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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.
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.
@article{ASENS_2017__50_2_357_0,
author = {Imbert, Cyril and Monneau, R\'egis},
title = {Flux-limited solutions for quasi-convex {Hamilton-Jacobi} equations on networks},
journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
pages = {357--448},
publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
volume = {Ser. 4, 50},
number = {2},
year = {2017},
doi = {10.24033/asens.2323},
mrnumber = {3621434},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.24033/asens.2323/}
}
TY - JOUR AU - Imbert, Cyril AU - Monneau, Régis TI - Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks JO - Annales scientifiques de l'École Normale Supérieure PY - 2017 SP - 357 EP - 448 VL - 50 IS - 2 PB - Société Mathématique de France. Tous droits réservés UR - http://geodesic.mathdoc.fr/articles/10.24033/asens.2323/ DO - 10.24033/asens.2323 LA - en ID - ASENS_2017__50_2_357_0 ER -
%0 Journal Article %A Imbert, Cyril %A Monneau, Régis %T Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks %J Annales scientifiques de l'École Normale Supérieure %D 2017 %P 357-448 %V 50 %N 2 %I Société Mathématique de France. Tous droits réservés %U http://geodesic.mathdoc.fr/articles/10.24033/asens.2323/ %R 10.24033/asens.2323 %G en %F ASENS_2017__50_2_357_0
Imbert, Cyril; Monneau, Régis. Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 2, pp. 357-448. doi : 10.24033/asens.2323. http://geodesic.mathdoc.fr/articles/10.24033/asens.2323/
Hopf-Lax formulas for semicontinuous data, Indiana Univ. Math. J., Volume 48 (1999), pp. 993-1035 (ISSN: 0022-2518) | MR | Zbl | DOI
Vanishing capillarity solutions of Buckley-Leverett equation with gravity in two-rocks' medium, Comput. Geosci., Volume 17 (2013), pp. 551-572 (ISSN: 1420-0597) | DOI | MR
Hamilton-Jacobi equations constrained on networks, NoDEA Nonlinear Differential Equations Appl., Volume 20 (2013), pp. 413-445 (ISSN: 1021-9722) | MR | Zbl | DOI
On a class of first order Hamilton-Jacobi equations in metric spaces, J. Differential Equations, Volume 256 (2014), pp. 2194-2245 (ISSN: 0022-0396) | MR | Zbl | DOI
A theory of -dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Ration. Mech. Anal., Volume 201 (2011), pp. 27-86 (ISSN: 0003-9527) | MR | Zbl | DOI
Hamilton-Jacobi equations for optimal control on junctions and networks, ESAIM Control Optim. Calc. Var., Volume 21 (2015), pp. 876-899 (ISSN: 1292-8119) | MR | mathdoc-id | DOI
Hamilton-Jacobi equations on networks as limits of singularly perturbed problems in optimal control: dimension reduction, Comm. Partial Differential Equations, Volume 40 (2015), pp. 652-693 (ISSN: 0360-5302) | DOI | MR
A Bellman approach for two-domains optimal control problems in , ESAIM Control Optim. Calc. Var., Volume 19 (2013), pp. 710-739 (ISSN: 1292-8119) | MR | Zbl | mathdoc-id | DOI
A Bellman approach for regional optimal control problems in , SIAM J. Control Optim., Volume 52 (2014), pp. 1712-1744 (ISSN: 0363-0129) | MR | Zbl | DOI
(Almost) everything you always wanted to know about deterministic control problems in stratified domains, Netw. Heterog. Media, Volume 10 (2015), pp. 809-836 (ISSN: 1556-1801) | DOI | MR
On Hopf's formulas for solutions of Hamilton-Jacobi equations, Nonlinear Anal., Volume 8 (1984), pp. 1373-1381 (ISSN: 0362-546X) | MR | Zbl | DOI
Optimal control problems on stratified domains, Netw. Heterog. Media, Volume 2 (2007), pp. 313-331 (ISSN: 1556-1801) | MR | Zbl | DOI
An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections, SIAM J. Numer. Anal., Volume 47 (2009), pp. 1684-1712 (ISSN: 0036-1429) | MR | Zbl | DOI
Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom., Volume 33 (1991), pp. 749-786 http://projecteuclid.org/euclid.jdg/1214446564 (ISSN: 0022-040X) | MR | Zbl
A comparison among various notions of viscosity solution for Hamilton-Jacobi equations on networks, J. Math. Anal. Appl., Volume 407 (2013), pp. 112-118 (ISSN: 0022-247X) | MR | Zbl | DOI
Viscosity solutions of Hamilton-Jacobi equations with discontinuous coefficients, J. Hyperbolic Differ. Equ., Volume 4 (2007), pp. 771-795 (ISSN: 0219-8916) | MR | Zbl | DOI
Uniqueness and error analysis for Hamilton-Jacobi equations with discontinuities, Interfaces Free Bound., Volume 6 (2004), pp. 329-349 (ISSN: 1463-9963) | MR | Zbl | DOI
, Applied stochastic analysis (New Brunswick, NJ, 1991) (Lecture Notes in Control and Inform. Sci.), Volume 177, Springer, Berlin, 1992, pp. 90-107 | DOI | MR
Cauchy problems for noncoercive Hamilton-Jacobi-Isaacs equations with discontinuous coefficients, Interfaces Free Bound., Volume 12 (2010), pp. 347-368 (ISSN: 1463-9963) | MR | Zbl | DOI
The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. Roy. Soc. Edinburgh Sect. A, Volume 111 (1989), pp. 359-375 (ISSN: 0308-2105) | MR | Zbl | DOI
, Calculus of variations and optimal control (Haifa, 1998) (Chapman & Hall/CRC Res. Notes Math.), Volume 411, Chapman & Hall/CRC, Boca Raton, FL, 2000, pp. 97-116 | MR | Zbl
Semicontinuous solutions of Hamilton-Jacobi-Bellman equations with degenerate state constraints, J. Math. Anal. Appl., Volume 251 (2000), pp. 818-838 (ISSN: 0022-247X) | MR | Zbl | DOI
Hamilton-Jacobi equations with discontinuous source terms, Comm. Partial Differential Equations, Volume 38 (2013), pp. 199-243 (ISSN: 0360-5302) | MR | Zbl | DOI
Eikonal equations in metric spaces, Trans. Amer. Math. Soc., Volume 367 (2015), pp. 49-66 (ISSN: 0002-9947) | MR | Zbl | DOI
A junction condition by specified homogenization and application to traffic lights, Anal. PDE, Volume 8 (2015), pp. 1891-1929 (ISSN: 2157-5045) | DOI | MR
Conservation laws with discontinuous flux, Netw. Heterog. Media, Volume 2 (2007), pp. 159-179 (ISSN: 1556-1801) | MR | Zbl | DOI
Representation formulas for solutions of the HJI equations with discontinuous coefficients and existence of value in differential games, J. Optim. Theory Appl., Volume 130 (2006), pp. 209-229 (ISSN: 0022-3239) | MR | Zbl | DOI
Metric viscosity solutions of Hamilton-Jacobi equations depending on local slopes, Calc. Var. Partial Differential Equations, Volume 54 (2015), pp. 1183-1218 (ISSN: 0944-2669) | DOI | MR
A new formulation of state constraint problems for first-order PDEs, SIAM J. Control Optim., Volume 34 (1996), pp. 554-571 (ISSN: 0363-0129) | MR | Zbl | DOI
Quasi-convex Hamilton-Jacobi equations posed on junctions: the multi-dimensional case (preprint arXiv:1410.3056 ) | MR
A Hamilton-Jacobi approach to junction problems and application to traffic flows, ESAIM Control Optim. Calc. Var., Volume 19 (2013), pp. 129-166 (ISSN: 1292-8119) | MR | Zbl | mathdoc-id | DOI
Generalized junction conditions for degenerate parabolic equations (preprint arXiv:1601.01862 ) | MR
Perron's method for Hamilton-Jacobi equations, Duke Math. J., Volume 55 (1987), pp. 369-384 (ISSN: 0012-7094) | MR | Zbl | DOI
Lectures at Collège de France (2013–2014)
Differential games, optimal control and directional derivatives of viscosity solutions of Bellman's and Isaacs' equations, SIAM J. Control Optim., Volume 23 (1985), pp. 566-583 | MR | Zbl | DOI
Metric viscosity solutions for Hamilton-Jacobi equations of evolution type, Adv. Math. Sci. Appl., Volume 24 (2014), pp. 333-351 (ISBN: 978-4-7625-0668-0, ISSN: 1343-4373) | MR
Existence theorems for general control problems of Bolza and Lagrange, Advances in Math., Volume 15 (1975), pp. 312-333 (ISSN: 0001-8708) | Zbl | MR | DOI
Transmission conditions on interfaces for Hamilton-Jacobi-Bellman equations, J. Differential Equations, Volume 257 (2014), pp. 3978-4014 (ISSN: 0022-0396) | MR | Zbl | DOI
Hamilton-Jacobi-Bellman equations on multi-domains, Control and Optimization with PDE constrains (Bredies, K. et al., eds.) (International Series of Numerical Mathematics), Volume 164 (2013), pp. 93-116 | MR | Zbl | DOI
Viscosity solutions of Eikonal equations on topological networks, Calc. Var. Partial Differential Equations, Volume 46 (2013), pp. 671-686 (ISSN: 0944-2669) | MR | Zbl | DOI
Viscosity Solutions of Hamilton-Jacobi Equations of Eikonal Type on Ramified Spaces (2006)
Optimality principles and representation formulas for viscosity solutions of Hamilton-Jacobi equations. I. Equations of unbounded and degenerate control problems without uniqueness, Adv. Differential Equations, Volume 4 (1999), pp. 275-296 (ISSN: 1079-9389) | MR | Zbl | DOI
Optimality principles and representation formulas for viscosity solutions of Hamilton-Jacobi equations. II. Equations of control problems with state constraints, Differential Integral Equations, Volume 12 (1999), pp. 275-293 (ISSN: 0893-4983) | MR | Zbl | DOI
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