Abstract
We investigate the controllability of an infinite-dimensional quantum system: a quantum particle confined on a Thick Quantum Graph, a generalisation of Quantum Graphs whose edges are allowed to be manifolds of arbitrary dimension with quasi-δ boundary conditions. This is a particular class of self-adjoint boundary conditions compatible with the graph structure. We prove that global approximate controllability can be achieved using two physically distinct protocols: either using the boundary conditions as controls, or using time-dependent magnetic fields. Both cases have time-dependent domains for the Hamiltonians.