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arXiv · 2003.14141

Unstructured Space-Time Finite Element Methods for Optimal Sparse Control of Parabolic Equations

Ulrich Langer·Olaf Steinbach·Fredi Tröltzsch·Huidong Yang
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Abstract

We consider a space-time finite element method on fully unstructured simplicial meshes for optimal sparse control of semilinear parabolic equations. The objective is a combination of a standard quadratic tracking-type functional including a Tikhonov regularization term and of the $L^1$-norm of the control that accounts for its spatio-temporal sparsity. We use a space-time Petrov-Galerkin finite element discretization for the first-order necessary optimality system of the associated discrete optimal sparse control problem. The discretization is based on a variational formulation that employs piecewise linear finite elements simultaneously in space and time. Finally, the discrete nonlinear optimality system that consists of coupled forward-backward state and adjoint state equations is solved by a semismooth Newton method.

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Ulrich Langer, Olaf Steinbach, Fredi Tröltzsch, Huidong Yang. 2020-03-31. Unstructured Space-Time Finite Element Methods for Optimal Sparse Control of Parabolic Equations. https://arxiv.org/abs/2003.14141

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