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 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 Quasi-optimality of an Adaptive Finite Element Method for an Optimal Control Problem

2011 ◽  
Vol 11 (2) ◽  
pp. 107-128 ◽  
Author(s):  
Roland Becker ◽  
Shipeng Mao
Keyword(s):  
Optimal Control ◽  
Finite Element ◽  
Posteriori Error ◽  
Adaptive Finite Element Method ◽  
Error Estimator ◽  
Adaptive Finite Element ◽  
Posteriori Error Estimator ◽  
A Posteriori Error ◽  
Data Approximation ◽  
Element Algorithm

Abstract We prove quasi-optimality of an adaptive finite element algorithm for a model problem of optimal control including control constraints. The quasi-optimility expresses the fact that the decrease of error with respect to the number of mesh cells is optimal up to a constant. The considered algorithm is based on an adaptive marking strategy which compares a standard residualtype a posteriori error estimator with a data approximation term in each step of the algorithm in order to adapt the marking of cells.

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