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Boundary observability of a numerical approximation of Maxwell’s system in a cube. (English) Zbl 1182.93030

Summary: We consider space discretization of Maxwell’s system by Yee’s scheme. For such a scheme, we first show that the boundary observability estimate is not uniform with respect to the mesh size. Using a discrete multiplier method, we prove a modified but uniform observability estimate and deduce that the Tychonoff regularization technique allows to recover the convergence of the discrete control to the continuous one. Note that we shortly describe a discrete differential calculus that is of its own interest.

MSC:

93B07 Observability
93B40 Computational methods in systems theory (MSC2010)
93C20 Control/observation systems governed by partial differential equations
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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References:

[1] C. Castro and S. Micu, Boundary controllability of a linear semidiscrete 1 —D wave equation derived from a mixed finite element method,Numer. Math. 102 (2006), 413–462. · Zbl 1102.93004 · doi:10.1007/s00211-005-0651-0
[2] C. Castro, S. Micu, and A. Münch, Numerical approximation of the boundary control of the 2 —D wave equation with mixed finite elements,IMA J. Numer. Analysis, to appear.
[3] M. Costabel, M. Dauge, and S. Nicaise, Singularities of Maxwell interface problems,M2AN Math. Model. Numer. Anal. 33 (1999), 627–649. · Zbl 0937.78003 · doi:10.1051/m2an:1999155
[4] M.M. Eller and J.E. Masters, Exact boundary controllability of electromagnetic fields in a general region,Appl. Math. Optim. 45 (2002), 99–123. · Zbl 0997.35099 · doi:10.1007/s00245-001-0030-x
[5] R. Glowinski, Ensuring wellposedness by analogy: Stokes problem and boundary control of the wave equation,J. Comput. Phys. 103 (1992), 189–221. · Zbl 0763.76042 · doi:10.1016/0021-9991(92)90396-G
[6] 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. · Zbl 0711.65084 · doi:10.1002/nme.1620270313
[7] 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,Japan J. Appl. Math. 7 (1990), 1–76. · Zbl 0699.65055 · doi:10.1007/BF03167891
[8] R. Glowinski and J.L., Lions, Exact and approximate controllability for distributed parameter systems,Acta Numerica (1996), 159-333.
[9] L.F. Ho.Observabilité frontière de l’équation des ondes, (Boundary observability of the wave equation),C. R. Acad. Sci. Paris Sér. I Math. 302 (1986), 443–446. · Zbl 0598.35060
[10] J.A. Infante and E. Zuazua, Boundary observability for the space semidiscretizations of the 1 —D wave equation,M2AN Math. Model. Numer. Anal. 33 (1999), 407–438. · Zbl 0947.65101 · doi:10.1051/m2an:1999123
[11] E. Isaacson and H.B. Keller,Analysis of Numerical Methods, John Wiley & Sons, Inc., New York-London-Sydney, 1966. · Zbl 0168.13101
[12] B.V. Kapitanov, Stabilization and exact boundary controllability for Maxwell’s equations.SIAM J. Control Optim. 32 (1994), 408–420. · Zbl 0827.35012 · doi:10.1137/S0363012991218487
[13] K.A. Kime, Boundary exact controllability of Maxwell’s equations in a spherical region,SIAM J. Control Optim. 28 (1990), 294–319. · Zbl 0693.93010 · doi:10.1137/0328016
[14] V. Komornik, Boundary stabilization, observation and control of Maxwell’s equations,Panamer. Math. J. 4 (1994), 47–61. · Zbl 0849.35136
[15] V. Komornik,Exact Controllability and Stabilization, The Multiplier Method, Research in Applied Mathematics, Masson, Paris, 1994. · Zbl 0937.93003
[16] J.E. Lagnese, Exact controllability of Maxwell’s equations in a general region,SIAM J. Control Optim. 27 (1989), 374–388. · Zbl 0678.49032 · doi:10.1137/0327019
[17] L. León and E. Zuazua, Boundary controllability of the finitedifference space semidiscretizations of the beam equation, (Special volume in the memory of J.L. Lions),ESAIM Control Optim. Calc. Var. 8 (2002), 827–862. · Zbl 1063.93025 · doi:10.1051/cocv:2002025
[18] J.L. Lions,Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués I, Recherches en Mathématiques Appliquées8, Masson, Paris, 1988.
[19] P. Monk, A mixed method for approximating Maxwell’s equations,SIAM J. Numer. Anal. 28 (1991), 1610–1634. · Zbl 0742.65091 · doi:10.1137/0728081
[20] P. Monk,Finite Element Methods for Maxwell’s Equations, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2003. · Zbl 1024.78009
[21] P. Monk and E. Süli, A convergence analysis of Yee’s scheme on nonuniform grids,SIAM J. Numer. Anal. 31 (1994), 393–412. · Zbl 0805.65121 · doi:10.1137/0731021
[22] A. Münch, A uniformly controllable and implicit scheme for the 1 –D wave equation,M2AN Math. Model. Numer. Anal. 39 (2005), 377–418. · Zbl 1130.93016 · doi:10.1051/m2an:2005012
[23] M. Negreanu and E. Zuazua, Uniform boundary controllability of a discrete 1 –D wave equation,Systems and Control Lett. 48 (2003), 261–280. · Zbl 1157.93324 · doi:10.1016/S0167-6911(02)00271-2
[24] M. Negreanu and E. Zuazua, Discrete Ingham inequalities and applications,SIAM J. Numer. Anal. 44 (2006), 412–448. · Zbl 1142.93351 · doi:10.1137/050630015
[25] S. Nicaise, Exact boundary controllability of Maxwell’s equations in heterogeneous media and an application to an inverse source problem,SIAM J. Control Optim. 38 (2000), 1145–1170. · Zbl 0963.93041 · doi:10.1137/S0363012998344373
[26] S. Nicaise and C. Pignotti, Boundary stabilization of Maxwell’s equations with spacetime variable coefficients,ESAIM Control Optim. Calc. Var. 9 (2003), 563–578. · Zbl 1063.93041 · doi:10.1051/cocv:2003027
[27] S. Nicaise and C. Pignotti, Energy decay rates for solutions of Maxwell’s system with a memory boundary condition,Collect. Math. to appear. · Zbl 1132.93037
[28] K.D. Phung, Contrôle et stabilisation d’ondes électromagnétiques,ESAIM Control Optim. Calc. Var. 5 (2000), 87–137. · Zbl 0942.93002 · doi:10.1051/cocv:2000103
[29] C. Pignotti, Observability and controllability of Maxwell’s equations,Rend. Mat. Appl. (7) 19 (1999), 523–546. · Zbl 0979.93057
[30] K. Ramdani, T. Takahashi, and M. Tucsnak, Uniformly exponentially stable approximations for a class of second order evolution equations: application to the optimal control of flexible structures,ESAIM Control Optim. Calc. Var. to appear. · Zbl 1126.93050
[31] D.L. Russell, The DirichletNeumann boundary control problem associated with Maxwell’s equations in a cylindrical region,SIAM J. Control Optim. 24 (1986), 199–229. · Zbl 0594.49026 · doi:10.1137/0324012
[32] L.R.T. Tébou and E. Zuazua, Uniform exponential long time decay for the space semidiscretization of a locally damped wave equation via an artificial numerical viscosity,Numer. Math. 95 (2003), 563–598. · Zbl 1033.65080 · doi:10.1007/s00211-002-0442-9
[33] N. Weck, Exact boundary controllability of a Maxwell problem,SIAM J. Control Optim. 38 (2000), 736–750. · Zbl 0963.93040 · doi:10.1137/S0363012998347559
[34] K.S. Yee, Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,IEEE Trans. Antennas and Propagation 14 (1966), 302–307. · Zbl 1155.78304
[35] X. Zhang, Exact internal controllability of Maxwell’s equations,Appl. Math. Optim. 41 (2000), 155–170. · Zbl 0952.93011 · doi:10.1007/s002459911009
[36] Q. Zhou, Exact internal controllability of Maxwell’s equations,Japan J. Indust. Appl. Math. 14 (1997), 245–256. · Zbl 1306.35129 · doi:10.1007/BF03167266
[37] E. Zuazua, Boundary observability for the finitedifference space semidiscretizations of the 2 –D wave equation in the square,J. Math. Pures Appl. 78 (1999), 523–563. · Zbl 0939.93016
[38] E. Zuazua, Optimal and approximate control of finitedifference approximation schemes for the 1 –D wave equation,Rend. Mat. Appl. (7) 24 (2004), 201–237. · Zbl 1085.49041
[39] E. Zuazua, Propagation, observation, and control of waves approximated by finite difference methods,SIAM Rev. 47 (2005), 197–243. · Zbl 1077.65095 · doi:10.1137/S0036144503432862
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