A Generalization Of Ekeland's Variational Principle In Uniform Space And Its Applications To Optimal Control Theory

This paper is devoted to study the optimal control problems in uniform space.In chapter 1, we review the history of optimal control theory and point out that there are still three basic problems in optimal control research, i.e., infinite time interval, unbounded controls, and function in system not being F-differentiable.We see that it is still a weakish in optimal control theory to overcome these difficulties. In the end of the chapter we give a synopsis of the paper.In chapter 2, we introduce a "integrable system" and make a research on its general nature. Especially we put forward a conception of "weak-smooth system", and this paper is mainly to discuss -the optimal control problems for weak-smooth system. With the conception, and example of " system", we see that the weak-smooth system not only exists, but also genuinely contains the systems discussed in many literatures(and we call these systems as "smooth system"). Hence, the optimal control problems for weak-smooth system includes the relevant ones forsmooth system, and thus concentrates the scattered problems and the deliberations--such asinfinite time interval, unbounded controls,and function in system not being F-differentiable,etc.In chapter 3, we set some new conceptions of G-derivative, and make a research on the relations among these new derivatives and the convexity of an operator. These new derivatives are equivalent to old one in Hilbert space, but this is not true in Banach space. Chapter 3 studies these problems and thus lays a foundation for discussing the optimal control theory in Banach space.In chapter 4, we give a general result of Ekeland's variational principle in uniform space. In addition, we define a control space Uad with a different structure from that in many literatures,i.e., replacing the Ekeland distance by an uniform topology. Endued with the new structure, Uad is free of uncompleted of the old structure under unbounded control situation. Meanwhile, the generalized Ekeland's principle can be used to probe into the infinite time interval and unbounded control problems. Chapter 4 is a focal point of the paper. And the generalization of Ekeland'sprinciple not only broadens its applications--just as we use it to solve the difficulty of controltheory in this paper, but also makes a develop sense in mathematical theory.In chapter 5. we discuss the patch perturbations of a integrable system. A distinction of it is that the derivative condition is loosed. Especially, we discuss the patch perturbations for a system that the function is only G-differentiable but not F-differentiable. This lays a foundation of discussing the optimal control problems for weak-smooth system. Chapter 5 is another focal point of the paper.In chapter 6, with the results of above chapters, we discuss the optimal control problems for weak-smooth system and draw a conclusion coinciding with that of smooth system, i.e., the Pontryagin's maximum principle (see Th 6.2.7). The result can be used in control problems whether the time interval is finite or infinite, the control set is bounded or unbounded, and also for systems with a weak differentiable function, i.e.,the function f is only strong continuously G -differentiable but not F-differentiable. So, Th 6.2.7 is a uniform theory of optimal control, and it has wide-ranging applications.