2019/02/15 by Dominik Hafemeyer, Hafemeyer, Dominik, Florian Mannel +5
Computer Science · Engineering · Mathematics · #26A45 #49J20 #49M25 #65N15 #65N30 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Nonlinear Partial Differential Equations #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.1902.05893
openalex publication_date 2019/02/15 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28
We consider an optimal control problem governed by a one-dimensional elliptic\nequation that involves univariate functions of bounded variation as controls.\nFor the discretization of the state equation we use linear finite elements and\nfor the control discretization we analyze two strategies. First, we use\nvariational discretization of the control and show that the L2- and\nL^\∞-error for the state and the adjoint state are of order mathcal\nO(h2) and that the L1-error of the control behaves like mathcal\nO(h2), too. These results rely on a structural assumption that implies that\nthe optimal control of the original problem is piecewise constant and that the\nadjoint state has nonvanishing first derivative at the jump points of the\ncontrol. If, second, piecewise constant control discretization is used, we\nobtain L2-error estimates of order \O(h) for the state and\nW1,\∞-error estimates of order \O(h) for the adjoint state.\nUnder the same structural assumption as before we derive an L1-error\nestimate of order \O(h) for the control. We discuss optimization\nalgorithms and provide numerical results for both discretization schemes\nindicating that the error estimates are optimal.\n