Optimal Assignment and Motion Control in Two-Class Continuum Swarms

We consider optimal swarm control problems where two different classes of agents are present. Continuum idealizations of large-scale swarms are used where the dynamics describe the evolution of the spatially-distributed densities of each agent class. The problem formulation we adopt is motivated by applications where agents of one class are assigned to agents of the other class, which we refer to as demand and resource agents respectively. Assignments have costs related to the distances between mutually assigned agents, and the overall cost of an assignment is quantified by a Wasserstein distance between the densities of the two agent classes. When agents can move, the assignment cost can decrease at the expense of a physical motion cost, and this tradeoff sets up a nonlinear infinite-dimensional optimal control problem. We show that in one spatial dimension, this problem can be converted to an infinite-dimensional, but decoupled, linear- quadratic (LQ) tracking problem when expressed in terms of the quantile functions of the respective agent densities. Solutions are given in the general one-dimensional case, as well as in the special cases of constant and periodically time-varying demands.