Control and stabilization of 2 × 2 hyperbolic systems on graphs

Serge Nicaise

doi:10.3934/mcrf.2017004

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2017, Volume 7, Issue 1: 53-72. Doi: 10.3934/mcrf.2017004

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Control and stabilization of 2 × 2 hyperbolic systems on graphs

Serge Nicaise^,

Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, F-59313 -Valenciennes Cedex 9, France

* Corresponding author
* Corresponding author

Received: May 31, 2015

Revised: January 31, 2016

Published: May 2017

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Abstract

We consider 2× 2 (first order) hyperbolic systems on networks subject to general transmission conditions and to some dissipative boundary conditions on some external vertices. We find sufficient but natural conditions on these transmission conditions that guarantee the exponential decay of the full system on graphs with dissipative conditions at all except one external vertices. This result is obtained with the help of a perturbation argument and an observability estimate for an associated wave type equation. An exact controllability result is also deduced.

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Mathematics Subject Classification: Primary: 35L50; Secondary: 93D15.

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