We address the following fundamental question: given a double integrator and a linear control that stabilizes it exponentially, is it possible to use the {\em 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 hence, a specific persistency of excitation property for nonlinear systems must be imposed; 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 simple, yet challenging case-studies.

10aRobotics1 aLor, A1 aChaillet, A1 aBesancon, G1 aChitour, Y uhttps://www.centropiaggio.unipi.it/publications/pe-stabilization-time-varying-systems-open-questions-and-preliminary-answers.html