  We develop a scheme to prepare a desired state or subspace in
high-dimensional Hilbert-spaces using repeated applications of a single static
projection operator onto the desired target and fixed unitary dynamics.
Benchmarks against other control schemes, performed on generic Hamiltonians and
on Bose-Hubbard systems, establish the competitiveness of the method. As a
concrete application of the control of mesoscopic atomic samples in optical
lattices we demonstrate the near deterministic preparation of Schr\"{o}dinger
cat states of all atoms residing on either the odd or the even sites.
