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A FEM for an optimal control problem subject to the fractional Laplace equation

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Abstract

We study the numerical approximation of linear–quadratic optimal control problems subject to the fractional Laplace equation with its spectral definition. We compute an approximation of the state equation using a discretization of the Balakrishnan formula that is based on a finite element discretization in space and a sinc quadrature approximation of the additionally involved integral. A tailored approach for the numerical solution of the resulting linear systems is proposed. Concerning the discretization of the optimal control problem we consider two schemes. The first one is the variational approach, where the control set is not discretized, and the second one is the fully discrete scheme where the control is discretized by piecewise constant functions. We derive finite element error estimates for both methods and illustrate our results by numerical experiments.

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Acknowledgements

We would like to thank Harbir Antil for giving a compact course on fractional PDEs as a preparation for this work. Furthermore, we would like to thank Johannes Pfefferer for many fruitful discussions during the preparation of this work. Further we thank Johannes Pfefferer and Constantin Christof for many helpful discussions during the revision of this work. Finally, we would like to thank the anonymous reviewers for their valuable suggestions to improve this work.

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Authors and Affiliations

  1. Institut für Angewandte Mathematik, Technische Universität Graz, 8010, Graz, Austria

    Stefan Dohr

  2. Zentrum Mathematik, Technische Universität München, 85748, Garching bei München, Germany

    Christian Kahle & Piotr Swierczynski

  3. Institut für Mathematik und Computergestützte Simulation, Universität der Bundeswehr München, 85577, Neubiberg, Germany

    Sergejs Rogovs

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  1. Stefan Dohr
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Correspondence to Christian Kahle.

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This work was supported by the International Research Training Group 1754, funded by the German Research Foundation (DFG), and the Austrian Science Fund (FWF).

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Dohr, S., Kahle, C., Rogovs, S. et al. A FEM for an optimal control problem subject to the fractional Laplace equation. Calcolo 56, 37 (2019). https://doi.org/10.1007/s10092-019-0334-3

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