Existence of Optimal Control for Dirichlet Boundary Optimization in a Phase Field Problem

Abstract

AbstractPhase field modeling finds utility in various areas. In optimization theory in particular, the distributed control and Neumann boundary control of phase field models have been investigated thoroughly. Dirichlet boundary control in parabolic equations is commonly addressed using the very weak formulation or an approximation by Robin boundary conditions. In this paper, the Dirichlet boundary control for a phase field model with a non-singular potential is investigated using the Dirichlet lift technique. The corresponding weak formulation is analyzed. Energy estimates and problem-specific embedding results are provided, leading to the existence and uniqueness of the solution for the state equation. These results together show that the control to state mapping is well defined and bounded. Based on the preceding findings, the optimization problem is shown to have a solution.