Robust feedback stabilization by means of Lyapunov-like functions determined by Lie brackets

Abstract

We use Lie brackets of unbounded vector fields to consider a dissipative relation that generalizes the differential inequality which defines classic control Lyapunov functions. Under minimal regularity assumptions, we employ locally semiconcave solutions of this extended relation, called in the following degree-k control Lyapunov functions, in order to design degree-k Lyapunov feedbacks, i.e. particular discontinuous feedback laws that stabilize the underlying system to a given closed target with compact boundary, in the sample and hold sense. We also prove that this feedback construction is robust when small measurement errors and external disturbances occur.