# American Institute of Mathematical Sciences

December  2011, 1(4): 469-491. doi: 10.3934/mcrf.2011.1.469

## Time-delayed boundary feedback stabilization of the isothermal Euler equations with friction

 1 Lehrstuhl 2 für Angewandte Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstr. 11, 91058 Erlangen, Germany, Germany

Received  December 2010 Revised  June 2011 Published  November 2011

We consider the isothermal Euler equations with friction that model the gas flow through pipes. We present a method of time-delayed boundary feedback stabilization to stabilize the isothermal Euler equations locally around a given stationary subcritical state on a finite time interval. The considered control system is a quasilinear hyperbolic system with a source term. For this system we introduce a Lyapunov function with delay terms and develop time-delayed boundary controls for which the Lyapunov function decays exponentially with time. We present the stabilization method for a single gas pipe and for a star-shaped network of pipes.
Citation: Martin Gugat, Markus Dick. Time-delayed boundary feedback stabilization of the isothermal Euler equations with friction. Mathematical Control & Related Fields, 2011, 1 (4) : 469-491. doi: 10.3934/mcrf.2011.1.469
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