A Riemann solver at a junction compatible with a homogenization limit

We consider a junction regulated by a traffic lights, with n incoming roads and only one outgoing road. On each road the Phase Transition traffic model, proposed in [6], describes the evolution of car traffic. Such model is an extension of the classic Lighthill-Whitham-Richards one, obtained by assuming that different drivers may have different maximal speed. By sending to infinity the number of cycles of the traffic lights, we obtain a justification of the Riemann solver introduced in [9] and in particular of the rule for determining the maximal speed in the outgoing road.Comment: 19 page

The Godunov Method for a 2-Phase Model

We consider the Godunov numerical method to the phase-transition traffic model, proposed in [6], by Colombo, Marcellini, and Rascle. Numerical tests are shown to prove the validity of the method. Moreover we highlight the differences between such model and the one proposed in [1], by Blandin, Work, Goatin, Piccoli, and Bayen.Comment: 13 page

The Aw-Rascle traffic model with locally constrained flow

We consider solutions of the Aw-Rascle model for traffic flow fulfilling a constraint on the flux at $x=0$. Two different kinds of solutions are proposed: at $x=0$ the first one conserves both the number of vehicles and the generalized momentum, while the second one conserves only the number of cars. We study the invariant domains for these solutions and we compare the two Riemann solvers in terms of total variation of relevant quantities. Finally we construct ad hoc finite volume numerical schemes to compute these solutions.Comment: 24 page

Stability and Optimization in Structured Population Models on Graphs

We prove existence and uniqueness of solutions, continuous dependence from the initial datum and stability with respect to the boundary condition in a class of initial--boundary value problems for systems of balance laws. The particular choice of the boundary condition allows to comprehend models with very different structures. In particular, we consider a juvenile-adult model, the problem of the optimal mating ratio and a model for the optimal management of biological resources. The stability result obtained allows to tackle various optimal management/control problems, providing sufficient conditions for the existence of optimal choices/controls.Comment: 22 pages, 7 figure

Polynomial Profits in Renewable Resources Management

A system of renewal equations on a graph provides a framework to describe the exploitation of a biological resource. In this context, we formulate an optimal control problem, prove the existence of an optimal control and ensure that the target cost function is polynomial in the control. In specific situations, further information about the form of this dependence is obtained. As a consequence, in some cases the optimal control is proved to be necessarily bang--bang, in other cases the computations necessary to find the optimal control are significantly reduced

Autonomous Vehicles Driving Traffic: The Cauchy Problem

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, 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.Comment: 32 page

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

Differential Equations Modeling Crowd Interactions

Nonlocal conservation laws are used to describe various realistic instances of crowd behaviors. First, a basic analytic framework is established through an "ad hoc" well posedness theorem for systems of nonlocal conservation laws in several space dimensions interacting non locally with a system of ODEs. Numerical integrations show possible applications to the interaction of different groups of pedestrians, and also with other "agents".Comment: 26 pages, 5 figure