Analysis of Optimal Control Problems with State Space Constraints Arising in Applications

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In this project, synthesis type sufficient conditions for optimality of controlled trajectories in optimal control problems with state-space constraints will be developed, including both local results in the form of a field of extremals around a reference trajectory and global results in the form of a regular synthesis. This research is strongly motivated by problems arising in applications and will be used to prove optimality of families of extremals for problems from two different application areas, namely (a) the minimization of the base transit time in semi-conductor devices in electronics and (b) the analysis of mathematical models for cancer chemotherapy when explicit upper limits on the number of cancer cells or lower limits for bone marrow are included. (a) In electronics, the problem of choosing the base-doping profile to minimize the base transit time in semi-conductor devices generally is solved numerically. In this project, the problem is formulated as an optimal control problem with state space constraints, and different models corresponding to both homojunction and heterojunction bipolar transistors are analyzed. For silicon bipolar transistors, explicit analytic solutions for general carrier diffusion coefficients will be derived, and their optimality will be proven by constructing a complete synthesis. (b) In many mathematical models for cancer chemotherapy the aim is to minimize the number of cancer cells at the end of a pre-determined fixed therapy interval while limiting the toxicity of the drugs through a penalty term. This, however, does not always prevent the number of cancer cells from rising to unacceptably high levels in between. It therefore is more realistic to consider models that incorporate explicit limits on the cancer cells as state-space constraints. The analysis of optimal controls for such models will be pursued.In this project, mathematical conditions will be developed that guarantee the optimality of controlled processes in the presence of strict limits. This theoretical research will be guided by two practical problems, one from electronics, the other biomedical. In the electronics topic, complete analytical solutions will be developed for a simplified model describing the speed of certain semiconductor devices. In the biomedical area, mathematical models for chemotherapy of cancer or other diseases will be analyzed in the presence of restrictions, for example, that bone marrow must not fall below a specified minimum level or that cancer cells are not allowed to increase beyond a maximum level. These topics also will be used to illustrate optimal control methods on an undergraduate level with more relevant and realistic problems than are currently available in most textbooks on the subject.