Asymptotic stability

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 of $x' = f(x)$, and let $x^* \in U$ be an equilibrium point. The point $x^*$ is asymptotically stable if it is Lyapunov stable 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.