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This paper studies the feasibility of optimal sparse boundary control of the bidomain-bath model in cardiac electrophysiology. We incorporated the Luo–Rudy phase I (LR-I) ventricular action potential model, which accounts for more biophysical details of cellular dynamics. The boundary optimal control framework involves a standard quadratic objective functional that is to be minimized and the L 1 -norm of the control accounting for the sparsity. The existence of the bidomain-bath model using the LR-I is demonstrated, and the derivation of the first-order optimality system is shown. Then we provide the well-posedness and the sparsity of the optimal control problem. The numerical realization of such a boundary control framework for a successful defibrillation study is demonstrated. [ABSTRACT FROM AUTHOR]

We present a virtual element method (VEM) for a nonlocal reaction–diffusion system of the cardiac electric field. For this system, we analyze an |$H^1$| -conforming discretization by means of VEM that can make use of general polygonal meshes. Under standard assumptions on the computational domain, we establish the convergence of the discrete solution by considering a series of a priori estimates and by using a general |$L^p$| compactness criterion. Moreover, we obtain optimal order space-time error estimates in the |$L^2$| norm. Finally, we report some numerical tests supporting the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract: This Note is devoted to the analysis of the null controllability of a nonlinear reaction–diffusion system, approximating a parabolic–elliptic system, modeling electrical activity in the heart. The uniform, with respect to the degenerating parameter, null controllability of the approximating system by a single control force acting on a subdomain is shown. The proof needs a precise estimate with respect to the degenerating parameter and it is done combining Carleman estimates and energy inequalities. [Copyright &y& Elsevier]

Abstract: In the present paper, an optimal control problem constrained by the tridomain equations in electrocardiology is investigated. The state equations consisting in a coupled reaction–diffusion system modeling the propagation of the intracellular and extracellular electrical potentials, and ionic currents, are extended to further consider the effect of an external bathing medium. The existence and uniqueness of solution for the tridomain problem and the related control problem is assessed, and the primal and dual problems are discretized using a finite volume method which is proved to converge to the corresponding weak solution. In order to illustrate the control of the electrophysiological dynamics, we present some preliminary numerical experiments using an efficient implementation of the proposed scheme. [Copyright &y& Elsevier]

Abstract: We analyze a finite volume scheme for the bidomain model, which is a degenerate reaction–diffusion system modeling electrophysiological waves in cardiac tissue. We establish existence and uniqueness of solutions to the finite volume scheme, and show that it converges to a weak solution of the bidomain model. The convergence proof is based on deriving series of a priori estimates and using a general compactness criterion. [Copyright &y& Elsevier]