I have read many versions of LaSalle's invariance principle. One of them is the following:
For $D\subset\mathbb{R}^{n}$ open let $f:D\to\mathbb{R}^{n}$ be $C^{1}(D;\mathbb{R}^{n})$. Consider $\dot{x}=f(x)$ and let $\Omega\subset{D}$ be compact and positively invariant and $V\in C^{1}(D;\mathbb{R})$ with $\langle\nabla{V},f\rangle\leq0$ on $\Omega$. Let $E:=\{x\in\Omega \vert\ \langle\nabla{V}(x),f(x)\rangle=0\}$ and let $M\subseteq E$ be the largest positively invariant set in $E$. The every solution starting in $\Omega$ appraches $M$ for $t\to\infty$.
I have often read this theorem with "$M$ positively invariant" and "$M$ invariant", and in many scriptums it is not clear whether invariant means that solutions stay in $M$ for $t\to\infty$ and $t\to -\infty$, or only for $t\to\infty$. I have seen both versions regarding that theorem.
My question is: If you are only interested in max. solutions on $[0,\infty[$, is the theorem true if $M$ is only positively invariant in the sense that "$x(0)\in M \Rightarrow x(t)\in M \text{ for all } t\geq0$"? I can't see where the proof for limit sets and LaSalle's principle would go wrong.