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Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations
About this Title
Olivier Alvarez, UMR 60-85, Université de Rouen, 76821 Mont-Saint Aignan cedex, France and Martino Bardi, Dipartimento di Matematica Pura ed Applicata, Università di Padova, via Trieste 63, 35121 Padova, Italy
Publication: Memoirs of the American Mathematical Society
Publication Year:
2010; Volume 204, Number 960
ISBNs: 978-0-8218-4715-2 (print); 978-1-4704-0574-8 (online)
DOI: https://doi.org/10.1090/S0065-9266-09-00588-2
Published electronically: November 5, 2009
Keywords: Singular perturbations,
differential games,
viscosity solutions,
dimension reduction,
stochastic games,
nonlinear parabolic equations,
Hamilton-Jacobi equations,
ergodic control,
stabilization,
homogenization,
averaging,
non-resonance,
optimal control,
controlled diffusion processes,
oscillating initial data.
MSC: Primary 35Bxx, 35Kxx, 93C70, 49N70; Secondary 49L25, 60J60, 91A23, 93E20
Table of Contents
Chapters
- 1. Introduction and statement of the problem
- 2. Abstract ergodicity, stabilization, and convergence
- 3. Uncontrolled fast variables and averaging
- 4. Uniformly nondegenerate fast diffusion
- 5. Hypoelliptic diffusion of the fast variables
- 6. Controllable fast variables
- 7. Nonresonant fast variables
- 8. A counterexample to uniform convergence
- 9. Applications to homogenization
Abstract
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.- V. I. Arnold and A. Avez, Problèmes ergodiques de la mécanique classique, Monographies Internationales de Mathématiques Modernes, vol. 9, Gauthier-Villars, Éditeur, Paris, 1967 (French). MR 0209436
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