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



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)


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.


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

2010 Mathematics Subject Classification

35Q70, 35Q84, 93C85

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