Definition:Lyapunov Function

Definition
Let $x_0$ be an equilibrium point of the system $x' = f \left({x}\right)$.

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


 * $V \left({x_0}\right) = 0$


 * $V \left({x}\right) > 0$ if $x \in U \setminus \left\{{x_0}\right\}$


 * $\nabla V \cdot f \le 0$ 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.