Consider the controlled system dx/dt=Ax+α(t)Bu where the pair
(A,B) is stabilizable and α(t) takes values in [0,1] and is
persistently exciting, i.e., there exist two positive constants μ,T such
that, for every t≥0, ∫tt+Tα(s)ds≥μ. In particular,
when α(t) becomes zero the system dynamics switches to an uncontrollable
system. In this paper, we address the following question: is it possible to
find a linear time-invariant state-feedback u=Kx, with K only depending on
(A,B) and possibly on μ,T, which globally asymptotically stabilizes the
system? We give a positive answer to this question for two cases: when A is
neutrally stable and when the system is the double integrator