Uniformly Lipschitz Optimal Feedback Control Problems

The present Ph.D. dissertation is concerned with uniformly Lipschitz optimal feedback control problems. In mathematical control theory, there are two kinds of control laws: open-loop controls and closed-loop controls. In the study of open-loop optimal control problems, a useful tool is the so-called "Pontryagin's Maximum Principle" which gives a set of necessary conditions to optimal open-loop controls. More precisely, it reveals a relationship among the optimal control, the corresponding optimal trajectory and its adjoint variable, by which one can obtain a representation of optimal open-loop controls in some sense. On the other hand, closed-loop controls are usually used to study the stabilizability of the given controlled system, and not very often to be used in the optimal control theory. This is because that the given optimal closed-loop control must be uniformly optimal for any initial condition (initial time and initial state) belonging to a given region. Hence, roughly speaking, an open-loop optimal control problem is of single-objective, while a closed-loop optimal control problem is of multi-objective. To our best knowledge, the investigation on how to describe the corresponding value function and how to obtain optimal feedback controls are far from complete. In this thesis, we study a special case of closed-loop controls: feedback control, which is a function just depending on the time variable and the current state variable. To ensure the existence and uniqueness of the state trajectory to the controlled system, it is usually required that a feedback control is uniformly Lipschitz with respect to the state variable. Therefore, this thesis is devoted to study uniformly Lipschitz optimal feedback control problems.The problem is discussed in a linear-quadratic framework. By the relationship between open-loop control problem and feedback control problem, we obtain the value function of the considered optimal control problem, which is different from that of the classical open-loop optimal control problem. Further, we derive a necessary condition to the ex-istence of optimal solutions: optimal feedback controls must be linear. Thus the original problem leads to a linear optimal feedback control problem with constrained feedback matrix. By solving this problem, we present a sufficient and necessary condition to the existence of an optimal linear feedback. Finally, the results are also obtained in an extended feedback control set |u| ≤ k|x|.