This paper is devoted to present and solve some optimal control problems, oriented to therapy, for a particular model of tumor growth. In the considered systems, the state is given by one or several functions that provide information on the cell population and also the tumor shape evolution and the control is a time dependent function associated to the therapy strategy (in practice, a cytotoxic drug). We first present and analyze the model (based on PDEs) and the related optimal control problems. The solutions are expected to provide the best therapy strategies for a given set of constraints (here, the cost or objective function is a measure of the number of cells at a given final time T). We also recall some mathematical techniques for solving the related optimization problems and we illustrate the behavior of the methods and the validity of the models with several numerical experiments. In view of the results, we are able to design appropriate strategies that, at least to some extent, are confirmed by real data. Finally, we present some conclusions and indications on future work.
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