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Exact Controllability of the 1-D Wave Equation on Finite Metric Tree Graphs

  • Sergei Avdonin
  • Yuanyuan ZhaoEmail author
Article

Abstract

In this paper, we consider initial boundary value problems and control problems for the wave equation on finite metric graphs with Dirichlet boundary controls. We propose new constructive algorithms for solving initial boundary value problems on general graphs and boundary control problems on tree graphs. We demonstrate that the wave equation on a tree is exactly controllable if and only if controls are applied at all or all but one of the boundary vertices. We find the minimal controllability time and prove that our result is optimal in the general case. The proofs for the shape and velocity controllability are purely dynamical, while the proof for the full controllability utilizes both dynamical and moment method approaches.

Keywords

Boundary control Controllability Tree graphs Wave equation 

Mathematics Subject Classification

35L05 35Q93 93B05 93C20 

Notes

Acknowledgements

The authors would like to thank the referees for their valuable comments which helped to improve the manuscript.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of Alaska FairbanksFairbanksUSA

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