Stability Results for Abstract Evolution Equations with Intermittent Time-Delay Feedback

Chapter
First Online:
Part of the Springer INdAM Series book series (SINDAMS, volume 22)

Abstract

We consider abstract evolution equations with on–off time delay feedback. Without the time delay term, the model is described by an exponentially stable semigroup. We show that, under appropriate conditions involving the delay term, the system remains asymptotically stable. Under additional assumptions exponential stability results are also obtained. Concrete examples illustrating the abstract results are finally given.

Keywords

Delay feedbacks Evolution equations Stabilization 
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References

  1. 1.
    Alabau-Boussouira, F., Cannarsa, P., Sforza, D.: Decay estimates for second order evolution equations with memory. J. Funct. Anal. 254, 1342–1372 (2008)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Alabau-Boussouira, F., Nicaise, S., Pignotti, C.: Exponential stability of the wave equation with memory and time delay. In: New Prospects in Direct, Inverse and Control Problems for Evolution Equations. Springer Indam Series, vol. 10, pp. 1–22. Springer, Berlin (2014)Google Scholar
  3. 3.
    Ammari, K., Nicaise, S., Pignotti, C.: Stabilization by switching time–delay. Asymptot. Anal. 83, 263–283 (2013)MATHMathSciNetGoogle Scholar
  4. 4.
    Bardos, C., Lebeau, G., Rauch, J.: Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim. 30, 1024–1065 (1992)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Dafermos, C.M.: Asymptotic stability in viscoelasticity. Arch. Rational Mech. Anal. 37, 297–308 (1970)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Dai, Q., Yang, Z.: Global existence and exponential decay of the solution for a viscoelastic wave equation with a delay. Z. Angew. Math. Phys. 65, 885–903 (2014)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Datko, R.: Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks. SIAM J. Control Optim. 26, 697–713 (1988)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Datko, R., Lagnese, J., Polis, M.P.: An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J. Control Optim. 24, 152–156 (1986)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Fragnelli, G., Pignotti, C.: Stability of solutions to nonlinear wave equations with switching time-delay. Dyn. Partial Differ. Equ. 13, 31–51 (2016)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Giorgi, C., Muñoz Rivera, J.E., Pata, V.: Global attractors for a semilinear hyperbolic equation in viscoelasticity. J. Math. Anal. Appl. 260, 83–99 (2001)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Guesmia, A.: Well-posedness and exponential stability of an abstract evolution equation with infinite memory and time delay. IMA J. Math. Control Inform. 30, 507–526 (2013)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Gugat, M.: Boundary feedback stabilization by time delay for one-dimensional wave equations. IMA J. Math. Control Inform. 27, 189–203 (2010)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Gugat, M., Tucsnak, M.: An example for the switching delay feedback stabilization of an infinite dimensional system: the boundary stabilization of a string. Syst. Control Lett. 60, 226–233 (2011)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Haraux, A., Martinez, P., Vancostenoble, J.: Asymptotic stability for intermittently controlled second-order evolution equations. SIAM J. Control Optim. 43, 2089–2108 (2005)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Kirane, M., Said-Houari, B.: Existence and asymptotic stability of a viscoelastic wave equation with a delay. Z. Angew. Math. Phys. 62, 1065–1082 (2011)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Komornik, V.: Exact Controllability and Stabilization, the Multiplier Method. RMA, vol. 36. Masson, Paris (1994)Google Scholar
  17. 17.
    Komornik, V., Loreti, P.: Fourier Series in Control Theory. Springer Monographs in Mathematics. Springer, New York (2005)MATHGoogle Scholar
  18. 18.
    Lagnese, J.: Control of wave processes with distributed control supported on a subregion. SIAM J. Control Optim. 21, 68–85 (1983)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Lasiecka, I., Triggiani, R.: Uniform exponential decay in a bounded region with L 2(0, T; L 2(Σ))-feedback control in the Dirichlet boundary conditions. J. Differ. Equ. 66, 340–390 (1987)CrossRefMATHGoogle Scholar
  20. 20.
    Lions, J.L.: Contrôlabilité exacte, perturbations et stabilisation des systèmes distribués, Tome 1. RMA, vol. 8. Masson, Paris (1988)Google Scholar
  21. 21.
    Liu, K.: Locally distributed control and damping for the conservative systems. SIAM J. Control Optim. 35, 1574–1590 (1997)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Nicaise, S., Pignotti, C.: Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM J. Control Optim. 45, 1561–1585 (2006)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Nicaise, S., Pignotti, C.: Asymptotic stability of second-order evolution equations with intermittent delay. Adv. Differ. Equ. 17, 879–902 (2012)MATHMathSciNetGoogle Scholar
  24. 24.
    Nicaise, S., Pignotti, C.: Stability results for second-order evolution equations with switching time-delay. J. Dyn. Differ. Equ. 26, 781–803 (2014)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Nicaise, S., Pignotti, C.: Exponential stability of abstract evolution equations with time delay. J. Evol. Equ. 15, 107–129 (2015)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Pata, V.: Stability and exponential stability in linear viscoelasticity. Milan J. Math. 77, 333–360 (2009)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences, vol. 44. Springer, New York (1983)Google Scholar
  28. 28.
    Pignotti, C.: Stability results for second-order evolution equations with memory and switching time-delay. J. Dyn. Differ. Equ. (2016, in press). doi:10.1007/s10884-016-9545-3Google Scholar
  29. 29.
    Rauch, J., Taylor, M.: Exponential decay of solutions to hyperbolic equations in bounded domains. Indiana Univ. Math. J. 24, 79–86 (1974)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Xu, G.Q., Yung, S.P., Li, L.K.: Stabilization of wave systems with input delay in the boundary control. ESAIM: Control Optim. Calc. Var. 12, 770–785 (2006)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Zuazua, E.: Exponential decay for the semi-linear wave equation with locally distributed damping. Comm. Partial Differ. Equ. 15, 205–235 (1990)CrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Dipartimento di Ingegneria e Scienze dell’Informazione e MatematicaUniversità di L’AquilaL’AquilaItaly

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