A fluid–structure interaction model with interior damping and delay in the structure

  • Gilbert PeraltaEmail author


A coupled system of partial differential equations modeling the interaction of a fluid and a structure with delay in the feedback is studied. The model describes the dynamics of an elastic body immersed in a fluid that is contained in a vessel, whose boundary is made of a solid wall. The fluid component is modeled by the linearized Navier-Stokes equation, while the solid component is given by the wave equation neglecting transverse elastic force. Spectral properties and exponential or strong stability of the interaction model under appropriate conditions on the damping factor, delay factor and the delay parameter are established using a generalized Lax-Milgram method.

Mathematics Subject Classification

35Q30 76D07 93D15 93D20 


Fluid–structure model Feedback delay Stability Generalized Lax-Milgram method 


  1. 1.
    Arendt W., Batty C.J.K.: Tauberian theorems and stability of one-parameter semigroups. Trans. Amer. Math. Soc. 360, 837–852 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Avalos G., Bucci F.: Rational rates of uniform decay for strong solutions to a fluid-structure PDE system. J. Differ. Equ. 258, 4398–4423 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Avalos G., Triggiani R.: The coupled PDE system arising in fluid-structure interaction, Part I: Explicit semigroup generator and its spectral properties. Contemp. Math. 440, 15–54 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Avalos G., Triggiani R.: Semigroup wellposedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction. Discr. Cont. Dyn. Syst. 2, 417–447 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Avalos G., Triggiani R.: Rational decay rates for a PDE heat-structure interaction: a frequency domain approach. Evol. Equ. Control Theory 2, 233–253 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Avalos G., Triggiani R.: Fluid structure interaction with and without internal dissipation of the structure : A contrast study in stability. Evol. Equ. Control Theory 2, 563–598 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Avalos G., Lasiecka I., Trigianni R.: Higher regularity of a coupled parabolic-hyperbolic fluid structure interactive system. Georgian Math. J. 15, 403–437 (2008)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Barbu V., Grujić Z., Lasiecka I., Tuffaha A.: Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model. Contemp. Math. 440, 55–81 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Barbu V., Grujić Z., Lasiecka I., Tuffaha A.: Smoothness of weak solutions to a nonlinear fluid-structure interaction model. Indiana U. Math. J. 57, 1173–1207 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Datko R.: Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedback. SIAM J. Control Optim. 26, 697–713 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Datko R., Lagnese J., Polis P.: An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J. Control Optim 24, 152–156 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Desch W., Fas̆angová E., Milota J., Propst G.: Stabilization through viscoelastic boundary damping: a semigroup approach. Semigroup Forum 80, 405–415 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Engel K.J., Nagel R.: One-Parameter Semigroups for Linear Evolution Equations, 2nd ed. Springer, Berlin (2000)zbMATHGoogle Scholar
  14. 14.
    Du Q., Gunzburger M.D., Hou L.S., Lee J.: Analysis of a linear fluid-structure interaction problem. Discr. Contin. Dyn. Syst. 9, 633–650 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Horn R.A., Johnson C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)CrossRefzbMATHGoogle Scholar
  16. 16.
    Kirane M., Said-Houari B.: Existence and asymptotic stability of a viscoelastic wave equation with delay. Z. Angew. Math. Phys. 62, 1065–1082 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    LasieckaLu Y.: Asymptotic stability of finite energy in Navier Stokes-elastic wave interaction. Semigroup Forum 82, 61–82 (2011)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lasiecka I., Lu Y.: Interface feedback control stabilization of a nonlinear fluid-structure interaction. Nonlinear Anal. 75, 1449–1460 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lasiecka I., Seidman T.: Strong stability of elastic control systems with dissipative saturating feedback. Syst. Control Lett. 48, 243–252 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lions J.L., Magenes E.: Non-Homogeneous Boundary Value Problems And Applications, Vol. 1. Springer, New York (1972)CrossRefzbMATHGoogle Scholar
  21. 21.
    Lyubich Y.I., Phong V.Q.: Asymptotic stability of linear differential equations in Banach spaces. Studia Matematica LXXXVII, 37–42 (1988)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Lu Y.: Stabilization of a fluid structure interaction with nonlinear damping. Control Cybern. 42, 155–181 (2013)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Lu Y.: Uniform decay rates for the energy in nonlinear fluid structure interaction with monotone viscous damping. Palest. J. Math. 2, 215–232 (2013)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Nicaise S., Pignotti C.: Stability and instability results of the wave equation with a delay term in boundary or internal feedbacks. SIAM J. Control Optim. 45, 1561–1585 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Peralta G., Propst G.: Stability and boundary controllability of a linearized model of flow in an elastic tube. ESAIM: Control Optim. Calc. Var. 21, 583–601 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Quarteroni A., Valli A.: Numerical Approximations of Partial Differential Equations. Springer, Heidelberg (2008)zbMATHGoogle Scholar
  27. 27.
    Sohr H.: The Navier-Stokes Equations: An Elementary Functional Analytic Approach. Birkhäuser, Berlin (2001)CrossRefzbMATHGoogle Scholar
  28. 28.
    Temam R.: Navier-Stokes Equations, Theory and Numerical Analysis. AMS Chelsea Publishing, Providence (2001)zbMATHGoogle Scholar
  29. 29.
    Tucsnak M., Weiss G.: Observation and Control for Operator Semigroups. Birkhäuser-Verlag, Basel (2009)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of the Philippines BaguioBaguio CityPhilippines

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