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

Barron, Emmanuel N.1
1 Department of Mathematics and Statistics, Loyola University Chicago, Chicago, IL 60660, USA
Comptes Rendus. Mathématique, Tome 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
Barron, Emmanuel N. 1

1 Department of Mathematics and Statistics, Loyola University Chicago, Chicago, IL 60660, USA 
@article{CRMATH_2013__351_19-20_737_0,
     author = {Barron, Emmanuel N.},
     title = {Semicontinuous viscosity solutions for quasiconvex {Hamiltonians}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {737--741},
     publisher = {Elsevier},
     volume = {351},
     number = {19-20},
     year = {2013},
     doi = {10.1016/j.crma.2013.09.021},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2013.09.021/}
}
TY  - JOUR
AU  - Barron, Emmanuel N.
TI  - Semicontinuous viscosity solutions for quasiconvex Hamiltonians
JO  - Comptes Rendus. Mathématique
PY  - 2013
SP  - 737
EP  - 741
VL  - 351
IS  - 19-20
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2013.09.021/
DO  - 10.1016/j.crma.2013.09.021
LA  - en
ID  - CRMATH_2013__351_19-20_737_0
ER  - 
%0 Journal Article
%A Barron, Emmanuel N.
%T Semicontinuous viscosity solutions for quasiconvex Hamiltonians
%J Comptes Rendus. Mathématique
%D 2013
%P 737-741
%V 351
%N 19-20
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2013.09.021/
%R 10.1016/j.crma.2013.09.021
%G en
%F CRMATH_2013__351_19-20_737_0
Barron, Emmanuel N. Semicontinuous viscosity solutions for quasiconvex Hamiltonians. Comptes Rendus. Mathématique, Tome 351 (2013) no. 19-20, pp. 737-741. doi : 10.1016/j.crma.2013.09.021. http://www.numdam.org/articles/10.1016/j.crma.2013.09.021/

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

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

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

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

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

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

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

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

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

Cité par Sources :