2017, 6(4): 487-516. doi: 10.3934/eect.2017025

Exact and approximate controllability of coupled one-dimensional hyperbolic equations

1. 

Laboratoire AMNEDP, Faculty of Mathematics, USTHB, Algiers, Algeria

2. 

Laboratoire de Mathématiques, Université de Bourgogne Franche-Comté, 16 route de Gray 25030 Besancon cedex, France

* Corresponding author: Farid Ammar Khodja

Received  December 2016 Revised  July 2017 Published  September 2017

We deal with the simultaneous controllability properties of two one dimensional (strongly) coupled wave equations when the control acts on the boundary. Necessary and sufficient conditions for approximate and exact controllability are proved.

Citation: Abdelaziz Bennour, Farid Ammar Khodja, Djamel Teniou. Exact and approximate controllability of coupled one-dimensional hyperbolic equations. Evolution Equations & Control Theory, 2017, 6 (4) : 487-516. doi: 10.3934/eect.2017025
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, On the influence of the coupling on the dynamics of single-observed cascade systems of PDE'S, Mathematical Control and Related Fields, 5 (2015), 1-30.

[3]

F. Alabau-Boussouira, M. Léautaud, Indirect controllability of locally coupled systems under geometric conditions, C. R. Acad. Sci. Paris, Ser. I, 349 (2011), 395-400. doi: 10.1016/j.crma.2011.02.004.

[4]

F. Alabau-Boussouira, 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.

[5]

F. Ammar Khodja, 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. doi: 10.1137/S0363012900366613.

[6]

F. Ammar Khodja, A. Benabdallah, M. Gonzá lez-Burgos, L. de Teresa, Recent results on the controllability of linear coupled parabolic problems: A survey, Math. Control Relat. Fields, 1 (2011), 267-306. doi: 10.3934/mcrf.2011.1.267.

[7]

F. Ammar Khodja, A. Benabdallah, M. Gonzá lez-Burgos, L. de Teresa, A new relation between the condensation index of complex sequences and the null controllability of parabolic systems, C. R. Math. Acad. Sci. Paris, 351 (2013), 743-746. doi: 10.1016/j.crma.2013.09.014.

[8]

F. Ammar Khodja, A. Benabdallah, M. Gonzá lez-Burgos, L. de Teresa, New phenomena for the null controllability of parabolic systems: Minimal time and geometrical dependence, J. Math. Anal. Appl., 444 (2016), 1071-1113.

[9]

S. Avdonin, A. Choque Rivero, 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.

[10]

S. A. Avdonin and S. A. Ivanov, Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems, Cambridge University Press, Cambridge, 1995.

[11]

S. Avdonin, W. Moran, Ingham-type inequalities and Riesz bases of divided differences, Int. J. Appl. Math. Comput. Sci., 11 (2001), 803-820.

[12]

C. Bardos, G. Lebeau, J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065. doi: 10.1137/0330055.

[13]

B. Dehman, J. Le Rousseau, M. Léautaud, Controllability of Two Coupled Wave Equations on a Compact Manifold, Arch. Rational Mech. Anal., 211 (2014), 113-187. doi: 10.1007/s00205-013-0670-4.

[14]

I. Gohberg and M. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Translations of Mathematical Monographs, Vol. 18, AMS, Providence, R. I. , 1969.

[15]

A. Ya. Kinchin, Continued Fractions, The University of Chicago Press, 1964.

[16]

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

[17]

I. Lasiecka & R. Triggiani, Carleman estimates and exact boundary controllability for a system of coupled, nonconservative second-order hyperbolic equations, In: "Partial Differential Equation Methods in Control and Shape optimization". G. Da Prato & J-P Zolésio (editors). Marcel and Dekker, INC (1997), 215-244.

[18]

J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisations de Systèmes Distribués, Masson, Paris, 1988.

[19]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhauser Advanced Texts: Basler Lehrbucher, Birkhauser Verlag, Basel, 2009.

[20]

D. Ullrich, Divided differences and systems of nonharmonic Fourier series, Proc. Amer. Math. Soc., 80 (1980), 47-57. doi: 10.1090/S0002-9939-1980-0574507-8.

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.

[2]

F. Alabau-Boussouira, On the influence of the coupling on the dynamics of single-observed cascade systems of PDE'S, Mathematical Control and Related Fields, 5 (2015), 1-30.

[3]

F. Alabau-Boussouira, M. Léautaud, Indirect controllability of locally coupled systems under geometric conditions, C. R. Acad. Sci. Paris, Ser. I, 349 (2011), 395-400. doi: 10.1016/j.crma.2011.02.004.

[4]

F. Alabau-Boussouira, 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.

[5]

F. Ammar Khodja, 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. doi: 10.1137/S0363012900366613.

[6]

F. Ammar Khodja, A. Benabdallah, M. Gonzá lez-Burgos, L. de Teresa, Recent results on the controllability of linear coupled parabolic problems: A survey, Math. Control Relat. Fields, 1 (2011), 267-306. doi: 10.3934/mcrf.2011.1.267.

[7]

F. Ammar Khodja, A. Benabdallah, M. Gonzá lez-Burgos, L. de Teresa, A new relation between the condensation index of complex sequences and the null controllability of parabolic systems, C. R. Math. Acad. Sci. Paris, 351 (2013), 743-746. doi: 10.1016/j.crma.2013.09.014.

[8]

F. Ammar Khodja, A. Benabdallah, M. Gonzá lez-Burgos, L. de Teresa, New phenomena for the null controllability of parabolic systems: Minimal time and geometrical dependence, J. Math. Anal. Appl., 444 (2016), 1071-1113.

[9]

S. Avdonin, A. Choque Rivero, 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.

[10]

S. A. Avdonin and S. A. Ivanov, Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems, Cambridge University Press, Cambridge, 1995.

[11]

S. Avdonin, W. Moran, Ingham-type inequalities and Riesz bases of divided differences, Int. J. Appl. Math. Comput. Sci., 11 (2001), 803-820.

[12]

C. Bardos, G. Lebeau, J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065. doi: 10.1137/0330055.

[13]

B. Dehman, J. Le Rousseau, M. Léautaud, Controllability of Two Coupled Wave Equations on a Compact Manifold, Arch. Rational Mech. Anal., 211 (2014), 113-187. doi: 10.1007/s00205-013-0670-4.

[14]

I. Gohberg and M. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Translations of Mathematical Monographs, Vol. 18, AMS, Providence, R. I. , 1969.

[15]

A. Ya. Kinchin, Continued Fractions, The University of Chicago Press, 1964.

[16]

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

[17]

I. Lasiecka & R. Triggiani, Carleman estimates and exact boundary controllability for a system of coupled, nonconservative second-order hyperbolic equations, In: "Partial Differential Equation Methods in Control and Shape optimization". G. Da Prato & J-P Zolésio (editors). Marcel and Dekker, INC (1997), 215-244.

[18]

J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisations de Systèmes Distribués, Masson, Paris, 1988.

[19]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhauser Advanced Texts: Basler Lehrbucher, Birkhauser Verlag, Basel, 2009.

[20]

D. Ullrich, Divided differences and systems of nonharmonic Fourier series, Proc. Amer. Math. Soc., 80 (1980), 47-57. doi: 10.1090/S0002-9939-1980-0574507-8.

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