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

Samer Dweik (1) , Guilherme Mazanti (1)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay

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.

Mots clés: Mean field games; non-autonomous optimal control; semi-concavity of the value function; $L^p$ estimate; MFG system

Domaines: Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]

Samer Dweik, Guilherme Mazanti. Sharp semi-concavity in a non-autonomous control problem and $L^p$ estimates in an optimal-exit MFG. 2018. ⟨hal-01962755v1⟩