Stability of Nonlinear Systems: The Lyapunov Method

Linear systems are stable if all eigenvalues of the matrix A lie in the left half-plane—a simple and convenient criterion. But real systems are often nonlinear: a pendulum, a chemical reactor, an electrical grid, a neural network. In 1892, Aleksandr Lyapunov proposed a universal method for analyzing the stability of nonlinear systems—without the need to solve differential equations. The method is based on searching for an "energy-like" function that decreases along the system's trajectories. This method has become the foundation of modern theory of nonlinear systems and nonlinear control.

Notions of Stability

System: ẋ = f(x), f(0) = 0 (the origin is an equilibrium point). Without loss of generality, we consider the stability of zero—other equilibria are reduced to zero by a change of variables.

Lyapunov stability. The equilibrium x = 0 is stable if for every ε > 0 there exists δ > 0 such that ||x(0)|| < δ implies ||x(t)|| < ε for all t ≥ 0. Small perturbations remain small.

Asymptotic stability. Stable AND x(t) → 0 as t → ∞. Perturbations not only remain small, but also die out.

Global asymptotic stability. Asymptotically stable for any initial condition. This is the strongest property.

Exponential stability. ||x(t)|| ≤ M·||x(0)||·e^{−λt} for some M, λ > 0. A clear rate of decay.

The Direct Lyapunov Method

Idea. Find a function V: ℝⁿ → ℝ that measures the system's "generalized energy." If energy decreases along trajectories, the system moves to equilibrium.

Lyapunov’s theorem. If there exists a continuously differentiable function V satisfying:

1. V(0) = 0, V(x) > 0 for x ≠ 0 (positive definite).
2. V̇(x) = ∇V(x)ᵀ·f(x) ≤ 0 for x ≠ 0 (V does not increase along trajectories).

Then x = 0 is Lyapunov stable. If, in addition, V̇(x) < 0 for x ≠ 0 (strictly decreasing), then asymptotically stable.

Global stability: Additionally, V(x) → ∞ as ||x|| → ∞ (radial unboundedness).

Searching for a Lyapunov Function

This is an “art,” but there are useful hints:

• Linear systems: V(x) = xᵀ·P·x with P > 0. V̇ = xᵀ·(AᵀP + PA)·x. For V̇ < 0, require Aᵀ·P + P·A = −Q < 0. This is the Lyapunov equation: for stable A and any Q > 0, there exists a unique solution P > 0.

• Mechanical systems: V = T + V_potential — the total energy, if it is positive definite.

• Systems with dissipation: often V = “energy”, V̇ ≤ 0.

Examples

1. Linear system ẋ = A·x with A = [−1, 1; 0, −2]. Eigenvalues −1, −2 — stable.

Solution of the Lyapunov equation Aᵀ·P + P·A = −I (Q = I): P = [0.625, 0.25; 0.25, 0.375]. det P = 0.234 − 0.0625 > 0, P > 0 — confirms stability.

2. Pendulum with friction: ẋ₁ = x₂, ẋ₂ = −sin(x₁) − x₂. Energy: V = (1 − cos x₁) + x₂²/2. V > 0 for (x₁, x₂) ≠ 0 in the neighborhood of zero. ∇V = (sin x₁, x₂). V̇ = sin(x₁)·x₂ + x₂·(−sin x₁ − x₂) = −x₂² ≤ 0.

V̇ = 0 only when x₂ = 0, but then (from the equations) ẋ₂ = −sin x₁, which leads away from x₂ = 0 except for the case sin x₁ = 0, i.e., x₁ = 0. By LaSalle's theorem → asymptotic stability in the neighborhood of (0, 0).

3. Nonlinear oscillator: ẍ + x + x³ = 0. V = x²/2 + x²·ẋ²/... (more precisely: “potential” V_pot = x²/2 + x⁴/4). V_total = ẋ²/2 + V_pot. V̇ = 0 (no dissipation) → stable, but not asymptotically. Adding friction −b·ẋ: V̇ = −b·ẋ² ≤ 0 → asymptotic stability.

LaSalle's Invariance Principle (LaSalle's Theorem)

If V̇ ≤ 0 and Ω = {x: V̇(x) = 0}, then trajectories converge to the largest invariant set M within Ω. If M = {0} — asymptotic stability. This allows working with "semi-definite" V̇.

Control Lyapunov Functions (CLF)

The problem of designing a controller that makes the system stable:

ẋ = f(x) + g(x)·u.

A CLF is a function V such that for any x ≠ 0 there exists u: V̇ = ∇Vᵀ·(f + g·u) < 0. Specifically: ∇Vᵀ·g = 0 → ∇Vᵀ·f < 0.

Sontag's universal formula: u(x) = −[L_f V + √((L_f V)² + (L_g V)⁴)] / L_g V, if L_g V ≠ 0; otherwise u = 0. This formula guarantees V̇ < 0.

Numerical Example: Stabilization of the Inverted Pendulum

Equation: θ̈ = (g/l)·sin θ + u/(m·l²). Linearization near θ = 0: θ̈ = (g/l)·θ + u/(m·l²). Without u — unstable.

Take V = θ²/2 + l²·θ̇²/(2g) (quadratic in phase coordinates). V̇ = θ·θ̇ + (l²/g)·θ̇·θ̈ = θ·θ̇·(1 + l/g·... — technical details). Selecting u = −K·(θ + 2·θ̇), one can make V̇ < 0 near 0 for sufficiently large K. Numerically: for g/l = 9.81, m·l² = 1, K = 30 — exponential stability with a rate of ~3 s⁻¹.

Real Applications

• Power systems. Analysis of synchronous stability of generators after a short circuit: Lyapunov functions based on the kinetic + potential energy of generators. Used to determine the “critical clearing time”.
• Robotics. Adaptive control with stability guarantees via CLF: it is proven that the robot reaches the desired trajectory even with unknown parameters (payload mass, friction).
• Biology. Stability analysis of population models (Lotka–Volterra, epidemiological models): a Lyapunov function shows which equilibrium (survival/extinction) the system converges to.
• Machine learning. Analysis of gradient descent convergence: the loss function L(θ) is a Lyapunov function, L̇ = −||∇L||² ≤ 0 for GD with a small step size.

Assignment. For the nonlinear oscillator with control: ẍ + x + x³ = u: (a) For u = 0, find a Lyapunov function V (for example, potential + kinetic energy). Show that V̇ = 0 → trajectories are periodic (stable, not asymptotically). (b) Design a control u = −K·x − D·ẋ (PD feedback) with suitable K, D. Check the CLF condition: V̇ < 0 in some neighborhood of zero. (c) Numerically simulate for x(0) = 2, ẋ(0) = 0 for u = 0 and for u = −2x − ẋ. Plot the phase portrait (x, ẋ). (d) How does the nonlinearity x³ affect the behavior compared to the linear oscillator ẍ + x = u?