In the control of many autonomous subsystems, such as autonomous vehicles or UAV networks, a centralized control may be hindered by the prohibitive complexity, limited communication bandwidth, or private information of subsystems. Therefore, it is desirable for the control center to coordinate the controls of subsystems by designing mechanisms such as pricing, which makes the local optimizations of subsystem dynamics also maximize the reward of the total system, namely the social welfare. The economics framework of mechanism design is employed for the coordination of the autonomous subsystems. To address the challenge of dynamics, which are not considered in conventional economics mechanism design, and the complexity of private information, the approaches of geometrization and machine learning are employed, by endowing different geometric structures to the problem. The theoretical framework is applied in the context of urban aerial mobility, where the numerical simulations show the validity of the proposed framework.