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.

中文翻译：

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内在稳定性：具有任何类型时延的动态网络和切换系统的稳定性

在现实世界的网络中，网络元素之间的交互本质上是延时的。这些时间延迟不仅会减慢网络速度，还会对网络动态产生不稳定影响，从而导致性能不佳。在用于机器学习等的计算网络中也是如此，其中时间延迟会增加网络的内存，但会降低网络的训练能力。然而，并非所有网络都会因时间延迟而不稳定。以前，已经表明，如果网络或高维动力系统本质上是稳定的，这是全局稳定性标准概念的更强形式，那么当向系统中引入恒定的时间延迟时，它会保持其稳定性。在这里我们展示了本质上稳定的系统，包括本质稳定的网络和一大类交换系统，即映射与时间相关的系统，在存在任何类型的时变时延的情况下保持稳定，无论这些时延是周期性的、随机的还是其他的。我们将这些结果应用于许多经过充分研究的系统，以证明内在稳定性的概念相对于其他方法在计算上成本低廉，并且可用于改进一些最著名的稳定性结果。我们还表明，本质上稳定的切换系统的渐近状态与系统的初始条件呈指数无关。我们将这些结果应用于许多经过充分研究的系统，以证明内在稳定性的概念相对于其他方法在计算上成本低廉，并且可用于改进一些最著名的稳定性结果。我们还表明，本质上稳定的切换系统的渐近状态与系统的初始条件呈指数无关。我们将这些结果应用于许多经过充分研究的系统，以证明内在稳定性的概念相对于其他方法在计算上成本低廉，并且可用于改进一些最著名的稳定性结果。我们还表明，本质上稳定的切换系统的渐近状态与系统的初始条件呈指数无关。