Control Localization in Networks of Dynamical Systems Connected via a Weighted Tree

For a network of dynamical systems coupled via an undirected weighted tree, we consider the problem of which system to apply control, in the case when only a single system receives control. We abstract this problem into a study of eigenvalues of a perturbed Laplacian matrix. We show that this eigenvalue problem has a complete solution for arbitrarily large control by showing that the best and the worst places to apply control have well-known characterization in graph theory, thus linking the computational eigenvalue problem with graph-theoretical concepts. Some partial results are proved in the case when the control effort is bounded. In particular, we show that a local maximum in localizing the best place for control is also a global maximum. We conjecture in the bounded control case that the best place to apply control must also necessarily be a characteristic vertex and present evidence from numerical experiments to support this conjecture.