Quantitative Fredholm backstepping and rapid stabilization
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866916022536110080 |
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| author | Gagnon, Ludovick Hayat, Amaury Marx, Swann Xiang, Shengquan Zhang, Christophe |
| author_facet | Gagnon, Ludovick Hayat, Amaury Marx, Swann Xiang, Shengquan Zhang, Christophe |
| contents | 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 $λ$. As a consequence, we obtain quantitative rapid stabilization and small-time null controllability results for a broad class of operators. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_17941 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quantitative Fredholm backstepping and rapid stabilization Gagnon, Ludovick Hayat, Amaury Marx, Swann Xiang, Shengquan Zhang, Christophe Optimization and Control Analysis of PDEs Functional Analysis 47A62, 93D15, 93C20, 93C25, 35Q93, 47D06 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 $λ$. As a consequence, we obtain quantitative rapid stabilization and small-time null controllability results for a broad class of operators. |
| title | Quantitative Fredholm backstepping and rapid stabilization |
| topic | Optimization and Control Analysis of PDEs Functional Analysis 47A62, 93D15, 93C20, 93C25, 35Q93, 47D06 |
| url | https://arxiv.org/abs/2605.17941 |