论文标题
扭转歧视线性时间流动系统的稳定性
Torsion Discriminance for Stability of Linear Time-Invariant Systems
论文作者
论文摘要
本文提出了一种新方法，以通过状态轨迹的扭转$τ（t）$来描述线性时间不变系统的稳定性。对于系统$ \ dot {r}（t）= ar（t）$，其中$ a $是可逆的，我们表明（1）如果存在一个可测量的套装$ e_1 $带有正面的lebesgue量度，则$ r（0）\ in e_1 $ in e_1 $ in e_1 $ in e_1 $ in e_1 $ in e_1 $ in $ \ lim \ lim \ lim \ lim \ lim \ lim \ to t \ to+++++\ \ \ \ \ fy或$ \ lim \ limits_ {t \ to+\ infty}τ（t）$不存在，那么系统的零解决方案稳定； （2）如果存在一个可测量的集合$ e_2 $，则具有正面的lebesgue度量，因此$ r（0）\ in E_2 $中的\表示$ \ lim \ limits_ {t \ to+\ to+\ infty}τ（t）=+\ infty $，那么系统的零解决方案是稳定的。此外，当$ a $类似于真实的对角线矩阵时，我们建立了轨迹$（i = 1,2，\ cdots）$之间的关系与零解决方案的稳定性之间的关系。
This paper proposes a new approach to describe the stability of linear time-invariant systems via the torsion $τ(t)$ 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 $\lim\limits_{t\to+\infty}τ(t)\neq0$ or $\lim\limits_{t\to+\infty}τ(t)$ 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 $r(0)\in E_2$ implies that $\lim\limits_{t\to+\infty}τ(t)=+\infty$, 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.
