Geodesic
Approximation of a Multivalued Solution of the Hamilton--Jacobi Equation
Matematičeskie zametki, Tome 108 (2020) no. 1, pp. 81-93.

Voir la notice de l'article provenant de la source Math-Net.Ru

The paper deals with the construction of a multivalued solution of the Cauchy problem for the Hamilton–Jacobi equation with discontinuous Hamiltonian with respect to the phase variable. The constructed multivalued solution is approximated by a sequence of continuous solutions of auxiliary Cauchy problems of the Hamilton–Jacobi equation with Hamiltonian which is Lipschitz with respect to the phase variable. The results of the study are illustrated by an example.
Keywords: Hamilton–Jacobi equation, multivalued solution, minimax/viscosity solution, viable set. 
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E. A. Kolpakova. Approximation of a Multivalued Solution of the Hamilton--Jacobi Equation. Matematičeskie zametki, Tome 108 (2020) no. 1, pp. 81-93. http://geodesic.mathdoc.fr/item/MZM_2020_108_1_a5/

[1] E. A. Kolpakova, “Postroenie neshevskogo ravnovesiya na osnove sistemy uravnenii Gamiltona–Yakobi spetsialnogo vida”, MTIP, 9:4 (2017), 39–53 | Zbl

[2] M. G. Crandall, H. Ishii, P. L. Lions, “User's guide to viscosity solutions of second order partial differential equations”, Bull. Amer. Math. Soc. (N.S.), 27:1 (1992), 1–67 | DOI | MR | Zbl

[3] A. I. Subbotin, Obobschennye resheniya uravnenii v chastnykh proizvodnykh pervogo poryadka. Perspektivy dinamicheskoi optimizatsii, In-t kompyuternykh issledovanii, M.–Izhevsk, 2003

[4] S. S. Kumkov, S. Le Ménec, V. S. Patsko, “Zero-sum pursuit-evasion differential games with many objects: survey of publications”, Dyn. Games Appl., 7:4 (2017), 609–633 | DOI | MR | Zbl

[5] M. Falcone, R. Ferretti, “Discrete time high-order schemes for viscosity solutions of Hamilton–Jacobi–Bellman equations”, Numer. Math., 67 (1994), 315–344 | DOI | MR | Zbl

[6] M. Quincampoix, P. Cardaliaguet, P. Saint-Pierre, “Numerical methods for differential games”, Stochastic and Differential Games: Theory and Numerical Methods, Birkhäuser, Basel, 1999, 177–247 | Zbl

[7] H. Ishii, “Hamilton–Jacobi equations with discontinuous hamiltonians on arbitrary open sets”, Bull. Fac. Sci. Engrg. Chuo Univ., 28 (1985), 33–77 | MR | Zbl

[8] A. S. Lakhtin, A. I. Subbotin, “Mnogoznachnye resheniya uravnenii s chastnymi proizvodnymi pervogo poryadka”, Matem. sb., 189:6 (1998), 33–58 | DOI | MR | Zbl

[9] M. Coclite, N. Risebro, “Viscosity solutions of Hamilton–Jacobi equations with discontinuous coefficients”, J. Hyperbolic Differ. Equ., 4 (2007), 771–795 | DOI | MR | Zbl

[10] A. S. Lakhtin, A. I. Subbotin, “Minimaksnye i vyazkostnye resheniya razryvnykh uravnenii s chastnymi proizvodnymi pervogo poryadka”, Dokl. AN, 359:4 (1998), 452–455 | Zbl

[11] S. I. Resnick, A Probability Path, Birkhäuser Boston, Boston, MA, 1998 | MR

[12] K. Kuratovskii, Topologiya, T. 1, Mir, M., 1966 | MR | Zbl