Target Controllability of Multi-Layer Networks with High-Dimensional Nodes

This paper studies the target controllability of multi-layer complex networked systems, in which the nodes are high-dimensional linear time invariant (LTI) dynamical systems, and the network topology is directed and weighted. The influence of inter-layer couplings on the target controllability of multi-layer networks is discussed. It is found that even if there exists a layer which is not target controllable, the entire multi-layer network can still be target controllable due to the inter-layer couplings. For the multi-layer networks with general structure, a necessary and sufficient condition for target controllability is given by establishing the relationship between uncontrollable subspace and output matrix. By the derived condition, it can be found that the system may be target controllable even if it is not state controllable. On this basis, two corollaries are derived, which clarify the relationship between target controllability, state controllability and output controllability. For the multi-layer networks where the inter-layer couplings are directed chains and directed stars, sufficient conditions for target controllability of networked systems are given, respectively. These conditions are easier to verify than the classic criterion.