Some Optimal Control Problems: Ponte Liya 's Principle And Its Application

This thesis is devoted to optimal control problem of three kinds of system. The thesis especially focus on the Pontryagin’s maximum principle, including the existence of optimal controls of some non-smooth system and its necessary conditions for an optimal control, the monotone control system under the background of biological model, and the maximum principle for the heat equation with memory.The thesis is divided into four chapters.In Chapter1, the optimal control theory is reviewed, and three systems to be discussed later on are introduced respectively, including research background, develop-ment, hot topics and the main difficulties. Meanwhile, the brief results of the thesis are explained.In Chapter2, an optimal control problem of some non-smooth system in general form is studied. The non-smoothness is cause by singularity and degeneracy. Approx-imation problems and the corresponding relaxed control problems are introduced to yield the maximum principle for optimal control. The main result of this chapter holds for only one optimal control, thus, the maximum principle can be hold for the situation of system containing a unique optimal control. An example is showed for this situation.In Chapter3, monotone control system of an biological model of growth type is studied. The information of optimal control is given by the using of maximum principle. To obtain the method of judging the monotone control system, the solution of the system is estimated, and the information is discussed to match the bang-bang property of monotone control. Essentially, this chapter is devoted to study the discriminant of monotone control systems in the view of optimal control theory.The final Chapter4is devoted to the optimal control problem of a heat equation with memory in anisotropic and non-homogeneous media. An maximum principle is given by using the method of homogeneous of variation. And the problem of the lack of translation invariance is solved.