You can to converte for integer number and compare.
Eg.:
$date_1 = date('Ymd');
$date_2 = '31_12_2011';
$date_2 = (int) implode(array_reverse(explode("_", $date_2)));
echo ($date_1 < $date_2) ? '$date_2 is bigger then $date_1' : '$date_2 is smaller than $date_1';