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

Abstract

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.