Cooperative (automated) vehicles have the potential to enhance traffic efficiency and fuel economy on urban roads, especially at signalized intersections. An optimal control approach to optimize the trajectories of cooperative vehicles at fixed-timing signalized intersections along an arterial is presented. The proposed approach aims to optimize throughput first, and then to maximize comfort while minimizing travel delay and fuel consumption. The proposed approach is flexible in dealing with both quadratic and more complex cost functions. Assuming fixed timing signal control in a cycle and vehicle-to-infrastructure communication, the red phase is taken into account in position constraints for vehicles that cannot pass the intersection in the green phase. Safety is guaranteed by constraining the inter-vehicle distance larger than some desired value. The approach is scalable and can be used for joint trajectory planning of one platoon approaching another stationary platoon. It can also be extended to multiple intersections with fixed signal plans. To verify the performances of the controlled platoon, simulation under three different traffic scenarios is conducted, namely: an isolated intersection with/without downstream vehicle queues, and platoon control at multiple intersections. Three baseline scenarios without control are also designed to compare performances in relation to both mobility and fuel consumption in each controlled scenario. The results demonstrate that the controlled vehicles generate plausible behavior under control objectives and constraints. Moreover, the consideration of downstream vehicle queues and the application at both an isolated signalized intersection and arterial corridors on urban roads verify the flexible characteristics of the control framework.