$4$ Languages and Supervisor

A system $G$ generates the language $L(G)\frac{?}{b}=ar(a^{**}b{a}^{**}).$ The marked language is $L_m(G)=a^{**}b{a}^{**},$

and the set of uncontrollable events is $E_{uc}=\{b\}.$ The language $L_{am}=\{a^{m}b{a}^n$:$mn0\}$

should be allowed.

$4\mathrm{.}1$ Is $\frac{?}{\text{b}\text{a}\text{r}}(L_{am})$ a regular language?

$4\mathrm{.}2$ Does a nonblocking supervisor $S$ exist, such that $L_m(\frac{S}{G})=L_{am}$ holds?

$4\mathrm{.}3$ Realize the supervisor $S$ as a Petri net.