Geodesic
On the structure of the singular set of a piecewise smooth minimax solution to the Hamilton-Jacobi-Bellman equation
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 2, pp. 198-205 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

The properties of a minimax piecewise smooth solution of the Hamilton-Jacobi-Bellman equation are studied. It is known the Rankine-Hugoniot conditions are necessary and sufficient conditions for the points of nondifferentiability (singularity) of the minimax solution. We generalize this condition and describe the dimension of smooth manifolds contained in the singular set of the piecewise smooth solution in terms of state characteristics that come to this set. New structural properties of the singular set are obtained in the case when the Hamiltonian depends on the impulse variable only.
Keywords: Hamilton-Jacobi-Bellman equation, minimax solution, singular set, piecewise smooth solution, Rankine-Hugoniot conditions.
Mots-clés : tangent space 
@article{TIMM_2015_21_2_a16,
     author = {A. S. Rodin},
     title = {On the structure of the singular set of a piecewise smooth minimax solution to the {Hamilton-Jacobi-Bellman} equation},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {198--205},
     year = {2015},
     volume = {21},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2015_21_2_a16/}
}
TY  - JOUR
AU  - A. S. Rodin
TI  - On the structure of the singular set of a piecewise smooth minimax solution to the Hamilton-Jacobi-Bellman equation
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2015
SP  - 198
EP  - 205
VL  - 21
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TIMM_2015_21_2_a16/
LA  - ru
ID  - TIMM_2015_21_2_a16
ER  - 
%0 Journal Article
%A A. S. Rodin
%T On the structure of the singular set of a piecewise smooth minimax solution to the Hamilton-Jacobi-Bellman equation
%J Trudy Instituta matematiki i mehaniki
%D 2015
%P 198-205
%V 21
%N 2
%U http://geodesic.mathdoc.fr/item/TIMM_2015_21_2_a16/
%G ru
%F TIMM_2015_21_2_a16
A. S. Rodin. On the structure of the singular set of a piecewise smooth minimax solution to the Hamilton-Jacobi-Bellman equation. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 2, pp. 198-205. http://geodesic.mathdoc.fr/item/TIMM_2015_21_2_a16/

[1] Bellman R.E., Dinamicheskoe programmirovanie, Izd-vo inostr. lit., Moskva, 1960, 400 pp. | MR

[2] L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, E.F. Mischenko, Matematicheskaya teoriya optimalnykh protsessov, Fizmatgiz, M., 1961, 384 pp. | MR

[3] Krasovskii N.N., Teoriya upravleniya dvizheniem, Nauka, M., 1968, 476 pp. | MR

[4] N.N. Subbotina, E.A. Kolpakova, T.B. Tokmantsev, L.G. Shagalova, Metod kharakteristik dlya uravneniya Gamiltona-Yakobi-Bellmana, Izd-vo UrO RAN, Ekaterinburg, 2013, 244 pp.

[5] Kolpakova E.A., “Obobschennyi metod kharakteristik v teorii uravnenii Gamiltona-Yakobi i zakonov sokhraneniya”, Tr. In-ta matematiki i mekhaniki UrO RAN, 16:5 (2010), 95–98 | MR

[6] Petrovskii I.G., Lektsii po teorii obyknovennykh differentsialnykh uravnenii, Izd-vo Mosk. un-ta, Moskva, 1984, 296 pp. | MR

[7] Rokafellar R., Vypuklyi analiz, Mir, M., 1973, 470 pp.

[8] Subbotin A.I., Obobschennye resheniya uravneniya v chastnykh proizvodnykh pervogo poryadka, In-t kompyuternykh issledovanii, M.; Izhevsk, 2003, 336 pp.

[9] Crandall G., Lions P.L., “Viscosity solutions of Hamilton-Jacobi equations”, Trans. Amer. Math. Soc., 277:1 (1983), 1–42 | DOI | MR | Zbl

[10] Subbotina N.N., Kolpakova E.A., “O strukture lokalno lipshitsevykh minimaksnykh reshenii uravneniya Gamiltona-Yakobi-Bellmana v terminakh klassicheskikh kharakteristik”, Tr. In-ta matematiki i mekhaniki UrO RAN, 15:3 (2009), 202–218