This study seeks to develop a cooperative platoon control for a platoon mixed with connected and autonomous vehicles (CAVs) and human-drive vehicles (HDVs), aiming to ensure system level traffic flow smoothness and stability as well as individual vehicles’ mobility and safety. Specifically, our study integrated/contributed the following technical approaches. First, the car-following behavior of human-drive vehicles is modeled by well-accepted Newell car-following models. Accordingly, an online curve matching algorithm is integrated to anticipate the aggregated response delay of the human-drive vehicles using real-time trajectory data. Built upon that, constrained One- or P-step MPC models are developed to control the movement of the CAV platoon upstream or downstream of the HDV platoon so that we can ensure both transient traffic smoothness and asymptotic stability of this sample mixed flow platoon, leveraging the communication and computation technologies equipped on CAVs. Considering the lack of the centralized computation facilities and severe changes of the platoon topology, this study develops a distributed algorithm to solve the MPCs according to the properties of the optimizers, such as solution uniqueness, sequentially feasibility, and nonempty interior point of the solution space. The convergence of the distributed algorithm as well as the stability of the MPC control is proved by both the theoretical analysis and the experimental study. Extensive numerical experiments based on the field data indicate that the distributed algorithm can solve the One-step and P-step MPCs efficiently. The cooperative MPC can dampen traffic oscillation propagation and stabilize the traffic flow more efficiently for the entire mixed flow platoon. Moreover, the cooperative platoon control scheme outperforms the other three control strategies, including the non-cooperative control strategy and a latest CACC strategy in literature.