Control of Continuum Swarm Systems via Optimal Control and Optimal Transport Theory
We consider problems of optimal motion control for multi-agent systems where assignments as well as motions have costs. In particular, we consider a demand distribution and a distribution of resource agents, which require and provide support respectively. We formulate a time-varying assignment problem which trades off two typically competing costs, namely an assignment cost which depends on distances between resource and demand, and a motion cost associated with moving resource agents to locations with lower assignment costs. We use the formalism of optimal transport theory, and the Wasserstein distance in particular, to capture assignment cost, while motion cost is captured by vehicular velocities over time. Both particle and continuum models for large-scale systems are considered, leading to infinite-dimensional nonlinear optimal control problems in general. We show how in the special case of one spatial dimension, the optimal control problem can be converted into an infinite dimensional Linear-Quadratic tracking problem by reparameterizing in terms of quantile functions, which can then be converted into a family of decoupled scalar Linear-Quadratic tracking problems. An analytic solution is provided in the general case. We investigate further two special cases where the demand distribution is static and where it is periodic in time. Explicit results and simulations are provided in each of these cases as well.