# Error estimates for the finite element approximation of bilinear boundary control problems

## TL;DR

This paper investigates finite element error estimates for bilinear boundary control problems, deriving optimality conditions, analyzing discretization methods, and validating results through numerical experiments.

## Contribution

It introduces new error estimates for finite element approximations of bilinear boundary control problems and compares discretization and postprocessing approaches.

## Key findings

- Finite element error estimates are established for both discretization methods.
- Numerical experiments confirm the theoretical error bounds.
- Postprocessing improves control approximation accuracy.

## Abstract

In this article a special class of nonlinear optimal control problems involving a bilinear term in the boundary condition is studied. These kind of problems arise for instance in the identification of an unknown space-dependent Robin coefficient from a given measurement of the state, or when the Robin coefficient can be controlled in order to reach a desired state. To this end, necessary and sufficient optimality conditions are derived and several discretization approaches for the numerical solution the optimal control problem are investigated. Considered are both a full discretization and the postprocessing approach meaning that we compute an improved control by a pointwise evaluation of the first-order optimality condition. For both approaches finite element error estimates are shown and the validity of these results is confirmed by numerical experiments.