Fredholm backstepping for critical operators and application to rapid stabilization for the linearized water waves

Abstract

Fredholm-type backstepping transformation, introduced by Coron and Lü, 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 |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 operator verifying 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 wave equation exhibiting an operator of critical order α = 3/2.

Rapid stabilization of a degenerate parabolic equation using a backstepping approach: The case of a boundary control acting at the degeneracy

“…To study the stabilization of the system (1), as already mentioned, we will use the backstepping method. We focus on the construction of an appropriate Fredholm transformation, in the spirit of [11], which has emerged as an alternative to the Volterra transformation during the last years (see also [12,13,28,16,17,15]). Let us emphasize that our main stabilization result can be obtained from the abstract theorem given in [1,Theorem 1.6], but the backstepping method has some additional advantages (see [16, Remark 2]) that make relevant the present study.…”

“…By using the change of variable y = x κ , then dx = dyx 1−κ κ and we obtain, using (15) for the first term and (8),…”