Optimal control of the bidomain system (I): The monodomain approximation with the Rogers–McCulloch model

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

“…4.2). In difference to the monodomain approximation considered in [11], the proof of the optimality conditions requires additional regularity of the minimizer (Φ tr ,Φ e ,Ŵ ,Î e ) in the case of the full bidomain system. 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.…”

“…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].…”

“…For spaces of Bochner integrable mappings, e.g. L 2 ( 0, T ) , W 1,2 (Ω) , we refer to the summary in [11], page 1542.…”

“…After the study of the monodomain approximation of the equations and a thorough stability and regularity analysis of weak solutions for the full bidomain equations, as contained in the previous papers [11,12], we are now in position to analyze the related control problems with respect to the existence of minimizers as well as to provide a rigorous proof of the first-order necessary optimality conditions.…”

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

“…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.…”