SIAM Journal on Control and Optimization, Volume 62, Issue 1, Page 297-325, February 2024.
Abstract. Sampled-data control of PDEs has become an active research area; however, existing results are confined to deterministic PDEs. Sampled-data controller design of stochastic PDEs is a challenging open problem. In this paper we suggest a solution to this problem for 1D stochastic diffusion-reaction equations under discrete-time nonlocal measurement via the modal decomposition method, where both the considered system and the measurement are subject to nonlinear multiplicative noise. We present two methods: a direct one with sampled-data controller implemented via zero-order hold device, and a dynamic-extension-based one with sampled-data controller implemented via a generalized hold device. For both methods, we provide mean-square [math] exponential stability analysis of the full-order closed-loop system. We construct a Lyapunov functional [math] that depends on both the deterministic and stochastic parts of the finite-dimensional part of the closed-loop system. We employ corresponding Itô’s formulas for stochastic ODEs and PDEs, respectively, and further combine [math] with Halanay’s inequality with respect to the expected value of [math] to compensate for sampling in the infinite-dimensional tail. We provide linear matrix inequalities (LMIs) for finding the observer dimension and upper bounds on sampling intervals and noise intensities that preserve the mean-square exponential stability. We prove that the LMIs are always feasible for large enough observer dimension and small enough bounds on sampling intervals and noise intensities. A numerical example demonstrates the efficiency of our methods. The example shows that for the same bounds on noise intensities, the dynamic-extension-based controller allows larger sampling intervals, but this is due to its complexity (generalized hold device for sample-data implementation compared to zero-order hold for the direct method).

中文翻译：

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一维随机抛物线偏微分方程的采样数据有限维观测器控制

SIAM 控制与优化杂志，第 62 卷，第 1 期，第 297-325 页，2024 年 2 月。
摘要。偏微分方程的采样数据控制已成为一个活跃的研究领域；然而，现有结果仅限于确定性偏微分方程。随机偏微分方程的采样数据控制器设计是一个具有挑战性的开放问题。在本文中，我们建议通过模态分解方法解决离散时间非局部测量下的一维随机扩散反应方程的这个问题，其中所考虑的系统和测量都受到非线性乘性噪声的影响。我们提出两种方法：一种是通过零阶保持设备实现采样数据控制器的直接方法，另一种是通过广义保持设备实现采样数据控制器的基于动态扩展的方法。对于这两种方法，我们提供全阶闭环系统的均方[数学]指数稳定性分析。我们构建了一个李亚普诺夫函数[数学]，它取决于闭环系统有限维部分的确定性和随机部分。我们分别采用相应的随机常微分方程和偏微分方程的 Itô 公式，并进一步将 [math] 与 Halanay 不等式相对于 [math] 的期望值结合起来，以补偿无限维尾部的采样。我们提供线性矩阵不等式 (LMI)，用于查找观察者维度以及采样间隔和噪声强度的上限，以保持均方指数稳定性。我们证明，对于足够大的观察者维度和足够小的采样间隔和噪声强度界限，LMI 总是可行的。数值示例证明了我们方法的效率。该示例表明，对于噪声强度的相同界限，基于动态扩展的控制器允许更大的采样间隔，但这是由于其复杂性（用于样本数据实现的广义保持设备与直接方法的零阶保持相比） ）。