Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Fredholm backstepping for critical operators and application to rapid
  stabilization for the linearized water waves   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 $\alpha=3/2$.

24
The categories of this paper are: dg-ga
20
The categories of this paper are: acc-phys
35
The categories of this paper are: mtrl-th
34
The categories of this paper are: chem-ph
1
The categories of this paper are: cs
29
The categories of this paper are: funct-an
27
The categories of this paper are: comp-gas
21
The categories of this paper are: adap-org
9
The categories of this paper are: nlin
18
The categories of this paper are: econ
0
The categories of this paper are: hep-ph
36
The categories of this paper are: cmp-lg
31
The categories of this paper are: ao-sci
33
The categories of this paper are: plasm-ph
3
The categories of this paper are: physics
19
The categories of this paper are: eess