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Description of stability for linear time‐invariant systems based on the first curvature
Author(s) -
Wang Yuxin,
Sun Huafei,
Huang Shoudong,
Song Yang
Publication year - 2019
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5896
Subject(s) - mathematics , lebesgue measure , invertible matrix , lebesgue integration , null set , zero (linguistics) , measure (data warehouse) , curvature , invariant measure , invariant (physics) , pure mathematics , mathematical analysis , set (abstract data type) , geometry , mathematical physics , computer science , linguistics , philosophy , database , ergodic theory , programming language
This paper focuses on using the first curvature κ ( t ) of trajectory to describe the stability of linear time‐invariant system. We extend the results for two and three‐dimensional systems (Wang, Sun, Song et al, arXiv:1808.00290) to n ‐dimensional systems. We prove that for a systemr ̇ ( t ) = A r ( t ) , (a) if there exists a measurable set whose Lebesgue measure is greater than zero, such thatlim t → + ∞ κ ( t ) ≠ 0 orlim t → + ∞ κ ( t ) does not exist for any initial value in this set, then the zero solution of the system is stable; (b) if the matrix A is invertible, and there exists a measurable set whose Lebesgue measure is greater than zero, such thatlim t → + ∞ κ ( t ) = + ∞ for any initial value in this set, then the zero solution of the system is asymptotically stable.