  In this paper, we prove the output feedback stabilization for the linearized
Korteweg-de Vries (KdV) equation posed on a finite domain in the case the full
state of the system cannot be measured. We assume that there is a sensor at the
left end point of the domain capable of measuring the first and second order
boundary traces of the solution. This allows us to design a suitable observer
system whose states can be used for constructing boundary feedbacks acting at
the right endpoint so that both the observer and the original plant become
exponentially stable. Stabilization of the original system is proved in the
$L^2$-sense, while the convergence of the observer system to the original plant
is also proved in higher order Sobolev norms. The standard backstepping
approach used to construct a left endpoint controller fails and presents
mathematical challenges when building right endpoint controllers due to the
overdetermined nature of the related kernel models. In order to deal with this
difficulty we use the method of [18] which is based on using modified target
systems involving extra trace terms. In addition, we show that the number of
controllers and boundary measurements can be reduced to one, with the cost of a
slightly lower exponential rate of decay. We provide numerical simulations
illustrating the efficacy of our controllers.
