Converse Lyapunov theorems for control systems with unbounded controls

TL;DR

This paper extends Lyapunov theorems to control systems with unbounded controls, establishing equivalences between controllability, stabilizability, and Lyapunov functions, including systems with chattering behaviors.

Contribution

It generalizes classical control Lyapunov theorems to systems with unbounded controls and impulsive extensions, covering polynomial and discontinuous controls.

Findings

Proves equivalence between controllability, stabilizability, and Lyapunov functions for unbounded control systems.

Includes systems with chattering behaviors and impulsive extensions.

Extends classical results to a broader class of control systems.

Abstract

In this paper, we extend well-known relationships between global asymptotic controllability, sample stabilizability, and the existence of a control Lyapunov function to a wide class of control systems with unbounded controls, which includes control-polynomial systems. In particular, we consider open loop controls and discontinuous stabilizing feedbacks, which may be unbounded approaching the target, so that the corresponding trajectories may present a chattering behavior. A key point of our results is to prove that global asymptotic controllability, sample stabilizability, and existence of a control Lyapunov function for these systems or for an {\em impulsive extension} of them are equivalent.