Networked agents with heterogeneous constraints: equilibria and their stability

In this paper, the problem of characterizing and analyzing multiagent systems with heterogeneous constraints is investigated. Heterogeneity of constraints reflects the difference of the internal mechanism taken by agents to update states. By considering heterogeneity, we introduce more diversity, namely, moderation and immoderation, to the agents compared to existing works. To incorporate heterogeneous constraints, discontinuous terms are involved into the vector field of the system dynamics. In view of this, we characterize the existence of an equilibrium point using a fixed point theorem for partially ordered sets. It is shown that an equilibrium point always exists in the presence of both moderate and immoderate agents. Then, global and local convergence to different equilibrium points under certain conditions are carefully studied using Lyapunov's indirect method and systems theory. We also investigate the homogeneous case where the agents are either moderate or immoderate. When all the agents are moderate, an equilibrium point is shown to exist uniquely, to which global asymptotic convergence can be guaranteed, or consensus is achieved globally and asymptotically, by resorting to the contractive property of the averaged operator. On the other hand, when all the agents are immoderate, it is interestingly observed that they polarize to a certain level under certain conditions. Finally, we present a few numerical examples to validate the theoretical findings.