Lyapunov functions and LaSalle invariance

Translate the equilibrium to zero and assume a locally Lipschitz autonomous vector field, with a continuously differentiable candidate V on a neighborhood. A positive-definite V is zero at the origin and positive at every other nearby point. Its derivative along motion is the gradient of V dotted with f. If that derivative is nonpositive in the neighborhood, sufficiently small sublevel sets trap solutions and prove Lyapunov stability: a small initial displacement stays small for all future time. Stability does not itself say the displacement vanishes.

If the derivative is strictly negative away from zero, the usual local Lyapunov theorem gives asymptotic stability. For a global conclusion, the hypotheses must hold globally and boundedness cannot be left implicit. A positive-definite, radially unbounded V has compact sublevel sets in finite dimensions; nonincrease then prevents escape, supplies forward existence for a smooth field, and supports global attraction under the appropriate strict or invariance argument. A candidate defined only inside one region cannot automatically prove a claim about every initial state.

Let K be a compact positively invariant subset of the domain, let V be continuously differentiable on a neighborhood of K, and suppose its orbital derivative is nonpositive on K. LaSalle's invariance principle states that every trajectory starting in K approaches the largest invariant subset M of the zero-derivative set E within K. Approaches means distance to M tends to zero; it does not generally promise convergence to a single point when M contains many states or recurrent motions.

The key step is invariance. A point can have zero derivative for an instant and immediately leave E, so it cannot belong to an entire invariant trajectory contained in E. For a damped oscillator, all states with zero velocity have zero instantaneous energy loss. Most have nonzero acceleration, so their motion immediately acquires velocity and dissipates. Keeping only the states that remain in the zero-loss set often reduces it to the equilibrium. Positive definiteness then supplies stability, and the reduced invariant set supplies attraction.

For position x and velocity v, consider x dot = v and v dot = −x−x cubed−v. Choose V as half v squared plus half x squared plus one quarter x to the fourth. The chain rule cancels the conservative terms and leaves minus v squared. The function is positive definite and radially unbounded, so every closed sublevel is compact and positively invariant. This gives forward completeness and the compact setting LaSalle needs, rather than assuming a bounded phase plot proves bounded motion.

Here E is the line v=0 intersected with the chosen sublevel. A trajectory staying in that line must also have v dot = −x−x cubed equal to zero. The only real possibility is x=0, so the largest invariant subset is the origin. Every initial state belongs to some compact sublevel, yielding global asymptotic stability. The proof gives no uniform exponential rate by itself. Linearization may provide a local rate, while other systems with zero Jacobian can converge only algebraically despite possessing a strict Lyapunov function.