Open AccessWell-posedness and optimal control for a Cahn–Hilliard–Oono system with control in the mass term
Author(s) -
Pierluigi Colli,
Gianni Gilardi,
Elisabetta Rocca,
Jürgen Sprekels
Publication year - 2022
Publication title -
discrete and continuous dynamical systems. series s
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.481
H-Index - 34
eISSN - 1937-1632
pISSN - 1937-1179
DOI - 10.3934/dcdss.2022001
Subject(s) - mathematics , optimal control , term (time) , logarithm , variable (mathematics) , pure mathematics , discrete mathematics , combinatorics , mathematical analysis , physics , mathematical optimization , quantum mechanics
The paper treats the problem of optimal distributed control of a Cahn–Hilliard–Oono system in \begin{document}$ {{\mathbb{R}}}^d $\end{document} , \begin{document}$ 1\leq d\leq 3 $\end{document} , with the control located in the mass term and admitting general potentials that include both the case of a regular potential and the case of some singular potential. The first part of the paper is concerned with the dependence of the phase variable on the control variable. For this purpose, suitable regularity and continuous dependence results are shown. In particular, in the case of a logarithmic potential, we need to prove an ad hoc strict separation property, and for this reason we have to restrict ourselves to the case \begin{document}$ d = 2 $\end{document} . In the rest of the work, we study the necessary first-order optimality conditions, which are proved under suitable compatibility conditions on the initial datum of the phase variable and the time derivative of the control, at least in case of potentials having unbounded domain.