Recently, Mao [19] initiates the study the mean-square exponential stabilization of continuous-time hybrid stochastic differential equations by feedback controls based on discrete-time state observations. Mao [19] also obtains an upper bound on the duration τ between two consecutive state observations. However, it is due to the general technique used there that the bound on τ is not very sharp. In this paper, we will be able to establish a better bound on τ making use of Lyapunov functionals. We will not only discuss the stabilization in the sense of exponential stability (as Mao [19] does) but also in other sense of H ∞ stability or asymptotic stability. We will not only consider the mean square stability but also the almost sure stability.

Feedback control based on discrete-time state observation for stochastic differential equations with Markovian switching was initialled by Mao (2013). In practice, various effects could cause some time delay in the control function. Therefore, the time delay is taken into account for the discrete-time state observation in this paper and the mean-square exponential stability of the controlled system IS investigated.This paper is devoted as a continuous research to Mao (2013).

Recently, a kind of feedback control based on discrete-time state observations was proposed to stabilize continuous-time hybrid stochastic systems in the mean-square sense. We find that the feedback control there still depends on the continuous-time observations of the mode. However, it usually costs to identify the current mode of the system in practice. So we can further improve the control to reduce the control cost by identifying the mode at discrete times when we make observations for the state. In this paper, we aim to design such a type of feedback control based on the discrete-time observations of both state and mode to stabilize the given hybrid stochastic differential equations (SDEs) in the sense of mean-square exponential stability. Moreover, a numerical example is given to illustrate our results.

This paper is devoted to the study of reproduction of asymptotic boundedness in the second moment and small moments of stochastic differential equations by the stochastic theta method. In addition, we illustrate that the asymptotic moment boundedness of the numerical solution stand-alone plays a key role in the study of numerical stationary distribution.