Finite state Nagent and mean field control problems
Abstract
We examine mean field control problems on a finite state space, in continuous time and over a finite time horizon. We characterize the value function of the mean field control problem as the unique viscosity solution of a HamiltonJacobiBellman 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 mean field control problem value function, with a convergence rate of order $1/\sqrt{N}$. Then, assuming convexity, we show that the limit value function is smooth and establish propagation of chaos, i.e. convergence of the $N$agent optimal trajectory to the unique limiting optimal trajectory, with an explicit rate.
 Publication:

arXiv eprints
 Pub Date:
 October 2020
 DOI:
 10.48550/arXiv.2010.11569
 arXiv:
 arXiv:2010.11569
 Bibcode:
 2020arXiv201011569C
 Keywords:

 Mathematics  Optimization and Control;
 Mathematics  Analysis of PDEs;
 Mathematics  Probability;
 35B65;
 35F21;
 49L25;
 49M25;
 60F15;
 60J27;
 91A12