At a stop deciding instant (t1), a first engine automatic-stop control permission decision is made on the basis of an instantly-decidable idle stop permitting condition. If the first engine automatic-stop control permission decision is 'permitted' at t1, a second engine automatic-stop control permission decision is started on the basis of a time-taking idle stop permitting condition, and an electric oil pump ON (drive) command (a test drive command) is issued. At t2 when it is detected that the electric oil pump performs an actual drive responding to the test drive command, an electric oil pump trouble decision is made 'normal' by deciding that the electric oil pump is not troubled by the abnormality of the electric system and by the biting trouble of a foreign matter. This decision has been ended before t3, at which the second engine automatic-stop control permission decision ends. At t3, an idle stop ON command is outputted by the 'permission' of the second engine automatic-stop control permission decision result and the 'normal' of the electric oil pump trouble decision.