A Priori Error Estimate of Stochastic Galerkin Method for Optimal Control Problem Governed by Stochastic Elliptic PDE with Constrained Control

2015 ◽  
Vol 67 (2) ◽  
pp. 405-431 ◽  
Author(s):  
Tongjun Sun ◽  
Wanfang Shen ◽  
Benxue Gong ◽  
Wenbin Liu
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Error Estimate ◽  
Control Problem ◽  
Galerkin Method ◽  
A Priori ◽  
Constrained Control ◽  
Elliptic Pde ◽  
A Priori Error Estimate ◽  
Stochastic Galerkin

scholarly journals A Priori Error Estimate of Stochastic Galerkin Method for Optimal Control Problem Governed by Random Parabolic PDE with Constrained Control

2016 ◽  
Vol 13 (05) ◽  
pp. 1650028 ◽  
Author(s):  
Benxue Gong ◽  
Tongjun Sun ◽  
Wanfang Shen ◽  
Wenbin Liu
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Galerkin Method ◽  
A Priori ◽  
Galerkin Approximation ◽  
Sufficient Optimality Conditions ◽  
Backward Euler ◽  
Parabolic Pde ◽  
Stochastic Galerkin

A stochastic Galerkin approximation scheme is proposed for an optimal control problem governed by a parabolic PDE with random perturbation in its coefficients. The objective functional is to minimize the expectation of a cost functional, and the deterministic control is of the obstacle constrained type. We obtain the necessary and sufficient optimality conditions and establish a scheme to approximate the optimality system through the discretization with respect to both the spatial space and the probability space by Galerkin method and with respect to time by the backward Euler scheme. A priori error estimates are derived for the state, the co-state and the control variables. Numerical examples are presented to illustrate our theoretical results.

scholarly journals Spectral Galerkin Approximation of Space Fractional Optimal Control Problem with Integral State Constraint

2021 ◽  
Vol 5 (3) ◽  
pp. 102
Author(s):  
Fangyuan Wang ◽  
Xiaodi Li ◽  
Zhaojie Zhou
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Jacobi Polynomials ◽  
A Priori ◽  
Galerkin Approximation ◽  
State Constraint ◽  
Diffusion Reaction ◽  
Lagrangian Multiplier ◽  
Adjoint State

In this paper spectral Galerkin approximation of optimal control problem governed by fractional advection diffusion reaction equation with integral state constraint is investigated. First order optimal condition of the control problem is discussed. Weighted Jacobi polynomials are used to approximate the state and adjoint state. A priori error estimates for control, state, adjoint state and Lagrangian multiplier are derived. Numerical experiment is carried out to illustrate the theoretical findings.

scholarly journals Error estimates of finite volume method for Stokes optimal control problem

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Lin Lan ◽  
Ri-hui Chen ◽  
Xiao-dong Wang ◽  
Chen-xia Ma ◽  
Hao-nan Fu
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Error Estimates ◽  
Finite Volume ◽  
Stokes Equations ◽  
A Priori ◽  
The State ◽  
Element Approximation ◽  
Norm Error

AbstractIn this paper, we discuss a priori error estimates for the finite volume element approximation of optimal control problem governed by Stokes equations. Under some reasonable assumptions, we obtain optimal $L^{2}$ L 2 -norm error estimates. The approximate orders for the state, costate, and control variables are $O(h^{2})$ O ( h 2 ) in the sense of $L^{2}$ L 2 -norm. Furthermore, we derive $H^{1}$ H 1 -norm error estimates for the state and costate variables. Finally, we give some conclusions and future works.

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