  Nascent quantum computers motivate the exploration of quantum many-body
systems in nontraditional scenarios. For example, it has become natural to
explore the dynamics of systems evolving under both unitary evolution and
measurement. Such systems can undergo dynamical phase transitions in the
entanglement properties of quantum trajectories conditional on the measurement
outcomes. Here, we explore dynamics in which one attempts to (locally) use
those measurement outcomes to steer the system toward a target state, and we
study the resulting phase diagram as a function of the measurement and feedback
rates. Steering succeeds when the measurement and feedback rates exceed a
threshold, yielding an absorbing-state transition in the trajectory-averaged
density matrix. We argue that the absorbing-state transition generally occurs
at different critical parameters from the entanglement transition in individual
trajectories and has distinct critical properties. The efficacy of steering
depends on the nature of the target state: in particular, for local dynamics
targeting long-range correlated states, steering is necessarily slow and the
entanglement and steering transitions are well separated in parameter space.
