Exact Controllability of the 1-D Wave Equation on Finite Metric Tree Graphs
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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.
KeywordsBoundary control Controllability Tree graphs Wave equation
Mathematics Subject Classification35L05 35Q93 93B05 93C20
The authors would like to thank the referees for their valuable comments which helped to improve the manuscript.
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