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Exact Controllability of the Time Discrete Wave Equation: A Multiplier Approach

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Applied and Numerical Partial Differential Equations

Part of the book series: Computational Methods in Applied Sciences ((COMPUTMETHODS,volume 15))

Abstract

n this paper we summarize our recent results on the exact boundary controllability of a trapezoidal time discrete wave equation in a bounded domain. It is shown that the projection of the solution in an appropriate space in which the high frequencies have been filtered is exactly controllable with uniformly bounded controls (with respect to the time-step). By classical duality arguments, the problem is reduced to a boundary observability inequality for a time-discrete wave equation. Using multiplier techniques the uniform observability property is proved in a class of filtered initial data. The optimality of the filtering parameter is also analyzed.

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Author information

Authors and Affiliations

  1. Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, CN-100080, Beijing, China

    Xu Zhang

  2. School of Mathematical Sciences, Beijing Normal University, CN-100875, Beijing, China

    Chuang Zheng

  3. Basque Center for Applied Mathematics, Bizkaia Technology Park, Building 500, ES-48160, Derio, Basque Country, Spain

    Enrique Zuazua

Authors
  1. Xu Zhang
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  2. Chuang Zheng
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  3. Enrique Zuazua
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Corresponding author

Correspondence to Xu Zhang .

Editor information

Editors and Affiliations

  1. Dept. Mathematics, University of Houston, Calhoun Road 4800, Houston, 77204-3008, U.S.A.

    W. Fitzgibbon

  2. Dept. Mathematics, University of Houston, Calhoun Road 4800, Houston, 77204-3008, U.S.A.

    Y.A. Kuznetsov

  3. Dept. Mathematical Information , University of Jyväsklyklä, Jyväskylä, 40351, Finland

    Pekka Neittaanmäki

  4. Numèrics en Enginyeria (CIMNE), Centre Internacional de Mètodes, C/ Gran Capitan s/n, Barcelona, 08034, Spain

    Jacques Périaux

  5. Dept. Mathematique, Université de Paris VI, place Jussieu 4 , Paris CX 05, 75252, France

    Olivier Pironneau

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Zhang, X., Zheng, C., Zuazua, E. (2010). Exact Controllability of the Time Discrete Wave Equation: A Multiplier Approach. In: Fitzgibbon, W., Kuznetsov, Y., Neittaanmäki, P., Périaux, J., Pironneau, O. (eds) Applied and Numerical Partial Differential Equations. Computational Methods in Applied Sciences, vol 15. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3239-3_17

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  • DOI: https://doi.org/10.1007/978-90-481-3239-3_17

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  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-3238-6

  • Online ISBN: 978-90-481-3239-3

  • eBook Packages: EngineeringEngineering (R0)

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