Towards uniform linear time-invariant stabilization of systems with persistency of excitation

Consider the controlled system dx/dt = Ax + alpha(t)Bu where (A,B) is stabilizable and alpha(t) takes values in [0,1] and is persistently exciting. In particular, when alpha(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, only depending on (A, B) and on the parameters of the persistent excitation, which globally exponentially 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.