On feedback stabilization for the Cahn–Hilliard equation and its numerical approximation

Abstract We consider the stabilization of solutions to the Cahn–Hilliard equation towards a given trajectory by means of a finite-dimensional static output feedback mechanism. Exponential stabilization of the controlled state around the target trajectory is proven using careful energy estimates and a spectral condition that characterizes the strength of the feedback. The analysis is general enough to allow for pointwise and distributed measurements and actuation. The main results are derived via arguments that carry over to appropriate discretization schemes, which allows us to establish corresponding exponential stabilization results also on the discrete level. The validity of our results and the importance of some of our assumptions are illustrated by numerical tests.