Synthesis of Cucker-Smale type flocking via Mean Field stochastic control theory: Nash equilibria

In this paper we study a controlled flocking model, where the state of each agent consists of both its position and its controlled velocity, by use of Mean Field (MF) stochastic control framework. We formulate large population stochastic flocking problem as a dynamic game problem in which the agents have similar dynamics and are coupled via their nonlinear individual cost functions. These cost functions are based on the Cucker-Smale (C-S) flocking algorithm in its original uncontrolled formulation. For this nonlinear dynamic game problem we derive a set of coupled deterministic equations approximating the stochastic system of agents as the population goes to infinity. Subject to the existence of a unique solution to this system of equations, the set of MF control laws for the system possesses an ε_N-Nash equilibrium property, where ε_N → 0 as the population size, N, goes to infinity. Hence, this model may be regarded as a controlled game theoretic formulation of the C-S flocking model in which each agent, instead of responding to an ad-hoc algorithm, obtains its control law from a game theoretic Nash equilibrium. Moreover, we retrieve the MF Linear-Quadratic-Gaussian (LQG) dynamic game solution for the C-S algorithm from the general nonlinear MF system of equations.