  Here we present an expanded analysis of a model for the manipulation and
control of observables in a strongly correlated, many-body system, which was
first presented in [McCaul et al., eprint: arXiv:1911.05006]. A field-free,
non-linear equation of motion for controlling the expectation value of an
essentially arbitrary observable is derived, together with rigorous constraints
that determine the limits of controllability. We show that these constraints
arise from the physically reasonable assumptions that the system will undergo
unitary time evolution, and has enough degrees of freedom for the electrons to
be mobile. Furthermore, we give examples of multiple solutions to generating
target observable trajectories when the constraints are violated. Ehrenfest
theorems are used to further refine the model, and provide a check on the
validity of numerical simulations. Finally, the experimental feasibility of
implementing the control fields generated by this model is discussed.
