Delay Feedback Stabilization For A Class Of Stochastic Differential Equations With Jumps

Stochastic differential equations with jumps are widely used in various fields,such as industry,finance,medical care and so on.In this paper,for a class of unstable stochastic differential equations with jumps,the problem of stabilizing them with linear time-delay feedback controllers is studied.The equation has unique solution,provided that the coefficients satisfy polynomial growth condition and Khasminskii type condition.By constructing the integral Lyapunov functional,the coefficient of matrix of the controller and the magnitude of the time delay are determined.Combined with an example,numerical simulation is carried out with matlab,which intuitively shows the feasibility of the theory.This article will be divided into the following parts: Firstly,it discusses the related research on the stability and stability analysis of stochastic systems and points out the innovation of this article.In the second chapter,the research object is stochastic differential equation with jump,the Khasminskii type condition and polynomial growth condition are given,and the existence and uniqueness of the solution of the equation are proved by using IT formula and Growall inequality.A linear controller is designed to determine the range of maximum and minimum eigenvalues,so that the solution of the equation is asymptotically stable after being controlled.Further considering the design of linear time-delay controller,the selected Lyapunov functional contains not only the quadratic term but also the quartic term,and the value of the term coefficient is determined,which is the key to prove the asymptotic stability of the solution.The difference between the proof process and the general method of stochastic differential equation stabilization studied lies in the treatment of the jump part,for example,when the key step is treated with H(?)lder inequality,Kunita inequality should be used for the jump part.In chapter 3,the research object is extended to stochastic delay differential equations with jumps,and the controller scope is expanded from finite dimension space to infinite dimension space.By designing polynomial coefficients and degrees and constructing integral Lyapunov functional,the polynomial will not show explosive growth trend,and the solution H? is obtained.It should be pointed out that the research results of this paper can also be applied to the field of automatic control of random processes with discontinuous sample paths due to disturbance.