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Approximation schemes and finite-difference operators for constructing generalized solutions of Hamilton-Jacobi equations. (English. Russian original) Zbl 0847.49026

J. Comput. Syst. Sci. Int. 33, No. 6, 127-139 (1995); translation from Izv. Ross. Akad. Nauk, Tekh. Kibern. 1994, No. 3, 173-185 (1994).
Summary: A first-order partial differential Hamilton-Jacobi equation and the guaranteed control problem which corresponds to it are examined. The generalized minimax solution of the Hamilton-Jacobi equation is the cost function of the control problem. The problem of constructing the cost function, the solution of which facilitates the finding of the optimal strategies, is investigated. Approximating (grid) schemes for the approximate computation of the minimax solution which use finite-difference operators based on the differential game theory and convex and nonsmooth analysis are proposed. The convergence of these schemes is substantiated. Illustrative examples are presented.

MSC:

49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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