  In this paper we study boundary controllability of the Korteweg-de Vries
(KdV) equation posed on a finite domain $(0,L)$ with the Neumann boundary
conditions:
  u_t+u_x+uu_x+u_{xxx}=0 in (0,L)x(0,T),
  u_{xx}(0,t)=0, u_x(L,t)=h(t), u_{xx}(L,t)=0 in (0,T),
  u(x,0)=u_0(x) in (0,L).
  We show that the associated linearized system
  u_t+(1+\beta)u_x+u_{xxx}=0 in (0,L)x(0,T),
  u_{xx}(0,t)=0, u_x(L,t)=h(t), u_{xx}(L,t)=0 in (0,T),
  u(x,0)=u_0(x) in (0,L)
  is exactly controllable if and only if the length $L$ of the spatial domain
$(0,L)$ does not equal to $-1$ or does not belong to set
  R_{\beta}:={\frac{2\pi}{\sqrt{3(1+\beta)}}\sqrt{k^{2}+kl+l^{2}}:k,l\in\mathbb{N}^{\ast}}\cup{\frac{k\pi}{\sqrt{1+\beta}}:k\in\mathbb{N}^{\ast}}
  and the nonlinear system is locally exactly controllable around a constant
steady state $\beta$ if the associated linear system is exactly controllable.
