Bilinear control of evolution equations with unbounded lower order terms. Application to the Fokker–Planck equation
We study the exact controllability of the evolution equation \begin{equation*} u^{\prime }(t)+Au(t)+p(t)Bu(t)=0 \end{equation*} where $A$ is a nonnegative self-adjoint operator on a Hilbert space $X$ and $B$ is an unbounded linear operator on $X$, which is dominated by the square root of $A$. The...
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Académie des sciences
2024-05-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.567/ |
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| author | Alabau-Boussouira, Fatiha Cannarsa, Piermarco Urbani, Cristina |
| author_facet | Alabau-Boussouira, Fatiha Cannarsa, Piermarco Urbani, Cristina |
| author_sort | Alabau-Boussouira, Fatiha |
| collection | DOAJ |
| description | We study the exact controllability of the evolution equation
\begin{equation*}
u^{\prime }(t)+Au(t)+p(t)Bu(t)=0
\end{equation*}
where $A$ is a nonnegative self-adjoint operator on a Hilbert space $X$ and $B$ is an unbounded linear operator on $X$, which is dominated by the square root of $A$. The control action is bilinear and only of scalar-input form, meaning that the control is the scalar function $p$, which is assumed to depend only on time. Furthermore, we only consider square-integrable controls. Our main result is the local exact controllability of the above equation to the ground state solution, that is, the evolution through time, of the first eigenfunction of $A$, as initial data.The analogous problem (in a more general form) was addressed in our previous paper [Exact controllability to eigensolutions for evolution equations of parabolic type via bilinear control, Alabau-Boussouira F., Cannarsa P. and Urbani C., Nonlinear Diff. Eq. Appl. (2022)] for a bounded operator $B$. The current extension to unbounded operators allows for many more applications, including the Fokker–Planck equation in one space dimension, and a larger class of control actions. |
| format | Article |
| id | doaj-art-316eda7c76ec4c54b24d06dcdb26c867 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-05-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-316eda7c76ec4c54b24d06dcdb26c8672025-02-07T11:21:12ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-05-01362G551154510.5802/crmath.56710.5802/crmath.567Bilinear control of evolution equations with unbounded lower order terms. Application to the Fokker–Planck equationAlabau-Boussouira, Fatiha0Cannarsa, Piermarco1Urbani, Cristina2https://orcid.org/0000-0003-3355-7879Laboratoire Jacques-Louis Lions, Sorbonne Université, 75005, Paris, Université de Lorraine, FranceDipartimento di Matematica, Università di Roma Tor Vergata, 00133, Roma, ItalyDipartimento di Scienze Tecnologiche e dell’Innovazione, Universitas Mercatorum, Piazza Mattei 10, 00186, Roma, ItalyWe study the exact controllability of the evolution equation \begin{equation*} u^{\prime }(t)+Au(t)+p(t)Bu(t)=0 \end{equation*} where $A$ is a nonnegative self-adjoint operator on a Hilbert space $X$ and $B$ is an unbounded linear operator on $X$, which is dominated by the square root of $A$. The control action is bilinear and only of scalar-input form, meaning that the control is the scalar function $p$, which is assumed to depend only on time. Furthermore, we only consider square-integrable controls. Our main result is the local exact controllability of the above equation to the ground state solution, that is, the evolution through time, of the first eigenfunction of $A$, as initial data.The analogous problem (in a more general form) was addressed in our previous paper [Exact controllability to eigensolutions for evolution equations of parabolic type via bilinear control, Alabau-Boussouira F., Cannarsa P. and Urbani C., Nonlinear Diff. Eq. Appl. (2022)] for a bounded operator $B$. The current extension to unbounded operators allows for many more applications, including the Fokker–Planck equation in one space dimension, and a larger class of control actions.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.567/ |
| spellingShingle | Alabau-Boussouira, Fatiha Cannarsa, Piermarco Urbani, Cristina Bilinear control of evolution equations with unbounded lower order terms. Application to the Fokker–Planck equation Comptes Rendus. Mathématique |
| title | Bilinear control of evolution equations with unbounded lower order terms. Application to the Fokker–Planck equation |
| title_full | Bilinear control of evolution equations with unbounded lower order terms. Application to the Fokker–Planck equation |
| title_fullStr | Bilinear control of evolution equations with unbounded lower order terms. Application to the Fokker–Planck equation |
| title_full_unstemmed | Bilinear control of evolution equations with unbounded lower order terms. Application to the Fokker–Planck equation |
| title_short | Bilinear control of evolution equations with unbounded lower order terms. Application to the Fokker–Planck equation |
| title_sort | bilinear control of evolution equations with unbounded lower order terms application to the fokker planck equation |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.567/ |
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