Based on the nonlinearly coupled mode equations (NLCME), the bistable steady characteristics and dynamic stability of linearly tapered nonlinear Bragg grating (LT-NLBG) have been investigated in detail. The results show that, when the device is tuned near an edge of "photonic band gap" (PBG), in contrast with the free-tapered grating, the negative-tapered grating enhances the switching-on threshold, but increases the on-off switching ratio, enlarges the stable regime, and strengthens the stability significantly. On the other hand, the positive-tapered grating decreases the switching-on threshold, but lower the on-off switching ratio, and worsen the stability remarkably.