RUI: Synthesis in Applications of Optimal Control

[9971747 . ledzewicz]该项目由两个部分组成，它们都解决了应用中出现的最优控制问题的规则综合问题。第一部分将在场论的背景下研究异常极值最优性的充分条件。将允许将异常轨迹纳入经典框架的微扰反馈控制律的推广用于构建围绕异常参考极值的局部正则综合。这些构造应该对该问题的值函数中最优异常极值和奇点（包括不连续点）之间的联系有所启发。期望这一部分的项目将通过提供补充经典结果的推广，并成为应用中的重要新工具，为微扰反馈控制、正则综合和最优性充分条件的理论和应用做出贡献。第二部分讨论了癌症化疗数学模型中一类低维非线性最优控制问题的精确和完整的解决方案。在该模型中，细胞周期是通过将细胞的不同阶段聚在一起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分析。到目前为止，这些模型只进行了数值分析，尚未得到最优控制结构的定性或解析结果。预计这些结果，从理论的角度来看也很有趣，将在安排癌症化疗方案的数学模型中得到应用。该项目由两个部分组成，这两个部分都解决了在应用中出现的最优控制问题的全面解决方案。第一种结构是由微扰反馈控制方案驱动的，这是一种经典的工程工具，特别是在飞机控制设计中。提出的建设旨在纳入所谓的异常轨迹，传统方案是无效的。期望这部分的工作能对理论和应用都有贡献，提供补充经典结果的概括，并成为应用中重要的新工具。第二部分将研究肿瘤化疗最优控制的项目模型。目标是使药物中的细胞抑制剂杀死的癌细胞数量最大化，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。预计这些结果，从理论的角度来看也很有趣，将在安排癌症化疗方案的数学模型中得到应用。

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.