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This paper develops distributed optimization-based, platoon-centered connected and autonomous vehicle (CAV) car-following schemes, motivated by the recent interest in CAV platooning technologies. Various distributed optimization or control schemes have been developed for CAV platooning. However, most existing distributed schemes for platoon centered CAV control require either centralized data processing or centralized computation in at least one step of their schemes, referred to as partially distributed schemes. In this paper, we develop fully distributed optimization based, platoon centered CAV platooning control under the linear vehicle dynamics via the model predictive control approach with a general prediction horizon. These fully distributed schemes do not require centralized data processing or centralized computation through the entire schemes. To develop these schemes, we propose a new formulation of an objective function and a decomposition method that decomposes a densely coupled central objective function into the sum of multiple locally coupled functions whose coupling satisfies the network topology constraint. We then exploit locally coupled optimization and operator splitting methods to develop fully distributed schemes. Control design and stability analysis is carried out to achieve desired traffic transient performance and asymptotic stability. Numerical tests demonstrate the effectiveness of the proposed fully distributed schemes and CAV platooning control.