Geodesic
On the structure of locally Lipschitz minimax solutions of the Hamilton–Jacobi–Bellman equation in terms of classical characteristics
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 15 (2009) no. 3, pp. 202-218
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Necessary and sufficient conditions for the minimax solution to the Cauchy problem for the Hamilton–Jacobi–Bellman equation are obtained as viability conditions for classical characteristics inside the graph of the minimax solution. Using this property, a representative formula for a one-dimensional conservation law in terms of classical characteristics is derived. An estimate of the numerical integration of the characteristic system is presented and errors of numerical realizations of representative formulas are determined for the conservation law and its potential equal to the minimax solution of the Hamilton–Jacobi–Bellman equation.
Keywords: Hamilton–Jacobi–Bellman equations, minimax/viscosity solutions, conservation laws, entropy solutions, method of characteristics. 
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N. N. Subbotina; E. A. Kolpakova. On the structure of locally Lipschitz minimax solutions of the Hamilton–Jacobi–Bellman equation in terms of classical characteristics. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 15 (2009) no. 3, pp. 202-218. http://geodesic.mathdoc.fr/item/TIMM_2009_15_3_a14/

[1] Subbotin A. I., Obobschennye resheniya uravnenii v chastnykh proizvodnykh pervogo poryadka. Perspektivy dinamicheskoi optimizatsii, Institut kompyuternykh issledovanii, M.–Izhevsk, 2003, 336 pp.

[2] Kurant R., Uravneniya s chastnymi proizvodnymi, Mir, M., 1964, 832 pp. | MR

[3] Bellman R., Dinamicheskoe programmirovanie, IL, M., 1960, 400 pp. | MR

[4] Pontryagin L. S., Boltyanskii V. G., Gamkrelidze R. V., Mischenko E. F., Matematicheskaya teoriya optimalnykh protsessov, Nauka, M., 1961, 392 pp. | MR | Zbl

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

[6] Krasovskii N. N., Subbotin A. I., Pozitsionnye differentsialnye igry, Nauka, M., 1974, 456 pp. | MR | Zbl

[7] Oleinik O. A., “O zadache Koshi dlya nelineinykh uravnenii v klasse razryvnykh funktsii”, Dokl. AN SSSR, 95:3 (1954), 451–454 | MR

[8] Tikhonov A. N., Samarskii A. A., “O razryvnykh resheniyakh kvazilineinogo uravneniya pervogo poryadka”, Dokl. AN SSSR, 99:1 (1954), 27–30 | Zbl

[9] Kruzhkov S. N., “Kvazilineinye uravneniya pervogo poryadka so mnogimi nezavisimymi peremennymi”, Mat. sb., 81(123):2 (1970), 228–255 | MR | Zbl

[10] Goritskii A. Yu., Kruzhkov S. N., Chechkin G. A., Kvazilineinye uravneniya s chastnymi proizvodnymi pervogo poryadka: obobschennye resheniya, udarnye volny, tsentrirovannye volny razrezheniya, Izd-vo MGU, M., 1994, 96 pp.

[11] Clarke F. N., Optimization and nonsmooth analysis, Wiley, New York, 1983, 293 pp. | MR | Zbl

[12] Subbotina N. N., “The method of characteristics for Hamilton–Jacobi equation and its applications in dynamical optimization”, Modern Mathematics and its Applications, 20 (2004), 2955–3091 | MR

[13] Kolpakova E. A., “Otsenki chislennoi approksimatsii optimalnogo rezultata dlya odnogo klassa singulyarno vozmuschennykh zadach optimalnogo upravleniya”, Problemy teoreticheskoi i prikladnoi matematiki, Tr. 39-i Vseros. molodezh. konf., 2008, 265–269

[14] Crandall M. G., Evans L. C., Lions P.-L., “Some properties of viscosity solutions of Hamilton–Jacobi Equations”, Trans. Amer. Math. Soc., 282 (1984), 487–502 | DOI | MR | Zbl

[15] Rockafellar R. T., Wets R. J.-B., Variational Analysis, Springer, New York, 1998, 734 pp. | MR | Zbl

[16] Varga Dzh., Optimalnoe upravlenie differentsialnymi i integralnymi uravneniyami, Nauka, M., 1977, 624 pp. | MR

[17] Sobolev S. L., Uravneniya matematicheskoi fiziki, Nauka, M., 1966, 444 pp. | MR