Control costs of long-range interacting multi-agent systems with noise perturbation

As an important stage of collective dynamics, consensus is widespread in real-world moving groups, where one of the long-term challenges is to optimize the control costs of reaching consensus. However, current research on the consensus of multi-agents systems (MASs) usually only considers direct interactions between agents and ignores long-range interactions (LRIs), which may influence or even determine the collective behavior of MASs. In our paper, a basic framework is developed for analyzing the control costs for reaching consensus of MASs with LRIs and noise perturbation. The theoretical estimate of the control cost from a time and energy perspective will depend on the network diameter, control parameters, noise strength, and network size, and LRIs can promote MASs to reach consensus. We provide some numerical simulations to test our theory. Both theoretical and numerical analyses reveal that there is a trade-off between time and energy costs regardless of whether MASs have LRIs or not, i.e., adjusting control parameters to reduce time cost inevitably leads to an increase in energy cost. Our results provide a new option for controlling MASs: designing control protocols based on LRIs which in some cases have superior performance to typical controllers.