CONTROLLER FOR ENGINE AND METHOD FOR CONTROLLING ENGINE

A controller for an engine includes processing circuitry configured to perform fuel cut-off for regenerating the filter by burning fine particulate matter collected in the filter. The processing circuitry is configured to perform an interruption process for interrupting the fuel cut-off in a case in which warming of the passenger compartment is requested when the fuel cut-off is being performed and an engine coolant temperature, which is a temperature of the engine coolant, becomes lower than or equal to a predetermined temperature after the fuel cut-off is started.