Optimal control for a reaction-diffusion model with tumor-immune interactions

The main objective of this paper is to consider an optimal distributed control problem for a reaction-diffusion model with tumor-immune interactions, which consists of a coupled system of reaction-diffusion equations for normal cells, tumor cells, immune cells and chemotherapeutic drug. Moreover, a suitable distributed control variable representing the concentration of cytotoxic drugs in medical treatment is introduced into the equation of chemotherapeutic drug. We first establish the well-posedness of the state system by combining of truncation method, Faedo-Galerkin method and maximum principle of second-order parabolic equations. Then we prove the existence of an optimal control, the Fr & eacute;chet differentiability of the control-to-state operator in a suitable functional analytic framework, and finally deduce the corresponding first- order necessary conditions of optimality by studying the corresponding linearized system and the backward adjoint system.