Set Input to State Stability for Multi-Agent Systems

In this paper, we investigate set Input to State Stability (set-ISS) in the context of multi-agent systems, specifically when agent interaction is spatial in nature. We review the definition of Input-to-State (ISS) Lyapunov functions with respect to sets, from which we provide a structural formulation of set-ISS that accommodates the local and per-agent nature of interacting systems. We argue that such a non-global characterization of set-ISS results in intuitive studies of multi-agent systems subject to external disturbances, resulting in superior understanding of collective and asymptotic behaviors. For the validation of our propositions, we consider a decentralized control law to reach swarm aggregation towards a bounded region of space. We demonstrate that our set-ISS structure, coupled with the requirement of physical occupancy in spatial interaction, connects with the classical notions of set-ISS, and yields both a fundamentally simplified analysis and superior insight into agent behavior compared to previous work.