Distributed Fiedler Vector Estimation With Application to Desynchronization of Harmonic Oscillator Networks

The Fiedler vector of a graph is the eigenvector corresponding to the algebraic connectivity, which is the second-smallest eigenvalue (counting multiple eigenvalues separately) of the corresponding Laplacian matrix. We propose a continuous-time distributed control protocol to drive the value of the state variables of a network toward the Fiedler vector, up to a scale factor. Our protocol is unbiased and robust with respect to the initial network state, but the knowledge of the algebraic connectivity is required. By means of the proposed control law, we design a local state feedback that achieves desynchronization on arbitrary undirected connected networks of diffusively coupled harmonic oscillators. We provide numerical simulations to corroborate the theoretical results.