Sparse stabilization and optimal control of the Cucker-Smale model
Marco Caponigro Massimo Fornasier Benedetto Piccoli Emmanuel Trélat
This article is mainly based on the work [7], and it is dedicated to the 60th anniversary of B. Bonnard, held in Dijon in June 2012.
    We focus on a controlled Cucker--Smale model in finite dimension. Such dynamics model self-organization and consensus emergence in a group of agents. We explore how it is possible to control this model in order to enforce or facilitate pattern formation or convergence to consensus. In particular, we are interested in designing control strategies that are componentwise sparse in the sense that they require a small amount of external intervention, and also time sparse in the sense that such strategies are not chattering in time. These sparsity features are desirable in view of practical issues.
    We first show how very simple sparse feedback strategies can be designed with the use of a variational principle, in order to steer the system to consensus. These feedbacks are moreover optimal in terms of decay rate of some functional, illustrating the general principle according to which ``sparse is better''. We then combine these results with local controllability properties to get global controllability results. Finally, we explore the sparsity properties of the optimal control minimizing a combination of the distance from consensus and of a norm of the control.
keywords: Cucker--Smale model $l_1$-norm minimization local controllability sparse optimal control. consensus emergence sparse stabilization
Benedetto Piccoli
To start a new journal in a fast growing scientific panorama is a serious challenge. We decided to start this new adventure because we felt the necessity of having an applied math journal covering an area of great interest and experiencing a big expansion.

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Keep right or left? Towards a cognitive-mathematical model for pedestrians
Mary J. Bravo Marco Caponigro Emily Leibowitz Benedetto Piccoli
In this paper we discuss the necessity of insight in the cognitive processes involved in environment navigation into mathematical models for pedestrian motion. We first provide a review of psychological literature on the cognitive processes involved in walking and on the quantitative one coming from applied mathematics, physics, and engineering. Then, we present a critical analysis of the experimental setting for model testing and we show experimental results given by observation. Finally we propose a cognitive model making use of psychological insight as well as optimization models from robotics.
keywords: Mathematical psychology pedestrian behavior multiscale models crowd dynamics experimental settings.
Traffic modeling and management: Trends and perspectives
Alexandre Bayen Rinaldo M. Colombo Paola Goatin Benedetto Piccoli
The present issue of Discrete and Continuous Dynamical Systems -- Series S is devoted to Traffic Modeling and Management. This subject dramatically developed in recent years. On one hand, the successes of the analytical theory of conservation laws have provided new tools to traffic researchers while, on the other hand, the requirements coming from the applications have grown dramatically. Remarkably, two of the papers that opened the way to this decades long development date the same year. In 1995 ``The Unique Limit of the Glimm Scheme'' by A. Bressan (Archive for Rational Mechanics and Analysis, 130, 3, 205--230) gave a basis for several well posedness results for 1D systems of conservation laws. In the same year, ``Requiem for High-Order Fluid Approximations of Traffic Flow'' by C. Daganzo (Transportation Research Part B: Methodological, 29B, 4, 277--287) posed serious criticisms to models studied at that time and started to fix minimal requirements for a traffic model to be seriously considered.

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A continuum-discrete model for supply chains dynamics
Gabriella Bretti Ciro D’Apice Rosanna Manzo Benedetto Piccoli
This paper is focused on continuum-discrete models for supply chains. In particular, we consider the model introduced in [10], where a system of conservation laws describe the evolution of the supply chain status on sub-chains, while at some nodes solutions are determined by Riemann solvers. Fixing the rule of flux maximization, two new Riemann Solvers are defined. We study the equilibria of the resulting dynamics, moreover some numerical experiments on sample supply chains are reported. We provide also a comparison, both of equilibria and experiments, with the model of [15].
keywords: networks conservation laws fluid-dynamic models finite difference schemes Supply chains
On fluido-dynamic models for urban traffic
Mauro Garavello Benedetto Piccoli
The aim of this paper is to address the following questions: which models, among fluido-dynamic ones, are more appropriate to describe urban traffic? While a rich debate was developed for the complicate dynamics of highway traffic, some basic problems of urban traffic are not always appropriately discussed. We analyze many recent, and less recent, models focusing on three basic properties. The latter are necessary to reproduce correctly queue formation at lights and junctions, and their backward propagation on an urban network.
keywords: fluido-dynamic models car traffic urban traffic networks.
Traffic circles and timing of traffic lights for cars flow
Yacine Chitour Benedetto Piccoli
In this paper we address the following traffic regulation problem: given a junction with some incoming roads and some outgoing ones, is it preferable to regulate the flux via a traffic light or via a traffic circle on which the incoming traffic enters continuously? More precisely, assuming that drivers distribute on outgoing roads according to some known coefficients, our aim is to understand which solution performs better from the point of view of total amount of cars going through the junction.
To deal with this problem we consider a fluid dynamic model for traffic flow on a road network. The model is that proposed in [9] and is applied to the case of crossings with lights and with circles. For the first we consider timing of lights as control and determine the asymptotic fluxes. For the second we extend and complete the model of [9] introducing some right of way parameters. Also in this case we determine the asymptotic behavior.
We then compare the performances of the two solutions. Finally, we can indicate which choice is preferable, depending on traffic level and control necessity, and give indications on how to tune traffic light timing and traffic circle right of way parameters.
keywords: Conservation laws traffic flow road networks.
Special issue from the launching meeting of networks and heterogeneous media
Benedetto Piccoli
The launching meeting of Networks and Heterogeneous Media took place on June 21-23 2006 in Maiori (Salerno, Italy). The meeting was sponsored by the American Institute of Mathematical Sciences, the Istituto per le Applicazioni del Calcolo of Roma, and the DIIMA of University of Salerno. For more information please click the "Full Text" above.
keywords: networks and heterogeneous media.
Special issue on Mathematics of Traffic Flow Modeling, Estimation and Control
Alexandre M. Bayen Hélène Frankowska Jean-Patrick Lebacque Benedetto Piccoli H. Michael Zhang
This Special Issue gathers contributions, most of which were presented at the Workshop ``Mathematics of Traffic Flow Modeling, Estimation and Control", organized at the Institute for Pure and Applied Mathematics of the University of California Los Angeles on December 7--9 2011.

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Cooperative controls for air traffic management
A. Marigo Benedetto Piccoli
We consider a model of cooperative control system, that is a system where different controls act with non conflictual purposes. For example one control is chosen in order to minimize some cost while another is designed for safety purposes. In our main application, a model of Air Traffic Management, one control minimizes the travel time while the other is applied to avoid crashes in a free flight environment.
The mathematical difficulties arise because of the use of two discontinuous feedbacks at the same time. We consider stratified feedbacks and solutions in Krasowskii sense. Two refinements of this concept are given to guarantee existence and good behavior of solutions.
keywords: optimal synthesis discontinuous ODEs. Cooperative control

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