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Convergence of a two-grid algorithm for the control of the wave equation. (English) Zbl 1159.93006

Summary: We analyze the problem of boundary observability of the finite-difference space semi-discretizations of the 2-d wave equation in the square. We prove the uniform (with respect to the mesh-size) boundary observability for the solutions obtained by the two-grid preconditioner introduced by R. Glowinski [J. Comput. Phys. 103, No. 2, 189–221 (1992; Zbl 0763.76042)]. Our method uses previously known uniform observability inequalities for low frequency solutions and a dyadic spectral time decomposition. As a consequence we prove the convergence of the two-grid algorithm for computing the boundary controls for the wave equation. The method can be applied in any space dimension, for more general domains and other discretization schemes.

MSC:

93B07 Observability
93C20 Control/observation systems governed by partial differential equations
35B37 PDE in connection with control problems (MSC2000)
35L20 Initial-boundary value problems for second-order hyperbolic equations

Citations:

Zbl 0763.76042
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References:

[1] Asch, M., Lebeau, G.: Geometrical aspects of exact boundary controllability for the wave equation-a numerical study. ESAIM Control Optim. Calc. Var. 3, 163-212 (1998) · Zbl 1052.93501 · doi:10.1051/cocv:1998106
[2] 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) · Zbl 0786.93009 · doi:10.1137/0330055
[3] Burq, N.: Contr\hat olabilité exacte des ondes dans des ouverts peu réguliers. Asymptot. Anal. 14, 157-191 (1997) · Zbl 0892.93009
[4] Burq, N.: Contr\hat ole de l’équation des ondes dans des ouverts comportant des coins. Bull. Soc. Math. France 126, 601-637 (1998) · Zbl 0937.35097
[5] Burq, N., Gérard, P.: Condition nécessaire et suffisante pour la contr\hat olabilité exacte des ondes. C. R. Acad. Sci. Paris Sér. I Math. 325, 749-752 (1997) · Zbl 0906.93008 · doi:10.1016/S0764-4442(97)80053-5
[6] Burq, N., Gérard, P., Tzvetkov, N.: Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds. Amer. J. Math. 126, 569-605 (2004) · Zbl 1067.58027 · doi:10.1353/ajm.2004.0016
[7] Castro, C., Micu, S.: Boundary controllability of a linear semi-discrete 1-D wave equa- tion derived from a mixed finite element method. Numer. Math. 102, 413-462 (2006) · Zbl 1102.93004 · doi:10.1007/s00211-005-0651-0
[8] Castro, C., Micu, S., Münch, A.: Boundary controllability of a semi-discrete 2-D wave equa- tion with mixed finite elements. Preprint.
[9] Glowinski, R.: Ensuring well-posedness by analogy; Stokes problem and boundary control for the wave equation. J. Comput. Phys. 103, 189-221 (1992) · Zbl 0763.76042 · doi:10.1016/0021-9991(92)90396-G
[10] Glowinski, R., Kinton, W., Wheeler, M. F.: A mixed finite element formulation for the bound- ary controllability of the wave equation. Int. J. Numer. Methods Engrg. 27, 623-635 (1989) · Zbl 0711.65084 · doi:10.1002/nme.1620270313
[11] Glowinski, R., Li, C. H., Lions, J.-L.: A numerical approach to the exact boundary controlla- bility of the wave equation. I. Dirichlet controls: description of the numerical methods. Japan J. Appl. Math. 7, 1-76 (1990) · Zbl 0699.65055 · doi:10.1007/BF03167891
[12] Glowinski, R., Lions, J.-L.: Exact and approximate controllability for distributed parame- ter systems. In: Acta Numerica, 1995, Cambridge Univ. Press, Cambridge, 159-333 (1995) · Zbl 0838.93014
[13] Ignat, L. I., Zuazua, E.: A two-grid approximation scheme for nonlinear Schrödinger equa- tions: dispersive properties and convergence. C. R. Math. Acad. Sci. Paris 341, 381-386 (2005) · Zbl 1079.65090 · doi:10.1016/j.crma.2005.07.018
[14] Infante, J. A., Zuazua, E.: Boundary observability for the space semi-discretizations of the 1-D wave equation. Math. Model. Numer. Anal. 33, 407-438 (1999) · Zbl 0947.65101 · doi:10.1051/m2an:1999123
[15] Lebeau, G.: Contr\hat ole de l’équation de Schrödinger. J. Math. Pures Appl. 71, 267-291 (1992) · Zbl 0838.35013
[16] Lebeau, G.: The wave equation with oscillating density: observability at low frequency. ESAIM Control Optim. Calc. Var. 5, 219-258 (2000) · Zbl 0953.35083 · doi:10.1051/cocv:2000109
[17] León, L., Zuazua, E.: Boundary controllability of the finite-difference space semi-dis- cretizations of the beam equation. ESAIM Control Optim. Calc. Var. 8, 827-862 (2002) · Zbl 1063.93025 · doi:10.1051/cocv:2002025
[18] Lions, J.-L.: Contr\hat olabilité exacte, perturbations et stabilisation de syst‘emes distribués. Tome 1, Recherches en Math. Appl. 8, Masson, Paris (1988) · Zbl 0653.93002
[19] Lions, J.-L., Magenes, E.: Probl‘emes aux limites non homog‘enes et applications. Vols. 1-2, Travaux et Recherches Math. 17, Dunod, Paris (1968) · Zbl 0165.10801
[20] Loreti, P., Mehrenberger, M.: An Ingham type proof for a two-grid observability theorem. ESAIM Control Optim. Calc. Var., to appear · Zbl 1157.35415 · doi:10.1051/cocv:2007062
[21] Macia, F.: Propagación y control de vibraciones en medios discretos y continuos. Tesis de doctorado, Univ. Complutense de Madrid (2002)
[22] Micu, S.: Uniform boundary controllability of a semi-discrete 1-D wave equation. Numer. Math. 91, 723-768 (2002) · Zbl 1002.65072 · doi:10.1007/s002110100338
[23] Miller, L.: Controllability cost of conservative systems: resolvent condition and transmutation. J. Funct. Anal. 218, 425-444 (2005) · Zbl 1122.93011 · doi:10.1016/j.jfa.2004.02.001
[24] Negreanu, M., Zuazua, E.: Uniform boundary controllability of a discrete 1-D wave equation. Systems Control Lett. 48, 261-279 (2003) · Zbl 1157.93324 · doi:10.1016/S0167-6911(02)00271-2
[25] Negreanu, M., Zuazua, E.: Convergence of a multigrid method for the controllability of a 1-D wave equation. C. R. Math. Acad. Sci. Paris 338, 413-418 (2004) · Zbl 1038.65054 · doi:10.1016/j.crma.2003.11.032
[26] Negreanu, M., Zuazua, E.: Discrete Ingham inequalities and applications. SIAM J. Numer. Anal. 44, 412-448 (2006) · Zbl 1142.93351 · doi:10.1137/050630015
[27] Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer Ser. Comput. Math. 23, Springer (1994) · Zbl 0803.65088
[28] Ramdani, K., Takahashi, T., Tenenbaum, G., Tucsnak, M.: A spectral approach for the exact observability of infinite-dimensional systems with skew-adjoint generator. J. Funct. Anal. 226, 193-229 (2005) · Zbl 1140.93395 · doi:10.1016/j.jfa.2005.02.009
[29] Russell, D. L., Weiss, G.: A general necessary condition for exact observability. SIAM J. Control Optim. 32, 1-23 (1994) · Zbl 0795.93023 · doi:10.1137/S036301299119795X
[30] Simon, J.: Compact sets in the space Lp(0, T ; B). Ann. Mat. Pura Appl. (4) 146, 65-96 (1987) · Zbl 0629.46031 · doi:10.1007/BF01762360
[31] Zuazua, E.: Boundary observability for the finite-difference space semi-discretizations of the 2-D wave equation in the square. J. Math. Pures Appl. 78, 523-563 (1999) · Zbl 0939.93016 · doi:10.1016/S0021-7824(98)00008-7
[32] Zuazua, E.: Propagation, observation, control and numerical approximation of waves ap- proximated by finite difference methods. SIAM Rev. 2, 197-243 (2005) · Zbl 1077.65095 · doi:10.1137/S0036144503432862
[33] Zuazua, E.: Control and numerical approximation of the wave and heat equations. In: Proc. ICM 2006, M. Sanz-Solé et al. (eds.), Vol. III, “Invited Lectures”, Eur. Math. Soc., 1389-1417 (2006) · Zbl 1108.93023
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