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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 deterministicdoi:10.1088/1367-2630/16/11/113038 fatcat:6rfhrowavff2nf6j3ef5iqpkcm