Optimal control for nonlocal reaction‐diffusion system describing calcium dynamics in cardiac cell

Abstract

The purpose of this paper is to introduce an optimal control for a nonlocal calcium dynamic model in a cardiac cell acting on ryanodine receptors. The optimal control problem is considered as a coupled nonlocal reaction‐diffusion system with a transmission boundary condition covering the sarcoplasmic reticulum and cytosolic domain. We establish the well‐posedness result of the adjoint problem using Faedo‐Galerkin approximation, a priori estimates, and compactness arguments. The numerical discretization of direct and adjoint problems is realized by using the implicit Euler method in time and the finite element for spatial discretization. Moreover, we obtain the stability result in the L2‐norm for the direct and the adjoint discrete problems. Finally, in order to illustrate the control of our calcium dynamic model, we present some numerical experiments devoted to constant and nonlocal diffusions using the proposed numerical scheme.