Uniform stabilization of some damped second order evolution equations with vanishing short memory

Louis Tebou

ESAIM: Control, Optimisation and Calculus of Variations (2014)

  • Volume: 20, Issue: 1, page 174-189
  • ISSN: 1292-8119

Abstract

top
We consider a damped abstract second order evolution equation with an additional vanishing damping of Kelvin–Voigt type. Unlike the earlier work by Zuazua and Ervedoza, we do not assume the operator defining the main damping to be bounded. First, using a constructive frequency domain method coupled with a decomposition of frequencies and the introduction of a new variable, we show that if the limit system is exponentially stable, then this evolutionary system is uniformly − with respect to the calibration parameter − exponentially stable. Afterwards, we prove uniform polynomial and logarithmic decay estimates of the underlying semigroup provided such decay estimates hold for the limit system. Finally, we discuss some applications of our results; in particular, the case of boundary damping mechanisms is accounted for, which was not possible in the earlier work mentioned above.

How to cite

top

Tebou, Louis. "Uniform stabilization of some damped second order evolution equations with vanishing short memory." ESAIM: Control, Optimisation and Calculus of Variations 20.1 (2014): 174-189. <http://eudml.org/doc/272762>.

@article{Tebou2014,
abstract = {We consider a damped abstract second order evolution equation with an additional vanishing damping of Kelvin–Voigt type. Unlike the earlier work by Zuazua and Ervedoza, we do not assume the operator defining the main damping to be bounded. First, using a constructive frequency domain method coupled with a decomposition of frequencies and the introduction of a new variable, we show that if the limit system is exponentially stable, then this evolutionary system is uniformly − with respect to the calibration parameter − exponentially stable. Afterwards, we prove uniform polynomial and logarithmic decay estimates of the underlying semigroup provided such decay estimates hold for the limit system. Finally, we discuss some applications of our results; in particular, the case of boundary damping mechanisms is accounted for, which was not possible in the earlier work mentioned above.},
author = {Tebou, Louis},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {second order evolution equation; Kelvin–Voigt damping; hyperbolic equations; stabilization; boundary dissipation; localized damping; plate equations; elasticity equations; frequency domain method; resolvent estimates; Kelvin-Voigt damping},
language = {eng},
number = {1},
pages = {174-189},
publisher = {EDP-Sciences},
title = {Uniform stabilization of some damped second order evolution equations with vanishing short memory},
url = {http://eudml.org/doc/272762},
volume = {20},
year = {2014},
}

TY - JOUR
AU - Tebou, Louis
TI - Uniform stabilization of some damped second order evolution equations with vanishing short memory
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2014
PB - EDP-Sciences
VL - 20
IS - 1
SP - 174
EP - 189
AB - We consider a damped abstract second order evolution equation with an additional vanishing damping of Kelvin–Voigt type. Unlike the earlier work by Zuazua and Ervedoza, we do not assume the operator defining the main damping to be bounded. First, using a constructive frequency domain method coupled with a decomposition of frequencies and the introduction of a new variable, we show that if the limit system is exponentially stable, then this evolutionary system is uniformly − with respect to the calibration parameter − exponentially stable. Afterwards, we prove uniform polynomial and logarithmic decay estimates of the underlying semigroup provided such decay estimates hold for the limit system. Finally, we discuss some applications of our results; in particular, the case of boundary damping mechanisms is accounted for, which was not possible in the earlier work mentioned above.
LA - eng
KW - second order evolution equation; Kelvin–Voigt damping; hyperbolic equations; stabilization; boundary dissipation; localized damping; plate equations; elasticity equations; frequency domain method; resolvent estimates; Kelvin-Voigt damping
UR - http://eudml.org/doc/272762
ER -

References

top
  1. [1] F. Alabau-Boussouira, Piecewise multiplier method and nonlinear integral inequalities for Petrowsky equation with nonlinear dissipation. J. Evol. Equ.6 (2006) 95–112. Zbl1117.34054MR2215634
  2. [2] F. Alabau and V. Komornik, Boundary observability, controllability, and stabilization of linear elastodynamic systems. SIAM J. Control Optim.37 (1999) 521–542. Zbl0935.93037MR1665070
  3. [3] W. Arendt, C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups. Trans. Amer. Math. Soc.306 (1988) 837–852. Zbl0652.47022MR933321
  4. [4] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization from the boundary. SIAM J. Control Optim.30 (1992) 1024–1065. Zbl0786.93009MR1178650
  5. [5] A. Bátkai, K.-J. Engel, J. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups. Math. Nachr.279 (2006) 1425–1440. Zbl1118.47034MR2269247
  6. [6] C.J.K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces. J. Evol. Equ.8 (2008) 765–780. Zbl1185.47043MR2460938
  7. [7] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups. Math. Annal.347 (2010) 455–478. Zbl1185.47044MR2606945
  8. [8] H. Brezis, Analyse fonctionnelle. Théorie et Applications. Masson, Paris (1983). Zbl1147.46300MR697382
  9. [9] N. Burq, Décroissance de l’énergie locale de l’équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel. Acta Math.180 (1998) 1–29. Zbl0918.35081MR1618254
  10. [10] G. Chen, Control and stabilization for the wave equation in a bounded domain. SIAM J. Control Optim.17 (1979) 66–81. Zbl0402.93016MR516857
  11. [11] G. Chen, S.A. Fulling, F.J. Narcowich and S. Sun, Exponential decay of energy of evolution equations with locally distributed damping. SIAM J. Appl. Math.51 (1991) 266–301. Zbl0734.35009MR1089141
  12. [12] G. Chen and D.L. Russell, A mathematical model for linear elastic systems with structural damping. Quart. Appl. Math. 39 (1981/1982) 433-454. Zbl0515.73033MR644099
  13. [13] F. Conrad and B. Rao, Decay of solutions of the wave equation in a star-shaped domain with nonlinear boundary feedback. Asymptotic Anal.7 (1993) 159–177. Zbl0791.35011MR1226972
  14. [14] S. Ervedoza, E. Zuazua, Uniform exponential decay for viscous damped systems. Advances in phase space analysis of partial differential equations. Progr. Nonlinear Differential Equ. Appl. vol. 78. Birkhäuser Boston, Inc., Boston, MA (2009) 95–112. Zbl1201.35046MR2664618
  15. [15] S. Ervedoza and E. Zuazua, Uniformly exponentially stable approximations for a class of damped systems. J. Math. Pures Appl.91 (2009) 20–48. Zbl1163.74019MR2487899
  16. [16] X. Fu, Longtime behavior of the hyperbolic equations with an arbitrary internal damping. Z. Angew. Math. Phys.62 (2011) 667–680. Zbl1264.35041MR2822347
  17. [17] X. Fu, Logarithmic decay of hyperbolic equations with arbitrary small boundary damping. Commun. Partial Differ. Equ.34 (2009) 957–975. Zbl1180.35104MR2560307
  18. [18] R.B. Guzmán and M. Tucsnak, Energy decay estimates for the damped plate equation with a local degenerated dissipation. Systems Control Lett.48 (2003) 191–197. Zbl1134.93397MR2020636
  19. [19] A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps. Portugal. Math.46 (1989) 245–258. Zbl0679.93063MR1021188
  20. [20] F.L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces. Annal. Differ. Equ.1 (1985) 43–56. Zbl0593.34048MR834231
  21. [21] V. Komornik, Rapid boundary stabilization of the wave equation. SIAM J. Control Optim.29 (1991) 197–208. Zbl0749.35018MR1088227
  22. [22] V. Komornik, Exact controllability and stabilization. The multiplier method, RAM. Masson and John Wiley, Paris (1994). Zbl0937.93003MR1359765
  23. [23] V. Komornik and V. Boundary stabilization of isotropic elasticity systems. Control of partial differential equations and applications (Laredo, 1994), vol. 174. Lect. Notes Pure and Appl. Math. Dekker, New York (1996) 135–146. Zbl0849.93029MR1364644
  24. [24] V. Komornik, Rapid boundary stabilization of linear distributed systems. SIAM J. Control Optim.35 (1997) 1591–1613. Zbl0889.35013MR1466918
  25. [25] V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation. J. Math. Pures Appl.69 (1990) 33–54. Zbl0636.93064MR1054123
  26. [26] J. Lagnese, Boundary stabilization of linear elastodynamic systems. SIAM J. Control Opt.21 (1983) 968–984. Zbl0531.93044MR719524
  27. [27] J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation. J. Differ. Equ.50 (1983) 163–182. Zbl0536.35043MR719445
  28. [28] J. Lagnese, Boundary Stabilization of Thin Plates, vol. 10. SIAM Stud. Appl. Math. Philadelphia, PA (1989). Zbl0696.73034MR1061153
  29. [29] I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Differ. Integral Equ.6 (1993) 507–533. Zbl0803.35088MR1202555
  30. [30] I. Lasiecka and R. Triggiani, Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions. Appl. Math. Optim.25 (1992) 189–224. Zbl0780.93082MR1142681
  31. [31] G. Lebeau, Equation des ondes amorties. Algebraic and geometric methods in mathematical physics (Kaciveli, 1993). Math. Phys. Stud. Kluwer Acad. Publ., Dordrecht 19 (1996) 73–109. Zbl0863.58068MR1385677
  32. [32] G. Lebeau and L. Robbiano, Stabilisation de l’équation des ondes par le bord. Duke Math. J.86 (1997) 465–491. Zbl0884.58093MR1432305
  33. [33] J.-L. Lions, Contrôlabilité exacte, Perturbations et Stabilisation des Systèmes Distribués, vol. 8 of RMA. Masson, Paris (1988). Zbl0653.93002MR963060
  34. [34] K. Liu, Locally distributed control and damping for the conservative systems. SIAM J. Control and Opt.35 (1997) 1574–1590. Zbl0891.93016MR1466917
  35. [35] Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation. Z. Angew. Math. Phys.56 (2005), 630–644. Zbl1100.47036MR2185299
  36. [36] K. Liu and B. Rao, Exponential stability for the wave equations with local Kelvin–Voigt damping. Z. Angew. Math. Phys.57 (2006) 419–432. Zbl1114.35023MR2228174
  37. [37] P. Martinez, Ph.D. Thesis, University of Strasbourg (1998). 
  38. [38] P. Martinez, Boundary stabilization of the wave equation in almost star-shaped domains. SIAM J. Control Optim.37 (1999) 673–694. Zbl0946.93025MR1675149
  39. [39] M. Nakao, Decay of solutions of the wave equation with a local degenerate dissipation. Israel J. Math.95 (1996) 25–42. Zbl0860.35072MR1418287
  40. [40] A. Osses, A rotated multiplier applied to the controllability of waves, elasticity, and tangential Stokes control. SIAM J. Control Optim.40 (2001) 777–800. Zbl0997.93013MR1871454
  41. [41] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Appl. Math. Sci. Springer-Verlag, New York (1983). Zbl0516.47023MR710486
  42. [42] K. D. Phung, Polynomial decay rate for the dissipative wave equation. J. Differ. Equ.240 (2007) 92–124. Zbl1130.35017MR2349166
  43. [43] K. D. Phung, Boundary stabilization for the wave equation in a bounded cylindrical domain. Discrete Contin. Dyn. Syst.20 (2008) 1057–1093. Zbl1156.35417MR2379488
  44. [44] J. Prüss, On the spectrum of C0-semigroups. Trans. Amer. Math. Soc.284 (1984) 847–857. Zbl0572.47030MR743749
  45. [45] J.P. Quinn, D.L. Russell, Asymptotic stability and energy decay rates for solutions of hyperbolic equations with boundary damping. Proc. Roy. Soc. Edinburgh Sect. A77 (1977) 97–127. Zbl0357.35006MR473539
  46. [46] D.L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Rev.20 (1978) 639–739. Zbl0397.93001MR508380
  47. [47] L.R. Tcheugoué Tébou, Sur la stabilisation de l’équation des ondes en dimension 2. C. R. Acad. Sci. Paris Ser. I Math.319 (1994) 585–588. Zbl0809.35050MR1298287
  48. [48] L.R. Tcheugoué Tébou, On the stabilization of the wave and linear elasticity equations in 2-D. Panamer. Math. J.6 (1996) 41–55. Zbl0853.35013MR1366607
  49. [49] L.R. Tcheugoué Tébou, On the decay estimates for the wave equation with a local degenerate or nondegenerate dissipation. Portugal. Math.55 (1998) 293–306. Zbl0918.35079MR1647163
  50. [50] L.R. Tcheugoué Tébou, Stabilization of the wave equation with localized nonlinear damping J.D.E.145 (1998) 502–524. Zbl0916.35069MR1620983
  51. [51] L.R. Tcheugoué Tébou, Well-posedness and energy decay estimates for the damped wave equation with Lr localizing coefficient, Commun. in P.D.E.23 (1998) 1839–1855. Zbl0918.35078MR1641733
  52. [52] L.R. Tcheugoué Tébou, Energy decay estimates for the damped Euler–Bernoulli equation with an unbounded localizing coefficient. Portugal. Math.61 (2004) 375–391. Zbl1072.35188MR2113555
  53. [53] L.R. Tcheugoué Tébou, On the stabilization of dynamic elasticity equations with unbounded locally distributed dissipation. Differ. Integral Equ.19 (2006) 785–798. Zbl1212.93260MR2235895
  54. [54] L. Tebou, Stabilization of the elastodynamic equations with a degenerate locally distributed dissipation. Syst. Control Lett.56 (2007) 538–545. Zbl1121.35016MR2332006
  55. [55] L. Tebou, Well-posedness and stabilization of an Euler–Bernoulli equation with a localized nonlinear dissipation involving the p-Laplacian. DCDS A32 (2012) 2315–2337. Zbl1242.93118MR2885813
  56. [56] L.R. Tcheugoué Tébou and E. Zuazua, Uniform exponential long time decay for the space finite differences semi-discretization of a locally damped wave equation via an artificial numerical viscosity. Numerische Mathematik95 (2003) 563–598. Zbl1033.65080MR2012934
  57. [57] L.T. Tebou and E. Zuazua, Uniform boundary stabilization of the finite differences space discretization of the 1 − d wave equation. Adv. Comput. Math. 26 (2007) 337–365. Zbl1119.65086MR2350359
  58. [58] M. Tucsnak, Semi-internal stabilization for a non-linear Bernoulli-Euler equation. Math. Methods Appl. Sci.19 (1996) 897–907. Zbl0863.93071MR1399056
  59. [59] A. Wyler, Stability of wave equations with dissipative boundary conditions in a bounded domain. Differential Integral Equ.7 (1994) 345–366. Zbl0816.35078MR1255893
  60. [60] E. Zuazua, Robustesse du feedback de stabilisation par contrôle frontière. C. R. Acad. Sci. Paris Ser. I Math.307 (1988) 587–591. Zbl0659.93055MR967367
  61. [61] E. Zuazua, Uniform stabilization of the wave equation by nonlinear boundary feedback. SIAM J. Control Optim.28 (1990) 466–477. Zbl0695.93090MR1040470
  62. [62] E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping. Commun. P.D.E.15 (1990) 205–235. Zbl0716.35010MR1032629
  63. [63] E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains. J. Math. Pures. Appl.70 (1991) 513–529. Zbl0765.35010MR1146833

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.