RichardKennaway, very interesting post. I actually specialized in control theory in graduate school, but didn't finish the program. I must object to what you've said here, in that control theory most certainly does make extensive use of Bayesian inferenence, under the name of the Kalman filter.

The Kalman filter is a way of estimating the paramaters of a system, given your observations and your knowledge of the system's dynamics. While it may not help you pick a good control input algorithm, and while the problems you listed there may not need such accurate estimation of the data, it is an integral part of finding out how much the system deviates from where you want it to be, and is used extensively in controls.