The Existence Of Periodic Solutions In Distribution For Stochastic Differential Equations With Markovian Switching

The theory of stochastic differential equation has been established with the development of science.With the abundance of knowledge and the deepening of cognition,it is found that the general stochastic differential equations can not apply to some specific circumstances.Such as,for some multi switch control systems with multiple states,they are difficult to be described by general stochastic differential equations.Based on this,Markovian switching stochastic differential equations are established.Owing to including both discrete states and continuous states,and the ability to switch freely between different states,Markovian chain can be used to describe the system well.Markovian switching stochastic differential equation has been widely used in aircraft control,network switching and other fields.On the other hand,for a differential equation system,it is necessary to study its periodic problem,so this paper will study and solve these problems.In this paper,we obtain the existence of random periodic solutions in distribution of stochastic differential equations with Markovian switching and establish the standard for the existence of periodic solutions in distribution.Finally,a law of large numbers for stochastic differential equations with Markovian switching is constructed by using a technique analogous to Halanay’s criterion.Lyapunov method is used to prove the existence and uniqueness of uniformly asymptotically stable periodic solutions for stochastic differential equations with Markovian switching in distribution based on these.This theorem also can be extended to stochastic functional differential equations with Markovian switching.In the end,an example is provided to illustrate the effectiveness of the theorem.