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In this paper we study the boundary controllability of the Gear-Grimshaw system posed on a finite domain (0, L), with Neumann boundary conditions:We first prove that the corresponding linearized system around the origin is exactly controllable in (L 2 (0, L)) 2 when h2(t) = g2(t) = 0. In this case, the exact controllability property is derived for any L > 0 with control functions h0, g0 ∈ H − 1 3 (0, T ) and h1, g1 ∈ L 2 (0, T ). If we change the position of the controls and consider h0(t) = h2(t) = 0 (resp. g0(t) = g2(t) = 0) we obtain the result with control functions g0, g2 ∈ H − 1 3 (0, T ) and h1, g1 ∈ L 2 (0, T ) if and only if the length L of the spatial domain (0, L) belongs to a countable set. In all cases the regularity of the controls are sharp in time. If only one control act in the boundary condition, h0(t) = g0(t) = h2(t) = g2(t) = 0 and g1(t) = 0 (resp. h1(t) = 0), the linearized system is proved to be exactly controllable for small values of the length L and large time of control T . Finally, the nonlinear system is shown to be locally exactly controllable via the contraction mapping principle, if the associated linearized systems are exactly controllable.1991 Mathematics Subject Classification. Primary 35Q53; Secondary 37K10, 93B05, 93D15.

This work deals with the local rapid exponential stabilization for a Boussinesq system of KdV-KdV type introduced by J. Bona, M. Chen and J.-C. Saut. This is a model for the motion of small amplitude long waves on the surface of an ideal fluid. Here, we will consider the Boussinesq system of KdV-KdV type posed on a finite domain, with homogeneous Dirichlet-Neumann boundary controls acting at the right end point of the interval. Our goal is to build suitable integral transformations to get a feedback control law that leads to the stabilization of the system. More precisely, we will prove that the solution of the closed-loop system decays exponentially to zero in the L 2 (0, L)-norm and the decay rate can be tuned to be as large as desired if the initial data is small enough.2010 Mathematics Subject Classification. Primary: 93B05, 93D15, 35Q53.

The aim of this work is to consider the controllability problem of the linear system associated to Korteweg-de Vries Burgers equation posed in the whole real line. We obtain a sort of exact controllability for solutions in L 2 loc (R 2 ) by deriving an internal observability inequality and a Global Carlemann estimate. Following the ideas contained in [26], the problem is reduced to prove an approximate theorem.2010 Mathematics Subject Classification. Primary: 35Q53, Secondary: 37K10, 93B05, 93D15.

A family of Boussinesq systems has been proposed to describe the bi-directional propagation of small amplitude long waves on the surface of shallow water. In this paper, we investigate the well-posedness and boundary stabilization of the generalized higher order Boussinesq systems of Korteweg-de Vries-type posed on a interval. We design a two-parameter family of feedback laws for which the system is locally well-posed and the solutions of the linearized system are exponentially decreasing in time.Date: 2018-08-27-a. 2010 Mathematics Subject Classification. Primary: 93B05, 93D15, 35Q53.