An abstract setting for the Fredholm backstepping transformation: self-adjoint case
Nov 8, 2025·,,,
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0 min read
Ludovick Gagnon
Amaury Hayat
Swann Marx
Shengquan Xiang
Christophe Zhang
Abstract
In this paper we address the existence of a Fredholm backstepping transformation for self-adjoint operators A. We prove, under assumptions on the control operator B and A, that there exists a Fredholm backstepping transformation for operators A of order strictly greater than 1. One of the main contribution of the paper is to exhibit the underlying isomorphism used to construct T. This isomorphism allows to deduce sharp estimates on ∥T∥ L(H;H) and ∥T^-1∥ L(H;H) with respect to the damping parameter λ > 0. This allow us to prove the finite-time stabilization for a large class of self-adjoint operators.
Type
Authors
Christophe Zhang
(he/him)
researcher
I am currently in secondment at Inria Paris, as a researcher in the CAGE team. I work on control, stabilization and optimal control problems in infinite dimension (PDEs and evolution equation), with a particular focus on constrained control.
I have also dabbled in data-driven modelling and control, in particular methods involving the Koopman operator.
More recently, I have taken an interest in reachability analysis. I co-supervised the PhD thesis of Ivan Hasenohr with Camille Pouchol and Yannick Privat, during which we developed computer-assisted proofs of reachability and non-reachability.