Torsion Discriminance for Stability of Linear Time-Invariant Systems

This paper proposes a new approach to describe the stability of linear time-invariant systems via the torsion $\tau(t)$ of the state trajectory. For a system $\dot{r}(t)=Ar(t)$ where $A$ is invertible, we show that (1) if there exists a measurable set $E_1$ with positive Lebesgue measure, such that $r(0)\in E_1$ implies that $\lim\limits_{t\to+\infty}\tau(t)\neq0$ or $\lim\limits_{t\to+\infty}\tau(t)$ 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 $r(0)\in E_2$ implies that $\lim\limits_{t\to+\infty}\tau(t)=+\infty$, then the zero solution of the system is asymptotically stable. Furthermore, we establish a relationship between the $i$th curvature $(i=1,2,\cdots)$ of the trajectory and the stability of the zero solution when $A$ is similar to a real diagonal matrix.