Optimal control for hitting a random point with gaussian distribution

Question

A particle $X$ starts from the origin $X_0=0$ of the real line and can move to the right or the left with speed $\pm 1$ and should hit a point, $\xi$, normally distributed (mean zero, and variance 1) on the same real line. For each control strategy $\nu$, we denote $\tau^\nu = \inf \{t: X^\nu_t = \xi \}$. What is the optimal strategy such that $E(\tau^\bar{\nu})=\inf_\nu E(\tau^\nu)$?