# Minimal time problem for discrete crowd models with a localized vector field

## TL;DR

This paper investigates the minimal time required to control a discrete crowd to a target configuration using a localized vector field, providing theoretical characterizations, an algorithm, and numerical simulations.

## Contribution

It introduces a novel characterization of minimal control time for discrete crowd models with localized vector fields, including an algorithm for computation.

## Key findings

- Characterization of minimal control time for discrete crowd models
- Development of an algorithm to compute control and minimal time
- Numerical simulations demonstrating the approach

## 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 characterize the minimal time for a discrete crowd model, both for exact and approximate controllability. This leads to an algorithm that computes the control and the minimal time. We finally present a numerical simulation.