Exact Controllability for Wave Equation on General Metric Graphs with Non-smooth Controls

Abstract

We study the exact controllability problem for the wave equation on a general 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. For such controls, we describe the state spaces for which our initial boundary value problem is well posed. The proofs for the shape and velocity controllability are purely dynamical, while the proof for the exact controllability utilizes both dynamical and spectral (moment method) approaches. The control time for this construction is determined by the chosen orientation and path decomposition of the graph.