Brief paper: Global stabilization of complex networks with digraph topologies via a local pinning algorithm

Authors:
Wenlian Lu;Xiang Li;Zhihai Rong
Affiliations:
Centre for Computational Systems Biology and School of Mathematical Sciences, Fudan University, Shanghai 200433, PR China;Adaptive Networks and Control Lab, Department of Electronic Engineering, Fudan University, Shanghai 200433, PR China;Department of Automation, Donghua University, Shanghai 201620, PR China
Venue:
Automatica (Journal of IFAC)
Year:
2010

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Abstract

This paper concerns the global stability of controlling a complex network with digraph topology to a homogeneous trajectory of the uncoupled system by the local pinning control algorithm. The derived stability condition indicates that the smallest real part of eigenvalues of the Laplacian sub-matrix corresponding to the unpinned vertices can be used to measure the stabilizability of a digraph with a given pinned vertex set. A pinned vertex set can stabilize a directed network to some unstable trajectories, for instance, to a chaotic trajectory of the uncoupled systems, if and only if the pinned vertex set can access all other vertices in the digraph. Furthermore, in the bigraph case, the analytical estimation of the stabilizability's lower bound suggests that an optimal pinning strategy should take not only the vertex degree, but also the shortest path between pairs of vertices into considerations.