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

Wenbin Liu, Wei Gong & Ningning Yan

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 pointwise 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.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-27-97, author = {Liu , WenbinGong , Wei and Yan , Ningning}, 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 pointwise 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 AU - Liu , Wenbin AU - Gong , Wei AU - Yan , Ningning 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, State-constraints, Fourth order variational inequalities, Nonconforming finite element method. AB -

In this paper, we study numerical methods for an optimal control problem with pointwise 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.

Liu , WenbinGong , Wei and Yan , Ningning. (2009). 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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