0 81 0 . 21 22 v 3 [ m at h . O C ] 1 8 M ay 2 00 9 On the stabilization of persistently excited linear systems ∗

We consider control systems of the type ẋ = 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., ∫ t+T t α(s)ds ≥ μ 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 .