Aubin / Cellina Differential Inclusions

Set-Valued Maps and Viability Theory
Erscheinungsjahr 2012
ISBN: 978-3-642-69512-4
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 264, 342 Seiten, eBook

Reihe: Grundlehren der mathematischen Wissenschaften

Zielgruppe

Research

Autoren/Hrsg.

Aubin, J.-P.

Cellina, A.

Inhaltsverzeichnis

0. Background Notes.- 1. Continuous Partitions of Unity.- 2. Absolutely Continuous Functions.- 3. Some Compactness Theorems.- 4. Weak Convergence and Asymptotic Center of Bounded Sequences.- 5. Closed Convex Hulls and the Mean-Value Theorem.- 6. Lower Semicontinuous Convex Functions and Projections of Best Approximation.- 7. A Concise Introduction to Convex Analysis.- 1. Set-Valued Maps.- 1. Set-Valued Maps and Continuity Concepts.- 2. Examples of Set-Valued Maps.- 3. Continuity Properties of Maps with Closed Convex Graph.- 4. Upper Hemicontinuous Maps and the Convergence Theorem.- 5. Hausdorff Topology.- 6. The Selection Problem.- 7. The Minimal Selection.- 8. Chebishev Selection.- 9. The Barycentric Selection.- 10. Selection Theorems for Locally Selectionable Maps.- 11. Michael’s Selection Theorem.- 12. The Approximate Selection Theorem and Kakutani’s Fixed Point Theorem.- 13. (7-Selectionable Maps.- 14. Measurable Selections.- 2. Existence of Solutions to Differential Inclusions.- 1. Convex Valued Differential Inclusions.- 2. Qualitative Properties of the Set of Trajectories of Convex-Valued Differential Inclusions.- 3. Nonconvex-Valued Differential Inclusions.- 4. Differential Inclusions with Lipschitzean Maps and the Relaxation Theorem.- 5. The Fixed-Point Approach.- 6. The Lower Semicontinuous Case.- 3. Differential Inclusions with Maximal Monotone Maps.- 1. Maximal Monotone Maps.- 2. Existence and Uniqueness of Solutions to Differential Inclusions with Maximal Monotone Maps.- 3. Asymptotic Behavior of Trajectories and the Ergodic Theorem.- 4. Gradient Inclusions.- 5. Application: Gradient Methods for Constrained Minimization Problems.- 4. Viability Theory: The Nonconvex Case.- 1. Bouligand’s Contingent Cone.- 2. Viable and Monotone Trajectories.- 3.Contingent Derivative of a Set-Valued Map.- 4. The Time Dependent Case.- 5. A Continuous Version of Newton’s Method.- 6. A Viability Theorem for Continuous Maps with Nonconvex Images..- 7. Differential Inclusions with Memory.- 5. Viability Theory and Regulation of Controled Systems: The Convex Case.- 1. Tangent Cones and Normal Cones to Convex Sets.- 2. Viability Implies the Existence of an Equilibrium.- 3. Viability Implies the Existence of Periodic Trajectories.- 4. Regulation of Controled Systems Through Viability.- 5. Walras Equilibria and Dynamical Price Decentralization.- 6. Differential Variational Inequalities.- 7. Rate Equations and Inclusions.- 6. Liapunov Functions.- 1. Upper Contingent Derivative of a Real-Valued Function.- 2. Liapunov Functions and Existence of Equilibria.- 3. Monotone Trajectories of a Differential Inclusion.- 4. Construction of Liapunov Functions.- 5. Stability and Asymptotic Behavior of Trajectories.- Comments.