  The Korteweg-de Vries (KdV) equation with the right Dirichlet control was
initially investigated more than twenty years ago. It was shown that this
system is small time, locally, exactly controllable for all non-critical
lengths and its linearized system is not controllable for {\it all} critical
lengths. Even though the controllability of the KdV system has been studied
extensively in the last two decades, the local controllability of this system
for critical lengths remains an open question. In this paper, we give a
definitive answer to this question. First, we characterize all critical lengths
and the corresponding unreachable space for the linearized system. In
particular, we show that the unreachable space is always of dimension 1.
Second, we prove that the KdV system with the right Dirichlet control is not
locally null controllable in small time. Third, we give a criterion to
determine whether the system is locally exactly controllable in finite time or
{\it not} locally null controllable in any positive time for {\it all} critical
lengths. Consequently, we show that there exist critical lengths such that the
system is not locally null controllable in small time but is locally exactly
controllable in finite time. These facts are surprising and distinct in
comparison with related known results. First, it is known that the
corresponding KdV system with the right zero Dirichlet is locally exactly
controllable in small time using internal controls. Second, the unreachable
space of the linearized system of the corresponding KdV system with the right
Neumann control might be of arbitrary dimension. Third, the KdV system with the
right Neumann control is locally exactly controllable in small time if the
corresponding unreachable space is of dimension 1.
