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<p>In this paper, we investigate the numerical approximation of optimal control problems for a class of stochastic partial differential equations (SPDEs) under partial observation. The system dynamics evolve in an infinite-dimensional Hilbert space and are driven by a cylindrical Wiener process, while observations are available in a finite-dimensional Euclidean space. We first derive a stochastic maximum principle (SMP) characterizing necessary conditions for optimality, with the associated adjoint processes formulated as backward SPDEs. We propose a numerical scheme based on the stochastic maximum principle that iteratively updates the control using stochastic gradient descent, combined with a particle filtering algorithm to estimate the conditional distribution of the system state. Numerical experiments illustrate the effectiveness of the proposed method.</p>

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Numerical Approximations for Partially Observed Optimal Control of Stochastic Partial Differential Equations

  • 期刊论文

  • 发表日期：

    2026年7月1日

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  • Feng Bao,
  • Yanzhao Cao,
  • Hongjiang Qian

摘要

In this paper, we investigate the numerical approximation of optimal control problems for a class of stochastic partial differential equations (SPDEs) under partial observation. The system dynamics evolve in an infinite-dimensional Hilbert space and are driven by a cylindrical Wiener process, while observations are available in a finite-dimensional Euclidean space. We first derive a stochastic maximum principle (SMP) characterizing necessary conditions for optimality, with the associated adjoint processes formulated as backward SPDEs. We propose a numerical scheme based on the stochastic maximum principle that iteratively updates the control using stochastic gradient descent, combined with a particle filtering algorithm to estimate the conditional distribution of the system state. Numerical experiments illustrate the effectiveness of the proposed method.