In a system, a predictor predicts a future trajectory of a rotational speed of a crankshaft during drop of the crankshaft's rotational speed in a forward rotational direction due to automatic stop of an engine. An engaging unit shifts, in response to when an engine restart condition is met during the drop of the crankshaft's rotational speed, a pinion to a ring gear for engaging the pinion with the ring gear. An engagement determiner determines, based on the future trajectory of the crankshaft's rotational speed, whether the pinion and the ring gear will be shifted to engagement state therebetween during reverse rotation of the output shaft. A controller controls, based on a result of the determination by the engagement determiner, at least one of: a start timing to shift the pinion to the ring gear by the engaging unit, and a start timing to energize the motor.