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>

scholarly journals Galerkin spectral method for elliptic optimal control problem with $L^2$-norm control constraint

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Zhen-Zhen Tao ◽  
Bing Sun
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Error Estimates ◽  
A Posteriori Error Estimates ◽  
Control Variable ◽  
Approximation Problem ◽  
Posteriori Error ◽  
Spectral Approximation ◽  
Norm Error

<p style='text-indent:20px;'>This paper is concerned with the Galerkin spectral approximation of an optimal control problem governed by the elliptic partial differential equations (PDEs). Its objective functional depends on the control variable governed by the <inline-formula><tex-math id="M1">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm constraint. The optimality conditions for both the optimal control problem and its corresponding spectral approximation problem are given, successively. Thanks to some lemmas and the auxiliary systems, a priori error estimates of the Galerkin spectral approximation problem are established in detail. Moreover, a posteriori error estimates of the spectral approximation problem are also investigated, which include not only <inline-formula><tex-math id="M2">\begin{document}$ H^1 $\end{document}</tex-math></inline-formula>-norm error for the state and co-state but also <inline-formula><tex-math id="M3">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm error for the control, state and costate. Finally, three numerical examples are executed to demonstrate the errors decay exponentially fast.</p>

scholarly journals Adaptive Semidiscrete Finite Element Methods for Semilinear Parabolic Integrodifferential Optimal Control Problem with Control Constraint

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Zuliang Lu
Keyword(s):  
Optimal Control ◽  
Finite Element ◽  
Optimal Control Problem ◽  
Control Problem ◽  
A Posteriori Error Estimates ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Adaptive Finite Element ◽  
Element Discretization ◽  
Semilinear Parabolic

The aim of this work is to study the semidiscrete finite element discretization for a class of semilinear parabolic integrodifferential optimal control problems. We derive a posteriori error estimates inL2(J;L2(Ω))-norm andL2(J;H1(Ω))-norm for both the control and coupled state approximations. Such estimates can be used to construct reliable adaptive finite element approximation for semilinear parabolic integrodifferential optimal control problem. Furthermore, we introduce an adaptive algorithm to guide the mesh refinement. Finally, a numerical example is given to demonstrate the theoretical results.

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.

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.

Morley FEM for a Distributed Optimal Control Problem Governed by the von Kármán Equations

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Sudipto Chowdhury ◽  
Neela Nataraj ◽  
Devika Shylaja
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Error Estimates ◽  
Lower Order ◽  
Control Variable ◽  
The State ◽  
Distributed Optimal Control ◽  
Von Kármán Equations ◽  
Adjoint Variables

AbstractConsider the distributed optimal control problem governed by the von Kármán equations defined on a polygonal domain of {\mathbb{R}^{2}} that describe the deflection of very thin plates with box constraints on the control variable. This article discusses a numerical approximation of the problem that employs the Morley nonconforming finite element method (FEM) to discretize the state and adjoint variables. The control is discretized using piecewise constants. A priori error estimates are derived for the state, adjoint and control variables under minimal regularity assumptions on the exact solution. Error estimates in lower-order norms for the state and adjoint variables are derived. The lower-order estimates for the adjoint variable and a post-processing of control leads to an improved error estimate for the control variable. Numerical results confirm the theoretical results obtained.

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