LQG homing in a finite time interval

Let X(t) be a controlled one-dimensional diffusion process having constant infinitesimal variance. We consider the problem of optimally controlling 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 fixed constant. The optimal control is obtained explicitly in the particular case when X(t) is a controlled Wiener process.