【作者】 门少平；

【导师】 罗学波；

【作者基本信息】 西北工业大学 ， 控制理论与控制工程， 2002， 博士

【摘要】 本文主要研究一致空间中的最优控制问题。 在第一章中，我们回顾了最优控制理论的发展历史，提出了当前研究中仍存在的三个基本问题——即无限时间区域问题、无界控制问题、以及系统函数不为F-可导问题。指出解决这些问题在最优控制理论中仍是较薄弱的环节，并介绍了本文所做的工作。 第二章提出了可积系统概念，研究了该系统的一般性质。特别是提出了弱光滑系统概念，本文的主要目标就是讨论该系统的最优控制问题。从弱光滑系统的概念及该章给出的Немыцкий系统的例子，可以看到弱光滑系统不但存在，而且真包含许多文献讨论的系统(本文称为“光滑系统”)为特例；进而，弱光滑系统的最优控制问题就包含了光滑系统的有关问题，其结果也使当前对一些零散的问题的研究——包括无限时域和无界控制问题等——得到了统一。 第三章提出了一些新的G-可微概念，并研究了算子的凸性与这些新的可微性之间的关系。这些“新的”微分概念在Hilbert空间中等价于已有的概念，但在Banach空间中将出现不等价问题。本章针对这些问题进行了研究，它为后面讨论Banach空间中的最优控制问题打下了一定的数学基础。 第四章给出了Ekeland变分原理在一致空间中的推广。在该章中，还定义了与以往文献中结构不同的允许控制集Uad。用一种一致结构代替以往文献中赋予Uad的Ekeland度量结构，从而解决了在讨论无界控制时Uad的原结构所遇到的不完备性问题；同时，我们对Ekeland原理的推广，也使该原理可用来讨论无限时间区域、无界控制条件下的最优控制问题。该章是本文的重点内容之一，其中对Ekeland原理的推广不仅使它的应用范围得到了拓广(本文在控制理论方面的应用正是这种拓广的一个实例)，而且在数学基础理论方面也有一定的发展意义。 第五章研究了可积系统的轨线变分问题。与以往的研究相比，这里对系统中函数的可微性要求都有所减弱。特别是首次对一类G-可微、但不为F-可微的系统的讨论，为下一章研究弱光滑系统最优控制的最大值原理奠定了基础。该章是本文的另一个重点内容。 第六章，在以上各章结果的基础上，研究了弱光滑系统的最优控制问题，得到了与光滑系统一致的结果——庞特李雅金类型的最大值原理(即定理6．2．7)。可以看到，我们的结11摘要果可用于有限时域和无限时域、有界控制和无界控制、并且对系统中函数f的可微性要求也有较大的减弱—f对状态变量x可以是仅有强连续的G一导数、不要求为F一可微。因此，定理6.2.7是最优控制问题研究中一个较统一的理论，有着较广‘泛的适用性。更多还原

【Abstract】 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.更多还原