  We consider control systems of the type $\dot x = A x +\alpha(t)bu$, where
$u\in\R$, $(A,b)$ is a controllable pair and $\alpha$ is an unknown
time-varying signal with values in $[0,1]$ satisfying a persistent excitation
condition i.e., $\int_t^{t+T}\al(s)ds\geq \mu$ for every $t\geq 0$, with
$0<\mu\leq T$ independent on $t$. We prove that such a system is stabilizable
with a linear feedback depending only on the pair $(T,\mu)$ if the eigenvalues
of $A$ have non-positive real part. We also show that stabilizability does not
hold for arbitrary matrices $A$. Moreover, the question of whether the system
can be stabilized or not with an arbitrarily large rate of convergence gives
rise to a bifurcation phenomenon in dependence of the parameter $\mu/T$.
