The Yakubovich S-Lemma Revisited: Stability and Contractivity in Non-Euclidean Norms
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| Format: | Preprint |
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2022
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| _version_ | 1866916090327597056 |
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| author | Proskurnikov, Anton V. Davydov, Alexander Bullo, Francesco |
| author_facet | Proskurnikov, Anton V. Davydov, Alexander Bullo, Francesco |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2207_14579 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The Yakubovich S-Lemma Revisited: Stability and Contractivity in Non-Euclidean Norms Proskurnikov, Anton V. Davydov, Alexander Bullo, Francesco Optimization and Control Systems and Control Classical Analysis and ODEs Dynamical Systems 34H05, 93C15 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. |
| title | The Yakubovich S-Lemma Revisited: Stability and Contractivity in Non-Euclidean Norms |
| topic | Optimization and Control Systems and Control Classical Analysis and ODEs Dynamical Systems 34H05, 93C15 |
| url | https://arxiv.org/abs/2207.14579 |