Consensus on complete Riemannian manifolds in finite time

In view of the fact that prototypes of Riemannian manifolds range from Euclidean surfaces to Lie groups, operator spaces, etc., the paper explores common consensus schemes for connected complete Riemannian manifolds. After studying the restriction exponential map, we introduce a local consensus protocol with switching communication topologies for the manifolds via the isometric mapping, based on one of the Euclidean consensus protocols. Equipped with the compression-decompression functions, we extend the local consensus protocol, and put forth a global consensus algorithm for any such manifolds. These consensus schemes can provide common solutions to the consensus problems arising from various nonlinear spaces or abstract spaces belonging to connected complete Riemannian manifolds. In particular, they are simple and concise, and can converge in finite time. Simulation on the unit sphere is provided to verify our consensus schemes, and to demonstrate the usage of our proposed techniques. © 2012.Sheng Chen, Peng Shi, Weigong Zhang, Lindu Zha