Section: New Results

Crowd motion modeled by Fokker-Planck constrained Nash games

Participants : Alfio Borzí [Univ. Wurzburg] , Abderrahmane Habbal, Souvik Roy [Univ. Wurzburg] .

Fokker-Planck-Kolmogorov (FPK) equations are PDEs which govern the dynamics of the probability density function (PDF) of continuous-time stochastic processes (e.g. Ito processes). In   [48] a FPK-constrained control framework, where the drift was considered as control variable is developed and applied to crowd motion.

In  [13] a new approach to modelling pedestrians' avoidance dynamics based on a Fokker–Planck (FP) Nash game framework is presented. In this framework, two interacting pedestrians are considered, whose motion variability is modelled through the corresponding probability density functions (PDFs) governed by FP equations. Based on these equations, a Nash differential game is formulated where the game strategies represent controls aiming at avoidance by minimizing appropriate collision cost functionals. The existence of Nash equilibria solutions is proved and characterized as a solution to an optimal control problem that is solved numerically. Results of numerical experiments are presented that successfully compare the computed Nash equilibria to the output of real experiments (conducted with humans) for four test cases.