From exact observability to identification of singular sources

We give general results about the identifiability of source terms for infinite-dimensional linear systems that are exactly observable. We allow the source term to be unbounded, i.e., not contained in the state space, but in one of a sequence of extended spaces. We show that the operator from the source term to the output function is bounded from below, in suitable norms. We apply the main result to a system described by the wave equation in a bounded (Formula presented.)-dimensional domain. We derive results of independent interest concerning the range of the input map of an exactly controllable system, when restricted to various spaces of smooth input functions.