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

Qiming Wang, Wanfang Shen, Wenbin Liu

TL;DR

The paper addresses stochastic parabolic optimal control problems with an integral state constraint under additive noise. It develops a time-implicit discretization and piecewise linear finite element space discretization, deriving a Lagrangian-based optimality system that couples forward and backward SPDEs with a constraint via a Lagrange multiplier. By introducing space-discrete FBSPDEs and a fully discrete auxiliary problem, the authors establish sharp a priori error estimates for the state, adjoint state, multiplier, and control, showing convergence orders of $O(h^4+\tau^2)$ for the primary variables and $O(h^2+\tau^2)$ for gradient-like terms. An efficient gradient projection algorithm is proposed to solve the discrete problem while preserving the integral constraint, with proven convergence under suitable step sizes. Numerical experiments in one and two dimensions validate the theoretical findings and demonstrate robustness with respect to the constraint parameter $\delta$.

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.

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

TL;DR

The paper addresses stochastic parabolic optimal control problems with an integral state constraint under additive noise. It develops a time-implicit discretization and piecewise linear finite element space discretization, deriving a Lagrangian-based optimality system that couples forward and backward SPDEs with a constraint via a Lagrange multiplier. By introducing space-discrete FBSPDEs and a fully discrete auxiliary problem, the authors establish sharp a priori error estimates for the state, adjoint state, multiplier, and control, showing convergence orders of for the primary variables and for gradient-like terms. An efficient gradient projection algorithm is proposed to solve the discrete problem while preserving the integral constraint, with proven convergence under suitable step sizes. Numerical experiments in one and two dimensions validate the theoretical findings and demonstrate robustness with respect to the constraint parameter .

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.

Paper Structure

This paper contains 8 sections, 13 theorems, 173 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Optimality condition system and space semi-discretization
  3. Space-time discretization and optimal error estimates
  4. Space-time discretization and error estimates of FBSPDEs
  5. Optimal error estimates of control problem
  6. A gradient projection algorithm
  7. Numerical experiments
  8. Conclusion

Key Result

Theorem 1

Suppose that the Slater's condition holds, i.e., there exists a $\tilde{U}(t)\in L^2_{\mathbb{F}}(\Omega;0,T;\mathcal{D})$ such that $G(\tilde{U}(t))<0$. Let $(X(t),U(t))$ be the solution of stochastic optimal control problem (reduced_object)-(reduced_domain), then there exists a real number $\mu\ge

Figures (4)

  • Figure 1: The profiles of numerical control (Left), expected numerical state (Middle) with ${\tau}={h}=\frac{1}{40}$ and convergence rates (Right) with ${\tau}={h}=\frac{1}{40},\frac{1}{45},\frac{1}{50},\frac{1}{60},\frac{1}{70}$ of Example \ref{['exm1']}.
  • Figure 2: The convergence rates with ${h}=\frac{1}{10},\frac{1}{15},\frac{1}{20},\frac{1}{25},\frac{1}{30}$, ${\tau}={h^2}$ (Left, Center-Left) and ${\tau}={h^4}$ (Center-Right, Right) of Example \ref{['exm1']}.
  • Figure 3: The profiles of expected numerical states with ${\tau}={h}=\frac{1}{40}$ and $\delta=0.2,0.1,-0.1,-0.2$ of Example \ref{['exm1']}.
  • Figure 4: The convergence rates with ${h}=\sqrt{2}\left(\frac{1}{40},\frac{1}{45},\frac{1}{50},\frac{1}{60},\frac{1}{70}\right)$, ${\tau}=\frac{h}{\sqrt{2}}$ (Left) and ${h}=\sqrt{2}\left(\frac{1}{10},\frac{1}{15},\frac{1}{20},\frac{1}{25},\frac{1}{30}\right)$, ${\tau}=\frac{h^2}{2}$ (Middle, Right) of Example \ref{['exm2']}.

Theorems & Definitions (27)

  • Theorem 1
  • proof
  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • proof
  • ...and 17 more