WIAS Preprint No. 3184, (2025)

First- and second-order optimality conditions in the sparse optimal control of Cahn--Hilliard systems


Authors

  • Colli, Pierluigi
    ORCID: 0000-0002-7921-5041
  • Sprekels, Jürgen
    ORCID: 0009-0000-0618-8604
  • Tröltzsch, Fredi

2020 Mathematics Subject Classification

  • 35K51 49K20 49N90 93C20

Keywords

  • Viscous Cahn--Hilliard equation, singular potential, optimal control, sparsity, first- and second-order optimality conditions

DOI

10.20347/WIAS.PREPRINT.3184

Abstract

This paper deals with the sparse distributed control of viscous and nonviscous Cahn--Hilliard systems. We report on results concerning first-order necessary and second-order sufficient optimality conditions that have recently established by the authors. The analysis covers both the cases when the nonlinear double well potential governing the evolution is of either regular or logarithmic type. A major difficulty originates from the sparsity-enhancing term in the cost functional which typically is nondifferentiable.

Appeared in

  • Proceedings of the RIMS Symposium on Evolution Equations and Related Topics, T. Fukao, ed., vol. 2324 of RIMS Kôkyûroku Bessatsu, Research Institute for Mathematical Sciences, Kyoto, 2025, pp. 7/1--7/22.

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