On the controllability of a nonlinear dispersive system in a weighted $L^2$-space

Abstract

Considered here is a model system derived by Gear and Grimshaw to describe the strong interaction of weakly nonlinear, long waves. It has the structure of a pair of Korteweg-de Vries equations coupled through both dispersive and nonlinear effects. Our aim is to investigate the controllability properties of the model, posed on a bounded interval, by means of distributed controls. When the control region is a neighborhood of the right end point of the interval, we prove the local exact controllability of the nonlinear problem in a well chosen weighted L2-spaces. The results are first established for the linearized system through a classical duality approach and then extended for the full system via a fixed point argument.