Stochastic asymptotical stability for stochastic impulsive differential equations and it is application to chaos synchronization

This paper studies the stochastic asymptotical stability of stochastic impulsive differential equations, and establishes a comparison theory to ensure the trivial solution's stochastic asymptotical stability. From the comparison theory, it can find out whether the stochastic impulsive differential system is stable just by studying the stability of a deterministic comparison system. As a general application of this theory, it controls the chaos of stochastic Lü system using impulsive control method, and numerical simulations are employed to verify the feasibility of this method.

In this paper, the asymptotical p-moment stability of stochastic impulsive differential equations is studied and a comparison theory to ensure the asymptotical p-moment stability of the trivial solution is established, which is important for studying the impulsive control and synchronization in stochastic systems. As an application of this theory, we study the problem of chaos synchronization in the Chen system excited by parameter white-noise excitation, by using the impulsive method. Numerical simulations verify the feasibility of this method.

Almost sure exponential stability and stochastic stabilization of impulsive stochastic differential delay equations

In this paper, the asymptotical p-moment stability of stochastic impulsive differential equations is studied, and a comparison theory to ensure the asymptotical p-moment stability for trivial solution of this system is established, from which we can find out whether a stochastic impulsive differential system is stable just from a deterministic comparison system. As an application of this theory, we control the chaos of stochastic Chen system using impulsive method, and a stable region is deduced too. Finally, numerical simulations verify the feasibility of our method.

In this article, the problems of p-moment exponential stability and almost surely exponential stability of impulsive stochastic delay differential equations with semi-Markov jump structure are examined. Based on Lyapunov-Razumikhin method, Razumikhin-type theorems on criteria ensuring pth-moment exponential stability and almost surely exponential stability of trivial solution of a class of impulsive stochastic differential delay equations with semi-Markov jump structure are developed. The distinguishing feature of this present paper is that the system remains in a state for a random amount of time with a generalized distribution before the jump, unlike in the case of Markov jump in which the duration of stay in a state before jump is exponentially distributed.

In most stochastic dynamical systems which describe process in engineering, physics and economics, stochastic components and random noise are often involved. Stochastic effects of these models are often used to capture the uncertainty about the operating systems. Motivated by the development of analysis and theory of stochastic processes, as well as the studies of natural sciences, the theory of stochastic differential equations in infinite dimensional spaces evolves gradually into a branch of modern analysis. In the analysis of such systems, we want to investigate their stabilities. This thesis is mainly concerned about the studies of the stability property of stochastic differential equations in infinite dimensional spaces, mainly in Hilbert spaces. Chapter 1 is an overview of the studies. In Chapter 2, we recall basic notations, definitions and preliminaries, especially those on stochastic integration and stochastic differential equations in infinite dimensional spaces. In this way, such notions as Q-Wiener processes, stochastic integrals, mild solutions will be reviewed. We also introduce the concepts of several types of stability. In Chapter 3, we are mainly concerned about the moment exponential stability of neutral impulsive stochastic delay partial differential equations with Poisson jumps. By employing the fixed point theorem, the p-th moment exponential stability of mild solutions to system is obtained. In Chapter 4, we firstly attempt to recall an impulsive-integral inequality by considering impulsive effects in stochastic systems. Then we define an attracting set and study the exponential stability of mild solutions to impulsive neutral stochastic delay partial differential equations with Poisson jumps by employing impulsive-integral inequality. Chapter 5 investigates p-th moment exponential stability and almost sure asymptotic stability of mild solutions to stochastic delay integro-differential equations. Finally in Chapter 6, we study the exponential stability of neutral impulsive stochastic delay partial differential equations driven by a fractional Brownian motion.

This manuscript is involved in the study of stability of the solutions of functional differential equations (FDEs) with random coefficients and/or stochastic terms. We focus on the study of different types of stability of random/stochastic functional systems, specifically, stochastic delay differential equations (SDDEs). Introducing appropriate Lyapunov functionals enables us to investigate the necessary conditions for stochastic stability, asymptotic stochastic stability, asymptotic mean square stability, mean square exponential stability, global exponential mean square stability, and practical uniform exponential stability. Some examples with numerical simulations are presented to strengthen the theoretical results. Using our theoretical study, important aspects of epidemiological and ecological mathematical models can be revealed. In ecology, the dynamics of Nicholson’s blowflies equation is studied. Conditions of stochastic stability and stochastic global exponential stability of the equilibrium point at which the blowflies become extinct are investigated. In finance, the dynamics of the Black–Scholes market model driven by a Brownian motion with random variable coefficients and time delay is also studied.

Most of the existing results on stochastic stability use a single Lyapunov function, but we shall instead use multiple Lyapunov functions in this paper. We shall establish the sufficient condition, in terms of multiple Lyapunov functions, for the asymptotic behaviours of solutions of stochastic differential delay equations. Moreover, from them follow many effective criteria on stochastic asymptotic stability, which enable us to construct the Lyapunov functions much more easily in applications. In particular, the well-known classical theorem on stochastic asymptotic stability is a special case of our more general results. These show clearly the power of our new results. Two examples are also given for illustration.

SummaryThis work investigates the problem of L2‐L∞ filtering for a class of stochastic nonlinear systems with nonuniform sampling. The sampled‐data filter developed in this paper is an impulsive differential system whose states change abruptly at every sampling instant. The resulting filtering error system is modeled as a stochastic nonlinear impulsive differential system. The goal is to propose a method for designing a target filter that ensures the stochastic asymptotic stability of the filtering error system and guarantees a prescribed L2‐L∞ performance. Based on a time‐varying Lyapunov functional, by virtue of a convex combination technique, a design method to achieve such a filter is formulated in the form of solving a set of linear matrix inequalities. The effectiveness of the proposed filtering strategy is shown via a numerical example of a stochastic Chua's circuit system.

In this paper, we study the exponential stability in the p th moment of mild solutions to impulsive stochastic neutral partial differential equations with memory. Sufficient conditions ensuring the stability of the impulsive stochastic system are obtained by establishing a new integral inequality. The results obtained here generalize and improve some well-known results.

Some Contributions to Stochastic Asymptotic Stability and Boundedness via Multiple Lyapunov Functions

In this paper, we study the time optimal control of a class of partial stochastic impulsive functional differential systems with pseudo almost periodic coefficients in the α-norm functional spaces. Firstly, the existence of p-mean piecewise pseudo almost periodic mild solutions for the impulsive stochastic control systems is investigated by utilising stochastic analysis, analytic semigroup, the fixed-point techniques with fractional power arguments. Secondly, the existence of time optimal pairs of a system governed by partial stochastic impulsive functional differential equations is also obtained. Finally, two examples are provided to illustrate the time optimal control problems of parabolic control system with pseudo almost periodic coefficients and impulses.

A Limit Theorem for the Solutions of Differential Equations with Random Right-Hand Sides

This study investigates the existence, stability, and controllability of multi-term stochastic fractional impulsive differential equations. By employing the contraction mapping principle, sufficient conditions ensuring stochastic stability are rigorously established. Two classes of systems—linear and nonlinear are analyzed in detail. Furthermore, the controllability of these sys-tems is demonstrated using the Gramian operator approach. To validate the theoretical findings, numerical simulations are performed in MATLAB, illustrating the effectiveness and applicability of the proposed results.

In contrast to previous research on periodic averaging principles for various types of impulsive stochastic differential equations (ISDEs), we establish an averaging principle without periodic assumptions of coefficients and impulses for impulsive stochastic fractional differential equations (ISFDEs) excited by fractional Brownian motion (fBm). Under appropriate conditions, we demonstrate that the mild solution of the original equation is approximately equivalent to that of the reduced averaged equation without impulses. The obtained convergence result guarantees that one can study the complex system through the simplified system. Better yet, our techniques dealing with multi-time scales and impulsive terms can be applied to improve some existing results. As for application, three examples are worked out to explain the procedure and validity of the proposed averaging principles.