-
Previous Article
An inverse problem for the pseudo-parabolic equation with p-Laplacian
- EECT Home
- This Issue
-
Next Article
Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints
April
2022, 11(2): 373-397.
doi: 10.3934/eect.2021004
Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains
| 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 |
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.
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 and Control Theory,
2022, 11
(2)
: 373-397.
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. |
| [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. |
| [3] |
A. V. Balakrishnan,
Superstability of systems, Applied Mathematics and Computation, 164 (2005), 321-326.
doi: 10.1016/j.amc.2004.06.052. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [9] |
M. Gugat, Exact controllability of a string to rest with a moving boundary, Control and Cybernetics, 48 (2019). |
| [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. |
| [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. |
| [12] |
V. Komornik,
Rapid boundary stabilization of the wave equation, SIAM J. Control Optim., 29 (1991), 197-208.
doi: 10.1137/0329011. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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. |
| [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. |
| [3] |
A. V. Balakrishnan,
Superstability of systems, Applied Mathematics and Computation, 164 (2005), 321-326.
doi: 10.1016/j.amc.2004.06.052. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [9] |
M. Gugat, Exact controllability of a string to rest with a moving boundary, Control and Cybernetics, 48 (2019). |
| [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. |
| [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. |
| [12] |
V. Komornik,
Rapid boundary stabilization of the wave equation, SIAM J. Control Optim., 29 (1991), 197-208.
doi: 10.1137/0329011. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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 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
| [1] |
Feng Zhou, Chunyou Sun. Dynamics for the complex Ginzburg-Landau equation on non-cylindrical domains I: The diffeomorphism case. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3767-3792. doi: 10.3934/dcdsb.2016120 |
| [2] |
Bopeng Rao, Laila Toufayli, Ali Wehbe. Stability and controllability of a wave equation with dynamical boundary control. Mathematical Control and Related Fields, 2015, 5 (2) : 305-320. doi: 10.3934/mcrf.2015.5.305 |
| [3] |
Arnaud Heibig, Mohand Moussaoui. Exact controllability of the wave equation for domains with slits and for mixed boundary conditions. Discrete and Continuous Dynamical Systems, 1996, 2 (3) : 367-386. doi: 10.3934/dcds.1996.2.367 |
| [4] |
Kim Dang Phung. Boundary stabilization for the wave equation in a bounded cylindrical domain. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 1057-1093. doi: 10.3934/dcds.2008.20.1057 |
| [5] |
Patrizia Donato, Florian Gaveau. Homogenization and correctors for the wave equation in non periodic perforated domains. Networks and Heterogeneous Media, 2008, 3 (1) : 97-124. doi: 10.3934/nhm.2008.3.97 |
| [6] |
Mohamed Ouzahra. Controllability of the semilinear wave equation governed by a multiplicative control. Evolution Equations and Control Theory, 2019, 8 (4) : 669-686. doi: 10.3934/eect.2019039 |
| [7] |
André da Rocha Lopes, Juan Límaco. Local null controllability for a parabolic equation with local and nonlocal nonlinearities in moving domains. Evolution Equations and Control Theory, 2022, 11 (3) : 749-779. doi: 10.3934/eect.2021024 |
| [8] |
Lucio Damascelli, Filomena Pacella. Sectional symmetry of solutions of elliptic systems in cylindrical domains. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3305-3325. doi: 10.3934/dcds.2020045 |
| [9] |
Patrick Martinez, Judith Vancostenoble. Exact controllability in "arbitrarily short time" of the semilinear wave equation. Discrete and Continuous Dynamical Systems, 2003, 9 (4) : 901-924. doi: 10.3934/dcds.2003.9.901 |
| [10] |
Umberto Biccari, Mahamadi Warma. Null-controllability properties of a fractional wave equation with a memory term. Evolution Equations and Control Theory, 2020, 9 (2) : 399-430. doi: 10.3934/eect.2020011 |
| [11] |
Abdelmouhcene Sengouga. Exact boundary observability and controllability of the wave equation in an interval with two moving endpoints. Evolution Equations and Control Theory, 2020, 9 (1) : 1-25. doi: 10.3934/eect.2020014 |
| [12] |
Alhabib Moumni, Jawad Salhi. Exact controllability for a degenerate and singular wave equation with moving boundary. Numerical Algebra, Control and Optimization, 2022 doi: 10.3934/naco.2022001 |
| [13] |
Yannick Privat, Emmanuel Trélat, Enrique Zuazua. Complexity and regularity of maximal energy domains for the wave equation with fixed initial data. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 6133-6153. doi: 10.3934/dcds.2015.35.6133 |
| [14] |
M. M. Cavalcanti, V.N. Domingos Cavalcanti, D. Andrade, T. F. Ma. Homogenization for a nonlinear wave equation in domains with holes of small capacity. Discrete and Continuous Dynamical Systems, 2006, 16 (4) : 721-743. doi: 10.3934/dcds.2006.16.721 |
| [15] |
K. Ravikumar, Manil T. Mohan, A. Anguraj. Approximate controllability of a non-autonomous evolution equation in Banach spaces. Numerical Algebra, Control and Optimization, 2021, 11 (3) : 461-485. doi: 10.3934/naco.2020038 |
| [16] |
G. Gentile, V. Mastropietro. Convergence of Lindstedt series for the non linear wave equation. Communications on Pure and Applied Analysis, 2004, 3 (3) : 509-514. doi: 10.3934/cpaa.2004.3.509 |
| [17] |
Leonid Kunyansky. Fast reconstruction algorithms for the thermoacoustic tomography in certain domains with cylindrical or spherical symmetries. Inverse Problems and Imaging, 2012, 6 (1) : 111-131. doi: 10.3934/ipi.2012.6.111 |
| [18] |
Chiu-Yen Kao, Yuan Lou, Eiji Yanagida. Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains. Mathematical Biosciences & Engineering, 2008, 5 (2) : 315-335. doi: 10.3934/mbe.2008.5.315 |
| [19] |
Umberto De Maio, Akamabadath K. Nandakumaran, Carmen Perugia. Exact internal controllability for the wave equation in a domain with oscillating boundary with Neumann boundary condition. Evolution Equations and Control Theory, 2015, 4 (3) : 325-346. doi: 10.3934/eect.2015.4.325 |
| [20] |
Jeong Ja Bae, Mitsuhiro Nakao. Existence problem for the Kirchhoff type wave equation with a localized weakly nonlinear dissipation in exterior domains. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 731-743. doi: 10.3934/dcds.2004.11.731 |
2021 Impact Factor: 1.169
Tools
Metrics
Other articles
by authors
[Back to Top]