Geometric Second-Order Laplacian Flow for Consensus on Lie Groups

In this work, the second-order Laplacian flow in Euclidean space which is a standard algorithm for consensus of double integrator systems is generalized to the abstract setting of Lie groups. For double integrator systems on a Lie group, by using gradients of Polar Morse functions whose critical points form a discrete subgroup, it is proved that consensus is achieved for almost all initial conditions of the agents whose connectivity is described by a nearest neighbor network. In this general framework, it turns out that the standard Euclidean second-order Laplacian flow and the Kuramoto oscillator are special cases in Euclidean space and the unit circle respectively.