Approximate and exact controllability of the continuity equation with a localized vector field

TL;DR

This paper investigates the controllability of a transport PDE related to crowd dynamics, demonstrating approximate controllability with Lipschitz controls and highlighting limitations for exact controllability due to regularity constraints.

Contribution

It establishes conditions for approximate controllability using Lipschitz controls and explores the regularity requirements for exact controllability in a crowd model PDE.

Findings

Approximate controllability is achievable with Lipschitz controls.

Exact controllability requires more regular controls, risking non-uniqueness.

Controllability depends on the ability of uncontrolled dynamics to cross the control set.

Abstract

We study the controllability of a Partial Differential Equation of transport type, that arises in crowd models. We are interested in controlling it with a control being a vector field, representing a perturbation of the velocity, localized on a fixed control set. We prove that, for each initial and final configuration, one can steer approximately one to another with Lipschitz controls when the uncontrolled dynamics allows to cross the control set. We also show that the exact controllability only holds for controls with less regularity, for which one may lose uniqueness of the associated solution.