  In this paper, we investigate optimal control problems for Allen-Cahn
variational inequalities with a dynamic boundary condition involving double
obstacle potentials and the Laplace-Beltrami operator. The approach covers both
the cases of distributed controls and of boundary controls. The cost functional
is of standard tracking type, and box constraints for the controls are
prescribed. We prove existence of optimal controls and derive first-order
necessary conditions of optimality. The general strategy is the following: we
use the results that were recently established by two of the authors for the
case of (differentiable) logarithmic potentials and perform a so-called "deep
quench limit". Using compactness and monotonicity arguments, it is shown that
this strategy leads to the desired first-order necessary optimality conditions
for the case of (non-differentiable) double obstacle potentials.
