American Institute of Mathematical Sciences

December  2019, 8(4): 687-694. doi: 10.3934/eect.2019032

Simultaneous controllability of two vibrating strings with variable coefficients

 1 University of Tunis El Manar, Faculty of Sciences of Tunis, Tunisia 2 University of Carthage, Polytechnic School of Tunisia, Tunisia

Received  January 2018 Revised  April 2019 Published  December 2019 Early access  June 2019

We study the simultaneous exact controllability of two vibrating strings with variable physical coefficients and controlled from a common endpoint. We give sufficient conditions on the physical coefficients for which the eigenfrequencies of both systems do not coincide and the associated spectral gap is uniformly positive. Under these conditions, we show that these systems are simultaneously exactly controllable.

Citation: Jamel Ben Amara, Emna Beldi. Simultaneous controllability of two vibrating strings with variable coefficients. Evolution Equations and Control Theory, 2019, 8 (4) : 687-694. doi: 10.3934/eect.2019032
References:
 [1] S. Avdonin, Simultaneous controllability of several elastic strings, Proc. CD of the Fourteenth International Symposium of Mathematical Theory of Networks and Systems, Perpignan, France, (2000), 19–23.http://www.math.ucsd.edu/ ~helton/MTNSHISTORY/CONTENTS/2000PERPIGNAN/CDROM/articles/B169.pdf. [2] S. Avdonin and M. Tucsnak, Simultaneous controllability in sharp time for two elastic strings, ESAIM: COCV, 6 (2001), 259-273.  doi: 10.1051/cocv:2001110. [3] S. Avdonin and W. Moran, Simultaneous control problems for systems of elastic strings and beams, Systems and Control Letters, 44 (2001), 147-155.  doi: 10.1016/S0167-6911(01)00137-2. [4] S. Avdonin and W. Moran, Ingham-type inequalities and Riesz bases of divided differences, Int. J. Appl. Math. Comput. Sci., 11 (2001), 803–820. https://www.researchgate.net/profile/Sergei_Avdonin/publication/265116566_Ingham-type_inequalities_and_Riesz_bases_of_divided_differences/links/546b80e70cf2397f7831c25b/Ingham-type-inequalities-and-Riesz-bases-of-divided-differences.pdf. [5] C. Baiocchi, V. Komornik and P. Loreti, Ingham type theorems and applications to control theory, Boll. Un. Mat. Ital., B2 (1999), 33–63.https://eudml.org/doc/194750. [6] C. Baiocchi, V. Komornik and P. Loreti, Généralisation d'un théorème de Beurling et application à la théorie de contrôle, C. R. Acad. Sci. Paris Ser. I Math., 330 (2000), 281-286.  doi: 10.1016/S0764-4442(00)00116-6. [7] C. Baiocchi, V. Komornik and P. Loreti, Ingham-Beurling type theorems with weakened gap conditions, Acta Math. Hungar., 97 (2002), 55-95.  doi: 10.1023/A:1020806811956. [8] J. N. J. W. L. Carleson and P. Malliavin, editors, The collected works of Arne Beurling, Volume 2, Birkhäuser, 1989. https://projecteuclid.org/euclid.die/1356060673 [9] M. S. P. Eastham, Theory of Ordinary Differential Equations, Van Nostrand ReinholdCompany, London, 1970. [10] M. V. Fedoryuk, Asymptotic Analysis, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-58016-1. [11] B. M. Levitan and I. S. Sargsjan, Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators, Americain Mathematical Society, Translation of Mathematical Monographs, 39, 197). [12] J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome 1 and 2, Masson, RMA, Paris, 1988. [13] D. L. Russel, The Dirichlet-Neumann boundary control problem associated with Maxwell's equations in a cylindrical region, SIAM J. Cont. Optim., 24 (1986), 199-229.  doi: 10.1137/0324012. [14] M. Tucsnak and G. Weiss, Simultaneous exact controllability and some applications, SIAM J. Cont. Optim., 38 (2000), 1408-1427.  doi: 10.1137/S0363012999352716.

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References:
 [1] S. Avdonin, Simultaneous controllability of several elastic strings, Proc. CD of the Fourteenth International Symposium of Mathematical Theory of Networks and Systems, Perpignan, France, (2000), 19–23.http://www.math.ucsd.edu/ ~helton/MTNSHISTORY/CONTENTS/2000PERPIGNAN/CDROM/articles/B169.pdf. [2] S. Avdonin and M. Tucsnak, Simultaneous controllability in sharp time for two elastic strings, ESAIM: COCV, 6 (2001), 259-273.  doi: 10.1051/cocv:2001110. [3] S. Avdonin and W. Moran, Simultaneous control problems for systems of elastic strings and beams, Systems and Control Letters, 44 (2001), 147-155.  doi: 10.1016/S0167-6911(01)00137-2. [4] S. Avdonin and W. Moran, Ingham-type inequalities and Riesz bases of divided differences, Int. J. Appl. Math. Comput. Sci., 11 (2001), 803–820. https://www.researchgate.net/profile/Sergei_Avdonin/publication/265116566_Ingham-type_inequalities_and_Riesz_bases_of_divided_differences/links/546b80e70cf2397f7831c25b/Ingham-type-inequalities-and-Riesz-bases-of-divided-differences.pdf. [5] C. Baiocchi, V. Komornik and P. Loreti, Ingham type theorems and applications to control theory, Boll. Un. Mat. Ital., B2 (1999), 33–63.https://eudml.org/doc/194750. [6] C. Baiocchi, V. Komornik and P. Loreti, Généralisation d'un théorème de Beurling et application à la théorie de contrôle, C. R. Acad. Sci. Paris Ser. I Math., 330 (2000), 281-286.  doi: 10.1016/S0764-4442(00)00116-6. [7] C. Baiocchi, V. Komornik and P. Loreti, Ingham-Beurling type theorems with weakened gap conditions, Acta Math. Hungar., 97 (2002), 55-95.  doi: 10.1023/A:1020806811956. [8] J. N. J. W. L. Carleson and P. Malliavin, editors, The collected works of Arne Beurling, Volume 2, Birkhäuser, 1989. https://projecteuclid.org/euclid.die/1356060673 [9] M. S. P. Eastham, Theory of Ordinary Differential Equations, Van Nostrand ReinholdCompany, London, 1970. [10] M. V. Fedoryuk, Asymptotic Analysis, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-58016-1. [11] B. M. Levitan and I. S. Sargsjan, Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators, Americain Mathematical Society, Translation of Mathematical Monographs, 39, 197). [12] J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome 1 and 2, Masson, RMA, Paris, 1988. [13] D. L. Russel, The Dirichlet-Neumann boundary control problem associated with Maxwell's equations in a cylindrical region, SIAM J. Cont. Optim., 24 (1986), 199-229.  doi: 10.1137/0324012. [14] M. Tucsnak and G. Weiss, Simultaneous exact controllability and some applications, SIAM J. Cont. Optim., 38 (2000), 1408-1427.  doi: 10.1137/S0363012999352716.
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