# Optimal finite element error estimates for an optimal control problem governed by the wave equation with controls of bounded variation

## TL;DR

This paper analyzes the convergence of a finite element method for an optimal control problem governed by the wave equation, focusing on controls of bounded variation and confirming theoretical rates through numerical experiments.

## Contribution

It provides new optimal error estimates for finite element discretization of wave equation control problems with bounded variation controls, including convergence analysis and numerical validation.

## Key findings

- Optimal convergence rates for state and control errors are established.
- Numerical experiments confirm the theoretical convergence rates.
- The control discretization approach effectively handles controls of bounded variation.

## Abstract

This work discusses the finite element discretization of an optimal control problem for the linear wave equation with time-dependent controls of bounded variation. The main focus lies on the convergence analysis of the discretization method. The state equation is discretized by a space-time finite element method. The controls are not discretized. Under suitable assumptions optimal convergence rates for the error in the state and control variable are proven. Based on a conditional gradient method the solution of the semi-discretized optimal control problem is computed. The theoretical convergence rates are confirmed in a numerical example.