We consider the problem of localization of control in dynamical systems coupled via a weighted tree, 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 must necessarily be a characteristic vertex and a pendant vertex, respectively. Some partial results are proved in the case of finite control. In particular, we show that a local maximum in localizing the best place for control is also a global maximum. We conjecture in the finite 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.
By: Ravindra Bapat, Chai Wah Wu
Published in: RC25466 in 2014
LIMITED DISTRIBUTION NOTICE:
This Research Report is available. This report has been submitted for publication outside of IBM and will probably be copyrighted if accepted for publication. It has been issued as a Research Report for early dissemination of its contents. In view of the transfer of copyright to the outside publisher, its distribution outside of IBM prior to publication should be limited to peer communications and specific requests. After outside publication, requests should be filled only by reprints or legally obtained copies of the article (e.g., payment of royalties). I have read and understand this notice and am a member of the scientific community outside or inside of IBM seeking a single copy only.
Questions about this service can be mailed to reports@us.ibm.com .