On approximate controllability of generalized KdV solitons

Abstract

We consider the approximate control of solitons in generalized Korteweg-de Vries equations. By introducing a suitable internal bilinear control on the equation, we prove that any soliton is locally null controllable, and moreover, any soliton can be accelerated to any particular positive velocity, after a suitable large amount of time. Precise estimates on the error terms and the rate of decay in the approximate null controllability result are also given. Our method introduces a new insight on the control of nonlinear objects, from the point of view of interaction and collision problems for nonlinear dispersive equations, recently developed by Y. Martel and F. Merle. It can be applied in principle, to several other models with soliton solutions.