Controllability analysis of descriptor multi-agent systems with multiplex signed networks

This paper studies the controllability of descriptor multi-agent systems with multiplex signed networks, where each layer contains an identical number of agents that have descriptor linear dynamics. First, a necessary and sufficient condition is presented for the controllability of general M-layer descriptor multi-agent systems. Moreover, the controllability of descriptor multi-agent systems is explored under two coupling modes: drive-response coupling and interdependent coupling. For each mode, the explicit forms of the bases for the eigenspace of the system matrix are qualitatively derived. Based on this, necessary and sufficient conditions for the controllability of two-layer descriptor multi-agent systems are established. Furthermore, the controllability of descriptor multi-agent systems with structurally balanced multiplex networks is studied. Some properties of structurally balanced multiplex networks from a graph-theoretic perspective are analysed. The obtained results show that a multiplex network is structurally balanced if and only if all intra-layer topologies are simultaneously structurally balanced and the inter-layer topology is structurally balanced. Leveraging this property, it is proven that the controllability of descriptor multi-agent systems with structurally balanced multiplex networks is equivalent to that under unsigned networks when the inter-layer topology is unsigned. Finally, several examples are given to illustrate the effectiveness of the theoretical results.