  Synchronization is an essential property of engineered and natural networked
dynamical systems. The Kuramoto model of nonlinear synchronization has been
widely studied in applications including entrainment of clock cells in brain
networks and power system stability. Synchronization of Kuramoto networks has
been found to be challenging in the presence of signed couplings between
oscillators and when the network includes oscillators with heterogeneous
natural frequencies. In this paper, we study the problem of minimum-set control
input selection for synchronizing signed Kuramoto networks. We first derive
sufficient conditions for synchronization in homogeneous as well as
heterogeneous Kuramoto networks using a passivity-based framework. We then
develop a submodular algorithm for selecting a minimum set of control inputs
for a given Kuramoto network. We evaluate our approach through a numerical
study on multiple classes of graphs, including undirected, directed, and cycle
graphs.
