Lyapunov's Direct Method
The direct method proves stability by exhibiting a positive-definite function whose derivative along trajectories is negative.
Proving Stability Constructively
Lyapunov's direct method (also called the second method) gives a concrete recipe for proving stability. Rather than analyzing trajectories, it asks the analyst to produce a single scalar function of the state with two properties, from which stability follows immediately. It is direct because it needs no solution of the dynamics.
The conditions
- V(x) is positive definite: V(0) = 0 at the equilibrium and V(x) > 0 for all other x nearby.
- The derivative of V along the system's trajectories is negative semidefinite: it never increases.
- If additionally that derivative is strictly negative away from the equilibrium, the equilibrium is asymptotically stable.
- If V grows without bound as the state grows and the conditions hold everywhere, stability is global.
The intuition
Think of V as a bowl-shaped landscape with its lowest point at the equilibrium. If the state always moves downhill on this landscape, it can only end up at the bottom. The negative derivative condition is exactly the statement that the state moves downhill, so it must converge to the equilibrium at the base of the bowl.
LaSalle's invariance principle
Often the derivative of V is only negative semidefinite, becoming zero on some set without the state being at equilibrium. LaSalle's invariance principle rescues these cases: the state converges to the largest invariant set where the derivative is zero. This frequently proves asymptotic stability when the strict-derivative condition alone would fail.
The central difficulty
The method is only as good as the candidate function chosen. There is no universal procedure to find a Lyapunov function, and failing to find one does not prove instability. For mechanical and electrical systems, physical energy is a natural first candidate. For general systems, quadratic forms, sum-of-squares optimization, and structured constructions such as backstepping provide systematic ways to build one.
When a Lyapunov function is found, it does more than certify stability: its level sets estimate the region of attraction, bounding how far from equilibrium the system can start and still converge.