 | Alvarez O. Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations / Alvarez O., Bardi M. - Providence: American Mathematical Society, 2010. - v, 77 p. - (Memoirs of the American Mathematical Society; Vol.204, N 960). - Bibliogr.: p.73-77. - ISBN 978-0-8218-4515-2; ISSN 0065-9266
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Chapter 1. Introduction and statement of the problem ............ 1
1.1. Introduction ............................................... 1
1.2. Stochastic differential games and the singular
perturbation problem ....................................... 6
1.3. The Bellman-Isaacs equations ............................... 8
Chapter 2. Abstract ergodicity, stabilization, and
convergence ......................................... 11
2.1. Ergodicity and the effective Hamiltonian .................. 11
2.2. Stabilization and the effective terminal cost ............. 15
2.3. The general convergence result ............................ 17
Chapter 3. Uncontrolled fast variables and averaging ........... 21
3.1. Ergodicity ................................................ 22
3.2. Stabilization ............................................. 23
3.3. Uniform convergence ....................................... 24
3.4. An explicit formula for the limit control problem ......... 26
Chapter 4. Uniformly nondegenerate fast diffusion .............. 29
4.1. Ergodicity ................................................ 29
4.2. Stabilization ............................................. 31
4.3. Uniform convergence ....................................... 33
Chapter 5. Hypoelliptic diffusion of the fast variables ........ 37
5.1. Ergodicity and stabilization .............................. 38
5.2. Uniform convergence ....................................... 39
Chapter 6. Controllable fast variables ......................... 41
6.1. Bounded-time controllability and ergodicity ............... 41
6.2. Stabilization and a formula for the effective initial
data ...................................................... 46
6.3. An explicit formula for the effective Hamiltonian and
the limit differential game ............................... 47
6.4. Uniform convergence ....................................... 50
6.5. The reduction order formula for the effective control
problem ................................................... 52
Chapter 7. Nonresonant fast variables .......................... 55
7.1. Ergodicity ................................................ 55
7.2. Stabilization ............................................. 57
7.3. Uniform convergence ....................................... 60
Chapter 8. A counterexample to uniform convergence ............. 61
Chapter 9. Applications to homogenization ...................... 67
9.1. Periodic homogenization of 1st order H-J equations ........ 67
9.2. Periodic homogenization of 2nd order equations ............ 69
Bibliography ................................................... 73
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