Journal of Evolution Equations

, Volume 18, Issue 3, pp 1471–1500 | Cite as

Boundary stabilization of quasilinear hyperbolic systems of balance laws: exponential decay for small source terms

  • Martin Gugat
  • Vincent Perrollaz
  • Lionel RosierEmail author


We investigate the long-time behaviour of solutions of quasilinear hyperbolic systems with transparent boundary conditions when small source terms are incorporated in the system. Even if the finite-time stability of the system is not preserved, it is shown here that an exponential convergence towards the steady state still holds with a decay rate which is proportional to the logarithm of the amplitude of the source term. The result is stated for a system with dynamical boundary conditions in order to deal with initial data that are free of any compatibility condition. The proof of the existence and uniqueness of a solution defined for all positive times is also provided in this paper.


System of balance laws Shallow water equations Telegraph equation Finite-time stability Dynamical boundary conditions Exponential stability Decay rate 

Mathematics Subject Classification

35L50 35L60 76B75 93D15 


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Martin Gugat
    • 1
  • Vincent Perrollaz
    • 2
  • Lionel Rosier
    • 3
    Email author
  1. 1.Department MathematikFriedrich-Alexander Universität Erlangen-Nürnberg (FAU)ErlangenGermany
  2. 2.Laboratoire de Mathématiques et Physique ThéoriqueUniversité de Tours, UFR Sciences et TechniquesToursFrance
  3. 3.Centre Automatique et Systèmes (CAS) and Centre de Robotique, MINES ParisTechPSL Research UniversityParis Cedex 06France

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