  We propose an optimal solution to a deterministic dynamic assignment problem
by leveraging connections to the theory of discrete optimal transport to
convert the combinatorial assignment problem into a tractable linear program.
We seek to allow a multi-vehicle swarm to accomplish a dynamically changing
task, for example tracking a multi-target swarm. Our approach simultaneously
determines the optimal assignment and the control of the individual agents. As
a result, the assignment policy accounts for the dynamics and capabilities of a
heterogeneous set of agents and targets. In contrast to a majority of existing
assignment schemes, this approach improves upon distance-based metrics for
assignments by considering cost metrics that account for the underlying
dynamics manifold. We provide a theoretical justification for the reformulation
of this problem, and show that the minimizer of the dynamic assignment problem
is equivalent to the minimizer of the associated Monge problem arising in
optimal transport. We prove that by accounting for dynamics, we only require
computing an assignment once over the operating lifetime --- significantly
decreasing computational expense. Furthermore, we show that the cost benefits
achieved by our approach increase as the swarm size increases, achieving almost
50\% cost reduction compared with distance-based metrics. We demonstrate our
approach through simulation on several linear and linearized problems.
