Efficient input placement for the optimal control of network moments

Philip Solimine, Anke Meyer-Baese

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.

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