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 permanent excitation condition i.e., $\int_t^{t+T}\al\geq \mu$ 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 real part of the eigenvalues of $A$ is non-positive. The stabilizability does not hold in general for matrices $A$ whose eigenvalues have positive real part. 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 $\mu/T$.