Optimal control of the bidomain system. III: Existence of minimizers and first-order optimality conditions (Q2842454)

The considered bidomain system representing the description of the electrical activity of the heart is the one introduced by \textit{L. Tung} in the dissertation [``A bi-domain model for describing ischemic myocardial D-C potential'', Ph.D. thesis, MIT-Boston, (1978)]. Optimal control problems related to this system have been investigated by the authors in two previous papers [\textit{K. Kunisch} and \textit{M. Wagner}, Nonlinear Anal., Real World Appl. 13, No. 4, 1525-1550 (2012; Zbl 1256.49009); ``Optimal control in the bidomain system (II): Uniqueness and regularity theorems'', Univ. of Graz, Institute for Mathematics and Scientific Computing, SFB-Report No. 2011-008 (2011)]. In the present article one concentrates on the existence of minimizers and first order necessary conditions. The extracellular excitation acts as control and should be small. In the performance functional the integrand is assumed to satisfy all needed regularity conditions. The first part of the paper is devoted to a discussion on weak solutions of bidomain system. Further the adjoint system is derived and one proves the existence and uniqueness of a weak solution for it. By means of classical variational methods, first order necessary optimality conditions for the stated optimal control problem are proved.