FREDHOLM BACKSTEPPING FOR CRITICAL OPERATORS AND APPLICATION TO RAPID STABILIZATION FOR THE LINEARIZED WATER WAVES

— Fredholm-type backstepping transformation, introduced by Coron and Lü, intensively developed over the last decade and has become a powerful tool for rapid stabilization. Its strength lies in its systematic approach, allowing to deduce rapid stabilization from approximate controllability. But limitations to prove the existence of a Fredholm backstepping transformation exist with the current approach for operators of the form |Dx|^α for α ∈ (1, 3/2]. We present here a new compactness/duality method which hinges on Fredholm’s alternative to overcome the α = 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 operators satisfying α > 1, a key step in the construction of the Fredholm backstepping transformation, where the usual methods only work for α > 3/2. The illustration of this new method is shown on the rapid stabilization of the linearized capillary-gravity water waves equation exhibiting an operator of critical order α = 3/2.