Study On The Stability Of Stochastic Nonlinear Systems With Impulse And Unbounded Time Delay

This paper studies the stability of two types of systems: one type of system is stochastic impulsive differential systems with discrete delays and infinite distributed delays,the second type of system is stochastic impulsive differential systems with bounded or unbounded distributed delays.By using the average dwell-time condition and the vector lyapunov function method and the Razumikhin technique,the stability criterion of these systems is obtained.The average dwell-time mainly imposes a constraint on the impulse.The method of vector lyapunov function is mainly to couple the components,reduce the workload and reduce the requirements of the system,large-scale systems can also be studied.Razumikhin technology mainly deals with time-delay terms and reduces the coupling between components.This article is mainly divided into two parts.The first part is to use the average dwell-time condition and the vector lyapunov function to study stochastic impulsive differential systems with discrete delays and infinite distributed delays.Two cases are considered,namely,unstable impulsive dynamics and stable impulsive dynamics.For these two cases,the results show that the stochastic differential system with discrete time delays and infinitely distributed time delays is stable,but the impulsive effect is unstable.On the basis of the relationship between the average dwell-time and the impulse,the average dwell-time is given a lower bound of makes the hybrid system stochastically globally exponentially stable;when the stochastic differential system with discrete time delay and infinitely distributed time delay is not stable,on the basis of the relationship between the average dwell-time and impulse,given an upper bound on the average dwell-time,the impulsive effect can successfully stabilize the system.At last,examples are given to verify the validity of the theory.The second part is the application of stochastic analysis technique and the average dwell-time method to consider bounded or unbounded distributed delay stochastic differential system with impulse,and a new vector type Razumikhin p-th moment exponential stability criterion is derived.The characteristics of this criterion show that the combination of Razumikhin technology and vector lyapunov function is used to solve this problem.The impulse may be stable or not.Because of Razumikhin technology,it does not need infinitely distributed time delay terms and The coupling of the time-delay terms.In the end,examples are given to verify the validity of the theory.