Stability Of Three Class Stochastic Nonlinear Control Systems With Poission Jump

Recently,the study of discrete stochastic dynamical systems based on stochastic differential or difference equations with ?Ito type has been a subject of intense activities.In real word,random phenomena and time-delay are inevitably encountered.However,the existence of random perturbations and delays affect the stability of system,hence,more and more scholars observe the importance of researching the stability of stochastic delayed dynamical systems.It has been recognized that Poisson process is a natural model for jump stochastic phenomena,so the stability analysis of discrete stochastic nonlinear delayed control systems(DSNDCSs)driven by Poisson processes is worth further investigating.This paper mainly focus on the stability of three kinds of DSNDCSs with Poisson processes.It should be pointed that a new effective method,namely compensated approach,to deal with the Poisson processes is proposed,which reduces the cost of the computation.Main content of this work are as follows.Firstly,the stability of DSNDCSs perturbed by Poisson processes is studied.Based on Lyapunov-Krasovskii functional and linear matrix inequality technique,some sufficient conditions to ensure the global asymptotical stability in the mean square sense are presented.At the same time,the explicit expression of the state feedback controllers is also derived.The numerical example verifies the theoretical outcome is effective.Secondly,considering that the history of the system would have an impact on the current state,another kind of time delay is introduced,that is the distributed delay.A theorem and the expression of state feedback controllers are respectively given to determine the global asymptotical stability of discrete stochastic nonlinear control systems(DSNCS)with Poisson processes and mixed time delays.Finally,because of the diversity of stochastic perturbations,the Brown process,Poisson processes and mixed time delays are taken into account in DSNCS.The controllability of DSNCS with mixed delays and Poisson process as well as Brown process is proved.The numerical simulations are given to illustrate the effectiveness of the theory.