Time and Energy Costs for Synchronization of Kuramoto-Oscillator Networks With or Without Noise Perturbation

In this paper, time and energy costs for achieving synchronization of the Kuramoto-oscillator network with or without noise perturbation are investigated. In order to achieve synchro-nization and optimize time and energy consumption, a novel switching controller is designed, which combines the advantages of both the proportional feedback control method and the finite-time control technology. Sufficient conditions for achieving synchronization are established, and the estimates of time and energy costs are obtained mathematically as well. Particularly, the theoretical analysis and simulating calculation show that there exists a trade-off between time and energy costs. That is to say, the energy consumption can be reduced by adjusting the control parameters, but the time cost will increase inevitably, and vice versa. Further, we find that for fixed weights of time and energy costs of the performance index, the optimal values of parameters can be chosen to minimize the total cost.