LQG Homing in a Finite Time Interval

Let X(t) be a controlled one-dimensional diffusion process havingconstant infinitesimal variance. We consider the problem of optimallycontrolling X(t) until time T(x)=min{T1(x),t1}, where T1(x) is the first-passage time of the process to a given boundary and t1 is a fixedconstant. The optimal control is obtained explicitly in the particularcase when X(t) is a controlled Wiener process