LaSalle's invariance principle
/ la-SAL /
Lyapunov's direct method proves asymptotic stability only when the energy strictly decreases everywhere — but in many real systems the energy sometimes holds steady, for instance when the velocity is momentarily zero. Does the system still settle down, or could it get stuck circling on a level set of energy? LaSalle's invariance principle answers exactly this, salvaging a conclusion when strict decrease fails.
The idea: suppose V is a Lyapunov function with V' less than or equal to zero (it never increases, but it is allowed to be flat in spots). Trajectories are trapped inside a bowl-shaped sublevel set, so they must approach the largest invariant set contained in the region where V' = 0. 'Invariant' means a set the flow never leaves; you find it by asking, of all the points where the energy momentarily stops dropping, which ones can the system actually stay among forever. Often that largest invariant set is just the single equilibrium, and then every trajectory converges to it — asymptotic stability, even though V was only non-increasing.
This is the workhorse for mechanical and control systems with damping. The classic case is a damped pendulum: energy decreases except at the instants the bob is momentarily at rest (velocity zero), where V' = 0; the only motion that can stay forever at zero velocity is sitting still at the bottom, so LaSalle concludes the pendulum must come to rest there. The principle requires bounded (precompact) trajectories to apply, which the trapping sublevel set usually supplies.
For x' = y, y' = -sin(x) - y with energy V = (1/2)y^2 + (1 - cos x), V' = -y^2 ≤ 0; it vanishes only when y = 0, but on that line the flow forces y' = -sin(x), so the system stays there only at x = 0 — LaSalle gives convergence to the bottom.
Energy is flat only on y = 0, and the only trajectory living there forever is the equilibrium, so all motions converge to it.
LaSalle does not need V' strictly negative — that is its whole point — but it does require trajectories to stay bounded, and it concludes convergence to an invariant SET, which only means asymptotic stability when that set is the single equilibrium.