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doi: 10.3934/dcdss.2021091
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## Boundary observability and exact controllability of strongly coupled wave equations

 1 Lebanese University, Faculty of sciences 1 and EDST, Khawarizmi Laboratory of Mathematics and Applications-KALMA, Hadath-Beirut, Lebanon 2 Lebanese International University, Department of mathematics and physics, Beirut, Lebanon, Lebanese University, Faculty of Business, Section 5, Nabateieh, Lebanon

* Corresponding author: Ali Wehbe

Received  March 2021 Revised  May 2021 Early access August 2021

In this paper, we study the exact controllability of a system of two wave equations coupled by velocities with boundary control acted on only one equation. In the first part of this paper, we consider the $N$-d case. Then, using a multiplier technique, we prove that, by observing only one component of the associated homogeneous system, one can get back a full energy of both components in the case where the waves propagate with equal speeds (i.e. $a = 1$ in (1)) and where the coupling parameter $b$ is small enough. This leads, by the Hilbert Uniqueness Method, to the exact controllability of our system in any dimension space. It seems that the conditions $a = 1$ and $b$ small enough are technical for the multiplier method. The natural question is then : what happens if one of the two conditions is not satisfied? This consists the aim of the second part of this paper. Indeed, we consider the exact controllability of a system of two one-dimensional wave equations coupled by velocities with a boundary control acted on only one equation. Using a spectral approach, we establish different types of observability inequalities which depend on the algebraic nature of the coupling parameter $b$ and on the arithmetic property of the wave propagation speeds $a$.

Citation: Ali Wehbe, Marwa Koumaiha, Layla Toufaily. Boundary observability and exact controllability of strongly coupled wave equations. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021091
##### References:
 [1] F. Alabau, Observabilité frontière indirecte de systèmes faiblement couplés, C. R. Acad. Sci. Paris Sér. I Math., 333 (2001), 645-650.  doi: 10.1016/S0764-4442(01)02076-6.  Google Scholar [2] F. Alabau-Bousouira, A two level energy method for indirect boundary obsevability and controllability of weakly coupled hyperbolic systems, SIAM J. Control. Optim, 42 (2003), 871-906.  doi: 10.1137/S0363012902402608.  Google Scholar [3] 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 [4] F. Alabau-Boussouira, Z. Wang and L. Yu, A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities, ESAIM Control Optim. Calc. Var., 23 (2017), 721-749.  doi: 10.1051/cocv/2016011.  Google Scholar [5] F. Ammar Khodja and A. Bader, Stabilizability of systems of one-dimensional wave equations by one internal or boundary control force, SIAM J. Control Optim., 39 (2001), 1833–1851 (electronic). doi: 10.1137/S0363012900366613.  Google Scholar [6] F. Ammar Khodja, A. Benabdallah and C. Dupaix, Null-controllability of some reaction-diffusion systems with one control force, J. Math. Anal. Appl., 320 (2006), 928-943.  doi: 10.1016/j.jmaa.2005.07.060.  Google Scholar [7] Y. Bugeaud, Approximation by Algebraic Numbers, vol. 160 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2004, doi: 10.1017/CBO9780511542886.  Google Scholar [8] S. Gerbi, C. Kassem, A. Mortada and A. Wehbe, Exact controllability and stabilization of locally coupled wave equations: Theoretical Results, Z. Anal. Anwend., 40 (2021), 67–96, arXiv e-prints, arXiv: 2003.14001. doi: 10.4171/ZAA/1673.  Google Scholar [9] V. Komornik, Exact Controllability and Stabilization, RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994, The multiplier method.  Google Scholar [10] V. Komornik and P. Loreti, Ingham-type theorems for vector-valued functions and observability of coupled linear systems, SIAM J. Control Optim., 37 (1999), 461-485.  doi: 10.1137/S0363012997317505.  Google Scholar [11] V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer Monographs in Mathematics, Springer-Verlag, New York, 2005.  Google Scholar [12] J.-L. Lions, Contrôlabilité Exacte Perturbations et Stabilisation de Systèmes Distribués. Tome 1., Contrôlabilité Exacte, Recherches en Mathématiques Appliquées, Masson, Paris, Milan, Barcelone, 1988. Google Scholar [13] P. Loreti and B. Rao, Compensation spectrale et taux de décroissance optimal de l'énergie de systèmes partiellement amortis, C. R. Math. Acad. Sci. Paris, 337 (2003), 531-536.  doi: 10.1016/j.crma.2003.08.009.  Google Scholar [14] N. Najdi, Etude de la Stabilisation Exponentielle et Polynômiale de Certains Systèmes d'équations Couplées par des Contrôles Indirects Bornés ou non Bornés, PhD thesis, Université de Valenciennes et Université Libanaise, 2016. Google Scholar [15] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983 doi: 10.1007/978-1-4612-5561-1.  Google Scholar [16] B. Rao and Z. Liu, A spectral approach to the indirect boundary control of a system of weakly coupled wave equations, Discrete and Continuous Dynamical Systems, 23 (2009), 399-414.  doi: 10.3934/dcds.2009.23.399.  Google Scholar [17] L. Toufayli, Stabilisation Polynomiale et Contrôlabilité Exacte Des Équations Des Ondes Par Des Contrôles Indirects et Dynamiques, PhD thesis, Université de Strasbourg, 2013. Google Scholar [18] A. Wehbe and W. Youssef, Indirect locally internal observability and controllability of weakly coupled wave equations, Differential Equations and Applications-DEA, 3 (2011), 449-462.  doi: 10.7153/dea-03-28.  Google Scholar [19] X. Zhang and E. Zuazua, Polynomial decay and control of a $1-d$ hyperbolic-parabolic coupled system, J. Differential Equations, 204 (2004), 380-438.  doi: 10.1016/j.jde.2004.02.004.  Google Scholar

show all references

##### References:
 [1] F. Alabau, Observabilité frontière indirecte de systèmes faiblement couplés, C. R. Acad. Sci. Paris Sér. I Math., 333 (2001), 645-650.  doi: 10.1016/S0764-4442(01)02076-6.  Google Scholar [2] F. Alabau-Bousouira, A two level energy method for indirect boundary obsevability and controllability of weakly coupled hyperbolic systems, SIAM J. Control. Optim, 42 (2003), 871-906.  doi: 10.1137/S0363012902402608.  Google Scholar [3] 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 [4] F. Alabau-Boussouira, Z. Wang and L. Yu, A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities, ESAIM Control Optim. Calc. Var., 23 (2017), 721-749.  doi: 10.1051/cocv/2016011.  Google Scholar [5] F. Ammar Khodja and A. Bader, Stabilizability of systems of one-dimensional wave equations by one internal or boundary control force, SIAM J. Control Optim., 39 (2001), 1833–1851 (electronic). doi: 10.1137/S0363012900366613.  Google Scholar [6] F. Ammar Khodja, A. Benabdallah and C. Dupaix, Null-controllability of some reaction-diffusion systems with one control force, J. Math. Anal. Appl., 320 (2006), 928-943.  doi: 10.1016/j.jmaa.2005.07.060.  Google Scholar [7] Y. Bugeaud, Approximation by Algebraic Numbers, vol. 160 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2004, doi: 10.1017/CBO9780511542886.  Google Scholar [8] S. Gerbi, C. Kassem, A. Mortada and A. Wehbe, Exact controllability and stabilization of locally coupled wave equations: Theoretical Results, Z. Anal. Anwend., 40 (2021), 67–96, arXiv e-prints, arXiv: 2003.14001. doi: 10.4171/ZAA/1673.  Google Scholar [9] V. Komornik, Exact Controllability and Stabilization, RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994, The multiplier method.  Google Scholar [10] V. Komornik and P. Loreti, Ingham-type theorems for vector-valued functions and observability of coupled linear systems, SIAM J. Control Optim., 37 (1999), 461-485.  doi: 10.1137/S0363012997317505.  Google Scholar [11] V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer Monographs in Mathematics, Springer-Verlag, New York, 2005.  Google Scholar [12] J.-L. Lions, Contrôlabilité Exacte Perturbations et Stabilisation de Systèmes Distribués. Tome 1., Contrôlabilité Exacte, Recherches en Mathématiques Appliquées, Masson, Paris, Milan, Barcelone, 1988. Google Scholar [13] P. Loreti and B. Rao, Compensation spectrale et taux de décroissance optimal de l'énergie de systèmes partiellement amortis, C. R. Math. Acad. Sci. Paris, 337 (2003), 531-536.  doi: 10.1016/j.crma.2003.08.009.  Google Scholar [14] N. Najdi, Etude de la Stabilisation Exponentielle et Polynômiale de Certains Systèmes d'équations Couplées par des Contrôles Indirects Bornés ou non Bornés, PhD thesis, Université de Valenciennes et Université Libanaise, 2016. Google Scholar [15] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983 doi: 10.1007/978-1-4612-5561-1.  Google Scholar [16] B. Rao and Z. Liu, A spectral approach to the indirect boundary control of a system of weakly coupled wave equations, Discrete and Continuous Dynamical Systems, 23 (2009), 399-414.  doi: 10.3934/dcds.2009.23.399.  Google Scholar [17] L. Toufayli, Stabilisation Polynomiale et Contrôlabilité Exacte Des Équations Des Ondes Par Des Contrôles Indirects et Dynamiques, PhD thesis, Université de Strasbourg, 2013. Google Scholar [18] A. Wehbe and W. Youssef, Indirect locally internal observability and controllability of weakly coupled wave equations, Differential Equations and Applications-DEA, 3 (2011), 449-462.  doi: 10.7153/dea-03-28.  Google Scholar [19] X. Zhang and E. Zuazua, Polynomial decay and control of a $1-d$ hyperbolic-parabolic coupled system, J. Differential Equations, 204 (2004), 380-438.  doi: 10.1016/j.jde.2004.02.004.  Google Scholar
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