Controllability for a Boussinesq system of BBM-BBM type with moving control

Abstract A family of Boussinesq systems was proposed by Bona, Chen and Saut to describe the two-way propagation of small amplitude gravity waves on the surface of water in a canal. Our work considers a class of these Boussinesq systems which couples two Benjamin–Bona–Mahony type equations posed on the one dimensional torus T . It is known that these type of dispersive equations have poor controllability properties and special control techniques must be employed to obtain positive results. One of these techniques consists of considering a control supported on a subset of the domain which is moving in time. We analyse the possibility to steer the components of the system to a given target when only one control of this type is available. By combining a moment approach and nonharmonic Fourier analysis, we first obtain the exact controllability of the linearized system in asymmetric Sobolev spaces for which the two components have different regularities. Then, by means of a contraction mapping principle, we establish the local exact controllability for the original nonlinear system in an asymmetric Sobolev space.