Synchronization on Riemannian Manifolds: Multiply Connected Implies Multistable

• 成果类型: Article
• 署名作者: Markdahl, Johan
• 署名单位: University of Luxembourg
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2020.3030849
• 发表日期: 2021
• 页码: 4311-4318
• 关键词: Agents and autonomous systems; consensus; network analysis and control; Nonlinear systems; optimization; Synchronization

摘要

This article concerns the evolution of multiagent systems on networks over Riemannian manifolds. The motion of each agent is governed by the gradient descent flow of a disagreement function that is a sum of (squared) distances between pairs of communicating agents. Two metrics are considered: geodesic distances and chordal distances for manifolds that are embedded in an ambient Euclidean space. We show that networks that, roughly speaking, are dominated by a large cycle yield multistable systems if the manifold is multiply connected or contains a closed geodesic that is of locally minimum length in a space of closed curves. This result summarizes previous results on the stability of splay or twist state equilibria of the Kuramoto model on S-1 and its generalization, the quantum sync model on SO(n). It also extends them to the Lohe model on U(n).