Cooperative Adaptive Cruise Control (CACC) in previous researches typically refers to the linear controller with a gap policy. The system could not be designed to fulfill multiple objectives. This inspires the concept of optimal control based CACC in this paper. The basic procedure of the proposed controller is to gather the information collected by each vehicle to the computation unit first, then plan the trajectory of all the followers by solving an optimal control problem, and dispatch the optimal motion command to each vehicle at last. This paper models CACC under optimal control framework. A numerical approach inspired by dynamic programming is adopted to solve the control problem. The stability of the proposed controller is thoroughly investigated in terms of both local stability and string stability. To verify the concept of controller, solution, and the analysis about stability, simulation is carried out. The simulation verifies that the numerical method is effective with respect to computation time. Both theoretical analysis and simulation proved that the proposed optimal control based CACC is both local stable and string stable. The low computation burden, local stability, and string stability together guarantee the future implementation of the proposed controller.