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 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.

The Yakubovich S-Lemma Revisited: Stability and Contractivity in Non-Euclidean Norms / Proskurnikov, Anton V.; Davydov, Alexander; Bullo, Francesco. - In: SIAM JOURNAL ON CONTROL AND OPTIMIZATION. - ISSN 0363-0129. - ELETTRONICO. - 61:4(2023), pp. 1955-1978. [10.1137/22M1512600]

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

Proskurnikov, Anton V.;
2023

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 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.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11583/2980093