Falcon 900B Captain reported one engine developed a thrust control problem; thrust control was stuck at full power. The engine was shut down at the top of descent.

Narrative

Today was a return leg from ZZZZ to ZZZ and ultimately back to ZZZ1. All aspects of the operation today were normal from the time the second in command (SIC) and I got to the airport; as well as after takeoff. There were no issues during the external or internal pre-flight checks leaving ZZZZ. The problem that we had arose about 40 minutes into the flight a little bit after we got established into cruise at FL400. I told the SIC I was going to get up briefly to close the cabin curtain; and check on the passengers to see if they needed anything. Upon returning to my seat; I noticed that the 'Computer 3' light had illuminated on the Master Warning Panel. She pointed this out to me once I sat down and asked; 'When did this happen?'. Her response was; 'Literally as soon as you sat back down in the chair; the light came on.' I looked up to make sure the switch didn't get bumped accidentally and checked the circuit breakers to make sure nothing was popped; or inadvertently pulled. Having this not be the case; I instructed the SIC to go ahead and pull out the checklist and to run through it together. The engine at this time was slowly increasing in power from our original cruise thrust setting of 93% N1; and the engine moved up to and stabilized at 98.6% and never exceeded 99% N1. It fluctuated slightly given the changes to deviation of ISA at altitude; it will always move the N1 slightly. Inter-stage Turbine Temperature (ITT) was normal and never exceeded 918C during this time. The engine had not run away; but rather appeared to be stuck at full power; and moving the thrust lever did absolutely nothing to change this situation. As per the checklist; we ran it step by step. Which included pulling the affected engine '#3' to idle; turn the Computer 'OFF' then back to 'Auto' to see if this would clear the issue. The engine briefly decelerated after turning the Computer back to 'Auto'; because the power lever was already at idle. Having seen this; I did my best to smoothly bring the thrust lever up to try and prevent shock cooling of the engine and stabilize the N1 without drawing too much attention that something was going on and freak out the passengers. The engine briefly bumped/vibrated twice; almost as if there was an onset of a compressor stall; but came back up to full power where it originally was stuck at before; around 98.7% N1 and 910C-918C on the ITT. The checklist then calls for the Computer to be placed into the 'Manual' position if the issue is not resolved. That was the end of the checklist. At this point; it had become clear that we had lost total control of the #3 Engine; and it was effectively going to be stuck at full power. Assessing the situation and where we were in the flight; how much fuel we had remaining I elected to continue the flight. I briefed the SIC thoroughly that we were going to continue to monitor the condition of the engine and that if it remains where it is at; it will be okay; because we still effectively needed the engine to remain at our cruise altitude if it was safe to do so and nothing else abnormal happened. However; I told her by the time we got to our Top of Descent; we ultimately would have no choice but to shut that engine down. I then instructed her to begin briefing the Engine Shutdown In-Flight checklist; and Approach and Landing with One Engine Inoperative Checklist. Thoroughly running through our brief of both checklists and having a game plan together; I told her that we were going to continue as normal until we begin our descent; and then start the process of shutting down the engine. After brief discussion of what we were going to do; I came up with the plan of by the time we got to the border and were talking to a controller back in the United States; that we were going to [request priority handling] and we were going to head back to ZZZ1. Given the fact we still had the fuel and the adequate safety margins to do so. As soon as I received Wi-Fi reception prior to our descent; Icalled the company as soon as I practically and safely could to notify that we were [requesting priority handling] and likely no longer feasible to go to ZZZ; and to please notify customs as soon as possible that we were deviating to ZZZ1 to meet the extent of our emergency. The dispatcher on duty got on it and I notified the director of operations as well. I also went to the back and quietly talked to the owner of the airplane; calmy describing the situation that we were having; that my plan was to shut down the engine; and we were going to ZZZ1. I mentioned to the owner that there was nothing to be worried about; the airplane will continue to fly just fine on two engines; and to please not be alarmed when we start descending towards the border that I will be shutting down the engine; reassuring that this is part of the plan and that everything will be okay. He gave me a thumbs up; and I went back to my seat in the flight deck. As briefed; we got our descent clearance as per usual to go down to FL240 around ZZZZZ intersection. Continuing to monitor the engine with extreme caution up to this point; the SIC and I executed our plan to shut down the #3 engine; and run the Engine Shutdown In-flight checklist step by step. While not ideal to shut down an engine that was already nearly at full power; it was the safest and best thing to do. I disconnected the Autopilot before descending; and after pulling the power to cutoff; I monitored the airplane handling and engine instruments for anything abnormal. The engine shut down the way it should; and uneventfully. I re-engaged the Autopilot once I had smoothly gotten the airplane trimmed out and stabilized. By the time we got down to 10;000 feet; I began to start electrically load shedding some of the items as the checklist called for and turned off the Windshield Heat once we descended into warmer air. The Cabin Power had already automatically Auto Load Shed as well. Upon switching over to ZZZ Approach on XXX.XX crossing the border; I instructed the SIC that I would take over the radio briefly to check in; and followed suit with our plan to [request priority handling]. The controller; while busy at this time; initially instructed us to descend from 8;000 ft to 6;000 ft; and I came back saying; 'Negative for Aircraft X. At this time; I would like to remain at 8;000 ft and we are [requesting priority handling]. I have lost control of one of my engines and had to shut it down. Requesting a deviation and clearance to ZZZ1.' The controller acknowledged swiftly and was helpful throughout the entire process. I gave the controller the normal required items of stating Souls on Board and Minutes of Fuel remaining. 10 souls; and at the time we had 90 minutes of gas remaining. The only other question I got asked by the controller was which engine was shut down. To which I responded 'It's our right engine'. We continued to just get headings almost all the way back to ZZZ1 and crossed over the ZZZ VOR before being handed off to the next Approach Controller that ended up giving us our visual approach clearance. The SIC and I had briefed that we would be doing the Visual Approach back to ZZZ1 backed up with the ILS; and allowing plenty of distance before turning final to be on speed; configuration and altitude for a stable approach. We left the Flaps at 20 degrees and briefed our additive of 5kts and 200ft to our landing distance as per the checklist. The landing was uneventful and touched down in the first 2;000 ft. Light to Moderate braking pressure was used; and exited XXR at 1 intersection; turning right and to go to customs. I had called for the Fire Trucks and Operations to inspect our airplane after we landed to make sure there was no damage to the right side of the aircraft; indication of a fire; or anything else abnormal. After we got the all clear; we continued taxiing into the ramp and completely ran the Shut Down Checklist; and determined we would not be running the APU. Just shut down theairplane completely. The passengers deplaned. The decision to come back to ZZZ1 was at the time I thought the safest option for us to do. Continuing to ZZZ would've left us stranded and strategically at a major disadvantage to resolve the passengers getting home; and getting the airplane worked on. Going to ZZZ2 or ZZZ3 was out of the question for many of the same reasons; and did not want to create more workload going to one of the busiest airports in the nation and still leaving us at a disadvantage of the logistics that would be required. Going back to ZZZ1 was the best option; as it still had customs (even If it meant waiting for an officer to come all the way from ZZZ3); we had an overflight permit; it was our home maintenance base; and all around what I thought was the safest option. The airplane was operated safely; and we had more than enough safety margins across the board to do so. However; with this being said; it may have been better to stop immediately at our intended airport of landing in ZZZ. Versus overflying many other airports to get back to home base. Our engine being shut down is more considered abnormal; but may have been too proactive by [requesting priority handling]. I also as pilot in command (PIC) should consider not getting up and leaving my duty station; even though things at the moment were 'okay'. Especially with a newer SIC.