Stabilization Of Highly Nonlinear Stochastic Functional Differential Equations With Markov Switching

In recent years,more and more attention has been paid to hybrid stochastic systems with continuous dynamics and discrete events intertwined.Due to the structure and parameters of these systems may change suddenly,people often use continuous-time Markov chain to model this phenomenon.These stochastic differential equations with Markov switching are widely used in various fields of engineering and science and technology.On the other hand,time delay and uncertainty are ubiquitous entrenched in everyday life.In the context of dynamic system,stochastic functional differential equations have also become a focal point in theoretical research.This paper focuses on the stabilization of highly nonlinear stochastic functional differential equations with Markov switching.Abandoning the traditional linear growth condition and retaining the local Lipschitz condition,the feedback control problems of highly nonlinear stochastic functional differential equations are discussed under the generalized Khasminskii condition.The main results of this paper is as follows:(1)The delay feedback control of hybrid stochastic differential equations with infinite delay is discussed.By selecting an appropriate phase space,the existence and uniqueness of the solution of the controlled system with Markov switching are guaranteed,and the asymptotic boundedness of the solution is obtained by using the properties of the constructed degenerate probability measure space.The author not only proves the mean square exponential stability of the controlled highly nonlinear equation,but also improves the similar results of the previous finite delay system by using more delicate estimation.Finally,according to the moment boundedness and mean square exponential stability of the solution,the almost surely exponential stability of the controlled functional equation is obtained by using Borel-Cantelli lemma.(2)Using the discrete state observation data,the feedback controller is designed to stabilize the highly nonlinear stochastic delay differential equation with Markov switching.In the proof,the moment of the solution on each observation period is estimated,and the asymptotic boundedness of the moment is obtained by recursive addition.The boundedness of the coefficients of the controlled equation in L2 is obtained by Lyapunov functional method,and the asymptotic stability of the moment is obtained by combining the uniform continuity of the mean square moment.Meanwhile,an explicit upper bound of discrete observation interval is given.Then the almost surely asymptotic stability of the controlled highly nonlinear equation is obtained by using the stochastic LaSalle theorem.Finally,three examples are given according to the difference of observability and controllability of the subsystems,and the role of Markov switching in feedback control is shown by examples.(3)The exponential stabilization of hybrid neutral stochastic differential equations with superlinear coefficients is further studied.Firstly,it is proved that the given control function can still make the coefficients of the controlled highly nonlinear equation satisfy the generalized Khasminskii-type condition.The asymptotic boundedness of the p-th moment of the solution of the controlled system is obtained by using the integral Lyapunov functional(p>2).Then,two stability criteria directly related to the equation coefficients are constructed through multiple M matrices,and the q-th exponential stability and almost surely exponential stability of the controlled system with Markov switching are obtained by using the inequality about the neutral term.Finally,a high-dimensional example is given to verify the previous theoretical results.(4)For highly nonlinear stochastic functional differential equations with Markov switching,the delay feedback control is designed based on discrete observation data to make the controlled system exponentially stable.The existence and uniqueness of the equation are obtained by using stopping time,random integral inequality and Gronwall inequality.Under the condition of moment boundedness of controlled high nonlinear systems,a Lyapunov functional with free parameter is constructed,and the boundedness of Lyapunov operator in L2 is obtained from the generalized Ito formula.On this basis,the author proves that the controlled system with Markov switching is moment exponentially stable and almost surely exponentially stable.Finally,the numerical simulation shows that the discrete observation time interval can be adjusted appropriately according to the difference of controller performance.