  We consider a general class of singular control problems with state
constraints. Budhiraja and Ross (2006) established the existence of optimal
controls for a relaxed version of this class of problems by using the so-called
`time-stretching' method and the J1-topology. We show that the weak-M1 topology
is better suited for establishing existence, since it allows to bypass the
time-transformations, without any additional effort. Furthermore, we reveal how
the time-scaling feature in the definition of the weak-M1 distance embeds the
time-stretching method's scheme. This case study suggests that one can benefit
from working with the weak-M1 topology in other singular control frameworks,
such as queueing control problems under heavy traffic.
