Fredholm transformation on Laplacian and rapid stabilization for the heat equation

2021/10/08 by Ludovick Gagnon, Amaury Hayat, Gagnon, Ludovick +5 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC) #math.AP #math.OC

paper · pdf · doi:10.48550/arxiv.2110.04028

57 pages

arxiv created 2021/10/08 · arxiv updated 2021/10/11

Abstract

We study 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 Fredholm transformations on Laplace operators in a sharp functional setting, which is the main objective of this work. Finally, we prove that the same Fredholm transformation also leads to the local rapid stability of the viscous Burgers equation.

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