# American Institute of Mathematical Sciences

March  2017, 7(1): 53-72. doi: 10.3934/mcrf.2017004

## Control and stabilization of 2 × 2 hyperbolic systems on graphs

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

* Corresponding author

Received  June 2015 Revised  February 2016 Published  December 2016

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.

Citation: Serge Nicaise. Control and stabilization of 2 × 2 hyperbolic systems on graphs. Mathematical Control & Related Fields, 2017, 7 (1) : 53-72. doi: 10.3934/mcrf.2017004
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