Stability Analysis Of Impulsive Stochastic Functional Differential Equations With Distributed Delay

In the real world,random and pulse phenomena exist objectively.Therefore,when investigating the long-term behavior of any dynamical system,We can not ignore their impact.Moreover,delay phenomenon is also widespread phenomenon,timedelay widely exists in the control system?circuit system and a great many real systems,it leads to instability and poor performance of the system,so to discuss the stability of differential equations with delay is very necessary.In the real system time-delay is varied,the discrete delay(usually called a constant delay)and distributed delay are natural and common,this requires a unified consideration of the two conditions,so that the so-called impulsive stochastic differential equation with a distributed(mixed)delay is formed.So,it's of great theoretical and practical significance to study the stability of this equation.In this paper,the stability of impulsive stochastic functional differential equations with distributed delay is studied.First of all,using the stochastic analysis theory,Lyapunov functional method,It?o formula we obtained the sufficient conditions for the stochastic stability of the equation and the sufficient conditions of uniform stability,secondly,by the establishment of the general form of the impulse integral differential inequality obtained sufficient conditions for the exponential stability of the equations,finally,this article also discussed the sufficiency condition with the application of impulsive stochastic differential equations with mixed delay,and numerical examples are given to illustrate the effectiveness and practicability of the obtained conditions.