On the stabilization of permanently excited linear systems

We consider control systems of the type x̃ = Ax+α(t)ub, where u ∈ R, (A; b) is a controllable pair and α is an unknown time-varying signal with values in [0; 1] satisfying a permanent excitation condition of the kind ∫^t+T<inf>t</inf> ε ≥ μfor 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 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 with an arbitrarily large rate of convergence gives rise to a bifurcation phenomenon in dependence of the parameter μ/T.