How to know the problem when i make this Full Backup Scenario?
If the database is down when the backup is supposed to start the backup can not run. If the database goes offline while the backup is running the backup will simply stop. It'll need to be restarted after the database comes back online. 

There won't be anything logged that there was a backup running when the database went offline. You'd need to look at the job history of the backup job to see if it was running when the database went offline. 

There's no way to guarantee that the backups run without issue.