Halliday & Matthiessen (2014: 570-1):
The other part of the time-phase system that has remained as a category of phase is that of ‘duration/inception’: ‘durative’ going on, contrasting with ‘inceptive/conclusive’, starting and stopping. For example:
||| I keep telling them || I give them the money || so long as they’ll leave. |||
||| The line needs to keep being shut down || to have mechanical work done on there. |||
||| If they stop performing their task, || they’re likely to be deprived of the opportunities [[ to dedicate themselves to intellectual work]] . |||
Of these, the ‘go on’ term takes the imperfective; starting and stopping take either, with little difference in meaning – except that stop requires imperfective; stop + perfective is now interpreted as a hypotactic clause complex of purpose, as in
||| α she stopped || ×β to think ||| ‘she stopped, in order to think’
There is also an inceptive-durative ‘start to go on’, as in they’ve taken to coming in at the back door instead of the front.