\begin{definition}[Definition:Lyapunov Function]
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 $x \in U$.
{{explain|What $\nabla$ means in this context.}}
If the inequality is strict except at $x_0$, then $V$ is '''strict'''.
\end{definition}