Lyapunov Stability — Mathematics Knowledge System

Overview

Lyapunov stability theory provides a rigorous, non-perturbative approach to determining whether an equilibrium is stable without solving the differential equations. The central idea is to find a Lyapunov function V(x) — a generalized energy-like scalar function — that decreases along trajectories. If such a function exists and satisfies appropriate sign conditions, stability (or asymptotic stability) is guaranteed. The LaSalle invariance principle extends this by allowing dV/dt to be merely nonpositive (rather than strictly negative), concluding convergence to the largest invariant set where dV/dt = 0. Together these tools form the 'direct method' of Lyapunov, which applies to nonlinear systems where eigenvalue analysis fails.

Intuition

Imagine a ball rolling in a bowl. The bowl's altitude function V(x,y) acts as potential energy: V is positive everywhere except the bottom, and as the ball rolls, V decreases. If you can find any such 'bowl-shaped' function that continuously decreases along every trajectory, the bottom of the bowl must be stable. You don't need to know the exact path of the ball; you just need to know that energy is always draining. LaSalle's twist: even if the energy only stops decreasing on some set S (rather than only at the bottom), the ball must eventually settle into the largest subset of S that is itself invariant under the dynamics — often just the equilibrium.

Formal Definition

Definition

Consider the autonomous system x' = F(x) with equilibrium x* = 0 (WLOG). A Lyapunov function is a C^1 function V: U -> R on a neighborhood U of 0.

V(0) = 0, \quad V(x) > 0 \text{ for } x \neq 0 \text{ (positive definite)}

Sign condition on V

\dot{V}(x) = \nabla V(x) \cdot F(x) \leq 0 \text{ for all } x \in U \setminus \{0\}

Lyapunov stability condition

\dot{V}(x) < 0 \text{ for all } x \in U \setminus \{0\} \implies x^* \text{ is asymptotically stable}

Asymptotic stability condition

E = \{x \in U : \dot{V}(x) = 0\}, \quad M = \text{largest invariant set in } E

LaSalle's invariance principle setup

Notation

Notation	Meaning
$V(x)$	Lyapunov function (generalized energy)
$\dot{V}(x)$	Time derivative of V along trajectories: nabla V · F(x)
$E$	Set where dV/dt = 0
$M$	Largest invariant subset of E (in LaSalle's principle)

Theorems

Theorem 1: Theorem 1

Let x^* = 0 be an equilibrium of \dot{ x } = F(x). If there exists a C^1 positive definite function V: U -> \mathbb{ R } with \dot{ V }(x) \leq 0 on U \setminus \{0\}, then x^* is (Lyapunov) stable. If additionally \dot{ V }(x) < 0 on U \setminus \{0\}, then x^* is asymptotically stable.

Theorem 2: Theorem 2

Let V be a C^1 positive definite function on a compact positively invariant region \Omega with \dot{ V }(x) \leq 0 on \Omega. Define E = \{x \in \Omega : \dot{ V }(x) = 0\} and let M be the largest invariant set in E. Then every solution starting in \Omega converges to M as t \to \infty.

Theorem 3: Theorem 3

If there exists a function V and a region U_0 adjacent to the origin such that V(0) = 0, V > 0 in U_0, and \dot{ V } > 0 in U_0, then the equilibrium x^* = 0 is unstable.

Worked Examples

1
Propose V(x) = x^2/2. This is positive definite: V(0) = 0 and V(x) > 0 for x ≠ 0.
$V(x) = \frac{x^2}{2}$
2
Compute the time derivative along trajectories: dV/dt = x * dx/dt = x * (-x^3) = -x^4.
$\dot{V} = x \cdot (-x^3) = -x^4$
3
Since -x^4 < 0 for all x ≠ 0, we have dV/dt < 0. By Lyapunov's theorem, x* = 0 is asymptotically stable.

✓ Answer

V(x) = x^2/2 is a Lyapunov function with negative definite derivative, so x* = 0 is asymptotically stable.

Practice Problems

Mediumproof writing

Use V(x,y) = x^2 + y^2 to analyze stability of dx/dt = -x + 2xy^2, dy/dt = -y - x^2*y.

Easyfree response

State the difference between Lyapunov stability and asymptotic stability, and give an example of each.

Common Mistakes

Common Mistake

Assuming dV/dt <= 0 alone implies asymptotic stability.

Non-positive dV/dt gives only Lyapunov stability (trajectories stay near x*). Strict negativity or LaSalle's invariance principle is required for asymptotic stability (convergence).

Common Mistake

Proposing V with V(x) >= 0 but V(x) = 0 at points other than the equilibrium.

V must be strictly positive definite: V(0) = 0 and V(x) > 0 for all x ≠ 0 in the neighborhood. A merely non-negative function does not qualify as a Lyapunov function.

Common Mistake

Applying a locally valid Lyapunov function to claim global asymptotic stability.

Global asymptotic stability requires a radially unbounded Lyapunov function (V(x) -> inf as |x| -> inf) defined on all of R^n, not just a neighborhood of the equilibrium.

Common Mistake

Concluding instability because no Lyapunov function was found.

Failure to find a Lyapunov function does not imply instability; it only means the method was inconclusive. Other tools (eigenvalue analysis, Chetaev's theorem) are needed to prove instability.

Quiz

Historical Background

Aleksandr Mikhailovich Lyapunov submitted his doctoral dissertation 'The General Problem of the Stability of Motion' to Kharkov University in 1892. In it he introduced two methods: the 'indirect method' (linearization/eigenvalue analysis) and the 'direct method' (energy-function approach). The direct method was largely forgotten in the West until Lefschetz translated it into English in 1947 and Zubov further developed it. Joseph LaSalle at RIAS (Research Institute for Advanced Studies) stated and proved the invariance principle bearing his name around 1960, greatly extending the applicability of Lyapunov's direct method to cases where the 'energy' derivative only vanishes on a set rather than being strictly negative.

1892
Lyapunov's doctoral dissertation introduces the direct and indirect methods of stability
Aleksandr Lyapunov
1947
Lefschetz translates Lyapunov's work into English, reviving Western interest
Solomon Lefschetz
1960
LaSalle states and proves the invariance principle, extending Lyapunov's direct method
Joseph LaSalle

Summary

A Lyapunov function V is a positive definite scalar function that decreases along trajectories; its existence certifies stability without solving the ODE.
Strict decrease (dV/dt < 0) gives asymptotic stability; mere non-increase (dV/dt <= 0) gives only Lyapunov (neutral) stability.
LaSalle's invariance principle: if dV/dt <= 0 on a compact invariant region, all trajectories converge to the largest invariant subset of {dV/dt=0}.
The method is nonlinear: unlike eigenvalue analysis it works far from equilibrium and does not require the fixed point to be hyperbolic.

References

BookStrogatz, S. H. Nonlinear Dynamics and Chaos. Westview Press, 2015. Chapter 7.
WebsiteWikipedia — Lyapunov stability