Sparse Control of Alignment Models in High Dimension

TL;DR

This paper demonstrates that in high-dimensional alignment models, nearly optimal sparse control can be achieved with high probability by acting on a single agent at each switching time, using low-dimensional random projections for decision-making.

Contribution

It introduces a method for controlling high-dimensional consensus models via low-dimensional random projections, enabling sparse and nearly optimal control strategies.

Findings

High probability of steering high-dimensional systems to consensus

Control achieved by acting on a single agent at each step

Use of random linear maps for low-dimensional system representation

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.