IMHO, the constraints here (as an armchair engineer) are more that the 'big rocket' isn't very responsive to requested changes in thrust, whereas the 'little nose rockets' may be responsive, but very weak compared to the mass of the thing they're trying to control.
In this model, you don't count the rate of thrust changes as a separate input because your control system's job is to determine the required thrust at each instant. You also don't count thrust if you have already counted the main engine throttle (as has Animats). So the primary point of the parent comment, that the system is severely underactuated and thus difficult to solve, is correct.
Sure you can, and must. velocity is a controllable and it's also very important that it be controlled as close to zero as possible when Y=0. Parent is correct, technically they care about 11 degrees of freedom, but rotation rate about the length is either a non issue or well regulated somehow.
I'd really like to see the controller they're using. Wonder if it's fancy, or fairly simple with some large matrices.
The controller for this almost has to be a predictive controller. You can't do this on feedback alone. The control goal is not stability; it's end state - stopped, vertical, and on target, all achieved at the same moment.
With enough fuel, you could do this step by step. Get into stable hover while vertical, then move slowly sideways over target, then descend slowly. NASA's Morpheus rocket works that way. Space-X is using much more aggressive control strategies; in the last seconds it's clear that all the goals are being sought simultaneously.
Landing should work much better on land, with a bigger, stationary target in a less windy place.
I agree the position at landing doesn't matter, but surely the rotation about the rocket's axis needs to be controlled so that it isn't spinning with any great speed - if it were, the RCS thrusters would have a hell of a time doing their job, as would the gimble of the main engine
Right, which is why the care about 11 degrees of freedom and not 12.