WIAS Preprint No. 2725, (2020)

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

Authors

• Colli, Pierluigi
• Gilardi, Gianni
• Sprekels, Jürgen

2010 Mathematics Subject Classification

• 35K45 35K90 35R11 49J20 40J05 49K20

Keywords

• Fractional operators, phase field system, nonconserved phase transition, optimal control, first-order necessary optimality conditions

DOI

10.20347/WIAS.PREPRINT.2725

Abstract

In their recent work Well-posedness, regularity and asymptotic analyses for a fractional phase field system'' (Asymptot. Anal. 114 (2019), 93--128), 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 Fréchet 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.