Minimal-time mean field games

Abstract

This paper considers a mean field game model inspired by crowd motion where agents want to leave a given bounded domain through a part of its boundary in minimal time. Each agent is free to move in any direction, but their maximal speed is bounded in terms of the average density of agents around their position in order to take into account congestion phenomena.After a preliminary study of the corresponding minimal-time optimal control problem, we formulate the mean field game in a Lagrangian setting and prove existence of Lagrangian equilibria using a fixed point strategy. We provide a further study of equilibria under the assumption that agents may leave the domain through the whole boundary, in which case equilibria are described through a system of a continuity equation on the distribution of agents coupled with a Hamilton-Jacobi equation on the value function of the optimal control problem solved by each agent. This is possible thanks to the semiconcavity of the value function, which follows from some further regularity properties of optimal trajectories obtained through Pontryagin Maximum Principle. Simulations illustrate the behavior of equilibria in some particular situations.2010 Mathematics Subject Classification. 91A13, 49N70, 49K15, 35Q91.

Sharp semi-concavity in a non-autonomous control problem and $$\pmb {L^p}$$ estimates in an optimal-exit MFG

“…The proof of Proposition 3.14 is omitted here since it can be obtained using analogous arguments to [60,Lemma 4.13 and Corollary 4.14], which provide similar results for the minimal-time problem (3.5).…”

“…We can now state our result on the existence of equilibria. The proof of Theorem 4.4 is based on the same fixed-point strategy used in [15,60]. Notice that the above theorem is slightly stronger than [60, Theorem 5.1] since existence of equilibria is obtained under weaker assumptions.…”

“…The distribution of players at time t ≥ 0 is described by a Borel probability measure ρ t ∈ P(Ω) and, as in [60], we assume that the interaction between players occur through their dynamics, the trajectory γ : [0, +∞) → Ω of a given player being described by the control system γ ′ (t) = k(ρ t , γ(t))u(t), where the control u : [0, +∞) → R d satisfies |u(t)| ≤ 1 for every t ≥ 0 and the function k : P(Ω) × Ω → [0, +∞) describes the maximal speed k(µ, x) an agent may have when their position is x and agents are distributed according to µ. A player chooses their control u in order to minimize the sum of their travel time to ∂Ω with a boundary cost g(z) on their arrival position z ∈ ∂Ω, which is a generalization of the time-minimization criterion of [60].…”

“…Contrarily to the classical approach for mean field games consisting on describing equilibria in terms of ρ t , we adopt here a Lagrangian approach, which amounts to describing the motion of agents as a measure on the set of all possible trajectories. This classical approach in optimal transport [7,64,65] has been used in some recent works on mean field games [9,15,20,24,60].…”

“…After this preliminary study of the optimal control problem, we turn to the analysis of the mean field game itself. We start by proving existence of equilibria in a Lagrangian setting and obtaining the corresponding MFG system of PDEs on ρ t and the value function ϕ using arguments similar to [60]. We then prove that an absolutely continuous initial distribution of agents with an L p density gives rise to an absolutely continuous distribution of agents at all positive times with a uniform bound on its L p norm (Theorem 4.14) and use this result and a limit procedure to obtain existence of equilibria and the corresponding MFG system for a less regular model to which the arguments of [60] do not apply (Theorem 4.18).…”