AN OPTIMAL CONTROL PROBLEM OF NONLOCAL PYRAGAS FEEDBACK CONTROLLERS FOR CONVECTIVE FITZHUGH-NAGUMO EQUATIONS WITH TIME-DELAY

We study an optimal control problem governed by a class of Pyragas-type nonlocal feedback controllers with time-delay for the convective FitzHugh-Nagumo equation. An optimal control problem is constrained by a coupled nonlinear system of parabolic partial differential equations. Well-posedness of the feedback control system and the differentiability of the control-to-state mapping are proved. Further, the main goal of the paper is to establish existence results for the distributed optimal kernels in the controllers. In particular, we take kernels as control functions, and we show that optimal kernels are the solution to the optimal control problem. Furthermore, we also derive the first-order necessary optimality conditions.