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 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.