  We prove existence and regularity for the solutions to a Cahn-Hilliard system
describing the phenomenon of phase separation for a material contained in a
bounded and regular domain. Since the first equation of the system is perturbed
by the presence of an additional maximal monotone operator, we show our results
using suitable regularization of the nonlinearities of the problem and
performing some a priori estimates which allow us to pass to the limit thanks
to compactness and monotonicity arguments. Next, under further assumptions, we
deduce a continuous dependence estimate whence the uniqueness property is also
achieved. Then, we consider the relating sliding mode control (SMC) problem and
show that the chosen SMC law forces a suitable linear combination of the
temperature and the phase to reach a given (space dependent) value within
finite time.
