Traffic signals, while serving an important function to coordinate vehicle movements through intersections, also cause frequent stops and delays, particularly when they are not properly timed. Such stops and delays contribute to significant amount of fuel consumption and greenhouse gas emissions. The recent development of connected and automated vehicle (CAV) technology provides new opportunities to enable better control of vehicles and intersections, that in turn reduces fuel consumption and emissions. In this paper, we propose a trajectory optimization method, PTO-GFC, to reduce the total fuel consumption of a CAV platoon through a signalized intersection. In this method, we first apply platoon-trajectory-optimization (PTO) to obtain the optimal trajectories of the platoon vehicles. In PTO, all CAVs in one platoon are considered as a whole, that is, all other CAVs follow the trajectory of the leading one with a time delay and minimum safety gap, which is enabled by vehicle to vehicle communication. Then, we apply gap-feedback-control (GFC) to control the vehicles with different speeds and headways merging into the optimal trajectories. We compare the PTO-GFC method with the other two methods, in which the leading vehicle adopts the optimal trajectory (LTO) or drive with maximum speed (AT), respectively, and the other vehicles follow the leading vehicle with a simplified Gipps' car-following model. Furthermore, we extend the controls into multiple platoons by considering the interactions between the two platoons. The numerical results demonstrate that PTO-GFC has better performance than LTO and AT, particularly when CAVs have enough space and time to smooth their trajectories. The reduction of travel time and fuel consumption shows the great potential of CAV technology in reducing congestion and negative environmental impact of automobile transportation.