Volume 38, Issue 1
Variational Discretization of A Control-Constrained Parabolic Bang-Bang Optimal Control Problem

Nikolaus von Daniels & Michael Hinze

J. Comp. Math., 38 (2020), pp. 14-40.

Published online: 2020-02

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  • Abstract

We consider a control-constrained parabolic optimal control problem without Tikhonov term in the tracking functional. For the numerical treatment, we use variational discretization of its Tikhonov regularization: For the state and the adjoint equation, we apply Petrov-Galerkin schemes in time and usual conforming finite elements in space. We prove a-priori estimates for the error between the discretized regularized problem and the limit problem. Since these estimates are not robust if the regularization parameter tends to zero, we establish robust estimates, which — depending on the problem’s regularity — enhance the previous ones. In the special case of bang-bang solutions, these estimates are further improved. A numerical example confirms our analytical findings.

  • Keywords

Optimal control, Heat equation, Control constraints, Finite elements, A-priori error estimates, Bang-bang controls.

  • AMS Subject Headings

49J20, 35K20, 49J30, 49M05, 49M25, 49M29, 65M12, 65M60

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

nvdmath@gmx.net (Nikolaus von Daniels)

michael.hinze@uni-hamburg.de (Michael Hinze)

  • BibTex
  • RIS
  • TXT
@Article{JCM-38-14, author = {Daniels , Nikolaus von and Hinze , Michael }, title = {Variational Discretization of A Control-Constrained Parabolic Bang-Bang Optimal Control Problem}, journal = {Journal of Computational Mathematics}, year = {2020}, volume = {38}, number = {1}, pages = {14--40}, abstract = {

We consider a control-constrained parabolic optimal control problem without Tikhonov term in the tracking functional. For the numerical treatment, we use variational discretization of its Tikhonov regularization: For the state and the adjoint equation, we apply Petrov-Galerkin schemes in time and usual conforming finite elements in space. We prove a-priori estimates for the error between the discretized regularized problem and the limit problem. Since these estimates are not robust if the regularization parameter tends to zero, we establish robust estimates, which — depending on the problem’s regularity — enhance the previous ones. In the special case of bang-bang solutions, these estimates are further improved. A numerical example confirms our analytical findings.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1805-m2017-0171}, url = {http://global-sci.org/intro/article_detail/jcm/13683.html} }
TY - JOUR T1 - Variational Discretization of A Control-Constrained Parabolic Bang-Bang Optimal Control Problem AU - Daniels , Nikolaus von AU - Hinze , Michael JO - Journal of Computational Mathematics VL - 1 SP - 14 EP - 40 PY - 2020 DA - 2020/02 SN - 38 DO - http://doi.org/10.4208/jcm.1805-m2017-0171 UR - https://global-sci.org/intro/article_detail/jcm/13683.html KW - Optimal control, Heat equation, Control constraints, Finite elements, A-priori error estimates, Bang-bang controls. AB -

We consider a control-constrained parabolic optimal control problem without Tikhonov term in the tracking functional. For the numerical treatment, we use variational discretization of its Tikhonov regularization: For the state and the adjoint equation, we apply Petrov-Galerkin schemes in time and usual conforming finite elements in space. We prove a-priori estimates for the error between the discretized regularized problem and the limit problem. Since these estimates are not robust if the regularization parameter tends to zero, we establish robust estimates, which — depending on the problem’s regularity — enhance the previous ones. In the special case of bang-bang solutions, these estimates are further improved. A numerical example confirms our analytical findings.

Nikolaus von Daniels & Michael Hinze. (2020). Variational Discretization of A Control-Constrained Parabolic Bang-Bang Optimal Control Problem. Journal of Computational Mathematics. 38 (1). 14-40. doi:10.4208/jcm.1805-m2017-0171
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