Internal rapid stabilization of a 1-D linear transport equation with a scalar feedback

Christophe Zhang

doi:10.3934/mcrf.2021006

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2022, Volume 12, Issue 1: 169-200. Doi: 10.3934/mcrf.2021006

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Internal rapid stabilization of a 1-D linear transport equation with a scalar feedback

Christophe Zhang^,

Chair for Applied Analysis (Alexander von Humboldt Professorship), Friedrich-Alexander Universität Nürnberg, Cauerstr. 11, 91058 Erlangen

^* Corresponding author: Christophe Zhang
^* Corresponding author: Christophe Zhang

Received: September 2020

Revised: November 2020

Early access: March 2021

Published: March 2022

This work was partially supported by ANR project Finite4SoS (ANR-15-CE23-0007), the French Corps des Mines, and the Chair of Applied Analysis, Alexander von Humboldt Professorship, Friedrich-Alexander Universität Nürnberg.

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Abstract

We use a variant the backstepping method to study the stabilization of a 1-D linear transport equation on the interval $ (0,L) $, by controlling the scalar amplitude of a piecewise regular function of the space variable in the source term. We prove that if the system is controllable in a periodic Sobolev space of order greater than $ 1 $, then the system can be stabilized exponentially in that space and, for any given decay rate, we give an explicit feedback law that achieves that decay rate. The variant of the backstepping method used here relies mainly on the spectral properties of the linear transport equation, and leads to some original technical developments that differ substantially from previous applications.

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Mathematics Subject Classification: 35L02, 93D15, 93B17, 93B30, 93D23.

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