  We study the backstepping stabilization of higher order linear and nonlinear
Schr\"odinger equations on a finite interval, where the boundary feedback acts
from the left Dirichlet boundary condition. The plant is stabilized with a
prescribed rate of decay. The construction of the backstepping kernel is based
on a challenging successive approximation analysis. This contrasts with the
case of second order pdes. Second, we consider the case where the full state of
the system cannot be measured at all times but some partial information such as
measurements of a boundary trace are available. For this problem, we
simultaneously construct an observer and the associated backstepping controller
which is capable of stabilizing the original plant. Wellposedness and
regularity results are provided for all pde models. Although the linear part of
the model is similar to the KdV equation, the power type nonlinearity brings
additional difficulties. We give two examples of boundary conditions and
partial measurements. We also present numerical algorithms and simulations
verifying our theoretical results to the fullest extent. Our numerical approach
is novel in the sense that we solve the target systems first and obtain the
solution to the feedback system by using the bounded invertibility of the
backstepping transformation.
