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

In this work, we continue our investigations of optimal control problems for the bidomain system. 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 [ Kunisch/Wagner 12 ] and [ Kunisch/Wagner 11 ] , 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. Let Ω ⊂ R be a bounded domain and T > 0 a fixed time horizon. Then the bidomain system, representing a well-accepted description of the electrical activity of the heart, is given by 01) ∂Φtr ∂t + Iion(Φtr,W )− div ( Mi∇Φi ) = Ii for almost all (x, t) ∈ Ω× [ 0 , T ] ; (1.1)