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We present a linear stability analysis of dispersion-managed (DM) solitons controlled by inline narrow-band filters. We show that the filters can destabilize the pulse if they are unsuitedly located in the dispersion map, which is contrary to the case of standard solitons in fibers with constant dispersion. We also show that for such an instability to take place the pulse energy and the dispersion-map strength should be significantly larger than those usually required for practical long-distance transmission. The filter-induced instability of DM solitons will be an issue in the operation of stretched-pulse mode-locked fiber lasers.