Exponential Stability and Delayed Impulsive Stabilization of Hybrid Impulsive Stochastic Functional Differential Systems

This paper is concerned with the stability and impulsive stabilization of hybrid impulsive stochastic functional differential systems with delayed impulses. Using the Razumikhin techniques and Lyapunov functions, some sufficient conditions for the pth moment exponential stability of the systems under consideration are established. Based on the derived stability results, impulsive controllers are designed to stabilize a given unstable linear or nonlinear hybrid stochastic delayed differential system. Different from the existing stability and impulsive stabilization results in the literature, the results obtained in this paper shown that the delayed part of impulses can make a contribution to the stability of systems. Three examples are provided to present the effectiveness and advantages of the proposed results.     

Impulses-induced p-exponential input-to-state stability for a class of stochastic delayed partial differential equations

This paper proposes a new stability concept called p-exponential input-to-state stability (pISS) for impulsive stochastic delayed partial differential equations (ISDPDEs). By employing the formula for the variation of parameters, the average impulsive interval approach and a new impulsive integral inequality, the sufficient conditions for pISS of ISDPDEs are derived. The issue of impulsive stabilisation to pISS of ISDPDEs is studied. It is unveiled that if the continuous stochastic delay partial differential equations may not be pISS, it can be stabilised by impulsive control. An example is given to illustrate our main results.     

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.     

A new unified input‐to‐state stability criterion for impulsive stochastic delay systems with Markovian switching

In this paper, the problems of the input‐to‐state stability (ISS), the integral input‐to‐state stability (iISS), the stochastic input‐to‐state stability (SISS) and the e^λt(λ>0)‐weighted input‐to‐state stability (e^λt‐ISS) are investigated for nonlinear time‐varying impulsive stochastic delay systems with Markovian switching. We propose one unified criterion for the stabilizing impulse and the destabilizing impulse to guarantee the ISS, iISS, SISS and e^λt‐ISS for such systems. We verify that when the upper bound of the average impulsive interval is given, the stabilizing impulsive effect can stabilize the systems without ISS. We also show that the destabilizing impulsive signal with a given lower bound of the average impulsive interval can preserve the ISS of the systems. In addition, one criterion for guaranteeing the ISS of nonlinear time‐varying stochastic hybrid systems under no impulsive effect is derived. Two examples including one coupled dynamic systems model subject to external random perturbation of the continuous input and impulsive input disturbances are provided to illustrate the effectiveness of the theoretic results developed.     

Stability of Solutions for Stochastic Impulsive Systems via Comparison Approach

This note studies stability problem of solutions for stochastic impulsive systems. By employing Lyapunov-like function method and It's formula, comparison principles of existence and uniqueness and stability of solutions for stochastic impulsive systems are established. Based on these comparison principles, the stability properties of stochastic impulsive systems are derived by the corresponding stability properties of a deterministic impulsive system. As the application, the stability results are used to design impulsive control for the stabilization of unstable stochastic systems. Finally, one example is given to illustrate the obtained results.     

Robust H∞ control of uncertain linear impulsive stochastic systems

This paper develops robust stability theorems and robust H_∞ control theory for uncertain impulsive stochastic systems. The parametric uncertainties are assumed to be time varying and norm bounded. Impulsive stochastic systems can be divided into three cases, namely, the systems with stable/stabilizable continuous‐time stochastic dynamics and unstable/unstabilizable discrete‐time dynamics, the systems with unstable/unstabilizable continuous dynamics and stable/stabilizable discrete‐time dynamics, and the systems in which both the continuous‐time stochastic dynamics and the discrete‐time dynamics are stable/stabilizable. Sufficient conditions for robust exponential stability and robust stabilization for uncertain impulsive stochastic systems are derived in terms of an average dwell‐time condition. Then, a linear matrix inequality‐based approach to the design of a robust H_∞ controller for each system is presented. Finally, the numerical examples are provided to demonstrate the effectiveness of the proposed approach. Copyright © 2007 John Wiley & Sons, Ltd.     

Stability of Neutral Impulsive Nonlinear Stochastic Evolution Equations with Time Varying Delays

In this paper, we consider the stability of mild solutions to neutral impulsive nonlinear stochastic evolution equations with time varying delays. With the non‐Lipschitz condition, the Lipschitz condition being taken as a special case, on the impulsive term, the existence, uniqueness and sufficient conditions for the exponential stability in the pth moment and the almost sure exponential stability of the mild solutions are derived by employing a fixed point approach. An example is provided to illustrate the efficiency of the obtained theorems.     

Exponential ultimate boundedness and stability of stochastic differential equations with impulese

The paper mainly studies globally pth moment exponentially ultimate boundedness and pth moment exponential stability of impulsive stochastic functional differential equations. By using the Lyapunov direct method of Razumikhin-type condition and the principle of comparison, some sufficient conditions for globally pth moment exponentially ultimate boundedness and globally pth moment exponential stability are presented. Theorems require the linear coefficients of the upper bound of Lyapunov differential operators are time-varying functions; this generalizes the previous results. When the time delay is not considered in the system, a unified criterion is given to achieve boundedness and stability when the system is disturbed by stabilizing impulse and destabilizing impulse. It shows that the stochastic differential equation may be unbounded or unstable, and it can be bounded or stable by adding appropriate impulsive perturbation. Finally, we use two examples to illustrate the validity of our 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.     

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.     

Stability analysis of impulsive stochastic Cohen–Grossberg neural networks driven by G-Brownian motion

This paper is devoted to study the stability of a class of impulsive stochastic Cohen–Grossberg neural networks driven by G-Brownian motion. By means of G-Lyapunov function, Borel–Cantelli lemma and inequality technique, a series of sufficient conditions on pth moment stability and quasi-sure stability with respect to a general decay function are established. A concrete example is given to illustrate the obtained results.     

p-Moment stability of stochastic differential equations with impulsive jump and Markovian switching

This paper introduces some new concepts of p-moment stability for stochastic differential equations with impulsive jump and Markovian switching. Some stability criteria of p-moment stability for stochastic differential equations with impulsive jump and Markovian switching are obtained by using Liapunov function method. An example is also discussed to illustrate the efficiency of the obtained results.     

Stability of impulsive stochastic functional differential systems in terms of two measures via comparison approach

Comparison principles for general impulsive stochastic functional differential systems are established.Employing the comparison principles and the theory of differential inequalities,stability and instability,involving two measures,of impulsive stochastic functional differential systems are investigated.Several stability and instability criteria are obtained,and two examples are also given to illustrate our results.     

Boundedness and stability analysis for impulsive stochastic differential equations driven by G-Brownian motion

In this article, the pth moment globally exponential ultimate boundedness, pth moment globally exponential stability, quasi sure globally exponential boundedness and quasi sure globally exponential stability are investigated for impulsive stochastic differential equations driven by G-Brownian motion. Using G-Lyapunov function methods and inequality techniques, some sufficient conditions are derived for the boundedness and stability. Comparing with the existing methods, the obtained results allow the corresponding impulse-free systems to be unstable and unbounded. An example is provided to show the effectiveness of the theoretical results.     

Exponential Stability Analysis for Stochastic Delayed Differential Systems with Impulsive Effects: Average Impulsive Interval Approach

This paper is concerned with the exponential stability analysis of stochastic delayed systems with impulsive effects. By using the average impulsive interval approach, and together with comparison lemma and Razumikhin techniques, sufficient conditions ensuring the moment exponential stability of the systems under consideration are established. A stability criterion for non‐delayed stochastic systems with impulses is also derived as a corollary. Compared with the existing stability results in the literature, which are usually based on the supremum or infimum of impulsive intervals, the results reported in this paper are less conservative. Two illustrative examples are provided to validate the effectiveness and advantages of our theoretical results.     

Exponential stability of stochastic high-order BAM neural networks with time delays and impulsive effects

In this paper, we consider the problem on exponential stability analysis of the stochastic impulsive high-order BAM neural networks with time delays. Through employing Lyapunov function method and stochastic bidirected halanay inequality, we constitute exponential stability of the stochastic impulsive high-order BAM neural networks with its estimated exponential convergence rate and feasible interval of impulsive strength. An example illustrates the main results.     

Mean-square exponential stability of impulsive stochastic time-delay systems with delayed impulse effects

This paper is concerned with the mean-square exponential stability problem for a class of impulsive stochastic systems with delayed impulses. The delays exhibit in both continuous subsystem and discrete subsystem. By constructing piecewise time-varying Lyapunov functions and Razumikhin technique, sufficient conditions are derived which guarantee the mean-square exponential stability for impulsive stochastic delay system. It is shown that the obtained stability conditions depend both on the lower bound and the upper bound of impulsive intervals, and the stability of system is robust with regard to sufficiently small impulse input delays. Finally, two examples are proposed to verify the efficiency of the proposed results.     

Mean square exponential stability of impulsive stochastic reaction-diffusion Cohen–Grossberg neural networks with delays

In this paper, we establish a method to study the mean square exponential stability of the zero solution of impulsive stochastic reaction-diffusion Cohen–Grossberg neural networks with delays. By using the properties of M-cone and inequality technique, we obtain some sufficient conditions ensuring mean square exponential stability of the zero solution of impulsive stochastic reaction-diffusion Cohen–Grossberg neural networks with delays. The sufficient conditions are easily checked in practice by simple algebra methods and have a wider adaptive range. Two examples are also discussed to illustrate the efficiency of the obtained results.     

Passive stochastic feedback stability-Part II. Applications

In Part I of this paper it was shown that the concept of passivity, used extensively in deterministic feedback stability, could be extended to stochastic feedback systems described through Markov theory to produce stochastic feedback stability theorems. To apply the theory developed in Part I, one is required to do a passivity analysis of the sub-systems of the feedback system. In this paper, six particular types of stochastic systems (sub-systems) are considered and a passivity analysis of each is performed. These systems have the appearance of standardnth-order linear differential systems with a random process entering at some point. The randomness is either white noise, impulsive noise or a Markov jump process. Once the passivity analysis has been made, feedback stability theorems can be derived for many feedback systems containing one or more of these six systems as sub-systems. Two such feedback systems are considered in the paper.     

Input‐to‐state stability of nonlinear stochastic time‐varying systems with impulsive effects

This paper considers the input‐to‐state stability, integral‐ISS, and stochastic‐ISS for impulsive nonlinear stochastic systems. The Lyapunov function considered in this paper is indefinite, that is, the rate coefficient of the Lyapunov function is time‐varying, which can be positive or negative along time evolution. Lyapunov‐based sufficient conditions are established for ensuring ISS of impulsive nonlinear stochastic systems. Three examples involving one from networked control systems are provided to illustrate the effectiveness of theoretical results obtained. Copyright © 2016 John Wiley & Sons, Ltd.