Uniform indirect boundary controllability of semi-discrete $ 1 $-$ d $ coupled wave equations

Abdeladim El Akri; Lahcen Maniar

doi:10.3934/mcrf.2020015

Article Contents

2020, Volume 10, Issue 4: 669-698. Doi: 10.3934/mcrf.2020015

This issue Previous Article Next Article The Kato smoothing effect for the nonlinear regularized Schrödinger equation on compact manifolds

Uniform indirect boundary controllability of semi-discrete $ 1 $-$ d $ coupled wave equations

Abdeladim El Akri^, and
Lahcen Maniar

Université Cadi Ayyad, Faculté des Sciences Semlalia, LMDP, UMMISCO (IRD- UPMC), Marrakech 40000, B.P. 2390, Maroc

^* Corresponding author: Abdeladim El Akri
^* Corresponding author: Abdeladim El Akri

The first author would like to thank S. Micu for fruitful discussions on several parts of this paper during his visit to Craiova University

Received: January 2019

Revised: October 2019

Early access: December 2019

Published: October 2020

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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.

Keywords:

Coupled wave equations,
uniform indirect exact boundary controllability,
space semi-discretization,
finite differences,
moment problem,
biorthogonal sequence,
filtered spaces.

Mathematics Subject Classification: Primary: 93B05, 35L05, 30E05; Secondary: 65M06.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[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.