On The General Stability Of Impulsive Stochastic Differential Equations With Markovian Switching

Random and pulse phenomena is common phenomena in the field of nature and society. Any dynamical system will be affected by them in its operation process. Sometimes even small random or pulse factors will have a greater impact on the system, such as the oscillation or instability of the system, and even the mutation of the system’s structure and parameters, which can bring great difficulties to the control of system in theory research and engineering practice. Therefore, the research of the stability of impulsive stochastic differential equations with Markov switching is not only has great theoretical significance, but also has important practical application value. When discussing the stability of an equation, we usually should consider its convergence speed. That is to say, in which rate the solution of equation will decay to equilibrium solution. Undoubtedly, exponential convergence is simple and ideal mode of convergence. However, in many cases, the equation is not exponential convergence, and may be lower than the convergence rate of exponent. In this case, we can only use the general functions such as polynomial and logarithmic to describe. Therefore, this paper mainly discusses the general stability of impulsive stochastic differential equations with Markov switching. Applying the theory of stochastic differential equations, Lyapunov function, Ito’s formula and Dini right upper derivative, this paper proposes the conditions of the general stability for impulsive stochastic differential equations with Markov switching. The whole thesis is divided into four chapters.In the first chapter, the research background and significance of the stability of impulsive stochastic differential equations with Markov switching are illustrated, and the overseas and domestic research status of stability and general stability of impulsive Stochastic differential equations is analyzed.In the second chapter, we mainly introduce some basic theory related to stochastic differential equations, and some common stability definitions and general stability definitions of stochastic differential equations.In the third chapter, we study the general stability of impulsive stochastic differential equation with Markov switching. Applying the theory of stochastic differential equations, Lyapunov function, Ito’s formula and Dini right upper derivative, we get some sufficient conditions of the general stability for impulsive stochastic differential equations with Markov switching.In the fourth chapter, the general stability conditions for stochastic delay differential equation and linear stochastic delay differential equation, which contain impulsive and Markov switching, are discussed by applying the results in the third chapter. Examples are given to demonstrate the effectiveness and practicality of the obtained conditions.