Optimal Control of the Mean Field Equilibrium for a Pedestrian Tourists’ Flow Model

Abstract

Art heritage cities are popular tourist destinations but for many of them overcrowding is becoming an issue. In this paper, we address the problem of modeling and analytically studying the flow of tourists along the narrow alleys of the historic center of a heritage city. We initially present a mean field game model, where both continuous and switching decisional variables are introduced to respectively describe the position of a tourist and the point of interest that it may visit. We prove the existence of a mean field game equilibrium. A mean field game equilibrium is Nash-type equilibrium in the case of infinitely many players. Then, we study an optimization problem for an external controller who aims to induce a suitable mean field game equilibrium.Keywords Tourist flow optimal control · mean field games · switching variables · dynamics on networks Mathematics Subject Classification (2010) 91A13 · 49L20 · 90B20 · 91A80

Origin-to-destination network flow with path preferences and velocity controls: A mean field game-like approach

“…Hereinafter, the flows (12) are sometimes called "estimated flows". Of course, a more precise formulation of them should consider the actual value of the control (and not only its sign) and estimate the real traverse time (something similar in this direction is made in [3]). Similarly, the mass ρ that satisfies (10) may be more precisely defined in order to represent the real dynamics of the agents.…”

“…This fact implies the possible discontinuity of the Hamiltonian associated to the value function and/or of the boundary data. Although the discontinuous HJB equations have been studied since the eighties (see e.g., [6,24]), in this paper instead of considering such equations, we will write, as in [3], optimality conditions in terms of the value functions for the exit-time/exit cost problem on each edge. The value functions (15) do not depend on the position θ e of the agents on the edge e ∈ p, because, as we are going to show, the optimal behavior of the agents is, for any traversed edge, to implement a constant control u e p ≥ 0 chosen when they enter the edges.…”

“…In [5], [3] a mean field game approach is implemented for studying the optimal behavior of agents flowing on a network having more than one target (vertices of the networks) to be reached (visited). In [4] an origin-destination model with path preferences dynamics as the one here presented is preliminary treated.…”

A mean field approach to model flows of agents with path preferences over a network

“…Fixed the noisy parameter β > 0, we then search for a fixed point of the multi-function ψ : X → X, with ρ → ρ ∈ ψ(ρ) where ρ is obtained as follows: (i) ρ is inserted in (16)-(22), the optimal control u is derived; (ii) u is inserted in (8) and the path preference vector z is obtained; (iii) ρ is derived from (12) after the computation of (11) and (14):…”

“…Macroscopic models, in contrast, focus on the overall behavior of pedestrian flows and are more applicable to investigations of extremely large crowds, especially when examining aspects of motion in which individual differences are less important [4]- [6]. In this paper, starting from the results in [7]- [8], we introduce a Mean Field (MF) approach to modeling and analytically studying the flow of daily agents along the street of a heritage city. Mean field games (MFG) theory goes back to the seminal work by Lasry-Lions [9] (see also [10]).…”