A note on diffusive solutions of the Lyapunov and Riccati inequalities for quasi-monotone (QM) mappings on cones

TL;DR

This paper explores the properties of quasi-monotone mappings on cones, extending classical stability concepts for matrices to more general settings involving diffusive solutions and Jordan algebraic methods.

Contribution

It introduces the concept of diffusive solutions for Lyapunov and Riccati inequalities in the context of quasi-monotone cone mappings, extending stability results beyond traditional matrix classes.

Findings

Extended D-stability to quasi-monotone cone mappings

Established diagonal Lyapunov stability for diffusive solutions

Connected results with Jordan algebraic methods for symmetric cones

Abstract

We consider three key properties of Metzler and nonnegative matrices and extensions of these to classes of self-dual proper convex cones. Specifically, we study mappings that are quasi-monotone (QM) with respect to a cone $K$ and discuss results extending D-stability, diagonal Lyapunov stability, and diagonal Riccati stability to this setting. Mappings that act diffusively with respect to the cone are used as generalisations of diagonal matrices. Relationships with recent results for symmetric cones obtained using Jordan algebraic methods are also discussed.