Stabilization under measurement noise: Lyapunov characterization

For systems affine in controls, Artstein´s theorem (1983) provides an equivalence, between continuous feedback stabilizability to an equilibrium and the existence of smooth control Lyapunov functions. This is one of the fundamental facts in nonlinear stabilization. The equivalence breaks down for general nonlinear systems, not affine in controls. One of the main results in this paper states that the existence of smooth Lyapunov functions implies the existence of, in general discontinuous, feedback stabilizers which are insensitive (or robust) to small errors in state measurements. Conversely, the existence of such stabilizers in turn implies the existence of smooth control Lyapunov functions. In a more general framework of systems under persistently acting disturbances, the existence of smooth Lyapunov functions turns out to be equivalent to the existence of (in general, discontinuous) feedback stabilizers which are robust with respect to small measurement errors and small additive external disturbances