A TRACON Controller reported a corporate jet did not comply with their heading assignment due to an aircraft malfunction and flew below the Minimum Vectoring Altitude.

Narrative

I was working a slow ZZZ final. I was previous planning for a different aircrafts approach. I turned Aircraft X to a 180 heading to join the ZZZ XXR Localizer. The Pilot readback the instruction correctly but continued on a 100 heading. I asked the Pilot if they were in the turn; they said yes but were indicating a 100 heading still. So I turned them again; continued turning them; alerted them they were going through the final into higher MVA. I gave altimeter; climbed them to 037 for MVA; kept turning them to join the localizer; re-cleared the aircraft for the approach; made sure they were established. I asked the Pilot what happened and they stated that the aircraft kept turning them left when they were trying to turn right. I should have climbed the aircraft immediately to 043 rather than just 037 when I realized the aircraft wasn't turning at all. I should have given a proper low altitude alert rather than just tell the Pilot I was going to have to do it if they didn't turn harder. I don't know what was happening inside the aircraft that was turning them left and preventing them from turning right but I know I should have climbed them and given proper phraseology for low altitude alert. I know the situation was unpreventable; but handling could have been better all in all.