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A theoretical investigation of Brockett’s ensemble optimal control problems

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Abstract

This paper is devoted to the analysis of problems of optimal control of ensembles governed by the Liouville (or continuity) equation. The formulation and study of these problems have been put forward in recent years by R.W. Brockett, with the motivation that ensemble control may provide a more general and robust control framework. Following Brockett’s formulation of ensemble control, a Liouville equation with unbounded drift function, and a class of cost functionals that include tracking of ensembles and different control costs is considered. For the theoretical investigation of the resulting optimal control problems, a well-posedness theory in weighted Sobolev spaces is presented for the Liouville and transport equations. Then, a class of non-smooth optimal control problems governed by the Liouville equation is formulated and existence of optimal controls is proved. Furthermore, optimal controls are characterised as solutions to optimality systems; such a characterisation is the key to get (under suitable assumptions) also uniqueness of optimal controls.

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Notes

  1. Given \(z\in {{\mathbb {R}}}\), we denote by [z] its entire part.

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Acknowledgements

The authors are indebted to the anonymous referee for her/his careful reading and the constructive remarks, which allowed to significantly improve the presentation of the paper, and for pointing out reference [12]. The authors are very grateful to Roger W. Brockett for encouraging discussions. The first author acknowledges partial support by SPARC Industries sarl, Luxemburg. The third author has been partially supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissement d’Avenir” (ANR-11-IDEX-0007), by the project BORDS (ANR-16-CE40-0027-01) and the programme “Oberwolfach Leibniz Fellows” by the Mathematisches Forschungsinstitut Oberwolfach in 2017.

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Authors and Affiliations

  1. Institut für Mathematik, Universität Würzburg, Emil-Fischer-Strasse 30, 97074, Würzburg, Germany

    Jan Bartsch & Alfio Borzì

  2. Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 blvd. du 11 Novembre 1918, 69622, Villeurbanne Cedex, France

    Francesco Fanelli

  3. Department of Mathematics, The University of Texas at Arlington, Mathematics, 411 South Nedderman Drive, Box 19408, Arlington, TX, 76019-0408, USA

    Souvik Roy

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  1. Jan Bartsch
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  2. Alfio Borzì
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  3. Francesco Fanelli
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  4. Souvik Roy
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Correspondence to Francesco Fanelli.

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Communicated by L. Ambrosio.

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Bartsch, J., Borzì, A., Fanelli, F. et al. A theoretical investigation of Brockett’s ensemble optimal control problems. Calc. Var. 58, 162 (2019). https://doi.org/10.1007/s00526-019-1604-2

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  • DOI: https://doi.org/10.1007/s00526-019-1604-2

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