RUI: Synthesis in Applications of Optimal Control

9971747 Ledzewicz该项目由两个部分组成，这两个部分都解决了应用中出现的最优控制问题的规则综合问题。在第一部分中，我们将在场论的背景下研究异常极值的最优性的充分条件。扰动反馈控制律的推广允许将异常轨迹合并到经典框架中，以构造围绕异常参考极值的局部正则合成。这些构造应该有助于揭示最优异常极值与问题的值函数中的奇点(包括间断)之间的联系。期望通过提供补充经典结果的推广，该项目的这一部分将有助于扰动反馈控制、正则综合和最优性充分条件的理论和应用，并成为应用中的重要新工具。第二部分讨论了癌症化疗数学模型中出现的一类低维非线性最优控制问题的精确而完整的解。在这些模型中，细胞周期是通过将不同的细胞阶段聚集到不同的隔间并使用阶段特异性杀伤剂来控制的。在这个项目中，将对两室和三室模型进行全面的分析。到目前为止，这些模型只进行了数值分析，但还没有得到关于最优控制结构的定性或分析结果。预计这些结果，从理论的角度来看也很有趣，将在癌症化疗方案调度的数学模型中得到应用。该项目由两个部分组成，都解决了在应用中出现的最优控制问题的完整解决方案的问题。第一种构造的动机是扰动反馈控制方案，这是工程中的一种经典工具，特别是在飞机控制设计中。拟议的建设旨在纳入传统方案无效的所谓异常轨道。预计该项目的这一部分将对理论和应用作出贡献，提供补充经典结果的概括，并成为应用中的重要新工具。在该项目的第二部分中，将研究癌症化疗的最优控制模型。其目标是最大限度地增加药物中的细胞抑制剂杀死的癌细胞数量，同时保持对正常组织的毒性可接受。一些以前只进行数值分析的模型将在本项目中进行研究，目的是获得完整的解析解。预计这些从理论角度来看也很有趣的结果将在安排癌症化疗方案的数学模型中找到应用。

9971747LedzewiczThe project consists of two parts which both address the question of a regular synthesis for optimal control problems arising in applications. In the first part sufficient conditions for optimality of abnormal extremals will be investigated in the context of field theory. A generalization of the perturbation feedback control law that allows incorporating abnormal trajectories into the classical framework will be used to construct a local regular synthesis around an abnormal reference extremal. These constructions should shed some light on the connections between optimal abnormal extremals and singularities (including discontinuities) in the value function for the problem. It is expected that this part of the project will contribute to both theory and application of perturbation feedback control, regular synthesis, and sufficient conditions of optimality by providing generalizations that complement the classical results and become important new tools in applications. The second part addresses precise and complete solutions to a class of nonlinear optimal control problems in low dimensions arising in mathematical models for cancer chemotherapy. In the models the cell cycle is controlled by clustering its different phases into compartments and using phase specific killing agents. In this project a full analysis of two- and three-compartment models will be pursued. So far these models were only analyzed numerically, but no qualitative or analytic results about the structure of optimal controls have been obtained. It is expected that these results, interesting also from a theoretical point of view, will find applications in mathematical models for scheduling cancer chemotherapy protocols.The project consists of two parts which both address the question of finding full solutions to optimal control problems arising in applications. The first construction is motivated by the perturbation feedback control scheme, a classical tool in engineering, especially in aircraft control design. The proposed construction aims to incorporate so-called abnormal trajectories for which the conventional scheme is ineffective. It is expected that this part of the project will contribute to both theory and application by providing generalizations that complement the classical results and become important new tools in applications. In the second part of the project models of optimal control of cancer chemotherapy will be investigated. The goal is to maximize the number of cancer cells that a cytostatic agent in the drug kills while keeping the toxicity to the normal tissue acceptable. Some of the models that were only analyzed numerically before will be investigated in this project with the aim of obtaining a full analytic solution. It is expected that these results, interesting also from a theoretical point of view, will find applications in mathematical models for scheduling cancer chemotherapy protocols.