There is a need to search on study approval date.

This is presumably .progressStatus[state='approved'].period.start

I think it needs a composite search parameter of state and period, to search on date of approval. This means a single progressStatus has that state and that period.
If it's not a composite then it could find a matching status and a matching period, but in different progressStatus-es.

It also seems helpful to be able to search on whether it is approved or not yet.
So we need another composite one on state and actual.

Propose
add composite search param on state|period
add composite search param on state|actual

Presumably actual and period are not completely independent. If the period start has passed then the status has happened. But then again maybe it was only a plan and it never happened.

Ideally you could search on .state, .date and .actual all at once, which may be impossible in FHIR (can't have a 3 way composite param).
Perhaps this area need to be re-modelled if it wants to be searched properly.

But for now suggest add the 2 params above.