Leader-Follower Cucker-Smale Type Flocking Synthesized via Mean Field Stochastic Control Theory

Abstract

In this paper we study a controlled Leader-Follower (L-F) flocking model (where the state of each agent consists of both its position and its controlled velocity) by use of the Mean Field (MF) Stochastic Control framework. We formulate the large population stochastic L-F flocking problem as a dynamic game problem. In this model, the agents have similar dynamics and are coupled via their nonlinear individual cost functions which are based on the uncontrolled Cucker and Smale (C-S) flocking algorithm. The cost of each leader is based on a trade-off between moving its velocity toward a certain reference velocity and a C-S type weighted average of all the leaders’ velocities. Followers react by tracking the C-S type weighted average of the velocities of all the leaders and followers. For this nonlinear dynamic game problem we derive two sets of coupled deterministic equations for both leaders and followers approximating the stochastic model in large population. Subject to the existence of unique solutions to these systems of equations we show that: (i) the set of MF control laws for the leaders possesses an \(\displaystyle\epsilon_N\)-Nash equilibrium property with respect to all other leaders, (ii) the set of MF control laws for the followers is almost surely \(\displaystyle \epsilon_N\)-Nash equilibrium with respect to all the other agents (leaders and followers), and (iii) \(\displaystyle \epsilon_N \rightarrow 0\) as the system’s population size, \(\displaystyle N\), goes to infinity. Furthermore, we analyze the MF system of equations for the leaders and followers with the linear coupling cost functions to retrieve similar MF equation systems in Linear-Quadratic-Gaussian (LQG) dynamic game problems.