Controlling a preselected subset of nodes (named “target control”) of complex networks with minimal control energy is a critically important physical issue. To address this issue, first, an energy cost function is established by designing an optimal controller, in which an input matrix $B$ is involved as a matrix variable to be determined so as to minimize the control cost. In particular, we integrate the design equations to obtain an equivalent expression of the cost function without solving the Riccati differential equation directly. Second, based on this expression, we derive its gradient with respect to $B$ by introducing some methods for matrix differentiation. Two different constraints, i.e., trace and positive element constraints, which are imposed on the matrix variable $B$ of the cost function, are considered. Last but not least, we propose two corresponding algorithms to solve these two different constraint optimization problems. Numerical examples in directed networks are provided to show the effectiveness of the proposed methods. This work suggests that the target control of complex networks can reduce both the minimum number of external control sources and the control cost.

中文翻译：

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输入可选的复杂网络的最优目标控制

用最少的控制能量来控制复杂网络的节点的预选子集（称为“目标控制”）是至关重要的物理问题。为了解决这个问题，首先，通过设计一个最优控制器来建立一个能量成本函数，其中一个输入矩阵$ B $作为要确定的矩阵变量涉及到“最小化”，以最小化控制成本。特别是，我们无需直接求解Riccati微分方程，就可以对设计方程进行积分以获得成本函数的等效表达式。其次，基于该表达式，我们得出其相对于$ B $通过介绍一些矩阵分化的方法。施加在矩阵变量上的两个不同的约束，即迹线约束和正元素约束$ B $考虑成本函数。最后但并非最不重要的一点是，我们提出了两种相应的算法来解决这两个不同的约束优化问题。提供了有向网络中的数值示例，以证明所提出方法的有效性。这项工作表明，复杂网络的目标控制可以减少外部控制源的最小数量和控制成本。