On the PE stabilization of time-varying systems: open questions and preliminary answers

We address the following question: given a double integrator and a linear control that stabilizes it exponentially, is it possible to use the same control input in the case that the control input is multiplied by a time-varying term? Such question has many interesting motivations and generalizations: 1) we can pose the same problem for an input gain that depends on the state and time; 2) the stabilization –with the same method–of chains of integrators of higher order than two is fundamentally more complex and has applications in the stabilization of driftless systems; 3) the popular backstepping method stabilization method for systems with non-invertible input terms. The purpose of this note is two-fold: we present some open questions that we believe are significant in time-varying stabilization and present some preliminary answers for a simple, yet challenging case-study. Our solutions are stated in terms of persistency of excitation.