We consider control systems of the type x˙=Ax+α(t)bu, where
u∈R, (A,b) is a controllable pair and α 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≥0, with
0<μ≤T independent on t. We prove that such a system is stabilizable
with a linear feedback depending only on the pair (T,μ) 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 μ/T