Lyapunov function

## [edit] 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\ .\\)