Boundary controllability of a Korteweg–de Vries-type Boussinesq system

The two-way propagation of a certain class of long-crested water waves is governed approximately by systems of equations of the Boussinesq type. These equations have been put forward in various forms by many authors and their higher-order generalizations arise when modeling the propagation of waves on large lakes, ocean, and in other contexts. Considered here is a class of such system which couple two higher-order Korteweg–de-Vries type equations. Our aim is to investigate the controllability properties of the linearized model posed on a periodic interval. By using the classical duality approach and some theorems on nonharmonic Fourier series, we prove that the system is exactly controllable in certain well-chosen Sobolev spaces by means of suitable boundary controls.