LaSalle's Invariance Principle

LaSalle's invariance principle extends Lyapunov's method to handle cases where $\dot{V}$ is only negative semi-definite, by analyzing the invariant structure of the set where $\dot{V} = 0$ .

Statement

Theorem7.9LaSalle's invariance principle

Let $\Omega \subset \mathbb{R}^n$ be a compact set that is positively invariant for $\mathbf{x}' = \mathbf{F}(\mathbf{x})$ (i.e., trajectories starting in $\Omega$ remain in $\Omega$ for all $t \geq 0$ ). Let $V: \Omega \to \mathbb{R}$ be $C^1$ with $\dot{V}(\mathbf{x}) \leq 0$ on $\Omega$ . Define $E = \{\mathbf{x} \in \Omega : \dot{V}(\mathbf{x}) = 0\}$ and let $M$ be the largest invariant subset of $E$ .

Then every solution starting in $\Omega$ approaches $M$ as $t \to \infty$ :

$\lim_{t \to \infty} d(\mathbf{x}(t), M) = 0.$

Proof

Since $V$ is non-increasing along trajectories and bounded below on $\Omega$ (compact), $V(\mathbf{x}(t))$ converges to some limit $c$ as $t \to \infty$ .

The $\omega$ -limit set $\omega(\mathbf{x})$ is nonempty (by compactness), contained in $\Omega$ , and invariant. For any $\mathbf{y} \in \omega(\mathbf{x})$ , there exists $t_n \to \infty$ with $\mathbf{x}(t_n) \to \mathbf{y}$ , so $V(\mathbf{y}) = \lim V(\mathbf{x}(t_n)) = c$ .

Therefore $V$ is constant ( $= c$ ) on $\omega(\mathbf{x})$ . Since $\omega(\mathbf{x})$ is invariant and $V$ is constant there, $\dot{V} = 0$ on $\omega(\mathbf{x})$ . Thus $\omega(\mathbf{x}) \subset E$ . Since $\omega(\mathbf{x})$ is invariant, $\omega(\mathbf{x}) \subset M$ (the largest invariant subset of $E$ ). $\blacksquare$

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Applications

ExampleDamped oscillator

$x'' + bx' + kx = 0$ ( $b, k > 0$ ). As a system: $\mathbf{x}' = (y, -kx-by)^T$ . Use $V = \frac{1}{2}(kx^2 + y^2)$ (energy).

$\dot{V} = kxy + y(-kx-by) = -by^2 \leq 0$ .

$E = \{y = 0\}$ . On $E$ : $x' = y = 0$ (so $x = \text{const}$ ) and $y' = -kx$ (so $x = 0$ for invariance). Therefore $M = \{(0,0)\}$ .

By LaSalle: all trajectories approach $(0,0)$ . Asymptotic stability is established even though $\dot{V}$ is only semi-definite.

ExampleNonlinear damping

$x'' + g(x') + h(x) = 0$ where $h(0) = 0$ , $xh(x) > 0$ for $x \neq 0$ , and $g(0) = 0$ , $yg(y) > 0$ for $y \neq 0$ .

$V = \int_0^x h(\xi)\,d\xi + \frac{1}{2}y^2 > 0$ . Then $\dot{V} = h(x)y + y(-g(y)-h(x)) = -yg(y) \leq 0$ .

$E = \{y = 0\}$ , and on $E$ : $y' = -h(x)$ , so invariance requires $h(x) = 0$ , hence $x = 0$ . So $M = \{(0,0)\}$ and the origin is asymptotically stable.

RemarkComparison with strict Lyapunov

Finding a strict Lyapunov function ( $\dot{V} < 0$ ) can be very difficult. LaSalle's principle allows the use of natural energy functions (which typically satisfy only $\dot{V} \leq 0$ ) and compensates by analyzing the invariant set structure. This makes it far more practical than requiring strict negative definiteness.