Consider the controlled system $dx/dt = Ax + \alpha(t)Bu$ where the pair $(A,B)$ 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 the parameters of the persistent excitation, 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.

10aRobotics1 aChaillet, A1 aChitour, Y1 aLor, A1 aSigalotti, M uhttp://www.centropiaggio.unipi.it/publications/uniform-stabilization-linear-systems-persistency-excitation-neutrally-stable-and-double