Differential Inclusions: Set-Valued Maps and Viability Theory

Title page......Page 1
Copyright page......Page 2
Epigraph......Page 3
Acknowledgments......Page 5
Table of Contents......Page 7
Introduction......Page 11
1. Continuous Partitions of Unity......Page 19
2. Absolutely Continuous Functions......Page 22
3. Some Compactness Theorems......Page 23
4. Weak Convergence and Asymptotic Center of Bounded Sequences......Page 25
5. Closed Convex Hulls and the Mean-Value Theorem......Page 28
6. Lower Semicontinuous Convex Functions and Projections of Best Approximation......Page 31
7. A Concise Introduction to Convex Analysis......Page 39
Introduction......Page 47
1. Set-Valued Maps and Continuity Concepts......Page 49
2. Examples of Set-Valued Maps......Page 56
3. Continuity Properties of Maps with Closed Convex Graph......Page 64
4. Upper Hemicontinuous Maps and the Convergence Theorem......Page 69
5. Hausdorff Topology......Page 75
6. The Selection Problem......Page 78
7. The Minimal Selection......Page 80
8. Chebishev Selection......Page 83
9. The Barycentric Selection......Page 87
10. Selection Theorems for Locally Selectionable Maps......Page 90
11. Michael's Selection Theorem......Page 92
12. The Approximate Selection Theorem and Kakutani's Fixed Point Theorem......Page 94
13. $\sigma$-Selectionable Maps......Page 96
14. Measurable Selections......Page 100
Introduction......Page 103
1. Convex Valued Differential Inclusions......Page 106
2. Qualitative Properties of the Set of Trajectories of Convex-Valued Differential Inclusions......Page 113
3. Nonconvex-Valued Differential Inclusions......Page 121
4. Differential Inclusions with Lipschitzean Maps and the Relaxation Theorem......Page 129
5. The Fixed-Point Approach......Page 137
6. The Lower Semicontinuous Case......Page 144
Introduction......Page 149
1. Maximal Monotone Maps......Page 150
2. Existence and Uniqueness of Solutions to Differential Inclusions with Maximal Monotone Maps......Page 157
3. Asymptotic Behavior of Trajectories and the Ergodic Theorem......Page 161
4. Gradient Inclusions......Page 168
5. Application: Gradient Methods for Constrained Minimization Problems......Page 173
Introduction......Page 182
1. Bouligand's Contingent Cone......Page 186
2. Viable and Monotone Trajectories......Page 189
3. Contingent Derivative of a Set-Valued Map......Page 198
4. The Time Dependent Case......Page 201
5. A Continuous Version of Newton's Method......Page 205
6. A Viability Theorem for Continuous Maps with Nonconvex Images......Page 208
7. Differential Inclusions with Memory......Page 214
Introduction......Page 223
1. Tangent Cones and Normal Cones to Convex Sets......Page 228
2. Viability Implies the Existence of an Equilibrium......Page 238
3. Viability Implies the Existence of Periodic Trajectories......Page 245
4. Regulation of Controled Systems Through Viability......Page 248
5. Walras Equilibria and Dynamical Price Decentralization......Page 255
6. Differential Variational Inequalities......Page 274
7. Rate Equations and Inclusions......Page 284
Introduction......Page 291
1. Upper Contingent Derivative of a Real-Valued Function......Page 294
2. Liapunov Functions and Existence of Equilibria......Page 300
3. Monotone Trajectories of a Differential Inclusion......Page 303
4. Construction of Liapunov Functions......Page 315
5. Stability and Asymptotic Behavior of Trajectories......Page 319
Comments......Page 332
Bibliography......Page 338
Index......Page 351