Space-Time Finite Element Discretization of Parabolic Optimal Control Problems with Energy Regularization

Ulrich Langer, Olaf Steinbach, Fredi Tröltzsch, Huidong Yang

Research output: Contribution to journalArticlepeer-review


In this paper, we analyze space-time finite element methods for the numerical solution of distributed parabolic optimal control problems with energy regularization in the Bochner space L2 (0, T; H-1 (Ω)). 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 forwardbackward primal and adjoint equations. Using Babuška'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 L2 -regularization and with a sparse optimal control approach.

Original languageEnglish
Pages (from-to)675-695
Number of pages21
JournalSIAM Journal on Numerical Analysis
Issue number2
Publication statusPublished - 9 Mar 2021


  • Discretization error estimates
  • Parabolic optimal control problems
  • Space-time finite element methods

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics
  • Numerical Analysis

Fields of Expertise

  • Information, Communication & Computing


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