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Networks and Heterogeneous Media (NHM)
 

Sparse control of alignment models in high dimension

Pages: 647 - 697, Volume 10, Issue 3, September 2015      doi:10.3934/nhm.2015.10.647

 
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Mattia Bongini - Technische Universität München, Fakultät Mathematik, Boltzmannstraße 3, D-85748 Garching, Germany (email)
Massimo Fornasier - Technische Universität München, Fakultät Mathematik, Boltzmannstraße 3, D-85748 Garching, Germany (email)
Oliver Junge - Technische Universität München, Fakultät Mathematik, Boltzmannstrasse 3, D-85748 Garching, Germany (email)
Benjamin Scharf - Technische Universität München, Fakultät Mathematik, Boltzmannstrasse 3, D-85748 Garching, Germany (email)

Abstract: 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.

Keywords:  Dimension theory, Poincaré recurrences, multifractal analysis.
Mathematics Subject Classification:  Primary: 58F15, 58F17; Secondary: 53C35.

Received: August 2014;      Revised: December 2014;      Available Online: July 2015.

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