Optimal control of a phase field system of Caginalp type with fractional operators

TL;DR

This paper investigates optimal control problems for fractional phase field systems of Caginalp type, establishing differentiability of the control-to-state map and deriving first-order optimality conditions.

Contribution

It extends previous work by analyzing distributed optimal control for fractional phase field systems, including differentiability and optimality conditions.

Findings

Control-to-state operator is Fréchet differentiable.

First-order necessary optimality conditions are derived.

The analysis applies to systems with spectral fractional operators.

Abstract

In their recent work `Well-posedness, regularity and asymptotic analyses for a fractional phase field system' (Asymptot. Anal. 114 (2019), 93-128; see also the preprint arXiv:1806.04625), two of the present authors have studied phase field systems of Caginalp type, which model nonconserved, nonisothermal phase transitions and in which the occurring diffusional operators are given by fractional versions in the spectral sense of unbounded, monotone, selfadjoint, linear operators having compact resolvents. In this paper, we complement this analysis by investigating distributed optimal control problems for such systems. It is shown that the associated control-to-state operator is Fr\'echet differentiable between suitable Banach spaces, and meaningful first-order necessary optimality conditions are derived in terms of a variational inequality and the associated adjoint state variables.