Finite state N-agent and mean field control problems

2020/10/22 by Cecchin, Alekos · 1 citation
#35B65 #35F21 #49L25 #49M25 #60F15 #60J27 #91A12 #Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC) #Probability (math.PR)

paper · doi:10.48550/arxiv.2010.11569

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 Hamilton-Jacobi-Bellman 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/√(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.

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• On Viscosity Solutions of Hamilton-Jacobi Equations in the Wasserstein space and the Vanishing Viscosity Limit

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