As an important part of cyberphysical systems (CPSs), multiple aerial drone systems are widely used in various scenarios, and research scenarios are becoming increasingly complex. However, planning strategies for the formation flying of aerial swarms in dense environments typically lack the capability of large-scale breakthrough because the amount of communication and computation required for swarm control grows exponentially with scale. To address this deficiency, we present a mean-field game (MFG) control-based method that ensures collision-free trajectory generation for the formation flight of a large-scale swarm. In this paper, two types of differentiable mean-field terms are proposed to quantify the overall similarity distance between large-scale 3-D formations and the potential energy value of dense 3-D obstacles, respectively. We then formulate these two terms into a mean-field game control framework, which minimizes energy cost, formation similarity error, and collision penalty under the dynamical constraints, so as to achieve spatiotemporal planning for the desired trajectory. The classical task of a distributed large-scale aerial swarm system is simulated by numerical examples, and the feasibility and effectiveness of our method are verified and analyzed. The comparison with baseline methods shows the advanced nature of our method.

As one of the important issues of multi-agent collaboration, the large-scale agents’ cooperative attack–defense evolution requires a large number of agents to make stress-effective strategies to achieve their goals in complex environments. Multi-agent attack and defense in high-dimensional environments (3D obstacle scenarios) present the challenge of being able to accurately control high-dimensional state quantities. Moreover, the large scale makes the dynamic interactions in the attack and defense problems increase dramatically, which, using traditional optimal control techniques, can cause a dimensional explosion. How to model and solve the cooperative attack–defense evolution problem of large-scale agents in high-dimensional environments have become a challenge. We jointly considered energy consumption, inter-group attack and defense, intra-group collision avoidance, and obstacle avoidance in their cost functions. Meanwhile, the high-dimensional state dynamics were used to describe the motion of agents under environmental interference. Then, we formulated the cooperative attack–defense evolution of large-scale agents in high-dimensional environments as a multi-population high-dimensional stochastic mean-field game (MPHD-MFG), which significantly reduced the communication frequency and computational complexity. We tractably solved the MPHD-MFG with a generative-adversarial-network (GAN)-based method using the MFGs’ underlying variational primal–dual structure. Based on our approach, we carried out an integrative experiment in which we analytically showed the fast convergence of our cooperative attack–defense evolution algorithm by the convergence of the Hamilton–Jacobi–Bellman equation’s residual errors. The experiment also showed that a large number of drones can avoid obstacles and smoothly evolve their attack and defense behaviors while minimizing their energy consumption. In addition, the comparison with the baseline methods showed that our approach is advanced.

We show that on a Sasakian 3-sphere the Sasaki–Ricci flow initiating from a Sasakian metric of positive transverse scalar curvature converges to a gradient Sasaki–Ricci soliton. We also show the existence and uniqueness of gradient Sasaki–Ricci soliton on each Sasakian 3-sphere.