A strict $H^1$-Lyapunov function and feedback stabilization for the isothermal Euler equations with friction

Markus Dick; Martin Gugat; Günter Leugering

doi:10.3934/naco.2011.1.225

Article Contents

2011, Volume 1, Issue 2: 225-244. Doi: 10.3934/naco.2011.1.225

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A strict $H^1$-Lyapunov function and feedback stabilization for the isothermal Euler equations with friction

1.
Lehrstuhl 2 für Angewandte Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Martensstr. 3, 91058 Erlangen

Received: September 30, 2010

Revised: December 31, 2010

Published: May 2011

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Abstract

We study the isothermal Euler equations with friction and consider non-stationary solutions locally around a stationary subcritical state on a finite time interval. The considered control system is a quasilinear hyperbolic system with a source term. For the corresponding initial-boundary value problem we prove the existence of a continuously differentiable solution and present a method of boundary feedback stabilization. We introduce a Lyapunov function which is a weighted and squared $H^1$-norm of the difference between the non-stationary and the stationary state. We develop boundary feedback conditions which guarantee that the Lyapunov function and the $H^1$-norm of the difference between the non-stationary and the stationary state decay exponentially with time. This allows us also to prove exponential estimates for the $C^0$- and $C^1$-norm.

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

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