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

Abstract

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.

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Correspondence to Lionel Rosier.

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Gugat, M., Perrollaz, V. & Rosier, L. Boundary stabilization of quasilinear hyperbolic systems of balance laws: exponential decay for small source terms. J. Evol. Equ. 18, 1471–1500 (2018). https://doi.org/10.1007/s00028-018-0449-z

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Keywords

  • 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