Various concepts of stability has been proposed for discrete event systems (DES's). In this paper, a new concept of stability, called L_f-stability, is proposed as an extension of the language stability proposed by Kumar et al. Intuitively, a DES G is defined to be L_f-stable w.r.t. a given prefix-closed language K if the generated language L (G) of the DES exists around K. We show a necessary and sufficient condition for a DES to be L_f-stable and to be L_f-stabilizable. We also provide an on-line algorithm for computing an optimal stabilizing supervisor in the sense of minimally restrictive.