Stability With General Decay Rate Of Stochastic Functional Differential Equations With Poisson Jumps

The stability study of stochastic system is not only an academic challenge to scholars,but also a great impetus to the development of science and technology.Since the establishment of the theory of stochastic differential equation,the stochastic system has attracted the attention of scholars and gained a lot of achievements.At present,stochastic differential equations have played an important role in mathematics,physics,mechanics,economics,biology and many other fields.In reality,because many systems may have hysteresis or random failures,sudden disturbances and so on,so the traditional stochastic differential equations are no longer suitable to describe such systems.In recent years,scholars have introduced a kind of stochastic functional differential equations with Poisson jumps.But at present,most of the stability studies of this kind equations focus on exponential stability and so on.Actually,not all the equation solutions of the system are exponential convergent.Therefore,according to the actual demand,some scholars put forward the concepts of the moment stability and the almost sure stability with general decay rate based on the definition of classical stability,it theoretically provides a new perspective for people to study the system.Therefore,from the perspective of general decay rate,this paper studies these p-moment stability and almost sure stability problems of stochastic functional differential equations with Poisson jumps.The main research results of this paper are as follows:1.In this paper,the definitions with general decay rate of p-moment stability and almost sure stability of stochastic functional differential equations with Poisson jumps are given.Based on Lyapunov function method,Dini derivative,It(?) formula and some inequalities,sufficient conditions of p-moment stability and almost sure stability of this equation with general decay rate are given respectively.2.Based on the sufficient conditions obtained in(1),two kinds of stability problems of stochastic differential equations with time-varying delays with Poisson jumps are discussed from the perspective of general decay rate,and the sufficient conditions of the two kinds of stability of the equation are obtained.3.The validity and practicability of the obtained conditions are verified by numerical examples.At the end of the paper,the work of the whole paper is summarized,and some problems in this direction which can be further discussed in the futhure are given.