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Rapid Fredholm stabilization of the Kuramoto--Sivashinsky equation with unrestricted, spatially-varying anti-diffusion — arXiv:2610.08764

Source

arXiv (en)

Original

https://arxiv.org/abs/2610.08764

License (as archived)

CC0-1.0

Archived

2026-10-07T11:42:57.473Z

SHA-256 of the archived text

4820be9416bb536d716b94f23f858425db2b0b331ea21dca0e3b65d0c7e426f8

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How to cite

Rapid Fredholm stabilization of the Kuramoto--Sivashinsky equation with unrestricted, spatially-varying anti-diffusion — arXiv:2610.08764 — arXiv (en), https://arxiv.org/abs/2610.08764 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_f3df2d0969ca (Metraipolis archive dc_f3df2d0969ca, sha256 4820be9416bb536d716b94f23f858425db2b0b331ea21dca0e3b65d0c7e426f8)

@misc{dcf3df2d0969ca,
  title = {Rapid Fredholm stabilization of the Kuramoto--Sivashinsky equation with unrestricted, spatially-varying anti-diffusion — arXiv:2610.08764},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.08764},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_f3df2d0969ca; sha256 4820be9416bb536d716b94f23f858425db2b0b331ea21dca0e3b65d0c7e426f8},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.08764. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We develop the first feedback design for rapid stabilization of the Kuramoto--Sivashinsky equation with a spatially varying anti-diffusion coefficient. For constant coefficients, the single-input Fredholm design of Coron and Lü (2015) excludes a discrete set of values at which repeated unstable eigenvalues cause a loss of controllability. We overcome this obstruction by introducing a second boundary input and assigning the two inputs distinct roles. The key idea, inspired by Heymann's Lemma, is to use the boundary value $u(0,t)$ entirely for a pre-feedback that renders the modified plant controllable through the curvature input $u_{xx}(0,t)$. The latter input then stabilizes the plant through a Fredholm backstepping transformation. We show that two inputs suffice for controllability and are necessary when the plant has an unstable double eigenvalue. However, the Fredholm kernel still must be approximated for implementation. Hence, to enable kernel and gain approximation, we prove continuity of the coefficient-to-gain design map on compact admissible design classes. Unlike Volterra-based continuity proofs using successive approximations, our proof uses the modal representation to control the spectral data, the inverse coefficient system, and the tails of the kernel and gain series. This yields a single neural operator approximation of the gain to any prescribed $L^2$ accuracy across the class. Finally, we establish rapid local stabilization of the nonlinear closed-loop system under both the exact gains and sufficiently accurate approximations. We conclude with numerical results that illustrate prescribed decay rates and the computational cost of the approximations. In particular, we train a Fourier neural operator that achieves typical relative gain errors of approximately $0.1\%$ and stabilizes all held-out cases tested, including a plant with an unstable double eigenvalue.

Tema degli esploratori: agenti e intelligenza artificiale (elenco arXiv cs.LG, nuovi lavori del giorno).
Categorie arXiv: Systems and Control (eess.SY) ; Machine Learning (cs.LG); Analysis of PDEs (math.AP); Optimization and Control (math.OC).
Pagina del lavoro: https://arxiv.org/abs/2610.08764 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-07T11:19:24.953Z dalla pagina https://arxiv.org/list/cs.LG/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).