It's not hijacked, the formulation is the same. For any layer there is a state-space formulation
h = Ah + Bx
y = Ch + Dx
where x is the input, y is the output, and h is the state. They use "h" instead of "s" for the state variables because they're called "hidden states" in the literature. edit: it is obnoxious they've flipped the convention for A/B/C/D which is the one thing controls people agree on (we can't even agree on the signs and naming of transfer function coefficients!).
Where this diverges from dynamical systems/controls is that they're proving that when x/h/y are represented with finite precision numbers, the model is limited in the problems it can represent (no surprise from controls perspective), and they prove this by using an equivalence to the state-space formulae that's consistent with evaluating it on massively parallel hardware.
The classical controls theory is not super applicable here, because what controls people care about (is the system stable, is its rise/fall time in bounds, what about overshoot, etc) is not what ML researchers care about (what classes of AI problems can be modeled and evaluated using this computational architecture).
Where this diverges from dynamical systems/controls is that they're proving that when x/h/y are represented with finite precision numbers, the model is limited in the problems it can represent (no surprise from controls perspective), and they prove this by using an equivalence to the state-space formulae that's consistent with evaluating it on massively parallel hardware.
The classical controls theory is not super applicable here, because what controls people care about (is the system stable, is its rise/fall time in bounds, what about overshoot, etc) is not what ML researchers care about (what classes of AI problems can be modeled and evaluated using this computational architecture).