Decentralized stabilization with symmetric topologies

A sparse matrix space is a vector space of matrices with entries that are either zero or arbitrary real. Letting the pattern of zero and arbitrary entries define an adjacency matrix (by setting the non-zero entries to one), we can attach a graph to such vector spaces and think of this graph as describing the decentralization structure of a control system. We want to determine whether this topology can sustain stable dynamics or, equivalently, whether the corresponding sparse matrix space contains stable matrices. We present in this paper a complete characterization the symmetric sparse matrix spaces that contain stable matrices.