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

Olivier Alvarez Université de Rouen, Mont-Saint Aignan, France
Martino Bardi Università di Padova, Padova, Italy
Available Formats:
Electronic ISBN: 978-1-4704-0574-8
Product Code: MEMO/204/960.E
List Price: $68.00 MAA Member Price:$61.20
AMS Member Price: $40.80 Click above image for expanded view Ergodicity, Stabilization, and Singular Perturbations for Bellman-Isaacs Equations Olivier Alvarez Université de Rouen, Mont-Saint Aignan, France Martino Bardi Università di Padova, Padova, Italy Available Formats:  Electronic ISBN: 978-1-4704-0574-8 Product Code: MEMO/204/960.E  List Price:$68.00 MAA Member Price: $61.20 AMS Member Price:$40.80
• Book Details

Memoirs of the American Mathematical Society
Volume: 2042009; 77 pp
MSC: Primary 35; 93; 49; Secondary 60; 91;

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.

• 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
• Requests

Review Copy – for reviewers who would like to review an AMS book
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Volume: 2042009; 77 pp
MSC: Primary 35; 93; 49; Secondary 60; 91;

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.

• 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
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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