{
  "id": "2004.06505",
  "version": "v1",
  "published": "2020-04-14T13:48:10.000Z",
  "updated": "2020-04-14T13:48:10.000Z",
  "title": "Weak KAM approach to first-order Mean Field Games with state constraints",
  "authors": [
    "Piermarco Cannarsa",
    "Wei Cheng",
    "Cristian Mendico",
    "Kaizhi Wang"
  ],
  "categories": [
    "math.AP",
    "math.OC"
  ],
  "abstract": "We study the asymptotic behavior of solutions to the constrained MFG system as the time horizon $T$ goes to infinity. For this purpose, we analyze first Hamilton-Jacobi equations with state constraints from the viewpoint of weak KAM theory, constructing a Mather measure for the associated variational problem. Using these results, we show that a solution to the constrained ergodic mean field games system exists and the ergodic constant is unique. Finally, we prove that any solution of the first-order constrained MFG problem on $[0,T]$ converges to the solution of the ergodic system as $T \\to +\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-14T13:48:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35D40",
      "35F21",
      "49J45",
      "49J53",
      "49L25"
    ],
    "keywords": [
      "first-order mean field games",
      "weak kam approach",
      "state constraints",
      "ergodic mean field games system"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}