An Essay On Optimal Controls Of Mean-field Stochastic Systems And Relevant Problems

Pontryagin’s maximum principle is an important optimization method in studying optimal control problems. Since the introduction of the mean-field backward stochastic differential equations, the optimal control problems of mean-field systems have attract-ed more attentions. Based on this, maximum principle of mean-field stochastic control systems is mainly investigated.This thesis is divided into eight chapters.Chapter1mainly introduces the original, background as well as development sta-tus of the optimal control theory, and presents the main results of our thesis.Chapter2considers an optimal control problem for fully coupled forward-backward stochastic differential equations (FBSDEs) of mean-field type in the case of controlled diffusion coefficient. Without the assumption of convex control domain, the necessary optimality conditions of Pontryagin’s type is established by virtue of a reduction method. The results of this chapter are new.Chapter3is devoted to the study of optimal control problems for partially coupled mean-field forward-backward stochastic systems, where the diffusion coefficient con-tains the control variable and the control domain is non-convex. By virtue of extended Ekeland’s principle and some estimates on the state and adjoint processes, a general s-tochastic maximum principle is established in the framework of mean-field theory. The results improve and generalize some known conclusions.Chapter4is intended to concern with partially observed optimal control problems for mean-field stochastic systems. Utilizing Girsanov’s theorem as well as spike vari-ational technique, a maximum principle of Pontryagin’s type under partial information is obtained. And this result generalizes some known conclusions.Chapter5is dealing with a partially observed optimal control problem described by mean-field forward and backward stochastic differential equations (FBSDEs). Uti-lizing Girsanov’s theorem as well as a spike variation, a maximum principle for partial-ly observed mean-field forward and backward stochastic systems is established. The result is new. In Chapter6, we take into account of the near-optimal control problems for mean-field singular stochastic systems, where the control domain is non-convex. By virtue of Ekeland’s variational principle and some estimates on the state and adjoint processes, necessary and sufficient conditions for near-optimality are established in the mean-field framework. This result generalizes part known conclusions.In Chapter7, a near-optimal control problem for mean-field forward-backward stochastic differential systems is investigated. With no restriction on the convexity of the control domain, we establish the necessary and sufficient near-optimality conditions in the form of Pontryagin’s type with the help of Ekeland’s variational principle. It is the first time that the near-optimal control theory is promoted to mean-field forward-backward stochastic systems.In the final Chapter8, a summary is presented on the main content of this dissertation.