  In this paper we study the exact controllability problem for the wave
equation on a finite metric graph with the Kirchhoff-Neumann matching
conditions. Among all vertices and edges we choose certain active vertices and
edges, and give a constructive proof that the wave equation on the graph is
exactly controllable if $H^1(0,T)'$ Neumann controllers are placed at the
active vertices and $L^2(0,T)$ Dirichlet controllers are placed at the active
edges. The proofs for the shape and velocity controllability are purely
dynamical, while the proof for the exact controllability utilizes both
dynamical and moment method approaches. The control time for this construction
is determined by the chosen orientation and path decomposition of the graph.

Optimization and Control