Consider a one-dimensional controlled process governed by the equation
where . The aim of the homing control problem is to minimize the expectation of a functional of the form
where is real and denotes the exit time from an interval for a solution starting from . In the particular case and the optimal control is found by means of the mathematical expectation of a geometric Brownian motion while the optimal process is shown to be a Bessel process. Conversely, if the uncontrolled process is a geometric Brownian motion, then the optimal control is found by means of an expectation of a Brownian motion.