## Section: New Results

### Crowd motion modeled by Fokker-Planck constrained Nash games

Participants : Alfio Borzí [Univ. Würzburg] , Paola Goatin, Abderrahmane Habbal, Souvik Roy [Indian Statistical Institute, Kolkata] .

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 [36] a FPK-constrained control framework, where the drift was considered as control variable is developed and applied to crowd motion.

We consider in [42] the extension of the latter framework to the case where two crowds (or pedestrian teams) are competing through a Nash game. The players strategies are the drifts, which yield two uncoupled FPK equations for the corresponding PDFs. The interaction is done through cost functions : each player would prefer to avoid overcrowding (w.r.t. the other one, hence the coupling) additionally to have her own preferred trajectory and obstacle avoidance. In this particular setting, we prove the existence and uniqueness of the Nash equilibrium (NE). The NE is computed by means of a fixed point algorithm and adjoint-state method is used to compute the pseudo-gradients. We finally present some numerical experiments to illustrate which dynamics may arise from such equilibria.