my $next_backup_file = "daily-incremental-backup." . 'date +%Y%m%d';

should be ...

my $next_backup_file = "daily-incremental-backup." . `date +%Y%m%d`;

##</code><code>##

my $youngest = "svnlook youngest $svn_repos";

should be....

my $youngest = `svnlook youngest $svn_repos`;


##</code><code>##

print 'gzip -9 $backups_dir/$next_backup_file';

should be

print `gzip -9 $backups_dir/$next_backup_file`;