Optimal Control of Stochastic Partial Differential Equations with Partial Observations: Stochastic Maximum Principles and Numerical Approximation

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Abstract. This work establishes a general stochastic maximum principle for systems of partially observed optimal control of semilinear stochastic partial differential equations in a nonconvex control domain. The state evolves in a Hilbert space driven by a cylindrical Wiener process and finitely many Brownian motions, while observations are in a Euclidean space having correlated noise. For the convex control domain and diffusion coefficients in the state being control-independent, numerical algorithms are developed to solve the partially observed optimal control problems using a stochastic gradient descent algorithm combined with finite element approximations and the branching filtering algorithm. Numerical experiments are conducted for demonstration.

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