---
tag: 05Y1JAM3Y5
kind: definition
name: Asymptotic stability
aliases: [asymptotically-stable-equilibrium]
msc: [34D20]
defines: 0NB689782W
uses: [092S6VBECQ, 0G4X36AE0X, 02903SSA48]
sources:
  - url: https://en.wikipedia.org/wiki/Lyapunov_stability
    locator: Definition for continuous-time systems
---

## Definition

Let $U$ be an open subset of $\mathbb{R}^n$, let $f : U \to \mathbb{R}^n$ be locally Lipschitz. Let $\varphi$ be the [[flow|02903SSA48]] of $x' = f(x)$, and let $x^* \in U$ be an [[equilibrium point|0G4X36AE0X]].
The point $x^*$ is asymptotically stable if it is [[Lyapunov stable|092S6VBECQ]] and there is an $\eta > 0$ with this property. For each $x_0 \in U$ with $\|x_0 - x^*\| < \eta$, $\varphi_t(x_0) \to x^*$ as $t \to \infty$.
If $U = \mathbb{R}^n$ and $\varphi_t(x_0) \to x^*$ for each $x_0 \in \mathbb{R}^n$, and $x^*$ is Lyapunov stable, then $x^*$ is globally asymptotically stable.

## Remarks

For $x' = -x$ on $\mathbb{R}$, the equilibrium $0$ is globally asymptotically stable, since $\varphi_t(x_0) = x_0 e^{-t}$.
The point $x^*$ is exponentially stable if there are $\eta, C, \alpha > 0$ with $\|\varphi_t(x_0) - x^*\| \le C e^{-\alpha t} \|x_0 - x^*\|$ for $\|x_0 - x^*\| < \eta$ and $t \ge 0$.
The attraction condition alone does not imply Lyapunov stability. There are planar examples where all solutions tend to $x^*$, but some first make a large excursion.

## Explanations

### L1 human

An equilibrium is asymptotically stable if nearby solutions stay near it and also approach it in the long run.
A ball at the bottom of a bowl with friction is a typical example.