A Fredholm Transformation for the Rapid Stabilization of a Degenerate Parabolic Equation

Abstract

This paper deals with the rapid stabilization of a degenerate parabolic equation with a right Dirichlet control. Our strategy consists in applying a backstepping strategy, which seeks to find an invertible transformation mapping the degenerate parabolic equation to stabilize into an exponentially stable system whose decay rate is known and as large as we desire. The transformation under consideration in this paper is Fredholm. It involves a kernel solving itself another PDE, at least formally. The main goal of the paper is to prove that the Fredholm transformation is well-defined, continuous and invertible in the natural energy space. It allows us to deduce the rapid stabilization.

Boundary stabilization of 1D hyperbolic systems

“…In the recent years several studies started to look at other more general linear transforms such as Fredholm transforms [52,49,46,154,155,48,75,74]. These transforms are more general, and therefore potentially more powerful, but they are not always invertible and proving the invertibility of the candidate transform becomes one of the main difficulties.…”

Boundary stabilization of one-dimensional cross-diffusion systems in a moving domain: Linearized system

“…The usual backstepping approach for PDE, presented in [40], searches for isomorphisms under the form of a Volterra transform of the second kind (see (4.3)), which are conveniently always invertible, among other advantages. Some attempt to introduce a generalized backstepping approach which does not necessarily rely on Volterra transforms have also been introduced in [20,17,27,49,50,18,26]. The Volterra approach has been used in many areas and for many systems in the last decades including parabolic equations (see for instance [5,21,24]), hyperbolic system (see for instance [39,46,3,2,29,30,19]), etc.…”

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

“…Let us now emphasize the main difficulties that arise compared to [16] (where the control was localized at point x = 1), that are of technical nature but make the present situation different and somehow trickier. Of course, the spectral properties of the underlying elliptic operator are the same in both cases, but the behaviour of the control operator changes: the asymptotic behaviour of the normal derivative of the eigenvectors at point x = 0 (which is related to the adjoint operator of our control operator) is different from the one at point x = 1 (see (24) and (25)).…”

“…Here, we consider the case where α ∈ (0, 1), so that our equation corresponds to what is called a weakly degenerate case according to the terminology of [8]. Notice that the stabilization by the backstepping method has been proved in [16] in the case where the control U is localized at point x = 1 (see also [7] for a result of finite-time stabilization with a distributed control, by a totally different technique relying on spectral tools). Here, the situation is different since we impose a Dirichlet control at point x = 0, where the degeneracy of the elliptic operator (x α ∂ x ) x holds, which turns out to be a more difficult question.…”

“…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.…”