A TRACON Controller reported a departing aircraft experiencing FMS; autopilot and autothrottle failures was not complying with ATC clearances.

Narrative

I was training a Certified Professional Controller in Training (CPC-IT). We had a good training session and complexity and volume were commensurate with this time of day. Aircraft X was on the ZZZZZ departure approaching the shoreline westbound when they indicated they needed to return to ZZZ for a maintenance issue and were not declaring an emergency or requesting assistance. The CPC-IT issued a heading westbound and 10000 ft. I discussed our plan to bring the aircraft back to ZZZ with the CPC-IT and alerted the Operations Supervisor (OS) of the situation. After a minute or two I asked the aircraft if they wanted delay vectors to troubleshoot or if they wanted to get right back in. The pilot indicated they wanted to get right back in. We issued a descent to 8000 ft. for traffic and a turn to head for the right downwind flow. Shortly after; the pilot stated they would probably need a visual approach to Runway XXL. This peaked my attention as the weather was marginal VFR with some pretty low clouds that would make a visual approach difficult but not impossible. I asked the pilot if they could provide more information as to the difficulties they were having and they stated that the FMS had failed; the autopilot had failed; they were flying the aircraft by hand and the autothrottle had stuck.At that point I [requested priority handling] for the flight crew and got the required info from them. I took over the position from the CPC-IT and issued multiple instructions. I was extremely conscious of my speech rate and inflection with any instructions to Aircraft X. Aircraft X read back a couple of vectors and altitude assignments but did not comply as expected. I pointed the aircraft out to radar since Aircraft X had not descended out of 8000 ft. roughly 5 miles after being issued the descent and I assumed they were busy troubleshooting. As the Aircraft X approached the downwind a different voice requested to continue on to ZZZ1. This required more coordination with radar/ZZZ/ZZZ1 and a new flight plan. The aircrew seemed very far behind the aircraft and weren't providing information about what was happening in the aircraft. More info could have been shared with ATC about the situation earlier. They seemed hesitant to [request priority handling] but the info they shared about multiple system failures would seem to dictate otherwise. If the aircrew were too busy to comply with ATC instructions they should have shared that. We moved multiple aircraft out of the way as we didn't know what they were doing or trust them to comply with instructions.