  This article is concerned with the internal feedback stabilization of the
phase field system of Cahn-Hilliard type, modeling the phase separation in a
binary mixture. Under suitable assumptions on an arbitrarily fixed stationary
solution, we construct via spectral separation arguments a feedback controller
having the support in an arbitrary open subset of the space domain, such that
the closed loop nonlinear system exponentially reach the prescribed stationary
solution. This feedback controller has a finite dimensional structure in the
state space of solutions. In particular, every constant stationary solution is
admissible.
