Stabilization Of Hybrid Stochastic Differential Equations And Applications

Recently, hybrid stochastic differential equations (SDEs) with markovian switching have gained considerable public concern because they can be widely used in modeling some systems which may experi-ence abrupt changes in their structures and parameters. In the theo-retical study of hybrid SDEs, stability analysis is one of the important issues. Under some reasonable conditions, this paper deals with the moment exponential stability of hybrid stochastic differential equa-tions in both linear and nonlinear cases with the help of Lyapunov functions and linear matrix inequalities (LMIs). This stability issue is divided into the following three aspects in detail:Firstly, we deal with the p(p > 2)th moment exponential stability of hybrid stochastic differential equations by feedback control based on the discrete-time state observations.Most of the pre-existing papers mainly focused on the mean-square exponential stability of hybrid stochastic differential equation-s, and we generally design a continuous-time feedback control in the drift part to make the given unstable SDEs stable. However, such a continuous-time feedback control needs one to continuously observe the state x(t) for all t ? 0, which is not realistic and costs much be-cause the state observations are discrete-t,ime. It, is nature to motivate us to design a discrete-time feedback control instead of designing the continuous-time feedback control in the drift part, Mao( [34]) is the first paper on this issue in this area. Based on the above analysis, in this part we consider the p(p > 2)th moment exponential stability of hybrid stochastic differential equations by the feedback control based on the discrete-time state observations.Secondly, we consider the p(p > 2)th moment exponential sta-bility of hybrid stochastic differential equations by feedback control based on the discrete-time state observations with a time delay.To make the unstable SDEs stable, the method we frequently used is to design a discrete-time feedback control in the drift part.However, some factors such as data transmission, time consuming often make the discrete-time feedback control yield a time delay in practice. Based on this fact, in this part we consider the effect of delay when designing the discrete-time feedback control in the drift part to stabilize the given SDE in the sense of p(p > 2)th moment exponential stability.Finally, we focus on the mean-square exponential stability of hybrid stochastic differential equations.Because the diffusion part includes the Brownian motion w(t),its research is more complex than the study of the drift part, for which reason, many papers only designed the feedback control in the drift part while not designed in the diffusion part. why not we design the feedback control in the diffusion part? Motivated by the above problem, in this part we design an discrete-time feedback control with a time delay in both the drift and diffusion part to stabilize the given SDE in the sense of mean-square exponential stability.