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Controllability of the semilinear wave equation governed by a multiplicative control
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 |
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.
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. |
[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. 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. |
[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 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. |
[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. |
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. |
[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. 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. |
[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 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. |
[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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