Definition:Lyapunov Function

Definition
Let $x_0$ be an equilibrium point of the system $x' = \map f x$.

Then a function $V$ is a Lyapunov function of the system on an open set $U$ containing the equilibrium :


 * $(1): \quad \map V {x_0} = 0$


 * $(2): \quad \map V x > 0$ if $x \in U \setminus \set {x_0}$


 * $(3): \quad \nabla V \cdot f \le 0$ for $x \in U$.

If the inequality is strict except at $x_0$, then $V$ is strict.

Also known as
The name is sometimes seen spelt Liapunov function.