A Spectral-based ISS small-gain theorem for boundary control systems with infinite couplings
Abstract
We study the input-to-state stability (ISS) of boundary control systems allowing for infinitely many boundary couplings. Using semigroup perturbation theory and the theory of positive linear operators on Banach lattices, we derive a spectral small-gain condition ensuring exponential ISS. We further investigate linear Boltzmann-type equations on an infinite network of intersecting circles, incorporating delays, scattering, and disturbances acting at the junction. For this class of systems, we prove that a spectral small-gain condition on the transmission operator matrix guarantees exponential ISS with respect to disturbances propagating through the network. Moreover, we derive explicit ISS estimates for {certain} classes of dynamical processes. Finally, we demonstrate the practical applicability of our results by considering two important classes of time-delayed transmission conditions.
Cite
@article{arxiv.2604.11031,
title = {A Spectral-based ISS small-gain theorem for boundary control systems with infinite couplings},
author = {Yassine El Gantouh and Jun Zheng and Guchuan Zhu and Dingshi Li},
journal= {arXiv preprint arXiv:2604.11031},
year = {2026}
}