具有有界时滞的脉冲随机泛函微分系统的有限时间稳定

本文研究了具有有界时滞的脉冲随机泛函微分系统的有限时间稳定和有限时间渐近稳定问题. 基于Lyap-unov函数、Razumikhin技巧以及平均脉冲区间条件, 本文建立了关于该系统有限时间稳定和有限时间渐近稳定的相关性准则. 最后, 给出例子说明结论的有效性.     

Stability criteria in terms of two measures for discrete systems

This paper studies stability problems for two kinds of discrete systems in terms of two different measures. The basic idea of this approach is to discuss the qualitative behaviour of a map h along solutions of discrete systems whose initial value are measured by a second map h₀. By doing this, one may deal with, in a unified way, several concepts and associated problems, which are usually considered separately. Stability criteria in terms of two measures are established employing the method of Lyapunov functions and Razumikhin techniques.     

Stability analysis and synthesis for linear impulsive stochastic systems

This paper presents a general framework for analyzing stability of linear impulsive stochastic systems (LISSs). Some simple mean square stability criteria for the three types of LISSs are firstly derived by analyzing an equivalent system. By exploring the hybrid characteristics of impulsive systems, the novel quasi‐periodic composite polynomial Lyapunov function and the time‐varying discretized Lyapunov function are developed, which leads to unified dwell‐time–based criteria for mean square stability and almost sure stability of LISSs without imposing the stability condition on continuous‐ and discrete‐time dynamics. Next, based on the established stability criteria, the synthesis problem of state‐feedback controller is solved. The computational complexity and the comparison with existing results on the deterministic systems are discussed. Finally, numerical examples are provided to illustrate the usefulness of the proposed results.     

pth moment and almost sure exponential stability of impulsive neutral stochastic functional differential equations with Markovian switching

In this paper, the problems on the pth moment and the almost sure exponential stability for a class of impulsive neutral stochastic functional differential equations with Markovian switching are investigated. By using the Lyapunov function, the Razumikhin-type theorem and the stochastic analysis, some new conditions about the pth moment exponential stability are first obtained. Then, by using the Borel–Cantelli lemma, the almost sure exponential stability is also discussed. The results generalise and improve some results obtained in the existing literature. Finally, two examples are given to illustrate the obtained results.     

Approximate controllability of nonlinear stochastic impulsive systems with control acting on the nonlinear terms

This paper is concerned with the approximate controllability of the stochastic impulsive system with control acting on the nonlinear terms. In the case that the nonlinear terms are dependent on the control, the control cannot be expressed explicitly and analysed. In this situation, we generate the control sequence by the approximate equations and give the properties of the control sequence and the driven solution sets. Some discussions on the assumptions are given to impose on the system. The Hausdorff measure is adopted to relax the requirement of the compactness condition. It is also shown that under some sufficient conditions the stochastic impulsive system is approximately controllable without the requirement of the controllability of the associated linear system. The results of this paper can be degraded into special cases and coincide with some existing ones.     

Stability and boundedness in terms of two measures for nonlinear impulsive control systems

In this paper we study stability and boundedness in terms of two measures for impulsive control systems. By using variational Lyapunov method, a new variational comparison principle and some criteria on stability and boundedness are obtained. An example is presented to illustrate the efficiency of proposed result.     

Stability of diffusion stochastic functional differential equations with Markov parameters

The paper justifies the second Lyapunov method for diffusion stochastic functional differential equations with Markov parameters, which generalize stochastic diffusion equations without aftereffect. Analogs of Lyapunov stability theorems, which generalize the results for systems with finite aftereffect, are proved. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 74–88, January–February 2008.     

Controllability of non-linear impulsive stochastic systems

In this article, we consider the finite-dimensional dynamical control systems described by non-linear impulsive stochastic differential equations. Sufficient conditions for the complete and approximate controllability of non-linear impulsive stochastic systems are formulated and proved under the natural assumption that the corresponding linear system is appropriately controllable.     

Moment exponential stability of stochastic nonlinear delay systems with impulse effects at random times

In this paper, we investigate the pth moment exponential stability for a class of impulsive stochastic functional differential equations with impulses at random times. The impulsive times considered in this paper are random times that are different from those investigated in the existing literature. By using the stochastic process theory, stochastic analysis theory, Razumikhin technique, and Lyapunov method, we obtain some new criteria of the pth moment exponential stability for the related system. Finally, some examples are provided to show the effectiveness of the theoretical results.     

New criteria for stochastic suppression and stabilization of hybrid functional differential systems

This paper establishes new criteria for stochastic suppression and stabilization of hybrid functional differential systems with general 1‐sided polynomial growth condition. For an unstable nonlinear hybrid functional differential system with general 1‐sided polynomial growth condition, 2 independent Brownian noise processes are used to perturb the system into the stochastic hybrid differential system. Theoretical analysis shows that one of the nonlinear diffusion terms may suppress the explosive solution of deterministic system, and the other one can make the perturbed hybrid system almost surely stable with general decay rate.     

Robust fuzzy control of nonlinear stochastic delay systems with impulsive effects

The problem of robust fuzzy control for a class of nonlinear fuzzy impulsive stochastic systems with time-varying delays is investigated. The nonlinear delay system is represented by the well-known T–S fuzzy model. The so-called parallel distributed compensation idea is employed to design the state feedback controller. Sufficient conditions for mean square exponential stability of the closed-loop system are derived in terms of linear matrix inequalities. Finally, a numerical example is given to illustrate the applicability of the theoretical results.     

Exponential ultimate boundedness of impulsive stochastic delay difference systems

This paper is concerned with the exponential ultimate boundedness problems for the impulsive stochastic delay difference systems. Several sufficient conditions on the global pth moment exponential ultimate boundedness are presented by using the Lyapunov methods and the algebraic inequality techniques, and the estimated exponential convergence rate and the ultimate bound are provided as well. As an application, the boundedness criteria are applied to a class of discrete impulsive stochastic neural networks with delays. The obtained results show that the impulses not only can stabilize an unstable stochastic difference delay system but also can make an unbounded stochastic difference delay system into a bounded system. Examples and simulations are also provided to demonstrate the effectiveness of the derived theoretical results.     

Existence and stability of impulsive stochastic coupled systems in infinite dimensions

This paper considers the exponential stability of a class of infinite-dimensional impulsive stochastic coupled systems. With the help of generalized Itô's formula for the mild solution of infinite-dimensional systems, we avoid limiting the domain of the mild solution. Then we use the combination of the Lyapunov function and graph theory to construct the Lyapunov function of the systems; the criteria of $p$-th moment exponential stability are obtained, which is related to the average impulsive interval $T_a$ and the connectivity of impulsive stochastic systems. In addition, noting that the existence may be affected by impulsive effects and stochastic perturbations, using the graph theory and the principle of contraction mapping, we get the condition that guarantees the existence and uniqueness, which is also related to the structure of the networks. Finally, we consider the stability of impulsive stochastic coupled heat equations and neural networks with reaction diffusion and give some numerical simulations to verify the theoretical results.     

一类脉冲随机系统的有限时间有界性分析与H∞控制

本文研究一类线性脉冲随机系统的有限时间有界性及H_∞控制问题.首先,利用Lyapunov函数和平均脉冲区间条件,建立了系统有限时间均方有界定理;其次,基于H_∞控制理论,获得了保证系统有限时间有界且满足一定的H_∞性能指标的判据;再次,针对有限时间H_∞控制问题,给出了状态反馈控制器存在的充分条件.最后,用一个数值例子表明了理论结果的有效性.     

On the pth moment integral input‐to‐state stability and input‐to‐state stability criteria for impulsive stochastic functional differential equations

In this paper, several new Razumikhin‐type theorems for impulsive stochastic functional differential equations are studied by applying stochastic analysis techniques and Razumikhin stability approach. By developing a new comparison principle for stochastic version, some novel criteria of the pth moment integral input‐to‐state stability and input‐to‐state stability are derived for the related systems. The feature of the criteria shows that time‐derivatives of the Razumikhin functions are allowed to be indefinite, even unbounded, which can loosen the constraints of the existing results. Finally, some examples are given to illustrate the usefulness and significance of the theoretical results.     

The maximum principle for stochastic differential systems with general cost functional

In this paper, under the framework of Fréchet derivatives, we study a stochastic optimal control problem driven by a stochastic differential equation with general cost functional. By constructing a series of first-order and second-order adjoint equations, we establish the stochastic maximum principle and get the related Hamilton systems.     

Finite-time annular domain stability of Itô stochastic impulsive systems with markovian jumping under asynchronous switching

ABSTRACT

This study examines the finite time annular domain stability (FTADS) and stabilisation of a class of Itô stochastic impulsive systems with asynchronous switching controller. The asynchronous switching means that the controller switching does not accurately coincide with system switching in delayed time interval. The design of the controller depends on the observed jumping parameters, which cannot be precisely measured in real-time because of switching delay. Our results apply to cases where some subsystems of the switched systems are not necessarily stable under the influence of input delay. When the subsystem is stable in the synchronous switching interval and unstable in the asynchronous case, a compromise among the average impulsive interval, the upper bound of delay, and the decay/increasing rate of Lyapunov function in the synchronous/asynchronous switching interval respectively is given. By the mode-dependent parameter approach (MDPA) and allowing the increase of the impulses on all the switching times, the extended FTADS criteria for Itô stochastic impulsive systems in generally nonlinear setting are derived first. Then, we focus on the case when the system in both synchronous and asynchronous switching intervals are unstable. By reaching a tradeoff among average impulsive interval, the upper bound of delay, the magnitude of impulses and the difference between the increasing rate of Lyapunov function in the synchronous and asynchronous switching interval, new sufficient conditions for existence of the state feedback controller are also developed by MDPA. In addition, we consider the effect of different impulsive strengths (harmful and beneficial impulses) and obtained less conservative results because the Lyapunov function may be non-decreasing during switching interval. Moreover, we extend the conclusion from nonlinear stochastic impulsive switching systems to linear case. Finally, we present two examples to illustrate the effectiveness of the results obtained in this study.     

Razumikhin stability theorems for a general class of stochastic impulsive switched time‐delay systems

This paper studies stability of a general class of impulsive switched systems under time delays and random disturbances using multiple Lyapunov functions and fixed dwell‐time. In the studied system model, the impulses and switches are allowed to occur asynchronously. As a result, the switching may occur in the impulsive intervals and the impulses can occur in the switching intervals, which have great effects on system stability. Since the switches do not bring about the change of the system state, we study two cases in terms of the impulses, ie, the stable continuous dynamics case and the stable impulsive dynamics case. According to multiple Lyapunov‐Razumikhin functions and the fixed dwell‐time, Razumikhin‐type stability conditions are established. Finally, the obtained results are illustrated via a numerical example from the synchronization problem of chaotic systems.