On approximate controllability of generalized KdV solitons

TL;DR

This paper demonstrates that generalized KdV solitons can be approximately controlled using internal bilinear controls, allowing for null controllability and velocity acceleration, with detailed error estimates provided.

Contribution

It introduces a novel control method for nonlinear solitons in KdV equations, including velocity control and precise error analysis.

Findings

Any soliton is locally null controllable.

Solitons can be accelerated to any positive velocity.

Error terms and decay rates are explicitly estimated.

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.