Optimal Distributed Control for Leader-Follower Networks: A Scalable Design

The focus of this paper is directed towards optimal control of multi-agent systems consisting of one leader and a number of followers in the presence of noise. The dynamics of every agent is assumed to be linear, and the performance index is a quadratic function of the states and actions of the leader and followers. The leader and followers are coupled in both dynamics and cost. The state of the leader and the average of the states of all followers (called mean-field) are common information and known to all agents; however, the local state of the followers are private information and unknown to other agents. It is shown that the optimal distributed control strategy is linear time-varying, and its computational complexity is independent of the number of followers. This strategy can be computed in a distributed manner, where the leader needs to solve one Riccati equation to determine its optimal strategy while each follower needs to solve two Riccati equations to obtain its optimal strategy. This result is subsequently extended to the case of the infinite horizon discounted and undiscounted cost functions, where the optimal distributed strategy is shown to be stationary. A numerical example with $100$ followers is provided to demonstrate the efficacy of the results.