Lyapunov Stability

Lyapunov stability theory provides a framework for determining the stability of equilibria without solving the differential equation, using energy-like functions.

Definitions of Stability

Definition7.1Stability in the sense of Lyapunov

An equilibrium $\mathbf{x}_0$ of $\mathbf{x}' = \mathbf{F}(\mathbf{x})$ is:

Stable if for every $\varepsilon > 0$ , there exists $\delta > 0$ such that $\|\mathbf{x}(0) - \mathbf{x}_0\| < \delta$ implies $\|\mathbf{x}(t) - \mathbf{x}_0\| < \varepsilon$ for all $t \geq 0$ .
Asymptotically stable if it is stable and additionally $\mathbf{x}(t) \to \mathbf{x}_0$ as $t \to \infty$ for initial conditions sufficiently close to $\mathbf{x}_0$ .
Unstable if it is not stable.

ExampleStability types

$x' = -x$ : $x = 0$ is asymptotically stable ( $x(t) = x_0 e^{-t} \to 0$ ).
$x' = 0$ : $x = 0$ is stable but not asymptotically stable.
$x' = x$ : $x = 0$ is unstable ( $x(t) = x_0 e^t \to \pm\infty$ ).
$x' = -x^3$ : $x = 0$ is asymptotically stable (but not exponentially: $x(t) \sim 1/\sqrt{2t}$ ).

Lyapunov Functions

Definition7.2Lyapunov function

A $C^1$ function $V: D \to \mathbb{R}$ defined on a neighborhood $D$ of an equilibrium $\mathbf{x}_0$ is a Lyapunov function if:

$V(\mathbf{x}_0) = 0$ and $V(\mathbf{x}) > 0$ for $\mathbf{x} \neq \mathbf{x}_0$ in $D$ (positive definite).
$\dot{V}(\mathbf{x}) = \nabla V \cdot \mathbf{F}(\mathbf{x}) \leq 0$ in $D$ (negative semi-definite along trajectories). If additionally $\dot{V}(\mathbf{x}) < 0$ for $\mathbf{x} \neq \mathbf{x}_0$ (negative definite), $V$ is a strict Lyapunov function.

Theorem7.1Lyapunov stability theorem

If there exists a Lyapunov function $V$ for the equilibrium $\mathbf{x}_0$ :

$\dot{V} \leq 0$ implies $\mathbf{x}_0$ is stable.
$\dot{V} < 0$ implies $\mathbf{x}_0$ is asymptotically stable.
$V > 0$ and $\dot{V} > 0$ implies $\mathbf{x}_0$ is unstable (Chetaev's theorem).

ExampleLyapunov function for a nonlinear system

For $x' = -x + 2y^2$ , $y' = -y$ , try $V = x^2 + y^2$ .

$\dot{V} = 2x(-x+2y^2) + 2y(-y) = -2x^2 + 4xy^2 - 2y^2 = -2x^2 - 2y^2 + 4xy^2$ .

Near the origin, $4xy^2$ is higher order than $-2(x^2+y^2)$ , so $\dot{V} < 0$ in a neighborhood. Asymptotically stable.

LaSalle's Invariance Principle

Theorem7.2LaSalle's invariance principle

If $V$ is a Lyapunov function with $\dot{V} \leq 0$ on a compact positively invariant set $\Omega$ , then every trajectory in $\Omega$ approaches the largest invariant subset of $\{x : \dot{V}(x) = 0\}$ .

RemarkPractical use

LaSalle's principle strengthens the Lyapunov theorem: even when $\dot{V}$ is only semi-definite ( $\leq 0$ ), asymptotic stability can be concluded if the set $\{\dot{V} = 0\}$ contains no complete trajectories other than the equilibrium. This is especially useful for mechanical systems with energy-like Lyapunov functions.