Pedestrian motion modelled by Fokker–Planck Nash games

Abstract

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.

On the SQH Method for Solving Differential Nash Games

“…However, the applicability of the proposed method seems not restricted to these models and appears to be a promising technique for solving infinite-dimensional differential Nash game problems that have been considered recently in different fields of applied mathematics; see, e.g., [10,28,31].…”

The Pontryagin maximum principle for solving Fokker–Planck optimal control problems

“…The Pontryagin maximum principle for solving Fokker-Planck… for all v ∈ V ad and w ∈ W ad , we analogously have that for almost all t 0 ∈ (0, T). Now, we have that the last line in (35) We remark that the (unusual) integral form of the PMP given in (34) results from the fact that the controls depend only on time variable, and so the needle variation. This is in contrast to the case where the control depends on both variables (x, t), see, e.g., [13,14,30] and references therein, in which case the needle variation is defined in Q.…”

“…In application, the model (2) is of central importance in statistical physics, e.g., in the study of Brownian processes, and in the study of biological systems [5]. Recently, it has attracted attention in the framework of modelling pedestrians' motion; see, e.g., [33][34][35] and references therein. Notice that in all these cases, a constant dispersion coefficient is usually considered.…”

“…obtain from the Average Value Theorem [29, Theorem 51] the following by taking the limit over k on both sides of the inequality(35) for all v ∈ K V and w ∈ K W and for almost all t ∈ (0, T) , renaming t 0 into t.…”

An anisotropic interaction model with collision avoidance

“…The applied mathematics community as well as engineers developed a variety of models that focus on different features, e.g. collision avoidance by defining side-stepping procedures [33], by minimizing appropriate cost functionals [4,52] or by computing collision times to anticipate a future collision and adjust the trajectories of the pedestrians involved accordingly [4,23,25].…”