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Volume 9, Issue 4
Finite Element Approximation of Optimal Control for the Heat Equation with End-Point State Constraints

G. Wang & L. Wang

Int. J. Numer. Anal. Mod., 9 (2012), pp. 844-875.

Published online: 2012-09

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  • Abstract

This study presents a new finite element approximation for an optimal control problem ($P$) governed by the heat equation and with end-point state constraints. The state constraint set $S$ is assumed to have an empty interior in the state space. We begin with building a new penalty functional where the penalty parameter is an algebraic combination of the mesh size and the time step. Based on it, we establish a discrete optimal control problem ($P_{h\tau}$) without state constraints. With the help of Pontryagin’s maximum principle and by suitably choosing the above-mentioned combination, we successfully derive error estimate between optimal controls of problems ($P$) and ($P_{h\tau}$), in terms of the mesh size and time step.

  • Keywords

Error estimate, optimal control problem, the heat equation, end-point state constraint, discrete.

  • AMS Subject Headings

35K05, 49J20, 65M60

  • Copyright

COPYRIGHT: © Global Science Press

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  • BibTex
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  • TXT
@Article{IJNAM-9-844, author = {G. Wang and L. Wang}, title = {Finite Element Approximation of Optimal Control for the Heat Equation with End-Point State Constraints}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2012}, volume = {9}, number = {4}, pages = {844--875}, abstract = {

This study presents a new finite element approximation for an optimal control problem ($P$) governed by the heat equation and with end-point state constraints. The state constraint set $S$ is assumed to have an empty interior in the state space. We begin with building a new penalty functional where the penalty parameter is an algebraic combination of the mesh size and the time step. Based on it, we establish a discrete optimal control problem ($P_{h\tau}$) without state constraints. With the help of Pontryagin’s maximum principle and by suitably choosing the above-mentioned combination, we successfully derive error estimate between optimal controls of problems ($P$) and ($P_{h\tau}$), in terms of the mesh size and time step.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/662.html} }
TY - JOUR T1 - Finite Element Approximation of Optimal Control for the Heat Equation with End-Point State Constraints AU - G. Wang & L. Wang JO - International Journal of Numerical Analysis and Modeling VL - 4 SP - 844 EP - 875 PY - 2012 DA - 2012/09 SN - 9 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/662.html KW - Error estimate, optimal control problem, the heat equation, end-point state constraint, discrete. AB -

This study presents a new finite element approximation for an optimal control problem ($P$) governed by the heat equation and with end-point state constraints. The state constraint set $S$ is assumed to have an empty interior in the state space. We begin with building a new penalty functional where the penalty parameter is an algebraic combination of the mesh size and the time step. Based on it, we establish a discrete optimal control problem ($P_{h\tau}$) without state constraints. With the help of Pontryagin’s maximum principle and by suitably choosing the above-mentioned combination, we successfully derive error estimate between optimal controls of problems ($P$) and ($P_{h\tau}$), in terms of the mesh size and time step.

G. Wang and L. Wang. (2012). Finite Element Approximation of Optimal Control for the Heat Equation with End-Point State Constraints. International Journal of Numerical Analysis and Modeling. 9 (4). 844-875. doi:
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