Optimal control of elliptic surface PDEs with pointwise bounds on the state

TL;DR

This paper develops a finite element approach for optimal control problems constrained by elliptic PDEs on surfaces, establishing convergence rates and extending Euclidean results to surface geometries.

Contribution

It extends finite element convergence results for PDE-constrained optimization from Euclidean domains to two-dimensional surfaces.

Findings

Proves convergence rates for discrete controls and states

Shows improved convergence with higher control regularity

Extends Euclidean PDE control results to surface settings

Abstract

We consider a linear-quadratic optimization problem with pointwise bounds on the state for which the constraint is given by the Laplace-Beltrami equation (to have uniqueness we add an lower order term) on a two-dimensional surface . By using finite elements we approximate the optimization problem by a family of discrete problems and prove convergence rates for the discrete controls and the discrete states. Furthermore, assuming (roughly spoken) a higher regularity for the control the order of convergence improves. This extends a result known in an Euclidean setting to the surface case.