Lower Bounded Control-Lyapunov Functions

Abstract

The well known Brockett condition - a topological obstruction to the existence of smooth stabilizing feedback laws - has engendered a large body of work on discontinuous feedback stabilization. The purpose of this paper is to introduce a class of control-Lyapunov function from which it is possible to specify a (possibly discontinuous) stabilizing feedback law. For control-affine systems with unbounded controls Sontag has described a Lyapunov pair which gives rise to an explicit stabilizing feedback law smooth away from the origin - Sontag’s “universal construction” of Artstein’s Theorem. In this work we introduce the more general “lower bounded control-Lyapunov function” and a “universal formula” for nonaffine systems. Our “universal formula” is a static state feedback which is measurable and locally bounded but possibly discontinuous. Thus, for the corresponding closed loop system, the classical notion of solution need not apply. To deal with this situation we use the generalized solution due to Filippov.