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arXiv · 2004.09504

Space-time finite element discretization of parabolic optimal control problems with energy regularization

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

We analyze space-time finite element methods for the numerical solution of distributed parabolic optimal control problems with energy regularization in the Bochner space $L^2(0,T;H^{-1}(\Omega))$. By duality, the related norm can be evaluated by means of the solution of an elliptic quasi-stationary boundary value problem. When eliminating the control, we end up with the reduced optimality system that is nothing but the variational formulation of the coupled forward-backward primal and adjoint equations. Using Babu\v{s}ka's theorem, we prove unique solvability in the continuous case. Furthermore, we establish the discrete inf-sup condition for any conforming space-time finite element discretization yielding quasi-optimal discretization error estimates. Various numerical examples confirm the theoretical findings. We emphasize that the energy regularization results in a more localized control with sharper contours for discontinuous target functions, which is demonstrated by a comparison with an $L^2$ regularization and with a sparse optimal control approach.

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Ulrich Langer, Olaf Steinbach, Fredi Tröltzsch, Huidong Yang. 2020-04-20. Space-time finite element discretization of parabolic optimal control problems with energy regularization. https://arxiv.org/abs/2004.09504

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