Reconfigurable intelligent surfaces (RIS) can actively perform beamforming and have become a crucial enabler for wireless systems in the future. The direction-of-arrival (DOA) estimates of RIS received signals can help design the reflection control matrix and improve communication quality. In this paper, we design a RIS-assisted system and propose a robust Lawson norm-based multiple-signal-classification (LN-MUSIC) DOA estimation algorithm for impulsive noise, which is divided into two parts. The first one, the non-convex Lawson norm is used as the error criterion along with a regularization constraint to formulate the optimization problem. Then, a Bregman distance based alternating direction method of multipliers is used to solve the problem and recover the desired signal. The second part is to use the multiple signal classification (MUSIC) to find out the DOAs of targets based on their sparsity in the spatial domain. In addition, we also propose a RIS control matrix optimization strategy that requires no channel state information, which effectively enhances the desired signals and improves the performance of the LN-MUSIC algorithm. A Cramer-Rao-lower-bound (CRLB) of the proposed DOA estimation algorithm is presented and verifies its feasibility. Simulated results show that the proposed robust DOA estimation algorithm based on the Lawson norm can effectively suppress the impact of large outliers caused by impulsive noise on the estimation results, outperforming existing methods.