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Torsion Discriminance for Stability of Linear Time-Invariant Systems

School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China
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Mathematics 2020, 8(3), 386; https://doi.org/10.3390/math8030386
Received: 12 February 2020 / Revised: 6 March 2020 / Accepted: 7 March 2020 / Published: 10 March 2020
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 ith curvature ( i = 1 , 2 , ) of the trajectory and the stability of the zero solution when A is similar to a real diagonal matrix. View Full-Text
Keywords: linear systems; stability; asymptotic stability; torsion; curvature linear systems; stability; asymptotic stability; torsion; curvature
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Wang, Y.; Sun, H.; Cao, Y.; Zhang, S. Torsion Discriminance for Stability of Linear Time-Invariant Systems. Mathematics 2020, 8, 386.

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