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Optimal Control of the Fokker–Planck Equation with Space-Dependent Controls

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Abstract

This paper is devoted to the analysis of a bilinear optimal control problem subject to the Fokker–Planck equation. The control function depends on time and space and acts as a coefficient of the advection term. For this reason, suitable integrability properties of the control function are required to ensure well posedness of the state equation. Under these low regularity assumptions and for a general class of objective functionals, we prove the existence of optimal controls. Moreover, for common quadratic cost functionals of tracking and terminal type, we derive the system of first-order necessary optimality conditions.

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Notes

  1. The probability flux describes the flow of probability in terms of probability per unit time per unit area.

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Acknowledgements

The authors wish to express their gratitude to Lars Grüne for suggesting them this very interesting subject and for many helpful comments. They would also like to thank Alfio Borzì for very helpful discussions and the referees for their valuable comments that helped to improve the manuscript. This work was partially supported by the EU under the 7th Framework Program, Marie Curie Initial Training Network FP7-PEOPLE-2010-ITN SADCO, GA 264735-SADCO, by the DFG Project Model Predictive Control for the Fokker–Planck equation, GR 1569/15-1, and by the INdAM through the GNAMPA Research Project 2015 “Analisi e controllo di equazioni a derivate parziali nonlineari.” Most of the results proved in this paper have been announced in a less general setting in a proceedings of the 2nd IFAC Workshop on Control of Systems Governed by Partial Differential Equations.

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Authors and Affiliations

  1. University of Bayreuth, 95440, Bayreuth, Germany

    Arthur Fleig

  2. Dyrecta Lab, 70014, Conversano, BA, Italy

    Roberto Guglielmi

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  1. Arthur Fleig
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  2. Roberto Guglielmi
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Correspondence to Arthur Fleig.

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Communicated by Roland Herzog.

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Fleig, A., Guglielmi, R. Optimal Control of the Fokker–Planck Equation with Space-Dependent Controls. J Optim Theory Appl 174, 408–427 (2017). https://doi.org/10.1007/s10957-017-1120-5

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