Lyapunov-like functions involving Lie brackets

TL;DR

This paper introduces degree-k control Lyapunov functions involving Lie brackets, extending classical concepts to broader systems and demonstrating their existence where standard functions do not, under weak regularity assumptions.

Contribution

It develops a generalized framework for control Lyapunov functions using Lie brackets, allowing for existence proofs in cases where traditional functions are absent.

Findings

Existence of smooth degree-k control Lyapunov functions where degree-1 functions do not exist.

Extension of the theory to systems with weak regularity assumptions.

Demonstration of global asymptotic controllability using the new functions.

Abstract

For a given closed target we embed the dissipative relation that defines a control Lyapunov function in a more general differential inequality involving Hamiltonians built from iterated Lie brackets. The solutions of the resulting extended relation, here called degree-k control Lyapunov functions (k>=1), turn out to be still sufficient for the system to be globally asymptotically controllable to the target. Furthermore, we work out some examples where no standard (i.e., degree-1) smooth control Lyapunov functions exist while a smooth degree-k control Lyapunov function does exist, for some k>1. The extension is performed under very weak regularity assumptions on the system, to the point that, for instance, (set valued) Lie brackets of locally Lipschitz vector fields are considered as well.