Indirect stabilization of hyperbolic systems through resolvent estimates

Roberto Guglielmi

doi:10.3934/eect.2017004

Article Contents

2017, Volume 6, Issue 1: 59-75. Doi: 10.3934/eect.2017004

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Indirect stabilization of hyperbolic systems through resolvent estimates

Roberto Guglielmi

Radon Institute for Computational and Applied Mathematics - RICAM, Altenbergerstraße 69, 4040 - Linz, Austria

Received: January 31, 2015

Revised: August 31, 2016

Published: March 2018

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Abstract

We prove a sharp decay rate for the total energy of two classes of systems of weakly coupled hyperbolic equations. We show that we can stabilize the full system through a single damping term, in feedback form, acting on one component only of the system (\emph{indirect stabilization}). The energy estimate is achieved by means of suitable estimates of the resolvent operator norm. We apply this technique to a wave-wave system and to a wave-Petrovsky system.

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Mathematics Subject Classification: Primary: 93D15, 35L53, 47A10; Secondary: 47D06.

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