The objective of the present numerical study is to increase mixing in turbulent flow behind a backward-facing step using a systematic feedback control method. Spatially and temporally varying blowing and suction with zero-net mass flow rate are provided at the step edge, based on the sensing of the spanwise distribution of the wall pressure fluctuations at a downstream location. The cost functional to be increased is the root-mean-square spanwise pressure-gradient fluctuations at the sensing location, which may be associated with mixing behind the backward-facing step. Given the cost functional, the actuation at the step edge is determined through the suboptimal feedback control procedure of Choi et al. (1993). Large-eddy simulations of turbulent flow are conducted at a Reynolds number of 5100 based on the step height and free-stream velocity. The results of suboptimal feedback controls are compared with those of non-feedback single-frequency actuations. In case of the suboptimal control, velocity and vorticity fluctuations substantially increase downstream of the backward-facing step as well as in the recirculation zone. As a result, the reattachment length is significantly reduced, as compared to those of uncontrolled flow and flow with single-frequency actuations. A simple open-loop control method is devised from the suboptimal feedback control result, producing nearly the same mixing enhancement as the feedback control.