Benedetto Piccoli

Traffic flow on a road network

This paper is concerned with a fluidodynamic model for traffic flow. More precisely, we consider a single conservation law, deduced from the conservation of the number of cars, defined on a road network that is a collection of roads with junctions. The evolution problem is underdetermined at junctions; hence we choose to have some fixed rules for the distribution of traffic plus optimization criteria for the flux. We prove existence of solutions to the Cauchy problem and we show that the Lipschitz continuous dependence by initial data does not hold in general, but it does hold under special assumptions.Our method is based on a wave front tracking approach [A. Bressan, Hyperbolic Systems of Conservation Laws. The One-dimensional Cauchy Problem, Oxford University Press, Oxford, UK, 2000] and works also for boundary data and time-dependent coefficients of traffic distribution at junctions, including traffic lights.

On the reachability of quantized control systems

In this paper, we study control systems whose input sets are quantized, i.e., finite or regularly distributed on a mesh. We specifically focus on problems relating to the structure of the reachable set of such systems, which may turn out to be either dense or discrete. We report results on the reachable set of linear quantized systems, and on a particular but interesting class of nonlinear systems, i.e., nonholonomic chained-form systems. For such systems, we provide a complete characterization of the reachable set, and, in case the set is discrete, a computable method to completely and succinctly describe its structure. Implications and open problems in the analysis and synthesis of quantized control systems are addressed.

Multiscale modeling of granular flows with application to crowd dynamics

In this paper a new multiscale modeling technique is proposed. It relies on a recently introduced measure-theoretic approach, which allows one to manage the microscopic and the macroscopic scale under a unique framework. In the resulting coupled model the two scales coexist and share information. This way it is possible to perform numerical simulations in which the trajectories and the density of the particles affect each other. Crowd dynamics is the motivating application throughout the paper.

Time-Evolving Measures and Macroscopic Modeling of Pedestrian Flow

This paper introduces a new model of pedestrian flow, formulated within a measure-theoretic framework. It consists of a macroscopic representation of the system via a family of measures which, pushed forward by some flow maps, provide an estimate of the space occupancy by pedestrians at successive times. From the modeling point of view, this setting is particularly suitable for treating nonlocal interactions among pedestrians, obstacles, and wall boundary conditions. In addition, the analysis and numerical approximation of the resulting mathematical structures, which are the principal objectives of this work, follow more easily than for models based on standard hyperbolic conservation laws.

Traffic Flow on a Road Network Using the Aw–Rascle Model

The article deals with a fluid dynamic model for traffic flow on a road network. This consists of a hyperbolic system of two equations proposed in Aw and Rascle (2000). A method to solve Riemann problems at junctions is given assigning rules on traffic distributions and maximizations of fluxes and other quantities. Then we discuss stability in L ∞ norm of such solutions. Finally, we prove existence of entropic solutions to the Cauchy problem when the road network has only one junction.

Pedestrian flows in bounded domains with obstacles

In this paper, we systematically apply the mathematical structures by time-evolving measures developed in a previous work to the macroscopic modeling of pedestrian flows. We propose a discrete-time Eulerian model, in which the space occupancy by pedestrians is described via a sequence of Radon-positive measures generated by a push-forward recursive relation. We assume that two fundamental aspects of pedestrian behavior rule the dynamics of the system: on the one hand, the will to reach specific targets, which determines the main direction of motion of the walkers; on the other hand, the tendency to avoid crowding, which introduces interactions among the individuals. The resulting model is able to reproduce several experimental evidences of pedestrian flows pointed out in the specialized literature, being at the same time much easier to handle, from both the analytical and the numerical point of view, than other models relying on nonlinear hyperbolic conservation laws. This makes it suitable to address two-dimensional applications of practical interest, chiefly the motion of pedestrians in complex domains scattered with obstacles.

Hybrid Necessary Principle

We consider a hybrid control system and general optimal control problems for this system. We suppose that the switching strategy imposes restrictions on control sets and weprovide necessary conditions for an optimal hybrid trajectory, stating a Hybrid Necessary Principle (HNP). Our result generalizes various necessary principles available in the literature.

MODELING CROWD DYNAMICS FROM A COMPLEX SYSTEM VIEWPOINT

This paper aims at indicating research perspectives on the mathematical modeling of crowd dynamics, pointing on the one hand to insights into the complexity features of pedestrian flows and on the other hand to a critical overview of the most popular modeling approaches currently adopted in the specialized literature. Particularly, the focus is on scaling problems, namely representation and modeling at microscopic, macroscopic, and mesoscopic scales, which, entangled with the complexity issues of living systems, generate multiscale dynamical effects, such as e.g. self-organization. Mathematical structures suitable to approach such multiscale aspects are proposed, along with a forward look at research developments.

Hybrid systems and optimal control

The aim of this paper is to treat some optimal control problems for a class of hybrid systems. We start providing a definition of hybrid system inspired by the concept introduced by Artstein (1995), who defines hybrid control in relation to stabilization problems for a classical control system. The same definition proved to be successful to tackle other stabilization problems. In this paper, we consider a class of systems with hybrid features and optimization problems for this class of systems. The word hybrid is motivated by the fact that these systems are characterized by the presence of both a continuous time evolution and a discrete time evolution. A trajectory for these systems evolves following some dynamical constraint and at some fixed or variable times (called location switching times) it jumps following the rules of a discrete time evolution. The definition of hybrid system we give is quite general and covers many interesting applications.

A General Phase Transition Model for Vehicular Traffic

An extension of the Colombo phase transition model is proposed. The congestion phase is described by a two-dimensional zone defined around an equilibrium flux known as the classical fundamental diagram. General criteria to build such a set-valued fundamental diagram are enumerated, and instantiated on several equilibrium fluxes with different concavity properties. The solution of the Riemann problem in the presence of phase transitions is obtained through the construction of a Riemann solver, which enables the definition of the solution of the Cauchy problem using wavefront tracking. The free-flow phase is described using a Newell-Daganzo fundamental diagram, which allows for a more tractable definition of phase transition compared to the original Colombo phase transition model. The accuracy of the numerical solution obtained by a modified Godunov scheme is assessed on benchmark scenarios for the different flux functions constructed.