  In this paper, we investigate how the self-synchronization property of a
swarm of Kuramoto oscillators can be controlled and exploited to achieve target
densities and target phase coherence. In the limit of an infinite number of
oscillators, the collective dynamics of the agents' density is described by a
mean-field model in the form of a nonlocal PDE, where the nonlocality arises
from the synchronization mechanism. In this mean-field setting, we introduce
two space-time dependent control inputs to affect the density of the
oscillators: an angular velocity field that corresponds to a state feedback law
for individual agents, and a control parameter that modulates the strength of
agent interactions over space and time, i.e., a multiplicative control with
respect to the integral nonlocal term. We frame the density tracking problem as
a PDE-constrained optimization problem. The controlled synchronization and
phase-locking are measured with classical polar order metrics. After
establishing the mass conservation property of the mean-field model and bounds
on its nonlocal term, a system of first-order necessary conditions for
optimality is recovered using a Lagrangian method. The optimality system,
comprising a nonlocal PDE for the state dynamics equation, the respective
nonlocal adjoint dynamics, and the Euler equation, is solved iteratively
following a standard Optimize-then-Discretize approach and an efficient
numerical solver based on spectral methods. We demonstrate our approach for
each of the two control inputs in simulation.
