Lyapunov Stability

Lyapunov's Direct Method: Stability Without Solving Equations

Aleksandr Mikhailovich Lyapunov and His Contribution

In 1892, the young St. Petersburg mathematician Aleksandr Lyapunov defended his dissertation "The General Problem of the Stability of Motion", which revolutionized the theory of differential equations. Lyapunov proposed analyzing stability not by solving the equations explicitly, but by finding a special auxiliary function (the Lyapunov function), whose behavior indicates the stability of the entire system.

An analogy from physics: a system is stable if its "energy" decreases along trajectories. Lyapunov generalized this observation to arbitrary dynamical systems by replacing actual energy with its mathematical analog—the Lyapunov function.

Definitions of Stability

Consider the system x' = f(x, t), with f(0, t) = 0 (an equilibrium point at the origin).

Lyapunov Stability: The equilibrium x* = 0 is stable if for any ε > 0 there exists δ > 0 such that |x(t₀)| < δ implies |x(t)| < ε for all t ≥ t₀.

Asymptotic Stability: Additionally: |x(t)| → 0 as t → ∞.

Instability: There exists ε > 0 such that for any δ > 0 there exists an initial condition with |x(t₀)| < δ and |x(T)| > ε for some T.

Intuition: a stable equilibrium is the "bottom of a well"; small perturbations do not lead the system far away. Asymptotically stable is "the bottom of a funnel": the system returns to equilibrium. Unstable is "the top of a hill": a small perturbation takes the system away.

Lyapunov Functions: Definition and Meaning

A Lyapunov function for the system x' = f(x) in the neighborhood of x* = 0 is a function V(x) such that:

1. V(x) > 0 for x ≠ 0, V(0) = 0 (positive definite),
2. V̇(x) = ∇V · f(x) = Σᵢ (∂V/∂xᵢ) fᵢ(x) ≤ 0 (the derivative along the trajectory is non-positive).

The derivative V̇ = dV/dt along the trajectory shows how V changes over time without explicitly finding x(t). If V̇ ≤ 0, the function V does not increase—trajectories cannot "escape" beyond the level V = const, and therefore the system is stable.

Lyapunov Stability Theorem: If there exists a Lyapunov function V with V̇ ≤ 0, then x* = 0 is stable. If V̇ < 0 (strictly), then x* is asymptotically stable.

Instability Theorem: If V is not positive definite, but V̇ > 0 in a neighborhood of zero on the set where V > 0, then x* is unstable.

Examples of Lyapunov Functions

Linear system x' = Ax with Re λᵢ < 0: We seek V = xᵀ P x (a quadratic form). From the condition V̇ = xᵀ(PA + AᵀP)x < 0, one needs PA + AᵀP = −Q for some positive definite Q. This is the Lyapunov equation—a linear matrix equation for P. For a stable A and a positive definite Q, there always exists a unique positive definite P.

Harmonic oscillator ẍ + ω²x = 0: Use the total energy V = ẋ²/2 + ω²x²/2. V̇ = ẋẍ + ω²xẋ = ẋ(ẍ + ω²x) = 0. The derivative is zero—energy levels are conserved. This is a center: stable but not asymptotically.

Nonlinear pendulum ẍ + sin x = 0: V = ẋ²/2 + (1 − cos x). V̇ = ẋẍ + ẋ sin x = ẋ(ẍ + sin x) = 0. Again a center—a conservative system.

Pendulum with friction ẍ + bẋ + sin x = 0: V = ẋ²/2 + (1 − cos x). V̇ = ẋ(ẍ + sin x) = ẋ(−bẋ) = −bẋ² ≤ 0. For b > 0, V̇ < 0 (except x = 0)—asymptotically stable. Friction "dissipates" energy, and the pendulum returns to equilibrium.

Detailed Example: Nonlinear Oscillator

System: ẋ = y − x³, ẏ = −x − y³.

We seek a Lyapunov function: Try V = (x² + y²)/2.

V̇ = x·ẋ + y·ẏ = x(y − x³) + y(−x − y³) = xy − x⁴ − xy − y⁴ = −x⁴ − y⁴ ≤ 0.

For (x, y) ≠ (0, 0): V̇ = −x⁴ − y⁴ < 0. Consequently, the origin is globally asymptotically stable—all trajectories tend to zero regardless of initial conditions!

LaSalle's Principle

Sometimes V̇ ≤ 0, but not strictly (V̇ = 0 on some set). The Barbălat–LaSalle principle refines the conclusion.

Formulation: Suppose V̇ ≤ 0 and the set M = {x : V̇ = 0} contains no complete trajectories except x* = 0 itself. Then x* = 0 is asymptotically stable.

Application: Pendulum with friction: V̇ = −bẋ² = 0 when ẋ = 0. This is not only the origin but also all points with x = ±π, ẋ = 0. However, for ẋ = 0, x ≠ 0, we have ẍ = −sin x ≠ 0—the system immediately leaves this set. By LaSalle's principle: the lower equilibrium is asymptotically stable. ✓

Question for thought: Why does a linear system x' = Ax with a stable matrix A always have a quadratic Lyapunov function? Is it possible to systematically construct Lyapunov functions for nonlinear systems?