**Memoirs of the American Mathematical Society**

2009;
77 pp;
Softcover

MSC: Primary 35; 93; 49;
Secondary 60; 91

Print ISBN: 978-0-8218-4715-2

Product Code: MEMO/204/960

List Price: $68.00

Individual Member Price: $40.80

**Electronic ISBN: 978-1-4704-0574-8
Product Code: MEMO/204/960.E**

List Price: $68.00

Individual Member Price: $40.80

# Ergodicity, Stabilization, and Singular Perturbations for Bellman-Isaacs Equations

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*Olivier Alvarez; Martino Bardi*

The authors study singular perturbations of optimal stochastic control problems and differential games arising in the dimension reduction of system with multiple time scales. They 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. The authors also construct an explicit example where the convergence is not uniform. Finally they give some applications to the periodic homogenization of Hamilton-Jacobi equations with non-coercive Hamiltonian and of some degenerate parabolic PDEs.

#### Table of Contents

# Table of Contents

## Ergodicity, Stabilization, and Singular Perturbations for Bellman-Isaacs Equations

- Chapter 1. Introduction and statement of the problem 18 free
- Chapter 2. Abstract ergodicity, stabilization, and convergence 1118
- Chapter 3. Uncontrolled fast variables and averaging 2128
- Chapter 4. Uniformly nondegenerate fast diffusion 2936
- Chapter 5. Hypoelliptic diffusion of the fast variables 3744
- Chapter 6. Controllable fast variables 4148
- 6.1. Bounded-time controllability and ergodicity 4148
- 6.2. Stabilization and a formula for the effective initial data 4653
- 6.3. An explicit formula for the effective Hamiltonian and the limit differential game 4754
- 6.4. Uniform convergence 5057
- 6.5. The reduction order formula for the effective control problem 5259

- Chapter 7. Nonresonant fast variables 5562
- Chapter 8. A counterexample to uniform convergence 6168
- Chapter 9. Applications to homogenization 6774
- Bibliography 7380