I've tried my luck with an exercise, but did not achieve anything yet. Maybe you could help me? :)
We have $x:\mathbb{R}_0^+\rightarrow\mathbb{R}^n,\ t\rightarrow x(t)$ solving
$\frac{dx}{dt}=-F(\nabla G(x(t))),\ t>0$ where $F:\mathbb{R^n}\rightarrow \mathbb{R}^n$ and $G:\mathbb{R^n}\rightarrow \mathbb{R}$. $G$ is cont. differentiable, and $F(0)=0$, and:
$F$ is cont. differentiable and has a positive semi-definite Jacobian;
or $F$ is Lipschitz continuous and fulfills $(F(x)-F(y))\cdot (x-y)\geq 0\ \forall \ x,y \in \mathbb{R}^n$;
then G is Lyapunov for any solution $x$.
Being Lyapunov means that $\frac{d}{dt}G(x(t))\leq 0$, but I don't know how to go on from here. Can someone give me a hint?