In this paper, we study the optimal control of the mean and variance of the network state vector. We develop an algorithm to optimize the control input placement subject to constraints on the state, which must be achieved at a given time threshold; seeking an input placement which moves the moment at minimum cost. First, we solve the state-selection problem for a number of variants of the first and second moment, and find solutions related to the eigenvalues of the systems' Gramian matrices. Our algorithm then uses this information to find a locally optimal input placement. This is a Generalization of the Projected Gradient Method (GPGM). We solve the problem for some common versions of these moments, including the mean state and versions of the second moment which induce discord, repel from a certain state, or encourage convergence. We then perform simulations, and discuss a measure of centrality based on the system flux -- a measure which describes what nodes are most important to optimal control of the average state.

中文翻译：

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用于优化控制网络矩的有效输入放置

在本文中，我们研究了网络状态向量均值和方差的最优控制。我们开发了一种算法来优化受状态约束的控制输入放置，这必须在给定的时间阈值下实现；寻求以最低成本移动时刻的输入位置。首先，我们解决了一阶和二阶矩的多个变体的状态选择问题，并找到与系统 Gramian 矩阵的特征值相关的解。我们的算法然后使用这些信息来找到局部最优的输入位置。这是投影梯度法 (GPGM) 的推广。我们解决了这些时刻的一些常见版本的问题，包括平均状态和第二时刻的版本，这些版本会引起不和谐、排斥某个状态或鼓励收敛。