Change of regularity in controllability and observability of systems of wave equations

Solutions of a system of wave equations are constructed for both homogeneous and inhomogeneous Dirichlet boundary conditions at every regularity level. We prove that boundary observability, and thus boundary exact controllability, at some regularity level is equivalent to boundary observability at all levels. The main ingredient is the ellipticity of a time-derivative on the Neumann trace of the solution, which is proved by microlocal techniques.