×

Max-min representations and product formulas for the viscosity solutions of Hamilton-Jacobi equations with applications to differential games. (English) Zbl 0526.35018


MSC:

35F20 Nonlinear first-order PDEs
35C99 Representations of solutions to partial differential equations
35D99 Generalized solutions to partial differential equations
49K35 Optimality conditions for minimax problems
47H99 Nonlinear operators and their properties
35F25 Initial value problems for nonlinear first-order PDEs
91A23 Differential games (aspects of game theory)
91A99 Game theory
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
49J35 Existence of solutions for minimax problems
35L60 First-order nonlinear hyperbolic equations

Citations:

Zbl 0469.49023
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Barles, G., Thèse de Doctorat 3ème Cycle, (Trans. Am. math. Soc. (1982-1983), Université de Paris IX-Dauphine), (to appear).
[2] Barron N. E., Evans L. C. & Jensen R., Viscosity solutions of Isaacs’ equations and differential games with Lipschitz controls, J. diff. Eqns; Barron N. E., Evans L. C. & Jensen R., Viscosity solutions of Isaacs’ equations and differential games with Lipschitz controls, J. diff. Eqns · Zbl 0548.90104
[3] Crandall, M. G.; Evans, L. C.; Lions, P. L., Some properties of viscosity solutions of Hamilton-Jacobi equations, (Mathematics Research Center TSR No. 2390 (June 1982), University of Wisconsin-Madison). (Mathematics Research Center TSR No. 2390 (June 1982), University of Wisconsin-Madison), Trans. Am. math. Soc., 282, 487-502 (1984) · Zbl 0543.35011
[4] Crandall, M. G.; Lions, P. L., Viscosity solutions of Hamilton-Jacobi equations, Trans. Am. math. Soc., 277, 1-42 (1983) · Zbl 0599.35024
[5] Crandall, M. G.; Lions, P. L., Two approximations of solutions of Hamilton-Jacobi equations, (Mathematics Research Center TSR No. 2431. Mathematics Research Center TSR No. 2431, Math. Comp. (September 1982), University of Wisconsin-Madison), (to appear). · Zbl 0874.49025
[6] Elliott, R. J.; Kalton, N. J., The existence of value in differential games, Mem. Am. math Soc., No. 126 (1972) · Zbl 0229.90061
[7] Evans L. C., Some max-min methods for the Hamilton-Jacobi equation, Indiana Univ. Math. J.; Evans L. C., Some max-min methods for the Hamilton-Jacobi equation, Indiana Univ. Math. J. · Zbl 0543.35012
[8] Evans, L. C.; Souganidis, P. E., Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations, (Mathematics Research Center TSR No. 2492. Mathematics Research Center TSR No. 2492, Indiana Univ. Math. J. (March 1983), University of Wisconsin-Madison), (to appear). · Zbl 0679.60040
[9] Fleming, W. H., The convergence problem for differential games, J. math. Analysis Applic., 3, 102-116 (1961) · Zbl 0113.14705
[10] Fleming, W. H., The convergence problem for differential games, II, (Advances in Game Theory, Ann. Math. Study, 52 (1964), Princeton University Press: Princeton University Press Princeton), 195-210 · Zbl 0137.14204
[11] Fleming, W. H., The Cauchy problem for degenerate parabolic equations, J. Math Mech., 13, 987-1008 (1964) · Zbl 0192.19602
[12] Fleming, W. H., Nonlinear partial differential equations probabilistic and game theoretic methods, (Problems in Nonlinear Analysis (1971), CIME, Ed. Cremonese: CIME, Ed. Cremonese Roma) · Zbl 0225.35020
[13] Friedman, A., Differential Games (1971), Wiley: Wiley New York · Zbl 0229.90060
[14] Friedman, A., Differential Games, (CBMS No. 18 (1974), American Mathematical Society: American Mathematical Society Providence) · Zbl 0208.39402
[15] Isaacs, R., Differential Games (1965), Wiley: Wiley New York · Zbl 0152.38407
[16] Krassovski, N.; Subbotin, A., Jeux Differentiels (1977), Mir Press: Mir Press Moscow
[17] Lions, P. L., Generalized Solutions of Hamilton-Jacobi Equations (1982), Pitman: Pitman Boston · Zbl 1194.35459
[18] Lions, P. L., Existence results for first-order Hamilton-Jacobi equations, Richerche Math. Napoli (1982-1983)
[19] Lions, P. L.; Nisio, M., A uniqueness result for the semigroup associated with the Hamilton-Jacobi-Bellman operator, Proc. Jpn. Acad., 58, 273-276 (1982) · Zbl 0516.93065
[20] Lions P. L., Papanicolaou G. & Varadhan S. R. S. (to appear).; Lions P. L., Papanicolaou G. & Varadhan S. R. S. (to appear).
[21] Lions P. L. & Souganidis P., Differential games, optimal control and directional derivatives of viscosity solutions of Bellman’s and Isaacs’ equations, SIAM J. Control Optim.; Lions P. L. & Souganidis P., Differential games, optimal control and directional derivatives of viscosity solutions of Bellman’s and Isaacs’ equations, SIAM J. Control Optim. · Zbl 0569.49019
[22] Souganidis, P. E., Existence of viscosity solutions of Hamilton-Jacobi equations, (Mathematics Research Center TSR No. 2488. Mathematics Research Center TSR No. 2488, J. diff. Eqns (March 1983), University of Wisconsin-Madison), (to appear) · Zbl 0598.35066
[23] Souganidis, P. E., Approximation schemes for viscosity solutions of Hamilton-Jacobi equations, (Mathematics Research Center TSR No. 2511 (April 1983), University of Wisconsin-Madison) · Zbl 0598.35066
[24] Souganidis P. E., Approximation schemes for viscosity solutions of Hamilton-Jacobi equations, J. diff. Eqns.; Souganidis P. E., Approximation schemes for viscosity solutions of Hamilton-Jacobi equations, J. diff. Eqns. · Zbl 0536.70020
[25] Subbotin, A., A generalization of the basic equation of the theory of differential games, Soviet Math. Dokl., 22, 358-362 (1980) · Zbl 0467.90095
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.