Title: Exponential Stability of the Heat Equation with Boundary Time-Varying Delays
Author(s): Mouataz Billah Mesmouli, Abdelouaheb Ardjouni, Ahcene Djoudi
Pages: 43-53
Cite as:
Mouataz Billah Mesmouli, Abdelouaheb Ardjouni, Ahcene Djoudi, Exponential Stability of the Heat Equation with Boundary Time-Varying Delays, Int. J. Anal. Appl., 11 (1) (2016), 43-53.

Abstract


In this paper, we consider the heat equation with a time-varying delays term in the boundary condition in a bounded domain of Rn, the boundary Γ is a class C2 such that Γ = ΓD∪ΓN, with ΓD∩ ΓN = ∅, ΓD 6= ∅ and ΓN 6= ∅. Well-posedness of the problems is analyzed by using semigroup theory. The exponential stability of the problem is proved. This paper extends in n-dimensional the results of the heat equation obtained in [11].

Full Text: PDF

 

References


  1. F. Ali Mehmeti, Nonlinear Waves in Networks, Mathematical Research, 80, Akademie Verlag, Berlin, 1994.

  2. R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM J. Control Optim., 26 (1988), 697–713.

  3. J. Hale and S. Verduyn Lunel, Introduction to Functional Differential Equations, volume 99 of Applied Mathematical Sciences, Springer-Verlag, New York, 1993.

  4. P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, 21, Pitman, Boston-London-Melbourne, 1985.

  5. T. Kato, Nonlinear semigroups and evolution equations, 19 (1967), 508–520.

  6. T. Kato, Linear and quasilinear equations of evolution of hyperbolic type, II ciclo:125–191, 1976.

  7. T. Kato, Abstract differential equations and nonlinear mixed problems, Lezioni Fermiane. [Fermi Lectures]. Scuola Normale Superiore, Pisa, 1985.

  8. J. L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Math ematiques, 17, Dunod, Paris, 1968.

  9. H. Logemann, R. Rebarber and G. Weiss, Conditions for robustness and nonrobustness of the stability of feedback systems with respect to small delays in the feedback loop, SIAM J. Control Optim., 34 (1996), 572–600.

  10. S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561–1585.

  11. S. Nicaise, J. Valein and E. Fridman, Stability of the heat and of the wave equations with boundary time-varying delays, Discrete Contin. Dyn. Syst. Ser. S. 2 (2009), 559–581.

  12. S. Nicaise, J. Valein and E. Fridman, Stability of the heat and of the wave equations with boundary time-varying delays, Discrete Contin. Dyn. Syst. Ser. S. 4 (2011), 693-722.

  13. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Math. Sciences. Springer-Verlag, New York, 1983.

  14. R. Rebarber and S. Townley, Robustness with respect to delays for exponential stability of distributed parameter systems, SIAM J. Control Optim., 37 (1999), 230–244.