A scalar-valued function V(x) defined on a neighbourhood of an equilibrium of a dynamical system that is positive-definite and whose derivative along system trajectories has a definite sign (typically nonincreasing), used as a certificate of stability without solving trajectories.

Principle

If V is positive-definite and its time derivative along solutions is negative-definite (or nonpositive), then the equilibrium is stable (or Lyapunov stable / asymptotically stable under stronger conditions).

Demonstration

For the linear system ẋ = −a x with a>0, choose V(x)=x^2: V>0 for x≠0 and dV/dt = 2x ẋ = −2a x^2 ≤ 0, proving asymptotic stability of the equilibrium at 0.

Misapplication

Using a candidate V that is not positive-definite or whose derivative changes sign, then claiming stability; or applying the concept without checking its domain of validity (local vs global).

Consequence

Provides a constructive, often coordinate-free method to prove stability or attractivity of equilibria; can yield regions of attraction and robustness certificates without integrating the system.

Reversal

Absence of a Lyapunov function of a given simple form does not imply instability; conversely, an energy-like function that increases along trajectories indicates instability or repulsion.

Boundary

Formulated for deterministic dynamical systems (continuous- or discrete-time) with sufficient regularity; existence may be only local and does not directly extend to systems with non-differentiable dynamics or certain stochastic formulations.

Semantic Tension

Related to but distinct from physical energy: both are scalar measures, but a Lyapunov function is a mathematical certificate for stability and need not correspond to conserved or dissipated physical energy.

Synthesis

A Lyapunov function is a scalar certificate defined near an equilibrium that, by being positive-definite and decreasing along trajectories, establishes stability properties without explicit solution of the dynamics.