Analysis and optimal boundary control of a nonstandard system of phase field equations

TL;DR

This paper studies a complex, nonlinear phase field model of Cahn-Hilliard type with nonhomogeneous boundary conditions, establishing well-posedness and deriving optimal boundary control conditions for the system.

Contribution

It extends previous models by analyzing a nonstandard boundary condition case, proving well-posedness, and formulating optimal boundary control problems with necessary conditions.

Findings

Proved well-posedness of the nonstandard boundary value problem.

Established existence of optimal controls for the boundary control problem.

Derived first-order necessary optimality conditions for the control.

Abstract

We investigate a nonstandard phase field model of Cahn-Hilliard type. The model describes two-species phase segregation and consists of a system of two highly nonlinearly coupled PDEs. It has been studied recently in the papers arXiv:1103.4585 and arXiv:1109.3303 for the case of homogeneous Neumann boundary conditions. In this paper, we investigate the case that the boundary condition for one of the unknowns of the system is of third kind and nonhomogeneous. For the resulting system, we show well-posedness, and we study optimal boundary control problems. Existence of optimal controls is shown, and the first-order necessary optimality conditions are derived. Owing to the strong nonlinear couplings in the PDE system, standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional will be of standard type.