This is about linear time-invariant, stationary problems.
According to Donald Kirk's book Optimal Control Theory, for minimum-fuel problems on page 299, I was told that if there exists a singular arc, then the controllability matrix is singular or the system matrix $A$ is singular. However, the controllability matrix being singular (or the system matrix $A$ being singular) does not imply a singular arc's existence, according to this source on MIT OCW.
On the other hand, for minimum-time problems, on page 296, the system is controllable if and only if there are no singular arcs (it is also mentioned by the MIT OCW lecture notes). How can one mathematically prove these?
I have seen the proofs of singular arcs being the sufficient condition, but how can one prove that it's a necessary condition for minimum-time (but not for minimum-fuel)?