Torsion Discriminance for Stability of Linear Time-Invariant Systems

Yuxin Wang, Huafei Sun, Yueqi Cao and Shiqiang Zhang
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Yuxin Wang: School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China
Huafei Sun: School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China
Yueqi Cao: School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China
Shiqiang Zhang: School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China

Mathematics, 2020, vol. 8, issue 3, 1-22

Abstract: This paper extends the former approaches to describe the stability of n -dimensional linear time-invariant systems via the torsion τ ( t ) of the state trajectory. For a system r ? ( t ) = A r ( t ) where A is invertible, we show that (1) if there exists a measurable set E 1 with positive Lebesgue measure, such that r ( 0 ) ∈ E 1 implies that lim t → + ∞ τ ( t ) ≠ 0 or lim t → + ∞ τ ( t ) does not exist, then the zero solution of the system is stable; (2) if there exists a measurable set E 2 with positive Lebesgue measure, such that r ( 0 ) ∈ E 2 implies that lim t → + ∞ τ ( t ) = + ∞ , then the zero solution of the system is asymptotically stable. Furthermore, we establish a relationship between the i th curvature ( i = 1 , 2 , ? ) of the trajectory and the stability of the zero solution when A is similar to a real diagonal matrix.

Keywords: linear systems; stability; asymptotic stability; torsion; curvature (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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