Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks

Imbert, Cyril; Monneau, Régis

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 2, pp. 357-448.

Nous étudions des équations de Hamilton-Jacobi posées sur des réseaux dans le cas d'Hamiltoniens quasi-convexes en la variable gradient et qui peuvent être discontinus en la variable d'espace au niveau des sommets. Nous prouvons d'une part qu'imposer une condition de jonction générale est équivalent à en imposer une de type contrôle optimal, qui ne dépend que des Hamiltoniens et d'un paramètre libre additionnel, le limiteur de flux. Nous introduisons d'autre part une méthode générale pour montrer des principes de comparaison. Cette méthode repose sur la construction d'une fonction sommet destinée à remplacer dans la méthode de dédoublement des variables la fonction quadratique habituelle. Nous présentons ensuite un large éventail d'applications, et notamment un résultat d'existence et d'unicité très général pour les équations de Hamilton-Jacobi quasi-convexes posées sur les réseaux.

We study Hamilton-Jacobi equations on networks in the case where Hamiltonians are quasi-convex with respect to the gradient variable and can be discontinuous with respect to the space variable at vertices. First, we prove that imposing a general vertex condition is equivalent to imposing a specific one which only depends on Hamiltonians and an additional free parameter, the flux limiter. Second, a general method for proving comparison principles is introduced. This method consists in constructing a vertex test function to be used in the doubling variable approach. With such a theory and such a method in hand, we present various applications, among which a very general existence and uniqueness result for quasi-convex Hamilton-Jacobi equations on networks.

Classification : 35F21, 49L25, 35B51.
Keywords: Hamilton-Jacobi equations, networks, quasi-convex Hamiltonians, discontinuous Hamiltonians, viscosity solutions, flux-limited solutions, comparison principle, vertex test function, homogenization, optimal control, discontinuous running cost.
Mot clés : Équations de Hamilton-Jacobi, réseaux, Hamiltoniens quasi-convexes, Hamiltoniens discontinus, solutions de viscosité, solutions à flux limité, principe de comparaison, fonction sommet, homogénéisation, contrôle optimal, coût instantané discontinu.

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