A boundary control problem for a possibly singular phase field system with dynamic boundary conditions

TL;DR

This paper establishes the existence of optimal controls for a phase field system with dynamic boundary conditions, including singular potentials, and derives necessary optimality conditions under certain assumptions.

Contribution

It extends control theory to phase field systems with possibly singular potentials and dynamic boundary conditions, providing existence and optimality conditions.

Findings

Existence of optimal control for general potentials, including singular cases.

Derivation of first order necessary optimality conditions for regular and logarithmic potentials.

Application of subdifferential calculus to handle non-differentiable potentials.

Abstract

This paper deals with an optimal control problem related to a phase field system of Caginalp type with a dynamic boundary condition for the temperature. The control placed in the dynamic boundary condition acts on a part of the boundary. The analysis carried out in this paper proves the existence of an optimal control for a general class of potentials, possibly singular. The study includes potentials for which the derivatives may not exist, these being replaced by well-defined subdifferentials. Under some stronger assumptions on the structure parameters and on the potentials (namely for the regular and the logarithmic case having single-valued derivatives), the first order necessary optimality conditions are derived and expressed in terms of the boundary trace of the first adjoint variable.