Sparse control of alignment models in high dimension

Mattia Bongini; Massimo Fornasier; Oliver  Junge; Benjamin  Scharf

doi:10.3934/nhm.2015.10.647

Article Contents

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: July 31, 2014

Revised: November 30, 2014

Published: August 2015

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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.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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