Optimality Conditions for Optimal Control of the Monodomain Model with Pointwise Control and State Constraints

This work deals with the theoretical study of the optimality conditions for the optimal control of the monodomain model arising in cardiac electrophysiology with mixed control-state constraints. The monodomain model is a coupled system of reaction–diffusion equation with cubic nonlinearity in the reaction term and an ordinary differential equation. The existence of optimal control is established, and the twice Fréchet differentiability of the control-to-state operator is discussed. The first-order necessary optimality condition is derived. Most significantly, a detailed proof of the sufficient second-order optimality condition is demonstrated with mixed control-state constraints.