Construction of a continuous minimax/viscosity solution of the Hamilton–Jacobi–Bellman equation with nonextendable characteristics

N. N. Subbotina ; L. G. Shagalova

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 20 (2014) no. 4, pp. 247-257

Résumé

The Cauchy problem for the Hamilton–Jacobi equation, which appears in molecular biology for the Crow–Kimura model of molecular evolution, is considered. The state characteristics of the equation that start in a given initial manifold bounded in the state space stay in a strip bounded in the state variable and fill a part of this strip. The values attained by the impulse characteristics on a finite time interval are arbitrarily large in magnitude. We propose a construction of a smooth extension for a continuous minimax/viscosity solution of the problem to the part of the strip that is not covered by the characteristics starting in the initial manifold.

Détail

Keywords: Hamilton–Jacobi–Bellman equations, method of characteristics, viscosity solutions, minimax solutions.

Bibliographie

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

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

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

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

[5] Kurzhanskii A. B., Nikonov O. I., “Evolyutsionnye uravneniya dlya puchkov traektorii sintezirovannykh sistem upravleniya”, Dokl. RAN, 333:5 (1993), 578–581 | MR

[6] Kurzhanski A. B., “Comparison principle for equations of the Hamilton–Jacobi type in control theory”, Proc. Steklov Inst. Math., 251, Suppl. 1, 2006, S185–S195 | MR | Zbl

[7] Kurzhanski A., Varaiya P., “The Hamilton–Jacobi equations for nonlinear target control and their approximation”, Analysis and Design of Nonlinear Control Systems, eds. A. Astolfi, L. Marconi, Springer-Verlag, New York, 2007, 77–90 | MR

[8] Saakian D. B., Rozanova O., Akmetzhanov A., “Dynamics of the eigen and the Crow-Kimura models for molecular evolution”, Phys. Rev. E (3), 78:4 (2008), 041908, 6 pp. | DOI | MR

[9] Subbotina N. N., Shagalova L. G., “O reshenii zadachi Koshi dlya uravneniya Gamiltona –Yakobi s fazovymi ogranicheniyami”, Tr. In-ta matematiki i mekhaniki UrO RAN, 17, no. 2, 2011, 191–208

[10] Subbotin A. I., Minimaksnye neravenstva i uravneniya Gamiltona–Yakobi, Nauka, M., 1991, 216 pp. | MR | Zbl

[11] Subbotin A. I., Generalized solutions of first order PDEs: The dynamical optimization perspective, Birkhauser, Boston, 1995, 312 pp. | MR

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

[13] Capuzzo-Dolcetta I., Lions P.-L., “Hamilton–Jacobi equations with state constraints”, Trans. Amer. Math. Soc., 318:2 (1990), 643–683 | DOI | MR | Zbl

[14] Subbotina N. N., Shagalova L. G., “Postroenie obobschennogo resheniya uravneniya, sokhranyayuschego tip Bellmana v zadannoi oblasti fazovogo prostranstva”, Tr. MIAN, 277, 2012, 243–256 | MR | Zbl

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

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

[17] Kruzhkov S. N., “Obobschennye resheniya nelineinykh uravnenii pervogo poryadka so mnogimi nezavisimymi peremennymi, I”, Mat. sb., 70(112):3 (1966), 394–415 | MR | Zbl

[18] Klark F., Optimizatsiya i negladkii analiz, Nauka, M., 1988, 280 pp. | MR

[19] Petrovskii I. G., Lektsii po teorii obyknovennykh differentsialnykh uravnenii, Nauka, M., 1970, 280 pp. | MR

[20] Elsgolts L. E., Differentsialnye uravneniya i variatsionnoe ischislenie, Nauka, M., 1969, 424 pp. | MR