Optimal trajectories in $L^1$ and under $L^1$ penalizations
Motivated by a MFG model where the trajectories of the agents are piecewise constants and agents pay for the number of jumps, we study a variational problem for curves of measures where the cost includes the length of the curve measures with the $L^1$ distance, as well as other, non-autonomous, cost...
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Académie des sciences
2024-07-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.583/ |
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| author | Dumas, Annette Santambrogio, Filippo |
| author_facet | Dumas, Annette Santambrogio, Filippo |
| author_sort | Dumas, Annette |
| collection | DOAJ |
| description | Motivated by a MFG model where the trajectories of the agents are piecewise constants and agents pay for the number of jumps, we study a variational problem for curves of measures where the cost includes the length of the curve measures with the $L^1$ distance, as well as other, non-autonomous, cost terms arising from congestion effects. We prove several regularity results (first in time, then in space) on the solution, based on suitable approximation and maximum principle techniques. We then use modern algorithms in non-smooth convex optimization in order to obtain a numerical method to simulate such solutions. |
| format | Article |
| id | doaj-art-1e908f264a594f56a5e61e32f825ff9c |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-07-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-1e908f264a594f56a5e61e32f825ff9c2025-02-07T11:21:51ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-07-01362G665769210.5802/crmath.58310.5802/crmath.583Optimal trajectories in $L^1$ and under $L^1$ penalizationsDumas, Annette0Santambrogio, Filippo1Institut Camille Jordan, Université Claude Bernard - Lyon 1; 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne cedex, FranceInstitut Camille Jordan, Université Claude Bernard - Lyon 1; 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne cedex, FranceMotivated by a MFG model where the trajectories of the agents are piecewise constants and agents pay for the number of jumps, we study a variational problem for curves of measures where the cost includes the length of the curve measures with the $L^1$ distance, as well as other, non-autonomous, cost terms arising from congestion effects. We prove several regularity results (first in time, then in space) on the solution, based on suitable approximation and maximum principle techniques. We then use modern algorithms in non-smooth convex optimization in order to obtain a numerical method to simulate such solutions.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.583/BV functionsnon-autonomous calculus of variationsregularitynon-smooth optimization |
| spellingShingle | Dumas, Annette Santambrogio, Filippo Optimal trajectories in $L^1$ and under $L^1$ penalizations Comptes Rendus. Mathématique BV functions non-autonomous calculus of variations regularity non-smooth optimization |
| title | Optimal trajectories in $L^1$ and under $L^1$ penalizations |
| title_full | Optimal trajectories in $L^1$ and under $L^1$ penalizations |
| title_fullStr | Optimal trajectories in $L^1$ and under $L^1$ penalizations |
| title_full_unstemmed | Optimal trajectories in $L^1$ and under $L^1$ penalizations |
| title_short | Optimal trajectories in $L^1$ and under $L^1$ penalizations |
| title_sort | optimal trajectories in l 1 and under l 1 penalizations |
| topic | BV functions non-autonomous calculus of variations regularity non-smooth optimization |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.583/ |
| work_keys_str_mv | AT dumasannette optimaltrajectoriesinl1andunderl1penalizations AT santambrogiofilippo optimaltrajectoriesinl1andunderl1penalizations |