Optimal Control for Maximum Instantaneous Convergence in Collective Migration Models

Abstract

This paper studies consensus tracking of collective migration models that involve the alignment force gathering agents and the tracking force matching target. Each agent's dynamics is controlled by the tracking strategy, which establishes a trade-off between the two forces through a convex combination. In order to drive the system to achieve consensus tracking with the maximum instantaneous convergence speed, an optimal control strategy is proposed that the agents whose alignment force is weaker than, or counteracts its tracking force sense only the target, and become leaders, while the others sense only their neighbours, and become followers. Interestingly, there exist some initial frameworks such that the optimal control strategy consists of letting all agents become followers, which is called ``inactivation principle'' in [Piccoli, Duteil and Scharf, Math. Models Methods Appl. Sci., 26(2), 2016] and means that the leaderless is better than the leader-follower structure for certain conditions. Both asymptotic and finite-time consensus tracking are investigated. Several numerical simulations show the effect of the optimal control strategy.