  The implementation of a combination of continuous weak measurement and
classical feedback provides a powerful tool for controlling the evolution of
quantum systems. In this work, we investigate the potential of this approach
from three perspectives. First, we consider a double-well system in the
classical large-atom-number limit, deriving the exact equations of motion in
the presence of feedback. Second, we consider the same system in the limit of
small atom number, revealing the effect that quantum fluctuations have on the
feedback scheme. Finally, we explore the behavior of modest sized Hubbard
chains using exact numerics, demonstrating the near-deterministic preparation
of number states, a tradeoff between local and non-local feedback for state
preparation, and evidence of a feedback-driven symmetry-breaking phase
transition.
