标题： 部分可观测随机偏微分方程最优控制的数值逼近 显示英文标题

标题： Numerical approximations for partially observed optimal control of stochastic partial differential equations

摘要： 本文研究了一类带有部分观测的随机偏微分方程最优控制问题的数值逼近。系统状态在Hilbert空间中演化，而观测值则以有限维空间$\rr^d$给出。我们首先建立此类问题的随机最大值原理（SMP），其中系统状态由一个圆柱形Wiener过程驱动。相应的伴随方程由倒向随机偏微分方程刻画。然后，我们开发了求解部分可观测最优控制问题的数值算法。我们的方法结合了受SMP指导的随机梯度下降法与粒子滤波算法来估计系统状态的条件分布。最后，通过数值实验展示了所提出算法的有效性。 显示英文摘要

摘要： In this paper, we study numerical approximations for optimal control of a class of stochastic partial differential equations with partial observations. The system state evolves in a Hilbert space, whereas observations are given in finite-dimensional space $\rr^d$. We begin by establishing stochastic maximum principles (SMPs) for such problems, where the system state is driven by a cylindrical Wiener process. The corresponding adjoint equations are characterized by backward stochastic partial differential equations. We then develop numerical algorithms to solve the partially observed optimal control. Our approach combines the stochastic gradient descent method, guided by the SMP, with a particle filtering algorithm to estimate the conditional distributions of the state of the system. Finally, we demonstrate the effectiveness of our proposed algorithm through numerical experiments.