Quantitative Fredholm backstepping and rapid stabilization

TL;DR

This paper develops a quantitative Fredholm backstepping method for self-adjoint and skew-adjoint operators, enabling rapid stabilization and null controllability with explicit estimates, overcoming classical limitations.

Contribution

It introduces an explicit isomorphism in Fredholm backstepping, allowing for sharp estimates and broad applicability beyond classical compactness-based approaches.

Findings

Established existence of Fredholm backstepping transformations for operators of order > 1.

Derived explicit bounds for the transformation norms related to decay rates.

Achieved quantitative rapid stabilization and small-time null controllability results.

Abstract

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 $\|T\|_{\mathcal{L}(H;H)}$ and $\|T^{-1}\|_{\mathcal{L}(H;H)}$ with respect to the decay rate $\lambda$. As a consequence, we obtain quantitative rapid stabilization and small-time null controllability results for a broad class of operators.