Mathematical Sciences: Abnormality in Optimization and Optimal Control Problems

在这个项目中，首席研究员将探索使用新技术来研究涉及非正则算子的最优化和最优控制问题中所谓的“异常”情况。这些技巧来自泛函分析，并已成功地应用于处理涉及欧拉-拉格朗日方程和庞特里亚金极大值原理的反常情况。主要研究者特别关心的是确定这些技术的极限，并获得比以前在推导一般极值问题的非平凡最优性条件时得到的结果更准确的结果。最优化和最优控制问题的最优性条件的推导和利用是第二次世界大战以来几十年来最成功的故事之一。这些结果为发展一系列提高生活质量的控制系统铺平了道路。涉及机器人系统的更新进展也可以追溯到这项工作。在这个项目中，首席研究员将研究传统理论不适用的非典型情况下的最优性条件。要做到这一点，需要使用强大的抽象泛函分析方法，并结合经典控制理论的透彻知识。

In this project the principal investigator will explore the use of new techniques to study what are called "abnormal" cases in optimization and optimal control problems that involve nonregular operators. These techniques come from functional analysis, and they have been applied successfully to treat abnormal cases involving the Euler-Lagrange equations and the Pontryagin Maximum Principle. The principal investigator is concerned specifically with determining the limits of these techniques and with obtaining sharper results than those obtained previously for the derivation of nontrivial optimality conditions for general extremal problems. The derivation and exploitation of optimality conditions for problems in optimization and optimal control have been one of the major success stories of the decades since World War II. These results have paved the way for the development of a host of control systems that have improved the quality of life. More recent advances involving robotic systems can be traced to this work as well. In this project the principal investigator will examine optimality conditions in atypical cases where the traditional theory is inadequate. To do so requires the use of powerful methods of abstract functional analysis combined with a thorough knowledge of classical control theory.