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

David Reber, Benjamin Webb

Abstract:

n 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 stabile, 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.