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**Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations.**
*(English)*
Zbl 1169.91317

The paper is concerned with the connections between the value of two-person, zero-sum differential games and the notion of viscosity solutions of Hamilton-Jacobi PDE introduced by M. G. Crandall and P.-L. Lions [Trans. Am. Math. Soc. 277, 1–42 (1983; Zbl 0599.35024)]. The authors prove that the upper and lower value functions are the unique viscosity solutions of the first order PDE with min-max and max-min type of nonlinearity (upper and lower Isaacs equations). Unlike previous papers of the authors and others, based on the Fleming and the Friedman definitions of upper and lower values for a differential game, this one follows the Elliot-Kalton definition. This results in a considerable simplification of the statements and proofs.

Another part of the paper is devoted to applications. The authors establish the representation formula, that enables one to consider the viscosity solution of a fairly arbitrary Hamilton-Jacobi equation as the solution of the upper Isaacs equation for some appropriate differential game. This representation formula is employed to prove results concerning the level sets of solutions of Hamilton-Jacobi equations with homogeneous Hamiltonians.

Another part of the paper is devoted to applications. The authors establish the representation formula, that enables one to consider the viscosity solution of a fairly arbitrary Hamilton-Jacobi equation as the solution of the upper Isaacs equation for some appropriate differential game. This representation formula is employed to prove results concerning the level sets of solutions of Hamilton-Jacobi equations with homogeneous Hamiltonians.

Reviewer: M. I. Gusev (MR0756158)