems with Markovian switching and with polynomial growth

This paper discusses the asymptotic stability and exponential stability of nonlinear stochastic differential systems with Markovian switching (SDSwMSs). The systems coefficients are assumed to satisfy local Lipschitz condition and polynomial growth condition. By applying some novel techniques, we propose some conditions under which such SDSwMSs are asymptotically stable and exponentially stable. Nontrivial examples are provided to illustrate our results.     

On Quadratic g-Evaluations/Expectations and Related Analysis

This work is concerned with coupling and exponential convergence rate for a class of Markovian switching jump-diffusion processes. The processes under consideration can be thought of as a number of jump-diffusion processes modulated by a Markovian switching device. For this class of processes, we construct some order-preserving couplings. Furthermore, by virtue of the coupling results, we also provide an estimate of exponential convergence rate for the Markovian switching jump-diffusion processes without Gaussian noise.     

Exponential Stability for Delayed Stochastic Bidirectional Associative Memory Neural Networks with Markovian Jumping and Impulses

In this paper, the problem of stability analysis for a class of delayed stochastic bidirectional associative memory neural network with Markovian jumping parameters and impulses are being investigated. The jumping parameters assumed here are continuous-time, discrete-state homogeneous Markov chain and the delays are time-variant. Some novel criteria for exponential stability in the mean square are obtained by using a Lyapunov function, Ito’s formula and linear matrix inequality optimization approach. The derived conditions are presented in terms of linear matrix inequalities. The estimate of the exponential convergence rate is also given, which depends on the system parameters and impulsive disturbed intension. In addition, a numerical example is given to show that the obtained result significantly improve the allowable upper bounds of delays over some existing results.     

Stability of infinite dimensional stochastic evolution equations with memory and Markovian jumps

A strong solutions approximation approach for mild solutions of stochastic functional differential equations with Markovian switching driven by Lévy martingales in Hilbert spaces is considered. The Razumikhin–Lyapunov type function methods and comparison principles are studied in pursuit of sufficient conditions for the moment exponential stability and almost sure exponential stability of equations in which we are interested. The results of [A.V. Svishchuk, Yu.I. Kazmerchuk, Stability of stochastic delay equations of Itô form with jumps and Markovian switchings, and their applications in finance, Theor. Probab. Math. Statist. 64 (2002) 167–178] are generalized and improved as a special case of our theory.     

Stability in distribution of neutral stochastic functional differential equations with Markovian switching

Stability in distribution of stochastic differential equations with Markovian switching and stochastic differential delay equations with Markovian switching have been studied by several authors and this kind of stability is an important property for stochastic systems. There are several papers which study this stability for stochastic differential equations with Markovian switching and stochastic differential delay equations with Markovian switching technically. In our paper, we are concerned with the general neutral stochastic functional differential equations with Markovian switching and we derive the sufficient conditions for stability in distribution. At the end of our paper, one example is established to illustrate the theory of our work.