Recently, a constructive method for the finite-dimensional observer-based control of deterministic parabolic PDEs was suggested by employing a modal decomposition approach. In this paper, for the first time we extend this method to the stochastic 1D heat equation with nonlinear multiplicative noise. We consider the Neumann actuation and study the observer-based as well as the state-feedback controls via the modal decomposition approach. We employ either trigonometric or polynomial dynamic extension. For observer-based control we consider a noisy boundary measurement. First, we show the well-posedness of strong solutions to the closed-loop systems. Then by suggesting a direct Lyapunov method and employing Itô’s formula, we provide mean-square $L^2$ exponential stability analysis of the full-order closed-loop system, leading to linear matrix inequality (LMI) conditions for finding the observer dimension and as large as possible noise intensity bound for the mean-square stabilizability. We prove that the LMIs are always feasible for small enough noise intensity and large enough observer dimension (for observer-based control). We further show that in the case of state-feedback and linear noise, the system is always stabilizable for noise intensities that guarantee the stabilizability of the stochastic finite-dimensional part of the closed-loop system with deterministic measurement. Numerical simulations are carried out to illustrate the efficiency of our method. For both state-feedback and observer-based controls, the trigonometric extension always allows for a larger noise than the polynomial one in the example.

最近，通过采用模态分解方法提出了一种基于有限维观测器的确定性抛物线 PDE 控制的构造方法。在本文中，我们首次将这种方法扩展到具有非线性乘法噪声的随机一维热方程。我们考虑 Neumann 驱动，并通过模态分解方法研究基于观察者的控制以及状态反馈控制。我们采用三角函数或多项式动态扩展。对于基于观察者的控制，我们考虑噪声边界测量。首先，我们展示了闭环系统强解的适定性。然后通过建议直接 Lyapunov 方法并使用 Itô 的公式，我们提供均方${\mathrm{大号}}^{\mathrm{2个}}$全阶闭环系统的指数稳定性分析，导致线性矩阵不等式 (LMI) 条件，用于寻找观察者维度和尽可能大的噪声强度边界，以实现均方稳定性。我们证明 LMI 对于足够小的噪声强度和足够大的观察者维度（对于基于观察者的控制）总是可行的。我们进一步表明，在状态反馈和线性噪声的情况下，系统对于噪声强度始终是稳定的，这保证了具有确定性测量的闭环系统的随机有限维部分的稳定性。进行数值模拟以说明我们的方法的效率。对于状态反馈和基于观察者的控制，