On dissipativity and stabilization of time-delay stochastic systems with switching control

This paper investigates dissipativity and stabilization problems for a class of stochastic systems with time delays. Several sufficient conditions on exponential dissipativity with respect to quadratic supply rates are derived via a Lyapunov functional approach. Passive and non-expansive property of time delay stochastic systems is also characterized. In addition, a switching controller has been developed for the time delay stochastic systems so that exponential stabilization can be achieved. A simulation example is finally given to illustrate the theoretical results.     

Pathwise Estimation of Stochastic Differential Equations with Unbounded Delay and Its Application to Stochastic Pantograph Equations

The existence and uniqueness of the global solution of stochastic differential equations with discrete variable delay is investigated in this paper, and the pathwise estimation is also done by using Lyapunov function method and exponential martingale inequality. The results can be used not only in the case of bounded delay but also in the case of unbounded delay. As the applications, this paper considers the pathwise estimation of solutions of stochastic pantograph equations.     

Stability analysis of highly nonlinear hybrid multiple-delay stochastic differential equations

Stability criteria for stochastic differential delay equations (SDDEs) have been studied intensively for the past few decades. However, most of these criteria can only be applied to delay equations where their coefficients are either linear or nonlinear but bounded by linear functions. Recently, the stability of highly nonlinear hybrid stochastic differential equations with a single delay is investigated in [Fei, Hu, Mao and Shen, Automatica, 2017], whose work, in this paper, is extended to highly nonlinear hybrid stochastic differential equations with variable multiple delays. In other words, this paper establishes the stability criteria of highly nonlinear hybrid variable multiple-delay stochastic differential equations. We also discuss an example to illustrate our results.     

Razumikhin-type theorems on general decay stability of stochastic functional differential equations with infinite delay

To the best of the authors’ knowledge, there are no results based on the so-called Razumikhin technique via a general decay stability, for any type of stochastic differential equations. In the present paper, the Razumikhin approach is applied to the study of both pth moment and almost sure stability on a general decay for stochastic functional differential equations with infinite delay. The obtained results are extended to stochastic differential equations with infinite delay and distributed infinite delay. Some comments on how the considered approach could be extended to stochastic functional differential equations with finite delay are also given. An example is presented to illustrate the usefulness of the theory.     

Razumikhin method and exponential stability of hybrid stochastic delay interval systems

This paper deals with the exponential stability of hybrid stochastic delay interval systems (also known as stochastic delay interval systems with Markovian switching). The known results in this area (see, e.g., [X., Mao, Exponential stability of stochastic delay interval systems with Markovian switching, IEEE Trans. Automat. Control 47 (10) (2002) 1604-1612]) require the time delay to be a constant or a differentiable function and the main reason for such a restriction is due to the analysis of mathematics. The main aim of this paper is to remove this restriction to allow the time delay to be a bounded variable only. The Razumikhin method is developed to cope with the difficulty arisen from the nondifferentiability of the time delay.     

Exponential synchronization of complex dynamical networks with markovian jump parameters and stochastic delays and its application to multi-agent systems

This paper studies the synchronization problem of complex dynamical networks with stochastic delay which switches stochastically among several forms of time-varying delays. Both the discrete and distributed delays are considered, as well as the Markovian jump parameters. The occurrence probability distribution of the stochastic delay is assumed to be known in prior. By utilizing the Lyapunov–Krasovskii stability theory and stochastic analysis techniques, some sufficient exponential synchronization criteria are obtained, which depend not only on the size of delays, but also on the occurrence probability distribution of the stochastic delay. Moreover, the main results are successfully extended to multi-agent systems with stochastic delay. Several numerical examples are given to illustrate the feasibility and effectiveness of the proposed methods.     

Stochastic evolution equations with general white noise disturbance

Existence and uniqueness theorems are proved for a general class of stochastic linear abstract evolution equations, with a general type of stochastic forcing term. The abstract evolution equation is modeled using an evolution operator (or 2-parameter semigroup) approach and this includes linear partial differential equations and linear differential delay equations. The stochastic forcing term is modeled by defining an Itô stochastic integral with respect to a Hilbert space-valued orthogonal increments process, which can be used to model both Gaussian and non-Gaussian white noise processes. The theory is illustrated by examples of stochastic partial differential equations and delay equations, which arise in filtering problems for distributed and delay systems.     

Stability of impulsive stochastic differential delay systems and its application to impulsive stochastic neural networks

This paper is concerned with the stability of n-dimensional stochastic differential delay systems with nonlinear impulsive effects. First, the equivalent relation between the solution of the n-dimensional stochastic differential delay system with nonlinear impulsive effects and that of a corresponding n-dimensional stochastic differential delay system without impulsive effects is established. Then, some stability criteria for the n-dimensional stochastic differential delay systems with nonlinear impulsive effects are obtained. Finally, the stability criteria are applied to uncertain impulsive stochastic neural networks with time-varying delay. The results show that, this convenient and efficient method will provide a new approach to study the stability of impulsive stochastic neural networks. Some examples are also discussed to illustrate the effectiveness of our theoretical results.     

中立型随机延迟微分方程分裂步θ方法的强收敛性

中立型随机延迟微分方程常出现在一些科学技术和工程领域中.本文在漂移系数和扩散系数关于非延迟项满足全局Lipschitz条件,关于延迟项满足多项式增长条件以及中立项满足多项式增长条件下,证明了分裂步θ方法对于中立型随机延迟微分方程的强收敛阶为1/2.数值实验也验证了这一理论结果.     

Exponential stabilization by delay feedback control for highly nonlinear hybrid stochastic functional differential equations with infinite delay

Given an unstable hybrid stochastic functional differential equation, how to design a delay feedback controller to make it stable? Some results have been obtained for hybrid systems with finite delay. However, the state of many stochastic differential equations are related to the whole history of the system, so it is necessary to discuss the feedback control of stochastic functional differential equations with infinite delay. On the other hand, in many practical stochastic models, the coefficients of these systems do not satisfy the linear growth condition, but are highly nonlinear. In this paper, the delay feedback controls are designed for a class of infinite delay stochastic systems with highly nonlinear and the influence of switching state.     

A Comparison of Stochastic Systems with Different Types of Delays

In this article, we investigate the effects of different types of delays, a fixed delay and a random delay, on the dynamics of stochastic systems as well as their relationship with each other in the context of a just-in-time network model. The specific example on which we focus is a pork production network model. We numerically explore the corresponding deterministic approximations for the stochastic systems with these two different types of delays. Numerical results reveal that the agreement of stochastic systems with fixed and random delays depend on the population size and the variance of the random delay, even when the mean value of the random delay is chosen the same as the value of the fixed delay. When the variance of the random delay is sufficiently small, the histograms of state solutions to the stochastic system with a random delay are similar to those of the stochastic model with a fixed delay regardless of the population size. We also compared the stochastic system with a Gamma distributed random delay to the stochastic system constructed based on the Kurtz's limit theorem from a system of deterministic delay differential equations with a Gamma distributed delay. We found that with the same population size the histogram plots for the solution to the second system appear more dispersed than the corresponding ones obtained for the first case. In addition, we found that there is more agreement between the histograms of these two stochastic systems as the variance of the Gamma distributed random delay decreases.     

Exponential stability for markovian jumping stochastic BAM neural networks with mode‐dependent probabilistic time‐varying delays and impulse control

In this article, an exponential stability analysis of Markovian jumping stochastic bidirectional associative memory (BAM) neural networks with mode‐dependent probabilistic time‐varying delays and impulsive control is investigated. By establishment of a stochastic variable with Bernoulli distribution, the information of probabilistic time‐varying delay is considered and transformed into one with deterministic time‐varying delay and stochastic parameters. By fully taking the inherent characteristic of such kind of stochastic BAM neural networks into account, a novel Lyapunov‐Krasovskii functional is constructed with as many as possible positive definite matrices which depends on the system mode and a triple‐integral term is introduced for deriving the delay‐dependent stability conditions. Furthermore, mode‐dependent mean square exponential stability criteria are derived by constructing a new Lyapunov‐Krasovskii functional with modes in the integral terms and using some stochastic analysis techniques. The criteria are formulated in terms of a set of linear matrix inequalities, which can be checked efficiently by use of some standard numerical packages. Finally, numerical examples and its simulations are given to demonstrate the usefulness and effectiveness of the proposed results. © 2014 Wiley Periodicals, Inc. Complexity 20: 39–65, 2015     

Stability of highly nonlinear hybrid stochastic integro-differential delay equations

For the past few decades, the stability criteria for the stochastic differential delay equations (SDDEs) have been studied intensively. Most of these criteria can only be applied to delay equations where their coefficients are either linear or nonlinear but bounded by linear functions. Recently, the stability criterion for highly nonlinear hybrid stochastic differential equations is investigated in Fei et al. (2017). In this paper, we investigate a class of highly nonlinear hybrid stochastic integro-differential delay equations (SIDDEs). First, we establish the stability and boundedness of hybrid stochastic integro-differential delay equations. Then the delay-dependent criteria of the stability and boundedness of solutions to SIDDEs are studied. Finally, an illustrative example is provided.     

Convergence and Mean-Square Stability of Exponential Euler Method for Semi-Linear Stochastic Delay Integro-Differential Equations

In this paper, the numerical methods for semi-linear stochastic delay integro-differential equations are studied. The uniqueness, existence and stability of analytic solutions of semi-linear stochastic delay integro-differential equations are studied and some suitable conditions for the mean-square stability of the analytic solutions are also obtained. Then the numerical approximation of exponential Euler method for semi-linear stochastic delay integro-differential equations is constructed and the convergence and the stability of the numerical method are studied. It is proved that the exponential Euler method is convergent with strong order $\frac{1}{2}$ and can keep the mean-square exponential stability of the analytical solutions under some restrictions on the step size. In addition, numerical experiments are presented to confirm the theoretical results.     

Approximate controllability of impulsive semilinear stochastic system with delay in state

Many practical systems in physical and biological sciences have impulsive dynamical behaviors during the evolution process that can be modeled by impulsive differential equations. This article studies the approximate controllability of impulsive semilinear stochastic system with delay in state in Hilbert spaces. Assuming the conditions for the approximate controllability of the corresponding deterministic linear system, we obtain the sufficient conditions for the approximate controllability of the impulsive semilinear stochastic system with delay in state. The results are obtained by using Banach fixed point theorem. Finally, two examples are given to illustrate the developed theory.     

Pathwise Stability of Degenerate Stochastic Evolutions

For linear stochastic evolution equations with linear multiplicative noise, a new method is presented for estimating the pathwise Lyapunov exponent. The method consists of finding a suitable (quadratic) Lyapunov function by means of solving an operator inequality. One of the appealing features of this approach is the possibility to show stabilizing effects of degenerate noise. The results are illustrated by applying them to the examples of a stochastic partial differential equation and a stochastic differential equation with delay. In the case of a stochastic delay differential equation our results improve upon earlier results.     

Dupire It\^{o}’s formula for the exponential synchronization of stochastic semi-Markov jump systems with mixed delay under impulsive control

This paper emphasizes the exponential synchronization for a class of stochastic semi-Markov jump systems with mixed delay via stochastic hybrid impulsive control. The impulsive sequence includes synchronous and asynchronous impulses with the impulsive gains being a sequence of stochastic variables. Inspired by the idea of average, a concept of ``average stochastic impulsive gain" is used to qualify the impulse intensity. Our approach expands Dupire functional It\^{o}$"s formula to the stochastic semi-Markov jump systems with mixed delay for the first time. Moreover, in view of the established Lyapunov functional, graph theory, and stochastic analysis theory, some exponential synchronization criteria for the systems are derived. The theoretical results are applied to a class of Chua"s circuit systems with semi-Markov jump and mixed delay. Some synchronization criteria for the circuit systems are provided. The simulation results verify the effectiveness of the theoretical results.     

Sufficient conditions for the eventual strong Feller property for degenerate stochastic evolutions

The strong Feller property is an important quality of Markov semigroups which helps for example in establishing uniqueness of invariant measure. Unfortunately degenerate stochastic evolutions, such as stochastic delay equations, do not possess this property. However the eventual strong Feller property is sufficient in establishing uniqueness of invariant probability measure. In this paper we provide operator theoretic conditions under which a stochastic evolution equation with additive noise possesses the eventual strong Feller property. The results are used to establish uniqueness of invariant probability measure for stochastic delay equations and stochastic partial differential equations with delay, with an application in neural networks.     

滞后型分段连续随机微分方程的稳定性

本文研究了滞后型分段连续随机微分方程的解析稳定性和数值稳定性问题.首先,利用伊藤公式等方法获得了解析解均方稳定的条件,其次,对于包括均方稳定和T-稳定在内的Euler-Maruyama方法的数值稳定性问题,运用不等式技术和随机分析方法获得了一些新的结果,证明了在一定条件下,Euler-Maruyama方法既是均方稳定又是T-稳定的,推广了随机延迟微分方程的数值稳定性结论.     

Razumikhin-type Theorems on Exponential Stability of Stochastic Functional Differential Equations with Infinite Delay

The main aim of this paper is to study the stability of the stochastic functional differential equations with infinite delay. We establish several Razumikhin-type theorems on the exponential stability for stochastic functional differential equations with infinite delay. By applying these results to stochastic differential equations with distributed delay, we obtain some sufficient conditions for both pth moment and almost surely exponentially stable. Finally, some examples are presented to illustrate our theory.