Using Arden's theoram to find the regular expression :
A = ε + A0 = ε0* = $0^*$
$B = A1 + B1 = 0^*1 +B1 = 0^*11^* $
As there are two final states, we should union both RE.
Note that $e+1^+=1^∗$
PS : Ardens theoram - Let P and Q be two regular expressions. If P does not contain null string, then R = Q + RP has a unique solution that is R = QP*
$L$ contains all binary strings where any $1$ is not followed by any $0$.