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

## TL;DR

This paper introduces a novel stability analysis method for linear time-invariant systems using the first curvature of trajectories, extending previous results to higher dimensions and providing new stability criteria.

## Contribution

It extends the use of trajectory curvature for stability analysis from 2D and 3D systems to n-dimensional systems, offering new criteria based on curvature limits.

## Key findings

- If the curvature limit does not tend to zero, the system is stable.
- If the curvature tends to infinity and the matrix is invertible, the system is asymptotically stable.
- The results generalize previous low-dimensional stability criteria to higher dimensions.

## Abstract

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, $\lim\limits_{t\to+\infty}\kappa(t)\neq0$ or $\lim\limits_{t\to+\infty}\kappa(t)$ 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, $\lim\limits_{t\to+\infty}\kappa(t)=+\infty$, then the zero solution of the system is asymptotically stable.