Volume 16, Issue 5
A Relaxation Approach to Discretization of Boundary Optimal Control Problems of Semilinear Parabolic Equations

B. Kokkinis

DOI:

Int. J. Numer. Anal. Mod., 16 (2019), pp. 731-744.

Published online: 2019-08

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  • Abstract

We consider an optimal boundary control problem described by a semilinear parabolic partial differential equation, with control and state constraints. Since this problem may have no classical solutions, it is reformulated in the relaxed form. The relaxed control problem is discretized by using a finite element method in space and a partially implicit scheme in time, while the controls are approximated by piecewise constant relaxed controls. We first state the necessary conditions for optimality for the continuous problem and the discrete relaxed problem. Next, under appropriate assumptions, we prove that accumulation points of sequences of optimal (resp. admissible and extremal) discrete relaxed controls are optimal (resp. admissible and extremal) for the continuous relaxed problem.

  • Keywords

Boundary optimal control, semilinear parabolic systems, state constraints, relaxed controls, discretization.

  • AMS Subject Headings

49K20, 49M25, 65K10, 65M60

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

bkok@math.ntua.gr (B. Kokkinis)

  • BibTex
  • RIS
  • TXT
@Article{IJNAM-16-731, author = {Kokkinis , B. }, title = {A Relaxation Approach to Discretization of Boundary Optimal Control Problems of Semilinear Parabolic Equations}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2019}, volume = {16}, number = {5}, pages = {731--744}, abstract = {

We consider an optimal boundary control problem described by a semilinear parabolic partial differential equation, with control and state constraints. Since this problem may have no classical solutions, it is reformulated in the relaxed form. The relaxed control problem is discretized by using a finite element method in space and a partially implicit scheme in time, while the controls are approximated by piecewise constant relaxed controls. We first state the necessary conditions for optimality for the continuous problem and the discrete relaxed problem. Next, under appropriate assumptions, we prove that accumulation points of sequences of optimal (resp. admissible and extremal) discrete relaxed controls are optimal (resp. admissible and extremal) for the continuous relaxed problem.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/13251.html} }
TY - JOUR T1 - A Relaxation Approach to Discretization of Boundary Optimal Control Problems of Semilinear Parabolic Equations AU - Kokkinis , B. JO - International Journal of Numerical Analysis and Modeling VL - 5 SP - 731 EP - 744 PY - 2019 DA - 2019/08 SN - 16 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/13251.html KW - Boundary optimal control, semilinear parabolic systems, state constraints, relaxed controls, discretization. AB -

We consider an optimal boundary control problem described by a semilinear parabolic partial differential equation, with control and state constraints. Since this problem may have no classical solutions, it is reformulated in the relaxed form. The relaxed control problem is discretized by using a finite element method in space and a partially implicit scheme in time, while the controls are approximated by piecewise constant relaxed controls. We first state the necessary conditions for optimality for the continuous problem and the discrete relaxed problem. Next, under appropriate assumptions, we prove that accumulation points of sequences of optimal (resp. admissible and extremal) discrete relaxed controls are optimal (resp. admissible and extremal) for the continuous relaxed problem.

B. Kokkinis. (2019). A Relaxation Approach to Discretization of Boundary Optimal Control Problems of Semilinear Parabolic Equations. International Journal of Numerical Analysis and Modeling. 16 (5). 731-744. doi:
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