On Binary Optimal Control in $H^s(0,T)$, $s < 1/2$
The function space $H^s(0,T)$, $s < 1/2$, allows for functions with jump discontinuities and is thus attractive for treating optimal control problems with discrete-valued control functions. We show that while arbitrary chattering controls are impossible, there exist feasible controls in $H^s(0,T)...
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| Format: | Article |
| Language: | English |
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Académie des sciences
2023-11-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.507/ |
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| _version_ | 1825206242094612480 |
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| author | Manns, Paul Surowiec, Thomas M. |
| author_facet | Manns, Paul Surowiec, Thomas M. |
| author_sort | Manns, Paul |
| collection | DOAJ |
| description | The function space $H^s(0,T)$, $s < 1/2$, allows for functions with jump discontinuities and is thus attractive for treating optimal control problems with discrete-valued control functions. We show that while arbitrary chattering controls are impossible, there exist feasible controls in $H^s(0,T)$ that have countably jump discontinuities with jump height one in each of countably many pairwise disjoint intervals. However, under mild assumptions, we show that certain types of jump discontinuities cannot be optimal. The derivation of meaningful optimality conditions via a direct variational argument using simple feasible perturbations remains a major challenge; as illustrated by an example. |
| format | Article |
| id | doaj-art-80e202ee847942acb3c7aa1a1c2a76de |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2023-11-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-80e202ee847942acb3c7aa1a1c2a76de2025-02-07T11:10:50ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-11-01361G91531154010.5802/crmath.50710.5802/crmath.507On Binary Optimal Control in $H^s(0,T)$, $s < 1/2$Manns, Paul0https://orcid.org/0000-0003-0654-6613Surowiec, Thomas M.1https://orcid.org/0000-0003-2473-4984TU Dortmund University, GermanySimula Research Laboratory, Oslo, NorwayThe function space $H^s(0,T)$, $s < 1/2$, allows for functions with jump discontinuities and is thus attractive for treating optimal control problems with discrete-valued control functions. We show that while arbitrary chattering controls are impossible, there exist feasible controls in $H^s(0,T)$ that have countably jump discontinuities with jump height one in each of countably many pairwise disjoint intervals. However, under mild assumptions, we show that certain types of jump discontinuities cannot be optimal. The derivation of meaningful optimality conditions via a direct variational argument using simple feasible perturbations remains a major challenge; as illustrated by an example.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.507/ |
| spellingShingle | Manns, Paul Surowiec, Thomas M. On Binary Optimal Control in $H^s(0,T)$, $s < 1/2$ Comptes Rendus. Mathématique |
| title | On Binary Optimal Control in $H^s(0,T)$, $s < 1/2$ |
| title_full | On Binary Optimal Control in $H^s(0,T)$, $s < 1/2$ |
| title_fullStr | On Binary Optimal Control in $H^s(0,T)$, $s < 1/2$ |
| title_full_unstemmed | On Binary Optimal Control in $H^s(0,T)$, $s < 1/2$ |
| title_short | On Binary Optimal Control in $H^s(0,T)$, $s < 1/2$ |
| title_sort | on binary optimal control in h s 0 t s 1 2 |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.507/ |
| work_keys_str_mv | AT mannspaul onbinaryoptimalcontrolinhs0ts12 AT surowiecthomasm onbinaryoptimalcontrolinhs0ts12 |