An application of Rantzer’s Dual Lyapunov Theorem to Decentralized Navigation

We provide a connection between Rantzer´s dual Lyapunov theorem that appeared in A. Rantzer (2001) with decentralized navigation functions (DNFs). 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 their goals provided that collisions between the team members tend to occur whenever agents are found sufficiently far from their desired destinations. This is derived based on the properties of the critical points of the DNF´s imposed by Rantzer´s Theorem. The result can be used as a new approach to guaranteed local-minima free decentralized control approaches.