On the homogenization of some non-coercive Hamilton--Jacobi--Isaacs equations

Martino Bardi; Gabriele Terrone

doi:10.3934/cpaa.2013.12.207

Article Contents

2013, Volume 12, Issue 1: 207-236. Doi: 10.3934/cpaa.2013.12.207

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On the homogenization of some non-coercive Hamilton--Jacobi--Isaacs equations

Martino Bardi^1, and
Gabriele Terrone^2,

1.
Dipartimento di Matematica, Università di Padova, via Trieste, 63; I-35121 Padova
2.
Instituto Superior Técnico, Universidade Técnica de Lisboa, Departamento de Matemática, Av. Rovisco Pais, 1049-001 Lisboa

Received: March 31, 2011

Revised: August 31, 2011

Published: December 2012

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Abstract

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.

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Mathematics Subject Classification: Primary: 35B27, 35F21, 49N70; Secondary: 49L25.

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