scholarly journals A posteriori error estimates of spectral method for nonlinear parabolic optimal control problem

2018 ◽  
Vol 2018 (1) ◽  
Author(s):  
Lin Li ◽  
Zuliang Lu ◽  
Wei Zhang ◽  
Fei Huang ◽  
Yin Yang
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Spectral Method ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Nonlinear Parabolic ◽  
Parabolic Optimal Control

scholarly journals Adaptive Finite Element Method for Optimal Control Problem Governed by Linear Quasiparabolic Integrodifferential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-26 ◽  
Author(s):  
Wanfang Shen ◽  
Hua Su
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
A Posteriori Error Estimates ◽  
Mathematical Formulation ◽  
Posteriori Error ◽  
Adaptive Finite Element Method ◽  
Adaptive Finite Element ◽  
A Posteriori Error ◽  
Quadratic Optimal Control

The mathematical formulation for a quadratic optimal control problem governed by a linear quasiparabolic integrodifferential equation is studied. The control constrains are given in an integral sense:Uad={u∈X;∫ΩUu⩾0,t∈[0,T]}. Then the a posteriori error estimates inL∞(0,T;H1(Ω))-norm andL2(0,T;L2(Ω))-norm for both the state and the control approximation are given.

Spectral Method Approximation of Flow Optimal Control Problems withH1-Norm State Constraint

2017 ◽  
Vol 10 (3) ◽  
pp. 614-638 ◽  
Author(s):  
Yanping Chen ◽  
Fenglin Huang
Keyword(s):  
Optimal Control ◽  
Control Problem ◽  
Spectral Method ◽  
Stokes Equations ◽  
Sparse Matrices ◽  
A Priori ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
State Constraint ◽  
A Posteriori Error

AbstractIn this paper, we consider an optimal control problem governed by Stokes equations withH1-norm state constraint. The control problem is approximated by spectral method, which provides very accurate approximation with a relatively small number of unknowns. Choosing appropriate basis functions leads to discrete system with sparse matrices. We first present the optimality conditions of the exact and the discrete optimal control systems, then derive both a priori and a posteriori error estimates. Finally, an illustrative numerical experiment indicates that the proposed method is competitive, and the estimator can indicate the errors very well.

scholarly journals A posteriori error estimates of hp spectral element method for parabolic optimal control problems

2022 ◽  
Vol 7 (4) ◽  
pp. 5220-5240
Author(s):  
Zuliang Lu ◽  
◽  
Fei Cai ◽  
Ruixiang Xu ◽  
Chunjuan Hou ◽  
...  
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Error Estimates ◽  
A Posteriori Error Estimates ◽  
Control Variable ◽  
Posteriori Error ◽  
Spectral Element ◽  
Element Approximation ◽  
Parabolic Optimal Control

<abstract><p>In this paper, we investigate the spectral element approximation for the optimal control problem of parabolic equation, and present a hp spectral element approximation scheme for the parabolic optimal control problem. For improve the accuracy of the algorithm and construct an adaptive finite element approximation. Under the Scott-Zhang type quasi-interpolation operator, a $ L^2(H^1)-L^2(L^2) $ posteriori error estimates of the hp spectral element approximated solutions for both the state variables and the control variable are obtained. Adopting two auxiliary equations and stability results, a $ L^2(L^2)-L^2(L^2) $ posteriori error estimates are derived for the hp spectral element approximation of optimal parabolic control problem.</p></abstract>

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