Optimal distributed control for a Cahn-Hilliard type phase field system related to tumor growth

The main objective of this paper is to consider an optimal distributed control problem for a Cahn-Hilliard type phase field system related tumor growth, which consists equation with two relaxation terms and reaction-diffusion suitable variable representing the concentration cytotoxic drugs in medical treatment. We prove well-posedness state its corresponding linearized system. Based on these results, we establish Fréchet differentiability control-to-state operator deduce first-order necessary optimality conditions variational inequality involving adjoint variables. Moreover, by establishing some priori estimates that are uniform parameter, implement asymptotical analysis solutions as well when parameter approaches zero.