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There has been much interest in studying the stability properties of discrete event dynamic systems (DEDS), and several definitions for stability with some kinds of stability analysis methods have been proposed, recently. However, they are too much theory-oriented and do not provide any sufficient understanding based on real applications. Therefore, in this paper, we explore the meaning of stability concepts in DEDS through the case study of a manufacturing system, the plasma etching system in semiconductor manufacturing processes. We apply the concepts of stability and stabilizability upon a general characterization of the stability properties of automata-theoretic DEDS models. The stability notion is expounded by considering the control-law synthesis of the plasma etching system in view of stabilizability.