  We present both, theory and an algorithm for solving time-harmonic wave
problems in a general setting. The time-harmonic solutions will be achieved by
computing time-periodic solutions of the original wave equations. Thus, an
exact controllability technique is proposed to solve the time-dependent wave
equations. We discuss a first order Maxwell type system, which will be
formulated in the framework of alternating differential forms. This enables us
to investigate different kinds of classical wave problems in one fell swoop,
such as acoustic, electro-magnetic or elastic wave problems. After a sufficient
theory is established, we formulate our exact controllability problem and
suggest a least-squares optimization procedure for its solution, which itself
is solved in a natural way by a conjugate gradient algorithm operating in the
canonical Hilbert space. Therefore, it might be one of the biggest advances of
this approach that the proposed conjugate gradient algorithm does not need any
preconditioning.
