Algorithms for sparse stable systems

We deal with the problem of designing stable sparse decentralized systems. Whether the communication structure of a decentralized system can sustain stable dynamics can be reduced to the study of whether a given vector space of sparse matrices contains stable (Hurwitz) matrices. In this paper, after a brief overview the main known results in the area, we derive methods to create sparse stable vector space (that is, vector spaces that contain stable matrices) recursively and in polynomial time in the dimension of the space. The approach relies on perturbation theory to prove stability and on graph theory to derive polynomial time algorithms.