Lyapunov Function

Definition

Consider a differentiable vectorfield $f:X \rightarrow X\ ,$ $x \mapsto f(x)\ ,$ $X \subset \mathbb{R}^n.$ A differentiable function $V:U \rightarrow \mathbb{R}\ ,$ defined on an open subset $U \subset X$ is called a Lyapunov function for $f$ on $U$ if the inequality $$\overset{\circ}{V}(x) := \nabla V(x)^T f(x) \, \leq 0$$ is satisfied for all $x \in U\ .$
$\overset{\circ}{V}$ defined as above is called the orbital differential of $V$ at $x\ .$

In other words, a Lypunov function is decreasing along the orbits of points in $U$ that are introduced by the flow corresponding to the vectorfield $f\ .$