Observations on robust diffusive stability and common Lyapunov functions

TL;DR

This paper investigates robust diffusive stability in coupled positive LTI systems, establishing conditions for stability via common and joint Lyapunov functions, and applies findings to population dynamics models.

Contribution

It introduces new sufficient conditions for robust diffusive stability using common and joint Lyapunov functions, extending previous results and applying them to Leslie matrices.

Findings

Common diagonal Lyapunov functions ensure RDS.

Joint linear copositive functions also suffice for RDS.

Results applied to population dynamics models with Leslie matrices.

Abstract

We consider the problem of robust diffusive stability (RDS) for a pair of coupled stable discrete-time positive linear-time invariant (LTI) systems. We first show that the existence of a common diagonal Lyapunov function is sufficient for RDS and highlight how this condition differs from recent results using linear copositive Lyapunov functions. We also present an extension of these results, showing that the weaker condition of \emph{joint} linear copositive function existence is also sufficient for RDS. Finally, we present two results on RDS for extended Leslie matrices arising in population dynamics.