Description of Stability for Linear Time-Invariant Systems Based on the First Curvature

This paper focuses on using the first curvature $\kappa (t)$ of trajectory to describe the stability of linear time-invariant system. We extend the results for two and three-dimensional systems [Y. Wang, H. Sun, Y. Song et al., arXiv:1808.00290] to $n$-dimensional systems. We prove that for a system $\dot{r}(t)=Ar(t)$, (i) if there exists a measurable set whose Lebesgue measure is greater than zero, such that for all initial values in this set, $\underset{t\to +\mathrm{\infty }}{\lim }\kappa (t)\mathrm{\ne }0$ or $\underset{t\to +\mathrm{\infty }}{\lim }\kappa (t)$ does not exist, then the zero solution of the system is stable; (ii) if the matrix $A$ is invertible, and there exists a measurable set whose Lebesgue measure is greater than zero, such that for all initial values in this set, $\underset{t\to +\mathrm{\infty }}{\lim }\kappa (t)=+\mathrm{\infty }$, then the zero solution of the system is asymptotically stable.