{"paper":{"title":"Description of Stability for Linear Time-Invariant Systems Based on the First Curvature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.OC","authors_text":"Huafei Sun, Shoudong Huang, Yang Song, Yuxin Wang","submitted_at":"2018-12-15T14:24:30Z","abstract_excerpt":"This paper focuses on using the first curvature $\\kappa(t)$ of trajectory to describe the stability of linear time-invariant system. We extend the results for two and three-dimensional systems [Y. Wang, H. Sun, Y. Song et al., arXiv:1808.00290] to $n$-dimensional systems. We prove that for a system $\\dot{r}(t)=Ar(t)$, (i) if there exists a measurable set whose Lebesgue measure is greater than zero, such that for all initial values in this set, $\\lim\\limits_{t\\to+\\infty}\\kappa(t)\\neq0$ or $\\lim\\limits_{t\\to+\\infty}\\kappa(t)$ does not exist, then the zero solution of the system is stable; (ii) i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.07384","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}