Stability Analysis Of Stochastic Differential Equations And Difference Equations With Markov Processes

With the deepening understanding of nature,people found that the ordinary differential equations are more adapted to the situation under ideal conditions,so the theory of stochastic differential equations has received more and more attention.Since Markov process contains both discrete and continuous states and can switch freely between different states,it can be well used to describe relatively complex differential equation systems.Therefore,stochastic differential equations with Markov process are a very important class of differential equations.The stability of stochastic differential equations has been a classical research direction,and in 1996,Roxin first proposed the finite-time stability under the stable equilibrium of the Lyapunov function.This concept has been widely used in the field of control once it was proposed.This thesis focuses on the finite-time stability of stochastic differential equations with Markov processes.Chapter 1 firstly introduces the background and significance of the thesis,briefly describes some recent related research results and introduces some main works done in this paper.Chapter 2 firstly introduces the model and some basic definitions of continuous-time stochastic differential equations with Markov process.Next,some important lemmas are given.Then sufficient conditions for the finite-time stability of these differential equations are given by using the Lyapunov function method and some better inequalities obtained from the Lemma.Finally,using Matlab software,arithmetic examples are given to verify the feasibility and superiority of the obtained conditions.Chapter 3focuses on the finite-time stability of discrete-time stochastic differential equations with Markov processes.Sufficient conditions for the finite-time stability of these differential equations are given by combining the improved inequality and Lyapunov function methods,and finally,examples are given to verify the feasibility and superiority of the obtained conditions.Chapter 4 concludes this thesis and points out some shortcomings and future research directions of this paper.