Optimal Control of a Fully Parabolic Attraction-Repulsion Chemotaxis Model with Logistic Source in 2D

This paper deals with a distributed optimal control problem for attraction-repulsion chemotaxis system which described the process of cells interacting with a combination of repulsive and attractive signal chemicals. We firstly show the global-in-time existence of solutions with some smallness conditions for chemotactic intensity chi and mu or the initial total mass parallel to u parallel to(L1) sufficiently small. Secondly, the existence of optimal controls is established. Then we prove that the control-to-state operator S is Frechet differentiable. Finally we derive the first-order necessary optimality condition for the optimal control problem.