THE YAKUBOVICH S-LEMMA REVISITED: STABILITY AND CONTRACTIVITY IN NON-EUCLIDEAN NORMS

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) = vertical bar vertical bar x vertical bar vertical bar(2) may serve as Lyapunov functions in stability problems. This paper presents a novel nonpolynomial S-Lemma that leads to constructive criteria for the existence of such functions defined by weighted lp 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.

Automated summary

This paper presents a novel nonpolynomial S-Lemma that provides constructive criteria for the existence of functions defined by weighted lp norms in the absolute stability of Lur'e-type systems. It introduces new absolute stability and absolute contractivity criteria, including a simple proof of the Aizerman and Kalman conjectures for positive Lur'e systems.