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

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 $c\mathrm{\ne }-1$ and $${\mathcal{R}}_c:=\{\frac{2\pi }{\sqrt{3(c+1)}}\sqrt{m^2+ml+m^2};m,l\in {\mathbb{N}}^{\ast }\}\cup \{\frac{m\pi }{\sqrt{c+1}};m\in {\mathbb{N}}^{\ast }\},$$ 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.