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 {ϕj}j=1N, N∈NN∈N, corresponding to the eigenvalues {λj}j=1N of the Laplace operator in Ω⊂RqΩ⊂Rq, q=2,3q=2,3. The stabilizing algorithm, we develop here, is applicable under the assumption that the system {∂ϕj∂n}j=1N is linearly independent on the part of the boundary where the control is applied.