Communications in Mathematical Sciences

Volume 22 (2024)

Number 3

Autonomous vehicles driving traffic: The Cauchy problem

Pages: 817 – 844

DOI: https://dx.doi.org/10.4310/CMS.2024.v22.n3.a9

Authors

Mauro Garavello (Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, Milano, Italy)

Francesca Marcellini (INdAM Unit, Dipartimento di Ingegneria dell’Informazione, Università di Brescia, Italy)

Abstract

This paper deals with the Cauchy problem for a PDE‑ODE model, where a system of two conservation laws, namely the Two-Phase macroscopic model proposed in [Rinaldo M. Colombo, Francesca Marcellini, and Michel Rascle, $\href{https://doi.org/10.1137/090752468}{\textrm{SIAM J. Appl. Math., 70(7):2652–2666, 2010}}$], is coupled with an ordinary differential equation describing the trajectory of an autonomous vehicle (AV), which aims to control the traffic flow. Under suitable assumptions, we prove a global-in-time existence result.

Keywords

$2 \times 2$ hyperbolic systems of conservation laws, mixed PDE-ODE system, continuum traffic models, autonomous vehicles, control problems

2010 Mathematics Subject Classification

35L65, 90B20

Received 30 November 2022

Received revised 14 August 2023

Accepted 7 September 2023

Published 4 March 2024