Communications in Mathematical Sciences

Volume 21 (2023)

Number 6

Fokker–Planck modeling of many-agent systems in swarm manufacturing: asymptotic analysis and numerical results

Pages: 1655 – 1677

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n6.a10

Authors

Ferdinando Auricchio (Department of Civil Engineering and Architecture, University of Pavia, Italy; and IMATI, National Research Council (CNR), Pavia, Italy)

Giuseppe Toscani (Department of Mathematics, University of Pavia, Italy; and IMATI, National Research Council, Pavia, Italy)

Mattia Zanella (Department of Mathematics, University of Pavia, Italy)

Abstract

In this paper we study a novel Fokker–Planck-type model that is designed to mimic manufacturing processes through the dynamics characterizing a large set of agents. In particular, we describe a many-agent system interacting with a target domain in such a way that each agent/particle is attracted by the center of mass of the target domain with the aim to uniformly cover this zone. To this end, we first introduce a mean-field model with discontinuous flux whose large-time behavior is such that the steady state is globally continuous and uniform over a connected portion of the domain. We prove that a diffusion coefficient, guaranteeing that a given portion of mass enters in the target domain, exists and that it is unique. Furthermore, convergence to equilibrium in 1D is provided through a reformulation of the initial problem involving a nonconstant diffusion function. The extension to 2D is explored numerically by means of recently introduced structure preserving methods for Fokker–Planck equations.

Keywords

swarm robotics, swarm manufacturing, multi-agent systems, Fokker–Planck equations

2010 Mathematics Subject Classification

35Q70, 35Q84, 93C85

The full text of this article is unavailable through your IP address: 172.17.0.1

This work has been written within the activities of the GNFM group of INdAM (National Institute of High Mathematics). M.Z. acknowledges partial support of MUR-PRIN2020 Project No. 2020JLWP23. The research of M.Z. was partially supported by MIUR, Dipartimenti di Eccellenza Program (2018–2022), and Department of Mathematics “F. Casorati”, University of Pavia.

Received 9 June 2022

Received revised 6 December 2022

Accepted 7 December 2022

Published 22 September 2023