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 approximate 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 into the control of nonlinear objects from the point of view of interaction and collision problems for nonlinear dispersive equations, recently developed by Martel and Merle [Ann. Math. ( 2), 174 ( 2011), pp. 757-857], [Invent. Math., 183 ( 2011), pp. 563-648]. It can be applied in principle to several other models with soliton solutions.