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

[404591]考虑一类广义最优控制问题，其中控制约束和/或状态约束是凸集中的包含形式。给出了数据函数可微时的辅助问题。一个新的术语预计会出现，因此，这是包络效应（由川崎引入）的概括。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。本文研究的核心思想是对时间变换进行适当的修正。Dubovitskii和a . a . Milyutin，以获得一个可以应用弱局部最优的二阶必要条件（在提案的第一部分中得到了发展）的新问题。本课题的第三部分是在存在纯状态不等式约束和控制集约束的情况下发展充分性理论。本研究涉及一个修正的Riccati方程，并研究Riccati方程、耦合点和二次变分正确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，最优控制理论是在20世纪50年代发展起来的，主要用于处理工程、自然资源和经济方面的问题。这个领域是优化的一个分支，可以描述如下：在满足给定要求的情况下，找到产生最小成本（或最大回报）的候选项。后者被称为约束。“一阶最优性条件”用于生成成本最小的可能候选项列表。在实践中，他们中的许多人并没有提供最低的成本。因此，需要进行另一次测试来筛选候选人并获得一个简短的名单。这个测试唤起了所谓的“二阶条件”。在这个项目中，最优控制问题的二阶最优性条件将得到非常一般的约束。事实上，最近在航空航天科学、工业过程控制、机器人和数学经济学中的几个问题都可以表述为具有这种一般约束（状态和/或控制集）的最优控制问题。然而，在这种复杂的情况下，目前还没有令人满意的二阶条件。因此，一旦完成，本研究将为工程师和经济学家选择最优策略提供新的工具。＊＊＊

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. ***