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