Learjet 60 Captain reported a pressurization malfunction during cruise flight resulting in a expedited descent to a lower altitude. The crew regained control over the pressurization system and continued to the destination.

Narrative

On Day 0 on a trip from ZZZ1 to ZZZ; Aircraft X experienced a loss of cabin pressure. The sequence of events are as follows.While at cruise at 41;000'; approximately 2.5 hours into the trip; we received an amber DUCT OV HT annunciator light followed by the temperature select needle falling to full cold on the Crew side of the temperature control panel. I instructed the First Officer to turn the temperature control knob for the crew to full cold; which he did; and the annunciator cleared. Moments later we received another amber DUCT OV HT annunciator light followed by the temperature select needle falling to full cold on the Cabin side of the temperature control panel. I instructed the First Officer to turn the temperature control knob for the cabin to full cold; which he did. At this point the cabin lost pressurization; the cabin climb rate gauge was pegged at full climb rate; and we could physically feel the loss of pressurization. The crew donned their oxygen masks; deployed the passenger masks; and executed an emergency descent according to memory items & SOP's. The first officer notified ATC of the pressurization issue and we were cleared down to 10;000'. During the descent I instructed the first officer to place the pressurization system into manual mode; MAN ALT down; and MAN RATE to the middle position. With the pressurization system in manual mode; we were able to regain control of the cabin and brought the cabin altitude down to a safe level. At this point EMER PRESS had been triggered; but CABIN ALT HIGH had not. With the cabin at a safe altitude; around 5000'; we slowed our descent rate and were able to remove our oxygen masks and regain normal communications with ATC. I informed the passengers that the situation was under control; they could remove their masks; and that once we leveled off at our assigned altitude we would come up with a game plan as to our options moving forward. I then informed ATC that the pressurization issue was under control and that once we leveled off at 10;000' we would inform them of our intentions based on remaining fuel and airport options. After we leveled off at 10;000'; based on fuel remaining; reduced power fuel burn; and weather conditions at ZZZ it was determined that we could continue to ZZZ as planned. Both ATC and the passengers were informed that we would be continuing to ZZZ. At this point the cabin was completely under control; and our annunciator panel was normal except for EMER PRESS. I instructed the first officer to get the QRH and pull up the checklist for EMER PRESS amber light. We followed the appropriate checklist and were able to stop emergency pressurization air flow; and clear the amber annunciator light (right bleed air switch to OFF). At this point our annunciator panel was normal. Pressurization mode remained in manual; and right bleed air switch remained off. We continued to ZZZ and landed safely without any further issues.