Control of a Boussinesq system of KdV-KdV type on a bounded interval

TL;DR

This paper investigates the exact controllability of a Boussinesq system modeling wave motion, identifying critical lengths where controllability fails and extending results to the full system with boundary feedback.

Contribution

It explicitly determines critical lengths for controllability failure and extends controllability results to the full Boussinesq system using boundary feedback techniques.

Findings

Identified all critical lengths for controllability failure.

Established controllability results with boundary controls.

Extended controllability to the full system with boundary feedback.

Abstract

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.