# A Contractive Approach to Separable Lyapunov Functions for Monotone Systems

## TL;DR

This paper develops methods for constructing separable Lyapunov functions for monotone systems that are also contractive, using weighted norms, and introduces an algorithm for their computation, with practical examples.

## Contribution

It introduces a novel approach to design contractive, separable Lyapunov functions for monotone systems using weighted norms and provides an SOS-based algorithm for their computation.

## Key findings

- Conditions for sum-separable Lyapunov functions using weighted one-norms.
- Conditions for max-separable Lyapunov functions using weighted infinity-norms.
- An SOS-based algorithm for computing these Lyapunov functions.

## Abstract

Monotone systems preserve a partial ordering of states along system trajectories and are often amenable to separable Lyapunov functions that are either the sum or the maximum of a collection of functions of a scalar argument. In this paper, we consider constructing separable Lyapunov functions for monotone systems that are also contractive, that is, the distance between any pair of trajectories exponentially decreases. The distance is defined in terms of a possibly state-dependent norm. When this norm is a weighted one-norm, we obtain conditions which lead to sum-separable Lyapunov functions, and when this norm is a weighted infinity-norm, symmetric conditions lead to max-separable Lyapunov functions. In addition, we consider two classes of Lyapunov functions: 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. In addition, we present an algorithm based on sum-of-squares programming to compute such separable Lyapunov functions. We provide several examples to demonstrate our results.