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We consider a particular phase field system which physical context is that oftumor growth dynamics. The model we deal with consists of a Cahn-Hilliard typeequation governing the evolution of the phase variable which takes into accountthe tumor cells proliferation in the tissue coupled with a reaction-diffusionequation for the nutrient. This model has already been investigated from theviewpoint of well-posedness, long time behavior, and asymptotic analyses assome parameters go to zero. Starting from these results, we aim to face arelated optimal control problem by employing suitable asymptotic schemes. Inthis direction, further assumptions have to be required. Mainly, we ought toimpose some quite general growth conditions for the involved potential and asmallness restriction for a parameter appearing in the system we are going toface. We provide existence of optimal controls and a necessary condition thatan optimal control has to satisfy has been characterized as well.

Optimal treatment for a phase field system of Cahn-Hilliard type modeling tumor growth by asymptotic scheme