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The Yakubovich S-Lemma Revisited: Stability and Contractivity in Non-Euclidean Norms

Anton V. Proskurnikov, Alexander Davydov, Francesco Bullo

TL;DR

This work generalizes the S-Lemma to non-Euclidean, norm-based Lyapunov functions by employing matrix log norms and weak pairings, enabling stability and contraction analysis for Lur'e systems with V(x)=||R x||^2_p. The authors derive a non-polynomial S-Lemma that links primal sector/slope constraints to dual log-norm conditions through a family of matrices P(τ)=P0−∑τjPj, providing tractable criteria for absolute stability and contractivity in ℓp norms (including weighted variants). Key contributions include a complete ℓ2 analysis with LMIs and KYP circle-criterion connections, a diagonal-weighted ℓ1/ℓ∞ framework leveraging Metzler structure, and new proofs of the Aizerman and Kalman conjectures for positive Lur'e systems. The results offer a unified, norm-general approach to stability and contraction that complements classical quadratic Lyapunov methods and has potential computational advantages in large-scale or non-Euclidean contexts. The framework extends to MIMO nonlinear blocks and motivates further work on bounded nonlinearities and efficient non-Euclidean norm validation algorithms.

Abstract

The celebrated S-Lemma was originally proposed to ensure the existence of a quadratic Lyapunov function in the Lur'e problem of absolute stability. A quadratic Lyapunov function is, however, nothing else than a squared Euclidean norm on the state space (that is, a norm induced by an inner product). A natural question arises as to whether squared non-Euclidean norms $V(x)=\|x\|^2$ may serve as Lyapunov functions in stability problems. This paper presents a novel non-polynomial S-Lemma that leads to constructive criteria for the existence of such functions defined by weighted $\ell_p$ norms. Our generalized S-Lemma leads to new absolute stability and absolute contractivity criteria for Lur'e-type systems, including, for example, a new simple proof of the Aizerman and Kalman conjectures for positive Lur'e systems.

The Yakubovich S-Lemma Revisited: Stability and Contractivity in Non-Euclidean Norms

TL;DR

This work generalizes the S-Lemma to non-Euclidean, norm-based Lyapunov functions by employing matrix log norms and weak pairings, enabling stability and contraction analysis for Lur'e systems with V(x)=||R x||^2_p. The authors derive a non-polynomial S-Lemma that links primal sector/slope constraints to dual log-norm conditions through a family of matrices P(τ)=P0−∑τjPj, providing tractable criteria for absolute stability and contractivity in ℓp norms (including weighted variants). Key contributions include a complete ℓ2 analysis with LMIs and KYP circle-criterion connections, a diagonal-weighted ℓ1/ℓ∞ framework leveraging Metzler structure, and new proofs of the Aizerman and Kalman conjectures for positive Lur'e systems. The results offer a unified, norm-general approach to stability and contraction that complements classical quadratic Lyapunov methods and has potential computational advantages in large-scale or non-Euclidean contexts. The framework extends to MIMO nonlinear blocks and motivates further work on bounded nonlinearities and efficient non-Euclidean norm validation algorithms.

Abstract

The celebrated S-Lemma was originally proposed to ensure the existence of a quadratic Lyapunov function in the Lur'e problem of absolute stability. A quadratic Lyapunov function is, however, nothing else than a squared Euclidean norm on the state space (that is, a norm induced by an inner product). A natural question arises as to whether squared non-Euclidean norms may serve as Lyapunov functions in stability problems. This paper presents a novel non-polynomial S-Lemma that leads to constructive criteria for the existence of such functions defined by weighted norms. Our generalized S-Lemma leads to new absolute stability and absolute contractivity criteria for Lur'e-type systems, including, for example, a new simple proof of the Aizerman and Kalman conjectures for positive Lur'e systems.
Paper Structure (25 sections, 13 theorems, 71 equations, 1 figure, 1 table)

This paper contains 25 sections, 13 theorems, 71 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Problem description and motivation
  3. Contributions
  4. Structure of the paper
  5. Technical preliminaries
  6. Log norms and weak pairings in normed spaces
  7. Non-polynomial 2-forms associated to a weak pairing
  8. Non-polynomial S-Lemma for general normed spaces
  9. Non-polynomial S-Lemma: a weak duality result
  10. The classic S-Lemma: the case of Euclidean norm $\ell_2$
  11. The S-Lemma for Metzler matrices in $\ell_1$ norm
  12. Some counterexamples
  13. Absolute stability and absolute contractivity of Lur'e systems
  14. Sector and slope constraints
  15. Problem setup
  16. ...and 10 more sections

Key Result

Proposition 2.1

\newlabelprop.M-l10 For every $n\times n$ Metzler matrix $M$, one has

Figures (1)

  • Figure 1: A Lur'e system is the feedback superposition of an LTI system and a nonlinearity

Theorems & Definitions (37)

  • Proposition 2.1
  • Remark 2.2
  • Theorem 3.1: Non-polynomial S-Lemma: Weak duality for Non-Euclidean norms
  • Proof 1
  • Remark 3.2: Conic constraints and more
  • Remark 3.3: Equivalent primal constraints
  • Remark 3.4: No constraints implies no gap
  • Lemma 3.5: The Yakubovich S-Lemma
  • Proof 2
  • Remark 3.6: Inhomogeneous quadratic functions
  • ...and 27 more