  In this paper, we investigate optimal boundary control problems for
Cahn-Hilliard variational inequalities with a dynamic boundary condition
involving double obstacle potentials and the Laplace-Beltrami operator. 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, which
follows the lines of the recent approach by Colli, Farshbaf-Shaker, Sprekels
(see the preprint arXiv:1308.5617) to the (simpler) Allen-Cahn case, is the
following: we use the results that were recently established by Colli, Gilardi,
Sprekels in the preprint arXiv:1407.3916 [math.AP] 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.
