Sharp semi-concavity in a non-autonomous control problem and Lp estimates in an optimal-exit MFG
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...
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Published in | Nonlinear differential equations and applications Vol. 27; no. 2 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
Cham
Springer International Publishing
2020
Springer Nature B.V |
Subjects | |
Online Access | Get full text |
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Summary: | 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. |
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ISSN: | 1021-9722 1420-9004 |
DOI: | 10.1007/s00030-019-0612-4 |