Feedback stabilization of the Cahn-Hilliard type system for phase separation

TL;DR

This paper develops a feedback control method to stabilize phase separation modeled by the Cahn-Hilliard system, ensuring exponential convergence to a stationary solution with a finite-dimensional controller.

Contribution

It introduces a spectral separation-based feedback stabilization approach for the nonlinear Cahn-Hilliard system with finite-dimensional controllers.

Findings

Achieves exponential stabilization of stationary solutions.

Supports arbitrary open subsets for controller placement.

Applicable to all constant stationary solutions.

Abstract

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.