The purpose of this paper is to investigate the controllability and the achievability of discrete event systems within a behavioral framework. Based on the notion of Willems' behavioral controllability, we introduce a new concept related to the controllability of discrete event systems. By using the controllability proposed here and the notion related to achievable behaviors, we show that the behavioral controllability for a given specification with respect to language is equivalent to the existence of a controller, so that an interconnected system satisfies the specification exactly. A proposed controller here is represented by the intersection of the behavior of a given plant and that of a given (controllable) specification. We also clarify that our controllers for a given specification fit the properties of well-known supervisory controllers proposed and developed by Ramadge and Wonham.
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