Let $B_t$ be a Brownian motion, let $\sigma > 0$ be fixed and let $X_t$ be a process with fixed beginning value $x_0$ that satisfies $dXt = u_tdt + \sigma X_tdB_t.$
Solve $E\left[\int_0^Tu^2dt+(X_T)^2\right] \rightarrow \min$
Hint: try a value function of the form $\phi(t)x^2$ and note that $1/(a \phi + \phi^2) = (1/a)[1/ \phi - 1/(\phi + a)]$.
I think I need to verify the Hint with a Bellman equation to solve the problem.
I think $u$ is a function of $t$.