In this paper, we treat the problem of uniform exact boundary controllability for the finite-difference space semi-discretization of the $ 1 $-$ d $ coupled wave equations with a control acting only in one equation. First, we show how, after filtering the high frequencies of the discrete initial data in an appropriate way, we can construct a sequence of uniformly (with respect to the mesh size) bounded controls. Thus, we prove that the weak limit of the aforementioned sequence is a control for the continuous system. The proof of our results is based on the moment method and on the construction of an explicit biorthogonal sequence.
Citation: |
[1] |
F. Alabau-Boussouira, A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems, SIAM J. Control Optim., 42 (2003), 871-906.
doi: 10.1137/S0363012902402608.![]() ![]() ![]() |
[2] |
F. Alabau-Boussouira and M. Léautaud, Indirect controllability of locally coupled wave-type systems and applications, J. Math. Pures Appl., 99 (2013), 544-576.
doi: 10.1016/j.matpur.2012.09.012.![]() ![]() ![]() |
[3] |
D. S. Almeida Júnior, A. J. A. Ramos and M. L. Santos, Observability inequality for the finite-difference semi-discretization of the 1-d coupled wave equations, Adv. Comput. Math., 41 (2015), 105-130.
doi: 10.1007/s10444-014-9351-6.![]() ![]() ![]() |
[4] |
S. Avdonin, A. Choque Rivero and L. de Teresa, Exact boundary controllability of coupled hyperbolic equations, Int. J. Appl. Math. Comput. Sci., 23 (2013), 701-709.
doi: 10.2478/amcs-2013-0052.![]() ![]() ![]() |
[5] |
H. Bouslous, H. El Boujaoui and L. Maniar, Uniform boundary stabilization for the finite difference semi-discretization of 2-D wave equation, Afr. Mat., 25 (2014), 623-643.
doi: 10.1007/s13370-013-0141-y.![]() ![]() ![]() |
[6] |
I. F. Bugariu, S. Micu and I. Rovenţa, Approximation of the controls for the beam equation with vanishing viscosity, Math. Comp., 85 (2016), 2259-2303.
doi: 10.1090/mcom/3064.![]() ![]() ![]() |
[7] |
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.
doi: 10.1007/s00211-005-0651-0.![]() ![]() ![]() |
[8] |
A. El Akri and L. Maniar, Indirect boundary observability of semi-discrete coupled wave equations, Electron. J. Differential Equations, 2018 (2018), 27 pp.
![]() ![]() |
[9] |
H. El Boujaoui, H. Bouslous and L. Maniar, Uniform boundary stabilization for the finite difference discretization of the 1-D wave equation, Afr. Mat., 27 (2016), 1239-1262.
doi: 10.1007/s13370-016-0406-3.![]() ![]() ![]() |
[10] |
S. Ervedoza and E. Zuazua, Numerical Approximation of Exact Controls for Waves, Springer Briefs in Mathematics, Springer, New York, 2013.
doi: 10.1007/978-1-4614-5808-1.![]() ![]() ![]() |
[11] |
H. O. Fattorini, Estimates for sequences biorthogonal to certain complex exponentials and boundary control of the wave equation, New Trends in Systems Analysis, Lecture Notes in Control and Inform. Sci., Springer, Berlin, 2 (1977), 111-124.
![]() ![]() |
[12] |
H. O. Fattorini and D. L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Ration. Mech. Anal., 43 (1971), 272-292.
doi: 10.1007/BF00250466.![]() ![]() ![]() |
[13] |
R. Glowinski and C. H. Li, On the numerical implementation of the Hilbert uniqueness method for the exact boundary controllability of the wave equation, C. R. Acad. Sci. Paris Sér. I Math., 311 (1990), 135-142.
![]() ![]() |
[14] |
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.
doi: 10.1007/BF03167891.![]() ![]() ![]() |
[15] |
L. Hörmander, The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis, Classics in Mathematics, Springer-Verlag, Berlin, 2003.
doi: 10.1007/978-3-642-61497-2.![]() ![]() ![]() |
[16] |
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.
doi: 10.1051/m2an:1999123.![]() ![]() ![]() |
[17] |
J.-L. Lions, Contrôlabilité Exacte Perturbations et Stabilisation de Systémes Distribués, Tome 1: Contrôlabilité Exacte, Recherches en Mathématiques Appliquées, 9. Masson, Paris, 1988.
![]() |
[18] |
P. Lissy, Construction of Gevrey functions with compact support using the Bray-Mandelbrojt iterative process and applications to the moment method in control theory, Math. Control Relat. Fields, 7 (2017), 21-40.
doi: 10.3934/mcrf.2017002.![]() ![]() ![]() |
[19] |
P. Lissy and I. Rovenţa, Optimal filtration for the approximation of boundary controls for the one-dimensional wave equation, Math. Comp., 88 (2019), 273-291.
doi: 10.1090/mcom/3345.![]() ![]() ![]() |
[20] |
S. Micu, Uniform boundary controllability of a semi-discrete 1-D wave equation, Numer. Math., 91 (2002), 723-768.
doi: 10.1007/s002110100338.![]() ![]() ![]() |
[21] |
S. Micu, Uniform boundary controllability of a semidiscrete 1-D wave equation with vanishing viscosity, SIAM J. Control Optim., 47 (2008), 2857-2885.
doi: 10.1137/070696933.![]() ![]() ![]() |
[22] |
S. Micu, I. Rovenţa and L. E. Temereancǎ, Approximation of the controls for the linear beam equation, Math. Control Signals Syst., 28 (2016), Art. 12, 53 pp.
doi: 10.1007/s00498-016-0161-x.![]() ![]() ![]() |
[23] |
W. Rudin, Real and Complex Analysis, Second edition, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974.
![]() ![]() |
[24] |
E. D. Sontag, Mathematical Control Theory. Deterministic Finite-Dimensional Systems, 2nd edition, Texts in Applied Mathematics, 6, Springer-Verlag, New York, 1998.
doi: 10.1007/978-1-4612-0577-7.![]() ![]() ![]() |
[25] |
L. T. 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.
doi: 10.1007/s10444-004-7629-9.![]() ![]() ![]() |
[26] |
R. M. Young, An Introduction to Nonharmonic Fourier Series, Pure and Applied Mathematics, 93. Academic Press, Inc., New York-London, 1980.
![]() ![]() |