B737 Captain reported an inflight engine shutdown following a series of compressor stalls on final approach. The flight continued on the approach and safely landed.

Narrative

During the ILS to XX R between approximately 2;500 and 3;000 feet we encountered a compressor stall on the number two engine. There was a loud ' banging ' noise accompanied by a decrease in N1. We had just broken out of a cloud layer and had the airport in sight. We simultaneously agreed that we should continue the approach and land. Memory items were completed. We configured the aircraft gear down; flaps 15; bugged the appropriate speed and hit the flap inhibit switch and I started the APU. We were stabilized and in a good position to land. Between 1;500 and 1;800 feet the N1 on engine 2 was at approximately 30%; an EGT was rising above 900c approaching red line and engine still making a banging noise. To prevent further damage and other issues; we shut the number 2 engine down using the QRC for engine severe damage. The first officer continued flying and did a great job. We were stable through landing. After landing; we cleared onto taxiway 1 and had Airport Rescue and Firefighting (ARFF) inspect the aircraft to make sure it was safe to continue to the gate. There were no abnormalities; so we continued to gate XX with ARFF vehicle following without further incident.