Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
E. A. Kolpakova, “A construction of Nash equilibrium based on system of Hamilton-Jacobi equations of special type,” Mat. Teor. Igr Pril. 9 (4), 39–53 (2017).
M. G. Crandall, H. Ishii, and P. L. Lions, “User’s guide to viscosity solutions of second order partial differential equations,” Bull. Amer. Math. Soc. (N. S.) 27 (1), 1–67 (1992).
A. I. Subbotin, Generalized Solutions of Partial Differential Equations of First Order. The Prospects of Dynamical Optimization (Inst. Komp’yut. Issled., Moscow–Izhevsk, 2003) [in Russian].
S. S. Kumkov, S. Le Ménec, and V. S. Patsko, “Zero-sum pursuit-evasion differential games with many objects: survey of publications,” Dyn. Games Appl. 7 (4), 609–633 (2017).
M. Falcone and R. Ferretti, “Discrete time high-order schemes for viscosity solutions of Hamilton–Jacobi–Bellman equations,” Numer. Math. 67, 315–344 (1994).
M. Quincampoix, P. Cardaliaguet, and P. Saint-Pierre, “Numerical methods for differential games,” in Stochastic and Differential Games: Theory and Numerical Methods (Birkhäuser, Basel, 1999), pp. 177–247.
H. Ishii, “Hamilton–Jacobi equations with discontinuous hamiltonians on arbitrary open sets,” Bull. Fac. Sci. Engrg. Chuo Univ. 28, 33–77 (1985).
A. S. Lakhtin and A. I. Subbotin, “Multivalued solutions of first-order partial differential equations,” Sb. Math. 189 (6), 849–873 (1998).
M. Coclite and N. Risebro, “Viscosity solutions of Hamilton–Jacobi equations with discontinuous coefficients,” J. Hyperbolic Differ. Equ. 4, 771–795 (2007).
A. S. Lakhtin and A. I. Subbotin, “Minimax and viscosity solutions of first-order discontinuous partial differential equations,” Dokl. Akad. Nauk 359 (4), 452–455 (1998).
S. I. Resnick, A Probability Path (Birkhäuser Boston, Boston, MA, 1998).
K. Kuratowski, Topology (Academic Press, New York–London, 1966), Vol. 1.
Funding
This work was supported by the Government of the Russian Federation (grant no. 211, contract no. 02. A03.21.0006).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kolpakova, E.A. Approximation of a Multivalued Solution of the Hamilton–Jacobi Equation. Math Notes 108, 77–86 (2020). https://doi.org/10.1134/S000143462007007X
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S000143462007007X