Distributed optimal control of a tumor growth treatment model with cross-diffusion effect

. In this paper, we examine an optimal control problem of a coupled nonlinear parabolic system with cross-diffusion operators. The system describes the density of tumor cells, effector-immune cells, circulating lymphocyte population and chemotherapy drug concentration. The distributed control has been taken for drug concentration to control the amount of drug to be injected and to evade the side effects of the drug. We prove the existence of a weak solution of the direct problem. Then, the existence of control for the proposed control problem is proved. Further, we derive the optimality conditions and also the existence of a solution of the adjoint problem. The finite element numerical method is implemented for the proposed control problem. Then, theoretical results are illustrated with the help of numerical experiments. Finally, the importance of control function and the cross-diffusion effect are studied for the proposed control problem using numerical computations.