A solution to the initial mean consensus problem via a continuum based Mean Field control approach

This paper presents a continuum approach to the initial mean consensus problem via Mean Field (MF) stochastic control theory. In this problem formulation: (i) each agent has simple stochastic dynamics with inputs directly controlling its state´s rate of change, and (ii) each agent seeks to minimize its individual cost function involving a mean field coupling to the states of all other agents. For this dynamic game problem, a set of coupled deterministic (Hamilton-Jacobi-Bellman and Fokker-Planck-Kolmogorov) equations is derived approximating the stochastic system of agents in the continuum (i.e., as the population size N goes to infinity). In a finite population system (analogous to the individual based approach): (i) the resulting MF control strategies possess an ε_N-Nash equilibrium property where ε_N goes to zero as the population size N approaches infinity, and (ii) these MF control strategies steer each individual´s state toward the initial state population mean which is reached asymptotically as time goes to infinity. Hence, the system with decentralized MF control strategies reaches mean-consensus on the initial state population mean asymptotically (as time goes to infinity).