Free Access
Issue
ESAIM: COCV
Volume 16, Number 2, April-June 2010
Page(s) 298 - 326
DOI https://doi.org/10.1051/cocv:2008071
Published online 19 December 2008
  1. H.T. Banks, K. Ito and C. Wang, Exponentially stable approximations of weakly damped wave equations, in Estimation and control of distributed parameter systems (Vorau, 1990), Internat. Ser. Numer. Math. 100, Birkhäuser, Basel (1991) 1–33. [Google Scholar]
  2. J.-P. Bérenger, A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114 (1994) 185–200. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
  3. E. Bogomolny, O. Bohigas and C. Schmit, Spectral properties of distance matrices. J. Phys. A 36 (2003) 3595–3616. [CrossRef] [MathSciNet] [Google Scholar]
  4. T.J. Bridges and S. Reich, Numerical methods for Hamiltonian PDEs. J. Phys. A 39 (2006) 5287–5320. [CrossRef] [MathSciNet] [Google Scholar]
  5. C. Castro and S. Micu, Boundary controllability of a linear semi-discrete 1-d wave equation derived from a mixed finite element method. Numer. Math. 102 (2006) 413–462. [CrossRef] [MathSciNet] [Google Scholar]
  6. C. Castro, S. Micu and A. Münch, Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square. IMA J. Numer. Anal. 28 (2008) 186–214. [CrossRef] [MathSciNet] [Google Scholar]
  7. L.C. Cowsar, T.F. Dupont and M.F. Wheeler, A priori estimates for mixed finite element methods for the wave equations. Comput. Methods Appl. Mech. Engrg. 82 (1990) 205–222. [CrossRef] [MathSciNet] [Google Scholar]
  8. S. Cox and E. Zuazua, The rate at which energy decays in a damped string. Comm. Partial Differ. Equ. 19 (1994) 213–243. [CrossRef] [MathSciNet] [Google Scholar]
  9. S. Cox and E. Zuazua, The rate at which energy decays in a string damped at one end. Indiana Univ. Math. J. 44 (1995) 545–573. [MathSciNet] [Google Scholar]
  10. S. Ervedoza and E. Zuazua, Perfectly matched layers in 1-d: Energy decay for continuous and semi-discrete waves. Numer. Math. 109 (2008) 597–634. [CrossRef] [MathSciNet] [Google Scholar]
  11. S. Ervedoza and E. Zuazua, Uniformly exponentially stable approximations for a class of damped systems. J. Math. Pures Appl. (to appear). [Google Scholar]
  12. S. Ervedoza, C. Zheng and E. Zuazua, On the observability of time-discrete conservative linear systems. J. Funct. Anal. 254 (2008) 3037–3078. [CrossRef] [MathSciNet] [Google Scholar]
  13. J. Frank, B.E. Moore and S. Reich, Linear PDEs and numerical methods that preserve a multisymplectic conservation law. SIAM J. Sci. Comput. 28 (2006) 260–277 (electronic). [CrossRef] [MathSciNet] [Google Scholar]
  14. R. Glowinski, Ensuring well-posedness by analogy: Stokes problem and boundary control for the wave equation. J. Comput. Phys. 103 (1992) 189–221. [CrossRef] [MathSciNet] [Google Scholar]
  15. R. Glowinski, W. Kinton and M.F. Wheeler, A mixed finite element formulation for the boundary controllability of the wave equation. Internat. J. Numer. Methods Engrg. 27 (1989) 623–635. [CrossRef] [MathSciNet] [Google Scholar]
  16. A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps. Portugal. Math. 46 (1989) 245–258. [MathSciNet] [Google Scholar]
  17. J.A. Infante and E. Zuazua, Boundary observability for the space semi discretizations of the 1-d wave equation. Math. Model. Num. Ann. 33 (1999) 407–438. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  18. A.E. Ingham, Some trigonometrical inequalities with applications to the theory of series. Math. Z. 41 (1936) 367–379. [CrossRef] [MathSciNet] [Google Scholar]
  19. S. Labbé and E. Trélat, Uniform controllability of semidiscrete approximations of parabolic control systems. Systems Control Lett. 55 (2006) 597–609. [CrossRef] [MathSciNet] [Google Scholar]
  20. G. Lebeau, Équations des ondes amorties, in Séminaire sur les Équations aux Dérivées Partielles, 1993–1994, École Polytechnique, France (1994). [Google Scholar]
  21. J.-L. Lions, Contrôlabilité exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome 1 : Contrôlabilité exacte, RMA 8. Masson (1988). [Google Scholar]
  22. F. Macià, The effect of group velocity in the numerical analysis of control problems for the wave equation, in Mathematical and numerical aspects of wave propagation – WAVES 2003, Springer, Berlin (2003) 195–200. [Google Scholar]
  23. A. Münch, A uniformly controllable and implicit scheme for the 1-D wave equation. ESAIM: M2AN 39 (2005) 377–418. [CrossRef] [EDP Sciences] [Google Scholar]
  24. M. Negreanu and E. Zuazua, Convergence of a multigrid method for the controllability of a 1-d wave equation. C. R. Math. Acad. Sci. Paris 338 (2004) 413–418. [CrossRef] [MathSciNet] [Google Scholar]
  25. M. Negreanu, A.-M. Matache and C. Schwab, Wavelet filtering for exact controllability of the wave equation. SIAM J. Sci. Comput. 28 (2006) 1851–1885 (electronic). [CrossRef] [MathSciNet] [Google Scholar]
  26. K. Ramdani, T. Takahashi and M. Tucsnak, Uniformly exponentially stable approximations for a class of second order evolution equations – application to LQR problems. ESAIM: COCV 13 (2007) 503–527. [CrossRef] [EDP Sciences] [Google Scholar]
  27. L.R. Tcheugoué Tébou and E. Zuazua, Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity. Numer. Math. 95 (2003) 563–598. [CrossRef] [MathSciNet] [Google Scholar]
  28. L.R. Tcheugoué Tebou and E. Zuazua, Uniform boundary stabilization of the finite difference space discretization of the 1-d wave equation. Adv. Comput. Math. 26 (2007) 337–365. [CrossRef] [MathSciNet] [Google Scholar]
  29. L.N. Trefethen, Group velocity in finite difference schemes. SIAM Rev. 24 (1982) 113–136. [CrossRef] [MathSciNet] [Google Scholar]
  30. R.M. Young, An introduction to nonharmonic Fourier series. Academic Press Inc., San Diego, CA, first edition (2001). [Google Scholar]
  31. E. Zuazua, Boundary observability for the finite-difference space semi-discretizations of the 2-D wave equation in the square. J. Math. Pures Appl. 78 (1999) 523–563. [CrossRef] [MathSciNet] [Google Scholar]
  32. E. Zuazua, Propagation, observation, and control of waves approximated by finite difference methods. SIAM Rev. 47 (2005) 197–243 (electronic). [CrossRef] [MathSciNet] [Google Scholar]

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