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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

* Corresponding author: Mokhtari Yacine

Received  July 2020 Revised  October 2020 Published  April 2022 Early access  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 and Control Theory, 2022, 11 (2) : 373-397. 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.

[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.

[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. 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.

[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.

[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.

[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 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.

[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. 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.

[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.

[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. 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.

[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.

[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.

[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 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.

[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 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
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