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Stabilization of Networked Hyperbolic Systems with Boundary Feedback

  • Markus DickEmail author
  • Martin Gugat
  • Michael Herty
  • Günter Leugering
  • Sonja Steffensen
  • Ke Wang
Chapter
Part of the International Series of Numerical Mathematics book series (ISNM, volume 165)

Abstract

We summarize recent theoretical results as well as numerical results on the feedback stabilization of first order quasilinear hyperbolic systems (on networks). For the stabilization linear feedback controls are applied at the nodes of the network. This yields the existence and uniqueness of a C 1-solution of the hyperbolic system with small C 1-norm. For this solution an appropriate L 2-Lyapunov function decays exponentially in time. This implies the exponential stability of the system. A numerical discretization of the Lyapunov function is presented and a numerical analysis shows the expected exponential decay for a class of first-order discretization schemes. As an application for the theoretical results the stabilization of the gas flow in fan-shaped pipe networks with compressors is considered.

Keywords

Quasilinear hyperbolic system Feedback stabilization Networked system Boundary control Lyapunov function 

Mathematics Subject Classification (2010)

35L50 93C20 

Notes

Acknowledgements

This work has been supported by DFG GU376/7-1 and HE5386/8-1.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Markus Dick
    • 1
    Email author
  • Martin Gugat
    • 1
  • Michael Herty
    • 2
  • Günter Leugering
    • 1
  • Sonja Steffensen
    • 2
  • Ke Wang
    • 3
  1. 1.Department of MathematicsUniversity of Erlangen-NurembergErlangenGermany
  2. 2.IGPM, Department of MathematicsRWTH Aachen UniversityAachenGermany
  3. 3.School of Mathematical SciencesFudan UniversityShanghaiChina

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