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Virtual Coupling (VC) is regarded as a breakthrough to the traditional train operation and control for improving the capability and flexibility in railways. It brings benefits as trains under VC are allowed to operate much closer to one another, forming a virtually coupled train set (VCTS). However, the safe and stable spacing between trains in the VCTS is a problem since there are no rigid couplers to connect them into a fixed formation, especially in high-speed scenarios. Due to the close spacing, the interference between trains becomes non-negligible as various maneuvers of the preceding train can significantly affect driving behaviors of the following train; this results in fluctuating spacing and therefore an unstable VCTS. Aiming at minimizing the interference and maintaining constantly safe spacing between trains in the VCTS, this paper presents a distributed model predictive control (DMPC) approach for solving the high-speed VCTS control problem. Particularly, the proposed control method focuses on the feasibility and stability of this problem, with considerations of the coupled constraint of safety braking distance and the individual constraints of speed limit variations and restricted traction/braking performance. To guarantee feasibility and stability, the terminal controller and invariant set of the DMPC are designed. For rigor, sufficient conditions of feasibility and stability are mathematically proved and derived. Based on the data of the Beijing-Shanghai high-speed railway line, numerical experiments are conducted to verify the correctness of derived sufficient conditions and the effectiveness of the proposed control method under interference and disturbances. [ABSTRACT FROM AUTHOR]