Optimal Control of a Free Boundary Problem: Analysis with Second Order Sufficient Conditions

TL;DR

This paper analyzes an optimal control problem involving a free boundary PDE system coupling bulk and surface equations, establishing second order optimality conditions under convex control constraints.

Contribution

It demonstrates twice continuous Fréchet differentiability of the control-to-state operator and establishes second order sufficient optimality conditions for the problem.

Findings

Control-to-state operator is twice Fréchet differentiable.

Existence of an optimal control satisfying second order conditions.

Improved regularity results for the state variables.

Abstract

We consider a PDE-constrained optimization problem governed by a free boundary problem. The state system is based on coupling the Laplace equation in the bulk with a Young-Laplace equation on the free boundary to account for surface tension, as proposed by P.\ Saavedra and L.\ R.\ Scott \cite{PSaavedra_RScott_1991}. This amounts to solving a second order system both in the bulk and on the interface. Our analysis hinges on a convex control constraint such that the state constraints are always satisfied. Using only first order regularity we show that the control to state operator is twice continuously Fr\'echet differentiable. We improve slightly the regularity of the state variables and exploit it to show existence of a control together with second order sufficient optimality conditions.