Optimal error estimates of the stochastic parabolic optimal control problem with integral state constraint

TL;DR

This paper establishes optimal strong error estimates for stochastic parabolic optimal control problems with integral state constraints, using discretization and a gradient projection algorithm, supported by numerical validation.

Contribution

It introduces a novel approach to derive optimal error estimates for stochastic parabolic control problems with integral constraints, including a convergence-proof gradient algorithm.

Findings

Optimal strong error estimates are derived for control, state, and adjoint variables.

A gradient projection algorithm with proven convergence rate is proposed.

Numerical experiments confirm the theoretical error bounds.

Abstract

In this paper, the optimal strong error estimates for stochastic parabolic optimal control problem with additive noise and integral state constraint are derived based on time-implicit and finite element discretization. The continuous and discrete first-order optimality conditions are deduced by constructing the Lagrange functional, which contains forward-backward stochastic parabolic equations and a variational equation. The fully discrete version of forward-backward stochastic parabolic equations is introduced as an auxiliary problem and the optimal strong convergence orders are estimated, which further allows the optimal a priori error estimates for control, state, adjoint state and multiplier to be derived. Then, a simple and yet efficient gradient projection algorithm is proposed to solve stochastic parabolic control problem and its convergence rate is proved. Numerical experiments are carried out to illustrate the theoretical findings.