  We propose and investigate a scheme based on Markovian feedback control that
allows for the preparation of single targeted eigenstates of a system of
bosonic atoms in a one-dimensional optical lattice with high fidelity. It can
be used for in-situ cooling the interacting system without particle loss, both
for weak and strong interactions, and for experimentally preparing and probing
individual excited eigenstates. For that purpose the system is assumed to be
probed weakly via homodyne detection of photons that are scattered
off-resonantly by the atoms from a structured probe beam into a cavity mode. By
applying an inertial force to the system that is proportional to the measured
signal, the system is then guided into a pure target state. The scheme is found
to be robust against reduced measurement efficiencies.
