Calculus of Variations and Geometric Measure Theory
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O. Alvarez - M. Bardi

Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations

created by bardi on 06 Sep 2007
modified on 12 Mar 2015


Published Paper

Inserted: 6 sep 2007
Last Updated: 12 mar 2015

Journal: Mem. Amer. Math. Soc.
Volume: 204
Pages: 1-88
Year: 2010


We study singular perturbations of optimal stochastic control problems and differential games arising in the dimension reduction of system with multiple time scales. We analyze the uniform convergence of the value functions via the associated Hamilton-Jacobi-Bellman-Isaacs equations, in the framework of viscosity solutions. The crucial properties of ergodicity and stabilization to a constant that the Hamiltonian must possess are formulated as differential games with ergodic cost criteria. They are studied under various different assumptions and with PDE as well as control-theoretic methods. We construct also an explicit example where the convergence is not uniform. Finally we give some applications to the periodic homogenization of Hamilton-Jacobi equations with non-coercive Hamiltonian and of some degenerate parabolic PDEs.

Keywords: Homogenization, singular perturbations, Ergodic control, nonlinear parabolic equations


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