Analysis and simulation of sparse optimal control of the monodomain model

Abstract

This paper concerns the sparse optimal control problem subject to the monodomain equations. Monodomain equations are coupled equations that model the electrophysiological wave propagation of the action potential in cardiac muscle. This model consists of a reaction-diffusion PDE coupled with an ODE. A non-smooth term is added to the cost in addition to the usual quadratic cost so that the optimal control exhibits sparsity. Such optimal controls play a significant role in determining the position of control devices. The existence of optimal control and the differentiability of the control-to-state operator is proved for two types of cost functions with non-smooth terms. The first-order necessary condition for optimality is derived. The numerical solutions are obtained using the finite element and projected gradient methods. Sparsity properties of the control are analyzed by varying regularization parameters. A gradient method with a primal-dual active set approach is also investigated to determine the optimal control.