SIAM Journal on Control and Optimization, Volume 59, Issue 1, Page 534-560, January 2021.
Consider a stochastic nonlinear system controlled over a possibly noisy communication channel. An important problem is to characterize the largest class of channels which admit coding and control policies so that the closed-loop system is stochastically stable. In this paper we consider the stability notion of (asymptotic) ergodicity. We prove lower bounds on the channel capacity necessary to achieve the stability criterion. Under mild technical assumptions, we obtain that the necessary channel capacity is lower-bounded by the log-determinant of the linearization, double-averaged over the state and noise space. We prove this bound by introducing a modified version of invariance entropy and utilizing the almost sure convergence of sample paths guaranteed by the pointwise ergodic theorem. Our results generalize those for linear systems and are in some cases more refined than those obtained for nonlinear systems via information-theoretic methods.

中文翻译：

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信息约束下受控随机非线性系统的遍历条件：体积增长方法

SIAM控制与优化杂志，第59卷，第1期，第534-560页，2021年1月。
考虑一个在可能嘈杂的通信信道上控制的随机非线性系统。一个重要的问题是表征允许编码和控制策略的最大通道类别，以使闭环系统随机稳定。在本文中，我们考虑（渐近）遍历性的稳定性概念。我们证明了达到稳定性标准所需的信道容量的下限。在温和的技术假设下，我们得出必要的信道容量受线性化的对数行列式的下限限制，在状态和噪声空间上进行两次平均。我们通过引入不变性熵的改进版本并利用逐点遍历定理所保证的几乎确定的样本路径收敛性来证明这一界限。