May  2022, 15(5): 1269-1305. doi: 10.3934/dcdss.2021091

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 Published  May 2022 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 and Continuous Dynamical Systems - S, 2022, 15 (5) : 1269-1305. 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.

[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.

[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.

[4]

F. Alabau-BoussouiraZ. 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.

[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.

[6]

F. Ammar KhodjaA. 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.

[7]

Y. Bugeaud, Approximation by Algebraic Numbers, vol. 160 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2004, doi: 10.1017/CBO9780511542886.

[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.

[9]

V. Komornik, Exact Controllability and Stabilization, RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994, The multiplier method.

[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.

[11]

V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer Monographs in Mathematics, Springer-Verlag, New York, 2005.

[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.

[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.

[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.

[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.

[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.

[17]

L. Toufayli, Stabilisation Polynomiale et Contrôlabilité Exacte Des Équations Des Ondes Par Des Contrôles Indirects et Dynamiques, PhD thesis, Université de Strasbourg, 2013.

[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.

[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.

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.

[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.

[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.

[4]

F. Alabau-BoussouiraZ. 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.

[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.

[6]

F. Ammar KhodjaA. 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.

[7]

Y. Bugeaud, Approximation by Algebraic Numbers, vol. 160 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2004, doi: 10.1017/CBO9780511542886.

[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.

[9]

V. Komornik, Exact Controllability and Stabilization, RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994, The multiplier method.

[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.

[11]

V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer Monographs in Mathematics, Springer-Verlag, New York, 2005.

[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.

[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.

[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.

[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.

[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.

[17]

L. Toufayli, Stabilisation Polynomiale et Contrôlabilité Exacte Des Équations Des Ondes Par Des Contrôles Indirects et Dynamiques, PhD thesis, Université de Strasbourg, 2013.

[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.

[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.

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