American Institute of Mathematical Sciences

Advanced
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  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

[1]

Alexander Khapalov. Controllability properties of a vibrating string with variable axial load. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 311-324. doi: 10.3934/dcds.2004.11.311
[2]

Jamel Ben Amara, Hedi Bouzidi. Exact boundary controllability for the Boussinesq equation with variable coefficients. Evolution Equations & Control Theory, 2018, 7 (3) : 403-415. doi: 10.3934/eect.2018020
[3]

Sergei A. Avdonin, Boris P. Belinskiy. Controllability of a string under tension. Conference Publications, 2003, 2003 (Special) : 57-67. doi: 10.3934/proc.2003.2003.57
[4]

M. Eller, Roberto Triggiani. Exact/approximate controllability of thermoelastic plates with variable thermal coefficients. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 283-302. doi: 10.3934/dcds.2001.7.283
[5]

Larissa V. Fardigola. Transformation operators in controllability problems for the wave equations with variable coefficients on a half-axis controlled by the Dirichlet boundary condition. Mathematical Control & Related Fields, 2015, 5 (1) : 31-53. doi: 10.3934/mcrf.2015.5.31
[6]

Chengming Cao, Xiaoping Yuan. Quasi-periodic solutions for perturbed generalized nonlinear vibrating string equation with singularities. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1867-1901. doi: 10.3934/dcds.2017079
[7]

Louis Tebou. Simultaneous controllability of some uncoupled semilinear wave equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3721-3743. doi: 10.3934/dcds.2015.35.3721
[8]

Sergei A. Avdonin, Boris P. Belinskiy. On the basis properties of the functions arising in the boundary control problem of a string with a variable tension. Conference Publications, 2005, 2005 (Special) : 40-49. doi: 10.3934/proc.2005.2005.40
[9]

Sergei Avdonin, Julian Edward. Controllability for a string with attached masses and Riesz bases for asymmetric spaces. Mathematical Control & Related Fields, 2019, 9 (3) : 453-494. doi: 10.3934/mcrf.2019021
[10]

Hai Huyen Dam, Kok Lay Teo. Variable fractional delay filter design with discrete coefficients. Journal of Industrial & Management Optimization, 2016, 12 (3) : 819-831. doi: 10.3934/jimo.2016.12.819
[11]

Fágner D. Araruna, Flank D. M. Bezerra, Milton L. Oliveira. Rate of attraction for a semilinear thermoelastic system with variable coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3211-3226. doi: 10.3934/dcdsb.2018316
[12]

Shikuan Mao, Yongqin Liu. Decay property for solutions to plate type equations with variable coefficients. Kinetic & Related Models, 2017, 10 (3) : 785-797. doi: 10.3934/krm.2017031
[13]

Takahiro Hashimoto. Nonexistence of weak solutions of quasilinear elliptic equations with variable coefficients. Conference Publications, 2009, 2009 (Special) : 349-358. doi: 10.3934/proc.2009.2009.349
[14]

Petronela Radu, Grozdena Todorova, Borislav Yordanov. Higher order energy decay rates for damped wave equations with variable coefficients. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 609-629. doi: 10.3934/dcdss.2009.2.609
[15]

Vo Van Au, Mokhtar Kirane, Nguyen Huy Tuan. Determination of initial data for a reaction-diffusion system with variable coefficients. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 771-801. doi: 10.3934/dcds.2019032
[16]

Shuang Liang, Shenzhou Zheng. Variable lorentz estimate for stationary stokes system with partially BMO coefficients. Communications on Pure & Applied Analysis, 2019, 18 (6) : 2879-2903. doi: 10.3934/cpaa.2019129
[17]

Qigui Yang, Qiaomin Xiang. Chaotic oscillations of linear hyperbolic PDE with variable coefficients and implicit boundary conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020335
[18]

Stephanie Flores, Elijah Hight, Everardo Olivares-Vargas, Tamer Oraby, Jose Palacio, Erwin Suazo, Jasang Yoon. Exact and numerical solution of stochastic Burgers equations with variable coefficients. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2735-2750. doi: 10.3934/dcdss.2020224
[19]

Bei Gong, Zhen-Hu Ning, Fengyan Yang. Stabilization of the transmission wave/plate equation with variable coefficients on $ {\mathbb{R}}^n $. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020068
[20]

Chiun-Chang Lee. Asymptotic analysis of charge conserving Poisson-Boltzmann equations with variable dielectric coefficients. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3251-3276. doi: 10.3934/dcds.2016.36.3251

2019 Impact Factor: 0.953

Tools

Metrics

  • PDF downloads (98)
  • HTML views (330)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences