---
tag: 0J6A752DTZ
kind: statement
role: theorem
name: Lyapunov's stability theorem
aliases: [lyapunov-direct-method]
msc: [34D20]
uses: [04P7B5XC22, 092S6VBECQ, 05Y1JAM3Y5]
sources:
  - url: https://en.wikipedia.org/wiki/Lyapunov_function
    locator: Basic Lyapunov theorems for autonomous systems
---

## Statement

Let $U$ be an open subset of $\mathbb{R}^n$, let $f : U \to \mathbb{R}^n$ be locally Lipschitz, and let $x^* \in U$ be an equilibrium point of $x' = f(x)$.
Let $W \subseteq U$ be an open neighbourhood of $x^*$ and let $V : W \to \mathbb{R}$ be a [[Lyapunov function|04P7B5XC22]] for $x^*$.
(1) Then $x^*$ is [[Lyapunov stable|092S6VBECQ]].
(2) If $V$ is a strict Lyapunov function, then $x^*$ is [[asymptotically stable|05Y1JAM3Y5]].
(3) If $W = U = \mathbb{R}^n$, $V$ is strict, and $V(x) \to \infty$ as $\|x\| \to \infty$, then $x^*$ is globally asymptotically stable.

## Remarks

Proof idea for (1): choose $\varepsilon$ so small that the sphere $S$ of radius $\varepsilon$ around $x^*$ lies in $W$. Let $m > 0$ be the minimum of $V$ on $S$. A solution that starts where $V < m$ keeps $V < m$, so it cannot reach $S$.
For the [[damped harmonic oscillator|0RNXVVTJTJ]] $x' = y$, $y' = -x - c y$ with $c > 0$, the energy $V = \frac{1}{2}(x^2 + y^2)$ has $\dot{V} = -c y^2 \le 0$. This gives stability, and [[LaSalle's invariance principle|0659J5X7B5]] gives asymptotic stability.
The method needs no solution formula. The difficulty is to find a suitable $V$.

## History

The method is named after Aleksandr Lyapunov, who introduced it in his study of the stability of motion.

## Explanations

### L1 human

If some energy-like quantity decreases along all motions near an equilibrium, the motions stay near it.
If the quantity decreases strictly, the motions must also run down to the equilibrium.