Our ability to coordinate the behavior in networks of complex dynamical systems is often challenged by the presence of noise affecting the individual dynamics and the communication links. In the literature, conservative global conditions guaranteeing the almost sure convergence toward the desired trajectory of a virtual node, the pinner, have been derived. In this article, we identify the minimal conditions on the individual dynamics, interconnection topology, and noise intensities, so that the network exponentially converges onto the pinner's trajectory. Specifically, we broaden the master stability function approach to deal with networks of coupled stochastic differential equations, and provide necessary and sufficient conditions for local exponential pinning controllability of networks of stochastic systems. Interestingly, our analyses show that noise can be either beneficial or detrimental for pinning controllability, depending on how it diffuses in each node. Our analytical findings are illustrated with representative numerical examples.

中文翻译：

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噪声复杂网络的随机钉扎控制

我们协调复杂动力系统网络行为的能力经常受到噪声的影响，这些噪声会影响各个动力系统和通信链路。在文献中，已经推导出了保证几乎确定地趋向虚拟节点的理想轨迹（pinner）收敛的保守全局条件。在本文中，我们确定了个体动力学，互连拓扑和噪声强度的最小条件，从而使网络以指数形式收敛到固定器的轨迹上。具体而言，我们拓宽了主稳定性函数方法，以处理耦合的随机微分方程网络，并为随机系统的网络局部指数固定控制提供了充要条件。有趣的是 我们的分析表明，噪声对钉扎可控性可能是有益的还是有害的，具体取决于噪声在每个节点中的扩散方式。我们的分析结果以具有代表性的数值示例进行了说明。