Impact of interaction forces in first order many-agent systems for swarm manufacturing

TL;DR

This paper analyzes the long-term behavior of a swarm of robots modeled by a nonlocal Fokker-Planck equation, demonstrating existence, uniqueness, and convergence to equilibrium, supported by numerical verification.

Contribution

It generalizes a previous Fokker-Planck model to include nonlocal discontinuous drift, proving key properties and equilibration rates for the new system.

Findings

Existence and uniqueness of global solutions are established.

Solutions converge to a quasi-stationary distribution at a quantifiable rate.

Numerical experiments confirm theoretical predictions and suggest extensions.

Abstract

We study the large time behavior of a system of interacting agents modeling the relaxation of a large swarm of robots, whose task is to uniformly cover a portion of the domain by communicating with each other in terms of their distance. To this end, we generalize a related result for a Fokker-Planck-type model with a nonlocal discontinuous drift and constant diffusion, recently introduced by three of the authors, of which the steady distribution is explicitly computable. For this new nonlocal Fokker-Planck equation, existence, uniqueness and positivity of a global solution are proven, together with precise equilibration rates of the solution towards its quasi-stationary distribution. Numerical experiments are designed to verify the theoretical findings and explore possible extensions to more complex scenarios.