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

We consider a particular phase field system which physical context is that of tumor growth dynamics. The model we deal with consists of a Cahn-Hilliard equation governing the evolution of the phase variable which takes into account the tumor cells proliferation in the tissue coupled with a reaction-diffusion equation for the nutrient. This model has already been investigated from the viewpoint of well-posedness, long-time behavior, and asymptotic analyses as some parameters go to zero. Starting from these results, we aim to face a related optimal control problem by employing suitable asymptotic schemes. In this direction, we assume some quite general growth conditions for the involved potential and a smallness restriction for a parameter appearing in the system we are going to face. We provide the existence of optimal controls and a necessary condition for optimality is addressed.