Fredholm transformation on Laplacian and rapid stabilization for the heat equations

Ludovick Gagnon (1, 2) , Amaury Hayat (3) , Shengquan Xiang (4) , Christophe Zhang (1)

1 SPHINX - Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization
2 IECL - Institut Élie Cartan de Lorraine
3 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
4 Département de Mathématiques - EPFL

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.

Mots clés: Fredholm transformation; backstepping; rapid stabilization; controllability

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Ludovick Gagnon, Amaury Hayat, Shengquan Xiang, Christophe Zhang. Fredholm transformation on Laplacian and rapid stabilization for the heat equations. 2021. ⟨hal-03319847v1⟩