Optimal Control of the Multiphase Stefan Problem
Abstract We consider the inverse multiphase Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free boundaries. Optimal control framework is pursued, where boundary heat flux is the control, and the optimality criteria...
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| Autor*in: |
Abdulla, Ugur G. [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2018 |
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| Schlagwörter: |
Inverse multiphase Stefan problem Parabolic free boundary problem |
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| Anmerkung: |
© Springer Science+Business Media, LLC, part of Springer Nature 2018 |
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| Übergeordnetes Werk: |
Enthalten in: Applied mathematics & optimization - Springer US, 1974, 80(2018), 2 vom: 03. Jan., Seite 479-513 |
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| Übergeordnetes Werk: |
volume:80 ; year:2018 ; number:2 ; day:03 ; month:01 ; pages:479-513 |
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| DOI / URN: |
10.1007/s00245-017-9472-7 |
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| 520 | |a Abstract We consider the inverse multiphase Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free boundaries. Optimal control framework is pursued, where boundary heat flux is the control, and the optimality criteria consist of the minimization of the $$L_2$$-norm declination of the trace of the solution to the Stefan problem from the temperature measurement on the fixed right boundary. The state vector solves multiphase Stefan problem in a weak formulation, which is equivalent to Neumann problem for the quasilinear parabolic PDE with discontinuous coefficient. Full discretization through finite differences is implemented and discrete optimal control problem is introduced. We prove well-posedness in a Sobolev space framework and convergence of discrete optimal control problems to the original problem both with respect to the cost functional and control. Along the way, the convergence of the method of finite differences for the weak solution of the multiphase Stefan problem is proved. The proof is based on achieving a uniform $$L_{\infty }$$ bound, and $$W_2^{1,1}$$-energy estimate for the discrete multiphase Stefan problem. | ||
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| 650 | 4 | |a Parabolic free boundary problem | |
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| 650 | 4 | |a Discrete optimal Stefan problem | |
| 650 | 4 | |a Convergence in functional and control | |
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10.1007/s00245-017-9472-7 doi (DE-627)OLC207264593X (DE-He213)s00245-017-9472-7-p DE-627 ger DE-627 rakwb eng 510 VZ Abdulla, Ugur G. verfasserin aut Optimal Control of the Multiphase Stefan Problem 2018 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract We consider the inverse multiphase Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free boundaries. Optimal control framework is pursued, where boundary heat flux is the control, and the optimality criteria consist of the minimization of the $$L_2$$-norm declination of the trace of the solution to the Stefan problem from the temperature measurement on the fixed right boundary. The state vector solves multiphase Stefan problem in a weak formulation, which is equivalent to Neumann problem for the quasilinear parabolic PDE with discontinuous coefficient. Full discretization through finite differences is implemented and discrete optimal control problem is introduced. We prove well-posedness in a Sobolev space framework and convergence of discrete optimal control problems to the original problem both with respect to the cost functional and control. Along the way, the convergence of the method of finite differences for the weak solution of the multiphase Stefan problem is proved. The proof is based on achieving a uniform $$L_{\infty }$$ bound, and $$W_2^{1,1}$$-energy estimate for the discrete multiphase Stefan problem. Inverse multiphase Stefan problem Optimal control Parabolic free boundary problem Method of finite differences Discrete optimal Stefan problem Convergence in functional and control Poggi, Bruno aut Enthalten in Applied mathematics & optimization Springer US, 1974 80(2018), 2 vom: 03. Jan., Seite 479-513 (DE-627)129095184 (DE-600)7418-4 (DE-576)014431300 0095-4616 nnns volume:80 year:2018 number:2 day:03 month:01 pages:479-513 https://doi.org/10.1007/s00245-017-9472-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 GBV_ILN_2018 GBV_ILN_4027 GBV_ILN_4323 AR 80 2018 2 03 01 479-513 |
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10.1007/s00245-017-9472-7 doi (DE-627)OLC207264593X (DE-He213)s00245-017-9472-7-p DE-627 ger DE-627 rakwb eng 510 VZ Abdulla, Ugur G. verfasserin aut Optimal Control of the Multiphase Stefan Problem 2018 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract We consider the inverse multiphase Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free boundaries. Optimal control framework is pursued, where boundary heat flux is the control, and the optimality criteria consist of the minimization of the $$L_2$$-norm declination of the trace of the solution to the Stefan problem from the temperature measurement on the fixed right boundary. The state vector solves multiphase Stefan problem in a weak formulation, which is equivalent to Neumann problem for the quasilinear parabolic PDE with discontinuous coefficient. Full discretization through finite differences is implemented and discrete optimal control problem is introduced. We prove well-posedness in a Sobolev space framework and convergence of discrete optimal control problems to the original problem both with respect to the cost functional and control. Along the way, the convergence of the method of finite differences for the weak solution of the multiphase Stefan problem is proved. The proof is based on achieving a uniform $$L_{\infty }$$ bound, and $$W_2^{1,1}$$-energy estimate for the discrete multiphase Stefan problem. Inverse multiphase Stefan problem Optimal control Parabolic free boundary problem Method of finite differences Discrete optimal Stefan problem Convergence in functional and control Poggi, Bruno aut Enthalten in Applied mathematics & optimization Springer US, 1974 80(2018), 2 vom: 03. Jan., Seite 479-513 (DE-627)129095184 (DE-600)7418-4 (DE-576)014431300 0095-4616 nnns volume:80 year:2018 number:2 day:03 month:01 pages:479-513 https://doi.org/10.1007/s00245-017-9472-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 GBV_ILN_2018 GBV_ILN_4027 GBV_ILN_4323 AR 80 2018 2 03 01 479-513 |
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10.1007/s00245-017-9472-7 doi (DE-627)OLC207264593X (DE-He213)s00245-017-9472-7-p DE-627 ger DE-627 rakwb eng 510 VZ Abdulla, Ugur G. verfasserin aut Optimal Control of the Multiphase Stefan Problem 2018 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract We consider the inverse multiphase Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free boundaries. Optimal control framework is pursued, where boundary heat flux is the control, and the optimality criteria consist of the minimization of the $$L_2$$-norm declination of the trace of the solution to the Stefan problem from the temperature measurement on the fixed right boundary. The state vector solves multiphase Stefan problem in a weak formulation, which is equivalent to Neumann problem for the quasilinear parabolic PDE with discontinuous coefficient. Full discretization through finite differences is implemented and discrete optimal control problem is introduced. We prove well-posedness in a Sobolev space framework and convergence of discrete optimal control problems to the original problem both with respect to the cost functional and control. Along the way, the convergence of the method of finite differences for the weak solution of the multiphase Stefan problem is proved. The proof is based on achieving a uniform $$L_{\infty }$$ bound, and $$W_2^{1,1}$$-energy estimate for the discrete multiphase Stefan problem. Inverse multiphase Stefan problem Optimal control Parabolic free boundary problem Method of finite differences Discrete optimal Stefan problem Convergence in functional and control Poggi, Bruno aut Enthalten in Applied mathematics & optimization Springer US, 1974 80(2018), 2 vom: 03. Jan., Seite 479-513 (DE-627)129095184 (DE-600)7418-4 (DE-576)014431300 0095-4616 nnns volume:80 year:2018 number:2 day:03 month:01 pages:479-513 https://doi.org/10.1007/s00245-017-9472-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 GBV_ILN_2018 GBV_ILN_4027 GBV_ILN_4323 AR 80 2018 2 03 01 479-513 |
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10.1007/s00245-017-9472-7 doi (DE-627)OLC207264593X (DE-He213)s00245-017-9472-7-p DE-627 ger DE-627 rakwb eng 510 VZ Abdulla, Ugur G. verfasserin aut Optimal Control of the Multiphase Stefan Problem 2018 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract We consider the inverse multiphase Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free boundaries. Optimal control framework is pursued, where boundary heat flux is the control, and the optimality criteria consist of the minimization of the $$L_2$$-norm declination of the trace of the solution to the Stefan problem from the temperature measurement on the fixed right boundary. The state vector solves multiphase Stefan problem in a weak formulation, which is equivalent to Neumann problem for the quasilinear parabolic PDE with discontinuous coefficient. Full discretization through finite differences is implemented and discrete optimal control problem is introduced. We prove well-posedness in a Sobolev space framework and convergence of discrete optimal control problems to the original problem both with respect to the cost functional and control. Along the way, the convergence of the method of finite differences for the weak solution of the multiphase Stefan problem is proved. The proof is based on achieving a uniform $$L_{\infty }$$ bound, and $$W_2^{1,1}$$-energy estimate for the discrete multiphase Stefan problem. Inverse multiphase Stefan problem Optimal control Parabolic free boundary problem Method of finite differences Discrete optimal Stefan problem Convergence in functional and control Poggi, Bruno aut Enthalten in Applied mathematics & optimization Springer US, 1974 80(2018), 2 vom: 03. Jan., Seite 479-513 (DE-627)129095184 (DE-600)7418-4 (DE-576)014431300 0095-4616 nnns volume:80 year:2018 number:2 day:03 month:01 pages:479-513 https://doi.org/10.1007/s00245-017-9472-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 GBV_ILN_2018 GBV_ILN_4027 GBV_ILN_4323 AR 80 2018 2 03 01 479-513 |
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10.1007/s00245-017-9472-7 doi (DE-627)OLC207264593X (DE-He213)s00245-017-9472-7-p DE-627 ger DE-627 rakwb eng 510 VZ Abdulla, Ugur G. verfasserin aut Optimal Control of the Multiphase Stefan Problem 2018 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract We consider the inverse multiphase Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free boundaries. Optimal control framework is pursued, where boundary heat flux is the control, and the optimality criteria consist of the minimization of the $$L_2$$-norm declination of the trace of the solution to the Stefan problem from the temperature measurement on the fixed right boundary. The state vector solves multiphase Stefan problem in a weak formulation, which is equivalent to Neumann problem for the quasilinear parabolic PDE with discontinuous coefficient. Full discretization through finite differences is implemented and discrete optimal control problem is introduced. We prove well-posedness in a Sobolev space framework and convergence of discrete optimal control problems to the original problem both with respect to the cost functional and control. Along the way, the convergence of the method of finite differences for the weak solution of the multiphase Stefan problem is proved. The proof is based on achieving a uniform $$L_{\infty }$$ bound, and $$W_2^{1,1}$$-energy estimate for the discrete multiphase Stefan problem. Inverse multiphase Stefan problem Optimal control Parabolic free boundary problem Method of finite differences Discrete optimal Stefan problem Convergence in functional and control Poggi, Bruno aut Enthalten in Applied mathematics & optimization Springer US, 1974 80(2018), 2 vom: 03. Jan., Seite 479-513 (DE-627)129095184 (DE-600)7418-4 (DE-576)014431300 0095-4616 nnns volume:80 year:2018 number:2 day:03 month:01 pages:479-513 https://doi.org/10.1007/s00245-017-9472-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 GBV_ILN_2018 GBV_ILN_4027 GBV_ILN_4323 AR 80 2018 2 03 01 479-513 |
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Optimal Control of the Multiphase Stefan Problem |
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Abstract We consider the inverse multiphase Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free boundaries. Optimal control framework is pursued, where boundary heat flux is the control, and the optimality criteria consist of the minimization of the $$L_2$$-norm declination of the trace of the solution to the Stefan problem from the temperature measurement on the fixed right boundary. The state vector solves multiphase Stefan problem in a weak formulation, which is equivalent to Neumann problem for the quasilinear parabolic PDE with discontinuous coefficient. Full discretization through finite differences is implemented and discrete optimal control problem is introduced. We prove well-posedness in a Sobolev space framework and convergence of discrete optimal control problems to the original problem both with respect to the cost functional and control. Along the way, the convergence of the method of finite differences for the weak solution of the multiphase Stefan problem is proved. The proof is based on achieving a uniform $$L_{\infty }$$ bound, and $$W_2^{1,1}$$-energy estimate for the discrete multiphase Stefan problem. © Springer Science+Business Media, LLC, part of Springer Nature 2018 |
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Abstract We consider the inverse multiphase Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free boundaries. Optimal control framework is pursued, where boundary heat flux is the control, and the optimality criteria consist of the minimization of the $$L_2$$-norm declination of the trace of the solution to the Stefan problem from the temperature measurement on the fixed right boundary. The state vector solves multiphase Stefan problem in a weak formulation, which is equivalent to Neumann problem for the quasilinear parabolic PDE with discontinuous coefficient. Full discretization through finite differences is implemented and discrete optimal control problem is introduced. We prove well-posedness in a Sobolev space framework and convergence of discrete optimal control problems to the original problem both with respect to the cost functional and control. Along the way, the convergence of the method of finite differences for the weak solution of the multiphase Stefan problem is proved. The proof is based on achieving a uniform $$L_{\infty }$$ bound, and $$W_2^{1,1}$$-energy estimate for the discrete multiphase Stefan problem. © Springer Science+Business Media, LLC, part of Springer Nature 2018 |
| abstract_unstemmed |
Abstract We consider the inverse multiphase Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free boundaries. Optimal control framework is pursued, where boundary heat flux is the control, and the optimality criteria consist of the minimization of the $$L_2$$-norm declination of the trace of the solution to the Stefan problem from the temperature measurement on the fixed right boundary. The state vector solves multiphase Stefan problem in a weak formulation, which is equivalent to Neumann problem for the quasilinear parabolic PDE with discontinuous coefficient. Full discretization through finite differences is implemented and discrete optimal control problem is introduced. We prove well-posedness in a Sobolev space framework and convergence of discrete optimal control problems to the original problem both with respect to the cost functional and control. Along the way, the convergence of the method of finite differences for the weak solution of the multiphase Stefan problem is proved. The proof is based on achieving a uniform $$L_{\infty }$$ bound, and $$W_2^{1,1}$$-energy estimate for the discrete multiphase Stefan problem. © Springer Science+Business Media, LLC, part of Springer Nature 2018 |
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Optimal Control of the Multiphase Stefan Problem |
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Poggi, Bruno |
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