Sharp semi-concavity in a non-autonomous control problem and estimates in an optimal-exit MFG
arXiv:1812.08714 · doi:10.1007/s00030-019-0612-4
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 density gives rise to an absolutely continuous distribution of agents at all positive times with a uniform bound on its norm. This is also used to prove existence of equilibria under fewer regularity assumptions on the dynamics thanks to a limit argument.
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