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# Abstracts

## Titles of Lectures

Mauro Garavello (University Milano-Bicocca)
Analytic theory for conservation laws on networks

Conservation laws on networks, in the last years, have attracted the interest of many mathematicians, due to the various applications, which vary from car traffic, to gas flow, crowd dynamics, blood circulatory systems, and so on. The focus of the lectures is on car traffic. Various traffic models and their extension to networks will be presented. There will be investigated the Cauchy problem on network and its solution, by using the wave-front tracking method, which based on the solution to Riemann problems.

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Guido Gentile (Sapienza University of Roma)
Macroscopic models for real-time traffic forecast: challenges in theory and practice

Traffic theory has largely evolved during the last decade from simple vertical queue models to complex systems of differential equations. What of this knowledge has been actually put into practice for off-line transport planning and real-time traffic management? Which are the challenges that researchers and developers are facing to translate the lessons learned so far into software solutions that can be applied on real networks?
How can big data be used to investigate travel demand and calibrate assignment models?
PTV Optima is a complete solution for traffic management and model-based forecast applied in several cities worldwide. We will present the theory underpinning the product and the practical challenges of some use cases.

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Raul Borsche (Technische Universität Kaiserslautern)
Numerical methods for conservation laws

This lecture covers the numerical treatment of scalar hyperbolic conservation equations. Modern discretization methods will be derived and analyzed. These methods will be based on finite differences or finite volume methods

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Daniele Vigo (University of Bologna)
Optimization Problems in Urban Logistics

The increase of pollution and congestion, together with the rapid increase of the demand for high performance delivery services, makes the management of urban logistics an extremely difficult challenge. During the last decade several new optimization algorithms were developed to handle the main characteristics of modern urban distribution systems, such as, for example the mix of pickup and delivery operations, the booking of parking slots, the consideration of dynamic travel times, and the use of electric and hybrid vehicles. The aim of this talk is to provide an overview of the state-of-the-art of urban logistics optimization algorithms and highlights some interesting research areas in this field.

## Abstracts of Invited Speakers

Pierre Degond (Imperial College London)
Metric versus topologic interactions
There is increasing evidence that interactions among social agents such asbirds or fish and to some extent, pedestrians, depend on topologic rather than metric proximity. In the latter case, the interaction intensity is a function of the metric distance, while in the former one, it depends on the proximity rank irrespective of the distance between the agents. Such topological interactions are scale-free and raise important questions that have been seldom addressed by mathematicians. In this talk, we will present one of the very first attempts to build kinetic models based on topological interaction rules and we will show that this results in completely new types of kinetic models which bring in important new open questions.
This is a joint work with Adrien Blanchet from the Toulouse School of Economics.

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Serge Hoogendoorn (Delft University of Technology)
Game theoretical approach to traffic modelling

Theory and modelling of traffic has been an active research field since the pioneering work of Greenshields in 1934. Many different modelling approaches have been put forward since then, some considering traffic flows as a continuum, and others aiming to describe and predict the behaviour of individual traffic participants. These so-called microscopic traffic flow models for the topic of this lecture.
Differential game theory has been successfully applied to model varies types of traffic flow, including car traffic with various levels of automation and pedestrian traffic. In this lecture, we will motivate the use of differential game theory and show its application to these traffic modes. We will discuss the model properties, including the so-called fundamental diagram of traffic and self-organised patterns that result when application of these models, and show the analogies with other model specifications. We will also briefly discuss the application of the approach to modelling shared space. Finally, we will discuss the calibration and validation of the models using different data sources.

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Jean-Patrik Lebacque (IFSTTAR)
Bidimensional modelling of traffic on large dense networks

Modelling traffic flow on very large networks presents two challenges. The first pertains to data: on large networks the proportion of sensor equipped links is usually very low, and streaming data (GPS, floating cars,...) will be insufficient to provide access to the full network traffic state. Modelling traffic on such a network also requires information on traffic light and on other traffic management measures, the full knowledge of which may not be readily available. Therefore a traffic model for large networks needs not to be very precise but should yield correct results based on scant data.The second challenge pertains to the shear size of the network, and the computational means that are required to implement a traffic flow model on such a network. On large networks it is also interesting to be able to describe and modelize very large-scale traffic patterns. The object of the presentation is to introduce some specialized tools in order to address these challenges.

The main idea proposed in the presentation is to approximate the surface network as a bidimensional medium, and to model traffic on this network as a bidimensional fluid. Such an approximation is justified in the case of dense networks. Some physical properties of the network carry over to the bidimensional medium: capacities, velocities, some characteristics of intersections, and global features such as isotropy or anisotropic (with privileged propagation directions) which depend on street geometry. Traffic flow is calculated by conservation equations, and the macroscopic properties of the medium, which carry over from the underlying network, induce the behavioral equations of the model. The proposed numerical schemes rely on a supply demand approach. Dynamic traffic assignment is considered in a reactive setting and results in assignment coefficients. Assignment coefficients contribute the route choice aspect to the behavioral equations of the model.

The presentation will start with a brief survey of static problems and related approaches in the literature. Then the main model and its discretization will be described, as well as some more specialized topics such as interaction between traffic in a bidimensional medium (dense surface network) and major arteries (on which traffic will be modeled as a GSOM flow). Some examples will conclude the presentation.
Joint work with: M.M. Khoshyaran (ETC), H. Haj-Salem (IFSTTAR), K. Sossoe (SystemX)

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Modeling pedestrian dynamics - from experiment to theory and back

We will give an introduction to empirical and theoretical approaches
for pedestrian dynamics and the motion of large crowds. A variety of
collective effects and self-organisation phenomena can be observed,
like the dynamical formation of lanes. An overview over various modeling approaches and results from large-scale experiments which
are relevant for their validation and calibration is given.
One focus will be on the effects of social groups on pedestrian dynamics, especially in evacuation scenarios. We propose an extension of the standard cellular automaton model for pedestrian dynamics, the floor model, which allows to include social groups. These social groups can have a considerable effect on the dynamics, e.g. on evacuation times. In order to test the model predictions we have performed laboratory exeriments of evacuations with different types and sizes of the social groups. Parameters that have been considered are (1) group size, (2) strength of intra-group interactions and (3) composition of the groups (adults, children and mixtures).

Joint work with C. von Krüchten and F. Müller

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Marie-Therese Wolfram (University of Warwick)
Self-organisation in pedestrian crowds

Self-organisation can be observed in many complex systems such as fish schools, bird flocks or pedestrian crowds. The emergence of behavioural patterns, such as velocity alignment or the formation of directional lanes, demonstrates how individual interactions can lead to complex macroscopic phenomena. The analysis of experimental data on large crowds, indicates that collision avoidance plays an important role in the individual behaviour. In this talk we focus on different modelling approaches to describe the collective dynamics of large pedestrian groups - in particular how collision avoidance can be included in a cellular automata and a kinetic approach. We discuss both approaches and analyse the behaviour of the corresponding meso- and macroscopic equations. Finally we illustrate the dynamics of the models with numerical experiments.
joint work with A. Festa and A. Tosin

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Daniel Work (University of Illinois, Urbana-Campaign)
Estimation and control of traffic in the era of automated vehicles

This talk will explore some new directions in estimation and control when the traffic stream is composed of a mix of human piloted and autonomous vehicles. First, we investigate the problem of modeling and estimating traffic flows in this mixed setting. A connection between the generalized Aw Rascle Zhang (ARZ) model and two-class traffic motivates the choice to describe the flow via a system of conservation laws. With the system dynamics defined, the traffic state is estimated via a fully nonlinear particle filtering approach, and results are compared to estimates obtained from a particle filter applied to a classical scalar conservation law description of traffic. Numerical experiments indicate that when the penetration rate of automated vehicles in the traffic stream is highly variable, the ARZ based estimator offers improved accuracy. Next, we explore the problem of controlling the human piloted traffic with only a small number of autonomous vehicles. We modify the experimental setting of Sugiyama et al. (2008) to measure the influence of a carefully controlled autonomous vehicle on human piloted vehicles. Even when the penetration rate of automated vehicles is as low as 5%, we show it is possible to reduce the presence of stop-and-go waves that can appear without the presence of a bottleneck. Our experiments imply that significant improvements in traffic fuel efficiency and safety may be achieved by means of very few mobile actuators in the traffic stream. The work presented is a collaborative project led by Benedetto Piccoli (Rutgers Camden), Benni Seibold (Temple), and Jonathan Sprinkle (Arizona).

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Michael Zhang (University of Illinois, Urbana-Campaign)
On the Generation Mechanisms of Stop-Start Waves in Traffic Flow

In this talk, we present several generation mechanisms of stop-start waves at highway merge bottlenecks from both a micro and a macro perspective. One of the generation mechanisms is traffic instability and non-convexity of the fundamental diagram, and the other is lane changing and the choice of lane change locations. We’ll compare and discuss the stop-start wave patterns generated by these two mechanisms and postulate the true generation mechanism in real traffic flow.

## Abstracts Contributed Talks

Fatima Al Reda (Paris-Sud University)
An instantanoeus Nash equilibrium model for crowd dynamics

We propose a microscopic crowd motion model based on game theoretic principles. The actual velocities are built as an instantaneous Nash equilibrium: each individual does its best considering the behavior of the neighbors that influence him (through an influence graph). We address theoretical and modeling issues in both:
1. All interactions are accounted for (non-oriented influence graph)
2. Interactions are asymmetric, based on the cone of vision of each individual (oriented influence graph).
We propose a numerical strategy to solve the problem when the influence graph is structured in a hierarchical way (acyclic graph). We also run some numerical simulations to show the difference between
this model and the purely granular model proposed in Maury and Venel [1], which is based on global optimization and symmetric interactions between individuals.
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Guillaume Costeseque (Inria Sophia-Antipolis)
Queue length estimation on urban corridors

In this talk, I will present a method to compute the length of queues generated by traffic lights on urban corridors. Queue length estimation is crucial for tuning the control strategies at signalized intersection.
The method we developed is built upon a Hamilton-Jacobi formulation of traffic flow models. We derive an optimization problem that reads as a MILP and that gives us the traffic condition on a link. The estimation method is available for the classical LWR model and for the modified LWR model with bounded acceleration.
We have compared the results of the estimations for both models using the NGSIM dataset on Lankershim Boulevard in Los Angeles, California.
This work has been done in collaboration with Edward S. Canepa from KAUST university, Kingdom of Saudi Arabia, and with Christian G. Claudel from University of Texas at Austin, USA.
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Emiliano Cristiani (IAC-CNR)
Sensitivity analysis of the LWR model for traffic forecast on large networks using Wasserstein distance

In this talk we present some recent results about the sensitivity of the LWR model on networks to its parameters and to the network itself. The quantification of sensitivity is obtained by measuring the Wasserstein distance between two LWR solutions corresponding to different inputs. Hence, we propose a numerical method to approximate the Wasserstein distance between two density distributions defined on a network.
We found a large sensitivity to the traffic distribution at junctions, the network size, and the network topology.
Joint work with M. Briani and E. Iacomini
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Edda Dal Santo (University of L'Aquila)
Phase transition models for vehicular traffic with point constraints on the flow

We present a general phase transition model that describes the evolution of vehicular traffic along a unidirectional one-lane road with no entrances and no exists and where overtaking is not allowed. Two different phases are taken into account, according to whether the traffic is low or heavy. The model is given by a scalar conservation law in the free-flow phase and by a system of two conservation laws in the congested phase. As a preliminary step towards the study of Cauchy problems, here we focus on Riemann problems in the case a unilateral local point constraint on the flow of the solutions is enforced. A point constraint accounts for inhomogeneities of the road that hinder the traffic flow, such as toll gates, traffic lights, speed bumps or construction sites.
Joint work with: M. D. Rosini (UMCS Lublin), N. Dymski (UMCS Lublin) and M. Benyahia (GSSI L’Aquila).
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Maria Laura Delle Monache (Inria Grenoble Rhône - Alpes)
Big data and the fundamental diagram

We propose novel methodologies to analyze traffic data from fixed sensors. The main aim is to understand traffic characteristics by looking at the fundamental diagram. The main novelties of our approach consist in the use advanced statistical methods to analyze traffic data. In particular, we analyze data from multiple location in United States and Europe to show different traffic regimes and how they are related even from different geographical location.
The achieved results open new perspectives in the interpretation of data from fixed sensors and the modeling of vehicular traffic.
Joint work with: Y. Chen, K. Chi, P. Goatin, K. Han, B. Piccoli and J. Qiu.
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Raul De Maio (SBAI - Sapienza University of Rome)
Modeling transports over networks with measures.

I will present  a   theory of  measure-valued solutions to transport equations on networks. The measure-valued setting makes our framework suitable to deal with multiscale flows of agents on networks in applications such as traffic management, distribution optimization and nonlinear transport. The building block of   our approach is the study  of a measure-valued linear transport equation in a bounded interval. For this problem, we give an explicit representation formula of the solution. Then we construct the global solution on the network by gluing all the measure-valued solutions on the arcs by means of appropriate distribution rules at the vertexes.
This is a joint work with F. Camilli (La Sapienza, Rome) and A.Tosin (Politecnico, Turin).
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Carlotta Donadello (University of Bourgogne Franche-Comté)
Point constraints in vehicle or pedestrian traffic models

We introduce and analyze a class of one-dimensional models with nonlocal (in
time and/or in space) point constraints for traffic flow through bottlenecks, such as
exits in the context of pedestrians traffic and reduction of lanes on a road under work
in vehicular traffic. We propose a theoretical analysis and discretization framework
that permits to include different data acquisition strategies; a numerical comparison
is provided. Nonlocal dependence of a constraint on the solution allows to model
the irrational behavior (“panic”) near the exit observed in dense crowds.
Existence and uniqueness of solutions is shown under suitable and “easy to check”
assumptions on the constraint operator. A numerical scheme for the problem, based
on finite volume methods, is designed, its convergence is proved and its validation
is done with an explicit solution. Numerical examples show that nonlocally con-
strained models are able to reproduce important features of the crowd dynamics
such as self-organization.
Joint work with: B. Andreianov, U. Razafison and M.D. Rosini
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Laurent Gosse (IAC-CNR)
Numerical treatment of a (2+2)-dimensional Vlasov-Poisson Fokker-Planck model of swarming

Following the methodology presented in [2], we consider a mean-field, kinetic Vlasov-Poisson Fokker-Planck system for modeling collective dynamics in 2+2 dimensions. The kinetic density f (t, x, v) satisfies,
$\partial_t f +v\cdot\nabla_x f-(\nabla U\ast\rho)\cdot\nabla_v f=\nabla_v\cdot [k v f+\sigma\nabla_v f ], \sigma, k\geq 0$
where U(x) is a Morse-type, attractive far away, but repulsive at short scale, potential. When simulating problems of such type in a bounded domain of R^2_x (like e.g., a closed room), the handling of the convolution product involving U may be delicate. We shall present a practical approach relying on two main ideas:
• reformulating the force field as a Poisson-type differential operator, yet allowing for an easy inclusion of boundary conditions;
• the setup, within a dimensional-splitting strategy, of numerical fluxes involving transport in both $x, v$ variables, as proposed in [3, Chap.12], which rely on Pagani’s eigenfunctions [5] (see also [1]).
Such fluxes were formerly used in the context of 1D aggregation equations.

[1] R. Beals, Partial-range completeness and existence of solutions to two way diffusion equations, J. Math. Phys. 22, 1981, 954–960.
[2] J. A. Carrillo, M. Fornasier, G. Toscani, and F. Vecil, Particle, Kinetic, and Hydrodynamic Models of Swarming, in Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, G. Naldi, L. Pareschi and G. Toscani Eds. Birkhauser (2010) 297–336.
[3] L. Gosse, Computing Qualitatively Correct Approximations of Balance Laws, Springer (2013) ISBN 978-88-470-2891-3.
[4] L. Gosse, N. Vauchelet, Numerical high-field limits in two-stream kinetic models and 1D aggregation equations, SIAM J. Scient. Comput. 38 (2016) A412–A434.
[5] Carlo D. Pagani, Studio di alcune questioni concernenti l’equazione generalizzata di Fokker-Planck, Boll. Un. Mat. Ital. (4) 3 (1970) 961–986.
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Stephan Knapp (University of Mannheim)
A pedestrian flow model with stochastic velocities: microscopic and macroscopic approaches

A well known phenomenon in crowds is the sudden presence of non-moving persons which look at their cell phones, reorientate themselves or watch at a point of interest. This leads to changes in the velocities of the surrounding neighbors, evasive maneuver and hence to bottlenecks depending on the crowd density. To include this behavior into a mathematical model, we consider the well-known social force model by D. Helbing and P. Moln´ar † . This, in the literature well studied microscopic model describes the movement of each pedestrian according to his/her destination as well as interaction and obstacle forces. We embed the mentioned stochastic behavior into the microscopic pedestrian model by redefining a time-discrete stochastic process which simultaneously implies a microscopic simulation algorithm. Due to the high computational costs of approximating the crowd density for a large number of people, we derive a hyperbolic scalar model which approximates the evolution of the crowd density. Finally, the microscopic and the scalar model are compared by numerical results in several examples.
Joint work with S. G¨ottlich and P. Schillen.
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Marjeta Kramar Fijavz (University of Ljubljana)
The semigroup approach to transport processes in networks

We consider linear transport equations taking place on the edges of a finite network with static or dynamic transmission conditions in the vertices. We present semigroup methods to show well-posedness and describe long-time behavior of the solutions. We also treat boundary control problems by imposing control in the vertices of the network.
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Nicolas Laurent-Brouity (INRIA)
Impact of GPS-enabled routing applications on traffic

My presentation will detail the impact of the increasing penetration of routing apps on road usage. To model the issue, we distinguish two distinct classes of drivers. The first category has full knowledge of the network and the traffic conditions, while the other only has limited knowledge. The framework applies both to manned vehicles in which human drivers follow app directions, and unmanned vehicles following shortest path algorithms. We show that the increased usage of GPS is overall beneficial for the road network of Los Angeles, with a decrease in average travel times and total vehicle miles traveled. However, this global increased efficiency in urban mobility has negative impacts as well, such that increase in traffic in cities bordering highway, due to the rerouting of application users aiming at avoiding congestion. Those observations should lead to research on new transportation policies, mitigating the effects of app-based routing on urban areas.
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Francesca Marcellini (University of Milano-Bicocca)
A Phase Transition Traffic Model at Junctions

We discuss a Phase Transition traffic model, where a scalar conservation law is coupled with a 2 × 2 system, see [1]. We consider the Riemann problem for such a model at a general junction with n incoming and m outgoing roads. We propose a Riemann solver at the junction which conserves both the number of cars and the maximal speed of each vehicle, which is a key feature of the Phase Transition model, see [2].
[1] R.M. Colombo, F. Marcellini, M. Rascle. A 2-Phase Traffic Model Based on a Speed Bound. SIAM Journal on Applied Mathematics, 70: 2652-2666, 2010.
[2] M. Garavello, F. Marcellini. The Riemann Problem at a Junction for a Phase Transition Traffic Model. Preprint 2016.
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Helene Ranetbauer (Radon Institute for Computational and Applied Mathematics, Austria)
On nonlinear PDE models for pedestrian flows

In this talk we present two non-linear convection-diffusion models, which describe the evolution of bidirectional and crossing flows. We start with a two-dimensional lattice based model for the two pedestrian densities and formally derive the corresponding nonlinear PDEs using Taylor expansion. We discuss existence and stability of solutions in either case and illustrate the dynamics with numerical simulations.
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Massimiliano Daniele Rosini (University of Ferrara)
Many particle approximation of macroscopic models for vehicular and pedestrian flows
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Elena Rossi (University of Brescia)
IBVPs for Hyperbolic Balance Laws in Traffic Modelling

In this talk, we overview recent analytic results concerning balance laws on bounded domains. We then specialize these statements in a non autonomous setting, suitable for applications to traffic dynamics, aiming in particular at the optimal management of vehicular flows. Indeed, the results obtained ensure the stability with respect to the flow of solutions to an IBVP for a scalar conservation law.
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Sheila Scialanga (University of Glasgow)
Decentralised control of interconnected systems: An interpolation based approach

The problem of constrained optimal control of large-scale systems is generally difficult to solve. A first approach is to control the entire network in a centralised fashion using a global (observed) state vector. However, such centralised scheme cannot easily integrate local constraints into the optimal control problem. It can also be computationally expensive due to high dimensions or impossible to solve for nonlinear systems. An alternative avenue is to view the large-scale system as a decentralised system consisting of a number of interconnected subsystems or neighbourhood systems forming an entity, where each subsystem or entity has independent local constraints, inputs and outputs. Then the decentralised constrained control problem is to design a controller for each subsystem, where each individual controller uses its local (observed) state vector to generate the local control for the interconnected system. Decentralised control has been the subject of research in different communities especially in the seventies, and in recent years due to the rapid development of sensing and communication technologies. Nowadays the interest for decentralised control in traffic networks is increased due to the recent advances in vehicle automation and communication technologies that allows for the deployment of V2X cooperative systems with decoupled constraints. The primary limitation of decentralised control is that the individual controllers do not (a-priori) coordinate their actions and behaviour, except if appropriate stability criteria are fulfilled. Consequently, local controllers may select individual actions that are locally optimal but that together result in global instabilities. Thus a coordination and connectivity mechanism is necessary to guarantee stability of the overall system. Concluding, the control of interconnected systems with constraints introduces a number of theoretical and computational challenges; on the other hand, considerable system improvements can be achieved in terms of constraint satisfaction, equity and fairness. In this work, we present a new framework for the decentralised control problem of largescale interconnected traffic systems with local constraints. The advantages of employing a decentralised control approach are dimensionality and well-structured decoupled information constraints. To this end, we extend an Interpolating based Control (IC) approach for the constrained decentralised control problem of large-scale interconnected systems. IC is a novel approach that incorporates the state and control constraints in the optimal control problem formulation and significantly reduces the computational effort compared to optimisation-based schemes such as Model Predictive Control (MPC). The main idea behind IC is to blend a global low-gain feedback controller that satisfies the control and state constraints with a user-chosen local high-gain controller that has its positively invariant set satisfying the constraints. The advantage here is that both low- and high-gain controllers can be determined off-line while the interpolation between them is performed on-line. For the interpolation a low dimensional Linear Programming (LP) problem is solved on-line at each time step. Proofs of recursive feasibility and asymptotic stability for the case of IC for interconnected systems are given. To investigate the performance of distributed IC, we compare the proposed decentralised control with centralised control for simple systems as well as large-scale interconnected traffic systems.
Joint work with Konstantinos Ampountolas.
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Silvia Siri (University of Genova)
Optimal control for regulating traffic in freeway systems via ramp metering with multiple objectives

In this work, ramp metering control is investigated for regulating traffic in freeway systems in case of different objectives, such as reduction of traffic congestion, minimization of traffic emissions, and maximization of safety conditions. By adopting a second-order traffic flow model, a multi-objective nonlinear optimal control problem is formulated. The numerical solution of this problem is found by applying a specific version of the feasible direction algorithm whose effectiveness is demonstrated by means of simulation results. In addition, the simulation results aim at evaluating whether and in which traffic conditions the considered objectives have a conflicting nature.
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Giuseppe Visconti (RWTH Aachen University)
Traffic flow models derived from a kinetic approach

Prediction and control of traffic have become important aspects of the modern world. A basic tool to analyze vehicular traffic problems is given by the relations between the macroscopic quantities of the flow: flux versus density diagrams (or fundamental diagrams) and speed versus density diagrams. Both show a two phase behavior with a sharp phase transition separating a free and a congested regime of the flow in which there is a large scattering of data.
In the current mathematical literature, there are different approaches to model traffic flow phenomena using PDEs: here we consider macroscopic and kinetic models. Macroscopic scale models provide a large-scale aggregate point of view neglecting the microscopic dynamics. Therefore, they study only the evolution of the macroscopic quantities related to traffic flow by means of (first or second-order) PDEs inspired by fluid dynamics laws. Although macroscopic equations can be treated easily, it is necessary to complete the equations with a closure law. This means that the fundamental diagram is given with an a-priori relation, derived from heuristic or physical arguments, which thus does not result from microscopic dynamics.
Since the collective behaviors are strictly linked to the microscopic interactions among vehicles, more refined closure laws can be obtained using a lower scale of representation. To this end, kinetic models, which are based on a statistical mechanics approach, provide quite naturally an aggregate description of traffic flow by modeling microscopic dynamics.
In this talk, we first present a recent spatially homogeneous kinetic model for traffic flow based on simple and reasonable microscopic binary interactions. This model provides fundamental diagrams resembling the qualitative structure of experimental surveys, including a realistic free flow phase and a phase transition characterized by a sharp decrease of the flux. The model can be generalized to the case of a mixture of vehicles, differing in length and maximum velocity, and which compete for the available space on the road. We show the result for two populations and observe that this allows to recover the scattering of data typically seen in experimental fundamental diagrams as a result of the heterogeneous composition of the flow. Then, we study the Fokker-Planck limit of the Boltzmann-type model in order to establish a link among microscopic dynamics and macroscopic effects.
Finally, since we are able to compute explicit analytical expressions of the time-asymptotic distribution of these kinetic models, we can derive a closure law for the macroscopic equations, which is thus obtained from the kinetic approach. The solutions of some relevant problems are compared with the solutions provided by the kinetic models in which we add a space dependence.
Joint work with M. Herty, G. Puppo, M. Semplice, A. Tosin