# 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 $$f\ .$$