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

TL;DR

This paper develops a spectral small-gain theorem for boundary control systems with infinite couplings, ensuring exponential input-to-state stability using advanced spectral and perturbation theories.

Contribution

It introduces a spectral small-gain condition for infinite boundary couplings and applies it to complex networked systems with delays and disturbances.

Findings

Spectral small-gain condition guarantees exponential ISS for infinite boundary control systems.

Explicit ISS estimates are derived for certain classes of dynamical processes.

The results are applicable to systems with time-delayed transmission conditions.

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.