Mean Field Mathematical Models for Excursionists Flow

in a Historic City Center

INDAM-GNAMPA Research Project, March 2017 - March 2018







This project deals with the problem of managing the excursionists flow in historic cities. Venice is considered as a case study. There, in high season, thousands of excursionists arrive by train in the morning; spend the day visiting different sites, moving along the narrow alleys of the city; finally, reach again the train station and leave in late afternoon.


We introduce a mean field model to study the possibility of directing the excursionists through different routes to avoid congestion. We cast the problem in two different approaches: the Mean Field Games (MFG) and the Dynamic Traffic Assignment (DTA).


Both approaches consider an optimal control problem with costs depending on the satisfaction of visiting the sites and on the congestions (the mean field) of the chosen path inside the city. Our goal is to study the possible dynamic equilibria.


The excursionists have more than one target to reach. This leads to the fact that a memory effect must be taken into account: different excursionists may occupy the same place at the same instant, but they may have different purposes, depending on which sites they have already visited.


Keywords: Optimal Control, Viscosity Solutions of Hamilton-Jacobi Equations, Transport Equations, Mean Field Games, Hysteresis and Memory Effects, Partial Differential Equations, Dynamic Traffic Assignment, Mathematical Programming, Dynamics on Networks, Pedestrian Flows, Jumping Processes, Hybrid Systems.





Fabio Bagagiolo (coordinator), University of Trento, Italy.

Silvia Faggian, University Ca' Foscari of Venice, Italy.

Rosario Maggistro, Polytechnic of Turin, Italy.

Raffaele Pesenti, University Ca' Foscari of Venice, Italy.




  • Paola Goatin, INRIA Sophia Antipolis, France.

  • Yves Achdou, Paris-Diderot, France.



  • Silvia Faggian and Raffaele Pesenti have visited Trento on July 10-12 2017.
  • Fabio Bagagiolo has visited Raffaele Pesenti in Venice on Monday June 5 2017.
  • Rosario Maggistro has visited the Department of Mathematics of Padova on Monday and Tuesday December 4, 5 2017.
  • Fabio Bagagiolo and Rosario Maggistro are going to visit Raffaele Pesenti and Silvia Faggian in Venice on Thursday and Friday February 1, 2 2018.


  • F. Bagagiolo, R. Pesenti: Non-memoryless pedestrian flow in a crowded environment with target sets, Preprint 2017, to appear in Annals of ISDG (International Society of Dynamics Games), vol. 15.