# Quantitative Fredholm backstepping and rapid stabilization

arXiv: https://arxiv.org/abs/2605.17941 · HTML: https://mlsanity.com/paper/2605.17941/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 May 2026.

## Built on by (most cited first, 1 of 1)

- [Rapid Fredholm stabilization of the Kuramoto--Sivashinsky equation with unrestricted, spatially-varying anti-diffusion](https://mlsanity.com/paper/2610.08764/) (2026)

## Abstract (arXiv)

In this paper, we address the existence of Fredholm backstepping transformations for self-adjoint and skew-adjoint operators A. Under suitable assumptions on the operator A and the possibly unbounded control operator B, we prove the existence of a Fredholm backstepping transformation for operators of order strictly greater than 1. This work overcomes two major limitations of the classical Fredholm backstepping framework. One of the main contributions is the explicit identification of the underlying isomorphism used in the construction of the transformation T, thereby bypassing the compactness arguments and Riesz basis mechanisms traditionally used in the literature. This explicit structure enables us to derive quantitative and sharp estimates for |T|L(H;H) and |T-1|L(H;H) with respect to the decay rate λ. As a consequence, we obtain quantitative rapid stabilization and small-time null controllability results for a broad class of operators.