Transportation in controlled industrial sites provides a conducive environment for technologies of Connected Automated Vehicles (CAV). Recent studies show that safe and efficient road sharing between CAVs and pedestrians is challenging. Besides safety issues, a significant loss of time occurs when pedestrians cross a stream of CAVs. Currently, many techniques have been employed to improve the coordination between CAVs and pedestrians. They focus on pedestrian detection, display of the intention of CAVs, and cooperative collision avoidance. However, one of the most significant sources of information that the pedestrian uses for her/his decision-making is the speed profile of CAVs. This paper aims to provide a safe and efficient pedestrian crossing at industrial sites through communicative crossing behavior. To this end, a suitable speed profile of the CAV is designed by assuming that pedestrians and CAV play a cooperative game to move as close as possible to their desired speed. First, a system analysis is proposed to derive the optimal decision and trajectory for each agent. Then, Deep Reinforcement Learning (DRL) is used to control the longitudinal speed of the CAVs. Compared with Model Predictive Control approach, DRL allows coping with unforeseeable pedestrian behaviors (e.g. long reaction time, varying ideal speed, stop in the middle, etc.). Simulations and experiments with real human testers based on immersive hamlet are performed. Results show that the proposed speed profile outperforms significantly the collision avoidance approach.