RUI: Synthesis in Applications of Optimal Control

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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.