## Section: New Results

### Macroscopic traffic flow models on networks

Participants : Régis Duvigneau, Nikodem Dymski, Paola Goatin, Nicolas Laurent-Brouty, Elena Rossi, Shuxia Tang, Alexandre Bayen [UC Berkeley, CA, USA] , Enrico Bertino [Politecnico Milano, Italy] , Guillaume Costeseque [Cerema, Nantes] , Alexander Keimer [UC Berkeley, CA, USA] , Antonella Ferrara [U Pavia, Italy] , Adriano Festa [Politecnico Torino, Italy] , Mauro Garavello [U Milano-Bicocca, Italy] , Thibault Liard [DeustoTech, Spain] , Benedetto Piccoli [U Rutgers, NJ, USA] , Giulia Piacentini [U Pavia, Italy] .

**Bounded acceleration.** In [50], we study a mathematical model accounting for the boundedness of traffic acceleration at a macroscopic scale that was introduced in [137]. Our model is built on a first order macroscopic PDE model coupled with an ODE describing the trajectory of the leader of a platoon accelerating at a given constant rate. We propose a Wave Front Tracking Algorithm to construct approximate solutions. We use this algorithm to prove the existence of solutions to the associated Cauchy Problem, and provide some numerical examples illustrating the solution behaviour.

This work was part of N. Laurent-Brouty's PhD thesis.

**Moving bottlenecks on road networks.** In [48], we generalize the Lighthill-Witham-Richards model for vehicular traffic coupled with moving bottlenecks introduced in [6] to the case of road networks.
Such models can be applied to study the traffic evolution in the presence of a slow-moving vehicle, like a bus.
At last, a numerical experiment is shown.

This work was part of N. Dymski's PhD thesis.

**Traffic control by autonomous vehicles.**
Autonomous vehicles (AVs) allow new ways of regulating the traffic flow on road networks.
Most of available results in this direction are based on microscopic approaches, where ODEs describe the evolution of regular cars and AVs.
In [49], we propose a multiscale approach, based on recently developed models for moving bottlenecks.
Our main result is the proof of existence of solutions for open-loop controls with bounded variation.

**Vehicle platooning in highway traffic.**
In [52], we consider a model describing the presence of a platoon of vehicles moving in the traffic flow.
The model consists of a coupled PDE-ODE system describing the interaction between the platoon and the surrounding traffic flow.
The scalar conservation law takes into account the main traffic evolution, while the ODEs describe the trajectories of the initial and final points of the platoon, whose length can vary in time. The presence of the platoon acts as a road capacity reduction, resulting in a space-time discontinuous flux function.
We describe the solutions of Riemann problems and design a finite volume numerical scheme
sharply capturing non-classical discontinuities.
Some numerical tests are
presented to show the effectiveness of the method.

This work is part of G. Piacentini's PhD thesis.

**Well-posedness of conservation laws on networks with finite buffers.**
In [51], we introduce a model dealing with conservation laws on networks and coupled boundary conditions at the junctions.
In particular, we introduce buffers of fixed arbitrary size and time dependent split ratios at the junctions, which represent how traffic is routed through the network,
while guaranteeing
spill-back phenomena at nodes.
Having defined the dynamics at the level of conservation laws, we lift it up to the Hamilton-Jacobi (H-J) formulation and write boundary datum of incoming and outgoing junctions as functions of the queue sizes and vice-versa. The Hamilton-Jacobi formulation provides the necessary regularity estimates to derive a fixed-point problem in a proper Banach space setting, which is used to prove well-posedness of the model.
Finally, we detail how to apply our framework to a non-trivial road network, with several intersections and finite-length links.

This work was realized in the framework of the IIP ORESTE and was part of N. Laurent-Brouty's PhD thesis.

**Traffic flow on multi-lane networks.**
In [28], we prove the well-posedness of a system of balance laws describing macroscopically the traffic
flow on a multi-lane road network. Motivated by real applications,
we allow for the the presence of space discontinuities both in the
speed law and in the number of lanes. This allows to describe a
number of realistic situations. Existence of solutions follows
from compactness results on a sequence of Godunov's
approximations, while ${L}^{1}$-stability is obtained by the doubling
of variables technique. Some numerical simulations illustrate the
behaviour of solutions in sample cases.

**Minimum time boundary controls.**
The paper [35] is motivated by the practical problem of controlling traffic flow by imposing restrictive boundary conditions.
For a one-dimensional congested road segment, we study the minimum time control problem of how to control the upstream
vehicular flow appropriately to regulate the downstream traffic into a desired (constant) free flow state
in minimum time. We consider the Initial-Boundary
Value Problem (IBVP) for a scalar nonlinear conservation law, associated
to the Lighthill-Whitham-Richards (LWR) Partial Differential Equation (PDE), where the left boundary condition, also treated as a valve for the traffic flow from the upstream,
serves as a control. Besides, we set absorbing downstream boundary conditions. We prove first a comparison principle for the solutions of the considered IBVP, subject to comparable initial, left and right boundary data, which provides estimates on the minimal time required to control the system. Then we consider a (sub-) optimal control problem and we give numerical results based on Godunov scheme. The article serves as a starting point for studying time-optimal boundary control of the LWR model and for computing numerical results.

This work was realized in the framework of the IIP ORESTE.

**Impact of on-line navigation devices in traffic flows.**
In [34], we consider a macroscopic multi-population traffic flow model on networks accounting for the presence of drivers (or autonomous vehicles) using navigation devices to minimize their instantaneous travel cost to destination. The strategic choices of each population differ in the degree of information about the system: while part of the agents knows only the structure of the network and minimizes the traveled distance, others are informed of the current traffic distribution, and can minimize their travel time avoiding the most congested areas. In particular, the different route choices are computed solving eikonal equations on the road network and they are implemented at road junctions.
The impact on traffic flow efficiency is illustrated by numerical experiments. We show that, even if the use of routing devices contributes to alleviate congestion on the whole network, it also results in increased traffic on secondary roads. Moreover, the generalized use of real-time information can even deteriorate the efficiency of the network.

**Uncertainty quantification in a macroscopic traffic flow model calibrated on GPS data.**
In [18], we analyze the inclusion of one or more random parameters into the deterministic Lighthill-Whitham-Richards traffic flow model and
use a semi-intrusive approach to quantify uncertainty propagation.
To verify the validity of the method, we test it against real data coming from vehicle embedded GPS systems, provided by Autoroutes Trafic .