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Regular Pairings for Non-quadratic Lyapunov Functions and Contraction Analysis

Anton V. Proskurnikov, Francesco Bullo

TL;DR

This work develops a unifying theory of regular pairings (a broad class of semi-inner products) on normed spaces to enable Lyapunov and contraction analysis beyond Euclidean spaces. It proves that the curve-norm derivative formula and Lumer's inequality are equivalent and shows they characterize regular pairings, unifying LG, upper JMT, and WP frameworks. The authors introduce polyhedral max pairings, provide computational tools for polyhedral norms via convex programs, and derive a contraction criterion that extends Demidovich-type results to non-Euclidean norms. Through detailed examples, including a biochemical reaction model, the paper demonstrates both the advantages and limitations of non-Euclidean norms for contraction, and outlines future directions for explicit regular pairings in broader norm families. Overall, regular pairings offer a powerful, flexible toolkit for stability and contraction analysis in complex, non-Euclidean settings.

Abstract

Recent studies on stability and contractivity have highlighted the importance of semi-inner products, which we refer to as pairings, associated with general norms. A pairing is a binary operation that relates the derivative of a curve's norm to the radius-vector of the curve and its tangent. This relationship, known as the curve norm derivative formula, is crucial when using the norm as a Lyapunov function. Another important property of the pairing, used in stability and contraction criteria, is the so-called Lumer inequality, which relates the pairing to the induced logarithmic norm. We prove that the curve norm derivative formula and Lumer's inequality are, in fact, equivalent to each other and to several simpler properties. We then introduce and characterize regular pairings that satisfy all of these properties. Our results unify several independent theories of pairings (semi-inner products) developed in previous work on functional analysis and control theory. Additionally, we introduce the polyhedral max pairing and develop computational tools for polyhedral norms, advancing contraction theory in non-Euclidean spaces.

Regular Pairings for Non-quadratic Lyapunov Functions and Contraction Analysis

TL;DR

This work develops a unifying theory of regular pairings (a broad class of semi-inner products) on normed spaces to enable Lyapunov and contraction analysis beyond Euclidean spaces. It proves that the curve-norm derivative formula and Lumer's inequality are equivalent and shows they characterize regular pairings, unifying LG, upper JMT, and WP frameworks. The authors introduce polyhedral max pairings, provide computational tools for polyhedral norms via convex programs, and derive a contraction criterion that extends Demidovich-type results to non-Euclidean norms. Through detailed examples, including a biochemical reaction model, the paper demonstrates both the advantages and limitations of non-Euclidean norms for contraction, and outlines future directions for explicit regular pairings in broader norm families. Overall, regular pairings offer a powerful, flexible toolkit for stability and contraction analysis in complex, non-Euclidean settings.

Abstract

Recent studies on stability and contractivity have highlighted the importance of semi-inner products, which we refer to as pairings, associated with general norms. A pairing is a binary operation that relates the derivative of a curve's norm to the radius-vector of the curve and its tangent. This relationship, known as the curve norm derivative formula, is crucial when using the norm as a Lyapunov function. Another important property of the pairing, used in stability and contraction criteria, is the so-called Lumer inequality, which relates the pairing to the induced logarithmic norm. We prove that the curve norm derivative formula and Lumer's inequality are, in fact, equivalent to each other and to several simpler properties. We then introduce and characterize regular pairings that satisfy all of these properties. Our results unify several independent theories of pairings (semi-inner products) developed in previous work on functional analysis and control theory. Additionally, we introduce the polyhedral max pairing and develop computational tools for polyhedral norms, advancing contraction theory in non-Euclidean spaces.
Paper Structure (19 sections, 11 theorems, 73 equations, 1 table)

This paper contains 19 sections, 11 theorems, 73 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Regular Pairings and Their Characterization
  4. Technical Definitions: Key Properties of a Pairing
  5. Main result: The Characterization Theorem and Regular Pairings
  6. Upper JMT Pairings as Regular Pairings
  7. Lumer-Giles (LG) Pairings
  8. Regular Pairings Associated to Weighted $\ell_p$ Norms on $\mathbb{R}^n$
  9. Regular Pairings for Polyhedral Norms
  10. Contraction Analysis Based upon Regular Pairings
  11. Contraction Criterion
  12. Example
  13. Conclusion
  14. A Lemma on the Differentiation of Curve Norms
  15. Proof of Theorem \ref{['thm.main']}
  16. ...and 4 more sections

Key Result

Lemma 2.6

For every WP $\left\llbracket{\cdot}, {\cdot}\right\rrbracket$ on $X$, the function $\|x\|:=\sqrt{\left\llbracket{x}, {x}\right\rrbracket}$ is a norm. In this norm, the function $\left\llbracket{\cdot}, {y}\right\rrbracket:X\to\mathbb{R}$ is Lipschitz continuous for every $y\in X$, and its Lipschitz

Theorems & Definitions (38)

  • Definition 2.1: The Induced Operator Norm and Logarithmic Norm
  • Definition 2.2: LG Pairing, or the Lumer-Giles SIP
  • Definition 2.3: JMT Pairings, or the James-Miličić-Tapia SIPs
  • Remark 2.4: Differences Between the LG and JMT Pairings
  • Definition 2.5: Weak Pairing
  • Lemma 2.6: Weak Pairings and Norms
  • Proof 1
  • Definition 3.1: Lumer's Property
  • Definition 3.2: Curve Norm Derivative
  • Definition 3.3: Partial Linearity
  • ...and 28 more