Stabilize-then-optimize: Feedback transformations as preconditioners in optimal control

What the math gives to ML

The paper gives a constructive preconditioning principle: before optimizing controls, apply a state-feedback coordinate change so that the uncontrolled state-to-control propagation is replaced by a closed-loop propagation with a much smaller operator norm. In the scalar recurrence example, the finite-horizon propagation norm grows as O(a^N) for an unstable multiplier a>1, while feedback can replace a by a stable closed-loop multiplier. This transfers naturally to recurrent and state-space neural networks by separating each recurrent update into a stabilizing feedback term K h_t and a learned residual/control v_t, then training in residual coordinates. A second transfer is to use the resulting control-to-state norm as a conditioning signal for adaptive gradient scaling.