Mathematical Sciences: Abnormal Minimizers and Discontinuous Value Functions in Optimal Control

数学科学：最优控制中的异常极小化器和不连续值函数

• 批准号: 9622967
• 负责人: Urszula Ledzewicz
• 金额: $ 6.5万
• 依托单位: Southern Illinois University at Edwardsville
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-06-15 至 1999-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9622967&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Abnormal; Minimizers; Discontinuous

9622967 Ledzewicz The project adresses two fundamental approaches to finding the solution of optimal control problems: 1) finding the optimal processes through an analysis of the Maximum principle and 2) finding the value-function as a solution of the Hamilton-Jacobi-Bellman equation. In this project the proposer investigates new nontrivial optimality conditions for abnormal trajectories and their role in connection with discontinuities in the value function. Most existing results for optimality do not apply to abnormal extremals or require continuity of the value function. The occurence of abnormal processes is related to the fact the surjectivity condition in the Lyusternik theorem is not satisfied and as a result the classical nontrivial results cannot be derived. While mathematically desirable, continuity of the value function relates to small-time local controllability and need not be satisfied for many control systems with free terminal time. Abnormal processes which satisfy the Maximum Principle do so regardless of the objective. Hence if they are really optimal this strongly hints that in a certain sense they are the only possible candidates to solve the problem whereas closeby trajectories fail to do so. Thus optimal abnormal processes somehow correspond to limiting or boundary-like behaviors of optimal trajectories which strongly correlates them with discontinuities in the value function. It is expected that the analysis of discontinuities in the value function coupled with existing theory provides a solution methodology to the general problem in optimal control. The proposal consists of two separate but closely related parts. The first part is a continuation of previous work and addresses further developments of the proposer's earlier theory of nontrivial optimality conditions for abnormal processes based on second order approximations. A high-order generalization of the Lyusternik theorem without surjectivity condition and high -order approximations to the constraint sets are investigated. Using these results extended nontrivial first and second order necessary conditions for optimality of both normal and abnormal problems in optimization and optimal control are analyzed. A generalization of the proposer's existing results to a nonsmooth setting coupled with direct approximations in the dual space by means of normal cones is being pursued. In the second part of the project the proposer investigates sufficient conditions for the optimality of abnormal trajectories. In particular, the proposer analyses the role which is played by optimal abnormal extremals in the construction of a regular synthesis. %%% There are two fundamental approaches to finding the solution of optimal control problems: 1) finding the optimal processes through an analysis of the Maximum principle and 2) finding the value-function as a solution of the Hamilton-Jacobi-Bellman equation. Both of these are in general rather difficult objectives to achieve and depend on the specifics of the problem under investigation. On the other hand, judging by the known examples the discontinuities of the value-function seem to relate rather directly to abnormal processes. Since both abnormal processes and discontinuities of the value function are indicating the limiting behaviors of optimal trajectories the proposer expects that there exists a relation between these two phenomena in general. In this project the proposer investigates optimal abnormal trajectories in optimal control and their role in connection with discontinuities in the value function. It is expected that the analysis of discontinuities in the value function coupled with existing theory provides a solution methodology to the general problem in optimal control. The proposal consists of two separate but closely related parts. The first part is a continuation of previous work and addresses further developments of the proposer's earlier theory of o ptimlity conditions for abnormal problems based on second order approximations. Using high-order approximations the proposer generalizes these results to obtain nontrivial first and second order necessary conditions for optimality of both normal and abnormal problems in optimization and optimal control. A generalization of the proposer's existing results to a nonsmooth setting coupled with direct approximations in the dual space by means of normal cones is being pursued. In the second part of the project the proposer investigates sufficient conditions for the optimality of abnormal trajectories. In particular, the proposer analyses the role which is played by optimal abnormal extremals in the construction of a regular synthesis, which is an essential part necessary for the complete solution of the optimal control problem. ***

该项目解决了寻找最优控制问题解的两种基本方法：1)通过对极大值原理的分析找到最优过程，2)找到作为Hamilton-Jacobi-Bellman方程解的值函数。在这个项目中，提议者研究了异常轨迹的新的非平凡最优性条件及其在价值函数中与不连续有关的作用。大多数现有的最优性结果不适用于异常极值或要求值函数的连续性。异常过程的出现与Lyusternik定理中的满性条件不满足而不能导出经典的非平凡结果有关。虽然在数学上是理想的，但值函数的连续性涉及小时间局部可控性，对于许多具有自由终端时间的控制系统不需要满足。无论目标如何，满足最大原则的异常过程都会这样做。因此，如果它们确实是最优的，这在某种意义上强烈暗示它们是解决问题的唯一可能的候选者，而靠近的轨迹则不能解决问题。因此，最优异常过程在某种程度上对应于与价值函数不连续密切相关的最优轨迹的极限或边界样行为。期望对值函数不连续点的分析与已有的理论相结合，为最优控制中的一般问题提供一种求解方法。该提案由两个独立但密切相关的部分组成。第一部分是先前工作的延续，并进一步发展了基于二阶近似的异常过程的非平凡最优性条件的早期理论。研究了无满射条件下Lyusternik定理的高阶推广和约束集的高阶逼近。利用这些结果，分析了优化和最优控制中正常问题和异常问题的非平凡一阶和二阶最优性的扩展必要条件。将提出者已有的结果推广到在对偶空间中使用法锥直接逼近的非光滑设置。在项目的第二部分，作者研究了异常轨迹最优性的充分条件。特别地，作者分析了最优异常极值在构造规则综合中所起的作用。找到最优控制问题的解有两种基本方法：1)通过对极大值原理的分析找到最优过程；2)找到作为Hamilton-Jacobi-Bellman方程解的值函数。一般来说，这两个目标都很难实现，而且取决于所调查问题的具体情况。另一方面，从已知的例子来看，价值函数的不连续似乎与异常过程直接相关。由于异常过程和值函数的不连续都表明了最优轨迹的极限行为，因此作者期望这两种现象之间一般存在联系。在这个项目中，提议者研究最优控制中的最优异常轨迹及其与值函数不连续的关系。期望对值函数不连续点的分析与已有的理论相结合，为最优控制中的一般问题提供一种求解方法。该提案由两个独立但密切相关的部分组成。第一部分是先前工作的延续，并进一步发展了基于二阶近似的异常问题的0最优条件理论。利用高阶近似推广了这些结果，得到了优化与最优控制中正常问题和异常问题的非平凡一阶和二阶最优性的必要条件。将提出者已有的结果推广到在对偶空间中使用法锥直接逼近的非光滑设置。在项目的第二部分，作者研究了异常轨迹最优性的充分条件。特别地，作者分析了最优异常极值在正则综合的构造中所起的作用，而正则综合是最优控制问题完全解的必要组成部分。＊＊＊