## 1 Introduction
In [20], the problem translated into the hybrid model paradigm [10] is solved through time or space regularization approaches. The models used in all the works described above are closely related to the one developed in this paper. They belong to the class of polytopic and switched models more recently investigated (see, e.g., [17; 21; 22; 23]). The main objective of this work is to propose a generic way for solving several control problems of interest associated with a large class of power converters. The idea is to formulate these problems in a general setting and propose solutions which are justified and qualified from a theoretical point of view using efficient theoretical and numerical tools. A crucial step consists in manipulating the original bilinear converter model in a way to allow exhibiting a constant dynamical matrix (i.e., independent of the control variable). The resulting model is closely related to the switched model considered in the literature (e.g, [15; 17]), but the main difference, which could appear as a detail but is central, resides in the possibility of writing the dynamical equation as a sum of a first term, linear with respect to the state, and a second one, which is a bilinear matrix function of the state and control variables. Moreover, the matrix associated with the linear part is Hurwitz. This important particularity allows solving control problems which cannot be easily solved when considering the original switched model. Associated with the modified switched model, it is also possible to define a relaxed model whose control variables are continuous and belong to the interval \([0,1]\). Among the main properties of this relaxed model, the following ones are central. Firstly, the model is convex with respect to the control variables. Secondly, the solution set of initial value problems of the switched model (trajectories) is dense in the solution set of initial value problems of the relaxed one for the \(C^{0}\) Whitney topology on the infinite interval \([0,\infty)\)[24; 25]. This last property is very important from a practical point of view. It means that a trajectory of the relaxed model can be arbitrarily approached by a trajectory of the switched one. A similar result is obtained in [26], where the connection between the two systems is used to solve optimal control problems. In the context of power converters, a control law developed from the relaxed model can be translated into a control compatible with the switched model using, for example a Pulse Width Modulator (PWM) of appropriate frequency or a hysteresis device [27]. This specificity can also be used to transform a variable frequency control into a control of a fixed desired frequency, simply by appropriately filtering the first one to deduce its associated duty-cycle, used after as the input of a PWM. Some care has to be taken to prevent undesirable effects such as steady state errors (see [28] for details). Finally, the approaches proposed in this paper could be extended in the context of hybrid model paradigm proposed in [20] with the associated time and space regularization techniques. This extension is not considered in this work and could be addressed in a near future. Due to space limitation, the case of stabilization and tracking problems is treated in two separate papers. The present paper deals with control laws design closely related to stabilization problems. It is organized as follows. The next section introduces the main assumptions and the different steps which lead to the general switched model considered in this paper. Intimately related to it, the relaxed model is also introduced and their relation is quickly invoked. Section 3 deals with the important problem of stabilization of a given equilibrium state. An important assumption is central for deriving the main result of the section. It is shown that this assumption is automatically satisfied for a large class of power converters, at least for the ones considered in all the works invoked above.