Boundary Controllability and Inverse Problem for the Wave Equation on Graphs

We study the boundary control problem for the wave equation on a planar graph. We suppose that the graph is a tree (i.e., it does not contain cycles), and on each edge a string equation (with a variable density) is given. The control is acting through the Dirichlet boundary conditions applied to the exterior vertices. The exact controllability is proved and a sharp estimate of the time of controllability is obtained. We prove also that the densities of the edges are uniquely determined by the Dirichlet-to-Neumann map given at all except one boundary vertices