Another look at the control properties of the Korteweg-de Vries equation

Abstract

This paper represents a new perspective in understanding the controllability of the Korteweg-de Vries (KdV) equation on unbounded domains. By studying the equation on both the right and left half-line with a single control input, we show that a class of solutions exists for which the KdV equation is exactly controllable. This is accomplished through the introduction of a method for explicitly characterizing controls arising from the Hilbert Uniqueness Method, referred to as operational controllability, which yields fundamental insights for proving exact controllability results for the KdV equation. This approach allows for explicitly characterizing both the control input and the controllable solutions. Furthermore, this concept holds significant potential for application to various nonlinear dispersive equations on the half-line and in bounded intervals.