Final-value systems with Gaussian inputs

A final-value system controls a response variable
r(t)
over a time interval
(O,T)
with the objective of minimizing the difference between a desired value
\rho
and the final response value
r(T)
. An ensemble of situations is considered, and the system input
i(t)
and the desired response
\rho
are random variables that are statistically related. Physical limitations of the element being controlled result in a maximum value constraint on the system velocity
r\prime (t)
. Earlier results suggest that a system consisting of an estimator followed by a "bang-bang" servo is approximately optimum. The estimator uses the input to produce an estimate
\rho^{\ast}
of the desired response and the servo results in a system velocity as large in magnitude as possible and with the same sign as the difference
\rho^{\ast} - r
. The present paper shows that this system is the true optimum when the joint distribution of the input and the desired response is Gaussian and the error criterion is minimization of the average of a nondecreasing function of the magnitude of the error.