I'm learning about graph-based distributed control and there's a problem called "the rendezvous problem" that uses the Laplacian matrix as the state matrix of the system. I have a graph with 4 nodes and already computed the Laplacian matrix, but I cannot figure out how to analytically derive the value to which the vector x converges when time goes to infinity.
What I was doing was computing the eigenvalues but having the eigenvalues I can only check if the system is stable or not, but then I still don't get how to get the point of convergence. The only thing I know about the Laplacian matrix is that it is time-invariant.
Can someone give me a hint on how to proceed? should I compute the eigenvectors and find the solution of the system? I mean its a good amount of work due to the size of the system.