Synchronization of heterogeneous linear networks with distinct inner coupling matrices

In this paper, we study synchronization of heterogeneous linear networks with distinct inner coupling matrices. Firstly, for synchronous networks, we show that any synchronous trajectory will converge to a corresponding synchronous state. Then, we provide an invariant set, which can be exactly obtained by solving linear equations and then used for characterizing synchronous states. Afterwards, we use inner coupling matrices and node dynamics to successively decompose the original network into a new network, composed of the external part and the internal part. Moreover, this new network can be proved to synchronize to the above invariant set by constructing the corresponding desired Lyapunov-like functions for the internal part and the external part respectively. In particular, this result still holds if the coupling strength is disturbed slightly. Finally, examples with numerical simulations are given to illustrate the validity and applicability of our theoretical results.