Numerical analysis for an optimal control of bidomain-bath model

This work is concerned with the study of the convergence analysis for an optimal control of bidomain-bath model. The bidomain-bath model equations describe the cardiac bioelectric activity at the tissue and bath volumes where the control acts at the boundary of the tissue domain. We establish the existence and convergence of the unique weak solution of the direct bidomain-bath model by a discrete Galerkin approach. The convergence proof is based on deriving a series of a priori estimates and using a general L2-compactness criterion. Moreover, the well-posedness of the adjoint problem and the first order necessary optimality conditions are shown. Comparing to the direct problem, the convergence proof of the adjoint problem is based on using a general L1-compactness criterion. The numerical tests are demonstrated which achieve the successful cardiac defibrillation by utilizing less total current. Finally, the robustness of the Newton optimization algorithm is presented for different finer mesh geometries.