Stabilization of bilinear sparse matrix control systems using periodic inputs

In this brief paper, using results from the theory of averaging of bilinear systems, we provide a graph theoretic characterization for the existence of periodic control inputs that stabilize a sparse matrix system to the origin. In particular, we introduce a class of extensions to the directed graph corresponding to a given sparse matrix system, which when contains a stable sparse matrix system implies that the original system is stabilizable using periodic inputs. When this condition holds, we provide a systematic procedure for designing such controllers. Our technical approach combines ideas from the theory of bilinear control systems and averaging theory with graph theoretic results.