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Boundary Controllability and Asymptotic Stabilization of a Nonlocal Traffic Flow Model

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Abstract

We study the exact boundary controllability of a class of nonlocal conservation laws modeling traffic flow. The velocity of the macroscopic dynamics depends on a weighted average of the traffic density ahead and the averaging kernel is of exponential type. Under specific assumptions, we show that the boundary controls can be used to steer the system towards a target final state or out-flux. The regularizing effect of the nonlocal term, which leads to the uniqueness of weak solutions, enables us to prove that the exact controllability is equivalent to the existence of weak solutions to the backwards-in-time problem. We also study steady states and the long-time behavior of the solution under specific boundary conditions.

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Acknowledgements

We acknowledge the travel funding provided by the “Bavaria California Technology Center” (BaCaTeC). N. De Nitti has been partially supported by the Alexander von Humboldt foundation and by the TRR-154 Project of the DFG. We thank G. M. Coclite and E. Zuazua for many helpful conversations on topics related to this work.

We also thank the referees for their constructive and valuable suggestions, which helped to improve the paper significantly. L. Pflug is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) Project-ID 416229255-CRC1411.

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Authors and Affiliations

  1. Institute of Transportation Studies (ITS), University of California, Berkeley, 94720, Berkeley, CA, USA

    Alexandre Bayen & Alexander Keimer

  2. Sorbonne Université, Université Paris-Diderot SPC, CNRS, INRIA, Laboratoire Jacques-Louis Lions, equipe Cage, Paris, France

    Jean-Michel Coron

  3. Department of Data Science, Chair of Applied Analysis (Alexander von Humboldt Professorship), Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstr. 11, 91058, Erlangen, Germany

    Nicola De Nitti

  4. Competence Unit for Scientific Computing, Martensstr. 5a, 91058, Erlangen, Germany

    Lukas Pflug

  5. Department of Mathematics, Chair of Applied Mathematics (Continuous Optimization), Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstr. 11, 91058, Erlangen, Germany

    Lukas Pflug

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  1. Alexandre Bayen
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  2. Jean-Michel Coron
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Bayen, A., Coron, JM., De Nitti, N. et al. Boundary Controllability and Asymptotic Stabilization of a Nonlocal Traffic Flow Model. Vietnam J. Math. 49, 957–985 (2021). https://doi.org/10.1007/s10013-021-00506-7

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