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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?

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1 Answer 1

up vote 2 down vote accepted

Compute: $$\frac{d}{dt}G(x(t))=\langle \nabla G(x(t)),\dot{x}(t)\rangle = \langle \nabla G(x(t)),-F(\nabla G(x(t)))\rangle \leq 0$$ because $\langle F(z),z\rangle \geq 0$ for every $z \in \mathbb{R}^n$. This follows from your assumptions by taking $z=y$ and $x=0$.

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