Stability of Paracontractive Open Multi-Agent Systems

In this paper, we examine networks consisting of multiple interacting agents that have the flexibility to join or leave the network at any moment, which we term open multi-agent systems (OMASs). Expanding upon the recently introduced theoretical framework for analyzing the dynamic characteristics of OMASs, we extend our study to encompass agents with vector states and discrete-time evolution. A key point of our work is the employment of the concept of 'open stability' w.r.t. the infinity norm, which naturally makes the distance between two points in the state independent of the number of agents. This obviates the necessity for distance normalization, as required by the standard Euclidean norm. Within this framework, the main contribution of our work is that of establishing sufficient conditions for the open stability of an OMAS, which include the boundedness of the arrival/departure process and the paracontractivity of the OMAS in the absence of arrivals/departures, thus generalizing existing results for contractive OMASs. To underscore the practical relevance of our theoretical framework, we present the formulation of the dynamic max-consensus protocol for OMASs. Through numerical simulations, we demonstrate the alignment of this protocol with the theoretical findings outlined in this manuscript.