Necessary Conditions For Stochastic Optimal Control Problems

Stochastic optimal control problems have a lot of practical applications in our life,such as physics, economics, biology and so on. This thesis studies the maximum principle,necessary conditions and the existence of optimal control problems.This Ph.D. thesis is divided into five chapters.In Chapter1, the related historical background and the current situation of stochasticoptimal control problems are introduced. And the main content of this thesis is also givenin this chapter.In Chapter2, we consider the second-order Taylor expansion for backward doublystochastic control system. The results are obtained under no restriction on the convexityof control domain. Moreover, the control variable is allowed in the drift coefficient andthe diffusion coefficient. The result is new. Some analysis about the second-order Taylorexpansion is also given.In Chapter3, a Pontryagin’s maximum principle for optimal control system governedby an ordinary differential equation with endpoint constraints is proved, under the assump-tion that the control domain has no linear structure. We first give a local linearization ofthe optimal control problem along the optimal control, and transform the original probleminto a new relaxed control problem. Then, we construct a penalty functional, and transformthe new problem to a series disturbed control problems. Finally, we use the approximatemethod to get the maximum principle of the original problem. This method is differentfrom the original ones, and it is a new way for solving this kind of problems.Chapter4is an extension of chapter3. We study the same problem in the stochasticcase under the same assumptions without the endpoint constraints. We also use the newperturbation method to solve this problem.In chapter5, we study the optimal control problem governed by semilinear parabolicequations, whose control variable is contained only in the boundary condition. Someconditions for existence of the optimal control are obtained, and this is a new way forstudying the existence problems.