A Class of Optimal Control Problems Governed by Caginalp Phase Field Models

Abstract

Caginalp’s phase field model was introduced to describe phase transition processes such as melting and solidification, but can also be more generally viewed as a model for non-isothermal phase separation processes. Core of the model is the Cahn-Hilliard equation, which was proposed to describe the so-called spinodal decomposition of binary alloys in a rapid cooling process, but can be utilized for phase-separating mixtures in a much broader range. The goal of this thesis is to set the theoretical framework for a numerical realization of an optimal control problem governed by Caginalp’s phase field model. In particular, this covers the derivation of Caginalp’s model from phase separation phenomena using basic thermodynamics, showing well-posedness of Caginalp’s model and proving the existence of an optimal control as well as deriving a first order necessary optimality condition.