Fredholm transformation on Laplacian and rapid stabilization for the heat equation

Abstract

We study the rapid stabilization of the heat equation on the 1-dimensional torus using the backstepping method with a Fredholm transformation. We prove that, under some assumption on the control operator, two scalar controls are necessary and sufficient to get controllability and rapid stabilization. This classical framework allows us to present the backstepping method with Fredholm transformations on Laplace operators in a sharp functional setting, which is the main objective of this work. Finally, we prove that the same Fredholm transformation also leads to the local rapid stability of the viscous Burgers equation.

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

“…We shall remark here that the bound r = 1 is sharp in the sense that for r ≥ 1 the unbounded operator A + BK does not anymore generate a strongly continuous semigroup in H r . We also underline that, while the isomorphism T depends on the regularity of the state space H r , the feedback law is, surprisingly, independent of r. This independence was already noticed in [19].…”

“…where A := L . As noted in [12,17,19], a good approach to this aim is to add a condition on T B, and to require instead the two following operator equalities…”

“…Note that (3.4) implies formally (3.3) and requiring (3.4) instead of (3.3) allows to deal with operator equations that are linear with (T, K). It also usually ensures the uniqueness of solution (see for instance [19]). In finite dimension, (3.3) corresponds to an equivalent formulation of the pole-shifting theorem [13,Section 2].…”

“…To be more precise, we consider the external pressure (the control) to be of the distributed control form P ext = B 1 (x)w 1 (t) + B 2 (x)w 2 (t). Notice that L has double eigenvalues (see Section 1.8), which means according to [19] two distributed controls are required to control/stabilize the system instead of one. Hence, for ease of notations we consider (1.6)…”

“…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.…”