Analysis of non scalar control problems for parabolic systems by the block moment method
This article deals with abstract linear time invariant controlled systems of parabolic type. In [9], with A. Benabdallah, we introduced the block moment method for scalar control operators. The principal aim of this method is to compute the minimal time needed to drive an initial condition (or a spa...
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Académie des sciences
2023-10-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.487/ |
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| author | Boyer, Franck Morancey, Morgan |
| author_facet | Boyer, Franck Morancey, Morgan |
| author_sort | Boyer, Franck |
| collection | DOAJ |
| description | This article deals with abstract linear time invariant controlled systems of parabolic type. In [9], with A. Benabdallah, we introduced the block moment method for scalar control operators. The principal aim of this method is to compute the minimal time needed to drive an initial condition (or a space of initial conditions) to zero, in particular in the case when spectral condensation occurs. The purpose of the present article is to push forward the analysis to deal with any admissible control operator. The considered setting leads to applications to one dimensional parabolic-type equations or coupled systems of such equations.With such admissible control operator, the characterization of the minimal null control time is obtained thanks to the resolution of an auxiliary vectorial block moment problem (i.e. set in the control space) followed by a constrained optimization procedure of the cost of this resolution. This leads to essentially sharp estimates on the resolution of the block moment problems which are uniform with respect to the spectrum of the evolution operator in a certain class. This uniformity allows the study of uniform controllability for various parameter dependent problems. We also deduce estimates on the cost of controllability when the final time goes to the minimal null control time.We illustrate how the method works on a few examples of such abstract controlled systems and then we deal with actual coupled systems of one dimensional parabolic partial differential equations. Our strategy enables us to tackle controllability issues that seem out of reach by existing techniques. |
| format | Article |
| id | doaj-art-b2006e59cbf74eb7b2235915820e813e |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2023-10-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-b2006e59cbf74eb7b2235915820e813e2025-02-07T11:10:23ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-10-01361G81191124810.5802/crmath.48710.5802/crmath.487Analysis of non scalar control problems for parabolic systems by the block moment methodBoyer, Franck0Morancey, Morgan1Institut de Mathématiques de Toulouse & Institut Universitaire de France, UMR 5219, Université de Toulouse, CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France.Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, FranceThis article deals with abstract linear time invariant controlled systems of parabolic type. In [9], with A. Benabdallah, we introduced the block moment method for scalar control operators. The principal aim of this method is to compute the minimal time needed to drive an initial condition (or a space of initial conditions) to zero, in particular in the case when spectral condensation occurs. The purpose of the present article is to push forward the analysis to deal with any admissible control operator. The considered setting leads to applications to one dimensional parabolic-type equations or coupled systems of such equations.With such admissible control operator, the characterization of the minimal null control time is obtained thanks to the resolution of an auxiliary vectorial block moment problem (i.e. set in the control space) followed by a constrained optimization procedure of the cost of this resolution. This leads to essentially sharp estimates on the resolution of the block moment problems which are uniform with respect to the spectrum of the evolution operator in a certain class. This uniformity allows the study of uniform controllability for various parameter dependent problems. We also deduce estimates on the cost of controllability when the final time goes to the minimal null control time.We illustrate how the method works on a few examples of such abstract controlled systems and then we deal with actual coupled systems of one dimensional parabolic partial differential equations. Our strategy enables us to tackle controllability issues that seem out of reach by existing techniques.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.487/ |
| spellingShingle | Boyer, Franck Morancey, Morgan Analysis of non scalar control problems for parabolic systems by the block moment method Comptes Rendus. Mathématique |
| title | Analysis of non scalar control problems for parabolic systems by the block moment method |
| title_full | Analysis of non scalar control problems for parabolic systems by the block moment method |
| title_fullStr | Analysis of non scalar control problems for parabolic systems by the block moment method |
| title_full_unstemmed | Analysis of non scalar control problems for parabolic systems by the block moment method |
| title_short | Analysis of non scalar control problems for parabolic systems by the block moment method |
| title_sort | analysis of non scalar control problems for parabolic systems by the block moment method |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.487/ |
| work_keys_str_mv | AT boyerfranck analysisofnonscalarcontrolproblemsforparabolicsystemsbytheblockmomentmethod AT moranceymorgan analysisofnonscalarcontrolproblemsforparabolicsystemsbytheblockmomentmethod |