# Finite element error analysis for measure-valued optimal control problems governed by a 1D wave equation with variable coefficients

## TL;DR

This paper analyzes finite element error estimates for measure-valued optimal control problems governed by a 1D wave equation with variable coefficients, including numerical validation.

## Contribution

It introduces three-level bilinear finite element discretizations for measure-valued controls in 1D wave equations and derives associated error estimates.

## Key findings

- Error estimates for the optimal state variable are established.
- Numerical results confirm the theoretical error bounds.
- The approach handles measure-valued controls effectively.

## Abstract

This work is concerned with the optimal control problems governed by a 1D wave equation with variable coefficients and the control spaces $\mathcal M_T$ of either measure-valued functions $L_{w^*}^2(I,\mathcal M(\Omega))$ or vector measures $\mathcal M(\Omega,L^2(I))$. The cost functional involves the standard quadratic tracking terms and the regularization term $\alpha\|u\|_{\mathcal M_T}$ with $\alpha>0$. We construct and study three-level in time bilinear finite element discretizations for this class of problems. The main focus lies on the derivation of error estimates for the optimal state variable and the error measured in the cost functional. The analysis is mainly based on some previous results of the authors. The numerical results are included.