Optimal control of elliptic PDEs at points

TL;DR

This paper studies an elliptic optimal control problem with pointwise state evaluations, analyzing discretization methods and providing error estimates supported by numerical experiments.

Contribution

It introduces two finite element discretization approaches for pointwise state constraints and derives a priori $L^2$ error estimates for the control.

Findings

Numerical results confirm the theoretical error estimates.

Different discretization approaches yield comparable accuracy.

The analysis highlights the importance of embedding the state space into continuous functions.

Abstract

We consider an elliptic optimal control problem where the objective functional contains evaluations of the state at a finite number of points. In particular, we use a fidelity term that encourages the state to take certain values at these points, which means our problem is related to ones with state constraints at points. The analysis and numerical analysis differs from when the fidelity is in the $L^2$ norm because we need the state space to embed into the space of continuous functions. In this paper we discretise the problem using two different piecewise linear finite element methods. For each discretisation we use two different approaches to prove a priori $L^2$ error estimates for the control. We discuss the differences between these methods and approaches and present numerical results that agree with our analytical results.