Consider system $\dot x = f(x)$ and let $\Omega\subset D$ be a positive invariant set. Let $V: D \rightarrow \mathbb R$ be a radially unbounded, positve definite function. Let the derivative fulfill
$$ \dot V(x) \leq -\alpha(x) \leq 0, $$
for all $x \in \Omega$, where $\alpha(x)$ is a positive semidefinite function. Can I say, analogously to the invariance principle, that all solutions approach the largest invariant set in $\mathcal{\tilde E} = \{x\in \Omega \mid \alpha(x)=0 \}$? This set is obviously larger than the original $\mathcal E = \{x\in \Omega \mid \dot V(x)=0 \}$ in Lasalle's Invariance Principle, since $\dot V(x) = 0$ implies $\alpha(x) = 0$ but the converse does not necessarily hold. Thus $\mathcal E \subset \mathcal{\tilde E}$. Is it correct to say, that the trajectories converge to the largest invariant set in $\mathcal{\tilde E}$? From my understanding, it should.
Thanks in advance!