On the optimal synchronization of oscillator networks via sparse interconnection graphs

Abstract — We consider the problem of synchronizing a given set of oscillators through the design of a conductance network, where the conductance connecting two oscillators models the amount of communication between them. Using optimal control theory, we formulate an optimization problem that addresses the trade-off between synchronization performance and con-ductance usage. Additionally, we promote the sparsity of the network by penalizing the number of interconnection links. We demonstrate that in the case of identical oscillators the optimization problem is convex and admits formulation as a semidefinite program. For non-identical oscillators that can be considered as perturbations around a central oscillator, we show that it is meaningful to design an optimal conductance network by assuming that all oscillators are identical to the central (average) oscillator. Finally, we derive explicit formulas for the optimal conductance values for some special problems. Index Terms — Convex relaxation, optimization, oscillator synchronization, reweighted `1 minimization, semidefinite pro