# American Institute of Mathematical Sciences

December  2020, 10(4): 669-698. doi: 10.3934/mcrf.2020015

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

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

* 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 Published  December 2019

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: Abdeladim El Akri, Lahcen Maniar. Uniform indirect boundary controllability of semi-discrete $1$-$d$ coupled wave equations. Mathematical Control & Related Fields, 2020, 10 (4) : 669-698. doi: 10.3934/mcrf.2020015
##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] A. El Akri and L. Maniar, Indirect boundary observability of semi-discrete coupled wave equations, Electron. J. Differential Equations, 2018 (2018), 27 pp.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] J.-L. Lions, Contrôlabilité Exacte Perturbations et Stabilisation de Systémes Distribués, Tome 1: Contrôlabilité Exacte, Recherches en Mathématiques Appliquées, 9. Masson, Paris, 1988. Google Scholar [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.  Google Scholar [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.  Google Scholar [20] S. Micu, Uniform boundary controllability of a semi-discrete 1-D wave equation, Numer. Math., 91 (2002), 723-768.  doi: 10.1007/s002110100338.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] W. Rudin, Real and Complex Analysis, Second edition, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974.  Google Scholar [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.  Google Scholar [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.  Google Scholar [26] R. M. Young, An Introduction to Nonharmonic Fourier Series, Pure and Applied Mathematics, 93. Academic Press, Inc., New York-London, 1980.   Google Scholar

show all references

##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] A. El Akri and L. Maniar, Indirect boundary observability of semi-discrete coupled wave equations, Electron. J. Differential Equations, 2018 (2018), 27 pp.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] J.-L. Lions, Contrôlabilité Exacte Perturbations et Stabilisation de Systémes Distribués, Tome 1: Contrôlabilité Exacte, Recherches en Mathématiques Appliquées, 9. Masson, Paris, 1988. Google Scholar [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.  Google Scholar [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.  Google Scholar [20] S. Micu, Uniform boundary controllability of a semi-discrete 1-D wave equation, Numer. Math., 91 (2002), 723-768.  doi: 10.1007/s002110100338.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] W. Rudin, Real and Complex Analysis, Second edition, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974.  Google Scholar [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.  Google Scholar [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.  Google Scholar [26] R. M. Young, An Introduction to Nonharmonic Fourier Series, Pure and Applied Mathematics, 93. Academic Press, Inc., New York-London, 1980.   Google Scholar
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