Comput Optim Appl (2011) 49: 149–178 DOI 10.1007/s10589-009-9280-3 Numerical solution for optimal control of the reaction-diffusion equations in cardiac electrophysiology Chamakuri Nagaiah · Karl Kunisch · Gernot Plank Received: 12 December 2008 / Published online: 8 August 2009 © Springer Science+Business Media, LLC 2009 Abstract The bidomain equations, a continuum approximation of cardiac tissue based on the idea of a functional syncytium, are widely accepted as one of the most complete descriptions of cardiac bioelectric activity at the tissue and organ level. Nu- merous studies employed bidomain simulations to investigate the formation of car- diac arrhythmias and their therapeutical treatment. They consist of a linear elliptic partial differential equation and a non-linear parabolic partial differential equation of reaction-diffusion type, where the reaction term is described by a set of ordinary dif- ferential equations. The monodomain equations, although not explicitly accounting for current flow in the extracellular domain and its feedback onto the electrical activ- ity inside the tissue, are popular since they approximate, under many circumstances of practical interest, the bidomain equations quite well at a much lower computational expense, owing to the fact that the elliptic equation can be eliminated when assuming that conductivity tensors of intracellular and extracellular space are related to each other. Optimal control problems suggest themselves quite naturally for this important class of modelling problems and the present paper is a first attempt in this direction. Specifically, we present an optimal control formulation for the monodomain equa- tions with an extra-cellular current, I e , as the control variable. I e must be determined such that electrical activity in the tissue is damped in an optimal manner. C. Nagaiah · K. Kunisch ( ) Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstr. 36, Graz, 8010, Austria e-mail: karl.kunisch@uni-graz.at C. Nagaiah e-mail: nagaiah.chamakuri@uni-graz.at G. Plank Institute of Biophysics, Medical University of Graz, Harrachgasse 21, Graz, 8010, Austria e-mail: gernot.plank@meduni-graz.at