  How to steer a given joint state probability density function to another over
finite horizon subject to a controlled stochastic dynamics with hard state
(sample path) constraints? In applications, state constraints may encode safety
requirements such as obstacle avoidance. In this paper, we perform the feedback
synthesis for minimum control effort density steering (a.k.a. Schr\"{o}dinger
bridge) problem subject to state constraints. We extend the theory of
Schr\"{o}dinger bridges to account the reflecting boundary conditions for the
sample paths, and provide a computational framework building on our previous
work on proximal recursions, to solve the same.
