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

TL;DR

It is shown that intrinsically stable networks and a broad class of switched systems remain stable in the presence of any type of time-varying time-delays whether these delays are periodic, stochastic, or otherwise, and it is proved that the asymptotic state of an intrinsically stable switched system is independent of both the system’s initial conditions and the presence.

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.

Figures (4)

• Figure 1: Left: The stable dynamics of the two-neuron Cohen-Grossberg network $(C,X)$ from Example \ref{['ConstantDelayExample']} is shown. Right: The unstable dynamics of the constant time-delayed version of this network $(C_D,X_3)$ is shown with the delay distribution given by the matrix $D$ in Equation \ref{['Dmatrix']}.
• Figure 2: Left: The dynamics of the intrinsically stable network $(C,X)$ from Example \ref{['ex:2']} is shown. Right: The stable dynamics of the constant time-delayed version of this network $(C_D,X_3)$ is shown with the delay distribution given by the matrix $D$ in Equation \ref{['Dmatrix']}. Both systems are attracted to the fixed point $\mathbf{x}^*=(-.386,1.595)$.
• Figure 3: Left: The dynamics of the switched network in Example \ref{['ex:switchednet']} is shown for two different initial conditions $\mathbf{x}^0=(2,3)$ (shown in blue and yellow) and $\mathbf{y}^0=(-2,-3)$ (shown in green and red). As the corresponding joint spectral radius $\overline{\rho}(S)$ of the network is less than 1, the orbits of these initial conditions converge to each other. Right: Modifying this switched network so that both $F,G\in M$ have the shared fixed point $\mathbf{0}$ results in a stable switched system with the globally attracting fixed point $\mathbf{0}$.
• Figure 4: Left: The dynamics of the intrinsically stable two-neuron Cohen-Grossberg network $(C,X)$ from Example \ref{['ex:2']} is shown in which the network has periodic time-varying time-delays. Right: The dynamics of the same Cohen-Grossberg network is shown in which the network has stochastic time-varying time-delays. Both systems are attracted to the fixed point $\mathbf{x}^*=(-.386,1.595)$ similar to the behavior shown in Figure \ref{['fig:2']}.

Theorems & Definitions (38)

• Definition 1
• Example 2.1
• Definition 2
• Theorem 1
• Definition 3
• Definition 4
• Example 3.1
• Theorem 2
• Definition 5
• Proposition 1