Connected and autonomous vehicles (CAV) is the development trend in the field of transportation systems. Recent studies show that the resources sharing between pedestrians and CAV is a big challenge. Considering traffic safety and efficiency at that sharing point not only requires a collision avoidance system but also more communicative behaviors of the CAV. More precisely, pedestrian needs to understand the intention of the incoming CAV whether it will cross first or not according to its speed profile. This paper uses the optimal trajectory control to provide CAV with a communicative behavior. A scenario where CAV and pedestrian cooperate together to cross a conflict zone is studied. A communicative CAV behavior is designed through an objective function. Hamiltonian analysis is used to derive the optimal control for the CAV. Based on Oculus virtual reality platform, the proposed approach is tested and the behavior of cars and pedestrians are studied. The tests show that this approach provides CAVs with a kind of automatic courtesy.