Mon

day3

Tuesday4

Wed

nesday5

Thursday6

Friday7

9:00-9:50

Seyfrie

dBa

rdi

9:00Caine

sCardaliaguet

Malhamé

9:50-10:40

Barbaro

Vizzari

9:45Degon

dHe

rty

Totzeck

10:40-11:00

Coffe

ebreak

Coffe

ebreak

10:30Co

ffeebreak

Coffe

ebreak

Coffe

ebreak

11:00-11:50

Achd

ouSchadschne

ider

10:50Ch

assagneu

xGo

ttlich

DiFrancesco

11:50-12:20

Mazanti

Tordeu

x11:35Santam

brogio

Duprez

Stivaletta

12:30-14:00

Lunch

Lunch

Lunch

Lunch

Lunch

14:00-14:50

Tono

nDa

amen

Maury

14:50-15:20

Bailo

Radici

Richardson

15:20-15:50

Coffe

ebreak

Duives

Gomes

15:50-16:10

Thieu

Coffe

ebreak

Coffe

ebreak

16:10-16:40

16:20Espo

sito

Pogodaev

Staritsyn

16:40-17:10

PosterSessio

nMaggistro

Fagioli

19:30

Dinn

erDinn

erDinn

erSocialdinner

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Freeafterno

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Monday 3rd, morningBottlenecks and rewards - adding fuel to the fire

Armin Seyfried, Forschungszentrum Jülich - Germany

The talk summarize the empirical state of knowledge on bottleneck flow and introduces an approachto describe crowd disasters. The approach combines quantities well known in natural sciences withconcepts of social psychology. It allows to describe crowds at bottlenecks in case of exceptional (life-threatening) or normal circumstances.

On the basis of empirical data, the influence of the spatial structure of the boundaries (width andlength of the bottleneck) and the motivation of pedestrians on flow and density will be presented. Thephenomenon of clogging and its effects on the flow will be discussed in connection with congestion,rewards, motivation and pushing. Positive effects of pillars in front of bottlenecks are critically ques-tioned by recent experiments. Results of two experiments including questionnaire studies connect flowand density with factors of social psychology like rewards, social norms, expectations or fairness.

Evacuations and disasters: modeling stressed crowds

Alethea Barbaro, Case Western Reserve University - USA

Psychologists have been arguing for many years that physics-based models are not good modelsfor humans, in part because they ignore a person?s psychological state. Yet these models persist inthe modeling literature. Here, I propose an emotional contagion model which serves as a compromisebetween the two communities. In this class of models, an agent?s stress level is an additional evolvingvariable which affects an agent?s movement dynamics. I will propose a multi-dimensional particlemodel and then formally derive a kinetic PDE from the model and address analytical results. I willthen discuss several variations of the model and related experimental work.

Mean field games in the context of crowd motion

Yves Achdou, Univ. Paris Diderot, Laboratoire Jacques-Louis Lions, UMR 7598, UPMC, CNRS, F-75205 Paris - France

Recently, an important research activity on mean field games (MFGs for short) has been initiatedby the pioneering works of Lasry and Lions: it aims at studying the asymptotic behavior of stochasticdifferential games (Nash equilibria) as the number n of agents tends to infinity. The field is nowrapidly growing in several directions, including stochastic optimal control, analysis of PDEs, calculusof variations, numerical analysis and computing, and the potential applications to economics andsocial sciences are numerous.The present talk will consist of a survey of different aspects related to modeling crowd motion withMFGs, which permits to take into account the rational anticipation of the agents. The followingaspects will be discussed:

• Models of congestion and weak solutions of the system of PDEs arising from the mean fieldgames models (results obtained with A. Porretta)

• Mean field games versus mean field type control (results obtained with M. Laurière)

• Numerical simulations with or without a common noise (results obtained with J-M. Lasry)

• Several populations of interacting agents

• Mean field games of control (results obtained with Z. Kobeissi)

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A minimal-time mean field game for crowd motion

Guilherme Mazanti, Laboratoire de mathématiques, Université Paris-Sud - France

Mean field games (MFGs) have been extensively studied since their introduction around 2006 bythe independent works of P. E. Caines, M. Huang, and R. P. Malhamé and J.-M. Lasry and P.-L.Lions. Such differential games with a continuum of indistinguishable agents have been proposed asapproximations of games with a large number of symmetric players, finding their applications in severaldomains. This talk considers a mean field game model for crowd motion in which agents want to leavea bounded domain through a part of its boundary in minimal time. Each agent is free to move in anydirection, but their maximal speed is assumed to be bounded in terms of the distribution of agentsaround their position in order to model congestion phenomena. With respect to most mean field gamemodels in the literature, the novelties here are the velocity constraint and the fact that the final timein the optimization criterion is not fixed, which are important features from a modeling point of viewfor crowd motion but bring several extra difficulties in the analysis.

After presenting the model and its motivation, we establish the existence of equilibria for thisgame using a fixed-point strategy based on a Lagrangian formulation. In the case where agents mayleave the domain through any point of its boundary, such equilibria are characterized by a system ofcoupled PDEs, called the MFG system, made of a continuity equation on the distribution of agentsand a Hamilton-Jacobi equation on the value function of the optimal control problem solved by eachagent. This characterization relies on further regularity properties of optimal trajectories obtainedfrom Pontryagin Maximum Principle, yielding in particular that the optimal trajectory of an agentconsists on moving with maximal speed on the opposite direction of the gradient of the value function.We also present some simulations in simple situations and conclude the talk by discussing other recentresults and ongoing work, including sufficient conditions ensuring the Lp regularity of the distributionof agents and the case where agents’ dynamics are stochastic.

This talk is based on joint works with Samer Dweik and Filippo Santambrogio.

Monday 3rd, afternoonAggregative behavior in crowds

Daniela Tonon, Université Paris Dauphine - France

In this talk, we consider a large number of agents aiming at aggregation, i.e. at converging toa common state. This situation can be modeled through time-dependent viscous Mean Field Gamesystems with local, decreasing and unbounded couplings. From the PDE viewpoint, several issues areintrinsic in this framework, mainly caused by the lack of regularizing effects induced by increasingmonotonicity of the coupling. Actually, non-existence, non-uniqueness of solutions, non-smoothness,and concentration are likely to arise. Even more than in the competitive case, the assumptionson the Hamiltonian, the growth of the coupling and the dimension of the state space affect thequalitative behavior of the system. We prove the existence of weak solutions that are minimizersof an associated non-convex functional, by rephrasing the problem in a convex framework. Underadditional assumptions involving the growth at infinity of the coupling, the Hamiltonian, and thespace dimension, we show that such minimizers are indeed classical solutions by a blow-up argumentand additional Sobolev regularity for the Fokker-Planck equation. These results are obtained in ajoint collaboration with Marco Cirant.

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Pedestrian Models based on Rational Behaviour

Rafael Bailo, Imperial College London - United Kingdom

Agent dynamics is a vast and prolific field. Following the paradigm set by Attraction-Repulsion-Alignment schemes, a myriad of models have been proposed to calculate the evolution of abstractagents, whether it be to develop CGI simulations of flocking (BOIDS - Reynolds 1987), to simulatethe motion of pedestrians (Social Force Model - Helbing, Molnar 1995) or to analyse the developmentof political opinions (Cucker, Smale 2007). While the emergent features of many agent systems havebeen described astonishingly well with force-based models, this is not the case for pedestrians. Manyof the classical schemes have failed to capture the fine detail of crowd dynamics, and it is unlikely thata purely mechanical model will succeed.

As a response to the mechanistic literature, we will consider a model for pedestrian dynamics thatattempts to reproduce the rational behaviour of individual agents through the means of anticipation.Each pedestrian undergoes a two-step time evolution based on a perception stage and a decision stage.We will discuss the validity of the model in regimes with varying degrees of congestion, ultimatelypresenting a correction to achieve realistic high-density dynamics.

A lattice model for active-passive pedestrian dynamics: a quest for drafting effects

T. K. Thoa Thieu, Gran Sasso Science Institute, L’Aquila - Italy

We consider a 2D lattice model in which two species, called “active” (A) and “passive” (P),perform a simple exclusion dynamics. The lattice is equipped with a number of sites, called “exitdoors”, through which particles can leave the system. The species A, unlike species P, is subject toa drift that encodes the awareness of the precise location of the exit doors. The two-species modelmimics the evacuation of a mix of pedestrian populations: pedestrians aware of the local geometry, andrespectively, pedestrians unaware of the local geometry. We use Monte Carlo simulations to investigatethe characteristic time scales occurring in a standard evacuation where both species attempt to findthe exit doors. We discuss, in particular, the onset of a peculiar "drafting" effect in the dynamics.Basically, the drafting phenomenon is observed at macroscopic scales, e.g. in aerodynamics or in fluiddynamics, and is responsible for reducing the overall effect of drag as well as for balancing the energyexpenditure of the moving objects. In this work, we discovered that “drafting” can occur also indiscrete states (say at a microscopic/mesoscopic level) in lattice-like gas dynamics. More precisely,our simulation results show that, in spite of the excluded volume interaction, the evacuation of speciesP is enhanced by the evacuation of species A.

This is joint work with Matteo Colangeli (University of L’Aquila, Italy), Emilio N. M. Cirillo(Sapienza University of Rome, Italy) and Adrian Muntean (Karlstad University, Sweden).

Measure solutions to a system of continuity equations driven by Newtonian non-local interactions

Antonio Esposito, University of L’Aquila - Italy

We prove global-in-time existence and uniqueness of measure solutions of a nonlocal interactionsystem of two species in one spatial dimension. For initial data including atomic parts we providea notion of gradient-flow solutions in terms of the pseudo-inverses of the corresponding cumulativedistribution functions, for which the system can be stated as a gradient flow on the Hilbert spaceL2(0, 1)2 according to the classical theory by Brézis. For absolutely continuous initial data we constructsolutions by using a minimising movement scheme in the set of probability measures. In addition weshow that the scheme preserves finiteness of the Lm-norms for all m ∈ (1,+∞] and of the second ordermoments. We then provide a characterisation of equilibria and prove that they are achieved (up totime subsequences) in the large time asymptotics. We conclude the paper constructing two examples

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of non-uniqueness of measure solutions emanating from the same (atomic) initial datum, showing thatthe notion of gradient flow solution is necessary to single out a unique measure solution.

Joint work with J. A. Carrillo, M. Di Francesco, S. Fagioli, M. Schmidtchen.

17:00 - Poster SessionAlasio Luca, Gran Sasso Science Institute - Italy

The role of a strong confining potential in a nonlinear Fokker-Planck equation

Ammenhäuser Sarah, University of Wuppertal - GermanyStatistical analysis of dynamic networks in pedestrian dynamics

Bonnet Benoît, Aix-Marseille Université - FranceSome problems in the modelling and optimal control of multi-agent systems

Favre Gianluca, University of Vienna - AustriaParticles relaxation with energy balance

Jaafra Yahya, University of L’Aquila - ItalyStationary solutions for Predator-Prey system with diagonal diffusion

Paolucci Alessandro, Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, L’Aquila- Italy

Delay effects in opinion formation

Sparnaaij Martijn, Delft University of Technology - NetherlandsDynamic spatial discretization for a mesoscopic pedestrian simulation model

Sylla Abraham, Institut Denis Poisson, Université de Tours - FranceTwo macroscopic models to reproduce self-organization near exits

Toumi Noureddine, Groupe d’études et de recherche en analyse des décisions - CanadaNon-local mean field modeling of crowd evacuation

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Tuesday 4th, morningConvergence of some deterministic Mean Field Games to aggregation and flockingmodels

Martino Bardi, Università di Padova - Italy

We consider some deterministic (i.e., first order) Mean Field Games with large discount factor inthe cost functional and cheap controls. We show that in the limit the mass distribution solves a non-local nonlinear continuity equation and the (rescaled) feedback producing the mean field equilibriumconverges to the gradient of the running cost, so that the limit dynamics follows the steepest descentof the running cost associated to the limit mass distribution.

For suitable choices of the underlying control system and running costs our results establish arigorous connection among Mean-Field Games and various agent-base models such as the aggregationequation, the nonlinear friction equation of granular flows, and some models of flocking such as Cucker-Smale.

Collaboration with P. Cardaliaguet.

What can machine learning bring to crowd analysis, modeling and simulation?Considerations and experiments

Giuseppe Vizzari, Università degli Studi di Milano-Bicocca - Italy

In this talk I’ll provide a perspective on the kind of contributions that machine learning can bringto the field of crowd analysis, modeling and simulation. While, in fact, deep learning approaches havebrought a sort of revolution within specific fields of application (e.g. computer vision and naturallanguage processing in particular), it is still not clear what could be the impact on crowd studies.After a brief introduction trying to frame the approaches within the different phases and tasks ofcrowd studies, I will provide some examples of application of machine learning approaches (some ofthem ongoing works, actually) related to crowd patterns analysis, multi-scale simulation, micro-scalesimulation.

Noise-Induced Stop-and-Go Dynamics in Pedestrian Single-file Motion

Andreas Schadschneider, University of Cologne - Germany

Laboratory experiments have revealed a transition from free flow to stop-and-go dynamics inpedestrian single-file motion. We propose a novel explanation of stop-and-go phenomena in pedestrianflows which does not require a phase transition as in the standard explanation for similar systems,especially vehicular traffic. As a toy model for this mechanism a model based on dynamics described bya deterministic first order equation with an additive coloured noise is proposed. Despite its simplicitythe model shows a very good agreement with empirical results. Furthermore it allows to elucidatethe role of inertia in pedestrian dynamics. We will demonstrate that many of the problems with 2ndorder (force-based) models are related to inertia effects.

A. Tordeux, A. Schadschneider, J. Phys. A49, 185101 (2016)

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Scaling, stability analysis and simulation of traffic flow continuum derivations offirst order OV following models

Antoine Tordeux, Bergische Universität Wuppertal - Germany

The collision-free optimal velocity (CFOV) model is a continuous first order microscopic traffic flowmodel [1]. The model has two predecessors for the interaction and includes a reaction time parameter.It describes unstable homogeneous streaming and stop-and-go wave phenomena as classical secondorder OV models but in a rigorous collision-free framework. Indeed, the first order model avoids byconstruction unbounded density and negative speed problems [3]. Such difficulties are specific to theinertial second order model class. They are largely investigated in the literature (see, e.g., [4]).

We derive a continuum traffic flow model from the microscopic CFOV model [2]. The macroscopicformulation results in first order convection-diffusion equations. In contrast to classical hyperbolicderivations of second order inertial microscopic models [5], the continuum derived from the first ordermodel is parabolic. The optimal velocity function regulates the convection while the diffusion termdepends on the reaction time and the spatial derivative of the density. The flow characteristics of theoutflow and inflow at the boundaries of a jam are asymmetric. Such a mechanism is responsible forinstability and the emergence of waves in the system.

We analyze the numerical stability of temporal and spatial Godunov schemes. The numerical con-ditions match the stability condition of the initial microscopic model for specific scaling of the reactiontime parameter [5]. Several simulation scenarios of the microscopic CFOV model and its macroscopicparabolic derivation with jammed, perturbed or random initial configurations are superposed. Theresults show that the dynamics of the microscopic model can be well captured by the macroscopicequations. Transitions to limit cycles with collisionfree stop-and-go dynamics are observed in theunstable case when the driver reaction time is high.

ReferencesA. Tordeux and A. Seyfried. Collision-free nonuniform dynamics within continuous optimal veloc-

ity models. Phys. Rev. E, 90:042812, 2014.A Tordeux, G Costeseque, M Herty, A Seyfried. From traffic and pedestrian follow-the-leader

models with reaction time to first order convection-diffusion flow models. SIAM Journal on AppliedMathematics 78:1, 63-79, 2018.

C. F. Daganzo. Requiem for second-order fluid approximations of traffic flow. Transport. Res. B:Meth., 29(4):277286, 1995.

D. Helbing. Improved fluid-dynamic model for vehicular traffic. Phys. Rev. E, 51:3164-3169,1995.

A. Aw and M. Rascle. Resurrection of “second order” models of traffic flow. SIAM Journal onApplied Mathematics, 60(3):916-938, 2000.

R. E. Wilson, P. Berg, S. Hooper, and G. Lunt. Many-neighbour interaction and non-locality intraffic models. Eur. Phys. J. B, 39(3):397-408, 2004.

A. Aw, A. Klar, M. Rascle, and T. Materne. Derivation of continuum traffic flow models frommicroscopic follow-the-leader models. SIAM Journal on Applied Mathematics, 63(1):259-278, 2002.

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Tuesday 4th, afternoonMulti-objective calibration of crowd models, and the required data collection

Winnie Daamen, Delft University of Technology - The Netherlands

Pedestrian traffic is usually very complex, presenting many situations such as bidirectional andcrossing flows, under various conditions, such as large-scale events and public transport facilities.To design suitable infrastructure and deploy and assess crowd management measures, reliable andaccurate pedestrian models need to be developed. Moreover, those models need to be applicable withdifferent pedestrian compositions, as pedestrian heterogeneity is known to considerably affect thecrowd dynamics. Many models already exist, but an extensive calibration is challenging, due to itsdependence on the situation and the need for data, but also on the wide range of behaviours coveredin such a model. Calibration is the optimisation of parameter values, such that the simulation resultsare as close as possible to the data used for calibration. The talk will address three elements of thecalibration process:

1. Data collection. While trajectory data from laboratory experiments become increasingly avail-able, empirical data from field observations and data describing the tactical and strategic pedes-trian behaviour are still scarce. The talk will show the variety of data sets and data collectiontechniques to acquire calibration data.

2. Choice of measure of performance. Researchers have shown that multiple (joint) objective func-tions are needed to come up with parameter estimates for generic models, that predict boththe microscopic and the macroscopic characteristics of pedestrian flows under a wide range ofconditions and base cases. The talk will address these measures-of-performance, and show theirbenefits and shortcomings.

3. Optimisation procedure, covering the objective functions, the measures of goodness-of-fit used tocompare the simulation results to the data and the optimisation algorithms to find the optimalparameter set. The talk will introduce objective functions and measures of goodness-of-fit andshow some results for calibration.

Deterministic particle approximation for scalar aggregation-diffusion equations withnonlinear mobility

Emanuela Radici, Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, L’Aquila -Italy

We describe the one dimensional dynamics of a biological population influenced by the presence ofa nonlocal attractive potential and a diffusive term, under the constraint that no over crowding canoccur. It is well known that this setting can be expressed by a class of aggregation- diffusion PDEswith nonlinear mobility. We investigate the existence of weak type solutions obtained as large particlelimit of a suitable nonlocal version of the follow-the-leader scheme, which is interpreted as the discreteLagrangian approximation of the target continuity equation. We restrict the analysis to nonnegativebounded initial data with finite total variation, away from vacuum and supported in a closed intervalwith zero-velocity boundary conditions. The main novelties of this work concern the presence of anonlinear mobility term and the non strict monotonicity of the diffusion function, thus, our resultapplies also to strongly degenerate diffusion equations. We also address the pure attractive regime,where we are able to achieve a stronger notion of solution. Indeed, in this case our scheme convergestowards the unique entropy solution to the target PDE as the number of particles tends to infinity.

This is a joint work with Marco Di Francesco and Simone Fagioli.

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Studying the impact of heterogeneity on the dynamics of high-density crowd move-ments

Dorine Duives, Delft University of Technology - The Netherlands

The management of crowded spaces has become more and more a necessity. Large crowds arisein transfer hubs, mass events and city centres. Sophisticated crowd monitoring systems in combi-nation with complex control techniques have been presented to alleviate unwanted dangerous crowdmovements. Yet, to act on the monitoring data, insights regarding the tipping points in the crowddynamics, such as capacity and the onset of flow breakdown, are needed for crowds.

Simulation studies by Campanella et al. (2009) and Yang et al. (2014) have studied flow breakdownphenomena and illustrate that heterogeneity severely influences the onset of breakdown. Daamen etal. (2012) furthermore shows that percentage of certain types of individuals (e.g. children and elderly)influences the capacity. However, these studies were unable to pinpoint to what extent local differencesin heterogeneity influence the shape of the fundamental diagram, the flow rate and the capacity.

This study aims to alleviate this issue and will study to what extent heterogeneity influencesthe movement behaviour of pedestrians under high densities. In particular, this research studies therelation between the fundamental diagram (i.e. speed, flow rate and density) and the distribution ofthe characteristics of the participants (i.e. age, gender and physical stature and reaction abilities)inside the infrastructure at a given point in time.

A large experiment was performed at Delft University of Technology. During these experiments theonset of crowd locks were studied. A mixed group of 140-160 participants was asked to simultaneouslywalk through an infrastructure. In total 24 runs were performed, 12 of which featured bi-directionalflows and 12 intersecting flows. During each run the participants were asked to perform severalassignments (i.e. do nothing, speed up, cross intentionally) which created instabilities in the flow thatpotentially could lead to flow breakdown. In contrast to earlier experiments, the trajectories of eachparticipant are matched with its personal details, reaction time and speed estimation abilities.

Joint work with Martijn Sparnaaij and Serge Hoogendoorn.

On the modeling of crowds in regions with moving obstacles via measure sweepingprocesses

Nikolay Pogodaev, Matrosov Institute for System Dynamics and Control Theory, SB RAS - Russia

In the talk, we discuss an approach for modeling crowd dynamics in a region with moving obstacles.As in most macroscopic models, we characterize the crowd by a probability mass distribution ρt onRd which changes in time, while the obstacles are described by a set-valued map t 7→ Ot whose valuesare open subsets of Rd. We look for a model satisfying the following natural assumptions: (i) theevolution of ρt obeys the mass conservation law:

∂tρt +∇x · (vtρt) = 0;

(ii) the obstacles carry zero mass: spt ρt ⊂ Rd \ Ot.= Dt; (iii) agents choose their velocity using full

information about the current state of the crowd: vt = V [ρt], where V is a certain function mappingmeasures to vector fields.

To combine these assumptions together in a single model we use the framework of measure sweepingprocesses (dynamical systems in the space of measures that describe how mass distributions evolvewhile pushed by moving sets). This notion, being a natural generalization of the classical Moreausweeping process, was introduced by S. Di Marino, B. Maury, andF. Santambrogio, who studied theevolution of a measure driven by a set Ct under the assumption that Ct is closed and convex for eacht. In our case, the driving set Dt is assumed to be just uniformly prox-regular for every t, whichallows to deal with a wide class of obstacles. The main result concerns the existence and uniqueness of

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solutions. To prove it, we adapt the classical catching-up scheme (for a perturbed Moreau’s sweepingprocess) to the problem in the space of measures.

Optimal control of the MFG equilibrium for a pedestrian tourists’ flow model

Rosario Maggistro, University Ca’ Foscari Venice - Italy

In the recent years, the continuous growth of tourists’ flow and the resulting overcrowding have ledsome heritage cities to seek solutions to manage this phenomenon. In this work we address the problemof modelling the daily tourists’ flow along the narrow alleys of a heritage city using a mean field gamesapproach. Network/switching is used to describe the tourists cost as a function of their position,taking into consideration whether they have already visited a site or not, i.e. allowing tourists to havememory of the past when making decisions. We assume for simplicity that tourists have only two mainattractions to visit, and we represent possible paths as a circular network containing three nodes: thetrain station where tourists arrive in the morning and to which they have to return in the evening; thefirst and second attraction. Moreover, we suppose that each tourist in his/ her evolution faces a costdepending on the satisfaction of visiting the sites and on the congestions (the mean field) of the chosenpath inside the city. The problem is then analysed in the framework of Hamilton-Jacobi/transportequations, as it is standard in mean field games theory, and an existence result of a mean field gameequilibrium is provided. This equilibrium is seen as a fixed point, over a suitable set of time-varyingdistributions, of a map which is obtained as a limit of a sequence of approximating functions. Each ofthese functions rather than choosing optimal controls, chooses ε− optimal controls and, along time,an ε− optimal stream. Then, we study a possible optimization problem for an external controllerwho aims to induce a suitable mean field game equilibrium. For that, we suppose that the externalcontroller (the city administration, for example) may act on the congestion functions, choosing themamong a suitable set of admissible functions.

This talk is based on joint works with Fabio Bagagiolo, Silvia Faggian and Raffaele Pesenti.

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Wednesday 5th, morningCommon session with the International Conference Dynamic, Games and Science

Graphon Mean Field Games and the GMFG Equations

Peter E. Caines, Dep. Electrical and Computer Engineering - McGill University - Montreal, QC -Canada

Very large networks linking dynamical agents are now ubiquitous and there is significant interestin their analysis, design and control. The emergence of the graphon theory of large networks and theirinfinite limits has recently enabled the formulation of a theory of the centralized control of dynamicalsystems distributed on asymptotically infinite networks [Gao and Caines, IEEE CDC 2017, 2018].Furthermore, the study of the decentralized control of such systems has been initiated in [Caines andHuang, IEEE CDC 2018] where Graphon Mean Field Games (GMFG) and the GMFG equations areformulated for the analysis of non-cooperative dynamical games on unbounded networks. In this talkthe GMFG framework will be first be presented followed by the basic existence and uniqueness resultsfor the GMFG equations, together with an epsilon-Nash theorem relating the infinite populationequilibria on infinite networks to that of finite population equilibria on finite networks.

Mathematical models of collective dynamics and self-organization

Pierre Degond, Department of Mathematics - Imperial College London - United Kingdom

In this talk, I will review some mathematical challenges posed by the modelling of collectivedynamics and self-organization. Then, I will focus on two specific problems, first, the derivation offluid equations from particle dynamics of collective motion and second, the study of phase transitionsand the stability of the associated equilibria.

Weak error expansion for Mean Field SDE

Jean-François Chassagneux, Laboratoire de Probabilités, Statistique et Modélisation - Université ParisDiderot - France

In this work, we study the weak approximation error by particle system of Mean Field SDE. Weprove an expansion of this error in terms of the number of particle. Our strategy of proof follows theapproach of Talay-Tubaro for weak approximation of SDE by an Euler Scheme. We thus considera PDE on the Wasserstein space (called the Master Equation in mean-field games literature) and,relying on smoothness properties of the solution, obtain our expansion. We also prove the requiredsmoothness properties under sufficient conditions on the coefficient function.

Equilibria and regularity in Mean Field Games with density penalization or con-straints

Filippo Santambrogio, Université Claude Bernard - Lyon 1 - France

Equilibria and regularity in Mean Field Games with density penalization or constraintsIn the talk, I will first present a typical Mean Field Game problem, as in the theory introduced by

Lasry-Lions and Huang-Caines-Malhamé, concentrating on the case where the game has a variationalstructure (i.e., the equilibrium can be found by minimizing a global energy) and is purely deterministic(no diffusion, no stochastic control). From the game-theoretical point of view, we look for a Nashequilibrium for a non-atomic congestion game, involving a penalization on the density of the playersat each point. I will explain why regularity questions are natural and useful for rigorously proving that

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minimizers are equilibria, making the connection with what has been done for the incompressible Eulerequation in the Brenier’s variational formalism. I will also introduce a variant where the penalizationon the density is replaced by a constraint, which lets a price (which is a pressure, in the incompressiblefluid language) appears on saturated regions. Then, I will sketch some regularity results which applyto these settings.

The content of the talk mainly comes from joint works with A. Mészáros, P. Cardaliaguet, and H.Lavenant.

Wednesday 5th, afternoonFaster is Slower effect in crowd evacuation: modeling issues

Bertrand Maury, Laboratoire de mathématiques, Université Paris-Sud - France

The so-called Faster is Slower (FiS) effect is commonly observed in real-life or experimental sit-uations. In the context of evacuation processes, it expresses that increasing the speed (or, moregenerally, the eagerness to egress) of individuals may induce a reduction of the flow through the exitdoor. We propose an investigation of the various ingredients which can be mobilized to reproducethis phenomenon. As for Social Force Models and Cellular Automata, an additional frictional termis the key ingredient to recover the FiS effect, but friction is not strictly necessary. We shall detailin particular how, in the context of non-frictional granular models, the pathologic character of anunderlying discrete Laplace operator can be identified as an alternative explanation of the FiS effect.

The effects of environment knowledge on evacuations in buildings under fire haz-ards: modelling and simulation

Omar Richardson, Karlstad University - Sweden

We present results on recent investigations in modelling evacuation dynamics within built envi-ronments. Inspired by both microscopic and macroscopic crowd models and practitioners’ evacuationevaluations, we investigate how the environment knowledge of occupants affects the evacuation of andthe interaction with other occupants. The effects of a lack of knowledge or even wrong knowledge areoften overlooked by crowd modellers, but can strongly impact both evacuation dynamics and evacu-ation time. In this work, we restrict ourselves to two agent species: residents, who are familiar withtheir surroundings and have determined the fastest way to the exit in advance, and visitors, who haveno inherent knowledge of the environment and have to rely exclusively on their (potentially) wiserneighbours to reach an exit.

We experiment on how mixtures of these populations affect the progress of the evacuation in acomplex geometry subject to

re emergencies. To this end, we emulate a source of fire that blocks corridors and creates smoke,that propagates throughout the environment. This hinders the occupants’ line of sight and exchangeof information.

Population dynamics are formulated by a set of microscopic ODE, where interaction betweenoccupants, walls, and fire hazards are modelled macroscopically through a set of PDE, giving rise to amultiscale crowd dynamics formulation. Among results are the expected leader-follower dynamics, ahigh observed crowd pressure associated with large numbers of residents, including incidental harmfuleffects caused by “stubbornness”, and the strong effect of the environment in the residence time ofthe visitors.

This is a joint work with Adrian Muntean (Karlstad University, Sweden) and Andrei Jalba (Eind-hoven University of Technology, The Netherlands).

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Parameter estimation for macroscopic pedestrian dynamics models

Susana Gomes, Warwick Mathematics Institute - United Kingdom

In this talk, we present a framework for estimating parameters in macroscopic models for crowddynamics using data from individual trajectories. We consider a model for the unidirectional flow ofpedestrians in a corridor which consists of a coupling between a density- dependent stochastic differen-tial equation and a nonlinear partial differential equation for the density. In the stochastic differentialequation for the trajectories, the velocity of a pedestrian decreases with the density according to thefundamental diagram. Although there is a general agreement on the basic shape of this dependence,its parametrization depends strongly on the measurement and averaging techniques used as well asthe experimental setup considered. We will discuss identifiability of the parameters appearing in thefundamental diagram, introduce optimisation and Bayesian methods to perform the identification,and analyse the performance of the proposed methodology in various realistic situations. Finally, wediscuss possible generalisations, including the effect of the form of the fundamental diagram and theuse of experimental data.

Impulsive control of moving ensembles of interacting agents

Maxim Staritsyn, Matrosov Institute for System Dynamics and Control Theory, SB RAS - Russia

We study an impulsive control problem for the continuity equation

∂t µt +∇ · (vt µt) = 0, t ∈ [0, T ],

driven by the nonlocal vector field

vt(x) = f0(x) +m∑

i=1ui(t) fi(x) + (g ? µt)(x),

where g ? µ denotes the convolution of a function g and a measure µ defined by

(g ? µ)(x) .=∫Rdg(x− y) dµ(y), x ∈ Rn,

and vector fields fi, g, i = 0, . . . ,m, enjoy the standard regularity properties. Controls u ∈ L∞([0, T ],Rm)are not subject to any geometric constraint, instead, they are assumed to be bounded in L1:∫ T

0

m∑i=1|ui(t)| dt ≤M.

The addressed model can be viewed as a “meanfield limit” of a dynamical system describing anensemble of interacting identical agents, which are influenced by (common for all agents) “shockimpacts”, i.e., control inputs whose intensity could be arbitrary high over short periods of time.

In the talk, we present a constructive description of this limiting impulsive continuity equationthrough the tools of finite-dimensional impulsive control, i.e., in terms of a discontinuous (singular)time reparameterization and differential equations driven by Borel measures. We discuss some basicproperties of the impulsive solutions and establish their connection with a prototypical continuityequation.

For a Mayer optimal control problem stated on solutions of the impulsive continuity equation, weprove a necessary optimality condition in the form of the Pontryagin Maximum Principle, and exhibitsome results of numeric analysis of certain relevant illustrative cases.

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Aggregation/diffusion models for opinion formation

Simone Fagioli, Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, L’Aquila - Italy

I will present some results on aggregation/diffusion equations arising in modelling opinion forma-tion phenomena. Those equations describe the dynamics for a certain density population and arecomposed of nonlinear diffusion equations with degenerate mobility combined with other effects, suchas transport driven by external forces (local potentials) and/or aggregation or repulsion induced bythe presence of non-local potentials. The equations are posed on a bounded real interval.

In case of fast-decay mobilities, namely mobilities functions under Osgood integrability condition,a suitable coordinate transformation is introduced. We observe that the coordinate transformationinduces a mass-preserving scaling on the density and we show that the rescaled density is the uniqueweak solution to a nonlinear diffusion equation with linear mobility. Moreover, the results obtainedfor the new density allow us to motivate the aforementioned change of variable and to state the resultsin terms of the original density without prescribing any boundary conditions. This is a joint workwith N. Ansini.

In the case of slow decay mobility we use a different approach, namely solutions are obtained aslarge particle limit of a suitable nonlocal version of the Follow-the-Leader scheme, which is interpretedas the discrete Lagrangian approximation of the target continuity equation. This approach is also usedto show existence in the multiple species opinion formation case (e.g. opinion leaders and followers)and to study the long time behavior both analytically and numerically. This is a joint work with E.Radici.

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Thursday 6th, morning(In)efficiency in mean field games

Pierre Cardaliaguet, Université Paris Dauphine - France

Mean field games (MFG) are dynamic games with infinitely many infinitesimal agents. In thisjoint work with Catherine Rainer (U. Brest), we study the efficiency of Nash MFG equilibria: Namely,we compare the social cost of a MFG equilibrium with the minimal cost a global planner can achieve.We find a structure condition on the game under which there exists efficient MFG equilibria and, incase this condition is not fulfilled, quantify how inefficient MFG equilibria are.

Numerical strategies for efficient control of large–scale particle systems

Michael Herty, RWTH Aachen University - Germany

We are interested in the derivation of optimality conditions for controlled interacting agent systems.We establish the relation between mean field optimality conditions and the optimality condition ofthe mean field control problem. This link is important for many recently published articles on controlstrategies for agent based systems since it establishes the precise relation between multipliers for theindividual agents and the probability density distribution of the multipliers in the mean field limit.The relation to different notions of differentiability are also shown.

Traffic flow models with non-local flux and extensions to networks

Simone Göttlich, University of Mannheim - Germany

We present a Godunov type numerical scheme for a class of scalar conservation laws with non-local flux arising for example in traffic flow modeling. The scheme delivers more accurate solutionsthan the widely used Lax-Friedrichs type scheme and also allows to show well-posedness of the model.In a second step, we consider the extension of the non-local traffic flow model to road networks bydefining appropriate conditions at junctions. Based on the proposed numerical scheme we show someproperties of the approximate solution and provide several numerical examples.

Control problems for crowd motion

Michel Duprez, Laboratoire Jacques Louis Lions, Sorbonne Universités, UPMC, CNRS - France

To describe the evolution of a crowd of agents, two main classes are widely used. In microscopicmodels, the position of each agent is clearly identified ; the crowd dynamics is described by a ordinarydifferential equation of large dimension, in which couplings of terms represent interactions betweenagents. In macroscopic models, instead, the idea is to represent the crowd by the spatial density ofagents; in this setting, the evolution of the density solves a partial differential equation, usually oftransport type. Nonlocal terms (such as convolutions) model interactions between agents.

In this talk, we will study a macroscopic model. Hence, the crowd will be represented by a measuredefined for each positive time on the whole space and in arbitrary dimension. The natural velocityfield (un-controlled) of the measure will be assumed Lipschitz and uniformly bounded. We will acton the velocity field in a fixed part of the space, which will be connected and non-empty. We willthen consider the continuity equation determined by the natural velocity of the crowd, the initialdata (initial configuration of the crowd) and an admissible control (modification of the velocity inthe control region). The system is a first approximations for crowd modeling, since the uncontrolledvector field is given, and it does not describe interactions between agents. Nevertheless, it is necessaryto understand control properties for such simple equations as a first step, before dealing with vector

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fields depending on the crowd itself. Even if the system is linear, the considered control problem isnon-linear which is the main difficulty of the present study.

We highlight in [Duprez, Morancey, Rossi 2018] that it is possible to steer a crowd near a finalgiven configuration thanks to a Lipschitz control when the un-controlled dynamic allows to cross thecontrol region. We prove that the exact controllability can occur only if we search a control withless regularity and, in this case, we can potentially lose the uniqueness of the solution to the system.We also give a characterization of the minimal time in [Duprez, Morancey, Rossi 2019]. The minimaltime to steer one initial configuration to another is related to the condition of having enough mass inthe control region to feed the desired final configuration. The construction of the control is explicit,providing a numerical algorithm for computing it. We will finally give some numerical simulations.

This is a joint work with M. Morancey and F. Rossi.

Thursday 6th, free afternoon

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Friday 7th, morningMin_LQG games and collective discrete choice problems

Roland Malhamé, Polytechnique Montréal - Canada

We introduce a novel class of finite horizon linear quadratic Gaussian games involving distinct po-tential finite destination states, interpreted as discrete choices under social pressure. The model pro-vides stylized interpretations of opinion swings in elections, the dynamics of discrete societal choices, aswell as a framework for achieving communication constrained group decision making in micro?roboticbased exploration.

Two distinct cases are considered: (i) The zero noise or “deterministic” case where agents are ini-tially randomly distributed over their range space; (ii) The fully stochastic case. Under mild technicalconditions, the existence of ε?Nash equilibria is established in both cases although these equilibriamay in general be multiple. The corresponding agent control strategies are of a decentralized natureand are characterized in each case by the fixed points of a specific finite dimensional operator. Indi-vidual agent destination choices are fixed at the outset in case (i), while by contrast, their probabilitydistribution evolves randomly along trajectories in case (ii), with a deterministic limit for the completepopulation as the latter grows to infinity.

This is joint work with Rabih Salhab and Jérôme Le Ny.

Optimal Control in the space of probability measures

Claudia Totzeck, Technische Universität Kaiserslautern - Germany

In this talk we discuss the relation of optimal control of interacting systems on the microscopicand mesoscopic level. As illustration we think of the example dogs herding sheep. In particular, weare interested in the underlying optimal control problem that models the fact of herding sheep to apredefined destination. As the mesoscopic model can be derived from the micro level via a mean-fieldlimit, the question arises if the optimal controls converge in the limit as well. We shall see that theanswer is affirmative and that an appropriate calculus in the space of probability measures helps inanswering this question. All theoretic results are underlined with numerical simulations.

Deterministic particle approximations of nonlinear transport equations

Marco di Francesco, University of L’Aquila - Italy

Abstract: Deterministic follow-the-leader Lagrangian particle schemes are well known to be a veryefficient tool to approximate traffic flow and pedestrian flow PDE models. I will recall a set of recentresults in collaboration with M. D. Rosini (Ferrara), S. Fagioli (L’Aquila), E. Radici (L’Aquila), G.Stivaletta (L’Aquila), and G. Russo (Catania) on the rigorous formulation via many particle limitsin this setting for a relatively wide set of problems including Cauchy problems and Initial-Boundary-Value problems for scalar conservation laws, second order models for traffic flow, the Hughes model forpedestrian movements, nonlocal transport equations with nonlinear mobility and external potentials.These results are based on BV estimates for the approximating piece-wise constant density. A relevantissue is that this scheme often allows to detect the L infinity - BV smoothing effect of a scalarconservation law with convex flux. In all cases, the scheme allows to solve the target PDE problem inthe classical entropy sense.

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Convergence of the follow-the-leader scheme for scalar conservation laws with spacedependent flux

Graziano Stivaletta, University of L’Aquila - Italy

We deal with the derivation of entropy solutions to Cauchy problems for a class of scalar conser-vation laws with space-density depending fluxes from systems of deterministic particles of follow-the-leader type. We consider fluxes which are product of a function of the density v(ρ) and a function ofthe space variable φ(x) which plays the role of an external drift term. These models can be applied inthe context of traffic flow (for example in situations in which the speed of the vehicles is also affectedby external factors, such as temporary road maintenance, or sudden turns or rises), but it has alsoa pretty wide range of potential applications in sedimentation processes, flow of glaciers, formationof Bose-Einstein condensates, etc. We cover four distinct cases in terms of the sign of φ, includingcases in which the latter is not constant. The main result is the convergence, up to a subsequence,of a suitable approximating density almost everywhere and in L1 on R× [0, T ] to the unique entropysolution for an arbitrary fixed time T > 0. The convergence result relies on a local maximum principleand on a uniform BV estimate for the approximating density.

Joint work with M. Di Francesco.

Friday 7th, free afternoon

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CROWDS: models and control

Scientific CommitteePeter E. CainesPaola GoatinSerge HoogendoornNaomi LeonardEmmanuel Trélat

Organizing CommitteeAlessandro GiuaMorgan MoranceyBenedetto PiccoliFrancesco RossiMarie-Therese Wolfram

For urgent matters: F. Rossi +39.338.82.46.909 Phone or WhatsApp

European emergency number: 112

Cover: MUCEM Marseille. Credit photo: Emma Adam.