Numerical Solution of a Parabolic Optimal Control Problem with Point-Wise State Constraints
The problem of optimal control over the system governed by the Dirichlet boundary value problem for a linear parabolic equation is constructed. Point-wise constraints are imposed on both control and state functions. The right-hand side of the equation is a control function in the problem. The object...
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| Format: | Article |
| Language: | English |
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Kazan Federal University
2016-03-01
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| Series: | Учёные записки Казанского университета. Серия Физико-математические науки |
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| Online Access: | http://kpfu.ru/portal/docs/F1561527295/158_1_phys_mat_6.pdf |
| _version_ | 1797968123060551680 |
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| author | A.V. Lapin A.A. Platonov |
| author_facet | A.V. Lapin A.A. Platonov |
| author_sort | A.V. Lapin |
| collection | DOAJ |
| description | The problem of optimal control over the system governed by the Dirichlet boundary value problem for a linear parabolic equation is constructed. Point-wise constraints are imposed on both control and state functions. The right-hand side of the equation is a control function in the problem. The objective functional contains an observation which is distributed in the space-time domain. Finite-difference approximation is constructed for the optimal control problem based on the Euler forward scheme for the state parabolic equation. The existence of its unique solution is proved. Constrained saddle point problem corresponding to the mesh optimal control problem is constructed. The existence of a solution for this saddle point problem and the converg ence of the generalized Uzawa iterative method are proved. The results of numerical experiments are given. |
| first_indexed | 2024-04-11T02:41:58Z |
| format | Article |
| id | doaj.art-27d655a99a004fb6b8f78fdfda3b4db7 |
| institution | Directory Open Access Journal |
| issn | 2541-7746 2500-2198 |
| language | English |
| last_indexed | 2024-04-11T02:41:58Z |
| publishDate | 2016-03-01 |
| publisher | Kazan Federal University |
| record_format | Article |
| series | Учёные записки Казанского университета. Серия Физико-математические науки |
| spelling | doaj.art-27d655a99a004fb6b8f78fdfda3b4db72023-01-02T18:49:10ZengKazan Federal UniversityУчёные записки Казанского университета. Серия Физико-математические науки2541-77462500-21982016-03-0115818189Numerical Solution of a Parabolic Optimal Control Problem with Point-Wise State ConstraintsA.V. Lapin0A.A. Platonov1Kazan Federal University, Kazan, 420008 RussiaKazan Federal University, Kazan, 420008 RussiaThe problem of optimal control over the system governed by the Dirichlet boundary value problem for a linear parabolic equation is constructed. Point-wise constraints are imposed on both control and state functions. The right-hand side of the equation is a control function in the problem. The objective functional contains an observation which is distributed in the space-time domain. Finite-difference approximation is constructed for the optimal control problem based on the Euler forward scheme for the state parabolic equation. The existence of its unique solution is proved. Constrained saddle point problem corresponding to the mesh optimal control problem is constructed. The existence of a solution for this saddle point problem and the converg ence of the generalized Uzawa iterative method are proved. The results of numerical experiments are given.http://kpfu.ru/portal/docs/F1561527295/158_1_phys_mat_6.pdfoptimal controlparabolic state equationstate constraintsfinite-difference approximationiterative method |
| spellingShingle | A.V. Lapin A.A. Platonov Numerical Solution of a Parabolic Optimal Control Problem with Point-Wise State Constraints Учёные записки Казанского университета. Серия Физико-математические науки optimal control parabolic state equation state constraints finite-difference approximation iterative method |
| title | Numerical Solution of a Parabolic Optimal Control Problem with Point-Wise State Constraints |
| title_full | Numerical Solution of a Parabolic Optimal Control Problem with Point-Wise State Constraints |
| title_fullStr | Numerical Solution of a Parabolic Optimal Control Problem with Point-Wise State Constraints |
| title_full_unstemmed | Numerical Solution of a Parabolic Optimal Control Problem with Point-Wise State Constraints |
| title_short | Numerical Solution of a Parabolic Optimal Control Problem with Point-Wise State Constraints |
| title_sort | numerical solution of a parabolic optimal control problem with point wise state constraints |
| topic | optimal control parabolic state equation state constraints finite-difference approximation iterative method |
| url | http://kpfu.ru/portal/docs/F1561527295/158_1_phys_mat_6.pdf |
| work_keys_str_mv | AT avlapin numericalsolutionofaparabolicoptimalcontrolproblemwithpointwisestateconstraints AT aaplatonov numericalsolutionofaparabolicoptimalcontrolproblemwithpointwisestateconstraints |