Optimal control of the bidomain system (IV): Corrected proofs of the stability and regularity theorems

Abstract

In a series of papers on optimal control problems for the monodomain as well as for the bidomain equations of cardiac electrophysiology, the authors studied existence of minimizers and derived first-order necessary optimality conditions. The analysis of these control problems is based on a regularity discussion for weak solutions, resulting in a stability estimate and a uniqueness theorem for the monodomain and bidomain system, respectively. Unfortunately, the authors recognized a serious error within the proof of these theorems. However, the present investigation shows that the assertions from [Kunisch/Wagner 12] and [Kunisch/Wagner 13a] can be maintained (with minor changes only) while the proofs must be subjected to considerable alterations. As a consequence, the optimization theorems from [Kunisch/Wagner 13b] allow for substantial improvements. Therefore, in the present paper we provide a refined regularity discussion of the bidomain system together with corrected proofs.