Boundary observability for the space semi-discretizations of the 1-d wave equation
ESAIM: Modélisation mathématique et analyse numérique, Tome 33 (1999) no. 2, pp. 407-438. 
@article{M2AN_1999__33_2_407_0,
     author = {Infante, Juan Antonio and Zuazua, Enrique},
     title = {Boundary observability for the space semi-discretizations of the $1-d$ wave equation},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {407--438},
     publisher = {EDP-Sciences},
     volume = {33},
     number = {2},
     year = {1999},
     mrnumber = {1700042},
     zbl = {0947.65101},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_1999__33_2_407_0/}
}
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Infante, Juan Antonio; Zuazua, Enrique. Boundary observability for the space semi-discretizations of the $1-d$ wave equation. ESAIM: Modélisation mathématique et analyse numérique, Tome 33 (1999) no. 2, pp. 407-438. http://www.numdam.org/item/M2AN_1999__33_2_407_0/

[1] M. Avellaneda, C. Bardos and J. Rauch, Contrôlabilité exacte, homogénéisation et localisation d'ondes dans un milieu non-homogène. Asymptotic Analysis 5 (1992) 481-494. | MR | Zbl

[2] C. Castro and E. Zuazua, Contrôle de l'équation des ondes à densité rapidement oscillante à une dimension d'espace. C. R. Acad. Sci. Paris 324 (1997) 1237-1242. | MR | Zbl

[3] R. Glowinski, Ensuring Weli-Posedness by Analogy; Stokes Problem and Boundary Control for the Wave Equation. J. Comput.Phys. 103 (1992) 189-221. | MR | Zbl

[4] R. Glowinski, C. H. Li and J. L. Lions, A numerical approach to the exact boundary controllability of the wave equation. (I).Dirichlet Controls: Description of the numerical methods. Jap. J. Appl. Math. 7 (1990) 1-76. | MR | Zbl

[5] R. Glowinski and J. L. Lions, Exact and approximate controllability for distributed parameter Systems. Acta Numerica (1996)159-333. | MR | Zbl

[6] J. A. Infante and E. Zuazua, Boundary observability for the space-discretizations of the 1 - d wave equation. C. R. Acad. Sci.Paris 326 (1998) 713-718. | MR | Zbl

[7] A. E. Ingham, Some trigonometrical inequalities with applications to the theory of series. Mathematische Zeitschrift 41 (1936) 367-379. | MR | Zbl

[8] E. Isaacson and H. B. Keller, Analysis of Numerical Methods. John Wiley & Sons (1966). | MR | Zbl

[9] V. Komornik, Exact controllability and stabilization. The multiplier method. John Wiley & Sons, Masson (1994). | MR | Zbl

[10] E. B. Lee and L. Markus, Foundations of Optimal Control Theory, The SIAM Series in Applied Mathematics. John Wiley & Sons (1967). | MR | Zbl

[11] J. L. Lions, Contrôlabilité exacte, stabilisation et perturbations de systèmes distribués. Tome 1. Contrôlabilité exacte. Masson,RMA8 (1988). | Zbl

[12] R. Vichnevetsky and J. B. Bowles, Fourier analysis of numerical approximations of hyperbolic equations. SIAM, Philadelphia(1982). | MR | Zbl