American Institute of Mathematical Sciences

Advanced
doi: 10.3934/eect.2021004

Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains

Mokhtari Yacine 1,2,,

1. 

Laboratoire de Mathématiques UMR 6623, Université de Bourgogne Franche-Comté, 16, route de Gray, 25030 Besançon cedex, France
2. 

University of Sciences and Technology Houari Boumedienne P.O.Box 32, El-Alia 16111, Bab Ezzouar, Algiers, Algeria

* Corresponding author: Mokhtari Yacine

Received  July 2020 Revised  October 2020 Published  January 2021

In this paper, we deal with boundary controllability and boundary stabilizability of the 1D wave equation in non-cylindrical domains. By using the characteristics method, we prove under a natural assumption on the boundary functions that the 1D wave equation is controllable and stabilizable from one side of the boundary. Furthermore, the control function and the decay rate of the solution are given explicitly.

Keywords: Wave equation, controllability, stability, non-cylindrical domains.
Mathematics Subject Classification: Primary: 93D15, 93B05, 35L05.
Citation: Mokhtari Yacine. Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains. Evolution Equations & Control Theory, doi: 10.3934/eect.2021004
References:
[1]

K. Ammari, A. Bchatnia and K. El Mufti, Stabilization of the wave equation with moving boundary, Eur. J. Control, 39 (2018), 35-38.  doi: 10.1016/j.ejcon.2017.10.004.  Google Scholar

[2]

K. Ammari, A. Bchatnia and K. El Mufti, A remark on observability of the wave equation with moving boundary, J. Appl. Anal, 23 (2017), 43-51.  doi: 10.1515/jaa-2017-0007.  Google Scholar

[3]

A. V. Balakrishnan, Superstability of systems, Applied Mathematics and Computation, 164 (2005), 321-326.  doi: 10.1016/j.amc.2004.06.052.  Google Scholar

[4]

C. Bardos and G. Chen, Control and stabilization for the wave equation Ⅲ: Domain with moving boundary, SIAM J. Control Optim., 19 (1981), 114-122.  doi: 10.1137/0319010.  Google Scholar

[5]

C. Castro, A. Munch and N. Cindea, Controllability of the linear one-dimensional wave equation with inner moving forces, SIAM J. Control Optim., 52 (2014), 4027-4056.  doi: 10.1137/140956129.  Google Scholar

[6]

L. Cui, X. Liu and H. Gao, Exact controllability for a one-dimensional wave equation in non-cylindrical domains, J. Math. Anal. Appl., 402 (2013), 612-625.  doi: 10.1016/j.jmaa.2013.01.062.  Google Scholar

[7]

L. Cui, Y. Jiang and Y. Wang, Exact controllability for a one-dimensional wave equation with the fixed endpoint control, Bound. Value Probl., 208 (2015), 1-10.  doi: 10.1186/s13661-015-0476-4.  Google Scholar

[8]

L. Cui, Exact controllability of wave equations with locally distributed control in non-cylindrical domain, Journal of Mathematical Analysis and Applications, 482 (2020), 123532, 17 pp. doi: 10.1016/j.jmaa.2019.123532.  Google Scholar

[9]

M. Gugat, Exact controllability of a string to rest with a moving boundary, Control and Cybernetics, 48 (2019). Google Scholar

[10]

M. Gugat, Optimal boundary feedback stabilization of a string with moving boundary, IMA Journal of Mathematical Control and Information, 25 (2008), 111-121.  doi: 10.1093/imamci/dnm014.  Google Scholar

[11]

B. H. Haak and D. T. Hoang, Exact observability of a 1-dimensional wave equation on a noncylindrical domain, SIAM J. Control Optim., 57 (2019), 570-589.  doi: 10.1137/17M112960X.  Google Scholar

[12]

V. Komornik, Rapid boundary stabilization of the wave equation, SIAM J. Control Optim., 29 (1991), 197-208.  doi: 10.1137/0329011.  Google Scholar

[13]

J. Le Rousseau, G. Lebeau, P. Terpolilli and E. Tré lat, Geometric control condition for the wave equation with a time-dependent observation domain, Analysis & PDE, 10 (2017), 983-1015.  doi: 10.2140/apde.2017.10.983.  Google Scholar

[14]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations. Recent progress and open questions, SIAM Rev, 20 (1978), 639-739.  doi: 10.1137/1020095.  Google Scholar

[15]

Rideau and P. Contrôle d'un, Assemblage de Poutres Flexibles par des Capteurs Actionneurs Ponctuels: Étude du spectre du système. Thèse, Ecole. Nat. Sup. des Mines de Paris, Sophia-Antipolis, France, 1985. Google Scholar

[16]

A. Sengouga, Exact boundary observability and controllability of the wave equation in an interval with two moving endpoints, Mathematical Control and Related Fields, 9 (2020), 1-25.  doi: 10.3934/eect.2020014.  Google Scholar

[17]

A. Shao, On Carleman and observability estimates for wave equations on time-dependent domains, Proc. Lond. Math. Soc., 119 (2019), 998-1064.  doi: 10.1112/plms.12253.  Google Scholar

[18]

H. Sun, H. Li and L. Lu, Exact controllability for a string equation in domains with moving boundary in one dimension, Electron. J. Diff. Equations, (2015), 1–7.  Google Scholar

[19]

E. Zuazua, Exact controllability for the semilinear wave equation in one space dimension, Ann. IHP, Analyse non Linéaire, 10 (1993), 109-129.  doi: 10.1016/S0294-1449(16)30221-9.  Google Scholar

show all references

References:
[1]

K. Ammari, A. Bchatnia and K. El Mufti, Stabilization of the wave equation with moving boundary, Eur. J. Control, 39 (2018), 35-38.  doi: 10.1016/j.ejcon.2017.10.004.  Google Scholar

[2]

K. Ammari, A. Bchatnia and K. El Mufti, A remark on observability of the wave equation with moving boundary, J. Appl. Anal, 23 (2017), 43-51.  doi: 10.1515/jaa-2017-0007.  Google Scholar

[3]

A. V. Balakrishnan, Superstability of systems, Applied Mathematics and Computation, 164 (2005), 321-326.  doi: 10.1016/j.amc.2004.06.052.  Google Scholar

[4]

C. Bardos and G. Chen, Control and stabilization for the wave equation Ⅲ: Domain with moving boundary, SIAM J. Control Optim., 19 (1981), 114-122.  doi: 10.1137/0319010.  Google Scholar

[5]

C. Castro, A. Munch and N. Cindea, Controllability of the linear one-dimensional wave equation with inner moving forces, SIAM J. Control Optim., 52 (2014), 4027-4056.  doi: 10.1137/140956129.  Google Scholar

[6]

L. Cui, X. Liu and H. Gao, Exact controllability for a one-dimensional wave equation in non-cylindrical domains, J. Math. Anal. Appl., 402 (2013), 612-625.  doi: 10.1016/j.jmaa.2013.01.062.  Google Scholar

[7]

L. Cui, Y. Jiang and Y. Wang, Exact controllability for a one-dimensional wave equation with the fixed endpoint control, Bound. Value Probl., 208 (2015), 1-10.  doi: 10.1186/s13661-015-0476-4.  Google Scholar

[8]

L. Cui, Exact controllability of wave equations with locally distributed control in non-cylindrical domain, Journal of Mathematical Analysis and Applications, 482 (2020), 123532, 17 pp. doi: 10.1016/j.jmaa.2019.123532.  Google Scholar

[9]

M. Gugat, Exact controllability of a string to rest with a moving boundary, Control and Cybernetics, 48 (2019). Google Scholar

[10]

M. Gugat, Optimal boundary feedback stabilization of a string with moving boundary, IMA Journal of Mathematical Control and Information, 25 (2008), 111-121.  doi: 10.1093/imamci/dnm014.  Google Scholar

[11]

B. H. Haak and D. T. Hoang, Exact observability of a 1-dimensional wave equation on a noncylindrical domain, SIAM J. Control Optim., 57 (2019), 570-589.  doi: 10.1137/17M112960X.  Google Scholar

[12]

V. Komornik, Rapid boundary stabilization of the wave equation, SIAM J. Control Optim., 29 (1991), 197-208.  doi: 10.1137/0329011.  Google Scholar

[13]

J. Le Rousseau, G. Lebeau, P. Terpolilli and E. Tré lat, Geometric control condition for the wave equation with a time-dependent observation domain, Analysis & PDE, 10 (2017), 983-1015.  doi: 10.2140/apde.2017.10.983.  Google Scholar

[14]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations. Recent progress and open questions, SIAM Rev, 20 (1978), 639-739.  doi: 10.1137/1020095.  Google Scholar

[15]

Rideau and P. Contrôle d'un, Assemblage de Poutres Flexibles par des Capteurs Actionneurs Ponctuels: Étude du spectre du système. Thèse, Ecole. Nat. Sup. des Mines de Paris, Sophia-Antipolis, France, 1985. Google Scholar

[16]

A. Sengouga, Exact boundary observability and controllability of the wave equation in an interval with two moving endpoints, Mathematical Control and Related Fields, 9 (2020), 1-25.  doi: 10.3934/eect.2020014.  Google Scholar

[17]

A. Shao, On Carleman and observability estimates for wave equations on time-dependent domains, Proc. Lond. Math. Soc., 119 (2019), 998-1064.  doi: 10.1112/plms.12253.  Google Scholar

[18]

H. Sun, H. Li and L. Lu, Exact controllability for a string equation in domains with moving boundary in one dimension, Electron. J. Diff. Equations, (2015), 1–7.  Google Scholar

[19]

E. Zuazua, Exact controllability for the semilinear wave equation in one space dimension, Ann. IHP, Analyse non Linéaire, 10 (1993), 109-129.  doi: 10.1016/S0294-1449(16)30221-9.  Google Scholar

Figure 1.  The curve $ (t,\alpha(t))_{t\geq0} $ in red and $ (t, \beta(t))_{t\geq0} $ in blue
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  An example of a boundary curves $ (t,\alpha(t)))_{t\geq0} $ and $ (t,\beta(t)))_{t\geq0} $ that do not satisfy assumption (10). The values of the solution are not defined on the green part of the characteristic lines lying under or above these curves
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  An example of a boundary curves $ (t,\alpha(t)))_{t\geq0} $ and $ (t,\beta(t)))_{t\geq0} $ that do not satisfy assumption (10). The values of the solution are not defined on the green part of the characteristic lines lying under or above these curves
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Zhi-Min Chen, Philip A. Wilson. Stability of oscillatory gravity wave trains with energy dissipation and Benjamin-Feir instability. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2329-2341. doi: 10.3934/dcdsb.2012.17.2329
[2]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825
[3]

Michael Schmidt, Emmanuel Trélat. Controllability of couette flows. Communications on Pure & Applied Analysis, 2006, 5 (1) : 201-211. doi: 10.3934/cpaa.2006.5.201
[4]

Valeria Chiado Piat, Sergey S. Nazarov, Andrey Piatnitski. Steklov problems in perforated domains with a coefficient of indefinite sign. Networks & Heterogeneous Media, 2012, 7 (1) : 151-178. doi: 10.3934/nhm.2012.7.151
[5]

Jian Yang, Bendong Lou. Traveling wave solutions of competitive models with free boundaries. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 817-826. doi: 10.3934/dcdsb.2014.19.817
[6]

Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Controllability of a 1-D tank containing a fluid modeled by a Boussinesq system. Evolution Equations & Control Theory, 2013, 2 (2) : 379-402. doi: 10.3934/eect.2013.2.379
[7]

Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020027
[8]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094
[9]

Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212
[10]

Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089
[11]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068
[12]

Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213
[13]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189
[14]

Habib Ammari, Josselin Garnier, Vincent Jugnon. Detection, reconstruction, and characterization algorithms from noisy data in multistatic wave imaging. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 389-417. doi: 10.3934/dcdss.2015.8.389
[15]

Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329
[16]

Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367
[17]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261
[18]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[19]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109
[20]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25

2019 Impact Factor: 0.953

Tools

Metrics

  • PDF downloads (30)
  • HTML views (74)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences