  In this paper, we design Dirichlet-Neumann boundary feedback controllers for
the Korteweg-de Vries (KdV) equation that act at the right endpoint of the
domain. The length of the domain is allowed to be critical. Constructing
backstepping controllers that act at the right endpoint of the domain is more
challenging than its left endpoint counterpart. The standard application of the
backstepping method fails, because corresponding kernel models become
overdetermined. In order to deal with this difficulty, we introduce the
pseudo-backstepping method, which uses a pseudo-kernel that satisfies all but
one desirable boundary condition. Moreover, various norms of the pseudo-kernel
can be controlled through a parameter in one of its boundary conditions. We
prove that the boundary controllers constructed via this pseudo-kernel still
exponentially stabilize the system with the cost of a low exponential rate of
decay. We show that a single Dirichlet controller is sufficient for exponential
stabilization with a slower rate of decay. We also consider a second order
feedback law acting at the right Dirichlet boundary condition. We show that
this approach works if the main equation includes only the third order term,
while the same problem remains open if the main equation involves the first
order and/or the nonlinear term(s). At the end of the paper, we give numerical
simulations to illustrate the main result.
