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

Oliver Mason

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.

A note on diffusive solutions of the Lyapunov and Riccati inequalities for quasi-monotone (QM) mappings on 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 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.
Paper Structure (7 sections, 7 theorems, 17 equations)

This paper contains 7 sections, 7 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Diagonal and D-stability
  5. Riccati stability
  6. Comments on relation to Jordan algebraic results
  7. A negative point about diagonal Riccati stability for switched systems

Key Result

Proposition 2.1

Let $K\subseteq V$ be a proper cone and $A \in L(V)$ be QM with respect to $K$. Then $A$ is stable if and only if there exists some $v \in \textrm{int}(K)$ with $-Av \in \textrm{int}(K)$.

Theorems & Definitions (10)

  • Proposition 2.1
  • Remark 3.1
  • Remark 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Proposition 3.6
  • Proposition 3.7
  • Proposition 3.8
  • Example 3.9