American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Absorption of characteristics by sonic curve of the two-dimensional Euler equations
  • DCDS Home
  • This Issue
  • Next Article
    Systems of coupled diffusion equations with degenerate nonlinear source terms: Linear stability and traveling waves
February  2009, 23(1&2): 571-604. doi: 10.3934/dcds.2009.23.571

Time discrete wave equations: Boundary observability and control

Xu Zhang 1, , Chuang Zheng 2, and  Enrique Zuazua 3,

1. 

Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100190
2. 

School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
3. 

Basque Center for Applied Mathematics (BCAM), Gran Via 35, 48009 Bilbao, Spain

Received  September 2007 Revised  February 2008 Published  September 2008

In this paper we study the exact boundary controllability of atrapezoidal time discrete wave equation in a bounded domain. Weprove that the projection of the solution in an appropriate filteredspace is exactly controllable with uniformly bounded cost withrespect to the time-step. In this way, the well-knownexact-controllability property of the wave equation can bereproduced as the limit, as the time step $h\rightarrow 0$, of thecontrollability of projections of the time-discrete one. By dualitythese results are equivalent to deriving uniform observabilityestimates (with respect to $h\rightarrow 0$) within a class ofsolutions of the time-discrete problem in which the high frequencycomponents have been filtered. The later is established by means ofa time-discrete version of the classical multiplier technique. Theoptimality of the order of the filtering parameter is alsoestablished, although a careful analysis of the expected velocity ofpropagation of time-discrete waves indicates that its actual valuecould be improved.
Keywords: filtering, multiplier technique, observability, wave equation, time discretization, Exact controllability.
Mathematics Subject Classification: Primary: 93B05; Secondary: 93B07, 35L05, 65M06.
Citation: Xu Zhang, Chuang Zheng, Enrique Zuazua. Time discrete wave equations: Boundary observability and control. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 571-604. doi: 10.3934/dcds.2009.23.571
[1]

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
[2]

Ali Wehbe, Marwa Koumaiha, Layla Toufaily. Boundary observability and exact controllability of strongly coupled wave equations. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1269-1305. doi: 10.3934/dcdss.2021091
[3]

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
[4]

Tianliang Yang, J. M. McDonough. Solution filtering technique for solving Burgers' equation. Conference Publications, 2003, 2003 (Special) : 951-959. doi: 10.3934/proc.2003.2003.951
[5]

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
[6]

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
[7]

Baowei Feng, Carlos Alberto Raposo, Carlos Alberto Nonato, Abdelaziz Soufyane. Analysis of exponential stabilization for Rao-Nakra sandwich beam with time-varying weight and time-varying delay: Multiplier method versus observability. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022011
[8]

Imen Benabbas, Djamel Eddine Teniou. Observability of wave equation with Ventcel dynamic condition. Evolution Equations and Control Theory, 2018, 7 (4) : 545-570. doi: 10.3934/eect.2018026
[9]

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
[10]

Tatsien Li, Bopeng Rao, Zhiqiang Wang. Exact boundary controllability and observability for first order quasilinear hyperbolic systems with a kind of nonlocal boundary conditions. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 243-257. doi: 10.3934/dcds.2010.28.243
[11]

José R. Quintero, Alex M. Montes. On the exact controllability and the stabilization for the Benney-Luke equation. Mathematical Control and Related Fields, 2020, 10 (2) : 275-304. doi: 10.3934/mcrf.2019039
[12]

Jamel Ben Amara, Hedi Bouzidi. Exact boundary controllability for the Boussinesq equation with variable coefficients. Evolution Equations and Control Theory, 2018, 7 (3) : 403-415. doi: 10.3934/eect.2018020
[13]

Mo Chen, Lionel Rosier. Exact controllability of the linear Zakharov-Kuznetsov equation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (10) : 3889-3916. doi: 10.3934/dcdsb.2020080
[14]

Irena Lasiecka, Roberto Triggiani. Global exact controllability of semilinear wave equations by a double compactness/uniqueness argument. Conference Publications, 2005, 2005 (Special) : 556-565. doi: 10.3934/proc.2005.2005.556
[15]

Peng Gao. Global exact controllability to the trajectories of the Kuramoto-Sivashinsky equation. Evolution Equations and Control Theory, 2020, 9 (1) : 181-191. doi: 10.3934/eect.2020002
[16]

Piermarco Cannarsa, Alessandro Duca, Cristina Urbani. Exact controllability to eigensolutions of the bilinear heat equation on compact networks. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1377-1401. doi: 10.3934/dcdss.2022011
[17]

Orazio Muscato, Wolfgang Wagner. A stochastic algorithm without time discretization error for the Wigner equation. Kinetic and Related Models, 2019, 12 (1) : 59-77. doi: 10.3934/krm.2019003
[18]

Olivier Goubet, Ezzeddine Zahrouni. On a time discretization of a weakly damped forced nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2008, 7 (6) : 1429-1442. doi: 10.3934/cpaa.2008.7.1429
[19]

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
[20]

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

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (60)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy