Captain reported loss of cabin pressure control and required an expedited descent with priority handling to keep cabin altitude at a safer level. The flight crew continued to destination airport.

Narrative

Flight uneventfully departed ZZZ for ZZZ1 with three passengers; continuing until initial descent into ZZZ1. Soon after initiating an ATC-directed normal descent from FL450; crew experienced physiological symptoms of a rapidly increasing cabin altitude. Crew immediately confirmed this by observing cabin altitude rising from approximately 8;000 feet at a rate of between 4;000 fpm and 6;000 fpm on the cabin pressurization controller; and moments later; a 'MASTER WARN' with 'CABIN ALT' CAS (Crew Alert System) annunciation.PIC called for; and crew executed; emergency checklist memory items for CABIN ALT (Item K2). Crew then observed cabin altitude maintaining approximately 15;000 feet; and referencing remaining printed items; noted 'procedure complete'. Crew contacted ATC; requesting a lower than previously cleared-to altitude; and requested priority handling. Crew was cleared incrementally to FL240; and descended uninterrupted to this altitude at an increased rate of approximately 3;000 fpm to 4;000 fpm. While in this descent; PIC elected to complete emergency checklist item K1(a); for 'Cabin Pressurization Controller Failure.' PIC was successful in regaining control of cabin pressurization with controller in 'manual' mode; lowering the cabin altitude to approximately 8;000 feet.Upon reaching FL240; crew again checked condition of passengers; receiving a 'thumbs up.' Crew discussed the situation; the condition of the aircraft; and possible courses of action; including diversion to ZZZ2 or continuation to ZZZ1; which were now approximately equidistant in time. Crew determined the pressurization system was now under control with cabin altitude remaining constant below 8;000 feet; and advised ATC and passengers of intention to continue to ZZZ1. Crew were cleared to lower altitude; and a shallow descent was initiated soon after; allowing for best control of the pressurization system.Crew completed Normal Checklist items; and conducted an uneventful visual approach and landing at ZZZ1. Due to the incident occurring while already in descent; as well as further timely and effective altitude and route clearances by ATC; the exercising of PIC emergency authority was deemed unnecessary.Upon deplaning; crew queried passengers about their physical and mental condition; and were advised that they had felt a change in cabin temperature and pressure that caused discomfort to their ears; and donned their oxygen masks when they dropped automatically. Passengers otherwise acted and appeared normally; though reporting that their ears were still slightly 'plugged.' All three passengers left the premises in their private vehicle soon after.PIC then notified Dispatch of the incident; and asked them to notify CP (Chief Pilot) and DOM (Director of Maintenance) to expect PIC's call after securing the aircraft. Crew then secured the aircraft as-normal; and after speaking with CP and DOM; completed a Discrepancy Log entry for a rapid decompression; as coordinated with DOM.