TRACON Controller reported an aircraft on approach deviated off course twice and flew below the MVA both times.

Narrative

Aircraft X was inbound to ZZZ1 on an IFR flight plan. He had requested the RNAV Y XXR approach via ZZZZZ1 with the previous Controller. We don't use ZZZZZ1 because it is outside of our airspace; so he was told to expect vectors. My trainee and I vectored him to final about 4 miles outside of ZZZZZ2. As he reached the final approach course from the west to the east; he went through final and began a left turn towards the north; which is the opposite of a turn to 'correct'. My trainee saw it and issued the appropriate low altitude alert; cancel approach clearance; etc. We vectored him to the right into lower terrain and re-sequenced him for another try. I explained to him about the higher terrain on the east side of the final and told him we can't allow any blow throughs. The next attempt he joined final as expected and was switched to Tower. Once he got to ZZZZZ3 (7 mi final) he again departed the approach course to the east into higher terrain. Again; my trainee caught it and called Tower to tell them to issue another alert. They said he was correcting and he landed without incident. The pilot later told the Operations Manager that he was experiencing issues with his avionics; and that he had no course guidance or descent information. It looked from my perspective that the aircraft was trying to go direct ZZZZZ1; as if it was still programmed for his previously denied request. The report that he didn't have proper course guidance was never told to us during the event. If he had told us he was having issues we could've suggested an alternate approach or course of action. The weather was OVC 2000 ft. not a time to be flying into high terrain without any guidance.