# Intrinsic Stability: Global Stability of Dynamical Networks and Switched Systems Resilient to any Type of Time-Delays

## TL;DR

This paper demonstrates that intrinsically stable dynamical networks and switched systems remain stable despite any form of time-varying delays, enhancing robustness in real-world and computational networks.

## Contribution

It proves that intrinsic stability ensures global stability under all types of time-delays, including stochastic and periodic, and shows this property is computationally efficient to verify.

## Key findings

- Intrinsic stability guarantees stability with any time-varying delays.
- The asymptotic state is exponentially independent of initial conditions.
- The approach improves existing stability results and is computationally inexpensive.

## Abstract

In real-world networks the interactions between network elements are inherently time-delayed. These time-delays can not only slow the network but can have a destabilizing effect on the network's dynamics leading to poor performance. The same is true in computational networks used for machine learning etc. where time-delays increase the network's memory but can degrade the network's ability to be trained. However, not all networks can be destabilized by time-delays. Previously, it has been shown that if a network or high-dimensional dynamical system is intrinsically stable, which is a stronger form of the standard notion of global stability, then it maintains its stability when constant time-delays are introduced into the system. Here we show that intrinsically stable systems, including intrinsically stable networks and a broad class of switched systems, i.e. systems whose mapping is time-dependent, remain stable in the presence of any type of time-varying time-delays whether these delays are periodic, stochastic, or otherwise. We apply these results to a number of well-studied systems to demonstrate that the notion of intrinsic stability is both computationally inexpensive, relative to other methods, and can be used to improve on some of the best known stability results. We also show that the asymptotic state of an intrinsically stable switched system is exponentially independent of the system's initial conditions.