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

  • Xu ZhangEmail author
  • Chuang Zheng
  • Enrique Zuazua
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 15)


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.

Key words

Exact controllability observability time discretization wave equation multiplier technique filtering. 


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Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Academy of Mathematics and Systems SciencesChinese Academy of SciencesBeijingChina
  2. 2.School of Mathematical SciencesBeijing Normal UniversityBeijingChina
  3. 3.Basque Center for Applied Mathematics, Bizkaia Technology Park DerioBasque Country, Spain

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