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

Abstract

This paper studies a mean field game inspired by crowd motion in which agents evolve in a compact domain and want to reach its boundary minimizing the sum of their travel time and a given boundary cost. Interactions between agents occur through their dynamic, which depends on the distribution of all agents.We start by considering the associated optimal control problem, showing that semi-concavity in space of the corresponding value function can be obtained by requiring as time regularity only a lower Lipschitz bound on the dynamics. We also prove differentiability of the value function along optimal trajectories under extra regularity assumptions.We then provide a Lagrangian formulation for our mean field game and use classical techniques to prove existence of equilibria, which are shown to satisfy a MFG system. Our main result, which relies on the semi-concavity of the value function, states 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. This is also used to prove existence of equilibria under fewer regularity assumptions on the dynamics thanks to a limit argument.

On the Asymptotic Nature of First Order Mean Field Games

“…for any s ∈ [1, +∞). As a consequence,ρ satisfies estimate (4.24) and, passing to the limit in (4.19), we get that the measure [26], in order to establish L p -estimates for the time evolving distributions describing equilibria in optimal-exit MFGs. Let us also point out that the absolute continuity of ρ(t), for all t ∈ [0, T ], has already been established in [30,Section 5], under more general assumptions than (A2), but without providing the L p -estimate (4.18).…”

“…The method of our proof is reminiscent of the techniques employed in nonatomic game theory (see [31,39,40]) and could also be useful in justifying the asymptotic nature of more sophisticated deterministic MFGs (see e.g. [26,37] for minimal-time MFGs and Remark 3.1(ii) for the state constrained MFG considered in [10]).…”

On the characterization of equilibria of nonsmooth minimal-time mean field games with state constraints

“…In this paper, we study MFG(K, m 0 ) in a Lagrangian setting, in which the evolution of agents is described by a measure Q ∈ P(C( Ω)) in the space of all continuous trajectories C( Ω). This classical approach in optimal transport has become widely used in the analysis of MFGs with deterministic trajectories in recent years (see, e.g., [13], [15], [21], [25], [26]). Note that the distribution m t of agents at time t ≥ 0 can be retrieved from Q using the evaluation map e t by m t = e t # Q.…”

First order Mean Field Games on networks

“…In the same vein, Mazanti and Santambrogio [23] obtain the existence of MFG equilibria for minimal time MFGs; in their problem, each agent aims to exit a given closed subset of a general compact metric space in minimal time with a bounded speed. (For some generalizations in the Euclidean setting, see also [16]). Moreover, in [3], Achdou et al prove the existence of a relaxed equilibrium in the case of deterministic MFG with control on the acceleration with state constraints; in this case, the strong controllability condition does not hold and the Hamiltonian fails to be convex or coercive.…”