  In this paper we study the optimal control of a parabolic initial-boundary
value problem of viscous Cahn-Hilliard type with zero Neumann boundary
conditions. Phase field systems of this type govern the evolution of diffusive
phase transition processes with conserved order parameter. It is assumed that
the nonlinear functions driving the physical processes within the spatial
domain 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 L1-norm, which leads to sparsity of
optimal controls. For such cases, we establish first-order necessary and
second-order sufficient optimality conditions for locally optimal controls. In
the approach to second-order sufficient conditions, the main novelty of this
paper, we adapt a technique introduced by E. Casas, C. Ryll and F. Tr\"oltzsch
in the paper [SIAM J. Control Optim. 53 (2015), 2168-2202]. In this paper, we
show that this method can also be successfully applied to systems of viscous
Cahn-Hilliard type with logarithmic nonlinearity. Since the Cahn-Hilliard
system corresponds to a fourth-order partial differential equation in contrast
to the second-order systems investigated before, additional technical
difficulties have to be overcome.
