To promote the implementation of realistic control over various complex networks, recent work has been focusing on analyzing energy cost. Indeed, the energy cost quantifies how much effort is required to drive the system from one state to another when it is fully controllable. A fully controllable system means that the system can be driven by external inputs from any initial state to any final state in finite time. However, it is prohibitively expensive and unnecessary to confine that the system is fully controllable when we merely need to accomplish the so-called target control — controlling a subnet of nodes chosen from the entire network. Yet, when the system is partially controllable, the associated energy cost remains elusive. Here we present the minimum energy cost for controlling an arbitrary subset of nodes of a network. We show the scaling behavior of the precise upper and lower bounds of the minimum energy in terms of the time given to accomplish control. For controlling a given number of target nodes, we further show that the associated energy over different configurations can differ by several orders of magnitude. When the adjacency matrix of the network is nonsingular, we can simplify the framework by just considering the induced subgraph spanned by target nodes instead of the entire network. Importantly, we find that energy cost could be saved by orders of magnitude as we only need the partial controllability of the entire network. Our theoretical results are all corroborated by numerical calculations, and pave the way for estimating the energy cost to implement realistic target control in various applications.

中文翻译：

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复杂网络目标控制的能源成本

为了促进对各种复杂网络的现实控制的实施，最近的工作一直集中在分析能源成本上。实际上，能源成本量化了在完全可控的情况下将系统从一种状态驱动到另一种状态所需的努力。完全可控的系统意味着系统可以在有限时间内由外部输入从任何初始状态驱动到任何最终状态。然而，当我们只需要完成所谓的目标控制——控制从整个网络中选择的节点子网时，将系统完全可控的限制是非常昂贵且不必要的。然而，当系统部分可控时，相关的能源成本仍然难以捉摸。在这里，我们提出了控制网络的任意节点子集的最小能量成本。我们根据完成控制的时间显示了最小能量的精确上限和下限的缩放行为。为了控制给定数量的目标节点，我们进一步表明，不同配置下的相关能量可以相差几个数量级。当网络的邻接矩阵是非奇异的时，我们可以通过只考虑目标节点而不是整个网络跨越的诱导子图来简化框架。重要的是，我们发现能源成本可以节省几个数量级，因为我们只需要整个网络的部分可控性。我们的理论结果都得到了数值计算的证实，并为估算能源成本以在各种应用中实施现实的目标控制铺平了道路。