We consider a setting in which one swarm of agents is to service or track a second swarm, and formulate an optimal control problem which trades off between the competing objectives of servicing and motion costs. We consider the continuum limit where large-scale swarms are modeled in terms of their time-varying densities, and where the Wasserstein distance between two densities captures the servicing cost. We show how this non-linear infinite-dimensional optimal control problem is intimately related to the geometry of Wasserstein space, and provide new results in the case of absolutely continuous densities and constant-in-time references. Specifically, we show that optimal swarm trajectories follow Wasserstein geodesics, while the optimal control tradeoff determines the time-schedule of travel along these geodesics. We briefly describe how this solution provides a basis for a model-predictive control scheme for tracking time-varying and real-time reference trajectories as well.

翻译：本文考虑一群智能体需要服务于或跟踪另一群智能体的场景，并构建了一个最优控制问题，该问题在服务成本与运动成本这两个相互竞争的目标之间进行权衡。我们采用连续体极限方法，将大规模集群建模为时变密度函数，并以两个密度之间的Wasserstein距离衡量服务成本。研究表明，这个非线性无限维最优控制问题与Wasserstein空间的几何结构密切相关，并在绝对连续密度及参考密度时不变的特殊情形下给出了新结论。具体而言，我们证明了最优集群轨迹遵循Wasserstein测地线，而最优控制权衡决定了沿这些测地线运动的时间调度。本文还简要阐述了该解决方案如何为跟踪时变及实时参考轨迹的模型预测控制方案提供理论基础。