Separability of Lyapunov Functions for Contractive Monotone Systems

TL;DR

This paper develops methods for constructing separable Lyapunov functions for monotone and contractive systems, facilitating stability analysis especially when state measurement is challenging.

Contribution

It introduces two classes of separable Lyapunov functions for monotone, contractive systems, including those based on system velocity, with practical examples.

Findings

Separable Lyapunov functions exist for these systems.

Separable functions can be based on state or velocity.

Examples demonstrate the applicability of the methods.

Abstract

We consider constructing Lyapunov functions for systems that are both monotone and contractive with respect to a weighted one norm or infinity norm. This class of systems admits separable Lyapunov functions that are either the sum or the maximum of a collection of functions of a single argument. In either case, two classes of separable Lyapunov functions exist: the first class is separable along the system's state, and the second class is separable along components of the system's vector field. The latter case is advantageous for many practically motivated systems for which it is difficult to measure the system's state but easier to measure the system's velocity or rate of change. We provide several examples to demonstrate our results.