December  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 and 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 and 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 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.

[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.  doi: 10.1137/S0363012900366613.

[6]

F. Ammar KhodjaA. BenabdallahM. Gonzá lez-Burgos and 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 KhodjaA. BenabdallahM. Gonzá lez-Burgos and 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 KhodjaA. BenabdallahM. Gonzá lez-Burgos and 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. AvdoninA. 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.

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

[12]

C. BardosG. Lebeau and 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. DehmanJ. Le Rousseau and 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. doi: 10.1007/978-3-7643-8994-9.

[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 and 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 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.

[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.  doi: 10.1137/S0363012900366613.

[6]

F. Ammar KhodjaA. BenabdallahM. Gonzá lez-Burgos and 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 KhodjaA. BenabdallahM. Gonzá lez-Burgos and 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 KhodjaA. BenabdallahM. Gonzá lez-Burgos and 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. AvdoninA. 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.

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

[12]

C. BardosG. Lebeau and 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. DehmanJ. Le Rousseau and 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. doi: 10.1007/978-3-7643-8994-9.

[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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