The determination of fixed controllable subspace in complex networks

Abstract

In general, a controllable subspace of complex networks consists of the non-fixed part and the fixed part. The non-fixed part changes with non-zero elements in the system matrices. The fixed part consists of all state vertices that can be controlled by any parameter in the controllable subspace, and is called a fixed controllable subspace. The fixed controllable subspace is considered as a stable and controllable part of the system. Actually, the fixed controllable subspace is a subspace of a state space with general properties, which can be controlled and is not affected by the change of parameters. In this paper, we show that the subspace of complex network is controllable when almost all of the parameters are free. This paper introduces the notion of fixed controllable subspace and applies it to the study of structural controllability. A few characteristics of the fixed controllable subspace are also derived by using the graph theory. In addition, we explore how the fixed part can be determined by using the well-known result of Hosoe's controllable subspace theory.