Long-Time Behavior Of Stochastic (Functional) Differential Equations And Related Research

This dissertation considers the long–time behavior of stochastic(functional)differen-tial equations and related research,including ergodicity,numerical ergodicity and feedback control.To be more precise,we consider the ergodicity of regime–switching stochastic functional differential equation with infinite delay,the numerical ergodicity of stochastic functional differential equation with finite delay and the approximation of invariant prob-ability measure,stabilisation by delay feedback control for neutral stochastic functional differential equations with infinite delay,stabilisation of hybrid system with different struc-tures by feedback control based on discrete–time observations.This dissertation includes the following six chapters:Chapter 1 introduces the research backgrounds and the development situation,some necessary notations,basic definitions,fundamental theorems,and fundamental inequalities.Chapter 2 considers a class of Markovian–switching stochastic functional differential equations with infinite delay.By using the estimate of exponential functionals of the finite Markov chain,we establish exponential ergodicity of corresponding Markov process in a Wasserstein distance under certain“averaging conditions”.Further,it follows from such an ergodicity property that a strong law of large numbers for additive functionals of the regime-switching diffusions is obtained.Finally,the ergodicity and the strong law of large numbers are applied to a long–run average type of stochastic optimization.Chapter 3 is concerned with stochastic functional differential equations with finite de-lay.Under one-sided Lipschitz condition on the drift coefficient,we first establish the strong convergence of“segment process”associated with the Backward Euler–Maruyama scheme on finite time interval[0,T].Then,it is demonstrated that the“segment process”is a Markov process and the corresponding discrete-time semigroup admits a unique numerical invariant probability measure.Finally,based on the strong convergence result of“segment process”,we reveal that the numerical invariant probability measure converges in the Wasserstein dis-tance to the underlying one.Chapter 4 considers a class of neutral-type stochastic functional differential equation-s with infinite delay and highly nonlinear coefficients(i.e.,coefficients do not satisfy the linear growth condition).Assume these systems are unstable.Main aim of this chapter is to design a delay feedback control to make this class of equation become stable with gen-eral decay rate.The general decay stability includes exponential stability and polynomial stability.Finally,to illustrate our results more clearly,this paper also introduces unstable scalar neutral-type stochastic integro-differential equations and discusses their exponential and polynomial stabilisation by delay feedback controls,respectively.Chapter 5 considers a class of hybrid system with different structures in different modes.In some modes,the coefficients satisfy the linear growth condition,while in the other modes,the coefficients are highly nonlinear.Assume these systems are bounded on qth moment but unstable,we aim to design a discrete–time feedback control,which is only put in a part of modes where the coefficients are highly nonlinear,such that the controlled system not only preserves the boundedness on qth moment but also is stable.The stabilities concerned include H_∞–stability and exponential stability in the moment,as well as almost sure stability.Finally,an example is given to illustrate these results.Chapter 6 makes a brief summary for this dissertation,list the main innovations and propose some questions for further thinking.