Differential Inclusions : Set Valued Maps and Viability Theory 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.