Well-posedness and optimal control for a viscous Cahn-Hilliard-Oono system with dynamic boundary conditions

TL;DR

This paper establishes well-posedness and derives optimality conditions for a viscous Cahn-Hilliard-Oono PDE system with dynamic boundary conditions, considering general potentials including quartic and logarithmic types.

Contribution

It proves well-posedness and develops first-order optimality conditions for a coupled PDE control problem with dynamic boundary conditions and general potentials.

Findings

Well-posedness of the PDE system is established.

First-order necessary conditions for optimal control are derived.

The analysis accommodates various potentials, including quartic and logarithmic.

Abstract

In this paper we consider a nonlinear system of PDEs coupling the viscous Cahn-Hilliard-Oono equation with dynamic boundary conditions enjoying a similar structure on the boundary. After proving well-posedness of the corresponding initial boundary value problem, we study an associated optimal control problem related to a tracking-type cost functional, proving first-order necessary conditions of optimality. The controls enter the system in the form of a distributed and a boundary source. We can account for general potentials in the bulk and in the boundary part under the common assumption that the boundary potential is dominant with respect to the bulk one. For example, the regular quartic potential as well as the logarithmic potential can be considered in our analysis.