Bilinear Optimal Control for a Nonlinear Parabolic Equation Involving Nonlocal-in-Time Term
We study a bilinear optimal control problem for an evolution equation with a nonlinear term that depends on both the state and its time integral. First, we establish existence and uniqueness results for this evolution equation. Then, we derive weak maximum principle results to improve the regularity...
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MDPI AG
2025-01-01
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| Online Access: | https://www.mdpi.com/2075-1680/14/1/38 |
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| author | Gisèle Mophou Arnaud Fournier Célia Jean-Alexis |
| author_facet | Gisèle Mophou Arnaud Fournier Célia Jean-Alexis |
| author_sort | Gisèle Mophou |
| collection | DOAJ |
| description | We study a bilinear optimal control problem for an evolution equation with a nonlinear term that depends on both the state and its time integral. First, we establish existence and uniqueness results for this evolution equation. Then, we derive weak maximum principle results to improve the regularity of the state equation. We proceed by formulating an optimal control problem aimed at steering the system’s state to a desired final state. Finally, we demonstrate that this optimal control problem admits a solution and derive the first- and second-order optimality conditions. |
| format | Article |
| id | doaj-art-c01fc9f51ad24452a20eaf40104129e6 |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-c01fc9f51ad24452a20eaf40104129e62025-01-24T13:22:13ZengMDPI AGAxioms2075-16802025-01-011413810.3390/axioms14010038Bilinear Optimal Control for a Nonlinear Parabolic Equation Involving Nonlocal-in-Time TermGisèle Mophou0Arnaud Fournier1Célia Jean-Alexis2Laboratoire L.A.M.I.A., Département de Mathématiques et Informatique, Université des Antilles, Campus Fouillole, 97159 Pointe-à-Pitre, FranceLaboratoire L.A.M.I.A., Département de Mathématiques et Informatique, Université des Antilles, Campus Fouillole, 97159 Pointe-à-Pitre, FranceLaboratoire L.A.M.I.A., Département de Mathématiques et Informatique, Université des Antilles, Campus Fouillole, 97159 Pointe-à-Pitre, FranceWe study a bilinear optimal control problem for an evolution equation with a nonlinear term that depends on both the state and its time integral. First, we establish existence and uniqueness results for this evolution equation. Then, we derive weak maximum principle results to improve the regularity of the state equation. We proceed by formulating an optimal control problem aimed at steering the system’s state to a desired final state. Finally, we demonstrate that this optimal control problem admits a solution and derive the first- and second-order optimality conditions.https://www.mdpi.com/2075-1680/14/1/38differentiabilitymaximum principlenonlocal-in-timeoptimal control problemparabolic partial differential equationSchauder’s fixed-point theorem |
| spellingShingle | Gisèle Mophou Arnaud Fournier Célia Jean-Alexis Bilinear Optimal Control for a Nonlinear Parabolic Equation Involving Nonlocal-in-Time Term Axioms differentiability maximum principle nonlocal-in-time optimal control problem parabolic partial differential equation Schauder’s fixed-point theorem |
| title | Bilinear Optimal Control for a Nonlinear Parabolic Equation Involving Nonlocal-in-Time Term |
| title_full | Bilinear Optimal Control for a Nonlinear Parabolic Equation Involving Nonlocal-in-Time Term |
| title_fullStr | Bilinear Optimal Control for a Nonlinear Parabolic Equation Involving Nonlocal-in-Time Term |
| title_full_unstemmed | Bilinear Optimal Control for a Nonlinear Parabolic Equation Involving Nonlocal-in-Time Term |
| title_short | Bilinear Optimal Control for a Nonlinear Parabolic Equation Involving Nonlocal-in-Time Term |
| title_sort | bilinear optimal control for a nonlinear parabolic equation involving nonlocal in time term |
| topic | differentiability maximum principle nonlocal-in-time optimal control problem parabolic partial differential equation Schauder’s fixed-point theorem |
| url | https://www.mdpi.com/2075-1680/14/1/38 |
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