In this lecture, we study optimal control problems for an Allen--Cahn
equation with nonlinear dynamic boundary condition involving the
Laplace--Beltrami operator. The nonlinearities both in the bulk and on the
boundary are assumed to be of double obstacle type, i.e., they involve the
subdifferential of the indicator function of the interval [-1,+1]. We
address both the cases of distributed and boundary controls.

Control problems involving nonlinearities of double obstacle type belong to
the class of "MPEC" problems. It is a well-established fact that for such
problems the standard constraint qualifications are violated, so that the
existence of Lagrange multipliers cannot be inferred from the standard
theory.

In this lecture, we take a new road of approach: in a recent paper by Colli
and Sprekels corresponding control problems involving logarithmic
nonlinearities in place of the indicator function were studied, and results
concerning existence and first-order necessary and second-order sufficient
optimality conditions were shown. Performing a "deep quench" approximation
(i.e., approximating the indicator function by logarithmic nonlinearities),
we are able to establish both the existence of optimal controls and
first-order necessary optimality conditions.

This is joint work with P. Colli (Pavia) and M. H. Farshbaf-Shaker (WIAS
Berlin).