  In the supervisory control framework of discrete-event systems (DES) with
infinite behavior initiated by Thistle and Wonham, a supervisor satisfying the
minimal acceptable specification and the maximal legal specification is
synthesized. However, this supervisor may incur livelocks as it cannot ensure
that the infinite behavior under supervision will always visit some marker
states. To tackle this problem, we propose the definition of markability by
requiring that all infinite cycles include at least one marker state. Then we
formulate the problem of $\omega-$nonblocking supervisory control of DES with
infinite behavior to synthesize an $\omega-$nonblocking (i.e. nonblocking,
deadlock-free and livelock-free) supervisor. An algorithm is proposed to
achieve $\omega-$nonblockingness by computing the supremal $*-$controllable,
$*-$closed, $\omega-$controllable and markable sublanguage. We utilize the
example of a robot as a running example.
