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2026
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| Online Access: | https://doi.org/10.5281/zenodo.19487533 |
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| _version_ | 1866902330111164416 |
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| author | Kaupp, Alexander |
| author_facet | Kaupp, Alexander |
| contents | <p>Abstract</p> <p>We study stability of linear dynamical systems generated by non-normal operators. Classical</p> <p>spectral criteria fail to predict transient amplification that may exceed operational limits. We</p> <p>introduce the Kaupp number</p> <p>K = \frac{\|X_0\| G_{\max}}{S}, \quad G_{\max} = \sup_{t \ge 0} \|e^{At}\|,</p> <p>where S denotes system capacity. We establish a resolvent-based lower bound</p> <p>G_{\max} \ge \sup_{\operatorname{Re} z > 0} \operatorname{Re}(z)\,\|(zI - A)^{-1}\|,</p> <p>and derive a frequency-domain formulation of the instability condition. A system fails when</p> <p>K \ge 1, even if all eigenvalues lie in the left half-plane. Constructive worst-case</p> <p>perturbations are obtained via singular value decomposition. Numerical examples</p> <p>demonstrate a fundamental gap between spectral stability and bounded response.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19487533 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Resolvent-Based Stability Criterion for Non-Normal Systems via the Kaupp Number Kaupp, Alexander <p>Abstract</p> <p>We study stability of linear dynamical systems generated by non-normal operators. Classical</p> <p>spectral criteria fail to predict transient amplification that may exceed operational limits. We</p> <p>introduce the Kaupp number</p> <p>K = \frac{\|X_0\| G_{\max}}{S}, \quad G_{\max} = \sup_{t \ge 0} \|e^{At}\|,</p> <p>where S denotes system capacity. We establish a resolvent-based lower bound</p> <p>G_{\max} \ge \sup_{\operatorname{Re} z > 0} \operatorname{Re}(z)\,\|(zI - A)^{-1}\|,</p> <p>and derive a frequency-domain formulation of the instability condition. A system fails when</p> <p>K \ge 1, even if all eigenvalues lie in the left half-plane. Constructive worst-case</p> <p>perturbations are obtained via singular value decomposition. Numerical examples</p> <p>demonstrate a fundamental gap between spectral stability and bounded response.</p> |
| title | Resolvent-Based Stability Criterion for Non-Normal Systems via the Kaupp Number |
| url | https://doi.org/10.5281/zenodo.19487533 |