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.

Fredholm transformation on Laplacian and rapid stabilization for the heat equation

“…There are mainly two ways to prove the existence of the transformation, either by direct methods [18,19] or, more commonly, by proving the existence of a Riesz basis. For the latter, we again distinguish two cases: either the Riesz basis is deduced directly by an isomorphism applied on an eigenbasis [17,50,51] or the existence of a Riesz basis follows by controllability assumptions and sufficient growth of the eigenvalues of the spatial operator allowing in particular to prove that the family is quadratically close to the eigenfunctions [16,20,21,27] (see Section 2.2 and Section 4 for a definition).…”

“…This condition is called in the literature the "T B = B condition" (here the function φ represent what is usually formally denoted B), which is now becoming a standard requirement for Fredholm type backstepping transformations, [16,17,20,21,27,51]. The aim of this section is to determine a precise candidate of {K n } n∈N * such that (i) The "T B = B condition" (4.88) holds, in a suitable space to be found.…”

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

“…The other results found in the literature were concerned with operators such that α ≥ 2, and in these case the Riesz basis properties was proved through the quadratically close criterion, thanks to the sufficient growth of the eigenvalues. Following the steps described in the previous section, the rapid stabilisation was obtained for the linearized bilinear Schrödinger equation [12], the KdV equation [16], the Kuramoto-Sivashinksi equation [17], a degenerate parabolic operator [20] and finally the heat equation for which the backstepping is proved in sharp spaces [19]. The variety of the PDEs for which this methodology can be applied tends to show that there exists an abstract theory for operators of order α > 1.…”

“…This abstract setting could allow to lift some difficult questions raised when trying to apply the backstepping with the Volterra transformation. One such difficult is seen for instance for degenerate parabolic equations ( [20]), where the Fredholm transformation lead to the study of well-known spectral properties of the Sturm-Liouville equation, whereas the PDE on the kernel of the Volterra transformation amounts to describe the propagation of bicharacteristics from a boundary satisfying a degenerate equation, a notoriously difficult problem.…”