WIAS Preprint No. 3190, (2025)

On the optimal control of viscous Cahn--Hilliard systems with hyperbolic relaxation of the chemical potential


Authors

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

2020 Mathematics Subject Classification

  • 35M33 49K20 49N90 93C20 37D35

Keywords

  • Optimal control, Cahn--Hilliard system, hyperbolic relaxation, first-order optimality conditions

DOI

10.20347/WIAS.PREPRINT.3190

Abstract

In this paper, we study an optimal control problem for a viscous Cahn--Hilliard system with zero Neumann boundary conditions in which a hyperbolic relaxation term involving the second time derivative of the chemical potential has been added to the first equation of the system. For the initial-boundary value problem of this system, results concerning well-posedness, continuous dependence and regularity are known. We show Fréchet differentiability of the associated control-to-state operator, study the associated adjoint state system, and derive first-order necessary optimality conditions. Concerning the nonlinearities driving the system, we can include the case of logarithmic potentials. In addition, we perform an asymptotic analysis of the optimal control problem as the relaxation coefficient approaches zero.

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