Analysis of Equilibrium Points and Convergent Behaviors for Constrained Signed Networks

This article is devoted to analyzing the equilibrium points and convergent behaviors for a constrained signed network with general topology containing a directed spanning tree, where the output of each agent is restricted by a constraint set. Different from unconstrained signed networks, the rooted subgraph and constraint sets are both critical for the theoretical analysis of the constrained signed network. By utilizing $H$-matrix theories, projection techniques, invariance principle, and an extended Barbalat's lemma, it is rigorously shown that the state of the constrained network globally asymptotically approaches the nonempty equilibrium set. Based on the equilibrium set and constraint sets, some notions and criteria are developed to explore the convergent behaviors of the constrained network, including interval bipartite consensus, bipartite consensus, global stability, and noninterior convergence. In sharp contrast to unconstrained signed networks, a constrained signed network may fail to achieve interval bipartite consensus or bipartite consensus even if the rooted subgraph is structurally balanced. Surprisingly, it is found that the constrained signed network under different initial conditions may exhibit different types of convergent behaviors. The theoretical results are illustrated by numerical examples.