Sparse optimal control of singular Allen-Cahn systems with dynamic boundary conditions

Jürgen Sprekels (WIAS and HU Berlin)

In this talk, we study an optimal control problem for the Allen-Cahn equation with logarithmic potential and dynamic boundary condition. In contrast to the related work of Colli-Sprekels (SICON 2015), the cost functional contains a nondifferentiable term that enhances sparsity.  Upon showing the differentiability of the control-to-state operator by means of the implicit function theorem, we derive first-order necessary conditions for local minimizers, which are used to establish conditions that characterize the occurrence of either sparsity in time or total sparsity. We also establish second-order sufficient conditions for local minimizers in terms of a coercivity condition for the second Fréchet derivative of the differentiable part of the cost functional on a “small“ critical cone. This is joint work with Fredi Tröltzsch (TU Berlin).