I am not aware of any substantiated report that any system on MH370 was 'shut down'. All we know was that various systems were no longer responding at certain times. We also know that the satcom was 'off' at least twice by the fact that it logged on twice. 'Off' in this sense means that the aircraft terminal was not in a logged on status (due to loosing the signal, being shut down, having power lost, etc.) but doesn't tell the reason it was 'off'.
we have a last Transponder return time, a last ACARS message time, a Last RTF time, several missed phone calls times, but not a single logoff or power down message. Nor do I believe any of those systems generate such a logoff messages.
Early efter disappearance it was said that it was shut down. I'm finding it hard to know the status of how the investigation looks at this today tough...
Final words from Flight MH370 came after systems shutdown | The Japan Times
Authorities have said someone on board the plane first disabled one of its communications systems — the Aircraft and Communications Addressing and Reporting System, or ACARS — about 40 minutes after takeoff. The ACARS equipment sends information about the jet’s engines and other data to the airline.
Around 14 minutes later, the transponder that identifies the plane to commercial radar systems was also shut down. The fact that both systems went dark separately offered strong evidence that the plane’s disappearance was deliberate.