scholarly journals Sparse Grid Approximation of the Riccati Operator for Closed Loop Parabolic Control Problems with Dirichlet Boundary Control

2021 ◽  
Vol 59 (6) ◽  
pp. 4538-4562
Author(s):  
Helmut Harbrecht ◽  
Ilja Kalmykov
Keyword(s):  
Boundary Control ◽  
Closed Loop ◽  
Dirichlet Boundary ◽  
Sparse Grid ◽  
Control Problems ◽  
Dirichlet Boundary Control ◽  
Grid Approximation

Energy corrected FEM for optimal Dirichlet boundary control problems

2018 ◽  
Vol 139 (4) ◽  
pp. 913-938 ◽  
Author(s):  
Lorenz John ◽  
Piotr Swierczynski ◽  
Barbara Wohlmuth
Keyword(s):  
Boundary Control ◽  
Dirichlet Boundary ◽  
Control Problems ◽  
Dirichlet Boundary Control ◽  
Boundary Control Problems

scholarly journals A convergent adaptive finite element method for elliptic Dirichlet boundary control problems

2018 ◽  
Vol 39 (4) ◽  
pp. 1985-2015 ◽  
Author(s):  
Wei Gong ◽  
Wenbin Liu ◽  
Zhiyu Tan ◽  
Ningning Yan
Keyword(s):  
Finite Element ◽  
Boundary Control ◽  
Dirichlet Boundary ◽  
Posteriori Error ◽  
Adaptive Finite Element Method ◽  
Control Problems ◽  
Adaptive Finite Element ◽  
Dirichlet Boundary Control ◽  
A Posteriori Error ◽  
Boundary Control Problems

Abstract This paper concerns the adaptive finite element method for elliptic Dirichlet boundary control problems in the energy space. The contribution of this paper is twofold. First, we rigorously derive efficient and reliable a posteriori error estimates for finite element approximations of Dirichlet boundary control problems. As a by-product, a priori error estimates are derived in a simple way by introducing appropriate auxiliary problems and establishing certain norm equivalence. Secondly, for the coupled elliptic partial differential system that resulted from the first-order optimality system, we prove that the sequence of adaptively generated discrete solutions including the control, the state and the adjoint state, guided by our newly derived a posteriori error indicators, converges to the true solution along with the convergence of the error estimators. We give some numerical results to confirm our theoretical findings.

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