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

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.