Studies On Optimal Control Problems Of Several Kinds Of Discrete-Time Stochastic Systems

This thesis mainly focuses on studies of optimal control problems of several kinds of discrete-time stochastic systems.Our present PH.D.thesis consists of eight chapters.Chapter one aims to introduce the history background,current situation of the discrete-time optimal control theory,and provide the main results of our thesis.Chapter two investigates the discrete-time indefinite mean-field stochastic linear-quadratic(LQ)optimal control problem.For the finite horizon problem,we characterize the existence of the open-loop optimal control for the LQ problem.Then,by introducing the linear matrix inequalities condition,we show that the in-definite LQ problem is well-posed if and only if the generalized difference Riccati equations(GDREs)are solvable.As for the infinite horizon problem,the maximal solution is employed to design the LQ optimal control and optimal value.Specifical-ly,we deduce that under the assumption of exact observability,the mean-field system is L~2-stabilizable if and only if the GDREs have a solution,which also is the maximal solution.Finally,some numerical examples are exploited to illustrate the validity of the obtained results.Chapter three considers a type of LQ optimal control and stabilization problems for the discrete-time forward-backward stochastic delayed system(FBSDS).By the solution to a kind of general forward-backward stochastic difference delayed equa-tions and stochastic maximum principle,we explicitly design the optimal control.In addition,an equivalent condition of the mean-square exponential stabilizability for the FBSDS is obtained by investigating the asymptotic behavior of the Riccati-ZXL difference equations.Finally,some numerical examples are exploited to illustrate the validity of the obtained results.Chapter four is concerned with the first-order and second-order necessary opti-mality conditions for discrete-time stochastic optimal control problems under weak-ened convexity assumptions.By virtue of a new discrete-time backward stochastic equation,we establish a more general and constructive first-order necessary optimal-ity condition in the form of a global stochastic maximum principle.Moreover,by introducing a new discrete-time backward stochastic matrix equation,the second-order multipoint necessary optimality conditions of singular controls are derived.Chapter five deals with the first-order and second-order necessary optimality conditions concerning the components for discrete-time stochastic systems under weakened convexity assumptions.By a new discrete-time backward stochastic e-quation,we establish a first-order necessary optimality condition concerning every component in the form of a global stochastic maximum principle.Besides,we also obtain an optimality condition concerning one component of vector control in the form of a global stochastic maximum principle and concerning the other in the form of linearized stochastic maximum principle.Moreover,using a new discrete-time backward stochastic matrix equation,we derive diverse second-order necessary opti-mality conditions of singular and quasi-singular controls concerning the components.Chapter six discusses two kinds of discrete-time stochastic singular optimal con-trols problems under weakened convexity assumption,in which the control variable has two components with the first being absolutely continuous and the second singu-lar.Based on a convex perturbation on the singular part,we derive a new stochastic maximum principle.Moreover,using the spike variation on the absolutely contin-uous one and the discrete-time backward stochastic matrix equation,we obtain the second-order necessary conditions of optimality for singular controls.Chapter seven considers a kind of singular optimal control problems for discrete-time stochastic system with recursive utilities under weaker convexity as-sumption.A new stochastic maximum principle for optimal singular control is estab-lished.Moreover,using two new adjoint equations and two new variational equations for backward stochastic difference equations,we obtain the second-order necessary condition for optimality of quasi-singular control.Chapter eight,the brief summary and research prospects are presented.