Sufficient Stability Conditions for Time-varying Networks of Telegrapher's Equations or Difference Delay Equations

TL;DR

This paper establishes sufficient stability conditions for networks of lossless telegrapher's equations with time-varying boundary conditions, applicable to microwave circuits and certain difference delay systems, using dissipativity and $L^p$-norm analysis.

Contribution

It provides explicit stability criteria based on dissipativity for time-varying telegrapher's networks and related difference delay systems, simplifying previous results.

Findings

Exponential stability conditions are valid for all $L^p$-norms.

Dissipativity of couplings ensures network stability.

Reproves and simplifies earlier stability results for difference delay systems.

Abstract

We give a sufficient condition for exponential stability of a network of lossless telegrapher's equations, coupled by linear time-varying boundary conditions. The sufficient conditions is in terms of dissipativity of the couplings, which is natural for instance in the context of microwave circuits. Exponential stability is with respect to any $L^p$-norm, $1\leq p\leq\infty$. This also yields a sufficient condition for exponential stability to a special class of linear time-varying difference delay systems which is quite explicit and tractable. One ingredient of the proof is that $L^p$ exponential stability for such difference delay systems is independent of $p$, thereby reproving in a simpler way some results from [Y. Chitour, G. Mazanti, and M. Sigalotti, $\it {Netw. Heterog. Media}$, 11 (2016), pp. 563--601].