An application of Rantzer's dual Lyapunov Theorem to decentralized formation stabilization

A connection between Rantzer's dual Lyapunov Theorem that appeared in [25] with decentralized formation stabilization of multiple single integrator kinematic agents is presented. A similar result for decentralized navigation to non-cooperative equilibria was recently provided by the authors in [7]. It is shown that when the agents' control law does not contain an element that forces them to cooperate with the rest of the team once they have reached their desired goal, global convergence cannot be guaranteed. A sufficient condition for this to happen is derived based on Rantzer's Theorem. In particular, it is shown that agents are driven towards the desired formation structure provided that collisions between the team members tend to occur whenever the formation potential of each agent is sufficiently large. This is derived based on the properties of the critical points of the proposed decentralized potential field-based control laws imposed by Rantzer's Theorem. The result can be used as a new approach to guaranteed local-minima free decentralized control approaches.