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Error estimates for the FEM approximation of optimal sparse control of elliptic equations with pointwise state constraints and finite‐dimensional control space
Author(s) -
Merino Pedro,
Nenjer Alexander
Publication year - 2020
Publication title -
optimal control applications and methods
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.458
H-Index - 44
eISSN - 1099-1514
pISSN - 0143-2087
DOI - 10.1002/oca.2608
Subject(s) - pointwise , optimal control , finite element method , a priori and a posteriori , mathematics , space (punctuation) , state space , approximation error , domain (mathematical analysis) , mathematical analysis , mathematical optimization , computer science , physics , statistics , operating system , philosophy , epistemology , thermodynamics
Summary In this work, we derive an a priori error estimate of orderh 2 | log ( h ) | for the finite element approximation of a sparse optimal control problem governed by an elliptic equation, which is controlled in a finite dimensional space. Furthermore, box‐constrains on the control are considered and finitely many pointwise state‐constrains are imposed on specific points in the domain. With this choice for the control space, the achieved order of approximation for the optimal control is optimal, in the sense that the order of the error for the optimal control is of the same order of the approximation for the state equation.