مقدمه‌ای بر نظریه شمول‌های دیفرانسیل (مطالعات تحصیلات تکمیلی در ریاضیات)

A differential inclusion is a relation of the form $\dot x \in F(x)$, where $F$ is a set-valued map associating any point $x \in R^n$ with a set $F(x) \subset R^n$. As such, the notion of a differential inclusion generalizes the notion of an ordinary differential equation of the form $\dot x = f(x)$. Therefore, all problems usually studied in the theory of ordinary differential equations (existence and continuation of solutions, dependence on initial conditions and parameters, etc.) can be studied for differential inclusions as well. Since a differential inclusion usually has many solutions starting at a given point, new types of problems arise, such as investigation of topological properties of the set of solutions, selection of solutions with given properties, and many others. Differential inclusions play an important role as a tool in the study of various dynamical processes described by equations with a discontinuous or multivalued right-hand side, occurring, in particular, in the study of dynamics of economical, social, and biological macrosystems. They also are very useful in proving existence theorems in control theory. This text provides an introductory treatment to the theory of differential inclusions. The reader is only required to know ordinary differential equations, theory of functions, and functional analysis on the elementary level. Chapter 1 contains a brief introduction to convex analysis. Chapter 2 considers set-valued maps. Chapter 3 is devoted to the Mordukhovich version of nonsmooth analysis. Chapter 4 contains the main existence theorems and gives an idea of the approximation techniques used throughout the text. Chapter 5 is devoted to the viability problem, i.e., the problem of selection of a solution to a differential inclusion that is contained in a given set. Chapter 6 considers the controllability problem. Chapter 7 discusses extremal problems for differential inclusions. Chapter 8 presents stability theory, and Chapter 9 deals with the stabilization problem. 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. Differential inclusions play an important role as a tool in the study of various dynamical processes described by equations with a discontinuous or multivalued right-hand side. This text acts as an introduction to the subject