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

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.