---
tag: 0659J5X7B5
kind: statement
role: theorem
name: LaSalle's invariance principle
aliases: [invariance-principle, krasovskii-lasalle-principle]
msc: [34D20, 34D23]
uses: [0J6A752DTZ, 04P7B5XC22, 05Y1JAM3Y5]
sources:
  - url: https://en.wikipedia.org/wiki/LaSalle%27s_invariance_principle
    locator: Global version
same_as:
  wikidata: Q3922068
  wikipedia: LaSalle's invariance principle
---

## Statement

Let $f : \mathbb{R}^n \to \mathbb{R}^n$ be locally Lipschitz with $f(0) = 0$, and let $V : \mathbb{R}^n \to \mathbb{R}$ be continuously differentiable.
Assume that $\dot{V}(x) = \nabla V(x) \cdot f(x) \le 0$ for all $x \in \mathbb{R}^n$, that $V(0) = 0$ and $V(x) > 0$ for $x \neq 0$, and that $V(x) \to \infty$ as $\|x\| \to \infty$.
Let $E = \{x \in \mathbb{R}^n : \dot{V}(x) = 0\}$.
Assume that the only solution $x(t)$ of $x' = f(x)$ with $x(t) \in E$ for all $t \ge 0$ is the solution $x(t) = 0$.
Then the origin is globally asymptotically stable for $x' = f(x)$.

## Remarks

The general principle: if $\dot{V} \le 0$, then each solution that is bounded for $t \ge 0$ approaches the largest invariant subset of $E$.
The principle allows $\dot{V} = 0$ outside the origin, where [[Lyapunov's stability theorem|0J6A752DTZ]] needs a strict Lyapunov function.
For the damped oscillator $x' = y$, $y' = -x - c y$ with $c > 0$, take $V = \frac{1}{2}(x^2 + y^2)$. Then $E = \{y = 0\}$, and a solution in $E$ has $y' = -x = 0$. So the origin is globally asymptotically stable.
A local version holds when the conditions on $V$ hold only near the origin.

## History

The principle is named after Joseph LaSalle. It is also named after Nikolay Krasovskii and Evgenii Barbashin.

## Explanations

### L1 human

Sometimes the energy only stops decreasing on a set of states, not only at the equilibrium.
LaSalle's principle says that the motion must still end at the equilibrium, if no motion can stay forever in that set.