Lyapunov function

Used in analyzing the stability of an equilibrium of a dynamical system. A Lyapunov function is a scalar-valued function of the state space that

1. decreases along trajectories of the systemFormally, for system $\dot{x} = g(x)$ a Lyapunov function $V(x)$ must satisfy $\nabla V \cdot g \le 0$ for all $x$. and
2. is positive everywhere but the equilibrium, where it decreases to zero.

If such a function can be constructed for a given system and equilibrium point, it demonstrates that the equilibrium is stable.

A trivial example: for gradient descent on a function $f(x)$ minimized at zero (e.g., an error rate or similar nonnegative loss function), the objective $f$ is itself a Lyapunov function of the gradient dynamics.