Internal rapid stabilization of a 1-D linear transport equation with a scalar feedback

Abstract

We use a variant the backstepping method to study the stabilization of a 1-D linear transport equation on the interval (0, L), by controlling the scalar amplitude of a piecewise regular function of the space variable in the source term. We prove that if the system is controllable in a periodic Sobolev space of order greater than 1, then the system can be stabilized exponentially in that space and, for any given decay rate, we give an explicit feedback law that achieves that decay rate. The variant of the backstepping method used here relies mainly on the spectral properties of the linear transport equation, and leads to some original technical developments that dier substantially from previous applications.

Fredholm transformation on Laplacian and rapid stabilization for the heat equation

“…As of late, the Fredholm transformation was introduced for the backstepping method as an alternative for certain limitations of the Volterra transformation. In particular, it seems better suited for internal stabilization problems [16,51]. The idea of using transformations remains the same, but proving the existence and invertibility of the transformation is generally more involved.…”

“…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).…”

“…In infinite dimension, the situation is more complex, but the same philosophy applies. The role of controllability in finding such backstepping transformations has been established for several important PDE models such as the transport equation [51], the KdV equation [20], Kuramoto-Sivashinsky equation [21], the linearized Schrödinger equation [16], the linearized Saint-Venant equation [17]. We highlight that the uniqueness equation T B = B was decisive in [16,17,51] to transform non-local terms emanating from distributed control functions into local terms for the operator equality, but was fundamentally used in an implicit way in [20,21] for boundary controls.…”

“…α > 3/4 is the case of the fractional Laplacian), meaning that for a growth of order |λ n | ∼ n s , 1 ≤ s ≤ 3/2 does not seem enough to prove that the family is quadratically close to the eigenbasis. An interesting open problem is therefore to apply the backstepping method with a Fredholm type transformation for spatial operator with eigenvalues with growth |λ n | ∼ n s , 1 < s ≤ 3/2, the first-order equation being excluded due to the positive answer [17,51] and its link to other transformation such as the Hilbert transform. However the case 1 < s ≤ 3/2 remains and this analysis may give some inspiration on the study of fractional Laplacian with the value of α lower than this threshold.…”

“…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.…”