{
  "id": "2010.11569",
  "version": "v1",
  "published": "2020-10-22T09:59:24.000Z",
  "updated": "2020-10-22T09:59:24.000Z",
  "title": "Finite state N-agent and mean field control problems",
  "authors": [
    "Alekos Cecchin"
  ],
  "categories": [
    "math.OC",
    "math.AP",
    "math.PR"
  ],
  "abstract": "We examine mean field control problems (MFCP) on a finite state space, in continuous time and over a finite time horizon. We characterize the value function of the MFCP as the unique viscosity solution of a HJB equation in the simplex. In absence of any convexity assumption, we exploit this characterization to prove convergence, as $N$ grows, of the value functions of the centralized $N$-agent optimal control problem to the limit MFCP value function, with a convergence rate of order $1/\\sqrt{N}$. Then, assuming convexity, we show that the limit HJB admits a smooth solution and establish propagation of chaos, i.e. convergence of the $N$-agent optimal trajectory to the unique limiting optimal trajectory, with an explicit rate.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-22T09:59:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B65",
      "35F21",
      "49L25",
      "49M25",
      "60F15",
      "60J27",
      "91A12"
    ],
    "keywords": [
      "mean field control problems",
      "finite state n-agent",
      "agent optimal control problem",
      "limit mfcp value function",
      "convergence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}