Vehicle and platoon dynamics models are considered for a cooperative platoon of longitudinal control with fuel or break time delay during a maneuvering process. Based on the models, a control policy is proposed based on local information between adjacent vehicles and global information of the leader vehicle. In order to avoid Slinky-Effects, the stable string conditions of a platoon are derived. Then, the individual vehicle stability is proved by using the Lyapunov-Razumikhin method, and the bound for fuel or brake time delay is calculated. On this basis, steady-state errors and the condition for preventing oscillations and collisions are analyzed. Finally, digital simulations are used to validate the above design. The comparative simulation results show that the proposed control strategy has preferable control effect.