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

Mathematical Control and Related Fields. 2018. Vol. 8. No. 2. P. 411–449.

Trautmann P., Vexler B., Zlotnik A.

This work is concerned with the optimal control problems governed by the 1D wave equation with variable coefficients and the control spaces $\mathcal M_T$ of either measure-valued functions $L^2(I,\mathcal M(\Omega))$ or vector measures $\mathcal M(\Omega,L^2(I))$. The cost functional involves the standard quadratic terms and the regularization term $\alpha\|u\|_{\mathcal M_T}$, $\alpha>0$. We construct and study three-level in time bilinear finite element discretizations for the 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.

Keywords: Wave equation, Волновое уравнение, метод конечных элементов, устойчивость, stability, finite element method, оптимальное управление, optimal control, оценки погрешности, error estimates, measure-valued control, vector measure control, мерозначное управление, управление - векторная мера

Publication based on the results of: Анализ погрешности численных методов решения задач математической физики на классах данных (2016)