Comptes Rendus
no. 19-20
Partial differential equations/Optimal control
Semicontinuous viscosity solutions for quasiconvex Hamiltonians
[Solutions de viscosité semicontinues des hamiltoniens quasi-convexes]

Présenté par : the Editorial Board

Emmanuel N. Barron1
1 Department of Mathematics and Statistics, Loyola University Chicago, Chicago, IL 60660, USA
Comptes Rendus. Mathématique, Volume 351 (2013) no. 19-20, pp. 737-741.

Le théorème principal reliant les hamiltoniens convexes et les solutions de viscosité semicontinues, due à Barron et Jensen, est étendu aux hamiltoniens quasi-convexes. Quelques applications sont indiquées.

The main theorem connecting convex Hamiltonians and semicontinuous viscosity solutions due to Barron and Jensen is extended to quasiconvex Hamiltonians. Some applications are indicated.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2013.09.021
Emmanuel N. Barron 1

1 Department of Mathematics and Statistics, Loyola University Chicago, Chicago, IL 60660, USA 
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Emmanuel N. Barron. Semicontinuous viscosity solutions for quasiconvex Hamiltonians. Comptes Rendus. Mathématique, Volume 351 (2013) no. 19-20, pp. 737-741. doi : 10.1016/j.crma.2013.09.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.09.021/

[1] O. Alvarez; E. Barron; H. Ishii Hopf–Lax formulas for semicontinuous data, Indiana Univ. Math. J., Volume 48 (1999), pp. 993-1035

[2] M. Bardi; I. Capuzzo-Dolcetta Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations, Birkhäuser Boston, Inc., Boston, MA, 1997

[3] G. Barles Solutions de viscosité des équations de Hamilton–Jacobi, Mathématiques et Applications, vol. 17, Springer, Paris, 1994

[4] E.N. Barron; R. Jensen Semicontinuous viscosity solutions of Hamilton–Jacobi equations with convex Hamiltonians, Comm. Partial Differential Equations, Volume 15 (1990) no. 12, pp. 1713-1740

[5] E.N. Barron; W. Liu Calculus of variations in L∞, Appl. Math. Optim., Volume 35 (1997), pp. 237-263

[6] E.N. Barron; W. Liu Semicontinuous solutions for Hamilton–Jacobi equations and the L∞ control problem, Appl. Math. Optim., Volume 34 (1996), pp. 325-360

[7] E.N. Barron; R. Jensen; W. Liu Hopf–Lax formula for ut+H(u,Du)=0: II, Comm. Partial Differential Equations, Volume 22 (1997), pp. 1141-1160

[8] H. Frankowska Lower semicontinuous solutions of Hamilton–Jacobi–Bellman equations, SIAM J. Control Optim., Volume 31 (1993) no. 1, pp. 257-272

[9] H. Ishii A generalization of a theorem of Barron and Jensen and a comparison theorem for lower semicontinuous viscosity solutions, Proc. R. Soc. Edinb. A, Volume 131 (2001) no. 1, pp. 137-154

[10] P. Soravia Discontinuous viscosity solutions to Dirichlet problems for Hamilton–Jacobi equations with convex Hamiltonians, Comm. Partial Differential Equations, Volume 18 (1993), pp. 1493-1514

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