Optimal trajectories in $L^1$ and under $L^1$ penalizations

Annette Dumas; Filippo Santambrogio

doi:10.5802/crmath.583

Article de recherche - Équations aux dérivées partielles, Théorie du contrôle

Optimal trajectories in

L^{1}

and under

L^{1}

penalizations
[Trajectoires optimales dans

L^{1}

et sous pénalisation de type

L^{1}

]

Annette Dumas ¹ ; Filippo Santambrogio ¹

¹ Institut Camille Jordan, Université Claude Bernard - Lyon 1; 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne cedex, France

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 657-692.

Résumé
Abstract

Motivé par un modèle de MFG dans lequel les trajectoires des agents sont constantes par morceaux et où les agents paient un coût dépendant du nombre de sauts, nous étudions un problème variationnel pour des courbes de mesures où le coût inclut la longueur de la courbe mesurée en termes de la distance $L^{1}$ , ainsi que d’autres termes non autonomes représentants des effets de congestion. Nous démontrons plusieurs résultats de régularité (d’abord en temps, puis en espace) sur la solution, en utilisant des techniques d’approximation et le principe du maximum. Ensuite, des algorithmes modernes d’optimisation convexe non lisse nous permettent d’obtenir une méthode numérique pour simuler de telles solutions.

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.

Reçu le : 2023-05-25
Révisé le : 2023-10-13
Accepté le : 2023-10-19
Publié le : 2024-07-09

DOI : 10.5802/crmath.583

Keywords: BV functions, non-autonomous calculus of variations, regularity, non-smooth optimization
Mot clés : fonctions BV, calcul des varations non-autonome, régularité, optimisation non lisse

Affiliations des auteurs :

Annette Dumas ¹ ; Filippo Santambrogio ¹

¹ Institut Camille Jordan, Université Claude Bernard - Lyon 1; 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne cedex, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Annette Dumas and Filippo Santambrogio},
     title = {Optimal trajectories in $L^1$ and under $L^1$ penalizations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {657--692},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.583},
     language = {en},
}

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Annette Dumas; Filippo Santambrogio. Optimal trajectories in $L^1$ and under $L^1$ penalizations. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 657-692. doi : 10.5802/crmath.583. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.583/

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