Citation Captain reported while in cruise the ACM overheat and EMER PRESS annunciators activated followed by the cabin altitude starting to increase. The Captain began an immediate descent to 9;000 feet to stabilize the cabin pressure and continued to their destination airport.

Narrative

Air Cycle Machine (ACM) overheat and EMER PRESS annunciators illuminated for a second around FL405. The QRH was performed. A few seconds later while leveling off at FL410 the same annunciators came on again for 3-4 seconds and the cabin altitude started to increase so I decided to start an decent. We asked for lower and ATC requested assistance for us before we had a chance to request it ourselves. The cabin altitude never exceeded 9000 feet. The passenger oxygen masks never deployed. I leveled off at the assigned altitude of 9000 feet. At that point we evaluated the condition of the airplane; weather; and fuel. The pressurization system was now functioning normally as the cabin altitude was now down to 2000 feet. We had the fuel to make to ZZZ and with everything on the airplane back to normal decided that continuing to ZZZ was the plan. ATC asked us if we needed any more help once leveled off. We informed them that everything was operating normally at this point. ATC asked if we would like to cancel our assistance request status. We did agree to cancel the status and the rest of the flight to ZZZ at 9000 feet was uneventful.