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

Simultaneous controllability of two vibrating strings with variable coefficients

Jamel Ben Amara 1, and  Emna Beldi 2,

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.

Keywords: Vibrating string, simultaneous controllability, variable coefficients.
Mathematics Subject Classification: 93B, 35P.
Citation: Jamel Ben Amara, Emna Beldi. Simultaneous controllability of two vibrating strings with variable coefficients. Evolution Equations & 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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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 Google Scholar

[9]

M. S. P. Eastham, Theory of Ordinary Differential Equations, Van Nostrand ReinholdCompany, London, 1970. Google Scholar

[10]

M. V. Fedoryuk, Asymptotic Analysis, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-58016-1.  Google Scholar

[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).  Google Scholar

[12]

J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome 1 and 2, Masson, RMA, Paris, 1988.  Google Scholar

[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.  Google Scholar

[14]

M. Tucsnak and G. Weiss, Simultaneous exact controllability and some applications, SIAM J. Cont. Optim., 38 (2000), 1408-1427.  doi: 10.1137/S0363012999352716.  Google Scholar

show all references

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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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 Google Scholar

[9]

M. S. P. Eastham, Theory of Ordinary Differential Equations, Van Nostrand ReinholdCompany, London, 1970. Google Scholar

[10]

M. V. Fedoryuk, Asymptotic Analysis, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-58016-1.  Google Scholar

[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).  Google Scholar

[12]

J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome 1 and 2, Masson, RMA, Paris, 1988.  Google Scholar

[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.  Google Scholar

[14]

M. Tucsnak and G. Weiss, Simultaneous exact controllability and some applications, SIAM J. Cont. Optim., 38 (2000), 1408-1427.  doi: 10.1137/S0363012999352716.  Google Scholar

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