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

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 $V(x)=\Vert x{\Vert }^2$ 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.

References in corpus (2)

• Contraction Theory for Nonlinear Stability Analysis and Learning-based Control: A Tutorial Overview
• Non-Euclidean Contraction Theory for Robust Nonlinear Stability

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• Perspectives on Contractivity in Control, Optimization, and Learning
• Exponential Stability of Parametric Optimization-Based Controllers via Lur'e Contractivity
• $k$-Contraction in a Generalized Lurie System
• Regular Pairings for Non-quadratic Lyapunov Functions and Contraction Analysis