# 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  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 & Control Theory, 2019, 8 (4) : 687-694. doi: 10.3934/eect.2019032
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##### References:
 [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] Jiangang Qi, Bing Xie. Extremum estimates of the $L^1$-norm of weights for eigenvalue problems of vibrating string equations based on critical equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020243 [11] 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 [12] 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 [13] 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 [14] 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 [15] 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 [16] 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 [17] 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 [18] 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 [19] 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 [20] 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

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