I came across this link that formulates the brachistochrone problem as an optimal control problem: brachistochrone. Instead of minimizing the time, I'd like - by using the same dynamics - to make the problem minimum distance between the two points A and B. So, in this case, is the following objective function correct?
$$ min\ J = \int_0^{t_f} v(t) dt\, $$ where $ t_f $ is free.