MinMax Mean-Field Team Approach for a Leader-Follower Network: A Saddle-Point Strategy

Abstract: This paper investigates a soft-constrained MinMax control problem of a leader-follower network. The network consists of one leader and an arbitrary number of followers that wish to reach consensus with minimum energy consumption in the presence of external disturbances. The leader and followers are coupled in the dynamics and cost function. Two non-classical information structures are considered: mean-field sharing and intermittent mean-field sharing, where the mean-field refers to the aggregate state of the followers. In mean-field sharing, every follower observes its local state, the state of the leader and the mean field while in the intermittent mean-field sharing, the mean-field is only observed at some (possibly no) time instants. A social welfare cost function is defined, and it is shown that a unique saddle-point strategy exists which minimizes the worst-case value of the cost function under mean-field sharing information structure. The solution is obtained by two scalable Riccati equations, which depend on a prescribed attenuation parameter, serving as a robustness factor. For the intermittent mean-field sharing information structure, an approximate saddle-point strategy is proposed, and its converges to the saddle-point is analyzed. Two numerical examples are provided to demonstrate the efficacy of the obtained results.