Minimal time for the continuity equation controlled by a localized perturbation of the velocity vector field

TL;DR

This paper investigates the minimal time required to control a crowd's configuration using localized velocity perturbations, providing explicit control constructions and numerical algorithms for both microscopic and macroscopic models.

Contribution

It offers a new characterization of minimal control time for crowd steering and explicit control construction methods applicable to different crowd models.

Findings

Minimal time depends on mass presence in control region.

Explicit control construction for crowd steering.

Numerical algorithms demonstrated with simulations.

Abstract

In this work, we study the minimal time to steer a given crowd to a desired configuration. The control is a vector field, representing a perturbation of the crowd velocity, localized on a fixed control set. We will assume that there is no interaction between the agents. We give a characterization of the minimal time both for microscopic and macroscopic descriptions of a crowd. We show that the minimal time to steer one initial configuration to another is related to the condition of having enough mass in the control region to feed the desired final configuration. The construction of the control is explicit, providing a numerical algorithm for computing it. We finally give some numerical simulations.