On the role of the eigenprojector of the laplacian matrix for reaching consensus in multiagent second-order systems

The problem of reaching consensus in multiagent second-order systems without a spanning outgoing tree in the dependency digraph is considered. A theorem stating that the asymptotic behavior of the system is uniquely determined by the eigenprojector of the Laplacian matrix of the dependency digraph is proved. The earlier results established by the author and by Ren and Atkins in their papers are further generalized. For the case in which the dependency digraph contains no spanning outgoing tree, a regularization method is proposed.