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.

💡 Research Summary

The paper investigates three classical stability properties—D‑stability, diagonal Lyapunov stability, and diagonal Riccati stability—originally known for Metzler and non‑negative matrices, and extends them to the setting of quasi‑monotone (QM) linear mappings defined on a finite‑dimensional Euclidean space equipped with a proper, self‑dual cone K. A linear operator A is called QM with respect to K if ⟨x,y⟩=0 for x,y∈K implies ⟨x,Ay⟩≥0. In the special case V=ℝⁿ, K=ℝⁿ₊, this condition coincides with A being Metzler. The authors first recall a known characterisation of stability for QM operators: A is stable (all eigenvalues have negative real parts) if and only if there exists a vector v≫0 (i.e., v∈int K) such that Av≪0.

The central novelty is the introduction of “diffusive” linear mappings D. A map D is diffusive with respect to K if it preserves the cone (D(K)⊆K) and, whenever ⟨x,y⟩=0 with x,y∈K, one also has ⟨x,Dy⟩=0. This property generalises diagonal positive matrices: on the non‑negative orthant, diffusive maps are exactly the diagonal matrices with non‑negative entries. The authors give concrete constructions for polyhedral cones and for cones obtained by orthogonal transformations of the orthant, showing that the class of diffusive maps is non‑empty and rich enough for the subsequent analysis.

Lemmas 3.3–3.5 establish that the QM property is preserved under adjoints, under multiplication by diffusive maps, and that an invertible diffusive map sends interior points of K to interior points. Using these facts, Proposition 3.6 proves a cone‑based version of D‑stability: if A is a stable QM operator and E is an invertible diffusive map, then EA remains QM and stable. This mirrors the classical result that multiplying a stable Metzler matrix by a positive diagonal matrix preserves stability.

To obtain diagonal Lyapunov solutions, the authors impose an additional structural condition on the cone, called Assumption D: for any two interior points v,w∈int K there exists a self‑adjoint diffusive map D with Dv=w. This assumption holds, for example, for cones that are orthogonal images of the non‑negative orthant. Under Assumption D, Proposition 3.7 shows that for a stable QM operator A there exists a symmetric positive‑definite diffusive map D such that AᵀD+DA≺0. In other words, a diagonal Lyapunov matrix (in the generalized sense of a diffusive map) always exists for stable QM systems on such cones.

The Riccati stability problem, originally posed by Verri­est, asks for the existence of matrices D≻0 and Q≻0 satisfying AᵀD+DA+Q BᵀD D B−Q≺0 when A is QM, B is K‑non‑negative, and A+B is stable. Proposition 3.8 extends the known diagonal Riccati result for Metzler‑B systems to the cone setting. The proof constructs D from Proposition 3.7 applied to A+B, then uses Assumption D to find a suitable Q, and finally builds a block operator M on V⊕V. By verifying that M is QM with respect to the product cone K×K and that −M(v,v) lies in the interior, the authors conclude M≺0, which yields the desired Riccati inequality.

The final subsection discusses the relationship with earlier work that employed Euclidean Jordan algebras and quadratic representations Pₐ. In symmetric cones, Pₐ plays the role of a diagonal matrix and previous results guarantee the existence of a∈int K such that AᵀPₐ+PₐA≺0, and analogous Riccati statements. However, for general self‑dual cones the quadratic representation does not satisfy the diffusive property, so the present approach based on diffusive maps provides a genuinely broader framework.

Overall, the paper successfully generalises three fundamental stability concepts from the classical non‑negative orthant to arbitrary proper self‑dual cones by introducing diffusive linear operators as a natural analogue of diagonal matrices. The results unify and extend earlier matrix‑theoretic findings, and they open the way for stability analysis of cone‑preserving dynamical systems beyond the realm of symmetric cones and Jordan algebras.