Edge-based strict Lyapunov functions for consensus with connectivity preservation over directed graphs

In this paper we address the edge-agreement problem with preserved connectivity for networks of first and second-order systems under proximity constraints and interconnected over a class of directed graphs. We provide a strict Lyapunov function that leads to establishing uniform asymptotic stability of the consensus manifold with guaranteed connectivity preservation. Furthermore, robustness of the edge-agreement protocol, in the sense of input-to-state stability with respect to external input disturbances, is also demonstrated. These results hold for directed-spanning-tree and directed-cycle topologies, which are notably employed, respectively, in leader–follower and cyclic-pursuit control.