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

* 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.

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. AmmariA. 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. AmmariA. 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. CastroA. 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. CuiX. 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. CuiY. 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 RousseauG. LebeauP. 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. AmmariA. 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. AmmariA. 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. CastroA. 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. CuiX. 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. CuiY. 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 RousseauG. LebeauP. 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 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]

Lucio Damascelli, Filomena Pacella. Sectional symmetry of solutions of elliptic systems in cylindrical domains. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3305-3325. doi: 10.3934/dcds.2020045

[2]

Oleg Yu. Imanuvilov, Jean Pierre Puel. On global controllability of 2-D Burgers equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 299-313. doi: 10.3934/dcds.2009.23.299

[3]

Teresa D'Aprile. Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains. Communications on Pure & Applied Analysis, 2021, 20 (1) : 159-191. doi: 10.3934/cpaa.2020262

[4]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242

[5]

Yanan Li, Zhijian Yang, Na Feng. Uniform attractors and their continuity for the non-autonomous Kirchhoff wave models. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021018

[6]

Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021015

[7]

Larissa Fardigola, Kateryna Khalina. Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition. Mathematical Control & Related Fields, 2021, 11 (1) : 211-236. doi: 10.3934/mcrf.2020034

[8]

Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270

[9]

Takiko Sasaki. Convergence of a blow-up curve for a semilinear wave equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1133-1143. doi: 10.3934/dcdss.2020388

[10]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

[11]

Patrick Martinez, Judith Vancostenoble. Lipschitz stability for the growth rate coefficients in a nonlinear Fisher-KPP equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 695-721. doi: 10.3934/dcdss.2020362

[12]

Izumi Takagi, Conghui Zhang. Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020400

[13]

Joel Kübler, Tobias Weth. Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3629-3656. doi: 10.3934/dcds.2020032

[14]

Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020052

[15]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[16]

Ludovick Gagnon, José M. Urquiza. Uniform boundary observability with Legendre-Galerkin formulations of the 1-D wave equation. Evolution Equations & Control Theory, 2021, 10 (1) : 129-153. doi: 10.3934/eect.2020054

[17]

Imam Wijaya, Hirofumi Notsu. Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1197-1212. doi: 10.3934/dcdss.2020234

[18]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074

[19]

Erica Ipocoana, Andrea Zafferi. Further regularity and uniqueness results for a non-isothermal Cahn-Hilliard equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020289

[20]

Linglong Du, Min Yang. Pointwise long time behavior for the mixed damped nonlinear wave equation in $ \mathbb{R}^n_+ $. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020033

2019 Impact Factor: 0.953

Article outline

Figures and Tables

[Back to Top]