Mathematical Sciences: Second-Order Optimality Conditions for Problems with Constraints

9404591 Zeidan考虑一般的最优控制问题，其中的控制和/或状态约束是凸集的包含形式。导出了数据函数与Lipschitz导数可微时的辅助问题。预计会出现一个新的术语，因此，这是(川崎引入的)包络效应的推广。结果将仅以原始数据表示。本文的第二部分给出了上述问题强局部极小性的新的二阶必要条件。这是一个长期存在的问题。本研究的核心思想是对A·Ya提出的时间转换进行适当的修改。Dubovitskii和A.A.Milyutin提出的弱局部最优性的二阶必要条件(在第一部分中得到)，以得到一个新的问题。本课题的第三部分发展了纯状态不等约束和控制集约束情况下的充分性理论。这项研究涉及一个修正的Riccati方程，并研究了Riccati方程、耦合点和二次变分的正定性这三个概念之间的联系。这个项目是最优控制理论的核心，最优控制理论是S在20世纪50年代发展起来的一门学科，主要是为了处理来自工程、自然资源和经济学的问题。这个领域是优化的一个分支，可以这样描述：找到一个在满足给定要求的同时产生最小成本(或最大回报)的候选者。后者被称为约束。“一阶最优条件”用来以最小的成本产生一份可能的候选名单。在实践中，许多条件并不能提供最低的成本。因此，需要另一种测试来筛选候选者并获得一份短名单。这种测试引起了所谓的“二阶条件”。在这个项目中，我们将得到具有非常一般约束的最优控制问题的二阶最优性条件。事实上，航空航天科学、工业过程控制、机器人学和数学经济学中的几个最新问题可以表示为具有这样的一般约束(状态和/或控制集)的最优控制问题。然而，在这种复杂的设置下，目前没有令人满意的二阶条件可用。因此，当这项研究完成后，将提供新的工具来帮助工程师和经济学家选择最优策略。***

9404591 Zeidan A general optimal control problem is considered in which the control and/or state constraints are of the form of an inclusion in a convex set. The accessory problem when the data functions are differentiable with Lipschitz derivatives will be derived. A new term is expected to appear and hence, this is a generalization of the envelope- like effect (introduced by Kawasaki). The results will be expressed only in terms of the original data. New second- order necessary conditions for strong local minimality of the above problem form the second part of this project. This is a long-standing problem. The key idea of this research is based on suitably modifying the time transformation, introduced by A. Ya. Dubovitskii and A. A. Milyutin, in order to obtain a new problem to which the second-order necessary conditions for weak local optimality (developed in the first part of the proposal) could be applied. The third part of this project is to develop the sufficiency theory for the case when pure state inequality constraints and control set constraints are present. This research involves a modified Riccati equation and the study of connections between the three concepts: Riccati equation, coupled points, and positive definiteness of the second variations. This project lies at the heart of optimal control theory, a subject which was developed in the 1950's mainly to deal with problems emanating from engineering, natural resources, and economics. This field, which is a branch of optimization, could be described as follows: Find a candidate that produces the smallest cost (or the largest return) while meeting given requirements. The latter are referred to as constraints. "First order optimality conditions'' serve to produce a list of possible candidates for the smallest cost. In practice, many of them do not furnish the least cost. Hence, there is a need for another test to screen the candidates and to obtain a short list. This test evok es what are known as "second order conditions''. In this project, second-order optimality conditions for optimal control problem with very general constraints will be derived. In fact, several recent problems in aerospace science, industrial process control, robotics and mathematical economics can be formulated as optimal control problems with such general constraints (state and/or control set). However, in this complicated setting no satisfactory second order conditions are presently available. Thus, when accomplished, this research will provide new tools to assist engineers and economists in the selection of optimal strategies. ***