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Fredholm transformation on Laplacian and rapid stabilization for the heat equations

Abstract : We revisit the rapid stabilization of the heat equation on the 1-dimensional torus using the backstepping method with a Fredholm transformation. We prove that, under some assumption on the control operator, two scalar controls are necessary and sufficient to get controllability and rapid stabilization. This classical framework allows us to present the backstepping method with the Fredholm transformation upon Laplace operators in a sharp functional setting, which is the major objective of this work, from the Riesz basis properties and the operator equality to the stabilizing spaces. Finally, we prove that the same Fredholm transformation also leads to the local rapid stability of the viscous Burgers equation.
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-03319847
Contributor : Shengquan Xiang Connect in order to contact the contributor
Submitted on : Friday, October 8, 2021 - 4:01:01 PM
Last modification on : Thursday, January 20, 2022 - 4:19:15 PM

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  • HAL Id : hal-03319847, version 2

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Ludovick Gagnon, Amaury Hayat, Shengquan Xiang, Christophe Zhang. Fredholm transformation on Laplacian and rapid stabilization for the heat equations. 2021. ⟨hal-03319847v2⟩

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