An apparatus for controlling a driving system for a vehicle having a driving source (E, MG) and a step automatic transmission (AT) for transmitting a driving force of the driving source (E, MG) to driving wheels (RL, RR), the apparatus comprising: control means for outputting a respective control command value to the driving source (E, MG) and the step automatic transmission (AT) in accordance with a running state of the vehicle; rotational speed control means for controlling a rotational speed of the driving source (E, MG) so that an input rotational speed of the step automatic transmission (AT) follows a target input rotational speed during an inertia phase during which the input rotational speed of the step automatic transmission (AT) varies in accordance with a shift; frictional engagement torque control means for controlling an engagement torque of a frictional engagement element (CL2) in the step automatic transmission (AT) during the inertia phase; driving force control means for controlling a driving force input to the step automatic transmission (AT) after the inertia phase in accordance with a sensed running state; and frictional engagement torque correction means comprising: sensing means for sensing a difference between the engagement torque before the completion of the inertia phase and the driving force input to the step automatic transmission (AT) after the completion of the inertia phase; and difference judgment means for judging whether or not said difference is equal to or greater than a predetermined value; wherein the frictional engagement torque correction means comprises learning control means for selectively correcting the engagement torque at a next shift so as to decrease said difference when the difference is equal to or greater than the predetermined value. control means for outputting a respective control command value to the driving source (E, MG) and the step automatic transmission (AT) in accordance with a running state of the vehicle; rotational speed control means for controlling a rotational speed of the driving source (E, MG) so that an input rotational speed of the step automatic transmission (AT) follows a target input rotational speed during an inertia phase during which the input rotational speed of the step automatic transmission (AT) varies in accordance with a shift; frictional engagement torque control means for controlling an engagement torque of a frictional engagement element (CL2) in the step automatic transmission (AT) during the inertia phase; driving force control means for controlling a driving force input to the step automatic transmission (AT) after the inertia phase in accordance with a sensed running state; and frictional engagement torque correction means comprising: sensing means for sensing a difference between the engagement torque before the completion of the inertia phase and the driving force input to the step automatic transmission (AT) after the completion of the inertia phase; and difference judgment means for judging whether or not said difference is equal to or greater than a predetermined value; sensing means for sensing a difference between the engagement torque before the completion of the inertia phase and the driving force input to the step automatic transmission (AT) after the completion of the inertia phase; and difference judgment means for judging whether or not said difference is equal to or greater than a predetermined value; wherein the frictional engagement torque correction means comprises learning control means for selectively correcting the engagement torque at a next shift so as to decrease said difference when the difference is equal to or greater than the predetermined value. <EOS>