Learjet 60 Captain reported a failure of the cabin pressure control system and performed a rapid descent and air turn back to make a precautionary landing at departure airport.

Narrative

We were performing a passenger flight from ZZZ-ZZZ1. Upon takeoff and early climb; the aircraft pressurization system was working normally. At approximately 11;000 ft. MSL; we received a 'press sys' cautionary light indicating the pressurization controller has malfunctioned. At roughly 12;000 ft. the system faulted and we completely lost pressurization indicated by the cabin climbing rapidly to our altitude and the emergency pressurization automatically turning on. We requested to descend to troubleshoot the problem and we were cleared to level off for about a minute.We were then cleared to descend to 10;000 ft. and shortly followed by 9;000 ft. We had to descend quickly due to the emergency pressure being extremely hot and upsetting the passengers. The system was supposed to be able to reset/turn off at 9;500 ft. but it did not until roughly 3;000 ft. on short vectors for the ILS back into ZZZ airport. Descending through about 5;000 ft.; we began to smell something burning but figured it was from the emergency air as there was no visible smoke inside of the airplane.For this reason; we requested priority vectors to the airport but did not [request priority handling]. We did try to reset the system numerous times but the system had completely faulted out and required a reset of aircraft power. I cycled the bleed air off numerous times without success until 3;000 ft. as well as cabin air and cabin dump below 10;000 ft.; all to no avail. We were able to land without incident and after resetting the cabin power; the pressurization system seemed to work properly evidenced by the lack of the fault lights that were present on the flight.