A TRACON Controller reported a small aircraft on approach in IMC repeatedly deviated from their course and descended below assigned and published altitudes flying below Minimum Vectoring Altitudes.

Narrative

I relieved another CPC and Developmental that had been training for approximately 90 minutes. They had just had an issue with this aircraft. The pilot turned and descended unexpectedly. The aircraft was trying to come to ZZZ and land. The weather conditions on the surface were IMC. The pilot was consistently unable to maintain altitude or assigned headings. There were multiple other aircraft on frequency in bound to ZZZ. I put Aircraft X on a vector to be last in my sequence. As the aircraft got closer to the airport I noticed him veer off the final. I issued reintercept vectors. Within 10 miles of the airport the pilot also went below safe altitudes. I issued multiple low altitude alerts and got the pilot to climb to a safe altitude and vacate the approach. I asked the pilot if he was having equipment issues or if there was something wrong with the aircraft. The pilot advised the issue was with the pilot himself. Knowing he was in an unsafe and scary situation; I tried to calm the pilot and reassure him by getting him to focus on maintaining the altitudes I assigned. I asked how much fuel too and he said 2 hours. I offered ZZZ1 as an alternate with better weather. The pilot asked for another approach at ZZZ so I gave him a heading to fly and periodically checked back in to make sure his altitude and heading were good. He asked if the weather was still IMC at ZZZ. I said yes and then he asked to try ZZZ1. I gave vectors and kept prompting the pilot to maintain a safe altitude. I issued more Low Altitude Alerts. The pilot did not report ZZZ1 in sight and asked to return to ZZZ2. I immediately had the pilot climb to safe altitudes and issued vectors away from high terrain. The pilot was able to climb safely and get on course. I advised the Center Sector of the issues before completing the handoff. Suggestion: Use vigilance for aircraft performance and when they are descending dangerously or not flying the correct path. The pilot did not seem capable of operating in IMC conditions.