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Networks and Heterogeneous Media

2015, Volume 10, Issue 3: 647-697. doi: 10.3934/nhm.2015.10.647
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Sparse control of alignment models in high dimension

  • 1.
    Technische Universität München, Fakultät Mathematik, Boltzmannstraße 3, D-85748 Garching
  • 2.
    Technische Universität München, Fakultät Mathematik, Boltzmannstrasse 3, D-85748 Garching
  • Received: 01 August 2014 Revised: 01 December 2014

  • Primary: 58F15, 58F17; Secondary: 53C35.

  • For high dimensional particle systems, governed by smooth nonlinearities depending on mutual distances between particles, one can construct low-dimensional representations of the dynamical system, which allow the learning of nearly optimal control strategies in high dimension with overwhelming confidence. In this paper we present an instance of this general statement tailored to the sparse control of models of consensus emergence in high dimension, projected to lower dimensions by means of random linear maps. We show that one can steer, nearly optimally and with high probability, a high-dimensional alignment model to consensus by acting at each switching time on one agent of the system only, with a control rule chosen essentially exclusively according to information gathered from a randomly drawn low-dimensional representation of the control system.

    Citation: Mattia Bongini, Massimo Fornasier, Oliver Junge, Benjamin Scharf. Sparse control of alignment models in high dimension[J]. Networks and Heterogeneous Media, 2015, 10(3): 647-697. doi: 10.3934/nhm.2015.10.647

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  • For high dimensional particle systems, governed by smooth nonlinearities depending on mutual distances between particles, one can construct low-dimensional representations of the dynamical system, which allow the learning of nearly optimal control strategies in high dimension with overwhelming confidence. In this paper we present an instance of this general statement tailored to the sparse control of models of consensus emergence in high dimension, projected to lower dimensions by means of random linear maps. We show that one can steer, nearly optimally and with high probability, a high-dimensional alignment model to consensus by acting at each switching time on one agent of the system only, with a control rule chosen essentially exclusively according to information gathered from a randomly drawn low-dimensional representation of the control system.


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