Optimal control of the bidomain system (II): Uniqueness and regularity theorems for weak solutions. Revised version

In the present paper, we continue our investigation of optimal control problems for the bidomain equations. In a previous work [ Kunisch/Wagner 11 ] , control problems involving the monodomain approximation of the bidomain system have been considered. Well-posedness of the problem formulation was proved, and first-order optimality conditions were derived. Turning now to the study of optimal control of the full bidomain system, the present work is focussed on the uniqueness, stability and regularity analysis for weak solutions of the system, thus establishing the well-posedness of the control-to-state mapping and examinating the possible gain of regularity for solutions, which satisfy the optimality conditions. At the same time, the stability estimate proved below forms itself the first part of the proof of the necessary optimality conditions for the problem (P) below. The investigation of the control problem will be continued in a subsequent publication with proving existence of global minimizers and completing the proof of the first-order optimality conditions. For a bounded domain Ω ⊂ R and T > 0, the bidomain system reads as follows: 01)