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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
Article
  • 64 Downloads

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.

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 

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References

  1. 1.
    F. Alabau-Boussouira, V. Perrollaz, L. Rosier, Finite-time stabilization of a network of strings, Math. Control Relat. Fields 5 (2015), no. 4, 721–742.MathSciNetCrossRefGoogle Scholar
  2. 2.
    G. Bastin, J.-M. Coron, B. d’Andréa-Novel, On Lyapunov stability of linearized Saint-Venant equations for a sloping channel, Networks and Heterogeneous Media 4 (2009), no. 2, 177–187.MathSciNetCrossRefGoogle Scholar
  3. 3.
    F. Bouchut, A. Mangeney-Castelnau, B. Perthame, J.-P. Vilotte, A new model of Saint Venant and Savage-Hutter type for gravity driven shallow water flows, C. R. Math. Acad. Sci. Paris 336 (2003), no. 6, 531–536.MathSciNetCrossRefGoogle Scholar
  4. 4.
    J.-M. Coron, R. Vazquez, M. Krstic, G. Bastin, Local exponential \(H^2\) stabilization of a \(2\times 2\) quasilinear hyperbolic system using backstepping, SIAM J. Control Optim. 51 (2013), no. 3, 2005–2035.MathSciNetCrossRefGoogle Scholar
  5. 5.
    R. Datko, J. Lagnese, M. P. Polis, An example of the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim. 24 (1986), no. 1, 152–156.MathSciNetCrossRefGoogle Scholar
  6. 6.
    L. Evans, R. Gariepy, Measure theory and fine properties of functions, Revised edition. Textbooks in Mathematics, CRS Press Boca Raton, FL, 2015CrossRefGoogle Scholar
  7. 7.
    J. de Halleux, C. Prieur, J.-M. Coron, B. d’Andréa-Novel, G. Bastin, Boundary feedback control in network of open channels, Automatica 39 (2003), 1365–1376.MathSciNetCrossRefGoogle Scholar
  8. 8.
    J. M. Greenberg, T.-T. Li, The effect of boundary damping for the quasilinear wave equations, J. Differential equations 52 (1984), 66–75.MathSciNetCrossRefGoogle Scholar
  9. 9.
    M. Gugat, Boundary feedbcak stabilization of the telegraph equation: Decay rates for vanishing damping term, Systems Control Lett. 66 (2014) 72–84.MathSciNetCrossRefGoogle Scholar
  10. 10.
    M. Gugat, M. Dick, G. Leugering, Gas flow in fan-shaped networks: Classical solutions and feedback stabilization, SIAM J. Control Optim., 49 (2011), 2101–2117.MathSciNetCrossRefGoogle Scholar
  11. 11.
    M. Gugat, G. Leugering, Global boundary controllability of the Saint-Venant system for sloped canals with friction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009) 257–270.MathSciNetCrossRefGoogle Scholar
  12. 12.
    M. Gugat, G. Leugering, K. Wang, Neumann boundary feedback stabilization for a nonlinear wave equation: A strict \(H^2\)-Lyapunov function, Math. Control Relat. Fields 7 (2017), no. 3, 419–448.MathSciNetCrossRefGoogle Scholar
  13. 13.
    V. Komornik, Rapid boundary stabilization of the wave equation, SIAM J. Control Optim. 29 (1991), 197–208.MathSciNetCrossRefGoogle Scholar
  14. 14.
    G. Leugering, E. J. P. G. Schmidt, On the modelling abd stabilization of flows in networks of open canals, SIAM J. Control Optim., 41 (2002), 164–180.MathSciNetCrossRefGoogle Scholar
  15. 15.
    A. Majda, Disappearing solutions for the dissipative wave equation, Indiana Univ. Math. J. 24 (1974/75), 1119–1133.Google Scholar
  16. 16.
    V. Perrollaz, Initial boundary value problem and asymptotic stabilization of the Camassa-Holm equation on an interval, J. Funct. Anal. 259 (2010), no. 9, 2333–2365.MathSciNetCrossRefGoogle Scholar
  17. 17.
    V. Perrollaz, L. Rosier, Finite-time stabilization of hyperbolic systems over a bounded interval, in 1st IFAC workshop on Control of Systems Governed by Partial Differential Equations (CPDE2013), 2013, 239–244.CrossRefGoogle Scholar
  18. 18.
    V. Perrollaz, L. Rosier, Finite-time stabilization of \(2\times 2\) hyperbolic systems on tree-shaped networks, SIAM J. Control Optim. 52 (2014), no. 1, 143–163.MathSciNetCrossRefGoogle Scholar
  19. 19.
    W. Rudin, Functional Analysis. McGraw-Hill Series in Higher Mathematics. McGraw-Hill, Inc., New York, 1973.zbMATHGoogle Scholar
  20. 20.
    Y. Shang, D. Liu, G. Xu, Super-stability and the spectrum of one-dimensional wave equations on general feedback controlled networks, IMA J. Math. Control and Inform. 31 (2014), 73–90.MathSciNetCrossRefGoogle Scholar
  21. 21.
    K. Yosida. Functional Analysis. Springer-Verlag, Berlin Heidelberg New York, 1978.CrossRefGoogle Scholar
  22. 22.
    E. Zeidler, Nonlinear Functional Analysis and its Applications. I. Fixed-Point Theorems, Springer-Verlag New York Inc., 1986.Google Scholar
  23. 23.
    G. Q. Xu, Stabilization of string system with linear bounded feedback, Nonlinear Analysis: Hybrid Systems 1 (2007) 383–397.MathSciNetzbMATHGoogle Scholar
  24. 24.
    C. Z. Xu, G. Sallet, Exponential stability and transfert functions of processes governed by symmetric hyperbolic systems, ESAIM Control Optim. Calc. Var. 7 (2002) 421–442.MathSciNetCrossRefGoogle Scholar

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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