  Fredholm-type backstepping transformation, introduced by Coron and L\"u, has
become a powerful tool for rapid stabilization with fast development over the
last decade. Its strength lies in its systematic approach, allowing to deduce
rapid stabilization from approximate controllability. But limitations with the
current approach exist for operators of the form $|D_x|^\alpha$ for $\alpha \in
(1,3/2]$. We present here a new compactness/duality method which hinges on
Fredholm's alternative to overcome the $\alpha=3/2$ threshold. More precisely,
the compactness/duality method allows to prove the existence of a Riesz basis
for the backstepping transformation for skew-adjoint operator verifying
$\alpha>1$, a key step in the construction of the Fredholm backstepping
transformation, where the usual methods only work for $\alpha>3/2$. The
illustration of this new method is shown on the rapid stabilization of the
linearized capillary-gravity water wave equation exhibiting an operator of
critical order $\alpha=3/2$.

Analysis of PDEs;Optimization and Control