  We consider a Boussinesq system of KdV-KdV type introduced by J. Bona, M.
Chen and J.-C. Saut as a model for the motion of small amplitude long waves on
the surface of an ideal fluid. This system of two equations can describe the
propagation of waves in both directions, while the single KdV equation is
limited to unidirectional waves. We are concerned here with the exact
controllability of the Boussinesq system by using some boundary controls. By
reducing the controllability problem to a spectral problem which is solved by
using the Paley-Wiener method introduced by the third author for KdV, we
determine explicitly all the critical lengths for which the exact
controllability fails for the linearized system, and give a complete picture of
the controllability results with one or two boundary controls of Dirichlet or
Neumann type. The extension of the exact controllability to the full Boussinesq
system is derived in the energy space in the case of a control of Neumann type.
It is obtained by incorporating a boundary feedback in the control in order to
ensure a global Kato smoothing effect.
