Necessary and Sufficient Conditions for Controllability and Essential Controllability of Directed Circle and Tree Graphs

The multi-agent controllability is intrinsically affected by the network topology and the selection of leaders. A focus of exploring this problem is to uncover the relationship between the eigenspace of Laplacian matrix and network topology. For strongly connected directed circle graphs, we elaborate how the zero entries in the left eigenvectors of Laplacian matrix $L$ arise. The topologies arising from left eigenvectors with zero entries are filtered to construct essentially controllable directed circle graphs regardless of the choice of leaders. We propose two methods for constructing a substantial quantity of essentially controllable graphs, with a focus on utilizing essentially controllable circle graphs as the foundation. For a special directed graph-OT tree, the controllability is shown to be related with its substructure-paths. This promotes the establishment of a sufficient and necessary condition for controllability. Finally, a method is presented to check the controllable subspace by identifying the left eigenvectors and generalized left eigenvectors.