Discontinuous stabilizing feedback using partially defined Lyapunov functions

We generalize earlier results on the construction of discontinuous feedback laws from smooth but partially defined control Lyapunov functions. The resulting feedback law is continuous at the origin and smooth except on a hypersurface of codimension 1. We provide a formula for the feedback law which is in a sense “universal”. The new results presented cover situations where trajectories of the closed loop system switch an infinite number of times between regions where smooth control Lyapunov functions exist. The conditions on the system vector fields can be verified without solving the differential equations and are therefore in the spirit of the “direct” methods of Lyapunov. Using a recently developed formula we are also able to guarantee certain bounds on the feedback controls provided that the Lyapunov property can be satisfied using controls values in the unit ball