An evolution mean field equation system of initial mean consensus behaviour: A stability analysis

The purpose of this paper is to study an evolution (i.e., forward in time) mean field equation system of a dynamic game initial mean consensus model. In this model: (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 long run average cost function involving a mean field coupling to the states of all other agents. The Evolution Mean Field (EMF) equation system of the continuum (i.e., as the population size N goes to infinity) version of this model consists of two coupled (forward in time) deterministic PDEs which are also coupled to a (spatially averaged) cost coupling function. The stationary equilibrium of the EMF equation system yields a mean-consensus behaviour in the system. The small perturbation stability of the EMF equation system around this stationary equilibrium solution is established. Hence, the EMF equation system provides a forward in time process which asymptotically in time converges to the stationary equilibrium solution from any given initial condition in a infinitesimal neighborhood of that equilibrium.