This paper presents a new control approach for mitigating Friction-Induced Vibration (FIV) issued from the mode-coupling mechanism. The key idea is to put the friction system onto a cascade of independent subsystems by using a first stage control. Then, a second stage control is defined and proven to be much more efficient for controlling both the transient and stationary regimes of the FIV. Asymptotic stabilization is formally demonstrated for the whole control scheme. Moreover, by numerical simulations, it is shown that the performances of the closed loop systems present an interesting robustness with respect to parameter uncertainty and to the saturation phenomenon which opens up promising practical perspectives for the proposed control technique.