  In this paper we study the optimal control of a parabolic initial-boundary
value problem of Allen--Cahn type with dynamic boundary conditions. Phase field
systems of this type govern the evolution of coupled diffuse phase transition
processes with nonconserved order parameters that occur in a container and on
its surface, respectively. It is assumed that the nonlinear function driving
the physical processes within the bulk and on the surface are double well
potentials of logarithmic type whose derivatives become singular at the
boundary of their respective domains of definition. For such systems, optimal
control problems have been studied in the past. We focus here on the situation
when the cost functional of the optimal control problem contains a
nondifferentiable term like the $L^1$-norm leading to sparsity of optimal
controls. For such cases, we derive second-order sufficient conditions for
locally optimal controls.
