Mean Field Analysis of Controlled Cucker-Smale Type Flocking: Linear Analysis and Perturbation Equations

Abstract This paper presents a linear analysis and perturbation equations in the synthesis of Cucker-Smale (C-S) type flocking via Mean Field (MF) stochastic control theory. In this model the state of each individual agent consists of both its position and its controlled velocity and all agents have similar stochastic dynamics. The agents are coupled via their nonlinear individual cost functions which are based on the C-S flocking algorithm in its original uncontrolled formulation. The MF system of equations approximates this stochastic system of individual agents as the population size goes to infinity. The key result is that C-S flocking behaviour may be obtained as a Nash dynamic competitive game equilibrium. After reviewing the case with linear cost coupling, we present the perturbation (i.e., linearized) equations of the nonlinear MF system of equations around the Gaussian solution of the linear cost coupling case.