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

Martin Gugat; Markus Dick

doi:10.3934/mcrf.2011.1.469

Article Contents

2011, Volume 1, Issue 4: 469-491. Doi: 10.3934/mcrf.2011.1.469

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

Martin Gugat^1, and
Markus Dick^1,

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

Received: December 2010

Revised: June 2011

Published: December 2011

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Abstract

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.

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Mathematics Subject Classification: 76N25, 35L50, 93C20.

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