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

9622967 Ledzewicz该项目介绍了两种寻找最优控制问题解的基本方法：1)通过最大值原理分析寻找最优过程；2)寻找作为哈密顿-雅可比-贝尔曼方程解的值函数。在这个项目中，提出者研究了异常轨迹的新的非平凡最优性条件，以及它们与值函数中的不连续性有关的作用。大多数现有的最优性结果不适用于异常极值，也不要求值函数的连续性。反常过程的发生与Lyusternik定理中的满足性条件不满足有关，因而不能得到经典的非平凡结果。虽然在数学上是可取的，但值函数的连续性与小时间局部可控性有关，对于许多具有自由终端时间的控制系统来说，不需要满足该连续性。满足最大值原理的异常过程不管目标是什么都会这样做。因此，如果它们真的是最优的，这强烈地暗示，在某种意义上，它们是解决问题的唯一可能的候选者，而邻近的轨迹无法做到这一点。因此，最优异常过程在某种程度上对应于最优轨迹的限制或类似边界的行为，这将它们与价值函数中的不连续性强烈地关联起来。值函数的不连续性分析与现有的理论相结合，可望为最优控制中的一般问题提供一种解决方法。该提案由两个独立但密切相关的部分组成。第一部分是前人工作的继续，并进一步发展了作者早期基于二阶近似的异常过程的非平凡最优性条件理论。研究了不含满射性条件的Lyusternik定理的高阶推广和对约束集的高阶逼近。利用推广的一阶和二阶非平凡的结果，分析了最优化和最优控制中正态和非正态问题的最优性必要条件。将作者已有的结果推广到非光滑环境，并借助法锥在对偶空间中进行直接逼近。在项目的第二部分，作者研究了反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正则综合中所起的作用。求最优控制问题的解有两种基本方法：1)通过极大值原理分析求出最优过程；2)求取值函数作为Hamilton-Jacobi-Bellman方程的解。一般而言，这两个目标都相当难以实现，并取决于正在调查的问题的具体情况。另一方面，从已知的例子来看，价值函数的不连续性似乎与异常过程有相当直接的关系。由于值函数的异常过程和不连续性都表明了最优轨迹的极限行为，因此作者预计这两种现象之间总体上存在一种关系。在这个项目中，提出者研究了最优控制中的最优异常轨迹，以及它们与值函数中的不连续性有关的作用。值函数的不连续性分析与现有的理论相结合，可望为最优控制中的一般问题提供一种解决方法。该提案由两个独立但密切相关的部分组成。第一部分是前人工作的继续，进一步发展了作者早先提出的基于二阶近似的异常问题的正则性条件理论。利用高阶逼近，推广了这些结果，得到了最优化和最优控制中正常和异常问题的最优性的一阶和二阶非平凡必要条件。将作者已有的结果推广到非光滑环境，并借助法锥在对偶空间中进行直接逼近。在项目的第二部分，作者研究了反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的完整解所必需的部分。***

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. ***