  Consider the controlled system $dx/dt = Ax + \alpha(t)Bu$ where the pair
$(A,B)$ is stabilizable and $\alpha(t)$ takes values in $[0,1]$ and is
persistently exciting, i.e., there exist two positive constants $\mu,T$ such
that, for every $t\geq 0$, $\int_t^{t+T}\alpha(s)ds\geq \mu$. In particular,
when $\alpha(t)$ becomes zero the system dynamics switches to an uncontrollable
system. In this paper, we address the following question: is it possible to
find a linear time-invariant state-feedback $u=Kx$, with $K$ only depending on
$(A,B)$ and possibly on $\mu,T$, which globally asymptotically stabilizes the
system? We give a positive answer to this question for two cases: when $A$ is
neutrally stable and when the system is the double integrator.
