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]

K. Ravikumar, Manil T. Mohan, A. Anguraj. Approximate controllability of a non-autonomous evolution equation in Banach spaces. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 461-485. doi: 10.3934/naco.2020038

[2]

Emanuela R. S. Coelho, Valéria N. Domingos Cavalcanti, Vinicius A. Peralta. Exponential stability for a transmission problem of a nonlinear viscoelastic wave equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021055

[3]

Huancheng Yao, Haiyan Yin, Changjiang Zhu. Stability of rarefaction wave for the compressible non-isentropic Navier-Stokes-Maxwell equations. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1297-1317. doi: 10.3934/cpaa.2021021

[4]

Peng Chen, Xiaochun Liu. Positive solutions for Choquard equation in exterior domains. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021065

[5]

Mohamed Ouzahra. Approximate controllability of the semilinear reaction-diffusion equation governed by a multiplicative control. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021081

[6]

Beom-Seok Han, Kyeong-Hun Kim, Daehan Park. A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3415-3445. doi: 10.3934/dcds.2021002

[7]

Tomoyuki Tanaka, Kyouhei Wakasa. On the critical decay for the wave equation with a cubic convolution in 3D. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021048

[8]

Jiacheng Wang, Peng-Fei Yao. On the attractor for a semilinear wave equation with variable coefficients and nonlinear boundary dissipation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021043

[9]

Hui Yang, Yuzhu Han. Initial boundary value problem for a strongly damped wave equation with a general nonlinearity. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021019

[10]

Yanling Shi, Junxiang Xu. Quasi-periodic solutions for nonlinear wave equation with Liouvillean frequency. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3479-3490. doi: 10.3934/dcdsb.2020241

[11]

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

[12]

Simão Correia, Mário Figueira. A generalized complex Ginzburg-Landau equation: Global existence and stability results. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021056

[13]

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, 2021, 41 (7) : 3109-3140. doi: 10.3934/dcds.2020400

[14]

Shoichi Hasegawa, Norihisa Ikoma, Tatsuki Kawakami. On weak solutions to a fractional Hardy–Hénon equation: Part I: Nonexistence. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021033

[15]

Zhengchao Ji. Cylindrical estimates for mean curvature flow in hyperbolic spaces. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1199-1211. doi: 10.3934/cpaa.2021016

[16]

Masahiro Ikeda, Ziheng Tu, Kyouhei Wakasa. Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021011

[17]

Pengyan Ding, Zhijian Yang. Well-posedness and attractor for a strongly damped wave equation with supercritical nonlinearity on $ \mathbb{R}^{N} $. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1059-1076. doi: 10.3934/cpaa.2021006

[18]

Jihoon Lee, Nguyen Thanh Nguyen. Gromov-Hausdorff stability of reaction diffusion equations with Robin boundary conditions under perturbations of the domain and equation. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1263-1296. doi: 10.3934/cpaa.2021020

[19]

Andrew Comech, Elena Kopylova. Orbital stability and spectral properties of solitary waves of Klein–Gordon equation with concentrated nonlinearity. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021063

[20]

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

2019 Impact Factor: 0.953

Article outline

Figures and Tables

[Back to Top]