Lyapunov function

[edit] Definition

Consider a differentiable vectorfield $f:X\to X,$ $x\mapsto f(x),$ $X\subset {\mathbb{R}}^n.$ A differentiable function $V:U\to \mathbb{R},$ defined on an open subset $U\subset X$ is called a Lyapunov function for $f$ on $U$ if the inequality$$\stackrel{\circ }{V}(x):=\mathrm{\nabla }V(x{)}^{T}f(x)\le 0$$ is satisfied for all $x\in U.$
$\stackrel{\circ }{V}$ 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 $f.$