Distributional Control of Ensemble Systems

The paper’s Theorem 1 (ensemble controllability implies pattern controllability) states the right claim and sketches the right route, but its proof contains a critical misstep: it asserts existence of preimages in P(M) that are close whenever the images in P(N) are close, invoking only continuity and surjectivity of the pushforward h#. That implication is generally false (continuity gives the forward direction). The remainder of the paper’s argument aligns with the standard and correct approach: realize the target output law as a pushforward, use ensemble controllability to steer the state field close in a probability metric, and pass to output measures by continuity. The candidate model’s proof supplies the missing ingredients (measurable selection for h and the correct continuity-in-probability to weak-law step), yielding a complete, rigorous argument.