Introduction to the Theory of Differential Inclusions
Series: Graduate Studies in Mathematics, Volume 41
Publisher: American Mathematical Society
Published: 2002
Place of publication: Providence, RI, USA
Language: eng
Format: Hardcover
Number of pages: 226
ISBN-13: 9780821829776
Copyright date: 2002
Table of contents: title: Machine generated contents note: Part 1. Foundations -- Chapter 1. Convex Analysis 3 -- 1.1. Convex sets 3 -- 1.2. Convex functions 13 -- 1.3. Differential properties of convex functions 24 -- 1.4. Problems 30 -- Chapter 2. Set-Valued Analysis 31 -- 2.1. Set-valued maps 31 -- 2.2. Derivatives of set-valued maps 37 -- 2.3. Lipschitzian approximations 41 -- 2.4. Extension theorem 44 -- 2.5. Fixed point theorems 47 -- 2.6. Convex processes 49 -- 2.7. Structure of a convex process 56 -- 2.8. Problems 61 -- Chapter 3. Nonsmooth Analysis 65 -- 3.1. Method of metric approximations 65 -- 3.2. Mordukhovich normal cone 67 -- 3.3. Separation theorem for nonconvex sets 72 -- 3.4. Nonsmooth calculus 75 -- 3.5. Lagrange multipliers 82 -- 3.6. Problems 83 -- Part 2. Differential Inclusions -- Chapter 4. Existence Theorems 87 -- 4.1. Background notes 88 -- 4.2. Lipschitzian differential inclusions 90 -- 4.3. Upper semi-continuous differential inclusions 96 -- 4.4. Discontinuous differential equations 103 -- 4.5. Existence of optimal solutions 106 -- 4.6. Dependence on initial conditions 109 -- 4.7. Discrete approximations 113 -- 4.8. Problems 116 -- Chapter 5. Viability and Invariance 119 -- 5.1. Monotone solutions to a differential inclusion 119 -- 5.2. Viability problem 122 -- 5.3. Invariant sets 127 -- 5.4. Existence of periodic solutions 130 -- 5.5. Pursuit in a differential game 132 -- 5.6. Problems ' 135 -- Chapter 6. Controllability 139 -- 6.1. Duality relation 139 -- 6.2. Controllability of convex processes 145 -- 6.3. Controllability at first approximation 147 -- 6.4. Controllability of some mechanical systems 152 -- 6.5. Problems 153 -- Chapter 7. Optimality 157 -- 7.1. Optimal solutions to discrete-time inclusions 157 -- 7.2. Optimal solutions to differential 4nclusions 160 -- 7.3. Time-optimal problem 165 -- 7.4. Problems 168 -- Chapter 8. Stability 171 -- 8.1. Lyapunov direct method 171 -- 8.2. Linear-selectionable differential inclusions 176 -- 8.3. Weak asymptotic stability of convex processes 185 -- 8.4. First approximation techniques 189 -- 8.5. Stability of a missile uniform motion 194 -- 8.6. Problems 196 -- Chapter 9. Stabilization 199 -- 9.1. Lyapunov functions for convex processes 200 -- 9.2. Stabilization problem 202 -- 9.3. Weak asymptotic stability and stabilizability 205 -- 9.4. Stabilizers for some mechanical systems 208 -- 9.5. Problems 210.