  We analyze numerically some macroscopic models of pedestrian motion such as
Hughes model [1] and mean field game with nonlinear mobilities [2] modeling
fast exit scenarios in pedestrian crowds. A model introduced by Hughes
consisting of a non-linear conservation law for the density of pedestrians
coupled with an Eikonal equation for a potential modeling the common sense of
the task. Mean field game with nonlinear mobilities is obtained by an optimal
control approach, where the motion of every pedestrian is determined by
minimizing a cost functional, which depends on the position, velocity, exit
time and the overall density of people. We consider a parabolic optimal control
problem of nonlinear mobility in pedestrian dynamics, which leads to a mean
field game structure. We show how optimal control problem related to the Hughes
model for pedestrian motion. Furthermore we provide several numerical results
which relate both models in one and two dimensions.
  References [1] Hughes R.L.: A continuum theory for the flow of pedestrians,
Transportation Research Part B: Methodological, 36, 507-535 (2000) [2] Burger
M., Di Francesco M., Markowich P.A., Wolfram M-T.: Mean field games with
nonlinear mobilities in pedestrian dynamics, Discrete and Continuous Dynamical
Systems. Series B. A Journal Bridging Mathematics and Sciences, 19, 1311-1333
(2014)
