## 1 Introduction  
The development of human society is inseparable from the exploitation and use of various resources. With land resources being gradually exhausted, attention has been turned towards the deep ocean. Numerous manganese nodules, deep-sea oil and gas, hydrothermal deposits, and gas hydrates exist in the submarine world. Thus, the efficient and safe exploration of such resources has become an urgent matter. Autonomous underwater vehicles (AUVs) are an auxiliary intelligent tool for ocean exploration, which play a key role in the application of ocean environment observation [1; 2], seabed geomorphology measurement [3], and military reconnaissance [4]. Owing to their power, cable-free autonomy, good masking performance, and wide search range, AUVs offer very wide application prospects in both the civil marine and coastal defense military fields. In particular, AUVs are essential in deep-sea underwater searching. As early as 1963, the \"Alvin\" and \"Kov\" underwater robot cooperation of the US to search for and salvage a lost hydrogen bomb in the Spanish trench was a successful case. In early 2014, the US used the \"Bluefin Tuna\" AUV to conduct a large-scale underwater search at a depth of 4500 m in the Indian Ocean for the missing Malaysia Airlines flight MH370. Ocean Infinity, which was founded in the US in 2017, has rapidly expanded into a leading global maritime technology company. In early 2020, Ocean Infinity launched Armada, which is a new ship technology and data company, and introduced fleet robotics. The Armada fleet can carry ROVs (remote operated vehicles) and AUVs, as well as a variety of other sensors or equipment, thereby replacing traditional support vessels for seabedmapping, oil field material transport, subsea construction support, salvage and rescue, military, and other activities. In 2011, researchers at the Ganz Artificial Life Lab in Austria unveiled the largest cluster of underwater unmanned vehicles in the world at the time: the CoCoRo AUV cluster. The project, which was funded by the European Union and led by [PERSON], consisted of 41 AUVs that could work together to accomplish tasks, with the main purpose of being used for underwater monitoring and searching. The cluster system was scalable, reliable, and flexible in terms of its behavioral potential. The researchers studied collective self-knowledge through experiments that were inspired by behavioral and psychological science, thereby allowing for the quantification of collective self-knowledge. Furthermore, the European Commission supported a project known as Smart and Networking Underwater Robots in Cooperation Meshes. The aim was to select, combine, and integrate different and heterogeneous communication technologies, components, and solutions to achieve the best performance for the management and control of underwater vehicles when the robots completed different missions and tasks. This project was completed by testing it in the sea. With the increasing complexity of AUV search tasks, multi-AUV (MAUV) systems have become an important research direction in the development of underwater vehicles when it is difficult for a single AUV to complete a task. MAUV systems can provide more solutions with higher work efficiency, a higher intelligence level, and better fault tolerance compared to single-AUV systems. MAUV systems (also known as swarm agents) have mainly arisen owing to limited technology, as the intelligence of a single agent cannot be extended. Thus, it is hoped that coordination and cooperation among multiple agents can deal with complex tasks that a single agent cannot handle. This concept has received significant attention in the scientific research and engineering circles [5; 6]. The consensus problem at the center of MAUV cooperative control has been developed over the past several years using various methods.