# Torsion Discriminance for Stability of Linear Time-Invariant Systems

- Authors
- Publication Date
- Mar 10, 2020
- Identifiers
- DOI: 10.3390/math8030386
- OAI: oai:mdpi.com:/2227-7390/8/3/386/
- Source
- MDPI
- Keywords
- Language
- English
- License
- Green
- External links

## Abstract

This paper extends the former approaches to describe the stability of n-dimensional linear time-invariant systems via the torsion &tau / ( 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 ) &isin / E 1 implies that lim t &rarr / + &infin / &tau / ( t ) &ne / 0 or lim t &rarr / + &infin / &tau / ( 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 ) &isin / E 2 implies that lim t &rarr / + &infin / &tau / ( t ) = + &infin / , 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.