Neumann boundary feedback stabilization for a nonlinear wave equation: A strict $H^2$-lyapunov function

Martin Gugat; Günter Leugering; Ke Wang

doi:10.3934/mcrf.2017015

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2017, Volume 7, Issue 3: 419-448. Doi: 10.3934/mcrf.2017015

This issue Previous Article Finite element approximation of sparse parabolic control problems Next Article Existence of optimal solutions to lagrange problem for a fractional nonlinear control system with riemann-liouville derivative

Neumann boundary feedback stabilization for a nonlinear wave equation: A strict $H^2$-lyapunov function

a.
Lehrstuhl 2 für Angewandte Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstr. 11,91058 Erlangen, Germany
b.
College of Science, Donghua University, Shanghai 201620, China

Received: June 30, 2015

Revised: December 31, 2016

Published: October 2017

This work was supported by DFG in the framework of the Collaborative Research Centre CRC/Transregio 154, Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks, project C03 and A05.

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Abstract

For a system that is governed by the isothermal Euler equations with friction for ideal gas, the corresponding field of characteristic curves is determined by the velocity of the flow. This velocity is determined by a second-order quasilinear hyperbolic equation. For the corresponding initial-boundary value problem with Neumann-boundary feedback, we consider non-stationary solutions locally around a stationary state on a finite time interval and discuss the well-posedness of this kind of problem. We introduce a strict $H^2$-Lyapunov function and show that the boundary feedback constant can be chosen such that the $H^2$-Lyapunov function and hence also the $H^2$-norm of the difference between the non-stationary and the stationary state decays exponentially with time.

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Mathematics Subject Classification: 76N25, 35L51, 35L53, 93C20.

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