If I have a Lyapunov candidate $V:[0,\infty)\rightarrow \mathbb{R}$ and I'm able to show that $$ V(t)\le k e^{-\eta t} V(0),\qquad \forall t\in[0,\infty) $$ can I conclude something about asymptotic\exponential stability?
I ask this question because on the definitions I read that $\dot{V}$ must be always negative (but in zero) to conclude that.