Optimal control of elliptic PDEs on surfaces of codimension 1

TL;DR

This paper develops and analyzes a finite element method for optimal control problems involving elliptic PDEs with integral objectives along hypersurfaces, providing error estimates and numerical validation.

Contribution

It introduces a discretization approach for hypersurface integral control problems, deriving a priori error estimates and comparing with point control scenarios.

Findings

Error estimates for control and state variables

Numerical results confirm theoretical error bounds

Comparison shows differences between hypersurface and point control

Abstract

We consider an elliptic optimal control problem where the objective functional contains an integral along a surface of codimension 1, also known as a hypersurface. In particular, we use a fidelity term that encourages the state to take certain values along a curve in 2D or a surface in 3D. In the discretisation of this problem, which uses piecewise linear finite elements, we allow the hypersurface to be approximated e.g. by a polyhedral hypersurface. This can lead to simpler numerical methods, however it complicates the numerical analysis. We prove a priori $L^2$ error estimates for the control and present numerical results that agree with these. A comparison is also made to point control problems.