In recent years, the research on space-air-ground integrated cross-domain multi-dynamic systems based on cloud control has attracted much attention. Through interconnection, intercommunication, and interoperability, space-air-ground integrated heterogeneous multi-dynamic systems realize information sharing and fusion, behavioral interaction and coordination, as well as task collaboration and cooperation. This considerably facilitates the system's functional complementation and multiplies the energy efficiency, thus significantly enhancing the system's capability of responding to complex environments and missions. This paper expounds on the research progress on the cooperative control and intelligent decision of space-air-ground integrated heterogeneous multi-dynamic systems. First, abstracts and components of the space-air-ground integrated network with the cloud architecture, as well as the related representative progress around the world, are provided. Then, from the perspective of cloud control and intelligence decision-making, state-of-the-art technologies related to space-air-ground integrated heterogeneous multi-dynamic systems are introduced, and solutions under the framework of cloud control and decision-making are given. Finally, the key technologies are summarized in terms of cluster management, cross-domain cooperative sensing, and control and decision-making, and the possible future research directions are discussed and prospected.