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

The first and the second author were supported by FWF and DFG through the International Research Training Group IGDK 1754 'Optimization and Numerical Analysis for Partial Differential Equations with Nonsmooth Structures'. The third has been funded within the framework of the Academic Fund Program at the National Research University Higher School of Economics in 2016-2017 (grant no. 16-01-0054) and by the Russian Academic Excellence Project '5-100'. He also thanks the Technical University of Munich for its hospitality in 2014-2015 years.
• 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 $L_{{{w}^{*}}}^{2}\left( I, \mathcal M\left( {\mathit \Omega } \right) \right)$ or vector measures $\mathcal M({\mathit \Omega }, L^2(I))$. The cost functional involves the standard quadratic tracking terms and the regularization term $α\|u\|_{\mathcal M_T}$ with $α>0$. 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.

Mathematics Subject Classification: Primary: 65M60, 49K20, 49M05, 49M25, 49M29; Secondary: 35L05.

 Citation: • • Figure 1.  Example 1: the reference solution $(\hat u, \hat y)$

Figure 2.  Example 1: errors as $h$ refines and $M = 2^{10}$

Figure 3.  Example 1: errors as $\tau$ refines and $N = 2^{10}$

Figure 4.  Example 2: the reference solution $(\hat u, \hat y)$

Figure 5.  Example 2: errors as $h$ refines and $M = 2^{10}$

Figure 6.  Example 2: errors as $\tau$ refines and $N = 2^{10}$

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