On the optimization of conservation law models at a junction with inflow and flow distribution controls

The paper proposes a general framework to analyze control problems for conservation law models on a network. Namely we consider a general class of junction distribution controls and inflow controls and we establish the compactness in $L^1$ of a class of flux-traces of solutions. We then derive the existence of solutions for two optimization problems: (I) the maximization of an integral functional depending on the flux-traces of solutions evaluated at points of the incoming and outgoing edges; (II) the minimization of the total variation of the optimal solutions of problem (I). Finally we provide an equivalent variational formulation of the min-max problem (II) and we discuss some numerical simulations for a junction with two incoming and two outgoing edges.Comment: 29 pages, 14 figure

Control Problems for Conservation Laws with Traffic Applications

Conservation and balance laws on networks have been the subject of much research interest given their wide range of applications to real-world processes, particularly traffic flow. This open access monograph is the first to investigate different types of control problems for conservation laws that arise in the modeling of vehicular traffic. Four types of control problems are discussed - boundary, decentralized, distributed, and Lagrangian control - corresponding to, respectively, entrance points and tolls, traffic signals at junctions, variable speed limits, and the use of autonomy and communication. Because conservation laws are strictly connected to Hamilton-Jacobi equations, control of the latter is also considered. An appendix reviewing the general theory of initial-boundary value problems for balance laws is included, as well as an appendix illustrating the main concepts in the theory of conservation laws on networks

Second order traffic flow models on road networks and real data applications

This thesis concerns macroscopic traffic models and data-driven models. In the first part we deal with the extension of Generic Second Order Models (GSOM) for traffic flow to road networks. We define a Riemann Solver at the junction based on a priority rule, providing an iterative algorithm able to build the solution to junctions with n incoming and m outgoing roads. The logic underlying our solver is the following: the flow is maximised respecting the priority rule, but the latter can be modified if the outgoing road supply exceeds the demand of the road with higher priority. We provide bounds on the total variation of waves interacting with the junction, giving explicit computations for intersections with two incoming and two outgoing roads. These estimates are fundamental to prove the existence of weak solutions to Cauchy problems on networks via Wave-Front-Tracking. GSOM are used to analyse traffic dynamics and their effects on the production of pollutant emissions. First we apply the proposed Riemann Solver to simulate traffic dynamics on diverge and merge junctions and on roundabouts obtained by combining these two types of intersection. Then, we propose a methodology to estimate the pollutant emissions deriving from traffic dynamics. The emission model is calibrated and validated using the NGSIM dataset of real trajectory data. Furthermore, we set up a minimisation problem aimed at finding the optimal priority rule for our Riemann Solver that reduces the emission rates due to the traffic dynamic. Finally, we analyse some chemical reactions which lead to the production of ozone, focusing on the effects on pollution of the presence of traffic lights on the road. Next, we introduce a macroscopic two-dimensional multi-class traffic model on a single road, aimed at including lane-changes and different types of vehicles. The multi-class model consists of a coupled system of conservation laws in two space dimensions. Besides the study of the Riemann problems, we present a Lax-Friedrichs type discretisation scheme and we recover the theoretical results by means of numerical tests. We then calibrate and validate the multi-class model with real trajectory data and we test its ability of simulating vehicles overtaking. Finally, we present a new methodology to recover mass movements from snapshots of its distribution. To this end we put in place an algorithm based on the combination of two methods: first, we use the dynamic mode decomposition to create a system of equations describing the mass transfer; second, we use the Wasserstein distance to reconstruct the underlying velocity field that is responsible for the displacement. We conclude this part with a real-life application: the algorithm is employed to study the travel flows of people in large populated areas using, as input, presence data of people in given region domains derived from the mobile phone network, at different time instants

Control Problems for Conservation Laws with Traffic Applications

Conservation and balance laws on networks have been the subject of much research interest given their wide range of applications to real-world processes, particularly traffic flow. This open access monograph is the first to investigate different types of control problems for conservation laws that arise in the modeling of vehicular traffic. Four types of control problems are discussed - boundary, decentralized, distributed, and Lagrangian control - corresponding to, respectively, entrance points and tolls, traffic signals at junctions, variable speed limits, and the use of autonomy and communication. Because conservation laws are strictly connected to Hamilton-Jacobi equations, control of the latter is also considered. An appendix reviewing the general theory of initial-boundary value problems for balance laws is included, as well as an appendix illustrating the main concepts in the theory of conservation laws on networks

A macroscopic traffic flow model with finite buffers on networks: Well-posedness by means of Hamilton-Jacobi equations

International audienceWe 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

Lois de conservation pour la modélisation des mouvements de foule

In this thesis, we consider nonclassical problems brought out by the macroscopic modeling of pedestrian flow. The first model consists of a conservation law with a discontinuous flux, the second is a mixed hyperbolic-elliptic system of conservation laws and the last one is a nonlocal equation. In the first chapter, we use the Hughes model in one space-dimension to represent the evacuation of a corridor with two exits. The model couples a conservation law with discontinuous flux to an eikonal equation. We implement the wave front tracking scheme, treating explicitly the solution nonclassical behavior at the turning point, to provide a reference solution, which is used to numerically test the convergence of classical finite volume schemes. In the second chapter, we model the crossing of two groups of pedestrians walking in opposite directions with a system of conservation laws whose flux depends on the two densities. This system loses its hyperbolicity for certain density values. We assist to the rising of persistent but bounded oscillations, that lead us to the recast of the problem in the framework of measure-valued solutions. Finally we study a nonlocal model of pedestrian flow in two space-dimensions. The model consists of a conservation law whose flux depends on a convolution of the density. With this model, we solve an optimization problem for a room evacuation with a descent method, evaluating the impact of the explicit computation of the cost function gradient with the adjoint state method rather than approximating it with finite differences.Dans cette thèse, on considère plusieurs problèmes issus de la modélisation macroscopique des mouvements de foule. Le premier modèle consiste en une loi de conservation avec un flux discontinu, le second est un système mixte hyperbolique-elliptique et le dernier est une équation non-locale. D'abord, on utilise le modèle de Hughes une dimension pour décrire l'évacuation d'un couloir avec deux sorties. Ce modèle couple une loi de conservation avec un flux discontinu à une équation eikonale. On implémente la méthode de suivi de fronts, qui traite explicitement le comportement de la solution non-classique au point de rebroussement, afin d'obtenir des solutions de référence. Elles serviront à tester numériquement la convergence de schémas aux volumes finis classiques. Ensuite, on modélise le croisement de deux groupes marchant dans des directions opposées avec un système de lois de conservation mixte hyperbolique-elliptique dont le flux dépend des deux densités. Le système perd son hyperbolicité pour certainement valeurs de densité. On assiste à l'apparition d'oscillations persistantes mais bornées, ce qui conduit à la reformulation du problème associé dans le cadre des mesures de probabilités. Finalement, on étudie un modèle non-local de trafic piétonnier en deux dimensions. Le modèle consiste en une loi de conservation dont le flux dépend d'une convolution de la densité. Avec ce modèle, on résout un problème d'optimisation pour une évacuation d'une salle avec une méthode de descente, évaluant l'impact du calcul explicite du gradient de la fonction coût avec la méthode de l'état adjoint plutôt que son approximation par différences finies