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 {t 1 (x),t 1 },where t 1 (x) is the first-passage time of the process to a given boundary and t 1 is a fixed constant. the optimal control is obtained explicitly in the particular case when x (t) is a controlled wiener process. © 2011 mario lefebvre.