Partial controllability of a leader-follower dynamic network

Generally speaking, controllability of multi-agent systems refer to transferring the remaining agents of such system from any arbitrary initial state to any final state by controlling dynamics of a small amount of agents under exchanged information between each other, which makes the system reflect the effect of a whole, that is the complete controllability. Comparing with the complete controllability of the general multi-agent systems, in reality, it may not be necessary to actuate all nodes to the desired configuration. This paper addresses the partial controllability of the discrete-time multi-agent system with a leader under fixed topology, proposes the concept of controllable node group. Some necessary and sufficient conditions are given for partial controllability of the discrete-time multi-agent system with a single leader. Moreover, the partial controllability criteria of some special cases are given. Finally, an numerical example and simulations are given to illustrate the correctness of the theoretical results.