Decentralized gain adaptation for optimal pinning controllability of complex networks

Coordinating ensembles of dynamical systems in a decentralized manner is of central importance when controlling complex networks. Pinning control is a much used technique where only a small fraction of the network nodes is directly controlled, with control gains being typically selected uniformly across the control layer. In this letter, we tackle the problem of optimally selecting the control gains of a pinning control strategy. Indeed, in networks of nonlinear dynamical systems fulfilling the so-called QUAD assumption, pinning controllability improves as the smallest eigenvalue \lambda -{1} of an extended Laplacian matrix increases. Based on this observation, we pose a constrained optimization problem on the network. Rather than solving it in a centralized fashion, we propose a fully decentralized multilayer approach. Specifically, one layer is used to evaluate the sensitivity of \lambda -{1} to the variation of the gains, while the second layer uses such an estimate to dynamically tune the control gains. The effectiveness of the approach is demonstrated via a representative example.