Local null controllability of a model system for strong interaction between internal solitary waves

Abstract

In this paper, we prove the local null controllability property for a nonlinear coupled system of two Korteweg–de Vries equations posed on a bounded interval and with a source term decaying exponentially on [Formula: see text]. The system was introduced by Gear and Grimshaw to model the interactions of two-dimensional, long, internal gravity waves propagation in a stratified fluid. We address the controllability problem by means of a control supported on an interior open subset of the domain and acting on one equation only. The proof consists mainly on proving the controllability of the linearized system, which is done by getting a Carleman estimate for the adjoint system. While doing the Carleman, we improve the techniques for dealing with the fact that the solutions of dispersive and parabolic equations with a source term in [Formula: see text] have a limited regularity. A local inversion theorem is applied to get the result for the nonlinear system.