## 1 Introduction  
Friction between two structures in contact can generate instabilities and self-sustained vibrations. This then results in unwanted noise conventionally called under the name of brake squeal in the automotive industry. Despite great progress in the modeling of brake squeal and numerical strategies for predicting friction-induced vibrations [1; 2; 3; 4], there is still an important issue to be able to reduce self-sustained vibrations and squeal noise. Research for reducing or removing brake squeal has been regularly performed for many years by applying structural modifications of the mechanical systems under study. Another strategy may consist in indirectly disturbing the dynamics of the mechanical system in the presence of instabilities in order to reduce and control the levels of the vibrations generated.  
First preliminary studies which are based on notions of passive or active control were initiated on simple phenomenological models subject to mono-instability [5; 6; 7]. In the field of nonlinear active control, linearization state feedback [8; 9; 10; 11] is one of the possible solutions to control such a system. The principle of this method is to apply an active control vector which makes it possible to compensate exactly the nonlinear behavior of the system. The use of this kind of control requires to perfectly know the nonlinear force. In the case of unknown phenomena, this approach can be coupled with the receptance method proposed by [PERSON] et al. [12] in the case of nonlinear systems. This method does not require the modeling of the system and may also be applied in the case of friction-induced vibration using partial poles placement as demonstrated by [PERSON] et al. [13]. Moreover, if the state vector cannot be measured, or in the case of partial poles placement, linearization feedback can be completed using nonlinear state observer introduced by [PERSON] [14] and applied in the case of friction-induced vibration by [PERSON] [15]. However, this approach is based on a model of the nonlinear structure and can show limits in terms of robustness. However, no study has yet considered the case of mechanical systems for which several instabilities can coexist with the generation of quasi-periodic transient and stationary self-excited vibrations. This aspect of the usefulness and feasibility of an active control to reduce not only periodic but also quasi-periodic vibrations of a mechanical system subjected to friction-induced vibration is one of the first contributions of the proposed study.  
This paper is organized as follows. The mechanical system under study is first presented. Then, the classical stability analysis and the prediction of nonlinear response of the mechanical systems subjected to friction-induced vibration are briefly discussed. In Section 3, the proposed active control design based on full linearization feedback is described. A complete numerical study is then undertaken to validate the use of the controller and to discuss more further on the robustness and the efficiency of linear and nonlinear controls for mechanical systems subjected to friction-induced vibration. More specifically, one of the contributions and novelties of the proposed study is to analyze quantitatively the contribution of linear and nonlinear parts in the control vector when the control is started exactly at the beginning of the self-oscillations.