Volume 27, Issue 1
A New Finite Element Approximation of a State-Constrained Optimal Control Problem

WenbinLiu, Wei Gongming & Ningning Yan

DOI:

J. Comp. Math., 27 (2009), pp. 97-114.

Published online: 2009-02

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  • Abstract

In this paper, we study numerical methods for an optimal control problem with point-wise state constraints. The traditional approaches often need to deal with the delta-singularity in the dual equation, which causes many difficulties in its theoretical analysis and numerical approximation. In our new approach we reformulate the state-constrained optimal control as a constrained minimization problems only involving the state, whose optimality condition is characterized by a fourth order elliptic variational inequality. Then direct numerical algorithms (nonconforming finite element approximation) are proposed for the inequality, and error estimates of the finite element approximation are derived. Numerical experiments illustrate the effectiveness of the new approach.

  • Keywords

Optimal control problem State-constraints Fourth order variational inequalities Nonconforming finite element method

  • AMS Subject Headings

49J20 65N30.

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COPYRIGHT: © Global Science Press

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@Article{JCM-27-97, author = {}, title = {A New Finite Element Approximation of a State-Constrained Optimal Control Problem}, journal = {Journal of Computational Mathematics}, year = {2009}, volume = {27}, number = {1}, pages = {97--114}, abstract = { In this paper, we study numerical methods for an optimal control problem with point-wise state constraints. The traditional approaches often need to deal with the delta-singularity in the dual equation, which causes many difficulties in its theoretical analysis and numerical approximation. In our new approach we reformulate the state-constrained optimal control as a constrained minimization problems only involving the state, whose optimality condition is characterized by a fourth order elliptic variational inequality. Then direct numerical algorithms (nonconforming finite element approximation) are proposed for the inequality, and error estimates of the finite element approximation are derived. Numerical experiments illustrate the effectiveness of the new approach.}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8562.html} }
TY - JOUR T1 - A New Finite Element Approximation of a State-Constrained Optimal Control Problem JO - Journal of Computational Mathematics VL - 1 SP - 97 EP - 114 PY - 2009 DA - 2009/02 SN - 27 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8562.html KW - Optimal control problem KW - State-constraints KW - Fourth order variational inequalities KW - Nonconforming finite element method AB - In this paper, we study numerical methods for an optimal control problem with point-wise state constraints. The traditional approaches often need to deal with the delta-singularity in the dual equation, which causes many difficulties in its theoretical analysis and numerical approximation. In our new approach we reformulate the state-constrained optimal control as a constrained minimization problems only involving the state, whose optimality condition is characterized by a fourth order elliptic variational inequality. Then direct numerical algorithms (nonconforming finite element approximation) are proposed for the inequality, and error estimates of the finite element approximation are derived. Numerical experiments illustrate the effectiveness of the new approach.
WenbinLiu, Wei Gongming & Ningning Yan. (2019). A New Finite Element Approximation of a State-Constrained Optimal Control Problem. Journal of Computational Mathematics. 27 (1). 97-114. doi:
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