A control apparatus for operating a fuel injector is provided. The control apparatus includes an electronic control unit configured to: identify when the engine is running under a fuel cut-off condition, and then perform a learning procedure to determine an actual value of energizing time that causes the fuel injector to inject a target fuel quantity. The learning procedure provides for the electronic control unit to perform several test injections with different energizing time values and measure an engine torque value caused by the test injection. The measured engine torque values and their correspondent energizing time values are used to extrapolate the actual value of the energizing time as the value that corresponds to a reference value of engine torque that is consistent with the target fuel quantity.