Description of Stability for Two and Three-Dimensional Linear Time-Invariant Systems Based on Curvature and Torsion

This paper focuses on using curvature and torsion to describe the stability of linear time-invariant system. We prove that for a two-dimensional system $\dot{r}(t)=Ar(t)$, (i) if there exists an initial value, such that zero is not the limit of curvature of trajectory as $t\to +\mathrm{\infty }$, then the zero solution of the system is stable; (ii) if there exists an initial value, such that the limit of curvature of trajectory is infinity as $t\to +\mathrm{\infty }$, then the zero solution of the system is asymptotically stable. For a three-dimensional system, (i) if there exists a measurable set whose Lebesgue measure is greater than zero, such that for all initial values in this set, zero is not the limit of curvature of trajectory as $t\to +\mathrm{\infty }$, then the zero solution of the system is stable; (ii) if the coefficient matrix is invertible, and there exists a measurable set whose Lebesgue measure is greater than zero, such that for all initial values in this set, the limit of curvature of trajectory is infinity as $t\to +\mathrm{\infty }$, then the zero solution of the system is asymptotically stable; (iii) if there exists a measurable set whose Lebesgue measure is greater than zero, such that for all initial values in this set, zero is not the limit of torsion of trajectory as $t\to +\mathrm{\infty }$, then the zero solution of the system is asymptotically stable.