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arXiv · 2604.11031

A Spectral-based ISS small-gain theorem for boundary control systems with infinite couplings

Yassine El Gantouh·Jun Zheng·Guchuan Zhu·Dingshi Li
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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.

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Yassine El Gantouh, Jun Zheng, Guchuan Zhu, Dingshi Li. 2026-04-13. A Spectral-based ISS small-gain theorem for boundary control systems with infinite couplings. https://arxiv.org/abs/2604.11031

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