In this article, we introduce a model and a control approach for herding a swarm of “follower” agents to a target distribution among a set of states using a single “leader” agent. The follower agents evolve on a finite state space that is represented by a graph and transition between states according to a continuous-time Markov chain (CTMC), whose transition rates are determined by the location of the leader agent. The control problem is to define a sequence of states for the leader agent that steers the probability density of the forward equation of the Markov chain. For the case, when the followers are possibly interacting, we prove local approximate controllability of the system about equilibrium probability distributions. For the case, when the followers are noninteracting, we design two switching control laws for the leader that drive the swarm of follower agents asymptotically to a target probability distribution that is positive for all states. The first strategy is open-loop in nature, and the switching times of the leader are independent of the follower distribution. The second strategy is of feedback type, and the switching times of the leader are functions of the follower density in the leader's current state. We validate our control approach through numerical simulations with varied numbers of follower agents that evolve on graphs of different sizes, through a 3-D multirobot simulation in which a quadrotor is used to control the spatial distribution of eight ground robots over four regions, and through a physical experiment in which a swarm of ten robots is herded by a virtual leader over four regions.

中文翻译：

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使用领导者聚集群的可控性和稳定性：均场方法

在本文中，我们介绍了一种模型和控制方法，用于使用单个“领导者”代理将一群“跟随者”代理人聚集到一组状态之间的目标分布。跟随者代理在有限状态空间上演化，该状态空间由图形表示，并根据连续时间马尔可夫链（CTMC）进行状态之间的转换，其过渡速率由领导者代理的位置确定。控制问题是为领导者定义状态序列，以控制马尔可夫链正向方程的概率密度。对于这种情况，当跟随者可能相互作用时，我们证明了系统关于平衡概率分布的局部近似可控性。在这种情况下，当关注者不互动时，我们为领导者设计了两个切换控制定律，这些定律将追随者群体渐进地驱动到对所有状态都为正的目标概率分布。第一种策略本质上是开环的，领导者的切换时间与跟随者的分布无关。第二种策略是反馈型，领导者的切换时间是领导者当前状态下跟随者密度的函数。我们通过对具有不同数量的跟随者代理的数值模拟进行数值模拟来验证我们的控制方法，这些跟随者在不同大小的图形上演化；通过3-D多机器人模拟（其中四旋翼用于控制八个地面机器人在四个区域上的空间分布）；以及一项物理实验，其中虚拟领导者在四个区域中聚集了十个机器人。