Boundary stabilization of the phase field system by finite-dimensional feedback controllers

We design here finite-dimensional stabilizing feedback Dirichlet boundary controllers for steady-state solutions to the phase field system. The feedback controllers are easily manageable from computational point of view since they are expressed in terms of the eigenfunctions { ϕ j } j = 1 N , N ∈ N , corresponding to the eigenvalues { λ j } j = 1 N of the Laplace operator in Ω ⊂ R q , q = 2 , 3 . The stabilizing algorithm, we develop here, is applicable under the assumption that the system { ∂ ϕ j ∂ n } j = 1 N is linearly independent on the part of the boundary where the control is applied.