Boundary control problem and optimality conditions for the Cahn–Hilliard equation with dynamic boundary conditions

Abstract

This paper is concerned with a boundary control problem for the Cahn–Hilliard equation coupled with dynamic boundary conditions. In order to handle the control problem, we restrict our analysis to the case of regular potentials defined on the whole real line, assuming the boundary potential to be dominant. The existence of optimal control, the Fréchet differentiability of the control-to-state operator between appropriate Banach spaces, and the first-order necessary conditions for optimality are addressed. In particular, the necessary condition for optimality is characterised by a variational inequality involving the adjoint variables.

Keywords:

• Cahn–Hilliard equation
• dynamic boundary conditions
• phase separation
• double-well potentials
• optimal control
• optimality conditions
• adjoint problem

AMS (MOS) Subject Classifications:

• 35K61
• 49J20
• 49K20
• 49J50

Acknowledgements

The current contribution originated from the work done by Andrea Signori for the preparation of his master thesis discussed at the University of Pavia on September 2017. Actuallly, the paper turns out to offer some improvement on the results there contained.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research of Pierluigi Colli is supported by the Italian Ministry of Education, University and Research (MIUR): Dipartimenti di Eccellenza Program (2018–2022) – Dept. of Mathematics ‘F. Casorati’, University of Pavia. In addition, PC gratefully acknowledges some other support from the GNAMPA (Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica) and the IMATI - C.N.R. (Istituto di Matematica Applicata e Tecnologie Informatiche "Enrico Magenes''), Pavia, Italy.