The problem of eliminating the effects of infinite cycles on asynchronous sequential machines is considered in a control theoretic context. The main objective is to develop state feedback controllers that stop infinite cycles in an existing asynchronous machine, while controlling the machine to match a prescribed model. Necessary and sufficient conditions for the existence of such controllers are derived in terms of an inequality condition between two numerical matrices. The results include an algorithm for the characterization of all infinite cycles of a given machine as well as an algorithm for the construction of appropriate controllers, whenever they exist.