Boundary controllability of the Korteweg-de Vries equation: The Neumann case

TL;DR

This paper investigates the boundary controllability of the Korteweg-de Vries (KdV) equation with Neumann boundary conditions on a finite interval, establishing controllability results in critical and non-critical cases using advanced control methods.

Contribution

It provides the first step towards understanding the critical set phenomenon for the KdV equation with Neumann boundary conditions and demonstrates controllability in the critical case for specific domain lengths.

Findings

KdV is controllable in the critical case when domain length is in set 

Exact controllability achieved in $L^2(0,L)$ for certain lengths

Utilizes return method and fixed point argument for control proof

Abstract

This article gives a necessary first step to understanding the critical set phenomenon for the Korteweg-de Vries (KdV) equation posed on interval $[0,L]$ considering the Neumann boundary conditions with only one control input. We showed that the KdV equation is controllable in the critical case, i.e., when the spatial domain $L$ belongs to the set $\mathcal{R}_c$, where $c\neq-1$ and $$ \mathcal{R}_c:=\left\{\frac{2\pi}{\sqrt{3(c+1)}}\sqrt{m^2+ml+m^2};\ m,l\in \mathbb{N}^*\right\}\cup\left\{\frac{m\pi}{\sqrt{c+1}};\ m\in \mathbb{N}^*\right\}, $$ the KdV equation is exactly controllable in $L^2(0,L)$. The result is achieved using the return method together with a fixed point argument.