Viscosity solutions and programmed iteration method for Isaacs equation

F. F. Nikitin

Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 2 (2014), pp. 84-92

Résumé

To solve zero-sum differential games Isaacs derived PDE of Hamilton–Jacobi type for value function. However, in many differential games the value function is not smooth. The theory of viscosity solutions overcomes non-smoothness of the value function by introducing generalized solutions of PDE. Programmed iteration method considers functional equation for the value function which is called generalized Isaacs–Bellman equation. In the paper connection between the theory of viscosity solutions and programmed iteration method is studied. It turns out that successive approximations utilized in programmed iteration method for finding solutions of generalized Isaacs–Bellman equation and any fixed point of value operators are corresponding viscosity super or sub-solutions of Isaacs equation. Bibliogr. 24.

Détail

Keywords: zero-sum differential games, viscosity solutions, Isaacs equation, programmed iteration method, value operators, value of differential game.

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