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On the homogenization of some non-coercive Hamilton-Jacobi-Isaacs equations. (English) Zbl 1264.35019
Summary: We study the homogenization of Hamilton-Jacobi equations with oscillating initial data and non-coercive Hamiltonian, mostly of the Bellman-Isaacs form arising in optimal control and differential games. We describe classes of equations for which pointwise homogenization fails for some data. We prove locally uniform homogenization for various Hamiltonians with some partial coercivity and some related restrictions on the oscillating variables, mostly motivated by the applications to differential games, in particular of pursuit-evasion type. The effective initial data are computed under some assumptions of asymptotic controllability of the underlying control system with two competing players.

MSC:
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35F21 Hamilton-Jacobi equations
49N70 Differential games and control
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
93B05 Controllability
35D40 Viscosity solutions to PDEs
91A23 Differential games (aspects of game theory)
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