An Euler-Poincaré Approach to Mean-Field Optimal Control

Mean-field dynamic systems are used to model collective behaviors among multi-agent systems. Different choices of interaction policies among agents lead to understandings of attraction behavior, alignment behavior and so on. Such systems are highly nonlinear, which hinders the further development of control strategies for them. In this paper, a geometric description of the mean-field optimal control problem is considered and the corresponding optimality conditions are derived following the Euler-Poincaré theory for ideal continuum motions. Comparing to Pontryagin maximum principle and Hamilton-Jacobi-Bellman strategies, our approach results in multiplier-free optimality conditions, which reduces computational complexities. To show its effectiveness, we numerically demonstrate a scenario where a multi-agent system splits from one cluster into two clusters.