Robust stability and controllability of stochastic differential delay equations with Markovian switching

In this paper, we investigate the almost surely asymptotic stability for the nonlinear stochastic differential delay equations with Markovian switching. Some sufficient criteria on the controllability and robust stability are also established for linear stochastic differential delay equations with Markovian switching.     

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.     

Stability in distribution of stochastic differential delay equations with Markovian switching

In this paper we discuss stochastic differential delay equations with Markovian switching. Such an equation can be regarded as the result of several stochastic differential delay equations switching from one to another according to the movement of a Markov chain. The aim of this paper is to investigate the stability in distribution of the equations.     

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.     

A note on almost sure asymptotic stability of neutral stochastic delay differential equations with Markovian switching

In this paper, we consider neutral stochastic delay differential equations with Markovian switching. Our key aim is to establish LaSalle-type stability theorems for the underlying equations. The key techniques used in this paper are the method of Lyapunov functions and the convergence theorem of nonnegative semi-martingales. The key advantage of our new results lies in the fact that our results can be applied to more general non-autonomous equations.     

Comparison theorem of one-dimensional stochastic hybrid delay systems

The comparison theorem of stochastic differential equations has been investigated by many authors. However, little research is available on the comparison theorem of stochastic hybrid systems, which is the topic of this paper. The systems discussed is stochastic delay differential equations with Markovian switching. It is an important class of hybrid systems.     

A note on stability in distribution of Markov-modulated stochastic differential equations with reflection

We extend the stability criterion in distribution as in Yuan and Mao [C. Yuan and X. Mao. Asymptotic stability in distribution of stochastic differential equations with Markovian switching, Stochastic Process. Appl. 103 (2003) 277–291] to multi-dimensional reflected Markov-modulated stochastic differential equations on a closed positive orthant.     

A note on attraction and stability of neutral stochastic delay differential equations with Markovian switching

This paper is concerned with neutral stochastic delay differential equations with Markovian switching (NSDDEs-MS). A kind of ψ ??function is introduced and some criteria on the attractor for the product of the ψ ??function are obtained. By the new criteria, the almost sure stability with the general decay rate of NSDDEs-MS could be examined, including the exponential stability and the polynomial stability. Finally, an example is provided to illustrate the applications of our results clearly.     

Stability of regime-switching stochastic differential equations

The main result is reduction of the asymptotic stability problem for a stochastic differential equation (SDE) with sufficiently rapid Markovian switching to the analogous wellstudied problem for the ??averaged?? SDE without switching. Applications to the switching stabilization problem and to ordinary differential equations (ODE) with switching are also considered.     

Stabilization of continuous-time hybrid stochastic differential equations by discrete-time feedback control

In this paper we are concerned with the mean-square exponential stabilization of continuous-time hybrid stochastic differential equations (also known as stochastic differential equations with the Markovian switching) by discrete-time feedback controls. Although the stabilization by continuous-time feedback controls for such equations has been discussed by several authors (see e.g. , , ,  and ), there is so far no result on the stabilization by discrete-time feedback controls. Our aim here is to initiate the study in this area by establishing some new results.     

On stability in distribution of stochastic differential delay equations with Markovian switching

This paper provides a new sufficient condition for stability in distribution of stochastic differential delay equations with Markovian switching (SDDEs). It can be considered as an improvement to the result given by Yuan C. et al. in [6].     

Exponential stability of stochastic delay interval systems with Markovian switching

In the past few years, a lot of research has been dedicated to the stability of interval systems as well as the stability of systems with Markovian switching. However, little research has been on the stability of interval systems with Markovian switching, which is the topic of this paper. The system discussed is the stochastic delay interval system with Markovian switching. It is a very advanced system and takes all the features of interval systems, Ito equations, and Markovian switching, as well as time lag, into account. The theory developed is applicable in many different and complicated situations so the importance of the paper is clear.     

Stabilization of hybrid neutral stochastic differential delay equations by delay feedback control

This paper is concerned with the problem of exponential mean-square stabilization of hybrid neutral stochastic differential delay equations with Markovian switching by delay feedback control. A delay feedback controller is designed in the drift part so that the controlled system is mean-square exponentially stable. We discussed two types of structure controls; that is, state feedback and output injection. The stabilization criteria are derived in terms of linear matrix inequalities.     

Attractor and Stochastic Boundedness for Stochastic Infinite Delay Neural Networks with Markovian Switching

First, we establish the stochastic LaSalle theorem for stochastic infinite delay differential equations with Markovian switching, from which some criterias on attraction are obtained. Then, by employing Lyapunov method and LaSalle-type theorem established above, we obtain some sufficient conditions ensuring the attractor and stochastic boundedness for stochastic infinite delay neural networks with Markovian switching. Finally, an example is also discussed to illustrate the efficiency of the obtained results.     

A note on the existence and uniqueness of mild solutions to neutral stochastic partial functional differential equations with non-Lipschitz coefficients

In this note, we study the existence and uniqueness of mild solutions to neutral stochastic partial functional differential equations under some Carathéodory-type conditions on the coefficients by means of the successive approximation. In particular, we generalize and improve the results that appeared in Govindan [T.E. Govindan, Almost sure exponential stability for stochastic neutral partial functional differential equations, Stochastics 77 (2005) 139–154] and Bao and Hou [J. Bao, Z. Hou, Existence of mild solutions to stochastic neutral partial functional differential equations with non-Lipschitz coefficients, Comput. Math. Appl. 59 (2010) 207–214].     

On the exponential stability in mean square of neutral stochastic functional differential equations

In [1 and 2], some efforts have been devoted to the investigation of exponential stability in mean square of neutral stochastic functional differential equations. However, the results derived there are either difficult to demonstrate in a straightforward way for practical situations or somewhat too restricted to be applied to general neutral stochastic functional differential equations, for instance, nonautonomous cases. In this paper, we shall establish some results which are more effective and relatively easy to verify to obtain the required stability.     

Robustness of general decay stability of nonlinear neutral stochastic functional differential equations with infinite delay

This paper establishes the existence-and-uniqueness theorem of neutral stochastic functional differential equations with infinite delay and examines the almost sure stability of this solution with general decay rate. This result may be used to examine almost sure robust stability. To illustrate our idea more carefully, we carefully discuss a scalar stochastic integro-differential equation with neutral type and its asymptotic stability, including the exponential stability and the polynomial stability.     

Stability of stochastic Markovian jump neural networks with mode-dependent delays

In this paper, the problem of stability analysis for a general class of uncertain stochastic neural networks with Markovian jumping parameters and mixed mode-dependent delays is considered. By the use of a new Markovian switching Lyapunov–Krasovskii functional, delay-dependent conditions on mean square asymptotic stability are derived in terms of linear matrix inequalities. Numerical examples are given to illustrate the effectiveness of the proposed approach.     

Stochastic stability of Ito differential equations with semi-Markovian jump parameters

In this note, the problem of stochastic stability for linear systems with jump parameters being semi-Markovian rather than full Markovian is further investigated. In particular, the system under consideration is described by Ito type nonlinear stochastic differential equations with phase type semi-Markovian jump parameters. Stochastic stability conditions are presented.     

Control of discrete-time hybrid stochastic systems

A realistic stochastic control problem for hybrid systems with Markovian jump parameters can have switching parameters in both the state and the measurement equations. Furthermore, both the `base' and jump states, in general, are not perfectly observed. There are only two existing controllers for this problem, both with complexity exponentially increasing with time. The authors present another control algorithm for stochastic systems with Markovian jump parameters. This algorithm is derived through the use of stochastic dynamic programming and is designed to be used for realistic stochastic control problems, i.e., with noisy state observations. This scheme has fixed computational requirements at each stage and a natural parallel implementation. Simulation results are used to compare the algorithm with previous schemes