Sufficient conditions for decentralized navigation functions based controllers using canonical vector fields

A combination of dual Lyapunov analysis and properties of decentralized navigation function based controllers is used to check the stability properties of a certain class of decentralized controllers for navigation and collision avoidance in multi-agent systems. The derived results yield a less conservative condition from previous approaches, which relates to the negativity of the sum of the minimum eigenvalues of the Hessian matrices at the critical points, instead of requiring each of the eigenvalues to be negative itself. This provides an improved characterization of the reachable set of this class of decentralized navigation function based controllers, which is less conservative than the previous results for the same class of controllers.