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This paper introduces a distributed adaptive formation control for large‐scale multi‐agent systems (LS‐MAS) that addresses the heavy computational complexity and communication traffic challenges while directly extending conventional distributed control from small scale to large scale. Specifically, a novel hierarchical game theoretic algorithm is developed to provide a feasible theory foundation for solving LS‐MAS distributed optimal formation problem by effectively integrating the mean‐field game (MFG), the Stackelberg game, and the cooperative game. In particular, LS‐MAS is divided into multiple groups geographically with each having one group leader and a significant amount of followers. Then, a cooperative game is used among multi‐group leaders to formulate distributed inter‐group formation control for leaders. Meanwhile, an MFG is adopted for a large number of intra‐group followers to achieve the collective intra‐group formation while a Stackelberg game is connecting the followers with their corresponding leader within the same group to achieve the overall LS‐MAS multi‐group formation behavior. Moreover, a hybrid actor–critic‐based reinforcement learning algorithm is constructed to learn the solution of the hierarchical game‐based optimal distributed formation control. Finally, to show the effectiveness of the presented schemes, numerical simulations and Lyapunov analysis is performed.

 

In this paper, a distributed swarm control problem is studied for large-scale multi-agent systems (LS-MASs). Different than classical multi-agent systems, an LS-MAS brings new challenges to control design due to its large number of agents. It might be more difficult for developing the appropriate control to achieve complicated missions such as collective swarming. To address these challenges, a novel mixed game theory is developed with a hierarchical learning algorithm. In the mixed game, the LS-MAS is represented as a multi-group, large-scale leader–follower system. Then, a cooperative game is used to formulate the distributed swarm control for multi-group leaders, and a Stackelberg game is utilized to couple the leaders and their large-scale followers effectively. Using the interaction between leaders and followers, the mean field game is used to continue the collective swarm behavior from leaders to followers smoothly without raising the computational complexity or communication traffic. Moreover, a hierarchical learning algorithm is designed to learn the intelligent optimal distributed swarm control for multi-group leader–follower systems. Specifically, a multi-agent actor–critic algorithm is developed for obtaining the distributed optimal swarm control for multi-group leaders first. Furthermore, an actor–critic–mass method is designed to find the decentralized swarm control for large-scale followers. Eventually, a series of numerical simulations and a Lyapunov stability proof of the closed-loop system are conducted to demonstrate the performance of the developed scheme. 

A bstract We study Ising Field Theory (the scaling limit of Ising model near the Curie critical point) in pure imaginary external magnetic field. We put particular emphasis on the detailed structure of the Yang-Lee edge singularity. While the leading singular behavior is controlled by the Yang-Lee fixed point (= minimal CFT $$ \mathcal{M} $$ M 2 / 5 ), the fine structure of the subleading singular terms is determined by the effective action which involves a tower of irrelevant operators. We use numerical data obtained through the “Truncated Free Fermion Space Approach” to estimate the couplings associated with two least irrelevant operators. One is the operator $$ T\overline{T} $$ T T ¯ , and we use the universal properties of the $$ T\overline{T} $$ T T ¯ deformation to fix the contributions of higher orders in the corresponding coupling parameter α . Another irrelevant operator we deal with is the descendant L_ 4 $$ \overline{L} $$ L ¯ _ 4 ϕ of the relevant primary ϕ in $$ \mathcal{M} $$ M 2 / 5 . The significance of this operator is that it is the lowest dimension operator which breaks integrability of the effective theory. We also establish analytic properties of the particle mass M (= inverse correlation length) as the function of complex magnetic field.