Numerical solution for optimal control of the reaction-diffusion equations in cardiac electrophysiology

Abstract

Bidomain model, Reaction-diffusion equations, Ionic model, Optimal control with PDE constraints, Existence and uniqueness, FEM, Rosenbrock type methods, NCG method,

The Effects of Control Domain Size on Optimal Control Problem of Monodomain Model

“…Kunisch and Wagner [4] proved that the control-to-state mapping is well-defined for the optimal control problem in (1). This allows us to rewrite the cost functional in (1) as…”

“…Optimal control problem of monodomain model was first proposed by Nagaiah et al [1] and has been subsequently studied in [2,3,4,5], with the control objective is to dampen the excitation wavefront of the transmembrane potential using optimal applied current. The optimal control problem of monodomain model is a nonlinear optimization problem that is constrained by the monodomain model that represents the electrical behavior of the cardiac tissue.…”

“…The nonlinear conjugate gradient method is first employed by Nagaiah et al [1] to solve the optimal control problem of monodomain model. Three variants of the nonlinear conjugate gradient method are chosen and compared, namely PolakRibière-Polyak (PRP) method [8,9], Dai-Yuan (DY) method [10] and Hager-Zhang (HZ) method [11].…”

Optimal control of the bidomain system (III): Existence of minimizers and first-order optimality conditions

“…In the literature, only a few studies related to the optimal control of the bidomain system are available as yet, mostly restricted to the monodomain approximation. We mention [1,5,10,[13][14][15][16] and refer to [11], page 1527, for a closer discussion. Numerical work concerning open-loop control of the bidomain equations with the goal of dampening of excitation and reentry waves has been realized in [10,[14][15][16].…”

Error estimates for a FitzHugh–Nagumo parameter-dependent reaction-diffusion system

“…A survey of various numerical techniques associated to the methods of lines, Runge-Kutta and operator splitting methods is presented in [27] (see also references within). Finally, in the very recent work of [34], numerical techniques for the solution of optimal control problems related to FHN systems are analyzed.…”