Infinite horizon optimal control problems with applications in biomedicine: models, optimality conditions, numerical solutions.

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At the present time, mathematical models with a fix finite time horizon are used for computation of optimal cancer treatment strategies. Their aim is mostly to eliminate all the tumor cells as fast as possible whereas the side-efects and the therapy costs should be minimized (Minimization of a damage functional). The drawback of this method lies in the side-effects onto the entire human body, which are nevertheless too high to be acceptable. These are the immune weakness, unnecessarilly killed or damaged healthy tissues as well as the increased resistance of tumor cells. On the contrary, the present research project deals with infinite horizon optimal control problems and their applications to bio-medicine, namely to the models of optimal cancer treatment. These should provide less aggressive stabilizing long term therapies. The essential aims of the project can be subdivided into two parts. The first part includes all the investigations which fall in the category of fundamental research and build the theoretical background for the second part. Here necessary optimality conditions as well as an existence theorem should be derived. The functional analytical formulation of the problem and the functional analytical methodology of the proof should play hereby the key role. Another very important issue is the development of a numerical method, called pseudospectral method, which is exactly "tailored" for the considered general setting of the optimal control problem. The second part includes the investigations which concern the biomedical applications. In this part, it is to find out which innovative cancer treatment strategies can be provided by the considered class of control problems and which concrete models are adequate. The choice of the perfomance criterium (functional in the objective) of the optimal control problem of a tumor growth model has a crucial influence on the optimal solution and, consequently, on the drug administration protocols and therapy itself. Besides the damage functional, in this project a stabilization functional, which constitutes the deviation of the process (human body) from a healthy equilibrium ("tumor-free" or "tumor- and normal cells coexist") of the dynamical system and stabilizes this simultaneously, should be alternatively considered. Applying the results from the first part of the project, numerical solutions should be computed by means of the pseudospectral method and their optimality has to be verified. The application of the OCMat software of the guest institution to the same problems of cancer treatment should enable the comparison of the optimal solutions and the qualitative analysis of the underlying dynamical systems. The structural comparison of optimal solutions to the problems with finite vs. infinite horizon is of great importance in view of the resulting therapies.