We prove the null controllability of a linearized Korteweg-de Vries equation with a Dirichlet control on the left boundary. Instead of considering classical methods, i.e. Carleman estimates, moment method etc., we use a backstepping approach which is a method usually used to handle stabilization problems. type controls are less preferred than piecewise continuous controls (or even L p type conditions), especially for stabilization problems. In [9], Coron and Cerpa proved rapid stabilization of the system (1.1) by using the backstepping method. But since they used some stationary feedback laws, this boundary condition problem is avoided. Recently, by using the (piecewise) backstepping approach, Coron and Nguyen proved the null controllability and semi-global finite time stabilization for a class of heat equations (see [23]). They showed how the use of the maximum principle leads to the well-posedness of the closed-loop system. Their method turns out to be a potential way to solve the local (or even semi-global) finite time stabilization problem for systems which can be rapidly stabilized by means of backstepping methods. At the same time, this method provides a visible way to get null controllability directly instead of using observability inequalities and the duality between controllability and observability.Initially the backstepping is a method to design stabilizing feedback laws in a recursive manner for systems having a triangular structure. See, for example, [15, Section 12.5]. It was first introduced to deal with finite-dimensional control systems. But it can also be used for control systems modeled by means of partial differential equations (PDEs) as shown first in [18]. For linear partial differential equations, a major innovation is due to Krstic and his collaborators. They observed that, when applied to the classical discretization of these systems, the backstepping leads, at the PDEs level (as the mesh size tends to 0), to the transformation of the initial system into a new target system which can be easily stabilized. This transformation is accomplished by means of a Volterra transform of the second kind. An excellent introduction to this method is presented in [39]. Krstic's innovation has been shown to be very useful for many PDEs control systems as, in particular, heat equations [45,4,23], wave equations [60], hyperbolic systems [25,27,36,20] [2, Chapter 7], Korteweg de Vries equations [9,21], and Kuramoto-Sivashinsky equations [44,22]. It was observed later on that for some PDEs more general transforms than Volterra transforms of the second kind have to be considered: see [19,21,22]). Recently, the backstepping method has been adapted to coupled systems, for example the Boussinesq system of KdV-KdV type [5]. For the case of finite dimensional control system and Krstic's backstepping, see [16].Krstic's backstepping requires solving a kernel equation. In the case of the heat equation, the kernel equation is a wave equation; however, in this paper the kernel equation turns out to be a third-order equ...
